Charge Renormalization and Charge Oscillation in Asymmetric Primitive Model of Electrolytes

Abstract

Debye charging method is generalized to study the linear response properties of the asymmetric primitive model for electrolytes. Analytic results are obtained for the effective charge distributions of constituent ions inside the electrolyte, from which all static linear response properties of the system follow. It is found that, as the ion density increases, both the screening length and the dielectric constant receive substantial renormalization due to ionic correlations. Furthermore, the valence of larger ion is substantially renormalized upward by ionic correlations, while those of smaller ions remain approximately the same. For sufficiently high density, the system exhibits charge oscillations. The threshold ion density for charge oscillation is much lower than the corresponding values for symmetric electrolytes. Our results agree well with large-scale Monte Carlo simulations, and find good agreement in general, except for the case of small ion sizes ($d = 4\,\AA$) near the charge oscillation threshold.

Appendices

Appendix 1: Mean Potential Acting on the Test Particle

In this appendix, we discuss how to compute $\psi ({\mathbf {r}}, Q)$, which is defined as the mean potential acting on Q at the center of the test particle, due to all the mobile charges $\{ q_i \}$ as well as the external charges $\rho ^\mathrm{ex}({\mathbf {r}})$.

To compute $\psi (\mathbf{0},Q)$, we shall first compute the total mean potential $\Phi ({\mathbf {r}}, Q)$ at ${\mathbf {r}}$, due to both the test ion fixed at the origin and all other mobile ions. $\psi (\mathbf{0},Q)$ can then be obtained from $\Phi ({\mathbf {r}}, Q)$ by subtracting the Coulomb potential due to Q itself and taking the local limit:

$$\begin{aligned} \psi (\mathbf{0},Q) = \lim _{{\mathbf {r}}\rightarrow 0} \left[ \Phi ({\mathbf {r}},Q) - \frac{Q}{4 \pi \epsilon r} \right] . \end{aligned}$$

(40)

Inside the contact surface $r < R_c$, no other ions can enter, and hence the potential $\Phi ({\mathbf {r}},Q)$ satisfies the Poisson equation:

$$\begin{aligned} -\epsilon \nabla ^2 \Phi ({\mathbf {r}},Q) = Q \, \delta ({\mathbf {r}}) + \rho _{ q}^\mathrm{ex}({\mathbf {r}}) , \quad r< R_c. \end{aligned}$$

(41a)

Outside the contact surface $r > R_c$, we shall use Eq. (17) to approximate the PMF of all constituent ions,¹ so that the ion number density of species $\mu$ is

$$\begin{aligned} n_{\mu }({\mathbf {r}}) = \bar{n}_{\mu }\,e^{-\beta K_{ \mu } * \Phi ({\mathbf {r}},Q)}. \end{aligned}$$

(41b)

To simplify the problem, we shall make the following local approximation for the kernels $\alpha$ and $K_{ \mu }$:

$$\begin{aligned} \hat{\alpha }({\mathbf {k}})= & {} \kappa _{ R}^2, \quad {\alpha }({\mathbf {x}}) = \kappa _{ R}^2\, \delta ({\mathbf {x}}). \end{aligned}$$

(41c)

$$\begin{aligned} \hat{K}_{\mu }({\mathbf {k}})\approx & {} q_{ R}(q_{\mu }), \quad K_{ \mu }({\mathbf {r}}) \approx q_{ R}(q_{\mu }) \, \delta ({\mathbf {r}}). \end{aligned}$$

(41d)

Correspondingly, the Green’s function in Eq. (1) is approximated by

$$\begin{aligned} \hat{G}_{ R}({\mathbf {k}}) \approx \frac{1}{k^2 + \kappa _{ R}^2}, \quad {G}_{ R}({\mathbf {r}}) \approx \frac{1}{4 \pi \epsilon \, r} e^{- \kappa _{ R} r}. \end{aligned}$$

(41e)

With this approximation, Eq. (5a) reduces to Eq. (6), as one can easily check. Such an approximation turns out be rather successful, as we shall demonstrate below. It is motivated by two considerations: (1) to preserve the large-scale feature of renormalized theory outside the hard core and (2) to make the analytic calculation feasible.

