Potential theory for gate adsorption on soft porous crystals

We demonstrate that an adsorption potential at the gate adsorption pressure of soft porous crystals (SPCs) based on the Polanyi's potential theory of adsorption shows a constancy to temperature. This was done using grand canonical Monte Carlo simulations and free energy analysis, which were carried out with a simplified stacked-layer SPC model. This finding implies that the characteristic curve obtained from an experimental gate adsorption isotherm on SPCs can be used to predict the temperature dependence of the gate-opening pressure, even though the potential theory of adsorption does not take into account the deformation of porous solids during the adsorption. We develop a modified potential theory for gate adsorption and show that the derived relation has a form that the Gibbs free energy change due to the host framework deformation per guest molecule, − ΔGhost/N, and a correction term, C, are added to the expression of the original potential theory of adsorption. The term C is not an empirical correction factor but is the difference of intermolecular interaction potential and entropy between the bulk liquid phase at the saturated state and the adsorbed phase, originating from spatial constraint of adsorbed guest molecules in the host. By evaluating the modified expression for gate adsorption using the simulation results, we demonstrate that the constancy of the adsorption potential to temperature results from a compensation effect between three terms: guest–host interaction potential per guest molecule, − ΔGhost/N and C, which have a temperature dependence.


Introduction
Porous metal -organic frameworks, also known as porous coordination polymers (PCPs), are a new class of nanoporous materials which have a wide range of crystal structures and host -guest properties. [1 -6] In particular, soft porous crystals (SPCs), [7] which were previously classified as the third-generation PCPs by Kitagawa and Kondo [4] and Kitagawa et al. [6], have attracted much attention because of their anomalous adsorption behaviour, called 'gate adsorption' [8] and 'breathing effect'. [9] Under this behaviour, guest molecules are adsorbed abruptly at a certain gas pressure accompanied by the deformation of the host framework. SPCs have various structural motifs such as one-dimensional chains, [10,11] two-dimensional (2D) interdigitated layers, [12,13] 2D elastic layered structures, [8,14] three-dimensional (3D) mutually interpenetrated frameworks, [15,16] 3D frameworks with breathing channels [9,17] and zeolitic frameworks. [18 -20] The high sensitivity of gate adsorption and the breathing effect to the gas pressure suggests potential applications such as gas storage, [21] separation, [22,23] adsorption heat pump [24] and molecular sensor [25]; however, the rational design of SPCs for a specific application is still challenging because it requires insight into the mechanisms of gate adsorption and the breathing effect. Thus, to shed light on the nature of adsorption-induced structural transitions, molecular simulation studies have been performed for various SPCs, [26 -34] but these transitions have not been completely understood yet.
Coudert et al. [35] first developed an osmotic thermodynamic model for gate adsorption and the breathing effect: they devised an analytical free energy analysis method to determine the change in the Helmholtz free energy of the host framework, DF host , during the adsorption-induced structural transition. Their method is simple and useful because it only requires fitting of a Langmuir isotherm to a plateau region of the experimental isotherm after the structural transition and integrating it to obtain the change in the grand potential of the guest, DV guest . The DF host value can be estimated such that the free energy change of the system, which is the sum of DF host , DV guest and PDV terms (P, gas pressure; DV, volume change of the host), becomes zero at the experimental gate adsorption pressure. Using their method, the temperature-loading phase diagrams of MIL-53 for Xe was successfully predicted, providing a good agreement with the experiment. [36] Neimark et al. [37,38] incorporated adsorption-induced stress exerted on SPCs into the osmotic thermodynamic model. They showed that the model provided a good description of the hysteresis phenomena experimentally observed during the breathing transitions for Xe and CH 4 adsorption on MIL-53, and concluded that the structural transition occurred when the stress reached a certain value that the host cannot resist.
