2 In the absence of interfacial forces we have I ′ ( d ) = 0, and the result reduces to Δ T = R$_{g}$T 2 m q$_{m}$ρ$_{l}$ ρ$_{i}$ , where the molar density of impurities is ρ$_{i}$ = N$_{i}$/d . This undercooling corresponds precisely to the classical colligative effect– the depression of the bulk freezing temperature due to dissolved impurities. Obviously, compared to the force free case, repulsive interfacial forces for which I ′ ( d ) < 0 result in a larger undercooling Δ T thereby enhancing the layer thickness. Conversely, attractive interfacial forces wherein I ′ ( d ) > 0 suppress the formation of a premelted liquid film. Indeed, for sufficiently large Δ T , Eq. (3) may have no solution showing that the attractive interfacial forces can dominate the colligative effect of the impurities and the premelted liquid layer will vanish. In order to calculate the equilibrium thickness d of a premelted layer for a given undercooling Δ T = T$_{m}$ − T , where T$_{m}$ is the bulk melting temperature of ice and T is the temperature of the system, we consider a three layer planar system. Layer I is given by the substrate, i.e., in the case of interfacial melting, the layer consists of the material which forms the porous medium, or of ice when grain boundary melting is considered. Layer II is the premelted quasi-liquid layer of thickness d and finally layer III consists of ice. In what follows we consider the complete free energy of the system and then focus on the individual complications and contributions in turn to show how known asymptotic results emerge from the theory. A. Interfacial melting and grain boundary melting: Planar interfaces II. THEORY 2. Contributions to the effective interfacial free energy I ( d ) 1. General theory in the presence of impurities The interfacial energy has two contributions; I ( d ) = F$_{dis}$ ( d ) + F$_{elec}$ ( d ), where F$_{dis}$ ( d ) is due to dispersion forces, acting between the three interfaces defining the system, and F$_{elec}$ ( d ) captures the effect of immobile charges at the interface. Although we compute the strength of the dispersion force interaction from the full frequency dependent theory [6], we take the small thick- ness limit in which only non-retarded forces contribute to F$_{dis}$ ( d ) which consequently can be written as It is assumed that layer II contains a certain amount of impurities. The Gibbs free energy of the system per unit area is given by where A$_{H}$ is the Hamaker constant for the given layered system [1, 4]. For symmetric systems, such as the ice- liquid-ice configuration, which occurs in the case of grain boundary melting, A$_{H}$ is positive and hence dispersion forces are attractive . At the interface between the ice and the porous matrix A$_{H}$ can assume both signs, depending on the material properties of the porous medium. F$_{dis}$ ( d ) = − A$_{H}$ 12 πd 2 , (4) G ( T, P, N$_{s}$, N$_{l}$, N$_{i}$ ) = µ$_{s}$ ( T, P ) N$_{s}$ + µ$_{l}$ ( T, P ) N$_{l}$ + µ$_{i}$ ( T, P ) N$_{i}$ + R$_{g}$T N$_{i}$ [ ln N$_{i}$ N$_{l}$ − 1 ] $^{+}$I( d ) , (1) Immobile surface charges with a given surface charge density q$_{s}$ on both interfaces which bound the quasi- liquid layer are screened by the counterions in the liquid. Making use of the Debye-H¨ uckel theory this leads to the contribution where T and P are temperature and pressure, µ$_{s}$ , µ$_{l}$ , µ$_{i}$ are the chemical potentials per mole of the solid, the liquid, and the impurity species, respectively, N$_{s}$ , N$_{l}$ , N$_{i}$ are the numbers of mole per unit area of the solid, the liquid, and the impurities, respectively. The term R$_{g}$T N$_{i}$ [ ln N$_{i}$ N$_{l}$ − 1 ] , where R$_{g}$ is the gas constant, corre- sponds to the mixing entropy of the impurities assuming low impurity concentration so that the ideal gas result applies. Finally, I ( d ) denotes the total effective interfa- cial free energy described in more detail below. Note that d is directly proportional to N$_{l}$ through N$_{l}$ = ρ$_{l}$d , where ρ$_{l}$ is the molar density of the liquid which is a constant in the calculation. The solid (ice) and the liquid (water) parts of the sys- tem can exchange material and hence the condition for thermodynamic equilibrium is given by where ϵ$_{0}$ is the vacuum permittivity, ϵ the relative per- mittivity of liquid water, and F$_{elec}$ ( d ) = 2 q 2 s κϵϵ$_{0}$ e − $^{κd}$, (5) κ − 1 = √ ϵϵ$_{0}$k$_{B}$T e $^{2}$N$_{A}$ ρ$_{i}$ = √ ϵϵ$_{0}$k$_{B}$T d e $^{2}$N$_{A}$ N$_{i}$ (6) ∂G ∂N$_{s}$ = ∂G ∂N$_{l}$ . (2) is the Debye length for monovalent ions, with the electron charge e , Avogadro’s number N$_{A}$ , and Boltzmann’s con- stant k$_{B}$ . The Debye length describes the characteristic decay of the ion field adjacent to the charged surface due to the screening by counter ions. A repulsive force be- tween two charged surfaces originates in the restriction Applying this criterion to the free energy given above, and using the fact that ∂µ$_{s,l}$/∂T ≡ − s$_{s,l}$ where s$_{s}$ ( s$_{l}$ ) is the specific entropy of the solid (liquid), we can approx- imate µ$_{l}$ − µ$_{s}$ ≈ q$_{m}$ Δ T T$_{m}$ , where q$_{m}$ is the latent heat of melting per mole and arrive at ρ$_{l}$q$_{m}$ Δ T T$_{m}$ = R$_{g}$T$_{m}$N$_{i}$ d − I ′ ( d ) . (3)