Dynamics of liquid crystals near isotropic-nematic phase transition and some contributions to density relaxation in non-equilibrium systems

A thesis Submitted for the Degree of

in the Faculty of Science

By

Prasanth P. Jose

Solid State and Structural Chemistry Unit INDIAN INSTITUTE OF SCIENCE BANGALORE-560012, INDIA January 2006

Declaration I here by declare that the matter embodied in this thesis entitled ”Dynamics of liquid crystals near isotropic-nematic phase transition and some contributions to density relaxation in non-equilibrium systems” is result of the investigation carried out by me in the Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, India, under the supervision of Professor Biman Bagchi. In keeping with the general practice of reporting scientific observations, due acknowledgement has been made whenever the work described is based on the findings of other investigators. Any omission which might have occured by oversight or error in judgement is regretted.

(Prasanth P Jose)

ii

Acknowledgements This thesis is the result of my work in the Indian Institute of Science, Bangalore. During my stay, I have been accompanied and supported by many people. It is a pleasant experience to express my gratitude to all of them. First and foremost, I would like to thank my thesis adviser Professor Biman Bagchi. He is not only a great mentor with deep vision but also and most importantly a kind person. He has been instrumental in introducing me to the research field of liquid state theory and molecular simulations. I also thank him for suggesting interesting problems and inspiring discussions. He has been a source of inspiration to me in the most important moments of making right decisions. It is a pleasure to work in the pleasant atmosphere of the Solid State and Structural Chemistry Unit. I thank former and present chairmen of SSCU department, Prof. M. S. Hegde, Prof. Biman Bagchi, and Prof. T. N. Guru Row for providing excellent facilities in the department. Many thanks go to faculty members of SSCU department whose assistance was vital for my research. I specially thank Prof. T. N. Guru Row, Prof. S. Yashonath, Prof. S. Ramasesha, and Prof. J. Gopalakrishan for their concern in my academic affairs. Office staff and the workshop staff have been helpful in my needs; I thank them for their help. I thank the Indian Institute of Science for the financial support and the Super Computer Education and Research Center for the computational facilities. I thank Dwaipayan and Suman for the discussions during our collaborative work. My former and present colleagues: Denny, Srinivas, Rajesh, Vinay, Vasanthi, Arnab, Prabhakar, Sarika, Swapan, Dwaipayan, Subrata, Suman, Sangeeta, Basavaraja, Biman, and Bharat have been supportive. I want to thank them all.

I thank my friends for being helpful during the years I spend in this Institute. Let me name a few: Glomin, Mahesh, Sivakumar, Feroz, Shaiju, Santo, late Suresh of IPC, Anirud, Jha, Hemanth, Pradeep, Madhavan, Murari, Krishnan, and Senthil. They have been helpful in my needs. I would like to thank my bachmates for their company during the stay in this institute, namely, Angshuman, Arup, Dinesh, Motin, Martha, Mousumi, Nagaveni, Pradip and Ujjal. I thank Mrs.(Dr.) Suhrita Bagchi for her care and concern. Finally, I thank my parents and brother for their constant support and encouragement.

iv

Preface This thesis consists of three parts and contains reports of theoretical investigations of relaxation dynamics of conserved and non-conserved variables in molecular systems. The systems studied are closely related to experimental ones. Investigations have been performed using various computational and theoretical techniques. Three different computer simulation methods have been employed to probe the time scales involved. THE FIRST PART of the thesis consists of 7 chapters on phase transitions of model nematogens (Bay-Berne ellipsoids) using molecular dynamics simulation (micro canonical ensemble) as the basic tool. These systems show various liquid crystalline phases under the variation of temperature and pressure. Chapter 1 gives a brief description of the phase transitions in liquid crystals, with emphasis on isotropic nematic phase transition (INPT). These mesophases exhibit marked anisotropy in their electric, magnetic and optical properties and are outcome of breakage of rotational and translational symmetry of the isotropic fluid phase. INPT is weakly first order, having significant second order characteristics (fluctuations) which play important role in the relaxation of the system. The basic theoretical formalism that describes the structure and dynamics of simple liquid has been described in this chapter and also earlier molecular dynamics experiments near INPT and in nematic liquid crystal reviewed. Chapter 2 presents a study on the static correlation functions and equilibrium structure of the Gay-Berne ellipsoids in general. The orientation dependent pair correlation function (gllm (r), where l and m are the standard spherical harmonic expansion coefficients) and isotropic structure factor (S(k)) show signatures of growth of static orientational order near INPT. This chapter also explores the signature of different liquid crystalline phases of the Gay-Berne ellipsoids in terms of S(k) and gllm (r). v

Chapter 3 addresses the problem of slow down of temporal relaxation of single particle and collective orientational order near INPT in terms of second rank orientational correlation function. The motivation for this study comes from recent Kerr relaxation experiments by Gottke et al. [J. Chem. Phys. 116, 360 (2002) and 116, 6339 (2002)] which revealed the existence of a pronounced temporal power law decay in the orientational relaxation (as probed by Kerr relaxation) near the INPT. Our studies reveal the emergence of such a short time power-law decay in the collective part of the second rank orientational time correlation function, when the density is very close to the transition density. At long times, the decay becomes exponential-like, as predicted by Landau-de Gennes mean field theory. In order to capture the microscopic essence of the dynamics of formation of pseudo-nematic domains inside the isotropic phase, a dynamic orientational pair correlation function (DOPCF) is introduced and calculated. The DOPCF exhibits power law relaxation when the pair separation length is below certain critical length. These results have been interpreted in terms of a mode coupling theory of orientational dynamics near the INPT. The relationship of the simulated power law with the one observed in a supercooled liquid near its glass transition temperature has been discussed. Chapter 4 reports further study on the analogies between the supercooled liquid and the pseudo-nematic domains in the pre-transition region of INPT at the level of the single particle dynamics. The translational mean square displacements (MSDs) show ordinary liquid like behaviour for a liquid near INPT; however, the rotational MSDs show prolonged sub-diffusive behavior, which is a signature of dynamical heterogeneity. This is similar to what observed in the supercooled liquids for the translational degrees of freedom. For the study of heterogeneous dynamics, the translational and rational non-Gaussian parameters (α2T and α2R respectively ) are calculated. α2R shows bimodal peaks which separates two time scales: the first one represents the relaxation of the orientation of a single ellipsoid inside an orientational cage of a pseudo-nematic domain; the second peak corresponds to the collective relaxation of a pseudo-nematic domain. The role of orientational confinement in the rotational dynamics is studied by analysing vi

the individual trajectories near INPT. Chapter 5 continues our study of the analogy between the supercooled liquid and a liquid near INPT, from perspective of wave vector dependent collective relaxation functions. The total intermediate scattering function (F (k, t)) show marked anisotropy in the structural relaxation as the system approaches INPT; this is reflected in the relaxation of the transverse (CT (k, t)) and the longitudinal (CL (k, t)) currents also. Another relaxation function which deserves study is the wave vector dependent relaxation of angular currents. The structural relaxation of the nematogens near INPT behaves similar to that of a simple liquid, however, the relaxation function shows anisotropy with respect to the direction of the wavevector. The Chapter 6 reports an investigation on the effect of I-N transition on viscoelasticity of the system and also the correlation of viscoelasticity with orientational relaxation. Although the viscosity undergoes a somewhat sharper than normal change near the I-N transition, it is not characterized by any divergence-like behaviour (like the ones observed in the supercooled liquid). The rotational friction, on the other hand, shows a sharper rise as the INPT is approached. Interestingly, the probability distribution of the amplitude of the three components of the tress tensor shows anisotropy near the I-N transition – similar anisotropy has also been seen in the deeply supercooled liquid. Frequency dependence of viscosity shows several remarkable features: (a) a weak, power law dependence on frequency (η 0 (ω) ∼ ω −α ) at low frequencies ; (b) a rapid increase in the sharp peak observed in η 0 (ω) in the intermediate frequency on approach to the INPT density. These features can be explained from the stress-stress time correlation function. The angular velocity correlation function also exhibits a power law decay in time. Chapter 7, presents a study on collective orientational relaxation in the liquid crystalline nematic phase of Gay-Berne ellipsoids. This is directly related to optical Kerr effect (OKE) signal. Transient optical Kerr signal obtained from molecular dynamics simulations reveal, two distinct power laws, with a cross-over region, in the decay of the orientational time correlation function at short to intermediate times (in vii

the range of a few ps to few ns). In addition, the simulation results seem to indicate an absence of any long time exponential decay component. Theoretical analysis suggests that the two power laws may originate from local fluctuations of the director (which is enhanced due to the proximity to the isotropic phase) and to density fluctuations leading to the smectic phase – both the processes are expected to involve small free energy barriers. There is evidence of pronounced coupling between orientational and spatial densities at intermediate wavenumbers. In addition to slow collective orientational relaxation, the single particle orientational relaxation is also found to exhibit slow dynamics in the long time. Similar results have been observed very recently in OKE experiments in the nematic phase. THE SECOND PART of the thesis systematically studies the relaxation of a one dimensional system through random walk of the carriers - in particular in the context of the recently reported anomalous dielectric relaxation and solvation dynamics observed in DNA (Phys. Rev. Lett. 88, 158101 (2002)). A master equation approach has been developed to mimic the dynamics of a collection of interacting random walkers in a closed and an open system and solved numerically using grand canonical Monte-Carlo technique. In this model, the random walkers interact through excluded volume interaction (single-file system); and the total number of walkers in the lattice can fluctuate because of exchange with a bath. In addition, the movement of the random walkers is biased by an external perturbation. Two models for the latter considered are: (1) an inverse potential (V ∝ 1r ), where r is the distance between the center of the perturbation and the random walker and (2) an inverse of sixth power potential (V ∝

1 ). r6

The calculated density of the walkers and the total energy show interesting dynamics. When the size of the system is comparable to the range of the perturbing field, the energy relaxation is found to be highly non-exponential. In this range, the system β

can show stretched exponential (e−(t/τs ) ) and even logarithmic time dependence of energy relaxation over a range of time. Introduction of density exchange in the lattice markedly weakens the non-exponentiality of the relaxation function, irrespective of the nature of perturbation. viii

THE THIRD PART of the thesis focuses on microscopic relaxation processes involved in the phenomenon of clustering of particles, suspended in a solution, due to the radiation pressure of a laser beam in the fundamental mode. Recent experimental studies have demonstrated the feasibility of using radiation pressure of a laser beam as a tool for cluster formation in solution (J. Phys. Chem. B, 102 1896 (1998), ibid, 103 1660 (1999)). Here we use Brownian dynamics simulation of solute particles under radiation pressure to study such aggregation. The force field generated by a laser beam in the fundamental mode is modeled as that of a two dimensional harmonic oscillator. The radial distribution function of the perturbed system show high inhomogeneities in the solute distribution. An explicit analysis of the nature of these clusters is carried out by calculating the density-density correlation functions in the plane perpendicular to beam direction g(r xy ); and along the direction of beam g(z). They give an average picture of shell structure formation in different directions. The relaxation time of the first shell structure calculated from the van Hove correlation function is found to be relatively large in the perturbed solution. The study on the dynamics of solute molecules during the cluster formation and dissolution gives a measure of collective relaxation, starting far away from equilibrium to return to the equilibrium distribution.

ix

x

List of Publications 1. Anomalous glassy relaxation near the isotropic-nematic phase transition, Prasanth P. Jose, Dwaipayan Chakrabarti and Biman Bagchi, Phys. Rev. E 71, 030701(R) (2005). 2. Anomalous viscoelasticity near the isotropic-nematic phase transition in liquid crystals, Prasanth P. Jose and Biman Bagchi, J. Chem. Phys. 121 , 6978 (2004). 3. In search of temporal power laws in the orientational relaxation near isotropic– nematic phase transition in model nematogens, Prasanth P. Jose and Biman Bagchi, J. Chem. Phys.120, 11256 (2004). 4. Density and energy relaxation in an open one-dimensional system, Prasanth P. Jose and Biman Bagchi, J. Chem. Phys. 120, 8327 (2004). 5. Formation of nano-clusters under radiation pressure in solution: A Brownian dynamics simulation study Prasanth P. Jose and Biman Bagchi, J. Chem. Phys. 116, 2556 (2002). 6. Universal power law in the orientational relaxation in thermotropic liquid crystals, Dwaipayan Chakrabarti, Prasanth P. Jose, Suman Chakrabarty and Biman Bagchi, Phys. Rev. Lett. (Accepted) .

xii

Contents I

DYNAMICS OF LIQUID CRYSTALS NEAR ISOTROPIC

TO NEMATIC PHASE TRANSITION

1

1 Introduction to liquid crystals

3

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Why and how liquid crystals form? . . . . . . . . . . . . . . . . . . . .

3

1.3

Classification of liquid crystals . . . . . . . . . . . . . . . . . . . . . . .

4

1.3.1

Nematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3.2

Smectic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3.3

Cholesteric

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.3.4

Columnar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4

Isotropic to nematic (I-N) phase transition . . . . . . . . . . . . . . . .

8

1.5

Molecular Field theories of liquid crystals . . . . . . . . . . . . . . . . .

9

1.5.1

Theory of Maier and Saupe . . . . . . . . . . . . . . . . . . . .

9

1.5.2

Theory of Onsager . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.5.3

Landau-de Gennes theory . . . . . . . . . . . . . . . . . . . . .

12

1.5.4

Pre-transition dynamics in Landau-de Gennes theory . . . . . .

15

1.5.5

Critical nature of the I-N transition . . . . . . . . . . . . . . . .

17

Optical Kerr effect measurements . . . . . . . . . . . . . . . . . . . . .

19

1.6

1.6.1

Optical Kerr effect experiments on the orientational ordering of nematogens . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.6.2

Experiments of short time collective orientational dynamics . . .

21

1.6.3

Mode coupling theory analysis . . . . . . . . . . . . . . . . . . .

24

1.6.4

Mode coupling theory of Li et al. . . . . . . . . . . . . . . . . .

28

xiii

CONTENTS

1.6.5 1.7

Hydrodynamics of liquid crystals . . . . . . . . . . . . . . . . .

30

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

Bibliography

33

2 Overview of the molecular simulations

39

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.2

Molecular models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.2.1

Gay-Berne ellipsoid model . . . . . . . . . . . . . . . . . . . . .

41

2.3

Static orientational pair correlation function . . . . . . . . . . . . . . .

46

2.4

Details of the simulations . . . . . . . . . . . . . . . . . . . . . . . . . .

50

2.5

Pair correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . .

50

2.5.1

Static pair correlation function near the I-N transition . . . . .

53

2.5.2

Pair correlation functions across the nematic-smectic and smectic-

2.6

solid phase transitions . . . . . . . . . . . . . . . . . . . . . . .

57

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Bibliography

63

3 Collective orientational relaxation

67

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

3.2

Dynamic orientational correlation functions

. . . . . . . . . . . . . . .

69

3.3

Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.4

Results and Discussions

. . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.5

Temperature dependence of relaxation . . . . . . . . . . . . . . . . . .

78

3.5.1

Orientational relaxation of a local director . . . . . . . . . . . .

80

Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

3.6

Bibliography

87 xiv

CONTENTS

4

Single particle rotational dynamics

89

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

4.2

Single particles dynamical correlation functions . . . . . . . . . . . . .

92

4.3

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

4.3.1

Orientational confinement near I-N transition . . . . . . . . . .

95

4.3.2

Single particle translation and incoherent intermediate scattering function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.3 4.4

98

Heterogeneous diffusion near I-N transition . . . . . . . . . . . . 105

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Bibliography

115

5 Wavenumber and frequency dependent collective relaxation in liquid crystals 5.1 5.2

119

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Collective correlation functions of density, current and angular current

123

5.3

Details of the simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.4

Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.5

5.4.1

Intermediate scattering function and dynamic structure factor . 128

5.4.2

Longitudinal currents . . . . . . . . . . . . . . . . . . . . . . . 134

5.4.3

Relaxation of the transverse currents . . . . . . . . . . . . . . . 138

5.4.4

Relaxation of the angular currents . . . . . . . . . . . . . . . . . 142

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Bibliography

147

6

151

Stress relaxation and viscosity 6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.2

Expressions of shear viscosity and rotational friction . . . . . . . . . . . 153

6.3

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.3.1

Shear viscoelasticity and stress-stress time correlation function . 155 xv

CONTENTS

6.4

6.3.2

Angular velocity auto correlation function . . . . . . . . . . . . 159

6.3.3

Rotational friction on the ellipsoids . . . . . . . . . . . . . . . . 161

6.3.4

Comparison between viscosity and rotational friction . . . . . . 162

Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Bibliography

165

7 Dynamics of nematic liquid crystals

167

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.2

Details of the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.3

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.3.1

Order parameter and equilibrium orientational pair correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.3.2

Single particle orientational relaxation . . . . . . . . . . . . . . 172

7.3.3

Collective orientational time correlation functions and Kerr relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.4

Role of fluctuations in multiple power law relaxation . . . . . . . . . . 182

7.5

Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Bibliography

II

187

RELAXATION IN A LINEAR LATTICE

8 Density and energy relaxation in an open one-dimensional system

189 191

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.2

The Generalized Master equation . . . . . . . . . . . . . . . . . . . . . 193

8.3

The details of Monte Carlo simulation . . . . . . . . . . . . . . . . . . 196

8.4

Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.4.1

8.5

Relaxation under Lennard-Jones potential . . . . . . . . . . . . 202

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Bibliography

207 xvi

CONTENTS

III

DYNAMICS IN OPTICAL TRAP

209

9 Formation of nanoclusters Under Radiation Pressure: A Brownian Dynamics Simulation Study

211

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

9.2

Modeling of the radiation pressure . . . . . . . . . . . . . . . . . . . . 213

9.3

System and Simulation Details . . . . . . . . . . . . . . . . . . . . . . . 215

9.4

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

9.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Bibliography

229

xvii

Part I DYNAMICS OF LIQUID CRYSTALS NEAR ISOTROPIC TO NEMATIC PHASE TRANSITION

1

Chapter 1 Introduction to liquid crystals 1.1

Introduction

Liquid crystals are the ’mesomorphic phases’ (mesomorph: of intermediate form) of matter that appear during the transition between an ordered solid and a disordered liquid [1–3]. Geometry and intermolecular forces play important role in the structure and stability of these phases. Mechanical properties of liquid crystals are similar to those of an ordinary liquid; they are viscous liquids and show coalescence of droplets. These phases show anisotropic optical and magnetic properties which are important for industrial applications such as displays, sensors etc. Liquid crystals were discovered in 1888 Friedrich Reinitzer, an Austrian botanist, while working with esters of cholesterol, natural substances in plants and animals. These materials exhibited a double melting transition that was reversible and repeatable on heating or cooling. As each of these substances were heated, the crystalline solid melted to form first an optically opaque liquid that transformed to a clear liquid at a distinct temperature.

1.2

Why and how liquid crystals form?

Despite the existence of several theoretical models for the understanding of formation of these mesophases, there still exist several mysteries unsolved. According to the general understanding, the requirement for the formation of liquid crystalline phase is a pronounced anisotropy in the intermolecular interaction. If the molecules are 3

1. INTRODUCTION TO LIQUID CRYSTALS

PRESSSURE

CRYSTAL

SMECTIC NEMATIC

ISOTROPIC

TEMPERATURE Figure 1.1: The schematic phase digram of a typical liquid crystalline system.

elongated and have flat faces, then liquid crystallinity is more likely to occur than otherwise [2]. Another feature of the molecules having liquid crystalline phases is that they have a fairly rigid backbone containing double bonds. Also, the existence of the strong dipoles and easily polarizable groups are important in the formation of liquid crystals. In the long molecules, the end groups have less importance in the formation of the liquid crystalline phase.

1.3

Classification of liquid crystals

Liquid crystals are broadly classified into two major categories according to their composition: (1) Lyotropic and (2) thermotropic [3, 4]. Lyotropic liquid crystals are multicomponent systems; one of them is an amphiphile ( they are long chain hydrocarbons with polar head groups attached to one end) and the other is water. Soap is one of the most common examples of an amphiphile. The dilute solution is the isotropic phase of this system. As concentration is increased, several 4

1.3 Classification of liquid crystals

mesophases are observed. Structure of these mesophases vary with the concentration of the solution. Thermotropic liquid crystals are generally organic or organometallic molecular systems and they are either rod-shaped (calamitic) or disc-shaped (discotic). For thermotropic liquid crystals, the way to induce phase transition is to vary the temperature; hence these materials are commonly called thermotropic. Liquid crystalline phases are obtained, when the solid phase of these compounds are melted by heating or by cooling of the isotropic phase. The structural classification of liquid crystals were fist proposed by G. Friedel [1, 3, 5]. According to structural classifications, there are more than 30 liquid crystalline phases exist in nature. In a broad sense, thermotropic liquid crystals are classified into four major mesophases according to their structure and symmetry. They are (a) nematic, (b) smectic, (c) cholesteric, and (d) columnar. Measogens that can form liquid crystalline phases are classified into two: they are calamitic and discotic. Calamitic liquid crystals are rod like molecules. They exhibit nematic, smectic, and cholesteric mesophases. The discotic molecules are disc like molecules. They exhibit nematic, cholesteric, and columnar phases. The first discotic liquid crystal was found in 1977 by the Indian researcher S. Chandrasekhar. This molecule has one central benzene ring surrounded by six alkyl chains.

1.3.1

Nematic

The name nematic is introduced by G. Friedel. It is named after a Greek word that refers to the thread like defects that are found in this phase. The simplest liquid crystalline phase is nematic, which forms due to the breakage of rotational symmetry. Optical birefringence experiments easily identify this phase from the optical anisotropy developed in the system. The transition from isotropic nematic phase is weakly first order in nature. The discontinuity in the first derivative of the free energy is not as remarkable as that observed in other strong first order transitions like solid-liquid and gas-liquid. Both calamitic and discotic molecules can form nematic phase. Disk-shaped mesogens can orient themselves in a layer-like fashion known as the discotic nematic 5

1. INTRODUCTION TO LIQUID CRYSTALS

Nematic

Isotropic

Smectic A

Cholesteric

Figure 1.2: A two dimensional schematic representation of the three liquid crystalline phases along with the isotropic phase of a calamitic liquid crystal.

phase. Nematics have fluidity similar to that of isotropic liquids, but, they can be easily aligned by an external magnetic or electric field. An aligned nematic has the optical properties of a uniaxial crystal and this makes them extremely useful in liquid crystal displays.

1.3.2

Smectic

The smectic phase has layered structure. Here the structure is more close to solid phase with one-dimensional long range order. It has, in addition to the orientational ordering of the nematic phase, translational ordering also. In this layered phase, there is no long-range correlation of density between the layers. This phase is named by G. Friedel and the name ’smectic’ originates the Greek word that means soap. Smectic phase shows mechanical properties, which are the reminiscent of soap. The inter layer spacing between the smectic layers can be measured in the X-ray scattering experiments. According to the arrangement of molecules in the single layer, the phase is further classified into different sub-classes. Some of the important phases, which are the sub-class of the smectic phase, are smectic A, smectic B and smectic C. Out of these 6

1.3 Classification of liquid crystals

three, in the smectic A phase, there exists no ordering within a particular layer and direction of the director of the system is perpendicular to the smectic layers. Among the smectic phases, smectic B possesses the highest orientational and positional ordering. The structure of smectic B phase is similar to that of the smectic A phase, except there is a hexagonal ordering of molecules within a layer. In the smectic C phase, there is no positional ordering within a layer and the direction of the director is not normal to the layers.

1.3.3

Cholesteric

Cholesteric phase is similar to the nematic phase; it has no positional ordering of the molecules. In contrast to the unidirectional nature of the nematic phase, in the cholesteric phase, the direction the average director changes in a regular fashion. When a chiral molecule (that has no mirror symmetry) is dissolved in a nematic liquid, the structure of the liquid undergoes a helical distortion. This distortion is also found in pure cholesterol esters (chiral molecules); hence this phase is called cholesteric. The helical pitch of the director is of the order of wavelength of light (few thousands of angstroms), which results in Bragg scattering of light and color effects. There are chiral discotic phases, similar to the chiral nematic phase, are also known.

1.3.4

Columnar

In the columnar phase molecules assemble into cylindrical structures to act as mesogens. Originally, these kinds of liquid crystals were called discotic liquid crystals because the columnar structures are composed of flat-shaped discotic molecules stacked onedimensionally. Since recent findings provide a number of columnar liquid crystals consisting of non-discoid mesogens, it is more common now to classify this state of matter and compounds with these properties as columnar liquid crystals. If the disks pack into stacks, the phase is called a discotic columnar. The columns themselves may be organized into rectangular or hexagonal arrays. As mentioned earlier the kind of mesophases that collection of a type particular molecule exhibits depends on type and symmetry of the intermolecular potential. 7

1. INTRODUCTION TO LIQUID CRYSTALS

Isotropic to nematic phase transition is viewed widely as the most simple phase transition, which may help us to understand other phase transition in liquid crystals. In this phase transition, rotational symmetry is broken without affecting translational symmetry. This thesis focuses on the study of relaxation dynamics near isotropic to nematic (I-N) phase transition, which is still not fully understood.

1.4

Isotropic to nematic (I-N) phase transition

If the aspect ratio is larger than certain critical value, a system of elongated molecules is known to undergo a phase transition at low temperatures from an orientationally disordered isotropic phase to an orientationally ordered nematic phase. This I-N transition is a weakly first order phase transition and is characterized by a large growth of orientational correlations near the transition point. Different aspects of the structure and the dynamics of isotropic to nematic phase transition has been extensively studied in different experiments [6–16] . Several theoretical methods such as molecular field theories [17, 18] and mean field approximations [1, 19, 20] have been developed to study the I-N transition. The Hamiltonian approach of Maier and Saupe [21, 22] is a particularly useful model of I-N transition. Density functional approaches [23–29] have been employed for the study of I-N transition in different systems. Computer simulations have studied I-N transition using different model systems such as hard rods [30–32], lattice models [33–35]. Although a first order phase transition, I-N transition shows fluctuations, which are comparable to that of a second order phase transition. There are several attempts to determine the critical exponents of this phase transition [13, 14]. An overview of theory and experiments on liquid crystal phases, which appear in solutions of elongated colloidal particles or stiff polymer can be found in the review of Vroege and Lekkerkerker [36] The order parameter of the system can be written in terms of three orthogonal vectors a, b, c associated with the molecule. Then the tensorial orientational order parameter of I-N transition is defined as [1, 2] Sijαβ =

1 < 3iα jβ − δαβ δij > 2 8

(1.1)

1.5 Molecular Field theories of liquid crystals

where α, β = x, y, z are indices referring to the laboratory frame, while i, j= a, b, c, and δαβ and δij are Kronecker delta symbols. Here the averaging is over the molecules of the system.

1.5 1.5.1

Molecular Field theories of liquid crystals Theory of Maier and Saupe

Maier-Saupe theory is a molecular field theory. Each molecule feels the average field generated by other molecules [1–3, 21, 22, 37]. This theory considered an orientation dependent van der Waal’s interaction between pairs of rod like molecules. It is also assumed that the distribution of center mass of the particles are spherically symmetric. Distribution of orientations of each molecule is accurately described by the average order parameter tensor (Sαβ ). It is also assumed that each molecule has a well defined long axis ν. The distribution function of f (ν) only needs to be considered. Then the effective potential felt by the molecule is given as 1 2 V (ν, S) = − A0 Sαβ (να νβ − δαβ ) 3 2

(1.2)

where the ν is the unit vector along the axis of the molecule and A0 is the quantity dependent only on the inter particle separation. The value of A0 is given by the expression A0 = a <

X

−6 > R1j

j

, where the sum is over all particle except 1, a is a constant, and angle brackets indicate the ensemble average. The probability distribution of the orientation of a molecule in the internal field is given by f (ν) = C e−V (ν,S)/kB T where 1/C =

R

dνe−V (ν,S)/kB T . This is self-consistent due to the condition that Sαβ =

Z

1 dννα νβ − δαβ )f (ν) 2 9

(1.3)

1. INTRODUCTION TO LIQUID CRYSTALS

In the case of uniaxial liquid crystals, this equation can be simplified where the direction is unique. S = 2πC

Z

π 0

1 3 ( cos2 θ − )e−V (θ,S)/kB T sinθdθ 2 2

(1.4)

where V (θ, S) = −A0 S( 32 cos2 θ − 12 ). The expression for C now becomes, 1/C = 2π

Z

π

e−V (θ,S)/kB T sinθdθ.

(1.5)

0

One trivial solution of equations 1.4 and 1.5 is S = 0, which belongs to the isotropic phase. The non-trivial part can be found by integration by parts and substituting equation 1.5, equation 1.4 can be written as 1 3 2 S = (ex /xD(x) − 1/x2 ) − 4 2

(1.6)

where x = 23 A0 /kB T and D is Darwin’s integral [38]. D(x) =

Z

x

2

ey dy 0

There are three solutions to the equation 1.4 for a range of temperature. The solution that gives the lowest free energy per particle is kB T 1 lnC F (T, S) = − A0 S 2 + 2 4π

(1.7)

The equation 1.4 is the derivative of free energy with respect to S at fixed T, hence the equation 1.4 naturally the extrima of F . Maier-Saupe theory predicts first order phase transition at temperature Tc defined by kB T = 0.22A0 . The value of the order parameter at the transition is Sc = 0.43, which is independent of A0 . The summary of the assumptions made in the Maier-Saupe theory are the following: (a) it assumes an attractive orientation dependent van der Waal’s interaction; (b) the configuration of the center of mass is not affected by the orientation dependent interaction; (c) it is a mean field theory where long range correlations are neglected. 10

1.5 Molecular Field theories of liquid crystals

1.5.2

Theory of Onsager

Theory of Onsager calculates the free energy of non-spherical molecules [1–3, 17]. For the case of rods with spherical caps of length L and diameter D with excluded volume interaction, Onsager theory can be applied as follows. The assumptions of the formalism are as follows. (a) The only important interaction between the rods corresponds to sterric repulsion, (b) the volume fraction of the rods Φ = c

1 πLD 2 4

is much less

than unity (c = concentration), and rods are very long L >> D. Onsager essentially computed a virial expansion (upto second order) of free energy F in concentration. The resulting expression for the free energy per particle is given by 1 F = F0 + kB T (log c + cβ1 + C(c2 )) 2 where β1 is the excluded volume. He has showed that how Mayer’s cluster theory may be used to find the equation of state of the system in terms of series expansion. The terms of this series include the orientational distribution function, which gives the fraction of molecules per unit solid angle having various orientations. It was shown that the volume fraction occupied by the rods Φ, at the transition is Φcnem = 4.5D/L At the same point the isotropic phase has the volume fraction Φciso = 3.3D/L According to this theory the I-N transition is first order. The main limitation is that the expansion is limited to second virial coefficient. Zwanzig proposed a simplified version of this model [3, 18] where particles have rectangular cross section and they can orient only in three mutually perpendicular directions. The length to breadth ratio is very high. Although these assumptions appear to be unrealistic, the essence of Onsager’s theory is captured in this model. The advantage of this model is that virial coefficients can be calculated up to much higher order. These treatments devised a method to treat each orientation as separate 11

1. INTRODUCTION TO LIQUID CRYSTALS

chemical species; this feature is retained by the subsequent theories. Zwanzig further showed that the first order phase transition pre-empts an orientational second order phase transition. Flory [39–41] and DiMarzio [42] developed another technique, which replaced the continuous space with cellular lattice. Here the evaluation of the partition function of the system reduces to counting of the number of ways N rods can be placed in a lattice. Cotter and Martire [43–45] and Lasher [46] has developed scaled particle theories. Alben [47] has developed a method to extend the counting method of the lattice model to continuous space. He has also introduced a van der Waals internal pressure term so that the transition take place at zero pressure. A treatment of the sterric model of the nematic liquid crystals based on the empirical equation of state is presented by Priest [48]. The influence of short-range orientational order may be estimated here in a more straightforward manner possible.

1.5.3

Landau-de Gennes theory

One of the most successful models of I-N transition is Landau-de Gennes mean-field theory, which predicts this transition to be first order for purely geometrical reasons [1, 49]. In fact, it is widely accepted experimentally that I-N transition is weakly first order. The free-energy of the system can be expanded in terms of the tensorial order parameter as 1 1 F = F0 + A(T )Qαβ Qβα + B(T )Qαβ Qβγ Qγα + O(Q4 ).... 2 3

(1.8)

(here the summation over the repeated indices is implied), where the terms are invariant under rotation of axes (x,y,z). There is no linear term in Q; this means that the state of minimum F is isotropic. The non vanishing term is of the order of Q3 since there is no relation between Qαβ and −Qαβ Qαβ



 −Q 0 0 =  0 −Q 0  0 0 2Q 12

(1.9)

1.5 Molecular Field theories of liquid crystals

has maximum alignment along the z direction of the coordinate system, while   Q 0 0 0  Qαβ =  0 Q (1.10) 0 0 −2Q

has maximum alignment in the x − y plane. These two states have identical free energies.

The phase transition is first order due to the cubical term in the expansion of the free energy. However the discontinuities observed in the density, heat content etc. is small, hence it is a weakly first order transition. The coefficient B(T ) is relatively small. When phase transition is approached, the pre-transitional effects are significant near I-N transition. The coefficients A(T ), B and C are in general functions of P and T . The model predicts a phase transition near the temperature where A(T ) vanishes. We assume the form of A(T ) as A(T ) = a(T − T ∗ )

(1.11)

The transition temperature itself is originally above T ∗ as obtained from the analysis following. B and C have no particular properties near T ∗ . For uniaxial liquid crystal Qαβ takes the form Qαβ = S(nα nβ − 21 δαβ ). Substituting this relation in the free energy expansion (Eq. 1.8) 1 2 1 F = F0 + AS 2 − BS 3 + CS 4 3 27 9

(1.12)

There is discontinuous phase transition at temperature Tc slightly above T ∗ as shown in the figure 1.3 The values that minimize the free energy can be located algebraically. The derivative of 1.12 is equated to zero to obtain the roots, 2 1 AS − BS 2 + CS 3 = 0. 3 3

(1.13)

The solutions of the equation are S=0 B (1 − 24β)1/2 S= 4C

isotropic nematic 13

(1.14)

1. INTRODUCTION TO LIQUID CRYSTALS

T**

F

T

c

T*

S

Figure 1.3: The Gibbs free energy of a nematogen near I-N transition is plotted against the order parameter. The cases of three special temperatures (T ∗∗ , T ∗ and Tc ) are shown. where β = AC/B 2 . Note that the third solution that corresponds to the maximum of the free energy has been discarded. Of the two solutions, one that corresponds to the transition temperature Tc is where the free energies of the isotropic and the nematic phase are equal. From equation 1.14 and 1.13, this point can be obtained as β =

1 27

Tc = T ∗ +

(1.15) 1 B2 . 27 aC

(1.16)

Above Tc the isotropic phase is stable and below Tc the nematic phase is stable. The value of the order parameter at the phase transition is Sc = B/3C. The difference in entropy between the two phases is found by differentiating Eq. 1.13 with respect to temperature. The resultant latent heat per volume is L=

aB 2 Tc . 27C 2

The temperature T ∗ corresponds to the limit of metastability of the isotropic phase. In principle, it is possible to supercool the isotropic liquid till T ∗ - at T ∗ the coefficient A in 14

1.5 Molecular Field theories of liquid crystals

the free energy changes sign. Similarly nematic phase becomes unstable when β > 1/24. This determines the temperature T ∗∗ = T ∗ + B 2 /24aC, which is the boundary of metastability of the nematic phase upon heating. The significance of the temperatures T ∗ and T ∗∗ are that they are the apparent ”critical points” for the isotropic and the nematic phase, respectively. Hence the susceptibilities and the correlation lengths which increase as the transition point is approached, will appear to be headed for divergence at a temperature slightly beyond the transition temperature Tc . There are several other theoretical investigations which extend this approach further. C. Fan and M. J. Stephen [50] calculated the corrections due to the fluctuations in light scattering and magnetic birefringence in the isotropic phase. Their investigation also included the effect external field on I-N transition. There exist a critical field beyond with the I-N transition is found to be second order.

1.5.4

Pre-transition dynamics in Landau-de Gennes theory

The stability of the nematic phase is considerable inside the isotropic phase due to the flatness of the free energy surface; the pseudo-nematic domains shows remarkable pretransitional stability. The life time and the nature of the relaxation can be obtained from the theory of the de Gennes [51]. The rate of relaxation of the order parameter is given by δQαβ ≈ Q˙ αβ . δt

(1.17)

The equation for the entropy source is given as 0 T S˙ = Q˙ αβ Φαβ + σαβ Aαβ ,

(1.18)

0 is the viscous where the Φαβ is the restoring force derived from the free energy, σαβ

stress, and Aαβ is the shear rate component. The expression for the restoring force Φαβ , Φαβ =

∂F = −A(T )Qαβ . ∂Qαβ 15

(1.19)

1. INTRODUCTION TO LIQUID CRYSTALS

The most general form of these equations compatible with rotational invariance and with the Onsager symmetry relations, is as follows 0 σαβ = 2ηAαβ + 2µQ˙ αβ

(1.20)

Φαβ = 2µAαβ + ν Q˙ αβ

(1.21)

where η, µ and ν have dimensions and magnitude of viscosity. At small wavevectors the fluctuations of the hydrodynamic velocity are very slow while those of Q are only semi-slow. Then the coupling between orientation and flow becomes ineffective. The shear rate component Aαβ becomes negligible. This makes the relaxation of the order parameter Qαβ exponential with a rate Γ(T ) =

A(T ) . ν

(1.22)

Since the A(T ) is very small near the transition, the restoring force is weak near Tc . The pseudo-nematic domains in the pre-transition region of the I-N transition relax exponentially. The exponential relaxation of the pseudo-nematic domain has been verified by several experimental groups. Earlier experimental studies well documented the success of the Landau-de Gennes formalism of orientational relaxation in the pretransition region of the I-N transition. Stinson and Lister observed the divergence of the magnetic birefringence and critical increase of slowing of fluctuations in the isotropic phase of a nematic liquid crystal - pmethoxy benzylidene p-n-butylaniline (MBBA)[7]. The Landau-de Gennes mean field model is found to be valid except near the critical region close to ordering temperature. They found non-Gaussian fluctuations in scattering volume. In an another experiment [8], they measured the correlation range for the fluctuations of orientational order parameter in MBBA. The correlation length diverges as (T − T ∗ )−1/2 when transition temperature is approached - this is in accordance with the Landau-de Gennes theory. In the experiment of Martinoty et al. [6], they measured the temperature dependence of flow birefringence of the MBBA which also agrees with Landau-de Gennes theory. Shen and coworkers in a series of experiments [10–12] used Optical Kerr relaxation to measure the relaxation of the orientational ordering of the nematic liquid crystal in the 16

1.5 Molecular Field theories of liquid crystals

pre-transition region as function of temperature. The temperature dependence of the orientational relaxation time and the non-linear refractive index are in good agreement with the predictions of Landau-de Gennes theory. The anomalous parts of the specific heat capacity and the isobaric thermal expansion coefficient of nematic liquid crystals above the transition point are calculated by Okano and Imura on the basis of the continuum fluctuation theory of de Gennes [52]. They have calculated the dynamic heat capacity of nematic liquid crystals just above the clearing point as an extension of their previous theory of static heat capacity. This theory explains semiquantitatively the available experimental results of anomalous ultrasonic absorption and dispersion [53] However, with the advent of the ultra short laser pulse with femtosecond resolution [54, 55], the optical Kerr effect experiments could probe into short time dynamics of liquid crystals. Fayer and coworkers have performed a series of experiments [56– 61] which shows highly non-exponential relaxation dynamics in the initial stage of collective orientational relaxation process in the isotropic phase of several nematogens near the isotropic nematic phase boundary. The inspiration of the work presented in the first part of the thesis comes from the optically heterodyne detected optical Kerr effect experiments of Fayer and coworkers [56–64] on liquid crystals and supercooled liquids. Details of these experiments have been discussed in the section 1.6.

1.5.5

Critical nature of the I-N transition

As mentioned earlier, the Landau-deGennes mean field theory predicted a first order I-N transition purely due to geometrical reasons. However, Monte-Carlo simulations of the rotationally invariant Maier-Saupe model yield a latent heat value lower than that predicted by the mean field theory [33, 34]. Experiments also yield a lower latent heat that predicted by the mean field theory [47]. These facts motivated several researchers to look into the critical nature of the I-N transition. The pronounced pretransitional phenomena, observed in the vicinity of the I-N transition, are due to the existence of a short-range orientational order of the mesogenic molecules, leading to the formation of pseudonematic domains in the isotropic liquid. 17

1. INTRODUCTION TO LIQUID CRYSTALS

The size of the domains increases as the temperature of the liquid approaches the nematic phase transition. The phenomenon can be quantitatively described by an intermolecular correlation length ξ obtained from the Landau-de Gennes theory of I-N transition, thus yielding the following temperature dependence of the correlation length 1/2  T∗ (1.23) ξ(T ) = ξ0 T − T∗ where the constant ξ0 is of the order of the molecular length and T ∗ denotes the temperature of the virtual second-order transition. As can be seen from this equation, for T = T ∗ the correlation length tends to infinity. Priest an Lubensky [65] analyzed the I-N transition on the basis of renormalization group theory [66, 67]. However, renormalization group theory does not provide a global picture of the behaviour of Qαβ and the explanation for why the I-N transition is more continuous than expected. Keyes has suggested a tricritical nature for the I-N transition [68]. In another experiment Keyes and Shane [14] have determined the gap exponent of the MBBA from the measurements of the induced birefringence in very strong magnetic fields. Their experiments determine the order parameter Q as a function of temperature T and magnetic field H in the isotropic phase. The relaxation can be expressed in the form B h h + ω = g( δ ) δ Q Q 

(1.24)

From the mean field theory of Landau-de Gennes, the free energy expansion is 1 1 1 F = F0 + AQ2 − BQ3 + Q4 − hQ, 2 3 4

(1.25)

where h = χa H 2 , with χa being the anisotropy of the magnetic susceptibility, and A = A0  with  = T − T ∗ . Minimizing (1.25) with respect to Q with g(x) = C − A0 /x2 , they get critical exponents δ = 3, ω = 1 and β = 1/2. This differs from the exponents for a tricritical point; the values of the exponents for the tricritical point are δ = 5, ω = 2 and β = 1/4. It is worth mentioning a peculiarity of the isotropic to nematic phase transition, namely, the fact that the region closest to the critical point T ∗ is inaccessible to experiment because the transition of the first order occurs at a temperature TN I somewhat 18

1.6 Optical Kerr effect measurements

higher than T ∗ . As a consequence, the determination of the critical exponents must be performed by using extrapolations, which, lead to diversity in the final results. This is one of the reasons that can explain why, despite numerous papers devoted to the I-N phase transition, the nature of the critical behavior in the vicinity of the transition is not properly understood yet.

