Individual ion species chemical potentials in the Mean Spherical Approximation

Abstract

The Mean Spherical Approximation (MSA) is a commonly used thermodynamic theory for computing the energetics of ions in the primitive model (i.e., charged hard-sphere ions in a background dielectric). For the excess chemical potential, however, the early MSA formulations (which were widely adopted) only included the terms needed to compute the mean excess chemical potential (or the mean activity coefficient). Other terms for the chemical potential μ_i of individual species i were not included because they sum to 0 in the mean chemical potential. Here, we derive these terms to give a complete MSA formulation of the chemical potential. The result is a simple additive term for μ_i that we show is a qualitative improvement over the previous MSA version. In addition, our derivation shows that the MSA’s assumption of global charge neutrality is not strictly necessary, so that the MSA is also valid for systems close to neutrality.

I. INTRODUCTION

The Mean Spherical Approximation (MSA)^1,2 has long been used to compute the energetics of model electrolyte solutions, such as the primitive model, where ions are charged hard spheres in a background dielectric. While it has thermodynamic inconsistencies,³ the power of the MSA has been its simple analytic formulas for thermodynamic properties, such as the electrostatic contributions to the Helmholtz, Gibbs, and internal energies, as well as the entropy and excess chemical potentials.

A recent study⁴ compared the MSA’s excess chemical potential for the individual ion species (denoted μ_i for species i) to Monte Carlo (MC) simulations and found that the MSA deviated significantly at high electrolyte concentrations. In that study, Gillespie et al.⁴ derived a new theory of the electrostatic excess chemical potential which corrected this deviation. The needed correction was adding a new term to an MSA-like excess chemical potential that was proportional to z_i, the valence of ion species i. In the commonly cited and commonly used literature of the MSA (reviewed, for example, in Ref. 5), terms proportional to z_i do not appear in μ_i.

The focus of this work is terms of this type within the MSA. In early MSA papers, such terms are mentioned (e.g., Ref. 6), but they were not explicitly written out. Rather, they were commonly lumped into a “+z_i · const.” term^6,7 because, by charge neutrality, such terms do not contribute to the mean excess chemical potential (or mean activity coefficient) of the electrolyte. Thus, the formulas given for μ_i are incomplete. Here, we derive the missing terms.

Having complete formulas for μ_i is important because often the incomplete formulas are used to study the energetics of individual ion species. For example, any calculation of ions in equilibrium (e.g., inhomogeneous electrolytes in an electrical double layer) requires, by the definition of chemical equilibrium in the grand canonical ensemble, an accurate value for μ_i for each ion species individually. Moreover, many theories apply the MSA formulas of a homogeneous system repeatedly to compute ion densities in inhomogeneous systems (e.g., classic density functional theory^8–10).

Our new terms that are proportional to z_i are derived from the early MSA papers that list the starting points, but not the final formulations.^1,3,6,11 Thus, we are building onto previous well-established results, adding back some lost pieces. When this extended MSA is compared to MC simulations, the result is a significant improvement in the excess chemical potentials.

This work is laid out as follows: First, in Sec. II A, we show how Blum’s original derivation of the MSA^1,2 is still valid close to, but away from, charge neutrality. This is necessary because calculating individual species chemical potentials involves adding a small amount of unbalanced charge to a charge neutral system. Next, in Sec. II B, we lay an additional foundation with a discussion of the MSA internal energy. In Sec. II C, we derive our main result for the individual species electrostatic excess chemical potentials [Eq. (63)]. Finally, in Sec. III, we compare the previous and new formulations to MC simulations to show the qualitative improvement of our extended MSA chemical potential compared over the standard MSA version. In Appendix A, an alternative derivation of our main result is made by using a neutralizing background.

II. THEORY

Throughout we generally employ the notation and results of Ref. 7.

A. The MSA for electrolytes close to neutrality

The MSA solution obtained by Blum for an asymmetric electrolyte is for a system with strict charge neutrality imposed.¹ However, to compute individual species chemical potentials, a single ion is added to the system and charge neutrality is violated. Here, we show that the usual MSA solution is still valid for systems close to neutrality. This also validates the use of MSA formulas in non-uniform system with local deviations from charge neutrality (e.g., Refs. ⁹ and ¹⁰).

We start with Blum’s solution¹ to the Baxter factorization functions Q_ij(r) of ion species i and j that define the direct correlation functions c_ij(r) by

$${\delta }_{ij}-{({\rho }_i{\rho }_j)}^{1/2}{\tilde{c}}_{ij}(k)=\sum _l{\tilde{Q}}_{il}(k){\tilde{Q}}_{jl}(k),$$ (1)

where the tilde denotes the Fourier transform and ρ_i is the density of species i. Finding the Q_ij(r) is possible for a system of Yukawa interaction potentials

$$\beta {\psi }_{ij}(r,Z)=\frac{\beta e^2{z}_i{z}_j}{4\pi \epsilon {\epsilon }_0}\frac{e^{-Zr}}{r}.$$ (2)

Here, β = 1/k_BT (with k_B being the Boltzmann constant and T the absolute temperature), e is the fundamental charge, ɛ is the dielectric constant of the system, and ɛ₀ is the permittivity of free space. Coulomb interactions are the limit Z → 0, a limit that will be important throughout.

The MSA enforces the boundary condition

$$c_{ij}(r)=-\beta {\psi }_{ij}(r,Z)(r>{\sigma }_{ij}),$$ (3)

where σ_ij = (σ_i + σ_j)/2, with σ_i the hard core diameter of species i. Therefore, the c_ij(r) always have the form

$$c_{ij}(r)=\left\{\begin{array}{cc}{c}_{Iij}(r),\hfill & r<{\sigma }_{ij},\hfill \\ -\beta {\psi }_{ij}(r,Z),\hfill & r>{\sigma }_{ij},\hfill \end{array}\right.$$ (4)

with the unknown function c_Iij(r) when the hard cores of the ions overlap. The other unknown correlation function h_ij(r) is related to c_ij(r) via the Ornstein–Zernike equation

$$h_{ij}(r)=c_{ij}(r)+\sum _l{\rho }_l\int c_{il}(r)h_{lj}(|\mathbf{r}-{\mathbf{r}}^{\prime }|)d{\mathbf{r}}^{\prime }$$ (5)

and its Fourier transform

$${\tilde{h}}_{ij}(k)={\tilde{c}}_{ij}(k)+\sum _l{\rho }_l{\tilde{c}}_{il}(k){\tilde{h}}_{lj}(k).$$ (6)

It has boundary conditions,

$$h_{ij}(r)=-1(r<{\sigma }_{ij}).$$ (7)

This is related to the pair distribution function g_ij(r) via

$$g_{ij}(r)=h_{ij}(r)+1.$$ (8)

