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Free Energy Analysis for Adsorption-Induced Structural Transition of Colloidal Zeolitic Imidazolate Framework-8 Particles Shuji Ohsaki, Satoshi Watanabe, Hideki Tanaka, and Minoru T. Miyahara J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b06836 • Publication Date (Web): 24 Aug 2017 Downloaded from http://pubs.acs.org on August 25, 2017

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The Journal of Physical Chemistry

Free Energy Analysis for Adsorption-Induced Structural Transition of Colloidal Zeolitic Imidazolate Framework-8 Particles Shuji Ohsaki, Satoshi Watanabe*, Hideki Tanaka, and Minoru T. Miyahara** Department of Chemical Engineering, Kyoto University, Katsura, Nishikyo, Kyoto 615-8510 Japan AUTHOR INFORMATION Corresponding Author *[email protected] (S.W.) **[email protected] (M.T.M.)

ABSTRACT

Particle size and shape of flexible metal-organic frameworks have been reported to change the gate adsorption phenomenon which is characterized by an abrupt increase in the adsorbed amount induced by structural transitions of a host framework, although a detailed mechanism has not yet been clarified. Here, we focus on zeolitic imidazolate framework-8 (ZIF-8) whose structural flexibility stems from the rotation of 2-methylimidazole linkers. We perform free

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energy analyses with the aid of adsorption simulations to understand the dependence of the gate adsorption phenomenon on the particle size and shape of ZIF-8. For the adsorption simulations, we construct a simulation cell referred to as the nanoparticle model, which has a ZIF-8 slab structure and gas phase regions. Our simulations demonstrate that a decrease in the slab width results in a reduction of the average adsorbed amount. This can be explained by the smaller amount adsorbed in the region close to the surface of the ZIF-8 structure, which reaches a thickness of 1.0 nm from the surface and is not altered by slab widths or linker rotational angles. In the free energy analysis, the gate adsorption of smaller sized ZIF-8 particle models occurs at higher pressures because of less stabilization by the gas adsorption. On the other hand, the crystal planes have little impact on the gate opening and closing pressures because of the similar free energy stabilization by adsorption, although variations in the crystal planes significantly alter the adsorption isotherms. Furthermore, we propose a model to predict the gate opening and closing pressures of ZIF-8 particles with arbitrary sizes. We find quantitative agreement between the proposed model and experiments, which suggests that the near-surface region with less adsorbed amounts is the origin of the particle size dependence of the gate adsorption phenomena.

TOC GRAPHICS

KEYWORDS Porous coordination polymers, Metal-organic frameworks, Gate adsorption, Free energy analysis, Molecular simulation


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1. Introduction Zeolitic imidazolate framework-8 (ZIF-8)1-4 consisting of zinc ions and 2-methylimidazolate (MIM) linkers is part of a family of porous coordination polymers (PCPs) or metal-organic frameworks (MOFs),5-6 which are promising for various applications such as gas storages,7-9 separations,10-12 catalysts,13-15 and drug delivery systems16-18 due to their high porosity, large surface area, and thermal/chemical stability. ZIF-8 is classified as a soft porous crystal (SPC)19 and exhibits intra-framework flexibility, which leads to peculiar adsorption phenomena henceforth referred to as gate adsorption.20-23 A typical isotherm for gate adsorption shows an abrupt increase in the adsorbed amount at a threshold gas pressure (gate opening pressure); and similarly an abrupt decrease at a gate closing pressure during the desorption process. The structural flexibility of ZIF-8 stems from a reorientation of MIM linkers, leading to the gate phenomena of Ar, N2, CO, and O2,24 and even enabling the high-efficiency kinetic separation of propane and propylene.25 In terms of industrial applications of SPCs, the control of the gate phenomenon is required for reasonable designs tailored to the usage environment.26 In this regard, decreasing the particle size is a promising approach as the particle size affects the gate opening/closing pressures.27-31 Along these lines, we have investigated the synthesis process of ZIF-8 particles using a microreactor, and successfully demonstrated the control of both particle size (51 nm to 1.8 µm) and shape (cube, chamfered cube, and rhombic dodecahedron) by simply adjusting the reactant concentration and the reaction temperature. We have further demonstrated that both the gate opening and closing pressures of N2 increase as the particle size decreases. Conversely, the change in the particle shape only shifts the gate opening pressure, while the gate closing pressure remains unchanged.32 However, the mechanism of the particle size and shape dependence of the gate phenomena has not been fully resolved.


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Computer simulations are an effective approach to reveal the mechanism of the gate phenomenon. To investigate the adsorption behaviors of ZIF-8, a key issue is how to deal with the flexibility of the ZIF-8 structure and the interaction potential between the host ZIF-8 framework and guest molecules. Han et al.33 calculated parameters for the interaction potential with an ab initio-based force field, and conducted adsorption simulations of H2 in ZIF-8. In this case, ZIF-8 was treated as a rigid framework by fixing the linkers at a certain angle. Their simulations successfully reproduced the experimental adsorption isotherms of H2 at 77 K and room temperature. However, the high computational cost associated with ab initio calculations is problematic. In this regard, general force fields such as universal force field (UFF)34 and DREIDING35 can be highly effective as interaction potentials reproduce the experimental adsorption isotherms for CH4, CO2, and N2 at room temperature.36-37 The incorporation of a rotational potential for the linkers of ZIF-8 enables one to deal with the flexibility of the ZIF-8 framework. This was demonstrated by molecular dynamics simulations of the diffusion behavior for H2, CH4, and CO2 in ZIF-8.38-39 However, the direct simulations of the gate phenomenon of ZIF-8 is difficult due to the difficulty in sampling the full phase space, which is particularly a problem when the adsorbates are densely packed in the pores. A possible solution to these issues is hybrid Monte Carlo/molecular dynamics (HMD) methods,40-41 in which Monte Carlo simulations for the molecular adsorption and MD simulations for the structural change are alternately implemented to directly address the gate phenomenon. Zhang et al.42 successfully demonstrated the utility of the HMD method by reproducing an experimental gate closing process for N2 at 77 K. However, the HMD method is not applicable for studying gate opening processes, which are induced by the structural transition from a metastable to a stable state. This is because the system in the simulation cannot surmount the activation energy required for the