1.6

Optical Kerr effect measurements

When a substance is placed in strong uniform electric field, it become birefringent and the optical axis corresponds to the direction of the line of the force - thus minimizing the energy of the molecule in the external field. This effect is discovered by Kerr (1875) [69, 70] and widely known as Kerr electro-optical effect. If there exists anisotropy in the polarizability, the maximum polarizability axis is aligned with the electric field. Optical Kerr effect is observed when light passes through a system of anisotropically polarizable molecules. There is a tendency for molecules to be aligned due to their interaction with the electric field of the radiation. The high frequency of the radiation allows only average effect to be felt by the molecules - this effect is known as optical Kerr effect. The alignment in the case of optical Kerr effect is such that the alignment of the molecule (axis of maximum polarizability) is perpendicular to direction of propagation of the light - the optical axis of the medium is the direction of propagation of light. The theory of optical Kerr effect is first developed by Buckingham [71].

1.6.1

Optical Kerr effect experiments on the orientational ordering of nematogens

Earlier experimental studies of optical Kerr effect confirm the validity of the Landaude Gennes theory in the pretransition region of nematic liquid crystals. Optical Kerr relaxation experiments of Shen and coworkers [10–12] have established the validity of the Landau-de Gennes theory in various liquid crystal samples. It is possible to calculate the time constant of the orientational relaxation from the Landau-de Gennes mean field theory. Since molecules are anisotropic, intense optical field can induce 19

1. INTRODUCTION TO LIQUID CRYSTALS

macroscopic anisotropy in the system. The resultant molecular ordering results in the anisotropy of the medium. If Q is the tensorial order parameter which describes the molecular alignment, then the optical susceptibility χij can be written as 2 χij = χδij + ∆Qij 3 where χ =

1 3

P

i

(1.26)

χi and the ∆χ is the anisotropy in χij with perfect molecular arrange-

ment such that 21 Qxx = −Qyy = −Qzz = 12 . In the absence of the field, in an isotropic medium, Qij =0 According to the Landau-de Gennes theory, the free energy of the ~ ~ 0 ) is medium in the presence of the intense field E(ω) and the weak probing field E(ω given by 1 1 F = F0 + a(T − T ∗ )QijQji + QijQjk Qki + .... 2 3 1 1 − χ(ω)Ei∗ (ω)Ej (ω) − χ(ω 0 )Ei∗ (ω 0 )Ej (ω 0 ) 4 4

(1.27)

The field induced molecular ordering then obeys the dynamic equation ν∂Qij /∂t = ∂F/∂Qij (See Eq.s 1.19 and 1.21). Here higher order terms in Q can be neglected. Since the probing pulse is week, the perturbation due to the probing pulse also can be neglected. By these assumptions the dynamical equations can be written in the form ν

∂Qij + a(T − T ∗ )Qij = fij (t) ∂t

(1.28)

where fij (t) = 61 ∆χ[Ei∗ Ej − 13 |E|2 δij ] and ν is the viscosity coefficient. The solution of the equation can be written as t

fij t−t0 0 e τ dt −∞ ν ν ν τ = = A a(T − T ∗ )

Qij (t) =

Z

(1.29) (1.30)

From Eq. 1.26, the linear birefringence, induced by a strong linearly polarized field along the ˆi is 2π 2π 2 ∆χ(Qii − Qjj ) = ∆χQij n 3 n Z 1 ∆χ t 2 0 −(t−t0 )/τ 0 Qii = |E| (t ))e dt 9 ν ∞

δni =

(1.31) (1.32)

If |E|2 (t) is a pulse shorter than or comparable with τ , then at sufficiently large time t, both Qii and δni will decay exponentially with a time constant τ . 20

1.6 Optical Kerr effect measurements

1.6.2

Experiments of short time collective orientational dynamics

Figure 1.4: Temperature dependent 5-CB data sets displayed on a log plot. Starting with the topmost curve, the temperatures for the data sets are 312.0, 312.9, 314.0, 315.0, 316.0, 318.0, 322.0, 324.0, 326.0, 329.0, 332.0, 336.0, 339.0, 344.0 K. (Taken from J. Chem. Phys., 116, 6339 (2002) )

Optical Kerr experiments measure the transient birefringence due to alignment of the molecules in the presence of strong laser pulse that is propagating through the medium [72]. The decay of the birefringence is probed by a variably delayed, weak polarized probe pulse as mentioned in the earlier section. The recent progress in generating the ultrashort laser pulses of duration few femtosecond [55, 73] has gave rise to the possibility of looking into short time dynamics of molecular liquid. In the recent study on nematogens in the isotropic phase using Optical-heterodynedetected optical Kerr effect (OHD-OKE) technique [58, 59], Fayer and coworkers have shown the deviation from the Landau-de Gennes mean field theory. The short time data obtained in the pretransition region of I-N transition show power law relaxation. 21

1. INTRODUCTION TO LIQUID CRYSTALS

Figure 1.5: OHD-OKE 5-CB data displayed on a log plot. (a) Data in the isotropic phase taken at 309 K, just above the phase transition. The longest time component is an exponential decay described by the Landau-de Gennes theory with a decay constant of 203 ns. Prior to the exponential decay is a power law, t−0.05 . The lines below the data are aids to the eye. At shorter time (>∼3 ps) is another power law, t−0.66 , the intermediate power law. There is also a very short time decay that is very fast for t < 3 ps. (b) Data in the nematic phase taken at 306 K, just below the phase transition. The longest time component in the nematic phase is a power law, t−0.54 , the final power law. At shorter times is another power law, t−0.38 , the intermediate power law. Between the two power laws is a crossover region. At very short time is a very fast decay that is like that in the isotropic phase. (This figure is taken from J. Phys. Chem. B 109, 6514 (2005))

The data obtained from these experiments are related to impulse response function of the system [74, 75], which is the time derivative of the polarizability-polarizability P (t) time correlation function CP (t)( that is, the Kerr signal ∝ − dCdt ). The decay ob-

served in this experiments is exponential on the longest time scale (beyond 100 nanosecond), which is well described by Landau-de Gennes theory [1, 19]. At short times, the decay is again exponential. However, at intermediate times (∼ 10 picoseconds 100 nanoseconds), the data can be fitted to a temporal power law with an exponent 22

1.6 Optical Kerr effect measurements

with a value less than unity. The value of the exponent is found to be independent of temperature, but to depend on the chemical identity of the nematogens (rather, on the aspect ratio), signaling (in the strict sense) a lack of universality of the type observed near a critical point. However, the values of exponent are all close to 2/3 and thus some sort of general behaviour appears to be present in the dynamics. The temperature dependence of the long time scale exponential decays is described by the Landaude Gennes theory of the randomization of pseudo-nematic domains that exist in the isotropic phase of liquid crystals near the I-N transition. The LdG exponential time constant is expected to have a temperature (T ) and viscosity (η) dependence given by [10, 51] τLdG

Vef∗ f η(T ) = kB (T − T ∗ )

(1.33)

where Vef∗ f is the approximate molecular volume, kB is the Boltzmann constant, and γ = 1 according to mean field theory. Gottke et al. also presented a naive non-self consistent mode coupling theory analysis which explained the emergence of the power law behaviour as a consequence of the growing correlation length near I-N transition [58, 59]. The aspect ratio dependence of power law relaxation also been studied [57]. In another study, Cang et al. [60] compared the dynamics observed over wide ranges of time and temperature between five supercooled liquids and four isotropic phase liquid crystals that were previously studied separately. OHDOKE measurements were employed to obtain the orientational relaxation dynamics over time scales from sub-picoseconds to tens of nanoseconds. For the supercooled liquids, the temperatures range from above the melting point down to close to TC , the mode coupling theory critical temperature. For the liquid crystals, the temperatures range from well above the I-N transition temperature TN I down to close to TN I . For time scales longer than those dominated by intramolecular vibrational dynamics (≥1 picosecond), the fundamental details of the dynamics are identical. All nine liquids exhibit a decay of the OHDOKE signal that begins (>1 ps) with a temperature-independent power law t−z , where z is somewhat less than or equal to 1. A crossover region, modeled as a 23

1. INTRODUCTION TO LIQUID CRYSTALS

second power law, follows the power law decay in both the supercooled liquids and liquid crystals. The longest time scale decay for all nine liquids is exponential. In another work, Cang, Novikov and Fayer [61] have done temperature scaling analysis using the same mode coupling theory scaling relationships (routinely employed for supercooled liquids) is applied to optical heterodyne detected optical Kerr effect data for four liquid crystals. The data cover a range of times from ≈ 1 picosecond to 100 nanoseconds and a range of temperatures from ≈ 50 K above the isotropic to nematic phase transition temperature TN I down to ≈ TN I . The slowest exponential component of the data obeys the Landaude Gennes theory for the isotropic phase of liquid crystals. It is also found that the liquid crystal data obey mode coupling theory scaling relationships, however, instead of a single scaling temperature TC as found for supercooled liquids, in the liquid crystals there are two scaling temperatures TCL (L for low temperature) and TCH (H for high temperature). In yet another experiment by Li, Wang and Fayer [64], orientational dynamics of a homeotropically aligned nematic liquid crystal, 40 -pentyl-4-biphenylcarbonitrile (5CB), is studied over more than six decades of time (500 femtoseconds to 2 picoseconds) using optical heterodyne detected optical Kerr effect experiments. In contrast to the dynamics of nematogens in the isotropic phase, the OKE data does not decay as a highly temperature-dependent exponential on the longest time scale, but rather, a temperature independent power law spanning more than two decades of time, the final power law, is observed. On short time scales (3 picoseconds to 1 nanoseconds) another power law, the intermediate power law, is observed that is temperature dependent.

1.6.3

Mode coupling theory analysis

Theoretically, two limits of orientational relaxation near I-N transition are understood clearly. The initial short time decay is expected to be exponential with a time constant (' 1-5 picoseconds) given by Enskog theory, extended to ellipsoids [76]. The long-time decay is given by the Landau-de Gennes theory which predicts an 24

1.6 Optical Kerr effect measurements

exponential decay of the second rank orientational correlation function LdG C20 = exp(−t/τLdG )

(1.34)

The Landau-de Gennes time constant τLdG shows a pronounced increase near the I-N transition. The physical origin of this slow down in relaxation is easily understood from a mean-field theory which gives the following expression for τLdG [77] τLdG =

S220 (k = 0) 6DR

(1.35)

where S220 (k) is the wave number dependent orientational structure factor in the intermolecular frame (with k parallel to the Z-axis), and the subscript ’220’ denotes the orientational correlation of spherical harmonics (Y`,m ) of rank `=2 and azimuthal number m=0. DR is the long-time rotational diffusion coefficient of the nematogens in the isotropic phase. As the I-N transition is approached, S220 (k = 0) becomes very large, reflecting the appearance of long-range orientational correlation near the phase transition. Note that S220 (k = 0) does not actually diverge because of the intervention of the (weakly) first order I-N phase transition. While the Landau-de Gennes mean field theory is expected to be valid in the long−1 time (at times much larger than 6DR ), the early short time decay of the orientational

time correlation function should be dominated by the short range local forces. Here the 0 0 decay rate should be close to 6DR , where DR is the short time limit of the rotational

diffusion coefficient. This rate is, for the systems studied by Fayer and coworkers, is of the order of 1011 s−1 or so. Note that the short time rotational diffusion coefficient 0 DR is expected to be much larger than the long-time diffusion coefficient, because

the former is determined by the local interactions while the latter contains the effects of slow density fluctuations. The long-time Landau-de Gennes exponential decay is expected to set in at times (much) longer than nanoseconds. It is the intermediate time range between, say, 10 picoseconds and 10 nanoseconds, which is of interest in this study. In this time range, the simple mean-filed theory is not expected to be valid because the memory function or the rotational friction should be a strong function of frequency. Mode coupling theory predicts interesting dynamical 25

1. INTRODUCTION TO LIQUID CRYSTALS

behaviour at such intermediate times. It suggests that due to rapid increase in the static two particle orientational correlation as the I-N transition is approached, the friction on individual rotors also increases rapidly. This effect is felt at intermediate times when the singular part of the friction is found to behave as [58, 59] √ Γsing (t) ≈ A0 / t

(1.36)

Orientational time correlation function is given as [58, 59] √ C20 (t) ≈ exp(a2 t)erf c(a t)

(1.37)

where erf c is a complementary error function and the quantity a is determined by a combination of orientational structure factor and rotational diffusion coefficient. As mentioned earlier, the above expression is predicted to be valid in the time window which is short compared to τLdG but long compared to (6DR )−1 . Complementary error function has the following asymptotic expansion 2

e−x √ erf c(x) ≈ πx



1 1.3 1.3.5 1− 2 + + + .... 2 2 2x (2x ) (2x2 )3



(1.38)

Thus, the leading term in the expansion varies as t−1/2 . As shown by Gottke et al., mode coupling theory prediction (Eq.4) seems to be capable of describing the experimentally observed decay over two orders of magnitude but not over the entire range of the observed slow decay [58, 59]. Therefore, not only does the mode coupling theory predict a weak time dependence at intermediate times, it also gives a clear physical picture of the origin of this slow decay. At short times, of the order of 1-10 picoseconds or less, orientational relaxation is determined by local dynamics. But at somewhat longer times (that is, beyond 10 picoseconds or so), orientational relaxation requires relaxation of the nearest neighbors. In quantitative terms, this implies that the friction on rotational motion is coupled to the isotropic and orientational density fluctuations of domains. However, fluctuations of these domains are slow to relax, especially near the I-N transition. As a consequence, the friction due to the presence of the non-local, pseudo-nematic density fluctuations 26

1.6 Optical Kerr effect measurements

Figure 1.6: The time derivative of the theoretical correlation function, C20 (t) Eq.1.37. Also shown is a t−0.63 power law (5-OCB). At short time, the derivative of the theoretical correlation function decays essentially as a power law. By the proper choice of the parameter, a, the decay can be made to match the short time portion of the experimental data. (Taken from J. Chem. Phys., 116, 6339 (2002) )

increases with time, that is with lowering frequency. The increase in friction at intermediate times is sufficiently fast to slow down the relaxation to a great extent. After certain time when the increase of friction slows down, the relaxation occurs with constant friction and we get back the exponential decay. This last stage is the Landau-de Gennes regime. Thus, mode coupling theory predicts two cross-over regions – one from a fast, short time exponential decay to a power-law regime of slow decay and a second one from the power law to, an even slower but exponential decay. The duration of the weak time dependence in the intermediate time is predicted to depend on the duration of the growth of the friction. This duration of the growth of the friction is correlated with the length of the long-range correlation present in the system. Thus, we expect the duration of the persistence of the power law to increase as the I-N transition is 27

1. INTRODUCTION TO LIQUID CRYSTALS

approached. Note that τLdG also increases simultaneously because S220 (k ≈ 0) becomes very large. In addition, since the Kerr relaxation experiments measure the time derivative of the polarizability-polarizability time correlation function, a very weak time dependence in experiments implies a nearly linear time dependence of the second rank, collective, orientational correlation function.

1.6.4

Mode coupling theory of Li et al.

Li et al. [78] have very recently developed a schematic mode coupling theory that has been successful in describing the dynamics of liquids near the I-N transition. A very general form for the kinetic equation of any autocorrelation is the following memory function equation [79] φ˙ 2 (t) = −

Z

t

M (t − t0 )φ2 (t0 ),

0

(1.39)

where M (t) is the memory function associated with φ2 (t), which according to Li et al. contains the contribution of the mode coupling and the Landau-de Gennes parts. In this specific case, φ2 (t) is the anisotropy of polarizability. Sjogren’s schematic mode coupling theory [80, 81] postulates a formula for mode coupling memory function that includes the coupling of the dynamics of the polarizability with the dynamics of density fluctuations in intermediate time scale. In the notations of Gotze, the memory function of the mode coupling theory can be expressed as MM CT (t) = Ω22 K2 (t).

(1.40)

where K2 (0)=1, and the time dependence of K2 (t) is expressed in terms of it’s memory function as K˙ 2 (t) = µK2 (t) −

Ω22

Z

t 0

dt0 m2 (t − t0 )K2 (t0 ),

(1.41)

where Ω2 is the frequency of the damped harmonic oscillator. The memory function of K2 (t) has been separated into a short time delta function part whose area is µ and a longer time part proportional to m2 (t). Let φ1 (t) denote the density correlator. The 28

1.6 Optical Kerr effect measurements

standard assumption about m2 (t) in schematic mode coupling theory is m2 (t) = κφ2 (t)φ1 (t)

(1.42)

where φ1 (t) is obtained using the standard F12 model for the density correlator [81, 82]. As described before (see section 1.5.4), Landau-de Gennes theory predicts exponential relaxation of the correlation function, in the long time, φ˙ 2 (t) = −Γφ2 (t)

(1.43)

The relaxation time Γ−1 ( Γ−1 = τLdG ) diverges as (T-TC )−1 as the critical temperature TC of the isotropic nematic transition is approached from above. Equation 1.43 is of the very general form 1.39, where memory function is of the form MLdG = Γδ(t)

(1.44)

According to Li et al. a simple way to get a combined theory is to assume that the total memory function of the correlator of interest is the sum of the mode coupling memory function and the Landau-de Gennes memory function. The total memory function M (t) thus can be written as M (t) = MM CT (t) + MLdG (t)

(1.45)

This is a conjecture is valid because the two theories describe very different effects that dominate on different time scales. It is reasonable to speculate that the appropriate way of combining the two effects is by combining their effect on the memory function. Using equations 1.39 - 1.45, they obtain a second order equation for the orientational correlation function as Z t 2 ¨ ˙ φ2 (t) = −(Ω + µ2 Γ)φ2 (t) − (µ2 + Γ)φ(t) − Ω2 dt0 m2 (t − t0 )φ˙ 2 (t0 ) 0 Z t − Ω22 Γ dt0 m2 (t − t0 )φ2 (t0 ) (1.46) 0

The memory function m2 is related to the density correlator φ1 (t) by equation 1.42 The equation for the density correlation function is given by the F12 model, Z t 2 2 ¨ ˙ dt0 m1 (t − t0 )φ˙ 1 (t0 ) φ1 (t) = −Ω1 φ1 (t) − µφ1 (t) − Ω1 0

m1 (t) = ν1 φ1 (t) + ν2 φ21 (t)

29

(1.47)

1. INTRODUCTION TO LIQUID CRYSTALS

There are a large number of parameters in these equations. In the absence of the memory terms, each equation is the equation for a damped harmonic oscillator with frequencies Ω1 and Ω2 , and the associated damping constants µ1 and µ2 . ν1 and ν2 are the amplitudes in the memory kernel m1 for the F12 model that describes the density, and κ is the coupling constant that reflects the strength of the coupling between the orientational correlation function, φ2 , and the density correlation function, φ1 . In addition there is the LdG decay constant, Γ with Γ−1 = τLdG with τLdG given in Eq. 1.33. They have obtained the fitting parameters using different fitting methods. The results show that the new schematic mode coupling theory is successful in reproducing the nematogen isotropic phase OHD-OKE data on all time scales and at all temperatures they have studied.

1.6.5

Hydrodynamics of liquid crystals

The hydrodynamics of molecular liquids is well studied. The inherent anisotropy of liquid crystal phases makes hydrodynamics of liquid crystal different from that of ordinary liquids. In addition to shear and sound wave modes, a new hydrodynamic mode appears in this theory, which is associated with fluctuations of the director. The foundation of the continuum model of nematic liquids crystals is laid by many. Oseen and Zocher [83, 84] gave first successful static theories of nematic state. These theories further modified by Frank [85] in terms of curvature elasticity. The hydrodynamic equation for nematic liquid crystal is formulated by Erickson [86, 87] and Leslie [88, 89].

1.7

Conclusion

As already discussed, the very nature of the I-N transition has drawn a lot of discussion. The well-known facts regarding the nature of I-N transition can be summarized as follows. Due to pure symmetry arguments of the order parameter Qαβ , I-N transitions is first order according to the well-known Landau-de Gennes theory. Experiments as well as simulations record a low value of the latent heat than that predicted by the mean 30

1.7 Conclusion

field theory. However, the analysis and experiments in the light of the theories of second order transition also fail to give a true picture of the I-N transition. Recent experiments of Fayer and coworkers shed light on some important features of the orientational relaxation of nematogens near the I-N transition, which is worthwhile examining. This thesis analyzes important transport properties and time correlation functions near IN transition of Gay-Berne fluid. The language used for the study of orientational relaxation is similar to that used in supercooled liquids and glass transition. We look at this problem as a glass transition that occurs mostly in orientational degree of freedom. The first part of this thesis addresses the microscopic relaxation processes that take place during orientational ordering that accompany liquid (isotropic) to nematic (only orientationally ordered) phase transition (I-N transition). Long time molecular dynamics simulations are used compute the time correlation functions, which may improve the present understanding of the I-N transition in the high frequency and large wave number limit. Note that the hydrodynamic description of this I-N transition is well addressed in the literature.

31

1. INTRODUCTION TO LIQUID CRYSTALS

32

Bibliography [1] P. G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974). 1.1, 1.3, 1.4, 1.5.1, 1.5.2, 1.5.3, 1.6.2 [2] M. J. Stephen and J. P. Straley, Rev. Mod. Phys. 46, 617 (1974). 1.1, 1.2, 1.4, 1.5.1, 1.5.2 [3] S. Chandrasekhar, Liquid Crystals (Cambridge University Press, Cambridge, 1977). 1.1, 1.3, 1.5.1, 1.5.2 [4] Critical Reports on Applied Chemistry: Thermotropic liquid crystals, edited by G. W. Gray (John Wiley & Sons, New York, 1987), Vol. 22. 1.3 [5] G. Friedel, Ann. Phys. (Paris) 19, 273 (1922). 1.3 [6] S. C. P. Martinoty and F. Debeauvais, Phys. Rev. Lett. 27, 1123 (1971).

1.4,

1.5.4 [7] T. W. Stinson and J. D. Lister, Phys. Rev. Lett. 25, 503 (1970). 1.4, 1.5.4 [8] T. W. Stinson and J. D. Lister, Phys. Rev. Lett. 30, 688 (1973). 1.4, 1.5.4 [9] S. K. Ghosh, E. Tettamanti, and A. Panatta, Phys. Rev. B 21, 1194 (1980). 1.4 [10] G. K. L. Wong and Y. R. Shen, Phys. Rev. Lett. 30, 895 (1973). 1.4, 1.5.4, 1.6.1, 1.6.2 [11] G. K. L. Wong and Y. R. Shen, Phys. Rev. A 10, 1277 (1974). 1.4, 1.5.4, 1.6.1 [12] E. G. Hanson, Y. R. Shen, and G. K. L. Wong, Phys. Rev. A 14, 1281 (1976). 1.4, 1.5.4, 1.6.1 33

BIBLIOGRAPHY

[13] C. Rosenblatt, Phys. Rev. A 27, 1234 (1983). 1.4 [14] P. H. Keyes and J. R. Shane, Phys. Rev. Lett. 42, 722 (1979). 1.4, 1.5.5 [15] Y. Poggi, P. Atten, and R. Aleonard, Phys. Rev. A 14, 466 (1976). 1.4 [16] J. M. Vaughan, Phys. Lett. A 58, 325 (1976). 1.4 [17] L. Onsager, Ann. N. Y. Acad. Sci. 51, 621 (1949). 1.4, 1.5.2 [18] R. Zwanzig, J. Chem. Phys. 39, 1714 (1963). 1.4, 1.5.2 [19] P. G. de Gennes, Phys. Lett. A 30, 454 (1969). 1.4, 1.6.2 [20] P. G. de Gennes, Mol. Cryst. Liq. Cryst. 12, 193 (1971). 1.4 [21] W. Maier and A. Saupe, Z. Naturf. A 13, 564 (1958). 1.4, 1.5.1 [22] W. Maier and A. Saupe, Z. Naturf. A 14, 882 (1959). 1.4, 1.5.1 [23] M. Baus, J. L. Colot, X. G. Wu, and H. Xu, Phys. Rev. Lett. 59, 2184 (1987). 1.4 [24] J. L. Colot, X. G. Wu, H. Xu, and M. Baus, Phys. Rev. A 38, 2022 (1988). 1.4 [25] H. Mulder and D. Frenkel, Mol. Phys. 55, 1193 (1985). 1.4 [26] Y. Singh, Phys. Rev. A 30, 583 (1984). 1.4 [27] J. P. Singh and Y. P. Singh, Phys. Rev. A 33, 2725 (1986). 1.4 [28] E. DeMiguel, L. F. Rull, K. E. G. M. K. Chalam, and F. V. Swol, Mol. Phys. 72, 593 (1991). 1.4 [29] A. Perera, G. N. Patey, and J. J. Weis, J. Chem. Phys. 89, 6941 (1988). 1.4 [30] A. Stroobants, H. N. W. Lekkerkerker, and D. Frenkel, Phys. Rev. A 36, 2929 (1987). 1.4 [31] J. A. C. Veerman and D. Frenkel, Phys. Rev. A 41, 3237 (1990). 1.4 34

BIBLIOGRAPHY

[32] P. Bolhuis and D. Frenkel, J. Chem. Phys. 106, 666 (1997). 1.4 [33] P. A. Labwohl and G. Lasher, Phys. Rev. A 6, 426 (1972). 1.4, 1.5.5 [34] P. A. Labwohl and G. Lasher, Phys. Rev. A 7, 2222 (1973). 1.4, 1.5.5 [35] Z. Zang, O. G. Mouristen, and M. J. Zuckermann, Phys. Rev. Lett. 69, 2803 (1992). 1.4 [36] G. J. Vroege and H. N. W. Lekkerkerker, Rep. Prog. Phys. 55, 1241 (1992). 1.4 [37] W. Maier and A. Saupe, Z. Naturf. A 15, 287 (1960). 1.5.1 [38] Handbook of mathematical Functions with Formulas Graphas and Mathematical Tables, edited by M. Abramowitz and I. Stegun (Dover Publications, INC., New York, 1964). 1.5.1 [39] P. J. Flory, Proc. R. Soc. Lond. A 234, 60 (1957). 1.5.2 [40] P. J. Flory, Proc. R. Soc. Lond. A 234, 73 (1957). 1.5.2 [41] P. J. Flory, J. Chem. Phys. 10, 31 (1942). 1.5.2 [42] E. A. DiMarzio, J. Chem. Phys. 35, 638 (1961). 1.5.2 [43] M. A. Cotter and D. E. Martire, J. Chem. Phys. 52, 1902 (1970). 1.5.2 [44] M. A. Cotter and D. E. Martire, J. Chem. Phys. 52, 1909 (1970). 1.5.2 [45] M. A. Cotter and D. E. Martire, J. Chem. Phys. 53, 4500 (1970). 1.5.2 [46] G. Lasher, J. Chem. Phys. 53, 4141 (1970). 1.5.2 [47] R. Alben, Mol. Cryst. Liq. Cryst. 13, 193 (1971). 1.5.2, 1.5.5 [48] R. G. Priest, Phys. Rev. A 8, 3191 (1973). 1.5.2 [49] L. D. Landau, Phys. Z.Sowjetunion 11, 26 (1937). 1.5.3 [50] C. Fan and M. J. Stephen, Phys. Rev. Lett. 25, 500 (1970). 1.5.3 35

BIBLIOGRAPHY

[51] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993). 1.5.4, 1.6.2 [52] H. Imura and K. Okano, Chem. Phys. Lett. 17, 111 (1972). 1.5.4 [53] H. Imura and K. Okano, Chem. Phys. Lett. 19, 387 (1973). 1.5.4 [54] S. Ruhman et al., J. Phys. Chem. 91, 2237 (1987). 1.5.4 [55] D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, IEEE J. Quant. Elec. 24, 443 (1988). 1.5.4, 1.6.2 [56] J. J. Stankus, R. Torre, and M. D. Fayer, J. Phys. Chem. 97, 9478 (1993). 1.5.4 [57] H. Cang, J. Li, and M. D. Fayer, Chem. Phys. Lett. 366, 82 (2002). 1.5.4, 1.6.2 [58] S. D. Gottke et al., J. Chem. Phys. 116, 360 (2002).

1.5.4, 1.6.2, 1.6.2, 1.6.3,

1.6.3, 1.6.3 [59] S. D. Gottke, H. Cang, B. Bagchi, and M. D. Fayer, J. Chem. Phys. 116, 6339 (2002). 1.5.4, 1.6.2, 1.6.2, 1.6.3, 1.6.3, 1.6.3 [60] H. Cang, J. Li, V. N. Novikov, and M. D. Fayer, J. Chem. Phys. 118, 9303 (2003). 1.5.4, 1.6.2 [61] H. Cang, J. Li, V. N. Novikov, and M. D. Fayer, J. Chem. Phys. 119, 10421 (2003). 1.5.4, 1.6.2 [62] J. J. Stankus et al., Chem. Phys. Lett. 194, 4193 (1992). 1.5.4 [63] G. Hinze, D. D. Brace, S. D. Gottke, and M. D. Fayer, Phys. Rev. Lett. 84, 2437 (2000). 1.5.4 [64] J. Li, I. Wang, and M. D. Fayer, J. Phys. Chem. B 109, 6514 (2005). 1.5.4, 1.6.2 [65] R. G. Priest and T. C. Lubensky, Phys. Rev. A 13, 4159 (1975). 1.5.5 [66] K. G. Wilson and J. Kogut, Phys. Rep. 12, 77 (1974). 1.5.5 36

BIBLIOGRAPHY

[67] S. keng Ma, Rev. Mod. Phys. 45, 589 (1973). 1.5.5 [68] P. H. Keyes, Phys. Lett. A 67, 132 (1978). 1.5.5 [69] J. Kerr, Phil. Mag. 50, 337 (1875). 1.6 [70] A. D. Buchingham and J. A. Pople, Proc. Phys. Soc. A 68, 905 (1955). 1.6 [71] A. D. Buchingham, Proc. Phys. Soc. London B 69, 344 (1956). 1.6 [72] G. R. Fleming, Chemical applications of ultrafast spectroscopy (Oxford university press, New York, 1986). 1.6.2 [73] J. A. Valdmanis, R. L. Fork, and J. P. Gordon, Opt. Lett. 10, 131 (1980). 1.6.2 [74] Y. Yan and K. A. Nelson, J. Chem. Phys. 87, 6240 (1987). 1.6.2 [75] Y. Yan and K. A. Nelson, J. Chem. Phys. 87, 6257 (1987). 1.6.2 [76] R. G. Cole and G. T. Evans, Ann. Rev. Phys. Chem. 37, 105 (1986). 1.6.3 [77] B. Bagchi and A. Chandra, Adv. Chem. Phys. 80, 1 (1991). 1.6.3 [78] J. Li, H. Cang, H. C. Andersen, and M. D. Fayer, private communication, 2005. 1.6.4 [79] W. Gotze, in Liquid Freezing and Glass Transition, edited by J. P. H. D. Levesque and J. Zinn-Justin (North Holland, Amsterdam, 1990). 1.6.4 [80] W. Gotze and M. Sperll, Phys. Rev. Lett. 92, 105701 (2004). 1.6.4 [81] L. Sjogren, Phys. Rev. A 33, 1254 (1986). 1.6.4, 1.6.4 [82] T. Franosch, W. Gotze, M. R. Mayr, and A. P. Singh, Phys. Rev. E 55, 3183 (1997). 1.6.4 [83] C. W. Oseen, Disc. Faraday Soc. 29, 883 (1933). 1.6.5 [84] H. Zocher, Disc. Faraday Soc. 29, 945 (1933). 1.6.5 37

BIBLIOGRAPHY

[85] F. C. Frank, Disc. Faraday Soc. 25, 19 (1958). 1.6.5 [86] J. L. Ericksen, Arch. Rotational Mech. Anal. 4, 231 (1960). 1.6.5 [87] J. L. Ericksen, Trans. Soc. Rheol. 5, 23 (1961). 1.6.5 [88] F. M. Leslie, Quart. J. Mech. Appl. Math. 19, 357 (1966). 1.6.5 [89] F. M. Leslie, Arch. Rotational Mech. Anal. 28, 265 (1968). 1.6.5

38

Chapter 2 Overview of the molecular simulations 2.1

Introduction

This chapter presents a brief introduction of simulations in molecular systems using different intermolecular interaction potentials. In particular, the Gay-Berne potential, which is an anisotropic extension of the well-known Lennard-Jones potential. This chapter also includes discussion on translational and rotational static pair correlations near the I-N transition.

2.2

Molecular models

Simulations of molecular systems have long been used to bridge the gap between the theory and the experiments. Unlike the atomic systems, the statistical mechanics of the molecular systems are still not well studied in simulations. This may be due to the difficulty in getting an effective potential that can fit different class of molecules which differs in their geometries. One of the successful approaches to study the molecular anisotropy is to include all the atomic interactions in the simulations. Earlier molecular dynamics and Monte Carlo simulations using elaborate molecular potentials could find the phase behaviour of liquid crystals [1, 2]. There are attempts to use the fully atomistic models to study the phase behaviour of nematic, smectic and discotic phases. These simulations were largely crippled by the small size of the system. Moreover, this approach has become 39

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

exceedingly difficult in different cases due to the complexity of intermolecular interactions. Another approach is to use the hard objects [3] such as hard spheres, ellipsoids, cylinders and spherocylinders to mimic molecular systems. According to the molecular field theories of Onsager and Zwanzig, a system of linear molecules with excluded volume interaction alone can form the orientationally ordered nematic phase [4, 5]. System of hard spheres have been extensively used for the study of freezing and phase behaviour of atomic systems. The results from these studies are used as test for the colloidal systems that behave nearly as hard spheres. The computer simulations studies of Alder and Wainwright [6], and Wood and Jacobson [7] proved that the hard spherical particles can undergo a first order freezing transition. Motivated by the success of these results, hard anisotropic objects have been used to study the phase behavior of anisotropic colloids. They show various liquid crystalline phases. According to Onsager, a system of thin rods at sufficiently high density forms nematic phase [5]. There exist three phases for the hard ellipsoids; they are isotropic, nematic and solid. The smectic phase is absent in the phase diagram of hard ellipsoids [8–10]. They also proved that pure hard core interaction is only required to form stable smectic and columnar phases. The phase diagram of the spherocylinders were studied by the several researchers. Frenkel and coworkers has been successful in constructing the phase diagram of spherocylinders for different aspect ratios at different densities [8, 11, 12]. Figure 2.2 shows the phase diagram of spherocylinders, which shows a rich phase behaviour, including the smectic phase. Long range intermolecular potential models play an important role in the molecular simulations. They include the attractive interactions, while calculating the phase behaviour and other molecular properties. For atomic systems, the most successful among them is the Lennard-Jones 12-6 intermolecular potential. The Lennard-Jones potential has been extended by Berne, Pechukas, and Berne [13, 14] to include anisotropy due to shape. Another important intermolecular potential which can model the anisotropy of the 40

2.2 Molecular models

Figure 2.1: Phase diagram of spherocylinders of aspect ratio L/D < 5. All two-phase regions are shown shaded. In the figure, the following phases can be distinguished: the low-density isotropic liquid, the high-density orientationally ordered solid, the low-L/D plastic solid and, for L/D > 3.7, the nematic and smectic-A phases. (Taken from J. Chem. Phys. 106, 666 (1997)) molecule is the Kihara potential [15].

2.2.1

Gay-Berne ellipsoid model

In 1972, Berne and Pechukas [13] proposed a Gaussian overlap intermolecular potential to account the anisotropy in the intermolecular interaction. How the Gaussian overlap model mimics a real molecule is shown in the figure 2.2. This model was devised primarily to give a simple expressions for the orientation dependence of the molecular interactions; it is not expected to mimic the distance dependence of the realistic potentials. The Berne-Pechukas intermolecular potential is similar to the Lennard-Jones 12-6 interaction potential with orientation dependent range and energy parameters. The Berne-Pechukas Gaussian overlap model was used in the simulations of Kushik and Berne [16] to study the stability of the nematic phase and for the illustration of the effect of cooperative reorientation on spectral line shapes. The Gay-Berne model of intermolecular potential is a modification of the Berne-Pechukas Gaussian overlap model [14]. The simulations presented in this work uses this important variant of 41

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

Figure 2.2: Modeling of the ethylbenzene. (Taken from J. Chem. Phys. 56, 4213 (1972) .) Berne-Pechukas model, which is known as Gay-Berne [14] ellipsoid model. This model maps a multisite Lennard-Jones model to a single site model. The difference between the Berne-Pechukas model and the Gay-Berne model manifests primarily in the simulations at low pressure. In a two dimensional system, the former condense to a phase where each ellipsoid forms the sides of a hexagonal lattice and the latter condense to a perfect nematic phase. The Gay-Berne potential shows same phase behaviour as that of Berne-Pechukas potential at high pressures where the repulsive sterric interactions 42

2.2 Molecular models

are important. The form of the modified inter-molecular Gay-Berne potential is, [14, 17] " 12  6 # σs σs − U = 4(ˆ r , ~ui , ~uj ) , r − σ(ˆ r, ~ui , ~uj ) + σs r − σ(ˆ r , ~ui , ~uj ) + σs

(2.1)

where rˆ is the unit vector that passes through the center of mass of a pair of molecules, ~ui and ~uj unit vectors that passes through the major axis of a pair of ellipsoidal molecules and  and σ give the strength and range of interaction. The σ can be defined as σ(ˆ r , ~ui , ~uj ) = σs



χ 1− 2



(~ui · rˆ + ~uj · rˆ)2 (~ui · rˆ − ~uj · rˆ)2 + 1 + χ(~ui · ~uj ) 1 − χ(~ui · uj )



,

(2.2)

where σs is double of the minor axis b, κ gives molecular elongation (aspect ratio), which is the ratio of end-to-end to side-to-side diameters, κ = σe /σs . In the expression for σ, χ can be given in terms of κ as χ = 

2

(ˆ r , ~ui , ~uj ) = 0 1 − χ (~ui · ~uj )

− 21

κ2 −1 κ2 +1

  2 χ0 (~ui · rˆ + ~uj · rˆ)2 (~ui · rˆ − ~uj · rˆ)2 1− + 2 1 + χ(~ui · ~uj ) 1 − χ(~ui · uj ) (2.3)

where 0 is the energy parameter and χ0 =

√ 0 √κ −1 κ0 +1

(κ0 = s /e gives the strength of

interaction which is side-to-side to end-to-end well depths). In general (ˆ r , ~ui , ~uj ) and χ0 are expressed as   µ  −ν/2 χ0 (~ui · rˆ + ~uj · rˆ)2 (~ui · rˆ − ~uj · rˆ)2 2 + (ˆ r , ~ui , ~uj ) = 0 1 − χ (~ui · ~uj ) 1− 2 1 + χ(~ui · ~uj ) 1 − χ(~ui · uj ) (2.4) χ0 =

κ0 1/µ − 1 κ0 1/µ + 1

(2.5)

The parameters ν and µ are originally adjusted so as to obtain a good fit to the four site Lennard-Jones potential with an overall elongation of 3:1. The exponents are fixed at µ = 2 and ν =1 in the original paper of Gay and Berne [14]. A phase diagram of the Gay-Berne fluid has been proposed by de Muguel et al. [18] for the standard parameters used. This is given in figure 2.3. The choice of the ratio κ is relatively straightforward, since for a real liquid crystal system, the length-to-breadth ratio of the constituent molecules must be equal to or greater than about 3:1. Accordingly, 43

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

Figure 2.3: The phase diagram of the Gay-Berne fluid as a function of temperature and density (Taken from Adv. Chem. Phys. 109, 39 (1999)) κ was set equal to 3 by Gay and Berne [14]. The selection of the parameters are by comparison of the potential with that of a pair of particles, each formed by four Lennard-Jones sites placed along a line, such that the length to-breadth ratio was 3:1. This gave κ0 as 5 and the exponents µ and ν as 2 and 1, respectively [14]. There are two problems with this composite Lennard-Jones model; first, the shape of a Gay-Berne particle is ellipsoidal whereas that for the four Lennard-Jones centers approximates to a spherocylinder. Secondly, the molecular structure of a typical nematogen, such as 4, 40 -dimethoxyazoxybenzene, is significantly more complex than that of the composite Lennard-Jones model. Despite these problems, the original parametrization of the GayBerne potential have proved to be remarkably successful. It has a relatively rich phase behavior with isotropic, nematic and smectic B phases having been clearly identified. 44

2.2 Molecular models

[18–20] In addition, there are claims to have observed a smectic A [20], a tilted smectic B [18] and a rippled smectic B phase [21]. The properties of the phases and the transitions between them also appear to be in reasonable agreement with those found for real mesogens. Such success has contributed these developments of models that include the introduction of dipolar forces [22], flexibility [23], the construction of more complex particle shapes based on a collection of Gay-Berne sites [24] and the extension to biaxial particles [25]. Berardi et al.

[26] have developed a biaxial version of

the Gay-Berne potential that can be used to model uniaxial anisotropic molecules. This biaxial potential can be used to deal with molecules with different attractive and repulsive contributions along their three axes. The Gay-Berne potential is mainly used for simulation of a system of identical particles. Cleaver et al. modified the BernePechukas original Gaussian overlap formalism to obtain an explicit expression for the potential of non-equivalent particles. de Miguel has presented a computer simulation study regarding the effect of system size on the isotropic-nematic transition in a molecular fluid model [27]. In this study, a system of 256, 500 and 864 molecules interacting through Gay-Berne inter molecular potential is analyzed along an isotherm using molecular dynamics in the NVT ensemble. In this system, the transition weakens as N gets larger and the transition shifts to higher densities. In another important work, de Muguel et al. [28] have studied the dependence of the phase behaviour of the Gay-Berne liquid on the anisotropy parameter κ0 . They have carried out constant NVT molecular dynamics simulations of the GayBerne fluid with exponents µ = 2 and ν = 1, molecular elongation κ = 3, and various values of the well-depth anisotropy parameter κ0 : 1, 1.25, 2.5, 5, 10, 25. They also studied the isotherm (T ∗ = 0.70), which turns out to be subcritical for κ0 = 1 and 1.25, and supercritical for the other values. For κ0 ≥ 5, the system shows a transition from the isotropic fluid to the smectic B phase. The estimated transition density shifting down slightly as κ0 increases. For κ0 ≤ 2.5 a transition from the isotropic liquid to the nematic phase is seen; the estimated transition density shifting up slightly as κ0 increases. Additionally, for κ0 = 2.5, they observe a transition from the nematic phase 45

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

to the smectic B: the nematic range is quite narrow for this case. Figure 2.4 shows the phase diagram of the Gay-Berne fluid obtained by de Muguel et al. [28] for limiting cases of the κ0 . Bates and Luckhurst have investigated the behavior of a Gay-Berne liquid crystal with the four parameters characterizing the potential chosen to have values similar to those expected for real mesogenic molecules [29]. Isothermal-isobaric Monte Carlo simulations have shown that the mesogen exhibits isotropic, nematic, smectic A and smectic B phases, depending on the choice of the pressure. They have also determined the phase diagram for this mesogen and have found that it is in good agreement with those found for real mesogens. This parametrization of the potential also seems to predict reasonably accurately the transitional properties, with the exception of the fractional volume change at the nematic-isotropic transition. Here, the Gay-Berne potential gives a value approximately ten times larger than that observed experimentally. It is likely that this failure of the potential results from the ellipsoidal shape of a particle. Gay-Berne ellipsoid model has been used for the study of the complex models also. Molecular dynamics simulations of liquid crystal molecules composed of two Gay-Berne particles connected by an eight-site Lennard-Jones alkyl chain have been carried out by Wilson [30]. This system exhibits the sequence of phases: isotropic liquid, smecticA, smectic-B. These simulations demonstrate the spontaneous growth of a smectic-A liquid crystal over a period of approximately 6 ns on cooling from the isotropic liquid. Model molecules are seen to remain flexible and able to change conformation in the smectic-A phase.

