Adsorption-Induced Structural Transition of an Interpenetrated Porous

Article pubs.acs.org/Langmuir

Adsorption-Induced Structural Transition of an Interpenetrated Porous Coordination Polymer: Detailed Exploration of Free Energy Profiles Hayato Sugiyama, Satoshi Watanabe, Hideki Tanaka, and Minoru T. Miyahara* Department of Chemical Engineering, Kyoto University, Katsura, Nishikyo, Kyoto 615-8510, Japan ABSTRACT: Specific types of coordination polymers show an adsorption-induced structural transition, or so-called “gate adsorption”, in which a host framework is said to change its structure from a “closed” nonporous phase to an “open” porous one for guest molecules. To identify the pathway for such a structural transition, we perform grand canonical Monte Carlo simulations for the adsorption of guest molecules in a host interpenetrated framework and calculate the free energy profiles of the structural changes in a complete threedimensional space. In addition to the open phase found in our previous analyses along a fixed one-dimensional path, we reveal the existence of another open configuration. Each of the two open phases yields the status of global minimum to the other depending on the external pressure, resulting in a two-step isotherm. Moreover, the shape of adsorption hysteresis associated with the structural transition can change depending on the energy barrier between a metastable and a stable state that the system can overcome. Our simulations not only give a comprehensive understanding of stepped isotherms observed empirically but also suggest that isotherms with hysteretic gate adsorption is closely related to the thermal fluctuation of the system.

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INTRODUCTION Porous coordination polymers (PCPs) or metal−organic frameworks (MOFs) have attracted much attention during the past decade.1−3 They are synthesized under mild conditions utilizing the self-assembly of metal ions as connectors and organic ligands as pillars. A number of combinations of the connector and pillar lead to enormous variations in their pore size and shape. Given their finely regular micropores with large specific surface areas, they are promising for such applications as gas storage, separation, and catalysis. In addition to these characteristics, what makes PCPs more attractive is the flexibility in their frameworks,4 of which several types have been observed including one-dimensional channels,5−7 twodimensional stacking layers and bilayers,8−13 pillared layers,14,15 one-dimensional linear-chain structures,16 interdigitated motifs,17−21 and interpenetrating frameworks.17,22−26 These PCPs show a peculiar adsorption phenomenon, a general feature of which is a stepwise increase in the adsorption amount at a certain pressure with a pronounced adsorption/desorption hysteresis. In this adsorption process, flexible PCPs are said to transform from a closed phase to an open phase at a certain adsorption pressure. This phenomenon is called “gate adsorption” and has potential applications in separation, molecular sensing, and actuators. For these applications, it is desirable to control adsorption/desorption pressures by tailored synthesis of PCPs. The gate-opening pressure changes depending on guest molecules as well as host frameworks, and a specific combination of host and guest can even result in adsorption isotherms with more than one step.27,28 This in turn © 2012 American Chemical Society

requires a deep understanding of the gate adsorption phenomenon. To clarify the mechanism of gate adsorption, the experimental approach is most straightforward. Although several stepped isotherms due to gate-opening phenomena have been reported,29,30 there have been a limited number of experimental studies on the transformation process of flexible PCP configurations during the gate adsorption process, probably because of the complexity of the phenomenon as well as the lack of appropriate experimental apparatuses. Takamizawa et al.5 have insisted that ethanol adsorption in Cu complex ([Cu2(bza)4(pyz)]n, bza = benzoate, pyz = pyrazine) shows the first-order phase transition judging from a slope of the adsorption enthalpy. Llewellyn et al.31 have also revealed that CO2 adsorption in MIL-53(Cr) shows coexistence phase of large pore phase and narrow pore phase at the transition pressure, which indicates the first-order phase transition. Millange et al.,32 by using time-resolved energy-dispersive Xray diffraction, have revealed that a flexible MOF (MIL-53) goes through an intermediate phase while replacing adsorbed water molecules with methanol molecules. One explanation they offered for this result is that, judging from its cell volume, the intermediate phase lies between a water phase and a methanol phase, containing a mixture of water and methanol. Kanoh et al. 13 have insisted that a CO 2 adsorption Received: December 22, 2011 Revised: February 25, 2012 Published: February 27, 2012 5093

