Comprehensive Modeling of Capillary Condensation in Open-Ended

Article pubs.acs.org/JPCC

Comprehensive Modeling of Capillary Condensation in Open-Ended Nanopores: Equilibrium, Metastability, and Spinodal Tatsumasa Hiratsuka, Hideki Tanaka,* and Minoru T. Miyahara* Department of Chemical Engineering, Kyoto University, Katsura, Nishikyo, Kyoto 615-8510, Japan S Supporting Information *

ABSTRACT: Capillary condensation within open-ended cylindrical nanopores is modeled thermodynamically and kinetically, in order to provide reasonable estimates of both the equilibrium phase transition pressure and capillary condensation pressure from a metastable state. The thermodynamic model is established by adding to the conventional Derjaguin−Broekhoff−de Boer model the effects of curvature on the fluid−solid interaction potentials and surface tension at the vapor−liquid interface. The equilibrium vapor−liquid phase transition pressure estimated via the thermodynamic model is in close agreement with that from the nonlocal density functional theory. In contrast to the equilibrium phase transition, capillary condensation from a metastable state cannot be estimated by the conventional thermodynamic models, because it inherently includes an activated process. To overcome this challenge, we propose a kinetic model of metastable capillary condensation for the first time. The model is based on the finding in our previous molecular simulation study, that metastable capillary condensation occurs when the corresponding rate constant reaches a critical value. The proposed kinetic model allows us to quantitatively reproduce the experimental capillary condensation pressures over a wide range of temperatures and mesopore sizes without computationally expensive molecular simulations.

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INTRODUCTION In mesopores (2−50 nm), the vapor−liquid and liquid−vapor phase transitions, known as capillary condensation and evaporation, respectively, occur at a pressure lower than the saturated vapor pressure P0 of the bulk fluid, owing to the curvature of the vapor−liquid interface1 and the van der Waals interactions between the fluid and solid.2,3 Thus, the capillary condensation/evaporation pressures depend strongly on not only the temperature4,5 but also the pore size.6,7 To effectively utilize the capillary condensation/evaporation phenomena in industrial applications, such as separation8 and characterization of porous materials,9 it is vital to understand the dependencies of the capillary condensation/evaporation pressures on the temperature and pore size, and develop methods to quantitatively estimate these relationships. Depending on the temperatures and pore sizes, capillary condensation/evaporation in open-ended cylindrical mesopores such as those in MCM-4110 and SBA-1511 are often accompanied by adsorption hysteresis, which is classified as the H1 type by the International Union of Pure and Applied Chemistry (IUPAC). 12,13 Many theoretical,1−3,14−17 experimental,18 and molecular simulation19−21 studies revealed that adsorption hysteresis is observed because capillary evaporation is a phase transition at thermodynamic equilibrium without nucleation, whereas capillary condensation often occurs from a metastable state with nucleation. The equilibrium phase transition is thermodynamically defined as a transition when the free energies of a multilayer adsorption state © 2017 American Chemical Society

(before condensation) and a condensed state (after condensation) are equal. On the other hand, capillary condensation is an activated process. Namely, there exists an energy barrier for this process, and the system is kinetically trapped in a local minimum of the free energy landscape.22 Thus, to completely understand the H1 type adsorption hysteresis and reasonably predict the temperature and pore size dependences of capillary condensation/evaporation behaviors, both thermodynamic and kinetic models are required. The equilibrium phase transition in a confined space has been widely studied, and many thermodynamic models have been proposed to describe it.2,3,14−16,23−32 A conventional thermodynamic model is the Derjaguin−Broekhoff−de Boer (DBdB) model.2,3,23 In this model, the equilibrium phase transition pressure, Peq, is determined as the pressure where the free energies of the vapor-like and liquid-like states are equal. In contrast to the Kelvin equation,1 the DBdB model takes into account the influence of the pore wall, resulting in better estimates than the Kelvin equation. However, the influence from the pore wall is expressed in the DBdB model as the interaction with a flat nonporous solid. In other words, the effect of pore curvature on the potential within the pore is not taken into account. Saam and Cole (SC)14,15 proposed a model similar to Received: September 28, 2017 Revised: November 8, 2017 Published: November 10, 2017 26877

DOI: 10.1021/acs.jpcc.7b09631 J. Phys. Chem. C 2017, 121, 26877−26886

Article

The Journal of Physical Chemistry C

adsorption process with the aid of GCMC and gauge cell MC37,38 simulations and also evaluated the dimensionless rate constant of capillary condensation k*(P) by applying the transient state theory (TST)39,40 to the obtained free energy landscape. Through a careful comparison with the experimental adsorption isotherms, we revealed that capillary condensation occurs at the pressure where k*(Pcond) = 10−17. Hence, we defined the critical rate constant for the realistic system kc* as 10−17 and demonstrated that the k*c value allows quantitative estimation of Pcond over a wide range of mesopore sizes and temperatures. These findings should help resolving the difficulties in modeling the spontaneous capillary condensation from a metastable state. Here, we propose both thermodynamic and kinetic models of capillary condensation that provide reasonable estimation of Peq and experimental values of Pcond, respectively, without using molecular simulations. These models take into consideration the effects of curvature on the interaction potentials from the pore wall and on the surface tension at the vapor−adsorbed film interface. The thermodynamic model of capillary condensation is compared with previously reported DBdB, SC, and SB models and NLDFT calculations, in order to systematically understand how the curvatures of the pore wall and adsorbed film influence the equilibrium phase transition. The kinetic model is based on the aforementioned finding that capillary condensation occurs when the rate constant reaches the critical rate value of k*c = 10−17. By comparison with the experimental adsorption isotherms of argon on open-ended mesoporous materials with different pore sizes at various temperatures, we demonstrate that the proposed model gives Pcond values that are in excellent agreement with the experimental counterparts. To the best of our knowledge, this is the first model that provides a reasonable description of the kinetically controlled capillary condensation over a wide range of temperatures and mesopore sizes, without resorting to computationally expensive calculations.

