## 1 Introduction  
Ultrafast membrane-based nanocalorimetry opens up exciting opportunities for materials science [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. This calorimetry makes it possible to measure the size-dependent properties of nanosized samples [1, 5, 9] and films as thin as about 0.1 nm [2, 3], as well as measure the melting and synthesis characteristics of a single-layer lamella [7, 8], and directly measure the desorption of polymer chains [12]. The ultrafast calorimetry method is described in detail in [10]. A review of microchip-based nanocalorimeters is presented in [11]. It is very important that ultrafast calorimetry allows nano- and microsamples to be heated or quenched at extremely high, controlled rates, creating non-equilibrium states under well-defined conditions, and to study the kinetics of phase transitions on a sub-millisecond time scale. A review of such experiments is presented in [14]. For example, crystallization and recrystallization kinetics [15, 16, 17, 18, 19, 20], as well as interfacial thermal conductance [21, 22], can be measured in ultrafast membrane nanocalorimeters during rapid melting and crystallization. It was shown that calorimetric measurements with controlled cooling and heating rates up to \(10^{8}\) K/s are possible using membrane sensors; in the experiment, an uncontrolled cooling rate of about \(10^{8}\) K/s was achieved [23]. However, the fundamental limitations on controlled cooling rates need to be investigated. Indeed, even if ultrafast heating can be controlled in a nanocalorimeter under quasi-adiabatic conditions, ultrafast cooling can only be performed under non-adiabatic conditions [23, 24, 25, 26]. In fact, the central hot zone of the membrane is used as a calorimetric cell. The heat transfer between the hot zone and the environment determines the maximum cooling rate achievable in an experiment. It was found that uniform ultrafast controlled cooling can be accomplished for nanogram samples if the measuring calorimetric cell is placed in heat-conducting nitrogen gas or helium gas [23; 24; 25; 26]. For several sensors of different size and geometry, it was shown that the maximum possible controlled cooling rate is proportional to the ratio \(\lambda/r_{0}\), where \(\lambda\) is the thermal conductivity of the surrounding gas and \(r_{0}\) is the radius of the central hot zone of the membrane [23; 24; 25; 26]. The possibility of increasing the maximum cooling rate with decreasing \(r_{0}\) has been successfully used in many experiments [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. The goal of this article is to answer the question: what is the maximum possible cooling rate in such experiments if \(r_{0}\) tends to zero? Indeed, on submicron scales, the mean free path of gas molecules \(l_{mfp}\) becomes comparable to \(r_{0}\). In fact, there is a jump in temperature \(\Delta T_{C}\) at the membrane/gas contact. The temperature jump \(\Delta T_{C}\) increases with the increase of the temperature gradient in the gas near the membrane hot zone; this temperature gradient is about \((T-T_{t})/r_{0}\), where \(T\) is the temperature of the hot zone, \(T_{t}\) is the temperature of the thermostat. Thus, for \(r_{0}\) on a submicron scale, a comprehensive analysis of the thermal processes occurring in the calorimetric sensor located in a heat-conducting gas is required.  
In the first part of the paper, we consider the thermal resistance at the membrane/gas interface and obtain the fundamental limitation on the maximum possible cooling rate in a membrane-based calorimeter with the calorimetric cell located in a heat-conducting gas. In the second part of the paper, the effect of the heat transfer from the central hot zone of the membrane to the periphery of the membrane is considered. Finally, it is shown that the controlled cooling rate in membrane calorimeters can reach up to ten billion K/s in helium gas.