Dependent Domain Model of Cylindrical Pores - The Journal of

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Dependent Domain Model of Cylindrical Pores Kunimitsu Morishige* Department of Chemistry, Okayama University of Science, 1-1 Ridai-cho, Kita-ku, Okayama 700-0005, Japan ABSTRACT: To examine the effects of pore corrugation on capillary condensation and evaporation of a gas in cylindrical pores, we measured the adsorption hystereses and subloops of O2 at 70 K on MCM-41 and of H2O at 295 K on SBA-15, and then calculated the adsorption hystereses, desorption scanning curves, and subloops of O2, N2, and H2O in the cylindrical pores with corrugation based on a dependent domain model. Here, the corrugated pores of the ordered mesoporous materials such as MCM-41 and SBA-15 are modeled as an assembly of pores that consists of alternating cylindrical domains with different diameters and the same length. In the cylindrical pores with corrugation, each domain is no longer independent of adsorption and desorption. Congruence/noncongruence between the two subloops taken in the same pressure range on adsorption and desorption branches depended not only on adsorbents but also on adsorbates. The dependent domain model can well account for the noncongruent subloops observed for O2, N2, and H2O in the ordered mesoporous materials with cylindrical pores. In addition, it is strongly suggested that there is a threshold amplitude of the pore corrugation below which capillary condensation and evaporation are no longer influenced by the corrugation and occur irreversibly in the whole space of a given pore at particular condensation and evaporation pressures, respectively.

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INTRODUCTION Capillary condensation of a gas in mesopores takes place at a pressure less than the saturation pressure of a bulk liquid that depends on the pore geometry and size, below the bulk critical temperature.1,2 It is very often accompanied by hysteresis; that is, capillary evaporation usually takes place at a pressure lower than the condensation with decreasing pressure. Characterization of mesoporous materials can be done through the proper analysis of the adsorption hysteresis.3 Ordered mesoporous materials such as MCM-414 and SBA-155 possess almost cylindrical pores arranged regularly in a silica matrix. At first many researchers thought that the materials can be used as model adsorbents with cylindrical pores in comparison with theories and simulations dealing with capillary condensation phenomena.6−17 However, it is now widely recognized that the mesopores of the ordered mesoporous materials have a variety of pore imperfections, depending on the kind of samples and preparations.18−24 Among the pore imperfections, corrugation is thought to affect the adsorption hysteresis most strongly.25−35 Here, the term “corrugation” represents a variation of the pore size along the pore axis in cylindrical pores. To examine the corrugation in cylindrical pores, besides the adsorption hysteresis, scanning curves and subloops are often measured.25−27,30,31,34,35 Adsorption and desorption scanning curves are obtained by reversing upon desorption and adsorption, respectively, the direction of change in the pressure. Subloops consist of performing adsorption/desorption cycles in the same pressure range on the adsorption and desorption branches. Although analysis of the experimental data is usually carried out according to the independent domain theory27,36,37 developed by Everett and subsequently other researchers, the © 2017 American Chemical Society

interpretation of the scanning curves and subloops obtained is often very difficult and sometimes confused.26,27,34,35,37−39 This is partly because the independent domain theory does not provide a framework for modeling scanning curves and subloops for cylindrical pores with corrugation. In the theory, a “pore domain” is assumed as a region of porous space in which capillary condensation and evaporation take place at well-defined pressure values, irrespective of the state of their neighboring pore entities. On the other hand, for cylindrical pores with corrugation, capillary condensation and evaporation in the individual wide and narrow sections of the single pore depend on the state of the neighbors.25−30 The independent domain theory indicates that two subloops taken in the same range of relative pressure on adsorption and desorption branches will be noncongruent even after correction due to the presence of the adsorbed film on the pore walls when the pore system does not behave as an assembly of independent porous domains.27,36,37,39 The theory, however, does not at all indicate the details of the noncongruency. In the corrugated pores of cylindrical shape, advanced adsorption26,27 and single pore blocking25−27 may take place on adsorption and desorption, respectively. It is almost certain that these effects affect significantly the adsorption hysteresis, scanning curves, and subloops. In previous studies,34,35 we have measured the adsorption hysteresis, desorption scanning curves, and subloops of N2 at 77 K on SBA-15, as well as the adsorption hysteresis and subloops of O2 at 70 K on COK-11 and MCM-41. COK11 possesses cylindrical pores similar to MCM-41.40 For O2 at 70 K on COK-11, the two subloops on the adsorption and Received: December 14, 2016 Published: February 22, 2017 5099

