0 0.5 1 1.5 2 N$_{i}$ [ mol/m$^{2}$] 0 20 40 60 T max  [K] q$_{s}$ = 0 q$_{s}$ = 0.1 q$_{s}$ = 0.25 q$_{s}$ = 0.5 q$_{s}$ = 0.75 ice-liquid-fused quartz system ice-liquid-gold system, with q$_{s}$ = 0 [q$_{s}$] = C/m 2 layer while for t > t$_{0}$ surface charges suppress layer for- mation. Considering that Eq. (27) applies to the t → 0 case while Eq. (28) to the t → ∞ case, this explains the essential difference between Eqs. (27) and (28) in terms of whether they support or suppress the formation of a liquid film. In order to isolate the dependence of d on the impu- rity concentration N$_{i}$ we consider Eq. (22) in the case where dispersion forces can be neglected ( a = 0) and the immobile interfacial charge density is small (i.e. q → 0). To lowest order d depends linearly on q , and hence we obtain From this result the sign of ∂d/∂N$_{i}$ can be shown to equal the sign of 7 d ( t, N$_{i}$ ; a = 0 , q → 0) = bN$_{i}$ t + bq t 2 ( N$_{i}$ − √$_{t}$ $_{c}$√$_{b}$ ) × e − cN$_{i}$ √ b/t $^{+}$O( q $^{2}$) . (29) S ≡ α $^{2}$(2 − α ) + e $^{α}$/β , (30) FIG. 3: The maximal undercooling Δ T max as a function of the impurity concentration N i for the same parameters as used for the calculation of d min in Fig. 2. The zero slope asymptote for N i → 0 is observed as expected from Eq. (25). The large Hamaker constant of the ice-liquid-gold system leads to low values of Δ T max . where α = cN$_{i}$ √ b/t and β = q/ ( bc $^{2}$N2 i ). We find S > 0 for all α > 0 as long as β < β$_{c}$ = e $^{4}$/ 32 ≃ 1 . 71. There- fore if N$_{i}$ > q/ ( β$_{c}$bc $^{2}$) the layer thickness d increases with N$_{i}$ for any given undercooling t . For lower impu- rity concentrations ( β > β$_{c}$ ) there is a finite interval for α which contains α$_{c}$ = 4, where S < 0. Therefore, at sufficiently low impurity concentration there exists an in- terval of intermediate undercooling t where d decreases when N$_{i}$ increases. This result implies that for large un- dercooling t the layer thickness d must always increase with N$_{i}$ which was precisely a conclusion obtained from Eq. (28). However, the result cannot be applied to the charge dominated case, Eq. (27), because the correspond- ing limit b → 0 renders Eq. (29) trivial. It is possible though to obtain S < 0 in the related limit b → 0, t → 0, if we assume that b/t approaches a large constant such that α > 2 holds. This result reveals an interesting consequence of the pres- ence of charges at the interface. Under appropriate con- ditions the layer thickness d decreases when the impurity concentration increases . This observation is contrary to the naive expectation based on the colligative effect and originates in the fact that for larger N$_{i}$ the electrostatic repulsion is screened for effectively [5]. For repulsive dispersion forces ( a < 0) the solution of Eq. (22) can be obtained asymptotically in the limit of large undercooling t → ∞ , whereas for attractive dis- persion forces ( a > 0) the premelted layer collapses to zero thickness for t larger than some finite value. The repulsive case is and seems to contradict Eq. (27) for two reasons. First, the negative sign of the term proportional to q in Eq. (28) indicates that immobile surface charges impede the for- mation of the premelted layer, whereas in Eq. (27) im- mobile surface charges alone are sufficient to stabilize a premelted layer. Second, while Eq. (27) implies that d de- creases when the impurity concentration N$_{i}$ is increased, the opposite conclusion follows from Eq. (28) where, due to the term ∝ q , the inhibition due to surfaces charges is reduced when N$_{i}$ increases. d ( t, N$_{i}$ ; a, q ) = $^{(}$− a ) 1 / 3 t 1 / 3 + bN$_{i}$ 3 t − $^{(}$− a ) 1 / $^{6}$q 3 $_{c}$√$_{N}$$_{i}$$_{t}$ 7 / 6 $^{+}$O ( 1 t 4 / 3 ) (28) Attractive dispersion forces The non-monotonic de- pendence of d on N$_{i}$ is illustrated in Fig. 4 where the full numerical solution of Eq. (22) is plotted for the case of the ice-liquid-fused quartz layered system. At low under- cooling Δ T the system displays characteristic colligative behavior as described by Eq. (23). The effect of immo- bile surface charges vanishes in the limit t → 0 as in- dicated by Eq. (29) and the layer thickness d increases when the impurity density N$_{i}$ increases. The layer thick- ness for small Δ T is obtained by requiring that the im- purity concentration ρ$_{i}$ = N$_{i}$/d equals the impurity con- centration which would be necessary in a bulk system to see a freezing point depression equal to the undercooling. When Δ T is increased, surfaces charges start to increase the layer thickness relative to the system without sur- face charges ( q$_{s}$ = 0). Interestingly, the resulting layer thickness for the system with the lower impurity density ( N$_{i}$ = 0 . 5 µ mol/m $^{2}$) becomes even larger than for the system with more impurities ( N$_{i}$ = 2 . 0 µ mol/m $^{2}$). This illustrates the result from the above analysis displaying The apparent contradiction can be reconciled by the following observations. A direct consequence of Eq. (22) is that immobile surface charges do not contribute to the film thickness d when the latter equals the Debye length, namely d$_{0}$ = 1 c 2 $_{N$_{i}$}$, in which case the surface charge term vanishes. The film thickness d$_{0}$ translates into an under- cooling t$_{0}$ = bc $^{2}$N2 i − ac $^{6}$N3 i . For an undercooling t < t$_{0}$ surface charges enhance the formation of a quasi-liquid