The Baxter factor functions are

$$Q_{ij}(r)=\left\{\begin{array}{cc}{q}_{ij}(r)-z_i{a}_j{e}^{-Zr},\hfill & r<{\sigma }_{ij},\hfill \\ -z_i{a}_j{e}^{-Zr},\hfill & r>{\sigma }_{ij},\hfill \end{array}\right.$$ (9)

where a_j is defined later in Eq. (56). To find q_ij(r), we start with Blum’s derivation¹ one step before an invocation of charge neutrality, namely, at his Eq. (2.23),

$$\begin{array}{ll}\hfill J_{ij}(r)& =Q_{ij}(r)+\sum _k{\rho }_k{\int }_{{\lambda }_{jk}}^{{\sigma }_{jk}}{Q}_{kj}(r^{\prime })J_{ik}(|r-r^{\prime }|)d{r}^{\prime }\hfill \\ \hfill & -e^{-Zr}\sum _k{\rho }_k{\int }_{{\sigma }_{jk}-r}^{\infty }{J}_{ik}(r^{\prime })A_{kj}{e}^{-Z{r}^{\prime }}d{r}^{\prime },\hfill \end{array}$$ (10)

where λ_jk = (σ_j − σ_k)/2,

$$J_{ij}(r)=2\pi {\int }_r^{\infty }t{h}_{ij}(t)dt$$ (11)

and

$$A_{ij}=z_i{a}_j.$$ (12)

The next step in Blum’s solution is to invoke the condition that the ion cloud around a central ion neutralizes that ion’s charge,

$$-z_i=4\pi \sum _k{z}_k{\rho }_k{\int }_0^{\infty }{r}^2{h}_{ik}(r)dr.$$ (13)

In the MSA, this relation about local charge neutrality around a central ion is true even without global charge neutrality (i.e., ∑_kz_kρ_k = 0). This was shown by Blum, where he derived Eq. (13) as Eq. (4.18) of Ref. 2. However, in the original MSA derivation, Blum¹ used Eq. (13) with h_ik(r) ↦ g_ik(r), which adds an unnecessary use of global charge neutrality.

Next, using integration by parts, we have

$$4\pi \sum _k{z}_k{\rho }_k{\int }_0^{\infty }{r}^2{h}_{ik}(r)dr=2\sum _k{z}_k{\rho }_k{\int }_0^{\infty }{J}_{ik}(r)dr,$$ (14)

so that, by Eq. (13),

$$\sum _k{\rho }_k{\int }_0^{\infty }{J}_{ik}(r)A_{kj}dr=-\frac{1}{2}{A}_{ij}.$$ (15)

This is Blum’s equation (2.25) of Ref. 1, and so we can take the limit Z → 0 just as he did to get Blum’s equation (2.26),

$$\begin{array}{ll}\hfill J_{ij}(r)& =Q_{ij}(r)+\frac{1}{2}{A}_{ij}+\frac{1}{2}{a}_j\delta q+\sum _k{\rho }_k{\int }_{{\lambda }_{jk}}^{{\sigma }_{jk}}{Q}_{kj}(r^{\prime })J_{ik}(|r-r^{\prime }|)d{r}^{\prime }\hfill \\ \hfill & +\sum _k{\rho }_k{\int }_0^{{\sigma }_{jk}-r}{J}_{ik}(r^{\prime })A_{kj}d{r}^{\prime }.\hfill \end{array}$$ (16)

This equation is the same as Blum’s, but without invoking global charge neutrality. However, charge neutrality was invoked once more in his analysis.

Blum next notes that, by Eqs. (7) and (11), for r < σ_ij,

$$J_{ij}(r)=\pi r^2+J_{ij}(0),$$ (17)

so that when r < σ_ij,

$$\begin{array}{ll}\hfill \pi r^2+J_{ij}(0)& =Q_{ij}(r)+\frac{1}{2}{A}_{ij}+\sum _k{\rho }_k{\int }_{{\lambda }_{jk}}^{{\sigma }_{jk}}{Q}_{kj}(r^{\prime })\left(\pi {(r-r^{\prime })}^2+J_{ik}(0)\right)d{r}^{\prime }\hfill \\ \hfill & +\pi \sum _k{\rho }_k{A}_{kj}{\int }_0^{{\sigma }_{jk}-r}\left(r^{\prime 2}+J_{ik}(0)\right)d{r}^{\prime }.\hfill \end{array}$$ (18)

The place where charge neutrality is invoked is in the evaluation of the last integral,

$$\begin{array}{ll}\hfill \sum _k{\rho }_k{A}_{kj}{\int }_0^{{\sigma }_{jk}-r}\left(r^{\prime 2}+J_{ik}(0)\right)d{r}^{\prime }& =\sum _k{\rho }_k{A}_{kj}{J}_{ik}(0)({\sigma }_{jk}-r)\hfill \\ \hfill & +\frac{a_j}{3}\sum _k{z}_k{\rho }_k{\left({\sigma }_{jk}-r\right)}^3.\hfill \end{array}$$ (19)

Expanding the cubic gives a new term for Eq. (18), namely,

$$-r^3\frac{\pi }{3}{a}_j\sum _k{z}_k{\rho }_k,$$ (20)

plus corresponding induced changes in the rⁿ (n = 0, 1, 2) terms. This new r³ term is asymptotically small as charge neutrality is approached.

Blum’s analysis continues (described in more detail in Ref. 2 than in Ref. 1) by using Eq. (18) to show that (with that one invocation of charge neutrality) the q_ij(r) in Eq. (9) [now denoted $q_{ij}^{\text{MSA}}(r)$] are quadratic in r,

$$q_{ij}^{\text{MSA}}(r)=(r-{\sigma }_{ij})q_{ij}^{\prime }+\frac{1}{2}{(r-{\sigma }_{ij})}^2{q}_j^{″}.$$ (21)

The quantities $q_{ij}^{\prime }$ and $q_j^{″}$ are defined later in Eqs. (52) and (53). Here, we see that with a small violation of charge neutrality the functional forms of q_ij(r) [and, therefore, the Q_ij (r, Z)] are altered only infinitesimally,

$$q_{ij}(r)=q_{ij}^{\text{MSA}}(r)-r^3\frac{\pi }{3}{a}_j\sum _k{z}_k{\rho }_k.$$ (22)

By Eq. (1), these small new terms in Q_ij(r, Z) produce a perturbation in the c_ij(r), which we denote δc_ij(r). By Eqs. (3) and (4),

$$\delta c_{ij}(r)=\left\{\begin{array}{cc}\delta c_{Iij}(r),\hfill & r<{\sigma }_{ij},\hfill \\ 0,\hfill & r>{\sigma }_{ij}.\hfill \end{array}\right.$$ (23)

Because, for r > σ_ij, the c_ij(r) remain unchanged, fixed by the MSA ansatz defined by Eq. (3). Next, we show that this perturbation does not significantly alter the excess Helmholtz free energy density A of the system.