Kanoh et al. applied the Hill equation, which was developed for demonstrating the allosteric effect on the binding of O 2 to haemoglobin (not a first-order phase transition but reaction equilibria), to reproduce the experimental adsorption isotherm of CO 2 on Elastic Layer-Structured Metal-Organic Framework-11 (ELM-11), taking advantage of the similarities between the enzyme -substrate complex and ELM-11 -CO 2 clathrate. [21] They found a linear correlation between the logarithm of the equilibrium constant obtained from the fitting of the Hill equation and the reciprocal temperature, which enables the prediction of the temperature dependence of the gate adsorption pressure. A linear correlation between the logarithm of the gate adsorption pressure and the reciprocal temperature was also reported in several papers, [10,39 -41] and the thermodynamic theory of the phenomenon has been well established with the aid of the molecular simulations in our previous work. [30] Moreover, Sircar et al. [42] reported that the 'universal adsorption theory' based on the Clausius -Clapeyron equation and the principle of corresponding states could quantitatively predict the gateopening pressure of Cu(2,5-dihydroxybenzonic acid) 2 (4,4 0bipyridine) [Cu(dhbc) 2 (4,4 0 -bpy)] for various adsorbates. However, more than two experimental adsorption isotherms at different temperatures are needed to predict the temperature dependence of the adsorption-induced structural transition using the above-mentioned theorems.
Recently, Yamazaki et al. [43] reported that, based on Polanyi's [44] potential theory of adsorption, a temperature-invariant 'characteristic curve' was obtained from the experimental adsorption isotherms on one of the SPCs [Cu (dhbc) 2 (4,4 0 -bpy)]. This suggests that the gate adsorption pressure at any temperature can be predicted by the characteristic curve obtained from only one experimental adsorption isotherm. This is quite surprising because the potential theory of adsorption assumes that porous solids do not deform during the adsorption. To the best of our knowledge, this is the first time that the potential theory of adsorption has been applied for the gate adsorption behaviour, though it has been used for describing various adsorption phenomena on general rigid porous materials. [45 -52] However, it is still unclear why the theory is applicable to a system showing the structural transition of the host framework by guest adsorption.
In order to address this point, in this study, we developed a thermodynamic theory of gate adsorption based on the Polanyi's potential theory of adsorption, and verified it via the results obtained from the molecular simulations for adsorption of Lennard-Jones (LJ) fluid on a simplified stacked-layer SPC model. The Polanyi's potential theory of adsorption is ordinarily applied when the adsorbed phase can be assumed to be identical to the saturated bulk liquid at a temperature between the triple point and the critical one, though it is frequently applied above the critical temperature, assuming a pseudo-vapour pressure to calculate the adsorption potential. [44][45][46] The reduced triple point temperature and the critical temperature of the LJ fluid are 0.7 [53] and 1.3 [54], respectively, (reduced temperature T * ¼ k B T/1; k B , the Boltzmann constant; T, temperature; 1, the LJ well-depth parameter), and we, therefore, discuss this issue over the temperature range of T * ¼ 0.8-1.2, which covers the theory's range of application. Figure 1 shows a simplified model of a stacked-layer SPC, which was used in our previous studies [30,32]; this consists of smeared-atom layers, which have uniform 2D solid density, and pillaring atoms located on one side of the layers. The interlayer width of the host framework is h, and it is assumed that the h value for all the neighbouring layers is the same.

