## 4 Maximum Possible Controlled Cooling Rate in the Membrane-Gas System  
Heat in gases is transferred by molecules moving at a speed sufficient to achieve high cooling rates. In fact, a cooling rate of about \(10^{8}\) K/s was achieved in the experiment [23]. The diffusion Fourier heat equation for gasses is a good approximation on a time scale that is much larger than the average time between collisions of gas molecules [28; 34]. Denoted by \(\tau_{m}=1_{mfp}/V_{m}\) the average time between collisions of gas molecules, where \(V_{m}=\sqrt{8k_{B}T/\pi m}\) is the average velocity of gas molecules [32; 33]. In fact, \(\tau_{m}=0.07\) ns, 0.16 ns, and 0.14 ns for hydrogen, helium, and nitrogen gases at \(V_{m}=1.78\) km/s, 1.26 km/s, and 0.476 km/s, respectively (see Table 3). The thermal perturbation in the gas around the  
Figure 2: Possible cooling rate \(R\) vs. the radius \(r_{0}\) of the hot zone for blackened (**a**) and bright (**b**) membrane surfaces for different gases. heated region of radius \(r_{0}\) relaxes during the thermal diffusion time \(\tau_{D}=r_{0}^{2}/D\). In fact, \(\tau_{D}=6.2\) ns, 5.3 ns, and 45 ns at \(D=1.6\cdot 10^{-4}\) m\({}^{2}\)/s, \(1.9\cdot 10^{-4}\) m\({}^{2}\)/s, and \(2.2\cdot 10^{-5}\) m\({}^{2}\)/s for hydrogen, helium, and nitrogen gases, respectively, at \(r_{0}=1\) um. The thermal parameters of the gases are available in [33] and collected in Table 3. Thus, \(\tau_{D}\) is much longer than the average time between collisions of gas molecules, \(\tau_{D}\gg\tau_{m}\), and about \(10^{2}\) collisions per molecule occurs during \(\tau_{D}\). In this case, the diffusion Fourier heat equation can be used [28; 34]. However, the limited rate of relaxation of thermal perturbations in the gas imposes the following limitation on the cooling rate: \(R<\Delta T/\tau_{D}\), where \(\Delta T\) is the interval of the temperature scanning. Assume \(\Delta T=1000\) K, then the cooling rate \(R\) is limited to about \(1.6\cdot 10^{11}\) K/s, \(2\cdot 10^{11}\) K/s, and \(2\cdot 10^{10}\) K/s for hydrogen, helium, and nitrogen gases, respectively. This limitation is not the most significant. The main restriction is associated with thermal relaxation of the membrane-gas system; this relaxation depends on the heat loss from the membrane, which mainly depends on the thermal conductivity of the gas \(\lambda\). Let us consider the temperature distribution \(T(t,r)\) in the lateral direction of the membrane for a ring-shaped heater. The temperature difference in the \(z\) direction is insignificant, since \(\lambda\) is much less than the membrane thermal conductivity \(\lambda_{m}\). Note that the temperature gradients near the central hot zone in the gas and membrane are approximately the same and equal to \((T-T_{t})/\tau_{0}\). Therefore, \(\partial T/\partial z\) is about \((\lambda/\lambda_{m})\partial T/\partial r\), as follows from the boundary condition for the heat flux at the membrane/gas interface. Thus, gradients along the \(z\) direction in the membrane can be neglected at \(\lambda<<\lambda_{m}\). In addition, the temperature distribution \(T(t,r)\) tends to zero at a distance of the order of \(r_{0}\), see below. Thus, the shape of the membrane at the periphery does not matter at \(r\gg r_{0}\). Therefore, we consider a boundary value problem with a homogeneous boundary condition \(T(t,b_{0})=0\), where \(b_{0}\) is the distance from the center of the membrane to the silicon frame, the temperature of which is equal to the temperature of the thermostat, and the temperature \(T(t,r)\) is measured from \(T_{t}\). Thus, for the membrane-gas system, we have an equation for \(T(t,r)\) in cylindrical coordinates, see Equation (9). The second term \((-T/\tau)\) on the right-hand side of Equation (9) describes the heat loss from the membrane to the gas.