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transition between the two states, and is consequently trapped in the metastable state. To avoid this difficulty, we used free energy analyses to understand the gate phenomena of SPCs.43-46 The free energy analysis traces the equilibrium states of the system by calculating the free energy landscape along structural variations for different pressures, thereby predicting adsorptioninduced structural transitions of SPCs. In our previous studies,47-50 we have performed the free energy analysis with the aid of grand canonical Monte Carlo (GCMC) simulations by using simple models to calculate adsorption isotherms for different types of frameworks and demonstrated that the activation energy for transition between the two minima in the free energy landscape is the origin of the wide hysteresis loop in the gate phenomenon. Additionally, we have applied the free energy analysis to an atomistic model of ZIF-8 and demonstrated that a gradual increase in the rotation angle of MIM linkers (up to 10.5°) was followed by a stepwise rotation from 10.5° to 25.5°. This reproduced the experimental isotherms accompanied with the gate opening and closing behaviors.51 However, the previous studies did not investigate the effects of the surface on the gate phenomenon, since the calculations used bulk crystal structures. Zhang et al.30 conducted the free energy analysis with GCMC simulations by using a ZIF-8 nanoparticle simulation cell, which they constructed by making cuts from a bulk crystal. As a result, their free energy analysis on the gate adsorption process demonstrated that the structural transition occurs at a significantly higher pressure in nanoparticles than in the bulk. This effect originates from reduced guest molecule loading close to the nanoparticle surface. However, their simulations did not account for kinetic transition processes hindered by energy barriers between metastable and stable states because they only focused on equilibrium transitions. Moreover, the effect of the particle shape on the gate phenomena is difficult to explore using their method due to the spherical approximation of the ZIF-8 structure. This resulted in a ZIF-8 structure with no


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defined crystal planes, which is an issue considering that our previous study showed that the particle shape can significantly impact the gate opening behavior rather than the closing one.32 In the present study, we constructed a slab of the ZIF-8 crystal with two flat surfaces cut along certain crystal planes, which was used for the free energy analysis with the aid of GCMC simulations to investigate the particle size and shape dependence of the gate opening and closing behaviors. Based on our simulation results, we developed a model to predict the experimental gate opening and closing pressures of ZIF-8 with varied particle sizes.

2. Simulations and methods 2.1 Simulation cell Figure 1 shows a schematic illustration of the simulation cell in which we set a slab of ZIF-8 crystal with its center placed at (x, y, z) = (0, 0, 0) and gas phase regions on both sides (y-axis direction) of the ZIF-8 slab. We constructed the ZIF-8 slab by cutting out a bulk crystal structure, which follows the experimental crystallographic data,3 along {100}- or {110}-plane. We chose {100}- and {110}-planes because ZIF-8 particles are composed of {100}- and/or {110}-planes to form a cube, a chamfered cube, or a rhombic dodecahedron.32, 52-53 The under-coordinated MIM linkers at the surface of the ZIF-8 crystal region were terminated with hydrogen atoms because a recent study reported that NH-terminated imidazole groups are on the surface of ZIFs.54 Notably, the different types of surface terminations have little effects on adsorption behavior.30 We varied the width of the ZIF-8 slab (defined as the particle width: L) in the ydirection from 4.8–9.6 nm while that of gas phase regions was kept constant to be 3.5 nm, and the details of the simulation cell size are summarized in Table 1. The rotational angle of MIM linkers (θMIM) was fixed in the range from 0° to 30°. Periodic boundary conditions were applied


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in all the x-, y-, and z-directions. Hereafter, we refer to the simulation cell shown in Fig. 1 as "nanoparticle model." As a reference to the nanoparticle model, we also used a bulk model without setting the gas phase regions, assuming an infinitely large particle. 2.2 Grand canonical Monte Carlo simulation We calculated adsorption isotherms of Ar in ZIF-8 by using the nanoparticle model with varied particle widths L, crystal planes, and θMIM as well as the bulk model at 87 K with the conventional GCMC method by using an in-house code. In the GCMC simulations, the constituent atoms of ZIF-8 were immobilized, and three trial movements (displacement, insertion, and removal) for the fluid particles were performed with equal probabilities. The system was equilibrated for 2.5×108 MC steps, after which data were collected for another 2.5×108 steps. The length of the Markov chain of 2.5×108 steps corresponds to more than 5.7×104 steps per particle for all of the cases investigated in the present study. The LennardJones (LJ) potential was used for Ar–Ar interactions with the following parameters: σff = 0.34 nm, εff/k = 119.8 K, and cut-off radius = 4.9σff. The interatomic interaction of Ar and the framework atoms of ZIF-8 were calculated from a scaled force field based on UFF,34 σUFF and αεUFF. The εUFF parameter was scaled following the approach by Fairen–Jimenez et al.,4 and the scaling factor, α, was determined to be α = 0.54 so that the calculated adsorption isotherm of Ar in the ZIF-8 bulk model with θMIM = 0° by using GCMC simulations reproduces the experimental adsorption isotherm in a low-pressure region at 87 K.51 The solid–fluid cross interaction parameters, σsf and εsf, were obtained from the Lorentz-Berthelot mixing rules (σsf = (σff +σUFF)/2，εsf = (εffαεUFF)1/2). The cut-off radius of the solid–fluid interactions was set to be 4.9σff. The calculated isotherms were used for the free energy analysis, the details of which are


