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Sep 13, 2013 - types suggested by de Boer were omitted in the IUPAC classification. A recent reappraisal of these classifications was made by Sing and...
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Condensation and Evaporation in Capillaries with Nonuniform Cross Sections Chunyan Fan, D. D. Do,* and D. Nicholson School of Chemical Engineering, University of Queensland, St. Lucia, Queensland 4072, Australia ABSTRACT: We present a Grand Canonical Monte Carlo simulation study of argon adsorption at 87 K in slit shape mesopores with nonuniform separation between the pore walls. We show that different wall curvatures can generate diverse hysteresis loop shapes, simply by tuning the wall curvature and hence the variation of width along the pore. In this way we are able to describe all the hysteresis loop shapes classified by de Boer in 1972 and later by the IUPAC in 1985, including the de Boer Types C and D, which were omitted from the IUPAC classification. In addition we observe other loop shapes not identified in either classification but which have been reported experimentally.

1. INTRODUCTION A major characteristic of adsorption hysteresis in mesoporous materials is the shape and size of the hysteresis loop. Early proposals to classify hysteresis loops and to relate them to pore structure were made by Everett1 and by de Boer (dB);2 the latter distinguished the five types of loop designated, A, B, C, D, and E, shown in Figure 1. Later, the IUPAC identified3 four loop types denoted by H1, H2, H3, and H4 based on experimental isotherms observed in disordered solids. The first 3 correspond to A, E, and B in the dB scheme, and H4 was associated with narrow slit-like pores, but some of the loop types suggested by de Boer were omitted in the IUPAC classification. A recent reappraisal of these classifications was made by Sing and Williams,4 who concluded that there was no immediate need to change the IUPAC classification. The origins of hysteresis loops have been discussed by Thommes.5 With the recent development of new materials, especially ordered mesoporous solids such as MCM-416 and SBA-15,7 many different hysteresis loops have been observed, especially Type C (also named “inverse H2”8)9,10 and a combination of Type H1 with Type C;11−14 a comprehensive review of hysteresis loops for ordered mesoporous materials has been presented by Horikawa et al.15 Advances in computational power have helped to make molecular simulation an important tool for the study of the mechanism of hysteresis, and various models have been examined, which consider not only pore geometry13,16−19 but also connectivity17,20,21 and pore morphologies,22−26 and which attempt to reproduce the hysteresis that is observed experimentally. Evidence from HRTEM,27 and recent work combining neutron scattering data with reverse Monte Carlo28,29 suggests that, in many porous carbon materials, the pores are formed by nonparallel graphene planes or planes that have curved rather than flat walls. In this work we examine the effects of changing the pore geometry on the shape of the hysteresis loops for three basic pore models: (i) pores with planar surfaces that are not parallel (wedge shaped pores), (ii) pores formed from convex curved surfaces, and (iii) pores formed from concave curved surfaces. We find that these simple pore models can generate all the loop © 2013 American Chemical Society

types identified in earlier classifications as well as types not previously recognized.

2. INTERACTION ENERGY We used argon as a model adsorbate with an intermolecular potential energy of interaction described by the 12-6 LennardJones (LJ) equation with collision diameter σff = 0.3405 nm and the reduced well-depth εff/k = 119.8 K, (where k is the Boltzmann constant). 3. PORE MODELS The open pore models are shown in Figure 2; the pore walls are finite in the y-direction and infinite in the x-direction. To construct a closed end pore model, we added a flat surface at the end. For pores with flat walls, the solid−fluid potential energy was calculated from the Bojan-Steele equation.30−32 The curved walls were modeled as segments of a cylinder, and the solid−fluid interaction was calculated from the following equation33 when an adsorbate molecule is inside the pore and otherwise set to zero. 2 φsf = 4π 2ρσ ε (I − I3) s sf sf 6

(1)

The functions I3 and I6 are given by 10 ⎛ r ⎞2 ⎤ 63 ⎛⎜ σsf ⎞⎟ ⎡ ⎢1 − ⎜ ⎟ ⎥ I6 = ⎝R⎠ ⎦ 128 ⎝ R ⎠ ⎣

−10

⎡ 9 ⎛ r ⎞2 ⎤ 9 F ⎢ − ; − ; 1; ⎜ ⎟ ⎥ ⎝R⎠ ⎦ 2 ⎣ 2 (2a)

