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PAPER Erich A. Müller et al. Molecular dynamics simulation of the mesophase behaviour of a model bolaamphiphilic liquid crystal with a lateral flexible chain

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Molecular dynamics simulation of the mesophase behaviour of a model bolaamphiphilic liquid crystal with a lateral flexible chain† Andrew J. Crane,a Francisco J. Martı´nez-Veracoechea,ab Fernando A. Escobedob and Erich A. Mu¨ller*a Received 25th February 2008, Accepted 14th May 2008 First published as an Advance Article on the web 24th June 2008 DOI: 10.1039/b802452a We present simulations of a coarse-grained model of a bolaamphiphile liquid crystal molecule with a grafted flexible side chain. The coarse-graining approach employed is based on minimising the attractions present in the system, on the premise that the most important features of the liquid structure stem from the balance between the close range repulsions and the strong directional forces typical of hydrogen bonding and association. The model consists of six fused rigid spheres, where the two end spheres have a significant attraction amongst themselves while the rest are repulsive in nature. A weakly self-attracting lateral chain consisting of fully flexible tangently bonded spheres is attached to one of the central spheres. A parametric study is made of the configurations of collectives of these molecules at temperatures that span from the isotropic fluid range down to the onset of crystallisation. The underlying rigid core molecules (with no side chain) are set up to exhibit a smectic liquid crystal behaviour. Upon increasing the number of spheres in the lateral chains from 1 to 12, the liquid regions exhibit a rich variety of self-assembled structures; for small number of lateral chain spheres columnar arrays of different cross sections (triangular, square, rectangular, hexagonal) are obtained and for the longer chains lamellar structures of different interlayer spacing are observed. We showcase and give a rational physical explanation for the global phase behaviour of the model, based on pertinent order parameters and apparent diffusivities of the several regimes encountered. Although no attempt has been made to fit the parameters of the model to real molecules, the model is inspired by the reported synthesis of a family of T-shaped polyphilic molecules (C. Tschierske, Chem. Soc. Rev., 2007, 36, 1930– 1970) where some of the above mentioned phases have been inferred from experimental measurements.

1 Introduction Recent literature is scattered with examples of ‘‘designer assemblies’’1,2 where molecules are specifically synthesized with the appropriate building blocks so they can self-assemble in predetermined ways. The range of uses of these self-assembling materials is enormous, from optical and electronic devices to more unconventional biomimetic applications and smart fluids applications in industry, to name just a few. The synthesis of these new soft materials is a chemical tour de force, and ways to understand and predict the phase behaviour and stability from the chemical constituents are most welcome. Computer simulation can play an important role in this aspect, as most of the larger and complex mesostructured phases found in biological and soft matter have roots in the intricate interplay of relatively simple interactions between like and unlike molecular segments. Unfortunately, the capabilities of present day computers do not allow the study of macromolecules on a level in which atomic detail is present. The timescales and system sizes required for

a Department of Chemical Engineering, Imperial College London, UK. E-mail: [email protected] b Department of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY, USA † Electronic supplementary information (ESI) available: Movie of self-assembly by quenching; movie of flipping mechanisms within the LC matrix; simulation details; details of chain behaviour and end-to-end distances. See DOI: 10.1039/b802452a

1820 | Soft Matter, 2008, 4, 1820–1829

self-assembly make studies prohibitively large for currently available and foreseeable computers (typically microsecond simulations on the order of 105 atoms). Detailed studies are thus rare and have only been performed for some ‘‘smaller’’ liquid crystals (LCs), oligomers or biomolecules. A recent review by Wilson3 highlights the difficulties involved in the detailed atomistic approaches to simulating these highly structured liquids. In spite of the above, most of the mesoscopic behaviour of soft matter can be captured in an accurate way by appropriately coarse-graining the atomistic level detail, whilst retaining overall size, energy and connectivity details that are relevant.4,5 Reaching the appropriate level of abstraction required is a subtle matter, as an undue simplification could mask the underlying physical chemistry. We use here an approach based on minimising the attractions in the systems, that is, basing our models on a soft repulsive interaction, on the premise that the more important details of the fluid structure stem from close range repulsions. In this context, the strong directional interactions, arising from hydrogen bonding and electrostatics, are treated as simple attractions. Further refinement can be obtained by including the dispersion interactions as a perturbation. This broad approach, as applied to simple fluids, has led to successful treatments of small organic mixtures, typical of the chemical and petrochemical industry, and is exemplified in the historical development of the SAFT equation of state.6 For larger and more complex macromolecules, an analogous detailed theoretical framework is not yet available, and one must rely on This journal is ª The Royal Society of Chemistry 2008

a combination of experimental observations combined with suitable molecular simulations to elucidate and interpret the experimental results. This type of approach has been successfully used in understanding phase segregation in related systems, such as polymer diblock molecules,7,8 liquid crystals,3 asphaltenes,9 and biological systems.10 A recent review11 of advances in the experimental synthesis and characterisation of functional liquid crystal assemblies highlighted the importance of a novel family of polyphilic molecules that consist of a rigid polyphenyl core with incompatible end and side groups. Typically the end groups are polar, with the ability to form multiple intermolecular hydrogen bonds. Hence the combined core and end groups are in essence a bolaamphiphilic unit.12 A recent series of papers by Tschierske et al. showcases the synthesis and characterisation of these bolaamphiphiles, with grafted side chains composed of alkanes,13 partially and totally perfluorinated chains14,15 or complex polyethoxy chains16,17 (for a review see ref. 18). An example of such a molecule is the grafted 4,40 -bis(2,3-dihydroxypropoxybiphenyl) depicted in Fig. 1. These unique T-shaped molecules all share the same broad aspects: a liquid crystal core where the end groups have been substituted, and a side chain of variable length grafted to it. The relative incompatibility between each molecular component causes these compounds to have rich phase behaviour, even as pure substances. Experimental analysis, through polarized light optical microscopy, differential scanning calorimetry and X-ray scattering, has indicated at least 11 unique self-assembled liquid structures that form depending on the length and chemical properties of the grafted chains. The phases observed are typically columnar phases where the lateral chains segregate into honeycombed cylinders of networks of bolaamphiphilic rigid units, or lamellar phases composed of lateralchain-rich and bolaamphiphilic-unit-rich layers. It has been suggested that the geometry of these phases is driven by both microsegregation of incompatible units and minimisation of free volume. Current descriptions are however inferred from indirect experimental observations, and as such would benefit from confirmation and further characterisation within a physically robust framework. In this sense, this work presents a molecular dynamics study of coarse-grained grafted liquid crystal bolaamphiphile analogues, in an attempt to elucidate the principles that originate complex mesophase behaviour, as well as gain an improved picture of their global phase diagrams.