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explained in section 2.3. The relation between the chemical potential of the system and the bulk fluid pressure, P, was obtained using the Johnson–Zollweg–Gubbins equation of state.55 The relative pressure, P/P0 (where P0 is the saturated vapor pressure), was calculated from the vapor– liquid coexistence curve of the LJ fluid obtained by Agrawal and Kofke56 to compare the GCMC adsorption isotherms with the experimental ones. 2.3 Free energy analysis Assuming a continuous ZIF-8 structure in an osmotic ensemble with a constant temperature, T, mechanical constraint exerted on the material, σ, and chemical potential of the adsorbed gas, µ, the free energy of the system, ΩOS, is given by57 Ω୓ୗ ሺߤ, ߪ, ܶ, ߠ୑୍୑ ሻ = ‫ܨ‬୦୭ୱ୲ ሺߠ୑୍୑ ሻ + ߪܸሺߠ୑୍୑ ሻ + Ωሺߤ, ܶ, ߠ୑୍୑ ሻ

(1)

where Fhost is the Helmholtz free energy of the ZIF-8 framework, V is the volume of the system, and Ω is the grand thermodynamic potential of the adsorbed gas. The grand thermodynamic potential can be calculated by integrating a continuous adsorption isotherm of the gas, Nad, as ఓ

Ωሺߤ, ܶ, ߠ୑୍୑ ሻ = −݇ܶܰୟୢ ሺߤ୧ୢ , ܶ, ߠ୑୍୑ ሻ − ‫׬‬ఓ ܰୟୢ ሺߤ ᇱ , ܶ, ߠ୑୍୑ ሻdߤ′ ౟ౚ

(2)

where k is Boltzmann constant. The first term on the right-hand side is the grand thermodynamic potential at a chemical potential of µid, which is sufficiently low enough for the adsorbed amount, Nad, at µid to essentially correspond to the ideal-gas value. The change in the osmotic free energy, ∆ΩOS, due to the rotation of MIM linkers from θMIM, 0 (= 0º) to θMIM is given by ∆Ω୓ୗ ሺߤ, ߪ, ܶ, ߠ୑୍୑ ሻ = Ω୓ୗ ሺߤ, ߪ, ܶ, ߠ୑୍୑ ሻ − Ω୓ୗ ൫ߤ, ߪ, ܶ, ߠ୑୍୑,଴ ൯ = ∆‫ܨ‬୦୭ୱ୲ ሺߠ୑୍୑ ሻ + ߪ൛ܸሺߠ୑୍୑ ሻ − ܸ൫ߠ୑୍୑,଴ ൯ൟ + ∆Ωሺߤ, ܶ, ߠ୑୍୑ ሻ

(3)


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where the second term σ{V(θMIM) – V(θMIM,0)} is zero because V(θMIM) is constant against the changes in θMIM. The change in the free energy of the ZIF-8 framework due to the rotation of the MIM linkers, ∆Fhost(θMIM), was determined in our previous study through a similar investigation using the bulk model;51 we used the resultant profile of ∆Fhost(θMIM) in the present study. Calculated adsorption isotherms were used to obtain ∆Ω(µ, T, θMIM).

3. Experimental methods ZIF-8 particles were synthesized following our previous study32 in which an aqueous solution of MIM (99%, Aldrich Chemical Co.) and an aqueous solution of zinc nitrate (98%, Aldrich Chemical Co.) were mixed using a central collision-type microreactor (channel width: 100 µm). Detailed experimental procedure of the ZIF-8 synthesis is described in the supporting information. Adsorption isotherms of Ar in the synthesized ZIF-8 particles at 87 K were measured with an automated adsorption apparatus, BELSORP-max (MicrotracBEL Corp., Japan), and a cryostat with a two-stage Gifford MacMahon refrigerator, which is the product of a joint development project among Iwatani Industrial Gases Corp., MicrotracBEL Corp., and our group. The ZIF-8 particles were evacuated at 423 K for over 1.5 h under a pressure of less than 0.1 mPa before each isotherm measurement. The cell temperature was kept within a ±0.005 K range during the adsorption measurements.

4. Results and discussion 4.1 Adsorption behaviors in nanoparticle model


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We simulated Ar adsorption isotherms at 87 K in the ZIF-8 nanoparticle model (L110 = 7.2 nm) with varied θMIM values in order to investigate the effect of the surface of ZIF-8 particles on the adsorption behavior. Figure 2a compares calculated isotherms of the nanoparticle model with those of the bulk model. Overall, the adsorbed amount increases suddenly at a relative pressure of ca. 10−3, which is followed by a gradual increase in the adsorbed amount. Additionally, larger θMIM values correspond to larger adsorbed amounts at a fixed relative pressure. This is due to the appearance of new adsorption sites, which result from rotations of the MIM likers. A remarkable feature observed in all θMIM conditions is a reversal in the adsorbed amounts of the nanoparticle model and the bulk model. In contrast to the bulk model which immediately reaches a plateau region, the isotherms of the nanoparticle model show a gradual increase in the adsorbed amount at low relative pressures, but surpass those of the bulk model at a relative pressure of ca. 0.1. Surface effects can explain the trends for the isotherms of the nanoparticle model. Figure 2b-e show snapshots of Ar adsorbed in the ZIF-8 nanoparticle model for different relative pressures during the adsorption process (where θMIM = 0°). At a low pressure of P/P0 = 8.4×10−4, most Ar molecules were adsorbed in the center of the ZIF-8 slab structure (Fig. 2b). Additional adsorption occurred in the center of the slab, while the region near the surface remained almost vacant at P/P0 = 2.2×10−3 (Fig. 2c). The increase in the gas pressure to P/P0 = 0.1 led to a filling of the space in the near-surface zone with Ar molecules (Fig. 2d). This, in turn, led to the bulk and nanoparticle models adsorbing similar amounts Ar at P/P0 ≈ 0.1 (see Fig. 2a). A further increase in the gas pressure finally induced the Ar adsorption on the surface of the ZIF-8 slab structure (Fig. 2e), leading to the reversal in the adsorbed amounts of the nanoparticle model against the bulk model. In order to quantitatively investigate surface effects, we calculated the number distribution of Ar in the nanoparticle model along the y-axis (ρnano) at P/P0 = 2.5×10−3.