Consequently, $\Phi ({\mathbf {r}},Q)$ satisfies the following nonlinear (and nonlocal) partial integro-differential equation:

$$\begin{aligned} -\epsilon \nabla ^2 \Phi ({\mathbf {r}},Q)= & {} \sum _{\mu } q_{\mu }\, \bar{n}_{\mu }\, e^{ -\beta q^R_{ \mu } * \Phi ({\mathbf {r}},Q)} + \rho _{ q}^\mathrm{ex}({\mathbf {r}}), \quad r >R_c. \end{aligned}$$

(41f)

Note that linearization of Eq. (41f) [together with Eq. (5a)] leads to Eq. (13). Equation (41f) is an improvement over the nonlinear PBE, and reduces to the latter if one approximate $K_{ \mu }$ by $q_{\mu } \delta ({\mathbf {r}})$. Additionally, $\Phi ({\mathbf {r}},Q)$ satisfies the standard electrostatic boundary conditions:

$$\begin{aligned}&\lim _{{\mathbf {r}}\rightarrow \infty } \Phi ({\mathbf {r}},Q) = 0, \end{aligned}$$

(42a)

$$\begin{aligned}&\Phi ({\mathbf {r}},Q), \frac{\partial }{\partial r} \Phi ({\mathbf {r}},Q) \,\, \text{ continuous } \text{ at } \,\, r = R_c. \end{aligned}$$

(42b)

We shall calculate the PMF up to the second order in Q. In view of Eq. (22b), we only need to solve Eqs. (41f) to the first order in Q and in $\phi _0$. We decompose $\Phi ({\mathbf {r}}, Q)$ into four parts:

$$\begin{aligned} \Phi ({\mathbf {r}},Q) = \phi ({\mathbf {r}}) + \phi ^\mathrm{h.c.}({\mathbf {r}}) + Q \left[ G({\mathbf {r}}) + \phi ^\mathrm{c}({\mathbf {r}}) \right] + O(Q^2) + O(\phi ^2), \end{aligned}$$

(43)

where $\phi ({\mathbf {r}})$ is the mean potential in the absence of the test ion, and satisfies the linear integro-differential equation (13). $\phi ^\mathrm{h.c.}({\mathbf {r}})$ arises due to the insertion of a neutral hard sphere. $G({\mathbf {r}})$ is independent of $\phi$, while $\phi ^\mathrm{c}$ is linear in $\phi$. Both $G({\mathbf {r}})$ and $\phi ^\mathrm{c}$ are independent of Q. All the ignored terms are at least quadratic either in Q or in $\phi$.

Let us set $Q = 0$ in Eq. (43), and obtain

$$\begin{aligned} \Phi ({\mathbf {r}},0) = \phi ({\mathbf {r}}) + \phi ^\mathrm{h.c.}({\mathbf {r}}) + O(\phi ^2). \end{aligned}$$

(44)

This satisfies Eqs. (41a) and (41f) with $Q=0$, and corresponds to inserting a neutral hard sphere at the origin. Expanding these equations to first order in $\phi$ and $\phi ^\mathrm{h.c.}$, and subtracting Eq. (13), we find

$$\begin{aligned} - \Delta \phi ^\mathrm{h.c.}({\mathbf {r}}) = \left\{ \begin{array}{ll} - \kappa _{ R}^2 \, \phi ^\mathrm{h.c.}({\mathbf {r}}), \quad &{} r > R_c;\\ \,\,\,\, \kappa _{ R}^2 \, \phi ({\mathbf {r}}), &{} r < R_c . \end{array} \right. \end{aligned}$$

(45a)

The equation satisfied by $G({\mathbf {r}})$ can be obtained by substituting Eq. (43) into Eqs. (41a) and (41f) and extracting the part that is linear in Q and independent of $\phi$:

$$\begin{aligned} \left\{ \begin{array}{ll} - \nabla ^2 G ({\mathbf {r}}) + \kappa _{ R}^2 \, G({\mathbf {r}}) = 0, \quad &{} r > R_c; \\ - \nabla ^2 G ({\mathbf {r}}) = \displaystyle { \frac{1}{\epsilon } \, \delta ({\mathbf {r}}) } , &{} r < R_c . \end{array} \right. \end{aligned}$$

(45b)

It is important to note that the function $G({\mathbf {r}})$ is different from the renormalized Green’s function $G_{ R}({\mathbf {r}})$ which satisfies Eq. (1). As one can see, $G({\mathbf {r}})$ explicitly takes into account the effects of hardcore repulsion, while $G_{ R}({\mathbf {r}})$ does not.

