In order to solve most transport problems, expressions for the flux, such as the Nernst–Planck equation, are incorporated into material balances or conservation equations. Here we derive a balance for a single species over a control volume of size ΔzΔxΔy as shown in Figure 4.3. The balance takes the form

(4.9)

The rate of accumulation in the control volume is

The net rate of material entering the control volume in the x-direction is

Similar equations can be written for the y– and z-directions. If R_i is the homogeneous rate of production per volume of species i, the rate of generation is

These equations combine to

In the limit of an infinitely small control volume,

The species material balance may be written more compactly in vector form as

(4.10)

Equation 4.10 represents a differential material balance on species i. Equation 4.10 along with the appropriate initial and boundary conditions will be used frequently and should be committed to memory. For a one-dimensional geometry, one initial condition and two boundary conditions must be specified to define completely a problem.

Let’s suppose for a moment that you would like to solve an electrochemical transport problem involving multiple ionic species in solution. How would you use Equation 4.10? What other equations are needed? A complete set of equations for n species in solution would include the material balance (Equation 4.10) for each of the species in solution, into which the corresponding flux equation (Equation 4.3) for each species has been substituted. Also, since the electrical forces between charged species in solution are quite large, and the mobilities of ions in an electrolyte solution are relatively high, significant separation of charge does not occur. Therefore, in the bulk electrolyte solution, electroneutrality is maintained as expressed by the electroneutrality equation:

(4.11)

Electroneutrality, of course, does not hold in the double layer, which can be modeled separately from the bulk if needed. If the velocity is known, we have a complete set of n + 1 equations (n material balance equations and the electroneutrality equation) for n concentrations and the potential. If the velocity is not known, we will also need the appropriate equations for fluid dynamics. Most frequently, we assume that the fluid flow equations are decoupled from the rest of the equation set and solve them independently. Also, we typically do not write a material balance for the solvent, and we assume that the reference velocity is essentially the same as that of the solvent.

The current density, i, is most often of great interest in the simulation of electrochemical systems. Consequently, a charge balance

(4.12)

can be substituted for one of the material balance equations. Equation 4.12 is not independent, but a weighted sum of the other material balance equations as follows:

Alternatively, one can solve the equation set above without the charge balance and then calculate the current density “postprocess” with Equation 4.4.

The Nernst–Planck description of the flux, when combined with material balances just described, results in a set of nonlinear differential equations that often do not have an analytical solution. However, some important simplifications are possible. In the absence of concentration variations, the potential follows Laplace’s equation, which is examined below. The most common simplifications for systems of varying composition involve elimination of the migration term in the flux equation. These, too, are considered below and include binary electrolytes, where the electric field is eliminated from the conservation equation through mathematical manipulation, and “excess supporting electrolyte,” where the effect of the electric field can be neglected. A form of the convective diffusion equation results for each of these cases, but for different reasons and with different implications. These and other classifications of electrochemical problems are summarized in Figure 4.4.

No Concentration Gradients

In situations where the concentration gradients can be neglected, the charge balance in Equation 4.12 can be simplified to yield

(4.13)

Equation 4.13, Laplace’s equation, is the only equation that needs to be solved since the concentrations and conductivity are known and constant. For a one-dimensional problem, the potential drop in solution for a system at constant concentration can be readily calculated if the solution potential at the two electrodes is known or if the current is known. This calculation is demonstrated in Illustration 4.1.

ILLUSTRATION 4.1

In Section 3.4, we used the equation  to calculate the potential drop in solution. What is the source of this equation, and what assumptions are implicit in its use? In that problem we assumed that the solution concentration was constant. The Laplace equation in one dimension is

This equation can be integrated twice to yield

where x = 0 at the zinc electrode (negative electrode: n) and x = L at the nickel electrode (positive electrode: p). Since we are looking for the potential difference, we arbitrarily set the potential in solution at x = 0 to be . Therefore, . In general, we will at a minimum need to specify the potential at one point in the domain when solving Laplace’s equation. To find A, we note that

The expression for ϕ becomes

We can now use this expression to determine the potential difference across the electrode:

This result is equivalent to the expression we used in Chapter 3, where Δx = L. Note that Laplace’s equation here relates the potentials in solution, ϕ₂.

In Chapter 3, we showed how to use the kinetic expressions to relate the solution potential, current, and potential of the metal at each electrode. What if the electrode reaction is very fast? For a large exchange-current density, the surface overpotential is zero and the potential difference between the metal and the solution is at the equilibrium value. Using the nomenclature from Chapter 3 relative to a zinc reference electrode,

In other words, the cell voltage is equal to the equilibrium voltage of the cell minus the ohmic drop (or overpotential). This expression was derived for fast kinetics.

