Soft potentials vary smoothly with distance. They may be simply repulsive such as the soft sphere potential (see figure II.2), or they may have additional attractive components as in the Lennard-Jones 12-6 potential (see figure n.3) for example. The origins of soft potentials have already been outlined in chapter II and so in
this section we will concentrate on the simulations that have been performed to date.
Single site soft potentials used in liquid crystal simulation must necessarily have an orientational dependent term and as such are all types of Comer potential [51]. Anisotropic soft potentials based on the Lennard-Jones 12-6 potential are used [36].
A continuous potential may be formed from the scalar Lennard-Jones plus a suitable anisotropic component. Luckhurst and Romano have investigated the behaviour of a system interacting via the potential;
Vlr = v0+va . pn.il]
A Lennard-Jones 12-6 form is chosen for V0. Assuming the particles are rigid and cylindrically symmetric the anisotropic potential VA, describing the interaction of these particles, may be expanded in a series of spherical harmonics. It should be noted that real mesogens do not possess such cylindrical symmetry however, this may not be a serious approximation. From a molecular cluster viewpoint, clusters of molecules are less anisometric than individual molecules. If the clusters are preserved at the transition then the shorter range anisotropic forces of individual molecules may not be so important in nematic phase formation. Truncating this expansion at the second-rank terms yields;
In [TIL 12] P2 is once again the second Legendre polynomial and P12 is the angle
between the symmetry axes of two particles. In essentially an extension of a previous simulation [52] that was confined to a lattice and considered only nearest neighbour interactions the scaling term w2 (^12) is chosen to take one of
two forms; K'2(n2) = —4Xe0f g - J ; [m.13] «" 2ini) = -4A.s0 ( / \1 2 ✓ f £ o | _ f£ o ] {r12J {rn J P . 14]
In the expressions p . 13] and p . 14] a 0 and s0 are the Lennard-Jones strength and range parameters and r12 is the particle separation. The parameter X is used to control the anisotropy and was given the value X = 0*15 selected after preliminary simulations: smaller values of X were not found to exhibit a liquid crystal phase above the melting point and additionally larger values of X gave orientationally ordered phases that were found to be stable right up to the boiling point. Results using the pair potentials of p . 13] and p . 14] were found to be qualitatively similar and to be in reasonable accord with the nematogen 4,4- dimethoxyazoxybenzene.
In the first simulation of a single-site soft potential with substantial anisotropy Kushick and Berne [53] used the Gaussian overlap model proposed by Berne and Pechukas [54] (see chapter II) in a simulation of 144 ellipses of axial ratios 2-to-l and 3-to-l. At the start of the simulation the principle axes of the ellipses are orientated by the action of a magnetic field. Thereafter the field is switched off
and the system is allowed to relax. An order parameter given by equation [111.15] is monitored to see if the induced orientational order is destroyed at various densities. In two dimensions;
In expression [III. 15] the u, and n represent the directions of the principle axis of each ellipse and the direction of the director respectively. It is to be noted that the average is taken over all the particles of the last configuration only. This necessarily leaves the results susceptible to the inherent fluctuations in the degree of alignment of the particles of an orientationally ordered phase. This effect would be further enhanced close to phase transitions. Nevertheless the results for the ellipses indicated a persistence of orientational ordering in a system at
p /p 0 = 0-75 with £ = 0*83. With knowledge of the solid-nematic transition
observed by Vieillard-Baron [7] occurring at p /p 0 = 0-87 for hard ellipses of
axial ratio 6-to-l, together with the observed melting density of hard discs (axial ratio 1) known to be p /p o = 0*79 the authors assumed, quite reasonably, that they had achieved stable rotational ordering below the density at which the solid is expected to form. By contrast an orientationally ordered state was observed for ellipses with axial ratio 2-to-l at p / p0 = 0-90 but this was not sustained at a density of p / p 0 = 0 - 8 0 . The higher density is thought to correspond to an
ordered solid phase.
