Lifshitz interaction can promote ice growth at water-silica interfaces

At air-water interfaces, the Lifshitz interaction by itself does not promote ice growth. On the contrary, we find that the Lifshitz force promotes the growth of an ice film, up to 1–8 nm thickness, near silica-water interfaces at the triple point of water. This is achieved in a system where the combined effect of the retardation and the zero frequency mode influences the short-range interactions at low temperatures, contrary to common understanding. Cancellation between the positive and negative contributions in the Lifshitz spectral function is reversed in silica with high porosity. Our results provide a model for how water freezes on glass and other surfaces.

Figure 1

Ice $\left({\varepsilon }_2\right)$ of thickness $L$ at the interface of water $\left({\varepsilon }_3\right)$ and silica $\left({\varepsilon }_1\right)$, illustrated here as three planar regions of infinite extent.

Figure 2

Permittivity as a function of frequency for ice, water, and different silica materials. The static values $\varepsilon \left(0\right)$ for ice and water are 91.5 and 88.2, respectively, using data from Elbaum and Schick [23]. For different ${\mathrm{SiO}}_2$ materials, the static values are 3.90, 2.62, and 1.69 using data from Malyi et al. [24] for volumes 44.53, 68.82, and $141.87{\AA }^3$, respectively (here extended to include phonon contributions), 3.80 from Grabbe [25], and 3.90 from data set 1 and data set 2 of van Zwol and Palasantzas [26]. The transition frequency, ${\zeta }_a\approx 1.60\times {10}^{16}$ rad/s, is where the permittivities of ice and water cross in the optical frequency region.

Figure 3

Spectral function $g(L,i{\zeta }_n)$ as a function of Matsubara frequency $\left({\zeta }_n\right)$ for a silica-ice-water system with $L=2.0$ nm thick ice film. We compare the result using different silica dielectric functions presented in Fig. 2. The zero frequency contributions for various silica materials (not shown in the figure) are at least one order of magnitude higher than the contributions from other Matsubara frequencies $\left(\approx -300\mathrm{nJ}/{\mathrm{m}}^2\right)$. All the curves vanish at the same point, corresponding to the transition frequency ${\zeta }_a\approx 1.60\times {10}^{16}$ rad/s.

Figure 4

Contributions to the Lifshitz free energy per unit area for a silica-ice-water $(V=68.82$ ${\AA }^3)$ system as a function of ice film thickness. Four different curves are shown: the total nonretarded energy, the contributions from the $n>0$ term to the retarded energy, the total retarded energy, and the contribution from the $n=0$ term alone.

Figure 5

Lifshitz free energy per unit area for silica-ice-water systems as a function of the ice film thickness using different silica dielectric functions presented in Fig. 2.