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We recorded the adsorbed amount of Ar in the xz plane at different y positions, and normalized ρnano with that in the bulk model (ρbulk). The red line in Fig. 3a shows the number distribution of Ar adsorbed in the nanoparticle model with L110 = 4.8 nm and θMIM = 0º, and the dotted line indicates the position of the particle surface at which we cut out the bulk model to construct the nanoparticle model. The nano-to-bulk ratio (ρnano/ρbulk) was almost unity in the central region of the ZIF-8 slab structure. It eventually started to decrease at y = 1.4 nm as indicated by the broken line, which demonstrates that surface effects dominate adsorption for depths from the surface less than or equal to 1.0 nm. Interestingly, ρnano/ρbulk of L110 = 7.2 and 9.6 nm (Fig. 3a) also showed a decrease from the depth of 1.0 nm. Consequently, the 1.0 nm depth remains unchanged against the particle widths. This is also the case for different θMIM values (Fig. 3b), in which the 1.0 nm depth with smaller adsorbed amount than that of the bulk model did not change with different θMIM values. We explain surface effects by dividing the adsorption isotherms of the nanoparticle model into bulk (central portion of the ZIF-8 slab) and near-surface zones (everything else). Figure 4 shows the adsorption isotherms of the nanoparticle model with L110 = 4.8 nm and θMIM = 0° (red triangles) and the contributions from both portions (green circles for the central portion from y = −1.4 to 1.4 nm and blue squares for the near-surface zone (y < −1.4 nm and y > 1.4 nm)) of the nanoparticle model as well as that of the bulk model with θMIM = 0° (a black solid line). Contributions from the central portion are almost the same as the isotherm of the bulk model, which suggests that the unique adsorption behavior of the nanoparticle model can be attributed to the near-surface zone. The adsorption behavior in the near-surface zone is due to a weak potential energy between an Ar molecule and the ZIF-8 framework in the nanoparticle model (Unano(y)). Figure 5 shows a calculated Unano(y) normalized with the bulk model (Ubulk(y)). Here, Unano(y) outside of the ZIF-8 slab crystal (y ≥ 2.4 nm for {110}-plane, y ≥


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2.5 nm for {100}-plane) is normalized with respect to the potential energy of the bulk model at the surface position, i.e., Ubulk(y = 2.4) for {110}-plane and Ubulk(y = 2.5) for {100}-plane. The nano-to-bulk ratio (Unano/Ubulk) on the {110}-plane (a green solid line) starts to deviate from 1 at about 1 nm from the surface. This corresponds with the Ar distribution trend shown in Fig. 3. The decrease in the potential energy in the near-surface zone, therefore, defines the depth of the surface effect on the adsorption behavior. This is in accordance with the 1.0 nm depth common to different particle widths (Fig. 3a). The decrease in potential energy for the ZIF-8 slab depends on the distance from the surface instead of on the slab width. Figure 6a shows the adsorption isotherms of the nanoparticle model (θMIM = 0º, {110}-plane) with varied particle widths, L110 = 4.8, 7.2, and 9.6 nm. As L110 decreases, the adsorbed amount decreases at intermediate relative pressures and increases at higher relative pressures than 0.079. The origin of this effect is the increase in surface-to-volume ratio typically associated with smaller particles, because the 1.0 nm depth with lower interaction potentials remains unchanged in respect to variations in L110. The crystal plane also affects the adsorption behavior (Fig. 6b), which can be accounted for by different Unano/Ubulk characteristics in the {110}- and {100}planes (Fig. 5). The adsorbed amount of the {100}-plane is smaller than that of {110}-plane at low relative pressures because of the lower Unano/Ubulk values of the {100}-plane than that of {110}-plane in the distance range from y = 1.5 to 2.0 nm, while it surpasses that of {110}-plane at a relative pressure of 7.6×10−3 due to the increase in Unano/Ubulk of the {100}-plane at y = 2.1 nm and 2.7 nm (on the surface of the particle). Recent studies have demonstrated crystal planespecific degradation behaviors of ZIF-8 particles in acidic solutions, which suggests that the chemical stability depends on the particle shape.58-59 In the present study, our simulations gain additional insights on the crystal plane-specific behavior related to the gas adsorption in ZIF-8


