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Adsorption, Capillary Bridge Formation, and Cavitation in SBA-15 Corrugated Mesopores: A Derjaguin−Broekhoff−de Boer Analysis Cedric J. Gommes* Department of Chemical Engineering, University of Liège B6A, Liège 4000, Belgium ABSTRACT: A Derjaguin−Broekhoff−de Boer analysis of adsorption and desorption in SBA-15 mesoporous silica is presented, using realistic geometrical models that account for the pore corrugation in these materials. The model parameters are derived from independent electron tomography and smallangle scattering characterization. A geometrical characteristic of the pore that is found to be important for adsorption is the corrugation length, lC, which describes the longitudinal size of the geometrical defects along a given pore. Capillary bridges are possible only for large values of lC. The results are explained in terms of two spinodal and two equilibrium pressures, characterizing the wide and the narrow sections of the pores. Simplified analytical expressions are obtained, which provide necessary conditions for bridge formation and for cavitation in terms of the radii of the narrow and wide sections of the pores, as well as of lC. Quite generally, the results show that the deviation of the pore shape from that of ideal cylinders is key to understanding adsorption and desorption in corrugated mesopores, notably in SBA-15.

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Monte Carlo molecular simulations,12,26−29 and specifically addressed the case of deviation from ideality in OMMs at both atomic12 and mesoscopic15 scales. Another approach is the mean-field lattice−gas model,30 which has recently been applied specifically to OMM-like structures.31 The questions addressed in these studies concern the effect of disorder on the condensation and evaporation pressures, the shape of hysteresis loops, the occurrence of cavitation or pore blocking upon desorption, and so forth. These experimental and theoretical studies have kindled a debate about the nature of the adsorption and desorption branches of the isotherms. It is usually acknowledged that the adsorption branch is metastable, while desorption takes place at the thermodynamic equilibrium. This is consistent with nonlocal density functional theory (NLDFT) calculations, suggesting that the condensation pressure in OMMs coincides with the liquid-like spinodal, while the evaporation pressure coincides with the equilibrium transition.32,33 However, this does not necessarily apply to disordered pores. In the case of locally constricted pores, the states reached upon desorption could be metastable because they rely on the occurrence of cavitation.23 Similarly, it has been argued that the spinodal may never be reached in disordered pores because geometrical defects can act as nucleation sites for capillary bridge formation, which would help to overcome free energy barriers.34,35 The conception that desorption states are stable and adsorption states are metastable is now being challenged for OMMs, based on the consideration of condensation and evaporation pressures10,36notably via NLDFT

INTRODUCTION The discovery of ordered mesoporous materials (OMMs) in the mid-1990s has offered unprecedented opportunities for experimentally studying the effect of confinement on phase equilibria.1 In particular, a large number of studies were devoted to analyzing capillary condensation and evaporation in these novel materials.2−6 The hope with these early studies was that OMMs would provide geometrically ideal experimental models to test and develop theories to characterize porous solids more accurately.7−10 However, it became progressively clear that the pore structure of OMMs such as SBA-1511 is not that of perfectly cylindrical mesopores, but that they are much more disordered, with a significant surface roughness as well as with a complementary porosity laterally connecting the main channels.12−18 Meanwhile, progress in nanofabrication techniques19 have enabled researchers to synthesize true model systems corresponding to the textbook cases of linear or constricted mesopores open at either one or both ends.20,21 Adsorption in this type of pores was considered to be well understood, at least qualitatively.22−24 However, the experimental results obtained on those systems are at odds with the common understanding of adsorption phenomena. The situation is best summarized by quoting the last sentence of one of those papers, namely, “We think that the determination of pore space characteristics from adsorption/desorption isotherms is a problem still unsolved”.21 Put less contentiously, that sentence points to the often overlooked importance of geometrical disorder for adsorption phenomena: minimal deviations from geometric ideality may dramatically change the way in which capillary condensation and evaporation occur. Many theoretical studies have been published concerning adsorption in disordered pores.25 Some were conducted with © 2012 American Chemical Society

Received: December 28, 2011 Revised: February 8, 2012 Published: February 10, 2012 5101

dx.doi.org/10.1021/la2051228 | Langmuir 2012, 28, 5101−5115

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calculations37as well as on the temperature-dependence of hysteresis loops.38 These seemingly incompatible results suggest that cavitation and bridge nucleation may depend in a very sensitive way on the geometrical details of the porous materials. It is therefore important to analyze them through material-specific geometrical models. In that respect, SBA-15 ordered mesoporous silica offers unprecedented opportunities because its mesopore structure has been investigated in great detail through a variety of techniques. In particular, the deviation of the mesopores shape from cylinders has been recently analyzed thoroughly with electron tomography (3DTEM) combined with image analysis.17,39 A representative rendering of the pore shape is shown in the inset of Figure 1. The average mesopore is about

solid phase when moving away from the pore center. Although this effect was initially interpreted in terms of a microporous corona surrounding each mesopore14 it is now acknowledged that it results from pore corrugation.17,18 A quantitative analysis of the scattering data along these lines enabled us to determine the values of Rm and σR given in Table 1, which are in excellent agreement with 3DTEM. The pore corrugation is also qualitatively supported by mercury intrusion porosimetry40 as well as by nitrogen sorption scanning.41 Using nitrogen adsorption as a characterization tool, however, leads to an underestimation of the actual pore size in SBA-15. The radii estimated from adsorption and desorption using a Derjaguin−Broekhoff−de Boer (DBdB)42−44 analysis of a cylindrical pore are given in Table 1. The theoretical adsorption and desorption isotherms calculated with R = 3.7 nm, in agreement with both 3DTEM and SAXS, are shown in Figure 1; they are clearly off the experimental isotherms. The fact that the pore size in SBA-15 is underestimated by adsorption, when compared to small-angle scattering, has also been reported by other authors.45,46 In the present paper, we show that these conflicting viewpoints can be reconciled if the adsorption−desorption analysis takes account of the actual corrugation of SBA-15. This type of approach has already been explored by others via the introduction of quenched disorder in the NLDFT to account for a microporous corona surrounding the main mesopores. This model, referred to as the quenched-solid DFT or QSDFT,16 has been used to analyze experimental adsorption data in SBA-15, notably in combination with in situ smallangle scattering measurements.47 However, the QSDFT is one-dimensional: it assumes that adsorption occurs uniformly along a given pore, which precludes the investigation of bridge formation and of cavitation. For these reasons, we explore here the potential of a DBdB approach42−44 using twodimensional pore models that account for the variability of the radius along a given pore. Moreover, a DBdB analysis relies only on macroscopic quantities, such as surface tension and disjoining pressure, and is therefore more prone to produce results that can be easily used to analyze experimental data. The structure of the paper is the following: In section 2, we present the general DBdB model and particularize it to the case of corrugated cylindrical pores. Section 3 presents the results for pores with both a cosine-shaped corrugation and with a more realistic Gaussian corrugation. The latter Gaussian model was used previously to analyze the 3DTEM data from which the values in Table 1 have been obtained. The results are then discussed qualitatively in section 4 by introducing a simplified pore model. This enables us to analytically derive the necessary conditions for bridge formation and cavitation. Finally, the general applicability of our results to SBA-15 and other mesoporous materials is discussed in section 5. A list of the main symbols and variables used in the analysis can be found at the end of the paper.

