Modeling of N2 Adsorption in MCM-41 Materials: Hexagonal Pores

Apr 9, 2009 - Low-temperature nitrogen adsorption in hexagonal pores and equivalent cylindrical pores is analyzed using nonlocal density functional th...
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Modeling of N2 Adsorption in MCM-41 Materials: Hexagonal Pores versus Cylindrical Pores Eugene A. Ustinov* Ioffe Physical Technical Institute, 26 Polytechnicheskaya, St. Petersburg 194021, Russia Received January 29, 2009. Revised Manuscript Received March 2, 2009 Low-temperature nitrogen adsorption in hexagonal pores and equivalent cylindrical pores is analyzed using nonlocal density functional theory extended to amorphous solids (NLDFT-AS). It is found that, despite significant difference of the density distribution over the cross-section of the pore, the capillary condensation/evaporation pressure is not considerably affected by the pore shape being slightly lower in the case of hexagonal geometry. However, the condensation/evaporation step in the hexagonal pore is slightly larger than that in the equivalent cylindrical pore because in the latter case the pore wall surface area and, hence, the amount adsorbed at pressures below the evaporation pressure are underestimated by 5%. We show that a dimensionless parameter defined as the ratio of the condensation/ evaporation step and the upper value of the amount adsorbed at the condensation/evaporation pressure can be used as an additional criterion of the correct choice of the gas-solid molecular parameters along with the dependence of condensation/evaporation pressure on the pore diameter. Application of the criteria to experimental data on nitrogen adsorption on a series of MCM-41 silica at 77 K corroborates some evidence that the capillary condensation occurs at equilibrium conditions.

1. Introduction Recent discovery of highly ordered mesoporous materials such as MCM-411,2 and SBA-153 has enabled verification of theories of capillary phenomena. Porous of MCM-41 and SBA-15 are usually modeled by cylindrical channels, having equivalent diameter determined with geometrical considerations4 for the honeycomb structure using interplanar d100 X-ray spacing and the primary mesopore volume Vp. Since the primary mesopore volume is used with the above technique, it is implied that the hexagonal and the equivalent cylindrical pore have the same cross-sectional area. Results of application of classical and molecular approaches to N2 and Ar adsorption in pores of the above silicates proved to be rather controversial. It is often claimed that the theory strongly suggests that the equilibrium transition occurs at desorption/evaporation and that the capillary condensation pressure corresponds to the vapor-like spinodal point.5-8 However, there is experimental evidence that the capillary condensation in cylindrical pores is the equilibrium phase transition, whereas the capillary evaporation is delayed during the course of desorption.9-14 It is important to establish the reason for such fundamental divergence in interpreting of *Corresponding author. E-mail: [email protected]. (1) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Nature (London) 1992, 359, 710. (2) Beck, J. S.; Vartuli, J. C.; Roth, W. J.; Leonowicz, M. E.; Kresge, C. T.; Schmitt, K. D.; Chu, C.T.-W.; Olson, D. H.; Sheppard, E. W.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. J. Am. Chem. Soc. 1992, 114, 10834. (3) Zhao, D.; Feng, J.; Huo, Q.; Melosh, N.; Fredrickson, G. H.; Chmelka, B. F.; Stucky, G. D. Science 1998, 279, 548. (4) Kruk, M.; Jaroniec, M.; Sayari, A. J. Phys. Chem. B 1997, 101, 583. (5) Ravikovitch, P. I.; Vishnyakov, A.; Neimark, A. V. Phys. Rev. E 2001, 64, 011602. (6) Ravikovitch, P. I.; Neimark, A. V. Colloids Surf., A 2001, 187-188, 11. (7) Ravikovitch, P. I.; Neimark, A. V. Langmuir 2002, 18, 1550. (8) Ravikovitch, P. I.; Neimark, A. V. Stud. Surf. Sci. Catal. 2000, 129, 597. (9) Qiao, S. Z.; Bhatia, S. K.; Zhao, X. S. Microporous Mesoporous Mater. 2003, 65, 287. (10) Morishige, K.; Ito, M. J. Chem. Phys. 2002, 117, 8036. (11) Morishige, K.; Nakamura, Y. Langmuir 2004, 20, 5403. (12) Ustinov, E. A.; Do, D. D.; Jaroniec, M. J. Phys. Chem. B 2005, 109, 1947. (13) Ustinov, E. A.; Do, D. D. Colloids Surf., A 2006, 272, 68. (14) Ustinov, E. A. Langmuir 2008, 24, 6668.

