Phase transition in a fluid of hard rods

We have emphasized that the asymmetry of the molecular shape is an important factor in determining whether or not a substance exhibits the liquid crystalline phase. Consider a fluid of long thin rods without any forces between them other than the one preventing their interpenetration. 

In Zwanzig s model the rods are rectangular parallelopipeds of length / and square cross-sectional area d. Assuming that the potential energy of interaction of N such hard rods is a sum of pair potentials, we may write 

Now for any given configuration of the system, there will be N(l) molecules pointing in the direction u(l), N(2) in the direction u(2) and N 3) in the direction u(3). Therefore becomes a function of the occupation numbers 

To evaluate this configurational integral, we make use of the maximum term method of statistical mechanics, i.e., in the limit N co, V co, but N/V = p = constant, can be approximated by the largest term t in the sum. The problem therefore reduces to one of maximizing / with respect to the occupation numbers. Using Stirling s approximation and introducing the mole fraction jc = N ol)/N of the various components. 

To proceed further, we need to know the dependence of the function on density, which can be obtained by resorting to the virial expansion. Now, the molecules having different orientations may be regarded as belonging to different species, so that in effect we have a multicomponent system. The virial expansion of the free energy of a gaseous mixture in ascending powers of the density is well known for our present purpose we use the form 

---

One prominent example of rods with a soft interaction is Gay-Berne particles. Recently, elastic properties were calculated [89,90]. Using the classical Car-Parrinello scheme, the interactions between charged rods have been considered [91]. Concerning phase transitions, the sohd-fluid equihbria for hard dumbbells that interact additionally with a quadrupolar force was considered [92], as was the nematic-isotropic transition in a fluid of dipolar hard spherocylinders [93]. The influence of an additional attraction on the phase behavior of hard spherocylinders was considered by Bolhuis et al. [94]. The gelation transition typical for clays was found in a system of infinitely thin disks carrying point quadrupoles [95,96]. In confined hquid-crystalline films tilted molecular layers form near each wall [97]. Chakrabarti has found simulation evidence of critical behavior of the isotropic-nematic phase transition in a porous medium [98]. 

The possibility of entropy-driven phase separation in purely hard-core fluids has been of considerable recent interest experimentally, theoretically, and via computer simulations. Systems studied include binary mixtures of spheres (or colloids) of different diameters, mixtures of large colloidal spheres and flexible polymers, mixtures of colloidal spheres and rods," and a polymer/small molecule solvent mixture under infinite dilution conditions (here an athermal conformational coil-to-globule transition can occur)." For the latter three problems, PRISM theory could be applied, but to the best of our knowledge has not. The first problem is an old one solved analytically using PY integral equation theory by Lebowitz and Rowlinson." No liquid-liquid phase separation... 

---