Statistical-Mechanical Theory on the Probability Distribution Function for the Net Charge of an Electrolyte Droplet

TL;DR

This paper develops a statistical-mechanical theory to derive the probability distribution function for the net charge of electrolyte droplets, incorporating electrostatic energy and ion distribution effects, providing analytical expressions for mean and variance.

Contribution

It introduces a novel theoretical framework for predicting charge fluctuations in electrolyte droplets, combining electrostatics and statistical mechanics.

Findings

Derived analytical expressions for average and variance of droplet charge.

Established the influence of ion concentration differences on charge distribution.

Provided a Gaussian approximation for the charge distribution.

Abstract

Droplets of electrolyte solutions in an insulating medium are ubiquitous in nature. The net charges of these droplets are normally nonzero, and they fluctuate. However, a theory on the probability distribution function for the net charge of droplets is lacking, so far. Thus, a statistical-mechanical theory of a charged droplet is developed with including the effect of the electrostatic energy of charging as well as the random distribution of ions. Then, the probability distribution function for the net charge of an electrolyte droplet is calculated assuming that it is generated from a macroscopic solution with the different cation and anion concentrations. Using the Gaussian approximation and Stirling's formula, the analytic results for the average and variance of the net charge of a droplet are obtained.