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experimental measurements. The main source of the divergence is uncertainty in the specific surface area of the reference solid. The surface area of nonporous silica determined with the BrunauerEmmett-Teller (BET) method is shown to be within an error of 35%15 depending on the pressure range taken for linearization. If, for example, the specific surface area is underestimated compared to real value, the gas-solid potential derived from the reference adsorption isotherm would be overestimated (in absolute term), which would lead to underestimation of the calculated capillary condensation/evaporation pressure, and vice versa. It was clearly shown12 that the dependence of condensation/evaporation pressure on the pore diameter significantly shifts nearly parallel along the pressure axis with change in the chosen surface area of the reference nonporous silica. This means that the appropriate choice of the reference surface area allows one to match the condensation and evaporation pressures determined theoretically with those determined experimentally in any combination, unless an additional condition is used. Thus, Qiao et al.9 used the classical Broekhoff-de Boer theory16 in the analysis of N2 adsorption in MCM-41 samples. At a given pore size, the authors took an analytical expression of the reference adsorption isotherm, which ensures good fitting of the adsorption isotherm in the region prior to phase transition. Subsequent calculation of the equilibrium phase transition pressure proved to be close to the capillary condensation pressure, which means that the adsorption branch corresponds to capillary equilibrium. The same conclusion was derived by Morishige et al.10,11 from analysis of the effect of temperature on the hysteresis. My analysis12 based on the nonlocal density functional theory (NLDFT) has shown that the experimental capillary condensation pressure can be excellently fitted by calculated equilibrium transition pressure at all pore sizes and that, most importantly, this is the only way to conform experimental and calculated data over the whole range of the pore diameter using the same set of the gas-solid molecular parameters. In particular, in the region of developed hysteresis one can (15) Jaroniec, M.; Kruk, M.; Olivier, J. P. Langmuir 1999, 15, 5410. (16) Broekhoff, J. C. P.; de Boer, J. H. J. Catal. 1967, 9, 8.

Published on Web 04/09/2009

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Article Table 1. Molecular Parameters for the N2-Silica System at 77.35 K

εff/k (K)

σff (nm)

dHS (nm)

99.92

0.3510

0.3576

-2 FSε(1) sf /k (K nm )

σ(1) sf (nm)

-2 FSε(2) sf /k (K nm )

σ(2) sf (nm)

ZS (nm)

(s) dHS (nm)

δ (nm)

S (m2 g-1)

