Hard Spherocylinders Research Articles

Computer simulation study of a flexible-rigid-flexible model for liquid crystals

Small molecules that form liquid crystals typically consist of a rigid core with flexible tails on one end or on both ends. To date, most computer simulation studies have used completely rigid models such as hard spherocylinders: cylinders, characterized by their length/diameter ratio L/D, with hemispherical end caps. We have studied a model consisting of spherocylinders with L/D = 4, with a flexible tail attached to each end. The tails are ‘ideal’ in the sense that they have no volume. Using Monte Carlo simulations the phase behaviour of this model was studied and, for comparison, the behaviour of hard spherocylinders with L/D = 4 without tails was studied as well. The addition of the tails is found to stabilize the smectic-A phase at a lower pressure, and the nematic phase disappears. In the smectic-A and crystal phases, the smectic layers are further apart when tails are added. The structure of the layers and the smectic-A–crystal transition pressure change only a little. For both models close to melting the crystal consists of ordered layers, but there is almost no correlation between particle positions in neighbouring layers. In fact, the layer coupling is so weak that in a long simulation the layers are found to glide over each other. As the pressure is increased the crystal gradually becomes more ordered and the crystalline layers ultimately ‘lock’ into place.

First-order nematic-smectic phase transition for hard spherocylinders in the limit of infinite aspect ratio

We report Monte Carlo simulations of the nematic-smectic phase transition for a system of hard spherocylinders with infinite length-to-diameter ratio. A finite-size scaling analysis suggests that this system undergoes a first-order phase transition. When combined with other simulations of the phase behavior of spherocylinders, these results suggest that the nematic-smectic phase transition is first-order for all aspect ratios. This appears to rule out the possibility of a tricritical point predicted by several density-functional theories.

Cell theory for the phase diagram of hard spherocylinders

Cell theory for the phase diagram of hard spherocylinders

Entropy-driven demixing in binary hard-core mixtures: From hard spherocylinders towards hard spheres

We present a computer simulation study of a binary mixture of hard spherocylinders with different diameters (D1,D2) and the same lengths (L15L25L). We first study a mixture of spherocylinders with lengths L 515D2 and D150, which can be regarded as a mixture of rodlike colloids and ideal needles. We find clearly an entropy-driven isotropic-isotropic (I-I) demixing transition in this mixture. In addition, we study a mixture of spherocylinders with diameter ratio D1 /D250.1 and we investigated the I-I demixing transition as a function of the length L of the particles. We observe a stable I-I demixing for all values of L in the range of 3<L/D2<15, but we could not reach the limit L50, i.e., the hard-sphere mixture with diameter ratio of 0.1. Striking agreement is found for L/D2515 with the results that follow from the second virial theory for infinitely elongated rods. For L/D252, we did not find a demixing transition till a total packing fraction of h50.581, which is higher than the packing fraction at which freezing occurs for a pure system of thick rods. Thus this result and the extrapolation of our finite-L data to L50 gives us a fingerprint that the fluid-fluid demixing transition in the binary hard-sphere mixture with a diameter ratio of 0.1 is metastable with respect to freezing or does not exist at all at densities below close packing.

Reaction-field and Ewald summation methods in Monte Carlo simulations of dipolar liquid crystals

The treatment of the long-range dipolar interactions in simulations of mesogens is examined. After a brief reformulation of the standard Ewald summation and reaction-field methods in the general context of electrostatics using Green functions, we report the results of Monte Carlo simulations of liquid crystalline phases for L/D = 5 hard spherocylinders (cylinder length Land diameter D) with central point dipoles oriented along the main axis of the cylinder. In the case of N = 1020 particles an equivalent description of the thermodynamic properties and the structure of the phases is obtained with both techniques. A good description of the dielectric constant of the surrounding continuum is achieved by using a simple selfconsistent iterative method based on the calculation of the dielectric constant within the cell. The reaction-field method allows a systematic study of the phase behaviour of the system to be made with relatively modest computational requirements. We make a preliminary assessment of the phase behaviour for this system. In the case of molecules with central longitudinal dipoles, the nematic phase is destabilized relative to the isotropic (I) and the smectic-A (SmA) phases when compared with the non-polar system; the nematic (N) phase disappears altogether when the temperature is lowered below an I–N–SmA triple point. Furthermore, there is no evidence of ferroelectricity although some short-range antiferroelectric ordering is seen. The destabilization of the nematic phase relative to the smectic phase is also seen for systems with central transverse dipoles, but contrasts with that for molecules with terminal longitudinal dipoles where the smectic-A phase is destabilized.

