Hard ellipsoids

Camp P J, Mason C P, Allen M P, Khare A A and Kofke D A 1996 The isotropic-nematic transition in uniaxial hard ellipsoid fluids coexistence data and the approach to the Onsager limit J. Chem. Phys. 105 2837-49... 

Frenkel D, Mulder B M and McTague J P 1984 Phase-diagram of a system of hard ellipsoids Phys. Rev.L 52 287-90... 

It has not proved possible to develop general analytical hard-core models for liquid crystals, just as for nonnal liquids. Instead, computer simulations have played an important role in extending our understanding of the phase behaviour of hard particles. Frenkel and Mulder found that a system of hard ellipsoids can fonn a nematic phase for ratios L/D >2.5 (rods) or L/D <0.4 (discs) [73] however, such a system cannot fonn a smectic phase, as can be shown by a scaling... 

Frenkel D and Mulder B 1985 The hard ellipsoid-of-revolution fluid. 1. Monte-Carlo simulations Mol. Phys. 55 1171-92... 

To make evaluations more definite, we use optical and microwave experimental data, as well as calculations of molecular dynamics of certain simple liquids which usually fit the experiment. Rotation is everywhere considered as classical, and the objects are two-atomic and spherical molecules, as well as hard ellipsoids. 

In simple single-site liquid crystal models, such as hard-ellipsoids or the Gay-Berne potential, a number of elegant techniques have been devised to calculate key bulk properties which are useful for display applications. These include elastic constants for nematic systems [87, 88]. However, these techniques are dependent on large systems and long runs, and (at the present time) limitations in computer time prevent the extension of these methods to fully atomistic models. 

The first of these was by Vieillard-Baron [5] who investigated a system of spherocylinders but failed to detect a liquid crystal phase primarily because the anisometry, L/D, of 2 was too small [37]. He also attempted to study a system of 2392 particles with the larger L/D of 5 but these simulations had to be abandoned because of their large computational cost. However, in view of the ellipsoidal shape of the Gay-Berne particles it is the behaviour of hard ellipsoids of revolution which is of primary relevance to us. 

It is clear that systems of hard ellipsoids exhibit an intriguingly simple phase behaviour with some resemblance to that of real nematogens. However, such systems cannot form smectic or columnar phases and in addition the phase transitions are not thermally driven as they are for real mesogens. As we shall see in the following sections the Gay-Berne potential with its anisotropic repulsive and attractive forces is able to overcome both of these limitations. 

Fig. 3a,b. The phase behaviour of a system of hard ellipsoids, both prolate and oblate, as a function of the ellipticity, alb, plotted against a the packing fraction, p b the scaled number density, p ... 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

There have been several simulations of discotic liquid crystals based on hard ellipsoids [41], infinitely thin platelets [59, 60] and cut-spheres [40]. The Gay-Berne potential model was then used to simulate the behaviour of discotic systems by Emerson et al. [16] in order to introduce anisotropic attractive forces. In this model the scaled and shifted separation R (see Eq. 5) was given by... 

The simplest model potentials that form liquid crystals are the hard ellipsoid fluid and the hard cylinder fluid [4]. Linear and angular momenta are constant between collisions so that very efficient molecular dynamics algorithms can be devised. Unfortunately, when transport coefficients are calculated external fields and thermostats are often applied. That means that the particles accelerate between collisions. The advantages of using hard body fluids is conse-... 

The first-rank and second-rank OTCFs are mostly studied because of their relevance to experiments. One of the early computational studies of orientational dynamics in the isotropic phase near the /-A transition is due to Allen and Frenkel [105]. In their molecular dynamics simulation study of a system of N = 144 hard ellipsoids of revolution, the slowdown of orientational dynamics on approaching the I-N transition was captured—in particular, in the time evolution of the... 

Everaers, R. and Ejtehadi, M. R. 2003. Interaction potentials for soft and hard ellipsoids. 

Cotter is based on the scaled particle theory. The alternative way is to use the so-called y-expansion of the hard-core free energy proposed by Gelbart and Barboy [44]. This is an expansion in powers of 7] /(1 - rj) which is much more reliable at high densities compared with the usual virial expansion in powers of rj. The y-expansion has also been used by Mulder and Frenkel [51 ] in the interpretation of the results of computer simulations for a system of hard ellipsoids. 

The form of the function efr ( ) is different in different versions of the smoothed-density approximation proposed by Somo-za and Tarazona [71, 72] and by Poniwier-ski and Sluckin [69, 73]. The density functional model of Somoza and Tarazona is based on the reference system of parallel hard ellipsoids that can be mapped into hard spheres. In the Poniwierski and Sluckin theory the effective weight function is determined by the Maier function for hard sphe-rocylinders and the expression for Ayr (p) is obtained from the Carnahan-Starling ex-... 

There is only a rudimentary microscopic theory for the chain length dependence of the rotational viscosity. The molecules are treated in these theories as hard rods or ellipsoids, and thus the increasing flexibility of the chains with increasing chain length is not taken into account. For hard ellipsoids Baalss and Hess [41] found that... 

At the high level of final state resolution provided by such experiments we can discern quantal interference effects. The more prominent feature for inelastic excitation is a rotational rainbow that arises by a mechanism similar to the intense scattering of the final velocity into certain directions (Section 2.2.5). Here too, the rainbow arises from different trajectories scattered into the same final state except that the state is specified not only by the direction of v but also by the rotational state of the molecule, NO in the case of Figure 10.9. This is a stereodynamic effect because the final state is determined not only by the impact parameter but also by the angle of approach, as shown for scattering by a hard ellipsoid in Figure 10.10. 

Figure 10.10 Scattering of an atom off a hard ellipsoid for a given impact parameter and angle of approach, y. The scattering angle is Q. The recoil momentum Ap corresponds to a change Aj = R x Ap in the rotational angular momentum of the ellipsoid. The two panels correspond to two different impact parameters that lead to scattering into the same final state.

FIGURE 1 The phase behaviour of hard ellipsoids as a fimctim of the aspect ratio x [2,9]. The reduced density p is defined such that the dmsity of regular close packing is equal to V2 fiv all x. The gr areas indicate the coexistence regions between the various diases, whidi are isotrr ic (I), nematic (N), solid (Cr) and plastic solid (PCr). The solid points indicate the simulation results. 

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