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particles caused by different potential profiles in the near-surface zone, and also demonstrate that the effect of the surface is enhanced by decreasing the particle size because of increased surfaceto-volume ratio. 4.2 Free energy analysis Figure 7a shows the osmotic free energy change, ∆ΩOS, calculated from the adsorption isotherms of the nanoparticle model (L110 = 4.8 nm) by using the Eqs (1−3). ∆ΩOS is plotted in units of kT per MIM linker. The balance between the destabilization induced by the rotation of the MIM linkers (∆Fhost) and the stabilization provided by the gas adsorption determine the free energy profile. ∆Fhost (broken line) shows a monotonous increase against the rotational angle, and therefore the most stable angle at zero pressure is θMIM = 0º. At low pressure (P/P0 = 0.015), the global minimum was slightly shifted to θMIM = 3º because of the increase in the adsorbed amount. Although a local minimum appeared at θMIM = 25.5º, the framework destabilization overwhelmed the stabilization in this low pressure region. As the gas pressure increased (P/P0 = 0.12), the global minimum further shifted from θMIM = 3º to 9º, while the linker angle of the local minimum remained unchanged. ∆ΩOS at θMIM = 25.5º did, however, become deeper. A further increase in the relative pressure provided more stabilization at θMIM = 25.5º, and consequently the most stable state switched from θMIM = 10.5º to 25.5º (P/P0 = 0.63). Figure 7b traces θMIM at the global minimum state for each relative pressure. θMIM gradually increases up to θMIM = 10.5º, followed by a stepwise increase in the rotational angle to θMIM = 25.5º, which demonstrates a thermodynamic structural transition. Our previous study51 using the ZIF-8 bulk model demonstrated that the gate closing process follows the thermodynamic equilibrium, while the gate opening one is the spontaneous structural transition from the metastable state (θMIM = 10.5º) to the stable one (θMIM = 25.5º) which occurs when the energy fluctuation of the system coincides


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with the activation energy between the two states. These results are also experimentally supported by adsorption measurements conducted with different equilibrium criteria. Changes in the gate opening pressure depend on the stringency of the equilibrium criteria, while the gate closing pressures remain unaltered (see the supporting information for the details of experimental data). Based on the above results, we calculated the gate opening and closing pressures of the nanoparticle model. The gate opening pressure was determined as the relative pressure at which the activation energy became the same as the energy fluctuation of the system, while the gate closing pressure was determined as the relative pressure at which the equilibrium structural transition occurred. Here, we applied 0.48 kT/MIM linker as the energy fluctuation of the system for the nanoparticle model by assuming that it is same as that for the bulk model.51 Additionally, we also assumed that the energy fluctuation remains constant in respect to variations in the particle width and the crystal plane. Figure 8a shows the calculated gate opening and closing pressures of the nanoparticle model with varied particle widths and crystal planes. Both the gate opening and closing pressures were higher than those of the bulk model, and increased as the particle width decreased. These results qualitatively agree with the experimental data.31-32 The nanoparticle model requires higher relative pressures for the structural transition to occur because the decrease in the amount adsorbed in the near-surface zone gives less stabilization than that in the bulk model (Fig. 8b). Surface effects are enhanced by a decrease in the particle width due to an increased surface-to-volume ratio, which results in the dependence of the gate pressures on the particle width. On the other hand, the nanoparticle models with {110}-plane (red triangles) and {100}-plane (blue squares) of similar particle widths showed almost the same gate pressures (Fig. 8a), which suggests that the crystal planes have little impact on the gate pressures. This is because variations in the crystal planes change the adsorption isotherms (Fig.


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6b) without changing the resultant free energy contributions. These are given by the integration of the isotherms using Eq. (2), and become similar as the lower stability of the {100}-plane (in respect to the {110}-plane) at low relative pressures is compensated by a reversal in the adsorbed amount above P/P0 = 7.6×10−3. The calculated results of unchanging gate pressures for different crystal planes agree with the experimental gate closing process in which the rhombic dodecahedron shape composed of {110}-plane and the cubic shape composed of {100}-plane show similar gate closing pressures.32 The gate opening process observed in the experiments, however, exhibits a crystal plane dependence in which the gate opening pressure of the rhombic dodecahedron is higher than that of the cube. This is not in accordance with the trend calculated in the present study. The disagreement most likely arises from the assumption that the fixed energy fluctuation is 0.48 kT/linker for different crystal planes. The energy fluctuation of the system may depend on the number of adsorbed Ar molecules as indicated in our previous study.51 It follows that the {110}-plane can possess smaller energy fluctuations than the {100}plane due to the smaller adsorbed amount of the {110}-plane in comparison with the {100}plane near the gate opening pressures (P/P0 ~ 0.5 in Fig. 6b). A system with lower energy fluctuations cannot surmount an activation energy that can be overcome by that with larger fluctuations at a certain pressure, resulting in a higher gate opening pressure. In this case, the calculated gate opening pressure of the {110}-plane would become higher than that of {100}plane, which is in accordance with the experimental trends. In this regard, the incorporation of the crystal plane dependence of the energy fluctuation accounts for the dependence of the gate opening pressure on the crystal planes. However, further investigation on the energy fluctuation is required for a more thorough quantitative understanding. 4.3 Prediction of the gate opening and closing pressures


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We demonstrated the effect of the particle width on the gate pressures based on calculating the free energy contribution of guest adsorption by integrating the isotherms of the nanoparticle model. However, we were not able to directly compare the calculated gate pressures with the experimental data because the width of the ZIF-8 slab structure was limited to 10 nm due to the large computational costs. Accordingly, the simulated isotherms of the ZIF-8 particles of submicron size were not available. Instead of conducting simulations with larger simulation cells, here we construct a model for estimating the isotherms of the ZIF-8 particles with an arbitrary particle size so that we can calculate their gate pressures from the free energy analysis. In the model, the adsorbed amount of ZIF-8 particles with a size of d, Nparticle(µ, θMIM, d), is given by ܰ୮ୟ୰୲୧ୡ୪ୣ ሺߤ, ߠ୑୍୑ , ݀ሻ = ߙୠ ܰୠ୳୪୩ ሺߤ, ߠ୑୍୑ ሻ + ሼ1 − ߙୠ ሽܰୱ୳୰୤ୟୡୣ ሺߤ, ߠ୑୍୑ ሻ