We can write an expression for the potential of the cell, V_cell, similar to the one shown in the illustration above, which includes the surface overpotentials. For a galvanic cell, the expression is

(4.14a)

Thus, for a galvanic cell, the maximum voltage is the equilibrium voltage, and the actual cell voltage is reduced from the equilibrium voltage by the amount needed to drive the surface reactions and the flow of current in solution. For an electrolytic cell,

(4.14b)

Here, the cell voltage is higher than the equilibrium voltage as additional power must be added to drive the reactions and the flow of current in solution. Integration of Laplace’s equation in one dimension is straightforward and is something that you should be able to do in Cartesian, cylindrical, and spherical coordinates. For two- and three-dimensional problems, solution of Equation 4.13 is more difficult and frequently results in a nonuniform current density at a surface. This issue and the use of kinetic expressions as boundary conditions are treated in more detail in Section 4.6.

Binary Electrolyte

A binary electrolyte is an electrolyte with only one type of anion and one type of cation. It is formed by dissolving a salt in a solvent to create an electrolyte solution. The neutral salt  dissolves in the solvent as follows:

(2-51)

where

(4.15)

As an example, for CaCl₂, ν₊ = 1 and ν₋ = 2. The resulting binary electrolyte is electrically neutral and, therefore, satisfies the electroneutrality equation (Equation 4.11). Maintaining electroneutrality during transport means that the fluxes of the two ions are not independent. In fact, electroneutrality couples the concentration of the two ions together and allows us to express concentration in terms of a single salt concentration,

(4.16)

Using this salt concentration, the Nernst–Planck equation, and the assumption of incompressible fluid (), one can write the material balance equations for the positive and negative species as follows:

(4.17a)

(4.17b)

We have assumed the absence of a homogeneous reaction. Equation 4.17b is subtracted from Equation 4.17a to yield

(4.18)

Equation 4.18 provides a relationship between the migration and diffusion terms in the material balance. This equation can be used to eliminate the potential from either one of the species balance equations to yield

(4.19)

where c is the concentration of the salt. Thus, the concentration follows the same equation as that for a neutral species. Here, the equivalent diffusion coefficient of the salt is given by

(4.20)

Equation 4.20 defines the diffusivity of the salt, which may often be more readily measured and reported than the diffusivities of the individual ions. Although the diffusivities may be quite different, the concentration of the two species will always be related to the stoichiometry of the neutral salt because of electroneutrality. The diffusivity of the salt, given by Equation 4.20, can be thought of as the weighted average of the diffusivities of the two ionic species.

Although the electric field has been eliminated mathematically in Equation 4.19 to facilitate solution of the problem, the physics have not changed; therefore, there is still a potential gradient in solution that leads to transport by migration. The effect of the potential can most readily be seen at the boundary. To illustrate this, consider the steady-state electrorefining of copper shown in Figure 4.5. Here Cu is oxidized at the anode and deposited on the cathode. The electrolyte is CuSO₄ and there is no convection. The solution contains only CuSO₄ and is therefore a binary electrolyte. For a one-dimensional system, Equation 4.19 reduces to

(4.21)

From Equation 4.21, it is clear that the concentration profile is linear between electrodes. Since only the copper reacts, the current is related by Faraday’s law to the flux of Cu²⁺ as follows:

(4.22)

As you can see, the copper(II) flux includes both migration and diffusion. At first glance, this creates a problem since we have not solved explicitly for the potential field. However, because sulfate does not react at the electrodes, its net flux must be zero; therefore, the contributions of migration and molecular diffusion must exactly offset each other. This balance allows us to use Equation 4.3 for the anion to relate the potential and concentration gradients:

(4.23)

We can now use Equation 4.23 to eliminate the potential gradient in Equation 4.22 in order to provide the boundary relation in terms of just the concentration gradient as needed. This process is illustrated with an example in Section 4.4. From this discussion, we see that solution of a problem involving a binary electrolyte can be facilitated by mathematically eliminating the electric field and using electroneutrality to express the equations in terms of a single salt concentration. The resulting equations can be solved directly for the concentration profile of the salt and appropriate fluxes. Migration does not show up explicitly in the equations; nonetheless, it is completely and properly accounted for in the above treatment.

Excess Supporting Electrolyte

In instances where a large amount of a salt that is not involved in the electrochemical reaction is added to the solution, the effect of the potential gradient on transport is diminished. The added salt that is not involved in the reaction is called a supporting electrolyte. There are several reasons for using a supporting electrolyte. Here we are concerned with the effect of the supporting electrolyte on migration. Returning to the electrorefining of copper discussed previously, we modify the system with the addition of sulfuric acid, H₂SO₄, which is a supporting electrolyte. If enough sulfuric acid is added to the solution so that migration can be neglected, then we have an excess supporting electrolyte. To repeat, the effect of the excess, nonreacting electrolyte is to reduce the electric field so that migration no longer contributes significantly to the transport of the reacting species. In contrast to the binary electrolyte, use of a supporting electrolyte implies that there must be at least three ionic species in solution.