Virial coefficients and equation of state of hard ellipsoids

The first five virial coefficients of hard ellipsoids have been evaluated numerically. Hard ellipsoids with length to breadth ratio in the range 3 to 10 were considered. Differences in the virial coefficients between prolate, oblate and biaxial ellipsoids with the same length to breadth ratio have been analysed. We were able to fit the virial coefficient data of hard ellipsoids to an empirical expression which contains only two non-sphericity parameters. This expression describes also correctly the virial coefficient of other convex bodies such as hard spherocylinders. Furthermore, computer simulations were performed for hard ellipsoids. It is found that for a given volume fraction and length to breadth ratio, the compressibility factor of the biaxial ellipsoid is smaller than that of the prolate spheroid, while the compressibility factor of the prolate spheroid is smaller than that of the oblate spheroid. However, differences in the compressibility factor for these three hard models were found to be small. This is surprising since their virial coefficients are quite different. An explanation for this is proposed. The accuracy of several analytical equations of state for hard convex bodies was analysed. Although these equations provide reasonable results they predict the same equation of state for prolate and oblate spheroids. Moreover these equations do not provide good predictions of virial coefficients for very elongated molecules. A new equation of state which uses the computed values of the first five virial coefficients of hard ellipsoids is proposed. This new equation of state is basically a truncated virial expansion, which uses Parsons-like scaling to estimate virial coefficients higher than the fifth. It is shown that this new equation of state performs better than those previously proposed for hard ellipsoids. Moreover it distinguishes between prolate, oblate and biaxial ellipsoids. The new equation of state describes satisfactorily the computer simulation data of hard spherocylinders.

Non-spherical molecules with spherical square-wells revisited: some possible new phase diagrams from density functional theory

A simple model fluid, in which molecules consist of hard non-spherical cores embedded in spherical square-wells, has been re-examined. Within the Gaussian approximation of Onsager–Parsons–Lee theory, for very narrow attractive square-wells, ordinary liquid–vapour coexistence is preempted by a direct transition between a low density disordered fluid phase and a nematic phase. This is consistent with recent simulation results on fluids of Gay–Berne particles, and of hard spherocylinders with generalized square-wells which probed the effect of changing the ratio of attraction range to molecular elongation. Moreover, coexistence between two nematic phases is found on further decreasing the width of square-well, analogous to the isostructural solid–solid transitions observed in fluids characterized by very short range isotropic interactions.

Non-spherical molecules with spherical square-wells revisited: some possible new phase diagrams from density functional theory

A simple model fluid, in which molecules consist of hard non-spherical cores embedded in spherical square-wells, has been re-examined. Within the Gaussian approximation of Onsager–Parsons–Lee theory, for very narrow attractive square-wells, ordinary liquid–vapour coexistence is preempted by a direct transition between a low density disordered fluid phase and a nematic phase. This is consistent with recent simulation results on fluids of Gay–Berne particles, and of hard spherocylinders with generalized square-wells which probed the effect of changing the ratio of attraction range to molecular elongation. Moreover, coexistence between two nematic phases is found on further decreasing the width of square-well, analogous to the isostructural solid–solid transitions observed in fluids characterized by very short range isotropic interactions.

Plastic crystal phases of hard dumbbells and hard spherocylinders

We compare the solid-fluid phase diagrams calculated from computer simulations of hard dumbbells and hard spherocylinders in the region of anisotropy where a plastic crystal phase is stable for both systems. When appropriate reduced units are used for the pressure and densities, the coexistence properties for these models are very similar. The existing evidence suggests that for both models the density change on freezing of the fluid into the plastic crystal phase becomes very small as the limit of stability of the plastic crystal phase is reached, but the transition remains first order.

Numerical study of the phase behavior of rodlike colloids with attractive interactions

We examine the influence of attractive interactions on the phase behavior of rodlike colloids. We model the rodlike particles by spherocylinders, for which the phase diagram, in the absence of attraction, is known for arbitrary aspect ratio. We consider the case that the attraction is due to depletion forces caused by the addition of nonadsorbing polymer. The range of this attraction is determined by the size of the polymer. If the radius of gyration of the polymer is small compared to the diameter of the rods, we can model the polymer-induced attraction by a suitable generalization of the square-well model for spherical particles. However, for longer ranged attractions, pairwise additive attractions lead to phase behavior that is very different from what is found when the nonadditivity of depletion forces is taken into account. In our simulations, we find evidence for demixing transitions in the isotropic, nematic and solid phases. We compare our simulation results with predictions based on the perturbation theory of Lekkerkerker and Stroobants [Nuovo Cimento D 16, 949 (1994)]. A crucial input in this theory is the so-called free-volume fraction of the hard spherocylinder reference system. In the work of Lekkerkerker and Stroobants, this quantity is estimated using scaled-particle theory. We test the validity of this approach by comparing it to numerical results for the free-volume fraction.