The authors further report that they had also observed stable nematic ordering with soft ellipsoids in three-dimensions with the parameters a = 3 • 5, p / p0 = 0•71
and £ = 0 • 85 using an appropriate expression for Q valid in three-dimensions that is in fact;
("-••*)• p11-16}
These were important results because they demonstrated the usefulness of the Gaussian overlap model in simulating orientationally ordered phases. However,
these were preliminary experiments. Only a small number of particles were simulated at any time. Problems were encountered whereby the system would enter a long-lived metastable state or configurational phase-space bottleneck from which it was difficult to escape.
Tsykalo and Bagmet [55] studied the nematic phase of the Beme-Pechukas potential and of two modifications of the potential; a WCA [56] type split of the potential and a version of the potential scaled by a function dependent on the orientation of the particles with respect to the intermolecular vector. Thus;
^scaledBP “ 2/- B- M’ a z(u1,u2,r) [HI. 17]. where a(u!,u2,f) is the range function of the Beme-Pechukas potential (see equation [11.15]). A potential of the form of [HI. 17] does indeed introduce the necessary intermolecular vector dependency but the well depth anisotropy is necessarily controlled by the shape anisotropy parameter %. Ideally a parameter independent of % which may be adjusted to vary the well depth anisotropy is preferred. These systems were studied over a range of shape anisotropy in the
side-to-side end-to-end interaction ratio; 3 • 5 > cjs/ a e > 2 • 0 at a variety of packing fractions; 0*50< rj <0*55, for only 1000 time steps after equilibrium. At sufficiently high packing fraction and shape anisotropy each system was found to possess residual nematic order after equilibration. The simulation results reported here cannot be considered accurate as only residual order is observed in systems that have been allowed to relax. Given sufficient length of run the residual order may eventually make way for a more stable thermodynamic phase with a different or perhaps even negligible degree of orientational ordering. Spontaneous order is not observed in this work, the importance of the work of Tsykalo and Bagmet therefore, lies in demonstrating the possibility of simulating anisotropic liquid crystalline phases using continuous potentials. Furthermore, although not elaborated on by the authors, the idea of including a well depth scaling term based on the relative orientation of the particles with respect to the intermolecular vector is introduced; an idea later expanded upon by Gay and Berne [57].
A nematic-isotropic transition was reported for a system of Beme-Pechukas centres [58] although in these studies again only a small number of particles were simulated with ensemble averages being collected only over the final 1000 simulation time steps after equilibrium had been attained. More recently the importance of large particle systems and long simulation runs over which the ensemble averaging is performed has been recognised. The results reported in the above publication are in qualitative agreement with experimental results and the tacit discrepancies reported therein are probably due to the aforementioned factors of system size and ensemble averaging.
In fact more recently it has been found that the nematic phase of the Beme- Pechukas potential is not thermodynamically stable. After extensive simulations it returns to the isotropic liquid phase [59].
Subsequently, several unrealistic features in the Beme-Pechukas potential were identified and in an attempt to correct for these undesirable features a cut and shifted rather than a simply scaled potential was employed by Gay and Berne [57]. The details of this Gay-Beme potential can be found in chapter II (see section n.2.3) where we have described the origins of the hybrid Gay-Beme Luckhurst-Romano potential which we have used in simulations of single-site calamitic and discotic liquid crystals and in a more ambitious multisite study of a calamitic mesogen; a discussion of some of the simulations to date using the Gay- Beme potential will suffice here.
i
The first successful simulation of a Gay-Beme fluid may be attributed to Adams et al [59]. Simulations were performed using the original parameterisation of Gay and Berne [57]; thus v = 1, p = 2, cjs / cre = 3 and es / se = 5. These values are chosen to fit the Gay-Beme potential to a linear array of 4-Lennard-Jones centres each separated by 2a0 /3. In these preliminary simulations an isotropic to nematic transition is identified and the approximate transition temperature located, the nematic phase becoming stable at T*< 1 • 7 for p* = 0-32, (p* = pGq / m, a 0 is the Lennard-Jones range parameter, m is the mass of the particle).