the DBdB model that takes into account the curvature effect on the interaction potentials, and their model is supposed to be more suitable for narrow pores than the DBdB model. Another important factor affected by the curvature in nanopores is the surface tension at the interface between the gas and adsorbed phases, although the effect is not taken into account in the DBdB and SC models. Sonwane and Bhatia (SB)24,25 modified the DBdB model by introducing not only the curvature-dependent potentials but also the following Tolman equation33 which describes the curvature-dependent surface tension γ∞ γ(rc) = 1 − δ∞/rc (1) where rc is the curvature radius of the cylindrical bubble, and γ∞ and δ∞ are the surface tension and the Tolman length for planar vapor−liquid interfaces, respectively. The authors reported that compared to the DBdB model, the SB model can better reproduce the curve of Peq versus pore size that is estimated from the nonlocal density functional theory (NLDFT) calculations. There are many other models in which the curvature effects on the solid−fluid interaction potentials and surface tension are taken into account, and their validity and advantages over the DBdB model have been discussed.16,26−32 However, the formulas and parameters employed to represent these two curvature-dependent variables change greatly from one model to another. As a result, there has been no systematic investigation on how the curvature-dependent interaction potentials and surface tension affect the equilibrium phase transition. Moreover, it is worth noting that all reported models that include the curvature-dependent surface tension16,24,25,27−32 employ only the first-order curvature correction to the surface tension, in common with the Tolman equation (eq 1). Koga et al.34 reported that the Tolman equation is valid only when the droplet/bubble radius is greater than 50σff, where σff is a Lennard−Jones (LJ) parameter for fluid molecules and on the order of molecular size. Block et al.35 calculated the surface free energy of cylindrical droplets/bubbles with different sizes using Monte Carlo (MC) simulations and density functional theory and demonstrated that the second-order curvature correction is required to represent the surface tension at the cylindrical vapor−liquid interface with a small radius. Thus, the thermodynamic model of equilibrium phase transition in a nanoscale pore should incorporate the second-order curvature correction to the surface tension. Contrary to the equilibrium phase transition, there is no known model that can properly estimate the kinetically controlled capillary condensation behavior. All previously r e p o r t e d m o d e l s c o n c e r n i n g c a pi l l a r y c on den s ation2,3,14−16,23,24,26−28,31,32 estimate the capillary condensation pressure Pcond as the vapor-like spinodal Pspi, where the energy barrier separating the vapor-like and liquid-like states disappears. While capillary condensation in sufficiently large pores (>5 nm) occurs at close to Pspi, for smaller pores it occurs below Pspi,19,20 leading to the deviation between the estimated and experimental Pcond. Moreover, the grand canonical Monte Carlo (GCMC) simulation, which is one of the most commonly used simulation techniques for the adsorption process, cannot directly estimate Pcond. The reason is that the density and energy fluctuations in MC-based simulations are far smaller than those in realistic systems, and thus, the system in the GCMC simulation is trapped in the local minimum of the free energy landscape. To overcome this difficulty and develop reliable ways to estimate the kinetically controlled capillary condensation behavior, in our previous study,36 we calculated the free energy change during the

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EXPERIMENT Argon Adsorption. The adsorption isotherms of argon on MCM-41C16, MCM-41C18, and SBA-15 materials at temperatures ranging from 56 to 90 K were measured volumetrically, with an adsorption apparatus consisting of an automatic gas adsorption instrument (BELSORP-18, Microtrac BEL, Japan) and a cryostat with a helium closed-cycle refrigerator. Note that “C16” and “C18” indicate the alkyl chain lengths of alkyltrimethylammonium bromides that were used as templates in the MCM-41 synthesis. The adsorbents were outgassed at 423 K for 2 h under a pressure below 0.1 mPa before the adsorption measurements. The temperature of the sample cell was kept within ±0.005 K during the measurements.

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MODELING Free Energy Change for Capillary Condensation. As shown in Figure 1a, we consider the capillary condensation process in a cylindrical mesopore, when the thickness of the cylindrical adsorbed film grows from t0 to t under a constant chemical potential μ. The pore is modeled by a single structureless layer of oxygen atoms, with the pore radius R defined as the distance between the pore center and the surface of oxygen atom of the pore wall. In analogy with the classical nucleation theory, the free energy change during the capillary condensation process, ΔW(μ, t0, t), is calculated by the sum of the following three factors: the work required to condense the 26878


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ΔW (μ , t0 , t ) = {(R − t0)2 − (R − t )2 )}(μ0 − μ)ρL πLz +

∫t

t

2(R − t )ρL ϕ(t )dt 0

+2{(R − t )γ(R − t ) − (R − t0)γ(R − t0)} (5)

Here, we employ the following function as the solid−fluid interaction potential ϕ(t)41 2 ⎡ 2 63 F ( − 4.5, − 4.5, 1.0; ξ ) ϕ(t ) = π 2ρε σ ⎢ s sf sf ⎣ 32 [(1 − ξ 2)R *]10

−3

F( −1.5, −1.5, 1.0; ξ 2) ⎤ ⎥ ⎦ [(1 − ξ 2)R *]4

(6)

where ξ = (R + σss/2)/(R + σss/2 − t) and R* = (R + σss/2)/σsf. ρs is the number density of oxygen atoms, σsf and εsf are the fluid− solid interaction parameters, σss is the interaction parameter for oxygen, and F(a, b, c; d) is the hypergeometric function. We use the following expression as the curvature-dependent surface tension for cylindrical interfaces35 Figure 1. (a) Capillary condensation process at a constant chemical potential μ. (b) Schematic ΔW(μ, t0, t) profile obtained from eq 5.

γ(rc) =

lc2 =

(2) t

2(R − t )ρL ϕ(t )dt

(3)

0

(7)

δ∞2 κ − 2 4γ∞

(8)

The values of parameters used in this study are listed in Table 1. The value of ρL at 87 K is obtained from the equation of state for LJ fluid proposed by Johnson et al.;42 γ∞ at 87 K is interpolated from the data for LJ fluid calculated by Janeček using long-range corrections.43 δ∞ and κ for the case with a cutoff distance of 5σff are interpolated from the theoretical data for LJ fluids calculated in a DFT study.44 Note that Broekhoff and de Boer2 already derived equations similar to eq 5. However, they ignored the impacts of curvature on the solid−fluid interactions and surface tension, even though these effects should be considerable in nanoscale pores. The model proposed here, which takes into account the curvature effects, is thus expected to provide a more reasonable free energy landscape for the capillary condensation process. Figure 1b shows the schematic ΔW(μ, t0, t) profile. The film thickness tα at the multilayer adsorption state α and tβ at the transient state β provide extreme values of ΔW(μ, t0, t), meaning that tα and tβ satisfy ∂ΔW(μ, t0, t)/∂t = 0. Thus, tα and tβ are determined as the solution of the following equation

ΔWfilm(μ , t0 , t ) = πLz{(R − t0)2 − (R − t )2 }(μ0 − μ)ρL

∫t

1 − δ∞/rc + 2(lc/rc)2

where the parameter lc is related to the Tolman length for planar vapor−liquid interfaces δ∞, surface tension for planar vapor− liquid interfaces γ∞, and the rigidity constant κ as35

vapor at a pressure lower than the saturated vapor pressure, ΔWfilm(μ, t0, t); the stabilization of the liquid film by the solid− fluid interaction, ΔWpot(t0, t); and the change in excess free energy on the gas−liquid interface, ΔWsurf(t0, t). We assume that the density of the adsorbed phase is constant and identical to that of the bulk liquid, ρL, and the surface tension between the vapor and the adsorbed phase γ(rc) depends on the curvature radius rc. Then, the above three factors are calculated as