beads are kept in a rigid linear configuration with an inter-bead distance of 0.7s, where s is a characteristic length of the model, defined to be roughly the diameter of a bead. This particular configuration resembles that used to model mesogenic molecules, where the aspect ratio of 4.5 is elongated enough to guarantee LC behaviour by itself.19–23 A flexible chain of length Nflx beads is attached to the third rigid bead of the rigid unit. In this work Nflx is varied from zero to twelve. The basic topology of this model molecule is depicted in Fig. 2. Bonded interactions between beads in the flexible chain and in the link between the flexible chain and the rigid unit are represented using a harmonic potential of the form 1 U har ¼ ksp ðr
ro Þ2 2

(1)

where r is the distance separating the beads, ro is the equilibrium separation and ksp is the spring constant. In the model an equilibrium distance, ro ¼ s, and spring constant, ksp ¼ 50(3/s2), were used. Here 3 is a characteristic energy of the model, defined in terms of the pair potentials between beads, described later. While this represents a completely flexible configuration, the reader is reminded that this is a coarse-grained representation, where each bead corresponds to a group of atoms. From this point of view, the flexibility shown by this model is analogous to the coarsegrained model typically used for polymer systems.24 In order to mimic bolaamphiphilic behaviour, the two end spheres of the rigid unit are defined to attract each other in a preferential way. The lateral chain beads are also defined to have self-attraction, although the strength of this is less than the end bead counterpart. Hence the beads in the model may be categorised into three distinct types. Type 1 beads, located at each extreme of the rigid unit, seek to represent strongly interacting hydrogen-bond-like sites. Type 2 beads, constituting the remainder of the rigid unit, represent weakly interacting sites typical of the polyphenyl core. Finally, type 3 beads, located in the lateral chain, represent medium-strength interaction sites,

2 Methodology 2.1 Molecular model A simple coarse-grained model for a bolaamphiphilic rod with flexible lateral chain is hereby implemented. In this model, six

Fig. 1 Structure of the core bolaamphiphile molecule that served as prototype for this study. R corresponds to a substitution of an alkane, perfluoroalkane or mixed alkane–perfluoroalkane chain. Adapted from ref. 14.

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Fig. 2 Cartoon of the molecules studied. Type 1 beads, corresponding to hydrogen-bond-like sites, are coloured orange; type 2 beads, corresponding to the polyphenyl core of the molecules, are coloured green; type 3 beads, corresponding to the grafted side chains, are coloured blue. All cross interactions are repulsive and all beads are the same diameter, s, and mass.

Soft Matter, 2008, 4, 1820–1829 | 1821

typical of alkane or perfluoroalkane chains. In line with the coarse-grained nature of our model, the non-bonded inter-bead interactions, are either soft short-range attractions or soft repulsions. To this end, all six bead pair potential energies were defined through the Lennard-Jones cut and shifted potential (LJCS) LJCS ðr; CAB Þ ¼ UAB




LJ LJ UAB ðrÞ
UAB ðCAB Þ 0

for for

r\CAB r$CAB

(2)

with LJ ðrÞ ¼ 43AB UAB

  sAB 12 sAB 6
r r

(3)

where A and B define the bead type, r is the bead separation, 3AB is an energy parameter defining the potential well depth, sAB is the length parameter defining the range of the potential, and CAB is the cut and shifted distance. Type 1 beads, representing the hydrogen bonding groups, are assigned an LJCS potential self interaction with 311 ¼ 3, s11 ¼ s and C11 ¼ 2s. No intrinsic saturation associated with these ‘‘hydrogen bonding’’ beads has been built in here. A molecule like that depicted in Fig. 1 shows a large hydrogen bonding capability, as each end group contains two O–H bonds and one –O– group, allowing in principle more than one hydrogen bond per end group. Steric hindrance will most likely limit the maximum number and type of these bonds in both the real molecule and the model. Type 3 beads have a selfinteraction with 333 ¼ 3/2, s33 ¼ s and C33 ¼ 2s, where 333 was chosen to reflect the weaker nature of this attraction. Finally, the type 2 bead self-interaction and all cross interactions are modelled with an LJCS potential with 3AB ¼ 3, sAB ¼ s and CAB ¼ 21/6s. This latter cut-off corresponds to the length at which the standard Lennard-Jones potential is at a minimum, consequently this potential represents a purely repulsive interaction (typically known as the Weeks–Chandler–Andersen potential). While the rather small value of C11 is an attempt to capture with a very simple model the behaviour of strongly interacting hydrogenbond forming sites, C33 has been set up to equal C11 for computer efficiency and to avoid the introduction of an additional parameter. All beads in the system are chosen to have the same mass. In principle, each bead accounts for a group of atoms. Consequently, if an accurate description of a particular model were desired, the properties of these beads would have to be mapped to experimental physical properties (e.g. radial distribution functions, densities, etc.). This has not been attempted in this work, as we are interested in presenting a proof-of-concept rather than a particular application. We wish to make the statement that the use of a simplistic, coarse-grained model helps to highlight the ‘‘crucial’’ characteristics that allow for the experimentally observed complex mesophase behaviour. By decreasing proportionally the strength of the interactions of all segments of the molecule, we can increase substantially the efficiency of the calculation, without changing the basic structure of the fluid phase, which will depend more on the interplay between the repulsive cores of the molecules and the directional and specific nature of the hydrogen bonds. The temperature, T* ¼ Tk/3, where k is Boltzmann’s constant, and pressure, P* ¼ Ps3/3, are expressed throughout this work in 1822 | Soft Matter, 2008, 4, 1820–1829

