## 5 Heat Transfer across Liquid-Solid Interfaces
### Evolution of Thermal Contact Conductance \(G\) during the Pre-Melting Process
This value exceeds the ITC associated with the gas thermal conduction at \(T_{int}\) about 433 K, where \(T_{int}\) is the temperature intermediate between \(T_{S}=430\) K and \(T_{m}=435\) K. The interfacial thermal conductance \(G_{LS}=5\cdot 10^{4}\) W/m\({}^{2}\)K can be represented as the sum of \(G_{g}(d)\) and \(G_{nf}(d)\), where \(G_{nf}(d)\) is about \(10^{4}\) W/m\({}^{2}\)K at \(d~{}=~{}17\)-50 nm, and \(G_{g}(d)\) is about \(4\cdot 10^{4}\) W/m\({}^{2}\)K at \(d~{}\leq 50\) nm, \(\sigma=0.6\), \(p=10^{5}\) Pa, \(T_{int}~{}=~{}433\) K, \(\delta=0.41\) um, and \(\varkappa_{g}(T_{int})=0.035\) W/m\(\cdot\)K [58]; see Equations (13) and (15). ITC parameters at the indium-membrane interface for the beginning and end of the pre-melting temperature range are collected in Table 4.  
Thus, in the process of pre-melting of pure indium, the distance \(d\) decreases from about ten micrometers to tens of nanometers, while the ITC increases by an order of magnitude. Note that at the beginning of the pre-melting process, the distance \(d\) significantly exceeds the membrane surface roughness \(R_{a}=0.2~{}\upmu\)m. Therefore, the surface of the sample may be in a pasty state rather than in a liquid state. The sample then gradually becomes liquid as the temperature approaches the melting point, at which the distance \(d\) becomes much smaller than the membrane surface roughness. Similar measurements were conducted during the melting of microdroplets of an aluminum alloy (AA7075) in contact with the membrane of the calorimetric sensor [25]. Using ultrafast membrane nanocalorimetry combined with high-speed IR thermography, the ITC was measured at heating-cooling rates ranging from \(10^{3}\) K/s to \(10^{5}\) K/s. The measurements were reproducible during many subsequent heating-cooling scans. A gradual increase in the interfacial thermal conductance by an order of magnitude with increasing temperature was found in the range from the solidus temperature \(T_{Sol}=805\) K to the liquidus temperature \(T_{Lig}=901\) K of the alloy. The ITC at the solidus temperature was about \(G_{pS}~{}=~{}3\cdot 10^{4}\) W/m\({}^{2}\)K. Then, as the sample was heated, the ITC gradually increased from \(3\cdot 10^{4}\) W/m\({}^{2}\)K at 805 K to \(G_{LS}=1.8\cdot 10^{5}\) W/m\({}^{2}\)K at the liquidus temperature \(T_{Lig}~{}=~{}901\) K [25]. After that, the melting process occurred at \(T_{Lig}\) with \(G_{LS}~{}=~{}1.8\cdot 10^{5}\) W/m\({}^{2}\)K. The pre-melting process occurring between \(T_{Sol}\) and \(T_{Lig}\) was volumetric with enthalpy consumption as shown by direct calorimetric measurements [25]. Near the solidus temperature \(T_{Sol}=805\) K, \(G_{pS}(d)=3\cdot 10^{4}\) W/m\({}^{2}\)K can be represented as the sum of \(G_{nf}(d)\) about \(10^{3}\) W/m\({}^{2}\)K and \(G_{S}(d)=2.9\cdot 10^{4}\) W/m\({}^{2}\)K at \(d\) about 100 nm, \(\sigma=0.6\), \(p=10^{5}\) Pa, \(T_{int}=820\) K, \(\delta=0.93~{}\upmu\)m, and \(\varkappa_{S}(T_{int})=0.057\) W/m \(\cdot\) K [58]; see Equations (13) and (15), where \(T_{int}\) is the temperature intermediate between \(T_{S}=805\) K and \(T_{m}=835\) K. Near the liquidus temperature \(T_{Lig}=901\) K, \(G_{LS}=1.8\cdot 10^{5}\) W/m\({}^{2}\)K can be represented as the sum of \(G_{nf}(d)=1.5\cdot 10^{5}\) W/m\({}^{2}\)K at \(d=6\)-20 nm and \(G_{g}(d)=3\cdot 10^{4}\) W/m\({}^{2}\)K at \(d\leq 20\) nm, \(\sigma=0.6\), \(p=10^{5}\) Pa, \(T=915\) K, \(\delta=1.1~{}\upmu\)m, and \(\varkappa_{S}(T_{int})=0.063\) W/m\(\cdot\)K [58]; see Equations (13) and (15). Thus, upon the pre-melting of aluminum alloy 7075, the distance \(d\) decreases from about 100 nm to 10 nm, while the ITC increases by an order of magnitude.