Temperature dependence of the static permittivity andintegral formula for the Kirkwood correlation factor ofsimple polar fluids

TL;DR

This paper derives an exact integral formula for the Kirkwood correlation factor and models the temperature dependence of static permittivity in polar fluids, successfully matching experimental data with minimal parameters.

Contribution

It introduces a new integral formula for the Kirkwood correlation factor and a simplified approach to model permittivity considering permanent and induced dipoles.

Findings

The theory matches experimental permittivity data for various polar fluids.

Using the Kirkwood superposition approximation simplifies modeling of dipole interactions.

Incorrect interpretation of $g_K$ can mislead about dipole alignment states.

Abstract

An exact integral formula for the Kirkwood correlation factor of isotropic polar fluids $g_K$ is derived from the equilibrium averaged rotational Dean equation, which as compared to previous approaches easily lends itself to further approximations. The static linear permittivity of polar fluids $\epsilon$ is calculated as a function of temperature, density and molecular dipole moment in vacuo for arbitrary pair interaction potentials. Then, using the Kirkwood superposition approximation for the three-body orientational distribution function, we suggest a simple way to construct model potentials of mean torques considering permanent and induced dipole moments. We successfully compare the theory with the experimental temperature dependence of the static linear permittivity of various polar fluids such as a series of linear monohydroxy alcohols, water, tributyl phosphate, acetonitrile, acetone, nitrobenzene and dimethyl sulfoxide, by fitting only one single parameter, which describes the induction to dipole-dipole energy strength ratio. We demonstrate that comparing the value of $g_K$ with unity in order to deduce the alignment state of permanent dipole pairs, as is currently done is in many situations, is a misleading oversimplification, while the correct alignment state is revealed when considering the proper interaction potential. Moreover we show, that picturing H-bonding polar fluids as polar molecules with permanent and induced dipole moments without invoking any specific H-bonding mechanism is in many cases sufficient to explain experimental data of the static dielectric constant. In this light, the failure of the theory to describe the experimental temperature dependence of the static dielectric constant of glycerol, a non-rigid polyalcohol, is not due to the lack of specific H-bonding mechanisms, but rather to an oversimplified model potential for that particular molecule.