In the Onsager theory for the phase transition from the isotropic fluid to the nematic liquid crystal phase, the Helmholtz free energy of a fluid of hard convex bodies (HCBs) is expressed as the sum of an entropy of a mixing-like term and an energy-like term (from the interaction of the HCBs). Whereas the Onsager theory expresses the interaction term in a virial expansion and determines the consequences of B2 alone, here we extend that treatment to incorporate B3 (with its attendant dependence on the mutual orientation of three HCBs). For HCBs (and specifically for Dooh ellipsoids) with large aspect ratios (5:1 or greater), the incorporation of B2 and B3 suffices to predict the variation ofthe order parameter (P2 [cos( 0)]) with density in accord with the Monte Carlo (MC) results of Allen and Wilson. As the aspect ratio decreases (from 5: 1) to more spherical molecules (say 3: 1), virial coefficients of higher order than B3 contribute to the interaction term and their effect is represented in part by the yexpansion (or resummation) theory proposed by Barboy and Gelbart. In thisy-expansionthird virial-Onsager theory, the predicted transition densities are in accord with the MC values of Frenkel and Mulder for prolate ellipsoids. Neither the y expansion nor the direct B2 and B3 theories find the phase diagram (i.e., transition density and order parameter regarded as a function of aspect ratio) to be symmetric for prolate and oblate ellipsoids. The dependence of B3 on the mutual orientation of the ellipsoids is also discussed and previous work is also addressed.
both values of[ are sufficiently large. This independence of [ is only expected i f the reduced potentials u at the same reduced distance r / R are compared in two different polyelectrolyte systems with the same value of aZC,.In a given polyelectrolyte system with 6 L 2, the counterions accumulate in the neighborhood of the polyion "surface", the fraction of accumulated counterions up to a given value of r / R increasing with [ but decreasing with increasing values of azCe.For high values of 6 ([ > 2) and relatively high equivalent concentrations, more than 50% of all counterions will be found at a distance of a few nanometers from the distance of closest approach (for Iz,I = 1).The scaling parameter x of the Poisson-Boltzmann equation may be considered as a screening length as (u),.., is always smaller than unity. Its value depends on aZCe, but if [ 2 2.5, it is practically independent of the charge parameter. It is of the same order of magnitude as, but somewhat larger than, the Debye length calculated taking into account only "noncondensed" counterions, Le., p,("')/[. The fraction of accumulated counterions at x = 1 increases with [ and decreases with a2Ce. For [ > 2 this fraction exceeds 60% and may become as high as 90% for large values of [ and small values of a2Ce (low concentrations and small value of a). The counterions within the screening length still occupy the smaller volume fraction of the cell volume. This distribution thus bears some resemblance with the "two-phase" model.The same conclusions are valid for a system containing a weak polyelectrolyte, the charge parameter of which may be changed by titration with a strong low molar mass base of acid. In the course of the titration the radius of the cell R remains the same whatever the degree of charge of the polyelectrolyte. This means that R is no longer a function of 6, and the [-dependence of u is determined only through IBI. Consequently, for constant a2Ce, at any position in the cell including the polyion "surface" at a, the reduced potential u will level off with increasing 5, thus displaying a condensation-like effect. This [-independence will, however, start at values considerably larger than . $ = 1 but depending only slightly on a2Ce. The Frank elastic constants for a nematic liquid crystal have been calculated by computer simulations for a fluid of hard ellipsoids and by the Poniewierski-Stecki method for ellipsoids with and without an attractive square well. Required for the Poniewierski-Stecki method is the direct correlation function c(1,2) and its dependence on the mutual molecular orientations. This was addressed using several models: the Parsons model, a two-term virial expansion, and the PY and HNC theories of Patey et al. In the Parsons model for c(1,2), the derived Frank elastic constants for hard ellipsoids were in agreement with the Poniewierski-Holyst values for hard spherocylinders. However, the agreement with simulation values was less satisfactory, as the simulation values exceeded the Parsons model values and the discrepa...
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