The Poisson–Boltzmann equation is a useful equation in many settings, whether it be to understand
physiological interfaces,
polymer science
Polymer science or macromolecular science is a subfield of materials science concerned with polymers, primarily synthetic polymers such as plastics and elastomers. The field of polymer science includes researchers in multiple disciplines includ ...
, electron interactions in a
semiconductor
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
, 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. The Poisson–Boltzmann equation is derived via
mean-field assumptions.
From the Poisson–Boltzmann equation many other equations have been derived with a number of different assumptions.
Origins
Background and derivation
The Poisson–Boltzmann equation describes a model proposed independently by
Louis Georges Gouy
Louis Georges Gouy (February 19, 1854 – January 27, 1926) was a French physicist. He is the namesake of the Gouy balance, the Gouy–Chapman electric double layer model (which is a relatively successful albeit limited model that describes the ...
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
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
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
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized ar ...
-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.
![Gouy-Chapman Simple Model](https://upload.wikimedia.org/wikipedia/commons/3/34/Gouy-Chapman_Simple_Model.jpg)
The Poisson–Boltzmann equation describes the
electrochemical potential
In electrochemistry, the electrochemical potential (ECP), ', is a thermodynamic measure of chemical potential that does not omit the energy contribution of electrostatics. Electrochemical potential is expressed in the unit of J/ mol.
Introductio ...
of ions in the diffuse layer. The three-dimensional potential distribution can be described by the
Poisson equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
where
*
is the local electric charge density in C/m
3,
*
is the dielectric constant (
relative permittivity
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insul ...
) of the solvent,
*
is the permittivity of free space,
* is the
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
.
The freedom of movement of ions in solution can be accounted for by
Boltzmann statistics
Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermod ...
. The
Boltzmann equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
is used to calculate the local ion density such that
where
*
is the ion concentration at the bulk,
*
is the work required to move an ion closer to the surface from an infinitely far distance,
*
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
,
*
is the temperature in
kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phy ...
s.
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 (e.g., 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 can be represented by
and
respectively.
These work equations can be substituted into the Boltzmann equation, producing two expressions
and
,
where ''e'' is the charge of an electron, 1.602 coulombs.
Substituting these Boltzmann relations into the local electric charge density expression, the following expression can be obtained
Finally the charge density can be substituted into the Poisson equation to produce the Poisson–Boltzmann equation.
Related theories
The Poisson–Boltzmann equation can take many forms throughout various scientific fields. In biophysics and certain surface chemistry applications, it is known simply as the Poisson–Boltzmann equation.
It is also known in
electrochemistry
Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference, as a measurable and quantitative phenomenon, and identifiable chemical change, with the potential difference as an outco ...
as Gouy-Chapman theory; in solution chemistry as
Debye–Huckel theory; in
colloid chemistry as
Derjaguin–Landau–Verwey–Overbeek (DLVO) theory.
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.
Solving analytically
Because the Poisson–Boltzmann equation is a
partial differential
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Par ...
of the second order, it is commonly solved
numerically; however, with certain geometries, it can be solved analytically.
Geometries
The geometry that most easily facilitates this is a planar surface. In the case of an infinitely extended planar surface, there are two dimensions in which the potential cannot change because of symmetry. Assuming these dimensions are the y and z dimensions, only the x dimension is left. Below is the Poisson–Boltzmann equation solved analytically in terms of a second order derivative with respect to x.