On the Hysteresis Loop of Argon Adsorption in Cylindrical Pores - The

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On the Hysteresis Loop of Argon Adsorption in Cylindrical Pores Phuong T. M. Nguyen, D. D. Do,* and D. Nicholson School of Chemical Engineering, University of Queensland, St. Lucia, QLD 4072 Australia ABSTRACT: We study the evolution of the hysteresis loop, its size and shape with the pore size, pore length, and closed end by GCMC simulation. It is shown that the mechanism of condensation and evaporation is essentially related to the fluid properties in the case of infinite cylindrical pores. In cylindrical pores of finite length, the hysteresis loop becomes smaller and then disappears when either the pore size or the pore length decreases according to the presence and movement of the fluid-vapor interface. A major deduction from this work is that by combining isotherms of pores of different sizes, lengths, and closed ends we can simulate adsorption of isotherms similar to those of solids with complex configurations, such as inkbottle pores. We conclude that not only the pore size but also the pore length and the affinity of the closed end play important roles in determining the shape of the adsorption isotherm and should be taken into account in the characterization of pore structure from gas adsorption.

1. INTRODUCTION According to classical thermodynamics, a physical adsorption isotherm should be reversible and the amount adsorbed should depend only on relative pressure P/P0 and not on the path along which the pressure is changed.1 It is however well-known that the adsorption and desorption branches of an isotherm are not always coincident, resulting in a hysteresis loop whose boundaries are the portions of the adsorption and desorption branches that do not overlap with each other. The presence of this loop and its size and shape depend on pore structure, surface chemistry, pore size, and temperature.2 The IUPAC classification distinguishes four types of hysteresis loops which are believed to be associated with different pore shapes and pore structures.3 Among them, the type H2 hysteresis loop which has a gradually increasing adsorption branch and a steeply decreasing desorption branch has been explained as being a consequence of the interconnectivity of a porous network, disordered solid, or the pore blocking effect in an inkbottle pore with a very narrow neck4 where the adsorbed fluid is held in the adsorbent until a critical blocking route is opened (e.g., through reaching a percolation threshold). In this paper, we study cylindrical pores with homogeneous surfaces and investigate in detail the effects of pore size, pore length, the presence or absence of a closed end, and the strength of the surface holding potential on the evolution of the hysteresis loop. Various shapes of hysteresis loops observed with complex solids can be observed with homogeneous cylindrical pores by accounting for appropriate distributions of pore size, pore length, surface strength, and closed end. This finding raises some reservations about the current interpretation of the shape, position, and size of the hysteresis loop in the characterization of pore structure. In order to gain a better understanding of the phase transition in the adsorbate, we explore the behavior of the core region of the pore for its density prior to condensation and prior to evaporation. Although it has been argued that condensation results from the instability of the adsorbed film and that evaporation is due to r 2011 American Chemical Society

the stretching of the liquid condensate beyond its metastable limit,5,6 our simulation results show that there is no significant change in the adsorbed film near the surface but rather that the state of the adsorbate in the core changes dramatically and that the core density approaches a threshold value just before condensation and just before evaporation. This density is found to be independent of pore size, implying that condensation and evaporation are associated with the fluid properties in the core. The evolution of the hysteresis loop as a function of pore size, pore length, surface strength, and closed end will be investigated and correlated with the microscopic behavior of the adsorbate in the pore.

2. INTERACTION ENERGY To describe the fluid-fluid interaction energy of argon, we use the 12-6 Lennard-Jones equation with the following molecular parameters, σff = 0.3405 nm and εff/k = 119.8 K.7,8 For the solid-fluid interaction energy, we consider two model solids of cylindrical shape. The first model is a cylindrical pore of infinite extent in the axial direction, and the second one is a finite cylindrical pore with either two open ends or one open end. For a sufficiently long cylindrical pore, the end effects are neglected and the solid-fluid potential energy can be calculated using the following equation (which depends only on the radial position of the adsorbate):9 jsf ¼ 4π2 Fs σsf 2 εsf ðI6 - I3 Þ

ð1Þ

where Fs is the surface density of solid atoms on the pore wall (for one graphene layer, Fs = 38.2 nm-2). The solid-fluid molecular parameters can be estimated from the Lorentz-Berthelot Received: November 26, 2010 Revised: December 20, 2010 Published: March 03, 2011 4706

dx.doi.org/10.1021/jp111254j | J. Phys. Chem. C 2011, 115, 4706–4720

The Journal of Physical Chemistry C

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Figure 1. Schematic diagram of a cylindrical pore and a gas particle inside the pore.

rule: 1 σsf ¼ ðσ ss þ σ ff Þ 2

εsf ¼ ðεss εf f Þ0:5

ð2Þ

while σss and εss are the collision diameter and the well depth of interaction energy of a carbon atom (σss = 0.34 nm and εss/ kB = 28 K). The functions I3 and I6 of eq 1 are given by 2 #-10 " 2 # " 63 σ sf 10 r 9 9 r I6 ¼ 1F - ; - ; 1; 128 R R 2 2 R ð3Þ 2 #-4 " 2 # " 3 σ sf 4 r 3 3 r I3 ¼ 1F - ; - ; 1; 4 R R 2 2 R