the standard reduced form that reflect the energy and length scales of the model. Simulations were performed in continuum space via molecular dynamics using the simulation suite DL_POLY.25 The following ensembles were used: the Micro-Canonical (NVE) ensemble where the number of particles (N), the total volume (V), and the total energy (E) are fixed; the Canonical (NVT) ensemble where N, V, and, the temperature T are fixed; the Isobaric–Isothermal ensemble (NPT) where N, T, and the pressure (P) are fixed; and the Isotension–Isothermal ensemble (NsT) where N, T and the components of the stress tensor (s) are fixed. In the NsT ensemble both the size and shape of the simulation box are allowed to change in order to satisfy the constraints imposed. Thus, the NsT ensemble has the advantage that it can ensure the equality of pressure in the three directions. Simulation details are provided in the ESI, section S3.† Phase structures and fluidities were studied through the computation of order parameters and mobility coefficients from simulation configuration data. This allowed a quantitative characterisation of the different phases and improved location of phase transition temperatures. A description of these quantities and the methods used to calculate them are outlined in the following section. Hysteresis in the systems was investigated through running an additional series of NsT simulations over phase transition temperature ranges. Whilst there was noticeable hysteresis, it is likely that this is due to the system sizes and simulation times. Phase transition temperatures in this work should be taken as a rough guide to the actual transition temperatures expected.

2.2 Metrics 2.2.1 Order parameter. To calculate order parameters, that provide a measure of structural order, an axis within the system that characterises the direction of order is required. For columnar/lamellar phases, this director may be viewed as the vector orthogonal to their layers. For our model, the set of unit vectors, {u1, u2,., uN}, associated with the bolaamphiphilic rigid unit axes was used to define the orientation of each molecule in the system. With this arrangement, the problem of determining the director was equivalent to finding a unit vector, n, that is maximally orthogonal to this set of molecular orientation vectors. Concisely this may be written as min(||Un||) subject to ||n|| ¼ 1 where U represents the N by 3 matrix containing the molecule orientation unit vectors, ui. The solution to this total least squares problem is obtained through determining the singular value decomposition (SVD) factorisation of matrix U. An outline of SVD terminology and methodology may be found in standard linear algebra textbooks; it is however sufficient to state here that the director, n, is found to be the right singular vector of U corresponding to the smallest singular value. We note that n does not correspond to the nematic director, but rather to a vector perpendicular to the planes formed by the unit vectors. With this director determined a range of order parameters becomes available. The planar order parameter and the planar This journal is ª The Royal Society of Chemistry 2008

orientational order parameters were used in this study and are defined below. 2.2.2 The planar order parameter. The planar order parameter, S2, measures how orthogonal the molecular orientation vectors are to the director vector, and is defined by the equation N P

S2 ¼

3 sin2 fj
2

j¼1

N

(4)

where fj is the angle between the director, n, and the orientation vector, uj, of the jth molecular rigid unit. This order parameter is analogous to the P2 order parameter that is frequently used to determine the orientation order of rod-like liquid crystalline systems.26 For an isotropic system where there is no correlation between the director and orientation vectors, S2 is defined such that it takes a value of 0. Conversely for an ordered system where all the molecular orientation vectors lie in the plane orthogonal to the director, S2 takes a value of 1. 2.2.3 The planar orientational order parameter. To monitor the geometry of the columnar phases as seen from an observer looking into the director vector (i.e. triangular/square/hexagonal columns), the planar orientational order parameter, jk, is defined through the equation   N 1 X    jk ¼  exp ikqj  (5)  N j¼1 where qj is the angle between the vector given by the projection of the orientation vector, uj, onto the director orthogonal plane, and a fixed arbitrary axis orthogonal to the director. The coefficient k takes the value 2, 4 and 6 for the linear, square and hexagonal planar orientational order parameters, respectively. As with the planar order parameter, it is designed so that values of the planar orientation order parameter close to 0 indicate low order, whilst values close to 1 indicate a high level of order. 2.2.4 The mobility coefficient. To quantify the mobility of the bolaamphiphile molecules in the various phases, mean squared bead displacements were calculated from the NVE simulations previously described. For a given number of time steps, s, each of time Dt, the mean squared bead displacement over time sDt, RsDt, was calculated through the equation

RsDt

N 6þN t
s P Pflx P 2 j;k jDrðiþsÞDt;iDt j j¼1 k¼1 i¼0   ¼ N 6 þ Nflx ðt
sÞ

(6)

j,k where Dr(i+s)Dt,iDt is the displacement of the kth bead in the jth molecule, between time iDt and (i + s)Dt, and t is the total number of time steps in the simulation. By determining the rate of change of this displacement with time, a quantity defined as the mobility coefficient was obtained. As the simulations were performed on coarse-grained models the reduced time does not have a direct link with real time, therefore this quantity is related in a nontrivial way to the diffusion coefficient of the system. Nevertheless it remains useful as a measure of the relative molecular mobility in the different phases. In determining the mobility coefficient it is

