4 H H ice mat li matrix A > 0 ice li mat ice matrix A < 0 r liquid FIG. 1: Sketch of the liquid pocket which forms between two matrix spheres of radius R . The amount of premelted liquid depends on the characteristic radius of curvature r of the ice- liquid interface as obtained from Eq. (16). Depending on the matrix material there is either always a premelted liquid layer (repulsive dispersion forces, A H < 0) or the premelted layer may collapse under certain conditions (attractive dispersion forces, A H > 0) and the liquid-ice interface forms an angle θ < π/ 2 with the matrix surface. For parsimony of presentation we show the neighboring matrix spheres of different materials while the present theory assumes porous media made up of a single material. where a$_{mat}$ is the surface area of the matrix particles per unit volume and d is obtained from Eq. (7) using the Hamaker constant A mat H for the ice-water-matrix lay- ered system and the appropriate surface charge density q mat s . The sphere number density of the matrix is n$_{s}$ = η/ (4 πR $^{3}$/ 3) and thus we obtain a$_{mat}$ = 4 πR $^{2}$n$_{s}$ = 3 η/R and The second contribution to the liquid fraction is due to grain boundary melting. Its importance depends sen- sitively on the grain size of the interstitial ice, which is a function of such factors as (1) the rate of cooling, (2) the undercooling Δ T , and (3) the time allowed for annealing. Even were we to attempt a complete computation of the coarsening rate of the ice grains in the matrix, there is no unique way to specify the initial value problem and hence to arrive at a concrete grain size as a function of the properties of the porous matrix. Thus, we introduce as an additional parameter the grain boundary density ρ$_{gb}$ , which is the surface area of grain boundaries per unit volume of the interstitial ice. We denote ˜ ρ$_{gb}$ = ρ$_{gb}$R as the related dimensionless density. The corresponding liquid fraction follows immediately A further curvature induced contribution to the liquid fraction brought out by CDF comes from the lines where a grain boundary and a matrix sphere meet. f mat l (Δ T, ρ$_{i}$ ) = 3 η $^{1}$− η × d (Δ T, ρ$_{i}$ ; A mat H , q mat s ) R − 1 . (14) f gblayer l (Δ T, ρ$_{i}$ ) = ˜ ρ$_{gb}$ × d (Δ T, ρ$_{i}$ ; A gb H , q gb s ) R − 1 , (15) f gbmat l (Δ T, ρ$_{i}$ ) = 3 ( $^{2}$− π 2 8 ) × η $^{1}$− η × ˜ ρ$_{gb}$ × [ r (Δ T, ρ$_{i}$ ) R ] 2 . (18) In the limit that the matrix particles are small on the scale of the linear scale of the ice grains, but that R is much larger than r (Δ T, ρ$_{i}$ ) the corresponding liquid fraction can be calculated using elementary geometry (see Appendix A) where A gb H is the Hamaker constant for the ice-liquid-ice layered system and q gb s is the surface charge density for the grain boundary. The thickness d of the quasi-liquid layer is again obtained from Eq. (7). We now deal with curvature induced premelting. Im- portant contributions to the liquid water fraction come from regions where the ice-liquid interface has a large curvature. In our idealized matrix these are the points where two matrix spheres are in contact. Following CDF [2] there is a liquid pocket forming at the contact point with a volume 2 πr $^{2}$R to lowest order in r , which is the radius of curvature of the ice-liquid interface (Fig. 1). Taking into account that the second principal radius of curvature is much larger than r , we can use Eq. (12) to relate r to the undercooling Δ T : Finally, curvature induced premelting gives rise to liq- uid veins along the lines where several grain boundaries meet. With the simplifying assumption that the ice grains are arranged in a cubic packing, the total length of veins per interstitial volume, or the density of veins ρ$_{veins}$ , is obtained from ρ$_{gb}$ as ρ$_{veins}$ = ρ 2 $_{gb}$/ 3. Given that the cross-sectional area of a vein in cubic packing is (4 − π ) r $^{2}$, we can calculate the corresponding contribu- tion to the liquid fraction as Using the number density n$_{s}$ of the sphere matrix and the matrix coordination number C we can express the corresponding fraction of liquid water with respect to the total interstitial volume: Therefore, the total liquid fraction is obtained as the sum of the five individual contributions which we have derived above; f mat l , f gblayer l , f contact l , f gbmat l , and f veins l . Thus far we have taken impurities into account in terms of a molar density ρ$_{i}$ or, in the case of the pre- with r (Δ T, ρ$_{i}$ ) from Eq. (16). f veins l (Δ T, ρ$_{i}$ ) = ρ$_{veins}$ × (4 − π $^{)}$× [ r (Δ T, ρ$_{i}$ )] 2 = 1 3 $^{(4}$− π $^{)}$× ˜ ρ 2 gb × ( r (Δ T, ρ$_{i}$ ) R ) 2 . (19) r (Δ T, ρ$_{i}$ ) = γ$_{sl}$ ρ$_{s}$ ( q$_{m}$ Δ T T$_{m}$ − R$_{g}$T$_{m}$ ρ$_{i}$ ρ$_{l}$ ) − 1 . (16) f contact l (Δ T, ρ$_{i}$ ) = C / $^{2}$× n$_{s}$ × 2 π [ r (Δ T, ρ$_{i}$ )] $^{2}$R $^{1}$− η = 3 2 × η $^{1}$− η × C 2 × [ r (Δ T, ρ$_{i}$ ) R ] 2 , (17) D. Implicit equation for the total liquid fraction