6.6: Debye–Hückel Theory
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Since it is nonlinear, it is not easy to solve the PBE, but for certain types of problems, we can make approximations to help. The Debye–Hückel approximation holds for small electrostatic potential or high temperature conditions such that
\[\dfrac{ze\Phi}{k_B T} \ll 1 \nonumber\]
This is the regime in which the entropy of mixing dominates the electrostatic interactions between ions. In this limit, we can expand the exponential in eq. (6.5.5) as \(\exp [-ze \Phi /k_B T] \approx 1 - ze \Phi /k_B T\). The leading term in the resulting sum drops because of the charge neutrality condition, eq. (6.5.4). Keeping the second term in the expansion leads to
\[\nabla^2 \Phi = \kappa^2 \Phi \label{eq6.6.1}\]
where
\[\kappa^2 = \dfrac{2e^2}{\varepsilon k_B T} I \nonumber\]
and the ionic strength, \(I\), is defined as
\[I = \dfrac{1}{2} \sum_i C_{0, i} z_i^2 \nonumber\]
Looking at eq. (\(\ref{eq6.6.1}\)), we see that the Debye–Hückel approximation linearizes the PBE. It is known as the Debye–Hückel equation, or the linearized PBE. For the case of the 1:1 electrolyte solution described by eq. , we again obtain eq. (\(\ref{eq6.6.1}\)) using \(\text{sinh} (x) \approx x\) as \(x \to \infty\), with
\[\kappa^2 = \dfrac{2z^2 e^2 C_0}{\varepsilon k_B T} = 8\pi z^2 C_0 \ell_B \nonumber\]
The constant \(\kappa\) has units of inverse distance, and it’s inverse is known as the Debye length \(\lambda_D = \kappa^{-1}\). The Debye length sets the distance scale over which the electrostatic potential decays, i.e., the distance over which charges are screened from one another. For the symmetric electrolytes
\[\lambda_D = \kappa^{-1} = \sqrt{\dfrac{\varepsilon k_B T}{2z^2 e^2 C_0}}\]
As an example: 1:1 electrolytes in \(\text{H}_2\text{O}\): \(\varepsilon = 80\); \(z_+ = -z_- = 1\); \(T = 300\ K\) leads to
\[\begin{array} {ll} {C_0 = 100\ mM} & {\lambda_D = 9.6\ \mathring{A}} \\ {C_0 = 10\ mM} & {\lambda_D = 30.4\ \mathring{A}} \end{array} \nonumber\]
\[\lambda_D (\mathring{A}) \approx 3.04 \cdot [C_0 (M)]^{-1/2} \nonumber\]
The Debye approximation holds for small electrostatic potentials relative to \(k_B T (r > \lambda_D)\). For instance, it’s ok for ion distribution about large protein or vesicle but not for water in a binding pocket.
The variation of Debye length with concentrations of electrolytes. Reprinted from P. Ghosh http://nptel.ac.in/courses/103103033/module3/lecture3.pdf.