Understanding the Different Approximations for Conservation of Charge in Electrolytes

Deciding how to model the charge transport in the electrolyte is usually the starting point when setting up an electrochemical model since this will determine what physics interface you will use when starting to build your model. Different theoretical descriptions of the electrolyte current density are included in COMSOL Multiphysics. They are applicable in different circumstances as discussed in this section.

The electric displacement field in a medium is related to the local charge density according to Gauss’s law, one of Maxwell’s equations:

In electrolytes, we can normally assume that the electrical permittivity is constant and equal to a bulk value:

Hence

In an electrolyte with ionic charge carriers, the charge density can be written as:

Hence

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:

Note that concentrated electrolyte systems, such as those in many batteries, use an extended concentrated species flux definition, based on the Maxwell-Stefan set of equations. This will result in a different set of equations to solve for, but the general principles and conclusion in this section will be the same.

Substituting the Nernst-Einstein relation for the electrical mobility of an ion we get:

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 with , but they are highly nonlinear and difficult to converge. Most often, further approximations can simplify the problem without compromising accuracy.

An important dimensional quantity occurring in the Poisson equation is:

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:

The constraint of electroneutrality can be used as a condition determine the electric potential in the Nernst-Planck equations, in place of the full Poisson equation. The Nernst-Planck equations with electroneutrality are used to describe current flow in the interface.

	The Tertiary Current Distribution, Nernst-Planck Interface

The current flow itself is given by:

From substitution of the Nernst-Planck expressions for Ni, the laws of conservation of mass and charge combine to automatically satisfy conservation of current.

We can simplify the system further by considering the arising expression for il in more detail:

Clearly, the right-most term is zero: that is, convection of an electroneutral solution does not cause current flow. The left-most term (diffusion current) also vanishes due to electroneutrality if the gradients of the charge carrying species are zero.

Even if this is not the case, however, this term is often much smaller than the central term (migration current), so long as the concentrations of the current-carrying ions do not vary markedly through the solution. Under conditions where the composition of the electrolyte can be considered nearly constant and current-carrying ions are not significantly depleted, the diffusion current can be assumed to contribute negligibly.

Hence, it follows that:

This expression for current density is used in the interface, and also the 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:

So long as this quantity does not vary markedly through the solution, the approximation of zero diffusion current is good. If the diffusivities and concentrations can be taken as constant, we can approximate that:

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.

The approximations used to derive the secondary current distribution expression place tighter constraints on the allowed system configurations, however. The ionic strength of the solution must remain near-constant for the constant conductivity approximation to be valid. Usually this is only the case for relatively high conductivity solutions.

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 interface.