Diffusion of Charged Species in Liquids

In this study the laws of mechanics for multi-component systems are used to develop a theory for the diffusion of ions in the presence of an electrostatic field. The analysis begins with the governing equation for the species velocity and it leads to the governing equation for the species diffusion velocity. Simplification of this latter result provides a momentum equation containing three dominant forces: (a) the gradient of the partial pressure, (b) the electrostatic force, and (c) the diffusive drag force that is a central feature of the Maxwell-Stefan equations. For ideal gas mixtures we derive the classic Nernst-Planck equation. For liquid-phase diffusion we encounter a situation in which the Nernst-Planck contribution to diffusion differs by several orders of magnitude from that obtained for ideal gases.

A A Some care must be taken with the interpretation of Eq. 2 since it applies to all Stokesian fluids and is therefore not limited to ideal gas mixtures (see Appendix B in ref. 21). Caution must also be used with the interpretation of Eq. 3 which is applicable to all ideal fluid mixtures. For example, Eq. 3 should provide reliable results for a liquid mixture of hexane and heptane, but it should not for a mixture of ethanol and water.
Here it is important to emphasize that Eq. 1 can only describe a fluid composed of non-interacting particles. To provide a reliable framework in which a fluid is considered to be composed of interacting particles ̶ as in the case of a liquid ̶ it is necessary to understand the process of Nernst-Planck diffusion in a fluid different from an ideal gas. This is the problem under consideration in this paper.
In this work we analyze the ion transport process from a fundamental point of view using an axiomatic mechanical perspective. Our analysis is for ideal fluid mixtures, and from the analysis we find that the classic Nernst-Planck relation is only valid for ideal gas mixtures. For ideal liquid mixtures we find an alternative expression in which the form is similar to that for ideal gas mixtures, but significant other terms appear in the final result.
In the following section we present a detailed analysis based on axiomatic statements for the mechanics of multi-component systems. Our objective is a search for clarity and rigor in terms of a diffusion equation that is applicable to both gas and liquid mixtures, i.e., applicable to fluid mixtures. We first derive a general equation for the species velocity, and from this we extract a general equation for the species diffusion velocity. This equation is the origin of Fick's law of diffusion; however, the derivation of Fick's law from the governing equation for the species diffusion velocity is complex. We begin our study of the general equation for the species diffusion velocity with a series of plausible restrictions that lead to a special form of the governing equation for the diffusion velocity. This provides a transport equation for the diffusion of charged molecules in fluids, and this general form can be used to clearly establish that the classic Nernst-Planck equation is only valid for ideal gas mixtures.

Analysis
The analysis in this study is presented in terms of axiomatic statements concerning the species A body illustrated in Fig. 1. There we have used V A (t) to identify the volume occupied by the species A body, and we have illustrated only the presence of one other species, species B; however, one should image the presence of many other species. We begin with a review of the axioms for mass and mechanics of multi-component systems, and then move on to explore the dominant terms in the species momentum equation. If one accepts the simplification that the electric field represents a specified force, the motion of both charged and uncharged species can be treated in terms of the axioms for mass and momentum that are given in the following paragraphs. It is crucial to understand that the following development is applicable to both gases and liquids.
Mass. We begin our study with the two axioms for mass given by The volume of the species A body is represented by V A (t) and the speed of displacement of the surface of this volume is v A ·n. Axiom I A can be used to derive the species A continuity equation by using the transport theorem and the divergence theorem (see Sec. 1.3.2 in ref. 22). This leads to

A A A A
The total continuity equation is obtained by summing Eq. 6 over all N species and imposing Eq. 5 in order to obtain (see page 83 in ref. 23) Here the total density, ρ , and the total mass flux, ρ v, are defined by The mass fraction of species A is defined according to and use of this definition with Eq. 8 provides a constraint for the mass fractions and a representation for the mass average velocity. These two results take the form The objective of this study is the development of an expression for the diffusive flux of species A. This requires that we decompose the species velocity according to Here we note that the mass diffusive flux is defined by Mechanics. The first of the four axioms associated with the species A body is the linear momentum principle given by (see page 85 in ref. 24)  Here the species stress vector, t A(n) , represents the force that the surroundings (into which n is directed) exert on species A within the species A body. We have used P AB to represent the diffusive force exerted by species B on species A and it is understood that