2.3

Static orientational pair correlation function

The study of molecular liquids differs from simple liquids in many aspects due to the anisotropy in the intermolecular potential. Static and dynamic correlation functions of system of N molecules have to consider the contribution not only from 3N translational degrees of freedom but also from 3N orientational degrees of freedom. The additional 46

2.3 Static orientational pair correlation function

Figure 2.4: Schematic representation of the phase diagram of Gay-Berne fluids in the temperature and density [(a), (c), and (e)] and pressure and temperature [(b), (d), and (f)] representations. Phase diagrams (a) and (b) correspond to the expected behavior of Gay-Berne fluids with large values of κ0 , where the nematic phase is only stable at high pressures (or temperatures). Phase diagrams (c) and (d) correspond to the case in which the nematic phase is stable up to the isotropic - nematic - smectic triple point (T0 ) below the critical temperature (Tc ), but with no nematic - vapor coexistence. Phase diagrams (e) and (f) correspond to Gay-Berne fluids with low values of κ0 . Nematic - vapor coexistence occurs for a range of temperatures between the vapor - nematic smectic B triple point (T2 ) and the vapor - isotropic - nematic triple point (T1 ). (Taken from reference [28]) 47

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

degrees of freedom may contribute to static and dynamic response of the system. I-N transition is accompanied by the change in thermodynamic parameters. Important thermodynamic parameters can be expressed microscopically in terms of the pair correlation function. Hence, it is worthwhile to study the variation of pair distribution functions, while the I-N transition takes place. The generalized coordinates of a system

x

θ1

φ

z

θ2

y Figure 2.5: given.

The schematic representation of the inter-molecular frame of reference

of linear molecules are positions ri , velocities vi and orientations Ωi and angular velocities ωi , where i = 1, ..N . The number of parameters required to represent a system of molecule is more than that of atomic liquid. In the case of the molecular systems the pair correlation function depend not only on the distance but also on the orientation of the molecules. The radial distribution function of non-spherical molecules in a laboratory fixed frame can be expressed in terms of average over delta function of positions and orientations as [31, 32], * N N 0 + XX 1 g(r, Ω, Ω0) = δ(r + rj − ri )δ(Ω − Ωi )δ(Ω0 − Ωj ) , Nρ i j

(2.6)

(the prime over the summation signifies that terms i = j are excluded from the sum) where ri is the translational coordinate in the arbitrary reference system and Ωi is the polar angle for the i-th linear molecule in the laboratory fixed reference frame. 48

2.3 Static orientational pair correlation function

This pair correlation function can be conveniently represented in an inter-molecular reference frame (See Fig.2.5) where z axis passes through the center of mass of two ellipsoids. The pair correlation function in this reference frame is given as g(r, ω, ω 0). It gives the joint probability to find any two ellipsoids separated by a distance r and with angular coordinates ω and ω 0 . This orientational correlation function can be expanded in spherical harmonics [31, 32]. g(r, ω, ω 0) =

X

gl1 l2 m (r)Yl1 ,m (ω)Yl2 ,m (ω 0 )

(2.7)

l1 ,l2 ,m

i.e., the ω = (θ, φ) and ω 0 = (θ 0 , φ0 ) are the angular coordinates in the inter-molecular frame. θ defines the angle, an ellipsoid makes with the z axis of the intermolecular frame and φ defines the angle the ellipsoid makes with the x-axis, in the x−y plane of the intermolecular frame. The angular pair correlation functions are obtained from the total pair correlation function using the orthogonal properties of the spherical harmonics [32]. These angular pair correlation functions can be calculated from simulation using the expression [29, 32] 

gl1 l2 m (r) = 16π 2 g(r, ω, ω 0)Yl1∗,m (ω)Yl∗2 ,m (ω 0 ) .

(2.8)

The distance dependence of angular pair correlation function gives the range of orientational order among the molecules in a system of nematogens. The structural changes that appears in various phases can be identified from the pair correlation function g(r) of the number density, which is given by * N N 0 + XX 1 g(r) = δ(r + rj − ri ). Nρ i j

(2.9)

g(r) obtained by integrating over the orientation dependence of g(r, Ω, Ω0) and assuming the isotropy of the liquid. S(k), the spatial Fourier transform of g(r), gives the structural changes in the reciprocal space (k is the reciprocal distance), which can be measured in experiments. The expression for the structure factor S(k) is given by [31] Z S(k) = 1 + ρ e−ik·r g(r)dr (2.10) For an isotropic liquid S(k) is same as S(k). 49

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

2.4

Details of the simulations

The simulations have been performed for a system of 576 Gay-Berne ellipsoids along the isotherm in the microcanonical (NVE) ensemble. The parameters used for the Gay-Berne potential is given in the section 2.2.1. The temperature of the system is set to the desired value initially by scaling at the regular intervals until the temperature of the system is fluctuating around the desired value. Approximately 0.5 × 106 steps are used for scaling the temperature. This is followed by another 0.5 × 106 steps to obtain the equilibrated configuration. Production steps starts from this initial configuration. 107 production steps are used to calculate the radial distribution functions and structure factors. Leap frog algorithm [33] is used to integrate the translational and rotational equation of motion. The same simulation procedure is used in the investigations described in first part of the thesis; different simulations vary only in their number of steps only. Mainly the studies are along an isotherm at T ∗ =1.0. In the first part of thesis, the density is varied from ρ∗ = 0.285 to 0.315 in different simulations along an isotherm to study the properties of the I-N transition.

2.5

Pair correlation functions

Various phases that appear in the simulation of the Gay-Berne fluid along an isotherm can be identified from the snapshots of the simulations box, given in the figure 2.6. Figure 2.7 shows the variation in the orientational order parameter given by Eq. 1.1, for the set of parameters of Gay-Berne, potential used in the simulations presented in this thesis, in a microcanonical (NVE) ensemble over the temperature and density range studied. Here Sijαβ is diagonalized and it’s highest absolute value is taken as the order parameter (S). The choice of the ensemble comes from the assumption that, the microcanonical ensemble generates true dynamics of the system [34]. The phase diagram of the Gay-Berne potential is sensitive to fluctuations. Depending on the selection of the ensemble, the phase boundary shifts. Hence the values of the orientational order parameter given in figure 2.7 identify various phases and phase boundaries at different state points, where the study of the dynamics has been studied. 50

2.5 Pair correlation functions

Figure 2.6: Snapshot of the various phases observed in the simulation box. Left side shows the orientations and the right side shows the ellipsoids. From the top to bottom various phase that appear successively are isotropic, nematic, Smectic and crystalline.

In the standard phase diagram used in this study (see Fig.2.3 ), there exist a triple point for the range of state points chosen. Along the isotherm at low temperature, while the isotropic to smectic transition takes place upon the increase of density, a sudden jump is observed in the orientational order parameter. At higher temperatures, along an isotherm, the variation of the order parameter is gradual than what observed at low temperature. A triple point can be identified in the figure 2.7 around temperature T ∗ ≈ 0.7 and density ρ∗ ≈ 0.3. At high temperatures, the nematic phase appears between the 51

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

1

Order parameter

0.8 0.6 0.4 0.2 0 0.4

1 0.2

Density

0.8 0

0.6

Temperature

Figure 2.7: The values of the orientational order parameter obtained from simulations of the micro canonical ensemble over the range of temperatures and densities studied are shown.

isotropic and smectic phases along the isotherm. Hence in case of Gay-Berne fluid the nematic phase appears because it is entropically favoured at high temperatures. In the phase diagram at high temperatures, along an isotherm, the width of the nematic phase increases in the density axis. The phase diagram is also found to be more sensitive to the variation in density rather than temperature. The state points chosen for the investigation is along an isotherm at temperature T ∗ = 1; the density is varied along this isotherm to probe the variation in static and dynamic correlation functions. In the studies along the isotherm (T ∗ =1) described in this thesis, the I-N transition appears near the density ρ∗ =0.315. The next state point studied is at ρ∗ = 0.32, which is in the nematic phase. As the density increases further, the smectic phase appears between ρ∗ =0.33 and ρ∗ = 0.34. Finally crystalline phase appears at ρ∗ =0.4. Various phases are identified through different order parameters. The isotropic nematic phase transition is identified by the variation of the orientational order parameter Sαβ (Eq. 1.1). 52

2.5 Pair correlation functions

1.6 1.4 1.2

g(r)

1 0.8 0.6 0.4 0.2 0 0

1

2

3

r*

4

5

6

Figure 2.8: The orientation averaged radial distribution function of nematogens near the I-N transition. The peak heights of g(r) increases as density increases from ρ∗ =0.285 to ρ∗ =0.315 on a grid of δρ∗ =0.005.

2.5.1

Static pair correlation function near the I-N transition

The growth of pair correlation of the number density across the I-N transition is visible in the radial distribution function, which is given in the figure 2.8 . The pair distribution function of the Gay-Berne liquid show the same characteristics as that of an atomic liquid. The peak height gradually increases as the density increases. The corresponding structure factor (Fig. 2.9) clearly shows the structural change that accompanying the I-N transition. Compared to the atomic liquid the structure factor shows a peak at intermediate wavenumber k ≈ 2π/a. This peak becomes clearer as the I-N transition occurs (ρ∗ varies from 0.285 to 0.315). Finally, in the nematic phase near the phase the boundary, the structure factor shows a plateau. This is an indication of the growth of anisotropy in the liquid due to the freezing of orientational degrees of freedom. Since the I-N transition is mainly a phase transition in the orientational degrees of freedom, a study on orientation dependent pair correlation function across I-N transition shows significant changes in their shape. Figure 2.10 shows various angular pair correlation functions in isotropic liquid and near the I-N transition. The growth of cor53

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

1.5 1 0.5

ρ*=0.29

ρ*=0.285

00

S(k)

1 0.5 0

ρ =0.295

ρ*=0.3

ρ*=0.305

ρ*=0.31

ρ*=0.315

ρ*=0.32

*

1 0.5 00 1 0.5 0 0

10

20

5

10

Figure 2.9: The orientation averaged structure factor of nematogens across the I-N transition.

relation length reflected in the non-zero value of the angular pair correlation function at large pair separation. Among the different angular pair correlation functions the l1 = 2, l2 = 2, m = 0 component shows the most significant change. This component get maximum contribution when the ellipsoids are aligned perfectly parallel (θ1 = π/2, θ2 = π/2 or θ1 = 0, θ2 = π ). Hence the coefficients of the Y20 (Ω)Y20 (Ω0 ) component in the expansion of the static and dynamic pair correlation function is most suitable for analysis of the I-N transition. Note that the other components also show the growth of the orientational correlation near the I-N transition: hence a complete analysis of the growth of the correlation length may include all the components of the angular pair 54

2.5 Pair correlation functions

1

g

(r)

220

0.5

g

420

g442(r)

gl

12

l m

(r)

0

−0.5

g440(r) g

(r)

420

−1

g

(r)

400

g200(r)

−1.5 0

1

2

1.5 g

3

distance

5

6

5

6

g (r) 400 (r)

220

1

4

g

(r)

442

g

(r)

440

0

gl

12

l m

(r)

0.5

−0.5 g

−1 g

−1.5 0

(r)

422

(r)

200

1

g

(r)

420

2

3

distance

4

Figure 2.10: The orientation dependent pair correlation function at the isotropic phase and at the transition point. Figure at he top shows the orientation dependent angular pair correlation functions are plotted against distance at density ρ∗ =0.285 in a dense isotropic liquid. The figure at the bottom shows the orientation dependent pair correlation functions plotted against distance near I-N transition ρ∗ =0.315

55

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

(r)

0 200

−0.5

g

(a)

−1 1

2

1

2

g220(r)

1

3

*

(b) r

4

ρ*=0.285 * ρ =0.290 ρ*=0.295 ρ*=0.30 ρ*=0.305 5 6 ρ*=0.31 ρ*=0.315

0.5 0 3

(c) r

*

4

5

6

g440(r)

0.4 0.2 0 0

2

*

4

6

r

Figure 2.11: Three angular pair correlation functions are plotted at different densities against distance of separation at T ∗ =1.0 along an isotherm: figure (a) shows g200 (r); figure (b) shows g220 (r); figure (c) shows g440 (r).

correlation function. As already mentioned, the I-N transition occurs above ρ∗ = 0.3 for all temperatures (T ∗ = 0.6 to 1.2) used in the simulations. Figures 2.11(a), 2.11(b) and 2.11(c) show the distance dependence of angular pair correlation functions plotted against position in a system of Gay-Berne ellipsoids at temperature 1.0 for the range of number densities ρ∗ = 0.285 - 0.315. The angular correlation function g200 (r) (plotted in figure 2.11(a)) 56

2.5 Pair correlation functions

shows the increase in the peak height as the I-N transition is approached. g200 (r) correlates distribution of the orientation of ellipsoids around a test sphere. Since the orientation of only one of the ellipsoid is taken into consideration in the calculation of g200 (r), it vanishes at finite intermolecular separation even in the nematic phase. But the sharpening of peaks of this correlation function shows the building up of orientational order when the nematic phase is approached along the density axis. The angular pair correlation function g220 (r) which measures the angular correlation between a pair of ellipsoids separated by a distance r, becomes non zero in the vicinity of I-N transition even at large intermolecular separations, as shown in figure 2.11(b). The growth in angular pair correlation is also visible in the higher order correlation functions which are shown in the figure 2.11(c). Note that these higher order correlations have non-negligible contribution in the series expansions of the pair correlation function g(r, ω, ω 0). Therefore, the truncation in the series expansion of angular pair correlation can create non-negligible error in the calcualtion of the total angular pair correlation function. Also note that for the system sizes simulated here, the correlation length of angular correlation is comparable to the size of the system. As the system approaches I-N transition whole system starts to behave like a single pseudo-nematic domain.

2.5.2

Pair correlation functions across the nematic-smectic and smectic-solid phase transitions

Nematic-smectic (N-S) and smectic-crystal (S-C) phase transitions, unlike the I-N transition, are accompanied by spacial structural changes. The radial distribution functions show these structural changes. Note that, the orientational order parameter varies from 0.6 to 0.9 when N-S and S-C phase transitions takes place in the increasing order of density along the isotherm at T ∗ = 1.0. Figure 2.12(a) shows the variation in the radial distribution function when N-S and S-C transitions take place. The curve, which appears significantly different from other curves in the figures 2.12(a), 2.12(b) and 2.12(c) is that of the crystalline phase. The N-S transition takes place at the density ρ∗ =0.335 (the transition is found to be continuous) and the S-C transition takes place at density 57

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

3

(a)

g(r)

2 1 0

g200(r)

0 −1

(b)

−2 2

g220(r)

(c)

1

0 0

1

2

3

distance

4

5

6

Figure 2.12: Three radial distribution functions are plotted across nematic-smecticcrystal phase transitions, along an isotherm (at T ∗ =1): (a) shows radial distribution function against distance; (b) shows the orientational pair correlation function g200 (r) against distance; (c) shows the orientational pair correlation function g200 (r) against distance. The successive curves are in the increasing order of modulus of peak height in the increasing order of density; from ρ∗ = 0.33 to 0.4 in grid of δρ∗ = 0.01.

ρ∗ =0.395. Unlike the N-S transition, the S-C transition is rapid and accompanied by sudden structural change. S-C transition is also accompanied by a small change in orientational order parameter. Note that in this case the orientation order grows beyond the system size. Figure 2.12(b) shows g200 (r) at different densities (ρ∗ = 0.33 to 0.4). This figure shows the coupling between the number density and the orientation of the ellipsoids. g200 (r) shows long distance correlation in nematic and smectic phases. A complete rotation of the molecules at these phases requires long-range number density 58

2.5 Pair correlation functions

fluctuations. Corresponding g220 (r) shows the growth of the long-range angular correlations at different densities across the N-S and S-C transitions along the isotherm at T ∗ = 1.0. Since the nematic and smectic phases sustain long-range orientational ordering only g220 (r) shows the strength of the orientational ordering in the simulations. The structural changes that accompany the N-S and S-C transitions can be clearly

2

3

ρ*=0.33

2

1 0 0 3

1 10

S(k)

20

10

20

ρ*=0.37

0 0 30 20

10

20

ρ*=0.38

10 10

20 *

0 0 100

10

20

ρ*=0.4

ρ =0.39 2 0 0

20

2

10 0 0 4

10

ρ*=0.36

1

20

0 0 4

ρ*=0.35

2

0 0 30

ρ*=0.34

50

10

20

0 0

10

20

Figure 2.13: The variation is structure factor when transition takes place from nematic liquid crystal crystalline state through the smectic crystalline phase

identified from the changes that appear in the shape of the structure factor (S(k)). 59

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

Figure 2.13 shows change in the structure factor across N-S and S-C transitions, along an isotherm at density T ∗ = 1, in the successive order of increase of density from ρ∗ =0.32 to 0.4. In a nematic phase, the structure of the liquid is characterized by a plateau near k ≈ 2π/a. Deeply inside the nematic phase, a peak appears near this wavenumber that grows as the density increases. The N-S transition take place when the density ρ∗ ≈ 0.33. Note that at this state point, the first peak grows above the second peak of S(k) in the figure 2.13. As the density further increases, the peak that corresponds to the separation between the smectic layers shows a rapid growth, which is sharp crystalline peak. However, the fluid like structure of other peaks remains the same in the smectic phase. At ρ∗ ≈ 0.39, which is the phase boundary of the smectic phase, the height of the first peak reduces. At ρ∗ = 0.39, due to the high density of the system, smectic layers starts to merge reducing the peak hight that corresponds to the interlayer separation. Here the first peak is shorter than the second peak. The structure factor S(k) at ρ∗ = 0.39, looks similar to S(k) near the N-S transition. When density increases further to ρ∗ ≈ 0.40 the system crystallizes. The smooth liquid like structure of S(k) completely replaced by the sharp peaks of the crystalline phase.

2.6

Concluding remarks

This chapter provides an overview of the structural changes that takes place in GayBerne liquid for the set of standard parameters used for the rest of the thesis. In this chapter, the variation in the order parameter is studied across three phase transitions, namely, isotropic to nematic, nematic to smectic and smectic to crystal phases of the Gay-Berne liquid. The values of the orientational order parameter helps in determining the phase boundaries of the I-N transition and other phase transitions and their location in the phase diagram as it appears in a micro-canonical ensemble. Note that the phase boundaries are sensitive to the change in the type of the ensemble chosen for the simulation. Various static pair correlation are recorded to find the effect of phase transition in the static orientational and translation structure of the liquid. While the orientationally averaged pair correlation function shows no unusual properties across 60

2.6 Concluding remarks

I-N transition, the orientation dependent pair correlation function indeed shows the growth of orientational order, which is rapid (order parameter varies from 0.0 - 0.6).

61

2. OVERVIEW OF THE MOLECULAR SIMULATIONS

62

Bibliography [1] M. P. Allen and M. R. Wilson, Comp. Mol. Design 3, 335 (1987). 2.2 [2] J. P. Bareman, G. Cardini, and M. L. Klein, Phys. Rev. Lett. 60, 2152 (1988). 2.2 [3] M. P. Allen, G. T. Evans, D. Frenkel, and B. M. Mulder, Adv. Chem. Phys. 86, 1 (1993). 2.2 [4] R. Zwanzig, J. Chem. Phys. 39, 1714 (1963). 2.2 [5] L. Onsager, Ann. N. Y. Acad. Sci. 51, 621 (1949). 2.2 [6] B. J. Alder and T. A. Wainwright, J. Chem. Phys. 27, 1208 (1957). 2.2 [7] W. W. Wood and J. D. Jacobson, J. Chem. Phys. 27, 1207 (1957). 2.2 [8] A. Stroobants, H. N. W. Lekkerkerker, and D. Frenkel, Phys. Rev. A 36, 2929 (1987). 2.2, 2.2 [9] A. Stroobants, H. N. W. Lekkerkerker, and D. Frenkel, Phys. Rev. Lett. 57, 1452 (1987). 2.2 [10] D. Frenkel, Liq. Cryst. 5, 3280 (1988). 2.2 [11] J. A. C. Veerman and D. Frenkel, Phys. Rev. A 41, 3237 (1990). 2.2 [12] P. Bolhuis and D. Frenkel, J. Chem. Phys. 106, 666 (1997). 2.2 [13] B. J. Berne and P. Pechukas, J. Chem. Phys. 56, 4213 (1972). 2.2, 2.2.1 63

BIBLIOGRAPHY

[14] J. G. Gay and B. J. Berne, J. Chem. Phys. 74, 3316 (1981).

2.2, 2.2.1, 2.2.1,

2.2.1 [15] T. Kihara, Adv. Chem. Phys. 5, 147 (1963). 2.2 [16] J. Kushick and B. J. Berne, J. Chem. Phys. 64, 1362 (1976). 2.2.1 [17] J. Crain and A. V. Komolkin, Adv. Chem. Phys. 109, 39 (1999). 2.2.1 [18] E. de Miguel, L. F. Rull, M. K. Chalam, and K. E. Gubbins, Mol. Phys. 75, 405 (1991). 2.2.1 [19] D. J. Adams, G. R. Luckhurst, and R. W. Phippen, Mol. Phys. 61, 1575 (1987). 2.2.1 [20] G. R. Luckhurst, R. A. Stephens, and R. W. Phippen, Liq. Cryst. 8, 451 (1990). 2.2.1 [21] R. Hashim, G. R. Luckhurst, and S. Romano, J. Chem. Soc. Faraday Trans. 91, 2141 (1995). 2.2.1 [22] K. Satoh, S. Mita, and S. Kondo, Liq. Cryst. 20, 757 (1996). 2.2.1 [23] G. L. Penna, D. Catalino, and C. A. Veracini, J. Chem. Phys. 105, 7097 (1996). 2.2.1 [24] M. P. Neal, A. J. Parker, and C. M. Care, Mol. Phys. 91, 603 (1997). 2.2.1 [25] D. J. Cleaver, C. M. Care, M. P. Allen, and M. P. Neal, Phys. Rev. E 54, 559 (1996). 2.2.1 [26] R. Berardi, C. Fava, and C. Zannoni, Chem. Phys. Lett. 236, 462 (1995). 2.2.1 [27] E. DeMiguel, Phys. Rev. E 47, 3334 (1993). 2.2.1 [28] E. deMiguel, E. M. delRio, J. T. Brown, and M. P. Allen, J. Chem. Phys. 105, 4235 (1996). 2.2.1, 2.4 64

BIBLIOGRAPHY

[29] M. A. Bates and G. R. Luckhurst, J. Chem. Phys. 110, 7087 (1999). 2.2.1, 2.3 [30] M. R. Wilson, Phys. Rev. E 107, 9654 (1997). 2.2.1 [31] J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). 2.3, 2.3, 2.3 [32] C. Grey and K. E. Gubbins, Theory of Molecular Fluids (Clarendon Press, Oxford, 1984). 2.3, 2.3, 2.3 [33] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987). 2.4 [34] S. Kammerer, W. Kob, and R. Schilling, Phys. Rev. E 56, 5450 (1997). 2.5

65

BIBLIOGRAPHY

66

Chapter 3 Collective orientational relaxation 3.1

Introduction

As already mentioned in the chapter 1, Fayer and coworkers [1, 1–5] observed power law decay from very short (few picosecond) to very long time scale (several hundred nanoseconds) as a function of temperature, using optical heterodyne detected Kerr effect. Several computer simulation studies [6, 7, 7–9] have earlier attempted to recover the power law in orientational relaxation near I-N transition. One of the first simulations to calculate second rank single particle and collective orientational time correlation function (OTCF) was performed by Allen et al. [7] using hard ellipsoids. While they could observe the critical slowing down of the dynamical correlation function near IN transition, but did not report any power law decay. Perera et al.

[8, 9] carried

out molecular dynamic simulation on a system of Gay Berne ellipsoids. This model undergoes an isotropic to nematic phase transition at a reduced density ρ∗ ' 0.315 and temperature T ∗ ' 1 ( where T ∗ = kb T /0 , while kb is the Boltzmann constant, T is the temperature and 0 is the energy constant used in the Gay-Berne potential) for aspect ratio (κ=3). They found second rank single particle and collective OTCF slow down appreciably near I-N transition. In another set of simulations of Allen et al. [7] calculated direct correlation functions of a system of Gay-Berne ellipsoids near I-N transition. Detailed molecular dynamic simulations of Ravichandran et al. [8, 9] found a sudden appearance of power law behaviour in the second rank single particle 67

3. COLLECTIVE ORIENTATIONAL RELAXATION

OTCF near I-N transition. The value of power law exponent was close to 0.56. These simulations also looked into the translational and rotational diffusion coefficient of a system of Gay-Berne ellipsoids. Vasanthi et al. [10] carried out molecular dynamics simulations to calculate diffusion of isolated tagged spheres in a sea of Gay-Berne ellipsoids. The diffusion is isotropic well inside the isotropic region of nematic liquid crystal. In the vicinity of the I-N transition the parallel and perpendicular component of (Dk and D⊥ ) diffusion coefficient decouples from each other. They proved that the D

anisotropy parameter ( D⊥k ) obeys a power law of the form (ρ∗c − ρ∗ )p . However, all the simulations reported till now have failed to detect the temporal power law in the decay of collective orientational correlation function, which, as already mentioned, is the experimental quantity one measures in Kerr relaxation experiments (actually one measures the time derivative of the second rank orientational time correlation function). Perhaps, the reason is that these early simulations were limited in the duration of the trajectory obtained. Or, may be, the intermolecular potential employed was inadequate. The non-observance of the power law decay in simulations has remained somewhat of a puzzle. In this work we carried out very long-time (approximately 50 ns in real time unit) molecular dynamics simulations of larger systems (than attempted earlier) to calculate the second rank collective OTCF of a system of Gay-Berne ellipsoids. We find that just prior to I-N transition, the relaxation of second rank collective OTCF (C2c (t), (defined in Eq.3.2) abruptly changes its decay behaviour from exponential to a pronounced power law decay which ranges from short to intermediate times. The decay again changes over to an exponential-like decay at longer times. In order to gain a microscopic understanding of the power law decay near the I-N transition, we have looked into the study of two other time correlation functions. First, we investigated the relaxation of the dynamic orientational pair correlation function (DOPCF) and, second we studied the OTCF of a local director. The DOPCF can probe orientational dynamics of a pair of ellipsoids at different length scales; this gives a more detailed description of the relaxation of the orientational order than the 68

3.2 Dynamic orientational correlation functions

collective OTCF. The DOPCF shows critical slow down of relaxation and the emergence of power law relaxation near I-N transition. The emergence of power law relaxation is also evident in the OTCF of the local director that is discussed in the appendix. Present simulations also explore the divergence of various equilibrium angular pair correlation functions, which measure the growth of correlation length near I-N transition.

3.2

Dynamic orientational correlation functions

Time dependent orientational correlation function gives a measure of the temporal decay of the memory of orientational order and one needs to define several such functions to describe various aspects of orientational dynamics. The single particle OTCF gives a temporal measure of loss of a single molecule’s memory of its orientation in the random potential created by the surrounding molecules in the liquid environment. The single particle OTCF of the rank l is defined as [7–9, 11, 12] P < i Pl (ˆ ei (0) · eˆi (t)) > s Cl (t) = P , < i Pl (ˆ ei (0) · eˆi (0)) >

(3.1)

where the eˆi is the unit vector or director associated with ith ellipsoid along the major axis of the ellipsoid and Pl is lth rank Legendre polynomial. The quantity measured in light scattering experiments is the collective OTCF [13]. The collective OTCF is defined as [7–9, 11, 14, 15]. P P < ei (0) · eˆj (t)) > i j Pl (ˆ Clc (t) = P P < i j Pl (ˆ ei (0) · eˆj (0)) >

(3.2)

This is a limiting case of the more general wavenumber (k) dependent orientational correlation function Clc (k, t) defined as P P < i j eik·rij Pl (ˆ ei (0) · eˆj (t)) > c Cl (k, t) = P P ik·rij < i je Pl (ˆ ei (0) · eˆj (0)) >

(3.3)

OHD-OKE spectroscopy allows one to measure directly in the time domain derivative

of the time correlation function [16, 17]. fOKE (t) = −

θ(t) ∂ C(t) kB T ∂t

69

(3.4)

3. COLLECTIVE ORIENTATIONAL RELAXATION

where the C(t) = hχ(k, t)χ(k, t)i and χ is the linear susceptibility of the liquid. In molecular dynamics simulation C(t) is given by is the second rank collective orientational correlation function [18]. This expression can be derived by using Kubo’s linear response theory. A wealth of information about the space dependence of orientational relaxation can be obtained by studying the two-particle correlation functions, such as molecular van Hove [14] correlation function. In order to study the orientational dynamics of a pair of particles we have defined a DOPCF. Orientational pair correlation function is obtained from the distinct part of Van Hove correlation function by expanding it in terms of spherical harmonics. Van Hove correlation function for a molecular liquid can be defined as 1 Gd (r, Ω, Ω0 , t) = N

*

N X N 0 X i

j

+

δ(r + rj (0) − ri (t))δ(Ω − Ωi (0))δ(Ω0 − Ωj (t)) , (3.5)

In a system of linear molecules, the van Hove correlation function can be represented in inter-molecular reference frame as Gd (r, ω, ω 0, t). This correlation function can be expanded in terms of spherical harmonics to get DOPCF. Gd (r, ω, ω 0 , t) =

X

ρGl1 l2 m (r, t)Yl1 ,m (ω)Yl2 ,m (ω 0 )

(3.6)

l1 ,l2 ,m

The DOPCF gives a direct measure of relaxation of orientational order at different shells around a linear molecule. The DOPCF can be calculated from the simulation using the expression

 Gl1 l2 m (r, t) = 16π 2 Gd (r, ω, ω 0, t)Yl1∗,m (ω)Yl∗2 ,m (ω 0 ) /ρ

(3.7)

In order to study the orientational relaxation of different cages, we can define orientational correlation function in terms of DOPCF. Let Cp (t) =

Gl1 l2 m (r, t) − Gl1 l2 m (r, t = ∞)    , Gl1 l2 m (r, t = 0) − Gl1 l2 m (r, t = ∞) r=R

where R is a fixed distance at which the relaxation is measured. 70

(3.8)

3.3 Simulation details

3.3

Simulation details

Long (≈ 50ns ) molecular dynamics simulations have been carried out for a system of 576 Gay-Berne ellipsoids [19] in a micro-Canonical ensemble. The value of κ0 used in the simulation is 5 [8]. The scaling used for moment of inertia is I ∗ = I/mσ0 2 . The density is scaled in the simulation as ρ∗ = ρσ0 2 and the temperature is scaled as T ∗ = kb T /0 . The equation of motion is integrated with reduced time (t∗ = (mσ02 /0 )1/2 ) steps with ∆t = 0.002 t∗ . The simulation starts from an equilibrated configuration of ellipsoids. Initial configuration of the ellipsoids is generated from a cubic lattice and then the simulation is run for two hundred thousand steps to obtain the equilibrium configuration. During the equilibration steps the temperature is scaled so that the system is in equilibrium at this particular temperature. The production steps starts from this equilibrated configuration. The production steps are run is for fifteen million steps to calculate statistically averaged single particle and collective static [15, 20, 21] and dynamic orientational correlation functions. When translated into argon units, this corresponds to a run time of 75 ns. We found that such long runs are indeed necessary to study decay of collective orientational correlation near the I-N transition.The ellipsoid used in the simulation has minor axis b = 0.5 and major axis a = 1.5 (in reduced units). The simulations are done at the state points near the pre-transition region of phase diagram, shown in figure 3.1 where the variation in temperature and the density employed is shown by arrows. The translational and rotational motions are solved using leap-frog algorithm. The calculated order parameter is shown in figure 3.2 where the order parameter is plotted against the density near I-N transition at three temperatures. The order parameter changes dramatically after density increases beyond 0.3. This is in accord with previous simulations [7–10].

3.4

Results and Discussions

The log-log plot of second rank single particle OTCF against reduced time at different densities ρ∗ =0.285 to 0.315 is shown in figure 3.3. The power law relaxation emerges 71

3. COLLECTIVE ORIENTATIONAL RELAXATION

1.2

1.0

Smectic B(t)

Smectic B

Nematic

Isotropic

.

T*

1.1

I−N

0.9 0.3

ρ*

0.35

0.4

.

Figure 3.1: The three liquid crystalline phases along with the isotropic phase are shown. 1 0.9 0.8 0.7

*

S(ρ )

0.6 0.5 0.4 0.3 0.2 0.1 0 0.24

0.26

0.28

0.3 Density

* (ρ)

0.32

0.34

0.36

0.38

Figure 3.2: The variation of order parameter (S(ρ∗ )) with density ρ∗ at different temperatures is shown in this figure. The continuous line gives the S(ρ∗ ) at T ∗ =1.0, the dash line gives the S(ρ∗ ) at T ∗ =1.1 and the dash-dot line gives S(ρ∗ ) T ∗ =1.2.

72

3.4 Results and Discussions

0 −0.5 −1

2

Cs(t)

−1.5 −2

−2.5 −3 −3.5 −4 0

ρ*=0.285 * ρ =0.290 ρ*=0.295 * ρ =0.30 * ρ =0.305 * ρ =0.31 * ρ =0.315 fit

1

*

y=−0.701(ln t )−0.633

2

3

4

ln t

5

6

7

*

Figure 3.3: The log-log plot of single particle OTCF (C2s (t)), (Eq. 3.1) at different densities is shown here. The linear region of the relaxation function signifies the emergence of a power law relaxation after ρ∗ =0.3. The linear portion of the C2s (t) is fitted to a straight line at density ρ∗ = 0.305. 1

ρ*=0.285 ρ*=0.295 ρ*=0.3 ρ*=0.305 ρ*=0.31 * ρ =0.315 power law fit

0.9 0.8 0.7

c

C2(t)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

500

1000

time (t*)

1500

2000

Figure 3.4: The second rank collective OTCF, (C2c (t), Eq. 3.2) is plotted against time at different densities and at constant temperature T ∗ = 1. As the density approaches I-N transition this correlation function shows power law relaxation. The regions where power law relaxation is dominant are fitted with the function y = 1.1 − 0.014 t0.58 at ρ∗ = 0.31 and y = 1.0 − 0.0016 t0.85 at ρ∗ = 0.315. 73

3. COLLECTIVE ORIENTATIONAL RELAXATION

−6

−7

2

ln (−dCc (t)/dt)

−6.5

−7.5 −8 −8.5 −9 0

1

2

3

4

*

ln t

5

6

7

8

Figure 3.5: The log of negative of the time derivative of second rank collective OTCF, (C2c (t) of Eq. 3.2), is plotted for densities ρ∗ = 0.31 (dashed line) and ρ∗ = 0.315 (continuous line) against log of time at T ∗ =1.0. The linear portion of the plot gives power law observed in the intermediate time scale of the simulation.

in the correlation function when the reduced density increases beyond ρ∗ = 0.3. In this region, the decay of the correlation function in the intermediate times can be represented by the function of the form X(t) ' b t−a , where a and b are constants. At higher densities the fast and slow relaxation processes are clearly separated by intermediate power law as indicated by the linear region of the curve (between 20 and 100 in reduced unit). In figure 3.3 the linear region of the log-log plot is fitted to a straight line, ln(X(t)) = b − a ln(t),

(3.9)

the exponent obtained from the fit is a = 0.70. As the density increases the single particle orientational relaxation becomes slower due to the strong coupling of rotational motion with the surrounding ellipsoids or due to an orientational caging effect of the ellipsoids. The formation of orientational cage or pseudo-nematic region surrounding an ellipsoid gives it the memory of the previous orientation resulting in the slow relaxation process. The collective relaxation of the ellipsoids that form an pseudo-nematic domain 74

3.4 Results and Discussions

is important near this state point (near 0.3 of the phase diagram shown in the figure 3.1). The collective OTCF function at densities ρ∗ ranging from 0.285 to 0.315 are plotted in figure 3.4. The collective OTCF also shows significant lengthening in the time scale of relaxation. The change in the time scale of the relaxation is of the order or hundreds when density changes from 0.285 to 0.315. . At very close to the I-N transition, the nature of the decay of collective orientational relaxation function changes abruptly. We note that this observation of slow down of the of relaxation of the collective orientational relaxation is in general agreement with Landau-de Gennes mean field theory at longer time scale. The emergence of the power law is evident in this correlation function at densities ρ∗ = 0.31 (the value of the exponent obtained is 0.58) and ρ∗ =0.315 (the value of the exponent obtained is 0.85) from the power law fit at the initial part of these correlation functions. In figure 3.5 we have plotted the logarithm of the time derivative of the second rank collective orientational correlation function (the experimentally measured quantity) against logarithm of time to further our comparison with the observed behaviour. The derivative has been plotted for two densities in the transition region. This figure brings out the crossover from the power law to the exponential decay at long times rather nicely. Also, this figure can be compared with figure 5 of Cang et al. [5]. Over more than two orders of magnitude, the similarity is rather remarkable. We have earlier discussed the explanation put forward by using the mode coupling theory of relaxation near the I-N transition. According to this interpretation, the power law decay arises because of the rapid increase in the rotational memory function as the frequency decreases. In other words, subsequent to the initial relaxation within the cage, any collective (or single particle) relaxation becomes coupled to the orientational density fluctuation, which becomes very slow near the I-N transition because of the emergence of the long-range orientational pair correlation. Thus, the reason for the power law decay in the collective and the single particle decay is nearly the same. However, the time scales of these two decays are vastly different — the former is 75

3. COLLECTIVE ORIENTATIONAL RELAXATION

much slower. The reason can be understood from the mean-field theory itself. Unlike the single particle dynamics, the collective orientation occurs in a collective manybody potential. For small density fluctuation, this potential can be approximated as a harmonic one. As the I-N transition is approached, the frequency of the harmonic potential decreases rapidly, slowing down relaxation in the well. 0.6 0.4 0.2

0.4

G

2,2,0

(r,t)

0

0.2 0 0.6

r

0.4 0.2 0 0

1

2

3

4

5

6

distance (r*) Figure 3.6: The dynamic orientational pair correlation function (DOPCF) (Eq. 3.7)is plotted at three different densities. In these figure continuous lines starting from top are plotted at time step of 1 in the time interval 1-10. Then dot-dash lines are plotted at a time step of 10 in the time interval 20-100. Finally the dashed lines are plotted at a time step of 100 in time interval 200-1000. All the lines arranged from top to bottom in the increasing order of time. The top sub-figure is at density ρ∗ = 0.295, middle sub-figure is at density ρ∗ = 0.305 and bottom sub-figure is at density ρ∗ = 0.315.

In many aspects orientational relaxation near I-N transition looks similar to that in the supercooled liquid near the glass transition [3, 4]. As discussed earlier, the reasons 76

3.4 Results and Discussions

0

ρ*=0.285 * ρ =0.290 ρ*=0.295 * ρ =0.30 ρ*=0.305 * ρ =0.31 ρ*=0.315 fit

*

−0.5

y=−0.79(ln t )−0.2

−1

ln Cp(t)

−1.5 −2

−2.5 −3 −3.5 −4 0

1

2

3

* 4

ln t

5

6

7

Figure 3.7: The log of pair time correlation function Cp (t) defined by Eq. 3.8 is plotted against logarithm of time at separation R equal to the first peak of DOPCF. The emergence of power is evident at ρ∗ =0.305. In this figure linear portion of ln Cp (t) versus ln t curve at density ρ∗ =0.305 is fitted to a straight line.

are also similar. Both are a reflection of a rapid increase in the memory function at low frequency. However, the reason for this rapid increase is different in the two cases. Now it is interesting to look into the orientational relaxation within various shells. This gives a close picture of the dynamics at varying length scales. Such information can be obtained from the dynamic pair correlation function. Figure 3.6 shows the behaviour of DOPCF G220 (r, t) at three densities at three different time scales. This gives a picture of slowing down of orientational relaxation near I-N transition. Note that in the top figure DOPCF at time t=0 converge to zero at large distance, signifying the absence of long-range orientational order. The picture of relaxation changes at higher densities. At ρ∗ =0.315, the DOPCF is non-zero at time t=0 even at long distance. At this density the slow orientation relaxation can be identified from the widely separated dash-dot (time scale of tens) lines and dashes lines (time scale separated by hundreds). The orientation relaxation at the first peak of DOPCF is denoted by Cp (t) (Eq. 3.8). In figure 3.7 the Cp (t) is plotted at R equal to first peak of DOPCF. When density is at 0.305 the sudden appearance of the power law is observed. In figure 3.7 the 77

3. COLLECTIVE ORIENTATIONAL RELAXATION

linear portion of the Cp (t) is fitted to a straight line. The exponent of the power law obtained from the slope of the fit is 0.79. The observation of power law in the first shell of the relaxation confirms that decay of pseudo nematic regions of size of first nearest neighbor distance also shows power law relaxation.

3.5

Temperature dependence of relaxation 0 −0.5 −1

2

Cs(t)

−1.5 −2

−2.5 −3 −3.5 −4 0

1

2

3

4

ln t

5

6

7

*

Figure 3.8: The log-log plot of the second rank single particle OTCF (C2s (t) in Eq. 3.1) at different temperatures and at constant density ρ∗ =0.315 are shown here. The continuous line gives C2s (t) at T ∗ =1.0, dash-dot line gives C2s (t) at T ∗ =1.1 and the dash line gives C2s (t) at T ∗ =1.2. The slope of the linear region varies with temperature showing temperature dependence of power law.

In figure 3.8, the log-log plot of single particle OTCF is shown at three different temperatures (T ∗ = 1.0, 1.1 and 1.2). At all temperatures this OTCF slows down significantly and takes long time to relax. The linear region of the second rank OTCF indicates the emergence of the power law. Note that when temperature increases the slope of the linear region of the second rank single particle OTCF decreases, showing the temperature dependence of power law exponent. In figure 3.9, the log-log plot of collective OTCF is plotted at three different temperatures (T ∗ = 1.0, 1.1 and 1.2). 78

3.5 Temperature dependence of relaxation

1 0.9 0.8 0.7

c

C2(t)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

200

400

600

800

1000

time (t *)

Figure 3.9: This figure shows the time dependence of collective second rank OTCF (C2c (t), defined in Eq. 3.2) at different temperatures and at constant density ρ∗ =0.315. The continuous line gives C2c (t) at T ∗ =1.0, dash-dot line gives C2c (t) at T ∗ =1.1 and the dash line gives C2c (t) at T ∗ =1.2.

−1 −1.5

ln Cp(t)

−2 −2.5 −3 −3.5 −4 0

1

2

3

ln t

4

5

*

Figure 3.10: The log of the pair time correlation function Cp (t) (Eq. 3.8) is plotted against log of time at three different temperature at density ρ∗ =0.305. The dash-dot line gives Cp (t) at T ∗ =1.0, the dashed line gives Cp (t) at density T ∗ =1.1 and the Cp (t) at T ∗ =1.2 is given by continuous line.

