Force-driven reversible liquid–gas phase transition mediated by elastic nanosponges

Nano-confined spaces in nanoporous materials enable anomalous physicochemical phenomena. While most nanoporous materials including metal-organic frameworks are mechanically hard, graphene-based nanoporous materials possess significant elasticity and behave as nanosponges that enable the force-driven liquid–gas phase transition of guest molecules. In this work, we demonstrate force-driven liquid–gas phase transition mediated by nanosponges, which may be suitable in high-efficiency heat management. Compression and free-expansion of the nanosponge afford cooling upon evaporation and heating upon condensation, respectively, which are opposite to the force-driven solid–solid phase transition in shape-memory metals. The present mechanism can be applied to green refrigerants such as H2O and alcohols, and the available latent heat is at least as high as 192 kJ kg−1. Cooling systems using such nanosponges can potentially achieve high coefficients of performance by decreasing the Young’s modulus of the nanosponge.

. Stress-strain curve obtained for ZTC and AC. V Hg and V 0 are the volume change of mercury and initial sample volume, respectively. At low pressure (< 20 MPa), V Hg quickly decreases by Hg impregnation of the interparticle spaces. The bulk modulus (K [MPa]) of the sample can be calculated from the linear part of the stress-strain curve at higher pressure (50-150 MPa), according to equation (2). The bulk moduli of ZTC and AC were determined to 0.70 and 1.7 GPa, respectively (the last value is considered an underestimate because of the presence of larger pores 1 ).

Supplementary Figure 2. Illustrations of the equipment for in situ adsorption measurement
under pressing. a A photo of a home-made press chamber. b An illustration of (a) showing the inside. Sample sheets are sandwiched by stainless plates and they are stacked to ensure the sample amount enough for the measurement. The sample stack can be mechanically pressed by a rotary manipulator. The chamber is connected to an automatic adsorption analyzer. c A photo of a whole measurement setup. The press chamber is immersed in a constant temperature bath highlighted by a red dashed square. The temperature is maintained by a circulation constant temperature water bath. d A photo of the inside of the constant temperature bath. Al beads are used as a heat medium. The press chamber is immersed in the Al beads, together with a coil for circulation temperature control.
The Al beads bath is covered by styrene foams. Note the press chamber is intentionally exposed for better view in (d).  When the adsorbed amount reaches a certain level, the adsorption room is switched to be a desorption room. Then, the refrigerant is desorbed (endothermic) by using an external high-temperature heat source (T H ). T H is usually about 350 K for water system. The vapour is then translated to liquid at a condenser, accompanied by the heat of condensation (exothermic). To keep the temperature of the condenser constant, an external middle-temperature heat source (usually the same as T M ) is used. Then, the liquid is transferred to the evaporator. In AHP, cooling is thus done by the evaporator, and therefore, AHP is categorized to the Refrigeration based on the Bulk Phase Transition (RBPT). Loosely speaking, adsorption and desorption in AHP play a role of mass transfer instead of a compressor and an expansion valve in air conditioners. AHP enables the use of water as a refrigerant, whereas AHP is much bulky than conventional air conditioners because of the limited adsorption capacity of nanoporous solids operated around P/P 0 = 0.1-0.5, and has not been used for compact air conditioners for private rooms and automobiles.   b Stress-strain curves measured on a ZTC/PTEF sheet. By using these data, COP is calculated at the vapour pressure of 2.9 kPa, based on equation (23). In this equation,  vap H and c re are 2453 kJ kg -1 and 4.184 kJ kg -1 K -1 , respectively. T H -T L is 5 K. c ns can be approximated by the value of the graphite (0.72 kJ kg -1 K -1 ). From equation (25), V 0 = w ns /ρ ns-ap = 0.001976w ns . w re-ads can be expressed by w ns , from the adsorption amount at 303 K on the pristine ZTC (point I; 51.1 mmol g -1 = 0.92 kg kg -1 ) as 0.92w ns . As described in equation (28), w re is the difference of w re-ads and w e . w e can be expressed by w ns , from the adsorption amount at 298 K on the compressed ZTC (point II;
The energy loss turns into heat and warms up the target room to be cooled. Accordingly, COP for ZTC working between 298 and 303 K at 2.9 kPa can be calculated as 17.3.

