APPLIED PHYSICS LETTERS

VOLUME 84, NUMBER 22

31 MAY 2004

Jamming phase diagram of colloidal dispersions by molecular dynamics simulations Anil Kumar and Jianzhong Wua) Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521-0425

共Received 4 February 2004; accepted 13 April 2004; published online 14 May 2004兲 We report a three-dimensional jamming phase diagram of a model colloidal system obtained from molecular dynamics simulations where the inter-colloidal forces are represented by the Derjaguin– Landau–Verwey–Overbeek potential. The jamming threshold is uniquely defined in terms of the critical volume fraction, the critical temperature, and the critical yield stress. The simulation results indicate that near the jamming transition the shear viscosity diverges following a critical-like scaling law as observed for realistic colloids. These results offer a convincing proof for unifying different nonequilibrium transitions in colloids under the concepts of jamming. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1759767兴

Under a wide variety of conditions, colloids and other soft-matter systems solidify without the formation of a crystalline order for reasons that appear quite unrelated to each other.1 Phrases such as gelation, coagulation, kinetic arrest, dynamic slowdown, and ergodic-nonergodic transition have been used to loosely describe the solidification phenomena. The resulted long-lived, but nonequilibrium fragile materials are of profound practical importance, being relevant to paint, pharmaceutical, food, and numerous other industries.2 The formation of amorphous materials rather than crystalline solids is sometimes undesirable, especially during the crystallization of the proteins in solution3,4 or in the fabrication of ordered nano-structures by self-assembly.5,6 However, prediction of the experimental conditions leading to desired ordered/disordered structures has been very difficult due to the lack of a clear understanding of the control parameters. A unifying framework to describe the phase behavior of various nonequilibrium states could be of immense importance. The off-equilibrium solidifications in colloids are characterized by the sudden arrest of the dynamics of the constituent particles.7–10 In addition, all these transitions show kinetic heterogeneities near the onset of solidification where the particle mobilities become heterogeneous in space and intermittent in time.11–13 These general observations led Liu and Nagel to conjecture that different transitions due to the kinetic arrest can be unified by a jamming phase diagram14 controlled by the thermodynamic variables that dictate the jammed state. For a one-component system, the three axes of the jamming phase diagram are density ␳, temperature T, and the applied load ␴, as the three axes of the jamming phase diagram. Indeed Trappe et al. verified the unified framework of jamming transitions experimentally in attractive colloidal systems like carbon black, poly共methylmethacrylate兲 and polystyrene.15,16 They found that the shear viscosity, elastic modulus, and yield stress follow a similar critical behavior as the jamming transition is approached. These observations enabled them to construct a jamming a兲

Author to whom correspondence should be addressed; electronic mail: [email protected]

phase diagram for the aforementioned attractive colloidal systems, supporting the concept of jamming. However, the experimental phase diagram differs significantly from that originally proposed by Liu and Nagel,14 having opposite curvatures everywhere. Very recently, Berthier and Barrat17 reported the ( ␴ ,T) plane of the jamming phase diagram from molecular dynamics simulations of binary Lennard-Jones mixture in a simple shear flow, which is very similar to that proposed by Liu and Nagel.14 But there have been no simulation studies reported on the full three-dimensional jamming phase diagram using conventional colloidal potentials. In this letter, we report a three-dimensional jamming phase diagram of a model colloidal system using molecular dynamics simulations. In contrast to most previous simulations reported in the literature that employ simple LennardJones-type potentials, we use a standard colloidal potential represented by Derjaguin–Landau–Verwey–Overbeek theory to describe the colloidal forces 6 V 共 r i j 兲 ⫽A/r 36 i j ⫹B exp共 ⫺ ␬ r i j 兲 /r i j ⫺C/r i j ,

共1兲

where r i j is the inter-particle separation. The parameters A⫽0.881, B⫽14.7624, C⫽3.6932, and ␬⫽2.0, all in appropriate reduced units, are chosen such that the potential mimics that between polystyrene particles as investigated by Trappe et al.15,16 But this simplified model system is different from the real colloidal systems in the following aspects. First, the attractive van der Waals interaction in our system is modeled by the (1/r 6 ) potential, but in real colloidal systems the attraction is much shorter ranged and sharply diminishing. Second, here the repulsive interaction is not essential to maintain the stability of the model system. In other words, the van der Waals attraction alone as shown in Eq. 共1兲 is insufficient to cause colloidal aggregations. Because of these differences, a quantitative agreement between simulation and experiments is not anticipated. However, we expect that the jamming transitions in the model system should be qualitatively similar to those corresponding to experimental observations. Different from the potential models used in other simulation studies, the potential given by Eq. 共1兲 has a longranged repulsion which suppresses a complete phase separa-

