5 Δ T , the impurity concentration N$_{i}$ , the dispersion forces as embodied in the Hamaker constant A$_{H}$ , and the sur- face density q$_{s}$ of immobile charges on the solid-liquid interfaces. This expression cannot be solved analytically in the general case. Hence, presently we analyze Eq. (7) analytically, to demonstrate that we can obtain known distinguished limits and facilitate intuition, and numer- ically, to provide specific ranges of results for particular physical systems of interest. We begin by rewriting this equation in terms of the reduced, or dimensionless, un- dercooling t = Δ T/T$_{m}$ as follows melted layer, a surface density of impurities N$_{i}$ = ρ$_{i}$d . Laboratory probes using scattering or related approaches involve relatively small samples with good thermal con- trol and a well defined (fixed for a given run) dopant level [1]. The temperature changes in these systems are quasi-static so that the sample is at coexistence. In the natural environment there are of course temperature gra- dients. Therefore, for applicability of our work there, the typical broad temperature gradients found for example in natural soils imply that a local region within will be nearly isothermal and thus isosaline. Clearly, there will be many environmental circumstances where this may be violated, but because the first test of such a theory should be of high precision we focus mainly on the simplest ex- perimental setting and exploit the fact that the impu- rity segregation coefficient of ice is small [9] so that the premelted liquid is enriched/diluted when the system is cooled/warmed. Hence it is prudent to introduce ρ$_{0}$ , the molar density of impurities in the interstitial water when the system is above the bulk melting temperature. Thus, because of the validity of the assumption that impurities are contained almost exclusively in the liquid fraction of the interstitial water it follows that where the constants b = R$_{g}$T$_{m}$/ ( ρ$_{l}$q$_{m}$ ), a = A$_{H}$/ (6 πρ$_{l}$q$_{m}$ ), and q = q 2 $_{s}$/ ( ϵϵ$_{0}$ρ$_{l}$q$_{m}$ ). For attractive dis- persion forces which suppress premelting we have a > 0, and for repulsive dispersion forces which promote pre- melting we have a < 0. All other quantities in Eq. (22) are positive. t = bN$_{i}$ d − a d 3 + q ( $^{1}$− 1 $_{c}$√$_{N}$$_{i}$$_{d}$ ) e − $_{c}$√$_{N}$$_{i}$$_{d}$ , (22) ρ$_{i}$ = ρ$_{0}$ f$_{l}$ , (20) d = bN$_{i}$t − $^{1}$, (23) 1. Maximal undercooling and minimal layer thickness The simplest case occurs when both dispersion and electrostatic forces are absent, i.e., a = 0 and q = 0. The thickness d of the liquid film is them simply given by thereby properly accounting for the increase in the im- purity concentration in the premelted liquid as the liquid fraction decreases. Due to the fact that the regions of pre- melted water are all interconnected, ρ$_{i}$ is homogeneous over the entire sample. Having now considered the essential physical phenom- ena we can formulate an equation for the total liquid fraction f tot l in the porous medium as a function of the undercooling Δ T and the impurity concentration ρ$_{0}$ in the completely melted sample. Combining all the contri- butions discussed in Section II C, we obtain representing the colligative effect of the dissolved impuri- ties which, due to the small segregation coefficient of ice, are strongly enriched in the liquid layer. The situation immediately becomes more complex if we take disper- sion forces into account. Let us first consider materials that have a > 0, which means that dispersion forces sup- press the formation of a premelted liquid film. Assuming q = 0 for the moment, Eq. (22) possesses a solution as long as the undercooling t does not exceed a critical value t$_{max}$ = 2 3 ( bN$_{i}$ ) 3 / 2 √$_{3}$$_{a}$ corresponding to a minimum film thick- ness d$_{min}$ = √ 3 a $_{bN$_{i}$}$. If the temperature is reduced such that t > t$_{min}$ the attractive dispersion forces dominate the colligative effect and the liquid film collapses to zero thickness. which is an implicit function for f tot l = f tot l (Δ T, ρ$_{0}$ ) and in the general case has no analytical solution. Moreover, the underlying equation (7) for the thickness d of the premelted layer, which is required for f mat l and f gblayer l , is itself an implicit function of d . This calls for a care- ful stepwise analysis first of Eq. (7) and subsequently of Eq. (21), which we perform in Sections III A and III B. f tot l = f mat l (Δ T, ρ$_{0}$/f tot l ) + f gblayer l (Δ T, ρ$_{0}$/f tot l ) + f contact l (Δ T, ρ$_{0}$/f tot l ) + f gbmat l (Δ T, ρ$_{0}$/f tot l ) + f veins l (Δ T, ρ$_{0}$/f tot l ) (21) III. ANALYSIS AND APPLICATION A. Thickness of the premelted layer For the most general form of Eq. (22), the maximal un- dercooling t$_{max}$ and corresponding minimal layer thick- ness d$_{min}$ at which the quasi-liquid layer collapses cannot be expressed analytically. We obtain the global maxi- mum ( d$_{min}$, t$_{max}$ ) by extremizing the right hand side of Eq. (22) with respect to d . Two limiting cases can be distinguished. For large impurity concentrations N$_{i}$ the contribution due to surface charges becomes negligible In Section II A we derived Eq. (7) for the thickness d of the premelted layer as a function of the undercooling