ð4Þ

In order to account for the pore end of a finite cylindrical pore, the solid-fluid potential energy between an adsorbate and a single wall finite cylindrical pore is obtained from the following equation (a detailed derivation is given in Appendix 1): Z 2π an σ 2n Iðθ, r, z, n, R, LÞ dθ ð5Þ jsf ¼ 4εsf Fs R sf n ¼ 3, 6 0

∑

where R is the pore radius, L is the pore length, r is the radial distance from the pore center to the adsorbate, z is the axial distance from the middle of the pore to the adsorbate, and an is a constant (a3 = -1 and a6 = 1). In the case of a multilayer cylinder, the potential energy is the summation of the potentials of concentric cylinders with radius Ri = R þ iΔ (i = 0, 1, ...), where Δ is the distance between two adjacent layers. In this paper, cylindrical pores with three graphitic layers are considered (Δ = 0.3354 nm). Figure 2 shows a typical solid-fluid potential profile for a finite cylinder pore with one graphene layer of the pore wall. The pore diameter is 2 nm, and the pore length is 3.4 nm. Equation 5 is valid for a finite cylindrical pore with two open ends. To develop a solid-fluid potential for a closed-end cylindrical pore, we add a flat surface at one end of the finite pore. The solidfluid potential interaction of an adsorbate inside the pore with this surface is modeled with the Steele 10-4-3 equation:10 " # 4 10 σ sf σ sf 4 2 2 σ sf jsf ¼ 2πεsf Fs σsf 5 h h 3Δðh þ 0:61ΔÞ3 ð6Þ

Figure 2. Solid-fluid potential energy profile of a finite cylindrical pore (D = 2 nm and L = 3.4 nm).

3. MONTE CARLO SIMULATION To acquire the data in the GCMC simulation, we use at least 100 000 cycles for the system to reach equilibrium and another 20 000 cycles for the statistics collection. Each cycle has 1000 moves and exchanges which include insertion and deletion with equal probability. In the equilibrium step, the displacement step length is adjusted at the end of every cycle by checking the displacement acceptance ratio. The cutoff radius is chosen as half of the simulation box length. We apply periodic boundary conditions at two ends of the pore to simulate a cylinder of infinite extent; a hard boundary is applied to the open ends of finite cylinders. For subcritical adsorption at 87.3 K, this hard wall boundary gives similar results to a finite pore connected to a bulk gas reservoir. In order to study the microscopic behavior of the adsorbate in the pore, we analyze pore density, local density, core size, and core density. 3.1. Pore Density. The pore density is defined as the amount of adsorbate in the pore per unit accessible volume:11 Fexcess ¼

Nexcess Vacc

ð7Þ

where Vacc is the accessible pore volume in which the solid-fluid potential is nonpositive12 and Nexcess is the excess amount of gas molecules in the pore, which is defined as the difference between the number of gas molecules inside the box (Nbox) and the hypothetical number of gas molecules occupying the accessible volume at the same density as the bulk gas density: Nexcess ¼ Nbox - FG Vacc

ð8Þ

where FG is the bulk gas density. 3.2. Local Density. To have a better picture of how the fluid is organized inside a pore, we calculate the local density distribution with the local density of a bin k at the distance r from the pore surface being defined as FðrÞ ¼

where h is the shortest distance from an adsorbate to this surface. 4707

ΔN ΔV

ð9Þ

dx.doi.org/10.1021/jp111254j |J. Phys. Chem. C 2011, 115, 4706–4720


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where ΔN is the number of particles falling between r and r þ Δr and ΔV is the differential volume at r. The above equation is applied along the radial direction for a sufficiently long cylinder. In the case of a finite cylinder, the local density is defined in terms of the radial direction as well as the axial direction: Fðr, zÞ ¼

ΔN ΔV ðr, zÞ

ð10Þ

where ΔN is the number of particles having coordinates falling between (r, r þ Δr) and (z, z þ Δz). In our computation, we use Δr = 0.01σff in the case of an infinite cylinder and Δr = 0.02σff, Δz = 0.05σff in the case of a finite cylinder. The local density distribution is calculated at the end of each cycle and averaged at the end of the sampling step. In the case of a finite cylindrical pore, the density profile is smoothed by averaging the local density distribution in the radius of 1.5σff in both radial direction and axial direction. 3.3. Accessible Core Size. In order to measure the core size of the pore, we calculate the accessible core size which is similar to the accessible pore size but takes account of the presence of the adsorbates. The accessible core volume (Vacc core ) is the accessible volume in which the summation of the solid-fluid potential and the fluid-fluid potential energy of a new particle with the particles existing in the system is nonpositive. In our computation, the accessible core volume is calculated at the end of every 1000 cycles and averaged at the end of the sampling step. This accessible core is assumed as a cylinder whose length is equal to the simulation cell length (Lcell); therefore, the core size is equal to the diameter of this cylinder and can be calculated as follows: rffiffiffiffiffiffiffiffiffiffiffi core Vacc core Dacc ¼ 2 ð11Þ πLcell

Figure 3. Adsorption of argon at 87.3 K in an infinite graphitic cylindrical pore with different pore diameters. The closed symbols and solid lines are the adsorption branches, and the opened symbols and dotted lines are the desorption branches. The isotherms of the 3.2, 3.6, 4.0, 4.4, 4.8, 5.2, 5.6, and 6.0 nm pores are shifted up 25, 50, 75, 100, 125, 150, 175, and 200 kmol/m3, respectively.