It is possible to obtain an expression for A in terms of the c_ij(r) via the quantity I = −βA. Specifically, Høye and Stell³ showed that

$$\begin{array}{ccc}\hfill I& =\hfill & \frac{1}{2}\sum _{i,j}{\rho }_i{\rho }_j({\tilde{c}}_{ij}(0)-{\tilde{c}}_{ij}^{\text{HS}}(0))\hfill \\ \hfill & \hfill & -\frac{1}{2}\frac{1}{{(2\pi )}^3}\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\int \{\ln [1-\rho \tilde{c}(k)]+\rho \tilde{c}(k)\}d\mathbf{k}\hfill \\ \hfill & \hfill & +\frac{1}{2}\frac{1}{{(2\pi )}^3}\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\int \{\ln [1-\rho {\tilde{c}}^{\text{HS}}(k)]+\rho {\tilde{c}}^{\text{HS}}(k)\}d\mathbf{k},\hfill \end{array}$$ (24)

where c(r) are matrices with matrix elements c_ij(r) and ρ is the vector of densities ρ_i. In this equation and throughout, the superscript HS denotes the reference system of hard spheres. By substituting c_ij(r) ↦ c_ij(r) + δc_ij(r) and using Eq. (6), one finds that the leading order error in I is

$$\begin{array}{ll}\hfill \delta I& =\frac{1}{2}\sum _{i,j}{\rho }_i{\rho }_j\delta {\tilde{c}}_{ij}(0)\hfill \\ \hfill & +\frac{1}{2}\frac{1}{{(2\pi )}^3}\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\int \frac{\rho \tilde{c}(k)}{1-\rho \tilde{c}(k)}\rho \delta \tilde{c}(k)d\mathbf{k}\hfill \\ \hfill & =\frac{1}{2}\sum _{i,j}{\rho }_i{\rho }_j\delta {\tilde{c}}_{ij}(0)+\frac{1}{2}\frac{1}{{(2\pi )}^3}\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\int \rho \tilde{h}(k)\rho \delta \tilde{c}(k)d\mathbf{k}\hfill \\ \hfill & =\frac{1}{2}\sum _{i,j}{\rho }_i{\rho }_j\delta {\tilde{c}}_{ij}(0)+\frac{1}{2}\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\int \rho h(r)\rho \delta c(r)d\mathbf{r}.\hfill \end{array}$$ (25)

For the last integral in expression (25), because h_ij(r) = −1 for r < σ_ij and δc_ij(r) = 0 for r > σ_ij, we have

$$\begin{array}{ll}\hfill \frac{1}{2}\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\int \rho h(r)\rho \delta c(r)d\mathit{r}& =-\frac{1}{2}\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\rho \rho {\int }_{r<{\sigma }_{ij}}\delta c(r)d\mathit{r}\hfill \end{array}$$ (26)

$$=-\frac{1}{2}\sum _{i,j}{\rho }_i{\rho }_j\delta {\tilde{c}}_{ij}(0),$$ (27)

where, in the last step, we used the generic identity

$$\tilde{f}(0)=\int f(r)d\mathbf{r}.$$ (28)

Thus,

$$\delta I=0.$$ (29)

From this, it follows that any change to A is second-order for small errors like δc_Iij(r) ∼ r³∑_kz_kρ_k. Consequently, the chemical potentials

$${\mu }_i=\frac{\partial A}{\partial {\rho }_i}$$ (30)

are the same as in the charge neutral case because, it being a derivative, any violation of charge neutrality is infinitesimal. (Appendix A shows that this is also true for a different but related case, where there is constant neutralizing background.) Similarly, the excess internal free energy density E is only perturbed to second-order since E = ∂βA/∂β.

Thus, we conclude that to leading order the MSA equations for the neutral electrolyte are also be valid away from neutrality, but close to it. Therefore, in the following, we can use the standard MSA equations to derive formulas for the individual species’ chemical potential.

B. MSA excess internal energy per particle

An important quantity we will need is u_i, the excess internal energy per particle of species i. Comparison of different formulations of this quantity away from charge neutrality requires a little care.

Høye and Blum⁶ stated that

$$\beta u_i=\frac{{\alpha }^2}{4\pi }{z}_i{N}_i+z_i\beta u^{*},$$ (31)

where α² = βe²/ɛɛ₀,

$$N_i=-\frac{\mathrm{\Gamma }{z}_i+\eta {\sigma }_i}{1+\mathrm{\Gamma }{\sigma }_i},$$ (32)

and

$$\beta u^{*}=-\frac{{\alpha }^2}{24}\sum _l{\rho }_l{\sigma }_l^2\left(N_l{\sigma }_l+\frac{3}{2}{z}_l\right).$$ (33)

Γ is the MSA screening length parameter given implicitly by

$$4{\mathrm{\Gamma }}^2={\alpha }^2\sum _i{\rho }_i{\left(\frac{z_i-\eta {\sigma }_i^2}{1+\mathrm{\Gamma }{\sigma }_i}\right)}^2,$$ (34)

with η = LP_n/2 for L = π/Δ,

$$\mathrm{\Delta }=1-\frac{\pi }{6}{\zeta }_3,$$ (35)

$${\zeta }_n=\sum _k{\rho }_k{\sigma }_k^n,$$ (36)

$$P_n=\frac{1}{\mathrm{\Omega }}\sum _k\frac{z_k{\rho }_k{\sigma }_k}{1+\mathrm{\Gamma }{\sigma }_k},$$ (37)

$$\mathrm{\Omega }=1+\frac{L}{2}\sum _k\frac{{\rho }_k{\sigma }_k^3}{1+\mathrm{\Gamma }{\sigma }_k}.$$ (38)

It is the z_iβu* in Eq. (31) that has generally been ignored and written off as “+z_i · const.” for subsequent derivations of μ_i.

A derivation of Eq. (31) is given in Appendix B, as Høye and Blum⁶ merely stated this non-trivial result. An important aspect of this derivation is that this u_i is the excess internal energy beyond the mean-field internal energy $u_i^{\text{MF}}$. Specifically,

$$\beta u_i=\frac{1}{2}\sum _j{\rho }_j\int h_{ij}(r)\beta {\psi }_{ij}(r,Z)d\mathbf{r}$$ (39)

and

$$\beta u_i^{\text{MF}}=\frac{1}{2}\sum _j{\rho }_j\int \beta {\psi }_{ij}(r,Z)d\mathbf{r}$$ (40)

$$\begin{array}{ll}\hfill & =\frac{1}{2}\sum _j{\rho }_j{\int }_{r<{\sigma }_{ij}}(\beta {\psi }_{ij}(r,Z)+c_{ij}(r))d\mathbf{r}-\frac{1}{2}\sum _j{\rho }_j{\tilde{c}}_{ij}(0),\hfill \end{array}$$ (41)

where the last term follows from Eqs. (3) and (28). While u_i is always finite, for a non-neutral Coulombic system, $u_i^{\text{MF}}=\infty$ because ${\tilde{c}}_{ij}(0)=\infty$; in a neutral system, $u_i^{\text{MF}}=0$. The sum $u_i+u_i^{\text{MF}}$ is the full internal energy per particle.

