Existence of the Dielectric Constant in Rigid-Dipole Fluids: The Direct Correlation Function

Abstract

The question of whether the dielectric constant ε exists (is well defined) for a finite fluid system of rigid dipolar molecules is reconsidered and reformulated. It is found that this question can most simply be expressed in terms of the behavior of the position‐ and orientation‐dependent direct correlation function c(r₁, ω₁; r₂, ω₂). It is shown that ε exists if c satisfies the following two conditions: (a) $\mathrm{c\sim -\phi /\; kT}$ for $|{\mathbf{r}}_1-{\mathbf{r}}_2\mathrm{|\; >\; \sigma }$ , where φ is the dipole‐dipole potential and σ is a length which is large microscopically but small macroscopically. (b) c(r₁, ω₁; r₂, ω₂) is of the form $c_8\mathrm{(|}{\mathbf{r}}_1-{\mathbf{r}}_2\mathrm{|)+}{\mathbf{F(r}}_1-{\mathbf{r}}_2\mathrm{):}\mathbf{e}{{(\omega }}_1)\mathbf{e}{{(\omega }}_2)$ for $|{\mathbf{r}}_1-{\mathbf{r}}_2\mathrm{|\; <\; \sigma }$ , where e(ω) is the unit vector with orientation ω. An explicit (and new) expression for ε in terms of c is automatically obtained; its applicability is ensured if the above conditions are satisfied. These results lend new intuition and insight into the question of the existence of ε, and suggest a promising approach for future investigations of this question.