Figure 1. Nitrogen adsorption (●) and desorption (○) isotherms measured on SBA-15 mesoporous silica; the inset shows a 3DTEM reconstruction of the pores in the very sample on which adsorption was measured.17 The red line is the isotherm obtained from a DBdB analysis of adsorption in a cylindrical pore with radius R = 3.7 nm.

7 nm in diameter, but the local value is variable along a given pore. A quantitative analysis based on a modeling of the graytone correlation function of electron tomograms enabled us to determine the average radius Rm, its standard deviation σR, as well as the length lC over which the radius fluctuations are correlated along a given pore.39 The latter length can be thought of as a characteristic size of the bulges seen in the inset of Figure 1. The values are reported in Table 1. We refer to the Table 1. Geometrical Characteristics of SBA-15 Mesoporous Silica, Obtained from Electron Tomography (3DTEM) Combined with Image Analysis, from Small-Angle X-ray Scattering, and from Nitrogen Adsorption−Desorption Isothermsa N2 adsorption−desorption Rm (nm) σR (nm) lC (nm)

electron tomography

scattering

condensation

evaporation

3.6 ± 0.2 1.2 ± 0.2 4.7 ± 2.8

3.7 0.6

2.8

3.2

b

b

b

b

b

a Rm: average radius; σR: amplitude of the corrugation; lC: corrugation length. bNot measurable.

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variability of the pore radius in terms of corrugation instead of roughness to emphasize that it consists of nanometer-scale rather than molecular-scale disorder. The pore corrugation in SBA-15 is largely supported by bulk characterization results. Small-angle X-ray scattering notably gives evidence for a progressive transition from pore space to

THE DERJAGUIN−BROEKHOFF−DE BOER MODEL

The DBdB model is equivalent to a local DFT under the assumption that the adsorbed phase exists only in the form of liquid at the same density as the saturated liquid.48,49 5102


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pressures P/P0. Minimizing eq 2 in the case of a flat surface yields indeed u(t) = RGT ln[P/P0].43 The function u(t) is physically equivalent to the disjoining pressure42,55,56 through the relation Π(t) = u(t)/Vm. For further purposes, we find it convenient to express it in the dimensionless form F(t) = u(t)/ (RGT). On the basis of a previous modeling of nitrogen adsorption on silica,53 we shall use the following relation

Accordingly, the grand thermodynamic potential is expressed as follows: Ω = γA +

∫V

d3x ρ(μ̃)[f (μ̃) + u(x) − μ]

P

(1)

where γ and A are the surface tension and the area of the liquid−vapor interface, μ̃ is the space-dependent intrinsic chemical potential of the fluid, ρ(μ̃ ) is its density, f(μ̃) is the Helmholtz free energy, u(x) is the solid−fluid interaction potential at point x, and μ is the chemical potential; μ and μ̃ are related to each other by μ̃ = μ − u(x). The integral in eq 1 is over both the liquid and the gas phases. Neglecting the contribution of the latter compared to the former, and assuming that the liquid phase is identical to the bulk liquid at saturation, leads to the following simplifications: The density is written as ρ = 1/Vm where Vm is the liquid molar volume at saturation, and the Helmholtz free energy is written as f = μ0−P0Vm, where μ0 and P0 are the chemical potential and pressure at saturation. Writing the chemical potential as that of an ideal gas eventually leads to Ω̅ =

⎡P⎤ u(x) 1 λA + d3x ln⎢ ⎥ − VP VP VF P R ⎣ 0⎦ GT

∫

F(t ) = F0 exp[−t /l] + Kt −m

(4)

with K = −0.0342, m = 2.62 (for t in nm), F0 = −9.97, and l = 0.216 nm. These values are derived from an empirical fit of nitrogen adsorption data on LiChrospher,57 which has also been used as a reference by other authors for adsorption studies on ordered mesoporous silica.36 Only pores with axial symmetry are considered here, which can be analyzed in cylindrical coordinates (see Figure 2). The

(2)

where RG is the ideal gas constant, T is the temperature, and Ω̅ =

Ω − Ω0 (VP/Vm)R GT

(3)

In these equations, Ω0 is the value of Ω for the pore entirely filled with liquid, VP is the total pore volume, VF is the volume of the pore free of liquid, and λ = γVm/(RGT) is a constant having the dimension of a length. For nitrogen at 77 K with24 Vm = 34.6 cm3/mol and γ = 8.85 mJ/m2, one has λ = 0.478 nm. Equation 2 is the expression of the grand thermodynamic potential in the DBdB approximation. The equilibrium configuration is found by optimizing the position of the vapor−liquid interface so as to minimize Ω̅ . Despite the crudeness of the approximations leading to eq 2, the DBdB model is found to be accurate for pores larger than 5 to 7 nm in diameter, depending on the authors.9,10,48,50 In addition to the applications that we have already mentioned, the general DBdB formalism has also been used to analyze the effect of the positive surface curvature on adsorption in areogels,51 the capillary condensation in aggregates of nanometer-sized particles,52,53 and very recently the adsorption-induced deformation of mesoporous solids.54 Expressing the thermodynamic potential Ω in the dimensionless form of eq 3 has several advantages. First, it is dimensionless and on the order of unity. Second, any value of Ω̅ ≥ 0 can be immediately said to correspond to a metastable state, because the completely filled pore would have a smaller potential and would therefore be more stable. Finally the integral in eq 2 is only over the free space of the pore, which spares us numerical difficulties related to the integration of u(x) close to a pore wall. The solid−fluid interaction energy u(x) could in principle be modeled as a Lennard-Jones potential, integrated over the total solid phase, which would notably enable us to account for the effect of surface curvature.48 However, we shall here make the so-called Derjaguin approximation, which consists in assuming that the field u(x) depends only on the distance t to the closest solid surface.55 This approximation enables one to derive the function u(t) directly from experimental thicknesses t(P/P0) of the layer adsorbed on a flat surface measured at a series of

Figure 2. Cylindrical coordinates (r,z) with z being the linear distance along a given pore; R(z) and r(z) are the radial positions of the pore surface and of the liquid−vapor interface; t is the shortest distance to the pore surface, which is generally different from R(z) − r(z).

pore has length L, and its shape is described through the function R(z), giving the distance from the pore axis to the wall at a distance z along the pore. The radial position of the liquid− vapor interface is given by the unknown function r(z), to be determined for any given pressure P/P0 by minimizing eq 2. With this specific pore shape, the thermodynamic potential takes the form Ω̅ =

2πλ VP

∫0

L

dz r(z) 1 + (dr /dz)2

⎡P⎤ π + ln⎢ ⎥ ⎣ P0 ⎦ VP −

2π VP

∫0

L

∫0

dz

L

∫0

dz r 2(z)

r(z)

dr rF(t[r , z , R(z)])

