6.8: Ion Distributions Near a Charged Sphere

Ion Distributions Near a Charged Sphere¹

Now let’s look at how ions will distribute themselves around a charged sphere. This sphere could be a protein or another ion. We assume a spherically symmetric charge distribution about ions, and a Boltzmann distribution for the charge distribution for the ions (\(i\)) about the sphere (\(j\)) of the form

\[\rho (r) = \sum_i ez_i C_{0, i} e^{-z_i e \Phi_j (r) /k_B T}\]

\(\Phi_j (r)\) is the electrostatic potential at radius \(r\) which results from a point charge \(z_j e\) at the center of the sphere. Additionally, we assume that the sphere is a hard wall, and define a radius of closest approach by ions in solution, \(b\). The PBE becomes

\[\dfrac{1}{r^2} \dfrac{d}{dr} \left (r^2 \dfrac{d\Phi}{dr} \right ) = \dfrac{1}{\varepsilon} \sum_i ez_i C_{0, i} e^{-z_i e \Phi_j (r) /k_B T} \nonumber\]

To simplify this, we again apply the Debye–Hückel approximation \((ze\Phi \ll k_B T)\), expand the exponential in eq. , drop the leading term due to the charge neutrality condition, and obtain

\[\rho (r) = -\sum_i C_{0, i} z_i^2 e^2 \Phi_j (r)/k_B T \label{eq6.8.2}\]

Then the linearized PBE is in the Debye–Hückel approximation is

\[\dfrac{1}{r^2} \dfrac{d}{dr} \left (r^2 \dfrac{d\Phi}{dr} \right ) = \kappa^2 \Phi \label{eq6.8.3}\]

As before: \(\kappa^2 = \lambda_D^{-2} = 2e^2 I/\varepsilon k_B T\). Solutions to eq. (\(\ref{eq6.8.3}\)) will take the form:

\[\Phi = A_1 \dfrac{e^{-\kappa r}}{r} + A_2 \dfrac{e^{\kappa r}}{r} \label{eq6.8.4}\]

To solve this use boundary conditions:

1. \(A_2 = 0\), since \(\Phi \to 0\) at \(r = \infty\).
2. The field at the surface of a sphere with charge \(z_j e\) and radius \(b\) is determined from
   \[4\pi b^2 E(b) = \dfrac{z_j e}{\varepsilon} \label{eq6.8.5}\]

Now, using

\[E(b) = -\dfrac{d\Phi}{dr}|_{r = b} \label{eq6.8.6}\]

Substitute eq. (\(\ref{eq6.8.4}\)) into RHS and eq. (\(\ref{eq6.8.5}\)) into LHS of eq. (\(\ref{eq6.8.6}\)). Solve for \(A_1\).

\[A_1 = \dfrac{z_j e e^{\kappa b}}{4\pi \varepsilon (1 + \kappa b)}\nonumber\]

So, the electrostatic potential for \(r \ge b\) is

\[\Phi (r) = \underbrace{\dfrac{z_j e}{4\pi \varepsilon_0 r}}_{\text{vacuum}} \dfrac{e^{-\kappa (r - b)}}{\varepsilon_r (1 + \kappa b)} \label{eq6.8.7}\]

Setting \(r = b\) gives us the surface potential of the sphere:

\[\Phi (b) = \dfrac{z_j e}{4\pi \varepsilon b (1 + \kappa b)}\nonumber\]

Note the exponential factor in eq. (\(\ref{eq6.8.7}\)) says that \(\Phi\) drops faster than \(r^{-1}\) as a result of screening. Now substitute eq. (\(\ref{eq6.8.7}\)) into eq. (\(\ref{eq6.8.2}\)) we obtain the charge probability density

\[\rho (r) = \dfrac{-\kappa^2 z_j e}{4\pi r} \dfrac{e^{-\kappa (r - b)}}{1 + \kappa b}\]

We see that the charge density about ion drops as \(e^{-\kappa (r - b)}/r\), a rapidly decaying function that emphasizes the strong tendency for ions to attract or repel at short range. However, the charge density between \(r\) and \(r + dr\) is \(4\pi r^2 \rho (r)\) and therefore grows linearly with r before decaying exponentially: \(r e^{-\kappa (r - b)}\). We plot this function to illustrate the thickness of the "ion cloud" around the sphere, which is peaked at \(r = \lambda_D\). Additionally, note, that the charge distribution around that ion is equal and opposite to the charge of the sphere "\(j\)".

\[\int_b^{\infty} \rho (r) 4 \pi r^2 dr = -z_j e\nonumber\]

It is also possible to calculate radial distribution functions for ions in the Debye–Hückel limit.² The radial pair distribution function for ions of type \(i\) and \(j\), \(g_{ij} (r)\), is related to the potential of mean force \(W_{ij}\) as

\[g_{ij} (r) = \exp [-W_{ij} (r) / k_B T]\]

If only considering electrostatic effects, we can approximate \(W_{ij}\) as the interaction energy \(U_{ij} (r) = z_i e\Phi_j (r)\). Using the Debye–Hückel result, eq. (\(\ref{eq6.8.7}\)),

\[U_{ij} (r) = \dfrac{z_i z_j e^2}{4\pi \varepsilon (1 + \kappa b)} \dfrac{e^{-\kappa (r - b)}}{r} \nonumber\]

Let’s look at the form of \(g(r)\) for two singly charged ions with \(\lambda_D = 0.7\ nm\), \(\epsilon = 80\), and \(T = 300\ K\). The Bjerrum length is calculated as \(\ell_B = e^2/4\pi \epsilon k_B T = 0.7\ nm\). Since the Debye–Hückel holds for \(ze\Phi \ll k_B T\), we can expand the exponential in eq. as

\[g_{ij} (r) = 1 - \chi_{ij} + \dfrac{1}{2} \chi_{ij}^2 + \cdots \nonumber\]

where we define \(\chi_{ij} = U_{ij} (r) /k_B T = \ell_B e^{-\kappa (r - b)} r^{-1} (1 + \kappa b)^{-1}\). The resulting radial distribution function for co- and counterions calculated for \(b = 0.15\ nm\) are shown below.

Readings

1. M. Daune, Molecular Biophysics: Structures in Motion. (Oxford University Press, New York, 1999), Ch. 16, 18.
2. D. A. McQuarrie, Statistical Mechanics. (Harper & Row, New York, 1976), Ch. 15.

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1. See M. Daune, Molecular Biophysics: Structures in Motion. (Oxford University Press, New York, 1999), Ch. 16.; D. A. McQuarrie, Statistical Mechanics. (Harper & Row, New York, 1976), Ch. 15.; Y. Marcus, Ionic radii in aqueous solutions, Chem. Rev. 88 (8), 1475-1498 (1988).
2. See D. A. McQuarrie, Statistical Mechanics. (Harper & Row, New York, 1976), Ch. 15.