Mathematical Description of Porous Electrodes

As we examine porous electrodes, we first consider a simple geometry consisting of straight cylindrical pores in an electrically conductive matrix as shown in Figure 5.1. The idea behind using a three-dimensional porous structure is to increase the amount of surface area in a given electrode volume. Let’s assume that we have an electrode that is 10 cm × 10 cm and 1 mm thick. The electrode is filled with an array of straight pores as shown in the figure. Assume that each pore has a diameter of 10 μm, and that the minimum distance between pores is 2.5 μm. That would give us 64 million pores in the 10 cm × 10 cm area. The internal surface area of each pore is about 0.0004 cm². This single pore area results in a total pore surface area greater than 25,000 cm², which is more than 250 times greater than the flat area of the original electrode. The area enhancement can be even higher for practical porous electrodes.

Previously, we assumed that the electrochemical reactions took place on the surface of the electrode and we treated the reactions appropriately as boundary conditions. The analysis of a porous electrode, however, is a bit more complex since the reaction takes place throughout the volume of the electrode. We could use our previous approach and perform a three-dimensional calculation on the actual geometry, including the individual pores, and apply the reaction boundary condition on each pore wall. Fortunately, a much more effective method, referred to as porous electrode theory, has been developed extensively by Newman. We will illustrate the method in one dimension since in most applications it is only variations in the thickness direction that are important. Situations that require multidimensional treatment are beyond the scope of this chapter.

Let’s begin with a discussion of the physical system of interest. As shown in Figure 5.1, we have a porous electrode attached to a current collector. We assume that the current collector is at a constant potential, and that the current density at the current collector is uniform. These assumptions are consistent with the one-dimensional treatment mentioned previously and, as a result, we don’t need to model the current collector. Our spatial coordinate system is defined as x = 0 at the back of the porous electrode (the side of the electrode in contact with the current collector) and x = L at the other side of the electrode, where L is the thickness of the electrode. Consistent with the notation introduced previously, i₁ and ϕ₁ are the current density and potential in the solid, electron-conducting phase; and i₂ and ϕ₂ represent the current density and potential in the electrolyte phase. The electrolyte fills the cylindrical pores. For the purposes of discussion, we assume that the electrode is the anode, although the relationships developed apply equally well to a cathode. The current associated with charging of the double layer is not considered in the following treatment. The extension of porous electrodes to electrochemical double-layer capacitors is made in Chapter 11.

At the back of the electrode (x = 0), all of the current is in the solid phase. Therefore,

(5.1)

where I/A is the specified current density in A·m⁻², and i₁ is based on the superficial area, A, of the electrode (100 cm² =0.01 m² for this case). At the other end of the electrode (front), the solid phase ends and all the current must be transferred in the electrolyte. Therefore,

(5.2)

In between x = 0 and x = L, the current is split between the solid and liquid phases. The following relationship must hold at any point in the electrode:

(5.3)

How does the current change from being entirely in the solid at the back of the electrode to being entirely in the electrolyte at the front of the electrode? The current varies due to electrochemical reaction at the interface between the solid and electrolyte phases, just as it did for the flat electrodes that we have considered up to now. In fact, we can use the same type of kinetic expression (e.g., linear, Tafel, or BV) that we used earlier to describe the reaction. The difference is that the reaction area is spread throughout the volume of the electrode in a porous electrode. That area is the surface area of the cylinders in our simplified electrode model (Figure 5.1). As we prepare to write a charge balance for the electrode, it is convenient to define a volumetric charge generation term,

(5.4)

where i_n is the current density normal to actual surface (A·m⁻²) and a is the surface area per volume (m⁻¹) or specific interfacial area. The volumetric charge generation rate has units of [C·m⁻³·s⁻¹)] or [A·m⁻³]. The current density, i_n, is the same current density that we have discussed previously in Chapter 3 and can be represented by, for example, a Tafel or BV expression. The surface area to which i_n corresponds is the interfacial area between the solid and electrolyte, and is equal to the combined surface area of the 64 million cylindrical pores in our example. The specific interfacial surface area, a, in our example is equal to the combined surface area divided by the superficial volume of the electrode:

We are now prepared to write charge balances for the current in solution (i₂) and the current in the solid phase (i₁). The balance for i₂ is

(5.5)

The positive sign on the generation term is consistent with the convention that anodic current is positive and represents the flow of current into the electrolyte. A similar expression can be written for i₁,

(5.6)

Note that the generation term is negative since charge leaves the solid phase when i_n is positive. It follows that

(5.7)

It is important to remember that both i₁ and i₂ are based on the same superficial area (10 cm × 10 cm in our example). Also, note that only two of these three equations are independent.

To describe completely the electrode, we also need material balance equations for the species in solution. In Chapter 4, a general material balance was provided.

(4.10)

This equation needs to be modified since our control volume now includes the electrolyte and solid phases. The term on the left is the rate of accumulation of species i in our control volume. We will continue to let c_i represent the actual concentration in the electrolyte (moles/volume of electrolyte). In the porous medium, only a fraction of the volume is occupied by the electrolyte; therefore, the amount of species i per control volume is c_iε, where ε is the volume fraction of the electrolyte. Furthermore, since the porosity may change with time and position, we must keep this term in the derivative. The flux, N_i, is the superficial molar flux and does not require a change in the differential material balance. Nevertheless, when the flux is expressed with the Nernst–Planck equation, for instance, effective transport properties must be used. Similar to our treatment of the flux,  is the generation rate based on the superficial volume that includes both solid and electrolyte phases. The form of the material balance for porous electrodes becomes

(5.8)

Heterogeneous reactions in the porous electrode affect the individual balances of reacting species, which are either consumed or generated by reaction at the surface. In the present macroscopic approach, we account for these terms as volumetric generation terms, similar to what was done above for the charge balance. Therefore, if we neglect generation terms other than those related to the electrochemical reaction:

(5.9)

where j_n is the species reaction rate at the surface in moles/(time–area). This rate is related to the current at the surface through Faraday’s law:

(5.10)

Note that j_n is positive when the current is positive (anodic) and the species of interest is a product of the anodic reaction (s_i < 0). You should review the definition of s_i and convince yourself that this is true. The species equation now becomes

(5.11)

This step completes the balance equations needed to describe a porous electrode. The unknowns are c_i for each of n species, ϕ₁ (potential in solid), and ϕ₂ (potential in electrolyte), for a total of n + 2. Thus, n + 2 equations are required: two charge balance equations, n − 1 species balances, and the electroneutrality equation.

We still need to write the equations that relate the currents and fluxes to their respective driving forces in order to substitute these into the relevant material balances. We will use the Nernst–Planck equation, although a similar procedure is suitable for other descriptions of the species flux. The expression after adjustment for the porous nature of the electrode is

(5.12)

where v is the actual molar-averaged velocity in the pores. Remember, the flux is based on the superficial area of the porous electrode, and not on the actual pore area. The porosity in the first two terms of the flux expression is usually combined with an additional correction to yield effective transport properties. Finally, ε and v are sometimes combined into a superficial velocity, which is based on the superficial area of the electrode and is essentially the volumetric flow rate divided by the superficial area. Effective properties are described below after a brief discussion of the characteristics of porous electrodes in order to broaden our view beyond the simplified electrode geometry considered previously. Note, however, that the material balances derived above are completely general and not restricted to the illustrative geometry consisting of straight cylindrical pores.

We have identified the equations necessary to describe the behavior of the porous electrode. Next, the methods of characterizing porous electrodes are presented, followed by a look at specific examples.