Finally, the equation satisfied by $\phi ^\mathrm{c}$ can be obtained by extracting the bilinear term (proportional to $Q \, \phi$) of Eqs. (41a) and (41f):

$$\begin{aligned} \left\{ \begin{array}{ll} - \nabla ^2 \phi ^\mathrm{c} ({\mathbf {r}}) + \kappa _{ R}^2 \, \phi ^\mathrm{c}({\mathbf {r}}) = {\epsilon }^{-1} { \beta ^2} \sum _{\mu } n_{\mu } q_{\mu } \left( q^R_{ \mu }\right) ^2 \left[ \phi ({\mathbf {r}}) + \phi ^\mathrm{h.c.}({\mathbf {r}}) \right] G({\mathbf {r}}), \quad &{} r > R_c; \\ - \nabla ^2 \phi ^\mathrm{c} ({\mathbf {r}}) =0 , &{} r < R_c. \end{array} \right. \nonumber \\ \end{aligned}$$

(45c)

The boundary conditions for $\phi ^\mathrm{h.c.}({\mathbf {r}})$, $G({\mathbf {r}})$ and $\phi ^\mathrm{c} ({\mathbf {r}})$ are identical to Eqs. (42).

Recall that $\phi ({\mathbf {r}}) = \phi _0 \, e^{i {\mathbf {k}}\cdot {\mathbf {r}}}$ is a plane wave, and can be expanded in terms of spherical harmonics $Y_{lm}$ using the well-known formula:

$$\begin{aligned} e^{i {\mathbf {k}}\cdot {\mathbf {r}}}= & {} 4 \pi \sum _{l=0}^{\infty } \sum _{m = -l}^l i^l j_l(kr ) \overline{Y_{lm}} (\hat{k}) Y_{lm} (\hat{r}) \nonumber \\= & {} j_0(k r) + \mathrm{anistropic}, \end{aligned}$$

(46)

where $j_l(k r)$ are the spherical Bessel functions of the first kind. Likewise, $\phi ^\mathrm{h.c.}({\mathbf {r}})$ can also be expanded in terms of spherical harmonics. For the present purpose, we only need the isotropic components of $\phi ^\mathrm{h.c.}({\mathbf {r}}), G(r)$ as well as $\phi ^\mathrm{c}({\mathbf {r}})$. Using Eq. (46) to extract the isotropic channel, Eq. (45a) together with the boundary condition, Eq. (42) can be readily solved:

$$\begin{aligned} \phi ^\mathrm{h.c.}(r) = \left\{ \begin{array}{ll} \phi _0\, f_1(kR_c, \kappa _{ R}R_c) \, k_0(\kappa _{ R} r), \quad &{} r> R_c; \\ \phi _0\left[ f_2 (kR_c, \kappa _{ R}R_c) + \frac{\kappa _{ R}^2}{k^2} j_0(kr) \right] , \quad &{} r < R_c, \end{array} \right. \end{aligned}$$

(47)

where the functions $f_1 (x,y), f_2( x, y)$ are defined as

$$\begin{aligned} f_1 (x,y)= & {} \frac{y^3 e^{y} ( \sin x - x \cos x)}{x^3(1 + y)}, \end{aligned}$$

(48a)

$$\begin{aligned} f_2( x, y)= & {} - \frac{y^2(y \sin x+ x \cos x)}{x^3(1 + y)}. \end{aligned}$$

(48b)