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important to realise that the mean squared displacement against time chart is composed of two distinct regions. At times less than the period of vibration of the beads, there is a steep increase in the mean squared bead displacement, due to localised vibrations of the beads. At longer times, the change of mean squared bead displacement with time is due to non-local movement. At high temperatures non-local bead movement dominates over any localised vibrations, consequently it is not possible to delineate these regions. At lower temperatures, however, where the mobility of beads is less, these two regions become clear. The gradient of the mean bead displacement in the longer time region is the appropriate one to use for the mobility coefficient, since non-local movement was of interest. As is seen from eqn (6), on increasing the interval length s, the sample of bead displacements used in the averaging procedure decreases. Hence for the longer time intervals, the value obtained for the mean squared displacement is less accurate, due to poorer statistics. To systematically determine the appropriate gradient in the longer time region, a least squares approach was used to fit a linear trend to a fixed time range of the mean squared displacement against time data, starting at a specific time. This process was repeated for different starting times, maintaining the same time range. To assess the quality of each linear regression the coefficient of determination was found (this is equivalent to the square of the Pearson correlation coefficient of the data set). The mobility coefficient was then assigned as the gradient of the linear regression with the maximum coefficient of determination.

3

Results

3.1 Global self-assembling structures As previously outlined the results are discussed in terms of Nflx, the number of lateral chain beads that characterises each particular molecule. A global phase diagram of the phases encountered when varying temperature and number of lateral chain beads is presented in Fig. 3. Further discussion on the observed phases is given below. 3.2 Bolaamphiphile For the case of a system with no lateral grafted chain, Nflx ¼ 0, the system behaves as a bolaamphiphilic liquid crystal. For T* $ 1.01 the observed phase is isotropic (Fig. 4a). For 0.79 # T* # 1.01 we observe the smectic phase (Fig. 4b). For T* < 0.79 the system crystallises as the mobility coefficient decreases several orders of magnitude below the isotropic value. The inclusion of attractions on the end segments favours the alignment in a smectic-like pattern, precluding the possibility of a nematic phase, for which we found no evidence. 3.3 Columnar phases If side chains are grafted to the bolaamphiphilic molecular core, their presence severely disrupts the parallel rod ordering. These side chains are moderately attractive to each other, so they will have a tendency of agglomerating in the same regions of space. Compounded with that propensity, the terminal beads of each bolaamphiphilic core will have the tendency of maintaining Soft Matter, 2008, 4, 1820–1829 | 1823

Fig. 3 Graphical representation of the phases encountered upon varying the reduced temperature for the different molecular conformations studied (varying the number of flexible grafted side chains, Nflx). Phase description: (A) isotropic; (B) smectic; (C) crystal; (D) triangular columnar crystal; (E) square columnar liquid crystal; (F) rhomboid columnar crystal; (G) hexagonal columnar liquid crystal; (H) rectangular columnar crystal; (I) out-of-plane bent rectangular columnar crystal; (J) lamellar in-plane isotropic liquid crystal; (K) lamellar in-plane smectic liquid crystal; (L) lamellar crystal.

Fig. 5 Snapshots of equilibrium configurations for liquid crystal columnar phases. Orange spheres represent the type 1 beads (hydrogen bonding), green spheres represent the type 2 beads (biphenyl core), blue spheres represent the type 3 beads (perfluoroalkane grafted chains). (a) to (d) correspond to fluid phases with Nflx ¼ 1, 2, 4 and 5 respectively.

3.3.1 Nflx ¼ 1. The simplest possible perturbation to the bolaamphiphilic core is the addition of a single side bead. For this scenario, where Nflx ¼ 1, equivalent to f ¼ 0.14, a triangular section columnar phase is observed for T* ¼ 0.50 (see Fig. 5a). If the attachment of the lateral chain is changed from the third position to second position of the rigid body, that is, to a bead adjacent to a type 1 bead, an imperfect smectic phase is observed. The triangular shaped columnar phase has also been experimentally confirmed for a similar T-shaped polyphilic molecule.17 Fig. 4 Snapshots of equilibrium configurations for molecules with no grafted chains (Nflx ¼ 0). (a) Isotropic phase at T* ¼ 1.3; (b) smectic phase at T* ¼ 1.0. Orange spheres represent the type 1 beads (hydrogen bonding), green spheres represent the type 2 beads (biphenyl core).

themselves close to each other. The overall result is the possibility of formation of columnar structures, where the walls of these structures are delimited by the cores and the inner part of the columns are ‘‘filled’’ with the side chains. Fig. 5 showcases some examples of the types of columnar structures we have encountered in our simulations, all of which echo experimentally observed conformations. The experimental behaviour of these systems has been described empirically with reference to the volume fraction of lateral chains with respect to the whole system. As this definition can be ambiguous for soft matter systems, below we define the volume fraction of lateral chains as f ¼ Vflx/Vmol. The conventions for the volumes and packing fractions used are detailed in the ESI, section S3.† 1824 | Soft Matter, 2008, 4, 1820–1829

