Langmuir 1998, 14, 1521-1524
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Capillary Coexistence and Criticality in Mesopores: Modification of the Kelvin Theory S. K. Bhatia* and C. G. Sonwane Department of Chemical Engineering, The University of Queensland, Brisbane, Queensland 4072, Australia Received August 5, 1997. In Final Form: January 27, 1998 The classical model of capillary equilibrium in cylindrical pores is modified here by the introduction of molecular concepts and the solid fluid interaction potential. The new approach accurately predicts capillary coexistence and criticality, with results quantitatively matching those from density functional theory for nitrogen adsorption, while also predicting condensation pressures in agreement with reported experimental findings for MCM-41. The larger critical pore size for nitrogen adsorption in these materials, however, suggests a modification of the potential function parameters, evaluated here from data for hydroxylated silica.
Introduction The classically accepted1-4 theory that, in pores above a critical size, adsorption is initiated as a surface layering, followed by an abrupt condensation transition at a characteristic pressure has, in recent years, been corroborated experimentally5,6 as well as theoretically by molecular simulations7 and density functional theory (DFT) computations.8-10 In pores below the critical size, however, a continuous pore filling adsorption is instead predicted,7-10 again consistent with classical assumptions.5 Notwithstanding this qualitative agreement, however, the molecular models have indicated quantitative deviations from the classical continuum theory in small pores ( 0, it follows that the equality
∂φ˜ vlγ∞(r - t + λ/2) ) ∂t (r - t - λ/2)3
(8)
provides the stability boundary, with a first-order condensation phase transition occurring at the corresponding solution tc when it exists. From the form of eq 8 it is readily anticipated that the inequality in eq 7 may only be satisfied for t < tc at sufficiently large pore radius r. Under such conditions the interface is stable and eq 6 may be used to estimate t at any pressure Pg. A condensation transition occurs at t ) tc, and eq 8 thus provides our modification of the classical Cohan relationship.1 Perhaps the most remarkable feature of the new development is the possibility of criticality evident from the form of the right-hand side of eq 8 which, as indicated above, suggests instability of the surface layering mechanism in pores below a critical size. Criticality must satisfy
d(∆G) d2(∆G) d3(∆G) ) ) )0 dN dN2 dN3
(9)
yielding the additional equality
∂2φ˜ 2γ∞vl(r - t + λ) ) ∂t2 (r - t - λ/2)4
(10)
where vm is the adsorbate molar volume at z ) t, and λ ) (2/3)1/2σff. For the curvature-dependent surface tension at the cylindrical meniscus, we obtain, following Defay et al.17 γ ) γ∞/[1 - λ/2(r - t)] where γ∞ is the value at a flat vapor-liquid interface. At the location z ) t (i.e. the vaporcondensate interface) the corresponding fluid phase must
Thus, surface layering can only occur in pores of radius r > rc, where rc and tc satisfy eqs 8 and 10, and adsorption in smaller pores must therefore follow the volume-filling mechanism without a phase transition. These pores are traditionally classified as micropores, and those larger than rc as mesopores and macropores.
(17) Defay, R.; Prigogene, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans: London, 1966.
(18) Ahn, W. S.; Jhon, M. S.; Pak, H.; Chang, S. J. Colloid Interface Sci. 1972, 38, 605.
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Langmuir, Vol. 14, No. 7, 1998 1523
Before presenting any computational results based on the present theory, we consider also the process of desorption in a filled cylindrical mesopore. In this case a hemispherical meniscus is present, and an analysis similar to that above yields
2γ∞vl(r - t)2 d(∆G) ) µg + dN (r - t - λ)[(r - t)(r - t - λ) + λσff/4] [µf(z,r) + φ˜ (z,r)] ) 0 (11) in which we have used the equilibrium condition
µf(z,r) + φ˜ (z,r) ) constant (r)
(12)
and the curvature-dependent surface tension for the hemispherical meniscus17 γ ) γ∞/[1 - λ/(r - t)]. In addition, radial gradients in adsorbate density are neglected. Here t represents the adsorbed layer thickness at the pressure Pg, following eq 6. Evaporation from the meniscus will occur when the corresponding pure fluid at the center is saturated, so that eq 11 yields
∫PP vg dP + φ˜ (r,r) ) 0
Figure 2. Capillary coexistence curve of nitrogen at 77.4 K for cylindrical pores: Kelvin equation, 1; modified Kelvin equation, 2; Broekhoff and de Boer, 3; NLDFT, 4; current work, 5.