The isotropic-nematic transition of dipolar spherocylinders: combining thermodynamic perturbation with Monte Carlo simulation

We attempt to predict the phase behaviour of elongated molecules possessing longitudinal point dipoles by treating the dipolar interaction as a perturbation on a reference system consisting of hard spherocylinders. Thermal averages of up to the fourth power of the total dipolar interaction are evaluated by Monte Carlo simulation in the NVT ensemble of the nonpolar spherocylinders over a range of densities spanning the isotropic-nematic transition density. From the evaluation of the first two terms in the high-temperature perturbation expansion of the free energy we find that the nematic phase is destabilized relative to the isotropic phase (when compared with the non-polar system) as one switches on a weak central dipolar interaction. In contrast, off-centre dipoles appear to have a stabilizing effect on the nematic phase for weak dipoles, but as the dipole strength increases the nematic is seen to be destabilized with respect to the isotropic phase.

The isotropic-nematic transition of dipolar spherocylinders: combining thermodynamic perturbation with Monte Carlo simulation

We attempt to predict the phase behaviour of elongated molecules possessing longitudinal point dipoles by treating the dipolar interaction as a perturbation on a reference system consisting of hard spherocylinders. Thermal averages of up to the fourth power of the total dipolar interaction are evaluated by Monte Carlo simulation in the NVT ensemble of the nonpolar spherocylinders over a range of densities spanning the isotropic-nematic transition density. From the evaluation of the first two terms in the high-temperature perturbation expansion of the free energy we find that the nematic phase is destabilized relative to the isotropic phase (when compared with the non-polar system) as one switches on a weak central dipolar interaction. In contrast, off-centre dipoles appear to have a stabilizing effect on the nematic phase for weak dipoles, but as the dipole strength increases the nematic is seen to be destabilized with respect to the isotropic phase.

The liquid crystalline phase behavior of dimerizing hard spherocylinders

The phase behavior of dimerizing (associating) rigid particles is studied by both theory and computer simulation. The model molecule comprises a hard spherocylinder of length L and diameter D with a terminal square well bonding site embedded in one of the hemispherical caps. This model mimics the properties of simple hydrogen bonding mesogens; for example, mesogens with a carboxylic acid end group which are capable of forming dimers. A recently proposed theory of the isotropic (I)-nematic (N) phase transition for long hard spherocylinders with an attractive site at one end [R. P. Sear and G. Jackson, Mol. Phys. 82, 473 (1994)] is extended to shorter molecules. In the original theory the free energy is truncated at the level of the second virial coefficient. We now include the higher virial coefficients in an approximate manner with a Parsons type scaling. The accuracy of the theory is demonstrated by comparison with novel Monte Carlo simulation data for the same model. Excellent agreement is found for densities, pressures and degrees of association especially at the liquid crystalline phase transition. In comparing the results for the L/D=5 associating system with those for its nonassociating analogue, the nematic phase is seen to be stabilized relative to the isotropic phase, while the nematic (N)-smectic-A (SmA) transition occurs at approximately the same density. The I-N transition for the dimerizing system is clearly first order, while the N-SmA is essentially continuous. The smectic-A phase has a monolayer structure which is similar to that formed by the nonassociating system. Furthermore, a system of otherwise nonmesogenic molecules with L/D=3 has a stable liquid crystal phase when dimerization is made possible with the inclusion of the terminal bonding sites. Rather than being a nematic phase, this phase is surprisingly found to have the layered structure of a smectic-A phase. We discuss our results in terms of the increase in the ‘effective’ aspect ratio as a result of association.

Chain and ring structures in smectic phases of molecules with transverse dipoles

Systems of hard spherocylinders with central transverse point dipoles are studied by Monte Carlo simulation. The dipoles are seen to stabilise the formation of the smectic-A phase relative to the nematic phase; the nematic phase disappears altogether at low temperatures. In the smectic-A phase the cores of the molecules align parallel to the layer normal with the dipoles in the plane. At high temperatures the dipoles are orientationally disordered; as the temperature is lowered ring-like domains appear, and then elongated antiferroelectric chain-like domains form. We analyse these results in the context of recent studies on two-dimensional vortex-antivortex systems.