1714 1017

0.271 0.394

13571 5151

0.0397 0.1554

0.1097 -0.0403

0.277 0.273

0.070 0.065

22.3 17.0

match the calculated equilibrium transition pressure and the spinodal condensation pressure with experimental evaporation and condensation pressures, respectively. However, at lower pore diameters the predicted condensation/evaporation pressures would be substantially underestimated. The first attempt to elucidate this issue with NLDFT without invoking any adjustable parameters was made recently.14 The idea was to reconstruct the reference N2 adsorption isotherm on silica surface by a combination of its low-pressure section (prior to capillary evaporation) on pore walls of MCM-41 samples and the high-pressure section on nonporous reference silica. The pore wall surface area was determined with the geometrical considerations4 using d100 spacing and accounting for negligibly small effect of surface curvature on the gas-solid potential.17 In that case, there was no need to use the BET method, but, again, the capillary condensation pressure was very close to that of equilibrium transition. The suggestion that the capillary condensation in the course of adsorption occurs at equilibrium conditions indirectly presents in the Kruk-Jaroniec-Sayari (KJS)18 approach developed for the pore size distribution analysis of mesoporous materials using the adsorption branch of the isotherm because it is based on a modification of the Kelvin equation derived for the evaporation from the cylindrical pore (i.e., for the equilibrium transition). While it is not in conformity with classical representations, analysis of experimental data unambiguously shows that the adsorption branch of the isotherm corresponds to the true thermodynamic equilibrium. This feature is still not completely understood and can only be explained with additional ad hoc assumptions such as constrictions, pore wall roughness, energetic heterogeneity, and so forth, which defies quantitative prediction. Sometimes, even the pore diameter is used as an adjusting parameter8 to make the classical scenario of capillary condensation/evaporation be obeyed. In this sense, it might be interesting to explore the effect of the pore shape on the condensation/ evaporation pressure. Pores of MCM-41 are known to be hexagonal rather than cylindrical. This issue was already investigated by means of molecular simulation,19,20 but density functional theory has not been applied to hexagonal pores so far. Coasne et al.20 have found that both the capillary condensation and evaporation occur at higher pressures in hexagonal pore than in the equivalent cylindrical pore. If this is the case, the difference between theoretical vapor-like spinodal pressure and experimental condensation pressure is even higher, which additionally supports the finding that the capillary condensation occurs at the equilibrium transition pressure. However, this conclusion should be taken with caution because Coasne et al.20 did not check their results against experimental data. The authors studied Ar adsorption at 77 K in fully and partly hydroxilated silica cylindrical and hexagonal pores with a diameter of 3.2 nm. In both cases, the onset of capillary condensation was found to be 0.12 p/p0 ( p0 is the saturation pressure), which is about two times lower than the experimental value. In the present study, We find it reasonable to consider N2 adsorption at 77 K in (17) Ustinov, E. A.; Do, D. D. J. Colloid Interface Sci. 2006, 297, 480. (18) Kruk, M.; Jaroniec, M.; Sayari, A. Langmuir 1997, 13, 6267. (19) Coasne, B.; Pellenq, R.J.-M. J. Chem. Phys. 2004, 120, 2913. (20) Coasne, B.; Renzo, F. D.; Galarneau, A.; Pellenq, R.J.-M. Langmuir 2008, 24, 7285.

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hexagonal pores of MCM-41 samples using NLDFT in conditions close to reality.

2. Computational Details We use nonlocal density functional theory extended to amorphous surfaces NLDFT-AS developed in our previous papers.12-14,17,21 The idea underlying this approach is to apply the same scheme developed for gas-gas interactions in the framework of classical NLDFT to the gas-solid interactions in the case of an amorphous solid. Such a scheme allows us to account for energetic heterogeneity and surface roughness inherent to the amorphous solid surface and substantially improve the fitting of experimental adsorption isotherms. The potential exerted by the amorphous solid is not uniform at the XY plane parallel to the surface as in the case of a crystalline surface such as that of graphitized carbon black. Therefore, on average, gas molecules approach closer to the surface, which requires accounting for the contribution of solid atoms to the decrease of the excluded volume (in terms of the van der Waals idea) in the molar excess Helmholtz free energy near the surface. The latter is negligible in the case of the crystalline surface because of extremely small adsorbed gas density at a distance less than 0.6 the gas-solid collision diameter from the surface due to strong repulsive forces. Further, since the repulsive term of the gas-solid potential is already accounted for via the excess Helmhotz free energy, we explicitly account for only the attractive term of the potential with the Weeks-Chandler-Andersen scheme (WCA)22 analogous to that used in the classical NLDFT. We also incorporated the surface roughness in the form of the error function, which describes the decay of the solid atom density over the transition zone of the gas-solid interface, with the standard deviation δ being the parameter of the surface roughness. The developed approach is not an extension of the NLDFT to a gas-solid binary mixture as it has subsequently been done in the framework of fundamental measure theory.23,24 In the latter case of the so-called quenched solid density functional theory (QSDFT), while not directly, it is implied that the gas-solid mixture is an ideal one, which requires additional proof. Analysis of application of the NLDFT-AS to the system, N2nonporous and porous silica, at 77 K has revealed the presence of a highly short-ranged term of the gas-solid potential. Besides, the long-ranged attractive term proved to be decaying as inverse forth-power distance from the surface, which means that the potential is exerted mainly by surface atoms. That is the reason for the negligibly small enhancement of the potential due to the surface curvature in pores having a diameter larger than 3 nm.17 The more complex structure of the potential is accounted for in the combined attractive potential defined as the sum of the longranged and the short-ranged terms.14 The first one is derived from the conventional 12-6 LJ potential using the WCA scheme. The second term was derived in a similar manner from the 16-8 potential, in which the repulsive and the attractive components decay as r-16 and r-8, respectively. All parameters found for the N2-silica system14 are listed in Table 1. Parameters ε, σ, and d are the potential well depth, the collision diameter, and the hard-sphere diameter, respectively; zS is the distance of the plane exerted potential (presumably, the layer of oxygen atoms) from the geometrical surface. The superscripts 1 and 2 refer to the long-ranged and the short-ranged terms of the (21) (22) (23) (24)