Luckhurst et al undertook more extensive simulations using a slightly differently parameterised version of the Gay-Beme. In particular the exponents v and p were exchanged so that the values v = 2 and p = 1 were employed. This change
does not effect the relative well depth minima for the side-by-side and end-to-end configurations, but it does help to stabilise the side-by-side configuration with respect to the X and T configurations. Thus it is thought that this parameterisation would be more likely to induce liquid crystal formation. The range and strength parameters, % and %’ respectively, retain their previous values:
thus a s/ a e = 3 and ss/e e =5. MD simulations performed with the above parameterisation demonstrate that the Gay-Beme model mesogen exhibits a diverse range of mesophases. In systems of N = 256 particles Luckhurst et al [60] identify nematic, smectic A, smectic B and a crystal phase in addition to the isotropic liquid in simulations performed at a density p* = 0-30, with the aid of distribution functions and computer graphics.
Visualisation of the shape of the Gay-Beme potential obtained by plotting the contour corresponding to f/GB(ui,u2,r) = 0 [60] shows that it is ellipsoidal in shape. It is therefore perhaps surprising that the Gay-Beme potential forms smectic liquid crystal phases because of the scaling arguments introduced by Frenkel [29]. However the argument proposed by Frenkel refers to hard particles only. It is therefore the long ranged anisotropic part of the Gay-Beme potential that is responsible for the formation of the observed more highly ordered mesophases. It should be noted that there is an internal well depth minimum in the Gay-Beme particle. This was also recognised by Gay and Berne [57]. However this is unlikely to cause problems in simulations of Gay-Beme particles since it is separated from the external potential minimum by an infinite potential barrier.
Extensive simulations of the Gay-Beme fluid have been performed by de Miguel and co-workers [61-67]. In particular the phase diagram of the Gay-Beme fluid
parameterised according to the original prescription of Gay and Berne [57] has been sketched out. With this parameterisation isotropic, nematic, smectic-B and a transition to a tilted smectic-B phase in addition to the crystalline phase are identified [64]. A smectic-A phase is not observed, although a density wave indicative of smectic-A layer structure was observed but found not to be stationary throughout the simulation. After sufficiently long runs the phase was in fact identified as smectic-B. Interestingly no smectic phase is observed along compression of an isotherm at T* = 1 • 25. This is thought to be due to the system becoming trapped in a metastable region of phase space preventing the onset of one-dimensional spatial order, as subsequent simulations along isochores [65] show that the smectic-B phase is stable at higher temperatures. Along isotherms T* < 0 • 80 the isotropic phase is seen to evolve directly into the smectic-B on compression of the system.
The observation of a tilted smectic-B phase [64] may be an artefact of compressing the system. In previous simulations of smectic-A and B phases the system director was always found to be aligned along the cubic simulation box diagonal [60]. It has been argued by Luckhurst et al that a diagonal orientation of the director allows the constraint that smectic layers must be commensurate with system periodic images to be more easily satisfied. It is quite possible therefore that once the director has been pinned in a certain direction say in a smectic-B phase further compression of the system may result in the observed tilted smectic-B phase, the director being unable to rotate in order to contain the correct number of layers composed of molecules normal to the layer surfaces commensurate with the system periodic images. However the observed tilt transition does appear to be weakly first order and has been observed in experimental systems [64].
In addition de Miguel et al have performed simulations on a WCA [56] split of the Gay-Beme potential [64]. The results indicated that the long range anisotropic attractive part of the potential does stabilise the onset on orientational order. For example along the isotherm T* = 0 -95 the isotropic-nematic transition is identified at the following reduced densities: full Gay-Beme potential, p* = 0 • 315; WCA Gay-Beme, p* = 0 • 335. However with the WCA Gay-Beme potential no smectic phases are observed indicating that the long range anisotropic attractive forces are vital to the formation of the smectic phase.
The same conclusions are reached in a study of the Gay-Beme mesogen parameterised to represent a real nematogen [68]. A methodology for projecting out the biaxiality of typical mesogens by performing Boltzmann weighted averages of the form;