ΔWpot(t0 , t ) = πLz

γ∞

and ΔWsurf (t 0 , t ) = πLz{2(R − t )γ(R − t ) − 2(R − t 0)γ(R − t 0)} (4)

where Lz is the pore length, μ0 is the chemical potential at the saturated vapor pressure P0 in the bulk system, and ϕ(t) is the solid−fluid interaction potential. Using eqs 2−4, we can write their sum ΔW(μ, t0, t) as

Table 1. Parameters Used to Calculate the Free Energy Change for Capillary Condensation of LJ Argon at 87 K

a

σff [Å]

σsf [Å]

εff/kB [K]

εsf/kB [K]

ρ*L a

γ*∞b

ρs [Å−2]

δ*∞c

κ*d

3.40

3.063

120.0

170.2

0.8316

1.063

0.153

−0.0956

−0.6852

ρL* = ρL × σ3ff; bγ∞ * = γ∞ × σ2ff/εff; cδ∞ * = δ∞/εff; dκ*= κ/εff. 26879


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=ϕ(t ) −

interaction potential (eq 6) and the second-order curvature correction to the surface tension (eq 7). To reveal how these effects influence the equilibrium phase transition behavior, we also estimate Peq by using three reported models (DBdB, SC, and SB) with the same parameters listed in Table 1. In these models and the currently proposed model, Peq is determined from the condition that the free energy change between the multilayer adsorption state and the condensation state (which is calculated from the combination of eqs 9 and 12) is equal to zero. However, the assumptions in each model about the impact of the curvature on the solid−fluid potential and surface tension are different, as mentioned in the Introduction. In the DBdB model, the surface tension is assumed to be constant with regard to curvature (γ(rc) = γ∞), and the solid−fluid interaction potential is estimated from the standard isotherms on the nonporous materials (i.e., estimated as the interaction between the adsorbates and a flat surface). Thus, we employ the classical Steele’s 10−4 potential for the flat surface

1 d {2(R − t )γ(R − t )} 2(R − t )ρL dt

−(μ0 − μ) = ϕ(t ) −

γ (R − t ) 1 dγ(rc) − ρL drc (R − t )ρL

rc = R − t

(9)

Eq 9 is a modified Kelvin equation for a cylindrical meniscus, in which the solid−fluid interaction (the first term on the righthand side) and the influence of curvature on the surface tension (the second and third terms on the right-hand side) are taken into account. The energy barrier ΔWa(μ) for capillary condensation at the chemical potential μ is the free energy change from the multilayer adsorption state α to the transient state β. Thus, using the obtained tα and tβ values, ΔWa(μ) is expressed as ΔWa(μ) = ΔW (μ , tα , tβ)

(10)

⎡ ⎛ ⎞10 ⎛ σsf ⎞4 ⎤ ⎢ 2 ⎜ σsf ϕ(t ) = 2πρε σ − ⎟ ⎜ ⎟⎥ s sf sf ⎢ ⎝ t + σss/2 ⎠ ⎥⎦ ⎣ 5 ⎝ t + σss/2 ⎠

Thermodynamic Equilibrium Phase Transition (Thermodynamic Model). The thermodynamic equilibrium phase transition point, μeq, is defined as the chemical potential where the free energy at the multilayer adsorption state α (before the phase transition) equals that at the condensation state (after the phase transition; i.e., t = R). The free energy difference between the two states at a chemical potential μ is given by ΔW(μ, tα, R). Namely, substituting t0 = tα and t = R into eq 5

which is the form of eq 6 in the limit of R → ∞. In the SC model, although the curvature effect on the surface tension is ignored, the solid−fluid interaction takes into account the curvature of a cylindrical pore and consequently depends on the pore size. Here, eq 6 is employed in the SC model to express the interaction potential from a cylindrical pore. The SB model is based on the concept that the solid−fluid interaction potential and surface tension are both affected by curvature. In particular, the surface tension is assumed to be in accordance with the classical Tolman equation (eq 1), which provides the first-order curvature correction to the surface tension. The differences among the models, and the expressions used in eqs 9 and 12 in the models are summarized in Table 2.

ΔW (μ , tα , R ) = (R − tα)2 (μ0 − μ)ρL πLz +

∫t

R


α

− 2(R − tα)γs(R − tα)

(11)

In this model, the hemispherical meniscus of the condensed phase at the pore mouth is ignored, because the pore length in cylindrical mesoporous materials is generally much longer than the pore diameter. Thus, the contribution of the meniscus at the pore mouth to the total free energy can be ignored. Since the thermodynamic equilibrium phase transition point is defined as when the chemical potential μ equals μeq (ΔW(μeq, tα, R) = 0), μeq is determined by the solution of μ satisfying the following equation (R − tα)2 (μ0 − μ)ρL +

∫t

Table 2. Key Features of Each Thermodynamic Model


− 2(R − tα)γs(R − tα) = 0

P0

= μeq − μ0

solid−fluid potential

surface tension

DBdB SC SB this study

flat (eq 14) cylinder (eq 6) cylinder (eq 6) cylinder (eq 6)

constant (γ(rc) = γ∞) constant (γ(rc) = γ∞) first-order correction (eq 1) second-order correction (eq 7)

Limit of Stability of the Adsorbed Film (Vapor-Like Spinodal Model). The vapor-like spinodal point, μspi, is defined as the limit of stability of the adsorbed film, where the energy barrier separating the multilayer adsorption state α and the condensed state disappears. When this energy barrier disappears, the state α coincides with the transient state β, which means that t = tα = tβ corresponds to the inflection point of the free energy profile ΔW(μ, t0, t). Thus, we can find μspi and tα (= tβ) at μspi from eq 9 (∂ΔW(μ, t0, t)/∂t = 0) and ∂2ΔW(μ, t0, t)/∂t2 = 0. The second-order differential equation can be more concretely written as

(12)

Note that when ignoring the interaction of the pore wall and the curvature effect on the surface tension (i.e., assuming ϕ(t) = 0 and γ(rc) = γ∞), eq 12 is reduced to the traditional Kelvin equation for hemispherical meniscus. The thickness tα at the multilayer adsorption state α is determined by eq 9. Therefore, the combination of eqs 7, 9, and 12 provides μeq and the equilibrium film thickness tα (at μeq). Assuming that the vapor phase behaves as an ideal gas, we can obtain the equilibrium phase transition pressure Peq by converting μeq to the pressure as

kBT ln

model

R

α

Peq

(14)