79

3. COLLECTIVE ORIENTATIONAL RELAXATION

0

*

ρ =0.285 * ρ =0.290 ρ*=0.295 * ρ =0.30 * ρ =0.305 * ρ =0.31 * ρ =0.315 fit

y=−0.344(ln t*)−0.58

−0.5

−1.5

2

Cd(t)

−1

−2

−2.5

−3 0

1

2

3

ln t

*

4

5

6

7

Figure 3.11: The log-log plot of second rank OTCF of the local director, (C2d (t), Eq.3.11 ) versus time at different densities and at constant temperature T ∗ =1 is shown in this figure. The linear portion of the curve found after ρ∗ =0.3 shows the emergence of power law. The linear fit of the second rank OTCF of the local director (C2d (t)) at ρ∗ =0.315 is shown in the figure.

Note that even at the high temperature 1.2 the decay of this correlation function is slow. This also point toward the fact that even the long time simulation used here is inadequate to probe the full decay of this correlation function. It is also evident from the phase diagram (Fig. 3.1) that transition point has only a small variation due to change of temperature. The log of the pair time correlation function Cp (t) in a system of density ρ∗ =0.305 is plotted against log of time at three different temperature in the figure 3.10.

3.5.1

Orientational relaxation of a local director

The definition of yet another OTCF can be done in terms of the local director the ellipsoids. This definition is important in the case of experiments where a small fraction of molecules are excited and the observation of the collective orientational relaxation of this fraction of molecules by probe pulse. Let there be a unit vector associated with each ellipsoid along the major axis. In a system of ellipsoids it is possible to define a 80

3.5 Temperature dependence of relaxation

0 −0.5 −1

d

C2 (t)

−1.5 −2

−2.5 −3 −3.5 −4 0

1

2

3

4

5

6

7

ln t *

Figure 3.12: The time dependence of second rank of the OTCF of the local director (C2d (t), defined in Eq. 3.11), is plotted at different temperatures and constant density ρ∗ =0.315. The continuous line gives C2d (t) at T ∗ =1.0, dash-dot line gives C2d (t) at T ∗ =1.1 and the dash line gives C2d (t) at T ∗ =1.2.

local director in terms of the sum of the unit vectors. Since there is no asymmetry along the axis in the interacting potential sum of unit vectors that initially has a direction will vanish due to the rotation of the directors. This rotational diffusion of the ellipsoid can be modeled in terms of a symmetric double well potential. The arbitrary initial direction of the ellipsoid is in the first well of this symmetric double-well potential and a rotation of π from the initial direction, which is indistinguishable from this initial direction is the other well of the symmetric double-well potential. For the π rotation the ellipsoid has to over come the potential barrier that is created by it’s neighbors. Correlation time can be defined in terms of the relaxation time of the initial direction of the resultant vector. This can also be defined in terms of principle of conservation of angular velocity in the system. The resultant orientation of the system is given by PN eˆi Uˆ = Pi=1 (3.10) | N ˆi | i=1 e

Hence the lth rank OTCF is defined as Cld (t) =

ˆ (t) · Uˆ (0)) > < P l (U ˆ (0) · Uˆ (0)) > < P l (U 81

(3.11)

3. COLLECTIVE ORIENTATIONAL RELAXATION

The angular velocity ω ~ is conserved locally in a system at equilibrium, so

PN

i=1

ω ~ i=

Constant. In a simulation to avoid the rotational flows in the box this constant is equal to zero. In the isotropic region all the ellipsoids are free to rotate individually, hence the direction of the resultant unit vector is changing fast. But near I-N transition the rotational motion of the individual ellipsoids are restricted due to the formation of pseudo-nematic domains. Hence the orientational relaxation of the resultant unit vector Uˆ slows down. Note that in the nematic region the molecules are aligned in one direction due to the restricted individual rotations. Therefore, the relaxation of each arbitrarily assigned unit vector associated with the ellipsoid also slows down, which results in the delay in the relaxation time the local director Uˆ . The long tail of the Cld (t) due to the forbidden rotation by formation of the nematic alignment which follows after the initial decay, results in power law behaviour. We believe that the power law relaxation of this second rank orientational time correlation function of the local director may be related to the experimental results [3, 4], as follows. Here for measuring the relaxation of local director, the ellipsoids are given an arbitrary initial direction. The relaxation time during which this average director looses the memory of the initial direction depends on the relaxation time of individual ellipsoids. It can be considered as lose of polarizability of a small region of the system of molecules due to random rotation. The log-log plot of second rank OTCF of the local director of a system of ellipsoids at different densities ρ∗ =0.285 to 0.315 at T ∗ = 1 near I-N transition is shown in the figure 12. This figure shows that the relaxation of the second rank OTCF of the local director exhibits pronounced power law decay over a rather long time interval. At higher densities (beyond ρ∗ = 0.305) a long tail appears in the second rank OTCF of the directors due to enhancement of collective interactions. After this particular density the change in the correlation function become dramatic and the relaxation become extremely slow. In figure 3.11 the linear potion of the second rank orientational time correlation function of the directors is fitted to a straight line at density 0.315. The power law exponent obtained from this fit is 0.34. In figure 3.12 the log-log plot of 82

3.6 Concluding Remarks

OTCF of the local director is shown at three different temperatures (T ∗ = 1.0, 1.1 and 1.2). This correlation function shows long linear region showing the emergence of power law.

3.6

Concluding Remarks

In this chapter results of extensive and long molecular dynamics simulations of orientational relaxation of model nematogens have been presented in a system of Gay-Berne ellipsoids, in an attempt to understand the origin of the experimentally observed temporal power law decay of the second rank orientational order parameter in the isotropic phase of a nematic liquid crystal. We have observed the well-known divergence of the angular pair correlation functions as the system goes toward I-N transition along the density axis. This divergence gives evidence of the formation of nematic domains inside the isotropic phase. The single particle second rank orientational time correlation function (OTCF) shows the emergence of a power law in the relaxation beyond the density 0.3. The power law exponent obtained from the linear fit of the second rank single particle OTCF is ∼ 0.7. Perhaps the most important result of this work is the finding that the second rank collective OTCF shows an abrupt emergence of a power law decay very close to the isotropic-nematic phase transition. In fact, over a substantial part of its short time decay, the correlation function decays almost linearly with time, predicting a very weak time dependence of the derivative of C2c (t) – it is the time derivative of C2c (t) is that measured in experiments. In the very long time, the decay becomes exponential-like, in agreement with the Landau-de Gennes mean-field theory (which describes relaxation in terms of randomization of pseudo-nematic domains). It is certainly heartening to observe the appearance of the power law in the decay of the collective correlation function. Note that all the earlier studies have failed to observe this power law decay because these early simulations were all limited to rather short times. In fact, the power law was found in the single particle correlation function because the latter decays on a much shorter time scale. 83

3. COLLECTIVE ORIENTATIONAL RELAXATION

Mode coupling theory analysis of this problem (presented earlier) suggests that not only the orientational pair correlation function must grow rapidly (with temperature or density) as the I-N transition is approached, but also the long wave number part of the Fourier transformed static pair correlation function must become very large to make a dominant contribution to the memory function [11, 13]. This happens at very small wavenumber. Thus, the success in capturing in simulations of many aspects of the interesting dynamical behaviour near the isotropic-nematic transition should be attributed to the carrying out of very long simulations. Another notable new result is the observed slow relaxation in the orientational pair correlation function. Over a limited range of time, this relaxation function can be fitted to a power law decay when the separation between the pairs is equal to the first maximum in the static pair correlation function. The existence of power law at the level of pair correlation function shows that even two-particle relaxation slows down at nearest neighbor level. Further work is required to fully understand the nature of these time correlation functions. As noted earlier, there are certain striking similarities between relaxation near I-N transition (in the isotropic phase) of a liquid crystalline system and that in supercooled liquids. [4, 5] It has been found that the same functional form can describe relaxation in both supercooled liquids and in isotropic liquid of liquid crystals. The nature of the relaxation function is similar in both the systems – short time decay, followed by a pronounced power law, which is again followed by a slow exponential-like decay. As the temperature is lowered toward the glass transition temperature (for supercooled liquids) or toward the I-N transition, the duration of the power law increases [5]. While the dynamics appear to be quite similar, there are also some differences, especially in the origin of the behaviour. In the case of liquid crystals, the behaviour is certainly due to the rapid growth in the equilibrium orientational pair correlation function, which is reflected in the rapid increase in the long wavenumber limit of the orientational structure factor. This is reflected in the large growth of S220 (k = 0) near the I-N transition. This growth also gives rise to slow down of the mean-field Landau-de 84

3.6 Concluding Remarks

Gennes relaxation time, τLdG , as given by Eq. 1.35 and discussed earlier. In the case of supercooled liquid, however, the slow down of relaxation at intermediate wavenumbers, when the wavenumber k ≈ 2π/σ, where σ is the molecular diameter. The mean-field theory again gives a decay at intermediate wavenumbers given by S(k)/D where S(k) is the static structure factor at intermediate wavenumbers while D is the self-diffusion coefficient. The large growth in relaxation time in supercooled liquid is due to a feedback mechanism [22–25] where slow down of density relaxation increases friction and the increase of friction causes further slowdown of the density relaxation. In MCT, the feedback mechanism is included through a self-consistent solution, which gives rise to the well-known power laws and the eventual divergence of the relaxation time in the mode coupling theory. It should also be noted here that the mode coupling theory breaks down completely at low temperatures because it predicts a critical point (and divergence of viscosity and relaxation time) at a temperature, which is about 30-50 K above the true glass transition temperature. However, prior to this breakdown, MCT is known to provide a satisfactory description of the relaxation behaviour. Therefore, it is clear that while the power law relaxation near I-N transition has a thermodynamic origin; the one in the supercooled liquid has a dynamic origin (note that MCT predicts a dynamic transition only). Nevertheless, experiments have unequivocally shown that the two relaxations are quite similar. The reason may be that the basic features simply cannot be too different. Both must have an initial fast decay (due to short range interactions), followed by a slow power-law decay (due to different reasons though) and both have a longtime exponential-like decay. Thus, the basic features of relaxations are the same and, therefore, can be described by the same functional form. Even with these observations, experimental observations of the similarity are indeed revealing. We have already discussed the mode coupling theory interpretation of the slow decay. It seems that the basic conclusions of MCT is in agreement with the simulation results. It will be interesting to use the two particle correlation functions directly from simulations and perform a more detailed comparison between theory and simulation 85

3. COLLECTIVE ORIENTATIONAL RELAXATION

results. In addition, we need a fully self-consistent MCT, which is not attempted yet. Also, even with our very long MD simulations, it is hard to study the crossover regions with definitive conclusions. For that to occur, we need to simulate much larger systems (may be something of the order of 5000 particles) for similarly long times. Also, we have not addressed the aspect ratio dependence of the power law behavior.

86

Bibliography [1] H. Cang, J. Li, and M. D. Fayer, Chem. Phys. Lett. 366, 82 (2002). 3.1 [2] A. Sengupta and M. D. Fayer, J. Chem. Phys. 102, 4193 (1995). 3.1 [3] S. D. Gottke et al., J. Chem. Phys. 116, 360 (2002). 3.1, 3.4, 3.5.1 [4] S. D. Gottke, H. Cang, B. Bagchi, and M. D. Fayer, J. Chem. Phys. 116, 6339 (2002). 3.1, 3.4, 3.5.1, 3.6 [5] H. Cang, J. Li, V. N. Novikov, and M. D. Fayer, J. Chem. Phys. 118, 9303 (2003). 3.1, 3.4, 3.6 [6] S. C. P. Martinoty and F. Debeauvais, Phys. Rev. Lett. 27, 1123 (1971). 3.1 [7] M. P. Allen and D. Frenkel, Phys. Rev. Lett. 58, 1748 (1987). 3.1, 3.2, 3.2, 3.3 [8] S. Ravichandran, A. Perera, M. Moreau, and B. Bagchi, J. Chem. Phys. 107, 8469 (1997). 3.1, 3.2, 3.2, 3.3, 3.3 [9] S. Ravichandran, A. Perera, M. Moreau, and B. Bagchi, J. Chem. Phys. 109, 7349 (1998). 3.1, 3.2, 3.2, 3.3 [10] R. Vasanthi, S. Ravichandran, and B. Bagchi, J. Chem. Phys. 115, 10022 (2001). 3.1, 3.3 [11] B. Bagchi and A. Chandra, Adv. Chem. Phys. 80, 1 (1991). 3.2, 3.2, 3.6 [12] D. J. Tildesley and P. A. Madden, Mol. Phys. 48, 129 (1983). 3.2 87

BIBLIOGRAPHY

[13] B. J. Berne and R. Pecora, Dynamic Light Scattering: With applications to Chemistry, Biology and Physics. (John Wiley & Sons, INC, New York, 1976). 3.2, 3.6 [14] J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). 3.2, 3.2 [15] W. B. Streett and D. J. Tildesley, Proc. R. Soc. Lond. A 355, 239 (1972). 3.2, 3.3 [16] Y. Yan and K. A. Nelson, J. Chem. Phys. 87, 6240 (1987). 3.2 [17] Y. Yan and K. A. Nelson, J. Chem. Phys. 87, 6257 (1987). 3.2 [18] D. Frenkel and J. P. McTague, J. Chem. Phys. 72, 2801 (1980). 3.2 [19] J. G. Gay and B. J. Berne, J. Chem. Phys. 74, 3316 (1981). 3.3 [20] M. P. Allen and M. A. Warren, Phys. Rev. Lett. 78, 1291 (1997). 3.3 [21] J. M. Haile and C. G. Grey, Chem. Phys. Lett. 76, 583 (1980). 3.3 [22] T. Geszti, J. Phys. C: Solid State Physics 16, 5805 (1983). 3.6 [23] E. Leutheusser, Phys. Rev. A 29, 2765 (1984). 3.6 [24] T. R. Kirkpatrick, Phys. Rev. A 31, 939 (1985). 3.6 [25] W. G¨otze and L. Sj¨ogren, Rep. Prog. Phys. 55, 241 (1992). 3.6

88

Chapter 4 Single particle rotational dynamics 4.1

Introduction

Although I-N transition has been a subject of intense study in the past 50 years, the molecular level studies of dynamics near this phase transition are rare. This may be because the focus of majority of the studies was on the long time hydrodynamic description of the system [1–3]. A study on single particle dynamics is certainly required for understanding the dynamics of nematogens, especially near the I-N transition, which is not well understood. According to the Landau - deGennes mean field theory [2] of the I-N transition, the order parameter of the system relaxes exponentially with time in the pre-transition region. This has been verified by several experimental studies [4–6]. However, recent studies of Fayer and coworkers (See section 1.6.1) using OHD-OKE on different liquid crystalline samples show striking similarities between the orientational relaxation in supercooled liquids and relaxation of nematogens near I-N transition. In these experiments, the OKE signal show an initial short time power law relaxation; which is followed by a slow exponential relaxation as predicted by the Landau-deGennes theory. Relaxation in supercooled liquids has been a subject of intense research in the past several years [7, 8]. An important problem in the dynamics of the supercooled liquids is to understand the origin of the non-exponential relaxation in the time correlation functions. The non-exponentiality in the relaxation may originate from the non-exponentiality of local relaxation processes or it may be due to the multiple locally 89

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

exponential relaxations with differing time constants. There are experimental evidence from NMR and Rayleigh scattering studies [9, 10] that the relaxation found in the supercooled liquids is not homogeneous. However, the experiments fail to identify the microscopic origin of the heterogeneous relaxation in the supercooled liquids. In supercooled liquids, the structural relaxation of the liquid shows a slow down of several orders of magnitude. Another important problem in the dynamics of supercooled molecular liquids is the role of orientational degrees of freedom. The translation rotation paradox in fragile glass forming liquid has been addressed by Stillinger et al. [11, 12]. Mode coupling theory of the glass transition in molecular liquids has been developed by many [13– 15]. Molecular dynamics simulations of the dumbbell systems have been used to study the role of orientational degrees of freedom in dynamics of supercooled liquids [16– 20]. The notable difference between the glass transition and the I-N transition in a molecular system is in the role played by the translational degree of freedom. During the glass transition the orientational degrees of freedom freeze because of the coupling between translational and rotational degrees of freedom. But during I-N transition, the translational degrees of freedom behaves rather similar to that of an ordinary liquid. However, there are several similarities between the glass transition and I-N transition. The description of appearance of the pseudo-nematic domains has similarities with the description of formation of heterogeneous domains of glassy and supercooled liquids [21–24]. Neither the theoretical analysis nor the experiments could identify the exact size and shape of the domains and also the transport coefficients such as viscosity and diffusion inside these domains. An analogous description can be used for pseudo nematic domains near I-N transition. Here the domains are due to the freezing of the rotation. The unanswered questions in a liquid near I-N phase transition are that, what acts as the domains walls; how similar are the mechanisms of freezing near IN transition and freezing in supercooled systems; do pseudo-nematic domains have a shape; if it is there, what is the approximate size and geometry. Hence, it is intuitive to see that the relaxation processes in the supercooled liquid and pre-transition region of 90

4.1 Introduction

nematogens are similar in the orientational degrees freedom. The pre-transition region of a system of nematogens may be used to understand the microscopic dynamics of orientation in supercooled liquids. The I-N transition is purely due to the arrest of orientation and hence can be called a ”orientational glass transition”. Note that, this system differs from glass transition of the rotors in a lattice, which is well known in the literature [25, 26]. During the I-N transition the translational degrees of freedom behaves like that of an ordinary liquid. Our long time simulation study using Gay-Berne ellipsoids near I-N phase transition also show a power law at the intermediate times, this study has been presented in the previous chapter (see also Ref. [24]). In this chapter, the microscopic aspects of single particle rotation and translation of the nematogens has been investigated using molecular dynamics simulations. Single particle translational dynamics shows anisotropy, which is captured in the single particle translational van Hove correlation function and incoherent intermediate scattering function. The orientation dependent single particle van Hove correlation function and the incoherent intermediate scattering function show dramatic slow down as I-N transition is approached. In supercooled liquids the dynamical heterogeneity gets reflected in both translational and rotational diffusion of the molecules. The best way to look at the dynamical heterogeneity of the translational and rotational diffusion is through the non-Gaussian parameter [27–29]. We have calculated a rotational non-Gaussian parameter of nematogens, which is analogous to translational non-Gaussian parameter. The calculated rotational non-Gaussian parameters of the nematogens at different state points across the I-N transition show bimodality near I-N transition. An analysis of the time scales comparing non-Gaussian parameter with that of the single particle second rank orientational correlation function and the orientational mean square displacement shows that, the first peak of the non-Gaussian parameter corresponds to dynamical heterogeneity within a pseudo nematic domain and the second peak corresponds to the dynamical heterogeneity of a pseudo nematic domain. The temperature dependent study on the single particle dynamics also confirms these results. The origin of the dynamical het91

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

erogeneity in rotation is further probed by examining the single particle trajectories that show confinement of the director in a double well potential, whose wells are separated by angle π. The plot of typical trajectories of the director at different state points show the existence of orientational jumps near I-N transition, which is a well-known feature of supercooled liquid. The single particle orientational van Hove correlation function quantifies the confinement of the particle inside the orientational cage. The rest of the chapter is organized as follows. Following section gives the basic expressions of the single particle correlation functions. Section 4.3 presents the results of molecular dynamics simulations and the analysis of the calculated correlation functions. The orientational confinement of nematogens is analyzed in the subsection 4.3.1. The orientation dependent structural relaxation of the nematogens are discussed in section 4.3.2. Subsection 4.3.3 look into the heterogeneity of the translational and rotational displacements of the molecules. The summary of this work is given in the section 4.4.

4.2

Single particles dynamical correlation functions

The important correlation functions, which quantify the single particle dynamics are the single particle van Hove correlation function and the incoherent intermediate scattering function [30–32]. Latter is obtained from the spatial Fourier transform of the former. The single particle molecular van Hove correlation function can be defined in a system of N molecules as 1 Gs (r, Ω, Ω0 , t) = N

*

N X i

+

δ(r + ri (0) − ri (t))δ(Ω − Ωi (0))δ(Ω0 − Ωi (t)) ,

(4.1)

where the ri and Ωi are position and the orientation of the i0 th molecule. This is an extension of the single particle van Hove correlation function of atomic liquids to molecular liquids. Here, the complexity arises from the angular dependence of Gs (r, Ω, Ω0 , t). which can be simplified by decomposing this correlation function in terms of spherical harmonics [33] (See also Sec.2.3). The coefficients of the expansion, 92

4.2 Single particles dynamical correlation functions

Gsll0 m (r, t) are taken as the orientation dependent single particle van Hove correlation functions. Gsll0 m (r, t) can be calculated from the simulations using the relation [33] Gsll0 m (r, t) = 16π 2 hGs (r, Ω, Ω0 , t)Ylm (Ω)Ylm (Ω0 )i,

(4.2)

where Ylm (Ω) is the spherical harmonic of order l and m. This gives a measure of the translational diffusion of the molecule along with the rotational diffusion. On averaging over angular coordinates, Gs (r, Ω, Ω0 , t) gives single particle van Hove correlation function of number density. The single particle van Hove correlation function of number density is given by 1 Gs (r, t) = N

* N X i

+

δ(r + ri (0) − ri (t)) ,

(4.3)

This definition can be extended to the angular displacement of a system of linear molecules whose total rotation can be represented by the angle between the initial direction and the final direction of the director [16, 20]; N X 2 δ(θ − θ(t)), Gs (θ, t) = N sinθ i

(4.4)

where θ is the angle between the initial orientation of the director and it’s orientation after time t. This measures the probability of finding the director at an angle θ on a later time t from its initial direction. The total probability of the finding the director at time t at an angle θ from the initial direction is (1/2)Gθs (θ, t)sinθdθ. Gs (θ, t) shows the nature of orientational diffusion of the molecular axis. The rotational confinement of the director can be identified from peak position of Gs (θ, t). It is also worthwhile to study single particle structural relaxation in the reciprocal space (k space). The incoherent intermediate scattering function of the molecular liquid is given by

 Fs (k, Ω, Ω0 , t) = e−ik·r δ(Ω − Ωi (0))δ(Ω0 − Ωi (t))

(4.5)

where k is taken as a scalar assuming the isotropy of the liquid. Unlike the case of translational degrees of freedom, there is no analogous reciprocal quantity in the 93

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

angular space. Fs (k, Ω, Ω0 , t) can be decomposed in terms of the spherical harmonics to obtain it’s orientation dependent coefficients Fs (k, Ω, Ω0 , t) =

X

0

F ll m (k, t)Ylm (Ω)Ylm (Ω0 )

(4.6)

ll0 m

0

The Fsll m (k, t) is the orientation dependent self incoherent intermediate scattering function [34]. The relaxation of this function has both orientational and translational 0

channels. This is the k space analogue of the Gsll0 m (r, t). The Fsll m (k, t) shows rich dynamical behavior when either the translational or rotational channels of relaxation 0

become slower. It will be interesting to study the dynamics of F ll m (k, t) near I-N transition, where the rotational degrees of freedom freeze when the transition is approached. The dynamical heterogeneity near I-N transition can be probed by non-Gaussian parameter of the translation and rotation [27, 32]. The non-Gaussian parameter for the translation (α2T ) is defined in terms of the second and fourth moments of the random displacements [27, 32] as α2T =

3h[r(t) − r(0)]4 i , 5h[r(t) − r(0)]2 i2

(4.7)

where the angular brackets stands for the ensemble average. The pre-factor 3/5 arises from the dimensionality of the system. Similarly the rotational non-Gaussian parameter can be defined as α2R =

4 ~ − φ(0)] ~ h[φ(t) i . 2 i2 ~ − φ(0)] ~ 2h[φ(t)

(4.8)

where the pre-factor 1/2 is for two dimensional systems [35]. The rotational random displacements are obtained from the simulations by integration of the angular velocity ω ~ [16, 20]; ~ i (t) = φ ~ i (0) + φ

Z

t

dt0 ω ~ i (t0 ),

(4.9)

0

Molecular dynamics simulations have been preformed for a system of 576 Gay-Berne ellipsoids in the state points across the I-N transition. For details of the simulations and the parameters used in the simulations see sections 2.2.1 and 2.4. Data collected from four millions simulations steps is used to calculate the correlation functions. The wavevectors for the study of the structural relaxation are chosen to be the integral multiples of 2π/L (L is the length of the simulation box). 94

4.3 Results and discussion

4.3 4.3.1

Results and discussion Orientational confinement near I-N transition

a

b

c

d

Figure 4.1: Typical trajectory of the unit vector of a selected particle.The four trajectories shown are at four different densities: (a) is in the isotropic phase at density ρ∗ = 0.285; (b) is near I-N transition at density ρ∗ = 0.31; (c) is near I-N transition at density ρ∗ = 0.315; (d) is in the nematic phase at density ρ∗ = 0.32.

One of the signatures of supercooled liquid is the existence of jump motions that results in diffusion and stress relaxation. Rotational jumps or flipping of the molecules can be identified in the pretransition region of the I-N transition by plotting the trajectory of the director of ellipsoids. Figure 4.1(a) shows typical trajectory of the director in the isotropic phase. Figure 4.1(b) shows a trajectory very near the I-N transition that shows the signature of rotational confinement and rapid flip of the director. Note that, now the transition between the neighboring wells takes place by large-angle an95

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

gular displacement ( π rotation) of the director. In figure 4.1(c), the orientational confinement become more pronounced, consequently, the flipping of the director is by the larger and faster angular displacements than the previous case. Figure 4.1(c) shows the typical trajectory of a director in the nematic liquid crystal. In the nematic liquid crystal the director of the system is confined to one of confining well for very long time. Note that these are representation of the typical trajectory, which majority of the particles in the system follows. The large-angle flipping of the director is also well known in the literature of supercooled liquids [16, 20, 36, 37]. However, near the I-N transition the origin of this phenomenon is rather entirely different from that observed in the supercooled liquid. In supercooled liquids, orientational jumps are due to the freezing translation and the subsequent development of the stress in the system, which acts as an orientational cage. Large angle rotation is the mechanism of relaxation of stress in the supercooled molecular liquids. In case of the I-N transition the stress developed in the system is due to the freezing of the orientational degrees of freedom. Hence the statement ”the I-N transition can be regarded as the supercooling only in the rotational degrees of freedom” can be partially justified. The flipping of the molecules can be further quantified in terms of the single particle orientational van Hove correlation function (Eq. 4.2). The formation and the relaxation of the orientational cage around a molecule can identified from the time dependence of Gθs (θ, t). Figure 4.2(a) shows Gθs (θ, t) of an isotropic liquid. The Gθs (θ, t) evolves as Gaussian at short time in a molecular liquid. As the pseudo-nematic domains begin to form, Gθs (θ, t) shows a short tail at large theta at short time, which is absent in Gθs (θ, t) of an isotropic liquid. Since the pseudo-nematic domains are long lived in a system near the I-N transition, Gθs (θ, t) takes a long to time relax to equilibrium. The orientational cage is statistical in nature and similar to a double well potential in shape. In figure 4.2(c) minima of Gθs (θ, t) corresponds to transition state of the double well potential. At long time, the probability for finding the particle in other well which is separated by angle of π increases. The double well formation can be identified from the change of shape of Gθs (θ, t); in figure 4.2(c), the peak grows above the equilibrium value. In addition, the 96

4.3 Results and discussion

4

5

3

3

θ

2

2

1 0

b

Gs(θ,t)

Gs(θ,t)

θ

4

a

1 50

100

0

150

50

θ

100

150

θ 10

Gθ(θ,t)

c

6 4

s

s

Gθ(θ,t)

8

d 5

2 0

50

100

0

150

θ

50

100

150

θ

Figure 4.2: The temporal evolution of orientational single particle van Hove correlation function is plotted for T ∗ =1 at four densities against angle: (a) at ρ∗ = 0.285 ;(b) at ρ∗ = 0.3 (c) at ρ∗ = 0.315 (d) ρ∗ = 0.32. The five successive lines shown are arranged in the decreasing order of the peak hight are at time t∗ =49.5, t∗ =99.5, t∗ =149.5, t∗ =199.5 and t∗ =249.5.

minima of the probability distribution, even at long time, remains below 1. Note that, figure 4.2(c) shows Gθs (θ, t) in the nematic phase, which also shows the double well formation. In the nematic phase, the flipping of nematogen is a rare event; however, once a nematogen flips it remains long time in the second well before it flips back again. As the well formation takes place, the rotational diffusion of the system slows down; hence the height of the first peak increases (compare the line at the same time interval in the figures 4.2(a) to 4.2(d) ) with density. Another characteristic of the double well is that, it reduces accessible angles of director of a molecule. In figure 4.2(d), for density ρ∗ = 0.32 (nematic phase), the region that separates the wells is flat as compared to that found near the I-N transition ρ∗ = 0.315. 97

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

Single particle translation and incoherent intermediate scattering function

0.01

2

s

4 π r G (r,t)

4.3.2

0.005

0 0

1

2

1

2

3

4

5

6

7

3

4

5

6

7

−4

5

2

s

4 π r G220(r,t)

x 10

0 0

distance

Figure 4.3: The figure shows the plot of single particle van Hove correlation function against time on the top and the orientation dependent van Hove correlation function against time in the bottom. The dashed lines is plotted for ρ∗ = 0.315 and the solid line is for ρ∗ = 0.285. The five successive curves are arranged such that the peak height decreases in the increasing order of time. These lines are between time t∗ = 2 and t∗ = 10 with the interval of δt∗ = 2.

The distance dependent and wave vector dependent single particle relaxation functions can probe into the structural changes that accompany the I-N transition. The single particle diffusion becomes slower as the density increases; this is the normal behavior of a simple liquid and is also true for a liquid near I-N transition also. Figure 4.3(a) show the single particle van Hove correlation function of a liquid in the isotropic phase and same liquid at a state point near I-N transition. Gs (r, t) at both the state points shows all the characteristics of an ordinary liquid. However, the orientation dependent van Hove correlation function, G220 s (r, t) shows more dramatic changes than Gs (r, t), while the I-N transition take place. In case of the I-N transition, relaxation of 98

4.3 Results and discussion

G220 s (r, t) depends on the orientational diffusion of the molecule also. In the isotropic liquid G220 s (r, t) relaxes faster than Gs (r, t) due to fast orientational relaxation. However, as I-N transition is approached, the orientational relaxation becomes slower due to the orientational caging. The major contribution of relaxation of G220 s (r, t) near I-N transition comes from the translational degrees of freedom. It is observed that, anisotropy develops in the single particle structural relaxation as I-N transition is approached. In order to quantify the growth of anisotropy as the system approaches the I-N transition, the orientation averaged incoherent intermediate scattering function Fs (k, t) is plotted at different densities in figure 4.4. The sub-figure given at the top shows Fs (k, t) of an isotropic liquid (ρ∗ = 0.285) and the sub-figure at the bottom (ρ∗ = 0.315) shows Fs (k, t) of a liquid near I-N transition. The semi-log plot of Fs (k, t) at smaller k, that is, at k = 2π/L, is linear, which hints at exponentiality of the single particle relaxation. As k becomes larger, relaxation becomes superdiffusive at short distance, where the MSD shows a ballistic region. Even in the isotropic phase, at k ≈ 2π/b, anisotropy in single particle relaxation exist in the isotropic Gay-Berne liquid. Anisotropy in the relaxation of Fs (k, t) is less at smaller and larger k values. In comparison with the isotropic liquid, near the I-N transition the anisotropy exists at all values of k. Notably near k ≈ 2π/b, it become much more pronounced. The Fs (k, t) can be well fitted to a stretched exponential function in both isotropic phase and near I-N transition. The stretched exponential function used for fitting is f (t) = e−(t/τ (k))

β(k)

,

(4.10)

where τ (k) and β(k) are respectively the k dependent time constant and exponent. The variation in τ (k) and β(k) shows the change of nature of the relaxation function on the variation of k. The value of β(k) remains close to 1 if the relaxation function is exponential. Figure 4.5 shows the variation of the fitting parameters of Fs (k, t) with a stretched exponential at successive k values, in an isotropic Bay-Berne liquid (open symbols) and the same liquid near the I-N transition (filled symbols). The log-log plot of the time constant τ (k) against k is shown in the subplot given at the top of the figure 4.5. This subplot shows a k −2 behavior at small values of k. The value of the 99

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

0

ln Fs(k,t)

−1

−2

−3

−4 0

2

4

2

4

time

6

8

10

6

8

10

0

ln Fs(k,t)

−1

−2

−3

−4 0

time

Figure 4.4: Semi-log plot of self intermediate scattering function against time for three set of wave vectors. The slowest relaxation is for the set of smallest k (k = 2π/L - L is the box length). The next slowest set among the shown is at k ≈ 2π/a and the fastest decaying set is at k ≈ 2π/b. One set consists of three lines that is for the three wavevectors in x, y and z directions with respect to the simulation box. The solid line represent Fs (k, t) along the x direction, the dashed line represent the Fs (k, t) along the y direction and the dash-dot line represent Fs (k, t) along the z direction. The plot, which is shown above is in isotropic phase at density ρ∗ = 0.285 and the plot shown below is near I-N transition at density ρ∗ = 0.315.

exponent of k decreases as the value of k increases. Near I-N transition, due to the anisotropy developed in the system, the value of τ (k) differs in different directions. However, the exponent of k remains the same. At large k the time constants τ (k) 100

4.3 Results and discussion

6

ln τ(k)

4 2 0

−2 −4 −1

0

1

2

ln k

β(k)

1.5

1

0

5

k

10

15

Figure 4.5: The fitting parameters of the single particle intermediate scattering functions to stretched the exponential function (Eq. 4.10). The graph at the top is a log-log plot that shows the wavevector dependence of the time constant τ (k). The plot in the bottom shows the exponent (β(k)) of the stretched exponential against wavevector k. In these plots, the fitting parameters that represented by the open symbols are at ρ∗ = 0.285, which is an isotropic liquid and the filled symbols are at ρ∗ = 0.315, which is near the I-N transition. Three different symbols give fitting parameters at three orthogonal directions: circle is for x direction; diamond is for y direction; triangle is for z direction

at different directions have same values; this is due to short distance ballistic motion of the molecules, which take place before the molecular collisions become active. In figure 4.5, the subplot given at the bottom shows the variation of the exponent β(k) of the stretched exponential fit against k. This plot shows change in the diffusive 101

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

motion at different length scales. Compared to the isotropic liquid, for a liquid near I-N transition, diffusion is slower and anisotropic and have a longer subdiffusive regime. The value of β(k) reduces till the k ≈ 2π/b; after that, the diffusion increases and finally it become superdiffusive at large k (short distances). This nature is common for a system of nematogens in the isotropic phase as well as near the I-N transition. For a liquid near the I-N transition, the subdiffusive regime is much more prolonged in two orthogonal directions. However, in one of the directions (in this case - y) the particle motion become superdiffusive for a low value of k than that of an isotropic phase. This may be explained in terms of formation of a global director in the system. Due to the emergence of the global director diffusion in different directions show anisotropy. In the case of linear molecules the value of the diffusion constant differs in the direction parallel to the director from that perpendicular to the director [38]. In this case, the director is more aligned toward the y direction (by inspection of the snapshot of the system) - hence it may be concluded that particle motion become superdiffusive along the direction of the global director at relatively small values of k. It is worthwhile at this point to compare the relaxation of the orientation dependent incoherent intermediate scatting function Fs220 (k, t) of nematogens in the isotropic phase with that of same liquid near the I-N transition. Figure 4.6 shows the Fs220 (k, t) for density ρ∗ = 0.285 (top) and at density ρ∗ =0.315 (bottom), the former is in the isotropic phase and the latter is near the I-N transition. In the isotropic phase, the relaxation of Fs220 (k, t) is mostly exponential, as shown by the straight lines in the semi-log plot. Comparing the nature of relaxation of Fs (k, t) and Fs220 (k, t), it can be concluded that, both functions show nearly exponential relaxation in the isotropic phase. Since Fs220 (k, t) also accounts for the orientational relaxation, in the isotropic phase Fs220 (k, t) relaxes faster than Fs (k, t) at all values of k. Near I-N transition, the relaxation of Fs220 (k, t) dramatically slows down due to the formation of the orientational cage. Here the relaxation of Fs220 (k, t) become highly non-exponential. The semi-log plot of Fs220 (k, t) against time is not linear near I-N transition. The fitting parameters of the Fs220 (k, t) at density ρ∗ = 0.285 (top) and at density ρ∗ = 0.315 102

4.3 Results and discussion

1

ln F220(k,t)

0

s

−1 −2 −3 −4 0

2

4

2

4

time

6

8

10

6

8

10

1

ln F220(k,t)

0

s

−1 −2 −3 −4 0

time

Figure 4.6: Semi-log plot of orientation dependent self intermediate scattering function against time is shown here. The slowest relaxation is for the smallest k (k = 2π/L - L is the box length). The next slowest among the shown is at k ≈ 2π/a and the fastest decay is at k ≈ 2π/b. F220 (k, t) for two densities are given: subplot on top shows the F220 (k, t) of an isotropic liquid and subplot in the bottom shows F220 (k, t) for a system near I-N transition.

(bottom) are shown in figure 4.7. Subplot of the figure 4.7 shown at the top gives the variation of fitting parameters of the stretched exponential fit; this in the isotropic phase. In this subplot, β(k) shows maxima at k ∼ 2π/a. The variation the exponent β(k) against k is oscillatory. The relaxation is superdiffusive for all values of k for which relaxation of Fs220 (k, t) is studied. Near the I-N transition Fs220 (k, t) shows pure 103

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

1.35

β(k)

ln τ(k)

1 0

−1

1.225

−2 0

1.1 0

2

4

6

8

1

ln k 10

2

12

k 1.5

α(k)

1

0.5

0 0

2

4

6

k

8

10

12

Figure 4.7: The fitting parameters of orientation dependent self intermediate scattering function are plotted against wavevector k. The plot at the top shows the stretched exponential exponent β(k) from the fit of F220 (k, t) in an isotropic liquid at ρ∗ = 0.285; the inset of this subplot shows log-log plot of corresponding τ (k) against k. The plot shown at the bottom, gives power low exponent α(k) from the fit of F220 (k, t) at ρ∗ = 0.315.

power law relaxation. The Fs220 (k, t) can be fitted to a power law of the form f (t) = a t−α(k) .

(4.11)

with the exponent α(k) varies with k. In the figure 4.7, the plot given in the bottom shows the variation of α(k) against k. The relaxation become faster as value of k increases and finally settle to a plateau at large k. Note that the Fs220 (k, t) relaxes in a comparable time scale as that of Fs (k, t). Interestingly, Fs220 (k, t) shows the power law 104

4.3 Results and discussion

relaxation. The long time exponential tail, which common in orientational relaxation, that follows the power law relaxation is absent here due to presence of fast relaxation channels through translation. The presence of the power law in the relaxation of Fs220 (k, t) near the I-N transition is due to the slow down of orientational relaxation.

4.3.3

Heterogeneous diffusion near I-N transition

Near isotropic-nematic transition, nematogens show heterogeneity in both translational and rotational dynamics.

Figure 4.8 shows the non-Gaussian parameter cal-

0.6

R

α2 (t)

0.4

0.2

0

−0.2

−1

10

0

10

1

Time

10

2

10

Figure 4.8: Semi-log plot of the rotational non-Gaussian parameter is plotted against time. Height of the peak increases with the increase density; lowest peak corresponds to the ρ∗ = 0.285; higher peaks are in the order of increase in the density in steps of δρ = 0.005; the highest peak corresponds to the density ρ∗ = 0.32.

culated from different simulations of the system near I-N transition along an isotherm (T ∗ = 1.0). Successive lines are arranged in the order of increase of the peak height as the density increases. Near the I-N transition the peak splits that shows a bi-modality in the non-Gaussian nature of the rotational displacements. The rise in the α2R (t) is continues until the density reaches ρ∗ = 0.32. On further increase of density the two peaks in α2R (t) get separated and moves apart (this feature is not given in the figure 105

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

1 0.9 0.8 0.7

2

Cs (t)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

20

40

60

80

100

Time

Figure 4.9: The second rank single particle time correlation function is plotted against time to show the short time relaxation. Time constant of the relaxation function increases with the increase density; smallest time constant corresponds to the ρ∗ = 0.285; Cs2 (t) with bigger time constants are in the order of increase in the density in steps of δρ = 0.005; the slowest Cs2 (t) corresponds to the density ρ∗ = 0.32. 2

10

1

10

0

<δ φ(t)2>

10

−1

10

−2

10

−3

10

−1

10

0

10

1

Time

10

2

10

Figure 4.10: Log-log plot of the rotational mean square displacement is plotted against time. Diffusion decreases as density increases; slowest diffusion corresponds to the density ρ∗ = 0.32; mean square displacements are in the order of decrease in the density with a δρ = 0.005 from bottom to top the plot.

106

4.3 Results and discussion

4.8). In the deep nematic state, due to large domain size, the second peak is difficult to locate in the simulation. The bimodality in the α2R (t) can be explained in terms of the orientational cage formation near I-N transition. The two time that corresponds peaks of the α2R (t), represent two distinct dynamical events: (1) the heterogeneous rotational diffusion a nematogen with in the orientational cage; (2) collective heterogeneous rotational diffusion of a pseudo-nematic domain. The features that appears in the α2R (t) may give a phenomenological way to distinguish the boundary of I-N transition. This also hints at the stability of the pseudo nematic domains that have random local director, which rotates non-destructively in this pre-transition region. This is in accordance with the weak first order nature of the I-N transition. The free energy cost for the formation of nucleus of nematic phase in side the isotropic low near I-N transition as compared to other strong first order transitions (solid-liquid, gas-liquid etc.). Hence, the nucleus of the nematic phase is relatively stable near I-N transition. In order to characterize further the observed heterogeneity in the rotational displacements of the molecules, we have calculated the single particle second rank orientational time correlation function (OTCF) (See Eq. 3.3). In figure 4.9, we have plotted the short time behavior of the second rank single particle OTCF . The second peak in the non-Gaussian parameter is located near the turning point of the single particle second rank OTCF. This is the crossover time in single particle second rank OTCF from the short time exponential dynamics to long time slow dynamics. The single particle second rank OTCF exhibits a power law relaxation near this turning point. Now it will be interesting to inspect the behavior of the calculated rotational mean square displacement (RMSD), which is shown in figure 4.10. As the density increases toward the I-N transition, the RMSD shows three distinct regions: one is the initial ballistic region; a subdiffusive region follows this; finally the diffusion sets in. The subdiffusive region becomes prolonged as the I-N transition is approached. This subdiffusive region can be understood in terms of diffusion of the orientational cage; in this time scale the fast diffusion of the individual nematogens equilibrates with the slow diffusion of the orientational cage. 107

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

0.3

0.2

2

αR(t)

0.25

0.15 0.1 0.05 0

−1

10

0

1

10

10

2

10

Time

Figure 4.11: Translational non-Gaussian parameter is given in a semi-log plot at different densities. Height of the peak increases with the increase density; lowest peak corresponds to the ρ∗ = 0.285; higher peaks are in the order of increase in the density in steps of δρ = 0.005; the highest peak corresponds to the density ρ∗ = 0.32.

2

10

1

10

0

<δ r(t)2>

10

−1

10

−2

10

−1

10

0

10

1

Time

10

2

10

Figure 4.12: Log-log plot of MSD at different densities are plotted. Diffusion decreases as density increases; slowest diffusion corresponds to the density ρ∗ = 0.32; mean square displacements are in the order of increase in the density with δρ = 0.005.