dx.doi.org/10.1021/la205063f | Langmuir 2012, 28, 5093−5100

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εff/kB = 148.1 K, where kB is the Boltzmann constant. The fluid−fluid interaction, ϕff, was modeled using the LJ 12−6 potential and written as

phenomenon in an elastic layer-structured metal−organic framework (ELM-11) is governed not by a physical adsorption but by a chemical reaction, assuming that a clathrate compound is formed between CO2 and ELM-11. Tanaka et al.19 have demonstrated that a gate-opening kinetic process can be explained by assuming an intermediate phase between a closed phase and an open phase for an interdigitated framework. However, the origin of the gate adsorption phenomenon is not completely understood based on experimental analyses. Recently, considerable efforts have been devoted to reproduce the gate adsorption isotherms and clarify the mechanism using computer simulations and theoretical approaches.33 Salles et al. have conducted molecular dynamics (MD) simulations of CO2 adsorption in MIL-53(Cr) and demonstrated that the number of CO2 molecules accommodated in the framework defines its structural form.34 DFT calculations by Walker et al. have demonstrated bistability of the MIL-53(Al) framework.35 Ghoufi and Maurin utilized a hybrid simulation technique, which combines the grand canonical Monte Carlo (GCMC) method for guest CO2 adsorption with MD for the volume change in host framework of MIL-53(Cr), and successfully captured the first structural transition but failed in the second one.36 Triguero et al. proposed a possible mechanism of the MIL-53 structural transition to be layer-by-layer shear and qualitatively reproduced a stepped adsorption isotherm experimentally observed.37 Taking a theoretical approach, Coudert et al. developed a general thermodynamic scheme for understanding the guest-induced structural transition.38 Neimark et al. proposed a stress-based model to describe the hysteretic structural transition of MIL-53 materials.39 However, the mechanism of the gate adsorption process is not yet fully clarified. Intrinsic difficulty involved in simulation studies is in a huge time-scale difference between the host framework and the guest molecules. In our previous study,40 we avoided the difficulty of analyzing the structural transition by combining GCMC simulations and a thermodynamic integration method to calculate a free energy landscape along a fixed transition coordinate. Our calculations demonstrated that the transition occurs under an equilibrium condition and that the equilibrium pressure for gate adsorption is determined by the balance between two conflicting contributions: an energy penalty in creating adsorption spaces in a host framework and energy stabilization given by the adsorption of guest molecules. Although the free energy analysis in our previous study was quite effective for grasping the physical essence of the structural transition,40 the transition path was hypothetically restricted to a diagonal or transverse direction in a mutually interpenetrating framework. Therefore, in the present study, we remove this limitation and examine free energy profiles in a complete threedimensional space to identify the structural transition pathway. We conduct GCMC simulations for the adsorption of guest molecules in a coarse-grained model interpenetrating framework and then calculate free energy profiles including the contributions of the host framework and the adsorbed molecules. We also discuss the relation between the energy barriers for adsorption and desorption and the width of a hysteresis loop specific to the gate adsorption.

