Phase-field interpretation of mercury porosimetry hysteresis: Capillary constrictions and connectivity-controlled trapping

Mercury intrusion--extrusion porosimetry probes capillary invasion in porous materials, but porograms are difficult to interpret when pore constrictions, pore-space accessivity, and irreversible mercury trapping contribute to hysteresis. We develop a quasi-static phase-field framework that treats porosimetry as a pressure-indexed sequence of capillary equilibrium configurations. An Allen--Cahn-type relaxation constructs stationary mercury--gas interfaces at prescribed Washburn-equivalent capillary pressures; the pseudo-time evolution serves only as a numerical continuation procedure, not as a time-resolved flow model. During depressurization, a connectivity-based irreversibility rule identifies mercury regions that lose hydraulic connection to the external reservoir and preserves them as trapped mercury, representing the residual-volume component of hysteresis. Because the formulation operates on a computational pore-space mask, it can be applied to sufficiently resolved mesh- or voxel-based geometries. The framework is demonstrated for bimodal alumina powder. Interparticle macropores are represented by random sphere packings, whereas the dominant intraparticle pore family in the mesopore range is represented by unduloid-shaped constricted capillaries. The unduloid parametrization links the intrusion--extrusion pressure ratio to throat and body dimensions, providing a structural interpretation of the pressure offset. Focused ion beam cross-section analysis with merge-tree descriptors provides a local microstructural consistency check for the inferred intraparticle pore scale. The resulting two-scale interpretation is consistent with the low-pressure macropore intrusion step, the high-pressure intrusion--extrusion offset of the intraparticle pore family, and the residual trapped mercury volume. These results show how constriction-controlled pressure shifts and connectivity-controlled retention can be separated within a general framework for interpreting porosimetry hysteresis.