DOI: 10.1021/acs.jpcc.6b12566 J. Phys. Chem. C 2017, 121, 5099−5107

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The Journal of Physical Chemistry C

the present treatment. See Figure 1 for a schematic illustration. A pore contains 100 domains and the assembly consists of

desorption branches were noncongruent; that is, the subloop on the adsorption branch was smaller in the change of the adsorbed amount compared to that on the desorption branch. Similarly, for N2 at 77 K on most kinds of SBA-15 samples, the two subloops on the adsorption and desorption branches were noncongruent. However, the subloops on the adsorption branches were larger in the change of the adsorbed amount than those on the desorption branches of the same samples, as opposed to O2 on COK-11. For other SBA-15 samples, the two subloops on the adsorption and desorption branches were almost congruent, suggesting that the amplitude of the corrugation in the cylindrical pores is negligibly small. The independent domain theory cannot at all explain the different behavior of noncongruence between the two subloops on adsorption and desorption branches for COK-11 and SBA-15. In the present study, we will further measure the adsorption hysteresis and subloops of H2O at 295 K on SBA-15 in order to examine the effect of adsorbate. In addition, we will measure the adsorption hysteresis and subloops of O2 at 70 K on MCM41 with pore size slightly larger than used previously, to confirm the previous results of O2 on COK-11. Then, a dependent domain model that provides a framework for modeling scanning curves and subloops for cylindrical pores with corrugation will be developed. The present study aims at examining the effects of the pore corrugation on the adsorption hystereses, desorption scanning curves, and subloops of O2, N2, and H2O in cylindrical pores based on the dependent domain model.

Figure 1. Schematic illustration of cylindrical pores consiting of alternating cylindrical sections with different diameters and the same length. The pore size distribution is composed of two components which characterize the diameter variations within the pores (σdomain) and between the pores (σpore).

10 000 pores. The total number of domains is then 106. Capillary condensation and evaporation pressures of each domain in an isolated state are related to Ddomain using the modified Kelvin equation.1,3 Capillary condensation and evaporation take place at the relative pressure

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EXPERIMENTAL SECTION Materials and Measurements. MCM-41(C22) was synthesized in the modified procedure of Beck et al.4 Dococyltrimethylammonium chloride (C22TMACl) was dissolved in deionized water at 373 K with stirring. Sodium silicate and pyrogenic silica were dissolved in deionized water at 373 K. Both solutions were mixed with stirring for 1 h at 373 K, and then the mixture was heated at 373 K for 24 h. The washed and then dried product was calcined at 823 K (heating rate 1 K/ min) for 6 h in air. Synthesis and characterization of SBA-15 have been described in a previous study.34 The SBA-15 samples were referred to as SPx-y-z, where x, y, and z represent the SiO2/P123 (surfactant) molar ratio, hydrothermal treatment temperature (in kelvin), and time (in days), respectively. The adsorption hystereses and subloops of O2 at 70 K were measured volumetrically on a homemade semiautomated instrument equipped with a closed cycle refrigerator.35 The adsorption hystereses and subloops of water at 295 K were measured gravimetrically on a Rubotherm (BEL Japan).41 Dependent Domain Model. The corrugated pores of the ordered mesoprous materials such as MCM-41 and SBA-15 are modeled as an assembly of pores that consists of alternating cylindrical sections (domains) with different diameters and the same length.27,30 The mean diameters (Dpore) of individual pores for MCM-41 and SBA-15 are supposed to follow a normal distribution with expected values of 4 and 8 nm, respectively, and standard deviation σpore. We assumed the same value of σpore (0.08 nm) for both types of pores. The diameter (Ddomain) of each domain within a given pore is also supposed to follow a normal distribution with its mean diameter and standard deviation σdomain. Therefore, the pore size distribution is composed of two components within the pores (σdomain) and between the pores (σpore). It is expected that the length of domain is a few nanometers,33 but it needs not be specified in

p /p0 = exp( −αK /(Ddomain /2 − t ))

