This is the Poisson equation relating the electrolyte potential to the distribution of charge carriers within the electrolyte. In its derivation we assumed that the only charge carriers are ions, and that the solvated ions and electric field do not alter the permittivity of the medium.
The mass transport of the charge carriers in aqueous systems is normally given by the Nernst-Planck equations. These equations neglect ion-ion interactions, and so they are only exact for infinitely dilute solutions:
The above expressions for the n species
i, together with the Poisson equation, give a set of
n+1 equations in
n+1 unknowns. These are the
Nernst-Planck-Poisson equations. They can be defined in COMSOL Multiphysics by coupling
Transport of Diluted Species with
Electrostatics, or by using the
Tertiary Current Distribution, Nernst-Planck interface with
Charge conservation model: Poisson, but they are highly nonlinear and difficult to converge. Most often, further approximations can simplify the problem without compromising accuracy.
This is the length across which electric fields are screened. It is called the Debye length. This is a very short length in electrolyte solutions: for a typical ionic strength, it is of the order of 1 nm.
Electroneutrality holds at distances much larger than 1 nm from a charged surface:
From substitution of the Nernst-Planck expressions for Ni, the laws of conservation of mass and charge combine to automatically satisfy conservation of current.
This expression for current density is used in the Secondary Current Distribution interface, and also the
Primary Current Distribution interface. The difference between these interfaces lies in the treatment of the electrode-electrolyte interfaces (see
Kinetics of Electrochemical Reactions below). From the above, the conductivity of the electrolyte
σl is given as:
The advantage of the ohmic expression for current density is that it is a linear relation of current density to electrolyte potential. It is only weakly nonlinear if σl is allowed to depend on a concentration solved for in a species transport interface. By comparison, the Nernst-Planck equations with electroneutrality can be highly nonlinear.
When the conductivity is large with respect to the current drawn, the electric field becomes negligible in solution. For negligible electric fields, a diffusion-only approximation may be used, where E = 0. This converts the Nernst-Planck equations into Fick’s laws, with a term for convective transport where necessary. Fick’s laws with convection and electrochemical boundary conditions are solved for in the
Electroanalysis interface.