3.3.2 Nflx ¼ 2. A richer phase behaviour is observed when considering Nflx ¼ 2, equivalent to f ¼ 0.24. The results for Nflx ¼ 2 are summarised in Fig. 6. Reviewing the diagram from high to low temperature, at T* ¼ 0.77 the transition from the isotropic state to that of a square columnar phase (Fig. 5b), is clearly indicated by an increase in the S2 and j4 order parameters. The value of j6 remains close to zero during this transition, indicating, as expected, that there is little hexagonal symmetry to the columnar structures. On decreasing the temperature further the values of S2, j4 and j6 remain relatively constant until around T* ¼ 0.47. At this point the value of j6 starts increasing, whilst j4 starts decreasing. Inspections of simulation movies reveal that below T* ¼ 0.47, the square columnar structures start to slant into a rhombus columnar structure. This behaviour clearly supports the trends in the order parameters observed. At these low temperatures, through deformation, the cross section areas of the columns are reduced whilst retaining the same cross sectional perimeter. This allows the lateral flexible chain particles to get closer in space, whilst the structure preserves the strongly This journal is ª The Royal Society of Chemistry 2008

Fig. 6 Order parameters (left ordinate) and mobility coefficient (right ordinate) for the Nflx ¼ 2 system as a function of temperature. Green line is the planar order parameter, S2; orange line is the square columnar order parameter, j4; purple line is the hexagonal columnar order parameter, j6; red line is the mobility coefficient.

shifting into a vacant site in the crystalline columnar structure at t* ¼ 7000. Lastly, Fig. 7c displays the behaviour of a molecule from the T* ¼ 0.47 simulation whose absolute displacement oscillates with a large amplitude. In the aforementioned movie this molecule is seen to be located outside the main columnar lattice, consequently the rigid rod has greater freedom to move due to weaker bonding of its type 1 beads (hydrogen bonding). The previous two examples represent typical defects that contribute to the non-zero mobility coefficient at low temperatures. To reduce the fluctuations observed in the mobility coefficient at lower temperatures, larger simulations or longer times could be used. This would allow the number of defects observed to reflect the true average expected at a given temperature. For this investigation however these results were deemed adequate, as the crystalline nature of the structures was not of primary interest. Between T* ¼ 0.47 and T* ¼ 0.77 the mobility coefficient increases linearly with increasing temperature. This provides clear evidence that there exists molecular mobility in this temperature range. Inspection of absolute displacements for molecules at T* ¼ 0.73 for example, show that whilst some molecules remain local in the liquid crystal structure, as shown in Fig. 8a, there is a significant proportion of molecules that are mobile in spite of the structured nature of the fluid phase, as displayed in Fig. 8b. Above T* ¼ 0.77 there is a steep increase in the mobility coefficient, associated with the transition to an isotropic phase, followed by a linear increase with temperature. In summary, the order parameter and mobility coefficient data suggest three phases present for the Nflx ¼ 2 compound; a crystalline rhomboidal columnar phase at temperatures below T* ¼ 0.47, a liquid crystalline square columnar phase at temperatures between T* ¼ 0.47 and T* ¼ 0.77, and finally an isotropic phase above T* ¼ 0.77.

bonded rigid rod network topology. With a further reduction of temperature below T* ¼ 0.17 the values of S2, j4 and j6 remain constant at 0.87, 0.73 and 0.70, respectively. Further information on the mobility of these phases may be garnered through looking at of the mobility coefficient, also displayed in Fig. 6. Between T* ¼ 0.13 and T* ¼ 0.47 the value of the mobility coefficient remains close to zero suggesting a solid crystalline phase. Fluctuations of the mobility coefficient, evident at lower temperatures, may be attributed to defects in the crystal structures. Inspection of the absolute displacements of individual molecules during a simulation provides evidence for this. Displayed in Fig. 7a is the absolute displacement, in units of s, for a typical molecule in a simulation at T* ¼ 0.47 (in the vicinity of the solid–LC transition). The oscillation here may be attributed to the molecule vibrating locally in the columnar structure. For a different molecule in the same simulation, a similar behaviour is shown in Fig. 7b, however here at t* ¼ 7000 the molecule undergoes a large displacement before settling back into a local oscillation. Isolation of this molecule in the simulation movie (see supplementary material S1†) clearly shows it

3.3.3 Nflx ¼ 3,4. For systems in which there are three or four beads in the lateral chain, only defective columnar shapes are formed. For example, with Nflx ¼ 4, equivalent to f ¼ 0.39, at T* ¼ 0.67, the columns in the LC region present cross sections of varying polygonal shapes wherein pentagons and hexagons are among the most regular ones observed (see Fig. 5c). In this case there is an

Fig. 7 Absolute displacements of selected individual molecules as a function of time for a system of Nflx ¼ 2 at T* ¼ 0.47, close to the LC– solid transition.

Fig. 8 Absolute displacements of selected individual molecules as a function of time for a system of Nflx ¼ 2 at T* ¼ 0.73, close to the isotropic–LC transition.

This journal is ª The Royal Society of Chemistry 2008

Soft Matter, 2008, 4, 1820–1829 | 1825

Fig. 9 Order parameters (left ordinate) and mobility coefficient (right ordinate) for the Nflx ¼ 5 system as a function of temperature. Line colours as in Fig. 6.