For a dilute bulk phase (ideal gas), Nk is very small and we have f ðNk , Nk Þ ¼1 ÆNk æ

In this paper, we use the term “core size” as a short name for the accessible core size. 3.4. Local Number of Particle Fluctuation. In order to determine the phase boundary between the adsorbed phase and the core gas phase, we analyze the number of particle fluctuation in each bin. The boundary between these two phases is the bin where the particle number fluctuation is maximum. In a grand canonical ensemble, the rate of change of the number of particles with pressure in a volume V is given by the following equation:13

3.5. Core Density. After determining the phase boundary from the analysis of the local number of particle fluctuations, the core density is calculated in the region where there is no variation in the fluctuation number of particles (core region) by the following equation:

DN f ðN, NÞ ¼ Dp ðN=V ÞkT

where ÆNcoreæ and Vcore are the average number of molecules and the volume of the core region, respectively.

ð12Þ

Thus, we define the particle number fluctuation as a measure of how fast the particle number varies with pressure: F ¼

ÆNNæ - ÆNæ2 f ðN, NÞ ¼ ÆNæ ÆNæ

ð13Þ

Similarly, by considering a bin k as a grand canonical system, the number of particle fluctuation for every bin is obtained with the total number of particles N being replaced by the number of particles in the bin k (Nk). Fk ¼

ÆNk Nk æ - ÆNk æ2 f ðNk , Nk Þ ¼ ÆNk æ ÆNk æ

ð14Þ

lim

Nk f 0

Fcore ¼

ÆNcore æ Vcore

ð15Þ

ð16Þ

4. RESULTS AND DISCUSSION We first discuss the results for infinitely long cylinders. They will be used as a reference in the subsequent discussion of the simulation results of finite cylinders to show the importance of the pore length on the behavior of the adsorption isotherm, especially the hysteresis loop, its size and shape, and the pressure at which evaporation occurs. 4.1. Infinitely Long Cylindrical Pore. For infinite cylindrical pores (with no end effects), the pore diameters are chosen in the range from 2.8 to 6.0 nm to investigate the evolution of the hysteresis loop, especially its size and the pressures at which the condensation and evaporation occur. 4708



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Figure 4. Snapshots of argon particles in graphitic cylindrical pores at 87.3 K before condensation and after evaporation.

Figure 5. The change of core size versus pressure (A) and critical core size just before condensation versus pore size (B) of the adsorption branch.

Figure 3 shows the isotherms of argon adsorption at 87.3 K in graphitic cylindrical pores. For the range of pore sizes considered, a number of features observed are listed below: (i) 2.8 nm is the critical pore diameter at which the hysteresis starts to appear for an infinitely long graphitic cylindrical pore. As the pore gets smaller, the sharp change in pore density becomes more gradual; the process of condensation is replaced by a pore filling process, typically observed for small micropores. (ii) For pore sizes greater than 2.8 nm, the adsorption isotherms show a hysteresis loop of type H1 in the IUPAC classification. (iii) The hysteresis loop area increases with increasing pore size. (iv) The reduced pressure (P/P0) at condensation shifts to a higher value with an increase in pore size. For pores having diameters greater than 5.2 nm, the evaporation occurs at P/P0 = 0.2, while, for smaller pores, the evaporation shifts to a lower pressure with a decrease in pore size. The reason for the evaporation to occur at P/P0 = 0.2 for large pores is that this is the pressure at which the density of the “core” has reached a threshold value below which a fluid cannot be sustained as a liquid-like phase. We discuss this in more detail below. To gain a general picture of how the molecules distribute in a pore just before condensation and just after evaporation, we

show in Figure 4 snapshots at the corresponding pressures for pores of different sizes. From this figure, it appears that, just before condensation, the size of the core is approximately the same for all pores. To quantify this, we determine the evolution of the size of the core as a function of pressure with the core size being defined in section 3.3. Figure 5A shows that the core size decreases with pressure and reaches a threshold value just before condensation. The threshold core size is plotted in Figure 5B as a function of pore size. For larger pores, the threshold core size is constant, suggesting that condensation is related to fluid properties while for smaller pores the threshold core size decreases with a decrease in pore size. Its dependence on pore size for small pores is due to the influence of the substrate which stabilizes the adsorbed layers. For pores larger than 4.8 nm, more than four layers are formed before condensation and the influence of the substrate diminishes as adsorption progresses. Therefore, the condensation is affected purely by the fluid properties, resulting in a constant core size just prior to condensation. Just before evaporation, we observe that there are about two layers of molecules remaining on the surface, irrespective of pore size (Figure 4). This is in agreement with the results from other GCMC simulation studies,14 NLDFT calculations,15 and experimental data16-19 for argon and nitrogen at various temperatures. Although the number of layers remaining after evaporation does not depend on the pore size, it depends on the surface strength. For a surface composed of three graphitic layers, two layers are 4709



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Figure 6. Critical core size (A) and film thickness (B) versus pore size just after evaporation.

