The Poisson–Boltzmann equation describes the distribution of the electric potential in solution in the presence of one or more charged surfaces. This distribution is important to determine how the electrostatic interactions will affect the molecules in solution. It is expressed as a differential equation of the electric potential ψ {\displaystyle \psi } , which depends on the solvent permitivity ε {\displaystyle \varepsilon } , the solution temperature T {\displaystyle T} , and the mean concentration of each ion species c i 0 {\displaystyle c_{i}^{0}} :
∇ 2 ψ = − 1 ε ∑ i c i 0 q i exp ( − q i ψ ( x , y , z ) k B T ) {\displaystyle \nabla ^{2}\psi =-{\frac {1}{\varepsilon }}\sum _{i}c_{i}^{0}q_{i}\exp \left({\frac {-q_{i}\psi (x,y,z)}{k_{B}T}}\right)}
The Poisson–Boltzmann equation is derived via mean-field assumptions. From the Poisson–Boltzmann equation many other equations have been derived with a number of different assumptions.
Origins
Background and derivation The Poisson–Boltzmann equation describes a model proposed independently by Louis Georges Gouy and David Leonard Chapman in 1910 and 1913, respectively. In the Gouy-Chapman model, a charged solid comes into contact with an ionic solution, creating a layer of surface charges and counter-ions or double layer. Due to thermal motion of ions, the layer of counter-ions is a diffuse layer and is more extended than a single molecular layer, as previously proposed by Hermann Helmholtz in the Helmholtz model. The Stern Layer model goes a step further and takes into account the finite ion size.
The Gouy–Chapman model explains the capacitance-like qualities of the electric double layer. A simple planar case with a negatively charged surface can be seen in the figure below. As expected, the concentration of counter-ions is higher near the surface than in the bulk solution.
The Poisson–Boltzmann equation describes the electrochemical potential of ions in the diffuse layer. The three-dimensional potential distribution can be described by the Poisson equation
∇ 2 ψ = ∂ 2 ψ ∂ x 2 + ∂ 2 ψ ∂ y 2 + ∂ 2 ψ ∂ z 2 = − ρ e ε , {\displaystyle \nabla ^{2}\psi ={\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}=-{\frac {\rho _{e}}{\varepsilon }},}
where
ρ e {\displaystyle \rho _{e}} is the local electric charge density in C/m3,
ε {\displaystyle \varepsilon } is the permittivity of the solvent, ψ is the electric potential. The freedom of movement of ions in solution can be accounted for by Boltzmann statistics. The Boltzmann equation is used to calculate the local ion density such that
c i = c i 0 ⋅ exp ( − W i k B T ) , {\displaystyle c_{i}=c_{i}^{0}\cdot \exp \left({\frac {-W_{i}}{k_{\mathrm {B} }T}}\right),}
where
c i 0 {\displaystyle c_{i}^{0}} is the ion concentration at the bulk,
W i {\displaystyle W_{i}} is the work required to move an ion closer to the surface from an infinitely far distance,
… excerpt ends here. Continue reading the full article.