I3 =

4 ⎡ 3 ⎛r⎞ ⎛ r ⎞2 ⎤ 3 ⎛⎜ σsf ⎞⎟ ⎡ 3 ⎢1 − ⎜ ⎟ ⎥ F ⎢ − ; − ; 1; ⎜ ⎟ ⎥ ⎝R⎠ ⎦ ⎣ 2 ⎝R⎠ ⎦ 4⎝ R ⎠ ⎣ 2 2 ⎤−4

(2b)

The pore walls consisted of three homogeneous layers with a constant surface density ρs of 38.2 nm−2 and a spacing Δ = 0.3354 nm to model a graphene surface, and the molecular parameters for a carbon atom in a layer are σss = 0.34 nm and Received: Revised: Accepted: Published: 14304

August 4, 2013 September 9, 2013 September 13, 2013 September 13, 2013 dx.doi.org/10.1021/ie402549z | Ind. Eng. Chem. Res. 2013, 52, 14304−14314

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Figure 1. de Boer classification of hysteresis in 1972.2

Figure 2. The models of carbon pores studied in this work: (a) wedge pore, (b) pore with convex surfaces, and (c) pore with concave surfaces.

εss/k = 28 K. The cross collision diameter and well-depth of the solid−fluid interaction energy were calculated by the Lorentz− Berthelot mixing rule.

5. RESULTS AND DISCUSSION 5.1. Pores with Open Ends. 5.1.1. Effects of Angle. A schematic diagram of a wedge pore is shown in Figure 3 with

4. MONTE CARLO SIMULATION We used GCMC simulation to obtain adsorption−desorption isotherms, with 105 cycles in both the equilibration and the sampling stages. Each cycle consisted of 1000 displacement moves and exchanges which included insertion and deletion with equal probability. In the equilibration stage, the maximum displacement length was initially set as half of the largest dimension of the box and was adjusted at the end of each cycle to give an acceptance ratio for displacement of 20%.34 The length of the simulation box in the x-direction was 10 times the collision diameter of argon and was determined by the pore dimensions in the other two directions. The gas reservoir had a length of 3 nm along the pore axis, and the dimensions in the other two directions were the same as those of the pore. A periodic boundary condition was applied in the x-direction, and the cutoff radius was 5 times the collision diameter. For a given pressure, the chemical potential was calculated from the equation of state of Johnson et al.35 and was used as the input in GCMC simulation. The 2D-density distribution is useful to understand the microscopic mechanisms of adsorption and desorption. To calculate the 2D-density distribution, the system was divided into bins in the z- and y-directions, and the density in these two-dimensional bins is defined as ρ (z , y ) =

⟨ΔN (z , y)⟩ LxΔzΔy

Figure 3. The wedge pore with planar walls and open ends.

two gas reservoirs connected to each end of the pore so that molecules within the pore maintain equilibrium between the pore and the surroundings. The open end pore with uniform width (i.e., α = 0 degree) of 3 nm and pore length of 20 nm is used as a reference. Its isotherm is shown in Figure 4 with solid line and the hysteresis loop is of Type H1. For a wedge pore with α = 2.5° the hysteresis loop changes from Type A (H1) to Type C of Figure 1. When the wedge angle is increased to 5°, the fraction of adsorption capacity associated with condensation is reduced at the expense of the reversible part after condensation, due to the movement of the meniscus toward the pore mouths. Most of the adsorbate in this reversible part is located in the larger end. To determine the mechanisms of adsorption and desorption, we present, in the right-hand panel of Figure 4, the 2D-density profiles at various points along the adsorption and desorption branches for the wedges angles of 2.5° and 5°. When the wedge angle is 2.5°, the pore walls are layered with adsorbate up to point A (just before the condensation), and the size of the gaslike core at the narrower end is small enough to induce condensation to fill most parts of the pore with adsorbate at point B. After the condensation, further increase in pressure fills the core space at the wider end (point C). On desorption, the

(3)

where ΔN(z,y) is the number of particles in the bin bounded by [z, z + Δz] and [y, y + Δy]. The bin size in both directions was chosen to be Δz = Δy = 0.1σf f. The local density was collected at the end of each cycle in the sampling stage, and the ensemble average was obtained at the end of the simulation. The results of 2D-density profile were smoothed by averaging the density within a sphere of 0.5σff radius. 14305