Figure 7. Local density distribution of argon 87.3 K in different pore sizes.

retained on the surface immediately after evaporation. For stronger surfaces, more layers remain after evaporation. To quantify the number of adsorbed layers after evaporation, we calculate the accessible volume of the core just after the evaporation (V*core); the thickness of

the liquid film is equal to the difference between the accessible volume of the empty pore and Vcore * . The core size just after evaporation increases with an increase in pore size (Figure 6A), while the film thickness is constant at about 1.9 collision diameter (Figure 6B), 4710



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Figure 8. Local density distributions before and immediately after condensation (A) and the fluctuation in the local number of particles just before condensation (B) in a 6.0 nm infinitely long cylindrical pore.

confirming that only two layers remain on the pore wall just after evaporation. In order to understand the microscopic picture of condensation and evaporation, we analyze the local density distributions during the adsorption and desorption. These are shown in Figure 7 for a number of pore sizes, where we see that the density of the core increases with pressure. Immediately before condensation, the density of the core is about 100 mol/m3. Interestingly, this density is independent of pore size, implying that the core density just before condensation is associated purely with the fluid properties. To determine the density of the core, we have to identify the interface separating the adsorbed layer and the core. To this end, we analyze the local fluctuation of particle number as a function of distance from the pore surface. We show this in Figure 8 for a pore of 6 nm diameter. The interface is defined as the position where the particle number fluctuation is maximum and the core density is calculated in the region where there is no variation in the fluctuation number of particles. Figure 9 shows the evolution of the core density for pores of various diameters during the adsorption and desorption processes.

Before condensation, the core density increases with pressure, and as the condensation pressure is approached, it increases sharply to about 100 mol/m3 at which condensation occurs (Figure 9A). Along the desorption path, we consider the density of the core as a function of pressure. This is shown in Figure 9B. For pores having diameters greater 5.2 nm, the core density at the point when the evaporation occurs is 30 kmol/m3. For pore diameters smaller than 5.6 nm, the corresponding core density is higher than 30 kmol/m3. This is due to the long-range effects of the solid-fluid interaction. So what is the significance of 100 and 30 kmol/m3 in condensation and evaporation, respectively? During adsorption, the adsorbed layer thickness increases with pressure and at the same time the density in the core increases dramatically even when the thickness of the adsorbed layer increases modestly. When the density of core has reached 100 mol/m3, condensation occurs independent of the pore size. This suggests that the condensation is associated with the fluid properties. Figure 10 shows the bulk fluid properties of argon at 87.3 K, as a plot of chemical potential versus density. The triangular symbols are obtained from NVT simulation; the 4711



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Figure 9. Core density before condensation (A) and before evaporation (B) of different pore sizes.

Figure 10. van der Waals loop of argon at 87.3 K: (A) bulk density on a linear scale; (B) bulk density on a logarithmic scale. The triangular symbols are obtained from NVT simulation, and the dotted dashed line is the equilibrium obtained from GEMC. The closed circle symbols with solid line and the open circle symbols with dashed line are the GCMC simulation results whose initial configuration is an empty simulation box and a saturated simulation box, respectively.

horizontal dotted-dashed line joins the equilibrium vapor and liquid density (obtained from GEMC simulation) at the equilibrium chemical potential. The closed circle symbols with solid line and the open circle symbols with dashed line are the GCMC simulation results whose initial configurations are an empty simulation box and a saturated simulation box, respectively. It is seen that the saturated vapor concentration is 100 mol/m3 (Figure 10), suggesting that, when the core density has reached this value, a phase change will occur in the same way as in the bulk fluid. This value of density is lower than the bulk gas density at which the bulk fluid changes its phase in GCMC simulation. The ease of the phase change of the confined fluid in the pore is perhaps due to the presence of nucleation sites in the adsorbed layers which induces condensation. On the other hand, when we consider the onset of evaporation, the critical concentration of the core prior to evaporation is 30 kmol/m3, which is equal to the liquid spinodal concentration of bulk argon. This density is lower than the bulk liquid density at equilibrium which is 35 kmol/m3. This result is in agreement with the suggestion that the confined liquid in the pore is in a metastable state and this state persists

until the liquid spinodal point is reached (the liquid condensate is stretched to its limit). As a result, a hysteresis loop is observed.14 Since the adsorption and desorption are sensitive to the pore diameter (Figure 3), we can combine isotherms of pores of different sizes to produce an isotherm whose shape is similar to a complex pore structure in Figure 11; the closed symbols are the isotherms of individual cylindrical pores, while the open symbols are the isotherms obtained by combining these isotherms in the same ratio for each pore size in the same plot. These combined isotherms are similar to those observed for inkbottle pores.20 This means that a particular hysteresis loop is not specific to the family of inkbottle pores. For example, by combining the isotherms of pores of different sizes (2.8-6.0 nm), we can reproduce five types of adsorption isotherms (Figure 11A-E) which are typical of the inkbottle model.20 Thus, it can be concluded that these hysteresis loops are not unique for inkbottle pores and, by analyzing the shape of the hysteresis loop, one cannot draw any conclusion about the pore structure. Another example to support this argument is that, if we combine isotherms of pores having diameters from 4.8 to 6 nm, we recover the type H2 hysteresis loop which has been attributed to the interconnectivity of a porous network, 4712



