Analytical Modeling of Effective Diffusivity in Micro-Porous Layers
Authors: Andisheh-Tadbir Mehdi , Hannach Mohamed El , Kjeang Erik , Bahrami Majid 
Published: February 2020

The membrane electrode assembly (MEA) used in polymer electrolyte fuel cells consists of a membrane, two electrodes (anode and cathode), and two gas diffusion layers (GDLs). The GDL is responsible for providing the pathways for transport of the reactant gases from the flow channels to the catalyst layers. Hence, the mass transport resistance of this layer should be minimized for high performance operation.
The GDL is typically a dual-layer carbon-based material composed of a macro-porous substrate, which usually contains carbon fibers, binder, and PTFE, and a thin delicate micro-porous layer (MPL), which is usually made of carbon nano-particles and PTFE. Spherical carbon nano-particles of 20-100 nm in diameter construct the complex structure of MPL. Small pore sizes and highly hydrophobic characteristics are the two main specifications of the MPL.
The effective diffusivity of the MPL is a key property in determining the fuel cell performance. However, since the MPL is a recently developed material, the number of published works with focus on its diffusivity is limited. Measuring the MPL diffusivity is a challenging task since the MPL needs a physical support and cannot be analyzed as a separate object. Similar to its measurements, modeling the MPL diffusivity is also a difficult task (Nanjundappa et al., Electrochimica Acta 110:349-57), since it involves the reconstruction of the complex structure numerically and solving the diffusion equations in the nano-scale pores where the continuum assumption may be invalid and Knudsen diffusion may prevail. Usually, complex numerical algorithms are employed to reconstruct a small portion of the MPL. This step is followed by a computationally intensive stage to solve the diffusion equations inside the void spaces of the reconstructed domain. Although this approach leads to reliable and accurate results, an analytical relationship that correlates certain design parameters to the MPL diffusivity could be helpful.
A fully analytical solution of the mass transport equation inside a randomly structured porous material is however not feasible. Simplifying assumptions are therefore required to derive an analytical model for predicting the transport properties of porous structures. The unit cell approach is one way of simplifying the structure. A unit cell is a hypothetical, periodic domain that represents the entire structure. Hence, optical observations can aid the selection of the optimal unit cell arrangement.
In this work a unit cell is chosen based on cross-sectional SEM images of a standard MPL material. Generally, two main regions can be observed in the MPL: i) clusters of spherical particles; and ii) pores. The considered unit cell, which is devised based on these observations, is shown in Figure 1. The analogy between the heat and mass transfer is employed to solve the diffusion equation in the unit cell and a relationship is proposed to calculate the effective MPL diffusivity. The obtained values from this relationship are compared with the experimental data available in the literature and another stochastic model which is under development in our group. The proposed relationship is able to predict the MPL diffusivity with an acceptable accuracy, i.e., with less than 10% deviation. The required input parameters of the model are particle diameter, MPL porosity, and average pore size. A parametric study is then performed to examine the effects of these parameters on the MPL diffusivity. The proposed relationship is expected to be useful to the fuel cell community, especially for the purposes of performance modeling. Furthermore, the results from the parametric study c provide valuable information on the effects of carbon particle size, the MPL porosity, and the MPL pore size distribution on its effective diffusivity.
Acknowledgments:
This research was supported by Ballard Power Systems and the Natural Sciences and Engineering Research Council of Canada.