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Figure 4. LHS: The isotherms for argon at 87 K in wedge pores with two open ends. The width of the narrow end is 3 nm, and the axial length is 20 nm. RHS: The 2D density profiles for α = 2.5° and 5°.

menisci at the two ends recede; more adsorbate is lost from the wider end because the solid−fluid interaction is lower there than at the narrower end. This withdrawal results in a gradual decrease in adsorbate density along the desorption branch (see points D and E) until point F is reached, where the liquid bridge is a thin biconcave lens. At this point the liquid bridge collapses, resulting in a steep evaporation. The capacity associated with this evaporation is so small that the hysteresis loop appears as Type C, in the dB classification. This type was omitted in the IUPAC scheme, possibly because no isotherms were found for adsorbents having a suitable wedge-pore topology. We will show below that Type C is not only specific to planar wedge pores but also to other pore-configurations. This suggests that there is a need to reconsider this loop type when hysteresis loops are classified in the future. [At the time of writing of this manuscript, we understand that Dr. M. Thommes, Professors Kaneko, Rouquerol, Sing, and coworkers were finalizing a new classification for IUPAC and that Type C will be included in the new classification as Type H2b, with the original Type H2 reassigned as Type H2a.] When the wedge angle is increased to 5°, the mechanisms of adsorption and desorption are the same as those just described giving a Type C hysteresis loop. The only difference is that the adsorptive capacity associated with the condensation is smaller, compared to the total capacity because the fraction of pore space filled with adsorbate immediately after the condensation is smaller. 5.1.2. Effects of Length. Pore length has a significant effect on the isotherm since the adsorbent potential well is deeper in narrower regions, due to overlap from opposite walls. To

examine this effect, the pore length was varied from 20 nm to 30 nm with the width of the narrow end fixed at 3 nm and the angle at 2.5°. When the pore length was 20 nm or less, most of the pore space is filled with adsorbate immediately after condensation. In longer pores, the solid−fluid potential becomes progressively weaker as pore width increases, and the curvature of the meniscus, after the condensation, decreases so that molecules at that interface will have fewer close neighbors, which weakens the fluid−fluid interactions. These effects are illustrated in Figure 5 with isotherms for two pore lengths of 20 nm and 30 nm. The pore length has no effect on the condensation and evaporation pressures because these depend only on the width at the narrow end. The difference between the two isotherms is the fraction of the adsorptive capacity associated with condensation, which is smaller for the longer pore. This is because after condensation, adsorbate fills a smaller fraction of the pore space in the longer pore, but the Type C hysteresis loop is retained for both pore lengths. This interpretation is supported by the 2D density profiles shown in Figure 6a for the 30 nm length pore. It is interesting, and significant, to note that the length of the section occupied by the condensed fluid immediately after condensation in 30 nm pore is the same as in the 20 nm pore (see Figure 6b). 5.1.3. Effects of Pore Size. Figure 7 shows the isotherms for pores with widths of 3 nm, 4 nm, and 4.8 nm at the narrow end, wedge angle 2.5°, and axial length 20 nm; other parameters are kept the same. The mechanisms for adsorption and desorption described above remain valid, but the hysteresis loop changes from Type C for the pore with a 3 nm narrow end to a mixture 14306

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Figure 5. Isotherms for argon at 87 K in wedge pores with two open ends; the width at the narrow end is 3 nm, and the angle is 2.5°.

Figure 7. Isotherms for argon at 87 K in wedge pores with wedge angle 2.5°, axial length 20 nm, and width at the narrow ends of 3 nm, 4 nm, and 4.8 nm.

of Type C (dB) and Type H1 (IUPAC), although rather more Type C than H1. As the narrower end is enlarged (typically greater than about 5 nm), the condensation occurs at the saturation pressure, resulting in the hysteresis loop of Type D of de Boer (see Figure 1). 5.1.4. Effects of Curvature. 5.1.4.1. Effects of Wall Curvature for Pores with Opening of the Same Width at Two Ends. This model serves as a reference. Figure 8 shows the isotherms for the widths of 3 nm at both ends and a length of 20 nm. For the pore with flat walls, this is extensively studied slit pore and for which the hysteresis loop is Type H1. For the strongly convex wall (open triangles) there is an early onset of adsorption to form a liquid bridge, where the walls are closely separated which enhances the adsorbent interaction due to potential overlap from the opposite walls. The thickness of the liquid bridge depends on the convexity of the walls; the smaller curvature giving a thicker liquid bridge. Adsorption then

