Mixing Plate-Like and Rod-Like Molecules With Solvent: A Test of Flory-Huggins Lattice Statistics

Boehm and Martire have shown that the Flory-Huggins (FH) lattice model applied to mixtures of squares and rigid rods in solvent on a two dimensional lattice gives different results depending on whether rods or squares are placed first onto the lattice. This correct derivation places the validity of the FH model itself into question since the final result should be independent of the order of placement. An analysis of the FH model in terms of Poisson statistics suggests an alternative formula for the probability of successfully placing a rectangle into an area partially filled with other rectangles, which when incorporated into the FH counting procedure gives the exact thermodynamic result for the tiling of squares (i.e., no solvent and no rods). An attempt to solve the order of placement problem is made by solving the problem of one square plus any number of rods and then generalizing the statistics so that they are consistent with this result. Equations are given for squares plus rods plus solvent in both two and three dimensions. For plates plus solvent in three dimensions a purely entropy driven phase transition between an anisotropic layered phase and an isotropic phase is obtained. This transition is analogous to the isotropic to nematic liquid crystal phase transition in rigid rods. Our equations, when augmented by energy considerations, are useful for calculating the equilibrium properties of discotic systems, polymer-layered silicate composites, and the adsorption of plate like molecules onto surfaces.