Simulation models and theory
The guest -guest interaction potential, f gg , was modelled with the 12-6 LJ potential; the potential parameters used were those of argon (s gg ¼ 0.341 nm and 1 gg /k B ¼ 119.8 K). The interaction potential between a guest molecule and each host layer, f gh , was calculated using the 10-4 LJ potential: where r h is the atomic number density of the host layer (which was set to be r * h ¼ r h s 2 gg ¼ 2:2), and z ik is the distance between the ith guest molecule and the kth layer. The LJ parameters for the guest -host layer interaction potential, s gh and 1 gh , were calculated according to the Lorentz -Berthelot mixing rules [(s gg þ s hh )/2 ¼ s gh and (1 gg 1 hh ) 1/2 ¼ 1 gh ], where the interaction parameters for the host layer were set to be s hh ¼ 0.34 nm and 1 hh /k B ¼ 28 K.  s hh h kl where h kl is the distance between the kth and the lth layers, which can be expressed as (l 2 k)h. The host layer -pillar interaction, f hp , and host pillar-pillar interaction, f pp , were calculated by the 10-4 LJ and the 12-6 LJ potentials, respectively. The LJ parameters of the pillar were the same as those of the guest molecule. The number density of the pillar was set to be 1/(100 s 2 gg ). The guest -pillar interaction was neglected because of the low density of the pillars. Thus, total inter-framework potential of the host, F hh , is expressed as where N L is the total number of the host layers, L x and L y are lengths of the layer in the x-y layer directions, d p is the length of the pillar (distance from a layer to an attached pillar) and r kl is the distance between the pillars located at different layers. The interlayer width, h 0 , at the degassed state was set to be 1.75s gg such that F hh can be minimised by tuning the pillar length d p . It is also worth noting that the work required to expand the interlayer spacing against the interlayer attractive force is increased as the pillar length is decreased, and finally for the system with h 0 ¼ 1.3s gg , the gate adsorption is definitely not observed because the global minimum of the free energy of the system does not switch from the closed state to the open state below the saturated vapour pressure [32] (see also Section 3.1 for details).
All the potentials (f gg , f gh , f hh , f hp and f pp ) were cut and shifted at the distance of 5 s gg . As shown in Figure 1, the simulation cell has seven unit cells, which contain one layer and an interlayer space. The size of the unit cell was set to be L x £ L y ¼ 10s gg £ 10s gg in the x-y layer directions and 1.60s gg 2 2.05s gg in the z-direction normal to the layers. Periodic boundary conditions were applied for all the directions.
Grand canonical Monte Carlo (GCMC) simulations were conducted to obtain the adsorption isotherms for various interlayer widths (1.60s gg 2 2.05s gg ) with a step of 0.01s gg . The reduced temperature, T * ¼ k B T=1 gg , was changed from 0.8 to 1.2. The length of the simulation run was at least 2.5 £ 10 7 steps and 2 £ 10 8 steps for the equilibration and sampling, respectively. The relation between the bulk gas pressure and the chemical potential was obtained from the Johnson-Zollweg-Gubbins [55] equation of state for the LJ fluid (LJ-EOS).
The thermodynamic states of the simplified stackedlayer SPC model at each gas pressure and interlayer width h were determined by calculating the osmotic free energy, V OS , as [35]: where m is the chemical potential of the adsorbed guest molecules, F host is the Helmholtz free energy of the host, P is the bulk gas pressure at m, V is the volume of the host with the interlayer width h and V guest is the grand thermodynamic potential of the adsorbed guest molecules. The grand thermodynamic potential can be calculated by integrating the GCMC adsorption isotherm, N(m, h), with respect to the chemical potential under constant T and V(h) as The osmotic free energy change, DV OS , with the change of the interlayer distance from h 0 to h at constant m is given by where DF host , DV and DV guest are the changes in the Helmholtz free energy of the host, the volume of the host and the grand thermodynamic potential, respectively. The Helmholtz free energy change of the host, DF host (h) ¼ F host (h) 2 F host (h 0 ), was approximated to the change in the total inter-framework potential, DF hh (h) ¼ F hh (h) 2 F hh (h 0 ), by assuming that the entropy change of the host is negligible. Figure 2 shows the typical osmotic free energy changes per unit cell, DV OS* ¼ DV OS /1 gg , as a function of the interlayer width and gas pressure at T * ¼ 1.0. The osmotic free energy profiles were obtained from Equation (6) using the GCMC adsorption isotherms. At zero pressure, the global minimum is located at h ¼ h 0 (closed state), and the system becomes unstable with opening the interlayer width from h 0 against the interlayer attractive force. The secondary minimum appears at h * ¼ h=s gg , 2 as the gas pressure increases, because the adsorbed guests stabilise the system. The closed state (h * 0 ¼ h 0 /s gg ¼ 1.75) and the open state (h * ¼ 1.99) become bistable at the