Poisson–Boltzmann equation
The Poisson–Boltzmann equation is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor, or more. It aims to describe the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution is important to determine how the electrostatic interactions will affect the molecules in solution. From the Poisson–Boltzmann equation many other equations have been derived with a number of different assumptions.
The Poisson–Boltzmann equation describes a model proposed independently by Louis George 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
where
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
where
The equation for local ion density can be substituted into the Poisson equation under the assumptions that the work being done is only electric work, that our solution is composed of a 1:1 salt (i.e. NaCl), and that the concentration of salt is much higher than the concentration of ions. The electric work to bring a charged cation or charged anion to a surface with potential ψ can be represented by and respectively. These work equations can be substituted into the Boltzmann equation, producing two expressions
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