−(μ0 − μ) − ϕ(t ) + (R − t )

(13)

dϕ(t ) 2 dγ(rc) + dt ρL drc

rc = R − t

2

+

Peq from Previous Models. As shown above, the proposed model takes into account the curvature effect on the solid−fluid 26880

R − t d γ(rc) ρL drc2

=0 rc = R − t

(15) DOI: 10.1021/acs.jpcc.7b09631 J. Phys. Chem. C 2017, 121, 26877−26886

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The Journal of Physical Chemistry C Capillary Condensation from a Metastable State (Kinetic Model). In general, an energy barrier exists during the capillary condensation process in open-ended nanopores, as mentioned above. Therefore, capillary condensation is a thermally activated process, and it is not thermodynamically but kinetically controlled. According to the transition state theory,39,40,45 the dimensionless rate constant k*(μ) is provided by the probability of finding the system at the top of the energy barrier, p(μ). The probability p(μ) is calculated as the ratio between the number of microscopic states at the energy barrier with the film width tβ and that of all microscopic states from t = 0 to t = tβ. Thus, k*(μ) = p(μ) is expressed using the free energy change (eq 5) and energy barrier (eq 10) for capillary condensation as k*(μ) = p(μ)

=

Figure 2. Thermodynamic equilibrium phase transition pressure Peq for argon at 87 K in the silica pore model as a function of pore diameter, calculated via the NLDFT,46 DBdB, SC, SB, and the proposed models. An enlarged view is shown in the inset.

⎡ D ΔW (μ) ⎤ exp⎣⎢ − L k aT ⎦⎥ z B ⎡

t

∫0 β 2πρL D(R − t )exp⎣⎢− LDz

ΔW (μ , tα , t ) ⎤ ⎦⎥dt kBT

a decreasing of the pore diameter D when D < 5 nm, which is reasonable because the curvature effects should increase with decreasing pore size. As seen in the inset of Figure 2, the Peq values for small pores from the DBdB model are higher than those from the SC model, which is due to the difference in the form of the solid−fluid interaction potentials. Eq 12, which describes the condition of the thermodynamic equilibrium phase transition, shows that a stronger attractive interaction potential reduces the chemical potential, resulting in a lower Peq value. Since the interaction potential energy from a cylindrical pore is stronger than that from a flat surface because of curvature, the Peq values from the SC model (in which the curvature-dependent interaction potential is taken into account) are lower than those from the DBdB model (in which the interaction potential for a flat surface is used). Comparing the results for small pores, we can see that the Peq values are in the following order: the proposed model > SB model > SC model, which is due to the different expressions of the curvature-dependent surface tension. Eq 12 demonstrates that the chemical potential at the thermodynamic equilibrium phase transition increases as the surface tension of the vapor−liquid interface at the multilayer adsorption state γs(R − tα) decreases. A lower surface tension stabilizes the multilayer adsorption phase on the pore wall and impedes the vapor−liquid phase transition within the pore, thereby leading to high Peq. Figure 3a shows the thickness of multilayer adsorption phase tα at Peq for argon from the SC, SB, and proposed models as well as the curvature radius of the cylindrical meniscus rc = D/2 − tα as a function of pore diameter. The curvature-dependent surface tensions for argon, calculated via eqs 1 and 7 are presented in Figure 3b. As seen in Figure 3a, rc is smaller than 2 nm in all models for pores with diameter below 5 nm. We can see in Figure 3b that the curvature-dependent surface tension γ(rc) decreases as rc decreases; more specifically, for rc < 2 nm, γ(rc) estimated from eq 7 (second-order curvature correction) decreases more rapidly than that from the Tolman equation (eq 1, first-order curvature correction). Therefore, Peq obtained from the proposed model, which accounts for the second-order curvature correction to the surface tension, is the highest. In contrast, Peq from the SC model, in which the surface tension is assumed to be a constant γ∞ independently of curvature, is the lowest among the three models. As mentioned above, the estimated Peq values are different among the models because of the curvature effects, especially for small pores. However, as seen in Figure 2, the differences are

(16)

where D is the pore diameter. Here, we assume that the characteristic length of condensation nucleus in nanopores is of the order of D and thus apply the free energy barrier and free energy change per D to eq 16. A more detailed description of the derivation of eq 16 can be found in the Supporting Information (SI). In ref 36, we have revealed that capillary condensation experimentally occurs at μeq if the rate constant at μeq is greater than the critical rate constant k*c = 10−17 (i.e., k*(μeq) > 10−17). Otherwise, capillary condensation is observed at the chemical potential μcond where k*(μcond) becomes equal to kc* = 10−17. We can thus estimate the capillary condensation pressure Pcond without molecular simulations according to the following scheme: 1. Determine μeq, tα, and tβ by eqs 7, 9, and 12. 2. Calculate the rate constant at μeq, k*(μeq), using eq 16. 3. If k*(μeq) > k*c = 10−17, the chemical potential of capillary condensation μcond is determined as μeq. 4. Otherwise, eq 16 is used to find the chemical potential μcond, which satisfies k*(μcond) = 10−17. 5. Assuming an ideal gas, calculate Pcond as kBT ln

Pcond = μcond − μ0 P0

(17)

36

We have also found that, in the GCMC simulation with 105 MC steps per molecule, capillary condensation occurs at the pressure PGCMC cond , where the rate constant reaches the critical value for the GCMC system of kc,GCMC * = 10−4. Hence, in addition to Pcond, we can also estimate PGCMC through the above scheme by cond using k*c,GCMC = 10−4 as the critical rate constant.