reduced pressure of P * ¼ Ps gg 3 /1 gg ¼ 6.8 £ 10 24 . According to the theory of equilibrium, the adsorption-induced structural transition occurs at this pressure (hereafter, equilibrium gate adsorption pressure: P Ã gate ¼ P gate s gg 3 /1 gg ). However, if the energy fluctuation of the system is smaller than the energy barrier, E eq A , located at h * ¼ 1.87 between the two stable states, the equilibrium structural transition should not be observed, i.e. a further increase in the bulk gas pressure is required to reduce the energy barrier, to switch the global minimum from the closed state to the open state, and, finally, to cause a spontaneous structural transition (see DV OS* at P * ¼ 1.25 £ 10 23 in Figure 2; the energy barrier is designated as E op A ). Then, in the desorption process, the system stays at the open state until the energy barrier, E cl A , becomes smaller than the energy fluctuation of the system (see DV OS* at P * ¼ 5.4 £ 10 24 in Figure 2). This should be the reason why a hysteresis is experimentally observed during the adsorption-induced structural transition in SPCs; however, it is worth noting that the gate-closing pressure is much closer to the equilibrium gate adsorption pressure than the gate-opening pressure; a detailed mechanism of the structural transition has been reported in our previous studies. [30,32] Therefore, for the sake of simplicity, in this study, we only provide a discussion based on the theory of equilibrium.

Theoretical gate adsorption isotherms
The resulting gate adsorption isotherm at T * ¼ 1.0 obtained from the free energy analysis using the GCMC data is plotted in Figure 3. The adsorption isotherm remains zero at less than P * gate ¼ 6.8 £ 10 24 , because the interlayer width of the closed state is smaller than the size of the guest molecule and the DV OS* value is always positive (see Figure 2). Then, at P * gate , the adsorption isotherm shows a steep rise, because the global minimum of the system shifts from the closed state to the open state (see Figure 2) and the gate-opening is induced. Namely, the gate adsorption isotherm obtained here is that traces the adsorption points where the DV OS* value becomes minimum for each pressure. The gate adsorption isotherms at different temperatures in the range between T * ¼ 0.80 and 1.2 were also obtained in the same manner and are plotted in Figure 3. It is clear that the gate adsorption pressure increases, and the adsorption amount decreases after the gate-opening as the temperature increases. The interlayer width h * at the gate adsorption pressure P Ã gate was increased from 1.75 ( ¼ h 0 * ) to 1.99 at T * ¼ 0.80-1.1, and to 1.98 at T * ¼ 1.2.

Potential theory for gate adsorption
The characteristic curves based on the Polanyi's [44] potential theory of adsorption were obtained from the theoretical gate adsorption isotherms obtained as above. The adsorption volume, V guest , was calculated from the number of adsorbed guest molecules, N, and the number density of the saturated bulk liquid LJ argon, r liq , at the adsorption temperature where the bulk liquid density was taken from Lotfi et al. [54]. The adsorption potential A(P), which corresponds to the compression work to transform the gas phase at pressure P to the adsorbed phase, was calculated according to Polanyi [44] as where m liq (P 0 ) is the chemical potential of the bulk liquid  at saturated vapour pressure P 0 , [54] and m gas (P) is that of the gas phase at pressure P that can be obtained from LJ-EOS. [55] The characteristic curves ðV * guest ¼ V guest =s 3 gg vs: A * ¼ A=1 gg Þ obtained from the simulated gate adsorption isotherms using Equations (7) and (8) are shown in Figure 4. The characteristic curves over the range of temperatures from T * ¼ 0.8 to 1.0 show good agreement with each other: the deviations of V * guest at A * (P 0 ) ¼ 0 are less than^2.4% from the centre value and those of the A * (P gate ) value are less than^0.64%. However, the characteristic curves at T * ¼ 1.1 and 1.2 have large deviations of V guest from these four characteristic curves, which should be because the compressibility of the adsorbate rapidly increases upon approaching the critical temperature T * ¼ 1.3 and the adsorbed phase is in a highly compressed state compared with that of the saturated bulk liquid at the adsorption temperature. We therefore re-evaluated the adsorption volume V guest according to Ozawa et al. [46] as where T b is the boiling temperature, r liq (T b ) is the number density of the saturated bulk liquid at T b (the data were taken from Lotfi et al. [54] in this study) and a ¼ 0.0025 is the thermal expansion coefficient of the superheated liquid. The re-calculated characteristic curves using Equations (8) and (9) are shown in Figure 5. The convergence of V * guest at A * (P 0 ) ¼ 0 was considerably improved and the deviations were decreased to less than^1.3% over the whole range of temperatures T * ¼ 0.8-1.2; however, the convergence just after the gate adsorption becomes worse, which suggests that the compressibility of the adsorbed phase depends on the gas pressure.
