Transverse Density Fluctuations around the Ground State Distribution of Counterions near One Charged Plate: Stochastic Density Functional View

Hiroshi Frusawa

doi:10.3390/e22010034

Abstract

We consider the Dean–Kawasaki (DK) equation of overdamped Brownian particles that forms the basis of the stochastic density functional theory. Recently, the linearized DK equation has successfully reproduced the full Onsager theory of symmetric electrolyte conductivity. In this paper, the linear DK equation is applied to investigate density fluctuations around the ground state distribution of strongly coupled counterions near a charged plate, focusing especially on the transverse dynamics along the plate surface. Consequently, we find a crossover scale above which the transverse density dynamics appears frozen and below which diffusive behavior of counterions can be observed on the charged plate. The linear DK equation provides a characteristic length of the dynamical crossover that is similar to the Wigner–Seitz radius used in equilibrium theory for the 2D one-component plasma, which is our main result. Incidentally, general representations of longitudinal dynamics vertical to the plate further suggest the existence of advective and electrical reverse-flows; these effects remain to be quantitatively investigated.

Highlights

Water-soluble materials often have surface chemical groups that are dissociated in a polar solvent
The macroion systems can be rephrased as inhomogeneous one-component ionic fluids in the presence of external fields—the one-component fluids of counterions can be regarded as the one-component plasma (OCP) [3,4,5]
In a single charged plate system, J (r0 ) is expressed as J (r0 ) = z0 /λ with z0 denoting the distance between the position r0 and the charged plate; Equation (6) implies that a large portion of the counterions are condensed within the thin electric double layer, which supports the observation that strongly coupled counterions behave like the 2D OCP

Summary

Introduction

Water-soluble materials often have surface chemical groups that are dissociated in a polar solvent Examples of such materials include mesoscopic particles, such as viruses, proteins, polyelectrolytes, membranes, and micelles, and macroscopic objects like a glass plate of sample cell [1,2]. This paper will address the strong coupling systems of counterions in the presence of one charged plate, focusing especially on the transverse dynamics of density fluctuations around the ground state that will be specified in Section 2.1 [8,9,10,11]. The main aim of this paper is to investigate the anisotropic fluctuation field n(r, t) of counterion density due to the coarse-grained dynamics of counterion density ρ(r, t) = ρ∞ (r) + n(r, t) around a ground state distribution ρ∞ (r), using the stochastic density functional equation.

Ground State of Counterion System in the Strong Coupling Limit

Imposing a Given Density Distribution ρ on the Grand Potential Ω

Stochastic Density Dynamics Obeying the Dean–Kawasaki Equation

Findings

Summary and Conclusions