Fluctuating Hydrodynamics and Debye-Hückel-Onsager Theory for Electrolytes

Abstract

We apply fluctuating hydrodynamics to strong electrolyte mixtures to compute the concentration corrections for chemical potential, diffusivity, and conductivity. We show these corrections to be in agreement with the limiting laws of Debye, Hückel, and Onsager. We compute explicit corrections for a symmetric ternary mixture and find that the co-ion Maxwell-Stefan diffusion coefficients can be negative, in agreement with experimental findings.Due to the long-range nature of Coulomb forces between ions it is well-known that electrolyte solutions have unique properties that distinguish them from ordinary mixtures [1]. Colligative properties, such as osmotic pressure, and transport properties, such as mobility, have corrections that scale with the square root of concentration [2,3,4,5]. This macroscopic effect has a mesoscopic origin, specifically, due to the competition of thermal and electrostatic energy at scales comparable to the Debye length.The traditional derivation of the thermodynamic corrections is by way of solving the Poisson-Boltzmann equation. For example, an approximate solution gives the Debye-Hückel limiting law for the activity coefficient [2,6]. The derivation of the transport properties, as developed by Onsager and co-workers [7,8,9], has a similar starting point but is much more complicated. Here, we present an alternative approach using fluctuating hydrodynamics (FHD) [10].This paper generalizes our previous derivation for binary electrolytes [11] to arbitrary solute mixtures, and, as an illustrative example, calculates transport properties for a ternary electrolyte. It should be noted that our FHD approach extends closely-related density functional Email address: donev@courant.nyu.edu (Aleksandar Donev) theory calculations of relaxation corrections to the conductivity [12] of binary electrolytes to account for advection, which enables us to also compute the electrophoretic corrections [2].First formulated by Landau and Lifshitz to predict light scattering spectra [13,14], more recently FHD has been applied to study various mesoscopic phenomena in fluid dynamics [10,15,16,17]. The current popularity of fluctuating hydrodynamics is due, in part, to the availability of efficient and accurate numerical schemes for solving the FHD equations [18,19,20,21,22,23,24,25,26]. While in this work we give analytical results for dilute electrolytes under simplifying assumptions, numerical techniques can in principle be used to compute the transport coefficients for moderately dilute solutions.From the work of Onsager et al. [7,8,9] we can obtain the Fickian diffusion matrix for sufficiently dilute electrolytes. For neutral multispecies mixtures, especially non-dilute ones, using binary Maxwell-Stefan (MS) diffusion coefficients (inverse friction coefficients) [27,28] is preferred because generally they are positive and depend weakly on concentration. In this paper we use our FHD formulation to calculate the effective macroscopic (renormalized) co-ion and counter-ion MS coefficients for binary and symmetric ter...

Fluctuating hydrodynamics of electrolytes at electroneutral scales

“…It can be confirmed that the electroneutral equations (3,4,7,18) are consistent with (31), which demonstrates that nothing special needs to be done in the electroneutral limit to handle the fluctuating diffusive fluxes except to include them in the right-hand side of the modified Poisson equation (18).…”

“…It is given by the solution of the modified elliptic equation (18), and not by the quasielectrostatic Poisson equation (2). In summary, the fluctuating electroneutral equations we consider in this work are given by (3,4,7,18).…”

“…For very dilute electrolyte solutions, in the absence of fluctuations the deterministic formulation reverts to the classical PNP equations for the composition, coupled to an incompressible NS equation for the fluid velocity. In recent work [2,3] we have demonstrated that the addition of thermal fluctuations renormalizes the PNP equations to reproduce the Debye-Hückel-Onsager theory for dilute solutions.…”

Electroosmosis in nanopores: Computational methods and technological applications

“…As for the fluid, we can update the continuity equation Eq. ( 4) for each species α with a stochastic term [218,226,227]…”

“…Such a formulation is interesting as it allows one to efficiently investigate fluctuationdriven behavior, both numerically and analytically. Importantly, this approach has been used to identify enhanced charge transport due to fluid fluctuations, that result in ion concentration corrections to diffusion, sometimes leading to negative diffu-sion [226] or enhanced electric conductivity [229]. In other contexts, giant instabilities may appear at interfaces with concentration gradients, that can be enhanced by applied electric fields [230].…”