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Figure 11. Different types of hysteresis loops obtained from adsorption in independent pores with a range of sizes. The closed symbols are the isotherms for individual cylindrical pores, while the open symbols are the isotherms obtained by combining these isotherms in the same ratio for each pore size in the same plot.

disordered solid, or pore blocking effect in an inkbottle pore (Figure 11F). These results perhaps can offer an explanation for the H2 hysteresis loop obtained experimentally for a network of parallel straight mesopores without lateral interconnection and with a wide range of pore size distribution.21 This is an interesting point in the pore characterization where pore structure is often deduced on the basis of the type of hysteresis loop. The contributions to triangular hysteresis loops that are caused by pore blocking in interconnected networks and those due to a combination of different pore sizes need to be delineated. This is a subject of our future investigation. Having discussed how the adsorption isotherm behaves with pore size for an infinitely long pore, we would like to investigate how the pore length would affect the isotherm.

4.2. Finite Cylindrical Pores. We believe that the pore length is an important parameter for evaporation because, in an infinitely long pore, the liquid condensate after condensation is an “infinitely” long column of liquid. This implies that the “infinite” fluid must be stretched beyond its stable limit (liquid spinodal point) before it can evaporate. On the other hand, in a finite pore, the presence of an interface between the bulk gas phase and the adsorbed phase will facilitate the mass transfer between these two phases. This will result in an evaporation process which is distinctly different from that in the infinitely long pore. The current practice for the determination of pore size is based on the premise that pores are infinitely long. Experimentally cylindrical pores with different morphologies have been successfully synthesized, for example, long channel morphology 4713



Figure 12. Adsorption isotherms of argon at 87.3 K in finite cylindrical pores of 4 nm diameter with different pore lengths. The closed symbols present adsorption branches, while the open symbols present desorption branches. The isotherms of 4.5, 5.0, 6.8, 13.6, 20.4, and 27.2 nm and infinite pore length are shifted up 15, 30, 45, 60, 75, 90, and 105 kmol/ m3, respectively.

of rope-like (above 1 μm), rodlike particles (700-900 nm), column-like particles (260 ( 50 nm), or even a shorter pore length of only 150 ( 50 nm with a pore diameter of 10-15 nm.22 Thus, pore length should be accounted for in pore characterization. The interplay between the pore size and the pore length could have a significant implication in the determination of the pore structure from the analysis of an adsorption isotherm. We have studied the effects of pore length on the adsorption of argon at 87.3 K in cylindrical pores. When the pore is extremely short (L = 4 nm), a reversible isotherm is observed, while, for longer pores, adsorption and desorption follow different paths, forming a hysteresis loop. This loop is larger and shifts to the left as the pore length increases. When the pore length is greater than 13.62 nm, the position of the hysteresis loop and its size are independent of pore length. Comparing this interesting asymptotic behavior with the hysteresis loop for the infinitely long cylinder, we observe that (i) the condensation pressure in the finite cylinder is the same as that for the infinitely long cylinder and (ii) the evaporation pressure in the finite cylinder is greater than that of the infinitely long cylinder; i.e., the evaporation branch for a finite cylinder falls between the adsorption and desorption branches of that for the infinitely long cylinder. The same behavior is also observed with a larger pore (D = 6 nm), as shown in Figure 13. This interesting observation raises a concern about the common practice, in the characterization of the porous structure of a solid, of using an infinitely long cylinder as a model for representing the equilibrium transition in the theoretical isotherm to match against the experiment desorption branch.23-25

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Figure 13. Adsorption isotherms of argon at 87.3 K in a finite cylindrical pore with D = 6 nm with different pore lengths. The closed symbols present adsorption branches, while the open symbols present desorption branches. The isotherms for 6.8 nm, 13.6 nm, and infinite pore length are shifted up 15, 30, 45, and 60 kmol/m3, respectively.

As an attempt to address this concern, we offer as an alternative that the desorption branch of the theoretical isotherm for a finite cylinder should be used to match against the experimental desorption branch to derive the pore length distribution of each pore size. This argument has merit because real mesoporous materials contain pores finite in length and are open to the adsorptive gas with an interface that facilitates the exchange of mass during the evaporation. In stark contrast to the infinitely long pore, for which the phase transition occurs when the fluid approaches a critical state quantified by the density of the core, the phase transition in a finite cylinder occurs gradually with the movement of the two menisci at the two open ends. The shape of these menisci depends on pore size and pore length. We analyze their behavior after condensation and before evaporation and show in Figure 14 the density profiles at the pressures just after condensation and just before evaporation for 4 nm cylindrical pores of different lengths with both ends open to the bulk gas. The meniscus is almost hemispherical after condensation, while its shape just after evaporation depends on the pore length. For the long pore of 13.6 nm length, the meniscus is not hemispherical but rather adopts a conical shape with a round end. This feature has been suggested by Broekhoff and de Boer.26 Therefore, the meniscus just before evaporation changes its shape from hemispherical for short pores to conical for long pores. The reason for this behavior is as follows: the longer the pore, the stronger is the interaction between the solid wall and the adsorbate and the greater the fluid-fluid interactions among molecules in a favorably packed confined liquid. This means that the confined liquid can remain 4714



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Figure 14. Density profile of argon 87.3 K in a 4 nm finite cylindrical pore with different pore length right after condensation and evaporation.

