8 0.01 0.1 1 10 T [K] 0.1 1 10 d [nm] q$_{s}$ = 0.0; N$_{i}$ = 0.5 q$_{s}$ = 0.0; N$_{i}$ = 2.0 q$_{s}$ = 0.5; N$_{i}$ = 0.5 q$_{s}$ = 0.5; N$_{i}$ = 2.0 [q$_{s}$] = C/m$^{2}$; [N$_{i}$] =  mol/m 2 0.01 0.1 1 10 T [K] 0.1 1 10 d [nm] q$_{s}$ = 0.0; N$_{i}$ = 0.5 q$_{s}$ = 0.0; N$_{i}$ = 2.0 q$_{s}$ = 0.5; N$_{i}$ = 0.5 q$_{s}$ = 0.5; N$_{i}$ = 2.0 [q$_{s}$] = C/m$^{2}$; [N$_{i}$] =  mol/m 2 FIG. 5: Thickness of the quasi-liquid layer for the ice/liquid/gold system ( A H = 1 . 573 × 10 $_{−}$21 J). In contrast to the ice/liquid/fused quartz system, Fig. 4, layer collapse occurs at thicknesses above 0 . 1 nm even for the cases with- out surface charges. This is due to the stronger attractive dispersion forces in the ice/liquid/gold system. Layer col- lapse for the cases with surface charges is altered only for the higher impurity density N i = 2 . 0 µ mol/m 2 relative to the ice/liquid/fused quartz system, because at d ≈ 1 nm disper- sion forces are weak compared to the electrostatic force. FIG. 4: Thickness d of the quasi-liquid layer for the ice/liquid/fused quartz system ( A H = 3 . 0 × 10 $_{−}$23 J) as a function of the undercooling Δ T . For moderate undercool- ing the system is dominated by the colligative effect. In this regime the layer thickness increases with impurity density N i . At intermediate undercooling the effect of immobile surface charges leads to a thickening of the quasi-liquid layer which is most effective for low impurity concentrations leading to a decrease of the layer thickness when the impurity density is increased. At large undercooling surface charge induced layer collapse is observed, while layer collapse due to dispersion forces alone occurs at layer thicknesses below 0 . 1 nm. Repulsive dispersion forces The result of the analysis following Eq. (29) can be tested on the ice/liquid/silicon system which has a Hamaker constant of A$_{H}$ = − 1 . $^{66}$× 10 − 21 J [4], the negative sign implying repulsive disper- sion forces. As shown in Fig. 6 the system displays the expected colligative behavior for low undercooling and an enhanced layer thickness for intermediate un- dercooling where the effect of surface charges enhances the colligative effect, which alone would yield a straight line, d ∝ N − 1 i . For intermediate undercooling the fig- ure reveals a transition from a regime of low impurity density where the layer thickness decreases when N$_{i}$ increases , to a regime of high impurity density where the colligative effect dominates and accordingly an in- crease of d is observed when N$_{i}$ increases . From the fig- ure the locus of the transition can be estimated to be Δ T$_{c}$ ≈ 10 − 20 K, d$_{c}$ ≈ 1 nm, and N$_{i,c}$ ≈ 2 . 5 µ mol/m $^{2}$. This is in good agreement with the analytical result ob- tained from Eq. (29), Δ T$_{c}$ = 9 . 6 K, d$_{c}$ = 1 . 25 nm, and N$_{i,c}$ = 3 . 97 µ mol/m $^{2}$. It should be noted that the an- alytical result is exact only in the limit q$_{s}$ → 0 but the complete solution of Eq. (22) shows that it applies quali- tatively and semi-quantitatively for surface charge densi- ties as high as q$_{s}$ = 0 . 5 C/m $^{2}$. For the ice/liquid/silicon system no layer collapse is observed. This results from the fact that dispersion forces, which dominate at low layer thicknesses, are repulsive and stabilize the quasi- liquid layer. However, at impurity densities N$_{i}$ smaller than those considered in Fig. 6, the Debye-H¨ uckel ex- pression for F$_{elec}$ generates an attractive force in the ice/liquid/silicon system with q$_{s}$ = 0 . 5 C/m 2 at suffi- the existence of an interval of intermediate undercooling where the layer thickness decreases when the impurity density increases. Recall that the reason for this is that at low impurity concentrations the electrostatic force is less effectively screened and hence the repulsion due to surface charges has a stronger effect, leading to a larger layer thickness than for higher impurity concentrations. When the system is undercooled further the collapse of the quasi-liquid layer can be observed in the figure. Here, because of the weakness of dispersion forces, the collapse is a consequence of the electrostatic force becoming at- tractive at low layer thicknesses. Collapse for the cases without surface charges cannot be observed in Fig. 4 as it occurs at thicknesses below 0 . 1 nm (c.f., Fig. 2). In order to see layer collapse at larger layer thicknesses a larger Hamaker constant is required, as provided by the ice-liquid-gold system. To see this, we plot solutions of Eq. (22) for this system in Fig. 5 for the same pa- rameters as considered in Fig. 4 for the ice-liquid-fused quartz system. Layer collapse is now observed at thick- nesses above 0 . 1 nm even for the configurations with- out surface charges. For the configurations with surface charges we observe that for N$_{i}$ = 0 . 5 µ mol/m 2 layer col- lapse is not altered by the stronger dispersion forces while for N$_{i}$ = 2 . 0 µ mol/m 2 the stronger attractive dispersion forces lead to a layer collapse at a larger layer thickness compared to the corresponding configuration considered in Fig. 4. Thus, in this case stronger dispersion forces drive layer collapse.