This distinction is important because, as shown by Høye and Stell,³ the Ornstein–Zernike equation [Eq. (5)] and its MSA boundary conditions [Eqs. (3) and (7)] allow for a simple statement of the full internal energy via Eq. (8),

$$\beta u_i+\beta u_i^{\text{MF}}\equiv \frac{1}{2}\sum _j{\rho }_j\int g_{ij}(r)\beta {\psi }_{ij}(r,Z)d\mathbf{r}$$ (42)

$$=-\frac{1}{2}\sum _j{\rho }_j\int (h_{ij}(r)+1)c_{ij}(r)d\mathbf{r}$$ (43)

$$=\frac{1}{2}\left(c_{ii}(0)+1\right)-\frac{1}{2}\sum _j{\rho }_j{\tilde{c}}_{ij}(0).$$ (44)

From this formulation, Høye and Stell³ derived the full Helmholtz free energy density (i.e., not subtracting off the Helmholtz of the mean-field contribution). Differentiating that, they found the chemical potential with the mean-field term ${\mu }_i^{\text{MF}}$ included,

$$\beta {\mu }_i+\beta {\mu }_i^{\text{MF}}=\beta u_i+\beta u_i^{\text{MF}}+\frac{1}{2}\sum _k{\rho }_k\left({\tilde{c}}_{ik}^{\text{HS}}(0)-{\tilde{c}}_{ik}(0)\right)$$ (45)

$$=2(\beta u_i+\beta u_i^{\text{MF}})+\frac{1}{2}\left(c_{ii}^{\text{HS}}(0)-c_{ii}(0)\right).$$ (46)

Equation (46) follows from Eq. (45) by applying Eq. (44).⁷

Before continuing on to the individual species excess chemical potentials, we note that in Sec. II A, the derivation of Eq. (29) includes all mean-field terms and those terms technically diverge due to the divergence of ${\tilde{c}}_{ij}(0)$ in the Z → 0 limit. However, the integral over the Coulomb interactions (for 0 < r < ∞) can be separated out as mean-field terms, and it is easily seen that that they have no influence upon δI and vice versa. This also means that mean-field terms do not interfere with the solution of the Ornstein–Zernike equation since they are not involved in correlations.

C. MSA individual species chemical potentials

The ion species’ excess chemical potential beyond the mean-field can be obtained from either Eq. (45) or (46) by subtracting off the mean-field contribution. From basic mean field theory, we know that

$${\mu }_i^{\text{MF}}=2u_i^{\text{MF}}.$$ (47)

It is then clear that Eq. (46) is more convenient to use because subtracting off $2u_i^{\text{MF}}$ is straightforward,

$$\beta {\mu }_i=2\beta u_i+\frac{1}{2}\left(c_{ii}^{\text{HS}}(0)-c_{ii}(0)\right).$$ (48)

All these terms are finite for both neutral and non-neutral systems as Z → 0.⁷

Using Eq. (45), on the other hand, is more complicated, even though it is equivalent to Eq. (46). That is because there is no convenient formulation for

$$\frac{1}{2}\sum _k{\rho }_k{\tilde{c}}_{ik}(0)-\beta u_i^{\text{MF}},$$ (49)

and, therefore, one cannot easily determine what remains after the infinities cancel. [See Ref. 7 for a discussion of Z⁻¹ terms in ${\tilde{c}}_{ik}(0)$ that lead to divergences without charge neutrality when Z → 0.]

Therefore, we use Eq. (48) to derive “+z_i · const.” terms. We start with⁷

$$\begin{array}{ll}\hfill 2\pi \left(c_{ii}^{\text{HS}}(0)-c_{ii}(0)\right)& =q_i^{″}-q_i^{\prime \prime \text{HS}}-z_i\sum _k{\rho }_k{a}_k{q}_{ik}^{\prime }\hfill \\ \hfill & +\sum _k{\rho }_k\left({\kappa }_{ik}^{\text{HS}}{q}_k^{\prime \prime \text{HS}}-{\kappa }_{ik}{q}_k^{″}\right),\hfill \end{array}$$ (50)

where

$$\begin{array}{ccc}\hfill {\kappa }_{ij}& \equiv \hfill & {\int }_{{\lambda }_{ji}}^{{\sigma }_{ij}}{q}_{ij}(r)dr\hfill \\ \hfill & =\hfill & -\frac{1}{2}{q}_{ij}^{\prime }{\sigma }_i^2+\frac{1}{6}{q}_j^{″}{\sigma }_i^3+\delta {\kappa }_{ij},\hfill \end{array}$$ (51)

$$q_{ij}^{\prime }=L\left({\sigma }_i+{\sigma }_j+\frac{L}{2}{\zeta }_2{\sigma }_i{\sigma }_j\right)-\frac{{\mathrm{\Gamma }}^2}{{\alpha }^2}{a}_i{a}_j,$$ (52)

$$q_j^{″}=2L\left(1+\frac{L}{2}{\zeta }_2{\sigma }_j\right)+L{P}_n{a}_j,$$ (53)

$${\zeta }_n=\sum _k{({\sigma }_k)}^n,$$ (54)

$$q_k^{\prime \prime \text{HS}}=2L\left(1+\frac{L}{2}{\zeta }_2{\sigma }_j\right),$$ (55)

and

$$a_i={\alpha }^2\frac{z_i-\eta {\sigma }_i^2}{2\mathrm{\Gamma }(1+\mathrm{\Gamma }{\sigma }_i)}.$$ (56)

The small correction to κ_ij from non-charge-neutrality is

$$\delta {\kappa }_{ij}=-\frac{\pi }{12}{a}_j\left({\sigma }_{ij}^4-{\lambda }_{ij}^4\right)\sum _k{z}_k{\rho }_k,$$ (57)

plus similar contributions from changes in rⁿ (n = 0, 1, 2) terms. Note that these terms contain only infinitesimally small δκ_ij corrections to standard MSA formulations. In the following, we will drop these, as they are arbitrarily small.

To derive the “+z_i · const.” terms, we evaluate Eq. (50) term by term. For the first terms in Eq. (50), by Eq. (A20) in Appendix 2 of Ref. 7, we have

$$q_i^{″}-q_i^{\prime \prime \text{HS}}=\delta q_i^{″}+z_i\delta q^{*},$$ (58)

with

$$\delta q_i^{″}=-\pi \frac{P_n}{\mathrm{\Delta }}{a}_i\mathrm{\Gamma }{\sigma }_i-{\pi }^2{\left(\frac{P_n}{\mathrm{\Delta }}\right)}^2\frac{{\alpha }^2}{\mathrm{\Gamma }}{\sigma }_i^2$$ (59)

and

$$\delta q^{*}\equiv \frac{\pi }{2}\frac{P_n}{\mathrm{\Delta }}\frac{{\alpha }^2}{\mathrm{\Gamma }}.$$ (60)

For the next term in Eq. (50), by Eq. (52), we have

$$\begin{array}{ll}\hfill \sum _k{\rho }_k{a}_k{q}_{ik}^{\prime }& =L{\sigma }_i\sum _k{\rho }_k{a}_k+\frac{L^2}{2}{\zeta }_2{\sigma }_i\sum _k{\rho }_k{\sigma }_k{a}_k\hfill \\ \hfill & -\frac{2{\mathrm{\Gamma }}^2}{{\alpha }^2}{a}_i\sum _k{\rho }_k{a}_k^2+L\sum _k{\rho }_k{\sigma }_k{a}_k.\hfill \end{array}$$ (61)

We now note that the first three terms all have coefficients σ_i or a_i before the sum. Thus, they cannot contribute any terms of the form z_i · const., where the constant is independent of i; only the last term can contribute such a term. This last term was evaluated in Eq. (A4) of Appendix 1 of Ref. 7, with the result

$$L\sum _k{\rho }_k{\sigma }_k{a}_k=L\frac{{\alpha }^2}{2\mathrm{\Gamma }}{P}_n=\delta q^{*}.$$ (62)

More details for evaluating Eq. (61) are given in Appendix C.

