Background and derivation

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] 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 in the Helmholtz model.^[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.^[4] 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.

A simple planar case for the Gouy–Chapman model

The Poisson–Boltzmann equation describes the electrochemical potential of ions in the diffuse layer. The three-dimensional potential distribution can be described by the Poisson equation^[4]

$${\displaystyle \nabla ^{2}\psi ={\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}=-{\frac {\rho _{e}}{\varepsilon _{r}\varepsilon _{0}}},}$$

where

• ${\displaystyle \rho _{e}}$ is the local electric charge density in C/m³,
• ${\displaystyle \varepsilon _{r}}$ is the dielectric constant (relative permittivity) of the solvent,
• ${\displaystyle \varepsilon _{0}}$ is the permittivity of free space,
• ψ is the electric potential.

The freedom of movement of ions in solution can be accounted for by Boltzmann statistics. The Boltzmann equation is used to calculate the local ion density such that

$${\displaystyle c_{i}=c_{i}^{0}\cdot e^{\frac {-W_{i}}{k_{\mathrm {B} }T}},}$$

where

• ${\displaystyle c_{i}^{0}}$ is the ion concentration at the bulk,^[8]
• ${\displaystyle W_{i}}$ is the work required to move an ion closer to the surface from an infinitely far distance,
• ${\displaystyle k_{\mathrm {B} }}$ is the Boltzmann constant,
• ${\displaystyle T}$ is the temperature in kelvins.

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.^[4] The electric work to bring a charged cation or charged anion to a surface with potential ψ can be represented by ${\displaystyle W^{+}=e\psi }$ and ${\displaystyle W^{-}=-e\psi }$ respectively.^[4] These work equations can be substituted into the Boltzmann equation, producing two expressions

$${\displaystyle c^{-}=c_{0}\cdot e^{\frac {e\psi (x,y,z)}{k_{B}T}}}$$

and ${\displaystyle c^{+}=c_{0}\cdot e^{\frac {-e\psi (x,y,z)}{k_{B}T}}}$,

where e is the charge of an electron, 1.602×10⁻¹⁹ coulombs.

Substituting these Boltzmann relations into the local electric charge density expression, the following expression can be obtained

$${\displaystyle \rho _{e}=e{(c^{+}-c^{-})}=c_{0}e\cdot \left[e^{\frac {-e\psi (x,y,z)}{k_{B}T}}-e^{\frac {e\psi (x,y,z)}{k_{B}T}}\right].}$$

Finally the charge density can be substituted into the Poisson equation to produce the Poisson–Boltzmann equation.^[4]

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.^[9] It is also known in electrochemistry as Gouy-Chapman theory; in solution chemistry as Debye–Huckel theory; in colloid chemistry as Derjaguin–Landau–Verwey–Overbeek (DLVO) theory.^[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.^[4]

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.^[4]

$${\displaystyle {\frac {d^{2}\psi }{dx^{2}}}={\frac {c_{0}e}{\varepsilon \varepsilon _{0}}}\cdot \left[e^{\frac {e\psi (x)}{k_{\mathrm {B} }T}}-e^{\frac {-e\psi (x)}{k_{\mathrm {B} }T}}\right]}$$

Analytical solutions have also been found for axial and spherical cases in a particular study.^[10] The equation is in the form of a logarithm of a power series and it is as follows:

$${\displaystyle {\frac {d^{2}\psi }{dr^{2}}}+{\frac {L}{r}}{\frac {d\psi }{dr}}=e^{\psi }-\delta e^{-\psi }}$$

It uses a dimensionless potential ${\displaystyle \psi ={\frac {e\Phi }{kT}}}$ and the lengths are measured in units of the Debye electron radius in the region of zero potential ${\displaystyle R_{eD}={\sqrt {\frac {kT}{4\pi e^{2}n_{e0}}}}}$ (where ${\displaystyle n_{e0}}$ denotes the number density of negative ions in the zero potential region). For the spherical case, L=2, the axial case, L=1, and the planar case, L=0.

