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ciently small layer thicknesses which results in a discon- tinuous drop in the layer thickness at a certain under- cooling. The layer, however, does not collapse to zero thickness but rather is stabilized at a smaller thickness due to the repulsive dispersion forces which dominate for very thin quasi-liquid layers. In this article we do not investigate this phenomenon of layer discontinuity which occurs in a regime where both the d -dependence of q$_{s}$ , the peculiarities of the Debye-H¨ uckel result, and other effects, have to be accounted for. 

## B. Volume fraction of premelted water 

Having analyzed premelting in planar layered systems, we now return to the implicit equation (Eq. 21) for the volume fraction of premelted liquid at a given undercool- ing t that we introduced in Section II D. In addition to being implicit for f$_{l}$ the equation uses the thickness of the premelted layer, which itself is obtained from an equation which does not have an analytical solution (Eq. 22). Ac- cordingly, we do not possess an analytical expression for f$_{l}$ in the general case. 

Low surface charge However, where immobile surface charges can be neglected ( q$_{s}$ = 0), Eq. (21) becomes more tractable. In the case where the matrix material has a 

negative Hamaker constant ( A mat H < 0), i.e., dispersion forces are repulsive and support the formation of a pre- melted layer, Eq. (21) becomes 

where 

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with ξ = γ$_{sl}$/ρ$_{s}$q$_{m}$ and where c$_{0}$ = R$_{g}$T$_{m}$ρ$_{0}$/q$_{m}$ρ$_{l}$ is, up to a constant factor, the density of impurities ρ$_{0}$ in the fully melted system. 

Obviously, Eq. (31) requires that t > c$_{0}$/f$_{l}$ must hold and thus, because 0 ≤ f$_{l}$ ≤ 1, when t < c$_{0}$ Eq. (31) is not applicable. The case t < c$_{0}$ corresponds to an undercooling which is smaller than the depression of the bulk freezing point due to the colligative effect originating from the presence of the impurities. Therefore, we must assume a liquid fraction f$_{l}$ = 1 when t < c$_{0}$ . 

While Eq. (31) does not posses a general analytical solution, it can be solved in the limit of low impurity concentration c$_{0}$ to yield 

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with ˜ t = A B t 5 / 3 where 

Note that the term ∝ c$_{0}$ is sufficient to obtain the cor- rect leading contribution due to impurities in both lim- iting cases t → 0 and t → ∞ . The factor ∝ c 2 0 becomes a constant in the limit t → 0 where g$_{2}$ vanishes. The coefficient belonging to c 3 0 vanishes as t − 7 / 3 in the limit t → ∞ and is thus subdominant to B t 2 . In the case where dispersion forces are attractive 

( A mat H > 0) the formation of a premelted layer requires an impurity concentration which is higher than the one which would be required to prevent the bulk liquid from freezing. Hence, in this case all the liquid water in the matrix pores must originate from liquid veins which form due to curvature induced premelting. In contrast to the premelted layer with A mat H > 0 these veins have formed when t > c$_{0}$/f$_{l}$ , a temperature for which the premelted layers are still collapsed and do not contribute to the liquid fraction. Thus, premelted layers do not actually form because at higher temperatures, beginning with the 

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