Ustinov, E. A.; Do, D. D. Appl. Surf. Sci. 2005, 252, 548. Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237. Kruk, M.; Jaroniec, M.; Sayari, A. Langmuir 1997, 13, 6267. Tarazona, P. Phys. Rev. A 1985, 31, 2672.

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Figure 1. Density distribution of nitrogen adsorbed in the cylindrical pore of 3 nm and equivalent hexagonal pore at p/p0 = 0.22 and T = 77 K. The relative value of the spinodal condensation pressure is 0.241 for the cylindrical pore and 0.223 for the hexagonal pore. For more details see the text. potential, respectively. The subscripts s and f denote the solid and the fluid, respectively. All details of the model can be found elsewhere.14 The correct parameters are presented in the first line of Table 1 and allow for proper fitting of the N2 adsorption isotherm on nonporous silica LiChrospher Si-100015 at the specific surface area of 22.31 m2/g. In the last line, we present the gas-solid molecular parameters obtained from the same reference adsorption isotherm at a surface area of 17 m2/g, which is useful to study the effect of the molecular parameters on adsorption in mesopores and the condensation/evaporation pressure. This issue is considered in section 3.2.

2.1. Calculation of Density Distribution over Hexagonal and Cylindrical Cross-Section . It is implied that the density does not change along the pore axis. In the absence of cylindrical symmetry, two variables are needed to describe the density distribution over cross-section of the pore. We used Cartesian coordinates and discretized the density profile along the x and y axes by 0.1dHS. The origin of coordinates was placed at the pore center. Calculations were accomplished in the segment of positive x and y, and the mirror image was used for the density distribution in the three other segments. To make a correct comparison, we used the same scheme and the same parameters including the grid of 10 points per the hard-sphere diameter for the hexagonal and cylindrical pore. The gas-solid potential was determined by numerical integration of the pair potential over the pore wall surface accounting for the pore shape. The calculations were performed using minimization of the grand thermodynamic function (an open system) along the adsorption and the desorption branch. The spinodal condensation was determined very carefully. In the first run, we used the relative pressure increment of 0.01 in the range from 0.01 to 1. Once the capillary condensation pressure was determined, the calculation was repeated around that point with the pressure increment of 0.0001. The capillary evaporation pressure was evaluated by equality of the grand thermodynamic potential corresponding to the adsorption and desorption branches.

