Nernst–Planck Equation

The most widely used expression for the flux in electrochemical systems is the Nernst–-Planck equation,

(4.3)

The flux of species, i, is the combination of three terms: migration, diffusion, and convection. The Nernst–Planck equation is similar to Equation 4.2 but adds a contribution that arises from the gradient in electrical potential called migration. For charged species, the force due to the electric field is equal to −z_iF∇ϕ. The charge on a species, z_i, can be positive or negative. The direction of the force changes for anions and cations because their charges are of opposite sign. Multiplying this driving force by the mobility of the ion, u_i, and the concentration, c_i, results in the net flux ascribed to migration. Migration is defined so that positive ions move from high to low potentials. We’ll see later that the mobility is related to the diffusivity of the ion.

The second contribution to the flux is the molecular diffusion as described by Fick’s law, which was discussed above. The last term is from the bulk movement of the electrolyte or convection. Equation 4.3 is frequently written for dilute solutions and is often referred to as dilute solution theory. When the concentration of ionic species is relatively small and the majority of the electrolyte is the solvent, the predominant interactions are between the ions and the solvent; ion–ion interactions are not important. Also, under dilute conditions, the average velocity in Equation 4.3 is well defined and independent of the minor species in solution. Under such conditions, v_mass = v_molar = v_solvent, and the balance equation for the solvent is not written since the concentration of the solvent is essentially constant.

The Nernst–Planck equation can also be used for more concentrated solutions. However, as concentration levels increase, it is necessary for transport coefficients to include the effect of ion–ion interactions (in addition to ion–solvent interactions) and for the reference velocity to be defined carefully. The potential in solution is perhaps best defined against a reference electrode under such conditions. As concentration increases, activity effects become more important since it is really the gradient of the electrochemical potential and not the concentration gradient that is the primary driving force for transport. Nevertheless, gradients in the activity coefficients can typically be neglected, even in moderately concentrated solutions.

More commonly, in electrochemical systems we are interested in the current density rather than fluxes of individual species. The electrical current in solution can be related to the species fluxes by Faraday’s law:

(4.4)

Substituting the Nernst–Planck equation into Equation 4.4 gives

(4.5)

The last term in Equation 4.5 is zero because we assume that the solution is electrically neutral. That is, the charge of anions is exactly balanced by the charge of the cations, which is a good assumption in most cases. It is evident from Equation 4.5 that the current density [A·m⁻²], like the species fluxes, is a vector. Also, since the current is the result of the transport of ions in solution, it includes contributions from both migration and diffusion.

Let’s consider a special case where there are no concentration gradients. Therefore, the second term on the right side of Equation 4.5 is also zero. In the absence of concentration gradients, the current density is proportional to the potential gradient multiplied by a term involving the mobility, concentration, and charge of the ions. Thus, Equation 4.5 reduces to Ohm’s law. Note that Ohm’s law is synonymous with the statement that the current is proportional to the gradient of the potential.

(4.6)

where the electrical conductivity is defined by

(4.7)

The SI unit for conductivity is [S·m⁻¹], where a siemen [S] is an inverse ohm [Ω⁻¹]. Note that the charge on each ion is squared so that conductivity is always a positive quantity. The electrical conductivity is the sum of the mobility of each ion weighted by its concentration in solution and represents the ability for charge to be transported in the presence of an electrical field.

In the absence of concentration gradients, the conductivity can be directly related to the resistance of the electrolyte. For example, consider one-dimensional transport between two flat plates of area A and separated by a distance L. Application of Equation 4.6 yields

(4.8a)

(4.8b)

(4.8c)

where R_Ω relates the total current to the difference in the potential. Also note that the conductance, G, is simply the inverse of resistance and has units of siemens.

Finally, remember that Ohm’s law only applies in the absence of concentration gradients; it must be modified if concentration gradients are present. Regardless, the electrical conductivity is an important transport property of the electrolyte.