proceeds by the advance of the two menisci toward the pore mouth. Desorption occurs via the withdrawal of the menisci toward the center, followed by the evaporation of the thin biconcave lens. The hysteresis is Type C for convex walls since this pore behaves like two wedge-like pores joined together. This is manifested in the 2D-density plots shown in the right panel of Figure 8. For pores with concave walls (open circles), the hysteresis loop is Type H1, similar to the pore with uniform width because here the condensation is induced by the formation of thin liquid bridges at the two ends at a pressure P*, followed immediately by complete filling of the pore because once the liquid bridges have formed the pore behaves as a closed pore which would be filled with adsorbate at this P*. This is substantiated in the top 2D-density plots in the right panel of Figure 8. On desorption menisci are formed at the

Figure 6. (a) The 2D density profiles of argon at 87 K in a wedge pore with two open ends; the pore width at the small end is 3 nm, the angle α equals 2.5°, and the pore length is 30 nm. (b) Comparison between the point C in (a) for a 30 nm pore and the point B in Figure 4b for a 20 nm pore. 14307

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Figure 8. LHS: The isotherms for argon at 87 K in the pores with two open ends, and curved pore walls, based on a uniform slit pore with width 3 nm and length 20 nm. RHS: The corresponding 2D density profiles.

Figure 9. LHS: The isotherms for argon at 87 K adsorption in the pores with two open ends and curved walls. The topologies of the pores are shown schematically, and the isotherms are shifted by n*40 Kmol/m3 where n is an integer number. RHS: The 2D density profiles for pores with concave walls.

pore mouths and recede into the pore interior, and at the same time the condensed fluid is stretched in the middle of the pore. When the menisci have reached deep enough into the pore, the condensed fluid evaporates by the mechanism described in our earlier publication as cavitation-like pore blocking.36 5.1.4.2. Effects of Wall Curvature in Pores with Different Sized Open Ends. Figure 9 shows a series of isotherms for pores with narrow and wide ends of 3 nm and 4.75 nm, respectively, and axial length 20 nm. The isotherms for pores having convex surfaces exhibit a Type C hysteresis loop in which the loop shifts to higher pressures and has a larger area as the curvature decreases. Adsorption in these pores is approximately equivalent to that of two wedge-like pores, with different angles, joined together.

When the pore walls are concave the hysteresis loop changes from Type C to Type H1 as the wall becomes more curved, passing through a loop shape which is a mixture of Type C and Type H1. This mixed shape has been observed experimentally,1,10,13 but as far as we are aware, it has not been reported previously in any simulation studies. The 2D-density plots for two pores of different curvature with concave walls are shown in the right panel of Figure 9. These plots show that, after layers of adsorbate have been formed on the surfaces along the adsorption branch, condensation occurs at the narrow end to form a thin liquid bridge, followed by filling of the remaining part by the advancing meniscus, which is similar to the process observed in a closed end pore.37,38 On the desorption branch, an interface is formed at the wider end and recedes as pressure 14308

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Figure 11b. A hysteresis loop was found for the longer pore, and it is notable that the loop area has increased for the reference pore again confirming pore length as an important parameter in determining the existence of a hysteresis loop. 5.2.2. Curved Pores with the Same Width at the Open and Closed Ends. The reference is a closed end pore with uniform width 3 nm and length 20 nm; its isotherm is shown in Figure 12 as a dashed line. When the walls are convex, there is competition between two processes: (1) adsorption starting from the closed end forms a meniscus which then advances as the pressure is increased; (2) a liquid bridge is formed at the center of the pore where the solid−fluid potential is strongest. The second process dominates in the chosen model, but it should be noted that when the width of the narrow space approaches molecular dimensions adsorbate would tend to be excluded from this region.39 The relative contribution of these two processes depends on the pressure marking the onset of the meniscus advance from the closed end and the pressure at which the liquid bridge is formed at the center of the pore. When the wall curvature is reduced (see the 2D density profile in Figure 12b), the liquid bridge is formed between A and B on the isotherm; further adsorption is contributed from two processes occurring in conjunction: (1) filling the cavity formed between the liquid bridge and the closed end by layering as in a closed pore (points B to C), followed by condensation in the cavity (points C to D); (2) filling the region between the liquid bridge and the open end (as in a closed end pore), via the advance of meniscus to the open end (points D to the saturation pressure). Desorption proceeds initially as in a closed end pore; i.e. a meniscus is formed at the pore mouth, which then recedes into the pore interior and at the same time, the fluid in the cavity section is stretched (points E and F). Because the fluid can be stretched more easily in the cavity section than in the center of the pore, where the holding potential is strongest, the fluid in the cavity evaporates (point F to G). If the pore walls have a stronger curvature (see the 2D density profile in Figure 12c), the onset of formation of the liquid bridge shifts to a much lower pressure and becomes reversible; the hysteresis loop is then caused only by the cavity formed between the liquid bridge and the closed end. In summary: closed end pores with convex walls behave like a composite of a closed end pore and an ink-bottle pore, and therefore the hysteresis loop is a combination of Type H1 for a closed end pore and Type H2 for an ink-bottle pore. The contribution of these two pore types to the hysteresis depends

