## 5 Heat Transfer across Liquid-Solid Interfaces
### Effective Gap Thickness at the Liquid-Solid Interface
For example, \(G_{LS}\) increases from \(2\cdot 10^{4}\) W/m\({}^{2}\)K to \(3.5\cdot 10^{4}\) W/m\({}^{2}\)K with a decrease in \(R_{a}\) from 1.2 \(\upmu\)m to 0.18 \(\upmu\)m for aluminum substrates, from \(3.5\cdot 10^{4}\) W/m\({}^{2}\)K to \(1.2\cdot 10^{5}\) W/m\({}^{2}\)K with a decrease in \(R_{a}\) from 1.5 \(\upmu\)m to 0.17 \(\upmu\)m for steel substrates, and from \(4\cdot 10^{4}\) W/m\({}^{2}\)K to \(2.6\cdot 10^{5}\) W/m\({}^{2}\)K with a decrease in \(R_{a}\) from 7.7 \(\upmu\)m to 0.07 \(\upmu\)m for copper substrates [23]. Thus, in the case of the liquid-solid interfaces, the effective thickness \(d\) is much less than the surface roughness \(R_{a}\); for example, see Table 3. In fact, liquids spread very efficiently over surface roughness even in the absence of good wetting [15; 16]. This interesting fact explains the significant difference between the surface roughness \(R_{a}\) and the effective distance \(d\) in experiments on the melting of various materials. However, in melting experiments, the distance \(d\) cannot tend to zero since the surface roughness usually has a multi-scale structure and liquids cannot spread over too small surface irregularities. The near-field thermal conductance \(G_{nf}(d)\) at nanometer distances \(d\) significantly exceeds the maximum possible value \(G_{S}^{max}\) associated with thermal conduction through the gas, see Figure 1. Moreover, as the distance \(d\) approaches the sub-nanometer scale, the ITC can be significantly increased due to the tunneling of low-frequency acoustic phonons. Indeed, \(G_{ph}(d)\) is about 10\({}^{7}\)-10\({}^{8}\) W/m\({}^{2}\)K at \(d<1\) nm [14]. However, phonon tunneling is usually not observed in melt-solidification experiments. In fact, liquids cannot spread over too-fine surface irregularities less than 1 nm in size. Thus, in melt-solidification experiments, the distance \(d\) is usually at least 1 nm or more. The main contributions to the ITC at melting experiments is made by the following two components: \(G_{nf}(d)\) and \(G_{S}(d)\), associated with the near-field thermal conductance and thermal conductance through the gas, respectively; see Figure 1.  
However, phonon tunneling was probably observed in \(G_{LS}\) measurements for water in contact with specially functionalized aluminum and gold substrates. Remarkably, for both hydrophilic and hydrophobic substrates, \(G_{LS}\) is of the order of 10\({}^{8}\) W/m\({}^{2}\)K [16]. Indeed, \(G_{LS}=10^{8}\) W/m\({}^{2}\)K and 1.8\(\cdot\)10\({}^{8}\) W/m\({}^{2}\)K for hydrophilic surfaces of gold and aluminum functionalized with 11-mercapto-1-undecanol and 2-[methoxy(polyethyleneoxy)-propyl]-trichlorosilane, respectively. The thickness of the functional layers was 1.1 nm and 0.7 nm for gold and aluminum, respectively. In the case of hydrophobic surfaces, \(G_{LS}\) is equal to \(5\cdot 10^{7}\) W/m\({}^{2}\)K and 6\(\cdot\)10\({}^{7}\) W/m\({}^{2}\)K for gold and aluminum functionalized with 1-octadecanethiol and octadecyl trichlorosilane, respectively. In both cases, the thickness of the functional layers was about 2.3-2.4 nm [16]. \(G_{S}^{max}\) determined by Equation (12) is about \(5.1\cdot 10^{4}\) W/m\({}^{2}\)K for nitrogen gas at \(d\to 0\), \(\sigma=0.6\), \(p=10^{5}\) Pa, and \(T=300\) K, and \(m=4.65\cdot 10^{-26}\) kg [58]. For saturated water vapor at 30C, the maximum possible value \(G_{S}^{max}=7.2\cdot 10^{3}\) W/m\({}^{2}\)K at \(d\to 0\), \(\sigma=1\), \(c_{v}=3k_{B}\), \(p=4.2\cdot\)10\({}^{3}\) Pa, \(T=303\) K, and \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Surface Roughness** & **ITC** & **ITC Component** & **ITC Component** & **Effective Thickness** \\ \(Ra\) & \(G_{LS}(d)\) & \(G_{S}^{max}\) & \(G_{nf}(d)\) & \(d\) \\ \(\upmu\)m & W/m\({}^{2}\)**K** & **W/m\({}^{2}\)K** & **W/m\({}^{2}\)K** & **nm** \\ \hline
7.7 & \(4.8\cdot 10^{4}\) & \(2.5\cdot 10^{4}\) & \(2.3\cdot 10^{4}\) & 20–62 \\ \hline
0.07 & \(9\cdot 10^{4}\) & \(2.5\cdot 10^{4}\) & \(6.5\cdot 10^{4}\) & 12–37 \\ \hline \hline \end{tabular}
\end{table}