(5)

where t[r,z,R(z)] is the distance from the point with cylindrical coordinates (z,r) to the closest point on the pore surface. In general, the functional t[r,z;R(z)] depends in a complex way on the values taken by R(z) in the vicinity of point (r,z). However, in the particular case where R(z) is a linear function of z, the relation simplifies to t=

R (z ) − r 1 + (dR /dz)2

(6)

In the general case where R(z) is not linear, eq 6 is still a good approximation close to the pore wall. It has to be stressed that F(t) decreases very rapidly with t so that the contribution of the last term in eq 5 is important only in the regions of the pore where r(z) is close to R(z). In those regions, eq 6 is a very good approximation. 5103


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lengths, capillary condensation occurs through the formation of a bridge across the pore, which progressively broadens as the pressure is increased. For an intermediate corrugation length, the shape of the liquid is that of a film that thickens uniformly until capillary condensation takes place. To thoroughly explore the configuration space, the minimization of Ω̅ was performed twice for each pressure P/P0: the first starting from a film configuration, and the second starting from a bridge configuration (see also the Appendix). The thermodynamic potentials of the states toward which the minimization algorithm converges in each case are referred to as Ω̅ 1 and Ω̅ 2, respectively. The values of Ω̅ 1 and Ω̅ 2 for various values of lC are plotted in Figure 4 as a function of P/P0. On the

The actual configuration of the liquid phase at any given pressure is obtained by minimizing the thermodynamic potential eq 5. The first term is proportional to the area of the liquid−vapor interface. It is minimum when the pore is filled with liquid; otherwise it tends to favor configurations with a smooth interface, i.e., a function r(z) that varies slowly with z. The second term, proportional to ln[P/P0], is negative and it reaches its minimal value for the largest possible value of r, i.e., for r = R(z). This accounts for the fact that the vapor does not condense spontaneously below the saturation pressure. The third term quantifies the solid−fluid interaction and it is positive; it is minimum for r(z) = 0, i.e., if the pore is completely filled with liquid. The actual configuration results from a complex compromise between these three different physical effects. The numerical procedure used to find both the local and the global minima of Ω̅ is explained in the Appendix. Those minima correspond to metastable and equilibrium configurations, respectively. We shall assume for now that the adsorption branch of the isotherms is metastable and the desorption is an equilibrium process.33 We shall analyze this assumption further in the Discussion and Conclusions section.

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RESULTS Cosine-Shaped Pores. The simplest corrugated pore that we shall consider is the cosine-shaped pore defined by R (z ) = R m +

⎡ z⎤ 2 σR cos⎢2π ⎥ ⎣ lC ⎦

(7)

where Rm is the average pore radius, lC is the corrugation length, and σR is the amplitude of the corrugation. The factor √2 in eq 7 enables us to think of σR as of the standard deviation of the pore radius distribution. This definition makes the comparison easier with pores having a more complex shape and with experimental data (see Table 1). Figure 3 shows the liquid configurations that correspond to local minima of eq 5 for a cosine-shaped pore with Rm = 3.7 nm,

Figure 4. Grand thermodynamic potential Ω̅ of various configurations as a function of P/P0: uniformly filled pore (black horizontal line), adsorbed film Ω̅ 1 (dashed red), and capillary bridge Ω̅ 2 (solid blue). The corrugation length lC is 16 nm (a), 25 nm (b), 40 nm (c), and 80 nm (d).

basis of these curves, we define four particular pressures. The two spinodal pressures PS1 and PS2 are the pressures beyond which the corresponding configurations become unstable. The first equilibrium transition pressure PE1 is defined as the pressure at which Ω̅ 1 = 0 or at which Ω̅ 1 = Ω̅ 2, whichever occurs at the highest pressure. The second equilibrium pressure PE2 is defined as Ω̅ 2(PE2/P0) = 0. These definitions are justified hereafter. The evolution illustrated in Figure 3 corresponds to the metastable states, i.e., to the adsorption branch of the isotherms. For lC = 16 nm (see Figure 4a), the system starts from a film configuration at low P/P0 and remains in this configuration until the first spinodal is reached, at PS1/ P0 ≃ 0.78. Beyond that pressure, the pore fills with liquid and Ω̅ = 0. Upon desorption, the pore remains filled until the equilibrium pressure PE1 is met, for which Ω̅ 1 = 0. At that point, the liquid takes suddenly the film configuration. Adsorption and desorption isotherms corresponding to that scenario are shown in Figure 5a−c. For slightly longer corrugation lengths (Figure 4 b), the adorption occurs in the same way. However, during desorption the equilibrium E2 is met before E1 because PE2 ≥ PE1. This means that the adsorbate takes a bridge configuration, i.e., that cavitation takes place. We use here the word cavitation in a very general sense, as the process by which a bubble (or cavity) forms in an otherwise filled pore. This definition is broader that the usual meaning of that word, which is often used exclusively

Figure 3. Metastable configurations of the adsorbed phase in cosineshaped pores with Rm = 3.7 nm, σR = 0.6 nm, and lC = 4, 12, and 50 nm (from left to right). The relative pressures are from top to bottom: P/P0 = 0.5, 0.72, 0.75, and 0.78. The solid is shown in gray and the liquid in blue.

σR = 0.6 nm, and three different values of lC. The pore being periodic, it is shown only over a length lC. For short corrugation lengths, the adsorbed film progressively fills the asperities of the pore surface when the pressure is increased, and capillary condensation occurs afterward. By contrast, for long corrugation 5104


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parameters σR and lC play a different role. For instance, there exists a critical corrugation amplitude below which capillary bridges are never observed upon adsorption, independently of the value of lC. This fact is analyzed in detail later, in particular when discussing Figure 13. Gaussian Pores. A more realistic pore shape is obtained by modeling the radius of the pore R(z) as a Gaussian random function R(z) = R m + σR

2 N

N

∑

cos[qnz − ϕn]

n=1

(8)

where qn and ϕn are random variables: ϕn is uniformly distributed in the interval [0,2π), and qn is drawn from a yet to be chosen probability distribution function. For large values of N, R(z) is Gaussian distributed, and the factor (2/N)1/2 ensures that its standard deviation is σR.60 The pore model that was used in ref 39 to analyze 3DTEM data yielding the values of Rm and σR given in Table 1 is exactly the one defined by eq 8. The particular statistical distribution chosen for q is a characteristic of the pore shape. In the present work, we shall consider the case where q obeys γ statistics, i.e., its probability density function is

Figure 5. Adsorption and desorption isotherms calculated for the cosine-shaped pore with Rm = 3.7 nm, σR = 0.6 nm, and lC = 4 nm (a), 12 nm (b), 18 nm (c), 25 nm (d), 40 nm (e), and 80 nm (f). The dashed red lines are the isotherms of the uniform cylindrical pore with the same value of Rm.

