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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 

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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 

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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 : 

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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: 

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with r (Δ T, ρ$_{i}$ ) from Eq. (16). 

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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. 

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) 

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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 

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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 . 

## D. Implicit equation for the total liquid fraction 

Thus far we have taken impurities into account in terms of a molar density ρ$_{i}$ or, in the case of the pre- 