Likewise, Eq. (45b) can be easily solved:

$$\begin{aligned} G(r) = \left\{ \begin{array}{ll} \displaystyle { \frac{e^{- \kappa _{ R}(r- R_c)}}{4 \pi (1 + \kappa _{ R}R_c) r}, } &{} r > R_c; \\ \displaystyle { \frac{1}{4 \pi \epsilon r} + \frac{\kappa _{ R}}{4 \pi \epsilon ( 1+ \kappa _{ R} R_c)} }, \quad &{} r < R_c. \end{array}\right. \end{aligned}$$

(49)

Finally Eq. (45c) will be discussed in Sect. 3 for the asymmetric electrolytes. For the symmetric electrolytes, $\phi ^\mathrm{c}(r)$ vanishes identically due to the symmetry reason.

To calculate $\phi ^\mathrm{c}(0)$, we only need to solve the isotopic channel of Eq. (45c). The isotropic component of r.h.s. of Eq. (45c) can be easily shown as

$$\begin{aligned}&\frac{ \phi _0 Q}{e } \theta (r - R_c) (m_{ R} - n_{ R}) \kappa _{ R}^2 (\kappa _{ R} b) \frac{e^{ \kappa _{ R}R_c}}{1+ \kappa _{ R}R_c} \nonumber \\&\quad \times k_0(\kappa _{ R} r) \big [ j_0(k r) + f_1(kR_c, \kappa _{ R}R_c) k_0(\kappa _{ R} r) \big ]. \end{aligned}$$

(50)

where $f_1(x,y)$ is defined in Eqs. (48), and $j_0(u) = {\sin u}/{u}, \quad k_0 (u) = {e^{-u}}/{u}$. $\phi ^\mathrm{c}({\mathbf {r}})$ can now be found using the standard Liouville method [19]:

$$\begin{aligned} \phi ^\mathrm{c}(0)&= \frac{2\, \phi _0 Q}{e}\, (m_{ R} - n_{ R}) \,(\kappa _{ R} b)\, \Psi _2 (kR_c, \kappa _{ R} R_c), \end{aligned}$$

(51)

$$\begin{aligned} \Psi _2 (x, y)&\equiv \frac{e^{2 y} }{4(1+y)^2 } \Big [ 2 f_1(x, y) \, \mathrm{E}_1 \left( 3 y \right) \nonumber \\&\quad +\, i \left( \frac{y}{x} \right) \big (\mathrm{E}_1 \left( 2 y + i x\right) -\mathrm{E}_1 \left( 2 y-i x \right) \big )\Big ], \end{aligned}$$

(52)

where $\mathrm{E}_1(z)$ is the exponential integral function [1], defined as

$$\begin{aligned} \mathrm{E}_1(z) = \int _{z}^{\infty } t^{-1} e^{-t} dt. \end{aligned}$$

(53)

$\mathrm{E}_1(z)$ has a logarithmic singularity at the origin $z = 0$, and a branch cut on the negative real axis. Consequently, the function $\Psi _2(x, y)$ as a function of complex variable x has two branch cuts on the imaginary axis: one from 2iy to $i \infty$, and the other from $-2 i y$ to $-i \infty$. These singularities have no influence on the leading-order asymptotics of mean potential by a charged hard sphere, or on the interaction between two charged spheres, as the leading-order asymptotics of the latter quantities are controlled by the pole $k = i \kappa _{ R}$, where the function $\Psi _2(k R_c, \kappa _{ R}R_c)$ is analytic.

Substituting this and Eqs. (47), (49) back into Eqs. (43) and (40), we find the potential acting on the charge Q (subtracting its bulk value):

$$\begin{aligned} \delta \psi (\mathbf{0},Q)&= \phi _0 \bigg [ 1+ \frac{\kappa _{ R}^2}{k^2} + f_2 (kR_c, \kappa _{ R}R_c) \nonumber \\&\quad + \frac{2 Q}{e}\, (m_{ R} - n_{ R})\, (\kappa _{ R} b)\,\Psi _2 (kR_c, \kappa _{ R} R_c) \bigg ]. \end{aligned}$$

(54)

Appendix 2: PMF of a Neutral Particle: “Contact Value Theorem”

In this appendix, we compute $\hat{K} ({\mathbf {k}}, Q)$, the effective charge density of a neutral hard sphere.

