Mean Square Particle Displacement and Diffusion Coefficients

For the study of liquid crystal phases it is important to be able to check on the fluidity of a phase. Herein lies the advantage in carrying out MD simulations which allow us to examine some of the dynamic properties of the systems under study. The mean square particle displacement provides a measure of the particle

self diffusion within a simulation. The gradient of a graph of mean square particle displacement against time t relates to the Einstein expression for the diffusion coefficient of molecules at long times (see section 11.6.1, equation fl.65). Results of the diffusion coefficients and their components resolved parallel and perpendicular to the director calculated by considering only the last two thirds of the extended runs are presented in table IV.4. In addition figures IV. 13a and IV. 13b illustrate the behaviour of the mean square displacement observed for the extended simulations (the whole run) recorded at the reduced temperatures (T *) = 5 • 0 and (T *) = 2 • 77.

In both figures the unresolved mean square diffusion shows a straight line behaviour at long time. This is characteristic of liquid like diffusion. In figure IV. 13a recorded from a run with ensemble average {Pi)~ 0-62 (well into the nematic phase) there does not appear to be a difference between the curves of mean square displacement resolved with respect to the system director. This is reflected in table IV.4 which shows that the particle self diffusion coefficients

D| * and D± * at this temperature is equal. At the lower temperature there is an obvious difference between the behaviour of the resolved components of the mean square displacement. The gradient of the component perpendicular to the director is greater than that parallel. Reference to table IV.4 reveals that the particle self diffusion coefficient in a direction perpendicular to the director is

(T *) 6 D * (P2) 5-3±0-2 57-84±002 0-49±0-06 5-0±0-2 51-74±0-02 0-62±0-02 2-88±008 19-30±0-02 0-889±0008 2-77±0-09 5-36±0-01 0-901±0-007 2-58±0-07 0-662±0001 0-911±0007 (T*) 6 ^ * 6D± * Dll* / D 1 * 5-3±0-2 19-44±0-05 19-20±0-02 101 5-0±0-2 17-90±003 16-92±0-02 106 2-88±0-08 6-15±0-02 6-67±002 0-92 2-77±0-09 l-301±0-004 2-028±0 004 0-64 2-58±0-07 0-1106±00002 0-2757±0-0009 0-40 Table IV.4

The reduced diffusion coefficients D * and components resolved parallel and perpendicular to the system director for the extended runs of disc-like parameterised single HGBLR centres. The fifth column presents the ratio of the parallel to perpendicular diffusion coefficients. The ensemble averaged order parameter (P2)

approximately one-and-a-half times greater than that parallel to the director. This indicates that the molecules move further and hence more freely in a direction perpendicular to the system director at this temperature.

Previous simulations of rod-like and disc-like mesogens have shown further displacement of the particles parallel and perpendicular to the director respectively, compared to other directions. Sometimes the coefficient of diffusion in the preferred direction is actually higher than for the isotropic phase just after the transition [7]. On moving through the isotropic-nematic phase

transition we would expect mean particle displacement perpendicular to the director to be greater than that parallel to the director. This obtains because in

the discotic nematic phase an arbitrary mesogen may move more easily perpendicular to the director as on average there is less likelihood of it encountering further discogens possessing an orientation perpendicular to itself impeding further motion in that direction. However in an isotropic phase there is an equal probability of all molecular orientations and thus an equal chance of other discogens presenting a large molecular surface to impede further displacement. Hence in the isotropic phase there is no preferred direction of displacement

However as clearly illustrated in figure IV. 13a, there is no difference between the resolved components of mean square displacement in the nematic phase of HGBLR centres. This can be explained by taking into account the spherical hard core of the HGBLR centres. Mesogens are prevented from diffusing further when they encounter the repulsive hard core of another mesogen. If we assume, in the nematic phase, that the longer range dispersive forces play no role in the

120 t 100 <*-< o 80

parallel

perpend

icular

Figure IV.13a

Total and resolved components with respect to the system director of particle mean square displacement for the extended runs of the system of disc-like HGBLR centres at (r * ) = 5-0. The resolved components are indicated, the unlabelled curve corresponds to the total mean square particle displacement.

12 T perpendicular a 8 ^ _H parallel 0 50 100 150 200 250

f*

Figure IV.13b

particle self diffusion process, then the orientated anisotropic attractive regions of HGBLR centres will not influence the diffusion process. The spherical hard core (for a fixed orientation of two HGBLR centres) then presents itself as an equal restriction to diffusion in all directions. Thus diffusion behaviour in the nematic phase is qualitatively the same as that expressed in the isotropic phase. We do observe an overall decrease in the magnitude of diffusion coefficient as the temperature is lowered.

As previously stated the radius of the spherical hard core depends on the fixed orientation of the HGBLR centres. However the difference between the minimum hard core radius and the maximum is small (see figures IV.2 and IV.3). Components of mean square displacement resolved with respect to the system director in the nematic phase essentially compare the ease of diffusion between side-side and end-end oriented centres. Examination of figures IV.2 and IV.3 shows that the HGBLR potential has the same hard core radius for these two orientations. Thus disc-like HGBLR centres trying to diffuse perpendicular to their symmetry axes would experience the same hard core radii presented by the surrounding molecules as they would experience trying to diffuse parallel to their symmetry axes.

A difference between the diffusion coefficients parallel and perpendicular to the director does emerge as the system enters the more highly ordered phase. This may be explained in terms of the environment that each particle finds itself. In the nematic phase each particle can be considered to be in identical surroundings as its neighbours: there is no positional ordering of the centres of mass, and on average the particles posses the same orientation. However when the particles enter the more highly ordered columnar phase, there is a distinction between

neighbours that belong to the same column and those that do not. Discs within columns are prevented from moving along the column axis because of the co­ operative motion that would be required from the discs above and below. It is far easier for the discs to diffuse perpendicular to the director.

This observation is in contrast to what is seen in experimental columnar systems. These have been described as orientated one-dimensional liquids [8]. The translational disorder of the molecules occurs within the columns themselves. However, high resolution X-ray studies on columnar systems have shown although the flat discotic cores are orientated with respect to each other, the hydrocarbon chains surrounding the cores exhibit practically isotropic scattering patterns. A detailed model of discotic liquid crystals would necessarily have to include the conformational degrees of freedom of the hydrocarbon chains [8]. Consideration of these effects may yield a model that more appropriately describes columnar discotic systems.