Poisson–Boltzmann equation
The Poisson–Boltzmann equation describes 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.
The Poisson–Boltzmann equation describes a model proposed independently by Louis Georges Gouy and David Leonard Chapman in 1910 and 1913, respectively.
[3] 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.
The Poisson–Boltzmann equation describes the electrochemical potential of ions in the diffuse layer.
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, and that the concentration of salt is much higher than the concentration of ions.
[9] Only minor modifications are necessary to apply the Poisson–Boltzmann equation to various interfacial models, making it a highly useful tool in determining electrostatic potential at surfaces.
In the case of an infinitely extended planar surface, there are two dimensions in which the potential cannot change because of symmetry.
Below is the Poisson–Boltzmann equation solved analytically in terms of a second order derivative with respect to x.
and the lengths are measured in units of the Debye electron radius in the region of zero potential
When using the Poisson–Boltzmann equation, it is important to determine if the specific case is low or high potential.
In the low-potential condition, the linearized version of the Poisson–Boltzmann equation (shown below) is valid, and it is commonly used as it is more simple and spans a wide variety of cases.
; however, the results that the equations yields are valid for a wider range of potentials, from 50–80mV.
As salt concentration increases, the Debye length decreases due to the ions in solution screening the surface charge.
The Poisson–Boltzmann equation can be applied in a variety of fields mainly as a modeling tool to make approximations for applications such as charged biomolecular interactions, dynamics of electrons in semiconductors or plasma, etc.
[13] The linearized Poisson–Boltzmann equation can be used to calculate the electrostatic potential and free energy of highly charged molecules such as tRNA in an ionic solution with different number of bound ions at varying physiological ionic strengths.
[14] Another example of utilizing the Poisson–Boltzmann equation is the determination of an electric potential profile at points perpendicular to the phospholipid bilayer of an erythrocyte.
This information is useful for many reasons including the study of the mechanical stability of the erythrocyte membrane.
The following expression utilizes chemical potential of solute molecules and implements the Poisson-Boltzmann Equation with the Euler-Lagrange functional:
where Finally, by combining the last three term the following equation representing the outer space contribution to the free energy density integral
These equations can act as simple geometry models for biological systems such as proteins, nucleic acids, and membranes.
[13] This involves the equations being solved with simple boundary conditions such as constant surface potential.
[13] An analytical solution to the Poisson–Boltzmann equation can be used to describe an electron-electron interaction in a metal-insulator semiconductor (MIS).
In metal-insulator semiconductor tunneling junctions, the electrons can build up close to the interface between layers and as a result the quantum transport of the system will be affected by the electron-electron interactions.
[16] Applying the following analytical solution of the Poisson–Boltzmann equation (see section 2) to MIS tunneling junctions, the following expression can be formed to express electronic transport quantities such as electronic density and electric current
Applying the equation above to the MIS tunneling junction, electronic transport can be analyzed along the z-axis, which is referenced perpendicular to the plane of the layers.
The electronic density and electric current can be found by manipulation to equation 16 above as functions of position z.
The permittivity of the solvent was assumed to be constant, resulting in a rough approximation as polar molecules are prevented from freely moving when they encounter the strong electric field at the solid surface.
Though the model faces certain limitations, it describes electric double layers very well.
