Population Balance Modeling of Pt Catalyst Nanoparticle Degradation in PEM Fuel Cells Accounting for Electrochemical Accessibility

Degradation of carbon-supported platinum (Pt/C) catalysts remains a major challenge limiting the durability of polymer electrolyte membrane (PEM) fuel cells. Platinum catalyst degradation leads to a progressive loss of electrochemically active surface area, resulting in reduced catalytic activity and overall fuel cell performance. Over the past decades, extensive research efforts, supported by the development of various accelerated stress tests (ASTs), have identified four primary mechanisms responsible for Pt catalyst degradation: (1) electrochemical dissolution of Pt particles, (2) Ostwald ripening under electrochemical conditions, involving the redeposition of dissolved Pt species from smaller to larger particles, (3) coagulation or coalescence of Pt nanoparticles on the carbon support, and (4) detachment of Pt particles induced by carbon corrosion. The dominant degradation pathway depends strongly on catalyst morphology, catalyst layer architecture, and operating conditions, particularly the applied potential and potential cycling protocol. For instance, high upper potential limits promote Pt dissolution and redeposition, while lower potentials favor particle coalescence through surface diffusion, and aggressive cycling accelerates carbon corrosion, leading to Pt detachment. Despite the mechanistic understanding, decoupling the contribution of various degradation mechanisms remains challenging, due to their complex interaction and simultaneous occurrence during fuel cell operation. Population balance equation (PBE) models have emerged as a powerful framework for describing the statistical evolution of catalyst particle size distributions (PSDs) under electrochemical cycling. By explicitly incorporating the physics of dissolution, redeposition, coagulation, and detachment, PBE models enable the quantification of competing degradation mechanisms. Recently developed PBE formulations have demonstrated good agreement with PSD evolution during fuel cell operation measured by scattering techniques or transmission electron microscopy. In PBE models, the Pt surface area is derived directly from the evolving particle size distribution by assuming that all geometrically available particle surface contributes equally to electrochemical activity. However, the actual electrochemically active surface area, as measured by cyclic voltammetry (CV), can be significantly lower than the geometrically available surface derived from scattering or microscopy techniques. Electrochemical accessibility is affected by additional degradation mechanisms that are overlooked in traditional PBE models validated on experimental data obtained from scattering or miscroscopy. In this work, we develop a PBE model for Pt catalyst nanoparticle degradation that explicitly distinguishes between structural PSD evolution and electrochemical surface accessibility. The model is based on a dimensionless PBE describing the time evolution of a normalized number-based PSD as a function of particle radius. The governing equation includes (i) a size-dependent advection term representing net particle growth or shrinkage due to dissolution and redeposition, (ii) a Smoluchowski-type coagulation kernel accounting for particle coalescence, and (iii) a detachment loss term representing the removal of particles from the catalyst layer. Model parameters are estimated by fitting the simulated evolution of the mean particle radius and distribution width to experimental in-operando small-angle X-ray scattering (SAXS) and CV data obtained during accelerated stress testing. To account for electrochemical accessibility of Pt nanoparticles, we introduce an electrochemical accessibility factor that maps the structural surface area predicted by the PBE to the ECSA measured by CV. This factor is decoupled from PSD evolution and can account for additional support corrosion, detachment, restructuring, and electrical isolation effects that are not captured by SAXS. This approach enables simultaneous consistency between SAXS-derived structural metrics and CV-derived ECSA trends without altering the underlying population balance dynamics. Results demonstrate that PBE-based models provide a robust framework for describing Pt nanoparticle size evolution observed by SAXS, but that electrochemical catalyst performance degradation cannot be fully described by traditional PBE. Explicitly accounting for electrochemical accessibility is essential to account for the actual catalyst electrochemical performance observed by CVs.