r matrix sphere grain boundary matrix A A’ ice a) b) R z r liquid h 13 FIG. 10: a) Geometry of a matrix sphere intersecting with a grain boundary which is located at distance z from the sphere center. b) Zoom on the curvature melting induced liquid “vein”. As r ≪ R the matrix surface can be considered as flat in the vicinity of the liquid vein. For the parameters under consideration the grain boundaries are always of zero thickness (i.e., collapsed), and hence only the collapse of the quasi-liquid layer at the ice-matrix interface is observed in f$_{l}$ (see Fig 7 for the case of the fused quartz matrix). When the tem- perature of the sample is decreased below the point of layer collapse, only curvature induced premelting con- tributes to f$_{l}$ . While dispersion forces are too weak in the fused quartz system to significantly influence the re- sults, in the case of the gold matrix dispersion forces are sufficiently strong to drive layer collapse, as opposed to the surface charge triggered layer collapse in the fused quartz system (see Fig. 8 for a comparison of the two cases). An important conclusion from Fig. 8 is that for high impurity concentration and/or low degree of poly- crystallinity the characteristic discontinuity of f$_{l}$ upon layer collapse is minute and most likely not resolved in experiments. Thus, transitions that would be dra- matic and easily detected at planar interfaces will be much more challenging to extract from measurements of the integrated/homogenized quantity f$_{l}$ . Finally, we study the influence of the degree of polycrystallinity on f$_{l}$ for the case of the silicon matrix where, due to the repulsive dispersion forces, layer collapse does not occur (Fig. 9). The results reveal an interesting consequence of the non-monotonic dependence of d on the impurity concentration. Although the actual impurity concentra- tion is higher in the single-crystalline sample than in the polycrystalline sample at all temperatures, the thickness of the premelted layer can be both larger and smaller in either case, depending on the temperature range. This peculiarity is obscured by the curvature induced contri- bution to f$_{l}$ providing further evidence that additional measurements are required f$_{l}$ to capture the physics con- trolling the sample microstructure. strength and range of the electrostatic force, the precise value of q$_{s}$ may be considered secondary. An important question for future research concerns the dynamical evolution of the porous matrix system in re- sponse to changes in the control parameters, in particu- lar the sample undercooling Δ T . In order to guarantee a homogeneous impurity concentration throughout a sam- ple, Δ T must be changed on a time scale that is slow compared to the typical impurity diffusion time. An even more ambitious question concerns impurity diffu- sion during layer collapse. The situation where the pre- melted quasi-liquid layer between the ice and the matrix collapses, which forces the impurities to migrate to re- maining liquid regions with curvature induced melting constitutes a challenging task for further investigations. We view our approach as having provided a thorough theory of premelting in a porous medium which can be applied to a plethora of experimental situations and ma- terial systems. For the sake of brevity and clarity, we presented examples of specific materials systems; gold, fused quartz, and silicon. However, we chose these ma- terials because their dielectric properties, as reflected in their Hamaker constants, span all the range of materials as diverse as sapphire, polystyrene, and polyvinylchlo- ride [4]. We did not explore the effect of varying the ma- trix sphere radius R , its roughness or its surface charge density q$_{s}$ . The value R = 500 µ m was chosen with the example of frozen soils in mind, while q$_{s}$ = 0 . 5 C/m 2 pro- vides a realistic order of magnitude as inferred from ex- perimental data for mineral-water interfaces at large pH [10]. However, not much is known experimentally about q$_{s}$ in the ice/liquid/matrix systems and the simplifying assumptions of symmetry (same charge at the ice-liquid and liquid-matrix interfaces) as well as independence of q$_{s}$ on the impurity concentration and layer thickness are not yet established experimentally on planar systems. Nevertheless, given the importance of the impurity con- centration dependent screening length for regulating the We consider a matrix sphere that is embedded at the interface between two ice grains. The grains are assumed to have a planar grain boundary and to be much larger than the matrix sphere. Let the distance between the sphere center and the grain boundary be z < R . Curva- ture induced melting leads to a circular “vein” along the line where the grain boundary and the sphere intersect (Fig. 10a). Because r is assumed to be much smaller than The authors have benefitted from discussions with J.G. Dash, S.S.L. Peppin, M. Spannuth, and E. Thomson at various stages of this work. We acknowledge the U.S. Na- tional Science Foundation Grant No. OPP0440841, the Department of Energy Grant No. DE-FG02-05ER15741, the Helmholtz Gemeinschaft Alliance “Planetary Evolu- tion and Life”, and Yale University for generous support of this research. Acknowledgments Appendix A: Melting volume at a grain boundary–matrix contact line