3 defines the surface normal. Thus, the surface energy, it- self a Gibbs free energy conjugate to area, is known in general to depend on the size of the system, the adsorp- tion of impurities and the crystallographic orientation [1] but the simplification of treating it as a constant is jus- tified both on the basis of experimental evidence for the first two dependencies and on the level of our analysis. The condition for thermodynamic equilibrium is again of the entropy of the ions as the surfaces are brought closer. Whereas ions of charge opposite to those of the surface are attracted to it, they are repelled from each other, and the increased proximity induced by decreas- ing the film thickness increases the free energy. Note the relatively complex d -dependence of F$_{elec}$ ( d ) which is due to the fact that κ − 1 is also a function of d . Thus, the richness of the role of impurities in the premelting of ice is a consequence of this and the efficient segregation of ions into the film; since nearly all of the impurities remain in the liquid, as the temperature changes the amplitude and range of the repulsive electrostatic interaction varies [5]. In principle, the situation could be even more com- plex when for example q$_{s}$ = q$_{s}$ ( d ). Indeed, when d is small the Debye-H¨ uckel expression is less accurate and steric effects become more important. However, to intro- duce such complications here exceeds our understanding of the situation based on present experimental evidence. and hence a calculation analogous to that in the Sec- tion II A yields the result or, for the general case of a surface with principal radii of curvature r$_{1}$ and r$_{2}$ , Using these expressions for I ( d ) = F$_{dis}$ ( d )+ F$_{elec}$ ( d ) in Eq. (3) we obtain an implicit equation for the thickness d of the premelted layer as a function of the undercooling Δ T : ∂G ∂N$_{s}$ = ∂G ∂N$_{l}$ , (10) q$_{m}$ Δ T T$_{m}$ − R$_{g}$T$_{m}$ ρ$_{i}$ ρ$_{l}$ = 2 γ$_{sl}$ ρ$_{s}$r , (11) q$_{m}$ Δ T T$_{m}$ − R$_{g}$T$_{m}$ ρ$_{i}$ ρ$_{l}$ = γ$_{sl}$ ρ$_{s}$ ( 1 r$_{1}$ + 1 r$_{2}$ ) . (12) ρ$_{l}$q$_{m}$ Δ T T$_{m}$ = R$_{g}$T$_{m}$N$_{i}$ d − A$_{H}$ 6 πd $_{3}$+ q 2 s ϵϵ$_{0}$ [ $^{1}$− 1 $_{c}$√$_{N}$$_{i}$$_{d}$ ] e − $_{c}$√$_{N}$$_{i}$$_{d}$ , (7) c = √ e $^{2}$N$_{A}$ ϵϵ$_{0}$k$_{B}$T$_{m}$ . (8) This result is the well known Gibbs-Thomson effect, i.e. the freezing point depression due to convexity of the ice- liquid interface, calculated here for the case where im- purities are present in the liquid water. The larger the undercooling Δ T , the smaller the radii of curvature of an ice-liquid interface in equilibrium. At the bulk freezing point, Δ T ↘ R$_{g}$T 2 $_{m}$ρ$_{i}$ /q$_{m}$ρ$_{l}$ , the radius of curvature di- verges and a planar interface is the stable configuration. where the constant c is This result is analyzed in detail in Section III A. We use the random close packing (RCP) of hard spheres as a model for the porous matrix. Apart from choosing the matrix material and hence calculating the properties which enter the determination of the Hamaker constant (see Section II A), the only free parameter char- acterizing the porous matrix which contains the partially premelted ice is the radius R of the hard spheres. The properties of RCP hard spheres which are required for our calculation are the sphere packing fraction η and the av- erage coordination number C . Recent estimates of these numbers [7] are η = 0 . 634 and C = 6. It should be noted, however, that the basic concept of RCP is not well de- fined mathematically, so these numbers vary slightly ac- cording to the protocol and algorithm used to determine the exact configuration [8]. In order to assess the effect of the ice-liquid interfacial curvature on the freezing temperature, we neglect the known anisotropy of the surface energy of ice [1] and consider a spherical ice crystal of radius r surrounded by undercooled water. As opposed to the planar case, the interfacial area is no longer a constant as it was in the treatment of Section II A. With this new complication we reconsider the Gibbs free energy G of the total system. The same holds for the numbers of moles N$_{s}$ , N$_{l}$ , and N$_{i}$ which are now taken for the total system and not per unit surface. For ease of notation we use the same symbols for these new quantities. All other quantities are as defined above. Whence, we have At a given undercooling Δ T we are interested in the fraction of the interstitial volume which is occupied by quasi-liquid water f$_{l}$ which we will more compactly refer to as the liquid fraction. The first contribution we con- sider comes from interfacial premelting at the surface of the porous matrix. In the limit where the premelted layer is much thinner than the radius of the matrix particles, d ≪ R , this fraction is well approximated by C. Premelting in a porous medium B. Curvature induced melting: The Gibbs-Thomson Effect f mat l (Δ T, ρ$_{i}$ ) = a$_{mat}$ $^{1}$− η × d (Δ T, ρ$_{i}$ ; A mat H , q mat s ) , (13) G ( T, P, N$_{s}$, N$_{l}$, N$_{i}$ ) = µ$_{s}$ ( T, P ) N$_{s}$ + µ$_{l}$ ( T, P ) N$_{l}$ + µ$_{i}$ ( T, P ) N$_{i}$ + R$_{g}$T N$_{i}$ [ ln N$_{i}$ N$_{l}$ − 1 ] + 4 πr $^{2}$γ$_{sl}$ , (9) where r is related to N$_{s}$ through N$_{s}$ = 4 π 3 r $^{3}$ρ$_{s}$ with the constant molar density ρ$_{s}$ . We note here that we under- stand that γ$_{sl}$ = γ$_{sl}$ ( r, N$_{i}$, φ ), where φ is the angle that