is reduced (for example to point E in Figure 9b). As the pressure is further decreased and the interface approaches the narrow end (point F in Figure 9b), the isotherm shows a kink, indicating that another process has come into play. Analyzing the 2D-density plots at various points, it is seen that the first stage of desorption is evaporation of the condensed fluid from the section close to the wider open end; this behavior is similar to that of a closed end pore, and, consequently, a Type H1 hysteresis loop results. However, at the kink (point F in Figure 9b), desorption has reached the narrow end where the solid− fluid potential is stronger, and therefore the condensed fluid is more strongly held and a lower pressure is required to remove this adsorbate. A comparison of the 2D-density plots in Figure 9b, for points B and F, where the density is the same on the adsorption and desorption branches, respectively, confirms that the adsorbate is more compact on the desorption branch. When pressure is reduced to point G, the remaining fluid has the shape of biconcave lens and is now thin enough for a sharp evaporation to occur. 5.2. Closed End. 5.2.1. Closed Small End. A diagram of this pore model is shown in Figure 10; the idealized (uniform)

Figure 10. Schematic diagram of a wedge pore with the small end closed.

closed end pore with angle α = 0° is the reference and has already been studied extensively.38 The isotherm of the pore having 3 nm width at the small closed end, 2.5° angle, and 20 nm axial length is presented in Figure 11a together with the reference isotherm. There is a hysteresis loop that has been discussed in detail in our previous publications37,38 (where its existence was confirmed by extremely long simulations). It can be seen that isotherm for the closed wedge pore is not only more gradual but also reversible. As the pore length is an important parameter affecting the existence of a hysteresis loop,37,38 we repeated the simulations with the axial length increased from 20 nm to 40 nm; the isotherms are shown in

Figure 11. The isotherms for argon adsorption at 87 K in wedge pores with the small end closed; the width of the small end SH = 3 nm, angle α = 2.5°, and the axial lengths are (a) L = 20 nm and (b) L = 40 nm. The isotherms obtained with angle α = 0° are shown for comparison. 14309

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Figure 12. LHS: Isotherms for argon at 87 K in pores with one closed end and curved walls, based on a uniform slit-shaped pore with pore width 3 nm and axial length 20 nm. The topologies of the pores are shown schematically. The reference isotherm for the uniform pore is shown as a dashed line. RHS: The 2D density profiles for the pores with convex walls.

of the pore (B), where the pore size is the largest. Further increase in pressure from this point will result in a steep condensation to fill the remaining space of the pore (C). The reason for this condensation (BC) is that once the interface has reached the center of the pore (B), the pore behaves as a tapered closed end pore and if the largest section is filled with adsorbates at B the remaining section must be filled because it is smaller. On desorption, a meniscus is formed at the pore mouth, as is the case for closed end pores, and while it starts to recede into the pore interior the condensed fluid in the pore is stretched such that it is going through a cavitation, a phenomenon which we call it cavitation-like pore blocking.36 5.2.3. Pores Closed at the Wider End. What we would like to show in this section is that the behaviors of a classical inkbottle pore are not unique, but rather other pore configurations can give similar behaviors. 5.2.3.1. Ink-Bottle Pore. A schematic diagram of an inkbottle pore is shown in Figure 14a. We chose two pore models: the first one had a neck width of 1.6 nm, which is smaller than the critical neck size for cavitation (≈3 nm for a slit shaped pore36), and the second with a neck width of 3.6 nm. For both pores, the cavity width was 7 nm, and the total length of the cavity and neck was 17 nm. To investigate the effects of the adsorptive capacities of the neck and the cavity, we chose cavity lengths of 4 nm, 7 nm, 10 nm, and 16 nm. The reference model is a wedge shaped pore, with its wide end closed, and wide and narrow ends with the same width as the corresponding inkbottle pore (Figure 14b). The isotherms for these two types of closed end pore are shown in Figure 15. The lengths of the cavity and the neck for the ink-bottle pore are labeled in the figures.

