In two-phase models of proton exchange membrane fuel cells (PEMFCs), the capillary pressure that drives liquid water through the porous catalyst layer (CL) and gas diffusion layer (GDL) is conventionally closed by the Leverett JJ-function, in which solid-surface wettability enters only through the static contact angle. Jiao and Li (2010) augmented this closure with a sliding angle θs\theta_s (a measure of dynamic wettability), using two heuristic multiplicative scalings chosen without pore-scale derivation or calibration. The open problem is to determine the correct functional dependence of capillary pressure on sliding angle — independent of static contact angle — for representative porous CL/GDL media: either derive and/or experimentally calibrate a validated sliding-angle-dependent capillary constitutive relation, or demonstrate with pore-scale or experimental evidence — for representative porous CL/GDL media, with surface roughness and pore structure controlled and spanning the distinct media and conditions necessary to support a general claim — that capillary pressure in such media carries no independent sliding-angle dependence. This constitutive question remains open.

Background

A proton exchange membrane fuel cell (PEMFC) converts hydrogen and oxygen to electricity; product water is generated inside the porous cathode catalyst layer (CL, a ~10 µm layer of platinum-on-carbon catalyst held together by ionomer, a polymer containing ionic groups that conducts protons) and must pass through the CL and the gas diffusion layer (GDL, a ~200 µm carbon-fiber porous sheet rendered hydrophobic by treatment with PTFE, polytetrafluoroethylene) into the flow channel to be removed. Excess retained liquid water blocks reactant transport ('flooding') and is a primary performance limitation, so two-phase models of PEMFC water management must represent how wetting forces drive liquid through these porous layers.

In continuum two-fluid PEMFC models, the coupling between gas and liquid water in a porous layer is expressed through the capillary pressure pc=pg−plqp_c=p_g-p_{lq} (gas pressure minus liquid-water pressure) as a constitutive function of liquid saturation. The standard closure is the Leverett JJ-function (Leverett, 1941), borrowed from petroleum engineering:

pc=σcos⁡(θc)(εK0)1/2J(slq)p_c = \sigma\cos(\theta_c)\left(\frac{\varepsilon}{K_0}\right)^{1/2} J(s_{lq})

where σ\sigma is surface tension, ε\varepsilon porosity, K0K_0 intrinsic permeability, slqs_{lq} liquid water volume fraction in pores, and θc\theta_c the static (equilibrium) contact angle — the single wettability parameter in the closure. Wettability of real GDL/CL materials is engineered by PTFE loading and by surface roughness or patterning.

The static contact angle, however, characterizes only equilibrium drop geometry. Dynamic wettability is quantified by the sliding angle θs\theta_s — the critical tilt at which a millimetric droplet begins to slide on an inclined surface — which is governed by contact angle hysteresis (the difference between advancing and receding contact angles) and pinning on surface roughness. Two surfaces can share the same θc\theta_c but very different θs\theta_s, so a closure based on θc\theta_c alone cannot distinguish a 'slippery' from a 'sticky' surface.

Jiao and Li (2010) proposed a modified Leverett closure in which the capillary term σcos⁡(θc)\sigma\cos(\theta_c) is multiplied by either a cosine scaling cos⁡(θs)/cos⁡(θspg)\cos(\theta_s)/\cos(\theta_s^{pg}) or a linear scaling [(90∘−θs)/90∘]/[(90∘−θspg)/90∘][(90^\circ-\theta_s)/90^\circ]/[(90^\circ-\theta_s^{pg})/90^\circ], normalized at a reference sliding angle θspg=60∘\theta_s^{pg}=60^\circ for a PTFE-coated GDL. Embedded in a three-dimensional steady-state two-fluid model of a single straight-channel cell, this closure predicted, e.g., roughly a threefold increase in cathode-CL liquid water fraction as θs\theta_s increased from 1∘1^\circ to 89∘89^\circ at fixed θc\theta_c. The authors themselves stressed that both scalings were arbitrary functional choices, made without derivation from pore-scale physics and without calibration against pore-network or experimental capillary-pressure data; the simulations used homogeneous effective properties (no explicit CL/GDL microstructure), and no experiments had ever varied θs\theta_s independently of θc\theta_c in porous PEMFC media.

Since then, related capabilities have grown along separate lines: dynamic-wettability models (sliding angle plus dynamic contact angles) are now used for water droplets in flow channels; pore-network models (capillary-pressure calculations on an idealized lattice of pore bodies and throats) and volume-of-fluid (VOF, an interface-tracking method for simulating two-phase flow) simulations deliver capillary-pressure–saturation curves for GDLs with heterogeneous or mixed static wettability; and experiments characterize sliding angles and contact angle hysteresis on GDL surfaces. None of these, however, establishes how — or whether — the sliding angle enters the macroscopic capillary pressure of the porous CL/GDL itself, which is the constitutive question the 2010 closure left open.

Problem Statement

Determine the correct functional dependence of capillary pressure on sliding angle θs\theta_s — independent of the static contact angle θc\theta_c — for representative porous PEMFC catalyst-layer and gas-diffusion media. Concretely: (i) derive, from pore-scale physics (pore-network modeling or direct two-phase simulation of realistic, heterogeneous, coated CL/GDL microstructures), the effective macroscopic capillary-pressure–saturation relation and how it depends on the sliding angle of the solid surfaces; or (ii) calibrate such a relation against experimental capillary-pressure data on media in which θs\theta_s is varied independently of θc\theta_c (with surface roughness and pore structure otherwise controlled), including identification of the reference parameters (such as the 60∘60^\circ PTFE-GDL normalization used by Jiao and Li); in either case, (iii) establish from this evidence whether each of the two heuristic multiplicative scalings of Jiao and Li (2010) is correct in form or requires modification, or whether capillary pressure in such porous media carries no independent sliding-angle dependence at all. A complete answer is either (a) a sliding-angle-dependent capillary constitutive relation for representative porous CL/GDL media, with its identified parameters, where the coverage that counts as validated must include at least one catalyst-layer and one gas-diffusion-layer medium, each with its surface roughness and pore structure reported and held fixed while θs\theta_s is varied, and the relation must reproduce independent capillary-pressure data (pore-scale simulation or measurement) not used to obtain it — together with a classification of each of the two named heuristic scalings of Jiao and Li (2010) separately, stating for each whether it is correct in form (the derived or calibrated θs\theta_s dependence reproduces that scaling up to the reference normalization θspg=60∘\theta_s^{pg}=60^\circ) or requires modification; or (b) a pore-scale/experimental demonstration, for the same media coverage, that capillary pressure in such media carries no independent sliding-angle dependence.

References

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