14 Now consider a random close packing with packing fraction η (the porous medium), which contains ice with two different crystallographic orientations being sepa- rated by a planar grain boundary. The volume fraction η is also the surface fraction of the grain boundary which is cut out in the form of circles with radii between 0 and R due to the presence of the matrix spheres. Given that the average surface area per intersecting sphere is R we can consider the matrix surface as being planar in the vicinity of the vein as shown in Fig. 10b. The angle between the grain boundary and the matrix sphere is Using elementary geometry we obtain the surface areas a sample of volume V$_{s}$ = A$_{s}$x$_{s}$ with a grain boundary perpendicular to the x -axis contains ηA$_{s}$/ ¯ a intersection circles of matrix spheres with the grain boundary, if A$_{s}$ is sufficiently large. If the sample contains the total grain boundary surface ρ ′ $_{gb}$V$_{s}$ this corresponds to a net surface A$_{s}$ = ρ ′ $_{gb}$V$_{s}$ / (1 − η ) and hence a density of intersection circles Accordingly, the volume of premelted water for the con- figuration in Fig. 10 is 2 πhA$_{tot}$ . Taking all possible dis- tances of the matrix sphere to the grain boundary into account yields an average volume ¯ v of premelted liquid as and the total surface area ¯ a = 1 R ∫ R 0 dzπ ( R 2 − z $^{2}$) = 2 3 πR 2 (A6) α = arccos ( z R ) . (A1) A = cot ( α 2 ) r 2 − ( π 2 − α 2 ) r 2 , (A2) α α A ′ = tan 2 r 2 − 2 r 2 (A3) A$_{tot}$ = A + A ′ = ( 2 sin α − π 2 ) r 2 . (A4) n$_{c}$ = ηρ ′ gb $^{(1}$− η )¯ a = ηρ$_{gb}$ ¯ a , (A7) ¯ vn$_{c}$ $^{1}$− η = 3 ( $^{2}$− π 2 8 ) × η $^{1}$− η × ˜ ρ$_{gb}$ × ( r R ) 2 . (A8) ¯ v = 1 R ∫ R 0 dz 2 πhA$_{tot}$ = 1 R R 0 dz 2 πR sin α ( 2 sin α − π 2 ) r 2 = 2 πr 2 ∫ R 0 dz ( $^{2}$− π 2 √ $^{1}$− z 2 R 2 = 2 πr $^{2}$R ( $^{2}$− π 2 8 ) . (A5) where ρ$_{gb}$ is the grain boundary density with respect to the interstitial volume. The volume of premelted liquid along the intersection circles (per unit volume of the sam- ple) is ¯ vn$_{c}$ . Or, relative to the interstitial volume: (2008). [8] S. Torquato, T.M. Truskett, and P.G. Debenedetti, Phys. Rev. Lett. 84 , 2064 (2000). [9] G.W. Gross, A. Gutjahr, and K. Caylor, J. Phys. (Paris) 48 , 527 (1987). [10] P.M. Dove and C.M. Craven, Geochim. Cosmochim. Acta 69 , 4963 (2005). [11] M. Elbaum and M. Schick, Phys. Rev. Lett. 66 , 1713 (1991). [12] L. Benatov and J.S. Wettlaufer, Phys. Rev. E 70 , 061606 (2004). [1] J.G. Dash, A.W. Rempel, and J.S. Wettlaufer, Rev. Mod. Phys. 78 , 695 (2006). [2] J.W. Cahn, J.G. Dash, and H. Fu, J. Cryst. Growth 123 , 101 (1992). [3] S.S.L. Peppin, A. Majumdar, and J.S. Wettlaufer, Proc. R. Soc. A, 466 , 177 (2010) [4] L.A. Wilen, J.S. Wettlaufer, M. Elbaum, and M. Schick, Phys. Rev. B 52 , 12426 (1995). [5] J.S. Wettlaufer, Phys. Rev. Lett. 82 , 2516 (1999). [6] I.E. Dzyaloshinskii, E.M. Lifshitz, and L.P. Pitaevskii, Adv. Phys. 10 , 165 (1961). [7] C. Song, P. Wang, and H.A. Makse, Nature 453 , 629