While the adsorption potential at the gate adsorption pressure A * (P gate ) is almost invariant to temperature in the whole temperature range without any modifications (the deviations are less than^2.0% from the centre value), which can well explain the experimental results reported by Yamazaki et al. [43]: the characteristic curves for methane adsorption on Cu(dhbc) 2 (4,4 0 -bpy) at T * ¼ 1.1 and 1.2 (1 gg / k B ¼ 148.1 K) are in good agreement. This is curious because the original potential theory of adsorption does not take into account the deformation of porous solids during the adsorption. We therefore constructed a model to derive a thermodynamic theory for the gate adsorption behaviour based on the Polanyi's potential theory. Figure 6 shows two isobaric-isothermal systems including an SPC crystal and a bulk gas phase. In one system (Figure 6(a), hereafter pretransition state), the SPC crystal is closed, and in the other system ( Figure 6    adsorbed in the SPC crystal, and therefore N t 2 N molecules are in the gas phase. The two systems are in equilibrium at pressure P gate , and thus the Gibbs free energies of the two systems are equal: This equation gives the following relation: where G pre host and G post host are the Gibbs free energy of the SPC crystals at the pre-and post-transition states, respectively; m gas (P gate ) is the chemical potential of the bulk gas phase at pressure P gate , and G guest is the Gibbs free energy of the adsorbed guest molecules. By defining the changes in the Gibbs free energy of the SPC crystal as DG host ¼ G post host 2 G pre host , Equation (11) can be rewritten as It should be noted that the first and second terms in the lefthand side of Equation (12) are not equal because we are considering the equilibrium relation between the pre-and post-transition states, not that between gas and guest in the post-transition state. If we assume that the adsorbed phase is similar to the bulk liquid phase as is the case of the original potential theory of adsorption, G guest can be expressed as where u gh is the guest-host interaction for one guest molecule. By substituting Equation (13) into Equation (12) and using Equation (8), we obtain Equation (14) indicates that, in the case of gate adsorption, the work for the host deformation per guest molecule, 2DG host /N, should be added to the expression of the original potential theory of adsorption: AðPÞ ¼ m liq ðP 0 Þ 2 m gas ðPÞ ¼ 2u gh . We tested Equation (14) using the GCMC data. The adsorption potentials for gate adsorption, A(P gate ), at each temperature were obtained from the results of the adsorption isotherms shown in Figure 3. On the other hand, the 2 u gh 2 DG host /N term of the right-hand side of Equation (14) was estimated separately. In particular, u gh and N were taken from the results of the GCMC simulations, and DG host was calculated by Equation (3) with the relation of DG host ¼ DF host þ P gate DV host < DF hh þ P gate DV host (DV host , change in the volume of SPC between the pre-and post-transition states). A comparison between A(P gate ) and 2 u gh 2 DG host /N is shown in Figure 7. The value of 2 u gh 2 DG host /N is almost independent of the temperature along with A(P gate ); however, a marked difference can be observed between the two values at all temperatures, indicating that Equation (14) is not adequate. This error possibly originates from the assumption made in the derivation of Equation (14) (the adsorbed molecules are in the bulk liquid state). This suggests that, as shown in Figure 