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# Fluctuation-enhanced electric conductivity in electrolyte solutions

Contributed by John B. Bell, August 31, 2017 (sent for review August 16, 2017; reviewed by Florence Suzy Baras and Annie Lemarchand)

## Significance

Using fluctuating hydrodynamics, we demonstrate that thermal fluctuations contribute to charge transport in binary electrolyte solutions. We show the existence of an enhancement, or renormalization, of the electric conductivity due to the coupling between fluctuations of charge and fluid velocity. This coupling results in nontrivial corrections to the classical Poisson–Nernst–Planck equations, which are of the order of the square root of the salt concentration and therefore significant even for dilute solutions. Our calculations predict a cation–anion cross-diffusion coefficient that is in quantitative agreement with experimental measurements. Our findings have important implications for the fields of both mesoscale hydrodynamics and electrolyte transport.

## Abstract

We analyze the effects of an externally applied electric field on thermal fluctuations for a binary electrolyte fluid. We show that the fluctuating Poisson–Nernst–Planck (PNP) equations for charged multispecies diffusion coupled with the fluctuating fluid momentum equation result in enhanced charge transport via a mechanism distinct from the well-known enhancement of mass transport that accompanies giant fluctuations. Although the mass and charge transport occurs by advection by thermal velocity fluctuations, it can macroscopically be represented as electrodiffusion with renormalized electric conductivity and a nonzero cation–anion diffusion coefficient. Specifically, we predict a nonzero cation–anion Maxwell–Stefan coefficient proportional to the square root of the salt concentration, a prediction that agrees quantitatively with experimental measurements. The renormalized or effective macroscopic equations are different from the starting PNP equations, which contain no cross-diffusion terms, even for rather dilute binary electrolytes. At the same time, for infinitely dilute solutions the renormalized electric conductivity and renormalized diffusion coefficients are consistent and the classical PNP equations with renormalized coefficients are recovered, demonstrating the self-consistency of the fluctuating hydrodynamics equations. Our calculations show that the fluctuating hydrodynamics approach recovers the electrophoretic and relaxation corrections obtained by Debye–Huckel–Onsager theory, while elucidating the physical origins of these corrections and generalizing straightforwardly to more complex multispecies electrolytes. Finally, we show that strong applied electric fields result in anisotropically enhanced “giant” velocity fluctuations and reduced fluctuations of salt concentration.

- fluctuating hydrodynamics
- electrohydrodynamics
- Navier–Stokes equations
- multicomponent diffusion
- Nernst–Plank equations

The interaction between ionic species and an externally imposed electric field is at the core of many electrokinetic problems and applications (1) such as electrophoresis. Studying these types of problems usually involves solving the Poisson–Nersnt–Planck equation, which assumes that the solution is ideal with no cross-diffusion between the different ions. In a recent publication (2), we presented a numerical scheme based on fluctuating hydrodynamics for simulating electrokinetic problems at mesoscopic scales where thermal fluctuations are nonnegligible. In this approach, the generalized Poisson–Nernst–Planck (PNP) equations are combined with the fluctuating Landau–Lifshitz Navier–Stokes equations, yielding a set of stochastic partial differential equations that can be solved either analytically or numerically. In this paper we use theoretical calculations to show that, in dilute electrolyte solutions under an applied electric field, there exists a coupling phenomenon between the fluctuations of local net charges and fluid velocity. This coupling results in an effective enhancement of the electric conductivity, which we refer to as “fluctuation-induced electroconvection.” This enhancement is similar to but distinct from the enhancement of mass diffusion (3⇓–5) associated with giant fluctuations (6, 7). Furthermore, we show that in the presence of an electric current there exists a coupling between the fluctuations of ion density and charge density that results in a reduction of the electric conductivity. We show that the renormalized conductivity is consistent with Onsager’s reciprocal relations provided that there exists a Maxwell–Stefan (MS) cross-diffusion coefficient between the cation and anion, as, indeed, is measured in experiments. The renormalized electrodiffusion equations are different from the starting fluctuating PNP equations to order square root of the salt concentration, which implies strong cross-coupling corrections even for dilute solutions. Nevertheless, at infinite dilution the enhancement of conductivity is consistent with the enhancement of the salt diffusion coefficient, even though the two originate from seemingly unrelated mechanisms, leading to the simple PNP equations in the limit of vanishing salt concentration. Finally, we show that the coupling produces an anisotropic enhancement of the momentum fluctuations of the fluid that scales as the square of the magnitude of the applied electric field.

## Problem Description

We model a homogeneous solution composed of a neutral solvent fluid (e.g., water) and two ionic solute species of opposite charge. We assume that a uniform electric field

The theoretical system we consider is infinite in all directions. The fluid is subjected to fluctuations in species mass flux and stress tensor consistent with the fluctuation–dissipation theorem (7). We use a low Mach approximation (8) and neglect density fluctuations, i.e., **1**, which result in enhanced velocity fluctuations through the term **2**. The calculations we carry out next closely resemble previous linearized or “one-loop renormalization” calculations of fluctuation-enhanced diffusivity in nonionic binary mixtures (3⇓–5).