108

4.3 Results and discussion

Corresponding non-Gaussian parameter of translational degrees of freedom is shown in the figure 4.11. The successive curves are arranged in the increasing order of the peak height: starting from density ρ∗ = 0.285 with a grid size of δρ∗ = 0.05; ending at density ρ∗ = 0.32. Compared to the rotational non-Gaussian parameter, translational non-Gaussian parameter shows monotonic increase in the height of the peak toward the I-N transition and also thereafter. Since the height of the peak of α2T (t) near the I-N transition show no significant growth, the dynamical heterogeneity near I-N transition may not be due to the supercooling of translation. The monotonic increase of the α2T (t) substantiate the argument that although the translational-rotational coupling is important for the I-N transition to occur, this transition is mostly due to the rotational freezing. The change in the α2T (t) in a supercooled system [29, 39] is clearly different from that of α2T (t) near I-N transition . In comparison with the α2R (t), α2T (t) shows only a single peak. Hence, the translation is not frozen during the orientational cage formation: that is, the movement of one nematogen from one orientational cage to another is not identified in translational diffusion. The increase of height and duration of the peak in α2T (t) is related only to the increase in the density of the system. The translational mean square displacement given in the figure 4.12 also supports above arguments . In this figure, the subdiffusive region of the mean square displacement is absent. The system used in our study exhibits sharper I-N transition along an isotherm than that along an isochore. It is difficult to pin point a transition temperature along an isochore. However, it is worthwhile to study this system along an isochore that may be close to experiments. In figure 4.13, α2R (t) is plotted at a temperature interval of 0.05 from T ∗ = 1.25 to 0.85 for density ρ∗ = 0.315. Note that the I-N transition is less sensitive to the variation of temperature than density. Here the second peak grows slowly. Most of the state points chosen for the simulation along an isochore are mostly in the pre-transition region. The plot shows a monotonic raise in the peak height of α2R (t) as the temperature decreases. Since state points chosen in the figure 4.13 are in the pre-transition region, the growth and dynamics of the pseudo-nematic 109

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

0.6

0.4

R

α2 (t)

a 0.2

0

−0.2

−1

10

0

10

Time

1

10

1

10

10

2

0.3

b

0.2

T

α2 (t)

0.25

0.15 0.1 0.05 0

−1

10

0

10

Time

10

2

Figure 4.13: The semi-log plot of the temperature dependence of the rotational and translational non-Gaussian parameters are shown in these figures. The successive lines given in these figures are arranged in the order of increase of the peak height from temperature T ∗ =0.85 to T ∗ =1.25 on grid of δT ∗ =0.05 at a constant density ρ∗ = 0.315.

domains are clearer in this figure. In comparison to the simulation along the isotherm, the rate of separation of the times scales between the peaks are slow along an isochore. However, the growth of heterogeneity is comparable to that found in the study along the isotherm (see figure 4.8). We may conclude that the temperature and density plays 110

4.4 Concluding remarks

similar role in the I-N transition in this system of Gay-Berne nematogens. Along the isochore, the change in the α2T (t) is not significant in comparison with the change the α2R (t). In figure 4.11, the α2T (t) is plotted at different temperatures starting from the temperature T ∗ = 1.25 to T ∗ = 0.85 at constant density ρ∗ = 0.315. From this plot following conclusions can be drawn: only rotational freezing is responsible for the formation of the pseudo nematic domains; the translational dynamical heterogeneity remains almost constant in the pre-transition region. It is also possible to conclude that, since the time scale of α2T (t) remains unchanged in the pre-transition region, the characteristics of the translational diffusion of nematogens near I-N transition are similar to that observed in the dense liquids.

4.4

Concluding remarks

This chapter contains results of molecular dynamics simulation of single particle orientational and translational relaxation in a system of 576 Gay-Berne ellipsoids. In the system simulated, the density has been chosen as the controlling variable. Different simulations are done across the I-N transition along an isotherm, with increase of density the I-N transition take place. These studies focus on the single particle dynamics of the system near I-N transition and it’s similarities with that of a supercooled liquid. In order to characterize the slowing down of the orientational relaxation process, we have examined the relaxation functions in both r and k space. This work further probes the role played by the orientational degrees of freedom in the slow down of relaxation process when the system approaches I-N transition. Analysis of the single particle director trajectory shows the formation of orientational caging in the system. Here the particles are confined in a double well potential with minima separated by angle π. The orientation of molecules near I-N transition exhibits fast π flipping. In the literature of the supercooled liquids this is jump motion is believed to be responsible for the removal non-ergodicity [40] of the orientational degrees of freedom. The rotational confinement is also observed in supercooled diatomic liquids [14, 16, 41]. The double well structure of the potential for the rotational confinement can be quantified from 111

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

the behaviour of single particle orientational van Hove correlation function. Near the I-N transition, the double well structure is responsible for the bimodality of the orientational van Hove correlation function (Gθs (θ, t)). The single particle spatial van Hove correlation function shows usual liquid like behavior. Interestingly, the relaxation of 220 the G220 s (r, t) become slow near I-N transition. Gs (r, t) in an isotropic liquid relaxes

fast as compared to Gs (r, t) due to presence of relaxation channels associated with rotational degrees of freedom. Due to the closure of these relaxation channels near the I-N transition G220 s (r, t) relaxes in the same time scale as that of Gs (r, t). The incoherent intermediate scatting function Fs (k, t) shows strong anisotropy in it’s relaxation near I-N transition. The fit of the Fs (k, t) to a stretched exponential function shows superdiffusive behaviour at large wavenumbers. Another the interesting observation is the fast the growth of the exponent β(k) of the stretched exponential fit with the increase of k in one of the direction near the I-N transition, which is rapid than that found in an isotropic liquid. While in the other two orthogonal direction the β(k) variation is slow. This anomalous behaviour may be related to the formation of the global director in the system. It shows that along the director, the ellipsoid motion is superdiffusive at large value of k in a system near I-N transition compared to an isotropic liquid. The orientation dependent self intermediate structure factor shows a stretched exponential relaxation in the isotropic phase, with the value of the β(k) greater than 1. This is due to formation of faster channels of relaxation of the Fs220 (k, t), through the orientational degrees of freedom. As the system approaches the I-N transition, the fast channels through the rotational degrees of freedom closes due to the freezing of the rotations. In this case Fs220 (k, t) show a power law relaxation with the exponent that starts at 0.2 for the lowest k value, crosses 1 around k ∼ 4 and finally reaches a plateau near k ∼ 8 with value of the exponent ∼ 1.5. We have calculated rotational non-Gaussian parameter to probe the non-Gaussian nature of rotational displacements near the I-N transition. The rotational non-Gaussian parameter on approach to I-N transition shows a bimodal structure that hints at two dynamical events in the relaxation process. The first one belongs to the heterogeneous 112

4.4 Concluding remarks

orientational diffusion of an ellipsoid inside an orientational cage. The second peak corresponds to the heterogeneous collective diffusion of the orientational cage. In the rotational α2R (t), the fist peak corresponds to the transformation from the ballistic to subdiffusive regime of the rotational mean square displacement and the second peak correspond to the transition from the subdiffusive to diffusive regime of rotational mean square displacement. This argument also supported by the second rank single particle orientational correlation function. In order to find dynamical heterogeneity in the translational degrees of freedom during the I-N transition, we have calculated the translational non-Gaussian parameter α2T (t). α2T (t) does not show any appreciable change while I-N transition take place. Subdiffusive regime is absent in the mean square displacement of the translation. The height of the peak of translational non-Gaussian parameter is enhanced as the I-N transition is approached along the density. However, this is not as pronounced as that observed normally in the supercooled liquids. A study on the non-Gaussian parameters of the system along an isochore by variation of the temperature shows similar characteristics as that observed along an isotherm. Since the I-N transition is more sensitive to the density, the study along the temperature axis shows pronounced pretransition effect in the rotational non-Gaussian parameter. The simulations clearly reveal the nature of the heterogeneity found in the system in the single particle dynamics.

113

4.

SINGLE PARTICLE ROTATIONAL DYNAMICS

114

Bibliography [1] S. Chandrasekhar, Liquid Crystals (Cambridge University Press, Cambridge, 1977). 4.1 [2] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993). 4.1 [3] M. J. Stephen and J. P. Straley, Rev. Mod. Phys. 46, 617 (1974). 4.1 [4] T. W. Stinson and J. D. Lister, Phys. Rev. Lett. 25, 503 (1970). 4.1 [5] G. K. L. Wong and Y. R. Shen, Phys. Rev. Lett. 30, 895 (1973). 4.1 [6] G. K. L. Wong and Y. R. Shen, Phys. Rev. A 10, 1277 (1974). 4.1 [7] M. D. Ediger, C. A. Angell, and S. R. Nagel, J. Phys. Chem. 100, 13200 (1996). 4.1 [8] W. Kob et al., Phys. Rev. Lett. 79, 2827 (1997). 4.1 [9] K. SchmidtRohr and H. W. Spiess, Phys. Rev. Lett. 66, 3020 (1991). 4.1 [10] M. T. Cicerone and M. D. Ediger, J. Chem. Phys. 103, 5686 (1995). 4.1 [11] F. H. Stillinger and J. A. Hodgdon, Phys. Rev. E 50, 2064 (1994). 4.1 [12] F. H. Stillinger and J. A. Hodgdon, Phys. Rev. E 53, 2995 (1996). 4.1 [13] R. Schilling and T. Scheidsteger, Phys. Rev. E 56, 2932 (1997). 4.1 [14] W. Gotze, A. P. Sing, and T. Voigtmann, Phys. Rev. E 61, 6934 (2000). 4.1, 4.4 115

BIBLIOGRAPHY

[15] S. H. Chong and W. Gotze, Phys. Rev. E 65, 051201 (2002). 4.1 [16] S. Kammerer, W. Kob, and R. Schilling, Phys. Rev. E 56, 5450 (1997). 4.1, 4.2, 4.2, 4.3.1, 4.4 [17] S. Kammerer, W. Kob, and R. Schilling, Phys. Rev. E 58, 2131 (1998). 4.1 [18] S. Kammerer, W. Kob, and R. Schilling, Phys. Rev. E 58, 2141 (1998). 4.1 [19] C. D. Michele and D. Leporini, Phys. Rev. E 63, 36701 (2001). 4.1 [20] C. D. Michele and D. Leporini, Phys. Rev. E 63, 36702 (2001). 4.1, 4.2, 4.2, 4.3.1 [21] S. D. Gottke et al., J. Chem. Phys. 116, 360 (2002). 4.1 [22] S. D. Gottke, H. Cang, B. Bagchi, and M. D. Fayer, J. Chem. Phys. 116, 6339 (2002). 4.1 [23] H. Cang, J. Li, V. N. Novikov, and M. D. Fayer, J. Chem. Phys. 119, 10421 (2003). 4.1 [24] P. P. Jose and B. Bagchi, J. Chem. Phys. 120, 11256 (2004). 4.1 [25] C. Renner, J. L. Lowen, and J. L. Barrat, Phys. Rev. E 52, 5091 (1995). 4.1 [26] M. Jimenez-Ruiz et al., Phys. Rev. E 52, 5091 (1995). 4.1 [27] A. Rahman, Phys. Rev. 136, 405 (1964). 4.1, 4.2 [28] J. N. Roux, J. L. Barrat, and J. P. Hansen, J. Phys.: Condens. Matter 1, 7171 (1989). 4.1 [29] T. Odagaki and Y. Hiwatari, Phys. Rev. A 43, 1103 (1991). 4.1, 4.3.3 [30] J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). 4.2 [31] U. Balucani and M. Zoppi, Dynamics of the liquid state (Claredon Press, Oxford, 1994). 4.2 116

BIBLIOGRAPHY

[32] J. P. Boon and S. Yip, Molecular hydrodynamics (McGraw-Hill International book company, New Delhi, 1980). 4.2, 4.2 [33] C. Grey and K. E. Gubbins, Theory of Molecular Fluids (Clarendon Press, Oxford, 1984). 4.2 [34] B. Bagchi and A. Chandra, Adv. Chem. Phys. 80, 1 (1991). 4.2 [35] A. H. Marcus, J. Schofield, and S. A. Rice, Phys. Rev. E 60, 5725 (1999). 4.2 [36] G. F. Signorini and J. L. Barrat, J. Chem. Phys. 92, 1294 (1990). 4.3.1 [37] P. Sindzingre and M. L. Klein, J. Chem. Phys. 96, 4681 (1992). 4.3.1 [38] R. Vasanthi, S. Ravichandran, and B. Bagchi, J. Chem. Phys. 115, 10022 (2001). 4.3.2 [39] P. P. Jose, D. Chakrabarti, and B. Bagchi, Phys. Rev. E 71, 030701 (2005). 4.3.3 [40] W. Gotze and L. Sjogren, Rep. Prog. Phys. 55, 241 (1992). 4.4 [41] C. D. Michele and D. Leporini, Phys. Rev. E 63, 036701 (2001). 4.4

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118

Chapter 5 Wavenumber and frequency dependent collective relaxation in liquid crystals 5.1

Introduction

Time correlation functions of collective number density and orientational fluctuations in the liquid play an important role in the study of collective relaxation in molecular liquids. The wavenumber dependent collective correlation functions of fluctuations bridge the gap between the single particle relaxation and the relaxation in the hydrodynamic limit (k → 0 and ω → 0) [1, 2]. These correlations functions can be measured in the experiments such as light scattering and inelastic neutron scattering [3–9]. In the hydrodynamic limit, the expressions of these correlation functions can be obtained from the linearized Navier-Stokes equations. This linearized hydrodynamic model of the simple liquids in equilibrium for small wavenumbers have been useful in interpreting the results obtained from the computer simulations and the experiments [3, 5, 10]. The time correlation function analysis of the liquid state includes expressions for the transport coefficients and the hydrodynamic behavior. Such an analysis includes the wavelengths and frequencies that are comparable to interatomic distances and microscopic relaxation rates. Deviations from the hydrodynamic behaviour often appear in this analysis above some critical wavenumber. The wavenumbers near this critical distance may be referred as intermediate wavenumbers. Even after many 119

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

years of investigation by different researchers, a complete microscopic understanding of the relaxation of collective fluctuation in liquids remains incomplete. At intermediate wavenumber, the correlation functions of collective fluctuations deviate from their hydrodynamic behavior; here the distance referred to is comparable to that of inter particle separation. The collective relaxation in the intermediate wavenumber regime can be treated in the framework of the generalized hydrodynamics [5, 7, 11, 12]. In contrast to simple liquids, the studies of molecular liquids are bound to be complex due to the presence of the internal degrees of freedom, which couple to the conserved modes and act as additional relaxation channels for fluctuation. Compared to simple liquids, there are several parameters of intermolecular potential, like aspect ratio and geometry of molecule, which may affect the microscopic relaxation of molecular liquids [13–15]. There are several attempts to study the relaxation of the hydrodynamic modes in molecular liquids, however, such an analysis near the I-N transition is found to be absent in the literature. It is interesting to study this aspect of I-N transition, especially in the context of the recent experimental results by Fayer and coworkers [16–21] (see chapter 1), where the orientational degree of freedom shows relaxation dynamics similar to that found in supercooled liquids. On the other hand, the experimental study of collective relaxation of other modes show the signature of the second order phase transition [22]. In the dynamic light scattering experiments near the I-N transition, the spectra of the orientational degrees of freedom mask dynamics of other modes. Mode-selective dynamic light scattering experiments of Takagi et al. [22] on 5CB and 7CB liquid crystal samples have studied the dynamics of modes usually hidden under fluctuation of the orientational mode. They have measured the thermal diffusion, orientational relaxation and propagating sound modes of those liquid crystal samples. Using photopyroelectric technique, Marinelly et. al [23] have measured the thermal conductivity and thermal diffusivity of aligned nCB samples. They have found that the thermal conductivity shows a discontinuity near the I-N transition, while, the thermal diffusivity shows a dip. 120

5.1 Introduction

The molecules used in the experiments of Fayer and coworkers are mostly rigid and rod like with aspect ratio between 3 and 4. Long time simulation studies and theoretical analysis on a system of Gay-Berne ellipsoids with aspect ratio 3 have confirmed the existence of the power law relaxation near the pre-transition region of the I-N transition - these simulations and the theoretical analysis are presented in the chapters 1 and 3. (See also ref. [17, 18, 24]). The single particle dynamics is a well-studied property in the literature of the supercooled liquids. The study of single particle dynamics in a system of nematogens, presented in the chapter 4 reveals the striking similarities of single particle orientational relaxation in a system near the I-N transition with that of a supercooled liquid [25]. In this chapter, the study of the nematogens is chiefly in the pre-transition region, where, even though there is an anisotropy due to the formation of pseudo nematic domains, they relax slowly to remove the anisotropy. However, as the orientational relaxation slows down close to the I-N transition, the relaxation of the anisotropy also slows down. Since the I-N transition is weakly first order, there are prolonged pretransitional effects, which enables the observation of the effect of orientational ordering in the structural relaxation for a range of state points across I-N transition. Since I-N transition occurs due the formation of the orientational cage that freezes rotation of the molecules, the angular momentum transferred to cage may propagate through the medium. As the I-N transition is approached, the exchange of the angular momentum with the cage gives rise to a strong back scattering region in the angular velocity auto correlation function (see figure 6.6). Hence an interesting property which is worthwhile to study is the wavenumber dependent propagation and transfer of the angular momentum. A generalized hydrodynamic description of the classical molecular liquids also involves the relaxation of the collective angular momentum fluctuations [26– 28]. The analysis by Aliawadi et al. [27, 28], on relaxation of the collective angular momentum is focused on it’s intrinsic part, which may be coupled to the conserved densities, namely, particle number, linear momentum density, and energy density. They have found that the longitudinal part of the spin angular momentum density relaxation 121

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

is not coupled to other variables. A system near the I-N transition, where orientational confinement is strong, seems to be an ideal system for the study of propagation of the intrinsic angular momentum. Investigations reported in this chapter are based on the simulation of a small class molecular liquids with aspect ratio ∼ 3 and can be approximated by the ellipsoids with two of the minor axis are equal. As already mentioned in the previous chapters this system undergoes an I-N transition as the density increases from ρ∗ =0.285 to ρ∗ =0.315, gradually along an isotherm at T ∗ = 1.0. The single particle relaxation dynamics in this system shows usual behavior as that of an ordinary liquid, except the presence of a marked anisotropy near I- N transition. Now, it is worthwhile to study the growth of the anisotropy in the level of collective dynamics at different wavenumbers. The present study on the relaxation of fluctuation of collective density and current modes focuses on the relaxation dynamics at different wavenumbers; one in the hydrodynamic limit (small wavenumber limit, k = 2π/L ) and other two limits are at wavenumbers k ∼ 2π/a (a is the major axis of the ellipsoid)- will be referred here after as intermediate wavenumbers and k ∼ 2π/b - (b is the minor axis of the ellipsoid)- will be referred here after as large wavenumbers. In this study, the intermediate scattering function calculated from the simulations is fitted to the linearized hydrodynamic model of the intermediate scattering function in order to find the thermodynamic and transport properties of the system. Thermodynamic and transport properties obtained from the fitting parameters show non-monotonic behavior at different state points across the I-N transition. The sound velocity obtained from the peak position of the Brillouin peak [5–7] shows that the adiabatic sound velocity increases across the I-N transition; except a deviation from it’s monotonic increase near the transition. The study on the transverse current-current correlation function shows the anisotropy of the transverse current fluctuation across the I-N transition for all the length scales. It can be concluded that the shear wave for all the wavenumbers propagates with different velocities in different directions as the I-N transition is approached. The transverse current-current correlation function shows large anisotropy in the relaxation even for 122

5.2 Collective correlation functions of density, current and angular current

the lowest wavenumber. The spectrum of the longitudinal current-current correlation function shows a single peak structure at small wavenumbers. The position of this peak corresponds frequency of the adiabatic sound propagation in the medium. In the intermediate wavenumbers an additional peak appears in the spectrum at lower frequencies, which is not present in atomic liquids. The study in the transverse and longitudinal part of the angular current-current correlation function shows weak wavenumber dependence in the relaxation of the collective angular momentum fluctuations. These correlation functions show a prolonged backscattering region at intermediate and small wavenumbers. The remaining sections of this chapter are organized as follows. Section 5.2 discusses the definition of the time correlation functions and their expressions in the hydrodynamic limit, which are used in the analysis. Details of simulations are presented in the section 5.3. The section 5.4 presents the results and discussions on the time correlation functions calculated from the simulations and their power spectra at different wavelengths. This section consists of four parts: subsection 5.4.1 presents a study on relaxation of the collective density fluctuations; subsection 5.4.2 presents a study on the relaxation of longitudinal currents; subsection 5.4.3 reports a study on the transverse current relaxation; results on relaxation of transverse and longitudinal current of intrinsic angular momentum are discussed in subsection 5.4.4. The summary of the work is presented in the section 5.5.

5.2

Collective correlation functions of density, current and angular current

For a system that consists of N linear molecules there are 3N coordinates of position and 2N coordinates of orientation. In this system, density is a function the positions as well as orientations (ρ = ρ(r, Ω)). The number density of the system ρ(r) can be obtained from ρ(r, Ω) by averaging over the orientations. The orientation of the molecule is not a conserved quantity. The microscopic densities for a molecular system 123

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

can be defined as [27] ρ(r, t) =

X a

u(r, t) =

X a

j(r, t) =

X a

and S(r, t) =

X a

a

δ(r − ra ),

(5.1)

a (t)δ(r − ra ),

(5.2)

pa (t)δ(r − ra ),

(5.3)

Sa (t)δ(r − ra ).

(5.4)

where ra and p are the center of mass position and momentum of the molecule at time t, Sa (t) is the angular momentum about the center of mass, with respect to the laboratory frame of reference and the a (t) is the energy that includes translational and rotational kinetic energy as well as total potential energy. Out of which, three densities are conserved quantities, they are number density (ρ(r, t)), energy density (u(r, t)) and momentum density (j(r, t)). In addition to this, molecular systems have an additional non-conserved density, which related to the internal angular momentum of the system with respect to the laboratory frame of reference [27]. The total angular momentum of the system is conserved for a system of molecules. The expression for the R total angular momentum is dr(r × j(r, t) + S(r, t)). The angular momentum density is independently conserved if the intermolecular potential is central.

The intermediate structure factor F (k, Ω, Ω0 , t) which measures the relaxation of density fluctuations in the molecular liquid is defined as [29], F (k, Ω, Ω0 , t) =

1 hρ(−k, Ω, 0)ρ(k, Ω0 , t)i. N

(5.5)

This correlation function also accounts for the correlation between the orientation and density at two points separated in space and time. This is a direct extension of the intermediate scattering function [5–7] of a monatomic liquid. On averaging over orientation, the F (k, Ω, Ω0 , t) gives the F (k, t), which is the intermediate scattering function of the system. In the theory of the linearized hydrodynamics, F (k, t) can be expressed as [3, 5] F (k → 0, t) = S(k → 0)[a e−t/τa + (1 − a) e−t/τb (cos(ω0 t) + b sin(ω0 t))], 124

(5.6)

5.2 Collective correlation functions of density, current and angular current

where S(k) is the static structure factor, a = (γ − 1)/γ (γ is the ratio of specific heats (Cp /Cv )), τa = 1/DT k 2 (DT is the thermal diffusivity), τb = 1/Γk 2 (Γ is the sound wave damping constant), ω0 = ck (c is the adiabatic sound velocity) and b = k((3Γ − Dv )/γc) (Dv is the kinematic viscosity). Dynamical structure factor is calculated from the F (k, t) by the Fourier transformation, 1 S(k, ω) = Re π

Z

F (q, t) e−iωt dt.

(5.7)

In this limit (k → 0), S(k, ω) is given by S(k) γ − 1 DT k 2 S(k → 0, ω) = π γ ω 2 + (DT k 2 )2   Γk 2 Γk 2 S(k) 1 + + π 2γ (ω − ck)2 + (Γk 2 )2 (ω + ck)2 + (Γk 2 )2   ω + ck ω − ck S(k) 1 b + + π 2γ (ω − ck)2 + (Γk 2 )2 (ω + ck)2 + (Γk 2 )2

(5.8)

As already mentioned, this expression is limited to wavenumbers k which are small enough to show the true collective behavior. The explicit information about the relaxation of the orientation density of the system is absent from F (k, t) and S(k, ω). Another important collective variable that describes the collective dynamics of the system is the current of the linear momentum [5–7]. The linear momentum current density of the system in the reciprocal space can be defined as (mass m = 1) j(k, t) =

X

vi (t) e−ik·r ,

(5.9)

i

where v is the velocity of the particle. The wavenumber dependent current-current correlation function has longitudinal and transverse parts. The current, j(k, t), can be split into longitudinal and transverse directions as ˆ · j(k, t), j(k, t) = kˆ kˆ · j(k, t) + (1 − kˆk) = jL (k, t) + jT (k, t),

(5.10) (5.11)

ˆ where k=k/k is the unit tensor. The longitudinal and transverse components are defined in terms of the direction of k. The jL (k, t) and jT (k, t) are the longitudinal and 125

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

transverse currents. The longitudinal current-current correlation function (CL (k, t)) is defined as CL (k, t) =

1 hjL (−k, 0) · jL (k, t)i. N

(5.12)

When the direction of k is along the z direction, CL (k, t) =

1 hjz (−k, 0)jz (k, t)i. N

(5.13)

Here the isotropy of the liquid is assumed. In other words the CL (k, t) can be written in terms of the velocity of the individual particle of the system as (mass m = 1). + * 1 X (5.14) CL (k, t) = vlx vmx e−ik(xl −xm ) . N lm In the single particle (large k) limit the expression of the CL (k, t) is given as    kB T k B T 2 2 − kB T k 2 t2 CL (k, t) = k t e 2m 1− . m m

(5.15)

Since the wavenumber is large in this limit the collective behavior becomes negligible and the CL (k, t) behaves as that of a free particles. Corresponding power spectra in the hydrodynamic limit is related to the S(k, ω), by the relation CL (k, ω) =

ω2 [S(k, ω)]. k2

(5.16)

The definition of the transverse current-current correlation function CT (k, t) is similar to the definition of CL (k, t). It is defined in terms of the longitudinal current as CT (k, t) =

1 hjT (−k, 0) · jT (k, t)i. N

(5.17)

In terms of the individual particle velocities the expression for CT (k, t) can be written for an isotropic liquid as CT (k, t) =

*

1 X vlx vmx e−ik(zl −zm ) N lm

+

,

(5.18)

here k is taken along the z direction. In the case of CT (k, t) the limiting expressions at the large wavenumbers for the CT (k, t) is given by   kB T 2 2 kB T e− 2m k t . CT (k, t) = m 126

(5.19)

5.2 Collective correlation functions of density, current and angular current

and the corresponding expression in the hydrodynamic limit is CT (k, t) = CT (k, 0)e−k

2 (η/ρm)t

,

(5.20)

where η is the shear viscosity and the ρ is the number density of the system. Hence in this limit, the CT (k, ω) shows exponential relaxation An important property that distinguishes the molecular fluids from that of atomic fluids is the collective angular momentum fluctuations. In this work we are interested in the intrinsic angular momentum. The intrinsic angular current S(r, t) of a system of particles is given by S(k, t) =

X

ω ~ i (t) e−ik·r ,

(5.21)

i

where ω is the angular velocity. Here after, the reference to angular momentum fluctuation means the intrinsic angular momentum fluctuations. The expression for the transverse and the longitudinal intrinsic angular momentum currents are similar to that of the linear momentum currents. The longitudinal part of the spin angular momentum correlation function CLav (k, t) can be written as CLav (k, t) =

1 hSz (−k, 0) · Sz (k, t)i, N

(5.22)

The explicit expression for the CLav (k, t) can be written for an isotropic liquid along k in the z direction CLav (k, t) =

*

1 X ωlx ωmx e−ik(xl −xm ) N lm

+

.

(5.23)

The CTav (k, t) shows how the angular momentum fluctuations propagate in longitudinal direction. Similarly the transverse part of the angular momentum correlation function CTav (k, t) can be written as CTav (k, t) =

1 hST (−k, 0) · ST (k, t)i. N

(5.24)

It can also be written explicitly in terms of the angular velocity of the particles in an isotropic liquid with k along the z direction as + * X 1 ωlx ωmx e−ik(zl −zm ) . CTav (k, t) = N lm

(5.25)

The CTav (k, t) shows how the angular momentum fluctuations propagate in transverse direction. 127

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

5.3

Details of the simulations

The molecular dynamics simulations in the micro-canonical ensemble of 576 particles are run for 5 million steps to generate trajectories. We have calculated the collective correlation functions across I-N transition from this data. The wavevectors are chosen to be in the x, y and z directions of the simulation box. The difference in the calculated correlation functions in different directions is a measure of the anisotropy developed in the system. More details of the simulations are given in the sections 2.2.1 and 2.4. Relaxation of wavenumber dependent correlation functions has been studied at three wavenumbers. The smallest wavenumber studied is at k = 2π/L, where L is the length of the simulation box. The intermediate wavenumber chosen for the study is at k ∼ 2π/a, where a is the length of the major axis of the ellipsoid. The largest wavenumber chosen is at k ∼ 2π/b, where b is the length of the minor axis of the ellipsoid.

5.4 5.4.1

Results and discussions Intermediate scattering function and dynamic structure factor

Figure 5.1 shows the total intermediate scattering function for all the three wavenumbers chosen for the study of relaxation, at two state points of the Gay-Berne liquid three figures on the left hand side ((a), (c) and (e)) show F (k, t) of an isotropic liquid and the figures on the right hand side ((b), (d) and (f)) show the corresponding F (k, t) near I-N transition. The three rows of figures are at three different wavenumbers chosen for the study of relaxation: figures (a) and (b) correspond to the smallest wavenumber of the system kL = 2π/L, which show the relaxation near the hydrodynamics limit (k → 0); in the figures (c) and (d), F (k, t) for the wavenumber ka ≈ 2π/a is shown; F (k, t) shown in the figures (e) and (f) are near kb ≈ 2π/b. The F (k, t) at k = kL shows hydrodynamic behavior. The rapid oscillation of the calculated F (k, t) at this wavenumber is related to the adiabatic sound propagation in the medium according to linearized hydrodynamics [3, 5, 7]. As the value of k becomes large, there is a reduction 128

5.4 Results and discussions

1 a

0.5

0 0

F(k,t)/S(k)

F(k,t)/S(k)

1

k=0.50 ρ*=0.285

10 time

F(k,t)/S(k)

F(k,t)/S(k)

k=1.99 ρ*=0.285

F(k,t)/S(k)

F(k,t)/S(k)

10 time

20

0.5

k=2.05 *

ρ =0.315

5 time

1 e

0 0

*

ρ =0.315

d

0 0

5 time

1

0.5

k=0.51

1 c

0 0

0.5

0 0

20

1

0.5

b

k=6.46 ρ*=0.285

2 time

4

f 0.5

0 0

k=6.16 ρ*=0.315

2 time

4

Figure 5.1: The calculated intermediate scattering function is plotted for three different directions of wavenumbers kx , ky and kz against time: shown respectively by solid, dashed and dash-dot lines. The figures (a), (c) and (e) are at density ρ∗ = 0.285 and figures (b), (d) and (f) are at density ρ∗ = 0.315. The figures (a) and (b) give the normalized F (k, t) at k = 2π/L; figures (c) and (d) give F (k, t) at k ≈ 2π/a and the figures (e) and (f) give F (k, t) at k ≈ 2π/b

129

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

0.6

1

b

a 0.4

S(k,ω)

S(k,ω)

k=0.50 ρ*=0.285

0.2 0 0

5

0.5

*

*

ρ =0.315

0 0

10

ω

3

k=0.51

5 *

ω

3

d

c 2

k=1.99 *

ρ =0.285

1 0 0

10

*

ω

4

k=2.05

S(k,ω)

S(k,ω)

2

*

ρ =0.315

1 0 0

20

10

S(k,ω)

*

ρ =0.285

10

*

ω

20

f

k=6.48

S(k,ω)

0 0

*

ω

10

e 2

10

20

30

k=6.16

5

0 0

*

ρ =0.315

10

20

30

*

ω

Figure 5.2: The dynamic structure factor is plotted at three different directions of wavenumbers against frequency - kx , ky and kz shown respectively by solid, dashed and dash-dot lines. The figures (a), (c) and (e) are at density ρ∗ = 0.285 and figures (b), (d) and (f) are at density ρ∗ = 0.315. The figures (a) and (b) give the S(k, ω) at k = 2π/L, the figures (c) and (d) give S(k, ω) at k ≈ 2π/a and the figures (e) and (f) show S(k, ω) at k ≈ 2π/b.

130

5.4 Results and discussions

10.5

2

10

a

1.9

b

γ

c

9.5 9

1.8

8.5 1.7

0.29

0.3 0.31 * ρ

8

0.29

0.3 0.31 * ρ

0.32

1.2

d

c

7 6

DT

Γ

8

0.32

0.7

5 4

0.29

0.3 0.31 * ρ

0.32

0.2

0.29

0.3 0.31 * ρ

0.32

Figure 5.3: Thermodynamic and transport properties obtained from the fit of F (k, t) at the lowest wavenumber to the hydrodynamic (k → 0) expression is shown here. All subplots (a), (b), (c) and (d) are plotted against density. The subplot (a) shows the ratio of the specific heats γ; the subplot (b) shows the sound velocity c; the subplot (c) shows the sound damping constant Γ; the subplot (d) shows the thermal diffusivity DT .

131

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

of collective effects in the dynamics. The property F (k, t) of an isotropic liquid and same liquid near I-N transition have two differences: (a) the relaxation is slow; (b) there is marked anisotropy in the relaxation. In the isotropic liquid, the transient anisotropy created by the intermolecular potential relaxes fast due to the reorientation of the molecules. As I-N transition is approached, the relaxation of the local anisotropy requires whole rotation of the pseudo nematic domain. Note that this relaxation process is not coupled strongly to number density, but, only to the orientational density. A wealth of information can be obtained from the dynamic structure factor (S(k, ω)) of the liquid. Figures 5.2(a) to 5.2(f) show the S(k, ω) that corresponds to the F (k, t) shown in figures 5.1(a) to 5.1(f). In figures 5.2(a) and 5.2(b), collective effects are visible at wavenumber k = kL with a well-defined Brillouin peak. The Brillouin peak in the dynamic structure factor vanishes between the wavenumbers k ≈ 1 and 1.5. The transition from elastic to viscous behavior of the fluid takes place between these wavenumbers. This transition in the Gay-Berne liquid is at a smaller wavenumber than that found in Lennard-Jones liquid. For a Lennard-Jones liquid this crossover wavenumber (km ) is at the minima of S(k). For a Gay-Berne liquid, the location of km does not vary appreciably at the I-N transition also. As the density increases the Reileigh’s peak becomes narrow, which indicates the freezing of the collective dynamics at high density. The anisotropy of F (k, t) is pronounced in the Gay-Berne liquid at wavenumbers k = ka and k = kb . From the figure 5.1, it is evident that the time constants of the relaxation of F (k, t) in an isotropic liquid and same liquid near I-N transition are comparable at the wavenumber k = ka . Relaxation of F (k, t) at k = kb shows a strong anisotropy in the relaxation in all the three different directions. The time constants of F (k, t) differ from each other in different directions. In addition to that the structure factors of the liquid also differ in the nature in different directions at this wavenumber. Near the I-N transition, since the rotation of the ellipsoids begins to freeze, the local anisotropy produced by the intermolecular potential is prolonged in space and time. F (k, t) is fitted to it’s hydrodynamic form (Eq. 5.6 ) at the lowest value of k. From 132

5.4 Results and discussions

the parameters of the fit of F (k, t), at k = kL , various thermodynamic parameters and transport coefficients can be obtained. Figure 5.3 shows various thermodynamic parameters and transport coefficients calculated from the fit of F (k, t) for k = kL . The ratio of the specific heats γ (γ = 1/1 − a) is plotted against density is shown in figure 5.3(a). γ shows a rapid increase at the transition point. In figure 5.3(d), the adiabatic sound velocity deduced from the position of the Brillouin peak is plotted against density. Except few points near the I-N transition, the adiabatic sound velocity c shows a monotonic increase. In figure 5.3(b), sound velocity shows a plateau like region very near the I-N phase boundary. Another transport coefficient that can be calculated from the fitting parameters is the sound wave damping constant, Γ (Γ = 1/τb k 2 ). The variation of Γ against density is shown in figure 5.3(c). Γ also shows a rapid growth near the I-N transition. The variation of thermal diffusivity (DT = 1/τa k 2 ) against density shown in the figure 5.3(d), which shows a dip near the I-N phase boundary. A similar feature was observed in the experiment of Marinelli et al. in the study of thermal diffusivity of aligned nCB samples [23]. A qualitative picture of the non-ergodicity in collective dynamics of a system of nematogens near the I-N transition can be obtained from the analysis presented above. In the isotropic phase of the Gay-Berne liquid, the relaxation of density is isotropic due the rapid rotation of the molecules. But, near the I-N transition due to the freezing of individual rotation of the molecules, the anisotropy persists for long, which makes the density relaxation anisotropic. This may be viewed in the perspective of supercooled liquids, where it take very long time to relax anisotropy. In supercooled liquids there is no direction for the anisotropy. Near the I-N transition, in the isotropic liquid, the anisotropy is due to the coupling of number density to the transient director of the system. In the case of the supercooled liquids the density relaxation also slows down several orders of magnitude [30, 31]. However, in the Gay-Berne liquid the number density, although anisotropic, relaxes in a comparable time scale as that of an isotropic liquid. In the case of a liquid in the pre-transition region of the I-N transition the non-ergodicity is due the slow relaxation of the director. 133

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

Longitudinal currents

CL(k,t)/CL(k,t=0)

1

a k=0.50

0.5

ρ*=0.285 0

−0.5 0

1 CL(k,t)/CL(k,t=0)

5.4.2

ρ*=0.315 0

−0.5 10 0

5 time

0.5

L

0 1

time

2

3

0

C (k,t)/C (k,t=0)

1

time

2

3

0.5

L

k=6.46

f

*

ρ =0.285

L

L

0

1 e

L

k=2.05 ρ*=0.315

1 C (k,t)/C (k,t=0)

10

d

L

L

C (k,t)/C (k,t=0)

k=1.99 ρ*=0.285

L

C (k,t)/C (k,t=0)

0.5

0 0

5 time

1 c

0.5

k=0.51

0.5

1

0

b

1 time

2

k=6.16 ρ*=0.315

0 0

1 time

2

Figure 5.4: The calculated longitudinal current-current correlation function is plotted for three different directions of wavenumbers kx , ky and kz against time: shown respectively by solid, dashed and dash-dot lines. The figures (a), (c) and (e) are at density ρ∗ = 0.285 and figures (b), (d) and (f) are at density ρ∗ = 0.315. The figures (a) and (b) give the normalized CL (k, t) at k = 2π/L; figures (c) and (d) give CL (k, t) at k ≈ 2π/a and the figures (e) and (f) give CL (k, t) at k ≈ 2π/b

134

5.4 Results and discussions

0.03

C (k,ω)/C (k,t=0)

a

k=0.50

L

ρ*=0.285

0.02

k=0.51

b

CL(k,ω)/CL(k,t=0)

0.04

ρ*=0.315

0.02

L

0.01

0 0 −3

10

ω

4 ρ*=0.285

10

ω

x 10

k=2.05

d

ρ*=0.315

2

−3

x 10

L

4

e

*

ω

0 0

50

−3

6 k=6.48 * ρ =0.285

4

f

ω*

50

k=6.16 ρ*=0.315

2

L

2 0 0

x 10

CL(k,ω)/CL(k,t=0)

0 0

C (k,ω)/C (k,t=0)

5*

CL(k,ω)/CL(k,t=0)

k=1.99

2

6

0 0 −3

c

CL(k,ω)/CL(k,t=0)

4

x 10

*5

ω*

0 0

50

*

50

ω

Figure 5.5: The calculated spectra of the longitudinal current-current correlation function is plotted at three different directions of wavenumbers against frequency - kx , ky and kz shown respectively by solid, dashed and dash-dot lines. The figures (a), (c) and (e) are at density ρ∗ = 0.285 and figures (b), (d) and (f) are at density ρ∗ = 0.315. The figures (a) and (b) give the CL (k, ω) at k = 2π/L, the figures (c) and (d) give CL (k, ω) at k ≈ 2π/a and the figures (e) and (f) show CL (k, ω) at k ≈ 2π/b.

135

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

25

ρ*=0.285 ρ*=0.31 ρ*=0.32

20

ωmax

15 10 5 0 0

2

4

k

6

8

10

Figure 5.6: The ωm of the CL (k, ω) is plotted against wavenumbers at three densities

Another important collective property of the system is the wavenumber dependent current of linear momentum. The longitudinal current-current correlation function CL (k, t) of the Gay-Berne fluid is plotted against time at two state points in figures 5.4(a) to 5.4(f): (1) with density ρ∗ = 0.285, which is an isotropic liquid; (2) with density ρ∗ = 0.315, which of the same liquid near the I-N transition. Due to the conservation of density, the longitudinal current has zero mean value. As a result, the zero frequency component is absent from the power spectra of the CL (k, t) [32, 33]. Hence negative values are inherent in CL (k, t). CL (k, t) at wavenumber k = kL is shown in figures 5.4(a) and 5.4(b). They show rapid oscillation at the characteristic frequency that corresponds to adiabatic sound propagation in the medium. In the intermediate wavenumber, at k = ka , (shown in figure 5.4(c) and 5.4(d).) CL (k, t) shows heavily damped oscillations. The damped oscillations of CL (k, t) also show anisotropy in near the I-N transition. Note that, here also presence of the sound wave can be located from the oscillation of CL (k, t). At wavenumber k = kb , F (k, t) does not show any oscillations. At k = kb , CL (k, t) shows a slow tail in the relaxation (See figures 5.4(e) and 5.4(f)). 136

5.4 Results and discussions

The analysis of the power spectra of CL (k, t) of linear molecules gives several interesting features which is absent in atomic liquids. Figures 5.5(a) to 5.5(f) show the spectra CL (k, ω) of the longitudinal current-current correlation function. At k = kL (figures 5.5(a) and 5.5(b)) CL (k, ω) shows a single peak structure. Position of this peak shifts to a higher frequency near the I-N transition. Location of this peak gives the frequency of adiabatic sound propagation in the medium. Because of the continuity equation, the intermediate structure factor F (k, t) is proportional to CL (k, t) . The dynamic structure factor of the liquid is related to CL (k, ω) by the relation [6, 7] k 2 CL (k, ω) = ω 2 S(k, ω).