1, the adsorbed guest molecules form a 2D-like liquid and have smaller coordination number than the 3D bulk liquid; therefore, the intermolecular interaction between the guest molecules is overestimated. Moreover, the entropy of the guest molecules should also be different from those of the bulk liquid, i.e. m liq (P 0 ) in Equation (14) should be replaced by m guest (P 0 0 ). Therefore, the equation can be rewritten as where m guest P 0 0 À Á is the chemical potential of the guest molecule excluding the contribution of the guest -host interaction at pressure P 0 0 , which is not the saturated vapour pressure but the pressure required for condensing the guest molecules between the host layers. Substituting m gas (P gate ) ¼ A(P gate ) 2 m liq (P 0 ) from Equation (14) into Equation (15), we obtain Using the thermodynamic state function, G ¼ mN ¼ U 2 TS þ PV (U, internal energy; S, entropy) and U ¼ 3k B TN/2 þ uN (u, intermolecular interaction potential per molecule), m liq (P 0 ) and m guest P 0 0 À Á can be  (8), 2u gh 2 DG host /N: the right-hand side of Equation (14) and 2u gh 2 DG host /N þ C: the right-hand side of Equation (16), which are reduced by 1 gg , at T * ¼ k B T=1 gg ¼ 0.8, 0.85, 0.9, 0.95, 1.0, 1.1 and 1.2. u gh is the guest-host interaction potential per guest molecule, DG host /N is the Gibbs free energy change due to the host framework deformation per guest molecule and C is a correction term originating from spatial constraint of adsorbed guest molecules in the host, respectively. The lines are guide to eyes.
represented as and m guest P 0 where u liq is the intermolecular interaction potential per molecule of the bulk liquid, u gg is the guest-guest interaction potential per molecule of the adsorbed phase, s liq and s guest are entropies per molecule of the bulk liquid and adsorbed phase, v liq and v guest are the volumes per molecule of the bulk liquid and adsorbed phase, respectively. Finally, the difference between Equations (17) and (18) can be arranged as where the P 0 v liq 2 P 0 0 v guest term was neglected because of its small quantity. By substituting Equation (19) into Equation (16), we obtain We tested the modified expression in Equation (20) by using the simulation data. The u liq and s liq values were taken from Johnson et al. [55]; u guest was obtained from the GCMC simulations. Then, the entropy of the guest molecule, s guest , was calculated using the following expression: All the terms in the right-hand side of Equation (21) were obtained from the GCMC simulations.
The obtained value of the right-hand side of Equation (20) is plotted as a function of the temperature in Figure 7, and the typical terms in Equations (17) - (21) at each temperature are listed in Table 1. The good agreement between the terms A(P gate ) and 2 u gh 2 DG host /N þ C (the error is , 4%) suggests the validity of Equation (20). Data in Table 1 clearly show that the C value mainly results from the large difference between u liq and u guest [because of the smaller coordination number of the guest molecules in the confined space (ca. 6) than that of the bulk liquid (ca. 12)] and not from the contribution of the entropy term, which is negligibly small. Thus, the inadequacy of Equation (14) can be attributed to the assumption that the adsorbed molecules are in the bulk liquid state, as stated above. Moreover, the constancy of the right-hand side of Equation (20) to temperature is due to a compensation effect between the three terms 2 u gh , 2 DG host /N and C, which show a temperature dependence.
