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The rotational dynamics of liquid crystals in nematic and isotropic phases have been investigated in experiments [2] and theoretical studies [3]. In particular, end-over-end rotational motion, which can be measured by dielectric relaxation experiments, is one of the characteristic dynamic properties of liquid crystals. The relaxation behaviour for end-over-end rotational motion of liquid crystalline molecules and for the realignment process by an external field in the nematic phase is very slow-mode, due to the orientational order in the long range. Therefore, a long time is required to find the average dynamic property in a molecular simulation.

The concept of retardation factor

where

Molecular reorientation in the nematic phase is expressed by the Wigner rotation matrix [18, 19]:

where

Figure 1. The first-rank orientational time correlation functions for the system with *N* of 8,788 in the nematic phase at *T ^{*}* of 1.50, 1.575 and 1.65.

where *b* is the angle between the molecule and director. In this study, the following formula is used to calculate the correlation function considering the fluctuation of the director,

where **n**(*t*) is the director at time *t*. Configuration data at each of 500.0 time steps for 600.0 *t ^{*}* to 1000.0

The first-rank orientational time correlation functions

Figure 2. Arrhenius plot of the relaxation time for the end-over-end rotational motion in the isotropic and nematic phases for the system at the reduced pressure *P ^{*}* of 2.0. The open circles indicate the values for the system with

Figure 2 shows the Arrhenius plot for the relaxation time *t _{||}* of the first-rank rotational correlation function

In liquid crystalline phases, not only the rotational motion, but also the translational motion is influenced by orientational order. In order to examine the translational dynamics in the nematic phase, the temperature dependence of the calculated translational diffusion coefficients is shown in Figure 3. Here the translational diffusion coefficients of parallel (

where

Figure 3. Arrhenius plot of the translational diffusion coefficients in the directions parallel to the director, *D*_{||}, and perpendicular to it, *D*_{|_}, in the nematic phase for the system at the reduced pressure *P*^{*} of 2.0 for the system with *N* of 8,788.

Figure 4 shows the retardation factor *g*_{||} in the nematic phase as a function of the Maier-Saupe strength parameter *s*(@*q/RT*) obtained from the simulation at constant pressure for the system of *N* = 8,788 together with two theoretical lines [4 - 6, 9] and the data obtained from the simulation at constant volume [8]. To our knowledge, experimental data on the retardation factor as a function of the second-rank orientational order parameter <*P _{2}*> are not available, although some data on the retardation factor as a function of temperature are. [21, 22] However, the mean field theory, which was successively described for the nematic phase by Maier and Saupe, can be used to explain the retardation factor

Therefore, the result from our simulation at constant pressure more closely reproduced the analytical expression by Coffey

Figure 4. Retardation factor *g*_{||} as a function of the Maier-Saupe strength parameter *s*(@*q/RT*) and the second rank orientational order parameter <*P*_{2}> for a nematic phase. The ordinary components for the relaxation times *t*_{0} are estimated from the extrapolation line for the data in the isotropic phase by assuming Arrhenius behaviour. The closed circles indicate the values obtained from the simulation in this study. The open circles show the values obtained from the molecular dynamics simulation at constant volume (*r*^{*} of 0.17 to 0.19) [8]. The dotted and solid lines are the theoretical curves by Meier-Saupe and Martin-Meier-Saupe solution, respectively.

[ 2] G. Williams,

[ 3] P.L. Nordio, R. Rigatti, U. Segre,

[ 4] W. Maier, A. Saupe,

[ 5] W. Maier, A. Saupe,

[ 6] W. Maier, A. Saupe,

[ 7] J.G. Gay, B.J. Berne,

[ 8] M.A. Bates, G.R. Luckhurst,

[ 9] A.J. Martin, G. Meier, A. Saupe,

[10] M.A. Bates, G.R. Luckhurst,

[11] H. Dominguez, E. Velasco, J. Alejandre,

[12] K.M. Aoki, M. Yoneya, H. Yokoyama,

[13] S. Melchionna, G. Ciccotti, B.L. Holian,

[14] H. C. Andersen,

[15] S. Nose,

[16] W.G. Hoover,

[17] G.R. Luckhurst, P.S.J. Simmonds,

[18] C. Zannoni and M. Guerra,

[19] C. Zannoni,

[20] W.T. Coffey, D.S. F. Crothers, Yu P. Kalmikov, J.T. Waldoron,

[21] G. Williams,

[22] R. Righini,

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