(4)

where Nbulk(µ, θMIM) and Nsurface(µ, θMIM) are amount adsorbed in the central portion of the ZIF-8 particles and in the near-surface zone, respectively. The coefficient of the first term on the righthand side, αb, is a ratio of the central portion of the particle that can be regarded as “bulk” in terms of adsorbed amount. We used the adsorption isotherms of the bulk model as Nbulk(µ, θMIM) because the adsorption isotherms of the central portion of the nanoparticle model were similar to those of the bulk model (see the green circles and black line in Fig. 4). Nsurface(µ, θMIM) was calculated from the adsorption isotherms in the near-surface zone of the nanoparticle model with the {110}-plane (blue squares in Fig. 4), as the crystal planes had little impact on the gate pressures. αb is given by Eq. (5) on the assumption that the ZIF-8 particle shape is spherical, ߙୠ =

ሺௗିଶ௧ొ౏ ሻయ

(5)

ௗయ


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where tNS is the depth of the near-surface zone and 1.0 nm, which is constant for different particle sizes and θMIM values as demonstrated in Fig. 3. For example, in the case of 100 nm particles, the ratio of the central portion of the particle (98 nm) and the near-surface zone (1.0 nm) are 94% and 6%, respectively. Therefore, the adsorbed amount of 100 nm ZIF-8 particles is calculated from Nparticle(µ, θMIM, 100 nm) = 0.94×Nbulk(µ, θMIM) + 0.06×Nsurface(µ, θMIM). In this manner, we estimated the adsorption isotherms with varied θMIM values, after which we obtained the free energy change of the system from Eqs. (1–3) by integrating the estimated isotherms. When we calculated ∆ΩOS from Eq. (3), we assumed that ∆Fhost is unchanged with respect to the particle sizes, which allowed us to investigate the effect of the gas adsorption on ∆ΩOS. On the basis of the free energy profile, we calculated the gate opening pressure as the relative pressure at which the activation energy becomes 0.48 kT/MIM linker, and the gate closing pressure as the relative pressure at which the equilibrium structural transition occurs. As shown in Fig. 9, the calculated gate opening and closing pressures of ZIF-8 particles of varied sizes agree fairly with both our experimental data and the study conducted by Tanaka et al.31 Here, we defined the gate opening and closing pressures as the pressures at the midpoints of the stepwise increase and decrease in the measured adsorption isotherms (see the supporting information). The agreement between our model and experiments suggests that the experimental dependence of the gate opening and closing pressures on the particle size is attributable to the decrease in the adsorbed amount near the surface. The depth of the near-surface zone is just 1.0 nm and accordingly the difference in the adsorbed amounts of the ZIF-8 particles with varied sizes is apparently small. However, the small difference will be accumulated in the free energy calculation, which results in the change in the gate pressures depending on the particle size.

4. Conclusions


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In the present study, we have performed free energy analysis with the aid the GCMC simulations using the nanoparticle and bulk model to investigate particle surface effects on the adsorption behavior of ZIF-8. Our investigation has yielded the following conclusions: 1. The amount adsorbed in the near-surface zone at low relative pressures is smaller than that in the central portion of the nanoparticle model. At high relative pressures, guest molecules start to adsorb on the surface, therefore leading to a reversal in the adsorbed amount of the nanoparticle model with respect to the bulk model. The effect of the surface on the adsorbed amount reaches a 1.0 nm depth from the surface, which is unaltered by changes in the particle width and the rotational angles of MIM linkers. 2. The crystal planes ({110}- and {100}-planes) also affect the adsorption isotherms because of the difference in potential energy between the adsorbed molecule and the ZIF-8 framework. 3. The free energy profiles demonstrate that the gate opening and closing behaviors occur at higher relative pressures for smaller particle widths. This is due to a decrease in the adsorbed amount near the surface, which is the result of larger surface-to-volume ratios for smaller ZIF-8 particles. On the other hand, the crystal planes ({110}- and {100}-planes) have little impact on the gate pressures because the free energy contribution by guest adsorption are similar although the variations in the crystal planes did lead changes in the adsorption isotherms. 4. The quantitative agreement between our proposed model and experimental data demonstrates that the near-surface zone with less adsorbed amount whose depth is 1.0 nm is the origin of the particle size dependence of gate adsorption.


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Figure Captions Figure 1. Schematic illustration of simulation cell, referred to as the nanoparticle model, with gas regions on both sides (y-axis direction) of the ZIF-8 slab crystal. The C atoms of the framework are shown in gray, N in purple, H in white, and Z in red. Figure 2. (a) Simulated Ar adsorption isotherms at 87 K in the nanoparticle model and the bulk model with varied θMIM values. Snapshots of Ar adsorbed in the ZIF-8 nanoparticle model at (b) P/P0 = 8.4×10−4, (c) P/P0 = 2.2×10−3, (d) P/P0 = 1.0×10−1, and (e) P/P0 = 5.0×10−1. The Ar atoms are shown in blue. Figure 3. Number distributions of Ar adsorbed in the nanoparticle model (ρnano) normalized with those in the bulk model (ρbulk) at P/P0 = 2.5×10−3. ρnano/ρbulk with (a) three different particle widths and (b) with varied θMIM values. Dotted lines and broken lines indicate the particle surface and the position at which ρnano/ρbulk starts to deviate from 1, respectively. Figure 4. Ar adsorption isotherms (87 K, θMIM = 0°) in the bulk model and the nanoparticle model (L110 = 4.8 nm, θMIM = 0°). Green circles and blue squares indicate contributions from the central portion (from y = −1.4 to 1.4 nm) and the near-surface zone (y < −1.4 nm and y > 1.4 nm). Figure 5. Potential energy between an Ar molecule and the ZIF-8 framework in the nanoparticle model (Unano) of {110}- and {100}-planes (L110 = 4.8 nm, L100 = 5.1 nm) normalized with that in the bulk model (Ubulk) as a function of the position of y-axis.