3. Results 3.1. Density Distribution in Cylindrical and Hexagonal Pores. A comparison of the N2 density distribution over the cross-section of the cylindrical pore of diameter 3 nm and the equivalent hexagonal pore is presented in Figure 1 in terms of 7452 DOI: 10.1021/la900369b

dimensionless density, i.e., Fd3HS. The equivalent diameter of the hexagonal pore is defined as that of the cylindrical pore having the same cross-sectional area. In both cases, the relative pressure is 0.22, which is slightly below the spinodal condensation pressure. The density of the bulk liquid nitrogen at 77 K and saturation pressure is 0.794. The color spectrum in the Figure changes from violet for the zero value to red for 1.6 according to the rainbow so that the bulk liquid density is designated by green color. The density distribution is substantially affected by the pore shape. In the case of the cylindrical pore, the adsorbed phase consists of cylindrical layers (two ones for the specified pressure), whereas the adsorbed phase in the equivalent hexagonal pore is much more structured. The highest density can be found near vertexes of the hexagon due to maximum enhancement of the potential at these regions. This results in the appearance of repulsive forces around these dense grains not only along the radius as in the case of the cylinder but also along the pore wall surface at a distance from the point of maximum density less than about one collision diameter. The consequence is that contrary to the cylindrical pore, the density oscillations in the hexagonal pore are observed both along the radius and the pore walls, which cause the formation of a coarse-grained structure of the adsorbed phase. Figure 2 shows the density distribution of nitrogen adsorbed in the cylindrical and the hexagonal pores of 3 nm at a relative pressure of 0.24 just after capillary condensation. The difference in the distributions is even larger than those in the previous case. The coarse-grained structure of nitrogen adsorbed in the hexagonal pore is clearly pronounced including the pore center region. In the case of cylindrical geometry the adsorbed phase consists of concentric cylindrical layers, which completely confirms the correctness of the results previously obtained with the one-dimensional version of NLDFT-AS. Since the difference in the density distribution is rather large, one can expect that the pore shape substantially affects the amount adsorbed at a specified pressure and the capillary condensation/evaporation pressure. In the next section, we will examine this issue. 3.2. Capillary Condensation and Evaporation in Hexagonal and Cylindrical Pores. Figure 3 presents the nitrogen adsorption isotherm at 77 K in the MCM-41 sample,23 having Langmuir 2009, 25(13), 7450–7456

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Figure 2. Density distribution of nitrogen adsorbed in the cylindrical pore of 3 nm and equivalent hexagonal pore of the same cross-sectional area at p/p0 = 0.24 and T = 77 K.

Figure 3. N2 adsorption isotherm in MCM-41 sample23 at 77 K.

The d100 spacing is 5.89 nm, and D is 5.96 nm.14 Symbols denote experimental data for adsorption (O) and desorption (b). The solid and dashed lines are calculated with the 2D NLDFT-AS for the hexagonal and cylindrical pore shape, respectively. The dash-dot line is the calculation for the gas-solid molecular parameters obtained with the reference surface area of 17 m2/g (see the last line of Table 1).

equivalent pore diameter of 5.96 nm, which was carefully determined in our previous paper.14 The key point of that study was that the reference N2 adsorption isotherm was evaluated using a combined analysis of the low-pressure section of N2-MCM-41 systems (below p/p0 = 0.1), interplanar d100 spacing, the pore volume determined accounting for the compressibility of adsorbed nitrogen, and the high-pressure section (above p/p0 = 0.1) of the N2 adsorption isotherm on nonporous silica Lichrosper Si-100015 at 77 K. Although that procedure is a bit involved, it allowed us to avoid invoking the BET approach. The obtained reference adsorption isotherm corresponds to that on Lichrosper Si-1000 having the specific surface area of 22.3 m2/g. Once this value was determined, the gas-solid molecular parameters were evaluated with NLDFT-AS by a fitting procedure. Vertical lines in Figure 3 correspond to the spinodal capillary condensation at higher pressures and the equilibrium phase transition at lower pressures. Interestingly enough, the isotherm calculated for the hexagonal pore (solid line) is very close to that calculated Langmuir 2009, 25(13), 7450–7456