for the creation of a bubble in a metastable liquid.58,59 We shall come back to this in the Discussion and Conclusions section. Going back to Figure 4b, the bridge in the equilibrium configuration eventually disappears at pressure PE1, below which the film configuration has the lowest Ω̅ . The corresponding adsorption−desorption isotherm is shown in Figure 5d. For even longer lC (Figure 4c), the spinodal pressure PS1 is lower than PS2, which means that a metastable capillary bridge forms upon adsorption (see isotherm in Figure 5e). For still longer lC (Figure 4d), S1 occurs at a lower pressure than E2. In these conditions there is an intermediate pressure range where the bridge configuration is stable, and over which the adsorption hysteresis disappears (see isotherm in Figure 5f). The evolution of capillary condensation and evaporation follows approximately the same trend when the corrugation amplitude σR is increased while keeping lC constant (see Figure 6a−f). Despite these similarities, the two geometrical

exp[−q/θ] g (q) dq = qk − 1 dq Γ[k]θk

(9)

with parameters θ and k. For k larger than 1, the distribution g(q) has a maximum at q = (k − 1)θ, which converts to a most probable corrugation length lC = 2π/((k − 1)θ). The parameter k is related to the breadth of g(q), and it is therefore a measure of the randomness of the pore structure. The smaller the k value, the more disordered the pore. In the limit k → ∞, the distribution g(q) becomes very peaked, and one recovers the case of the cosine-shaped pore. The configuration of the liquid in Gaussian pores as a function of P/P0 is illustrated in Figure 7 for two different corrugations lengths. The results are qualitatively similar to those obtained with cosine-shaped pores. For short corrugation lengths, the liquid is a film that covers the pore wall and progressively fills the asperities of the surface. When capillary condensation eventually occurs, the entire pore fills uniformly. By contrast, in the case of long corrugation lengths, capillary condensation occurs via the formation of bridges, followed by the lateral displacement of the menisci. The corresponding adsorption and desorption isotherms are shown in Figure 8. For any value of lC, the isotherms are calculated as averages over 8 independent realizations of the Gaussian pore, each having length 5 × lC. Contrary to the case of cosine pores, bridge formation and cavitation in Gaussian pores do not lead to risers in the adsorption and desorption isotherms. This is a consequence of the polydispersity of the model that makes these phenomena occur over broader pressure ranges. The common feature of the cosine-shaped and Gaussian pores is that the hysteresis shifts progressively toward higher pressures when lC increases, and that it becomes narrower and more sloped. The results of the calculations on Gaussian and cosine pores are summarized in Figure 9a,b, showing the four pressures PS1, PS2, PE1, and PE2 for pores with Rm = 3.7 nm and σR = 0.6 nm, and for increasing corrugation lengths lC. These pressures are well-defined for cosine pores. However, for Gaussian pores, any local minimum or maximum of R(z) may lead to a bridge

Figure 6. Adsorption and desorption isotherms calculated for the cosine-shaped pore with Rm = 3.7 nm, lC = 25, and σR = 0.2 nm (a), 0.4 nm (b), 0.5 nm (c), 0.6 nm (d), 0.7 nm (e), and 0.9 nm (f). The dashed red lines are the isotherms of the uniform cylindrical pore with the same value of Rm. 5105


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Figure 7. Metastable (a−e) and equilibrium (f−i) configurations of the adsorbate in Gaussian pores with Rm = 3.7 nm, σR = 0.6 nm, k = 5, lC = 6 nm (left), and lC = 50 nm (right).

Figure 8. Adsorption and desorption isotherms calculated for the Gaussian pore with Rm = 3.7 nm, σR = 0.6 nm, k = 5, and lC = 4 nm (a), 12 nm (b), 18 nm (c), 25 nm (d), 40 nm (e), and 80 nm (f). The dashed red lines are the isotherms of the uniform cylindrical pore with the same value of Rm.

or a bubble with particular characteristic pressures. Therefore, for Gaussian pores, the pressures PS2 and PE1 were determined as the upper and lower limits of the hysteresis loop. The pressure PS1 was determined as the lowest pressure at which capillary bridges are observed upon adsorption, and PE2 was determined as the largest pressure at which cavitation occurs upon desorption.

Figure 9. Influence of corrugation length lC on adsorption and desorption mechanisms in (a) Gaussian pores, (b) cosine-shaped pores, and (c) the simple two-section pores shown in Figure 10. In all cases, Rm = 3.7 nm and σR = 0.6 nm. The curves S1, S2, E1, and E2 are the spinodal and equilibrium transition pressures of the narrow and wide sections of the pores (see text). For the Gaussian pores, the errors are the standard errors of the mean calculated over eight realizations of length 5 × lC each.

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ANALYSIS OF THE RESULTS WITH A SIMPLIFIED PORE MODEL Figure 9 puts clearly in evidence the two different adsorption regimes observed when discussing Figure 5 for long and short corrugation lengths, respectively. It is only for long values of lC that capillary bridges form during adsorption and that cavitation occurs during desorption. By contrast, for short corrugation lengths, the asperities of pore surface are progressively

smoothed out by the adsorbate at low relative pressure; capillary condensation and evaporation then occur uniformly throughout the entire pore. The aim of the present section is to use a simplified model to analytically determine the lCdependence of the curves in Figure 9, and to analyze how they are affected by the corrugation amplitude σR. 5106


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It is tempting to analyze the cutoff corrugation length between the two regimes in terms of the total curvature of the pore surface. This approach would be tantamount to analyzing bridge formation in terms of a Plateau−Rayleigh instability,56 which would put the cutoff corrugation length at lR = 2πRm. In the case of Figure 9 with Rm = 3.7 nm, this would mean a cutoff length around 23 nm, which is a reasonable value. On the other hand, it is well understood that disjoining forces stabilize thin films and prevent capillary instabilities.61 In that context, another useful concept to analyze the cutoff would be the healing length.56,62 The latter represents the characteristic length of surface asperities that are smoothed out by an adsorbed film of a given thickness t. The healing length lh is generally of the same order of magnitude as t, and it is therefore much smaller than lR. Therefore, the overall transition between the two regimes extends from very short values of l C (comparable to lh) to approximately lR. In the present section, we propose a simplified pore model that enables us to analyze the adsorption−desorption results in the limits of lC → lh as well as for lc ≫ lR. The model is sketched in Figure 10; it consists of a pore with alternating

Figure 11. Thermodynamic potential for adsorption in a cylindrical pore with R = 3.7 nm, for P/P0 = 0.6 (□), 0.71 (◇), 0.81 (Δ) and 0.9 (○). The values P/P0 = 0.71 and 0.81 are the equilibrium and spinodal pressures, respectively. The inset shows the different contributions to Ω̅ in eq 10: surface energy (+), disjoining forces (solid line), and pressure term (same symbols as in the main figure).

the value 0, which occurs at the equilibrium transition pressure P/P0 ≃ 0.71. We shall refer hereafter to the spinodal (or condensation) pressure in a uniform pore of radius R as Pc(R). It is obtained by solving43,44 ∂ 2Ω̅

∂Ω̅ = 0 and ∂r

∂r 2

=0

(11)

simultaneously for r and P. The equilibrium (or evaporation) pressure, Pe(R), is the solution of43,44 ∂Ω̅ = 0 and ∂r

Figure 10. Simplified corrugated pore model, consisting of alternating narrow and wide sections with lengths and radii L1, R1 and R2, L2. The thickness of the adsorbed film is assumed to be constant in each section.