g
2γ∞vl(r - t)2 (r - t - λ)[(r - t)(r - t - λ) + λσeff/4]
(13)
which, in conjunction with eq 6, provides the capillary coexistence curve. Along this curve eq 11 also provides di(∆G)/dNi ) 0, i g 1, so that a first-order phase transition is predicted. As will be seen eqs 6 and 13 also predict criticality at a particular pore radius, with no solution for pores below this size. Clearly such pores will not display an evaporative transition but instead undergo continuous desorption. Results and Discussion On the basis of the above model, computations were first performed for predicting the capillary coexistence curve for nitrogen at 77.4 K in MCM-41, a predominantly siliceous mesoporous material with parallel hexagonal pores. The solid fluid interaction parameters in eq 2 were obtained by fitting standard isotherm data19 for the adsorption of nitrogen on nonporous hydroxylated silica, yielding σfs ) 3.606 Å and fs/k ) 63.00 K, which were assumed to hold for MCM-41 as well. For nitrogen the BWR equation of state is used, along with σff ) 3.681 Å, ff/k ) 91.5 K, and γ∞ ) 8.88 × 10-3 Nt/m1. The capillary coexistence curve is now readily obtained by combining eqs 1, 2, 6, and 13 and is depicted in Figure 2. Also shown in the figure is the result of Ravikovitch et al.9 using nonlocal DFT, with parameters also obtained by fitting the same data,19 indicating the remarkable accuracy of our computationally far less demanding approach. In comparison the result from the Broekhoff and De Boer model16 has larger deviation, while the original Kelvin result and its modification to include the layer thickness1 are far less accurate. However, perhaps the most impressive feature of the present theory is its ability to predict criticality, which is absent in these formulations. For the parameters indicated above, eqs 6, 8, and 10 yield the solution r ) rc ) 7.5 Å, and for smaller pores stable surface layering cannot occur. The existence of criticality, as predicted by the theory, is most clearly evident from Figure 3, depicting the variation of d2(∆G)/dN 2 with thickness (19) Gobet, J.; Kovats, E. Ads. Sci. Technol. 1984, 1, 285.
Figure 3. Variation of [d2(∆G)/dN2] with thickness for cylindrical pores. Numbers beside the curves refer to pore diameter.
t. For pores below 15 Å diameter d2(∆G)/dN 2 is always negative, and the vapor-condensate interface is unstable. Additionally, while predicting criticality for pores of 15 Å diameter, our computations also showed that eqs 6 and 13 have no solution for pores smaller than 19 Å diameter, with d(∆G/dN) for the evaporative transition always being positive, in a manner similar to that depicted in Figure 3. Thus, a hemispherical meniscus cannot be sustained in pores smaller than this larger latter size of 19 Å diameter, which must therefore desorb as well as adsorb by the volume filling mechanism. This result is in remarkable agreement with the value of 18 Å diameter predicted by DFT9 for the critical pore size for capillary coexistence. In such pores our theory would therefore suggest absence of hysteresis. While predicting the absence of hysteresis in pores smaller than 19 Å diameter, our theory, as well as nonlocal DFT,9 indicates hysteresis in larger pores, with differences in condensation and evaporation pressures. Recent experiments with nitrogen adsorption on MCM-41,20
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however, demonstrate absence of hysteresis up to about 35 Å diameter, consistent with earlier results. While the new theory has been validated against the DFT, this result indicates that the potential model used or its parameters may need modification for MCM-41. This requirement of an improved potential model for MCM-41 has been recognized in the recent work of Maddox et al.,21 who account for heterogeneity of the solid surface. Nevertheless, it was found that our theory does correctly predict observed condensation pressures, as illustrated in Figure 4. In this figure we depict the reported20 variation of condensation pressure with pore diameter, and the predictions of the classical theories as well as our own (eqs 6 and 8). For the data the open squares are obtained with diameter calculated from the X-ray diffraction lattice parameter20 and assuming a wall thickness of 10 Å,22 while the filled squares correspond to a variable wall thickness depending on pore size as estimated by Kruk et al.20 The issue of wall thickness of MCM-41 has not yet been satisfactorily resolved, and we shall address this question in conjunction with our adsorption experiments in a subsequent article. Nevertheless, it is clear that our theory performs remarkably better than the earlier classical models in interpreting the data. Summary In summary, it is evident from the present exposition that with the introduction of molecular concepts the (20) Kruk, M.; Jaroniec, M.; Sayari, A. J. Phys. Chem. B 1997, 101, 583. (21) Maddox, M. W.; Olivier; J. P.; Gubbins, K. E. Langmuir 1997, 13, 1737. (22) Zhao, X. S.; Lu, G. Q.; Millar, G. J. Ind. Eng. Chem. 1996, 35, 2075.
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Figure 4. Variation of condensation pressure with pore diameter for nitrogen on MCM-41 at 77.4 K: modified Cohan equation, -‚-; Broekhoff and de Boer, ‚‚‚; current work, ;; data of Kruk et al.,20 0, 9.
classical or continuum approach can be resurrected to accurately predict qualitatively as well as quantitatively many of the phenomena hitherto predicted only by the computationally intensive simulation or density functional theory methods. The new model should find considerable use in modeling adsorption phenomena, from the point of view of both characterization and process analysis. LA9708804
