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The Poisson–Boltzmann equation describes a model proposed independently by Louis Georges Gouy and David Leonard Chapman in 1910 and 1913, respectively. [3] 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. [4]
Although this equation has solid theoretical justification, it is computationally expensive to calculate without approximations. A number of numerical Poisson-Boltzmann equation solvers of varying generality and efficiency have been developed, [10] [11] [12] including one application with a specialized computer hardware platform. [13]
The most popular model to describe the electrical double layer is the Poisson-Boltzmann (PB) model. This model can be equally used to evaluate double layer forces. Let us discuss this model in the case of planar geometry as shown in the figure on the right.
The Debye–Hückel theory was proposed by Peter Debye and Erich Hückel as a theoretical explanation for departures from ideality in solutions of electrolytes and plasmas. [1] It is a linearized Poisson–Boltzmann model, which assumes an extremely simplified model of electrolyte solution but nevertheless gave accurate predictions of mean activity coefficients for ions in dilute solution.
The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation. This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved unless the collision term in f is known.
The Poisson–Boltzmann equation plays a role in the development of the Debye–Hückel theory of dilute electrolyte solutions. Using a Green's function, the potential at distance r from a central point charge Q (i.e., the fundamental solution ) is φ ( r ) = Q 4 π ε r , {\displaystyle \varphi (r)={\frac {Q}{4\pi \varepsilon r}},} which is ...
where V schr is the quantum correction potential, z is the direction perpendicular to the interface, n q is the quantum density from the Schrödinger equation which is equivalent to the converged Monte Carlo concentration, V p is the potential from the Poisson solution, V 0 is the arbitrary reference potential far away from the quantum region ...
The model is based on the Poisson-Boltzmann equation, which is an expansion of the original Poisson's equation. Solvation Models (SMx) and the Solvation Model based on Density (SMD) have also seen wide spread use. SMx models (where x is an alphanumeric label to show the version) are based on the generalized Born equation. This is an ...