(1)

K = γVL /RT

(2)

where γ and VL are the surface tension and molar volume of the liquid, respectively, R is the gas constant, T is the temperature, t is the thickness of the adsorbed layer just before condensation and just after evaporation, and α is a parameter specifying the degree of metastability that controls the width of adsorption hysteresis. α takes a value between 1 and 2 for condensation, while α = 2 for evaporation. Evaporation takes place at equilibrium. The values of α for condensation were determined from broadly adjusting to the width of the adsorption hysteresis observed experimentally. The values of K for O2 at 70 K,42,43 N2 at 77 K,10,12 and H2O at 295 K44 were calculated from the values of γ and VL. We assumed the same t value for condensation and evaporation of a fluid for simplicity, although the t value for condensation is expected to be slightly larger than that for evaporation and also the t values depend on the relative pressure at which capillary condensation and evaporation take place. The values of t for O2 in MCM-41 and for N245 and H2O44 in SBA-15 were fixed according to the literature. Table 1 summarizes the values of the parameters of the modified Kelvin equation. In the cylindrical pore that consists of alternating domains with different diameters and the same length, each domain is no longer independent.27 Capillary condensation and evaporation in a given domain are influenced by the state of the neighboring domain. Upon adsorption, a porous domain will be filled at (p/ Table 1. Parameters of the Modified Kelvin Equation α(condensation) K/nm t/nm 5100

O2(70 K)

N2(77 K)

H2O(295 K)

1.7 0.819 0.7

1.4 0.481 1.0

1.2 0.530 0.3


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The Journal of Physical Chemistry C p0)condensation when its two neighbors are still empty. If one of the neighbors is filled, then the domain will be filled at (p/ p0)evaporation (advanced adsorption). Upon desorption, a domain will be emptied at (p/p0)evaporation when at least one of the neighbors is empty. If its two neighbors are still filled, the domain will not be emptied until one of the neighbors evaporates (single pore blocking). Under these conditions, we calculated adsorption hystereses, desorption scanning curves, and subloops as a function of σdomain for O2 at 70 K in the corrugated pores of ⟨Dpore⟩ = 4 nm, and N2 at 77 K and H2O at 295 K in the corrugated pores of ⟨Dpore⟩ = 8 nm, respectively.

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EXPERIMENTAL RESULTS O2 on MCM-41. Figure 2 shows the adsorption hystereses and subloops for O2 at 70 K on MCM-41 and COK-11. The

Figure 3. Hysteresis loops, desorption scanning curves, and subloops for nitrogen at 77 K on SP45-373-1 (a), SP60-373-1 (b), SP75-373-1 (c), SP45-373-14 (d), SP60-373-14 (e), and SP75-373-14 (f).34 Dashed lines denote the subloops on the adsorption branches that were corrected for the adsorbed film on the pore walls39 for comparison. The isotherms of (b), (c), (e), and (f) are displaced rightward by 0.20, 0.45, 0.25, and 0.45, respectively.

Figure 2. Hysteresis loops and subloops for oxygen at 70 K on COK1135 (a), MCM-41(C18)pH35 (b), and MCM-41(C22) (c). Dashed lines denote the subloops on the adsorption branches that were corrected for the adsorbed film on the pore walls39 for comparison. The isotherms of (b) and (c) are displaced rightward by 0.15 and 0.20, respectively.

increasing SiO2/P123 ratio, the hysteresis loops were wider and the extent of congruence between the two subloops became worse, regardless of the hydrothermal treatment time. With prolonged hydrothermal treatment at 373 K, the extent of congruence between the two subloops tended to become better, except for the sample with high SiO2/P123 ratio. The subloops for SP45-373-14 and SP60-373-14 were congruent, and those for SP45-373-1 were nearly congruent. Although the returning, conversing, and crossing types of desorption scanning curves that were classified by Tompsett et al.38 were all observed, there is not necessarily a clear relationship between the extent of congruence between the two subloops and the type of scanning curves. The crossing type of desorption scanning curve was observed for SP45-373-1, and the conversing type of desorption scanning curves was observed for three kinds of SBA-15 with the prolonged hydrothermal treatment, namely SP45-373-14, SP60-373-14, and SP75-37314. The returning type of desorption scanning curves was observed for two kinds of SBA-15 with the relatively short hydrothermal treatment, that is, SP60-373-1 and SP75-373-1. Figure 4 shows the adsorption hystereses and subloops for H2O at 295 K on four kinds of SBA-15 samples. The hysteresis loops of H2O are wider than those of N2, suggesting a high nucleation barrier for water condensation in the pores. For the sample (SP75-373-14) that was synthesized with initial SiO2/ P123 ratio of 75 and the prolonged hydrothermal treatment of