We shall now outline a method for the PMF of a neutral hard sphere inside an electrolyte, Eq. (18). As illustrated in Fig. 1, all the ions are excluded from the region $r < R_c$. Hence, the partition function of the total system is

$$\begin{aligned} Z= & {} \int \prod _i d^3 {\mathbf {r}}_i \theta (r_i - R_c) e^{-\beta H } = \int _{R_c}^{\infty } \prod _i dr_i \int \,\, \prod _i d^2 {{\hat{{\mathbf {r}}}}}_i e^{-\beta H }, \end{aligned}$$

(55)

where ${{\hat{{\mathbf {r}}}}}_i$ is the unit vector parallel to $\mathbf{r}_i$. The total free energy is

$$\begin{aligned} F = - k_{ B} T \log {Z} = F_0 + \Delta F(\mathbf{0} , 0, R_c). \end{aligned}$$

(56)

where $F_0$ is the free energy of the unperturbed electrolyte, while $\Delta F(\mathbf{0} , 0, R_c)$ is the free energy of insertion of the neutral hard sphere [c.f. Eq. (11)]. Now let us take the differential of F with respect to $R_c$:

$$\begin{aligned} \frac{\partial \Delta F}{\partial R_c} d R_c= & {} d R_c \frac{k_{ B} T}{Z} \sum _i \int d^2 \hat{r}_i \prod _{j, j\ne i} \int d^3 {\mathbf {r}}_j \, \theta (r_j - R) \, \left( e^{-\beta H} \right) _{r_i = R} \nonumber \\= & {} k_{ B} T \, d R_c \oint d^2 \hat{r} \Big \langle \sum _i\delta ({\mathbf {r}}_i - {\mathbf {r}}) \Big \rangle \nonumber \\= & {} k_{ B} T \, d R_c \,\oint d^2 \hat{r} \sum _{\mu } n_{\mu }({\mathbf {r}}) , \end{aligned}$$

(57)

where $n_{\mu }({\mathbf {r}})$ is the average ion number density of species $\mu$, and the integral $\oint d^2 \hat{r}$ is over the contact surface. Equation (57) is a variation of the contact value theorem [9], which gives an exact relation between the particle number density for hard sphere systems and the pressure acting on a hard wall. It is important to note that, in Eq. (57), we have used the fact that the Hamiltonian is independent of $R_c$. This would not be correct if there are image charge effects.

We can now use to Eq. (41b) to express $n_{\mu }({\mathbf {r}})$ in terms of $\Phi ({\mathbf {r}},0)$, given in Eq. (44). Substituting the result back into Eq. (57), linearizing in terms of $\Phi ({\mathbf {r}},0)$, using approximation Eq. (41d), and further integrating over the contact surface, we can express the r.h.s of Eq. (57) as a linear functional of $\Phi ({\mathbf {r}},0)$. Finally integrating over the radius of contact surface $R_c$, we obtain the PMF of a neutral hard sphere (which is part of $\Delta F$ that depends on $\Phi$):

$$\begin{aligned} U (\mathbf{0}, 0, R_c) = - 4 \pi \sum _{ \mu } \bar{n}_{\mu }\,q_{\mu }^R \int _0^{R_c} d R_c \, R_c^2 \langle \Phi ({\mathbf {r}},0) \rangle _\mathrm{cont}, \end{aligned}$$

(58)

where $\langle \,\cdot \, \rangle _\mathrm{cont}$ means average over the contact surface. We reemphasize that this result is applicable only if the hard sphere has the same dielectric constant as the solvent.