Supplementary Figure 8. Example of adsorption isotherms on an imaginary nanosponge.
The properties of the nanosponge are given as an inset. V ns and E are the same as those of ZTC, while ρ ns and c ns are the same as those of graphite. Thus, the imaginary nanosponge is not far from actual materials. Based on equation (23), larger w re affords better COP. Therefore, the gap in the adsorption values at the points I and II shown in the Supplementary Fig. 7a should be larger. Herein, an imaginary adsorption isotherms are shown, which is obtained by modifying Supplementary Fig.   7a, by the following two assumptions: (i) at 303 K and 2.9 kPa, the adsorption amounts in the pristine and compressed ZTCs are the same as those at 298 K, and (ii) the isotherms have a flat plateau after water uptake is completed. The latter feature is seen in mesoporous silicas with hydrophilic uniform mesopores 2 , and can be considered as an achievable target also in elastic nanosponge materials. Additionally, the H 2 O uptake pressures at different temperatures are adjusted according to the fact that the H 2 O uptake occurs at the same P/P 0 regardless of temperature 2 .
Consequently, the isotherms shown here can be an achievable target. In this case, w re is 0.36w ns , and where q i is the atomic charge, ε 0 the vacuum permittivity (8.8542 × 10 -12 C 2 N -1 m -2 ), r ij the interatomic distance, and σ ij and ε ij the LJ parameters. The interaction potentials were calculated with the cross-interaction parameters obtained from the Lorentz-Berthelot mixing rules, and were truncated at a cut-off distance of 1.89 nm. The atomic charges in each of the ZTC models were obtained by the dispersion corrected density functional theory (DFT-D3) calculations with the PBE functional and DZVP-MOLOPT basis set and Mulliken population analyses, using the CP2K software package. 3 We used the TIP4P water model 4 and the universal force field (UFF) 5 was applied to calculate the LJ interaction terms for the ZTC atoms (see Supplementary Table 2 where  is a given chemical potential. The first term in the right-hand side is the grand thermodynamic potential at chemical potential  id , which is low enough so that the adsorption amount N A at point A is essentially the ideal gas value: where k is the Boltzmann constant. The grand thermodynamic potential  EG along the desorption branch (line EG) is calculated by integrating the Gibbs adsorption isotherm N EG from point E at chemical potential  r . Point E can be arbitrarily chosen if  eq <  r . The  E value is obtained by integration along two reversible paths, namely, a supercritical adsorption isotherm N HI at temperature T 2 (line HI), and a path at the constant chemical potential  r (line IE), which connects the two isotherms at T 1 and T 2 . The integration of the supercritical adsorption isotherm from  id (point H) to  r (point I) is expressed as: and then, the integration along the path IE at the constant chemical potential  r from T 2 to T 1 is expressed as: where N IE is the number of particles and E IE is the sum of the potential energy and the kinetic energy (3NkT/2) along the line IE. Therefore, the grand thermodynamic potential  EG is obtained as: Finally, the true phase equilibrium is determined as the point of intersection between the grand thermodynamic potentials,  AC and  EG .
By the GCMC method, we generated an adsorption isotherm of H 2 O at 298 K and a supercritical adsorption isotherm at 700 K, and a path at the constant chemical potential  r * =  r / O = 69.15, which connects the two isotherms. We performed the GCMC simulations at seven different temperatures to obtain the path at the constant chemical potential of  r * = 69.15: 650 K, 600 K, 550 K, 500 K, 450 K, 400 K and 350 K. The H 2 O adsorption isotherms of the ZTC models with and without compression by the GCMC method were obtained as well (Supplementary Fig. 3b).  Table 4). The LJ potential was applied for the interaction between CNT and the two diamond slabs. The TIP4P model (see Supplementary Table   5) was assumed for the H 2 O molecule and the SHAKE method was used to constrain the bonds of

MD Simulation for the translation of adsorbed H 2 O into H 2 O vapour by mechanical pressing
The MD simulation for forced H 2 O desorption from ZTC by compression was performed using the LAMMPS software package 7 . We cut out a structure from the 2×2×2 supercell of ZTC model previously developed by Nishihara et al. 9

Calculation of COP of RBPT using HFC-134a or water
The change from the state 1 to 2 in Fig. 6a is an isentropic process (S 1 = S 2 ). A refrigerant is saturated vapour at the state 1. It is turned into superheated vapour (state 2) by a compressor with an external work (W in ), which corresponds to the enthalpy change (H 2 -H 1 ). The change from the state 2 to 3 is an isobaric process (P 2 = P 3 ). The refrigerant is turned into saturated liquid by a condenser 20 at which the refrigerant exhausts the heat (Q H ) to outside of a target room to be cooled. Q H corresponds to the enthalpy change (H 3 -H 2 ). The change from the state 3 to 4 is an isenthalpic process (H 3 = H 4 ). An expansion valve (Fig. 6a) decreases the pressure of the refrigerant, and turns it into a mixture of gas and liquid. In this process, the evaporation heat and Joule-Thomson effect decrease the temperature. This is an irreversible process. The change from the state 4 to 1 is an isobaric process (P 4 = P 1 ). The refrigerant is turned into saturated vapour by an evaporator at which the refrigerant gains the heat (Q L ) from the target room, meaning that the room is cooled. Q L corresponds to the enthalpy change (H 1 -H 4 ).
The coefficient of performance (COP) is defined as follows; COP (11) When the pressure (P 1 ) of a refrigerant at the state 1 is assumed, the corresponding enthalpy of the saturation vapour (H 1 ) can be determined from the steam table of the refrigerant 10,11 . Then, by assuming the pressure (P 2 ) at the state 2, the corresponding enthalpy of the superheated vapour (H 2 ) is determined as a result of an isentropic change from state 1. The enthalpy (H 3 ) of saturated liquid at the state 3 is also determined from the assumption of pressure (P 3 = P 2 ). The enthalpy (H 4 ) at state 4 is equal to H 3 .
In Thus, COP eventually falls in 5.4, almost the same level as that in HFC-134a, along with the penalty of its huge system volume. This is why water is not used in conventional air conditioners.
When the pressures at the states 2&3 are increased to 1.3 kPa, COP decreases to 4.7.