results for COK-11 and MCM-41(C18)pH have been reproduced from our previous study.35 COK-11 has a porous structure similar to that of MCM-41,40 although the properties of the pore walls of COK-11 may differ from those of MCM-41. MCM-41(C18)pH was synthesized using stearyltrimethylammonium chloride (C18TMACl) as a template with pH adjustment.46 As the pore size of MCM-41(C22) is larger than those of COK-11 and MCM-41(C18)pH, the hysteresis loop of MCM-41(C22) is wider than those of the latter two samples. The two subloops on the adsorption and desorption branches for MCM-41(C18)pH are almost congruent, whereas those for COK-11 and MCM-41(C22) are noncongruent. This strongly suggests that for MCM-41(C18)pH the amplitude of the corrugation in the cylindrical pores is negligibly small, while for COK-11 and MCM-41(C22) the pores are corrugated. For COK-11 and MCM-41(C22), the subloops on the adsorption branches are smaller in the change of the adsorbed amount than those on the desorption branches. N2 and H2O on SBA-15. Figure 3 shows the adsorption hystereses, desorption scanning curves, and subloops for N2 at 77 K on six kinds of SBA-15 samples that were synthesized with initial SiO2/P123 ratios of 45, 60, and 75, respectively, and hydrothermally treated at 373 K for 1 or 14 days.34 P123 surfactant was used as a template in the synthesis. All these data have been also reproduced from our previous study.34 With 5101


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Figure 5. Hysteresis loops, desorption scanning curves, and subloops calculated for oxygen at 70 K in two types of independent cylindrical pores with mean pore diameter of 4 nm and σpore of 0.08 nm: (a) perfect and (b) imperfect cylindrical pores. The isotherms of (b) are displaced rightward by 0.15.

Figure 4. Hysteresis loops and subloops for water at 295 K on SP60373-1 (a), SP60-373-14 (b), SP75-373-1 (c), and SP75-373-14 (d). Dashed lines denote the subloops on the adsorption branches that were corrected for the adsorbed film on the pore walls39 for comparison. The isotherms of (b), (c), and (d) are displaced rightward by 0.30, 0.60, and 0.90, respectively.

14 days, the two subloops on adsorption and desorption branches are almost congruent. This is in sharp contrast with the noncongruence between the two subloops of N2 for the same sample. Similarly, for the sample (SP60-373-1) with initial SiO2/P123 ratio of 60 and the hydrothermal treatment of 1 day, the two subloops of H2O are almost congruent, although the two subloops of N2 are noncongruent. On the other hand, both the subloops of N2 and H2O on SP60-373-14 are congruent. This clearly indicates that congruence/noncongruence between two subloops on adsorption and desorption branches depends on the adsorbate as well. For the sample (SP75-373-1) with initial ratio of 75 and the hydrothermal treatment of 1 day, the two subloops of H2O are noncongruent. In any event, when for N2 and H2O on SBA-15 the two subloops on adsorption and desorption branches are noncongruent, the subloops on the adsorption branches are always larger in the change of the adsorbed amount than those on the desorption branches, as opposed to the subloops for O2 on MCM-41 and COK-11.

pore diameters are normally distributed about 4 nm with σpore of 0.08 nm. Capillary condensation and evaporation pressures were calculated according to the modified Kelvin equation using the parameters in Table 1. In the case of imperfect cylindrical pores, the desorption pressures are supposed to be normally distributed about the original values with a standard deviation of 0.01. The shapes of the hysteresis loops, desorption scanning curves, and subloops are determined by the distributions of the condensation pressures versus the evaporation pressures for all pores in the system. Therefore, the shapes of subloops observed experimentally could be always reproduced on the basis of the independent domain model in principle, if the two subloops on adsorption and desorption branches are congruent. The congruent subloops of O2 on MCM-41(C18)pH strongly suggest that capillary condensation and evaporation of O2 on this substrate take place independently. Dependent Domain Model. O2 on MCM-41. Figure 6 shows the theoretical adsorption hystereses, desorption scanning curves, and subloops as a function of σdomain for O2 at 70 K in the corrugated pores with a mean pore diameter of 4