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Appl. Phys. Lett., Vol. 84, No. 22, 31 May 2004

FIG. 1. 共a兲 Shear viscosity ␩ vs inverse temperature at different colloidal volume fractions, ␾⫽0.1共䊉兲, 0.15共䊏兲, 0.2共䉱兲, 0.3共⽧兲, 0.4共⫹兲. The solid lines are the fits to the power law, ␩ ⫽ ␩ s (1/T c ⫺1/T) ⫺ ␷ T , with ␷ T ⬇0.155. 共b兲 T c vs 1/␾ c at zero applied stress.

tion when the system is quenched into the coexistence region resulting in a nonequilibrium state.18 –20 A potential that is purely attractive at long range 共as in the Lennard-Jones potential兲 has no mechanism to suppress the long-range fluctuations leading to the complete phase separation.18 Moreover, we believe that a long-ranged repulsive potential plays an important role in the slowing down dynamics of colloidal systems, especially at low volume fractions. In the jammed state, where the particles are kinetically trapped and the relative motion of colloidal particles and the suspending fluid are minimal, hydrodynamic interactions will not be of importance and can be safely ignored. Microcanonical ensemble molecular dynamics simulations were performed for a system of 512 particles at different volume fractions and temperatures. The equations of motion were integrated using the velocity Verlet algorithm.21 Periodic boundary conditions were imposed in all three dimensions. Equilibration of the system was tested by monitoring the total energy and other quantities like the meansquared displacement 共MSD兲. When the total energy is a constant, and the MSD curves show no dependence on the initial time, the system is claimed equilibrium and the dynamic properties are then calculated. The central quantity of

A. Kumar and J. Wu

FIG. 2. 共a兲 The yield stress ␴ y vs volume fraction ␾ c at the reduced temperature T⫽0.20. The line is a fit to Eq. 共5兲. 共b兲 ␴ y vs 1/T c at volume fraction ␾⫽0.40. The continuous line is a fit to Eq. 共6兲.

our study is the shear viscosity calculated using the Green– Kubo relation22

␩⫽

1 k B TV

冕

⬁

0

dt 具 J v 共 0 兲 J v 共 t 兲 典 ,

共2兲

where k B is the Boltzmann constant and J v is the momentum flux related to the particle momentum p, mass m, and interparticle force F i j J v⫽

兺j

p xj p yj m

⫹

r xi j F iyj . 兺 i⬎ j

共3兲

In Eq. 共3兲 i and j are the particle indices and x and y stand for the x and y components of momentum and force. Figure 1共a兲 shows the shear viscosity plotted against 1/T at different volume fractions of colloidal particles. The shear viscosity diverges as the transition temperature is approached and it follows a power-law behavior of the form15

␩ ⫽ ␩ s 共 1/T c ⫺1/T 兲 ⫺ ␷ ␾ .

共4兲

We found that the exponent ␷ ␾ ⬇0.155 by best fitting of the simulations results. For jamming phase diagram of polystyrene dispersions, Trappe et al.15 reported ␷ ␾ ⬇0.13 which agrees well with the simulation results. Using the value of T c obtained by fitting the simulation results to Eq. 共4兲 at differ-

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Appl. Phys. Lett., Vol. 84, No. 22, 31 May 2004

A. Kumar and J. Wu

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Figure 3 shows the full three-dimensional jamming phase diagram. The overall shape and behavior is similar to that obtained in the experiments.15,16 The simulation results support the concept of jamming and the existence of a jamming phase diagram in the case of attractive colloids. As observed in experiments, the shape of the surface separating the jammed and unjammed states is concave everywhere in contrast to the original proposal of jamming phase diagram by Liu and Nagel.14 To summarize, the results from molecular dynamic simulations confirm the existence of a jamming phase diagram in colloidal systems. The dynamical properties follow a critical-type behavior during the jamming transition. This work is sponsored in part by the University of California Research and Development Program 共Grant No. 69757兲, by Lawrence Livermore National Laboratory 共Grant No. MSI-04-005兲, and by the National Science Foundation 共Grant No. CTS-0340948兲. FIG. 3. The jamming phase diagram obtained from Figs. 1共b兲, 2共a兲, and 2共b兲. The points are from the simulation data. The lines are guides to the eye.