Figure 15. Parametric map for a three-layer graphitic cylindrical pore: square symbols present the reversible isotherms, and circle symbols present the hysteresis isotherms.

in a metastable state longer until a lower reduced pressure is reached and capillary evaporation occurs.14 We have seen that the pore size and the pore length affect the hysteresis loop in such a way that the loop becomes smaller and disappears when the pore size and/or pore length is decreased. It is useful to develop a parametric map to delineate the region of hysteresis and the region of reversibility. This is shown in Figure 15. The vertical dashed line is for a 2.8 nm pore, which has been found to be the critical pore size for a cylinder of finite length. If the isotherm for this pore size is reversible for an infinitely long cylinder, the isotherms for finite cylinders of any length are also reversible. Thus, the boundary between the two regions is only for a pore size greater than 2.8 nm, and this is shown as the solid line in Figure 15, resulting from our extensive simulations of pores of different sizes and lengths. The closed symbols are for simulations where we observe reversibility and the open symbols for those where a hysteresis loop is observed.

Figure 16. Adsorption isotherm of argon at 87.3 K in a finite cylindrical pore with D = 4.0 nm and L = 6.8 nm and with different surface strengths of the closed-end surface. Closed symbols are for adsorption branches and open symbols for desorption branches. The isotherms of pores, whose affinities at the closed-end surface are 26.7, 32.7, 37.8, 50, and 100% of the affinity three-layer graphene surface, are shifted up 15, 30, 45, 60, and 75 kmol/m3, respectively.

4.3. A Cylindrical Pore with Closed End. Here, we address two questions: (a) How does closing one end of the pore affect the adsorption isotherm? (b) What is the effect of changing the surface adsorption strength of the closed end? It is believed that the adsorption in a cylindrical pore with closed end is reversible because the closed end acts as a strong site for adsorption and as a result the adsorbed phase originates there and moves from the closed end to the pore mouth as the pressure is increased and that the reverse path is followed for desorption.27 However, a hysteresis loop has been observed experimentally, for example, for the adsorption of nitrogen at 77.2 K in a closed-end silicone cylindrical pore.28 One possible reason for the hysteresis could be that the adsorption affinity of 4715



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Figure 17. Density profile of argon before condensation in a closed-end cylindrical pore with different affinities of the closed end.

Figure 18. Adsorption isotherms of argon at 87.3 K in a closed-end cylindrical pore of length L = 20.43 nm with different pore diameters. Closed symbols are for adsorption branches, and open symbols are for desorption branches. The isotherms for the 4.4, 4.8, 5.2, 5.6, and 6.0 nm pores are shifted up 20, 40, 60, 80, and 100 kmol/m3, respectively.

the closed end is weaker than that of the pore wall, making the closed end relatively repulsive rather than attractive compared to the pore wall. To substantiate this hypothesis, we study the effects of the affinity of the closed end by varying its surface strength. Specifically, we study the cases when its surface strength is the same or weaker than that of a graphite surface, because if it is stronger we would expect the closed end is a strong site, facilitating reversibility between the adsorption and desorption paths. Figure 16 shows argon adsorption isotherms at 87.3 K in a number of closed-end pores having a diameter of 4 nm and a length of 6.8 nm. The affinities of these closed-end surfaces are 26.7, 32.7, 37.8, 50, and 100% of the affinity of a three-layer graphene surface. Also plotted in the same figure is the adsorption isotherm for a pore having the same dimensions and two open ends, for comparison. When the affinity of the closed end is the same as the graphitic pore, the isotherm is reversible (the top curve in Figure 16). However, when the affinity of the closed end is reduced, we observe an evolution of the hysteresis loop, as seen in the four middle isotherms

in Figure 16. A common factor among the isotherms of the closedend pores is that the desorption branches are almost the same, with the evaporation occurring at P/P0 = 0.22 regardless of the affinity of the closed end. This can be explained as follows: once the pore is filled and the desorption process starts, argon particles evaporate from the hemispherical interface at the pore mouth and this interface recedes toward the closed end, irrespective of the strength of the closed end. The strength of the closed end only comes into play when this interface reaches close to the end. If the affinity of the closed end is the same as the graphitic wall, we see an adsorbed film at the closed end. On the other hand, when the closed end is a weakly adsorbing surface, we observe a depletion of the adsorbed layer at the closed end, due to the fact that the closed end is not strong enough to form an adsorbed layer. The increase in the thickness of the adsorbed layer before the condensation with the affinity of the closed end is shown in Figure 17. When comparing the desorption branches of closed-end pores with desorption from the two-open-ended pores, we observe an earlier evaporation from the open-ended pore because of the presence of two open ends compared to one open end in the closed-end pore. The adsorption branch of the closed-end pore shifts to a higher pressure when the closed end is weaker in strength, and it approaches that of the two-open-end pores. This is explained as follows. When the affinity of the closed end is the same as the graphitic wall, the adsorbates fill the pore from the closed end and the resulting meniscus moves from the closed end to the pore mouth as the pressure is increased. However, when the closed end is weaker, the adsorbate prefers to fill the pore by radial layering from the pore wall in a manner similar to the case of the two-open-end pore. Therefore, there is a competition between two processes: radial layering from the pore wall and filling from the closed end. Therefore, as the affinity of the closed end decreases, the process of layering from the pore wall dominates the process and the adsorption branch shifts to the right and approaches that of the two-open-end pores. It is clear that the important parameters affecting the behavior of the isotherms are the pore size, the pore length, and the closed end (and its affinity). Similar to what was discussed earlier with pores having two open ends, we investigate here the effects of pore size and length on adsorption in pores having only one open end. First, the effects of pore size are studied for pores having one open end. We keep the pore length of 20.43 nm and vary the pore size from 4 to 6 nm. The argon adsorption isotherms of these pores are presented in Figure 18. A number of features are observed as follows: (i) The adsorption isotherm changes from reversible to irreversible and the hysteresis loop is larger as the pore size increases. (ii) For pore sizes greater than 4 nm, the adsorption and desorption are gradual and they form type H1 hysteresis loops. 4716