⎧⎛ ⎞12 ⎛ ⎞6⎫ ⎪ σ ⎪ σ ϕff (rff ) = 4εff ⎨⎜ ff ⎟ − ⎜ ff ⎟ ⎬ ⎪⎝ rff ⎠ ⎝ rff ⎠ ⎪ ⎩ ⎭

(1)

where rff is the intermolecular distance. The potential cutoff distance was set to be 5σff. Figure 1 shows the simulation box of the mutually interpenetrating structure consisting of two identical cubic-

Figure 1. Schematic illustration of the simulation box of a mutually interpenetrating jungle gym framework.

lattice rigid frameworks, each of which we call a jungle gym (JG) structure modeled with uniform structureless rods. The simplified JG framework is the same as that used in our previous studies.40−42 The center-to-center distance between two adjacent parallel rods in a JG was defined as the pore width, d, which was set to be 3.2σff. A minimum constituent unit of a JG is a cube with a volume of d3. The simulation box contains 5 × 5 × 5 cubes for all simulations. The position of one JG relative to the other is defined by the position R = (Rx*, Ry*, Rz*), where Rx* = Rx/σff and Rx is the relative gap between two JGs along the x-coordinate. The configuration where an intersection of rods in one JG framework is located in the middle of a cube in the other JG is set to be R = (0, 0, 0). This configuration is referred to as the “middle float” configuration. Note that the complete overlap of two JGs, which is impossible in reality, is defined as the position R = (d*/2, d*/2, d*/2), where d* = d/σff. The fluid−rod interaction was modeled using the LJ “11−5” potential, which was derived by a curvilinear integral of eq 1. It is given by 11 ⎧ ⎫ ⎛ σsf ⎞5⎪ 3πεsf ρ l σsf ⎪ 21 ⎛ σsf ⎞ ⎨ ⎜ ⎟ −⎜ ⎟ ⎬ ϕsf (rsf ) = ⎪ 32 ⎝ rsf ⎠ 2 ⎝ rsf ⎠ ⎪ ⎭ ⎩

■

(2)

where rsf is the distance between the center of a rod and that of a fluid molecule, ρl is the line density of molecules in a rod, and σsf and εsf are the LJ cross parameters for the fluid−rod interaction given by the Lorentz−Berthelot mixing rules. In the

MODEL AND SIMULATION Potential Models. We used Lennard-Jones (LJ) methane as the fluid molecule with the parameters of σff = 0.381 nm and 5094


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Figure 2. (left) Cross-sectional Rx*−Ry* surfaces of three-dimensional free energy profiles at Rz* = 0.0 with each contour at the base of the graphs. The free energy per simulation box is normalized by kBT. (right) Profiles along the diagonal direction on the Rx*−Ry* surface for the total free energy, ΔΩtotal (triangles), the direct free energy, ΔΩdir (squares), and the fluid free energy, ΔΩfld (circles). Both columns show results at (a) a low relative pressure of 0.025, (b) an intermediate relative pressure of 0.092, and (c) a high relative pressure of 0.30.

present study, the constituent molecule of the JG rods was benzene, with the LJ parameters of σss = 0.5349 nm and εss/kB = 412.3 K, aligned with line density, ρl, of 1/σss. The potential energy between a fluid molecule and the interpenetrating JG framework was given by the summation of the fluid−rod interaction potentials (eq 2). A triply overlapped interaction potential at an intersection of three rods was corrected by twice subtracting the LJ 12−6 potential interaction between a constituent molecule of a rod and a fluid molecule. The calculated potential energy profile in the framework was tabulated to speed up the calculation. Simulation Method. We conducted GCMC simulations of the adsorption of fluid molecules in the host framework. In each simulation, three trial types (translation, insertion, and removal) were performed 20 000 times per molecule while the JG framework was fixed. The system was equilibrated for the first 10 000 steps of the total MC run. The force acting on a JG was averaged over the last 10 000 steps. This was required for the thermodynamic integration method described in the next

section to calculate free energy profiles. Periodic boundary conditions were imposed for all directions. The overall density, ρ*, of fluid adsorbed in the framework is ρ* = ρσff 3

with ρ =

⟨N ⟩ V

(3)

where ⟨N⟩ is the ensemble average of the number of molecules in the framework and V is the volume of the simulation box including the volume of the rods. The pressure, p, is given by the equation

⎛ p⎞ μ − μs = kBT ln⎜⎜ ⎟⎟ ⎝ ps ⎠

(4)

where T is the temperature and μs the chemical potential corresponding to the saturated vapor pressure, ps, at T = 148.1 K. We set μs to be −10.81kBT.43 All simulations were carried out at the temperature of T* = kBT/εff = 1.0. 5095


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fluid contribution, Ffld, is given by the ensemble average of the forces from the adsorbed molecules acting on a JG framework. It is calculated using eq 8, which is derived from the differential of the fluid−rod interaction potential equation, eq 2.