ent volume fractions, we were able to get the jamming phase diagram at zero applied stress as shown in Fig. 1共b兲. The third dimension of the jamming phase diagram is defined by the applied stress, which affects the formation of the solid network spanning the system.15,22 An increase in the applied stress causes the jammed systems to yield and flow and thereby shift the value of ␾ c toward higher values and T c toward lower values. In order to find the dependence of ␾ c and T c on the applied stress, we carried out NPTensemble molecular dynamics simulations where the number of particles (N), pressure ( P), and temperature (T) are fixed. A shear stress is applied by introducing nonzero offdiagonal elements in the pressure tensor. The behavior of shear viscosity is monitored at different applied stresses. This enabled us to calculate the yield stress at different volume fractions and also at different temperatures. The dependence of the yield stress, ␴ y , on volume fraction, ␾, and temperature, T, are also well described by critical-like behavior

␴ y ⫽ ␴ ␾共 ␾ ⫺ ␾ c 兲 ␮␾,

共5兲 ␮T

␴ y ⫽ ␴ T 共 1/T⫺1/T c 兲 .

共6兲

By fitting the simulation results as shown in Figs. 2共a兲 and 2共b兲, we obtained the values of the exponents ␮ ␾ ⬇0.66 and ␮ T ⬇0.93. These exponents are different from those reported by Trappe et al. ( ␮ ␾ ⬇3.4 and ␮ T ⬇2.4兲.15 It appears that the exponents are nonuniversal, depending on the specific interaction potential between colloidal particles.

Jamming and Rheology, edited by A. J. Liu and S. R. Nagel 共Taylor & Francis, New York, 2001兲. 2 D. F. Evans and H. Wennerstrom, The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet 共Wiley-VCH, New York, 1999兲. 3 A. McPherson, Crystallization of Biological Macromolecules 共CSHL, New York, 1999兲. 4 O. Galkin and P. G. Vekilov, Proc. Natl. Acad. Sci. U.S.A. 97, 6277 共2000兲. 5 J. Z. Zhang, Z. L. Wang, J. Liu, S. Chen, and G. Liu, Self-Assembled Nanostructures 共Kluwer, New York, 2002兲. 6 Y. Yin and Y. Xia, Adv. Mater. 共Weinheim, Ger.兲 13, 267 共2001兲. 7 E. Stiakakis, D. Vlassopoulos, B. Loppinet, J. Roovers, and G. Meier, Phys. Rev. E 66, 051804 共2002兲. 8 P. N. Segre, V. Prasad, A. B. Schofield, and D. A. Weitz, Phys. Rev. Lett. 86, 6042 共2001兲. 9 D. Pontoni, S. Finet, T. Narayanan, and A. R. Rennie, J. Chem. Phys. 119, 6157 共2003兲. 10 K. Dawson, G. Foffi, G. D. McCullagh, F. Scortino, P. Tartagalia, and E. Zaccarelli, J. Phys.: Condens. Matter 14, 2223 共2002兲. 11 M. D. Ediger, Annu. Rev. Phys. Chem. 51, 99 共2000兲. 12 K. Vollmayr-Lee, W. Kob, K. Binder, and A. Zippelius, J. Chem. Phys. 116, 5158 共2002兲. 13 W. Kob, C. Donati, S. J. Plimpton, P. H. Poole, and S. C. Glotzer, Phys. Rev. Lett. 79, 2827 共1997兲. 14 A. J. Liu and S. R. Nagel, Nature 共London兲 396, 21 共1998兲. 15 V. Trappe, V. Prasad, L. Cipelletti, and D. A. Weitz, Nature 共London兲 411, 772 共2001兲. 16 V. Prasad, V. Trappe, A. D. Dinsmore, P. N. Segre, L. Cipelletti, and D. A. Weitz, Faraday Discuss. 123, 1 共2003兲. 17 L. Berthier and J.-L. Barrat, J. Chem. Phys. 116, 6228 共2002兲. 18 B. D. Butler and J. H. M. Hanley, J. Sol-Gel Sci. Technol. 15, 161 共1999兲. 19 A. M. Puertas, M. Fuchs, and M. E. Cates, Phys. Rev. Lett. 88, 098301 共2002兲. 20 A. M. Puertas, M. Fuchs, and M. E. Cates, Phys. Rev. E 67, 031406 共2003兲. 21 W. C. Swope, H. C. Andersen, P. H. Berns, and K. R. Wilson, J. Chem. Phys. 76, 637 共1982兲. 22 J. P. Hansen and I. R. McDonald, Theory of Simple Fluids 共Academic, London, 1986兲. 1

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