6.8: Ion Distributions Near a Charged Sphere
- Page ID
- 294299
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
- \(A_2 = 0\), since \(\Phi \to 0\) at \(r = \infty\).
- 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
- M. Daune, Molecular Biophysics: Structures in Motion. (Oxford University Press, New York, 1999), Ch. 16, 18.
- D. A. McQuarrie, Statistical Mechanics. (Harper & Row, New York, 1976), Ch. 15.
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- 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).
- See D. A. McQuarrie, Statistical Mechanics. (Harper & Row, New York, 1976), Ch. 15.