Free Energy Calculation. The total free energy of a system, which includes the host framework and guest molecules, was obtained using a thermodynamic integration method. For force calculations in the free energy analysis method, we conducted a GCMC simulation to obtain an equilibrium state with a fixed gap, R, between two JGs. In the thermodynamic integration method, the grand free energy change, ΔΩij, from position Ri to Rj, is given by44 R j ⎛ ∂Ω ⎞

ΔΩij = −

∫R

=−

∫R

⎜

⎟

⎝ ∂R ⎠

i

Rj

Ffld(rsf ) =

■

⟨F(R)⟩ dR

i

j

≈ − ∑ ⟨F(R k)⟩ΔR k (5)

where ⟨F(R)⟩ is the mean force from the surroundings acting on a JG against its movement along the transition coordinate. We calculated the mean force, ⟨F(Rk)⟩, from a GCMC simulation run with a fixed gap, Rk. We then shifted the relative position of a JG by ΔR along one coordinate, fixing the two others. We set the increment of the gap, ΔR, to be 0.02σff. Repeating the simulation with varied gap distances between the two JGs along the Rx, Ry, and Rz directions, we obtained a free energy profile of the interpenetrating JG for a given μ by integrating the mean forces of a series of equilibrium simulations. By scanning along all x, y, and z directions instead of assuming a transition path as in the previous study,40 the calculated free energy profile covered all possible configurations in the complete three-dimensional space. The above procedure was applied to various settings in μ. It should be noted that the total free energy obtained by our approach is essentially the same as that from the osmotic ensemble applied by Coudert et al.38 because our system has no volume change accompanied by the structural transition displacing the relative position between two JG frameworks. The mean force is the sum of the direct interaction between two JGs, Fdir, and the fluid contribution, Ffld. The interaction potential between two JGs, Udir(R), is composed of the sum of the potentials between a rod of finite length, L, and a rod of infinite length with parallel alignment, Wparallel, and skew, Wskew: Wparallel(rss) =

Figure 3. Adsorption amount profiles along a diagonal line on a cross section at Rz* = 0.0 for three different relative pressures of 0.30 (squares), 0.092 (circles), and 0.025 (triangles). The displacement is identical to the horizontal axis of the free energy profiles in the right column of Figure 2. The middle float configuration is allocated at displacement 0. The two dotted lines with arrows indicate the side contact configurations with a reduced displacement of 0.28.

11 ⎧ ⎫ ⎛ σ ⎞5⎪ 3πεssρ l 2 σssL ⎪ 21 ⎛ σss ⎞ ⎨ ⎜ ⎟ − ⎜ ss ⎟ ⎬ ⎪ 32 ⎝ rss ⎠ 2 ⎝ rss ⎠ ⎪ ⎭ ⎩ (6)

Wskew(rss) =

3πεssρ l 2 σss2 2

∫θ

10 ⎧ θ2 ⎪ 21 ⎛ σss ⎞

1

amount at three different pressures plotted against the displacement of one JG relative to the other. As seen in Figure 2a, the MF phase is the most stable state at the low pressure of p/ps = 0.025. The MF structure has no space to accommodate guest molecules (Figure 3). Shifting the relative position of the JGs creates adsorption space for guest molecules by sacrificing the stability of the framework itself. Given that the adsorbed amount is small at this low pressure of p/ps = 0.025 (Figure 3), the destabilization of the framework predominates the stabilization by guest adsorption, resulting in the global minimum at the MF structure. At an intermediate pressure of p/ps = 0.092 (Figure 2b), four local minima, as indicated by the dotted arrows, appear in the total free energy profile because of the stabilization provided by increased amounts of adsorbed molecules at R = (±0.2, ± 0.2, 0.0). We refer to this