⎫ ⎡P⎤ 1 ⎧ ⎨λ2πr + ln⎢ ⎥πr 2 − 2πI(r ; R )⎬ ⎣ P0 ⎦ πR2 ⎩ ⎭ ⎪

⎪

⎪

⎪

(12)

In both cases, Ω̅ is given by eq 10. The theoretical isotherms plotted in Figure 1 were obtained by solving eq 11 for the adsorption branch, and eq 12 for the desorption. In the limit of large mesopores, the DBdB condensation and evaporation pressures converge toward the values obtained from Kelvin’s equation,24 namely,

narrow and wide sections with lengths L1 and L2, and radii R1 and R2. It is reminiscent of the models used very recently by Fan et al.,28 only with axial symmetry. Before proceeding with its analysis, we shall briefly review some aspects of the classical DBdB analysis of adsorption in a uniform cylindrical pore,43,44 if only to clarify the language and introduce some useful notations. For a cylindrical pore with radius R, the DBdB thermodynamic potential given by eq 5 takes the simple form Ω̅ (r ) =

Ω̅ = 0

⎡ P (R ) ⎤ −λ ln⎢ c ⎥≃ R ⎣ P0 ⎦

and

⎡ P (R ) ⎤ −2λ ln⎢ e ⎥≃ R ⎣ P0 ⎦

(13)

A systematic comparison of the DBdB condensation and evaporation pressures with other approaches can be found notably in ref 33. The main results obtained on cosine and Gaussian pores can be understood qualitatively by considering the simplified corrugated pore model shown in Figure 10. To simplify the analysis, we shall further assume that the radial positions of the film surface, r1 and r2, are constant along each section. From eq 5, the corresponding free energy is written as

(10)

where r is the radial position of the vapor−adsorbate interface and I(r;R) = ∫ 0r F(R−r)r dr. The potential Ω̅ (r) is plotted in Figure 11 for the particular value R = 3.7 nm and various relative pressures. When the pressure is progressively increased starting from low values, the system remains in the local minimum close to r ≃ 3 nm until the spinodal pressure is reached at P/P0 ≃ 0.81. Beyond that pressure, the local minimum ceases to exist, and the system moves to the global minimum at r = 0, which is the configuration where the pore is completely filled. Upon desorption, the system always adopts the configuration that is the global minimum. Therefore the pore empties suddenly when the local minimum of Ω̅ reaches

⎫ ⎡P⎤ L ⎧ Ω̅ (r1, r2) = 1 ⎨λ2πr1 + ln⎢ ⎥πr12 − 2πI(r1; R1)⎬ Vp ⎩ P ⎣ 0⎦ ⎭ +

⎪

⎪

⎪

⎪

⎫ ⎡P⎤ L2 ⎧ λ ⎨λ2πr2 + ln⎢ ⎥πr22 − 2πI(r2 ; R2)⎬ + 2π|r12 − r22| Vp ⎩ ⎣ P0 ⎦ ⎭ Vp ⎪

⎪

⎪

⎪

(14)

The first two terms are the contributions of each section and the last term approximates the extra surface area of the film 5107


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wide section of the pore empties at pressure PE2, which is therefore the equilibrium cavitation pressure. The other equilibrium pressure PE1 is the pressure below which the local minimum for finite r1 and r2 becomes lower than the one on the r2 axis. This is the pressure at which the narrow section of the pore empties. The formation of a capillary bridge upon adsorption and the occurrence of cavitation upon desorption depend on the relative values taken by PS1, PS2, PE1, and PE2. The latter depend a priori in a complex way on R1, R2, L1, and L2. The actual functional dependence can be derived by noting that eq 14 can be rewritten as

when the narrow and wide sections meet. Note that the surface area π|r12 − r22| is counted twice because there are two such surfaces in a length L1 + L2. Of course the estimation of the extra surface area term is oversimplified, but this crude approximation is sufficient for our present purpose. Examples of the thermodynamic potential Ω̅ corresponding to eq 14 are given in Figure 12 for three different pressures.

⎪ L ⎧ Ω̅ (r1, r2) = 1 ⎨ λ2πr1 + Vp ⎪ ⎩

+

⎫ ⎛ ⎡P⎤ ⎪ λ⎞ ⎜⎜ln⎢ ⎥ − 2 ⎟⎟πr12 − 2πI(r1; R1)⎬ ⎪ P L ⎣ ⎦ ⎝ 0 1⎠ ⎭

⎫ ⎛ ⎡P⎤ ⎪ ⎪ L2 ⎧ λ ⎞ ⎨λ2πr2 + ⎜⎜ln⎢ ⎥ + 2 ⎟⎟πr22 − 2πI(r2 ; R2)⎬ ⎪ ⎪ Vp ⎩ L2 ⎠ ⎝ ⎣ P0 ⎦ ⎭

(16)

in any physically relevant situation where r1 ≤ r2. In the context of the simplified pore model, the formation of a capillary bridge occurs with a constant value of r2 (see Figure 12), for which the second term in eq 16 is therefore irrelevant. Comparing eq 16 with eq 10 it becomes apparent that the pressure of bridge formation, PS1, is given by

Figure 12. Thermodynamic potential of the simplified pore model of Figure 10 with R1 = 3.5 nm, R2 = 4.5 nm, L1 = L2 = 40 nm (top) and L1 = L2 = 5 nm (bottom), for P/P0 = 0.6, 0.72, and 0.8 (left to right). The semitransparent red horizontal surface is the plane Ω̅ = 0.

⎡P ⎤ ⎡ P (R ) ⎤ λ ln⎢ S1 ⎥ = ln⎢ c 1 ⎥ + 2 L1 ⎣ P0 ⎦ ⎣ P0 ⎦

The top and bottom rows of the figure are typical of long and short corrugation lengths, respectively. In the latter case, the dominant contribution to Ω̅ is the term proportional to r22 − r12. Therefore, the minimum of Ω̅ is always found on the diagonal r1 = r2 except for very low relative pressures. This means that the asperities of the pore surface fills with adsorbate at low relative pressure. This situation corresponds to the limit where lC is very close to the healing length lh. The relevant approximation of the potential in this case is

where Pc(R1) is the condensation pressure of the narrow section, considered as an isolated pore. The pressure PS1 is larger than Pc(R1), due to the surface area of the two menisci that have to be created; this effect is accounted for by the term 2λ/L1. Once capillary bridges exist, the collapse of the remaining bubble between the bridges occurs with r1 = 0. The first term in eq 16 can therefore be neglected during that process. Accordingly, the pressure PS2 at which the bubble collapses is given by

⎫ ⎡P⎤ L1 L + L2 ⎧ ⎨λ2πr + ln⎢ ⎥πr 2 − 2πI(r ; R1)⎬ Ω̅ (r ) = 1 L1 + L 2 Vp ⎩ ⎣ P0 ⎦ ⎭ ⎪

