## 4 Maximum Possible Controlled Cooling Rate in the Membrane-Gas System
\[\frac{\partial T}{\partial t}=D_{m}\bigg{(}\frac{\partial^{2}T}{\partial r^{2 }}+\frac{1}{r}\frac{\partial T}{\partial r}\bigg{)}-\frac{T}{\tau}+\Phi(t,r), \tag{9}\]  
where \(D_{m}=\lambda_{m}/\rho_{m}c_{m}\) is the thermal diffusivity of the membrane, \(\tau=\rho_{m}c_{m}d_{m}/\alpha\), \(\alpha=\lambda/r_{0}\), \(\Phi(t,r)=HF(t,r)/\rho_{m}c_{m}d_{m}\), and \(HF(t,r)\) is the surface density of the heat flux generated by the heater [35]. The solution to the problem can be represented as a series expansion:  
\[T(t,r)=\sum_{m=1}^{N}J_{0}(\mu_{n}r/b_{0})\psi_{n}(t), \tag{10}\]  
where \(J_{0}(\mu_{n}r/b_{0})\) are the orthogonal Bessel functions of the first kind and of zero order, \(\{\mu_{n}\}\) is a monotonously increasing sequence of positive roots of the equation \(J_{0}(\mu_{n})=0\) for \(n=1,2,3\ldots\), and \(N\) is the number of terms in the series expansion sufficient to achieve the required accuracy. Note that the structure within the hot zone can be accurately calculated if the number \(N\) is large compared to \(b_{0}/b_{1}\), where \(b_{1}\) is the thickness of the thinnest heater structure. Good accuracy can be achieved with \(N\geq\)1000 for \(b_{1}=\)1/3 um and \(b_{0}=350\) um. Further calculations are performed at \(N=\)2000. In fact, the difference in the results for \(N=\)1000 and 2000 was insignificant. The functions \(\psi_{n}(t)\) can be obtained from the Fourier components of Equation (9). \[\partial\psi_{n}(t)/\partial t+\Big{(}\tau^{-1}+\tau_{n}^{-1}\Big{)}\psi_{n}(t )=B_{n}(t), \tag{11}\]  
where \(\tau_{n}=1/D_{m}(\mu_{n}/b_{0})^{2}\) and  
\[B_{n}(t)=\frac{2}{b_{0}^{2}J_{1}(\mu_{n})^{2}}\int_{0}^{b_{0}}J_{0}(\mu_{n}r/b_ {0})\Phi(t,r)rdr. \tag{12}\]Consider the solution \(T(t,r)\) for a ring heater. Suppose \(\Phi(t,r)=q(t)F(r)\), where \(F(r)=1\) in the ring \(r\in(r_{1},r_{0})\) and \(F(r)=0\) outside this ring. Thus,  
\[B_{n}(t)=\frac{q(t)}{\rho_{m}c_{m}d_{m}}\frac{2r_{0}{I_{1}(\mu_{n}r_{0}/b_{0})}- 2r_{1}{I_{1}(\mu_{n}r_{1}/b_{0})}}{\mu_{n}b_{0}{I_{1}(\mu_{n})}^{2}}, \tag{13}\]  
where \(q(t)=P(t)/\pi\big{(}r_{0}^{2}-r_{1}^{2}\big{)}\) and \(P(t)\) is the heater power. Therefore, with a zero initial condition \(T(t,r)=0\) at \(t\leq 0\), we obtain a solution to Equation (11):  
\[\psi_{n}(t)=\int_{0}^{t}B_{n}\big{(}t^{\prime}\big{)}\text{exp}\Big{(}-\Big{(} \tau^{-1}+\tau_{n}^{-1}\Big{)}\big{(}t-t^{\prime}\big{)}\Big{)}dt^{\prime}. \tag{14}\]  
For model calculations, in order to obtain approximately constant heating and cooling rates, we take the power \(P(t)\) increasing as \((t/t_{0})^{\beta_{1}}\) during heating and decreasing as \(1-((t-t_{0})/t_{1})^{\beta_{2}}\) during cooling, where \(t_{0}\) and \(t_{1}\) are the duration of temperature scanning during heating and cooling, respectively, \(\beta_{1}\) and \(\beta_{2}\) are positive parameters less than 1. Thus, we obtain the temperature distribution \(T(t,r)\) calculated at \(R=10^{8}\) K/s and \(3\cdot 10^{8}\) K/s, see Figure 3. The calculations were carried out for helium gas with \(\lambda=0.156\) W/m\(\cdot\)K\(\,\)at \(r_{0}=1\) um, \(r_{1}=2r_{0}/3\), \(d_{m}=1\) um, \(\lambda_{m}=3\)W/m\(\cdot\)K, \(D_{m}=1.3\cdot 10^{-6}\) m\({}^{2}\)/s, \(\rho_{m}c_{m}=2.3\cdot 10^{6}\) J/m\({}^{3}\)K, and the scanning temperature range \(\Delta T=1000\) K. The temperature profile \(T(t_{0},r)\) was calculated at the end of heating for \(t_{0}=10.8\) us and \(3.65\) us at \(R=10^{8}\) K/s and \(3\cdot 10^{8}\) K/s, respectively. Thus, controlled cooling is possible at \(R=10^{8}\) K/s and even \(3\cdot 10^{8}\) K/s.