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RESULTS AND DISCUSSION Thermodynamic Equilibrium Phase Transition. Figure 2 shows the thermodynamic equilibrium phase transition pressure Peq for argon at 87 K in the cylindrical silica pore with a smooth wall as a function of pore diameter, calculated via the DBdB, SC, SB, and the current thermodynamic models, together with those from the NLDFT calculations by Neimark and Ravikovitch.46 Note that, in the NLDFT calculations, the solid−fluid interaction potential was calculated via eq 6.46 The Peq values for argon obtained from each model are almost the same for pores larger than ∼5 nm. However, the deviations progressively increase with 26881


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Figure 3. (a) Pore size dependence of the thickness of multilayer adsorption phase tα for argon at the equilibrium phase transition pressure Peq at 87 K, obtained from the SC, SB, and proposed models, and curvature radius of the cylindrical meniscus rc = D/2 − tα. (b) First- and second-order curvature corrections to the surface tension γ(rc)/γ∞ for argon at 87 K calculated by eqs 1 and 7, respectively. γ∞ is the surface tension of the planar vapor−liquid interface.

adsorption region do not depend on the expression of γ(rc), resulting in the same N values estimated from the SC, SB, and proposed models. On the other hand, the curvature effect on the interaction potential from a pore wall has some impact on the width of the adsorption film. More specifically, the adsorption amount in the multilayer adsorption region estimated from the DBdB model is smaller than those from the other models, which is because the interaction potential from a flat surface is weaker than that from a curved wall. Capillary Condensation from a Metastable State. To verify the proposed kinetic model, we estimated the dimensionless rate constant k*(P) for a variety of pore sizes and compared the results with those calculated using molecular simulation. Figure 5 shows the pressure dependences of the rate constants

quite small as long as the same parameters are used. Consequently, all models could successfully reproduce the NLDFT data over a wide range of pore sizes, revealing that although the curvatures of the pore and vapor−liquid interface influence the fluid−solid interaction potential and surface tension, they do not strongly affect the equilibrium phase transition behavior in a confined space. It is worth noting that the four models can provide not only Peq but also a full adsorption isotherm including a multilayer adsorption region. Figure 4 shows the argon adsorption

Figure 4. Full adsorption isotherms of argon at 87 K including multilayer adsorption and capillary condensation regions for pore diameters of 2.7, 3.7, 4.7, and 5.7 nm, calculated via the DBdB, SC, SB, and proposed models. Ntotal is the adsorption amount when the pore is completely filled with liquid.

Figure 5. Pressure dependences of the rate constant k* for argon at 87 K for pore diameters of 2.7, 3.7, 4.7, and 5.7 nm calculated with the proposed model and the gauge cell MC simulations.36 Triangles and squares indicate k* at the equilibrium phase transition pressure Peq and vapor-like spinodal Pspi estimated by the gauge cell MC simulations, respectively (k*(Peq) = 10−48 and 10−72 for D = 4.7 and 5.7 nm, respectively, are outside of the graph). An enlarged view of the data for pore diameter of 3.7 nm is shown in the inset.

isotherms at 87 K for pore diameter of D = 2.7, 3.7, 4.7, and 5.7 nm calculated using eq 9 with the concept of the DBdB, SC, SB, and proposed models. In the calculation, the film width tα was converted to the amount adsorbed N using the bulk liquid density. The results reveal that the amounts in the multilayer adsorption region estimated from the SC, SB, and proposed models are almost the same, which suggests that the curvature effect on the surface tension does not affect the multilayer adsorption behavior at all. Eq 9 demonstrates that the adsorption film width is determined by both surface tension and its gradient (second and third terms on the right-hand side, respectively). As seen in Figure 3b, the curvature-dependent surface tension γ(rc) is in the following order: planar surface > first-order correction > second-order correction, while the gradient of γ(rc) is in the reverse order, which suggests that the increase of the third term with increasing curvature can be canceled by the decrease of the second term. Thus, the amounts adsorbed in the multilayer

for pore diameters of 2.7, 3.7, 4.7, and 5.7 nm calculated via the proposed model, together with those from the gauge cell MC simulations37,38 calculated previously.36 The comparison demonstrates that the two approaches are in good agreement for all studied pore sizes. This result indicates that we can estimate the metastable capillary condensation pressure using the proposed model with the critical rate constant, as shown in the inset of Figure 5. As mentioned above, the critical rate constant for the realistic system, k*c = 10−17, provides a way to estimate the experimental capillary condensation pressures, while that for the GCMC system, kc,GCMC * = 10−4, could predict the capillary 26882


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The Journal of Physical Chemistry C condensation pressures in the GCMC simulation PGCMC cond . Figure 6 shows the PGCMC values for argon at 87 K on cylindrical smooth cond

Figure 6. Pore size dependence of the capillary condensation pressure in 36 the GCMC simulation PGCMC cond for argon at 87 K and that estimated by the proposed model with k*c,GCMC = 10−4.

pore models with different pore diameters calculated by the GCMC simulations,36 together with those estimated by the proposed model using k*c,GCMC = 10−4. Clearly, the proposed model successfully reproduces the pore size dependence of the capillary condensation pressures in the GCMC system. This agreement suggests that the experimental capillary condensation can also be predicted by the proposed model with k*c = 10−17. Comparison with Experiments. The triple point of the fluid confined in a nanoscale pore is lower than that of the bulk fluid, with larger triple point depression in smaller pores.47 Since the proposed model describes the vapor−liquid phase transition, not the vapor−solid one, we first determined the triple point of argon confined in the MCM-41C16, MCM-41C18, and SBA-15 materials in a manner similar to that of Morishige et al.48 The details of the method and results are provided in the SI. We found that the triple point of argon for MCM-41C16 is below 56 K (sublimation does not occur in the measured temperature range), while those for MCM-41C18 and SBA-15 are 67 and 70 K, respectively. Figure 7 depicts the argon adsorption isotherms on MCM-41C16 in the temperature range of 56−90 K and those on MCM-41C18 and SBA-15 at temperatures ranging from the triple point to 90 K. All the phase transitions shown in Figure 7 are vapor−liquid phase transitions (condensation/evaporation). To determine the experimental capillary condensation and evaporation pressures (Pcond and Pevap, respectively), the regions of the multilayer adsorption state (before condensation), adsorption step (during condensation), and condensation state (after condensation) were fitted by linear functions, as shown in the inset of Figure 7c. The beginning and end of capillary condensation/evaporation were determined as the intersection of the fitting lines. Consequently, Pcond and Pevap were determined at the midpoint of these two points, and their errors were defined as the pressure difference between the two points. The temperature dependences of Pcond and Pevap of argon for MCM-41C16, MCM-41C18, and SBA-15 are presented in Figure 8. The saturated vapor pressure P0 of the supercooled liquid at temperatures below the triple point of bulk argon (83.8 K) was estimated by the extrapolation of the vapor−liquid phase diagram. Figure 8 also shows the equilibrium phase transition pressure Peq, the vapor-like spinodal pressure Pspi, and the kinetically controlled capillary condensation pressure Pcond estimated by the proposed models for pore diameters of 3.8, 4.5, and 6.6 nm as a function of temperature. When estimating these transition pressures, the temperature dependences of the LJ liquid density ρL and surface tension γ∞ were taken into account by using the equation of state by Johnson et al.42 and the

