Poisson–Boltzmann equation

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Gouy-Chapman Simple Model
Linearized vs Full Potential Forms of the Poisson-Boltzmann Equation
File:MIS tunneling junction schematic.jpg
MIS tunneling junction schematic
File:Self-consistent average potential vs z.jpg
self-consistent average potential vs z

The Poisson–Boltzmann equation is a partial differential equation that describes the electrostatic potential in a fluid containing charged particles. It is a fundamental equation in the field of electrostatics and is widely used in physical chemistry, biophysics, and materials science to model the behavior of electrolyte solutions, colloids, and biomolecules.

Formulation[edit | edit source]

The Poisson–Boltzmann equation is derived from the Poisson equation and the Boltzmann distribution. It combines the principles of electrostatics and statistical mechanics to account for the distribution of ions in a solution. The equation is given by:

<math> \nabla^2 \psi(\mathbf{r}) = -\frac{\rho(\mathbf{r})}{\epsilon} </math>

where:

  • <math>\nabla^2</math> is the Laplacian operator,
  • <math>\psi(\mathbf{r})</math> is the electrostatic potential,
  • <math>\rho(\mathbf{r})</math> is the charge density,
  • <math>\epsilon</math> is the dielectric constant of the medium.

In the context of an electrolyte solution, the charge density <math>\rho(\mathbf{r})</math> is related to the concentration of ions by the Boltzmann distribution:

<math> \rho(\mathbf{r}) = \sum_i z_i e c_i^0 \exp\left(-\frac{z_i e \psi(\mathbf{r})}{k_B T}\right) </math>

where:

  • <math>z_i</math> is the valence of ion species <math>i</math>,
  • <math>e</math> is the elementary charge,
  • <math>c_i^0</math> is the bulk concentration of ion species <math>i</math>,
  • <math>k_B</math> is the Boltzmann constant,
  • <math>T</math> is the temperature.

Applications[edit | edit source]

The Poisson–Boltzmann equation is used to model a variety of systems, including:

Numerical Solutions[edit | edit source]

Solving the Poisson–Boltzmann equation analytically is often challenging due to its nonlinearity. Therefore, numerical methods such as finite difference, finite element, and boundary element methods are commonly employed. These methods discretize the equation and solve it iteratively to obtain the electrostatic potential.

Related Pages[edit | edit source]

See Also[edit | edit source]

References[edit | edit source]

External Links[edit | edit source]

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Contributors: Prab R. Tumpati, MD