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

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

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

• Electrolyte solutions: Describing the distribution of ions around a charged object.
• Colloids: Understanding the stability and interactions of colloidal particles.
• Biomolecules: Modeling the electrostatic interactions in proteins, nucleic acids, and other biological macromolecules.

Numerical Solutions

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

• Poisson equation
• Boltzmann distribution
• Electrostatics
• Electrolyte
• Colloid
• Biomolecule
• Finite difference method
• Finite element method
• Boundary element method

See Also

• Debye–Hückel theory
• Gouy–Chapman model
• Nonlinear Poisson–Boltzmann equation