(5.26)

Brillouin peak of S(k, ω) from relation Eq. 5.26, appears at the same position of S(k, ω) obtained from the temporal Fourier transform of F (k, t). The calculation of the S(k, ω) from the CL (k, ω) is more economic since CL (k, t) decays faster than F (k, t). At the intermediate wavenumbers, the CL (k, ω) shows a double peak structure which is shown in figures 5.5(c) and 5.5(d). At the intermediate wavenumbers a second peak appear at low frequencies. The second peak of CL (k, ω) in the figures 5.5(c) and 5.5(d) is by the shift of the single peak found at small wavenumbers to high frequencies. At large wavenumber (k = kb ) the peak corresponds to the adiabatic sound propagation in the medium vanishes, as shown in the figures 5.5(e) and 5.5(f). At small wavenumbers CL (k, ω) shows a single peak followed by a broad spectra that shows the reduction of the frequency dependence of CL (k, ω) as compared to intermediate wavenumbers. The growth and the dispersion of the two peaks in the CL (k, ω) can be understood from ωm versus wavenumber plot shown in the figure 5.6. In this figure, the three lines plotted are at three different densities: lowest density is that of isotropic liquid (ρ∗ = 0.285); intermediate density (ρ∗ = 0.31) is very near the I-N transition; the highest number density is at ρ∗ = 0.32, which is in the nematic phase. The trend followed in all three curves is similar. The initial rise corresponds to the shifting of the single peak found at low wavenumbers to higher frequencies. At higher wavenumbers a new peak begin to grow at low frequencies. The sudden fall in the ωm shown in figure 5.6 is at the wavenumber where the first peak become taller than the second peak. The 137

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

linear growth of the ωm shown the figure 5.6 after the fall corresponds to the growth of the low frequency peak that appear at large wavenumbers. Figure 5.6 also shows a linear dispersion of the sound velocity peak that appear at high frequencies and low wavenumbers. Note that as the height of the low frequency peak of CL (k, ω) increases as wavenumber increases: meanwhile height of the peak at high frequency decreases.

5.4.3

Relaxation of the transverse currents

Figure 5.7 shows the transverse part of the linear momentum current-current correlation function (CT (k, t)) at two state points: (a) in an isotropic liquid with number density ρ∗ = 0.285 (figures 5.7(a), 5.7(c) and 5.7(e) ) and same liquid near the I-N transition at ρ∗ = 0.315 (figures 5.7(b), 5.7(d) and 5.7(f) ). At k = kL , CT (k, t) take long time to relax (shown in the figures 5.7(a) and 5.7(b)), this shows the ability of the system to propagate shear waves of long wavelength. Near the I-N transition at, ρ∗ = 0.315, the relaxation of CT (k, t) becomes highly anisotropic. In this case CT (k, t) shows a deeper minima along kz , but it relaxes fast along the kx (decay completes at t∗ = 4). This suggests that the strong coupling of the CT (k, t) with the global director at k = kL . The CL (k, t), unlike the CT (k, t) does not show any anisotropy in the relaxation at k = kL . For k = ka , (shown in the figures 5.7(c) and 5.7(d)) the relaxation of CT (k, t) is fast: here the decay completes below t∗ = 1. At this wavenumber also the CT (k, t) shows marked anisotropy. Here the CT (k, t) shows three different regions of relaxation: at the short time the CT (k, t) is a Gaussian; this is followed by a fast exponential relaxation, which is up to 80% of the decay; the final phase of the relaxation shows a non-exponential tail with comparatively longer life time. Even though there is an anisotropy in the relaxation (show in figure 5.7(d)) the total life time of the correlation remains the same. Note that at k = ka , the system is in the viscoelastic regime, where memory effects plays an important role in the relaxation. The final 20% decay of the CT (k, t) is the duration where the system show the memory effects which appears as a tail in CT (k, t). CT (k, t) at k = kb ( figures 5.7(e) and 5.7(f) ) shows similar relaxation dynamics as that found in the intermediate wavenumbers. The collective memory effects in this case have shorter life time and appear after 90% of the decay is 138

5.4 Results and discussions

1 C (k,t)/C (k,t=0)

a

b

*

*

ρ =0.315

T

T

ρ =0.285 0 0

5

time

10

0

15

0

C (k,t)/C (k,t=0)

c

*

ρ =0.285

0.5 time

*

ρ =0.315 0 0

1

1

time

T

f 0.5

k=6.16 *

ρ =0.315

T

*

C (k,t)/C (k,t=0)

T

k=6.46 ρ =0.285

T

0.5 time

1 e

0 0

15

k=2.05

0.5

1

0.5

10

d

T

0 0

time

T

k=1.99

T

0.5

5

1

T

C (k,t)/C (k,t=0)

1

C (k,t)/C (k,t=0)

k=0.51

0.5

T

k=0.50

0.5

T

C (k,t)/C (k,t=0)

1

0.5

0 0

time

0.5

Figure 5.7: The calculated transverse current-current correlation function is plotted for three different directions of wavevectors kx , ky and kz against time: shown respectively by solid, dashed and dash-dot lines. The figures (a), (c) and (e) are at density ρ∗ = 0.285 and figures (b), (d) and (f) are at density ρ∗ = 0.315. The figures (a) and (b) give the normalized CT (k, t) at k = 2π/L; figures (c) and (d) give CT (k, t) at k ≈ 2π/a and the figures (e) and (f) give CT (k, t) at k ≈ 2π/b

139

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

0.06

k=0.50

0.04

ρ*=0.285

0.02

k=0.51

0.04

ρ*=0.315

0.02

0 0 −3

x 10

5 *

ω

−3

x 10 C (k,ω)/C (k,t=0)

CT(k,ω)/CT(k,t=0)

*

ρ =0.285

10

ω

k=2.05

4

ρ*=0.315

2

T

2

5 * d

6

T

k=1.99

4

0 0

10

c

6

0 0 −3

x 10

*

ω

0 0

50

−3

x 10

e

4

C (k,ω)/C (k,t=0)

k=6.48 *

ρ =0.285

0.01

2

2

*

ω

50

f k=6.16 ρ*=0.315

T

0.005

0 0

4

T

CT(k,ω)/CT(k,t=0)

b

CT(k,ω)/CT(k,t=0)

CT(k,ω)/CT(k,t=0)

a

0.06

0 0

20 40

*

ω

0 0

50

*

ω

50

Figure 5.8: The calculated spectra of the transverse current-current correlation function is plotted at three different directions of wavevectors against frequency - k x , ky and kz shown respectively by solid, dashed and dash-dot lines. The figures (a), (c) and (e) are at density ρ∗ = 0.285 and figures (b), (d) and (f) are at density ρ∗ = 0.315. The figures (a) and (b) give the CT (k, ω) at k = 2π/L, the figures (c) and (d) give CT (k, ω) at k ≈ 2π/a and the figures (e) and (f) show CT (k, ω) at k ≈ 2π/b. The inset of the subplot (e) shows the spectral function of single particle velocity autocorrelation function.

140

5.4 Results and discussions

0.7

0.6

k∼1/2

0.5

*

ωm

a

0.4

0.3

0.2 0.28

0.29

0.3 *

0.31

ρ

0.32

3 2.5

b

*

ωm

2 1.5 *

ρ =0.285 * ρ =0.31 * ρ =0.32

1 0.5 0 0

1

2

k

3

4

Figure 5.9: Figure (a) shows the ωm , which is the frequency at the peak of the transverse current spectrum, plotted at k = kL against density. Figure (b) shows the ωm versus k plot at three different state points which shows the dispersion of the shear waves.

over. At higher wavenumbers the CT (k, t) shows similar characteristic as that of single particle velocity autocorrelation function. The spectra of the CT (k, t), gives the information about the frequency dependence of the shear wave relaxation. Figure 5.8 show the spectra at state points and the 141

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

wavenumbers same as that shown in the figure 5.7. At the low wavenumber, the position of the ωm in CT (k, ω) is at ω ∗ ≈ 2.6. The value of the peak frequency ωm increases as the wavenumber increases and finally become a constant approximately at k = 3. The anisotropy of CT (k, ω) in Gay-Berne liquid at the I-N transition arises from low frequency part of the spectra at small wavenumbers. The frequency response shown in figure 5.8(b) is slightly broader than that observed in 5.8(a). At k = ka (shown in figure 5.8(c) and 5.8(d) ) CT (k, ω) shows the presence of anisotropy in the shear wave propagation near the I-N transition. Note that in this case the anisotropy appears even at high frequencies. The CT (k, ω) at the wavenumber k = ka similar to the spectra of the single particle velocity autocorrelation function, which is shown in the inset of figure 5.8(e). By analyzing the data at different state points the frequency of maximum response ωm can be obtained for the shear waves. Figure 5.9(a) shows the growth of the ωm at k ∼ 1/2 as density increases. Note that near the I-N transition the slope of the line ωm versus k plot increases. The plot showing the value of the ωm against k at three wavenumbers are shown in the figure 5.9(b). The linear dispersion relation ωm = ct (k −kt ) can be used to find the velocity of shear wave ct and kt (kt is the critical wavenumber above which the shear waves are not supported [33, 34] by the system). A linear fit of ωm against k at the small wavenumbers at three state points give the value of the ct and kt : at density ρ∗ = 0.285 in the isotropic phase ct = 0.87 and kt = 0.16 ; at density ρ∗ = 0.31, very near to the I-N transition, ct = 1.4 and kt = 0.15; in the nematic phase at density ρ∗ = 0.32, ct = 1.2 and kt = 0.07. Note that very near to the I-N transition the velocity of the shear wave increases.

5.4.4

Relaxation of the angular currents

The relaxation of the collective angular momentum fluctuations shows that they are short lived even in a dense system of linear molecules. Figure 5.10(a) shows the transverse collective angular momentum correlation function (CTav (k, t)) for a dense system of Gay-Berne ellipsoids in the isotropic phase (ρ∗ = 0.285). Compared to the linear momentum current fluctuation, the noticeable feature observed in the case of angular 142

1

0.8

0.8

0.6

Cav(k,t)

1

a

0.2

0

0 0.5

1 time

1.5

2

0 1

0.8

0.8

0.6

Cav(k,t)

1

c

0.5

0.6

1 time

1.5

2

1.5

2

d

0.4

L

L

0.4 0.2

0.2

0

0

0

b

0.4

0.2 0

Cav(k,t)

0.6

T

0.4

T

Cav(k,t)

5.4 Results and discussions

0.5

1 time

1.5

2

0

0.5

1 time

Figure 5.10: The transverse and longitudinal parts of the angular currents are plotted at two densities. The subplots (a) and (c) give the CTav (k, t) and CLav (k, t) at ρ∗ = 0.285 and (b) and (d) give the CTav (k, t) and CLav (k, t) at ρ∗ = 0.315, which is near I-N transition. Each plot contains three lines, which represent the correlation function at three chosen wavevectors: line at k = kL is represented by the solid line; line at k = ka is represented by the dashed line; line at k = kb is represented by the dashed-doted line.

momentum currents is that, the major part of relaxation take place before t∗ =1. In the case of CT (k, t) and CL (k, t), the fast relaxation is at the large wavenumbers. The wavenumber dependence of the relaxation is weak in this case. At all the wavenumbers the time constant of the relaxation does not vary appreciably. At k = ka and k = kb , CT (k, t) shows backscattering. Similar behavior is also observed in the case of single particle angular velocity autocorrelation function (see Fig. 6.6). At k = kL , the final 5% of the decay of CTav (k, t) shows a short power law tail of the form t−α with exponent α ∼ 0.82. Figure 5.10(b) shows CTav (k, t) at ρ∗ = 0.315, where it shows a prolonged 143

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

backscattering region with a deeper minima than at ρ∗ = 0.285. At ρ∗ = 0.315 the tail of the CTav (k, t) is absent for the wavenumber k = kL . CTav (k, t) relaxes exponentially at k = kL for ρ∗ = 0.315. The longitudinal angular momentum correlation function CLav (k, t) at ρ∗ = 0.285 shown in the figure 5.10(c). This correlation function is very similar to CTav (k, t). At both k = ka and k = kb , CLav (k, t) show backscattering. According to the generalized hydrodynamic theory of Aliawadi et al.[27] the longitudinal spin density does not couple to hydrodynamic variables of the system and it decays exponentially. However, CLav (k, t) calculated from the simulations shows exponential relaxation only in the case of k = kL . At k = ka and k = kb , CLav (k, t) shows a back scattering that indicate the exchange of the angular momentum between the neighbors. At density ρ∗ = 0.315, CLav (k, t) shows more prolonged back scattering region that of the CTav (k, t). However, the origin of the back scattering demands more investigation on the origin of coupling of angular currents to other degrees of freedom. Note that in the case of CTav (k, t), it couples to the collective fluctuation of the linear currents [27]. This explicit analysis of the angular current shows that the angular current does not propagate in the system, rather due to its strong coupling to the linear momentum currents it dissipates fast even in a system that have strong orientational ordering.

5.5

Concluding remarks

In this chapter we have presented a study of I-N transition from the perspective of molecular hydrodynamics using molecular simulations. In a system of 576 particles the smallest wavenumber that can be studied is approximately is k ∼ 1/2 in the range of the number densities ρ∗ = 0.285 to ρ∗ = 0.32. The calculation finds a fairly well defined Brillouin peak in the range of the wavenumbers studied. These simulations show the role of density fluctuation and the orientational ordering in development of anisotropy in the density relaxation near the I-N transition. However, at the large wavenumbers the calculation shows only marginal evidence for the existence and propagation of the density fluctuations. The main results of this work are that it shows the effect of the 144

5.5 Concluding remarks

formation of the director on the relaxation of fluctuations of the collective variables of the system. A fit of intermediate scattering function to the linearized hydrodynamic model at small wavenumbers (k = 2π/L) show the variation of the thermodynamic as well as transport properties across I-N transition. The ratio of the specific heats shows a discontinuity near the I-N transition. The thermal diffusion shows a dip; a similar behavior is reported in the recent experiments of Marinelly et al.[23]. The sound dispersion coefficient also shows a divergence like behavior near I-N transition. The sound velocity calculated from the peak position if the Brillouin peak deviate from its monotonic increase near I-N transition. The longitudinal current-current correlation function also shows large anisotropy. At large wavenumbers the first peak that appears at low wavenumber disappears from the spectra of longitudinal current-current correlation function. The transverse current-current correlation function shows that there exists a strong anisotropy in the shear wave propagation at all wavenumbers. As the molecular dynamics simulations are the only technique which can measure transverse momentum current fluctuations, these studies bring light on the role played by the transverse current in the I-N transition. The maximum value of frequency at which the shear wave propagates also shifts to higher values during I-N transition. The longitudinal current correlation function shows a double peak structure at intermediate wavenumbers k ∼ 2π/a. This low frequency wave appears at large wavenumbers. The correlation function of the intrinsic angular momentum is calculated for the first time near the I-N transition. These correlation functions hint at the transient nature of the collective angular momentum fluctuations. Both longitudinal and transverse part of the angular momentum correlation function show a weak wavenumber dependence in the relaxation. This hints at the true local nature of the collective angular momentum current correlation functions. They dissipate at a distance comparable to that of inter particle separation. This work demonstrates the spatial and temporal propagation of anisotropy of the molecular potential in the medium at different state points across the I-N transition. 145

5. WAVENUMBER AND FREQUENCY DEPENDENT COLLECTIVE RELAXATION IN LIQUID CRYSTALS

Thus during the I-N transition the system shows the non-ergodicity due to the formation of director, that slowly relaxes due to the mutual coupling of the orientation within a pseudo nematic domain.

146

Bibliography [1] L. van Hove, Phys. Rev. 95, 249 (1954). 5.1 [2] R. Zwanzig, Ann. Rev. Phys. Chem. 16, 67 (1965). 5.1 [3] B. J. Berne and R. Pecora, Dynamic Light Scattering: With applications to Chemistry, Biology and Physics. (John Wiley & Sons, INC, New York, 1976). 5.1, 5.2, 5.4.1 [4] J. R. D. Copley and S. W. Lovesey, Rep. Prog. Phys. 38, 461 (1974). 5.1 [5] J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). 5.1, 5.2, 5.2, 5.4.1 [6] U. Balucani and M. Zoppi, Dynamics of the liquid state (Claredon Press, Oxford, 1994). 5.1, 5.2, 5.2, 5.4.2 [7] J. P. Boon and S. Yip, Molecular hydrodynamics (McGraw-Hill International book company, New Delhi, 1980). 5.1, 5.2, 5.2, 5.4.1, 5.4.2 [8] A. Rahman, Phys. Rev. 130, 1334 (1963). 5.1 [9] A. Rahman, K. S. Singwi, and A. Sjolander, Phys. Rev. 126, 986 (1962). 5.1 [10] R. D. Mountain, Rev. Mod. Phys. 38, 205 (1966). 5.1 [11] I. M. de Schepper et al., Phys. Rev. A 38, 271 (1988). 5.1 [12] C. H. Chung and S. Yip, Phys. Rev. 182, 182 (1969). 5.1 [13] G. I. A. Stegeman and B. P. Stoicheff, Phys. Rev. Lett. 21, 202 (1968). 5.1 147

BIBLIOGRAPHY

[14] V. Volterra, Phys. Rev. 180, 156 (1969). 5.1 [15] C. H. Chung and S. Yip, Phys. Rev. A 4, 928 (1971). 5.1 [16] J. J. Stankus, R. Torre, and M. D. Fayer, J. Phys. Chem. 97, 9478 (1993). 5.1 [17] S. D. Gottke et al., J. Chem. Phys. 116, 360 (2002). 5.1 [18] S. D. Gottke, H. Cang, B. Bagchi, and M. D. Fayer, J. Chem. Phys. 116, 6339 (2002). 5.1 [19] H. Cang, J. Li, and M. D. Fayer, Chem. Phys. Lett. 366, 82 (2002). 5.1 [20] H. Cang, J. Li, V. N. Novikov, and M. D. Fayer, J. Chem. Phys. 118, 9303 (2003). 5.1 [21] H. Cang, J. Li, V. N. Novikov, and M. D. Fayer, J. Chem. Phys. 119, 10421 (2003). 5.1 [22] S. Takagi and H. Tanaka, Phys. Rev. Lett. 93, 257802 (2004). 5.1 [23] M. Marinelli, F. Mercuri, U. Zammit, and F. Scudieri, Phys. Rev. E 58, 5860 (1998). 5.1, 5.4.1, 5.5 [24] P. P. Jose and B. Bagchi, J. Chem. Phys. 120, 11256 (2004). 5.1 [25] P. P. Jose, D. Chakrabarti, and B. Bagchi, Phys. Rev. E 71, 030701 (2005). 5.1 [26] S. R. deGroot and P. Mazur, Non-equilibrium thermodynamics (North-Holand, Amsterdam, 1964). 5.1 [27] N. K. Alliawadi, B. J. Berne, and D. Forster, Phys. Rev. A 3, 1462 (1971). 5.1, 5.2, 5.2, 5.4.4 [28] N. K. Alliawadi, B. J. Berne, and D. Forster, Phys. Rev. A 3, 1472 (1971). 5.1 [29] B. Bagchi and A. Chandra, Adv. Chem. Phys. 80, 1 (1991). 5.2 148

BIBLIOGRAPHY

[30] W. Gotze, in Liquid Freezing and Glass Transition, edited by J. P. H. D. Levesque and J. Zinn-Justin (North Holland, Amsterdam, 1990). 5.4.1 [31] W. Gotze and L. Sjogren, Rep. Prog. Phys. 55, 241 (1992). 5.4.1 [32] D. Levesque, L. Verlet, and J. Kurkijavri, Phys. Rev. A 92, 1690 (1972). 5.4.2 [33] A. Rahman and F. H. Stillinger, Phys. Rev. A 10, 368 (1974). 5.4.2, 5.4.3 [34] M. M. G. Alemany and L. J. Gallego, Phys. Rev. B 58, 685 (1998). 5.4.3

149

BIBLIOGRAPHY

150

Chapter 6 Stress relaxation and viscosity 6.1

Introduction

The orientational dynamics of the rod-like molecules near the isotropic-nematic phase transition shows certain anomalies, the most notable being the pronounced temporal power law decay in the collective orientational correlation function discovered recently by Fayer and coworkers [1–6] as discussed in previous chapters. A mode coupling theory analysis qualitatively explains the observed power law decay in terms of large and rapidly growing orientational correlation near the I-N transition [5, 6]. Theoretical analysis has earlier quantified this growth of orientational pair correlation in an adequate manner [7]. Several computer simulations have investigated the orientational dynamics near the I-N transition [8–13]. Earlier studies have found the signature of slow down of orientational relaxation in general. Very long time MD simulation of Jose et al. in a system of Gay-Berne ellipsoids has observed the power law relaxation of the collective orientational correlation function [13] (see chapter.3). Simulation studies on the single particle orientational dynamics in a system of Gay-Berne ellipsoids show dynamical heterogeneity near isotropic nematic phase boundary. In this study, the density relaxation shows anisotropy. Details of the study on the single particle dynamics can be found in chapter 4. We have also studied dynamics of the collective relaxation of conserved and non-conserved variables, namely, density, linear momentum current and angular momentum current, near I-N phase boundary (see chapter 5). These studies 151

6.

STRESS RELAXATION AND VISCOSITY

show that, due to the formation a global director near I-N transition, an anisotropy develops in the collective relaxation. An important transport property of the liquid is its shear viscosity. In the nematic phase, the shear viscosity is anisotropic. In a recent molecular dynamics simulation of Gay-Berne ellipsoids (of aspect ration 3), Smondyrev et al. [14] have calculated the temperature dependence of rotational and shear viscosities. They obtained temperature variation of Miesowicz viscosities [15] which are defined as combinations of the components of shear viscosity but now defined in the nematic phase with respect to the direction of the director. However, Smondyrev et al. did not study shear viscosity across the I-N transtion. Since the orientational correlation function shows anomalous behavior near the IN transition, it is natural to ask about the variation of shear viscosity. In this work the shear viscosity of a system of Gay-Berne ellipsoids is calculated along the density axis near I-N transition. The shear viscosity of the system shows a rise as the I-N transition is approached along the density axis. However, shear viscosity does not diverge near I-N transition. The emergence of the I-N transition is marked by the anisotropy in the stress distribution function. Different components of the stressstress correlation function also behave differently due to the anisotropy caused by the orientational ordering. The angular velocity autocorrelation function shows a power law with exponent -1.2 near I-N transition. This result differs from the hydrodynamic prediction of -5/2 indicating that asymptotic long time decay has not been attained in simulations [16–19]. Tang et al. have investigated the important aspects of viscoelasticity of hard ellipsoids [20]. In this work, both generalized kinetic theory and computer simulations were used for the study of the viscoelastic response. These authors observed several anomalies in the frequency dependence of the real part of the frequency dependent viscosity. Firstly, the viscosity decreases with frequency (ω) at low frequencies, here the frequency dependence is very weak and can be described by a power law. Secondly, the viscosity shows a pronounced minimum at intermediate frequencies, which is not 152

6.2 Expressions of shear viscosity and rotational friction

observed in atomic liquids. The authors explained the latter result as due to the coupling between the stress and the collective orientational relaxation. Tang et al. also considered the applicability of the Debye-Stokes-Einstein relation between rotational diffusion and viscosity. The present simulation study have found similar behaviour near the I-N transition for a system of Gay-Berne ellipsoids, except that both the anomalies get attenuated near the I-N transition. Comparison with the stress time correlation function (TCF) clearly reveals the origin of these anomalies. The weak ω dependence of viscosity at low ω can be traced back to a power law dependence of the stress-stress TCF while the peak is due to the negative region in the stress-stress TCF. We find that the hydrodynamic relation between the rotational friction and the viscosity breaks down completely on approach to the I-N phase transition. The organization of the rest of the chapter is as follows. In the next section gives the basic theoretical expressions required for the computation of the viscoelasticity of the system. Section 6.3 present the results and discuss their significance. Section 6.4 concludes with a few remarks.

6.2

Expressions of shear viscosity and rotational friction

Consider a system of N nematogens. The instantaneous state of the system in the phase space is specified by position ri , linear momentum pi , orientation Ωi and angular momentum gΩi , where i =1,...N. The αβ component of shear viscosity of this system in the isotropic phase is given by the relation [21]. Z ∞ V dthσαβ (t)σαβ (0)i ηαβ = kB T 0

(6.1)

where V is the volume of the system, kB is the Boltzmann constant, T is the temperature, and σαβ is the shear stress tensor [14, 21–23]. Shear stress tensor is defined as

σαβ

1 = V

X pαi pβj i

m

+

153

XX i

j>i

rαij fβij

!

(6.2)

6.

STRESS RELAXATION AND VISCOSITY

where pαi is the α component of the momentum of i0 th molecule and rij and fij are relative position and force between a pair of molecules, respectively. The indices α, β = x, y, z and α 6= β The normalized stress-stress correlation function of the system is given by the expression σ Cαβ (t) =

hσαβ (t)σαβ (0)i hσαβ (0)σαβ (0)i

(6.3)

In this system the frequency dependent viscosity is defined as ηαβ (ω) =

V kB T

Z

∞

dthσαβ (t)σαβ (0)i cos ωt

(6.4)

0

Viscoelasticity of the system is given by frequency dependent viscosity ηαβ (ω) [20, 24]. Another important correlation function, which is a direct measure of the building up of orientational order in the system is angular velocity autocorrelation function. The angular velocity autocorrelation function of a system of ellipsoids is given as Cω (t) =

hωi (t) · ωi (0)i , hωi (0) · ωi (0)i

(6.5)

where ωi is the angular velocity of the ith molecule. The relation between the rotational friction and the angular velocity autocorrelation function in the Laplace plane is given by the relation [25, 26] Cω (z) =

6.3

kB T I[z + ζR (z)]

(6.6)

Results and Discussion

The simulation starts from an equilibrated configuration of 576 ellipsoids. Initial configuration of the ellipsoids is generated from a cubic lattice and then the simulation is run for two hundred thousand steps to obtain the equilibrium configuration. During the equilibration steps the temperature is scaled so that the system is in equilibrium at this particular temperature. Four million production steps are used for calculation of viscosities. More details of the simulations and the parameters used are provided in the sections 2.2.1 and 2.4. 154

6.3 Results and Discussion

6.3.1

Shear viscoelasticity and stress-stress time correlation function

It has been discussed elsewhere that the orientational correlation functions slow down dramatically as the I-N transition is approached from low densities [5, 6, 13]. The variation of shear viscosity near the I-N transition is another subject of interest. It is known that viscosity does not exhibit any divergence near the I-N transition. It is also known that the viscosity becomes anisotropic near the transition. However, a detailed study of the emergence of the anisotropy and the variation of viscosity seems to be absent in the literature. 3.5 η xy ηxz η

3

yz

2.5

ηαβ

ηavg 2

1.5 1 0.5 0 0.2

0.22

0.24

0.26

0.28

ρ*

0.3

0.32

0.34

Figure 6.1: The shear viscosity of a system of Gay-Berne ellipsoids is plotted against density in the Isotropic phase. When the I-N transition is approached, various components of viscosity increase and they become anisotropic.

Figure 6.1 show the variation of shear viscosity with density. There are several points of interest about this figure. First, the increase in viscosity with density becomes sharper as the I-N transition is approached. However, this growth is rather mild. Second point to note is that beyond the transition point (ρ∗ = 0.315), the viscosity becomes anisotropic. The emergence of anisotropy can be better understood by studying the probability distribution of the mean square stress. In figure 6.2 we show 155

6.

STRESS RELAXATION AND VISCOSITY

.01

*

Probability (P(σαβ))

.005

0

.004 .002

.008 .004 0 −1500

−1000

−500

0

* * V σαβ

500

1000

1500

Figure 6.2: The distribution of three components of stress are plotted against stress: σxy is given by the dashed line; σxz is given by the solid line; σyz is given by the dasheddotted line. The top sub-figure is for density ρ∗ =0.20 (Isotropic phase) and the middle sub-figure is for density ρ∗ =0.315 (Near I-N transition) and the bottom sub-figure is for density ρ∗ =0.34 (Nematic phase).

the distribution at three densities. Note that the distribution is fully isotropic at low densities but become anisotropic as the transition is approached. This anisotropy is important in determining the viscous response of the liquid. Note that the stress tensor also becomes anisotropic in the deeply supercooled liquid [27]. Although this analogy is rather nice, the origin of those two power laws is quite different. In figure 6.3 we show the decay of the stress-stress TCF. Note the negative region and the slow long time decay. The viscoelasticity of the liquid can be of considerable interest. In figure 6.4 we show the frequency dependence of the viscosity. There are two unusual features of this figure. (a) The fall of viscosity with frequency is non-linear and unusually weak at low frequencies. It is clear that the frequency dependence obeys 156

6.3 Results and Discussion

1 0.5

Cαβ(t)

0

σ

0.5 0

0.5 0 0

0.5

*

1

1.5

time(t ) Figure 6.3: Components of the normalized stress-stress time correlation function (the dashed line gives the xy component, the dashed-dotted line gives the xz component, the dotted line gives the yz component and the solid line gives the average) is plotted against time here at three densities. The top sub-figure is at density ρ∗ =0.20 (Isotropic phase) and the middle sub-figure is at density ρ∗ =0.315 (Near I-N transition) and the bottom sub-figure is at density ρ∗ =0.34 (Nematic phase).

a power law with an exponent less than unity. (b) There is a pronounced peak near a value of frequency which is ≈ 20. The explanation, lies in the time dependence of the stress-stress time correlation function (TCF) shown in figure 6.3. Note the slow decay in the long time. In an interesting paper, Zwanzig pointed out the correlation between the slow long time decay of the stress-stress TCF and a non-analytic frequency dependence of the viscosity [28]. That is, if the long time tail decays as t−3/2 , then the frequency dependence is given 157

6.

STRESS RELAXATION AND VISCOSITY

12

12

ρ*=0.285 ρ*=0.290 * ρ =0.295 ρ*=0.3 ρ*=0.305 ρ*=0.31 * ρ =0.315

10

10 ′

ηαβ(ω)

8

8

6

αβ

η′ (ω)

4

6

2

0

0.5

1

1.5

2

2.5

3

ln(ω*)

4 2 0 0

50

100

150

200

frequency(ω*)

250

300

350

Figure 6.4: The frequency dependent viscosity is plotted against the frequency at different densities near I-N transition. The inset gives the plot ηαβ (ω) against ln(ω). This highlights the low frequency characteristics of ηαβ (ω).

0.04 0.03

0.01

αβ

Cσ (t)

0.02

0

−0.01 −0.02

0.5

1

1.5

2

2.5

*

time(t )

3

3.5

4

4.5

5

Figure 6.5: The tail of the stress-stress correlation function is fitted to the function y = 0.0031 t−1.1 (the solid line gives the fit and the dashed line gives the stress-stress correlation function) at density ρ∗ = 0.315. The exponent α obtained from the fit is 1.1.

158

6.3 Results and Discussion

by η(ω) = η(0) − Aω 1/2 + O(ω)

(6.7)

When we fit the long time part of the stress TCF to a form t−α , we find a value of 1.1 for the exponent α. The fit is shown in figure 6.5. The value of the exponent does provide an explanation for the weak frequency dependence at small frequencies of ηαβ (ω). The peak in ηαβ (ω) at intermediate frequencies seems to originate from the negative σ (t) becomes more region in the stress-stress TCF. Note that the negative region of Cαβ

pronounced as the I-N transition is approached, so does the peak in ηαβ (ω).

6.3.2

Angular velocity auto correlation function 1 *

Cω(t)

0.4

−0.02 −0.03 −0.04

ω

0.6

−0.01

C (t)

0.8

ρ =0.285 * ρ =0.29 * ρ =0.295 ρ*=0.3 ρ*=0.305 ρ*=0.31 * ρ =0.315

−0.05 −0.06 −0.07 −0.08

0.2

0.15

0.2

0.25

t*

0.3

0.35

0.4

0 −0.2 0

1

2

Time(t*)

3

4

5

Figure 6.6: The angular velocity autocorrelation function is plotted for different densities against time. As the density of the system increases the depth of back scattering region of the angular velocity autocorrelation function becomes deep. The back scattering region is highlighted in the inset.

In figure 6.6, we show the time dependence of the angular velocity auto correlation functions. Note the slow decay in the long time, following the negative dip at short times. The rise of the angular velocity autocorrelation function after the minimum has 159

6.

STRESS RELAXATION AND VISCOSITY

0 −0.01 −0.02

C (t)

−0.03

ω

−0.04 −0.05 −0.06 −0.07 −0.08

1

2

3

4

5

Time(t*)

6

7

8

9

10

Figure 6.7: The tail of the angular velocity autocorrelation function fitted to a powerlaw (the continuous line gives the fit and the dashed line gives the stress-stress correlation function). The exponent α obtained from the fit is 1.2 at density ρ∗ = 0.315.

been fitted to a function of the following form

hω(0)ω(t)i ≈ −a1 exp(−a2 t) − a3 t−a4

(6.8)

The fit has been shown in figure 6.7. The power law exponent is about 1.2. Hydrodynamic prediction of long time power law exponent is much larger, equal to 5/2 [16–19]. This point deserves further study. It is possible that the long time decay observed in simulation is not the true hydrodynamic long time decay – instead an intermediate time decay one often finds in the mode coupling theory (MCT) which arises from the coupling with the density mode. The last point can be further substantiated if one uses the MCT form of the friction [25, 26] in Eq.6.6. One finds nearly perfect agreement at long times, substantiating the logic presented here. However, a more complete calculation is required to fully reproduce the angular velocity correlation curve. 160

6.3 Results and Discussion

200

R

ζ (z=0)

150

100

50 0.285

0.29

0.295

0.3

0.305

0.31

Density(ρ*)

0.315

Figure 6.8: The zero-frequency friction is plotted against density. The friction rapidly increases as I-N transition is approached along the density axis. 200 180

ρ*=0.285 * ρ =0.29 ρ*=0.295 ρ*=0.3 * ρ =0.305 ρ*=0.31 * ρ =0.315

160 140

ζR(z)

120 100 80 60 40 20 0 0

0.5

1

*

1.5

Laplace fequency(z )

2

Figure 6.9: The frequency dependent friction is plotted for different densities.

6.3.3

Rotational friction on the ellipsoids

We have discussed earlier that the single particle orientational correlation function exhibits power law decay behaviour at intermediate times. In figure 6.8 we show the 161

6.

STRESS RELAXATION AND VISCOSITY

density variation of the zero frequency friction at the temperature T ∗ =1.0. Note the rapid rise in the friction as the transition density is approached. Mode coupling theory predicts that due to the emergence of the long-range orientational density correlation, the rotational friction should show strong frequency dependence. Here the frequency is the Laplace frequency. In figure 6.9, we show the frequency dependence of rotational friction at several densities. Note the increasingly faster rise of the friction at low frequencies as the I-N transition is approached.

6.3.4

Comparison between viscosity and rotational friction 140 130 120

ζ /η

110

R

100 90 80 70 60 0.285

0.29

0.295

0.3

0.305

Density(ρ*)

0.31

0.315

Figure 6.10: The ratio of zero frequency friction to the average viscosity is plotted against density across the I-N transition.

While the friction shows a sharp increase near the I-N transition, the same is absent for viscosity. In figure 6.10, the ratio of ζR /η is plotted. The figure shows a hydrodynamic-like, density independent (or, weakly dependent) behaviour at low densities, which gives away to a strong variation as the I-N transition point is approached. This is the manifestation of the strong growth of friction near the I-N transition where viscosity shows only weak dependence on density. Thus, the Debye-Stokes relation between rotational friction and viscosity breaks down near the I-N transition. 162

6.4 Concluding Remarks

6.4

Concluding Remarks

Molecular dynamics simulations of a system of Gay-Berne ellipsoids of aspect ratio 3 near its isotropic-nematic phase transition have been carried out. The viscoelastic properties of the system are calculated. While the rotational friction shows a sharp rise in its value as the isotropic-nematic phase transition is approached, the viscosity shows only a mild increase. The frequency dependent viscosity shows interesting anomalies – a power-law fall at low frequencies and a sharp peak at the intermediate frequency. The observed behaviour for Gay-Berne fluids near the I-N transition is yet to be understood from a quantitative theory. The anomalies in the viscoelastic properties can be qualitatively understood from the stress-stress time correlation function which shows a negative dip at short times and a power law decay at long times. The stress becomes anisotropic near the transition region as expected. One of the motivations of the present study was the recent work by Cang et al. who showed that Kerr response near the I-N transition is surprisingly close to that in supercooled liquids [2, 3]. We find that although the orientational correlation function indeed shows power law decay, viscosity and viscoelasticity do not show similar behaviour. This is because the origin of the slow decay is completely different in the two cases. In supercooled liquid, the slow decay is believed to arise from a non-linear feedback mechanism between the density and the stress relaxation, where the slow density relaxation occurs at nearest neighbour distance (near the first peak of the static structure factor). In the present system, the slowness arises from the long-range orientational correlation. However, the strong similarity between the two phenomena is really intriguing. The main reason appears to be the rapid growth in the value of the friction and the memory function of the stress time correlation function. Mode coupling theory √ shows that the rotational friction grows approximately as 1/ z at low frequencies. This gives rise to the power law decay. A similar rapid growth in the friction also takes place in the supercooled liquid. While the theory of Tang et al.[20] provides a framework to understand many aspects of the relaxation behaviour reported here, 163

6.

STRESS RELAXATION AND VISCOSITY

it will be worthwhile to develop a mode coupling theory to understand the effect of orientational relaxation on viscoelasticity.

164

Bibliography [1] J. J. Stankus, R. Torre, and M. D. Fayer, J. Phys. Chem. 97, 9478 (1993). 6.1 [2] H. Cang, J. Li, and M. D. Fayer, Chem. Phys. Lett. 366, 82 (2002). 6.1, 6.4 [3] H. Cang, J. Li, V. N. Novikov, and M. D. Fayer, J. Chem. Phys. 118, 9303 (2003). 6.1, 6.4 [4] A. Sengupta and M. D. Fayer, J. Chem. Phys. 102, 4193 (1995). 6.1 [5] S. D. Gottke et al., J. Chem. Phys. 116, 360 (2002). 6.1, 6.3.1 [6] S. D. Gottke, H. Cang, B. Bagchi, and M. D. Fayer, J. Chem. Phys. 116, 6339 (2002). 6.1, 6.3.1 [7] A. Perera, G. N. Patey, and J. J. Weis, J. Chem. Phys. 89, 6941 (1988). 6.1 [8] M. P. Allen and D. Frenkel, Phys. Rev. Lett. 58, 1748 (1987). 6.1 [9] M. P. Allen and M. A. Warren, Phys. Rev. Lett. 78, 1291 (1997). 6.1 [10] S. Ravichandran, A. Perera, M. Moreau, and B. Bagchi, J. Chem. Phys. 106, 1280 (1998). 6.1 [11] S. Ravichandran, A. Perera, M. Moreau, and B. Bagchi, J. Chem. Phys. 107, 8469 (1997). 6.1 [12] R. Vasanthi, S. Ravichandran, and B. Bagchi, J. Chem. Phys. 115, 10022 (2001). 6.1 [13] P. P. Jose and B. Bagchi, J. Chem. Phys. 120, 11256 (2004). 6.1, 6.3.1 165

BIBLIOGRAPHY

[14] A. M. Smondyrev, G. B. Loriot, and R. A. Pelcovits, Phys. Rev. Lett. 75, 2340 (1995). 6.1, 6.2 [15] M. Miesowicz, Nature (London) 158, 27 (1946). 6.1 [16] B. J. Berne, J. Chem. Phys. 56, 2164 (1972). 6.1, 6.3.2 [17] J. A. Montgomery and B. J. Berne, J. Chem. Phys. 66, 2161 (1977). 6.1, 6.3.2 [18] B. Cichocki and B. U. Felderhof, Physica A 213, 465 (1995). 6.1, 6.3.2 [19] C. P. Lowe, D. Frenkel, and A. J. Maters, J. Chem. Phys. 103, 1582 (1995). 6.1, 6.3.2 [20] S. Tang, G. T. Evans, C. P. Mason, and M. P. Allen, J. Chem. Phys. 102, 3794 (1995). 6.1, 6.2, 6.4 [21] J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). 6.2, 6.2 [22] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987). 6.2 [23] J. Crain and A. V. Komolkin, Adv. Chem. Phys. 109, 39 (1999). 6.2 [24] W. G. Hoover et al., Phys. Rev. A 22, 1690 (1980). 6.2 [25] B. Bagchi and A. Chandra, Adv. Chem. Phys. 80, 1 (1991). 6.2, 6.3.2 [26] B. J. Berne and R. Pecora, Dynamic Light Scattering: With applications to Chemistry, Biology and Physics. (John Wiley & Sons, INC, New York, 1976). 6.2, 6.3.2 [27] S. Bhattacharyya and B. Bagchi, Phys. Rev. Lett. 89, 025504 (2002). 6.3.1 [28] R. Zwanzig, Proc. Natl. Acad. Sci. (U.S.A) 78, 3296 (1981). 6.3.1

166

Chapter 7 Dynamics of nematic liquid crystals 7.1

Introduction

Till now we have discussed structure and dynamics in the isotropic phase and near the I-N phase boundary. In this chapter we shall address dynamics in the Nematic phase. Orientationally ordered but positionally disordered nematic liquid crystalline phase is formed by elongated (rod-like) molecules with aspect ratio typically larger than three. Nematic phase, in a system of GB nematogens, exists over a small density range, bounded on the lower density side by the isotropic phase and on the higher density side by the smectic or crystalline solid phase. This phase exists over a wider temperature range, reflecting the importance of the repulsive part of the intermolecular interaction potential in the stability of the phase, as envisaged by Onsager and Zwanzig [1, 2] in their seminal studies. The transition from the isotropic to the nematic phase is weakly first order, driven by large growth in the orientational correlation length gives this transition somewhat of a second order character. The nematic to the smectic phase transition is also weakly first order [3, 4] and shows similar characteristics of I-N transition [5, 6]. Therefore, nematic phase is well known for its static fluctuations, reflected in its large electric and magnetic birefringence and large scattering of light. Dynamic properties of the nematic phase has drawn a great deal of attention [7–9] with emphasis on electric field and flow induced behaviour. Much of the studies have been directed to long time (radio frequency range) dynamics where the relaxation is exponential with the relevant time constant comes from the director fluctuations. Another 167

7. DYNAMICS OF NEMATIC LIQUID CRYSTALS

property of particular interest is the macroscopic instability phenomena of the domain and the pattern formation when the nematic phase is driven at high voltage with frequency larger than certain voltage dependent critical frequency, ωc , which could be in the inverse ms range. Dielectric relaxation studies show that orientational relaxation (with parallel and perpendicular components) occurs at smaller time scales (microwave frequencies). A much slower component is found for orientation around the axis for those nematogens, which have a significant dipole moment along the long axis. The latter motion, which can be observed in NMR experiments, occur in the radiowave frequencies and may originate from π rotation of the molecule. This is a rare process. Such processes have recently been observed in computer simulation studies [10–12]. Dynamics of liquid crystals have also been studied by acoustic attenuation. All the experimental studies discussed above employ frequency domain measurements and largely restricted to measuring low frequency of dynamic response of the nematic phase. With the advent of femtosecond (fs) laser spectroscopy, it is now possible to study the relaxation directly in the time domain on a much faster time scale using, for example, optical Kerr effect (that is, the disappearance of an initially created optical birefringence). As already discussed in the chapter 1, such studies have been carried out recently by Fayer and coworkers [13–17], both in the isotropic and in the nematic liquid crystalline phases. It has been observed that both the isotropic and the nematic phase exhibit multiple temporal power laws in the short time. While the long time decay is exponential in the isotropic phase with a time constant close to that given by Landau-deGennes theory. This exponential decay is found to be missing in the nematic phase. The origin of the short-to-intermediate power law in the isotropic phase near its coexistence with the nematic phase can be understood in terms of mode coupling theory, which gives an expression for the rotational memory function in terms of equilibrium orientational pair correlation among the nematogens [16]. According to this picture, as the length of this pair correlation undergoes rapid increase when the nematic phase is approached from the isotropic side; a divergence-like behaviour develops in the 168

7.1 Introduction

wavenumber (k) dependence at small wavenumber (k → 0). Since the mode coupling theory (MCT) expression of memory function involves a wavenumber integration over the orientational correlation function (originally orientational dynamic structure factor [18]), the memory function shows an anomalous frequency (z) dependence and behaves like z −α (here z is the Laplace frequency) over a frequency range near low frequency. It is easy to show that such a memory function gives rise to a power law in the decay of the orientational time correlation function. This power law behaviour is more clear in the transient Kerr effect which measures the time derivative of the collective polarization time correlation function [19, 20], given by χ(t) = −θ(t)

dα(t) dt

(7.1)

where α is the total polarizability-polarizability time correlation function (at equilibrium) of the system. θ (t) is the step function. The above mode coupling theory makes several other predictions, which can be tested against experiments and/or simulations. The time dependence of the first and second rank orientational correlation functions should be vastly different because the orienting field that localizes the director is different. Second, the single particle orientational time correlation function should also exhibit power law decay, but on smaller time scales. Simulations have verified both. In addition, the z −α growth of the memory function at small z has been observed in simulations. The long time decay, as mentioned above, is exponential in the isotropic phase. However, the experimental results seem to indicate a different and somewhat surprising scenario in the nematic phase [21]. OKE experiments find no signature of the exponential decay. This is in contrast to the earlier frequency domain measurements, which finds exponential decay in the longer time (longer than nanosecond, for example, for MBBA [7]). In order to understand the dynamics of the orientational relaxation in the nematic liquid crystalline phase, we have carried out a series of simulations where we compute transient optical Kerr signal for model prolate ellipsoids, which show isotropic-nematicsmectic phase transitions. We had earlier presented results on short-to-intermediate 169

7. DYNAMICS OF NEMATIC LIQUID CRYSTALS

time power laws in the transient OKE in the isotropic phase near the I-N phase boundary, see chapter 3. Those simulations produced results in qualitative agreement with the experiments; in addition to the short time power laws, the long time decay was exponential. The situation is found to be different for nematic liquid crystals. Two important observations can be made from the present simulations. First, the transient optical Kerr time correlation function for the system of prolate ellipsoids (with aspect ratio equal to three) shows, two distinct power laws, with a crossover region, which can also be described by a power law. Second, we also find the absence of long time exponential decay component. The absence of the long time decay signal in the transient OKE could possibly be attributed to the very sluggish nature of the long time decay of the orientational time correlation function which makes it difficult to detect the decay in the OKE because it measures the time derivative of the collective polarizability-polarizability time correlation function. Theoretical analysis suggests that the two power laws may originate from local fluctuations of the director (which is enhanced due to the proximity to the isotropic phase) and from the density fluctuations leading to the smectic phase - both the processes are expected to involve small free energy barriers. Simulations reveal the evidence for pronounced coupling between orientational and spatial densities. This coupling is usually small in normal conditions but it is large and slow in the present case. In addition to slow collective orientational relaxation, the computed single particle orientational relaxation is also found to exhibit slow dynamics in the long time. The power law decay in the nematic phase is contrasted to the one observed in the isotropic phase near the isotropic-nematic phase boundary [22]. The multiple power law relaxation observed in the simulated Kerr Relaxation signal is collective in origin and can be related the three Frank elastic constants of the nematic liquid crystal [23, 24]. As the magnitude of these elastic constants changes, the duration of the power law relaxation also varies. The rest of this chapter is arranged as follows. The details of the simulations is given in the following section (section 7.2). The results of the simulations are given in the section 7.3. Properties of the static fluctuations are given in 7.3.1. The properties of 170

7.2 Details of the simulation

the collective orientational correlation functions and the Kerr signals obtained from the simulations are discussed in the section 7.3.3. Single particle orientational correlation functions are discussed in the section 7.3.2. A theoretical analysis of the observed power laws are given in the section 7.4. The summary of this study is presented in the section 7.5.