⎪

⎪

⎪

(17)

⎡P ⎤ ⎡ P (R ) ⎤ λ ln⎢ S2 ⎥ = ln⎢ c 2 ⎥ − 2 L2 ⎣ P0 ⎦ ⎣ P0 ⎦

(15)

which results from eq 14 with r1 = r2 = r, noting also that I(r;R2) ≪ I(r;R1). The latter approximation results from the observation that I(r) decreases very steeply close to the pore surface, as also shown in the inset of Figure 11. The expression of Ω̅ is identical to that of a cylindrical pore; only the relevant radius is that of the narrow section R1, and the disjoining pressure term is reduced by a factor L1/(L1 + L2). The situation is quite different for long corrugation lengths (Figure 12, top row). At low pressure, the minimum of Ω̅ is found at finite values of r1 and r2, which corresponds to a film covering the surfaces of both narrow and wide sections of the pore. At spinodal pressure PS1 the minimum shifts to the r2 axis, which means that a bridge forms and fills the narrow section. When the pressure is further increased beyond the second spinodal PS2, the local minimum along the r2 axis disappears, and the only minimum left is r1 = r2 = 0, corresponding to the pore filled with adsorbate. The equilibrium pressure PE2 is the pressure for which the free energy surface is tangent to the plane Ω̅ = 0 on the r2 axis, i.e., for r1 = 0. Upon desorption, the

(18)

where Pc(R2) is the condensation pressure of the wide section, considered as an isolated pore. The pressure PS2 is smaller than Pc(R2) because the surface area of the existing menisci destabilizes that configuration. The two curves PS1 and PS2 obtained through eqs 17 and 18 are plotted in Figure 9c, assuming L1 = L2 = lC/2. These curves are qualitatively similar to those obtained with the cosine and Gaussian pores. However, the lC-dependence is overestimated in the case of PS2 because the assumption of constant r2 does not hold once a bridge has formed. Once a bridge has formed, the collapse of the remaining bubble occurs through the lateral displacement of the menisci. This should occur at pressure Pe(R2), which corresponds to the classical case of a pore closed at its end.24 In the following, we shall therefore assume PS2 = Pe(R2). An interesting situation is observed for lC ≃ 15−20 nm. Over that range of corrugation lengths PS1 ≥ PS2, which means that the bubble resulting from the bridge formation is unstable when the bridge is created. Therefore, the overall pore-filling pressure 5108


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is PS1, in agreement with what is observed for the cosine-shaped pore. The condition for observing a bridge at all upon adsorption is that PS1 be lower than PS2. This leads to

2

⎡ P (R ) ⎤ λ ≤ ln⎢ e 2 ⎥ L1 ⎣ Pc(R1) ⎦

(19)

which notably requires Pe(R2) ≥ Pc(R1)

(20)

as a necessary conditions. This last condition imposes a minimum difference between R1 and R2 for a capillary bridge to be possible upon adsorption. We shall come back to this shortly. The analysis of desorption proceeds in the same way. Starting from r1 = r2 = 0 and lowering the pressure, the system adopts the configuration that is the global minimum of Ω̅ . The equilibrium pressure PE2 at which the wide segment empties is given by ⎡P ⎤ ⎡ P (R ) ⎤ λ ln⎢ E2 ⎥ = ln⎢ e 2 ⎥ − 2 L2 ⎣ P0 ⎦ ⎣ P0 ⎦

Figure 13. Number of capillary bridges observed in cosine (top) and Gaussian (botton) pores with Rm = 3.7 nm, and various corrugation amplitudes σR and lengths lC. The number of bridges is the maximum calculated over the total pressure range, both for adsorption (●) and desorption (red ○). The two solid lines in the horizontal plane are eq 19 (black) and eq 23 (red), with R1 = Rm − √2σR, R2 = Rm + √2σR, and L1 = L2 = lC/2.

(21)

This is the equilibrium cavitation pressure. It is lower than the evaporation pressure of the wide section Pe(R2) because of the surface area of the two menisci that have to be created. Once the wide section of the pore has emptied, the narrow section empties at a pressure PE1 satisfying ⎡P ⎤ ⎡ P (R ) ⎤ λ ln⎢ E1 ⎥ = ln⎢ e 1 ⎥ + 2 L1 ⎣ P0 ⎦ ⎣ P0 ⎦

of the pore should be twice are wide as the narrow section: R2 ≥ 2R1. In smaller mesopores for which disjoining forces are not negligible, the threshold is less severe; the value found for Rm = 3.7 nm is σR ≥ 0.45 nm, but there exists no simple analytical expression. No capillary bridge can ever form upon adsorption for smaller σR. The case of desorption is quite different: bridges can in principle be observed for any corrugation amplitude σR if the corrugation length lC is long enough. This is consistent with the observation that two-step desorption branches are far more common experimentally than two-step adsorption branches. In the case of Gaussian pores (Figure 13 bottom), the conditions are not as sharp as for the cosine pores. This results from the randomness and polydispersity of the former. However, also in that case, eqs 19 and 23 with R1 = Rm − √2σR and R2 = Rm + √2σR are found to provide reliable estimates of the critical corrugation amplitude and/or lengths beyond which capillary bridges can form upon adsorption and desorption.

(22)

It is interesting to note that for short corrugation lengths, one has PE1 ≥ PE2 so that the overall pressure of capillary evaporation is PE2. The condition for cavitation to occur at all upon desorption is that PE1 be lower than PE2, which leads to ⎡ P (R ) ⎤ ⎛1 1 ⎞ 2λ⎜ + ⎟ ≤ ln⎢ e 2 ⎥ L2 ⎠ ⎣ Pe(R1) ⎦ ⎝ L1

(23)

This equation provides an explicit lower bound for the corrugation lengths L1 and L2, in terms of the radii of the narrow and wide section R1 and R2, below which cavitation cannot occur. Alternatively, it can also be understood as a lower bound for the difference between R1 and R2, for a specified corrugation length. In order to assess the validity of eq 19 and eq 23, we plot in Figure 13 the number of capillary bridges observed in various cosine and Gaussian pores, both upon adsorption and desorption over the relative pressure range from 0.5 to 0.9. These data were calculated by varying both lC and σR, while keeping Rm constant. In the case of the Gaussian pores, the number of bridges is normalized by the number of corrugation lengths in the calculation domain. The formation of capillary bridges and the occurrence of cavitation are fairly well predicted by eqs 19 and 23 assuming R1 = Rm − √2σR and R2 = Rm + √2σR. This shows that the model of Figure 10 captures the essence of the physics of adsorption in corrugated pores. An interesting feature is that capillary bridges are possible upon adsorption only for corrugation amplitude σR larger than a given threshold, which can be calculated through eq 20. In the case of large mesopores, for which Kelvin’s approximation holds (see eq 13), the limit is found to be that the wide section