Low-potential vs high-potential cases

When using the Poisson–Boltzmann equation, it is important to determine if the specific case is low or high potential. The high-potential case becomes more complex so if applicable, use the low-potential equation. 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.^[11] ${\displaystyle \psi =\psi _{0}e^{-\mathrm {K} x}}$

Low-potential case conditions

Strictly, low potential means that ${\displaystyle e\left\vert \psi \right\vert \ll k_{\mathrm {B} }T}$; however, the results that the equations yields are valid for a wider range of potentials, from 50–80mV.^[4] Nevertheless, at room temperature, ${\displaystyle \psi \leq \mathrm {25mV} }$ and that is generally the standard.^[4] Some boundary conditions that apply in low potential cases are that: at the surface, the potential must be equal to the surface potential and at large distances from the surface the potential approaches a zero value. This distance decay length is yielded by the Debye length ${\displaystyle \lambda _{D}}$ equation.^[4]

$${\displaystyle \mathrm {K} ={\sqrt {\frac {2c_{0}e^{2}}{\varepsilon \varepsilon _{0}k_{\mathrm {B} }T}}}}$$

$${\displaystyle \lambda _{D}=\mathrm {K} ^{-1}}$$

As salt concentration increases, the Debye length decreases due to the ions in solution screening the surface charge.^[12] A special instance of this equation is for the case of ${\displaystyle 25^{\circ }C}$ water with a monovalent salt.^[4] The Debye length equation is then:

$${\displaystyle \lambda _{D}={\frac {\mathrm {0.304nm} }{\sqrt {c_{0}\mathrm {\frac {L}{mol}} }}}}$$

These equations all require 1:1 salt concentration cases, but if ions that have higher valence are present, the following case is used.^[4]

$${\displaystyle \mathrm {K} ={\sqrt {{\frac {e^{2}}{\varepsilon \varepsilon _{0}k_{\mathrm {B} }T}}\sum c_{i}{Z_{i}}^{2}}}}$$

Conditions

In low potential cases, the high potential equation may be used and will still yield accurate results. As the potential rises, the low potential, linear case overestimates the potential as a function of distance from the surface. This overestimation is visible at distances less than half the Debye length, where the decay is steeper than exponential decay. The following figure employs the linearized equation and the high potential graphing equation derived above. It is a potential-versus-distance graph for varying surface potentials of 50, 100, 150, and 200 mV. The equations employed in this figure assume an 80mM NaCl solution.

Potential versus distance for varying surface potentials of 50, 100, 150, and 200 mV. The equations employed in this figure assume an 80mM NaCl solution.

Physiological applications

The Poisson–Boltzmann equation can be applied to biomolecular systems. One example is the binding of electrolytes to biomolecules in a solution. This process is dependent upon the electrostatic field generated by the molecule, the electrostatic potential on the surface of the molecule, as well as the electrostatic free energy.^[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. It is shown that electrostatic potential depends on the charge of the molecule, while the electrostatic free energy takes into account the net charge of the system.^[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 takes into account both the glycocalyx and spectrin layers of the erythrocyte membrane. This information is useful for many reasons including the study of the mechanical stability of the erythrocyte membrane.^[15]

Electrostatic free energy

The Poisson–Boltzmann equation can also be used to calculate the electrostatic free energy for hypothetically charging a sphere using the following charging integral:

$${\displaystyle \Delta G^{\text{el}}=\int ^{\tau }qU(\tau ')\,d\tau '}$$

where ${\displaystyle \tau q}$ is the final charge on the sphere

The electrostatic free energy can also be expressed by taking the process of the charging system. The following expression utilizes chemical potential of solute molecules and implements the Poisson-Boltzmann Equation with the Euler-Lagrange functional:

$${\displaystyle \Delta G^{\text{el}}=\int _{V}\left(kT\sum _{i}c_{i}^{\infty }\left[1-\exp \left({\frac {-z_{i}qU}{kT}}\right)\right]+p^{f}U-{\frac {-\varepsilon ({\boldsymbol {\nabla }}U)^{2}}{8\pi }}\right)dV}$$

Note that the free energy is independent of the charging pathway [5c].

