Theory of polar fluids

Abstract

The theory of fluids of polar polarizable molecules is extended. The graph theoretical formalism established previously is enriched by the proof of two new identities. It is shown that the conventional high-frequency dielectric constant ε ∞ plays a natural rˆole in the graphical formalism. Grand canonical s-point correlation functions involving arbitrary integral powers of the dipole moment M are defined. For s = 1 and 2 and low powers of M explicit expressions are given in terms of quantities which occurred in the previous graph theoretical treatment. Expressions for the mean-squared dipole moment of arbitrary macroscopic sub-volumes of a system are given. Specialized to spheres, agreement is found with an older result for a sphere in vacuum, but disagreement with two expressions that have been proposed for a sphere embedded in a larger system. Positive definiteness conditions are derived both from fluctuation theorems and the hard-core conditions which must be imposed on point-dipole models. The positive definiteness conditions are transformed to different representations, and it is shown that the conditions from the two sources combine in a natural way. Several specific inequalities are derived from the positive definiteness conditions. One of them is found to be identical with a convexity theorem obtained by a different method. Another inequality reduces in the non-polar limit to a bound on the dielectric constant previously obtained by Prager et al. using electrostatic variational principles.