■

DISCUSSION AND CONCLUSIONS The pore structure in SBA-15 mesoporous silica can be characterized through a variety of techniques, most notably small-angle scattering, electron microscopy and tomography, and vapor adsorption. However, using the pore radius obtained from small-angle scattering to estimate capillary condensation and evaporation leads to a significant overestimation of the relevant pressures. This is clear from Figure 1, and similar findings have also been repeatedly reported by other authors.37,45,46 This discrepancy has notably lead several researchers to challenge the general view that the adsorption branch of the isotherms is metastable and they have proposed to analyze it instead as an equilibrium process.10,36 All the mentioned analyses are based on the assumption that the pores of the material can be approximated by ideal cylinders. To gain insight into these issues, we have analyzed adsorption and desorption in realistic geometrical models of 5109


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SBA-15. Our geometrical models are built on electron tomography characterization17,39 and they notably account for the significant pore corrugation of the material. The values of the average radius Rm and of the corrugation amplitude σR derived from scattering and from electron tomography are very similar (see Table 1). Another important geometrical parameter of the pores is their corrugation length lC, which can only be obtained through electron tomography (lC ≃ 5 nm). Because this value is based on a local observation and it might be subject to sampling uncertainty, we have investigated a series of corrugation lengths ranging from lC = 4 nm to lC = 80 nm. Our results show that the adsorption and desorption processes are very dependent on the corrugation length lC and amplitude σR. Two regimes can be identified. For short-lC, healing lengths effects56,62 are dominant: the adsorbed film progressively smoothes out the pore wall asperities, and capillary condensation and evaporation occur uniformly all along the pore. By contrast, for long-lC, metastable or stable capillary bridges can be observed. There is an intermediate-lC regime where the spinodal pressure of the narrow section of the pore is larger than that of the wide section (see Figure 9b for lC ≃ 15 nm). In that intermediate regime, the pore corrugation acts as nucleation sites for the formation of unstable bridges that eventually lead to capillary condensation. This is globally in line with previous studies,23,34,35,63 but we have shown that this regime is relevant only for a limited lC range. The lowering of capillary condensation pressure that results from this nucleation effect is fairly well described by eq 17 as a function of the pore characteristics. The shape of the adsorption and desorption isotherms are quite different in the two lC regimes (see Figure 8). For short corrugation lengths, the shape of the adsorption hysteresis is not modified by the corrugation: both the adsorption and desorption branches are steep, only they are shifted toward lower pressures compared to ideal cylindrical pores. When lC is progressively increased, while keeping other characteristics constant, the hysteresis loop shifts toward larger pressures; it also becomes more slanted and narrower. Globally, the types of hysteresis loops that we obtained with the DBdB model are very similar to those obtained recently by other authors based on grand canonical Monte Carlo calculations.28 When comparing the experimental isotherms measured on SBA-15 (see Figure 1) with the theoretical isotherms in Figure 8, it appears that adsorption in SBA-15 is typical of the short-lC regime. In the limit of extremely short corrugation lengths, we have shown that the adsorption and desorption isotherms can be calculated approximately through eq 15, using the radius of the narrow sections of the pore and a reduced disjoining pressure term. The dashed lines in Figure 14 are the adsorption and desorption isotherms calculated in that way, using R1 = Rm − √2σR and L1/(L1 + L2) = 0.5 with the values of Rm and σR derived from small-angle scattering (see Table 1). Compared to the cylindrical pore model, the agreement with the experimental isotherms is clearly improved (compare with Figure 1). The short-lC approximation is to be compared with the so-called corona model14 that is sometimes used to analyze adsorption data in SBA-1546,64 notably with the QSDFT model.16,47 Our analysis suggests that the corona is an approximation that is valid only in the short-lC limit. The overestimation of the width of the hysteresis loop in Figure 14 results from the prohibitive thermodynamic cost of nucleating a capillary bridge in the short-lC limit. To reduce the width of the hysteresis, it is necessary to consider a pore model with a finite lC, which reduces the free energy barrier for bridge nucleation. The

Figure 14. Comparison of experimental and theoretical adsorption isotherms in SBA-15 (same data as in Figure 1): the dashed line is calculated in the short-lC limit through eq 15, and the solid line is obtained with with the Gaussian pore model with Rm = 3.7 nm, σR = 0.8 nm, k = 10, and lC = 4 nm. The inset shows a particular realization of the Gaussian pore model.

solid line in Figure 14 was obtained from a Gaussian pore model with lC = 4 nm and σR = 0.8 nm. These values lead to a better agreement with the experimental data, in particular to a much narrower hysteresis. The value of lC was chosen to be comparable to the one obtained from electron tomography, and the value of σR is intermediate between the one obtained from electron tomography and small-angle scattering (see Table 1). A particular realization of the Gaussian pore model with these values of the parameters is shown in the inset of Figure 14. Slightly different sets of parameters lead to roughly similar adsorption and desorption isotherms. A better agreement between the experimental and calculated desorption isotherms is obtained using corrugation lengths of the order of 2 nm, or using smaller values of k. However, these parameters lead to asperities at the pore wall that are much smaller than the size limit beyond which the DBdB approximation is expected to hold. We therefore did not attempt to reach a better agreement with the data. However, our results clearly hint at the importance of pore corrugation for analyzing adsorption and desorption in SBA-15. To further support the plausibility of the Gaussian pore model shown in the inset of Figure 14, we calculate its specific surface area in the Appendix and find the value 470 m2/g. This value is very close to the mesopore surface area of 420 m2/g calculated through a t-plot.17 For comparison, the surface area of the noncorrugated pore with the same radius is only 290 m2/g, and the Brunauer− Emmett−Teller (BET) surface area (including micropores) is 660 m2/g. The reader is referred to ref 65 for a comprehensive geometrical model of SBA-15 that accounts for both mesopore corrugation and complementary porosity. Globally, the present study supports the view that adsorption and desorption in SBA-15 are indeed dominated by geometrical heterogeneities or defects, which contribute to shift the condensation pressure to significantly lower values and to narrow the hysteresis loop. This observation questions the general applicability of commonly used methods for the determination of pore-size distributions, and calls for the development and use of material-specific geometrical models.16,51−53,66,67 From a material’s point of view, the fact that the particular SBA-15 sample considered in the present study has a short corrugation length lC need not be a general characteristic of that material. In particular, adsorption isotherms of the type of Figure 8e,f, which are typical of long corrugation lengths, are often described in SBA-15 materials after long hydrothermal treatments.68−71 Hydrothermal treatments are expected to reduce 5110


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materials features with a short lC but to enhance those with long lC through a Rayleigh instability.56 However, the latter process is much slower than the former because they are both limited by diffusion, and diffusion has to take place over over longer distances in the latter case.72 Adsorption isotherms typical of long lC are therefore expected only after long hydrothermal treatments. Corrugated pores with long corrugation lengths are expected also in a variety of porous materials beyond OMMs. For example, the present analysis is also relevant to adsorption is porous silicon73,74 and alumina75 obtained by nanofabrication. In the context of catalysis also, it has recently been demonstrated through electron tomography that the pores in mesoporous Y-zeolites obtained by steaming or acid-leaching consist of linear mesopores76 with some local constriction and a small tortuosity.77 Yet another example is that of multiwalled carbon nanotube bundles, which are held together by dispersive forces; the inter-nanotube mesopores in the bundles are expected to have slowly variable sections, i.e., very long lC. In this case also, the adsorption isotherms is similar to Figure 8 f, with a very narrow and sloped hysteresis loop.78 Finally, it has to be noted that the present analysis is based on thermodynamic considerations. From our results, there is no reason to question the metastability of the adsorption states, provided the geometrical heterogeneities of the materials are explicitly taken into account. By contrast, the conditions that we have derived for cavitation are necessary but not sufficient. In the conditions of corrugation amplitude and length in which equilibrium cavitation is possible, it is yet unclear in what circumstances metastable cavitation,59 pore blocking, or equilibrium cavitation21 is expected.79 It is, however, likely that the occurrence of these phenomena depends on subtle geometrical characteristics of the material. We hope that our present work can contribute to improve the general understanding of that subject.