THEORETICAL RESULTS Independent Domain Model. For an assembly of perfect cylindrical pores with a distribution of diameters and the same length, the desorption scanning curves of a fluid condensed in the pores will be a straight line across to a hysteresis loop.27,37,38 The two subloops taken in the same relative pressure range on adsorption and desorption branches are reversible and congruent because the plots of the evaporation pressures versus the condensation pressures in individual pores are almost linear.27,37 Even if the restriction of perfect cylinders and the same length with respect to the pore geometry is removed, capillary condensation and evaporation in individual pores can still take place independently. Wootters and Hallock37 have for the first time shown that in this case subloops are not reversible anymore, and yet two subloops on adsorption and desorption branches taken in the same pressure range will always be congruent because the same pores participate in both the subloops on adsorption and desorption branches. Figure 5 shows the theoretical adsorption hystereses, desorption scanning curves, and subloops of O2 at 70 K in two types of independent cylindrical pores mentioned above. The

Figure 6. Hysteresis loops, desorption scanning curves, and subloops calculated for oxygen at 70 K in dependent cylindrical pores with mean pore diameter of 4 nm and different σdomain values. The values of σdomain are 0.04 (a), 0.08 (b), 0.11 (c), and 0.14 nm (d), respectively. The isotherms of (b), (c), and (d) are displaced rightward by 0.15, 0.30, and 0.45, respectively.

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The Journal of Physical Chemistry C nm and σpore of 0.08 nm. The shapes of desorption scanning curves and subloops changed remarkably with increasing σdomain. At small σdomain values of 0.04−0.08 nm, the subloops on adsorption branches are smaller in the change of the adsorbed amount than those on desorption branches of the same σdomain. With increasing σdomain, the amplitude of the subloop on the adsorption branch increases, whereas that on the desorption branch decreases. Therefore, the noncongruent subloops observed for O2 at 70 K on MCM-41(C22) and COK-11 could be accounted for by the small amplitude of the pore corrugation in these adsorbents. Figure 7 shows the visualizations of filled domains in one of the dependent cylindrical pores with small σdomain values during

Figure 8. Visualizations of filled domains (closed red squares) for oxygen at 70 K in one of the dependent cylindrical pores with mean pore diameter of 4 nm and σdomain of 0.14 nm: (a) subloop on adsorption branch and (b) subloop on desorption branch. The subloops were taken in the p/p0 range 0.254−0.307. The values of p/ p0 are given to the right of the images.

pore because capillary condensation proceeds gradually in the neighborhood of many nucleation sites (narrow domains) in the pore (Figure 8a). Capillary evaporation pressures of most of the domains in the pore are still larger than the condensation pressures of the narrow domains in the pore under the conditions of large σdomain. Desorption and adsorption occur at both ends of the discrete rods of liquid. As desorption from the completely filled rods of liquid occurs only at entrances of the pores, on the other hand, it results in formation of single rods of liquid unfilled in the neighborhood of the entrances rather than the discrete rods of liquid (Figure 8b). Then, adsorption occurs at both ends of the single rods of liquid. The number of the rods of liquid participating in the subloop taken on the adsorption branch is larger than that of the rods of liquid participating in the subloop taken on the desorption branch. This may result in the smaller amplitude of the subloop on the desorption branch, as opposed to the noncongruent subloops observed for O2 on COK-11 and MCM-41(C22). At σdomain = 0.11 nm, the two subloops on adsorption and desorption branches are nearly congruent. This resembles the subloops observed on MCM-41(C18)pH. However, both the adsorption and desorption branches for the corrugated pores are shifted into lower p/p0 compared to those for the pores with no corrugation because of the appearance of advanced adsorption and single pore blocking, being inconsistent with the temperature dependence of the adsorption hysteresis observed for O2 on MCM-41(C18)pH.35 The desorption branch of this sample corresponds to equilibrium. It is apparent that the congruent subloops obtained on the basis of the dependent domain model with a certain critical value of σdomain are not able to account for the congruent subloops for MCM41(C18)pH. In addition, the temperature dependence of the adsorption hysteresis has indicated that the desorption branch of COK-11 appears at a pressure lower than equilibrium,35 in accord with the occurrence of single pore blocking in the substrate. N2 on SBA-15. Figure 9 shows the theoretical adsorption hystereses, desorption scanning curves, and subloops as a function of σdomain for N2 at 77 K in the corrugated pores with a mean pore diameter of 8 nm and σpore of 0.08 nm. Similarly, the shapes of desorption scanning curves and subloops changed remarkably with increasing σdomain. In addition, the width of the hysteresis loop increased with increasing σdomain. At relatively small σdomain values, the subloops on adsorption branches are smaller in the change of the adsorbed amount than those on desorption branches of the same σdomain. With increasing σdomain,