for the equivalent cylindrical (dashed line) pore. The small difference in the amount adsorbed is due to the larger (by 5%) pore wall surface area in the former case. That is the reason why the amount adsorbed in the cylindrical pore is slightly underestimated up to the capillary condensation. The difference between the spinodal condensation and the equilibrium transition pressures for the two cases is even smaller. This observation does not conform to that of Coasne et al.20 The authors have made their computations with a molecular simulation technique and found that the capillary condensation/evaporation pressure is markedly larger in the case of the hexagonal pore, whereas our result is completely the opposite. The reason of such discrepancy is not quite clear and requires additional analysis. It should be mentioned that the equivalent cylindrical pore can be alternatively chosen from the condition of the same pore wall surface area or the same volume to surface ratio. In the former case, the predicted amount adsorbed in the low-pressure region would not change due to the larger (by 5%) equivalent pore diameter, but the capillary condensation/evaporation pressure shifts toward higher pressures. In the latter case, the equivalent diameter is less by about 5%, and therefore, the capillary condensation/evaporation pressure shifts toward lower values. The shortcoming of the above definitions is that they lead to the incorrect pore volume and the predicted adsorption capacity of the sample. An interesting feature is that the experimental capillary condensation pressure nearly coincides with the theoretical equilibrium phase transition pressure. It was already mentioned in section 1 that this result was obtained many times earlier, but this time, no adjusting parameters were used. Nevertheless, we find it interesting and important to make an attempt at adjusting the gas-solid molecular parameters to match capillary condensation/evaporation pressures obtained theoretically for relatively large pores showing the developed hysteresis with those obtained experimentally. The only condition is a good fitting of the reference nitrogen adsorption isotherm on nonporous silica at a different value of its specific surface area. Thus, we explored the case of 17 m2/g for the LiChrospher Si-1000 sample,15 instead of the more grounded 22.3 m2/g. The corresponding molecular parameters are presented in Table 1. The dash-dot line in Figure 3 is the calculation of the isotherm for the same hexagonal pore of 5.96 nm. As seen in Figure 3, the hysteresis DOI: 10.1021/la900369b

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Figure 4. Capillary evaporation (1) and condensation (2) pressure versus pore diameter for N2 adsorption at 77 K in MCM-41 samples.23 The

surface area of the reference silica taken for calculations is 22.31 m2/g (a) and 17 m2/g (b). Calculated curves are plotted for hexagonal (3,4,7,8) and cylindrical (5,6) pores.

Figure 5. The lower (1) and larger (2) amounts adsorbed at the evaporation step and the limiting nitrogen density at the saturation pressure in the hexagonal (1-3) and cylindrical (4) pore versus the equivalent pore diameter at 77 K predicted by the 2D NLDFT-AS.

loop has shifted toward lower pressures, and now its position nearly coincides with that of the experimental hysteresis loop. However, the condensation/evaporation step has drastically reduced. This is because the gas-solid potential is stronger than that in the previous case, which provokes the condensation earlier but, at the same time, increases the amount adsorbed on the pore walls at the same pressures below condensation. One can see that this change extremely worsens the fitting of the isotherm, with the calculated amount adsorbed being highly overestimated to the left of the hysteresis loop. What it means is that imposing of a pore size distribution (PSD) function can only worsen the agreement. It is clear from the fact that any isotherm can be precisely fitted by a combination of Heaviside step functions (compare the case of Horvath-Kawazoe method), which means that the imposition of a PSD results in a smoother isotherm compared to the local isotherm. Hence, the experimental adsorption isotherm presented in Figure 3 cannot be described theoretically along with the hysteresis loop, unless the condition of proper fitting of the reference adsorption isotherm is relaxed. Another way to overcome this difficulty is quite questionable and involves adjusting the pore diameter of MCM41 samples.8 Figure 4 is another illustration of the problem under discussion. As seen from Figure 4, experimental data on the capillary condensation of nitrogen in pores of MCM-41 samples are fitted 7454 DOI: 10.1021/la900369b

Figure 6. Dimensionless capillary evaporation (1) and capillary condensation (2) steps determined from series N2 adsorption isotherms at 77 K on MCM-41 samples23 versus capillary evaporation and condensation, respectively. (3) Prediction for the equilibrium phase transition; (4) the same for lower specific surface area S of 17 m2/g; (5) the dimensionless capillary condensation step calculated at S = 17 m2/g.

