## 5 Heat Transfer across Liquid-Solid Interfaces  
### Effective Gap Thickness at the Liquid-Solid Interface  
The interface between the contacting materials can change significantly during melting and solidification. If the surface roughness approaches the submicron scale during melting, near-field thermal radiation can play a significant role in the overall heat transfer. Consider a thermal contact between a molten sample and a solid substrate. It can be expected that for the liquid-solid interfaces, the effect of substrate surface roughness on ITC will be strongly suppressed. Indeed, liquid samples can spread well over surface irregularities. Moreover, liquids can spread very effectively over surface roughness, even in the absence of good wetting. For example, at the mercury-silicon interface, \(G_{LS}\) is about \(10^{6}\) W/m\({}^{2}\)K for mercury microdroplets at \(T=450\) K and a low gas pressure \(p\) of about 60 Pa [15]. In this case, the interfacial thermal conductance through the gas is negligible. Actually, \(G_{S}^{max}\) determined by Equation (12) is about 25 W/m\({}^{2}\)K for nitrogen gas at \(\sigma=0.6\), \(p=60\) Pa, and \(T=450\) K. However, the interfacial thermal conductance of about \(10^{6}\) W/m\({}^{2}\)K can be achieved due to the near-field thermal conductance \(G_{nf}(d)\). The near-field thermal conductance \(G_{nf}(d)\) can be approximately estimated between the values \(G_{nf}^{max}(d)/100\) and \(G_{nf}^{max}(d)/10\). Then, from Equation (7), we obtain that \(G_{nf}(d)\) is about \(10^{6}\) W/m\({}^{2}\)K at \(d\) in the range of 1-5 nm. Thus, the effective gap \(d\) at the interface between mercury and silicon substrate can be estimated to be a few nanometers. Data on the interfacial thermal contact of mercury and silicon are collected in Table 2.  
In the case of liquid-solid interfaces, the effective thickness \(d\) is much less than the substrate surface roughness \(R_{a}\), as follows from \(G_{LS}\) measurements for molten copper droplets cooling on cold copper substrates [23]. Furthermore, the effect of surface roughness on the ITC of liquid-solid interfaces is rather weak. Indeed, \(G_{LS}\) increases only by a factor of 2 (from \(4.8\cdot 10^{4}\) to \(9\cdot 10^{4}\) W/m\({}^{2}\)K) with a decrease in surface roughness \(R_{a}\) by two orders of magnitude (from 7.7 to 0.07 \(\upmu\)m) [23]. In this experiment, the gas component \(G_{S}(d)\) can contribute to the total ITC no more than the maximum possible value \(G_{S}^{max}=2.5\cdot 10^{4}\) W/m\({}^{2}\)K at \(d\to 0\), \(\sigma=0.6\), \(p=10^{5}\) Pa, and \(T=1500\) K. In fact, \(G_{S}(d)\) is approximately equal to \(G_{S}^{max}\) at \(d<100\) nm since the dependence \(G_{S}(d)\) saturates at \(d\) less than 100 nm; for ex \begin{table}
\begin{tabular}{c c c c c c} \hline
**ITC** & **ITC Component** & **ITC Component** & **Effective Thickness** & **Temperature** & **Gas Pressure** \\ \(G_{LS}(d)\) & \(G_{S}^{max}\) & \(G_{nf}(d)\) & \(d\) & **T** & \(p\) \\ W/m\({}^{2}\)K & W/m\({}^{2}\)K & W/m\({}^{2}\)K & **nm** & **K** & **Pa** \\ \hline \(10^{6}\) & 25 & \(10^{6}\) & 1–5 & 450 & 60 \\ \hline \end{tabular}
\end{table} 
Table 2: Thermal contact parameters of the mercury-silicon interface at low gas pressure. ample, see Figure 1. The near-field component \(G_{nf}(d)\) can be approximately estimated between the values \(G_{nf}^{max}(d)/100\) and \(G_{nf}^{max}(d)/10\). Thus, \(G_{nf}(d)=G_{LS}-G_{S}^{max}\) is about \(2.3\cdot 10^{4}\) W/m\({}^{2}\)K at \(d=20\)-62 nm for substrates with surface roughness \(R_{a}\)=7.7 \(\upmu\)m. Additionally, \(G_{nf}(d)=G_{LS}-G_{S}^{max}\) is about \(6.5\cdot 10^{4}\) W/m\({}^{2}\)K at \(d=12\)-37 nm for substrates with \(R_{a}=0.07\)\(\upmu\)m. Data on ITC parameters for copper-copper interfaces at different surface roughness are collected in Table 3.  
Similar results were obtained when measuring \(G_{LS}\) for molten nickel cooled on cold substrates of copper, aluminum, and stainless steel with different roughness [23].