It is frequently reported or implied that the role of the supporting electrolyte is to reduce the electric field by carrying the current. This conclusion is in error since, at steady state, the flux of any nonreacting species must be zero, and the supporting electrolyte is by definition nonreacting. Therefore, only the fluxes of the reacting species contribute to the current (see Equation 4.4). That said, the presence of the supporting electrolyte does diminish the magnitude of the potential gradient. To explore this further, consider a system at steady state with no bulk flow where any transport of the supporting electrolyte due to migration must be offset by diffusion. Concentration gradients of the nonreacting species must be connected to those of the minor reacting species in order to preserve electroneutrality. Therefore, concentration gradients of the supporting species cannot be large. Furthermore, the conductivity of the supporting electrolyte is high. These two facts together make it impossible to support a large potential gradient in solution. As a result, migration can be neglected for the reacting species when its concentration is significantly less than that of the supporting electrolyte. Convection can be used to minimize concentration gradients and to increase the rate of transport of the reacting species.

Returning to the Nernst–Planck equation, we neglect migration due to the supporting electrolyte and express the flux as

(4.24)

If we further assume that the diffusivity is constant, the resulting material balance is

(4.25)

Finally, in the absence of bulk flow, we are left with Fick’s second law:

(4.26)

Note that, in contrast to the binary electrolyte, Equations 4.25 and 4.26 include both the diffusivity and the concentration of the individual minor species of interest. There are many techniques available for solution of Equation 4.26. What’s more, there are many physical situations where Fick’s second law is appropriate.

Figure 4.6 shows results from a simulation for a problem similar to the copper electrorefining considered earlier. In both cases, copper is consumed at the anode and deposited at the cathode. This particular simulation only considers transport by diffusion and migration. Because there is no convection, mass transport limitations occur at very low current densities, limiting the range of results. In spite of the limited range, the results illustrate the influence of the supporting electrolyte. The current density at a given cell voltage is higher for the system with the supporting electrolyte when mass transfer is not limiting. This increase is because the potential drop in solution is very small in the presence of the supporting electrolyte, and the entire applied voltage goes to driving the reactions at the electrodes (see Figure 4.6c). In contrast, a considerable ohmic drop is observed without the supporting electrolyte (Figure 4.6b). At just under 0.04 V in this simulation, the curves cross as the mass-transfer limiting current (∼10 A·m⁻²) is approached for the system with the excess electrolyte. The mass-transfer limit in the presence of the sulfuric acid is half of the limiting value without the acid because migration does not contribute to the transport when there is excess electrolyte. As the mass-transfer limit is approached, the concentration of copper(II) ions at the cathode decreases, leading to a sharp increase in the cathodic overpotential (Figure 4.6c). As illustrated by this simulation, excess electrolyte increases the current that can be achieved at a particular cell voltage as long as the system is operated below the mass-transfer limit. Note that the mass-transfer limit in a practical system will be much higher than the values that pertain to this simulation due to convection in the practical system. Excess supporting electrolytes are used in both industrial processes and analytical experiments. For industrial processes, a reduction in ohmic loss is the principal reason. Supporting electrolyte makes it possible to achieve the same current at a lower cell potential. This result is true in the region of operation where kinetics is important. The supporting electrolyte may actually reduce the magnitude of the limiting current by eliminating the contribution of migration to the transport of the minor species. Industrial use of supporting electrolytes may also be driven by the lower cost of the supporting electrolyte relative to the minor salt of interest. In addition, supporting electrolyte can be used to overcome difficulties associated with a low solubility of the reacting component. For electroanalytical methods (Chapter 6), the goal is the elimination of migration so that transport is controlled by diffusion and analysis is greatly simplified.

ILLUSTRATION 4.2

A common experiment is a potential step study under diffusion control, where the diffusion of a reactant is limiting. As shown in the figure below, the concentration at the surface approaches zero at time zero when the potential step is applied. Assuming no convection and neglecting migration, Fick’s law applies (Equation 4.1). Substituting this into the material balance equation, Equation 4.10, gives

There are several ways to solve this equation, including the use of Laplace transforms or Fourier analysis. Here we pursue a similarity solution. For convenience, we drop the i subscript since we are only dealing with the transport of a single species. We begin by introducing a new variable,  which transforms the partial differential equation to an ordinary differential equation:

with the following BCs,  This ODE may be readily solved using reduction in order. The solution is

where the error function is defined as

The concentration profiles under these conditions are shown in the figure. For electrochemical systems, we are generally interested in the response of the current or potential. If we consider that the diffusing species undergoes reduction at the surface, the flux of the oxidized species to the surface can be related to the current with Faraday’s law:

The error-function solution above can be differentiated to obtain an expression for the concentration gradient at the electrode surface. Substituting that gradient into the expression for the current density, i, yields

Here we see that the current depends inversely on the square root of time. This relationship is known as the Cottrell equation (see Chapter 6), and this dependence on time is indicative of mass-transfer control. This simple analytical relationship applies in situations where migration can be neglected.