# Three-Dimensional Infinite Cluster Function as a Descriptor of Through-Plane Effective Conductivity in Porous Electrodes of Membrane Electrode Assemblies

## TL;DR

This paper studies how the 3D structure of porous electrodes affects their conductivity, showing that connectivity quality, not just quantity, determines performance.

## Contribution

The study introduces a new descriptor, the three-dimensional infinite cluster function, to evaluate through-plane conductivity in MEA electrodes.

## Key findings

- OCF morphology shows the highest normalized conductivity due to vertically coherent channels.
- MFM1 underperforms due to laminated constrictions despite high spanning-cluster fraction.
- Conductivity magnitude depends on percolating-skeleton quality, not just phase amount.

## Abstract

Through-plane electronic transport in porous membrane electrode assembly (MEA) electrodes is governed by the three-dimensional (3D) connectivity of the conducting phase. Here, we quantify the role of the spanning-cluster fraction P∞, defined as the fraction of conducting-phase voxels that belong to the z-spanning connected component in a finite reconstructed volume, on effective conductivity using scanning electron microscopy (SEM)-informed 3D reconstructions of four archetypal morphologies: a granular catalyst layer (CL), labeled CL1; a fibrous gas diffusion layer (GDL), labeled GDL1; an open-cell foam (OCF); and a micro-fibrous non-woven (MFM), labeled MFM1. Each morphology is reconstructed on a 150×150×150 voxel grid, and z-spanning connectivity is identified with a 26-neighbor flood-fill algorithm. Steady-state conduction is solved by a finite-volume method (FVM) with an imposed potential difference between the z-faces and no-flux lateral boundaries. Although all samples exhibit through-thickness connectivity, the normalized conductivity σeff/σbulk varies widely, from ≈0.134 (MFM1) to ≈0.706 (OCF). The corresponding (P∞,σeff/σbulk) pairs are 0.996,≈0.306 for CL1, 0.999,≈0.303 for GDL1, 0.997,≈0.706 for OCF, and 0.901,≈0.134 for MFM1. OCF exhibits the highest response due to vertically coherent channels, whereas MFM1 underperforms due to laminated constrictions; CL1 and GDL1 lie in an intermediate regime with nearly isotropic skeletons. Overall, the results show that while a z-spanning connected component is required for measurable conduction, the magnitude of σeff is dictated by percolating-skeleton quality (bottlenecks, cross-sectional constrictions, and pathway alignment) rather than phase amount alone. The proposed descriptors therefore enable percolation-aware screening metrics for designing and comparing MEA-relevant GDL and CL microstructures.