9θ

⎨ ⎜ ⎟ cos ⎪ 32 ⎝ rss ⎠ ⎩

⎫ ⎛ σ ⎞4 ⎪ − ⎜ ss ⎟ cos3 θ ⎬ dθ ⎪ ⎝ rss ⎠ ⎭

(8)

RESULTS AND DISCUSSION Three-Dimensional Free Energy Map. Figure 2a−c shows the cross-sectional Rx*−Ry* surfaces of the calculated three-dimensional total free energy profiles at Rz* = 0 (left column) and the free energy profiles along the diagonal directions on Rx*−Ry* surfaces extracted from the threedimensional profiles (right column) for three different pressures. The total free energy, ΔΩtotal, is the sum of the direct free energy from a JG, ΔΩdir, and the free energy from the adsorbed molecules, ΔΩfld. We set the base point of the free energy to be the middle float (MF) configuration of R = (0, 0, 0); thus, the free energies in Figure 2 are the differences from those of the MF configuration. Figure 3 shows the adsorbed

dR

k=i

12 ⎧ ⎫ ⎛ σ ⎞6⎪ 3πεsf ρ l ⎪ ⎛ σsf ⎞ ⎨231⎜ ⎟ − 160⎜ sf ⎟ ⎬ 64 ⎪ ⎝ rsf ⎠ ⎝ rsf ⎠ ⎪ ⎩ ⎭

(7)

where rss is the rod−rod distance, θ is the angle between the perpendicular line from the finite rod to the infinite rod and a line connecting an arbitrary point on the finite rod and the foot of the perpendicular on the infinite rod, and θ1 and θ2 are the angles when setting the arbitrary point to be an end and the other end of a finite rod. We used numerical differentiation of the potentials to calculate the direct interaction force, Fdir. The 5096


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Figure 4. Cross-sectional Rx*−Ry* surfaces of three-dimensional free energy maps for four Rz* positions at a relative pressure of 0.30. At (a) Rz* = 0.00, the middle float (MF) configuration is located at Rx* = Ry* = 0.0 (pink point), and the side contact (SC) configurations are shown at the four sites indicated by the dotted blue arrows. At (b) Rz* = 0.12, the peaks of the SC phase still occur at the same Rx*, Ry* coordinates. At (c) Rz* = 0.16, the diagonal contact (DC) configurations are located at the sites indicated by the red arrows (Rx* = Ry* = Rz* = 0.16). At (d) Rz* = 0.20, SC phases indicated by the dotted blue arrows reappear.

configuration, which is the same for each of the four minima because of symmetry, as the “side contact (SC)” configuration. The SC structure is a newly discovered configuration through a complete three-dimensional search carried out in the present study. A further increase in the pressure (Figure 2c) yields more adsorption, resulting in a global minimum for the SC configuration in place of the MF configuration. Figure 4 shows cross sections of the total free energy profiles for different Rz* values at the same relative pressure of p/ps = 0.30. In addition to the SC configuration corresponding to the four local minima as seen in Figure 4a at R = (±0.2, ± 0.2, 0.0) and Figure 4d at R = (±0.2, 0, 0.2) and (0, ±0.2, 0.2), the shift of the JG configuration in a diagonal direction creates another stable state, which is observed as four minima in Figure 4c. This stable configuration corresponds to a position of R = (0.16, 0.16, 0.16) and is named the “diagonal contact (DC)” configuration, which is identical to the contact configuration found in the previous study.40 Figure 5 schematically illustrates the three stable configurations with guest molecules. For clarity, the thickness of the JG framework is shown to be thinner, whereas the size of the adsorbed guest molecule is larger. The MF configuration (Figure 5a) has no adsorption space in a