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Figure 19. The schematic of the pore filling (A) and emptying (B) process in a closed-end cylindrical pore.

shown that the adsorption and desorption follow the same mechanism of filling from the closed end. However, there is a hysteresis in the movement of the meniscus in the desorption process compared to the adsorption. Instead of growing quickly as the pressure increases, the meniscus moves toward the closed end more gradually during the desorption process. This can be explained as follows. At high density, the confined liquid is in the metastable state and the favorable fluid-fluid interactions maintain this state until a lower pressure is reached. When either the pore size or the pore length is increased, the adsorbate particles in the confined liquid have a larger volume to rearrange to achieve a better packing. Therefore, the metastable state becomes more favorable and it is maintained in the desorption branch until a lower pressure is reached. Thus, similar to the finite cylindrical pore with two open ends, a hysteresis loop appears and increases in size when either the pore size or the pore length increases. The reason for this is the persistence of the metastable state of the confined liquid in the desorption branch because of the favorable fluid-fluid interactions.

Figure 20. Adsorption isotherms of argon at 87.3 K in closed-end cylindrical pores (D = 6 nm) with different pore lengths (L). Closed symbols are for adsorption branches, and open symbols are for desorption branches. The isotherms of pore whose lengths are 13.6 and 20.4 nm are shifted up 20 and 40 kmol/m3, respectively.

(iii) Both adsorption and desorption branches shift to higher relative pressures as the pore size increases. (iv) There is no sudden change in the amount adsorbed in the pore. Therefore, no capillary condensation or evaporation occurs during the adsorption and desorption processes. Second, the effects of pore length are studied by varying the pore length of a 6 nm cylindrical pore from 6.8 to 20.4 nm (Figure 20). When the pore is short (L = 6.8 nm), the adsorption isotherm is reversible. When the pore is longer, the adsorption and desorption follow different branches, forming a hysteresis loop. The longer the pore, the larger the size of the observed hysteresis loop without the occurrence of condensation or evaporation because there is no sudden change in the amount adsorbed. No matter how long the pore is, the mechanism of pore-filling is a gradual filling process from the closed end toward the pore mouth which is reversed on desorption (Figure 19). However, when the pore is longer than 4 nm, the desorption branch does not follow the same path as adsorption and there is then a hysteresis loop. Figure 21 shows different density configurations of the system in adsorption and desorption at the same relative pressure for a 6 nm closed-end pore with a graphitic strength closed end. It is

5. CONCLUSION A number of interesting features emerge from this study of argon adsorption at 87.3 K in graphitic mesoporous cylindrical pores with different pore configurations. In an infinitely long cylindrical pore, the evolution of the hysteresis loop with pore size can be better understood from an analysis of the microscopic behavior of the fluid just before condensation and evaporation. It is found that the mechanism of condensation and evaporation is directly related to the behavior of the fluid in the central core of the pore. By comparing this with the behavior of the bulk fluid, we have shown that condensation occurs when the core density is about the same as the saturated vapor density of the bulk fluid, while for evaporation the density of the core is equal to the liquid spinodal density just before evaporation. This result is in agreement with the suggestion that the metastable state of the confined liquid is preserved on desorption until a sudden evaporation occurs at a pressure lower than the equilibrium transition pressure. Similar behavior is also observed in finite pores with two open ends or with one open end. The effect of pore length is similar to that of pore size, in that the hysteresis loop becomes smaller and then disappears when either the pore size or pore length decreases. We have presented a parametric map delineating the hysteresis region and reversible region as a function of pore size and length. A hysteresis loop has been observed not only in the case of two-open-end pores but also for pores with one open end, but there is a subtle difference in the sharpness of the hysteresis loop, which is much shallower than that for infinitely long pores or two-open-end pores. For pores with one open end, we have explored the effects of the surface strength of the closed end and have found that the pore-filling results from the competition between two processes: one is the radial layering from the pore wall until the point when 4717



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Figure 21. The density profile of argon in a closed-end cylindrical pore of 6 nm diameter changes with pressure during adsorption and desorption.

