There are many instances when both gas and liquid fill the void volume of a porous electrode. Here, we introduce important concepts to describe these two fluids contained in the pores. A key aspect of porous media relevant for our studies is capillarity or capillary action. The capillarity can be understood from the ability of a fluid to flow against a gravitational or other body force. The wetting of a paper towel by wicking of the fluid against gravity is a familiar example. Let’s consider a simple situation of a small capillary (thin tube) in a fluid as shown in Figure 5.9. Here surface tension causes the fluid to rise in the capillary.

Figure 5.10 shows the contact angle where there is an intersection of the liquid, gas, and solid. If the contact angle is less than 90°, the material is said to be hydrophilic or wetting. In this case, the meniscus will be concave as shown in Figure 5.9. Conversely, for large contact angles, the material is said to be hydrophobic, and the meniscus is convex. The height of the fluid in the capillary depends on the contact angle, θ, and the surface tension, γ.

(5.43)

Typically, gravitational forces are not significant in porous electrodes; and Equation 5.43 is not applicable. On the other hand, surface tension and the contact angle are important. We define a capillary pressure, which represents the difference in pressure between the nonwetting (nw) and wetting (w) phases. For instance, at the top of the column of water in Figure 5.9, the pressure of the gas (nonwetting) will be higher than the pressure in the adjacent liquid (wetting phase). The liquid pressure can be determined from hydrostatics to be below that of the gas by a value of ρgh. Thus, the capillary pressure, p_c, is

(5.44)

For a wetting fluid with a contact angle less than 90°, the capillary pressure is positive, meaning the gas pressure is above that of the liquid. Conversely, for contact angles greater than 90°, the liquid pressure will be larger than that of the gas. Hydrophilic materials will have a positive capillary pressure and naturally wick up the fluid. In contrast, hydrophobic materials will have a negative capillary pressure and pressure must be applied to wet the material.

For many electrodes, the entire void volume is filled with the electrolyte. In others, gas occupies some of the volume. Often, this partially filling with fluid is by design, such as with low-temperature fuel cells where the void volume is filled with one or more immiscible fluids. The degree to which these fluids fill the void volume is quantified by the saturation level, given the symbol S_i.

(5.45)

Since the porous medium typically consists of a range of pore sizes, there are a range of capillary pressures as given by Equation 5.44. Again, this is determined principally from the surface tension, contact angle, and pore size distribution. For a partially filled porous body, the distribution of fluid will be described by this capillary pressure. Assuming the contact angle and surface tension are constant, for a wetting fluid the smallest pores will have the largest capillary pressure and will be filled first. As more liquid is made available, larger and larger pores will fill. If the distribution of pore size is known, we can relate the fill level with the capillary pressure as shown in Figure 5.11. The pore size distribution becomes the key factor in determining the shape of this curve.

ILLUSTRATION 5.4

The separator of a phosphoric acid fuel cell consists of a matrix of micrometer-sized silicon carbide particles filled with phosphoric acid. The function of the separator is to keep the fuel and oxidant separate. Invariably, there will be small pressure differences between these gases. Estimate the capillary pressure in the matrix if the particle size is 1.5 μm and the porosity is 0.5. Assume that the contact angle is zero and the surface tension is 0.02 N·m⁻¹.

This pressure represents the maximum pressure that the matrix can resist before gas breakthrough occurs.