by the theoretical dependence of equilibrium transition pressure on the pore diameter, no matter whether we use the cylindrical or hexagonal shape of the pore. Note, however, that the equivalent cylindrical pore always gives a slightly larger condensation and evaporation pressure compared to those of the hexagonal pore. Figure 4 also depicts the p/p0 - D dependence for stronger potential evaluated for a lower specific surface area of the reference silica. For pores larger than 5.5 nm, the predicted spinodal condensation and equilibrium transition pressures do fit the experimental capillary condensation and evaporation pressure, respectively. However, at a lower pressure range the theory substantially underpredicts the experimental reversible capillary step. Moreover, this seems to be thermodynamically inconsistent because the reversible capillary step is believed to be the equilibrium transition and therefore must coincide with the theoretical pressure of capillary evaporation. 3.2.1. Capillary Condensation and Evaporation Steps. The correlation between the theoretical and experimental condensation/evaporation pressure is widely used as a criterion of correctness of the approach. However, this criterion is not sufficient to ensure proper fitting of the experimental adsorption and desorption isotherms. An additional criterion can be Langmuir 2009, 25(13), 7450–7456

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Figure 7. Effect of the number of intervals per hard-sphere diameter on the equilibrium transition pressure (a) and the amount adsorbed at this pressure at its lowest point (b). The equivalent pore diameter is 3.282 nm.

proposed as the ratio of the condensation step Δacond and the amount adsorbed in the completely filled pore acond at the condensation pressure, i.e., Δacond/acond. Similarly, for the evaporation step the criterion can be defined as Δaevap/aevap. These dimensionless values can change from zero to unity and depend on the respective condensation and equilibrium transition pressure. Below we consider these dependences. Some features associated with the equilibrium transition step determined with the NLDFT-AS for hexagonal and cylindrical pores as well as the average density of nitrogen at the saturation pressure are shown in Figure 5. As seen in Figure 5, the first order phase transition cannot occur (theoretically) in pores smaller than 2 nm. At a larger pore size, the transition step is determined by the separation between curves 1 for the adsorption branch and 2 for the desorption branch of the isotherm. At the equilibrium transition pressure, the amount adsorbed in the partly filled cylindrical pore is lower than that in the hexagonal pore of the same diameter because of lower pore wall surface area (compare branch 1 and the closest dashed line in Figure 5). However, the density corresponding to the onset of capillary evaporation pressure is nearly the same in the filled pores of different shape. Therefore the capillary evaporation step is larger in the case of the cylindrical pore. The density at the saturation pressure in pores of different shape but the same equivalent diameter is also nearly the same. The distance between curves 2 and 3 along the y axis is the increase of density in the completely filled pore due to compressibility of the adsorbed nitrogen. This feature is persistently ignored and ascribed exclusively to adsorption at a ghost external surface, which may cause inaccurate determination of the pore volume, size, and surface. The dimensionless relationships defined above are presented for the series of MCM-41 samples in Figure 6. One can see in Figure 6 that the experimental dependence Δacond/acond - pcond/ p0 is well described by the dependence Δaevap/aevap - pevap/p0 predicted by the theory for the equilibrium transition (the solid line). This again corroborates the suggestion that capillary condensation seems to be an equilibrium phase transition. Lines 4 and 5 are dependences for the capillary evaporation and the capillary condensation step, respectively, calculated for the overestimated gas-solid potential obtained by fitting the reference N2-silica system for the smaller specific surface area of 17 m2/g. As it was already mentioned, in this case we managed to superimpose theoretical capillary condensation/evaporation pressure with the experimental value for pores larger than 5.5 nm. If the model were correct in terms of classical representations at least for the large pores, dashed line 5 would describe Langmuir 2009, 25(13), 7450–7456