obvious competition between the pentagonal and hexagonal columns as regular pentagons themselves cannot tessellate a plane and hexagonal columns provide too much void space to be filled by the side chains, thus packing frustration is evident. 3.3.4 Nflx ¼ 5. For the case of a five bead lateral chain, Nflx ¼ 5, equivalent to f ¼ 0.44, the system exhibits an LC phase where hexagonal columns are abundant. The results for the Nflx ¼ 5 are summarised in Fig. 9. On reviewing the diagram passing from high to low temperature, the transition from the isotropic to hexagonal columnar phase (Fig. 5d) is clearly delineated by a jump from near zero values of S2 and j6 to values of around 0.72 and 0.36 respectively at T* ¼ 0.87 (Fig. 9). This hexagonal ordering has been inferred from experimental observations of T-shaped bolaamphiphiles.18 As supplementary material (S2)† we include a movie of the quenching process for this molecule. Two things are readily apparent from the movie. In the first place, it is seen from the slideshow how precipitously the system achieves a general ordering. The system relaxes, finding minimum energy configurations as hexagonal-columnar structures. It is also clear from the movie that the system is, at this temperature, a liquid phase, as the movement of individual

molecules is evident. On decreasing the temperature further, both S2 and j6, slowly increase until around T* ¼ 0.60, where there is a steep increase in j4. Below this temperature the j4 parameter slowly increases, whilst S2 and j6 slowly decrease. Visual inspection of the molecular dynamics simulation (not included in the supplementary material) indicates that this result is due to a flattening of the hexagonal columns into a more rectangular shape. Fig. 10 shows the variation of the fj and qj population distributions [see eqn (4) and (5)] with temperature, and provides a good indicator of structure. The fj population distribution shows changes that would otherwise be difficult to pick out from inspection of the order parameters alone. The angle fj spans between the rigid core of each individual molecule and the system director, that is generally the vector aligned with the columnar axes. At isotropic conditions, the orientation of the bolaamphiphilic molecular cores is essentially random, the director is therefore ill-defined, and the distribution of fj is sinusoidal, reflecting the lack of orientational correlation between the bolaamphiphilic cores and director. At temperatures below the isotropic–LC transition, T* ¼ 0.87, there is a clear indication that a large proportion of the molecules are, at any given time, at 90 from the director, i.e. aligned in planes which are roughly orthogonal to the director. Since the phase is a liquid one, the peak shown is broad. The transition to the solid phase is characterised by a narrowing of the distribution, as is evident below T* ¼ 0.60, corresponding to the LC–solid transition. At T* ¼ 0.80 the single peak at 90 splits into three peaks centred at approximately 72 , 90 , 108 , indicating the structure relaxes from a state having columns formed by parallel stacked hexagons to a form where a proportion of the rigid rods are not directed in the plane orthogonal to the director. The outcome of this change is to retain the hydrogen bonding network but effectively reduce the cross sectional area of the column in the plane orthogonal to the director. This reduction of cross sectional area allows the lateral chain units to arrange closer in space and hence bond more effectively. Inspection of the population distribution for the angles qj between the molecules in a given plane orthogonal to the director also shows the changes associated with these above mentioned phase transitions. In the isotropic region, as stated before the orientation of the bolaamphiphilic molecular cores is

Fig. 10 Population plots as a function of temperature for the system Nflx ¼ 5 for (a) variation of the angle fj between the molecules’ rigid core and the column director n; (b) variation of the angle qj between the molecules’ rigid core and an arbitrary (fixed) vector in the plane orthogonal to the column director n.

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essentially random, therefore the director is ill-defined, and the distribution of fj shows no angular dependence. However, in the LC region, six distinct equispaced peaks are shown with a period of 60 , corresponding to the angles expected in the regular hexagonal structure. Upon crystallisation, below T* ¼ 0.60, the peaks sharpen and at the lowest temperature range studied, corresponding to the secondary crystalline structure, two of the peaks tend to merge. This, along with the fj distribution, suggests a structure that is collapsing towards a herringbone flattened rectangular columnar structure, whose layers have a resemblance to the pages of a partially open book. As with Nflx ¼ 2, further insight into the phase behaviour of the system may be obtained by looking at the variation of the mobility coefficient with temperature. At temperatures below T* ¼ 0.60 the mobility coefficient is very close to zero. The fluctuations at lower temperature may be rationalised in the same way as previously discussed. On increasing the temperature above T* ¼ 0.60, the mobility coefficient begins a roughly linear increase with temperature, until the isotropic transition at T* ¼ 0.87. The non-zero values of the mobility coefficient in this region clearly indicate that the phases have some fluidity, and hence point towards liquid crystalline behaviour. Inspection of the qj population distribution in this temperature range (Fig. 10b), where the distribution is noticeably diffuse on comparison to temperatures below T* ¼ 0.60, supports the assertion that this phase is relatively mobile (and also rationalises the relatively low value of j6). The population of rigid rod units at angles intermediate to those associated with the hexagonal column, may be due to molecule flipping between hexagonal column sides. At T* ¼ 0.87 the onset of the isotropic phase is associated with a significant jump in the mobility coefficient, followed in the isotropic phase by a linear increase with temperature. To summarise, the information here indicates the presence of four distinct phases, an isotropic phase above T* ¼ 0.87, a LC phase between T* ¼ 0.60 and T* ¼ 0.87, and two crystalline type phases, one between T* ¼ 0.43 and T* ¼ 0.60 and one below T* ¼ 0.43. Similar columnar behaviour is also observed for Nflx ¼ 6 and Nflx ¼ 7 but the structures formed are riddled with an increasing number of defects. 3.4 Lamellar phases For side chains of eight or more beads long, the columnar structure breaks down and the system presents a lamellar structure. This is in synchronicity with the results of experiments on related T-shaped molecules, where lamella are formed for systems with large side substituents.18 Unique from the simulations is the observation that the lamella are bridged by LC cores. For Nflx ¼ 8, equivalent to f ¼ 0.56, at T* ¼ 0.66 a lamellar phase with bridges one-molecule long is observed (Fig. 11a). We report that for Nflx $ 8 the system dynamics slow down and equilibrating phases becomes increasingly difficult. We have not observed regular columnar structures for higher values of Nflx (only defective phases with no periodicity). We do not preclude the existence of other structures, and note that it would require longer runs and bigger unit cells than the ones used here. For the even longer chain length Nflx ¼ 11, equivalent to f ¼ 0.64, at T* ¼ 0.50 a lamellar phase with bridges two-molecules long is observed (Fig. 11b). When the flexible chains are removed the This journal is ª The Royal Society of Chemistry 2008