Figure 7. Visualizations of filled domains (closed red squares) for oxygen at 70 K in one of the dependent cylindrical pores with mean pore diameter of 4 nm and σdomain of 0.04 nm: (a) adsorption branch and (b) subloop on adsorption branch. The subloop was taken in the p/p0 range 0.278−0.326. The values of p/p0 are given to the right of the images.

adsorption and desorption processes of oxygen at 70 K. For the corrugated pores with small σdomain values, capillary condensation in a given pore takes place at a burst when the pressure attains the capillary condensation pressure of the narrowest domain in the pore (see Figure 7a). The liquid formed in the narrowest domain acts as a nucleus for capillary condensation in other domains of the pore because capillary condensation in the neighboring domains occurs at the capillary evaporation pressures of these domains and the capillary evaporation pressures of other domains in the pore are all lower than the capillary condensation pressure of the nucleation site under the conditions of small σdomain values. Therefore, the individual pores will be suddenly filled at the pressures determined by the condensation pressures of the narrowest domains in the pores. Desorption of the completely filled rods of liquid from the individual pores occurs at the entrances and thus in the beginning leaves single rods of liquid unfilled in the neighborhood of the entrances because of single pore blocking (Figure 7b). The number of the single rods of liquid participating in the subloop taken on the desorption branch is larger than that participating in the subloop taken on the adsorption branch. The unfilled rods of liquid are easily converted into the completely filled rods of liquid in the pores by advanced adsorption. This may result in the larger amplitude of the subloop on the desorption branch, in accord with the noncongruent subloops observed for O2 on COK-11 and MCM-41(C22). Figure 8 shows the visualizations of filled domains in one of the dependent cylindrical pores with relatively large σdomain values during adsorption and desorption processes of oxygen at 70 K. For the corrugated pores with large σdomain, capillary condensation first gives rise to discrete rods of liquid in a given 5103


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SP75-373-1 seem to be well reproduced by the dependent domain models with σdomain values of 0.64 and 1.12 nm, respectively. H2O on SBA-15. Figure 10 shows the theoretical adsorption hystereses, desorption scanning curves, and subloops as a

Figure 9. Hysteresis loops, desorption scanning curves, and subloops calculated for nitrogen at 77 K in dependent cylindrical pores with mean pore diameter of 8 nm and different σdomain values. The values of σdomain are 0.48 (a), 0.64 (b), 0.80 (c), 0.96 (d), and 1.12 nm (e), respectively. The isotherms of (b), (c), (d), and (e) are displaced rightward by 0.20, 0.40, 0.60, and 0.80, respectively. Figure 10. Hysteresis loops, desorption scanning curves, and subloops calculated for water at 295 K in dependent cylindrical pores with mean pore diameter of 8 nm and different σdomain values. The values of σdomain are 0.64 (a), 0.80 (b), 0.90 (c), 0.96 (d), and 1.12 nm (e), respectively. The isotherms of (b), (c), (d), and (e) are displaced rightward by 0.20, 0.40, 0.60, and 0.80, respectively.