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Figure 6. Calculated Ar adsorption isotherms (87 K, θMIM = 0°) in the nanoparticle model with (a) varied particle widths, L110 = 4.8, 7.2, and 9.6 nm, and with (b) different crystal planes (L110 = 7.2 nm, L100 = 6.8 nm). Figure 7. (a) Osmotic free energy change of the nanoparticle model with the particle width of L110 = 4.8 nm for different relative pressures. (b) MIM linker angles at the thermodynamic equilibrium state at each relative pressure. Figure 8. (a) Calculated gate pressures from the free energy analysis of nanoparticle model with varied particle widths and crystal planes. (b) Grand thermodynamic potential of the adsorbed Ar, Ω, calculated by integrating adsorption isotherms of the bulk model (θMIM = 0°) and the nanoparticle model (θMIM = 0°, L110 = 4.8 nm). Figure 9. Comparison between estimated gate pressures of ZIF-8 particles with varied sizes using our proposed model and experimental data. Table 1. Details of the simulation cell size used for the nanoparticle model.

ASSOCIATED CONTENT Supporting Information. The synthesis of ZIF-8 particles; Adsorption isotherms measured with different criteria for equilibrium; Adsorption isotherms of Ar in ZIF-8.

ACKNOWLEDGMENT


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This work is financially supported in part by a Grant-in-Aid for Challenging Exploratory Research (No. 26630391), Grant-in-Aid for JSPS Fellows (No. 15J08218), and Hosokawa Powder Technology Foundation. REFERENCES

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Figure, Table


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 1. Schematic illustration of simulation cell, referred to as the nanoparticle model, with gas regions on both sides (y-axis direction) of the ZIF-8 slab crystal. The C atoms of the framework are shown in gray, N in purple, H in white, and Z in red.


29


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Page 30 of 48

Figure 2. (a) Simulated Ar adsorption isotherms at 87 K in the nanoparticle model and the bulk model with varied θMIM values. Snapshots of Ar adsorbed in the ZIF-8 nanoparticle model at (b) P/P0 = 8.4×10−4, (c) P/P0 = 2.2×10−3, (d) P/P0 = 1.0×10−1, and (e) P/P0 = 5.0×10−1. The Ar atoms are shown in blue.


30

Page 31 of 48

ρnano/ρbulk [–]

(a) 1.5 4.8 nm

7.2 nm

9.6 nm

1.0 0.5 0 0

(b) 1.5 ρnano/ρbulk [–]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.0 2.0 3.0 4.0 Position of y-axis [nm]

5.0

1.0

θMIM 0.5 0 0

0° 6° 12°

18° 24° 1.0 2.0 Position of y-axis [nm]

3.0

Figure 3. Number distributions of Ar adsorbed in the nanoparticle model (ρnano) normalized with those in the bulk model (ρbulk) at P/P0 = 2.5×10−3. ρnano/ρbulk with (a) three different particle widths and (b) with varied θMIM values. Dotted lines and broken lines indicate the particle surface and the position at which ρnano/ρbulk starts to deviate from 1, respectively.


31


Amount adsorbed [cm3 /g]

600 500 400

nanoparticle model central portion near-surface zone bulk model

300 200 100 0 10−4

10−3 10−2 10−1 Relative pressure P/P0 [–]

100

Figure 4. Ar adsorption isotherms (87 K, θMIM = 0°) in the bulk model and the nanoparticle model (L110 = 4.8 nm, θMIM = 0°). Green circles and blue squares indicate contributions from the central portion (from y = −1.4 to 1.4 nm) and the near-surface zone (y < −1.4 nm and y > 1.4 nm).

1.2 1.0 Unano/Ubulk [–]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 48

0.8

{110}-plane

0.6

{100}-plane

0.4 0.2 0 0

1.0 2.0 Position of y-axis [nm]

3.0

Figure 5. Potential energy between an Ar molecule and the ZIF-8 framework in the nanoparticle model (Unano) of {110}- and {100}-planes (L110 = 4.8 nm, L100 = 5.1 nm) normalized with that in the bulk model (Ubulk) as a function of the position of y-axis.


32

Page 33 of 48

Amount adsorbed [cm3/g]

500 (a) 400 300 200

L110 = 4.8 nm L110 = 7.2 nm L110 = 9.6 nm

100 0 10−4 500


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



100

(b)

400 300 200 100 0 10−4

L110 = 7.2 nm L100 = 6.8 nm 10−3 10−2 10−1 Relative pressure P/P0 [–]

100

Figure 6. Calculated Ar adsorption isotherms (87 K, θMIM = 0°) in the nanoparticle model with (a) varied particle widths, L110 = 4.8, 7.2, and 9.6 nm, and with (b) different crystal planes (L110 = 7.2 nm, L100 = 6.8 nm).


33

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

MIM linker angles, θMIM [degree] Free energy change ΔΩOS [kT/linker]


Page 34 of 48

4 ΔFhost (P/P0 = 0) P/P0 = 0.015 P/P0 = 0.12 P/P0 = 0.63

3 2 1 0

−1

0

10

20

θMIM [degree]

30

30

20

10

0 10−4


100

Figure 7. (a) Osmotic free energy change of the nanoparticle model with the particle width of L110 = 4.8 nm for different relative pressures. (b) MIM linker angles at the thermodynamic equilibrium state at each relative pressure.


