## 2 Membrane-Based Ultrafast Nanocalorimeter  
The considered membrane-based calorimeter consists of an amorphous silicon nitride membrane with a thin-film resistive heater and a thermocouple located in the center of the membrane, see Figure 1. The central hot zone of the membrane is used as the calorimetric cell. The sample to be measured is placed in the center of this zone, next to the measuring hot junction of the thermocouple, see Figure 1. The thermocouple, heater, and electrical leaders are formed by p- and n-type polysilicon tracks with suitable thermoelectric properties and resistivity. For the protection and electrical insulation, the measuring circuit is covered with a submicron amorphous silicon nitride layer so that the total membrane thickness \(d_{m}\) is about 1 \(\upmu\)m. The membrane is attached to a silicon frame at a distance of \(b_{0}=350\)\(\upmu\)m from the center of the membrane. The frame is mounted on a standard TO-5 housing. The calorimetric cell is located in a thermostat with controlled temperature \(T_{t}\) and gas pressure \(p\).  
The heat generated by the heater in the hot zone of the membrane is transferred to the surrounding gas and membrane. Consider the transfer of heat to the gas. Let \(\alpha\)  
Figure 1: Schematic sectional view of a membrane-based nanocalorimeter, not to scale.  
denote the coefficient of heat transfer from the membrane surface to the surrounding gas. This coefficient, measured in W/m\({}^{2}\)K, describes the heat loss from the central hot zone of the membrane to the gas. The heat losses are mainly associated with the thermal conductivity \(\lambda\) of the surrounding gas [23; 24; 25; 26]. This heat loss is proportional to \(\frac{\lambda(T-T_{t})}{L}\) where \(L\) is the characteristic length of the problem. Since \(r_{0}\) is small with respect to the distance \(L_{t}\) from the hot zone to the thermostat, the heat loss is proportional to \(\frac{\lambda(T-T_{t})}{r_{0}}\). In fact, the hot zone can be considered as a spherical (point) heat source embedded in the gas. Consequently, \(\alpha=\lambda/r_{0}\) with good accuracy for a radius \(r_{0}\) of tens of microns or less [23; 24; 25; 26]. The convective and radiative contributions to the heat losses are negligible for such a small \(r_{0}\), see Appendix A. Thus, the cooling rate \(R\) of the hot zone of the membrane depends on the heat transfer parameter \(\alpha=\lambda/r_{0}\) at \(L_{t}\gg r_{0}\gg l_{mfp}\).  
Typically, nitrogen gas or helium gas is used to cool the membrane [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. In differential scanning calorimetry, the calorimetric cell loaded with the sample and the reference unloaded cell are maintained at the same temperature \(T(t)\) in the central hot zone; this temperature is controlled at the desired scanning rate \(R(t)\). The difference in input power required to match the temperature of the loaded cell to the temperature of the reference cell is equal to the power \(P_{S}(t)\) absorbed or released from the sample being measured [27]. This study aims to estimate the maximum possible controlled cooling rate \(R_{max}\) for membrane calorimeters. Thus, we estimate \(R_{max}\) for an unloaded cell, considering the limiting case when the heat capacity of the sample is negligible.