on the relative importance of the two processes discussed above. For pores with concave walls, the H1 hysteresis loop becomes larger and shifts to higher pressure and becomes more like Type H2, compared to the closed end pore of uniform width. The mechanisms of adsorption and desorption in these pores can be seen in the 2D-density plots in Figure 13. For adsorption, the process is started with the formation of adsorbed layers on the walls and the formation of a meniscus at the closed end (point A). This meniscus advances to the middle

Figure 13. The 2D density profiles of argon at 87 K in a closed end pore with concave walls; the two ends have the same width of 3 nm, and the axial length is 20 nm. 14310

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Figure 14. Schematic diagram of the ink-bottle pore and the wedge shaped pore closed at the wide end.

adsorbed layers to leave a narrow enough core for condensation to occur. (4) The relative lengths of the cavity and the neck are reflected in the relative adsorptive capacities, i.e. the loop height is smaller when the cavity is shorter. When the neck size exceeds the critical size, some notable features emerge; especially on the adsorption branch (right panel of Figure 15). By increasing the cavity length from 4 nm to 16 nm, the hysteresis loop evolves in an interesting manner: the condensation occurs in a single stage for a short cavity but changes to two stages and then finally to a single stage for a longer cavity. To interpret these different features related to the length of the cavity, we present 2D-density plots in Figure 16. Here it can be seen that, when the cavity is short, condensation occurs simultaneously in the neck and in the cavity. However, when the cavity size is increased, adsorbate condenses in the neck, followed by condensation in the cavity, giving a single step on the adsorption branch. On further increase in the length of the cavity the neck becomes too short for condensation to occur; after the formation of adsorbed layers, the remaining part of the pore behaves as a closed end pore and is filled by the advancing meniscus. On desorption, the cavitation-like pore blocking phenomenon, first noted in ref 36 was observed for all cavity lengths, but when the neck length is small the evaporation occurs at a higher pressure. 5.2.3.2. Pores with Curved Walls, Closed at the Wider End. Curvature of the pore walls can affect the hysteresis loop of pores with the wider end closed, in the same way as it does for open end pores discussed in Section 5.1.4. We present, in Figure 17, a series of isotherms for closed end pores, with wall curvature varying from convex to concave, and a narrow open end of width 1.6 nm. A number of features may be noted: (1) Condensation occurs at a higher pressure when the pore wall changes from convex to concave, as a consequence of the reduction in the adsorbent field strength.

Figure 15. The adsorption isotherms for argon at 87 K in ink-bottle pores. The wide end (BH) = 7 nm, the width of the narrow ends are (a) SH = 1.6 nm and (b) SH = 3.6 nm; the total pore length L = 17 nm. The lengths of cavity and neck were varied as indicated in the figures.

A number of features may be noted for the pore with the narrower neck size (1.6 nm): (1) The hysteresis loops are of type H2 for the ink bottle pore and for the closed wedge pore. (2) Evaporation is via cavitation, which occurs at the same pressure, for all the cavity lengths except 4 nm, when the cavitation shifts to a lower pressure because of the adsorbent field stabilization effect in the small cavity. (3) Condensation shifts to a higher pressure for the longer cavities because a higher pressure is required to build enough

Figure 16. The 2D density profiles of (a) the 1st isotherm, (b) the 2nd isotherm, and (c) the 4th isotherm from the top in Figure 15b. 14311