the amplitude of the subloop on the adsorption branch increases, whereas that on the desorption branch first decreases and then increases. A comparison of Figures 3 and 9 indicates that the noncongruent subloops observed for SBA-15 could be well accounted for by the dependent domain model with relatively large σdomain. The hysteresis loops, desorption scanning curves, and subloops for N2 on SP60-373-1 and SP75-373-1 can be qualitatively reproduced by the dependent model with σdomain values of 0.64 and 1.12 nm, respectively. Similarly, the dependent model with σdomain of 0.96 nm can reproduce qualitatively the subloops for N2 on SP75-373-14, although the conversing type of the desorption scanning curve on this substrate cannot be accounted for by the dependent model with the same σdomain. In any event, the large amplitude of corrugation in the pore walls of SBA-15 as well as the small amplitude of pore corrugation in MCM-41 is consistent with the general view3 of pore imperfections for MCM-41 and SBA15. With σdomain of 0.52 nm the two subloops are almost congruent (not shown). However, the dependent domain model with such a critical value of σdomain does not seem to account for the congruent subloops observed for two kinds of SBA-15 (SP45-373-14 and SP60-373-14). It is very unlikely that the two kinds of SBA-15 samples possess incidentally such a critical amplitude of corrugation in the pore walls. In addition, the shape of the congruent subloops for SP45-373-14 is considerably different from that for SP60-373-14. This could be explained only on the basis of the independent domain models with different distributions of the condensation pressures versus the evaporation pressures, similar to O2 on MCM-41(C18)pH. Crossing, conversing, and returning types of desorption scanning curves38 can be reproduced based on the dependent domain models of cylindrical pores with small, medium, and large σdomain values, respectively. With large σdomain, there are many discrete rods of liquid in the individual pores before measuring the desorption scanning curve. As desorption occurs from both ends of the discrete rods of liquid in the pores, it is less influenced by pore blocking compared to desorption from the completely filled rods of liquid in the case of adsorption hysteresis. This will result in the returning type of desorption scanning curve. The returning type of desorption scanning curves and noncongruent subloops for N2 on SP60-373-1 and

function of σdomain for H2O at 295 K in the corrugated pores with a mean pore diameter of 8 nm and σpore of 0.08 nm. Similarly, the shapes of desorption scanning curves and subloops changed remarkably with increasing σdomain. With increasing σdomain, the amplitude of the subloop on the adsorption branch increases, whereas that on the desorption branch decreases. The relative amplitude of subloops on adsorption and desorption branches is reversed with increasing σdomain. The noncongruent subloops of water observed for SP75-373-1 seem to be well reproduced by the dependent domain model with σdomain of 1.12 nm. Similarly, the noncongruent subloops observed for N2 on this substrate were well accounted for by the dependent domain model with the same σdomain. With σdomain of 0.90 nm, the two subloops are almost congruent. However, the dependent domain model with such a critical value of σdomain does not seem to account for the congruent subloops observed for three kinds of SBA-15 samples (SP60-373-1, SP60-373-14, and SP75-373-14). It is very unlikely that the three kinds of samples possess incidently the same amplitude of pore corrugation. The noncongruent subloops of nitrogen observed for SP60-373-1 and SP75-37314 were well accounted for by the dependent domain models with σdomain values of 0.64 and 0.96 nm, respectively. On the other hand, the dependent domain models with these σdomain values give rise to noncongruent subloops for H2O, being inconsistent with the congruent subloops observed for H2O on the two substrates. It is strongly suggested that the congruent subloops of H2O on the three kinds of SBA-15 could be accounted for only by the independent domain models with different distributions of the capillary condensation pressures versus the evaporation pressures in the individual pores.

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DISCUSSION It is expected that, for cylindrical pores with relatively small amplitude of corrugation, capillary condensation and evapo5104


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N2 adsorption at 77 K in the cylindrical pores of diameter 8 nm, the threshold amplitude of the pore corrugation can be estimated on the basis of the critical value (0.52 nm) of σdomain. The effect of single pore blocking is seen the most strongly when a liquid is confined between two of the narrowest domains in a given pore. Since a pore contains 100 domains with a normal distribution of the diameter, the two of the narrowest domains would have domain diameters between D − 2σdomain and D − 3σdomain. Therefore, the threshold amplitude (3σdomain) of the pore corrugation may amount to ∼1.5 nm. The adsorption hysteresis of N2 at 77 K is routinely used in measurements of a pore size distribution of mesoporous materials. All the methods of pore size analysis are based on the independent domain model. Accurate pore size distributions would be obtained through the proper analysis of the desorption branch when for the mesoporous materials examined the corrugation of the cylindrical pores is smaller than the threshold amplitude.

