Poisson–Boltzmann equation

The Poisson–Boltzmann equation is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor, 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. 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 George Gouy 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 in the Helmholtz model. The Stern Layer model goes a step further and takes into account the finite ion size.

Boltzmann equation

In physics, specifically non-equilibrium statistical mechanics, the Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in thermodynamic equilibrium. It was devised by Ludwig Boltzmann in 1872. The classic example is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random (and biased) transport of particles. In the modern literature the term Boltzmann equation is often used in a more general sense and refers to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.

The equation arises not by statistical analysis of all the individual positions and momenta of each particle in the fluid; rather by considering the probability that a number of particles all occupy a very small region of space (mathematically written d³r, where d means "differential", a very small change) centered at the tip of the position vector r, and have very nearly equal small changes in momenta from a momentum vector p, at an instant of time.