content_image/418992.jpg content_image/418991.jpg content_image/418987.jpg content_image/418998.jpg content_image/418999.jpg
and hence we recover the result 

for the case q = 0 which we derived above. On the other hand, it can be shown that in the limiting case N$_{i}$ → 0 the values d$_{min}$ and t$_{max}$ follow from the surface charge dominated limit ( b = 0 and a = 0) which has 

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In Fig. 2 we plot the minimal thickness d$_{min}$ of the quasi-liquid layer corresponding to the maximal under- cooling t$_{max}$ as a function of the impurity concentration N$_{i}$ for different values of the surface charge q$_{s}$ . We use a Hamaker constant A$_{H}$ = 3 . 0 × 10 − 23 J which is the re- sult obtained by Wilen et al. [4] for the ice/liquid/fused quartz layered system. The analytical results for N$_{i}$ → 0 and N$_{i}$ → ∞ are characterized by the N$_{i}$ -dependencies of the asymptotic behavior. Note that the minimal thick- ness in the dispersion force dominated limit ( q$_{s}$ = 0) is below 0 . 1 nm and therefore equal to the dimension of a single molecule. Hence, the notion of a quasi-liquid layer can no longer be applied and for the ice/liquid/fused quartz system layer collapse can only be observed exper- imentally at low impurity concentrations and it is always triggered by surface charges. This is seen directly from Eq. (22) where the contribution of the surface charge for sufficiently small d becomes negative , indicating attrac- tive forces, which suppress liquid layer formation. In or- der to observe a collapse of the quasi-liquid layer driven by dispersion forces, the latter have to be more strongly attractive. We have included in Fig. 2 the case of the ice-liquid-gold system which has a Hamaker constant of A$_{H}$ = 1 . $^{573}$×10 − 21 J [4], and hence dispersion forces are indeed strong enough to yield a liquid-layer collapse at thicknesses of the order of those calculated for the sur- face charge dominated case. The contrast between the two scenarios will be studied further in the second part of this section (Figs. 4 and 5). In Fig. 3 the corresponding results for the maximal undercooling Δ T$_{max}$ = T$_{m}$t$_{max}$ are plotted. As we expect from Eqs. (24) and (25), Δ T$_{max}$ increases with surface charge q$_{s}$ resulting in a q$_{s}$ -dependent offset for N$_{i}$ → 0. Moreover, for all q$_{s}$ , Δ T$_{max}$ increases with impurity concentration N$_{i}$ . A large Hamaker constant, for example of the ice/liquid/gold sys- tem, results in a low Δ T$_{max}$ as shown by Eq. (24), where Δ T$_{max}$ ∝ A − 1 / 2 H . There are interesting experimental con- sequences of Fig. 3. Obviously, measuring Δ T$_{max}$ for just one given impurity concentration N$_{i}$ is not sufficient to determine the surface charge q$_{s}$ at the interface. This follows from the fact that any two curves intersect, and hence measurements for at least two different impurity concentrations are required. 

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## 2. Layer thickness for a given undercooling 

When q = 0, Eq. (22) can still be solved analytically but because the resulting expression is lengthy and not particularly instructive we present the solution only in the small t limit of moderate undercooling, 

This result clearly illustrates the fact that the presence of attractive dispersion forces ( a > 0) acting across the system suppresses the formation of a quasi-liquid layer, whereas repulsive dispersion forces ( a < 0) tend to in- crease the thickness of the premelted layer. 

When surfaces charges on the interfaces of the quasi- liquid layer with the substrate/ice are involved, Eq. (22) becomes transcendental and has no analytical solution. In order to get some insight in the effect of surface charges, we consider several special cases. First, let us as- sume that for moderate undercooling the system is dom- inated by surface charges. This scenario corresponds to setting a = 0, b = 0 in Eq. (22) and considering only the leading contributions as d approaches infinity. The solution is then 

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