1 10 T [K] 10 -6 10 -5 10 -4 10 -3 f$_{l}$ f$_{l}$ tot f$_{l}$ layer f$_{l}$ curv f$_{l}$$^{tot}$ = f$_{l}$ curv f$_{l}$$^{tot}$;  $_{0}$ = 0.1 mol/m 3 q$_{s}$ = 0.5 C/m 2 fused quartz matrix $_{0}$ = 0.01 mol/m 3 $_{gb}$ = 1  m -1 10 veins, the sample completely liquifies. Therefore, the cor- responding equation for f$_{l}$ is which has a simple analytical solution. Again, we exam- ine the leading order contributions in the limiting cases t → 0 and t → ∞ to find f$_{l}$ = B ( t − c$_{0}$/f$_{l}$ ) 2 , (37) f$_{l}$ $^{t}$→ 0 −→ B t 2 + 2 c$_{0}$ t − c 2 0 B + O ( t ) and (38) f$_{l}$ $^{t}$→∞ −→ c$_{0}$ t + √$_{Bc}$$_{0}$ t 3 / 2 + B 2 t 2 $^{+}$O ( 1 t 5 / 2 ) , (39) where the latter relation is an extension of Eq. (34) which is undefined in the limit t → ∞ and A → 0. Non-zero surface charge In the general case with im- mobile surface charges ( q$_{s}$ = 0) we solve Eq. (21) numer- ically. In Fig. 7 we show results for a matrix of fused quartz with a grain size of R = 500 µ m, which may serve as a model for frozen soils. The surface charge density is chosen to be q$_{s}$ = 0 . 5 C/m 2 which has the correct order of magnitude for mineral/water interfaces at high pH [10]. The ice which fills the voids of the porous ma- trix is taken to be polycrystalline with a grain bound- ary density ρ$_{gb}$ = 1 µ m − $^{1}$. A comparison of Eqs. (14) and (15) shows that for the given ρ$_{gb}$ the surface area provided by grain boundaries is by a factor 100 larger than the surface area of the porous matrix. Therefore, we expect the impurity concentration in the sample to be very low when grain boundaries are melted and, ac- cording to Eq. (24), the maximal undercooling Δ T$_{max}$ for which quasi-liquid interfacial water at the grain bound- aries is stable should be low. An upper limit for Δ T$_{max}$ can be calculated by assuming that all the impurities in the sample are located at the premelted grain boundaries. For the values of ρ$_{0}$ from Fig. 7 we use N$_{i}$ = ρ$_{0}$/ρ$_{gb}$ to obtain a range N$_{i}$ = 0 . 01 − 0 . 1 µ mol/m 2 and, with the aid of Eq. (24) this corresponds to a range in Δ T$_{max}$ of 0 . 01 − 0 . 26 K. The range of Δ T$_{max}$ uses the Hamaker constant A$_{H}$ = 3 . 3 × 10 − 22 J computed from complete dispersion force theory [6] in the limit of thin layers with the data for the dielectric functions from Elbaum and Schick [11]. Including a typical surface charge density of q$_{s}$ = 0 . 01 C/m 2 for the grain boundary [12] lowers these values to Δ T$_{max}$ = 0 . $^{01}$− 0 . 20 K. Hence, the grain boundaries are always collapsed to zero film thickness for the range of Δ T considered in the figure. A promi- nent feature of f$_{l}$ = f tot l with ρ$_{0}$ = 0 . 01 mol/m 3 in the fused quartz matrix is the discontinuity at Δ T = 22 . 6 K where f$_{l}$ drops by almost an order of magnitude. This is the signature of the collapse of the premelted layer at the ice/fused quartz interface described in detail in Sec- tion III A. In the strongly undercooled, layer collapsed system, the only contribution to f$_{l}$ comes from curva- ture induced melting; f curv l = f contact l + f gbmat l + f veins l (cf. Eq. 21), which discontinuously increases at the un- dercooling where the premelted layer collapses. This is In Fig. 8 many of the same conditions as in Fig. 7 are considered ( R = 500 µ m, q$_{s}$ = 0 . 5 C/m $^{2}$, ρ$_{0}$ = 0 . 1 mol/m $^{3}$) the principal difference being that the pore ice is single-crystalline ( ρ$_{gb}$ = 0). We find that f tot l in the fused quartz matrix is qualitatively the same as for because, due to the efficient rejection of impurities by the ice lattice, the impurity concentration in the remain- ing liquid increases discontinuously when the liquid layers collapse. At low undercooling curvature melting becomes increasingly important exceeding the contribution from interfacial premelting f layer l = f mat l . However, at larger undercooling, close to the location of layer collapse, cur- vature induced melting yields only a minute contribution to f$_{l}$ . The importance of curvature induced melting at low undercooling is reflected in Eq. (34) where the cur- vature term B/t 2 dominates for t → 0. In the opposite limit of large undercooling the analytical result Eq. (39) applies. The leading term c$_{0}$/t originates from the col- ligative effect and curvature induced melting enters only through the subdominant term. Hence, for large under- cooling the principal contribution to the liquid fraction is the colligative effect. However, liquid water is located in regions of high curvature; contact of matrix spheres, contact lines of grain boundaries and matrix spheres and liquid veins. Figure 7 also shows f$_{l}$ in the same system with a higher impurity concentration ( ρ$_{0}$ = 0 . 1 mol/m $^{3}$). Here the collapse of the liquid layer is moved to a lower temperature (Δ T ≈ 110 K) so that no discontinuity is observed in the range of undercooling considered in the figure. FIG. 7: Liquid fraction f l for water in a porous matrix of fused quartz with a grain size R = 500 µ m as a function of the undercooling Δ T . The result for the low impurity density ρ 0 = 0 . 01 mol/m 3 (with respect to the interstitial volume) shows a discontinuity where the quasi-liquid layer between the ice and the matrix collapses. This is accompanied by an increase of the impurity concentration in the remaining (curvature melting induced) liquid which creates a discon- tinuous increase of f curv l . For the higher impurity density ρ 0 = 0 . 1 mol/m 3 the discontinuity is not visible in the figure because it occurs at large undercooling (Δ T ≈ 110 K).