For the last term in Eq. (50), we use the definition of the κ_ij in Eq. (51) to see that all summands in an expansion of this term will have a power of σ_i as a coefficient and, therefore, cannot lead to terms of the form z_i · const. Therefore, the third term in Eq. (50) contributes only to the previously derived MSA chemical potential.

Combining these results, we have that the two δq* terms in Eqs. (60) and (62) cancel; only $2z_i\beta u_i^{*}$ from the 2βu_i term in Eq. (48) remains beyond the standard MSA excess chemical potential formulation. Therefore,

$${\mu }_i={\mu }_i^{\text{MSA}}+2z_i{u}^{*},$$ (63)

where u* is given by Eq. (33) and ${\mu }_i^{\text{MSA}}$ is the previous MSA excess chemical potential given by⁷

$$\beta {\mu }_i^{\text{MSA}}=-\frac{{\alpha }^2}{4\pi }\left[\frac{z_i^2\mathrm{\Gamma }}{1+\mathrm{\Gamma }{\sigma }_i}+\eta {\sigma }_i\left(\frac{2z_i-\eta {\sigma }_i^2}{1+\mathrm{\Gamma }{\sigma }_i}+\frac{\eta {\sigma }_i^2}{3}\right)\right].$$ (64)

We note that u* = 0 when all the ions have the same size (i.e., σ_i = σ for all i), so our results are identical to the standard MSA results in the restricted primitive model. Moreover, nothing about the MSA quantities like Γ are changed; their formulas are the same as previous standard MSA formulas since we are only restoring chemical potential terms that were dropped when only the mean chemical potential (activity coefficient) were sought.

To derive Eq. (63), we used the existing MSA formulation, but where we considered that there was a small (infinitesimal) violation of charge neutrality so that it is valid to take the derivative in Eq. (30). Appendix A derives Eq. (63) for a different but related system, where there is a constant neutralizing background charge density, instead of a violation of charge neutrality. Thus, our main result [Eq. (63)] can be obtained in two very different ways. In Sec. III, we will show that it agrees very well with simulation results for μ_i.

In Appendix C, Eq. (61) is verified in more detail. Specifically, with a_i given by Eq. (56), it is possible to separate out a constant from its x²/(1 + x) = x − 1 + 1/(1 + x) part, where x = Γσ_i. This could potentially change the +z_i · const. term in μ_i calculated from Eq. (61) (and potentially in ${\mu }_i^{\text{MSA}}$ as well). That this uncertainty about a_i does not change result (63) is shown in Appendix C.

III. RESULTS AND DISCUSSION

We verify Eq. (63) by comparing it to the chemical potentials from grand canonical MC simulations of homogeneous fluids of charged, hard spheres. This is then a direct comparison between identical systems.

The details of the simulations are given in Ref. 4. To summarize, each ion species’ total chemical potential m_i is the sum of ideal gas, hard-sphere, and electrostatic components,

$$\beta m_i=\ln {\rho }_i+\beta {\mu }_i^{\text{HS}}+\beta {\mu }_i.$$ (65)

The last term (μ_i) is what the MSA computes and is our focus. In MC simulations, the total chemical potential m_i for each species must be specified, not the densities ρ_i. In the MSA, the reverse is true. Therefore, in the simulations, values for the m_i are iterated until the desired ion concentrations ρ_i are achieved, using a well-established algorithm.¹² Since both the m_i and the ρ_i are now known, so is $\beta m_i-\ln {\rho }_i=\beta {\mu }_i^{\text{HS}}+\beta {\mu }_i$. Thus, once the hard-sphere component is determined, so is the desired μ_i (by subtraction). The hard-sphere component ${\mu }_i^{\text{HS}}$ is computed from the acceptance/rejection ratio of attempts to insert uncharged hard sphere “ions,” as previously described.¹³ This process works until, at high enough ion packing fractions, the correlation lengths between ions become too long to be contained in reasonably sized simulation box.

The first comparisons are shown in Fig. 1. Our extended MSA μ_i from Eq. (63) (solid lines) gives significantly better values for both μ₊ and μ₋ compared to ${\mu }_{+}^{\text{MSA}}$ and ${\mu }_{-}^{\text{MSA}}$ from Eq. (64) (dashed lines). In fact, for many cases the results are qualitatively better. For example, for monovalent anions (z_i = −1), μ₋ has a minimum in both the simulations and our formulation, while the MSA is always monotonic for both ions.

FIG. 1.

Comparison of our extended MSA μ_i from Eq. (63) (solid lines) to both the standard MSA formulation Eq. (64) (dashed lines) and MC simulations (symbols). Each panel shows the results for μ_i vs ρ₊ for two electrolytes with different ion valences z₊ and z₋, but the same cation sizes. One electrolyte is depicted with cations and anions in green and blue colors, respectively, and the other electrolyte is depicted with cations and anions in black and red colors, respectively. The dielectric constant values is always ɛ = 78.45 and T = 298.15 K. The anion diameter is always 0.3 nm, while the cation diameter σ₊ is indicated in each panel.

Also, for 2:2 electrolytes with large ions, both ${\mu }_{+}^{\text{MSA}}$ and ${\mu }_{-}^{\text{MSA}}$ are qualitatively incorrect; the cation curves (black dashed lines in the top row of Fig. 1) follow the anion simulation results (red symbols) while the anion curves (red dashed lines) follow the cation simulation results (black symbols). Our extended MSA μ₊ and μ₋ split much more correctly, although there is sometimes a minimum for μ₋ that is not present in the simulation results. This maybe where Eq. (63) breaks down, or it maybe that Eq. (63) is too early to reveal a minimum that occurs at higher concentrations in the simulations. Currently, the simulated concentrations are not high enough to make the distinction, so this will need to be explored in future work.

Figure 2 shows that our Eq. (63) also works extremely well at different dielectric constants ɛ. Here, μ₊ vs ɛ is shown for both small cations [Fig. 2(a)] and large cations [Fig. 2(b)]. As in Fig. 1, the previous MSA formulations and our extended version give very similar results when the cations are small [Fig. 2(a)]. However, the two are quite different when the cations are large compared to the anions [Fig. 2(b)]. In fact, the previous MSA formulation’s error becomes larger with larger size asymmetry. This can be seen in Fig. 2(b) by noting that the dashed black line is much farther from the solid black line (the 0.9 nm cation case) than the dashed and solid blue curves are from each other (the 0.6 nm cation case).

