Bubbles in highly porous media: Clogging and unclogging at constrictions

Abstract: Gas bubble transport through highly porous transport layers (PTLs) is a key process in electrochemical devices such as proton exchange membrane water electrolyzers, where bubbles generated at catalyst surfaces must migrate through complex porous networks. To understand this process, we focus on model systems, namely the motion of single, paired and multiple bubbles in capillaries and study these by combining analytical modeling, three-dimensional color-gradient lattice Boltzmann simulations, and X-ray radiography. For single bubbles, we derive an analytical expression for the critical Bond number separating passage from clogging and show that, in the low deformation regime, it accurately predicts this transition in circular capillaries. Extending the study to bubble pairs, we uncover additional clogging and unclogging pathways, including hydrodynamic unclogging driven by pressure buildup in the interbubble film, and coalescence-induced clogging and unclogging. By mapping our results as functions of confinement ratio and Bond number, we define distinct dynamical regimes that control bubble passage. Experiments on bubble chains rising through highly porous nickel foams confirm the predicted clogging and unclogging mechanisms.

• The paper develops analytical, lattice Boltzmann, and X-ray methods showing that single-bubble passage is governed by a critical Bond number, with the model working best under moderate confinement.
• Bubble pairs introduce hydrodynamic and coalescence-induced unclogging mechanisms that can enable passage even when an isolated bubble would clog, with moderate-confinement transitions organized by dimensionless scaling laws.
• Experiments in 95%-porous nickel foams broadly confirm the predicted clogging and unclogging regimes, providing practical guidance for managing gas transport and pressure losses in water electrolyzers.

Motivation and scope

Gas bubbles generated at catalyst surfaces must traverse the porous transport layer (PTL) of devices such as proton exchange membrane water electrolyzers, where their accumulation degrades mass transport, electrical conductivity, and pressure losses. The paper by Beunen et al. (2603.28511) addresses a fundamental subproblem of this transport: whether a bubble passes or clogs at a single constriction between pores, and how the presence of additional trailing bubbles modifies this outcome. Rather than attacking the full heterogeneity of a PTL directly, the authors reduce the problem to a circular capillary with an abrupt cylindrical constriction and study it through three complementary approaches: an analytical pressure-balance model, three-dimensional color-gradient lattice Boltzmann simulations, and X-ray radiography of bubble chains rising through hydrophilic nickel foams.

Dimensionless framework

The geometry is characterized by a confinement ratio $\mathrm{C} = d/(2R)$, relating the constriction diameter $d$ to the bubble radius $R$, with the constriction length fixed at $\mathrm{C}_l = l/(2R) = 0.2$ so that a bubble is never fully contained within the throat. For paired bubbles, a bubble offset ratio $\mathrm{B}_{\text{offset}} = r_{21}/(2R)$ quantifies the interbubble spacing. Driving is expressed through the Bond number $\mathrm{Bo}$, the ratio of buoyancy to capillary forces. Two derived quantities organize the results: the confinement Bond number $\mathrm{Bo}\,\mathrm{C}^2$, which replaces the bubble length scale by the constriction length scale, and the offset Bond number $\mathrm{Bo}\,\mathrm{B}_{\text{offset}}^2$. This framework proves essential for collapsing state diagrams across confinement regimes and for mapping experimental conditions onto simulation predictions.

Analytical model for the critical Bond number

Neglecting viscous contributions and assuming uniform background pressure, the model balances gravity-induced pressure against the Laplace pressure difference between the two spherical caps of a deforming bubble. Volume conservation combined with the condition that resistance is maximal when the front cap radius equals $d/2$ yields a closed-form expression for the critical Bond number $\mathrm{Bo}_{\text{cr}}$ depending solely on $d$0, obtained via Cardano's method following Zhang et al.'s approach for pressure-driven droplet flow. A companion estimate gives the dimensionless passage time $d$1, which diverges as $d$2.

Two assumptions bound the validity of this result: viscous forces are neglected, which fails near the transition; and perfectly spherical cap deformation is assumed, which breaks down when driving forces are comparable to surface tension forces or when geometric pinning forces non-spherical shapes. These limitations are borne out in the simulations, as discussed below.

Numerical method

Simulations use a D3Q19 color-gradient lattice Boltzmann scheme with BGK collision ($d$3), a perturbation operator implemented via Kupershtokh's exact difference scheme to impose surface tension and buoyancy through a continuum-surface-force formulation, and Latva-Kokko–Rothman recoloring with $d$4 to maintain immiscibility while minimizing spurious currents. Wetting is imposed via Akai et al.'s boundary condition at an equilibrium contact angle of $d$5, with halfway bounce-back no-slip walls. Bubbles are tracked using the Hoshen-Kopelman algorithm on the color field, enabling automatic detection of coalescence and breakup events. The domain is $d$6 lattice sites with $d$7, $d$8, and $d$9; surface tension is varied over two decades to span $R$0, and each run lasts $R$1 time steps.

Single-bubble dynamics

Three states emerge: clogging, passage, and passage with breakup. Four confinement regions are identified: no confinement ($R$2), weak confinement ($R$3), moderate confinement ($R$4), and strong confinement ($R$5). The analytical $R$6 accurately separates clogging from passage in the moderate-confinement regime, but underestimates the transition for weak confinement—attributed to neglected viscous effects—and becomes inapplicable under strong confinement, where only clogging and breakup occur and deformation is strongly non-spherical.

Passage times diverge as $R$7 approaches the predicted critical value for $R$8, consistent with the analytical asymptote, but violate it in the breakup regime. Deviations from the analytical passage-time curve arise from both viscous dissipation (underestimating times) and non-spherical cap deformation: asymmetric left-cap deformation reduces the rear Laplace pressure and increases the required Bond number, whereas uneven right-cap deformation increases the opposing Laplace pressure and shortens passage.