Let us calculate the PMF of a neutral hard sphere $U(0,0,R_c)$ using Eq. (58). For this purpose, we need the mean potential $\Phi ({\mathbf {r}}, 0)$ in the presence of a neutral hard sphere, Eq. (44), with $\phi ^\mathrm{h.c.}({\mathbf {r}})$ given by Eq. (47). Averaging over the contact surface is trivial, because we have already thrown out the anisotropic parts. The result is

$$\begin{aligned} \left\langle \Phi ({\mathbf {r}},0) \right\rangle _\mathrm{cont} = \phi _0 \big [ j_0(k R_c) + f_1(k R_c, \kappa _{ R} R_c) k_0(\kappa _{ R} R_c) \big ]. \end{aligned}$$

(59)

Substituting this into Eq. (58) and further back into Eq. (18), we find the effective charge distribution $\hat{K}({\mathbf {k}}, 0)$ for a neutral hard sphere:

$$\begin{aligned} \hat{K}({\mathbf {k}}, 0)= & {} - 4 \pi R_c^3 \left( \sum _{\alpha } \rho _{\alpha } q_{\alpha }^{R}\right) \Psi _0(k R_c, \kappa _{ R} R_c). \end{aligned}$$

(60)

where the function $\Psi _0(x, y)$ is defined as

$$\begin{aligned} \Psi _0(x, y)&= \int _0^1 \big [ \, j_0(x t) + f_1(x t, y t) \, k_0(yt) \, \big ] t^2\, dt \nonumber \\&= \frac{1}{x^4 y} \Big [ -x \, \text {Ci} \left( x +{x}{y}^{-1}\right) \left( x \cos (x y^{-1}) - y \sin \left( x y^{-1}\right) \right) \nonumber \\&\quad + x \, \text {Ci} \left( x y^{-1} \right) \left( x \cos \left( x y^{-1}\right) -y \sin \left( x y^{-1}\right) \right) \nonumber \\&\quad - x\, \text {Si}\left( x + x y^{-1} \right) \left( x \sin \left( x y^{-1}\right) +y \cos \left( x y^{-1}\right) \right) \nonumber \\&\quad + x\, \text {Si}\left( x y^{-1}\right) \left( x \sin \left( x y^{-1}\right) +y \cos \left( x y^{-1}\right) \right) \nonumber \\&\quad - y \left( \left( x^2+2 y\right) \cos (x)+x (y-2) \sin (x)\right) +2 y^2 \Big ], \end{aligned}$$

(61)

where $\text {Ci}(x)$ and $\text {Si}(x)$ are the cosine integral and sine integral functions:

$$\begin{aligned} \text {Ci}(z)= & {} - \int _z^{\infty } \frac{\cos t}{t} {dt}, \quad \text {Si}(z) = \int _0^z \frac{\sin t}{t} {dt}. \end{aligned}$$

(62)

Both $\text {Ci}(z)$ and $\text {Si}(z)$, and hence $\Psi _0(k R_c, \kappa _{ R} R_c)$ as well, are entire functions.

Using Eqs. (8) and (31), we can rewrite Eq. (60) into the following dimensionless form:

$$\begin{aligned} \frac{1}{e} \hat{K}({\mathbf {k}}, 0)= & {} - \frac{\kappa _0^2 R_c^3}{b} \frac{\left( m_Rn - n_R m \right) }{mn(m+n)} \Psi _0(kR_c, \kappa _{ R}R_c), \end{aligned}$$

(63)

where $b = e^2/4 \pi \epsilon T$ is the Bjerrum length. If both the inserted particle and the constituent ions are point-like, $R_c \rightarrow 0$, and $\Psi _0(0,0) = 1/3$, and hence

$$\begin{aligned} \hat{K}({\mathbf {k}}, 0 ) \rightarrow - \frac{4 \pi }{3} R_c^3 \sum _{\mu } \bar{n}_{\mu }q_{\mu }^{R}. \end{aligned}$$

(64)

Appendix 3: Symmetric Electrolytes

In this appendix, we apply the general method to the simple case of symmetric electrolytes, where positive/negatives ions have the charges $\pm q$ and hard sphere of the diameter d. The charged renormalization and the oscillations in symmetric electrolytes have been studied by various authors at different times. In particular, Eq. (69a) was obtained previously by Hall [7]. Other results were also obtained by Kjellander [12] some time ago. Nonetheless, we discuss the case of the symmetric electrolytes for the purpose of illustrating the formalism.