FIG. 2.

Comparison of our extended MSA μ₊ from Eq. (63) (solid lines) to both the standard MSA formulation Eq. (64) (dashed lines) and MC simulations (symbols) as dielectric constant ɛ varies. In all panels, only cation results are shown. The anion diameter is always 0.3 nm, while the cation diameter σ₊ is indicated on each set of curves. Each panel shows the results for two 1:1 electrolytes with ρ₊ = 1 M. (a) σ₊ = 0.15 nm is shown with black lines/symbols and σ₊ = 0.45 nm with blue. (b) σ₊ = 0.6 nm is in blue and σ₊ = 0.9 nm in black.

IV. CONCLUSIONS

We have extended the standard MSA theory to fully account for individual ion species excess chemical potentials [Eq. (63)]. To verify the accuracy of this theory, in the figures, we directly compared against simulation for at least 28 unique electrolytes (i.e., different combinations of z₊:z₋, σ₊, and ɛ) over a very wide range of concentrations (1 μM to $>1$ M). These included challenging combinations of trivalents (z₊ = 3), very large ions (σ₊ = 0.9 nm), and small and large dielectric constants (20 ≤ ɛ ≤ 120). The excellent results of our extended MSA give us confidence not only that our two derivations (in the main text and Appendix A) are correct, but also that Eq. (63) is useful for real-world applications whenever the primitive model of electrolytes is a valid model (e.g., ionic liquids).

APPENDIX A: MSA OF ELECTROLYTES IN A UNIFORM NEUTRALIZING BACKGROUND

On a macroscopic scale, an electrolyte of uniform ion densities has to fulfill charge neutrality. If this is obtained by adding a uniform background charge, then the chemical potential of single ions may be derived in a slightly different way than in Sec. II C. For such a background, a high density of charged point particles spread uniformly throughout the electrolyte can be used. Let these particles have density ρ₀, valence z₀, and hard core diameter σ₀ = 0. Then, the limit ρ₀ → ∞ and z₀ → 0 is considered with z₀ρ₀ finite. The background will interact with the regular ions, but they will remain uncorrelated. With high density they will give a corresponding high contribution to pressure, but this does not influence correlations in the limit considered.

The usual MSA solution for charged hard spheres cannot be applied directly to this system. The reason is that it is restricted to additive hard spheres. To that point, charges also have to stay outside the hard cores from other charged particles. But, as we will argue and find below, the MSA system may be transformed to represent a system with a uniform background.

The internal energy per particle u_i of component i from the MSA solution of charged hard spheres [Eqs. (31)–(33)] is

$$\beta u_i=\frac{{\alpha }^2}{4\pi }{z}_i{N}_i+z_i\beta u^{*},$$ (A1)

with N_i given by Eq. (32) and

$$\beta u^{*}=-\frac{{\alpha }^2}{24}\sum _l{\rho }_l{\sigma }_l^2\left(N_l{\sigma }_l+\frac{3}{2}{z}_l\right).$$ (A2)

The internal energy density of the ionic fluid is⁷

$$\beta E=\sum _i\beta {\rho }_i{u}_i=\frac{{\alpha }^2}{4\pi }\sum _i{\rho }_i{z}_i{N}_i$$ (A3)

$$=-\frac{{\alpha }^2}{4\pi }\left(\mathrm{\Gamma }\sum _i\frac{{\rho }_i{z}_i^2}{1+\mathrm{\Gamma }{\sigma }_i}+\frac{\pi }{2\mathrm{\Delta }}\mathrm{\Omega }{P}_n^2\right).$$ (A4)

where Δ is given by Eq. (35), P_n by Eq. (37), and Ω by Eq. (38). Charge neutrality requires

$$\sum _i{z}_i{\rho }_i=0,$$ (A5)

by which the quantity u* does not contribute to E. The Helmholtz free energy per unit volume A is given by⁷

$$\beta A=\beta E+\frac{{\mathrm{\Gamma }}^3}{3\pi }.$$ (A6)

Since

$$\frac{\partial (\beta A)}{\partial \mathrm{\Gamma }}=0,$$ (A7)

Equation (A6) gives the thermodynamic relation ∂(βA)/∂β = E, as required. [We verify Eq. (A7) below.]

With relation (A7), the chemical potentials can be found as

$$\beta {\mu }_i=\frac{\partial (\beta A)}{\partial {\rho }_i}=\frac{\partial (\beta E)}{\partial {\rho }_i}=\beta u_{0i}+\delta {\mu }_{0i},$$ (A8)

where

$$\beta u_{0i}=\frac{{\alpha }^2}{4\pi }{z}_i{N}_i$$ (A9)

and

$$\delta {\mu }_{0i}=-\frac{{\alpha }^2}{4\pi }\eta {\sigma }_i\frac{\eta {\sigma }_i^2(\mathrm{\Gamma }{\sigma }_i-2)+3z_i}{3(1+\mathrm{\Gamma }{\sigma }_i)}.$$ (A10)

Due to charge neutrality, the u* term does not appear here and has no influence.

Now consider the system with a uniform background. The background is then also present inside the hard sphere ions and effectively adds a charge to them. The influence of this charge on its surroundings is the same as for a point charge at the center of each ion. This is a well-known property of Coulomb interaction. However, there will be an additional interaction between the given point charge with valence z_l and the part of the background inside the hard core diameter σ_l. This is proportional to the last term of the quantity u* in Eq. (A2). Moreover, the first term of u* represents the volume of a sphere of diameter σ_l. The background charge inside it is proportional to this volume. Thus, there is reason to expect that the quantity u* has something to do with the neutralizing background.

Next, we show that the system with a background charge can be transformed into an effective MSA problem. This is done by adding the background charge inside each sphere to its given point charge with valence z_i. Then, the particle gets an effective valence of

$$z_{ei}=z_i+\frac{\pi }{6}{z}_0{\rho }_0{\sigma }_i^3$$ (A11)

for ion species i = 1, 2, 3, …, n, where n is the number of components apart from the background. Since part of the background has been added to the particle charges, its average density from the remaining ions has been reduced in magnitude (with fixed z₀, the ρ₀ can be greater or less than 0). Its effective density becomes

$${\rho }_{e0}={\rho }_0\left(1-\frac{\pi }{6}\sum _l{\sigma }_l^3{\rho }_l\right).$$ (A12)

Charge neutrality requires

$$\sum _i{\rho }_i{z}_i+{\rho }_0{z}_0=0.$$ (A13)

Equations (A11)–(A13) then give

$$\sum _i{\rho }_i{z}_{ei}+{\rho }_{e0}{z}_0=0.$$ (A14)

Thus, we have obtained an effective MSA system where charge neutrality is fulfilled. The main part of its internal energy is given by Eq. (A4) with z_i ↦ z_ei. It can be noted that there is no contribution from the background when z₀ → 0. The situation is the same for the quantities Ω and P_n since σ₀ = 0.