Fig. 11 Snapshots of equilibrium configurations for lamellar phases. Colouring as in Fig. 5. (a) Nflx ¼ 8; (b) Nflx ¼ 11.

system shows resemblance to the ‘‘extended hexagons’’ proposed in the experimental paper.18 However, not all the ‘‘hexagons’’ are of the same size and they present perforations in the lateral walls. 3.4.1 Nflx ¼ 12. During the simulations for the twelve bead system, Nflx ¼ 12, equivalent to f ¼ 0.66, at T* ¼ 0.83, a lamellar phase was observed composed of a flexible-chain-rich layer and a rigid-unit-rich layer. In this phase the rigid units were distributed more or less isotropically within the plane of the rigid-unit domain (see Fig. 12a). When the temperature was lowered to 0.5 # T*#0.73, the lamellar structure persisted. However, in the rigid-unit rich domain a transition was observed, where smectic order was observed within the layers (as can be seen in Fig. 12b). Additionally, a few molecules were observed to lie perpendicular to the rest. Equilibration from direct quenching was relatively slow, making it in some cases necessary to run the system for up to 107 integration steps in order to see the formation of the lamellar phase. Following the initial simulations, a study of the variation of temperature on the twelve bead system was undertaken, as summarised in Fig. 13. For this system the S2, j4 and j2 order parameters were found to be the most effective at monitoring the temperature dependence of the phase behaviour. Above T* ¼ 0.90 the values of S2 and j2 are close to zero in the isotropic phase. As the temperature is reduced below T* ¼ 0.90, S2 increases to a value of around 0.56, whilst j4 and j2 remain close to zero. On inspection of the simulation movie (not included) the phase is clearly seen to exhibit a lamellar structure with isotropic arrangement of rods in the lamellar plane, as previously

Fig. 12 Snapshots of equilibrium configurations for lamellar phases for Nflx ¼ 12. Details of the lateral chains have been omitted for clarity. The snapshots look into the layers of the lamella from the side-chain rich mesophase, i.e. a view orthogonal to that of Fig. 11. (a) Isotropic phase at T* ¼ 0.83; (b) smectic phase at T* ¼ 0.6.

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Fig. 13 Order parameters (left ordinate) and mobility coefficient (right ordinate) for the Nflx ¼ 12 system as a function of temperature. Green line is the planar order parameter, S2; purple line is the linear columnar order parameter, j2; orange line is the square columnar order parameter, j4; red line is the mobility coefficient.

discussed. A further reduction in the temperature below T* ¼ 0.80 leads to a further increase of S2 to around 0.9, and an increase of j4 and j2 to 0.66 and 0.73, respectively. The change in j2 is associated with the alignment of the rigid rod units in the lamellar plane; inspection of the simulation snapshots reveal this to be the in-plane smectic. The fact that j2 is not closer to 1, and j4 is relatively close to 1, is due to the presence of defects where the rigid rod units of the bolaamphiphile are perpendicular to the in-lamellar-plane smectic director between the smectic layering. Inspection of the qj population distribution, in Fig. 14 clearly demonstrates this to be the case, with two weak peaks found to be exactly mid-way between the main 180 separated peaks. As the temperature is reduced to T* ¼ 0.47, j2 is found to dip slightly before returning to 0.67, whilst S2 and j4 slowly increase to 0.96 and 0.91, respectively. Below T* ¼ 0.47 all the order parameters stay the same. Studying the mobility coefficient reveals four main regions. Below T* ¼ 0.47 the mobility coefficient is very close to zero, indicating a crystalline structure. Between T* ¼ 0.47 and T* ¼ 0.80, there is a slight positive gradient in the mobility coefficient with increasing temperature indicating some fluidity to the phase. Inspection of Fig. 14

supports this with a clear transition from sharply defined qj population distribution below T* ¼ 0.47 to a more spread out distribution above this temperature. On going above T* ¼ 0.80 there is a jump in the mobility coefficient, and then the gradient of mobility coefficient with temperature increases from that in the T* ¼ 0.47 to T* ¼ 0.80 range. Above T* ¼ 0.90, with the onset of the isotropic phase there is another increase in the gradient. To summarise, for this chain length, the order parameter and mobility coefficient analysis point towards four distinct phases; a crystalline lamellar phase below T* ¼ 0.47, a liquid crystalline lamellar phase with in-lamellar-plane smectic ordering of the bolaamphiphile rigid rods between T* ¼ 0.47 and T* ¼ 0.80, a liquid crystalline lamellar phase with isotropic ordering of the bolaamphiphile rigid rods between T* ¼ 0.80 and T* ¼ 0.90, and an isotropic phase above T* ¼ 0.90. The temperature and the mesophase have a strong influence on the conformations of the lateral chains. Here we have studied only infinitely flexible chains, however, it is expected that the actual flexibility will depend on the chemical nature of the chains (e.g. alkane-like, perfluorinated or semi-perfluorinated). A detailed description of the observed configurations of these lateral chains is included in the ESI (section S4).†

4

Conclusions

We have shown here, as a matter of proof-of-concept, that one may successfully employ coarse-graining techniques that reduce the overall attraction of the components of the molecules and simplify the overall geometry to model very complex fluid selfassembly. The key aspect of the method is to retain in the CG model the key physical elements of the original molecule, in terms of rigidity and flexibility, relative volume fractions and relative intramolecular characteristic energies. The success of such a minimalistic model in qualitatively representing the experimentally observed phase behaviour of the T-shaped amphiphiles, strongly indicates that the complex behaviour observed is fairly independent of the specific details of the liquid crystal molecule and rests on larger scale issues that seem to induce the formation of complex mesophases: 1) the presence of a rigid section with strongly attractive sites at both ends and 2) a lateral chain, with unlike interactions towards the

Fig. 14 Population plots as a function of temperature for the system Nflx ¼ 12 for the (a) variation of the angle fj between the molecules’ rigid core and the director n, in this case, orthogonal to the lamella; (b) variation of the angle qj between the molecules’ rigid core and a random (fixed) vector in the plane of the lamella.