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(2) Evaporation is via cavitation, which is similar to the classical ink-bottle pores whose necks are smaller than the critical size (Figure 15a). When the open end has a width of 3.6 nm (greater than the critical size), it is interesting to enquire whether the wall curvature will affect the features that were observed earlier for the ink-bottle pores. The left panel of Figure 18 shows the isotherms for closed end pores with varying curvature. It can be seen that all the features observed for the corresponding inkbottle pores are reproduced viz.: (1) the hysteresis loops are either type H1 or type H2, some of which display a two-stage condensation; (2) evaporation is by the cavitation-like pore blocking mechanism. To provide a better insight into the adsorption processes, we chose three pores and present their respective 2D-density plots in the right panel of Figure 18. The bottom plots of the right panel of Figure 18 are for the pore with convex walls, and the closest distance between the two walls is smaller than the open end; the 2D density profiles are shown in Figure 18d. On the adsorption branch, molecular layering occurs up to point A, and then the condensation is induced in the region where the pore walls are close to each other (point B). As pressure is increased beyond this point, the pore behaves like a closed pore with a cavity formed between

Figure 17. The adsorption isotherms for argon at 87 K in the pores with curved pore based on a wedge pore, in which the wider end, BH = 7 nm and the narrower end, SH = 1.6 nm; the axial length L = 17 nm. The schematic topologies are illustrated on the graphs.

Figure 18. LHS: The adsorption isotherms for argon at 87 K in the pores with curved pore walls, based on the wedge pore. The wider end, BH = 7 nm, the narrower end, SH = 3.6 nm, and the total pore length L = 17 nm. The topologies of the pores are shown schematically on the graphs. RHS: The 2D density profiles for pores with curved walls. 14312

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the liquid bridge and the closed end. The process of adsorption is then to build further layers in the cavity up to the point where condensation occurs (point D). On desorption, evaporation occurs via a classical cavitation mechanism if the two walls are close enough to form a neck narrower than the critical size but occurs by cavitation-like pore blocking when the two convex walls are further apart (see Figure 18c). The 2D-density plots for a closed end pore with concave walls are shown in Figure 18b. Instead of a two-stage condensation as for the strongly convex walls, the isotherm shows Type H1 hysteresis, with advanced condensation and advanced evaporation.36 The mechanisms of adsorption and desorption are as follows: in adsorption there is a layering process, and the pore is progressively filled by the advancing meniscus until point B is reached; when pressure is further increased, a liquid bridge is formed at the narrow opening which then facilitates condensation in the cavity. This has been referred to as advanced condensation for the ink-bottle pore with the neck size larger than the critical size. The desorption begins with evaporation from the opening of the pore; at point D a meniscus is formed and the fluid inside the pore is stretched, and as the interaction between the solid and the fluid and inside the pore is weaker than at the opening, the condensed fluid evaporates via a cavitation-like pore blocking mechanism, which is viewed as advanced evaporation in the ink-bottle pore model.

Figure 20. Evolution of hysteresis for different pores developed based on a uniform closed end pore.

hysteresis loop is Type C, while for concave walls it can be either type C or type H1, depending on the relative size of the two ends. The variety of hysteresis loops for pores with one end closed is summarized in Figure 20. For pores having the same widths at both ends, the loop area is larger when the pore walls have low curvature and is of Type H1 for concave walls but either Type H1 or H2 for convex walls. For nonuniform closed end pores closed at the narrow end, the hysteresis loop is Type H1; however, when the wider end is closed, the hysteresis is similar to that found in ink-bottle pores, i.e. the evaporation occurs either through cavitation (Type H2) or pore blocking (Type H1), depending on the size of the opening. In summary, all the types of hysteresis loop proposed by de Boer, as well as new types which were not included in any previous classification, can be successfully described by molecular simulation using the simple pore models proposed in this paper. Although these models cannot fully describe the physical complexity of a real adsorbent, they do show features that would be expected to be present in adsorbent materials and show that simple modifications to the basic slit pore model can lead to valuable insights into the nature of mesoporous adsorption.

6. SUMMARY OF THE HYSTERESIS LOOP In Figures 19 and 20, we summarize the evolution of hysteresis obtained in an independent pore having two ends open and one end closed.

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Figure 19. Evolution of hysteresis for different pores developed based on a uniform open end pore.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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7. CONCLUSIONS The effects of curvature on the shape of adsorption isotherms and the form of the hysteresis loop have been investigated in simple slit pores either open at both ends or with one end closed. In open pores of uniform width (Figure 19), the hysteresis loop is type H1 in the IUPAC classification (type A in the de Boer classification) but becomes Type C or D when the width is varied linearly along the pore. For convex pore walls, the

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