AA
In the last term in Eq. 15 we have indicated the possibility that species A entering or leaving the species A body owing to chemical reaction may have a velocity ⁎ v A different than the continuum velocity v A . The second axiom is the angular momentum principle that takes the form Returning to Eq. 15 we note that Axiom I B can be used to develop Cauchy's lemma given by (see Section 203 in ref. 28 in which t A(−n) represents the force that species A exerts on the surroundings. Cauchy's fundamental theorem for the species stress vector is given by (see Lecture 5 in ref. 24).
At this point we can use the continuity equation given by Eq. 6 to obtain and this allows us to express Eq. 22 in the form Both Eq. 22 and Eq. 24 can be found in the literature; however, locating them requires some effort because of the lack of a uniform nomenclature (see Sec. The analysis of Axiom II B is rather long; however, after some algebra one finds that Eq. 17 leads to the symmetry of the species stress tensor indicated by (see Lecture 5 in This is identical in form to Cauchy's second law of motion (see page 187 in ref. 36). At this point it seems clear that there is general agreement concerning the form of the species linear momentum equation that represents the governing equation for the species velocity, v A . In the study of diffusion processes, it is the diffusion velocity, u A , that is important, and in the following paragraphs we develop the governing equation for this velocity.

Diffusion Velocity
The total linear momentum equation is obtained by summing Eq. 22 over all species and imposing the axioms given by Eqs 18 and 19. This leads to in which the total stress tensor and the total body force are defined by Here τ is the viscous stress tensor, and p is the local thermodynamic pressure given by

Slattery Callen Slattery Callen
Here m represents the mass of the closed system preferred by Callen 39 . Use of Eq. 28 with Eq. 26 allows us to express the total momentum equation in the form It is important to recognize that the application of Eqs 29 and 33 requires an assumption of local thermodynamic equilibrium (see Sec. 3.4 of ref. 38), and this should be satisfactory for many, but not all, mass transfer processes. The use of Eq. 32 in Eq. 22 leads to the following form of the species A momentum equation Here one should note that the diffusive force is constrained by Eq. 16. At this point it will be convenient to use Eq. 7 with Eq. 26 to obtain the following form of the total momentum equation The analogous form of the species momentum equation is given by Eq. 24 and repeated here as At this point we wish to use Eqs 38 and 39 to derive the governing equation for the mass diffusion velocity, u A . We begin by multiplying Eq. 38 by the mass fraction ω A leading to and we subtract this result from Eq. 39 to obtain the desired governing differential equation for the mass diffusion velocity, u A , given by Plausible restrictions associated with this governing equation for u A are given by (see Sec. 1.2 in ref. 21) and are listed here as A A Use of this representation for the partial pressure indicates that our analysis is restricted to ideal fluid mixtures and it allows us to express Eq. 50 in the form It is important to keep in mind that Eq. 50 represents a special form of the governing equation for the diffusion velocity, u A , subject to the restrictions indicated by Eqs 44 through 49. In addition, this equation for the diffusion velocity is restricted by the thermodynamic results indicated by Eqs 29 and 33. Finally, we remind the reader that Eq. 50 is a continuum result that is not restricted to either gases or liquids but is restricted to Stokesian fluid mixtures.
We begin our analysis of Eq. 52 by neglecting the pressure diffusion term. This represents a satisfactory simplification for many diffusion processes; however, it is not acceptable for centrifugation processes (see page 776 in ref. 23  and we are ready to analyze the term on the right hand side.

Diffusive Force
From dilute gas kinetic theory we know that P AB|gas is given by where it is understood that all the terms on the right side are associated with a liquid mixture. It should be clear that Eq. 55 represents an empirical expression for the diffusive force, P AB | liq , and that D AB represents an empirical coefficient to be determined by experiment. We can generalize Eqs 54 and 55 to the single form  In Appendix A we show that for dilute solutions we obtain the result given by and D A is a mixture diffusion coefficient given by We refer to J A as the mixed-mode diffusive flux since it is made up of the molar concentration, c A , and the mass diffusion velocity, u A .
Use of Eq. 60 in Eq. 59 allows us to express the mixed-mode diffusive flux according to At this point we make use of the analysis given in Appendix B where we demonstrate the well-known result given by (see page 776 in ref. 23) In order to represent the second term on the right hand side of Eq. 67 in terms of what is usually called Nernst-Planck diffusion, we introduce the product RT and arrange that term to obtain the following result

Conclusions
In this paper we have derived an ion transport equation for ideal fluid mixtures, and we have shown that the classic Nernst-Planck equation applies only to ideal gas mixtures. For ideal liquid mixtures, the laws of mechanics, the laws of electrostatics, and the thermodynamic representation for the partial pressure all lead to a new result. Non-ideal gases and liquids can be analyzed using the approach presented in this paper; however, that subject has not been explored in this work 50,51 . Greek Letters ρ A = mass density of species A, kg/m 3 ρ = total mass density, kg/m 3 τ = viscous stress tensor, N/m 2 τ A = viscous stress tensor for species A, N/m 2 ω A = ρ A /ρ , mass fraction of species A Ψ = electric potential function, V