Figure 7. Experimental adsorption isotherms of argon on (a) MCM41C16 at 56−90 K, (b) MCM-41C18 at 67−90 K, and (c) SBA-15 at 70−90 K. The inset of (c) is the enlarged view for SBA-15 at 87 K, using a linear scale for the relative pressure. The lowest measured temperatures are higher than the triple point of argon confined in each adsorbent.

data by Janeček,43 respectively. The ρL and γ∞ values for the supercooled liquid were estimated from the extrapolation of the liquid properties. The pore diameters were chosen so that the Peq value estimated from the NLDFT46 calculation coincided with the experimental results for evaporation at 87 K, since capillary evaporation, which occurs via a receding meniscus from the pore mouth without nucleation, is the thermodynamic equilibrium phase transition. The experimental Pevap values are successfully reproduced for all pores by Peq estimated via the proposed thermodynamic model. Figure 8c demonstrates that the experimental capillary condensation pressures for SBA-15 with relatively large pores can be roughly estimated by the vapor-like spinodal pressures Pspi. However, the deviations between Pspi and experimental Pcond increase markedly with decreasing pore sizes, as shown in Figure 8a,b. On the other hand, the Pcond values estimated by the proposed kinetic model with kc* = 10−17 are in excellent agreement with the experimental counterparts at all temperatures and pore sizes. As seen in Figure 5, the rate constant at the equilibrium phase transition pressure k*(Peq) increases with decreasing the pore size, which corresponds to the fact that the energy barrier at Peq is lower with decreasing pore 26883


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Figure 8. Temperature dependences of the experimental capillary condensation/evaporation pressures of argon for (a) MCM-41C16, (b) MCM41C18, and (c) SBA-15; temperature dependences of the pressures of the equilibrium phase transition Peq(Model), vapor-like spinodal Pspi(Model), and kinetically controlled capillary condensation Pcond(Model) for argon estimated by the proposed model with pore diameters of (a) 3.8, (b) 4.5, and (c) 6.6 nm. The error bars of the experimental values Pcond(Exp.) and Pevap(Exp.) indicate the beginning and end of the respective processes (see Figure 7c).

size. Consequently, Pcond gets close to Peq and away from Pspi for smaller pores, and thus, the kinetic model for metastable capillary condensation is required to correctly predict Pcond. Since the conventional thermodynamic theory cannot estimate the kinetically controlled capillary condensation pressure, to the best of our knowledge, this is the first model that allows one to estimate both the equilibrium capillary evaporation and metastable capillary condensation behaviors over a wide range of pore sizes and temperatures without molecular simulations. Application: Evaluation of Pore Size Distribution. We confirmed in the previous sections that the proposed models correctly estimates the relationship between the pore size and capillary condensation/evaporation behaviors. This naturally suggests that these models are useful for evaluating the pore size distribution (PSD) from the experimental capillary condensation/evaporation pressures. Figure 9a,b shows the adsorption isotherms on the enlarged MCM-41-type material for argon at 87 K and for nitrogen at 77 K, respectively, which were measured by Sayari et al.49,50 Figure 9c shows the corresponding PSDs

evaluated by applying the relationship between the pore size and capillary condensation/evaporation pressures to the experimental adsorption and desorption branches, in which the desorption branch was assumed to be the thermodynamic equilibrium path. The adsorption branch Pcond was estimated using kc* = 10−17. For nitrogen at 77 K, the pore size dependences of Peq and Pcond were calculated by using the proposed models with the following parameters:19,42,43 σff = 3.615 Å, σsf = 3.17 Å, εff/kB = 101.5 K, εsf/ kB = 147.3 K, ρL* = 0.8178, and γ∞ * = 0.9996. The details of the method for evaluating the PSD, in which the proposed model is applied to the step-by-step analysis of the adsorption/desorption branches, can be found in the SI. It is clear that all PSDs calculated from two gases and both branches are in outstanding agreement. Moreover, the peaks of the PSDs are close to the corresponding mean pore diameter estimated on the basis of XRD analysis.49 These results demonstrates the validity and usefulness of the proposed model. Namely, it can reasonably estimate the relationship between the pore size and phase transition pressures for any adsorbates. Consequently, it can assess the PSDs of mesoporous materials showing H1 type adsorption hysteresis, independent of the adsorbate and the adsorption/desorption paths. Note that all of the PSDs shown in Figure 9 are obtained based on the assumption that the pores have “smooth” cylindrical walls. As mentioned in several papers,51,52 the ordered mesoporous silica materials can have rough pore walls, which affect the resulting PSD.53,54 In fact, Smith et al. estimated the pore size of the SBA-15 materials by using XRD analysis, which takes into account surface heterogeneity at the atomic scale, and concluded that the NLDFT analysis, which also assumes the smooth pore wall, overestimates mesopore diameters by 10%.55 Therefore, it is important to grasp the definition of the resulting pore diameter and keep in mind that what kind of assumptions are made for evaluating PSD.

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CONCLUSION The vapor−liquid phase transition in open-ended mesoporous silica materials was modeled not only thermodynamically but also kinetically. We first compared various models (DBdB, SC, SB, and the proposed thermodynamic models) with the same parameters for the equilibrium vapor−liquid phase transition of argon in the smooth silica pores, in order to systematically understand the curvature effect on the equilibrium phase transition behavior. The attractive solid−fluid interaction potentials can be enhanced by the curvature of the pore wall. Consequently, the condensed phase inside the pore becomes more stable, and the equilibrium phase transition is facilitated.

Figure 9. Adsorption isotherms of (a) argon at 87 K and (b) nitrogen at 77 K on the enlarged MCM-41 material.49,50 (c) Pore size distribution obtained by applying the proposed model to the adsorption/desorption isotherms of argon and nitrogen. The dashed vertical line indicates the mean pore diameter estimated from XRD data.49 26884


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Nishihara of Tohoku University, for providing the SBA-15 samples and are also thankful to Mr. Mori for his assistance with the adsorption isotherm measurements. This work was financially supported by Grant-in-Aid for JSPS Fellows 14J01101, Grant-in-Aid for Scientific Research (B) 24360318, (B)17H03097 and JST, CREST Grant Number JPMJCR1324, Japan.