Figure 22. The adsorbate is inside the pore (A) or outside the pore (B); differential area at point A (C).

the capillary condensation occurs, and the other is the advance of the interface originating from the closed end. When the affinity of the closed end decreases, the process of layering from the pore wall is the dominant mechanism for condensation and hysteresis results when the pore size is greater than the critical pore size. One interesting point arising from this simulation study is that the adsorption and desorption branches are sensitive to pore size, pore length, and the number of pore ends. For example, by combining isotherms of pores of different sizes, lengths, and closed ends, we can generate isotherms with sufficient variation to mimic the isotherms of solids with complex configurations such as inkbottle pores. The type H2 hysteresis loop, which has been believed to be a result of complex pore structures, has also been reproduced. Therefore, not only pore size but also pore

length and affinity of the closed end, in the case of a closed-end pore, should be taken into account in the characterization of pore structure from adsorption isotherms. Finally, it has been a practice to match the experiment desorption branch to the equilibrium transition branch of the theoretical isotherm of an infinitely long cylindrical pore to derive the pore size distribution. Here, we argue that the experiment desorption branch should be matched against the desorption branch of the theoretical isotherm of a finite cylinder instead. This will introduce extra degrees of freedom in the characterization process because a question is then raised about the choice of the pore length and whether the pore has two open ends or one open end. This is a subject of our future correspondence. 4718



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’ APPENDIX 1 The solid-fluid interaction energy between an adsorbate located at D(r,z) and a finite cylindrical pore of pore radius R and pore length L in eq 5 is derived as shown in Figure 22. The interaction energy between an adsorbate located at D(r,z) with all the atoms in an annulus of the finite cylindrical pore of radius R is the integration of the interaction energy between 0 that adsorbate with a differential area rdθdz located at A (Figure 22B): ( " 6 #) 12 σ σ dj ¼ Fs Rdz0 dθ 4ε ð17Þ d d or dj ¼ 4εFs Rdz0 dθ

∑ an σ n n ¼ 3, 6 ðd 2 Þ 2n

ð18Þ

where an is a constant (a3 = -1 and a6 = 1) and d is the distance between A and D: d2 ¼ ðz - z0 Þ2 þ R 2 þ r 2 - 2Rr cos θ

and

ð19Þ

2n

ð20Þ

j ¼ 4εFs R Z

2π

(Z

0

∑ an σ2n n ¼ 3, 6

z0 ¼ L=2

dz0 2 n 0 2 2 z0 ¼ -L=2 ½ðz - z Þ þ R þ r - 2Rr cos θ

) dθ ð21Þ

Therefore, j ¼ 4εFs R

∑ an σ n ¼ 3, 6

Z

2π

2n

Iðθ, R, r, n, z, LÞ dθ

ð22Þ

0

where Iðθ, r, z, n, R, LÞ ¼

ðR 2

Hðx2 , nÞ - Hðx1 , nÞ þ r 2 - 2Rr cos θÞn - 0:5

Hðx2 , nÞ - Hðx1 , nÞ Gn - 0:5

ð23Þ

G ¼ R 2 þ r 2 - 2Rr cos θ

ð24Þ

¼

Z Hðx, nÞ ¼ ¼

x ð2n - 1Þ

dx ð1 þ x2 Þn

n-1

ð2n - 1Þð2n - 3Þ:::ð2n - 2j þ 1Þ ∑ n-j j j ¼ 1 2 ðn - 1Þðn - 2Þ:::ðn - jÞð1 þ x2 Þ þ

ð2n - 3Þ!! 2n - 1 ðn - 1Þ!

arctanðxÞ

ð25Þ

x2 ¼

L -z 2 pffiffiffiffi G

" # 2048 1 1 Iðθ, r, z, 6, R, LÞ ¼ þ 11 ðL - 2zÞ11 ðL þ 2zÞ11 " # 49152 2 1 1 G þ þ 13 ðL - 2zÞ13 ðL þ 2zÞ13 " # 229376 4 1 1 G þ þ oðG6 Þ ð27Þ 5 ðL - 2zÞ15 ðL þ 2zÞ15

σ ∑ an 2 n 0 2 n ¼ 3, 6 ½ðz - z Þ þ R þ r 2 - 2Rr cos θ

To find the total potential energy between an adsorbate located at D and all the solid atoms on the cylinder, we integrate the above equation with respect to z0 and θ:

Although eq 23 can be used for any nonzero value of G, it is important that we derive the analytical Taylor series of eq 25 for small values of G because of the singularity of the two terms in the right-hand side of eq 25. Thus, when 0 e G , 1, we use the following Taylor series: " # 32 1 1 Iðθ, r, z, 3, R, LÞ ¼ þ 5 ðL - 2zÞ5 ðL þ 2zÞ5 " # 384 2 1 1 G þ þ 7 ðL - 2zÞ7 ðL þ 2zÞ7 " # 1024 4 1 1 G þ ð26Þ þ oðG6 Þ 3 ðL - 2zÞ9 ðL þ 2zÞ9

If we substitute d2 of eq 19 into eq 18, we get dj ¼ 4εFs Rdz0 dθ

L - -z 2 pffiffiffiffi ; x1 ¼ G

In our computation, when G is smaller than 10% of the collision diameter of the adsorbate, we use the Taylor series in eqs 26 and 27; otherwise, eq 23 is used. This solid-fluid potential energy calculation is valid for a particle either inside or outside of a cylindrical pore.

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On the Hysteresis Loop of Argon Adsorption in Cylindrical Pores - The

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