34

Page 35 of 48

Gate pressure, P/P0 [–]

(a) 0.7

(Ads.) (Des.) {110}-plane (Ads.) (Des.) {100}-plane

0.6 0.5 0.4

Gate opening of bulk model Gate closing of bulk model

0.3 4

5

6 7 8 Particle width, L [nm]

9

10

(b) 0

Ω [kT/MIM linker]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−2

−4 bulk model nanoparticle model −6 10−4

10−3 10−2 10−1 Relative pressure P/P0 [–] Figure 8. (a) Calculated gate pressures from the free energy analysis of nanoparticle model with varied particle widths and crystal planes. (b) Grand thermodynamic potential of the adsorbed Ar, Ω, calculated by integrating adsorption isotherms of the bulk model (θMIM = 0°) and the nanoparticle model (θMIM = 0°, L110 = 4.8 nm).


35


1.0 0.9 0.8 0.7


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 48

0.6 0.5 Proposed model 0.4 Ads. Des.

0.3 Experiment (Ads.) (Des.) Present study (Ads.) (Des.) Tanaka et al.27 0.2 1 10 100 Particle size [nm]

1000

Figure 9. Comparison between estimated gate pressures of ZIF-8 particles with varied sizes using our proposed model and experimental data.

Table 1. Details of the simulation cell size used for the nanoparticle model. Exposed crystal plane, i-plane

Width of ZIF-8 slab in y direction (Li ) [nm]

Width of gas phase in y direction [nm]

Width in x direction

Width in z direction

{110}-plane

4.8, 7.2, 9.6

3.5

4.8

3.4

{100}-plane

5.1, 6.8, 8.5

3.5

3.4

3.4


36

Page 37 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TOC Graphic


37


Page 38 of 48

TOC

Nanoparticle Gas phase

Less adsorption near surface

P/P0

1.0 nm

Gate closing pressure [–]

Bulk

Particle size vs Gate pressure

Free energy analysis

Bulk vs Nanoparticle Amount adsorbed [cm3/g]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Proposed model Experiments

Particle size [nm]


Page 39 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47


Figure 1.

Gas region 3.5 nm

ZIF‐8 crystal region 4.8–9.6 nm

Gas region 3.5 nm y z

x



Figure 2.

600


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Page 40 of 48

(a)

500 400 300 200 100 0 10−4

MIM = 0° MIM = 12° MIM = 24°

nanoparticle model bulk model

10−3 10−2 10−1 Relative pressure P/P0 [–] (b) (c) (d) (e) Surface of particles


100

Page 41 of 48

Figure 3.

(a) 1.5 ρnano/ρbulk [–]

4.8 nm

7.2 nm

9.6 nm

1.0 0.5 0 0

(b) 1.5 ρnano/ρbulk [–]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47


1.0 0.5 0 0

1.0 2.0 3.0 4.0 Position of y‐axis [nm]

MIM

0° 6° 12°

5.0

18° 24°

1.0 2.0 Position of y‐axis [nm]


3.0


Figure 4.

600 Amount adsorbed [cm3/g]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Page 42 of 48

500 400

nanoparticle model central portion near‐surface zone bulk model

300 200 100 0 10−4



100

Page 43 of 48

Figure 5.

1.2 1.0 Unano/Ubulk [–]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47


0.8

{110}‐plane

0.6

{100}‐plane

0.4 0.2 0 0

1.0 2.0 Position of y‐axis [nm]


3.0


Figure 6.


500

(a)

400 300 200

L110 = 4.8 nm L110 = 7.2 nm L110 = 9.6 nm

100 0 10−4

500 Amount adsorbed [cm3/g]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Page 44 of 48


100

(b)

400 300 200 100 0 10−4

L110 = 7.2 nm L100 = 6.8 nm 10−3 10−2 10−1 Relative pressure P/P0 [–]


100

Page 45 of 48

Figure 7.

MIM linker angles, MIM [degree] Free energy change ΔΩOS [kT/linker]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47


4 ΔFhost (P/P0 = 0) P/P0 = 0.015 P/P0 = 0.12 P/P0 = 0.63

3 2 1 0

−1

0

10

20

MIM [degree]

30

30

20

10

0 10−4



100


Figure 8.


(a) 0.7

(Ads.) (Des.) {110}‐plane (Ads.) (Des.) {100}‐plane

0.6 0.5 0.4

Gate opening of bulk model Gate closing of bulk model

0.3 4

5

6 7 8 Particle width, L [nm]

9

10

(b) 0 Ω [kT/MIM linker]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Page 46 of 48

−2

−4 bulk model nanoparticle model −6 10−4

10−3 10−2 Relative pressure P/P0 [–]


10−1

Page 47 of 48

Figure 9.

1.0 0.9 0.8 0.7


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47


0.6 0.5 0.4

Proposed model Ads. Des.

0.3 Experiment (Ads.) (Des.) Present study (Ads.) (Des.) Tanaka et al.27 0.2 1 10 100 Particle size [nm]


1000


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Page 48 of 48

Table 1.

Exposed crystal plane, i‐plane

Width of ZIF‐8 slab in y direction (Li) [nm]

Width of gas phase in y direction [nm]

Width in x direction

Width in z direction

{110}‐plane

4.8, 7.2, 9.6

3.5

4.8

3.4

{100}‐plane

5.1, 6.8, 8.5

3.5

3.4

3.4


Free Energy Analysis for Adsorption-Induced ... - ACS Publications

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