3. Temperature-dependent structures of ice surfaces

Growth rates of crystal surfaces strongly depend on their microscopic structures[16, 17]. In the case of snow crystals, they grow in the temperature conditions near the melting temperature(273.15K). It is well known that the ice surfaces are covered with the thin liquid like layer (we call it the quasi-liquid layer(qll)) in this temperature range. This problem was first considered by Faraday[18] in 1850, and has been studied intensely over the past ten years[19]. This is a kind of the structural phase transition occurring on the crystal surface, which is called the surface melting transition and is not peculiar to ice crystals. This phenomenon generally occurs on the many kinds of crystals. In this decade, many theoretical and experimental studies have been published to clarify the structures and the physical properties of surfaces that undergo surface melting[20,21,22].

Here, let us consider the thermodynamic aspect about the surface melting very briefly[23,24]. The existence of a qll on the surface below the melting temperature is disadvantageous in the sense of the free energy of bulk liquid phase. Namely, the qll can exist on the surface when the wettability parameter, Ё , defined by the equation, i|(w{i/w), is positive for the ice/water system: wherei is the surface tension (specific surface free energy) of ice without a qll, w the surface tension of water and i/w the interfacial tension between ice and water.

Though the existence of a qll below the melting temperature is disadvantageous in the sense of bulk free energy of the liquid phase, it rather becomes an advantage in the sense of surface free energy (namely, iw+i/w) [23,24]. Since the total free energy of the surface system covered with the qll is given as a sum of the surface free energy and the bulk free energy, an equilibrium thickness of the layer may be determined by minimization of the total free energy of the surface system. The thickness is very large at the melting temperature and decreases with falling the temperature, because increasing the instability of the bulk liquid phase (the qll) in comparison with the stable phase of bulk ice. As the thickness becomes equal to one of a monomolecular layer of ice at a certain temperature TI/II, the model of surface with the qll should be valid only for temperatures TTI/II. However, the layer thickness, being thinner than that of a monolayer at temperatures below TI/II, would be regarded as the ice surface which is geometrically irregular. This may correspond to the case of the so-called roughening transition, in which the surface may be uneven on a molecular level[19]. The degree of irregularity may decrease with falling temperature and become very small at another particular temperature TII/III, which is the so-called roughening temperature. Below TII/III, the surface is expected to be singular or smooth with weak adsorption of water molecules, which is similar to the low-temperature crystal/vapor interface of other materials. We should emphasize again that, on a molecular scale, ice surface structures depend sensitively on the temperature. This provides a qualitative description of the temperature dependence of ice surface structures near the melting point, and an ice crystal is one of the best materials in which to observe these interesting phenomena.

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