The renormalized charges of the positive and the negative ions remain opposite to each other: $- q_{ R}(- q)= q_{ R}(q) \equiv q_{ R}.$ Consequently, the r.h.s. of (45c) vanishes identically. This means $\phi ^\mathrm{c}$ vanishes identically, to the first order in $\phi$. Likewise, the r.h.s. of Eq. (58) vanishes identically. Hence, $U (\mathbf{0}, 0, R_c) = 0$.

The total potential $\Phi ({\mathbf {r}}, Q)$ can be obtained by substituting Eqs. (47), (49) back into Eq. (43). Using Eq. (40), we further calculate $\psi (\mathbf{0},Q)$, the mean potential acting on Q:

$$\begin{aligned} \psi (\mathbf{0},Q)= & {} \phi _0 \left[ 1+ \frac{\kappa _{ R}^2}{k^2} + f_2 (kR_c, \kappa _{ R}R_c) \right] + \frac{Q \, \kappa _{ R}}{4 \pi \epsilon ( 1 + \kappa _{ R} R_c)}. \end{aligned}$$

(65)

We now use Eq. (21) to compute $\delta \psi (\mathbf{0},Q)$, and use Eq. (22b) and the fact that $\hat{K}({\mathbf {k}}, 0)$ vanishes to obtain the effective charge distribution $\hat{K}({\mathbf {k}}, Q)$ for the test particle:

$$\begin{aligned} \hat{K}({\mathbf {k}}, Q) = Q \left\{ 1+ \frac{\kappa _{ R}^2}{k^2} -\frac{\kappa _{ R}^2 \left[ \kappa _{ R} \sin (k R_c)+ k \cos (k R_c) \right] }{k^3 (1 + \kappa _{ R} R_c )} \right\} + O(Q^3). \end{aligned}$$

(66)

which is an entire function of k. Recall that we have calculated $\hat{K}({\mathbf {k}}, Q)$ up to the order of $Q^2$; hence, the ignored terms are at least of order $Q^3$. The renormalized charge can be obtained using Eq. (5d):

$$\begin{aligned} {Q_{ R}(Q)} = \hat{K}(i \kappa _{ R}, Q) = \frac{ {Q} \,e^{\kappa _{ R} R_c}}{1+ \kappa _{ R} R_c} + O\left( Q^3\right) . \end{aligned}$$

(67)

The fact that there is no term of order of $Q^2$ is actually enforced by the charge inversion symmetry: $Q_{ R}(-Q) = - Q_{ R}(Q)$.

We may apply Eq. (66) to the constituent ions with $R_c = d, Q=q$, and further apply Eq. (5a) to calculate the linear response kernel $\alpha$:

$$\begin{aligned} \hat{\alpha }({\mathbf {k}})= & {} \kappa _0^2 \left[ 1+ \frac{\kappa _{ R}^2}{k^2} + f_2 (kd, \kappa _{ R}d) \right] , \end{aligned}$$

(68)

where $f_2(x,y)$ was already defined in Eqs. (48).

The renormalized Debye length can be obtained via Eqs. (5) and (67):

$$\begin{aligned} \left( \frac{\kappa _{ R}}{\kappa _0} \right) ^2 = \frac{e^{\kappa _{ R} d}}{1+ \kappa _{ R} d} =\frac{q_{ R}}{q}, \end{aligned}$$

(69a)

where $q_{ R}$ is the renormalized charge of the positive constituent ion. This is a self-consistent equation for the renormalized inverse Debye length $\kappa _{ R}$. We can also calculate the renormalized dielectric constant:

$$\begin{aligned} \frac{\epsilon _{R}}{\epsilon } = 2- & {} \frac{1}{2} \kappa _{ R} d - e^{- \kappa _{ R} d} \left[ (1+ \kappa _{ R} d) - \frac{ 1}{2} \sinh \kappa _{ R} d \right] . \end{aligned}$$

(69b)

Careful analysis of Eq. (69a) indicates a critical value $\kappa _0^*$ defined by

$$\begin{aligned} \kappa _0^* d = \sqrt{2} \left( 2+\sqrt{3}\right) e^{-\frac{1}{2}-\frac{\sqrt{3}}{2}} \approx 1.3465, \end{aligned}$$

