## 5 Heat Transfer across Liquid-Solid Interfaces
### Evolution of Thermal Contact Conductance \(G\) during the Pre-Melting Process
At distances \(d\) less than 100 nm, the interfacial thermal conductance \(G_{LS}\) can be represented as the sum of \(G_{g}(d)\) and \(G_{nf}(d)\), where the near-field component \(G_{nf}(d)\) is estimated as \(G_{nf}\in\left(G_{nf}^{max}/100,G_{nf}^{max}/10\right)\), and \(G_{g}(d)\) is approximately equal to \(G_{g}^{max}\), see Figure 1. In this case, \(G_{nf}\) is estimated with an uncertainty of about 50%. The uncertainty of \(G_{g}^{max}\) is about 45%, mainly due to the uncertainty of \(\sigma\) of about 30%. The total error in estimating \(d\) is about 80% because ITC is measured with an uncertainty of about 50%. However, the accuracy of the estimate \(d\) is quite satisfactory since the distance \(d\) changes by orders of magnitude in melting experiments (note the logarithmic scales in Figure 1). Measurements on a millisecond time scale made it possible to separate the influence of the pre-melting and melting processes on the ITC. These measurements are conducted in calorimetric microchips, in which the sample is placed on an amorphous silicon nitride membrane with a surface roughness of about 200 nm [66]. This roughness, if necessary, can be reduced by chemical etching to about 1-10 nm [67; 68]. The roughness of the sample may be greater than that of the membrane at the beginning of sample melting. However, as the sample melts, this roughness decreases, and, finally, the interfacial gap thickness \(d\) can be much smaller than the membrane roughness. Consider experiments on the melting of microsamples of pure indium in contact with the membrane of a calorimetric sensor [26]. Indium microsamples were heated from 320 K to 550 K (above the melting point \(T_{melt}=430\) K) at a rate of about \(2\cdot 10^{3}\) K/s and cooled back to 320 K at the same rate. In these experiments, the position and shape of the sample remained stable after the first melt, and the measurements were completely reproducible during subsequent heating-cooling scans. It was found that the melting process is divided into two stages: the pre-melting process and the actual melting process. At the beginning of the pre-melting process (at \(T_{melt}-20\) K), the ITC sharply increases from \(G_{SS}~{}=~{}2.7\cdot 10^{3}\) W/m\({}^{2}\)K to \(G_{pS}=4.1\cdot 10^{3}\) W/m\({}^{2}\)K. This jump in ITC is associated with a change in the surface of the sample, as indicated by calorimetric measurements [26]. Further, within 2.6 ms in the temperature range \(\left(T_{melt}-20\text{ K},T_{melt}\right)\), a volumetric pre-melting process occurs with an enthalpy consumption, which is clearly seen from measurements of the enthalpy absorbed by the pre-melting sample [26]. ITC gradually increases from \(G_{pS}=4.1\cdot 10^{3}\) W/m\({}^{2}\)K to \(G_{LS}=5\cdot 10^{4}\) W/m\({}^{2}\)K during this pre-melting process. After that, the actual melting occurs for 1.6 ms at \(T_{melt}=430\) K. The ITC remains constant (\(G_{LS}~{}=~{}5\cdot 10^{4}\) W/m\({}^{2}\)K) during this melting process [26]. In this experiment, \(G_{SS}=2.7\cdot 10^{3}\) W/m\({}^{2}\)K corresponds to the ITC for the solid-solid interface between the indium sample and the calorimeter membrane. This value (\(2.7\cdot 10^{3}\) W/m\({}^{2}\)K) can be attributed to the heat conduction through the nitrogen gas at \(d~{}=~{}11.5\) um, \(\sigma=0.6\), \(p=10^{5}\) Pa, \(T_{int}=420\) K, \(\delta=0.39\) um, and \(\varkappa_{g}(T_{int}~{})\) = 0.034 W/m \(\cdot\) K [58]; see Equations (13) and (15), where \(T_{int}\) is the temperature intermediate between the sample temperature \(T_{S}=410\) K and the membrane temperature \(T_{m}=430\) K. Similarly, \(G_{pS}(d)=4.1\cdot 10^{3}\) W/m\({}^{2}\)K can be attributed to the heat conduction through the nitrogen gas at \(d=7.4\) um and \(T_{int}=420\) K. The near-field component is negligible at \(d=7.4\) um. However, with subsequent heating, the ITC gradually increases and reaches the value \(G_{LS}=5\cdot 10^{4}\) W/m\({}^{2}\)K at the melting point.