7.2

Details of the simulation

The procedures for the simulations are same as that followed in the previous chapters. The details of the simulations and the parameters of the potential used are given in the sections 2.2.1 and 2.4. The simulation starts from an equilibrated configuration of ellipsoids. Initial configuration of the ellipsoids is generated from a cubic lattice and then the simulation is run for two hundred thousand steps to obtain the equilibrium configuration. During the equilibration steps the temperature is scaled so that the system is in equilibrium with this particular temperature. The data from 50 million production steps are used to calculate the correlation functions.

7.3 7.3.1

Results and Discussion Order parameter and equilibrium orientational pair correlation functions

Before addressing the nature of the orientational relaxation in the nematic liquid crystal, we look into the pair distribution function and structure factor of the GB liquid. Figure 7.1 shows the variation of the orientational order parameter near the isotropicnematic transition at three different temperatures for the model Gay-Berne liquid (aspect ratio =3). The earlier reported analysis of simulations is performed near the isotropic-nematic (I-N) phase boundary, but still on the isotropic side. In the present work, the simulations are in the nematic phase. As already mentioned, the I-N coexistence is rather narrow in the density plane, but quite wide in the temperature plane. The strength of the coupling between the orientational density (δΩ) and the number density (δρ) can be obtained from the range of the g200 (r). The figure 7.2a 171

7. DYNAMICS OF NEMATIC LIQUID CRYSTALS

1 0.9 0.8 0.7

Nematic

0.6 0.5 0.4 0.3 0.2 0.1 0 0.2

0.3

*

ρ

0.4

Figure 7.1: The order parameter is plotted against density at two different temperatures (solid line is at T ∗ = 1 and dashed line is at T ∗ =1.25). Note the relatively narrow nematic range

shows identical structure for g200 (r) for all the state points. The g200 (r) is found to be non-vanishing even at large inter particle separation. This shows long range coupling between orientation and density. Figure 7.2(b) shows the equilibrium orientational correlation function, g220 (r), which gives the spatial dependence of the orientational correlation between two molecules that are separated by a distance r. g220 (r) become long ranged as the isotropic-nematic phase transition (at ρ∗ = 0.315) is approached from below (in density). In the nematic phase also g220 (r) is long ranged and nondecaying as expected. Figure 7.3 shows the equilibrium isotropic structure factor of the liquid both in the nematic and in smectic phase (dotted line). Note that the sharp peak in the smectic phase is at the value of the wavenumber that corresponds to the inter-layer separation. Even in the nematic phase, there is considerable smectic order (shown by the first peak of the S(k)) in the nematic phase.

7.3.2

Single particle orientational relaxation

Figure 7.4 shows log-log plot of the second rank single particle OTCF, C2s (t), (see Eq. 3.1) in the nematic phase. The inset of the figure 7.4 shows the time derivative of the 172

7.3 Results and Discussion

g200(r)

0 −0.5

a

−1 *

0

0

1

2

3 *

6

b

g

220

(r)

r 1

*

ρ =0.32, T =1.0 * * ρ =0.34, T =1.25 4 * * 5 ρ =0.33, T =1.0

0.5

0

0

1

2

3 *

4

5

6

r

Figure 7.2: The static orientational pair correlation functions g200 (r) and g220 (r) are plotted against inter particle distance in the nematic phase.

second rank single particle OTCF. After the initial fast exponential decay C2s (t) also shows the power law relaxation in the intermediate time scale. The tail of C2s (t) decays exponentially. Figure 7.5 shows the log-log plot of the single particle first rank OTCF, which shows slow exponential decay in the long time after the initial fast relaxation is over. There are noticeable qualitative difference between first rank single particle OTCF and the second rank single particle OTCF; while the second rank correlation function clearly shows the existence of a plateau and a power law time dependence in the intermediate time, the decay of the first rank correlation function can be fitted to single exponential decay. The time constants of the exponential relaxation function at the three state points are 184 for the case (a), 655 for the case (b) and 575 for the case (c). The well 173

7. DYNAMICS OF NEMATIC LIQUID CRYSTALS

2.5 ρ*=0.32,T*=1.0 ρ*=0.34,T*=1.25 ρ*=0.33,T*=1.0 * ρ*=0.36,T =1.0

2

S(k)

1.5

1

0.5

0 0

5

10

15

k

20

Figure 7.3: The structure factor S(k) is plotted against the wave numbers k at the state points where simulation is done. Note that a pure smectic A phase is distinguished from nematic phase by sharp peak at inter layer separation length

0

−0.5

s

−1

ln(−dC2(t)/dt)

ln

(Cs (t)) 2

0

−1.5

−5

−10 −2

0

2 ln (time)

4

−2

−2.5 −2

0

2

ρ*=0.32,T*=1.0 ρ*=0.34,T*=1.25 ρ*=0.33,T*=1.0

4

6

ln (time)

Figure 7.4: Log-log plot showing the second rank single particle orientational correlation function against time. The inset shows the log-log plot of the derivative of second rank single particle orientational correlation function against time

174

7.3 Results and Discussion

0 −0.5

ln (Cs (t))

−1 1

*

ρ*=0.32,T =1.0 ρ*=0.34,T*=1.25 * ρ*=0.33,T =1.0

−1.5 −2 −2.5 −3 −2

0

2

time

4

6

8

Figure 7.5: Log-log plot showing the first rank single particle orientational correlation function against time.

known π flip of the molecules with up-down symmetry is responsible for the decay of the first rank correlation function. During the flip, the orientational cage that surrounds the molecule has to be rearranged. This gives rise to the cooperativity in the relaxation process in a length scale of few molecular diameters. Note that the correlations appear in the g200 (r) is has a range r ∗ ≈ 6, which shows that density fluctuation of the range r ∗ ≈ 6 may require for the π flip of a molecule to occur. The restoring forces generated while these rearrangements take place are related to the Frank elastic constants of the nematic liquid crystal.

7.3.3

Collective orientational time correlation functions and Kerr relaxation

The collective orientational relaxation is slow in the case of the nematic liquid crystal. Figure 7.6 shows the C2c (t) (Eq. 3.2) against time. In the figure 7.6 during the time interval of 1000, C2c (t) relaxes to ∼ 30% - 40 % of it’s initial value. Along the isotherm the time constant of the C2c (t) increases at higher densities as expected. The C2c (t) show several interesting features in it’s short time decay before the long time relaxation set in. 175

7. DYNAMICS OF NEMATIC LIQUID CRYSTALS

1

ρ*=0.32, T*=1.0 ρ*=0.34, T*=1.25 * * ρ =0.33, T =1.0

0.95 0.9

0.8

c

C2(t)

0.85

0.75 0.7 0.65 0.6 0.55 0

200

400

*

600

800

1000

t

Figure 7.6: The collective second rank orientational correlation function in nematic phase is plotted against time at three state points.

Multiple portions of the C2c (t) which differ in their characteristics of relaxation is visible in the plot of time derivate of C2c (t) versus time. Also note that, the time derivative of the C2c (t) gives the transient signal that observed the OHD-OKE experiments. The relaxation of the transient Kerr signal intensity calculated from the simulations at three state points of the nematic liquid crystal is shown in the figures 7.7 and 7.8: (a) near the isotropic phase (T ∗ = 1.0, ρ∗ =0.32); (b) near smectic phase at a higher density and temperature (T ∗ = 1.25, ρ∗ =0.34); (c) near smectic phase at higher density than the case (a) (T ∗ = 1.0, ρ∗ =0.33). Figures 7.7 and 7.8 show the transient Kerr signal calculated at different time interval and duration. It is computationally expensive to calculate the full decay of the Kerr signal with high resolution. Hence figure 7.7 shows the short time part of the Kerr signal given in 7.8 at a high resolution. The common feature of subfigures 7.8(a) 7.8(b) and 7.8(c) is the three flat regions in the transient Kerr signal. The flat regions of almost zero slopes separate the initial and final regions. In figure 7.8(a) a log-log plot the time dependence of the Kerr signal is shown for the state point (a). The inset of figure 7.7(a) shows the corresponding short time part of 176

7.3 Results and Discussion

log of Kerr signal −arb. units

a

−1

0

1

2

3

4

2

3

4

2

3

4

b

−1

0

1

c

−1

0

1

*

t

Figure 7.7: The calculated Kerr signal intensity in nematic phase at short time with high resolution is plotted against time at three state points. Subplot (a) is at ρ∗ =0.32 and T ∗ =1, (b) is at ρ∗ =0.34 and T ∗ = 1.25 and ρ∗ =0.32 and T ∗ = 1.0. The thick straight lines in the figures are linear fit at flat portions

the relaxation. Note that the Kerr signal decay is highly non-exponential. Different flat regions of the transient Kerr signal can be fitted to multiple power laws (∝ t−α ), with exponents, α = 0.14, 0.014 and 0.56 in the successive order they appear in time. In the long time, the Kerr signal has become very small due to the slow decay that 177

7. DYNAMICS OF NEMATIC LIQUID CRYSTALS

log of Kerr signal −arb. units

a

0

1

2

3

4

5

6

4

5

6

4

5

6

b

0

1

2

3

c

0

1

2

3 t*

Figure 7.8: The calculated Kerr signal intensity in nematic phase (long time data with low resolution) is plotted against time at three state points. Subplot (a) is at ρ∗ =0.32 and T ∗ =1, (b) is at ρ∗ =0.34 and T ∗ = 1.25 and ρ∗ =0.32 and T ∗ = 1.0. The thick straight lines in the figures are linear fit to the data at the flat regions

appears in C2c (t) (see figure 7.6) in the long time. The time intervals at which the Kerr signal intensity show power laws deserve special attention. In contrast to the power law observed near the isotropic-nematic phase boundary, here, in the nematic phase, not only does the power law decay set in early, 178

7.3 Results and Discussion

but, in addition, the power law is of short duration. In this case (a) the first power law lasts for t∗ ≈ 1 - 20, the intermediate one lasts for t∗ ≈ 45 - 200 and the final one lasts for t∗ ≈ 300 - 650. Corresponding second rank collective orientational time correlation function (OTCF) shows (see figure 7.6) multiple turning points marking different regions of power law relaxation. Here the evolution of the collective orientational relaxation of ellipsoids, which is shows Gaussian relaxation in the initial stage, changes to a final slow relaxation through the intermediate stages. This can be viewed as the result of slow development of cooperativity in the collective orientational relaxation. The tail of the C2c (t) relaxes due to the rotation of the whole nematic domain. In figures 7.7(b) and 7.8(b), we plot the calculated transient Kerr signal at a different temperature and density which is away from the isotropic phase or deeper into the nematic than the case (a) (T ∗ = 1.25, ρ∗ = 0.34). The figure 7.6 shows the corresponding second rank collective OTCF. The different regimes in figure 7.7b and can be fitted to three different power laws with values of the exponent α = 0.13, 0.015 and 0.26 in the order they appear in time. The first exponent remains the same as that of the case (a), however, the values the other two exponents are differ. The basic features of the transient Kerr signal are quite similar. Here also the signal becomes very weak at long times due to the slow decay of the second rank collective OTCF (shown in the figure 7.6). Note that the time interval in which power laws dominate the relaxation is essentially the same - the initial power law lasts between t∗ ≈ 1 and 16. The second power is between t∗ ≈ 16 and 100 and the last power law is between t∗ ≈ 160 and 400. The power law relaxation is short lived in this case. Finally, in figures 7.7c and 7.8c we plot the calculated Kerr signal in the case (c) (T ∗ = 1.0, ρ∗ = 0.33). The three different temporal regions can be fitted to different power laws, with the value of the exponent α = 0.14, 0.1 and 0.24, is in the same order as that in the previous cases. In this case the second and the first power laws have comparable slope. By comparing transient Kerr signal obtained in the case (a), (b) and (c) one may conclude that the in the nematic phase there exist three different regions power law relaxation. The exponent of the first power law is somewhat similar in all 179

7. DYNAMICS OF NEMATIC LIQUID CRYSTALS

the tree cases – all being close to ∼ 0.14. The last power law is somewhat stronger. However, the values of the exponent cannot be taken very seriously because much longer simulations (with probably much larger system sizes) are required to draw any conclusive evidence about values of these exponents. Note that the second power law is actually a crossover region - shorter in figures 7.8(a) and 7.8(b) and only seems to be a genuine power law in the figure 7.8(c). The long time exponential tail of the C2s (t) is absent in the case of calculated C2c (t), which can be captured only in very long time simulations. Now, we compare the time scale at which the intermediate power law appears in the single particle OTCF it is in the range t∗ ≈ 5 - 15 which is approximately same time scale as that of the first power law in the collective second rank OTCF. Hence the first power law in the Kerr signal intensity arises from the interaction of the ellipsoids in the confining orientational cage. However, the nature of the orientational cage in both cases differs. It is also important to note that these temporal power laws appear at relatively short times. The first one sets in within a couple of pico seconds, while the longest one in about 100 ps. Thus, these power laws could arise from the fluctuations, which are local in origin because long wavelength director fluctuations are expected to be slower. Two obvious candidates are fluctuations leading to (i) the local orientational melting to the isotropic phase since the free energy difference between the two phases is small, and (ii) local ordering to smectic phase which is also of comparable free energy. As noted earlier, both the transitions are weakly first order. The figure 7.9 shows several snapshots of position and orientation of the system at various temperatures and densities, in the nematic phase. The state points ranges from near isotropic-nematic boundary 7.9(a) to the smectic phase 7.9(d). Figures 7.9(b) and 7.9(c) are closer to the nematic-smectic boundary. However, one clearly sees both isotropic and smectic phase like domains in figures 7.9(a), 7.9(b) and 7.9(c). These domains introduce a strong coupling to the density fluctuations and may be responsible for the anomalous power law behaviour observed in Kerr relaxation. It is worthwhile to place these results in the perspective of the earlier frequency domain experiments. 180

7.3 Results and Discussion

(a)

(b)

(c)

(d)

Figure 7.9: This figures show the snapshots of the unit vectors of the ellipsoids in the system, that show the Smectic fluctuation present in the system. The snapshots are taken at four state points: subplot (a) is at ρ∗ =0.32 and T ∗ =1.0; subplot (b) is at ρ∗ =0.34 and T ∗ =1.25; subplot (c) is at ρ∗ =0.34 and T ∗ =1.0; subplot (d) is at ρ∗ =0.36 and T ∗ =1.0.

First, the nematogens in the study have ellipsoidal (up-down) symmetry. So, rotation about the long axis does not give rise to any relaxation of the time correlation function. Therefore, this model cannot describe the short time decay observed in NMR and dielectric relaxation of long nematogens, like MBBA. In those experiments, even the short time is in the range of nanosecond or longer and some processes are occurring even in the microsecond range. In sharp contrast, the relaxations observed here are in the pico second domain; the shorter power law starts within a couple of picoseconds. We find that Kerr signal becomes quite weak by about 2 ns. The last power law appears by about 200 ps. The slow tail of the Kerr signal arises from the random rotation of the whole nematic domain, which in case of the simulations, is due to the rotation of the global director of the system. Some of the features observed in simulations seem to be similar to the recent ex181

7. DYNAMICS OF NEMATIC LIQUID CRYSTALS

perimental results on transient Kerr effect reported by Fayer and coworkers [21] who also reported multiple power laws in the short time. In the experiments also, the first power law sets in about 3 ps and lasts to about 30 ps, which is in semi-quantitative agreement with our simulations. The exponent of the first power law is temperature dependent and is in the range 0.38 to 0.48 as temperature is varied from 298 to 306 K. The last power law starts about 300 ps and lasts till 30 ns. The exponent of the last power law is larger and is about 0.55. Note that all the above exponents for the Kerr signal and not for the time correlation function. However, the onset of power law is not distinctly visible in the time correlation function. As discussed in the Introduction, the mode coupling theory can provide a semiquantitative explanation for the emergence of power law in isotropic phase near the I-N phase boundary in terms of the growing correlation length of orientational correlations among the nematogens. However, no such explanation is available in the present case. Since these power laws are in the relatively shorter time, it is likely that, the reason for the power laws lie in local fluctuations. The following scenario can be envisaged for the first power law. Subsequent to the passage of the optical pulse, the molecules are now locally oriented somewhat randomly to the direction of the director because the electric field of the propagating pulse is uncorrelated to the direction of the director. Thus, we can consider this laser induced local reorientation (from the original direction from the pulse arrives) to be akin to a local orientational melting - that is, nematic to isotropic transition. The subsequent relaxation back to the nematic phase occurs in the effective field of the director, which favors the nematic alignment. This is expected to be a slow process.

7.4

Role of fluctuations in multiple power law relaxation

As mentioned earlier, the nematic phase, especially the Gay-Berne system with aspect ratio 3, is close both to the isotropic and the smectic phases in the phase plane. The proximity to the isotropic phase allows large fluctuations in the director. Such local 182

7.4 Role of fluctuations in multiple power law relaxation

fluctuations of the director often lead locally to the formation of the isotropic phase within the orientational cage. The first power law observed may be viewed as due to the fluctuation leading to the formation of the isotropic phase locally leading to the relaxation of the orientational cage. And, it is also well known that the nematic phase exhibits fluctuations to smectic phase. A picture of such fluctuations is already shown in figure 7.9 – it is found that these fluctuations are quite stable which give rise to first peak in the structure factor (See figure 7.3). Note that smectic phase has increased orientational ordering, in addition to it’s one-dimensional order. For GayBerne ellipsoids, the smectic phase is a free energy minimum which not too far in the order parameter space. It is possible to express the free energy surface of a nematic phase in terms of fluctuations of these two order parameters (for orientation and for one dimensional spatial ordering). Such a free energy description already exists. The total free energy surface of the Smectic A phase can be written as [5–7, 25] F = Fel + Fsm ,

(7.2)

where Fel is the elastic free energy of the unperturbed nematic state and Fsm is the free energy of the smectic A phase. The Fel may be expanded in terms of the Frank elastic constants K1 , K2 and K3 [7, 9, 26]. Z 1 d3 k Fel = [(K1 k⊥ 2 + K3 kk 2 )δns 2 + (K2 k⊥ 2 + K3 kk 2 )δnt 2 ] 2 (2π)3

(7.3)

where K1 , K2 and K3 are respectively splay twist and bend elastic constants; δns and δnt are splay and twist fluctuations of the director n0 , parallel and perpendicular to the k⊥ respectively. For a Smectic A phase this term is to added to the smectic Landau-Ginzburg free energy Fsm introduced by de Gennes [5, 6]. Z 1 Fsm = d3 r[a0 | ψ |2 + b+ | ψ |4 +(∇ + iq0 n)ψ ∗ i ⊗ Γij ⊗ (∇ − iq0 n)ψj ] 2

(7.4)

where a0 ∝ t (t = (T − TN S )/TN S ), b is a constant, q0 is the wave vector of the density wave, Γij is the inverse mass tensor, ψ is the density wave order parameter for the nematic to smectic transition and ⊗ stands for the tensor product. ψ is related to the 183

7. DYNAMICS OF NEMATIC LIQUID CRYSTALS

density ρ(r) of the system through the relation ρ(r) = ρ0 +

P

n [<

ψn > eink·r + c.c.]

(ψ =< ψ > e−ik.r ). The elements of Γij are ξ⊥0 2 ,ξ⊥0 2 and ξk0 2 - ξ⊥0 and ξk0 are the bare correlation lengths perpendicular and parallel to n respectively. As the nematic smectic transition is approached, the correlation lengths diverge with critical exponents ν⊥ and νk : ξ⊥ 2 ∝ t−ν⊥ and ξk 2 ∝ t−νk . Theoretical studies have led to the conclusion that while nematic to smectic transition could be second order in the absence of the coupling of the smectic order parameter, the density wave ψ, coupling to the nematic director makes this transition weakly first order [3, 27]. Recent experimental studies seem to confirm this because one observes non-zero latent heat of the transition. One of the predicted effects of the coupling between the nematic order parameter (the director) and the smectic density wave is the hardening of the Franck’s elastic force constants, K2 and K3 . The second and the third power law regions of the relaxation functions become prolonged as the chosen state point in the simulation becomes closer to the nematicsmectic phase boundary. Note that in nematic liquid crystals the values of these force constants do not differ by an order of magnitude; so the relaxation process mediated by them will not differ much in their time scale. Note that the final relaxation does not include any local effects and may be exponential for a simulation of large system size whose orientational correlation length is comparable to a domain size of the nematic liquid crystal. These are the long time dynamics discussed by de Gennes[7] and are clearly outside the scope of the simulations reported here. Optical Kerr effect measures the time derivative of the collective polarizability, which is the small wavenumber (that is, k≈ 0) limit of the Fourier transform of the polarizability of the system. In the nematic phase, when the orientational order has largely been formed, one measures the fluctuation around the director. Infinitesimally small fluctuations of are of course expected to decay fast, within a few hundred fs, or at most a few ps, and primarily exponential. The long time decay requires local fluctuations of the director, which, as discussed above, are coupled to collective properties of the system. In view of the pre-transition effects existing in the nematic phase, such power law decay is perhaps not surprising, but as mentioned above; quantitative 184

7.5 Concluding Remarks

understanding is yet to be evolved.

7.5

Concluding Remarks

Let us first summarize the main results of this chapter. We have carried out extensive molecular dynamics simulations of transient optical Kerr effect in a system of prolate ellipsoids (with aspect ratio equal to three) in the nematic phase. Our simulations reveal two distinct power laws in the decay of the time derivative of the second rank collective orientational time correlation function that is measured in the optical Kerr effect experiments. However, the simulations failed to detect any long time exponential component. We find that the absence of the long time decay signal in the simulated transient OKE could possibly be attributed to the very sluggish nature of the long time decay, which is difficult to identify in the OKE signal calculated from the simulations; because, as mentioned earlier, it measures the time derivative of the collective polarizability time correlation function. The simulation results have been compared with recent experimental results [21]. Theoretical analysis suggests that the two power laws may originate from fluctuations of the local director (which is enhanced due to the proximity to the isotropic phase) and final power law marks the emergence of cooperativity in the relaxation process from intermediate length scales to the relaxation of the director of the system. The single particle relaxation, on the other hand, does exhibit an exponential decay in the long time (figures 7.4 and 7.5). The present simulations probe the linear response of the system in the sense that no electric field is included in the simulations. It will certainly be worthwhile to carry out a simulation with the electric field explicitly taken into account in the preparation of the initial state, for several reasons. First, the linear response may have limitations near the phase transition. Second, it might be easier to capture the fluctuations that couple to Kerr relaxation in such a direct approach. To conclude, we stress that the present simulation results (which appear to be in agreement with recent optical Kerr effect experiments [21]) reveal the importance of orientational fluctuations in the nematic phase. These results are quite different from 185

7. DYNAMICS OF NEMATIC LIQUID CRYSTALS

the earlier dynamical studies on nematic that focused on the long time exponential decay. Future work shall concentrate on molecules with larger aspect ratio where the nematic phase is more stable than in the present case and where one may recover the exponential decay even in the Kerr relaxation.

186

Bibliography [1] L. Onsager, Ann. N. Y. Acad. Sci. 51, 621 (1949). 7.1 [2] R. Zwanzig, J. Chem. Phys. 39, 1714 (1963). 7.1 [3] B. I. Halperin, T. C. Lubensky, and S. K. Ma, Phys. Rev. Lett. 32, 292 (1974). 7.1, 7.4 [4] C. Dasgupta and B. I. Halperin, Phys. Rev. Lett. 47, 1556 (1981). 7.1 [5] P. G. de Gennes, Solid State Commun. 10, 753 (1972). 7.1, 7.4, 7.4 [6] B. R. Patton and B. S. Andereck, Phys. Rev. Lett. 69, 1556 (1992). 7.1, 7.4, 7.4 [7] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993). 7.1, 7.1, 7.4, 7.4, 7.4 [8] S. Chandrasekhar, Liquid Crystals (Cambridge University Press, Cambridge, 1977). 7.1 [9] M. J. Stephen and J. P. Straley, Rev. Mod. Phys. 46, 617 (1974). 7.1, 7.4 [10] R. Vasanthi, S. Ravichandran, and B. Bagchi, J. Chem. Phys. 115, 10022 (2001). 7.1 [11] C. D. Michele and D. Leporini, Phys. Rev. E 63, 36702 (2001). 7.1 [12] P. P. Jose, D. Chakrabarti, and B. Bagchi, Phys. Rev. E 71, 030701 (2005). 7.1 [13] J. J. Stankus, R. Torre, and M. D. Fayer, J. Phys. Chem. 97, 9478 (1993). 7.1 [14] H. Cang, J. Li, and M. D. Fayer, Chem. Phys. Lett. 366, 82 (2002). 7.1 187

BIBLIOGRAPHY

[15] H. Cang, J. Li, V. N. Novikov, and M. D. Fayer, J. Chem. Phys. 118, 9303 (2003). 7.1 [16] S. D. Gottke et al., J. Chem. Phys. 116, 360 (2002). 7.1 [17] S. D. Gottke, H. Cang, B. Bagchi, and M. D. Fayer, J. Chem. Phys. 116, 6339 (2002). 7.1 [18] B. Bagchi and A. Chandra, Adv. Chem. Phys. 80, 1 (1991). 7.1 [19] Y. Yan and K. A. Nelson, J. Chem. Phys. 87, 6257 (1987). 7.1 [20] R. Torre, F. Tempestini, P. Bartolini, and R. Raghini, Phil. Mag. B 77, 645 (1998). 7.1 [21] J. Li, I. Wang, and M. D. Fayer, J. Phys. Chem. B 109, 6514 (2005). 7.1, 7.3.3, 7.5 [22] P. P. Jose and B. Bagchi, J. Chem. Phys. 120, 11256 (2004). 7.1 [23] F. C. Frank, Disc. Faraday Soc. 25, 19 (1958). 7.1 [24] C. W. Oseen, Disc. Faraday Soc. 29, 883 (1933). 7.1 [25] A. Linhananta and D. E. Sullivan, Phys. Rev. A 44, 8189 (1991). 7.4 [26] P. M. Chaikin and T. C. Lubensky, Principles of condensed matter physics (Cambridge University Press, Cambridge, 1998). 7.4 [27] I. Lelidis, Phys. Rev. Lett. 86, 1267 (2001). 7.4

188

Part II RELAXATION IN A LINEAR LATTICE

189

Chapter 8 Density and energy relaxation in an open one-dimensional system 8.1

Introduction

Relaxation dynamics of interacting particles in a one-dimensional system is often difficult to understand because the traditional coarse-grained descriptions (such as hydrodynamics or time dependent mean-field type approximations) fail in this case. The latter is because of the existence of long-range correlations mediated through the excluded volume interaction. In such cases, random walk models have often proved to be successful in describing the non-exponential relaxation commonly observed in onedimensional systems [1, 2]. Recently, several experimental studies have reported energy relaxation in onedimensional or quasi one-dimensional systems such as DNA [3–5]. These experiments have employed the time dependent fluorescence Stokes shift (TDFSS) technique to gather a quantitative measure of the time scales involved. In one of these experiments, Bruns et al.[3] have calculated structural relaxation of DNA oligonucleotides. They found that the red shift of the fluorescent spectrum follows logarithmic time dependence. The experiment described above is an example where dimensionality plays an important role in the energy and number density relaxations. In this work, a random walk model for the carriers in a one-dimensional channel to mimic the relaxations of energy and number density found in experiments is described. There were several theoretical studies devoted to random walk model in 191

8. DENSITY AND ENERGY RELAXATION IN AN OPEN ONE-DIMENSIONAL SYSTEM i

i+1

i+2

i+3

i+4

i+5

Bath Figure 8.1: Schematic representations of different transitions at lattice sites of a linear lattice.

one-dimensional systems [6–12]. Many recent studies based on random walk model in one-dimensional systems have explored transport and other dynamical properties such as, DC conductivity, frequency dependent conductivity, effect of bias on diffusion, space and time dependent probability distribution of random walkers, mean residence time and mean first passage time of random walkers etc. [12–19]. The work presented here is based on a master equation for random walk in a singlefile system (a one-dimensional system that does not allow particles to pass through each other) that allows exchange of the particles with the bath (see figure 8.1 for a schematic illustration). Two model potentials have been employed with different characteristics to study the effects of perturbation on the number density and energy relaxation processes. The potentials used here are (1) the inverse of distance potential (e.g., a Coulomb’s field generated by a trapped charge in the lattice) and (2) inverse of sixth power of distance (this is a short range interaction compared to Coulomb interaction and is taken as the attractive part of the Lennard-Jones interaction). Energy and number density relaxation functions in a one-dimensional channel without number density fluctuation are calculated and then compared with that of a channel, where particle number density fluctuates due to exchange with the bath. It is observed that number density fluctuation in the channel makes energy relaxation exponential. The rest of this chapter is organized as follows. Section 8.2 gives the description of the master equation used in the simulation. The details of the Monte Carlo simulations are given in the section 8.3. Results of the simulations are analyzed in the section 8.4. 192

8.2 The Generalized Master equation

Section 8.5 presents the conclusions.

8.2

The Generalized Master equation

Let the total number density of lattice sites in a system consisting of a linear lattice and a bath be NT = NL + NB , where NL is the number density of lattice sites of the lattice and NB is the number density of accessible bath sites. The time dependent probability PiL (t) of finding a particle in the i th site of the lattice at time t is given by the following master equation N L 0 X dPiL (t) = wm,i (t)PmL (t) − wi,m (t)PiL (t) dt m=1

−PiL (t)

NB X

wi,j (t) +

j=1

NB X

wj,i (t)PjB (t),

(8.1)

j=1

(the prime on the summation signifies that the term i = m is to be omitted from the sum) where wm,i (t) gives the transition probability of the particle from site m to i per unit time. The last two terms of the master equation introduce number density fluctuation in the lattice by allowing exchange of particles with the bath. In the master equation, the index j runs over the bath sites and P B j is the probability of finding a particle in the j th bath site. This master equation can be simplified by assuming that lattice sites in the bath are numerous ( NB >> NL ) and the number density of walkers in the bath is much larger than that in the lattice. These assumptions allow us to perform an averaging over the bath sites. Summation in the last two terms of the equation 8.1 can be omitted by substituting the properties of the bath with that of an ideal bath defined below. The sum of the transition probability wi,j (t) to the bath P B sites can be replaced by the term wout (t) ( N j=1 wi,j (t) = wout (t)), since wi,j (t) depends only on the temperature, size and number density of particles in the system (discussed

in detail later in this section). This ideal bath has infinite capacity to absorb particles. Similarly, when lattice absorbs a particle, this bath site acts as a supplier of infinite number density of particles. 193

8. DENSITY AND ENERGY RELAXATION IN AN OPEN ONE-DIMENSIONAL SYSTEM

The transition probabilities from all lattice sites to the bath are equal and assumed to be independent of the instantaneous state of the bath. Therefore, wj,i can be replaced P B B by win and N j=1 Pj can be replaced by Pbath . The resultant master equation is simpler

and is given by

N L 0 X dPiL (t) wm,i (t)PmL (t) − wi,m (t)PiL (t) = dt m

−wout PiL (t) + win (t)Pbath .

(8.2)

If only nearest neighbor exchanges are allowed, the master equation can be written as dPiL (t) L L = wi+1,i (t)Pi+1 (t) + wi−1,i (t)Pi−1 (t) dt −wi,i+1 (t)PiL (t) − wi,i−1 (t)PiL (t) −wout (t)PiL (t) + win (t)Pbath .

(8.3)

At equilibrium, this system represents a grand canonical ensemble. The grand-canonical Monte Carlo method is used for insertion and removal of the particles in the randomly selected sites, where probability (Pµ,L,T ) of the system having number density of particles N is [20] Pµ,L,T ∝

LN e−β[ E−µN ] , ΛN N !

(8.4)

where β is the inverse of the Boltzmann constant times absolute temperature ( kB1T ), q h2 Λ (= ) is the thermal de-Broglie wavelength, L is the length of the linear 2πmkB T lattice, µ is the chemical potential and E is the potential energy of the linear lattice.

Particles of this system obey the Boltzmann distribution at equilibrium; hence the transition probability wi,i+1 (t) can be calculated from the total energy cost of the move. The hopping probability is calculated for a particle from one site to another using Metropolis scheme [20]. Hopping probability of a random walker from one site to the neighboring site of the lattice is given by wi,i+1 = min[1, e−β∆E ]. 194

(8.5)

8.2 The Generalized Master equation

The transition probabilities for absorption and desorption from the lattice can be obtained [20] so as to satisfy the detailed balance at equilibrium as Q e−β ∆E ], (N + 1) N β ∆E e ], = min[1, Q

win = min[1, wout where Q is given by

L Λ

(8.6) (8.7)

eβµ . A non-trivial aspect of this master equation is that the

transition probability win (t) varies with time in response to the number density fluctuations. This explicit time dependence of win (t) poses formidable difficulty in obtaining an explicit analytic solution of the problem. In this system of interacting particles, the transition probability to neighboring sites depends on the instantaneous probability of that site being occupied. In a one-dimensional channel with hard rod interactions between the carriers, mobility of each carrier is restricted to a portion of the lattice. The particles move in the linear lattice under the influence of a chosen potential, which is at a fixed position of the lattice. Hence the energy cost for hopping in the lattice is given by ∆E =



E1 (transition to vacant site) ∞ (transition to occupied site)

1 1 ) for the potential 1 and E1 = K2 ( x1i 6 − xi+1 where E1 = K1 ( x1i − xi+1 6 ) for the potential

2. For absorption of particles to the lattice at ith site ∆E =



E2 (creation of vacant site) ∞ (creation of occupied site)

where E2 = K1 x1i for potential 1 and E2 = K2 x1i 6 for potential 2. For the desorption of particles from the ith site, ∆E = E2 .

(8.8)

Q in this system is constant and equal to the initial number density of the particles in the lattice. This assumption specifies the value of the chemical potential of this linear lattice. 195

8. DENSITY AND ENERGY RELAXATION IN AN OPEN ONE-DIMENSIONAL SYSTEM

8.3

The details of Monte Carlo simulation

The Monte Carlo simulations are carried out on a linear lattice, with 50% (on the average) of the lattice sites are occupied by the random walkers. Closed boundary condition is used to explore the size effect on the relaxation. In this one-dimensional lattice, the adjacent sites are equally spaced and all the sites are identical in energy in the absence of the perturbation. Here inhomogeneity in the lattice is generated by the perturbation of potential introduced in the lattice. In addition, there is a dynamic disorder in the lattice that originates from the instantaneous rearrangement of the interacting random walkers. The initial configuration of the random walkers in an unperturbed lattice is chosen from a random distribution, such that no two particles occupy the same site. The inverse potential arise from the Coulomb interaction; hence the magnitude of the biasing potential is calculated from the interaction between the charge of the carriers (q1 ) and the charge at the center of bias (q2 ) in a medium of dielectric constant . The constant K1 for the potential energy can be calculated as K1 =

q1 q2 . 

Here both the perturbing charge and the charge of the carrier have

magnitude of one electron and their signs are opposite. Assuming high screening effect, the value of  is taken as equal to that of water, that is 80. The distance between the adjacent sites in the linear lattice is 4˚ A(approximately the vertical distance between two DNA base pairs in the double helix). All simulations were conducted near room temperature (300K). The value of K1 used in the simulation is -1.7kB T . Compared to Coulomb’s interaction, the Lennard-Jones interaction is short ranged; hence the constant K2 is kept high to extend the range of the potential. The value of K2 used is -10 kB T . The non-equilibrium Monte Carlo simulation starts from a randomly chosen initial configuration. Then a perturbing potential is introduced in the lattice at time t = 0. Subsequent relaxation of energy is monitored. The simulation is repeated with different initial configurations and the results are averaged. In the Monte Carlo simulation, movements are chosen randomly such that no two events can occur at the same time. In this simulation, one Monte Carlo step is equivalent to one unit of time. The total 196

8.4 Results and discussions

potential energy of the lattice at any instant of time when perturbed by Coulomb potential is given as E(t) =

X

κ1

i

1 Pi (t), xi

(8.9)

where κ1 is a constant. Similarly, for the simulations that use the attractive LennardJones potential as the perturbation, the instantaneous potential energy of the lattice is given by E(t) =

X

κ2

i

where κ2 is a constant.

1 Pi (t), x6i

(8.10)

For comparison of relaxation function for different system sizes, a dimensionless energy relaxation function S(t) is calculated, which can be defined as S(t) =

E(t) − E(∞) , E(0) − E(∞)

(8.11)

where E(t) is the instantaneous energy at time t and E(∞) is the average energy of the lattice, in equilibrium with perturbation. Density relaxation is measured in terms of a dimensionless quantity C(t) = where X(t) is defined as X(t) =

P

i

X(t) − X(∞) , X(0) − X(∞)

(8.12)

xi Pi (t).