Free energy barriers for interlayer diffusion in the smectic-A phase of hard spherocylinders

Computer simulations are used to investigate the recent prediction (R. van Roij et al., 1995, Phys. Rev., E52, R1277) of transverse interlayer order in the smectic-A phase of hard spherocylinders. For short rods, and close to the melting point of the smectic-A phase, a fraction of about one per cent of the particles are in the transverse interlayer state. For longer rods and further away from the melting transition a much lower fraction is found. Possible implications for interlayer diffusion and end-over-end rotational diffusion are discussed.

Free energy barriers for interlayer diffusion in the smectic-A phase of hard spherocylinders

Computer simulations are used to investigate the recent prediction (R. van Roij et al., 1995, Phys. Rev., E52, R1277) of transverse interlayer order in the smectic-A phase of hard spherocylinders. For short rods, and close to the melting point of the smectic-A phase, a fraction of about one per cent of the particles are in the transverse interlayer state. For longer rods and further away from the melting transition a much lower fraction is found. Possible implications for interlayer diffusion and end-over-end rotational diffusion are discussed.

Tracing the phase boundaries of hard spherocylinders

We have mapped out the complete phase diagram of hard spherocylinders as a function of the shape anisotropy L/D. Special computational techniques were required to locate phase transitions in the limit L/D→∞ and in the close-packing limit for L/D→0. The phase boundaries of five different phases were established: the isotropic fluid, the liquid crystalline smectic A and nematic phases, the orientationally ordered solids—in AAA and ABC stacking—and the plastic or rotator solid. The rotator phase is unstable for L/D⩾0.35 and the AAA crystal becomes unstable for lengths smaller than L/D≈7. The triple points isotropic-smectic-A-solid and isotropic-nematic-smectic-A are estimated to occur at L/D=3.1 and L/D=3.7, respectively. For the low L/D region, a modified version of the Gibbs–Duhem integration method was used to calculate the isotropic-solid coexistence curves. This method was also applied to the I-N transition for L/D&amp;gt;10. For large L/D the simulation results approach the predictions of the Onsager theory. In the limit L/D→∞ simulations were performed by application of a scaling technique. The nematic-smectic-A transition for L/D→∞ appears to be continuous. As the nematic-smectic-A transition is certainly of first order nature for L/D⩽5, the tri-critical point is presumably located between L/D=5 and L/D=∞. In the small L/D region, the plastic solid to aligned solid transition is first order. Using a mapping of the dense spherocylinder system on a lattice model, the initial slope of the coexistence curve could even be computed in the close-packing limit.

The liquid-crystalline phase behaviour of hard spherocylinders with terminal point dipoles

We examine the effect of dipolar interactions on the liquid-crystalline phase behaviour of L/D = 5 hard spherocylinders with a terminal point dipole. The hard spherocylinder consists of a cylinder of length L and diameter D with hemispherical caps on each end; the point dipole is located at the centre of the hemispherical cap (2.5D from the centre of the spherocylinder) and is oriented along the principal molecular axis. The phase transitions exhibited by this system are studied using the isothermal - isobaric Monte Carlo (MC-NPT) technique. As for systems with central dipoles, the terminal dipole is seen to slightly destabilize the nematic (orientationally ordered) phase relative to the isotropic phase when compared with the non-polar hard spherocylinders. More interestingly, the smectic (layered) phase is destabilized in the terminal dipole case, and is only seen at the very highest densities. This is in stark contrast to what is seen for systems with central point dipoles in which the smectic phase is stabilized relative to the nematic phase due to the strong anti-parallel dipolar interactions. We do not find any evidence for ferroelectric or anti-ferroelectric ordering in these systems.

Smectic-A Phase of a Bidisperse System of Parallel Hard Rods and Hard Spheres

A binary system of parallel hard spherocylinders and hard spheres are considered to examine influence of addition of spherical molecules to the pure system of rod-like molecules, especially the influence on the structure of the smectic-A phase. Free energy of the system is calculated through the second virial approximation, and nematic–smectic-A transition is estimated by stability analysis of nematic state with sinusoidal density fluctuation. Constant pressure Monte Carlo simulation is also used to confirm a prediction of the stability analysis which prediction indicates that the addition of spherical molecules induces a smectic-A phase as a microphase separation between the spheres and the spherocylinders.

RESEARCH NOTE A method for predicting the virial coefficients of hard linear homonuclear molecules

This paper proposes a simple method for predicting the virial coefficients, up to the fifth, for hard linear homonuclear molecules. The method uses the virial coefficients for hard spherocylinders and hard linear tangent spheres to obtain the virial coefficients of linear fused hard spheres. The results are in very good agreement with the known values for the latter kind of molecules. Thus the method is suitable for predictive purposes when virial coefficients are not known, which makes it useful for the derivation of certain equations of state.