ration take place independently in the individual pores. The temperature dependence of the adsorption hysteresis for O2 on MCM-41 has clearly indicated that the amplitude of the pore corrugation in MCM-41(C18)pH is distinctly smaller than those in COK-11 and MCM-41(C22). However, calculations based on the dependent domain model suggest that in this case the subloop on the desorption branch would be significantly larger in the change of the adsorbed amount than that on the adsorption branch, as opposed to the congruent subloops observed for MCM-41(C18)pH. A comparison of Figures 2 and 6 suggests that the amplitude of the pore corrugation in MCM-41(C18)pH is smaller than that corresponding to σdomain of 0.08 nm. Therefore, it is most probable that, in the individual pores of MCM-41(C18)pH with such a small amplitude of pore corrugation, capillary evaporation of O2 would be no longer influenced by the corrugation and take place independently at their particular pressures. The dependent domain model with σdomain of less than 0.52 nm for N2 on SBA-15 suggests that the subloop on the desorption branch would be larger in the change of the adsorbed amount than that on the adsorption branch. However, such subloops have not been observed experimentally for N2 adsorption on SBA-15.34,39 Instead, congruent subloops were observed. This suggests that, in the individual pores with corrugation of amplitude corresponding to σdomain of less than 0.52 nm, capillary condensation and evaporation of N2 are no longer influenced by the corrugation and take place independently at their particular condensation and evaporation pressures, respectively. A comparison of the experimental and theoretical subloops for N2 adsorption suggests that SP60-373-1, SP75-373-1, and SP75-373-14 possess pore corrugation of amplitude corresponding to σdomain of 0.64, 1.12, and 0.96 nm, respectively. However, the two SBA-15 samples (SP60-373-1 and SP75-37314) with corrugation of amplitude corresponding to σdomain of 0.64−0.96 nm showed congruent subloops for H2O adsorption. This strongly suggests that capillary condensation and evaporation of water in silica pores are influenced less by the pore corrugation compared to N2. With σdomain of 0.90 nm, the theoretical subloops for H2O were almost congruent. However, the dependent domain model with such a critical value of σdomain does not seem to account for the congruent subloops of H2O observed for the three kinds of SBA-15 (SP60-373-1, SP60-373-14, and SP75-373-14). The congruent subloops of H2O on the three kinds of SBA-15 could be explained only by the independent domain models with the different distributions of the condensation pressures versus the evaporation pressures. Even in the individual pores with relatively high amplitude of corrugation, capillary condensation and evaporation of H2O could take place independently at their particular condensation and evaporation pressures, respectively. The dependent domain model of cylindrical pores was able to account for the noncongruent subloops on adsorption and desorption branches observed for capillary condensation and evaporation of O2, N2, and H2O in the ordered mesoporous materials. In addition, the comparison of the experimental and theoretical subloops strongly suggests that there is a threshold amplitude of pore corrugation below which capillary condensation and evaporation are no longer influenced by the corrugation and irreversibly occur in the whole space of a given pore at the particular condensation and evaporation pressures, respectively. The threshold amplitude of the pore corrugation may depend on the pore size, adsorbate, and temperature. For

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CONCLUSIONS Congruence/noncongruence between the two subloops taken in the same pressure range on adsorption and desorption branches depended not only on the adsorbents but also on the adsorbates. All the noncongruent subloops observed for O2, N2, and H2O on MCM-41, COK-11, and SBA-15 could be well accounted for by the dependent domain model of cylindrical pores. In addition, it is strongly suggested that there is a threshold amplitude of pore corrugation below which capillary condensation and evaporation are no longer influenced by the pore corrugation and occur independently in the whole space of a given pore at the particular condensation and evaporation pressures, respectively.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +81-86-256-9494. Fax: +81-86-256-9757. E-mail: [email protected]. ORCID

Kunimitsu Morishige: 0000-0003-3874-5115 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS We thank Ms. A. Tezuka for her technical assistance in the preparation of MCM-41(C22) samples. REFERENCES

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