The above expression can be rewritten into separate free energy terms based on different contributions to the total free energy

$${\displaystyle \Delta G^{\text{el}}=\Delta G^{\text{ef}}+\Delta G^{\text{em}}+\Delta G^{\text{mob}}+\Delta G^{\text{solv}}}$$

where

• Electrostatic fixed charges = ${\displaystyle \Delta G^{\text{ef}}=\int _{V}{\frac {p^{f}U}{2}}dV}$
• Electrostatic mobile charges = ${\displaystyle \Delta G^{\text{em}}=\int _{V}{\frac {\sum _{i}c_{i}z_{i}qU}{2}}dV}$
• Entropic free energy of mixing of mobile species = ${\displaystyle \Delta G^{\text{mob}}=kT\int _{V}c_{i}\ln {\frac {c_{i}}{c_{i}^{\infty }}}dV}$
• Entropic free energy of mixing of solvent = ${\displaystyle \Delta G^{\text{solv}}=kT\int _{V}\sum _{i}c_{i}^{\infty }\left[1-\exp \left({\frac {-z_{i}qU}{kT}}\right)\right]dV}$

Finally, by combining the last three term the following equation representing the outer space contribution to the free energy density integral

$${\displaystyle \Delta G^{\text{out}}=\Delta G^{\text{em}}+\Delta G^{\text{mob}}+\Delta G^{\text{solv}}}$$

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. These approximations are useful in fields such as colloid chemistry.^[13]

Materials science

An analytical solution to the Poisson–Boltzmann equation can be used to describe an electron-electron interaction in a metal-insulator semiconductor (MIS).^[16] This can be used to describe both time and position dependence of dissipative systems such as a mesoscopic system. This is done by solving the Poisson–Boltzmann equation analytically in the three-dimensional case. Solving this results in expressions of the distribution function for the Boltzmann equation and self-consistent average potential for the Poisson equation. These expressions are useful for analyzing quantum transport in a mesoscopic system. 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] Certain transport properties such as electric current and electronic density can be known by solving for self-consistent Coulombic average potential from the electron-electron interactions, which is related to electronic distribution. Therefore, it is essential to analytically solve the Poisson–Boltzmann equation in order to obtain the analytical quantities in the MIS tunneling junctions.^[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

$${\displaystyle f_{1}f^{0}-f_{0}+{\frac {eE_{z}\tau _{0}}{m}}{\frac {\partial f_{0}}{\partial v_{z}}}\left(1-e^{\frac {-\tau }{\tau _{0}}}\right)-\int _{0}^{t}{\frac {e}{m}}e{^{\frac {t-\tau '}{\tau _{0}}}}\nabla \rho [r-v(t-t')]\times {\frac {\partial f_{0}}{\partial v}}dt'}$$

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. An n-type junction is chosen in this case with a bias V applied along the z-axis. The self-consistent average potential of the system can be found using

$${\displaystyle \rho \rho _{1}+\rho _{2}}$$

where

• ${\displaystyle \rho _{1}\approx {\frac {aE_{z}}{2\lambda _{D1}}}e^{-\lambda _{D1}z}}$ and
• ${\displaystyle \rho _{2}\approx {\frac {ne{\sqrt {\pi }}G(i\lambda _{D1})e^{{\frac {-t}{\tau _{0}}}-\lambda _{D1}z}}{3{\sqrt {3}}\varepsilon _{0}\varepsilon _{r}\lambda _{D1}}}\left(1-e^{1-{\sqrt {\frac {2ne^{2}t^{2}}{m\varepsilon _{0}\varepsilon _{r}}}}}\right)}$

λ is called the Debye length.

The electronic density and electric current can be found by manipulation to equation 16 above as functions of position z. These electronic transport quantities can be used to help understand various transport properties in the system.