In the equilibrium simulation, the system is allowed to equilibrate with perturbation for 10,000 steps to get the initial configuration. The fluctuations in total energy of the system can be defined as F (t) = < E(0)E(t) >. Here a dimensionless energy fluctuation relaxation function can be defined as S(t) =

8.4

F (t) − F (∞) . F (0) − F (∞)

(8.13)

Results and discussions

Figure 8.2 shows the energy relaxation function for different system sizes obtained from the non-equilibrium simulations of a one-dimensional channel without particle fluctuation. At short times, the potential energy of the lattice relaxes faster as a 197

8. DENSITY AND ENERGY RELAXATION IN AN OPEN ONE-DIMENSIONAL SYSTEM

1 0.9 0.8 0.7

S(t)

0.6

1 0.8 0.6 0.4

0.5 0.4 0.3 0.2

0.2 0 0

1

2

3

4

5 5

x 10

0.1 0 0

1

2

3

4

5

5

x 10

time

Figure 8.2: The energy relaxation function S(t) is plotted for four different system sizes under the Coulomb potential: NL =50 (dashed line), NL =100 (dotted line), NL =150 (dash-dot line), and NL =200 (solid line)in a closed system. The inset shows the fits of the energy relaxation function for NL =200 using functions f1 (t) (dotted line) and f2 (t) (dash-dot line).

response to the newly created center of perturbation. This results in the accumulation of random walkers near the center of the biasing field, which slows down the relaxation rate. At the same time, channels far from the center of perturbation remain active due to the reduction in the number density of particles in that region. These effects together contribute to the observed slowing down of the relaxation process after the fast initial decay. As the system size increases, the lowest possible energy accessible to the system obviously becomes lower; hence the relaxation becomes progressively slow as the system size increases. However, this effect becomes insignificant beyond a limiting size of the lattice when the strength of perturbation becomes negligible. To analyze the nature of relaxation in the system, relaxation the function is fitted to different functional forms. One of the best fitting functions is a sum of an exponential and a stretched exponential, and has the form f1 (t) = b1 e−t/τ1 + b2 e−(t/τ2 ) 198

β

(8.14)

8.4 Results and discussions

(with constraints b1 + b2 = 1 and 0 ≤ b1 , b2 ≤ 1). In order to compare with the nature of relaxation with the results obtained by Bruns et al.[3], a logarithmic function of the form f2 (t) = 1 − c1 log10 (at) + c2 e−t/γ

(8.15)

is also used to fit the relaxation function. The inset of figure 8.2 shows the fits of the energy relaxation function with f1 (t) and f2 (t) when size of the system is NL =200. The fitting parameters obtained for the function f1 (t) are b1 = 0.20, τ1 = 1.4 × 105 , τ2 = 3.4 × 105 , β = 0.52. The fitting parameters for the function f2 (t) are c1 = 0.16 a = 2.5 c2 = 0.25 and γ = 7.1 × 103 . It is evident from the figure that the stretched exponential function and the logarithmic function both give good representation of the energy relaxation function in a one-dimensional lattice without number density fluctuations in presence of the Coulomb potential. The time dependence of stretched β

exponential [21, 22] relaxation, is given by the function of the form S(t) = S0 e−(t/τs ) , where 0 < β < 1. Theoretical explanation of the origin of stretched exponential relaxation in the condensed matter have been addressed by many [23–26], including Huber and coworkers in a series of papers [27–29]. Their model is based on the following simple master equation approach dPi (t) X = wm,i Pm (t) − wi,m Pi (t) dt m6=i

(8.16)

where wm,i gives the transition probability of the particle from site m to i. Stretched exponential relaxation can arise when there is a continuum of relaxation channels and the probability of any single channel being open is much less than unity. Figure 8.3 shows the number density relaxation function (which is a measure of particle aggregation) for different sizes of a system without number density fluctuations. For NL =200, a double exponential f3 (t) = c1 e−t/γ1 + c2 e−t/γ2

(8.17)

(with constraints c1 + c2 = 1 and 0 ≤ c1 , c2 ≤ 1) (shown in the inset of figure 8.3) fit of the number density relaxation function reveals the presence of two distinct time scales 199

C(t)

8. DENSITY AND ENERGY RELAXATION IN AN OPEN ONE-DIMENSIONAL SYSTEM

1

1

0.9

0.8

0.8

0.6

0.7

0.4

0.6

0.2 0 0

0.5

1

2

3

4

5 6

x 10

0.4 0.3 0.2 0.1 0 0

1

2

time

3

4

5

6

x 10

Figure 8.3: The number density relaxation function C(t) is plotted in a closed system for four system sizes under the Coulomb potential: NL =50 (dashed line), NL =100 (dotted line), NL =150 (dash-dot line) and NL =200 (solid line). The inset shows double exponential fit (solid line) of C(t) for NL =200.

in the number density relaxation. The fitting parameters obtained for the double exponential (f3 (t)) fit of number density relaxation function are c1 = 0.90, γ1 = 1.4 × 106 , γ2 = 8.3 × 104 . We now turn to the case when the number density of walkers in the lattice can fluctuate due to the exchange with the bath. Figure 8.4 gives the energy relaxation function plotted for different system sizes for this case. Note that, in this case, the energy relaxation is faster than the non-fluctuating case. The exchange of particles with bath is equivalent to opening up of many wider channels of relaxation that dominate the relaxation process. The log S(t) versus t plot in the inset of figure 8.4 shows straight lines, which is an evidence of exponential relaxation. In this case, walkers can bypass the obstacle caused by hard rod interaction in the path by exchange of particles with the bath. Figure 8.5 shows the number density relaxation function of an open system for different sizes. As system size increases, the effect of the Coulomb field on the distribution of carriers decreases. Hence the carrier number density oscillates around 200

8.4 Results and discussions

1

0

0.9 −1

0.7

S(t)

0.6 0.5 0.4 0.3 0.2

log S(t)

0.8

−2

−3 X(t)=(E(t)−E(∞))/(E(0)−E(∞))

−4 0

200

400

600

800

1000

time

0.1 0 0

500

1000

time

1500

2000

Figure 8.4: The energy relaxation function is plotted for different system sizes (NL =50 (solid line), NL =100 (dash-dot line), NL =150 (dotted line), and NL =200 (dashed line)) under the perturbation of Coulomb potential in an open system. The log of S(t) versus t plot in the inset shows straight lines due to the exponential relaxation.

1.5

1

C(t)

0.5

0

−0.5

−1 0

500

1000

time

1500

2000

Figure 8.5: The number density relaxation function with number density fluctuation C(t) is plotted here for four system sizes (NL =50 (solid line), NL =100 (dash-dot line), NL =150 (dotted line), and NL =200 (dashed line)) under the Coulomb potential.

201

8. DENSITY AND ENERGY RELAXATION IN AN OPEN ONE-DIMENSIONAL SYSTEM

a mean value due to the number density fluctuations and the effect of perturbation in the number density relaxation remains feeble and short lived. The energy fluctuation 1

1

0.9

0.8

0.8

0.6

0.7 0.4

S(t)

0.6

0.2

0.5 0.4

0

2

4

6

8

10 4

0.3

x 10

0.2 0.1 0 0

2

4

6

time

8

10 4

x 10

Figure 8.6: The energy fluctuation relaxation function is plotted for a closed system for four sizes (NL =50 (solid line), NL =100(dash-dot line), NL =150 (dotted line), and NL =200 (dashed line)) in equilibrium under the perturbation of the Coulomb potential. The inset shows the fit of the energy relaxation function for NL =200 using the functions f1 (t) (dotted line) and f2 (t) (dash-dot line).

relaxation function for different sizes of a system with constant number density and that is in equilibrium with perturbation is shown in figure 8.6. This relaxation function shows very slow non-exponential decay. The inset of the figure 8.6 shows the energy relaxation function obtained is well fitted by f1 (t) and f2 (t). The fitting parameters obtained for f1 (t) are b1 = 0.2, τ1 = 3.7 × 103 , τ2 = 3.6 × 104 , β = 0.34 and the fitting parameters obtained for f2 (t) are c1 = 0.21 a = 0.06 c2 = 0.13 and γ = 6.9 × 103 . Note that the energy relaxation function in non-equilibrium and energy fluctuation relaxation function in equilibrium show logarithmic time dependence.

8.4.1

Relaxation under Lennard-Jones potential

The pronounced non-exponential decay found in non-equilibrium energy relaxation function for the Coulomb potential is also found in the case of the attractive part of 202

8.4 Results and discussions

Lennard-Jones potential. Figure 8.7 shows the energy relaxation function for differ1 0.9 0.8 0.7

S(t)

0.6

1 0.8 0.6 0.4

0.5 0.4 0.3 0.2

0.2 0 0

0.5

1

1.5

2 4

x 10

0.1 0 0

0.5

1

time

1.5

2

4

x 10

Figure 8.7: The dimensionless energy relaxation function S(t) of a one-dimensional channel with constant number density for different system sizes (NL =50 (solid line), NL =100 (dash-dot line), NL =150 (dotted line), and NL =200 (dashed line)) under the perturbation of Lennard-Jones potential is shown here. The inset shows the fit of the energy relaxation function for NL =200 using function f1 (t) (dotted line) and f2 (t) (dash-dot line).

ent sizes of a system without number density fluctuation perturbed by the attractive Lennard-Jones potential. The fits of the energy relaxation function with f1 (t) and f2 (t) are shown in the inset of the figure 8.7. The fitting parameters obtained for f1 (t) are b1 ' 0, b2 ' 1.0, τs = 1.1 × 103 , β = 0.62 and for f2 (t) are c1 = 0.14, a = 5.6 × 101 , c2 = 0.58, and γ = 1.3 × 103 . It is evident from the figure that, in the case of short-ranged interaction, the appropriate function, which can fit relaxation function, is stretched exponential. Here due to the greater strength of the potential near the perturbation center, the initial relaxation is driven and faster. The corresponding number density relaxation function for different sizes of this system is shown in figure 8.8. The energy relaxation function (S(t) versus t) of a linear lattice with particle number density fluctuation is plotted for different system sizes in the figure 8.9. The inset of the figure 8.9 shows log of S(t) versus t plot of the energy relaxation 203

8. DENSITY AND ENERGY RELAXATION IN AN OPEN ONE-DIMENSIONAL SYSTEM

1 0.9 0.8 0.7

C(t)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

x 10

time

Figure 8.8: The number density relaxation function of the lattice under the perturbation of Lennard-Jones potential for different system sizes (NL =50 (solid line), NL =100 (dash-dot line), NL =150 (dotted line), and NL =200 (dashed line)) for a closed system is shown in the figure. 1

0

0.9

−0.5 −1 ln S(t)

0.8

−1.5

0.7

−2

S(t)

0.6

−2.5 −3 0

0.5 0.4

500

1000

1500

2000

time

0.3 0.2 0.1 0 0

500

1000

time

1500

2000

Figure 8.9: The energy relaxation function (S(t)) is plotted for different system sizes (NL =50 (solid line), NL =100 (dash-dot line), NL =150 (dotted line), and NL =200 (dashed line))with particle number density fluctuations under Lennard-Jones potential. The log of S(t) versus t plot in the inset shows straight lines due to the exponential relaxation.

204

8.4 Results and discussions

function, which show straight lines that is a signature of exponential relaxation. It is clear from the figures 8.4 and 8.9 that the number density fluctuation in the system makes the energy relaxation nearly exponential irrespective of the nature and range of the perturbing potential. The equilibrium energy fluctuation relaxation function of a 1 0.9 0.8 0.7

S(t)

0.6

1 0.8 0.6 0.4

0.5 0.4 0.3 0.2

0.2 0 0

1

2

3

4

5

5

x 10

0.1 0 0

2

4

6

time

8

10 5

x 10

Figure 8.10: The energy fluctuation relaxation function is plotted for system in equilibrium under the perturbation of the Lennard-Jones potential for different system sizes (NL =50 (solid line), NL =100 (dash-dot line), NL =150 (dotted line), and NL =200 (dashed line)). The inset shows the fits of the energy relaxation function for NL =200 using functions f1 (t) (dotted line) and f2 (t) (dash-dot line).

system under the Lennard-Jones perturbation, is plotted in the figure 8.10 for different system sizes. The energy fluctuation in this system is mostly from the region where potential is weak. The fit of the energy fluctuation relaxation with functions f1 (t) and f2 (t) are shown in the inset of the figure 8.10. The fitting parameters for the function f1 (t) are b1 = 0.42, τ1 = 8.8 × 103 , τ2 = 1.6 × 105 , β = 0.73 and for the f2 (t) are c1 = 0.14 a = 5.6 × 101 c2 = 0.58 and γ = 1.3 × 103 . Finally, note that the stretched exponential fit of relaxation function is more appropriate in this case due to the short range of the potential. It is found that the logarithmic time dependence for the energy relaxation is not a good representation of the relaxation function in this case. 205

8. DENSITY AND ENERGY RELAXATION IN AN OPEN ONE-DIMENSIONAL SYSTEM

8.5

Conclusion

The work in this chapter has demonstrated that the cooperative dynamics of random walkers in a simple one-dimensional channel can give rise to a highly non-exponential relaxation when number density fluctuations are not allowed. In these simulations, two perturbing potentials (the Coulomb potential and the Lennard-Jones potential) have been used to study the effects of perturbing potential on the relaxation process. The energy relaxation under these potentials can be approximately described by a stretched exponential function (in general) in a closed system. The variation in the time scale of relaxation under these two well-known potentials can be understood in terms of the difference in the range of these potentials. The Coulomb potential, being long ranged (in comparison with the Lennard-Jones interaction), shows comparatively strong nonexponentiality in the energy relaxation. Under the Coulomb potential, the exponent β of the non-equilibrium energy relaxation function is 0.52 while under Lennard-Jones potential it is 0.73. The simulations seem to agree with the results of Bruns et al.[3] in showing a logarithmic time dependence of the energy relaxation function under the Coulomb potential. However, under the short-ranged Lennard Jones potential, the energy relaxation function does not show logarithmic time dependence. In the smaller sized systems, the relaxation is found to be faster and as the system size increases, relaxation slows down as expected. The number density fluctuations in this one-dimensional channel found to make the relaxation function decay faster and exponentially. When number density fluctuation is allowed, random walkers overcome the resistance of the hard rod interactions by moving in and out of the linear lattice such that the random walkers experience no major hindrance to their flow. In this case, the system behaves like a system of weakly interacting particles without the need for strong cooperativity for relaxation. It is worth noting that interactions found in nature are numerous, but the models of asymptotic time dependence of relaxation commonly found in nature are limited in number. However, the parameters of the relaxation function depend on the form of the biasing potential and the nature of interaction between the carriers. 206

Bibliography [1] I. Oppenheim, K. E. Schuler, and G. H. Weiss, Stochastic processses in chemical physics: The Master equation (MIT Press, Cambridge, 1977). 8.1 [2] N. G. van Kampen, Stochastic processses in physics and chemistry (North Holland, Amsterdam, 2001). 8.1 [3] E. B. Bruns et al., Phys. Rev. Lett. 88, 158101 (2002). 8.1, 8.4, 8.5 [4] S. K. Pal, L. Zhao, T. Xia, and A. H. Zewail, Proc. Natl. Acad. Sci. (U.S.A) 100, 13746 (2003). 8.1 [5] D. Andreatta et al., J. Am. Chem. Soc. 127, 7270 (2005). 8.1 [6] S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). 8.1 [7] Non-equilibrium statistical mechanics in one dimension, edited by V. Privman (Cambridge University Press, Cambridge, 1997). 8.1 [8] S. Alexander, J. Bernasconi, W. R. Schneider, and R. Orbach, Rev. Mod. Phys. 53, 175 (1981). 8.1 [9] V. F. E. Barkai and J. Klafter, Phys. Rev. E 61, 1641 (2000). 8.1 [10] D. L. Huber, Phys. Rev. B 15, 533 (1977). 8.1 [11] P. M. Richards, Phys. Rev. B 16, 1393 (1977). 8.1 [12] R. L. R. P. M. Richards, Phys. Rev. B 21, 3740 (1980). 8.1 [13] K. W. Yu and P. M. Hui, Phys. Rev. A 33, 2745 (1986). 8.1 207

BIBLIOGRAPHY

[14] R. Pitis, Phys. Rev. B 48, 4196 (1993). 8.1 [15] G. Zumofen and J. Klafter, Phys. Rev. E 51, 2805 (1995). 8.1 [16] A. Bar-Haim and J. Klafter, J. Chem. Phys. 109, 5187 (1998). 8.1 [17] C. Rodenbeck, J. Karger, and K. Hahn, Phys. Rev. E 55, 5697 (1997). 8.1 [18] P. H. Nelson and S. M. Aurbach, J. Chem. Phys. 105, 9235 (1999). 8.1 [19] M. S.Okino et al., J. Chem. Phys. 111, 2210 (1999). 8.1 [20] B. S. D. Frenkel, Understanding molecular simulation: From algorithms to applications (Academic Press, San Diego, 1996). 8.2, 8.2, 8.2 [21] R. Klohlrausch, Ann. Phys. (Leipzig) 12, 393 (1847). 8.4 [22] G. Williams and D. C. Watts, Trans. Faraday. Soc. 66, 80 (1970). 8.4 [23] W. Gotze, in Liquid Freezing and Glass Transition, edited by J. P. H. D. Levesque and J. Zinn-Justin (North Holland, Amsterdam, 1990). 8.4 [24] W. Gotze and L. Sjogren, Rep. Prog. Phys. 55, 241 (1992). 8.4 [25] M. O. Vlad and M. C. Mackey, J. Math. Phys. 36, 1834 (1995). 8.4 [26] M. W. Cohen and G. S. Grest, Phys. Rev. B 24, 4091 (1981). 8.4 [27] D. L. Huber, D. S. Hamilton, and B. Barnett, Phys. Rev. B 16, 4642 (1977). 8.4 [28] D. L. Huber, Phys. Rev. B 31, 6070 (1985). 8.4 [29] D. L. Huber, Phys. Rev. B 53, 6544 (1996). 8.4

208

Part III DYNAMICS IN OPTICAL TRAP

209

Chapter 9 Formation of nanoclusters Under Radiation Pressure: A Brownian Dynamics Simulation Study 9.1

Introduction

When radiation is scattered by a medium, a part of its momentum is transferred to target particles. This is purely a mechanical force, which comes into effect when radiation is not coherently interacting. This force is known in literature as radiation pressure. Recent experimental studies have demonstrated the feasibility of using radiation pressure of a laser beam as a tool for cluster formation in solution. In this chapter we describe the Brownian dynamics simulation of solute molecules under the perturbation induced by laser radiation. Here the force field generated by a laser beam in the fundamental mode is modeled as that of a two dimensional harmonic oscillator. The radial distribution function of the perturbed system gives indication of high inhomogeneities in the solute distribution. An explicit analysis of the nature of these clusters is carried out by calculating the density-density correlation functions in the plane perpendicular to beam direction g(r xy ) and along the direction of beam g(z); they give an average picture of shell structure formation in the different directions. The relaxation time of the first shell structure calculated from the van Hove correlation function is found to be relatively large in the perturbed solution. This is the signature of formation of stable nano-clusters in the presence of the radiation field. This study on the dynamics 211

9. FORMATION OF NANOCLUSTERS UNDER RADIATION PRESSURE: A BROWNIAN DYNAMICS SIMULATION STUDY

of solute molecules during the cluster formation and dissolution gives the duration of collective relaxation from away from the equilibrium to an equilibrium distribution. This relaxation time is found to be large for a perturbed solution. The study of cluster formation in the solution is a subject of great current interest [1, 2]. Often the route used is a chemical reaction, followed by aggregation [3]. Recently, however, an interesting technique has been developed where radiation pressure is used to selectively bring together specific particles and form a cluster. The merit of this technique is that clusters of desired shape can be formed, which gives this technique a special advantage. This work presents a Brownian dynamics simulation of such cluster formation in a uniform solution. This study reveals microscopic aspects of such cluster formation. Interaction of electromagnetic radiation with atoms and molecules via absorption and emission gives insight into the structure and dynamics of inner degrees of freedom. If the incoming radiation is not resonantly interacting, this interaction gives rise to a pure mechanical pressure which could be independent of the internal structure of atoms and molecules. The radiation pressure is the mechanical pressure exerted by radiation due to the partial transfer of the momentum of radiation while it is reflected or refracted. Alternatively, radiation pressure can be defined microscopically as the force experienced by a particle, when the dipole induced by external field on the particle tries to minimize its energy by re-positioning. In 1909, Debye carried out a complete study of radiation pressure on spherical particles of arbitrary size and optical constants [4, 5]. A general derivation of the magnitude of force generated by the radiation can be done by the principle of conservation of momentum of incident radiation [5, 6]. There are two kinds of forces arising from the scattering of radiation. One is along the direction of propagation of radiation and the second one gives the transverse component, which is due to the scattering of a beam of radiation with an asymmetric distribution of intensity by the target particle. This creates a net force on the particle due to the difference in the momentum transferred at different parts of the target particle. The first experimental evidence of radiation pressure using experiments after the advent of 212

9.2 Modeling of the radiation pressure

the lasers can be found in the celebrated work of Ashkin, where he reported acceleration of single nano-particle by radiation pressure. His successive works confirmed the effect of radiation pressure on huge biomolecules, latex particles etc. [7, 8]. Recently Masuhara et al. [9–17] have carried out extensive experimental work on nano-particle formation in the solution by a radiation field. They have also reported an analysis of the potential arising from the radiation, which is found to be, harmonic in nature. These experiments look into the classical many-body problem of cluster formation under an inhomogeneous force field. Another interesting aspect of the problem is the dynamics of formation of such clusters under the radiation field and dissolution of them when radiation is switched off. This work pursues a different aspect of the problem, namely the statistical behavior of a collection of interacting molecules under the stable external force field created by a laser beam. The organization of the rest of the chapter is as follows. Section 9.2 gives the details of modeling of radiation pressure as a harmonic potential. The details of the Brownian dynamics simulation are given in section 9.3. Results obtained from the simulation are described in the section 9.4 Some concluding remarks are presented in the section 9.5.

9.2

Modeling of the radiation pressure

Here the interest is particularly in the incoherent interaction of the molecules with radiation. For a coherent beam of radiation having Gaussian intensity distribution in the plane of cross-section, the force field experienced by a molecule can be modeled analogous to that of a two dimensional harmonic oscillator with force constant k. When the electromagnetic radiation interacts with a dipole, it experiences a Lorentz force [18] p ~ ~ E ~ + 1 ∂~ × E, F~ = (~ p.∇) c ∂t

(9.1)

~ is the electric field in the plane of beam cross-section and p~ is the dipole where E ~ E ~ = 1 ∇E ~ 2− moment. This expression can be rearranged using the vector identity E.∇ 2

~ × (∇ ~ × E) ~ and p~ = αE, ~ where α is assumed to be the positive polarizability [18] E ! ~ × B) ~ ∂( E 1 1 ~ 2+ ∇E . (9.2) F~ = α 2 c ∂t 213

9. FORMATION OF NANOCLUSTERS UNDER RADIATION PRESSURE: A BROWNIAN DYNAMICS SIMULATION STUDY

Due to the heavy mass of the scatterer the force in the direction of beam can be neglected; hence the total force can be approximated by the first term of equation 9.2 1~ 2 F~ ' α ∇E . 2

(9.3)

It is evident from this equation that the force due to radiation depends on the gradient of the magnitude of electric field in the transverse direction of the beam. Here we are assuming that the laser beam is in the T EM00 mode [19]. Hence the intensity distribution function is a Gaussian in the x − y plane with the direction of the beam propagation is along the z direction. The intensity of the beam as experienced by the j th molecule is uj = u0 exp(

2 −2~r0j ), w02

(9.4)

2 where r0j = x2j + yj2 , r0j is the displacement of j th molecule from the radiation axis in

the x-y plane, w0 is the spot size which is much larger than r0j and u0 is the intensity at the center of the beam. The magnitude of electric field of this Gaussian beam is −~r2

described by E = E0 exp( w00j2 ). The force experienced on j th molecule by this stable potential well created by this laser beam can be derived by substituting this relation in to Eq: 9.3

2

2 −2~r0j 2~r0j F~j (r) = −αE0 2 2 exp( ). w0 w0

Since w0 >> |~r0j |, the Gaussian can be expanded in powers of 2 2~r0j 2~r0j F~j (r) = −αE0 2 2 (1 − ....), w0 w0 2

(9.5) 2~r0j 2 , w0 2

(9.6)

within the 00 th order approximation it can be written as ~r0j F~j (r) ' −2αE0 2 2 , w0

(9.7)

F~j (r) ' −k~r0j ,

(9.8)

or,

where k=

2αE0 2 . w0 2 214

(9.9)

9.3 System and Simulation Details

Therefore, the force due to the radiation pressure of a laser beam in the fundamental mode can be approximated by a harmonic potential. The magnitude of the electric field can be calculated from the power of the laser [19].

9.3

System and Simulation Details

In the model simulated here, the solute molecules move in the solution under the force field generated by the sum of mutual interactions and the external field generated by the radiation pressure. Therefore, the motion of the solvent is analogous to that of a Brownian particle which moves under the random forces in a potential well. Hence Brownian dynamics simulation [20, 21] is the suitable method for tracing the dynamics of solute molecules in the solution. In the Brownian dynamics simulation, the solute molecules are selectively simulated as moving under the friction or resistance generated by solvent molecules. In this simulation the equation of advancements of the position is obtained by integrating the single particle Langevin equation; hence all the hydrodynamic effects are neglected. Simulation is confined to the parallel rays of the laser beam since the difference in the angle between any two rays falling on different part of a single particle is negligible. In addition, the entire box of simulation occupies a small volume in comparison with the region of cluster formation in the experimental arrangements used [9–17]; hence the approximation used here is justified. Other details of the simulation are as follows. A system of 500 molecules is selected for simulation. The simulation is carried out inside a cubical box (the schematic diagram of the simulation box is given in the figure 9.1), which is placed in the positive quadrant of the coordinate system. Hence the position coordinates along x, y and z axes vary from zero to l, where l is the length of the simulation box. In this simulation, the box-length is computed from the density of solute molecules and it is approximately 14σ (σ is the molecular diameter) in this simulation. Intermolecular interaction is modeled through Lennard-Jonnes potential with a cut-off at rij = 2.5σ. 215

9. FORMATION OF NANOCLUSTERS UNDER RADIATION PRESSURE: A BROWNIAN DYNAMICS SIMULATION STUDY

Axis of Radiation

a

14 a 14 Z

Y

0

0 0

14

X

Figure 9.1: The schematic diagram of the simulation box is shown in the figure. The z disc and the cylinder used for calculating g(r xy ) and g(rij ) are schematically drawn here. The thickness of the disc and the diameter of the cylinder are equal (a = σ).

The inter molecular interaction is given by "   6 # 12 σ σ vij = 4 . − rij rij

(9.10)

Hence the total potential energy Vi on ith molecule due to the inter-molecular interaction is given by Vi =

X

vij .

(9.11)

j

Here radiation is passing symmetrically through the center of the x − y plane of the simulation box and its direction of propagation is along the z axis. Therefore, the Gaussian intensity distribution of radiation is situated symmetrically around the center 216

9.3 System and Simulation Details

of the x−y plane. The radiation force from this Gaussian beam is modeled as generated by a two dimensional harmonic potential and it can be written as 1 2 Vjrad = kr0j . 2

(9.12)

The total potential energy is given by Vtotal =

X

Vj + Vjrad .

(9.13)

j

By integrating the single particle Langevin equation, time evolution of the position is obtained as ~r(t + 4t) = ~r(t) +

D f~4t ~ + R(4t), kB T

(9.14)

where f~ is the total deterministic force, D is the diffusion constant, and kB is the Boltzmann constant. The random displacements coming from the solvent molecules √ are sampled from a Gaussian distribution whose mean is zero and variance is 2Dt in all the three directions (x, y and z). The force constant k is expressed as a dimensionless quantity as k ∗ = kσ 2 / and its value used in this simulation is 80.0. All other quantities used in the simulation are in reduced units given in terms of the Lennard-Jones parameters. They are density ρ∗ = ρσ 3 , temperature T ∗ = kB T /, distance r ∗ = r/σ, and p time t∗ = /mσ 2 t. The simulation is performed at a high concentration of the solute

molecules at ρ∗ = 0.2 and at T ∗ = 1.2. Here this external potential has the symmetry

of a cylinder. That is, all the molecules situated on the curved surface of a cylinder, which is centered at the radiation axis, will experience a force equal in magnitude toward the center. Hence the bins that are used for recording the density are concentric cylindrical shells centered at the radiation axis. These shells are constructed such that all of them have equal volume. The simulation starts from an FCC configuration and it is equilibrated with 20000 simulation steps. Periodic boundary condition is used in all the three directions. The equation of motion is integrated with the time step t∗ =0.005. Positions of solute molecules are recorded (without perturbation) in the next 105 steps. From positions, the radial distribution function (g(r)) is calculated. g(r) is defined in 217

9. FORMATION OF NANOCLUSTERS UNDER RADIATION PRESSURE: A BROWNIAN DYNAMICS SIMULATION STUDY

terms of delta function of positions as [22] + * N X X 1 δ(~r + ~rj − ~ri ) . g(~r) = Nρ i6=j

(9.15)

g(~r) = g(r) in a homogeneous system. The conventional radial distribution function gives the distribution of the molecules in a homogeneous solution. Since the application of radiation pressure creates a strong inhomogeneity in the solution, here the conventional g(r) may be poor in representing the distribution of molecules on x − y plane and in the z direction explicitly. To overcome this difficulty, we have calculated density-density correlation function in the x − y plane which can be defined in terms of delta function as 1 g(~rxy ) = Nρ

*

N XX i6=j

+ δ(~rxy + ~rjxy − ~rixy )

,

(9.16)

|∆~ z |<0.5σ

where ~rxy = ~r − ~z (or ~rxy is the projection of displacement in the x − y plane and ∆~z is the projection of displacement between two molecules along the z axis). In the homogeneous solution, this distribution function is related to the g(r) by the relation lim|∆~z|→0 g(r xy ) = g(r) since all the directions are equivalent in a homogeneous solution. Similarly, a distribution function along the z axis can be defined as * + N X X 1 , g(~z) = δ(~z + ~zj − ~zi ) xy Nρ i6=j

|∆~r

(9.17)

|<0.5σ

where ~z is the projection of displacement along the z axis and ∆~r xy is the projection of displacement between two molecules in the x − y plane. Similar to the previous case, g(z) is also related to the conventional g(r) by the relation lim|∆~rxy |→0 g(z) = g(r). For the calculation of the g(~rxy ) and g(~z), we have chosen the limits |∆~z| < 0.5σ and |∆~rxy | < 0.5σ respectively, rather than the |∆~z| → 0 and |∆~rxy | → 0 for better convergence. For calculating the correlation function g(r xy ) on the x − y plane, we have considered a disc (shown schematically in figure 9.1) which lies in the x − y plane around each molecule, whose axis is along the z direction and having a thickness of one molecular diameter. The distribution of molecules whose center of mass lies in this disc gives the planar distribution function around each molecule in the x − y plane. 218

9.4 Results and Discussion

Similarly, for calculating the linear distribution function g(z), we have considered a cylinder whose diameter is that of one molecule with reference molecule inside this cylinder (shown schematically in figure 9.1). The density-density distribution function of molecules whose center of mass lies in this cylinder is calculated. These steps are repeated with perturbation to get the corresponding correlation functions. The distinct part of van Hove correlation function is defined as [22] 1 Gd (~r, t) = N

*

N XX i6=j

δ(~r + ~rj (0) − ~ri (t))

+

(9.18)

which gives the information about average lifetime of shell structure around each molecule. The van Hove correlation functions in the unperturbed and the perturbed states are calculated for comparing the lifetime of shell structures in these states. Since the perturbation is introduced by a position dependent external field, the two-body correlation function is not a good representation of spatial distribution of molecules. Hence the density distribution of molecules in perturbed state is calculated from the stored positions, which gives the spatial distribution of density. For monitoring the variation of density with time while the transformation takes place from the non-equilibrium to an equilibrium state immediately after the release of external perturbation, simulation starts from an equilibrated configuration with perturbation that is preserved till the end of the simulation. The simulation is performed for 10000 steps without perturbation and using the preserved positions as starting configuration. During this time, the density is monitored in the interval of 10 consecutive steps. Next initial configuration is generated by running the simulation for 1000 steps (in the initial equilibrated configuration) starting from the preserved positions. These simulation steps are repeated over 100 runs and the output data are averaged to obtain good statistics and to smooth the time evolution curve of density. In a similar way, transformation from unperturbed to perturbed state is also recorded. 219

9. FORMATION OF NANOCLUSTERS UNDER RADIATION PRESSURE: A BROWNIAN DYNAMICS SIMULATION STUDY 6

5

g(r)

4

3

2

1

0 0

1

2

3

r*

4

5

6

7

Figure 9.2: The radial distribution functions of the solute molecules in the presence (solid line) and absence (dashed line) of the radiation pressure are given. Note that the solid line goes below 1.0 indicating the inhomogeneity in the distribution.

9.4

Results and Discussion

Figure 9.2 shows the g(r) of 500 solute molecules. The dashed curve represents the g(r) without any radiation and the solid curve gives the g(r) in the presence of radiation pressure (RP). Change in the radial distribution function in the figure 9.2 follows from the clustering of the solute molecules near the radiation axis. The g(r) with perturbation shows a very high peak which is much higher than that of an ordinary solution. Since in the solution the solute molecules are in low density, the second peak of the g(r) is not much visible in the absence of RP. However, with the RP, the second peak becomes visible giving clear indication of the formation of second shell structure. Note that between the first and the second peak the density is much higher than the average density, which shows the random structure of the clusters. The g(r xy ) and the g(z) are separately plotted in figure 9.3. In the absence of the perturbation, g(r xy ) and g(z) shows similarity that allows both curves to coincide. The dotted (g(r xy )) and the bold ((g(z)) lines in the figure 9.3 coincide almost one over the other, which 220

9.4 Results and Discussion 6

5

g(z) & g(r xy)

4

3

2

1

0 0

1

2

3

r*

4

5

6

Figure 9.3: The radial distribution functions g(r xy ) and the g(z) are plotted to find the average density-density distribution in the x − y plane and in the z direction, respectively. The dotted line represents g(r xy ) without perturbation. The dotted line almost coincides with the bold line which is g(z) without RP. When perturbation is applied, the change in the g(r xy ) is shown in dot-dash line. The corresponding change in the g(z) is shown as dashed line. Note that the g(r xy ) under perturbation goes below 1 at shorter distances indicating the formation of inhomogeneity in the x − y plane. The shell structure formation in the z direction under perturbation is indicated by the second peak in the dashed curve.

is a signature of a homogeneous solution. Under the influence of the inhomogeneous perturbation due to the RP, the first peaks of the g(r xy ) (dash-dot curve in figure 9.3) and the g(z) (dash curve in figure 9.3) rise to very high values and their second peaks are also become visible. This indicates explicit shell structure formation in the x − y plane and in the z direction, but their structures considerably differ from each other. The structural difference can be inferred from the heights of the first peak and the second peak positions of the g(z) and the g(r xy ) with and without the RP. The g(r xy ) with RP goes below one at a distance of about 5 - this indicates high inhomogeneity in the distribution of molecules in the x − y plane. The g(z) behaves similar to that of unperturbed case, but the peaks become more visible and this function decays to g(z) = 1.6 instead of g(z) = 1 at large distances due to the formation of a high density 221

9. FORMATION OF NANOCLUSTERS UNDER RADIATION PRESSURE: A BROWNIAN DYNAMICS SIMULATION STUDY

3.5

3

Relative Density− ρ(r0)/ρ(0)

2.5

2

1.5

1

0.5

0

0

1

2

3

4

* 0

5

6

r

Figure 9.4: Variation in the relative density plotted from the imaginary radiation axis (along the z-axis at the center of the x-y plane) to the sides of the simulation box in a perturbed solution. In the presence of RP the density shows inhomogeneity.

14

12

10

y/σ

8

6

4

2

0

0

2

4

6

x/σ

8

10

12

14

Figure 9.5: A snapshot of molecules in the simulation box projected onto the x − y plane with radiation pressure. Circles have approximately the same size as that of the molecules.

222

9.4 Results and Discussion

14

12

10

y/σ

8

6

4

2

0

0

2

4

6

x/σ

8

10

12

14

Figure 9.6: A snapshots of molecules in the simulation box projected onto the x − y plane without radiation pressure.

region along the radiation axis. This fact is also supported by density distribution from the radiation axis radially outwards which is shown in figure 9.4. Figure 9.4 gives the arrangement of clusters in the radiation field. The inhomogeneity in the force field is reflected in the rearrangement of density in the perturbed state. These clusters are visible in the snapshot of the molecules in the simulation box. Figure 9.5 gives the snapshot after equilibration with perturbation. Here the clusters are in the center of the simulation box in comparison with the clusters in figure 9.6 which is the snapshot taken without perturbation. Figure 9.7 gives the van Hove correlation function in an inhomogeneous solution; this gives the information about the collective dynamics of the molecules in equilibrium in the presence of RP. In figure 9.7 Gd (r, t) is plotted from time 0.5 t∗ in the successive intervals of 0.5 t∗ . This shows how fast the local shell structure of liquid relaxes in the presence of the RP. Figure 9.8 gives the corresponding van Hove correlation function in the absence of RP. These curves are plotted from time 0.05 t∗ in the successive intervals of 0.05 t∗ . The time interval used in the figure 9.7 is ten times larger than that used in the figure 9.8. Hence the shell structure breaks down slowly in the perturbed state, 223

9. FORMATION OF NANOCLUSTERS UNDER RADIATION PRESSURE: A BROWNIAN DYNAMICS SIMULATION STUDY

0.7

0.6

0.5

Gd(r,t)

0.4

0.3

0.2

0.1

0

0

1

2

3

*

4

5

6

r

Figure 9.7: Distinct part of the van-Hove correlation function is calculated and plotted against the position in the perturbed solution. Here the successive curves starting near r = 0 from bottom to top are separated by the time interval of 0.5 t∗ .

0.35

0.3

0.25

d

G (r,t)

0.2

0.15

0.1

0.05

0

0

1

2

3

r*

4

5

6

Figure 9.8: Distinct part of the van-Hove correlation function is calculated and plotted against the position in the unperturbed solution. Here the successive curves starting near r = 0 from bottom to top are separated by time the interval of 0.05 t∗ .

224

9.4 Results and Discussion

1.2

1

0.8

C(t)

0.6

0.4

0.2

0

−0.2

0

0.5

1

1.5

2

t*

2.5

3

3.5

4

4.5

Figure 9.9: The variation in time of the shell structures is plotted in perturbed and unperturbed systems. Here C(t) is plotted against time at r value equal to the peak position of the first shell. The circles give the C(t) without RP and stars give C(t) with RP. The shell structure breaks down slowly in the presence of RP, which signifies the existence of relatively stable clusters than those formed in the unperturbed solution.

which essentially means the stability of the clusters in the presence of the RP. Figure 9.9 gives the plot of C(t), which is defined as C(t) =

Gd (r, t) − Gd (r, t = ∞) , Gd (r, t = 0) − Gd (r, t = ∞)

(9.19)

against time at r equal to the peak position of first shell. The stars in the figure 9.9 show C(t) in the perturbed state and the circles show the C(t) in the unperturbed state. This plot gives a quantitative comparison between the life times of clusters in perturbed and unperturbed solution; the relaxation time with RP is approximately ten times greater than that of without RP. The plot of density variation with time gives a clear picture of the dynamics of density fluctuations and the relaxation of solute molecules. The density(relative) variation with time at different points in the simulation box immediately after the application of the RP is plotted in figure 9.10. Density fluctuations at r0∗ = 0.0, 1.7, 2.5, 3.0, 3.5, 4.0, 4.3, 4.6, 5.0, 5.3, 5.5, 5.8, 6.0, 6.3, 6.6 away from the radiation axis in radial direction (of cylinder representing the 225

9. FORMATION OF NANOCLUSTERS UNDER RADIATION PRESSURE: A BROWNIAN DYNAMICS SIMULATION STUDY

4.5

4

Relative Density −ρ(r0,t)/ρ(0)

3.5

3

2.5

2

1.5

1

0.5

0

0

2

4

6

8

10 *

12

14

16

18

20

t

Figure 9.10: Density variation at the center of the radiation axis with time after RP is applied. Different lines are plotted for representing the density variation at the different positions. This set of lines give the density variation with time from the radiation axis at distances r0∗ = 0.0, 1.7, 2.5, 3.0, 3.5, 4.0, 4.3, 4.6, 5.0, 5.3, 5.5, 5.8, 6.0, 6.3, 6.6 (successively from top to bottom near t∗ = 2) away from the radiation axis.

beam) show similar relaxation and their relaxation times also do not appreciably differ. It is now interesting to compare the relaxation of the solute molecules in perturbed state with that in the unperturbed state. Figure 9.11 shows the corresponding density variation in the system after the RP is removed. Here, due to the absence of RP, the solution is in a lower pressure than it is in the previous case. The density of solute is found to have shorter relaxation time in this case. Here also density fluctuations at distances r0∗ = 0.0, 2.5, 3.5, 4.3, 5.0, 5.5, 6.0, 6.5 are plotted; successive lines show similarity in relaxation.

9.5

Conclusion

In conclusion, we have demonstrated the formation of nano-clusters under, the radiation pressure. These clusters are found to be more stable than that formed in an unperturbed solution. We have found that even though the force field resulting from 226

9.5 Conclusion

3

2

0

Relative Density−ρ(r ,t)/ρ(0)

2.5

1.5

1

0.5

0

0.5

1

1.5

2

2.5 *

3

3.5

4

4.5

5

t

Figure 9.11: Density variation at the center of the radiation axis with time after RP is removed. This set of lines give density variation with time from the radiation axis at distances r ∗ = 0.0, 2.5, 3.5, 4.3, 5.0, 5.5, 6.0 and 6.5 (successively from top to bottom at t∗ = 0) away from the radiation axis.

the RP is confined to x − y plane, it modifies the shell structure in the z direction. The g(z) is still found to be homogeneous but a strong inhomogeneity exists in the g(r xy ). The van Hove correlation calculated in the unperturbed state relaxes approximately ten times faster in the unperturbed state - this substantiates the stability of clusters in the perturbed state. Moreover, we could observe the rich dynamical behavior during the formation and the dissolution of clusters, and could observe the variation in the relaxation time in the perturbed and the unperturbed states. It is interesting to note the oscillations in local density when the electro-magnetic field is turned on or off. These oscillations are manifestations of the viscoelasticity of the liquid. If the linear response is valid, then one could possibly describe these oscillations by using the dynamic structure factor of the liquid [22]. Such a calculation is non-trivial because the system becomes inhomogeneous in the presence of the position dependence of the field. Thus, one would require to use the density functional theory in real (that is, position) space. Such a calculation is computationally intensive, but 227

9. FORMATION OF NANOCLUSTERS UNDER RADIATION PRESSURE: A BROWNIAN DYNAMICS SIMULATION STUDY

may be worthwhile to perform.

228

Bibliography [1] M. Lewerenz, B. Schilling, and J. P. Toennies, J. Chem. Phys. 102, 8191 (1995). 9.1 [2] H. Tanaka, J. Chem. Phys. 107, 3734 (1997). 9.1 [3] S. Jayanetti et al., J. Chem. Phys. 115, 954 (2001). 9.1 [4] P. Debye, Ann. Physik (4) 30, 57 (1909). 9.1 [5] M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, New York and London, 1969). 9.1 [6] J. P. Gordon, Phys. Rev. A 8, 14 (1973). 9.1 [7] A. Ashkin, Phys. Rev. Lett. 24, 156 (1970). 9.1 [8] A. Ashkin and J. M. Dziedzic, Appl. Phys. Lett. 19, 283 (1971). 9.1 [9] J. Hotta, K. Sasaki, and H. Masuhara, J. Am. Chem. Soc. 118, 11968 (1996). 9.1, 9.3 [10] . Hoffkens et al., Mol. Cryst. Liq. Cryst. 283, 165 (1996). 9.1, 9.3 [11] J. Hoffkens et al., Langumir 13, 414 (1997). 9.1, 9.3 [12] J. Hoffkens et al., J. Am. Chem. Soc. 119, 2741 (1997). 9.1, 9.3 [13] K. Sasaki, M. Tsukima, and H. Masuhara, Appl. Phys. Lett. 71, 37 (1997). 9.1, 9.3 [14] J. Won et al., Appl. Phys. Lett. 75, 1506 (1997). 9.1, 9.3 229

BIBLIOGRAPHY

[15] P. Borowicz, J. Hotta, K. Sasaki, and H. Masuhara, J. Phys. Chem. B 102, 1896 (1998). 9.1, 9.3 [16] J. Hotta, K. Sasaki, H. Masuhara, and Y. Morishima, J. Phys. Chem. B 102, 7687 (1998). 9.1, 9.3 [17] T. A. Smith et al., J. Phys. Chem. B 103, 1660 (1999). 9.1, 9.3 [18] Y. R. Shen, The Principles of Nonlinear Optics (John Wiley and Sons, New York, 1984). 9.2, 9.2 [19] A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge University Press, Cambridge, 1991). 9.2, 9.2 [20] M. P. Allen and D. J. Tildesley, Computer simulation of liquids (Clarendon Press, Oxford, 1987). 9.3 [21] D. L. Ermak and J. A. McCammon, J. Chem. Phys. 69, 1352 (1978). 9.3 [22] J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1986). 9.3, 9.3, 9.5

230