Figure 5. Three types of configurations in a cube with adsorption molecules: (a) middle float (MF) configuration at R = (0, 0, 0), (b) diagonal contact (DC) configuration at R = (0.16, 0.16, 0.16), and (c) side contact (SC) configuration at R = (0.20, 0.20, 0). Red and blue frames represent two individual JG structures. Green spheres represent adsorbed molecules.

cube. The DC configuration (Figure 5b) has an adsorption space that accommodates one molecule per unit cube. The SC configuration (Figure 5c) has two adsorption spaces in a unit cube for two guest molecules. Further search of the free energy, however, found no stable states other than the three configurations described above. These results demonstrate that the global minimum state changes depending on the 5097


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relative pressures higher than p/ps = 0.045. Further increases in the relative pressure make the SC phase more stable than the DC phase at a pressure of p/ps = 0.25. The MF and DC phases are thus in thermodynamic equilibrium at p/ps = 0.045. Subsequently, the DC and the SC phases are in equilibrium at p/ps = 0.25. This result demonstrates that the MF-to-DC transition occurs first, after which the DC-to-SC transition follows. It should be noted that the MF-to-SC transition never occurs under equilibrium conditions because the MF configuration has already transformed into the DC configuration at the equilibrium pressure between the MF and SC configurations. Tracing the adsorbed amount for the global minimum configurations yields a two-step adsorption isotherm, as shown in Figure 6b. The mechanism of the stepped adsorption is quite different from that of a type V adsorption isotherm associated with multiple steps according to the IUPAC classification of adsorption behavior. In the case of gate adsorption, the steps of the isotherm originate from the adsorption-induced structural transition of a porous solid structure as shown in Figure 6. On the other hand, the steps of a type V isotherm result from a phase transition of confined fluids. In this way, gate adsorption can occur under thermodynamic equilibrium conditions. Our calculations suggest that even a gate-adsorption isotherm, reported to have only one step, may show a second step in a region of much higher pressure. Hysteresis of the Gate Adsorption. In the previous section, we calculated the adsorption isotherm in equilibrium. However, the system must overcome an energy barrier between two minima to reach the equilibrium. If the energy barrier is appreciably high, the host framework will stay in a metastable state within a certain finite time scale. Here we define the energy barrier as the difference in free energy between a metastable state and a saddle point that exists on a pathway from the metastable to the stable state. To estimate an energy barrier, we examined all the possible pathways from a metastable state to the stable one in the calculated 3D free energy profile and calculated a free energy difference between a metastable state and the maximum along each pathway. The pathway that gives the minimum free energy difference must go through a saddle point, and accordingly the energy barrier is obtained. Note that the energy barriers for adsorption and desorption are the same at an equilibrium pressure because the free energy profile is bistable. Figure 7a shows the energy barriers for the three possible transitions: MF-to-DC, MF-toSC, and DC-to-SC. The energy barrier for the DC-to-SC transition at equilibrium is the highest among the three, whereas that for the MF-to-SC transition is the lowest. The energy barrier for adsorption decreases as the relative pressure increases because the system is stabilized by the adsorbing guest molecules. Similarly, the energy barrier for desorption decreases as the pressure decreases. The threshold height for the energy barrier that the system can overcome would be closely related to the thermal fluctuation of the system. However, because a theoretical basis to the structural transition phenomenon in a grand canonical ensemble has not yet fully established, setting the threshold height would involve arbitrariness. Here we show in Figure 7b an example of an isotherm in which the threshold energy barrier is set to be 10kBT, resulting in hysteresis loops at the first and second steps. The limit point of the adsorption (desorption) branch of the isotherm corresponds to the pressure at the intersection of an adsorption (desorption)

external pressure. At low pressures, the MF configuration is the most stable, whereas at higher pressures, the SC or DC configuration becomes the global minimum state because of the stabilization provided by the adsorption of guest molecules. The question, then, is how and when the system transforms from the MF configuration to another stable configuration as the pressure increases. This will be discussed in the next section. Structural Transition Pathway. Figure 6a shows the free energy of the three stable configurations plotted against the