## Nonequilibrium Fluctuations

We use linearized fluctuating hydrodynamics to compute the spectrum of the steady-state concentration and velocity fluctuations. We first define the fluctuations **1** yields**2** and, as in ref. 7, we apply a double curl operator to eliminate the pressure term. We obtain, in Fourier space,** k**. In that case, the

**4**becomes

Taking the Fourier transform of Eq. **3** and combining it with Eq. **5**, we obtain that the vector

The complete expression for

## Enhancement of Electric Conductivity

The electroconvective coupling results in a net charge flux. From Eq. **1**, we can write the average charge flux as

We first examine the advective charge flux **11** using Eq. **7**, and using the fact that the Schmidt number in liquids is large, **13** we expand to leading order in **13** can be interpreted as a difference of Stokes–Einstein coefficients for a sphere of radius

The flux **8**, becomes

Both **13** and **15** show that the deterministic linear response that is obtained by ensemble averaging the equations is not the bare response expressed by the conductivity

## Renormalized Transport Coefficients

The renormalization of the electric conductivity is connected to the renormalization of the diffusion coefficient that results from giant fluctuations (3⇓–5). In ref. 5, a calculation very similar to the one performed above is carried out for the renormalization of the diffusion coefficient in a nonionic mixture, and it is found that diffusion is renormalized^{∗} by

For infinitely dilute solutions (^{†}; it is worth noting that in the fully nonlinear diffusion model studied in ref. 13 the only noise term is the stochastic stress and all diffusion arises from advection by thermal velocity fluctuations.

For finite **16** do not satisfy the Nernst–Einstein relation so the renormalized PNP equation must be corrected to leading order in

To give a more physical interpretation of the cross-diffusion coefficient, we link the renormalized Fickian diffusion matrix to a renormalized MS diffusion matrix (14). The MS diffusion coefficients can be physically interpreted as inverse friction coefficients between pairs of distinct species. For a very dilute solution, it has been assumed when writing Eq. **1** that the (bare) MS cross-diffusion coefficient between the two ionic species, ** D** with nonzero off-diagonal coefficients. Introducing the renormalized MS diffusion coefficients

Using the complete formulas for the electrophoretic (**18** to unequal ions. With parameters of water (molecular mass ^{2}/s and ^{2}/s), where ^{2}/s, in very good agreement (within 10% difference) with published experimental measurements (12, 15, 16).

## Enhancement of Velocity Fluctuations

The fluctuation-induced electroconvection derived in this paper is associated with a corollary phenomenon, namely, the enhancement of velocity fluctuations in the direction of the electric field, as shown by the expression of ** k** and

For small wavenumbers (large length scales), the structure factor for wavevectors orthogonal to

We note that, on the other hand, the fluctuations of

## Concluding Remarks

In summary, using a fluctuating hydrodynamics formulation, we show that there exists a coupling between the fluctuations in charge density and fluid velocity that is proportional to the applied electric field. This coupling leads to an effective enhancement, or renormalization, of the measured electric conductivity of an ionic mixture. This enhancement is comparable to the enhancement of the diffusion coefficients that results from giant fluctuations, in that the enhancement coefficients match in the limit of infinite dilution. For finite dilution, the renormalization of mass diffusivity and electric conductivity are different. This shows that, although we started from a diagonal Fickian diffusivity matrix, renormalizing the fluctuating PNP equations yields an off-diagonal Fickian diffusion term, itself linked with a nonzero renormalized cross-diffusion MS coefficient between the two counterions, in good agreement with experimental coefficients reported in the literature. In fact, in our prior work (2) we demonstrated that results from Debye–Huckel theory, including the nonanalytic Debye–Huckel correction to the internal energy, can be obtained from a fluctuating hydrodynamics theory of dilute electrolyte solutions. The present work further demonstrates that fluctuating hydrodynamics provide a generalizable and systematic approach to derive corrective transport coefficients such as the electrophoretic and the relaxation term. Finally, for large electric fields, the applied field can significantly amplify the velocity fluctuations and suppress fluctuations of salt concentration. We expect this phenomenon to be observable experimentally and by molecular dynamics simulations.

The theory developed here can readily be extended in a number of important directions. First, the assumption of dynamically identical ions can be removed so that a more direct comparison with experimental measurements for different salts can be performed, including polyvalent salts. It is also important to consider solutions with one ion and two counterions, such as for example solutions of NaCl and KCl in water. Although beyond the scope of the present paper, such extensions reveal that the surprising experimental observation of negative MS diffusion coefficients (18, 19) between coions (20) can be explained by fluctuating hydrodynamics and renormalization. Here we considered only strong electrolytes but the generalization to weak electrolytes is possible by using fluctuating hydrodynamics for reactive fluids (21). Finally, we started here with fluctuating hydrodynamics equations based on the PNP equations, i.e., we assumed an ideal solution with no cross-diffusion, so our starting equations had only one mobility coefficient per ion, instead of one MS coefficient per pair of ions. The renormalized equations, on the other hand, have cross-diffusion and also a nonideal Debye–Huckel contribution to the free energy density. This suggests that a more proper theory should start from the more complete equations, allowing for a nonzero bare MS cross-coefficient

## Acknowledgments

We thank Burkhard Duenweg and Mike Cates for illuminating discussions about linear response theory and renormalization. This work was supported by the US Department of Energy (DOE), Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics Program under Award DE-SC0008271 and Contract DE-AC02-05CH11231. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US DOE under Contract DE-AC02-05CH11231.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: JBBell{at}lbl.gov.

Author contributions: J.-P.P., J.B.B., A.D., and A.L.G. designed research; J.-P.P., A.J.N., A.D., and A.L.G. performed research; and J.-P.P., J.B.B., A.D., and A.L.G. wrote the paper.

Reviewers: F.S.B., Laboratory Interdisciplinaire Carnot de Bourgogne; and A.L., CNRS, Université Pierre et Marie Curie.

The authors declare no conflict of interest.

↵

^{∗}Quantitatively, assigning the experimental self-diffusion coefficient of the ions to*D*_{enh}and*η*provides estimates of the length-scale*a*on the order of the ionic diameter.↵

^{†}The renormalization of diffusion originates from the velocity fluctuations and their coupling with a concentration gradient, while the renormalization effect studied here results from charge density fluctuations and their coupling with the electric field.

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