experimental points 1 (solid circles), which is definitely not the case. 3.3. Accuracy of Calculations with the 2D NLDFT-AS. The exact solution for the hexagonal pore can be obtained only in the framework of the two- or three-dimensional model. In this case, the computation consumes much more time than in the case of the 1D task. For this reason, we discretized the density profile along the x and y axes by quite large spacings of 0.1dHS. In this context, it is important to estimate the discretization error. To this end, we have calculated the adsorption isotherm in the hexagonal pore having a perimeter of 30dHS for different number N of equal intervals per hard-sphere diameter. The equivalent pore diameter is 3.252 nm (i.e., 9.094dHS). For each N, we determined the set of the Tarazona weighted functions w0, w1, and w2 of the smoothed density approximation (SDA)24 by its analytical averaging over all intervals corresponding to vector r at the XY plane from a given lattice site. This ensured the necessary condition of zero values of the sum of w1 and w2 over the area of a circle of radius r = 2dHS. The summation of w0 gives unity, as required. A similar procedure was realized for discretizing the attractive pair potential according to the WCA scheme. In doing so, we preserved the correctness of the Tarazona SDA and did not disturb the attractive component of intermolecular potential. It has turned out that the smallest number of intervals per the hard-sphere diameter is five. Smaller numbers lead to instability of the iteration procedure. Therefore, we traced the effect of N on the adsorption isotherm in the range from 5 to 20, which corresponds to the grid spacing from 0.2dHS to 0.05dHS. Some results are presented in Figure 7. For example, Figure 7a shows the relative equilibrium transition pressure calculated at different number of intervals N. As seen in the Figure, the equilibrium transition pressure is nearly the same at N g 6. Note that the scale of the pressure axis is very small. The discretization error in the range of N from 6 to 20 is only 0.1%. However, at N = 5, the calculated pressure pevap/p0 substantially drops. Hence, the value of grid spacing 0.1dHS (for N = 10) seems to be quite reasonable. Figure 7b depicts the amount adsorbed at the lower point of the equilibrium transition step. Again, the discretization error is negligibly small at N g 6. The whole N2 adsorption isotherms calculated with the numbers N equal to 6 and 20 completely merge at a larger scale.

4. Conclusions In the current work, nitrogen adsorption in hexagonal and cylindrical pores was analyzed with the nonlocal density functional theory extended to amorphous surfaces (NLDFT-AS) and DOI: 10.1021/la900369b

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adjusted to the case of 2D density distribution over the crosssection pore area. The same 2D technique was applied to the adsorption in cylindrical pores to make a correct comparison of these two cases. The density distribution of nitrogen at 77 K was found to be drastically different in the hexagonal and the equivalent cylindrical pores at a specified pressure. In the latter case, the cylindrical symmetry is maintained, and the adsorbed phase structure represents a number of concentric cylindrical layers. In the former case, the structure of the adsorbed phase is coarse-grained, which resulted from the hexagonal shape of the wall and the enhancement of the gas-solid potential at vertexes of the hexagon. Nevertheless, the adsorption isotherms calculated for the hexagonal and the equivalent cylindrical pores are surprisingly similar and show only a small decrease of the amount adsorbed on the pore wall in the latter case due to the smaller surface area. We also indicate that the spinodal condensation and the equilibrium transition pressure slightly increase with the replacement of the hexagonal pore by the equivalent cylindrical

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pore. An intriguing result is that the best fit of experimental nitrogen adsorption isotherms in MCM-41 samples with the NLDFT-based approach extended to hexagonal pores corresponds to the assumption that the capillary condensation occurs at equilibrium conditions. The effect of grid spacing along coordinates at the cross-section plane on the discretization error is examined. This error appeared to be insignificant at spacings smaller than one-sixth the hard-sphere diameter. Therefore, the large upper limit of the grid spacing is a result of the precise procedure of discretization of the Tarazona smoothed density approximation and the Weeks-Chandler-Andersen perturbation scheme for the attractive intermolecular potential. This dramatically reduces the computational cost and makes the 2D NLDFT-AS approach applicable to similar tasks of adsorption in confined volumes. Acknowledgment. Support from the Russian Foundation for Basic Research is gratefully acknowledged.

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