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aforementioned rigid part, hence inducing, as a consequence of packing frustration considerations, the formation of complex structures. Although no effort was made in this work to map the parameters of the molecules to the actual prototype T-shaped amphiphiles, we were able to account for most of the reported experimentally observed phases and rationalise the formation of them in terms of simple metrics and physical insight. If a more direct mapping were required for a particular molecule, the present model has sufficient independent variables (sizes, number and energy of the different segments) to be able to give quantitative information.

Acknowledgements Partial financial support for this work has been given by the U.K. Engineering and Physical Sciences Research Council (EPSRC), grant EP/E016340, ‘‘Molecular Systems Engineering’’, and U.S. Department of Energy, grant no. DE-FG02-05ER15682. Fruitful discussions with Prof. George Jackson are gratefully acknowledged.

References 1 J. H. Fuhrhop and J. Ko¨ning, Membranes and Molecular Assemblies: The Synkinetic Approach, Royal Society of Chemistry, Cambridge, UK, 1994. 2 L. F. Lindoy and I. M. Atkinson, Self Assembly in Supramolecular Systems, Royal Society of Chemistry, Cambridge, UK, 2000. 3 M. R. Wilson, Int. Rev. Phys. Chem., 2005, 24, 421–455. 4 S. O. Nielsen, C. F. Lopez, G. Srinivas and M. L. Klein, J. Phys.: Condens. Matter, 2004, 16, R481–R512. 5 F. Mu¨ller-Plathe, ChemPhysChem, 2002, 3, 754–769. 6 W. G. Chapman, K. E. Gubbins, G. Jackson and M. Radosz, Fluid Phase Equilib., 1989, 52, 31–38; W. G. Chapman, K. E. Gubbins, G. Jackson and M. Radosz, Ind. Eng. Chem. Res., 1990, 29, 1709– 1721; For a review see: E. A. Mu¨ller and K. E. Gubbins, Ind. Eng. Chem. Res., 2001, 40, 2193–2211.

This journal is ª The Royal Society of Chemistry 2008

7 F. J. Martinez-Veracoechea and F. A. Escobedo, J. Chem. Phys., 2006, 125, 104907. 8 M. A. Horsch, Z. L. Zhang and S. C. Glotzer, J. Chem. Phys., 2006, 125, 184903. 9 B. Aguilera-Mercado, C. Herdes, J. Murgich and E. A. Mu¨ller, Energy Fuels, 2006, 20, 327–338. 10 H. D. Nguyen and C. K. Hall, J. Am. Chem. Soc., 2006, 128, 1890– 1901. 11 T. Kato, N. Mizoshita and K. Kishimoto, Angew. Chem., Int. Ed., 2006, 45, 38–68. 12 J. H. Fuhrhop and T. Wang, Chem. Rev., 2004, 104, 2901– 2938. 13 M. Ko¨lbel, T. Beyersdorff, X. H. Cheng, C. Tschierske, J. Kain and S. Diele, J. Am. Chem. Soc., 2001, 123, 6809–6818. 14 X. H. Cheng, M. Prehm, M. K. Das, J. Kain, U. Baumeister, S. Diele, D. Leine, A. Blume and C. Tschierske, J. Am. Chem. Soc., 2003, 125, 10977–10996. 15 X. H. Cheng, M. K. Das, U. Baumeister, S. Diele and C. Tschierske, J. Am. Chem. Soc., 2004, 126, 12930–12940. 16 B. Chen, U. Baumeister, G. Pelzl, M. K. Das, X. Zeng, G. Ungar and C. Tschierske, J. Am. Chem. Soc., 2005, 127, 16578–16591; B. Chen, X. Zeng, U. Baumeister, G. Ungar and C. Tschierske, Science, 2005, 307, 96–99; B. Chen, X. B. Zeng, U. Baumeister, S. Diele, G. Ungar and C. Tschierske, Angew. Chem., Int. Ed., 2004, 43, 4621–4625. 17 F. Liu, B. Chen, U. Baumeister, X. Zeng, G. Ungar and C. Tschierske, J. Am. Chem. Soc., 2007, 129, 9578–9579. 18 C. Tschierske, Chem. Soc. Rev., 2007, 36, 1930–1970. 19 J. G. Gay and B. J. Berne, J. Chem. Phys., 1981, 74, 3316– 3319. 20 M. Whittle and A. J. Masters, Mol. Phys., 1991, 72, 247–265. 21 G. V. Paolini, G. Ciccotti and M. Ferrario, Mol. Phys., 1993, 80, 297–312. 22 F. Affouard, M. Kroger and S. Hess, Phys. Rev. E, 1996, 54, 5178–5186. 23 P. Tian, D. Bedrov, G. D. Smith and M. Glaser, J. Chem. Phys., 2001, 115, 9055–9064. 24 G. S. Grest, M. D. Lacasse, K. Kremer and A. M. Gupta, J. Chem. Phys., 1996, 105, 10583. 25 www.ccp5.ac.uk/DL_POLY/. For a recent review see: W. Smith, Mol. Simul., 2006, 32, 933–1121. 26 E. de Miguel, E. M. del Rio and F. J. Blas, J. Chem. Phys., 2004, 121, 11183–11194.

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