Isotropic potentials, orientational distribution function

Figure I. Schematic representation of statistical orientational distribution functions in isotropic (I) and nematic (N) potentials. Dashed curves depict adjusted distributions under electric field poling.

So far we have been considering the motion of a test polymer in an isotropic environment. We now consider a slightly different problem how does the orientational distribution function of polymers change under external fields such as a potential field / ( ) or a velocity gradient K. Let (ti t) be the probability that an arbitrarily chosen polymer is in the direction u. Since each polymer feels the external field as in eqn (8.15), the time evolution of W( t) can be described by ... 

If the symmetry axes of the rest state orientation distribution function coincide with the principal axes of deformation (always true for an initially isotropic network), then conditions 1-5 apply for all admissible potentials U(t,x) ... 

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. 

In this chapter we consider the very simplest approach to the molecular theory of liquid crystals. We shall approach the theory phenomenologically, treating the problem of the existence of the nematic phase as an order-disorder phenomenon. Using the observed symmetry of the nematic phase we shall identify an order parameter and then attempt to find an expression for the orientational potential energy of a molecule in the nematic liquid in terms of this order parameter. Such an expression is easily found in the mean field approximation. Once this is accomplished, expressions for the orientational molecular distribution function are derived and the thermodynamic functions simply calculated. The character of the transformation from nematic liquid crystal to isotropic fluid is then revealed by the theory, and the nature of the fluctuations near the transition temperature can be explored. 

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