As indicated earlier, there is an additional contribution to the internal energy. This is the interaction between the ionic point charges z_i and the background inside each hard sphere ion. With charge density that follows from Eq. (A11), this Coulomb energy Δu_i is

$$\beta \mathrm{\Delta }{u}_i=\frac{{\alpha }^2}{4\pi }4\pi \underset{0}{\overset{{\sigma }_i/2}{\int }}\frac{1}{r}{z}_i{\rho }_0{z}_0{r}^{2}dr$$ (A15)

$$=\frac{{\alpha }^2}{4\pi }\frac{\pi }{2}{z}_i{\sigma }_i^2{z}_0{\rho }_0.$$ (A16)

This gives the additional internal energy density of

$$\beta \mathrm{\Delta }E=\sum _i\beta {\rho }_i\mathrm{\Delta }{u}_i=\frac{{\alpha }^2}{4\pi }\frac{\pi }{2}\sum _l(z_l{\rho }_l{\sigma }_l^2)z_0{\rho }_0.$$ (A17)

One sees that this term is similar to the z_l term of Eq. (A2). Altogether the internal energy is

$$E=E_e+\mathrm{\Delta }E,$$ (A18)

where E_e is Eq. (A4), with z_i↦z_ei. With this modification, the Helmholtz free energy density (A6) will remain the same. The reason is that z_ei and ΔE do not contain the parameter Γ, and so the partial derivative (A7) still vanishes. Thus, the chemical potentials are given by

$$\beta {\mu }_i=\frac{\partial (\beta A)}{\partial {\rho }_i}=\frac{\partial (\beta E)}{\partial {\rho }_i}.$$ (A19)

By differentiation, where z_ei and ΔE are kept constant, Eq. (A8) is recovered. Then, one can differentiate with respect to z_ei to get

$$\frac{\partial (\beta E)}{\partial z_{el}}=-2\frac{{\alpha }^2}{4\pi }\left[\mathrm{\Gamma }\frac{{\rho }_l{z}_{el}}{1+\mathrm{\Gamma }{\sigma }_l}+\frac{\pi }{2\mathrm{\Delta }}{P}_n\frac{{\rho }_l{\sigma }_l}{1+\mathrm{\Gamma }{\sigma }_l}\right]$$ (A20)

$$=2\frac{{\alpha }^2}{4\pi }{\rho }_l{N}_l.$$ (A21)

With charge neutrality (A13), expression (A11) can be rewritten as

$$z_{el}=z_l-\frac{\pi }{6}{\sigma }_l^3\sum _k{z}_k{\rho }_k.$$ (A22)

Then,

$$\beta \delta {\mu }_{1i}\equiv \sum _l\frac{\partial (\beta E)}{\partial z_{el}}\frac{\partial z_{el}}{\partial {\rho }_i}$$ (A23)

$$=-2\frac{{\alpha }^2}{4\pi }{z}_i\frac{\pi }{6}\left(\sum _l{\rho }_l{\sigma }_l^2{N}_l{\sigma }_l\right).$$ (A24)

Likewise, with charge neutrality, we have

$$\beta \delta {\mu }_{2i}\equiv \frac{\partial (\beta \mathrm{\Delta }E)}{\partial {\rho }_i}$$ (A25)

$$=-2\frac{{\alpha }^2}{4\pi }{z}_i\frac{\pi }{6}\left(\sum _l{\rho }_l{\sigma }_l^2\frac{3}{2}{z}_l\right),$$ (A26)

where the limit of zero background charge is used (i.e., z₀ρ₀ → 0). Adding these together gives

$$\beta \delta {\mu }_{1i}+\beta \delta {\mu }_{2i}=-2z_i{u}^{*}$$ (A27)

with u* given by Eq. (A2). Thus, with expressions (A8), (A10), and (A27), the individual species chemical potentials become

$$\beta {\mu }_i=\beta u_{0i}+\beta \delta {\mu }_{0i}-2z_i{u}^{*}.$$ (A28)

This is the identical to our main result, Eq. (63).

Finally, we verify Eq. (A7). By partial differentiation one first finds with expression (A4),

$$\begin{array}{ll}\hfill \frac{\partial (\beta E)}{\partial \mathrm{\Gamma }}& =-\frac{{\alpha }^2}{4\pi }\sum _i\frac{{\rho }_i{z}_i^2}{{(1+\mathrm{\Gamma }{\sigma }_i)}^2}-\frac{{\alpha }^2}{4\pi }\frac{\pi }{2\mathrm{\Delta }}\left(2P_n\frac{\partial }{\partial \mathrm{\Gamma }}(\mathrm{\Omega }{P}_n)-\frac{\partial \mathrm{\Omega }}{\partial \mathrm{\Gamma }}{P}_n^2\right).\hfill \end{array}$$ (A29)

With some algebra, one further finds

$$\frac{\partial (\mathrm{\Omega }{P}_n)}{\partial \mathrm{\Gamma }}=-\sum _i\frac{{\rho }_i{\sigma }_i^2{z}_i}{{(1+\mathrm{\Gamma }{\sigma }_i)}^2},$$ (A30)

$$\frac{\partial \mathrm{\Omega }}{\partial \mathrm{\Gamma }}=-\frac{\pi }{2\mathrm{\Delta }}\sum _i\frac{{\rho }_i{\sigma }_i^4}{{(1+\mathrm{\Gamma }{\sigma }_i)}^2},$$ (A31)

$$\frac{\partial (\beta E)}{\partial \mathrm{\Gamma }}=-\frac{{\alpha }^2}{4\pi }\sum _i\frac{{\rho }_i\left(z_i^2-2\eta {\sigma }_i^2{z}_i+{\eta }^2{\sigma }_i^4\right)}{{(1+\mathrm{\Gamma }{\sigma }_i)}^2}$$ (A32)

$$=-\frac{{\mathrm{\Gamma }}^2}{\pi },$$ (A33)

where the last equality follows from expression (34) for Γ. Thus, with expression (A6) for A, we have

$$\frac{\partial (\beta A)}{\partial \mathrm{\Gamma }}=\frac{\partial (\beta E)}{\partial \mathrm{\Gamma }}+\frac{{\mathrm{\Gamma }}^2}{\pi }=0,$$ (A34)

which is Eq. (A7).

APPENDIX B: DERIVATION OF u*

Since Høye and Blum⁶ only stated the result for the internal energy per particle [Eq. (31)] with N_i defined by Eq. (32) and u* defined by Eq. (33), we offer a short derivation here. We use Blum’s review of the MSA derivation in Ref. 2 as our guide.