On the other hand, the surface tension of the vapor−liquid interface decreases with curvature, resulting in the stabilization of the multilayer adsorption state (i.e., the state before condensation). Consequently, the equilibrium phase transition is hindered. However, we also revealed that the curvatures of the pore wall and vapor−liquid interface do not drastically affect the equilibrium phase transition. Thus, all four models with different assumptions about the influence of curvature successfully reproduced the results of NLDFT calculations over a wide range of pore sizes, as long as the same parameters were employed. Subsequently, we modeled the metastable capillary condensation process with a focus on its kinetic nature. The rate constants of capillary condensation as a function of pressure k*(P) estimated with the proposed model are in close agreement with those from the molecular simulations for different pore sizes, which ensures the validity of the proposed kinetic model. By finding the pressure where the rate constant equals the critical value k*c = 10−17, which was found in our previous work,36 we successfully predicted the experimental capillary condensation pressures for a variety of pore sizes and temperatures, which is impossible with the traditional thermodynamic capillary condensation theory because capillary condensation is inherently kinetically controlled owing to the associated energy barrier. Thus, the proposed models provide the first quantitative prediction of the relationship between the pore size and both capillary condensation/evaporation pressures without molecular simulations. Finally, the potential application of the proposed models was demonstrated by successfully evaluating the PSD for the enlarged MCM-41 type materials, using the adsorption/ desorption isotherms of argon and nitrogen at 87 and 77 K, respectively. The results suggest that the proposed models can be a powerful tool for easily assessing the PSD of materials exhibiting H1 type adsorption hysteresis, regardless of the adsorption/desorption branches, temperatures, and adsorbates.

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(1) Cohan, L. H. Sorption hysteresis and the vapor pressure of concave surfaces. J. Am. Chem. Soc. 1938, 60, 433−435. (2) Broekhoff, J. C. P.; de Boer, J. H. Studies on pore systems in catalysts: IX. Calculation of pore distributions from the adsorption branch of nitrogen sorption isotherms in the case of open cylindrical pores A. Fundamental equations. J. Catal. 1967, 9, 8−14. (3) Broekhoff, J. C. P.; de Boer, J. H. Studies on pore systems in catalysts: X. Calculations of pore distributions from the adsorption branch of nitrogen sorption isotherms in the case of open cylindrical pores B. Applications. J. Catal. 1967, 9, 15−27. (4) Burgess, C. G. V.; Everett, D. H.; Nuttall, S. Adsorption hysteresis for porous materials. Pure Appl. Chem. 1989, 61, 1845−1852. (5) Morishige, K.; Fujii, H.; Uga, M.; Kinukawa, D. Capillary critical point of argon, nitrogen, oxygen, ethylene, and carbon dioxide in MCM41. Langmuir 1997, 13, 3494−3498. (6) Llewellyn, P. L.; Grillet, Y.; Schüth, F.; Reichert, H.; Unger, K. K. Effect of pore size on adsorbate condensation and hysteresis within a potential model adsorbent: M41S. Microporous Mater. 1994, 3, 345− 349. (7) Ravikovitch, P. I.; Wei, D.; Chueh, W. T.; Haller, G. L.; Neimark, A. V. Evaluation of pore structure parameters of MCM-41 catalyst supports and catalysts by means of nitrogen and argon adsorption. J. Phys. Chem. B 1997, 101, 3671−3679. (8) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Phase separation in confined systems. Rep. Prog. Phys. 1999, 62, 1573. (9) Thommes, M. Physical adsorption characterization of nanoporous materials. Chem. Ing. Tech. 2010, 82, 1059−1073. (10) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Ordered mesoporous molecular sieves synthesized by a liquid-crystal template mechanism. Nature 1992, 359, 710−712. (11) Zhao, D.; Feng, J.; Huo, Q.; Melosh, N.; Fredrickson, G. H.; Chmelka, B. F.; Stucky, G. D. Triblock copolymer syntheses of mesoporous silica with periodic 50 to 300 angstrom pores. Science 1998, 279, 548−552. (12) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A. Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity (Recommendations 1984). Pure Appl. Chem. 1985, 57, 603−619. (13) Thommes, M.; Kaneko, K.; Neimark, A. V.; Olivier, J. P.; Rodriguez-Reinoso, F.; Rouquerol, J.; Sing, K. S. W. Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report). Pure Appl. Chem. 2015, 87, 1051−1069. (14) Cole, M. W.; Saam, W. F. Excitation spectrum and thermodynamic properties of liquid films in cylindrical pores. Phys. Rev. Lett. 1974, 32, 985−988. (15) Saam, W. F.; Cole, M. W. Excitations and thermodynamics for liquid-helium films. Phys. Rev. B: Condens. Matter Mater. Phys. 1975, 11, 1086−1105. (16) Celestini, F. Capillary condensation within nanopores of various geometries. Phys. Lett. A 1997, 228, 84−90. (17) Ravikovitch, P. I.; Domhnaill, S. C. O.; Neimark, A. V.; Schüeth, F.; Unger, K. K. Capillary hysteresis in nanopores: Theoretical and experimental studies of nitrogen adsorption on MCM-41. Langmuir 1995, 11, 4765−4772. (18) Morishige, K. Nature of adsorption hysteresis in cylindrical pores: Effect of pore corrugation. J. Phys. Chem. C 2016, 120, 22508−22514.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b09631. Theoretical background of transition state theory (TST), determination of the triple point of the fluid confined in nanopores, and evaluation of pore size distribution (PSD) (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (H.T.) *E-mail: [email protected]; Phone: +81 75 383 2672; Fax: +81 75 383 2652 (M.T.M.) ORCID

Tatsumasa Hiratsuka: 0000-0003-0706-3955 Hideki Tanaka: 0000-0003-2146-9177 Minoru T. Miyahara: 0000-0001-8883-5851 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to Prof. Matsumoto of Toyohashi University of Technology, Dr. Kubo of Hiroshima University, Prof. Shimojima and Prof. Ohkubo of the University of Tokyo, for providing the MCM-41 samples, and Prof. Kyotani and Dr. 26885