These findings confirm that the gate adsorption pressure at any temperature in the range of T * ¼ 0.8-1.2 can be predicted by converting experimental gate adsorption data at one temperature into A(P gate ) based on the original potential theory of adsorption, though the correct thermodynamic expression for the gate adsorption should be Equation (20). The conversion can be made by assuming the ideal gas law as A(P gate ) ¼ k B T ln(P 0 /P gate ); however, it gives a worse convergence (the deviations are less than^6.5% from the centre value, see Table S1, Supplemental Data) than that (^2.0%) using m liq (P 0 ) 2 m gas (P gate ) (Equation (8)). Therefore, it is desirable to calculate the adsorption potential using k B T ln( f 0 /f gate ), where f 0 and f gate are fugacities at the saturated vapour pressure and the gate adsorption pressure, according to the relation k B T ln( f 0 /f gate ) ¼ m liq (P 0 ) 2 m gas (P gate ). Then, it is worth noting that the difference between the terms A (P gate ) and 2 u gh 2 DG host /N þ C slightly increases with increasing temperature above T * ¼ 1.0, which should be because of the neglect of the P 0 v liq 2 P 0 0 v guest term in the definition of C (Equation (19)). We thus calculated C m ¼ C þ P 0 v liq 2 P gate v guest by assuming P 0 0 ¼ P gate , and confirmed that a perfect agreement between the terms A(P gate ) and 2 u gh 2 DG host /N þ C m was obtained (the errors are less than^0.53%) over the whole range of temperatures T * ¼ 0.8-1.2 (see Table S2 and Figure S1, Supplemental Data). It should also be noted that, in our modified potential theory for gate adsorption, the temperature dependence of the internal energy of the host, U host , and the entropy change, DS host (T) , due to the host deformation, are neglected by assuming DG host ¼ DF host þ P gate DV host < DF hh þ P gate DV host . If the DS host value is too large for a real SPC, then DG host should change with the temperature, according to the thermodynamic state function DF host (T) ¼ DU host 2 TDS host (T). Therefore, the constancy of A (P gate ) to temperature should not be observed experimentally. Moreover, if the guest molecules are highly confined in the host framework (e.g. only one guest molecule is isolated in a narrow cavity of the host), the V guest values after the gate opening at different temperatures would show some deviations because the r liq value changes with temperature.

Conclusions
In this work, we assessed the characteristic curves transformed from the theoretical gate adsorption isotherms at several temperatures, which were obtained from the GCMC simulations for the simplified stacked-layer SPC model and free energy analysis, based on the Polanyi's potential theory of adsorption. The obtained theoretical characteristic curves showed a good agreement with each other; this can well demonstrate the experimental observations reported by Yamazaki et al. [43] and suggests that a characteristic curve obtained from an experimental gate adsorption isotherm can be used to predict the gate adsorption pressure at any temperature.
We also developed a modified thermodynamic theory for gate adsorption on the basis of Polanyi's potential theory of adsorption (Equation (20)) and verified it using the simulation results for the simplified stacked-layer SPC model. The derived relation shows that the work for the host deformation per guest molecule 2 DG host /N and the correction term C originating from the spatial constraint of adsorbed guest molecules in the host framework are added to the expression of the original potential theory of adsorption. The thermodynamic formulation and extension of the potential theory of adsorption to the SPC system are unprecedented. Moreover, the introduction of the dimensionality effect on the adsorbed phase (the correction term C), which can also have an impact to raise controversial feature of the application of the original potential theory of adsorption for the rigid porous materials with narrow adsorption space, should give a new physical insight into the confinement of fluid in a lowdimensional pore.
By evaluating the modified expression for gate adsorption with the use of the simulation results, we confirmed that the constancy of the adsorption potential A (P gate ) to temperature results from a compensation effect between the three terms with a temperature dependence: guest -host interaction potential per molecule, 2 DG host /N and C. However, if the entropy change due to the host deformation is significant in a real SPC, the DF host value would change with the temperature, and thus the characteristic curves obtained from the experimental gate adsorption isotherms on the SPC at different temperatures would no longer coincide with each other.
The next challenge is to assess the applicability of the modified potential theory for gate adsorption to understand the process of adsorption-induced structural transitions in SPCs through meta-stabilised and activated states, which cause hysteresis phenomenon. Several related investigations using a real SPC are currently in progress.

Disclosure statement
No potential conflict of interest was reported by the authors.

Supplemental data
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