We start with Eq. (39), the definition of excess internal energy beyond the mean-field contribution. With Coulombic interactions,

$$\int h_{ij}(r)\beta {\psi }_{ij}(r,0)d\mathbf{r}=z_i{z}_j{\alpha }^2\int t{h}_{ij}(t)dt$$ (B1)

$$=z_i{z}_j\frac{{\alpha }^2}{2\pi }{J}_{ij}(0)$$ (B2)

for J_ij(r) defined in Eq. (11). Therefore,

$$\beta u_i=\frac{{\alpha }^2}{4\pi }{z}_i{B}_i,$$ (B3)

where

$$B_i=\sum _k{z}_k{\rho }_k{J}_{ki}(0).$$ (B4)

By Eq. (B.40) of Ref. 2 [and the equivalent Eq. (2.31) of Ref. 1], the B_i are given implicitly by

$$N_i=B_i+\frac{\pi }{4\mathrm{\Delta }}\left[{\chi }_2+\frac{2}{3}\sum _k{\rho }_k{\sigma }_k^3{B}_k\right],$$ (B5)

where Δ is from Eq. (35) and

$${\chi }_2=\sum _k{z}_k{\rho }_k{\sigma }_k^2.$$ (B6)

Since we know N_i from Eq. (32), we can solve for the B_i. The matrix associated with this linear system has a unique structure that allows for an analytic solution. Specifically, Eq. (B5) in matrix form is

$$\left(I+\left(\begin{array}{ccc}\hfill {\xi }_1\hfill & \hfill \cdots \hfill & \hfill {\xi }_n\hfill \\ \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill \\ \hfill {\xi }_1\hfill & \hfill \cdots \hfill & \hfill {\xi }_n\hfill \end{array}\right)\right)\vec{B}=\vec{N}-\left(\begin{array}{c}\hfill \chi \hfill \\ \hfill ⋮\hfill \\ \hfill \chi ,\hfill \end{array}\right),$$ (B7)

where I is the identity matrix, χ = πχ₂/4Δ and ${\xi }_k=\pi {\rho }_k{\sigma }_k^3/6\mathrm{\Delta }$. The solution of this is

$$B_i=\frac{r_i+r_i{\sum }_k{\xi }_k-{\sum }_k{r}_k{\xi }_k}{1+{\sum }_k{\xi }_k}$$ (B8)

for r_k = N_k − χ. Substituting in for r_k, χ, and ξ_k gives Eq. (31) after some algebra.

APPENDIX C: EVALUATION OF THE SUM ${\sum }_k{\rho }_k{a}_k{q}_{ik}^{\prime }$

The sum $\beta u_{0i}=(z_i/4\pi ){\sum }_k{\rho }_k{a}_k{q}_{ik}^{\prime }$ in Eq. (61), via the definition of $q_{ik}^{\prime }$ in Eq. (52), is given by

$$\beta u_{0i}=\frac{z_i}{4\pi }\left\{\frac{\pi }{\mathrm{\Delta }}\left[{\sigma }_i{S}_1+\left(1+\frac{\pi }{2\mathrm{\Delta }}{\zeta }_2{\sigma }_i\right)S_2\right]-\frac{2{\mathrm{\Gamma }}^2}{{\alpha }^2}{a}_i{S}_3\right\}$$ (C1)

for the S_i defined below.

To evaluate the sum

$$S_1=\sum _k{\rho }_k{a}_k,$$ (C2)

we rewrite the a_k given by Eq. (56) as

$$a_k=\frac{{\alpha }^2}{2\mathrm{\Gamma }}\left[z_k-\frac{z_k\mathrm{\Gamma }{\sigma }_k}{1+\mathrm{\Gamma }{\sigma }_k}-\frac{\pi }{2\mathrm{\Delta }}{P}_n{\sigma }_k^2+P_n\frac{\pi }{2\mathrm{\Delta }}\frac{\mathrm{\Gamma }{\sigma }_k^3}{1+\mathrm{\Gamma }{\sigma }_k}\right].$$ (C3)

Then,

$$S_1=\frac{{\alpha }^2}{2\mathrm{\Gamma }}\left[0-\mathrm{\Gamma }\mathrm{\Omega }{P}_n-\frac{\pi }{2\mathrm{\Delta }}{P}_n{\zeta }_2+P_n(\mathrm{\Omega }-1)\mathrm{\Gamma }\right]$$ (C4)

$$=\frac{{\alpha }^2}{2\mathrm{\Gamma }}\left[-\frac{\pi }{2\mathrm{\Delta }}{P}_n{\zeta }_2-P_n\mathrm{\Gamma }\right],$$ (C5)

with P_n and Ω given by Eqs. (37) and (38).

With a_k of form (56),

$$S_2=\sum _k{\rho }_k{\sigma }_k{a}_k$$ (C6)

$$=\frac{{\alpha }^2}{2\mathrm{\Gamma }}[\mathrm{\Omega }{P}_n-P_n(\mathrm{\Omega }-1)]$$ (C7)

$$=\frac{{\alpha }^2}{2\mathrm{\Gamma }}{P}_n.$$ (C8)

Finally, by Eq. (14) of Ref. 7,

$$S_3=\sum _k{\rho }_k{a}_k^2={\alpha }^2.$$ (C9)

Adding everything together, we find

$$\begin{array}{ccc}\hfill \beta u_{0i}& =\hfill & \frac{z_i}{4\pi }\left\{\frac{\pi }{\mathrm{\Delta }}\left[{\sigma }_i{S}_1+\left(1+\frac{\pi }{2\mathrm{\Delta }}{\zeta }_2{\sigma }_i\right)S_2\right]-\frac{2{\mathrm{\Gamma }}^2}{{\alpha }^2}{a}_i{S}_3\right\}\hfill \\ \hfill & =\hfill & \frac{z_i}{4\pi }\left[\frac{\pi {\alpha }^2}{2\mathrm{\Delta }\mathrm{\Gamma }}(-\mathrm{\Gamma }{\sigma }_i+1)P_n-2{\mathrm{\Gamma }}^2{a}_i\right]\hfill \\ \hfill & =\hfill & \frac{z_i}{4\pi }\left[\frac{\pi {\alpha }^2}{2\mathrm{\Delta }\mathrm{\Gamma }}(-\mathrm{\Gamma }{\sigma }_i+1)-2{\mathrm{\Gamma }}^2{\alpha }^2\frac{z_i-\eta {\sigma }_i^2}{2\mathrm{\Gamma }(1+\mathrm{\Gamma }{\sigma }_i)}\right]\hfill \\ \hfill & =\hfill & \frac{{\alpha }^2}{4\pi }{z}_i\left(N_i+\frac{\eta }{\mathrm{\Gamma }}\right).\hfill \end{array}$$ (C10)

Here, the last term of this result is the same as the last sum of Eq. (61), while the first one is βu_i − z_iβu*.

For the analysis of the remaining terms of (61), we only give a brief outline. We note that the last terms of Eqs. (45) and (46) differ only by βu_0i plus another term; this follows from Eqs. (35) and (43) of Ref. 7. This other term, by that reference’s Eq. (A17), is a z_i · const. term that was neglected in Ref. 7. The remaining terms, except for another z_i · const. term in its Eq. (A20), of its Eq. (44) (which computes βμ_i − βu_i) were included in their ${\mu }_i^{\text{MSA}}$. This latter z_i · const. term, like z_iδq* of Eq. (58), exactly compensates the last sum of Eq. (61), from which Eq. (63) follows.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.