Bubble pairs: new clogging and unclogging pathways

Adding a trailing bubble produces five distinct states beyond the single-bubble cases: ordinary clogging and passage, plus coalescence-induced clogging (a trailing bubble catches up and merges before passage, producing a larger bubble that blocks the throat), coalescence-induced unclogging (merger produces a larger bubble that can overcome the barrier), and hydrodynamic unclogging (pressure buildup in the interbubble film pushes the leading bubble through without coalescence). The existence of these pathways demonstrates that collective effects qualitatively alter single-bubble predictions: a configuration that clogs in isolation may pass when a second bubble follows.

Under no confinement, pairs pass except at $R$9 with $\mathrm{C}_l = l/(2R) = 0.2$0, where instant coalescence induces clogging. Under weak confinement, hydrodynamic unclogging appears for sufficiently large spacing ($\mathrm{C}_l = l/(2R) = 0.2$1), so that even at low Bond numbers passage remains possible if bubbles are well separated. Under moderate confinement, the state space collapses cleanly onto the confinement Bond number axis: passage and clogging separate at $\mathrm{C}_l = l/(2R) = 0.2$2, slightly below the maximum analytical critical value $\mathrm{C}_l = l/(2R) = 0.2$3 at $\mathrm{C}_l = l/(2R) = 0.2$4, again reflecting violation of the spherical cap assumption. Below this threshold, unclogging probability increases with both confinement and offset Bond numbers, and the separation between clogging and unclogging follows a power law,

$\mathrm{C}_l = l/(2R) = 0.2$5

with only four outliers among the unclogging states. Notably, the type of unclogging cannot be predicted from these two dimensionless numbers alone under moderate confinement. Under strong confinement, the passage/clogging boundary shifts to $\mathrm{C}_l = l/(2R) = 0.2$6, with unclogging possible down to $\mathrm{C}_l = l/(2R) = 0.2$7; here the offset Bond number does discriminate between mechanisms, with coalescence-induced unclogging dominating below offset Bond numbers near ten and hydrodynamic unclogging emerging above. Breakup is nearly ubiquitous during passage under strong confinement.

Passage-time analysis shows that a trailing bubble accelerates the leading bubble substantially, and hydrodynamic unclogging enables passage for $\mathrm{C}_l = l/(2R) = 0.2$8 combinations inaccessible to single bubbles. The collapse of data across offset ratios confirms that inertial effects are negligible in this regime.

Experimental validation

X-ray radiography at 150 frames per second with 0.06 mm pixel size was applied to air bubble chains rising through HMDSO-functionalized nickel foams of 95% open porosity (average pore diameters 2.3 mm and 1.4 mm). Temporal differentiation of the gas-fraction field distinguishes mobile from immobile (clogging) bubbles, and individual bubbles are tracked along their paths. Five experimental cases were mapped onto the simulated state diagrams using estimated constriction diameters between $\mathrm{C}_l = l/(2R) = 0.2$9 and $\mathrm{B}_{\text{offset}} = r_{21}/(2R)$0, derived from a Kelvin-cell idealization of the foam.

The comparison is broadly consistent: Case 1 (moderate confinement, high offset Bond number) falls in the predicted unclogging region, and tracking indicates hydrodynamic rather than coalescence-driven unclogging, matching simulations. Cases 2 and 3 show intermittent pore clogging and unclogging consistent with weak-to-moderate confinement predictions, independent of spacing. Cases 4 and 5, at smaller pore diameter and lower confinement Bond number ($\mathrm{B}_{\text{offset}} = r_{21}/(2R)$1), exhibit persistent clogging as predicted, with unclogging occurring only after clustering of multiple bubbles into aggregates spanning several pore diameters—a behavior suggestive of coalescence-induced unclogging but not directly resolvable at the achieved resolution.

Limitations and open questions

Several limitations should be weighed against the results. The analytical $\mathrm{B}_{\text{offset}} = r_{21}/(2R)$2 neglects viscosity and assumes spherical caps, restricting its quantitative accuracy to the moderate-confinement, low-deformation regime; it systematically underestimates the transition under weak confinement and fails entirely under strong confinement. The experimental validation relies on estimating constriction diameters from an idealized Kelvin-cell foam model rather than measured pore-throat distributions, and the radiographic resolution does not permit direct observation of pairwise coalescence events inside the foam, leaving the coalescence-induced unclogging pathway supported only indirectly by clustering observations. Simulations initialize the leading bubble adjacent to the constriction, justified by negligible inertia but not representative of full pore-network trajectories. Open questions include whether the power-law unclogging boundary retains its form for wetting conditions other than $\mathrm{B}_{\text{offset}} = r_{21}/(2R)$3, how the state diagrams extend to chains longer than two bubbles, and what spatial and temporal resolution is required to resolve coalescence dynamics experimentally within porous structures.

Conclusion

This work establishes a dimensionless framework—confinement ratio, Bond number, and their derived confinement and offset Bond numbers—that organizes the passage, clogging, breakup, and unclogging behavior of bubbles at constrictions. Single-bubble passage is governed by a closed-form critical Bond number accurate under moderate confinement, while bubble pairs introduce hydrodynamic and coalescence-mediated unclogging pathways absent from classical single-bubble descriptions, including a predictive power-law boundary for unclogging under moderate confinement. X-ray measurements in highly porous nickel foams confirm the principal regimes and provide evidence that gas traverses such media through successive unclogging events. The results offer a quantitative basis for anticipating two-phase transport limitations in electrolyzers and related electrochemical devices.