Calculation of COP of RMPTA
Based on the prototype RMPTA system shown in Fig. 6b, a continuous refrigeration cycle can be designed as shown in Fig. 6e and f, and more details are illustrated in Supplementary Fig. 6.
Supplementary Fig. 6a is  (12) where w ns [kg] is the mass of the nanosponge, and w re-ads [kg] is the amount of refrigerant adsorbed in nanosponge at T H . Similarly, a sensible heat Q' sh is supplied from the outside to the condenser, whereas Q' sh is not involved in the calculation of COP. Since temperatures of the two chambers are changed, adsorption and desorption occur at the evaporator and the condenser, respectively (Supplementary Fig. 6b to c). For simplifying, it is assumed that adsorption/desorption occur after the temperatures of the condenser and the evaporator become T H and T L , respectively. Adsorption and desorption are associated by the generation of heats, Q c and Q' c , respectively, as shown in Supplementary Fig. 6c. Q c increases the temperature of the target room, but the following process offsets Q c . By adsorption/desorption, the pressures of the evaporator and condenser become P 2 and P 3 , respectively. The order of the pressures is P 2 < P 1 < P 3 . Next, as shown in Supplementary Fig.   6d, the nanosponge in the evaporator is compressed to desorb the refrigerant, to increase the pressure up to P 1 . By this desorption process, the heat (Q c ) is moved from the target room to the evaporator, and the temperature increase at Supplementary Fig. 6c is compensated. The necessary work for this process is W 1 [J], which a part of W 1 turns into heat (Q f1 ) to warm up the target room by the internal friction of the nanosponge if it is not perfectly elastic. In the condenser, the nanosponge is freely expanded, and adsorption occurs. The generated heat (Q' c ) is discharged to the outside. When the pressure becomes P 1 , the valve is opened, and the nanosponge in the evaporator is further compressed to the limit ( Supplementary Fig. 6e). . This is the amount of refrigerant which is used to indeed cool the target room, and is one of the most important parameter to determine COP.
Supplementary Fig. 6e is actually the same as the initial state ( Supplementary Fig. 6a). By repeating the cycle shown in Supplementary Fig. 6, continuous refrigeration is achieved.
Q L can be obtained as follows: where  vap H is the enthalpy change of the refrigerant from liquid to gas [J kg -1 ]. Note that the enthalpy change upon adsorption/desorption is almost the same as that of bulk phase transition.
The total work for the cycle is the sum of W 1 and W 2 , and it is denoted as W ns , while the total heat loss by the internal friction (Q f ) is obtained by Q f = Q f1 + Q f2 . By using equations (12) and (13), COP for this cycle can be described as follows; In equation (14), W ns depends on the elasticity of nanosponge. where F and S are force and cross-section area of nanosponge, respectively. The definition of ε d is as follows: where x is displacement and L 0 is the initial length of nanosponge.
W ns is the sum of the work which is necessary to deform blank nanosponge (W b ) and the work against the force derived from adsorption-induced pressure (W ai ). (18) By assuming that the Young's modulus is unchanged by the inclusion of refrigerant (see Supplementary Fig. 4), W b is expressed as follows: where V 0 is the initial volume of nanosponge [m 3 ].
W ai is described as follows: (20) where f ai is the adsorption-induced pressure which can be experimentally determined as shown in When a part of W b (the ratio is φ f ) is lost as Q f , Q f = φ f W b (0 ≤ φ f ≤ 1). As shown in Supplementary   Fig. 7b, φ f can be obtained from the hysteresis in stress-strain curves of a nanosponge material.
Accordingly, equation (14) can be deformed as follows:

The impact of nanosponge bulk modulus on COP
In equation (23), w re-ads , V 0 , w re can be expressed by w ns , using physical properties of nanosponge. Hereafter, the approximate relation between COP, E, and ε d is obtained when water is used as a refrigerant. In equation (23),  vap H and c re are 2453 kJ kg -1 and 4.184 kJ kg -1 K -1 , respectively. T H -T L is assumed to be 20 K. For c ns , the value of graphite (0.72 kJ kg -1 K -1 ) can be assumed. φ f is measured to be 0.25 in a ZTC/PTFE sheet, and this value is used for the rough calculation here.
Thus, equation (23)  To obtain W ai given by equation (21) In equation (34), ρ re is 1000 kg m -3 . ρ ns-ap of general porous materials is at most ca. 1000 kg m -3 . V ns can be assumed as at most about 0.003 m 3 kg -1 . P is assumed as 2.9 kPa (the same as that in Supplementary Fig. 7 and 8). φ ns is assumed to be 1. Accordingly, COP can be approximately expressed by E and ε d , as shown below: