As noted in Chapter 4, a uniform current distribution is often desired. Generally, the calculation of the current distribution for porous electrodes is not amenable to analytical solutions. There are a few exceptions—most notably the one-dimensional treatment of a porous electrode in the absence of concentration gradients, which is described by the secondary current distribution. In contrast to what was examined in Chapter 4, here we are interested in how the current is distributed through the thickness of the porous electrode.

Consider the porous electrode shown in Figure 5.5. Let κ_eff be the conductivity of the electrolyte and σ_eff the conductivity of the solid phase. For convenience, we will drop the subscripts in the treatment that follows, but you should remember that these are effective properties. The current density in the electrolyte is i₂ and the current density in the solid phase is i₁. There are also two potentials of interest: ϕ₂, the potential of the electrolyte, and ϕ₁, the potential of the solid. We will simplify the analysis by neglecting concentration gradients, and assuming steady state. We also assume that the superficial current density, I/A, is constant.

In the absence of concentration gradients, two charge balance equations are required to solve the problem. From Equation 5.5,

(5.21)

Assuming linear kinetics (see Equation 3.30), this equation in one dimension becomes

(5.22)

We will also use Equation 5.7

(5.23)

which can be integrated to yield (see Equation 5.3)

(5.24)

Since there are no concentration gradients, we can use Ohm’s law for both the electrolyte and solid phases. As a reminder, σ and κ are effective values for the conductivity.

(5.25)

(5.26)

Initially, we will develop an analytical solution for the current density in solution. To do this, we first differentiate Equation 5.22 to yield

(5.27)

Since we are after the current density, we can use Equations 5.24–5.26 to eliminate the potential from this equation and obtain an expression where the only independent variable is i₂:

(5.28)

where I/A is constant and equal to the applied superficial current density at the electrode. Equation 5.28 is an ordinary differential equation with i₂ as the only unknown. We now define the following dimensionless variables:

(5.29)

(5.30)

(5.31)

(5.32)

Substituting these into Equation 5.28 and simplifying yields:

(5.33)

This second order, ordinary differential equation can be solved analytically to yield i^* as a function of z. Two boundary conditions are required. Note that x = 0 is the back of the electrode at the current collector. The original boundary conditions for i₂ and the dimensionless equivalents are

(5.34)

and

(5.35)

The solution is

(5.36)

This current density can also be differentiated to give a dimensionless local reaction rate:

(5.37)

Use of the dimensionless values allows us to examine the shape of the current distribution and the local reaction rate as a function of two parameters, ν and , without having to insert values for each of the variables in the dimensional equation. For this exercise to be meaningful, we must first examine the physical significance of each of these two parameters. The parameter ν² is the ratio of the ohmic and the kinetic resistances, and includes contributions from both the solid and liquid phases to the ohmic resistance. It is essentially the inverse of the Wa for linear kinetics that was presented earlier in Chapter 4 adapted to the porous electrode. The only reason that this ratio is defined as ν² rather than ν is to avoid having to write the square root of ν repeatedly in the solution of the differential equation. Small values of ν² are controlled by kinetic resistance and result in a uniform or near-uniform distribution of current through the thickness of the porous electrode, similar to what you learned in Chapter 4. In contrast, ohmic resistance controls the current behavior at high values of ν², leading to a nonuniform distribution. The second parameter, K_r, is simply the ratio of the conductivity of the electrolyte (κ) to that of the solid matrix (σ). Limiting behavior is reached as K_r → 0 (electrolyte limited) or K_r → ∞ (solid limited). It is common to have systems where the solid-phase conductivity is significantly greater than that of the electrolyte. With this background, we will now examine the behavior of i₂ and the local reaction rate in their dimensionless form.

Figure 5.6 shows the dimensionless current density in solution (a) and the local reaction rate (b) as a function of position in the electrode at different values of ν² for the case where the conductivity of the solid is an order of magnitude larger than that of the electrolyte, K_r = 0.1. For small values of ν² (kinetically limited), the reaction rate is flat throughout the electrode (ai_n is constant) and the current in solution varies roughly linearly with position. The linear behavior is expected since, for this case,  is equal to a constant. As the value of ν² is increased, the reaction rate becomes nonuniform and is preferred at the front of the electrode due to the additional ohmic resistance of the electrolyte associated with penetration of the electrode. The nonuniform reaction rate is reflected in the solution current as shown in the figure. If the solid-phase conductivity is high compared to the solution phase, then the reaction would shift toward the electrode/separator interface (x = L). If the conductivities are reversed, (σ < κ), the reaction would shift toward the current collector.