(70)

such that for $\kappa _0 > \kappa _0^*$, there is no real root for Eq. (69a). What happens is that a pair of real roots collide with each other at $\kappa _0^*$ and bifurcate into the complex plane. Consequently, the renormalized charge $q_{R}$ and the renormalized dielectric constant $\epsilon _{R}$ also become complex valued. While our local approximation is not really applicable if $\kappa _0$ is sufficiently close to $\kappa _0^*$, it seems very natural to argue that one should take the real parts of Eqs. (2) and (4) if the relevant parameters become complex. Therefore, in the regime $\kappa _0 > \kappa _0^*$, mean potential decays in an oscillatory fashion, and the system exhibits charge oscillation. The corresponding $\kappa _{ R}^*$ at the threshold is

$$\begin{aligned} \kappa _{ R}^* d= & {} 1 + \sqrt{3} \approx 2.732. \end{aligned}$$

(71a)

Fig. 4

Renormalized parameters for dense symmetric electrolyte. Left Renormalized vs. bare inverse Debye length. Middle: the imaginary part of $\kappa _{ R} d$ in the charge oscillation regime. Symbols MC simulation results. Solid curves our analytic results Eq. (67). The dashed curves are, respectively: black, the generalized Debye–Huckel by Lee and Fisher [3]; red and blue results using other approaches, LMPB and MSA, both from the reference [6]. The other popular theory HNC does not yield a closed-form result. Right Renormalized dielectric constant, for which we have not found any previous analytic result. Simulation details are discussed in a separate publication [16] (Color figure online)

We simulated $1:-1$ electrolytes with three different ion sizes: $d = 5, 7.5 , 10 \,\AA$ respectively, and determined all the renormalized parameters. The simulation method is described elsewhere. [16] As shown in Fig. 4, our MC results seem to agree with Eqs. (69) remarkably well, both below and above the threshold of charge oscillation. A general argument due to Kjellander and Mitchell shows that ${\epsilon _{ R}^*}$ should vanish at the threshold. This contradicts our result Eq. (69b), which gives $\epsilon _{ R}^* \approx 0.64$. We note that near the threshold, the local approximation Eq. (41e) breaks down, and therefore Eq. (69b) cannot be trusted. In any case, simulations near the threshold are very difficult, and we have no reliable results to report so far.

The real space version of the effective charge distribution can also be calculated, by Fourier transforming Eq. (66):

$$\begin{aligned} K({\mathbf {r}}, Q) =Q \, \delta ({\mathbf {r}}) + Q \frac{\kappa _{ R}^2 (\kappa _{ R} R_c - \kappa _{ R} r+1)}{4 \pi (1 + \kappa _{ R} R_c ) r} \theta (R_c - r), \end{aligned}$$

(72)

where $\theta (R_c - r)$ is the Heaviside step function. The first term (Dirac delta function) is clearly due to the bare charge of the test particle. The second term is positive and monotonically decreasing, and vanishes identically outside the contact surface ($r >R_c$). It is the diffusive part due to charge correlations. Note that even though the second term is singular (diverges as $r^{-1}$) at the origin, it does not generate any singularity in the mean potential. The mean potential, which is related to $\hat{K}({\mathbf {k}}, Q)$ via Eq. (3), can also be explicitly calculated:

$$\begin{aligned} \phi ({\mathbf {r}}, Q) = \left\{ \begin{array}{ll} \frac{Q_{ R}\, e^{-\kappa _{ R} r}}{4 \pi \epsilon r}, &{} r> R_c; \\ \frac{Q}{4 \pi \epsilon r} - \frac{Q \kappa _{ R} }{4 \pi \epsilon (1+ \kappa _{ R} R_c)}, \quad \quad &{} 0< r< R_c, \end{array} \right. \end{aligned}$$

(73)

which has the exact form of the screened Coulomb potential outside the contact surface. Since $Q_{ R}$ is given by Eq. (67) and always has the same sign as the bare charge, we see that there is no charge inversion in the symmetric electrolytes at the level of our approximation.