## 4 Heat Transfer through the Gas Gap  
In experiments on melting and solidification, the interfacial gas gap depends on the state of the melting substance and the roughness of the contacting bodies. The effective thickness \(d\) of this gap can be in the range of 1-100 nm for liquid-solid interfaces. Note that the mean free pass \(l_{mfp}\) of nitrogen gas molecules is on the order of 100 nm at \(T>300\) K [58]. Thus, in order to estimate the interfacial thermal conductance \(G_{g}\) associated with heat transfer through the gas, we consider the case of heat exchange between two plates separated by a thin gap of thickness \(d\ll l_{mfp}\). Since the experiments considered below were conducted in air at elevated temperatures, we make estimates at \(T>300\) K for nitrogen gas, which is the main component of air. Consider two plates heated to different temperatures \(T_{1}\) and \(T_{2}\), with \((T_{1}-T_{2})\ll T\), where \(T\) is the intermediate of \(T_{1}\) and \(T_{2}\). Note that the heat transfer at the boundary due to natural convection in gases is negligibly small compared to the thermal conductivity of gases at a microscale distance [59]. The convective component of heat transfer in gases arises in a gravitational field in the presence of temperature gradients due to the temperature dependence of the gas density. The convective component decreases with the decreasing characteristic length \(L\) of the problem. The ratio of the convective to conductive components of heat transfer is equal to the Nusselt number \(\text{Nu}=\left(\text{Gr})^{1/4}f(\text{Pr})\)[60]. The Prandtl number Pr and the value of the function \(f(\text{Pr})\) for gases are about 1 [58; 60]. The Grashof number for gases is approximately equal to \(\text{Gr}=L^{3}g(T_{1}-T_{2})/\
u^{2}T\), where \(g\) is the gravitational acceleration, and \(\
u\) is the kinematic viscosity of the gas [60]. Thus, the Nusselt number decreases proportionally to \(L^{3/4}\) as \(L\) decreases. Nu is about \(0.8\cdot 10^{-2}\) at \(L=1\) um, \((T_{1}-T_{2})/T=0.1\), and \(\
u=1.6\cdot 10^{-5}\) m\({}^{2}\)/s for nitrogen gas at normal pressure and room temperature [58]. Therefore, the convective contribution to heat transfer is less than 1% at \(d<1\) um (this contribution is about 4% at \(L\) about 10 um). Let us first assume that the gas molecules interact with the plates under conditions of complete accommodation, i.e., the molecules reflected from a body have the same temperature as the surface of this body. The amount of heat transferred by the gas molecules between the plates is proportional to the number of molecules striking a unit area of the surface per unit time \(Z=nV_{m}/4\), where \(n=p/k_{B}T\) is the number of molecules per unit volume at pressure \(p\) and \(V_{m}=\sqrt{8k_{B}T/\pi m}\) is the average velocity of gas molecules of a mass \(m\). Thus, \(Z=p/\sqrt{2\pi mk_{B}T}\)[61]. The heat flux carried away by gas molecules from the surface is equal to \(q_{g}=Z(c_{v}+k_{B}/2)T\), where \(c_{v}\) is the heat capacity per one molecule at a constant volume. This result is based on the assumption of a Maxwellian velocity distribution [61]. Then, the thermal conductance \(G_{g}=\Delta q_{g}/(T_{1}-T_{2})\) between plates 1 and 2 due to the gas can be represented by Equation (10). \[G_{g}=(c_{v}+\frac{k_{B}}{2})\frac{p}{\sqrt{2\pi mk_{B}T}}. \tag{10}\]  
Thus, \(G_{g}\) does not depend on \(d\) since \(d\) is much smaller than the transverse dimension of the plates and \(d\ll l_{mfp}\). Consider the case of incomplete accommodation of gas molecules with the surfaces of the plates. Then, there are discontinuities of the temperature at the surfaces. The molecules rarely interact with each other but interact mainly with the surfaces of the plates at \(d<l_{mfp}\). Thus, it is necessary to clarify the concept of gas temperature \(T_{g}\). Usually, \(T_{g}\) is defined as the average energy \(E_{g}\) of molecules at a given point in the gas, and the relationship between the gas temperature and energy is the same as for large gas volumes [61; 62].