IV. SUMMARY AND CONCLUSION 12 1 10 T [K] 10 -5 10 -4 10 -3 f$_{l}$ f$_{l}$$^{tot}$;  $_{gb}$ = 0 f$_{l}$$^{layer}$;  $_{gb}$ = 0 f$_{l}$$^{curv}$;  $_{gb}$ = 0 f$_{l}$$^{tot}$;  $_{gb}$ = 1 f$_{l}$$^{layer}$;  $_{gb}$ = 1 f$_{l}$$^{curv}$;  $_{gb}$ = 1 $_{0}$ = 0.07 mol/m 3 silicon matrix q$_{s}$ = 0.5 C/m 2 [ $_{gb}$] =  m -1 FIG. 9: Liquid fraction f l for a matrix consisting of silicon spheres with radius R = 500 µ m as a function of the under- cooling Δ T . Two different cases concerning the ice, which is contained in the matrix, are considered: polycrystalline ice ( ρ$_{gb}$ = 1 µ m $^{$_{−}$1}$) and single-crystalline ice ( ρ$_{gb}$ = 0). The polycrystalline sample always has a higher liquid fraction, the difference with the single-crystalline sample being most pronounced at small undercooling. In this regime f l in the polycrystalline sample is almost entirely made up of f curv , which mainly contains contributions from liquid veins in the ice. An interesting consequence of the non-monotonic depen- dence of the premelted layer thickness on the impurity den- sity N i can be observed: although N i is always larger in the single-crystalline sample than in the polycrystalline sample, the curves for f layer l (which are proportional to the layer thick- nesses) intersect. In this work we construct a theory for the premelting of ice in a porous matrix (Section II), which is modeled as a random close packing of monodisperse spheres. The the- ory includes premelting at the matrix-ice interface, grain boundary premelting, as well as curvature and impurity induced premelting. The latter includes premelting in liquid veins, at contact lines of the matrix with a grain boundary, and at the contact points between two matrix spheres. The matrix is characterized by the sphere ra- dius R and the matrix material, which enters the theory via the Hamaker constant A$_{H}$ for the ice/liquid/matrix layered system. In addition to dispersion forces at the ice-matrix interface, which, depending on the matrix ma- terial, either stabilize or destabilize the premelted quasi- liquid layer, we include the effect of electrostatic forces at the interface due to adsorbed surface charges at the interface along the lines of our previous work Ref. 5. The theory includes the effect of charged impurities in the premelted water both on the thickness of the premelted layers (grain boundaries, ice-matrix interface) as well as on the curvature induced melting. The main idea of the present approach is to make use of the small segrega- tion coefficient of ice which leads to an enrichment of impurities in the premelted liquid (making the impurity concentration a function of the fraction of premelted liq- uid) and of the fact that regions of premelted liquid in a sample are all interconnected and therefore the impurity concentration is homogeneous. This approach allowed us to derive an implicit equation for the liquid fraction f$_{l}$ as a function of the undercooling Δ T , Eq. (21), which we analyze for general parameters by considering limiting cases where analytical solutions are possible (Section III). To understand each contribution we began by perform- ing a careful study of interfacial melting, analyzing an implicit equation for the thickness d of the premelted layer as a function of Δ T , Eq. (7), for general parame- ters. In particular, we addressed the discontinuous col- lapse of the premelted layer under certain conditions and the non-monotonic dependence of d on the impurity con- centration (cf. Ref. 5). However, in the general case both Eq. (7) and Eq. (21), the latter of which makes use of the former, have to be solved simultaneously through a nu- merical scheme. In so doing we chose physically relevant parameters of interest to a wide community of scientists. For the porous matrix we use R = 500 µ m and considered two materials (fused quartz and gold) that yield attrac- tive dispersion forces at the ice/matrix interface, thereby inhibiting the formation of a premelted quasi-liquid layer, and one material (silicon) which enhances the premelted layer. The surface charge at the interface is taken to be q$_{s}$ = 0 . 5 C/m $^{2}$. The impurity concentrations lie in the range of 0 . 01 − 0 . 1 mol/m 3 defined with respect to the interstitial volume, and we consider both single- and polycrystalline ice. We calculate f$_{l}$ for Δ T in the range of 0 . $^{5}$− 60 K and obtain liquid fractions which range be- tween 10 − 6 and 10 − 3 of the interstitial volume. proportional to the corresponding layer thicknesses, in- tersect at an undercooling Δ T$_{int}$ = 1 . 38 K. Thus for Δ T < Δ T$_{int}$ the larger impurity density of a single crys- tal sample leads to a smaller layer thickness than the polycrystalline case. It is only for Δ T > Δ T$_{int}$ that the usual increase of the layer thickness with increasing im- purity concentration is observed. This complex behavior provides further evidence for the observation (Fig. 8), that measuring the integrated/homogenized quantity f$_{l}$ alone does not yield satisfactory information about the microscopic structure and behavior of the sample, in par- ticular the rich and interesting temperature dependence of the thickness of the premelted layer in a frozen matrix containing ice with different degrees of polycrystallinity. By parity of reasoning, inclusion of just one or two mi- croscopic phenomena in a theory that is intended to pre- dict the effective medium behavior of a porous medium, will provide irrelevant and misleading predictions unless the proper asymptotic limits of a complete theory are assured.