Classical Density Functional Theory of Ionic Solutions

Abstract

The basic structure of classical density functional theory (DFT) is reviewed from a rather general perspective. The treatment is then specialized to ionic solutions, describing the various possible extensions beyond the Poisson–Boltzmann level, that DFT offers, such as excluded volume effects, non-electrostatic interactions, connectivity (polymers) and ion correlations. The last effects are discussed rather thoroughly, with several explicit illustrations.

Appendix: The Hole Corrected Debye–Hückel Theory

We include a summary of the Debye–Hückel hole (DHH) theory, closely following the original descriptions [29–34]. This entails a correlation-corrected theory of the one-component plasma, OCP. In the OCP, one charged species is treated explicitly, whereas the other serves as a neutralizing background charge. Let us assume that the explicit ions carry a charge $q$ (valency $z$), and that the bulk density is $n_b$. We will furthermore put a “tagged” particle at the origin. The (configurational) free energy can then be expressed as a functional of the ion density surrounding our central ion:

$$\begin{aligned} \beta {\fancyscript{F}}[n(\mathbf{r})] = \int \ n(\mathbf{r}) (\ln n(\mathbf{r}) + 1)\,{\mathrm {d}}\mathbf{r}\;+\; z^2l_B \int \ {\varDelta n}(\mathbf{r}) \left( \frac{1}{2} \int \frac{ {\varDelta n}(\mathbf{r}^{\prime }) }{ |\mathbf{r} - \mathbf{r}^{\prime }| } d\mathbf{r}^{\prime } + \frac{1}{|\mathbf{r}|}\,{\mathrm {d}}\mathbf{r} \right) \nonumber \\ \end{aligned}$$

(2.31)

where $\varDelta n(\mathbf{r})=n(\mathbf{r})-n_{b}$ is the deviation of the ion density from its bulk value. Minimizing this functional leads to the anticipated Boltzmann distribution:

$$\begin{aligned} n(r) = n_{b}e^{-\beta q\psi (r)} \end{aligned}$$

(2.32)

where the potential $\psi$ is given by:

$$\begin{aligned} \psi (r) = \frac{q}{4\pi \varepsilon _{0}\varepsilon _r} \left( \frac{1}{r} + \int \frac{{\varDelta n}(\mathbf{r}^{\prime })}{|\mathbf{r} - \mathbf{r}^{\prime }|}\,{\mathrm {d}}\mathbf{r}^{\prime } \right) \end{aligned}$$

(2.33)

Upon linearisation, we end up at the Debye–Hückel level:

$$\begin{aligned} {\varDelta n}(r) = \frac{-\kappa ^{2} \exp ( -\kappa r)}{4\pi r}, \qquad \beta \kappa ^{2} = \frac{n_{b}q^{2}}{\varepsilon _{0}\varepsilon _r}. \end{aligned}$$

(2.34)

The electrostatic coupling strength, $\varGamma$, is a convenient dimensionless quantity, defined as:

$$\begin{aligned} \varGamma = \frac{l_Bz^{2}}{a} \end{aligned}$$

(2.35)

where $a={\bigl (4\pi n_{b}/3\bigr )}^{-1/3}$ measures the radius of a spherical volume per particle. The (potential) energy per particle, $u$, can be neatly expressed in terms of $\varGamma$:

$$\begin{aligned} \beta u = - \frac{\sqrt{3}}{2} \, \,\varGamma ^{3/2} \end{aligned}$$

(2.36)

A charge integration gives us the electrostatic free energy per particle, $f$:

$$\begin{aligned} \beta f = \beta \int \limits _{0}^{\varGamma } \frac{{\mathrm {d}}\varGamma ^{\prime }}{\varGamma ^{\prime }} \ u(\varGamma ^{\prime }) = - \frac{1}{\sqrt{3}} \varGamma ^{3/2} \end{aligned}$$

(2.37)

At this Debye–Hückel level, the radial distribution function is:

$$\begin{aligned} g_{\mathrm {DH}}(r) = 1 - \frac{\kappa ^{2}}{4\pi n_{b}}\frac{1}{r}\exp (-\kappa r) \, \, . \end{aligned}$$

(2.38)

As Nordholm pointed out [31], $g_{\mathrm {DH}}(r)$ becomes negative at small $r$, which of course is unphysical. In the DHH theory, this is avoided via the introduction of a correlation hole, of radius $h$, surrounding the central ion:

$$\begin{aligned} g(r) = {\left\{ \begin{array}{ll} 0, &{} \qquad r < h \\ 1 - {\displaystyle {\frac{h}{r}}} \exp \bigl (-\kappa (r - h)\bigr ), &{} \qquad r \ge h \end{array}\right. } \, \, . \end{aligned}$$

(2.39)

The size of the “hole” is adjusted, such that it contains one ion, assuming a uniform bulk density $n_b$:

$$\begin{aligned} n_{b} \int \limits _{0}^{\infty } {\mathrm{d}}r \ 4\pi r^{2} \bigl ( 1 - g(r) \bigr ) = 4 \pi n_{b} \left[ \frac{h^{3}}{3} + \int \limits _{h}^{\infty } {\mathrm{d}}r \ rh \exp \bigl (-\kappa (r - h)\bigr ) \right] = 1. \end{aligned}$$

(2.40)

Thus:

$$\begin{aligned} h = \kappa ^{-1} \left[ {\left( 1 + (3\varGamma )^{3/2} \right) }^{1/3} - 1 \right] . \end{aligned}$$

(2.41)

With the introduction of such a hole, the electrostatic energy becomes:

$$\begin{aligned} \beta u = -\frac{1}{4} \left[ {\left( 1 + (3\varGamma )^{3/2} \right) }^{2/3} - 1 \right] . \end{aligned}$$

(2.42)

whereas a charge integration gives:

$$\begin{aligned} \beta f = \frac{1}{4} \left( 1 + \frac{2\pi }{3\sqrt{3}} + \ln \left( \frac{\omega ^{2} + \omega + 1}{3} \right) - \omega ^{2} - \frac{2}{\sqrt{3}}\arctan \left( \frac{2\omega +1}{\sqrt{3}} \right) \right) \, \, , \end{aligned}$$

(2.43)

where

$$\begin{aligned} \omega = {\left( 1 + (3\varGamma )^{3/2} \right) }^{1/3} \, \, . \end{aligned}$$

(2.44)

The DHH theory has proven remarkably accurate for a range of different coupling strengths [33], and model systems [29, 30, 32, 34].