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The Journal of Physical Chemistry C (19) Neimark, A. V.; Ravikovitch, P. I.; Vishnyakov, A. Adsorption hysteresis in nanopores. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2000, 62, R1493−R1496. (20) Vishnyakov, A.; Neimark, A. V. Studies of liquid-vapor equilibria, criticality, and spinodal transitions in nanopores by the gauge cell Monte Carlo simulation method. J. Phys. Chem. B 2001, 105, 7009−7020. (21) Coasne, B.; Galarneau, A.; Di Renzo, F.; Pellenq, R. J. M. Molecular simulation of nitrogen adsorption in nanoporous silica. Langmuir 2010, 26, 10872−10881. (22) Neimark, A. V.; Vishnyakov, A. Phase transitions and criticality in small systems: Vapor-liquid transition in nanoscale spherical cavities. J. Phys. Chem. B 2006, 110, 9403−9412. (23) Derjaguin, B. A theory of capillary condensation in the pores of sorbents and of other capillary phenomena taking into account the disjoining action of polymolecular liquid films. Acta Physicochim. URSS 1940, 12, 181−200. (24) Sonwane, C. G.; Bhatia, S. K. Adsorption in mesopores: A molecular-continuum model with application to MCM-41. Chem. Eng. Sci. 1998, 53, 3143−3156. (25) Bhatia, S. K.; Sonwane, C. G. Capillary coexistence and criticality in mesopores: Modification of the Kelvin theory. Langmuir 1998, 14, 1521−1524. (26) Ustinov, E. A.; Do, D. D.; Jaroniec, M. Equilibrium sdsorption in cylindrical mesopores: A modified Broekhoff and de Boer theory versus density functional theory. J. Phys. Chem. B 2005, 109, 1947−1958. (27) Qiao, S. Z.; Bhatia, S. K.; Zhao, X. S. Prediction of multilayer adsorption and capillary condensation phenomena in cylindrical mesopores. Microporous Mesoporous Mater. 2003, 65, 287−298. (28) Qiao, S. Z.; Bhatia, S. K.; Nicholson, D. Study of hexane adsorption in nanoporous MCM-41 silica. Langmuir 2004, 20, 389− 395. (29) Miyahara, M.; Kanda, H.; Yoshioka, T.; Okazaki, M. Modeling capillary condensation in cylindrical nanopores: A molecular dynamics study. Langmuir 2000, 16, 4293−4299. (30) Kanda, H.; Miyahara, M.; Higashitani, K. Condensation model for cylindrical nanopores applied to realistic porous glass generated by molecular simulation. Langmuir 2000, 16, 6064−6066. (31) Kowalczyk, P.; Jaroniec, M.; Terzyk, A. P.; Kaneko, K.; Do, D. D. Improvement of the Derjaguin-Broekhoff-de Boer theory for capillary condensation/evaporation of nitrogen in mesoporous systems and its implications for pore size analysis of MCM-41 silicas and related materials. Langmuir 2005, 21, 1827−1833. (32) Pellenq, R. J. M.; Coasne, B.; Denoyel, R. O.; Coussy, O. Simple phenomenological model for phase transitions in confined geometry. 2. Capillary condensation/evaporation in cylindrical mesopores. Langmuir 2009, 25, 1393−1402. (33) Tolman, R. C. The effect of droplet size on surface tension. J. Chem. Phys. 1949, 17, 333−337. (34) Koga, K.; Zeng, X. C.; Shchekin, A. K. Validity of Tolman’s equation: How large should a droplet be? J. Chem. Phys. 1998, 109, 4063−4070. (35) Block, B. J.; Das, S. K.; Oettel, M.; Virnau, P.; Binder, K. Curvature dependence of surface free energy of liquid drops and bubbles: A simulation study. J. Chem. Phys. 2010, 133, 154702. (36) Hiratsuka, T.; Tanaka, H.; Miyahara, M. T. Mechanism of kinetically controlled capillary condensation in nanopores: A combined experimental and Monte Carlo approach. ACS Nano 2017, 11, 269− 276. (37) Neimark, A. V.; Vishnyakov, A. Gauge cell method for simulation studies of phase transitions in confined systems. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2000, 62, 4611−4622. (38) Neimark, A. V.; Vishnyakov, A. A simulation method for the calculation of chemical potentials in small, inhomogeneous, and dense systems. J. Chem. Phys. 2005, 122, 234108. (39) Eyring, H. The activated complex in chemical reactions. J. Chem. Phys. 1935, 3, 107−115. (40) Chandler, D. Introduction to modern statistical mechanics; Oxford University: New York, 1987; pp 234−269.

(41) Tjatjopoulos, G. J.; Feke, D. L.; Mann, J. A. Molecule-micropore interaction potentials. J. Phys. Chem. 1988, 92, 4006−4007. (42) Johnson, J. K.; Zollweg, J. A.; Gubbins, K. E. The Lennard-Jones equation of state revisited. Mol. Phys. 1993, 78, 591−618. (43) Janeček, J. Long range corrections in inhomogeneous simulations. J. Phys. Chem. B 2006, 110, 6264−6269. (44) Wilhelmsen, O.; Bedeaux, D.; Reguera, D. Tolman length and rigidity constants of the Lennard-Jones fluid. J. Chem. Phys. 2015, 142, 064706. (45) Frenkel, D.; Smit, B. Understanding molecular simulation: From algorithms to applications, 2nd ed.; Academic Press: New York, 2002; pp 431−450. (46) Neimark, A. V.; Ravikovitch, P. I. Capillary condensation in MMS and pore structure characterization. Microporous Mesoporous Mater. 2001, 44−45, 697−707. (47) Morishige, K.; Kawano, K. Freezing and melting of water in a single cylindrical pore: The pore-size dependence of freezing and melting behavior. J. Chem. Phys. 1999, 110, 4867. (48) Morishige, K.; Uematsu, H.; Tateishi, N. Comparative study of liquid-solid phase transition of nitrogen in controlled pore glass and SBA-15: Accurate pore size of conventional mesoporous materials. J. Phys. Chem. B 2004, 108, 7241−7246. (49) Sayari, A.; Liu, P.; Kruk, M.; Jaroniec, M. Characterization of large-pore MCM-41 molecular sieves obtained via hydrothermal restructuring. Chem. Mater. 1997, 9, 2499−2506. (50) Kruk, M.; Jaroniec, M. Accurate method for calculating mesopore size distributions from argon adsorption data at 87 K developed using model MCM-41 materials. Chem. Mater. 2000, 12, 222−230. (51) Muroyama, N.; Ohsuna, T.; Ryoo, R.; Kubota, Y.; Terasaki, O. An analytical approach to determine the pore shape and size of MCM-41 materials from X-ray diffraction data. J. Phys. Chem. B 2006, 110, 10630− 10635. (52) Coasne, B.; Hung, F. R.; Pellenq, R. J. M.; Siperstein, F. R.; Gubbins, K. E. Adsorption of simple gases in MCM-41 materials: The role of surface roughness. Langmuir 2006, 22, 194−202. (53) Neimark, A. V.; Lin, Y.; Ravikovitch, P. I.; Thommes, M. Quenched solid density functional theory and pore size analysis of micro-mesoporous carbons. Carbon 2009, 47, 1617−1628. (54) Jagiello, J.; Olivier, J. P. A simple two-dimensional NLDFT model of gas adsorption in finite carbon pores. Application to pore structure analysis. J. Phys. Chem. C 2009, 113, 19382−19385. (55) Smith, M. A.; Ilasi, M. G.; Zoelle, A. A novel approach to calibrate mesopore size from nitrogen adsorption using X-ray diffraction: An SBA-15 case study. J. Phys. Chem. C 2013, 117, 17493−17502.

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Comprehensive Modeling of Capillary Condensation in Open-Ended

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