1 10 T [K] 10 -5 10 -4 10 -3 f$_{l}$ f$_{l}$ tot f$_{l}$ layer f$_{l}$ curv f$_{l}$$^{tot}$ = f$_{l}$ curv $_{0}$ = 0.1 mol/m 3 fused quartz matrix (x 10) q$_{s}$ = 0.5 C/m 2 gold matrix $_{gb}$ = 0 11 the polycrystalline ice (see Fig. 7) in that it does not show liquid layer collapse on the scale of the plot for ρ$_{0}$ = 0 . 1 mol/m $^{3}$. When replacing the matrix with gold, which has stronger attractive dispersion forces at the ice- gold interface, we see a collapse of the premelted layer at Δ T ≈ 7 K. This is accompanied by, as in the case of layer collapse for the fused quartz matrix at ρ$_{0}$ = 0 . 01 mol/m 3 (Fig. 7), a discontinuous increase of the liquid fraction f curv l . As for the fused quartz matrix, for sufficiently low undercooling that the premelted layer has collapsed, the only contribution to f tot l comes from curvature induced melting; f tot l = f curv l . While, on the scale of the plot f tot l does not display a discontinuity at the point of layer collapse, there is a discontinuity Δ f$_{l}$ which can be seen by expanding the full result in the limit of large impurity concentration c$_{0}$ where t$_{coll}$ is the undercooling at layer collapse and f layer l the liquid fraction from interfacial melting at a temper- ature slightly above collapse t = t$_{coll}$ − δt , with δt > 0. Clearly, Δ f$_{l}$ → 0 for c$_{0}$ → ∞ . Δ f$_{l}$ = 1 2 × f layer l × ( B c$_{0}$t$_{coll}$ ) 1 / 2 $^{+}$O ( 1 c 3 / 2 0 ) , (40) FIG. 8: Liquid fraction f l for the gold matrix compared to the fused quartz matrix as a function of the undercooling Δ T for ρ 0 = 0 . 1 mol/m $^{3}$. Importantly, although the Hamaker constants for the two materials differ by almost two orders of magnitude, which results in a layer collapse for the gold matrix at Δ T ≈ 7 K, the curves for the total liquid fraction are almost identical. Note that the curves for fused quartz have been shifted for clarity. In particular, the discontinu- ity of f tot l at the location of layer collapse is too small to be resolved in the figure, which results from the high impurity concentration, see Eq. (40). The strength of dispersion forces is important only for thin premelted layers, i.e. in the imme- diate vicinity of layer collapse. The discontinuity for fused quartz is not visible in the figure because it occurs at large undercooling (Δ T ≈ 103 K). An important result seen in Fig. 8 is that although the Hamaker constants for gold and fused quartz differ by almost two orders of magnitude, the results for f$_{l}$ are almost identical over a large range of Δ T ; the deviation increases from 0 . 02 % to 0 . 4 % at layer collapse and is 6 % for Δ T = 60 K. This is because dispersion forces are only relevant for thin premelted layers and hence at tem- peratures slightly above layer collapse. However, when layer collapse occurs, according to Eq. (40) the high im- purity concentration, combined with single-crystallinity (i.e. small B ), insures that the Debye length and the discontinuity of f$_{l}$ are small. Therefore, the deviation between the results for the two different matrix materi- als remains small. The conclusions drawn from Fig. 8 are important to note for experimental determination of f$_{l}$ in different materials because they imply that even though the underlying microscopic behavior of two sam- ples may fundamentally differ (interfacial premelting vs. collapsed premelted layer) this may not be resolved in the integrated/homogenized quantity f$_{l}$ . Spatially resolved measurements may be required to detect characteristic transitions such as the collapse of interfacially premelted layers. ice ( ρ$_{gb}$ = 1 µ m − $^{1}$). As expected, the polycrystallinity leads to a higher liquid fraction f$_{l}$ . In particular, for small undercooling Δ T , where curvature induced melting is enhanced and hence liquid veins greatly contribute, f$_{l}$ consists almost entirely of f curv l . The values of f curv l for the single crystal ice, for which contributions derive only from the contact points between matrix spheres, are al- ways by at least an order of magnitude lower than for the polycrystalline ice case. This is not as obvious as it may first appear. Due to the higher liquid fraction in polycrys- talline ice the impurity concentration ρ$_{i}$ is smaller than for the single crystal case. Thus, polycrystalline ice has a smaller ice-water radius of curvature (see Eq. 16). Conse- quently, the volume of liquid contained for example at the contact point between two matrix spheres, is smaller for the polycrystalline case than for the single crystal case. It is the grain boundary density ρ$_{gb}$ , which causes B to be larger for the polycrystalline ice, that insures f curv l will be also larger, as indicated by Eq. (34). In addition to the cases studied in Section III A, another interesting manifestation of the non-monotonic dependence of the premelted layer thickness on the impurity concentration for the systems with interfacial charges can be seen in Fig. 9. Although the impurity concentration ρ$_{i}$ and the impurity density N$_{i}$ are always larger in the single-crystal case, the liquid fractions f layer l , which from Eq. (14) are Effect of polycristallinity Finally, we study the in- fluence of the degree of polycrystallinity of the inter- stitial ice for the case where the porous matrix con- sists of silicon, which leads to dispersion forces that are repulsive ( A$_{H}$ < 0) and prevent the interfacially pre- melted layer from collapsing. As in the previous exam- ples the grain size of the matrix is R = 500 µ m and as in Fig. 7 the grain boundaries are collapsed. Thus, Fig. 9 shows the curves for f$_{l}$ = f tot l , the interfacial melting ( f layer l ) and curvature induced melting ( f curv l ) contribu- tions for single-crystalline ( ρ$_{gb}$ = 0) and polycrystalline