In situations where a local solid-phase reactant is not “consumed,” the distribution for high ν² values implies that a significant fraction of the electrode is not being used and that similar performance could be obtained with a thinner electrode. In contrast, for a battery electrode with a solid-phase reactant, the results for high ν² would indicate, for example, that the outside of the electrode would discharge first, followed by the regions closer to the middle, leading to increased resistance at lower states of charge.

The asymmetry shown in Figure 5.6 is the result of the assumption that K_r = 0.1. If the conductivity of the solid and the electrolyte were identical, the reaction distribution would be symmetrical. Figure 5.7 shows the current distribution for a porous electrode with the solid-phase conductivity equal to that of the solution phase. Because of the equal resistance in the solid and electrolyte phases, the current distribution is symmetrical, neither preferring the front nor the back of the electrode. The reaction is more nonuniform at higher values of ν², shifting away from the center to the front and the back of the electrode.

It is useful to estimate a penetration depth for an electrode. Penetration depth is not an issue for small values of ν², since such a system is under kinetic control and reaction takes place throughout the entire electrode. Here we consider the situation where K_r → 0 and behavior is limited by transport of current through the electrolyte. With these assumptions, the current changes monotonically from a high value at the front of the electrode to a low value at the back of the electrode. The expression for the local reaction rate, Equation 5.37, can be simplified to

(5.38)

We will use a practical approach to the penetration depth by defining it in terms of the reaction rate at the back of the electrode relative to that at the front. The approximate rate at the front of the electrode (z = 1) is

(5.39)

Similarly, at the back of the electrode (z = 0):

(5.40)

The ratio of the rate at the back over that at the front is

(5.41)

A penetration depth can be estimated by solving this equation for the ν value that corresponds to a specified relative rate of reaction. The penetration depth, L_p, can then be back calculated from the definition of ν, remembering that we have assumed 1/σ ∼ 0. In other words, given the rest of the parameters, we are seeking the value of L that will yield the ν value calculated from Equation 5.31. This process is shown in Illustration 5.3 that follows.

ILLUSTRATION 5.3

You are designing an electrode for a lithium-ion battery for operation at room temperature. The solid-phase electronic conductivity is much larger than that of the electrolyte, κ = 0.1 S·m⁻¹. If it is desired to keep the reaction rate at the back of the electrode no less than 30% of the front, what is the maximum thickness of the electrode? Additional kinetic data are α_a = α_c = 0.5, i_o = 100 A·m⁻², a = 10⁴ m⁻¹. Starting with Equation 5.41,

Equation 5.31 is

σ is assumed to be infinity; thus, the only unknown is L_p. Solving for L_p yields

We are also interested in the internal resistance of the electrode. This resistance is nothing more than the change in potential with current density. For linear kinetics the resistance of a porous electrode is

(5.42)

The internal resistance of the electrode as a function of thickness is shown in Figure 5.8. It is important to recognize that this resistance is a combination of ohmic drop in the solution and solid phases as well as kinetic resistance. For a large electrode thickness, the value of ν² is high, and the reaction is nonuniform. Here the resistance increases with thickness. For thinner electrodes, the reaction is more uniform but there is relatively little surface area available for the reaction. Increasing the thickness makes more surface area available for reaction, and thus, the resistance decreases. There is a value of L, the electrode thickness where the resistance is minimized. These effects are shown in Figure 5.8 for two values of K_r.

Similar analytical solutions are available for Tafel kinetics. Although the use of Tafel kinetics is more appropriate for a large number of problems, the linear and Tafel results are qualitatively similar. We have presented the linear results because they have fewer parameters and are easier to understand. Tafel kinetics is explored in Problem 5.9 at the end of the chapter. There is an important difference between the two solutions. As we can see from Equation 5.36, there are only two parameters for linear kinetics: K_r and ν. Although the exchange-current density is part of ν, the current, I, does not impact the dimensionless current distribution. This result in completely analogous to what we saw for the Wa number in Chapter 4. For linear kinetics, the exchange-current density is a parameter, and the current distribution is independent of the average current density as long as it is small. In contrast, for Tafel kinetics the exchange-current density is not a parameter and has been replaced with the average current density. Since Tafel kinetics applies when the current density is much larger than the exchange-current density, this difference makes sense.