## 3 Interfacial Thermal Resistance at the Membrane/Gas Interface
\[G_{max}=(f+1)k_{B}\frac{\sigma}{(2-\sigma)}\frac{p}{\sqrt{2\tau mk_{B}T}}, \tag{6}\]  
where \(m\) and \(f\) are the mass and number of degrees of freedom of the gas molecule, respectively. \(k_{B}=1.38\cdot 10^{-23}\) J/K is Boltzmann constant [30]. In fact, \(G_{max}\) is equal to the thermal contact conductance \(G_{C}=\lambda/\delta\). Indeed, from Equations (2), (5) and (6) we obtain the equality \(G_{C}=G_{max}\) taking into account the equation for the mean free pass \(l_{mfp}=\
u\sqrt{\pi m/2k_{B}T}\)[32]. Thus, \(R_{max}\) for an unloaded cell can be estimated as  
\[R_{max}=\frac{2G_{max}(T-T_{t})}{\rho_{m}c_{m}d_{m}}, \tag{7}\]  
where \(d_{m}=1\) um is the membrane thickness and \(\rho_{m}c_{m}=2.3\cdot 10^{6}\) J/m\({}^{3}\)K is the volumetric heat capacity of the membrane at the density \(\rho_{m}=3.2\) g/cm\({}^{3}\) and the specific heat capacity \(c_{m}=0.71\) J/gK [33]. Suppose \(T-T_{t}=500\) K, \(\sigma=1\) for a blackened membrane surface, and \(r_{min}=\delta\). Then the rate \(R_{max}\) can be estimated for several gases, see Table 1. A similar Table 2 is represented for \(\sigma\) corresponding to a bright surface. Data on \(\lambda\), \(\
u\), and \(l_{mfp}\) for gases are available in [33] and are collected in Tables 1-3. The temperature jump distance \(\delta\) is calculated using Equation (2). The results of measurements of the thermal accommodation parameter of various gasses on several material are summarized in [30]. A significant scatter of these results shows that uncontrolled defects of the material surface also affect the parameter \(\sigma\)[30]. Since our goal is not to accurately calculate the cooling rate, but to estimate the maximum possible rate, we use the available data for bright and blackened surfaces. In fact, \(\sigma\) depends not so much on the material but significantly on the quality of the surface, therefore in Table 2 we use the data available for some bright solid surfaces and several gases [30]. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Gas** & \begin{tabular}{c} **Number of Degrees of** \\ **Freedom of Gas Molecules** \\ \(f\) \\ \end{tabular} & \begin{tabular}{c} **Mean Free Path of** \\ **Gas Molecules** \\ \end{tabular} & \begin{tabular}{c} **Temperature Jump** \\ **Distance** \\ \end{tabular} & \begin{tabular}{c} **Thermal** \\ **Conductivity** \\ \end{tabular} & \begin{tabular}{c} **Maximum Possible** \\ **Parameter** \\ \end{tabular} & \
\begin{tabular}{c} **Maximum Interfacial** \\ **Thermal Conductance** \\ \end{tabular} &
\begin{tabular}{c} **Maximum Cooling** \\ **Rate** \\ \(f\) \\ \end{tabular} \\
**Dimensionless** & \begin{tabular}{c} **\(\mu\)m** \\ \end{tabular} & \begin{tabular}{c} **\(\mu\)m** \\ \end{tabular} & \begin{tabular}{c} **W/Wm-K** \\ \end{tabular} & \begin{tabular}{c} **M/Wm-K** \\ \end{tabular} & \begin{tabular}{c} **M/Wm-K** \\ \end{tabular} & \
\begin{tabular}{c} **M/Wm-K** \\ \end{tabular} &
\begin{tabular}{c} **W/Wm-K** \\ \end{tabular} \\ \hline H\({}_{2}\) & 5 & 0.126 & 0.21 & 0.192 & 0.90 & 0.90 & \(3.9\cdot 10^{8}\) \\ \hline He & 3 & 0.200 & 0.38 & 0.156 & 0.41 & 0.42 & \(1.8\cdot 10^{8}\) \\ \hline N\({}_{2}\) & 5 & 0.068 & 0.11 & 0.026 & 0.24 & 0.24 & \(1.04\cdot 10^{8}\) \\ \hline CO\({}_{2}\) & 6 & 0.045 & 0.07 & 0.017 & 0.22 & 0.22 & \(0.96\cdot 10^{8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermal and contact parameters of gases for a blackened membrane at 300 K and \(10^{5}\) Pa.