Figure 6. (a) Free energy curves of the three stable configurations as a function of the relative pressure. The solid line shows a transition path in thermodynamic equilibrium. (b) Adsorption isotherm (solid black line) of the overall density of adsorption molecules following the equilibrium transition path in (a). Chained, dotted, and broken lines represent the calculated adsorption densities of the SC, DC, and MF configurations, respectively.

relative pressure to investigate a pathway for the gate adsorption phenomenon in terms of the relative stability among the three. The total free energy of the MF structure remains constant because it has no space to accommodate guest molecules. Accordingly, ΔΩfld does not contribute to the total free energy. We set the constant free energy of the MF phase to be zero as a base point of the free energy landscape. As seen in Figure 6a, the free energy of the MF phase is the least among the three because steric repulsion between the rods is dominant in the DC and SC phases to make space for guest adsorptions at lower pressures of p/ps = 0.01−0.045. As the relative pressure increases, the free energies of the two other contact configurations decrease because the adsorption of guest molecules stabilizes the system. As a result, the free energy of the DC phase becomes lower than that of the MF phase under 5098


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sional search. There exist three stable states in all when considering the configuration without guest molecules (the middle float configuration). The global minimum state among the three configurations changes depending on the external pressure. Comparisons of free energy among the three yielded a transition path of the framework configuration from the middle float to the side contact by way of the diagonal contact configuration. Tracing the adsorbed amount of guest molecules for the global minimum configurations resulted in a two-step adsorption isotherm caused by two structural transitions of the host framework. Thus, our simulation demonstrates the possible appearance of a second step at a region of much higher pressure, even for interpenetrated PCPs showing onestep gate-adsorption isotherms. Furthermore, by setting a threshold height for the energy barrier for adsorption and desorption, hysteretic phenomena in adsorption and desorption can occur at the two transition pressures. Depending on the energy barrier and the threshold height that the system can overcome, the width of hysteresis loops associated with structural transitions can dramatically change.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by a Grant-in-Aid for Scientific Research (B) from MEXT. We thank the JSPS Research Fellowship and the Global Center of Excellence (GCOE) Programs.

Figure 7. (a) Energy barriers for three types of structural transition: MF-to-DC (blue), MF-to-SC (red), and DC-to-SC (green). Thick and thin curves represent the energy barriers for adsorption and desorption, respectively. (b) Isotherms for adsorption (solid black line) and desorption (broken gray line) with a threshold energy barrier height of 10kBT. Chained, dotted, and broken lines represent the calculated adsorption densities of the SC, DC, and MF configurations, respectively.

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REFERENCES

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energy barrier curve and the threshold height. Thus, hysteresis loops of gate adsorption isotherms would result from the existence of an energy barrier between a metastable and stable state. A characteristic feature of the isotherm is that the equilibrium pressure of the structural transition is close to the desorption branch rather than the adsorption branch because the slope for the desorption energy barrier is steeper than that for the adsorption energy barrier. Setting a low threshold height means that the system cannot make a transition until the energy barrier becomes sufficiently low. Thus, the system with a low threshold cannot overcome the energy barrier, resulting in a larger adsorption hysteresis, and vice versa. In this way, the shape of the isotherm strongly depends on the height of the energy barrier and the threshold height that the system can overcome, which is related to the thermal fluctuation of the system.

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CONCLUSION In the present study, we investigated the gate adsorption phenomenon of a coarse-grained mutually interpenetrating framework by conducting a three-dimensional free energy analysis based on GCMC simulations. In addition to the diagonal contact configuration found in a previous study,40 we newly revealed a stable configuration, which we named as the side contact configuration, through a complete three-dimen5099


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