Mixtures Of Hard Rods Research Articles

Rigorous statistical mechanical results for one-dimensional bounded soft rod systems and for soft rod mixtures

We obtain the exact partition functions for modified soft rod systems, thus exploring mixing and end effects. A hypergeometric function appears in each partition function. For a soft and hard rod mixture, we exhibit the excess Helmholtz free energy of mixing in the thermodynamic limit. Using the centrally important integral I( D, M′, L, b 1), closely related to the confluent hypergeometric function 1 F 1, we have sketched the evaluation of the soft rod partition function in a form containing 1 F 1 rather than a Bessel function as derived earlier by us [6]. Again using I( D, M′, L, b 1) but with altered indices, we have evaluated the partition function for soft rods with hard and soft boundaries in terms of the 1 F 1 function. We have recognized that the sum of these two partition functions, multiplied by N′ equals the partition function of a system of soft rods subject to periodic boundary conditions. Conceptually, then, periodic boundary conditions in the soft rod system are equivalent to imposition of either hard or soft boundary conditions. Additionally, we have noted that soft boundary conditions are equivalent to a two-phase system composed of a bound monolayer and a free bulk phase. Again using I( D, M′, L, b 1), we have obtained the exact partition function of a mixture of hard and soft rods by a modification of the integral indices. In order to derive results of potential utility in solution theory, we have obtained the thermodynamic limit of this partition function, and have derived the excess Helmholtz free energy of mixing at constant number density and temperature. At high and low temperatures, this excess function assumes greatly simplified mathematical forms. Finally, we have obtained the exact partition function for a finite binary mixture of soft rods of different force constants, using I( D, M′, L, b 1) to generate an intermediate result which led to a partition function containing a hypergeometric function. We hope that the results of this work, which demonstrated intriguing common features of functional form in the various partition functions, will be of evaluating approximate statistical mechanical theories, in the theory of solutions, and in the theory of small systems.

Time-displaced correlation functions in an infinite one-dimensional mixture of hard rods with different diameters

Time-displaced conditional distribution functions are calculated for an infinite, one-dimensional mixture of equal-mass hard rods of different diameters. The kinetic equation that describes the time dependence of the one-particle total distribution function is found to be non-Markovian, in contrast with the situation in systems of identical rods. The correlation function does not contain any isolated damped oscillation, except for systems of equal-diameter rods with discrete velocities. Thus, we generalize the one-component results of Lebowitz, Percus, and Sykes, removing some nontypical features of that system.

Physical clusters in an imperfect vapor

We examine the general problem of calculating the equation of state in terms of physical clusters, and direct attention to the two main components of the problem: (i) formulating a satisfactory definition of the physical cluster concept; (ii) developing a method to account for interference between clusters and to assess the importance of interference. For (i) we use Stillinger’s definition, which is based on the distance between molecular centers. To account for interference, we treat the clusters in a one-dimensional vapor as a mixture of hard rods, each of which assumes its most probable length. We find that neglect of interference induces errors in the second virial coefficient down to a very low temperature, which depends on the strength of the pair attraction. Thus, we conclude that interference between clusters can never be neglected in a calculation of the equation of state when ordinary dispersion forces operate between molecules.

Lattice Models for Thermotropic Liquid Crystals

Abstract The utility of the DiMarzio-Alben lattice model for investigating thermotropic liquid crystals is evaluated, using a system of straight, inflexible rods with hard cores and nearest neighbor attractions on a simple cubic lattice. In the most general case, treated in the Bragg-Williams approximation, the rods, of length-to-breadth ratio x, are divided into a core portion of c segments and two “tail” portions of (x - c)/2 segments each and there are seven different segmental attractive energies: w cc, w ct, w tt, w ee, w cc, w ct, w tt, where c and t denote core and tail segments, respectively, unprimed and primed energies are for interactions between parallel and perpendicular rods, respectively, and w ee is for the end-to-end interaction between the last segments of two parallel rods. In addition, a special case of the model is also treated in the quasi-chemical or Bethe approximation and the Bragg-Williams approach is extended to simple mixtures of hard rods of two different lengths with solvent-...

Scaled particle theory for nonadditive hard spheres. General solution in one dimension

Two general solutions of scaled particle theory for a binary mixture of hard rods interacting with arbitrary nonadditive diameters are presented. These solutions agree very well with the exact solution.

Lattice model for a binary mixture of hard rods of different lengths. Application to solute induced nematic→isotropic transitions

A statistical mechanical treatment for two-component mixtures of hard rods placed on a simple cubic lattice is presented. Each mixture is composed of two types of rods of different length-to-breadth ratios; and the pressure-to-temperature ratios are chosen so that the system is anisotropic when only longer rods are present but isotropic when composed of only shorter rods. Such solutions can be induced to undergo a first-order anisotropic-isotropic phase transition, with a small two phase region, by increasing the concentration of the shorter rods. The dependence of this transition on the parameters of the model is described and deviations from ideal solution behavior are discussed. Recent experimental data are cited and compared with the results of the lattice calculation.