Static and dynamic theory of polarization under internal and directing electric fields: Fixed-charge and fixed-potential conditions

TL;DR

This paper develops a continuum theory for the static and dynamic polarization behavior of polar fluids under different boundary conditions, revealing how fixed charge or potential constraints influence polarization fluctuations and correlations.

Contribution

The authors introduce a cross-coupled free energy functional that reproduces Onsager and Kirkwood results and derive dynamic equations for polarization correlations under various boundary conditions.

Findings

Polarization fluctuations depend on boundary conditions (fixed charge vs. fixed potential).

Nonlocal polarization correlations inversely proportional to system volume are found under fixed potential.

Surface charge fluctuations and Stern layer potential drops significantly influence dielectric response.

Abstract

We present a continuum theory on statics and dynamics of polar fluids, where the orientational polarization ${\bi p}_1$ and the induced polarization ${\bi p}_2$ are governed by the Onsager directing field ${\bi E}_d$ and the Lorentz internal field $\bi F$, respectively. We start with a dielectric free energy functional $\cal F$ with a cross term $\propto \int\hspace{-0.5mm} d{\bi r}~{\bi p}_1\cdot{\bi p}_2$, which was proposed by Felderhof $[$J. Phys. C: Solid State Phys. {\bf 12}, 2423 (1979)$]$. With this cross-coupling, our theory can yield the theoretical results by Onsager and Kirkwood. We also present dynamic equations using the functional derivatives $\delta {\cal F}/\delta {\bi p}_i$ to calculate the space-time correlations of ${\bi p}_i$. We then obtain analytic expressions for various frequency-dependent quantities including the Debye formula. We find that the fluctuations of the total polarization drastically depend on whether we fix the electrode charge or the applied potential difference between parallel metal electrodes. In the latter fixed-potential condition, we obtain a nonlocal (long-range) polarization correlation inversely proportional to the cell volume $V$, which is crucial to understand the dielectric response. It is produced by nonlocal charge fluctuations on the electrode surfaces and is sensitive to the potential drops in the Stern layers in small systems. These nonlocal correlations in the bulk and on the surfaces are closely related due to the global constraint of fixed potential difference. We also add some results in other boundary conditions including the periodic one, where nonlocal correlations also appear.