■

Table 2. Main Symbols and Variables

Pe(R) PE1PE2 PS1,PS2 q R(z) R1, R2 RG Rm r(z) r1,r2 T t u(x), u(t) Vm VF VP x z ϕm γ λ μ μ̃ Ω Ω̅ Ω0

for (24)

Π(t) ρ(μ̃ ) σR

and the radii ri are the variables over which Ω̅ is minimized. Assuming a piecewise linear function for r(z) is tantamount to approximating the vapor−liquid interface by a series of conical frustums. The first term of eq 5, proportional to the area of the interface is therefore written as πλ VP

N−1

∑

liquid−vapor interface area dimensionless disjoining pressure empirical constants adsorbate free energy density wave-vector probability distribution of the Gaussian pore integral of the disjoining pressure corrugation length lengths of the narrow and wide sections of the simple pore model number of random modes used for the Gaussian pore model relative vapor pressure condensation pressure in a cylindrical pore of radius R evaporation pressure in a cylindrical pore of radius R equilibrium transition pressures of the narrow and wide sections of the pore spinodal transition pressures of the narrow and wide sections of the pore random wave vector used for the Gaussian pore model local pore radius radii of the narrow and wide sections of the simple pore model perfect gas constant average pore radius radial position of the liquid−vapor interface radial position of the liquid−vapor interface in the narrow and wide sections of the simple pore model absolute temperature thickness of the adsorbed liquid film fluid−solid interaction potential molar volume of the liquid adsorbate volume of the pore unoccupied by the liquid total volume of the pore three-dimensional coordinate in the pore linear coordinate along the pore axis random phase used for the Gaussian pore model adsorbate surface tension constant (γVm)/(RGT) chemical potential intrinsic chemical potential grand thermodynamic potential dimensionless grand thermodynamic potential grand thermodynamic potential of the pore filled with liquid disjoining pressure adsorbate density standard deviation of pore radius

P/Po Pc(R)

In principle, the minimization of Ω̅ could be done through an Euler−Lagrange equation,80 which would transform the problem into solving a differential equation for r(z). However, the appearance of capillary bridges would lead to infinite values of r(z), which would make the problem difficult to handle numerically. The direct minimization of a discretized version of eq 5 was therefore preferred. Given a series of discrete abscissae zi and corresponding radii ri, the function r(z) is modeled as a piecewise linear function as

Ω̅ A =

A F(t) F0, l, m, K f(μ̃) g(q) dq

N

Numerical Procedure for Local and Global Minimization of Ω̅

zi ≤ z ≤ zi + 1

meaning

I(r;R) lC L1, L2

APPENDIX

z −z z − zi r(z) = ri i + 1 + ri + 1 zi + 1 − zi zi + 1 − zi

symbol

first occurrence eq eq eq eq eq

1 4 4 1 8

eq 10 eq 7 Figure 10 eq 8 eq 2 eq 11 eq 12 Figure 4 Figure 4 eq 8 Figure 2 Figure 10 eq 3 eq 7 Figure 2 Figure 11

eq 3 Figure 2 eq 1 eq 3 eq 2 eq 1 eq 1 Figure 2 eq 8 eq 1 eq 2 eq 1 eq 1 eq 1 eq 3 eq 3

eq 1 eq 7

The second term of eq 5, proportional to the free volume of the pore is

(ri + ri + 1) (ri − ri + 1)2 + (z i − z i + 1)2

Ω̅ V =

N−1 π ⎡P⎤ ln⎢ ⎥ ∑ (z i + 1 − z i)(ri2 + riri + 1 + ri2+ 1) 3VP ⎣ P0 ⎦ i=1

i=1 (25)

(26) 5111


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The last term, depending on the disjoining forces through F(t), is approximated by a trapezoidal integration over the length of the pore, which leads to π Ω̅ F = VP

configurations, respectively, are illustrated in the top two rows of Figure 15.

N−1

∑

(z i + 1 − z i)(Ii + Ii + 1) (27)

i=1

where Ii is the following integral ∫ 0r dr rF(t) evaluated at point i. Using eqs 4 and 6, the integral can be written as ⎧ 1 − (1 − r /R )1 − m I = R2K (αR )−m ⎨ 1−m ⎩ ⎪

⎪

Figure 15. Various steps involved in the global minimization of Ω̅ , from top to bottom: (1) local minimum starting from a film configuration, (2) local minimum starting from a bridge configuration, (3) same as 2, after possible bridge removal, and (4) same as 3 after possible bubble removal. The values of Ω̅ from top to bottom are: 0.004, 0.010, 0.003, and −0.002. Particular case of a Gaussian pore with Rm = 3.7 nm, σR = 0.6 nm, k = 5, lC = 25 nm, for P/P0 = 0.7.

1 − (1 − r /R )2 − m ⎫ ⎬ + α2l 2F0 − 2−m ⎭ ⎪

⎪

⎡ R ⎤⎧ ⎡ r ⎤⎫ ⎛ r⎞ × exp⎢ −α ⎥⎨1 − ⎜1 − α ⎟ exp⎢α ⎥⎬ ⎣ ⎦ ⎣ l ⎦⎭ ⎝ ⎠ l ⎩ l

(28)

with α = 1/(1 + (dR/dz)2)1/2. Using eqs 25, 26, and 27, and for given zi, Ri, and αi, the grand thermodynamic potential is obtained as a function of the radial positions ri: Ω̅ (r1, r2, ..., rN) for any given pressure P/P0. The minimization proceeds as follows. The points zi are set to be evenly spaced along the pore, with about N = 200 points over a corrugation length lC. For cosine-shaped pores, the total length of the pore L is chosen to be lC; for a Gaussian pore, the domain is extended over 5×lC, which amounts to a total of 1000 points. The minimization starts at low relative pressure, say P/P0 = 0.2, for which the overall shape of the interface is expected to follow closely that of the pore wall;53 the radii are initialized to ri = Ri − t, where t is the solution of F(t) = ln[P/P0] corresponding to the thickness of the layer adsorbed on a flat surface. The values of ri are then optimized using a conjugated gradient algorithm81 with constraints 0 ≤ ri ≤ Ri. The convergence is greatly facilitated by the tridiagonal structure of Hessian matrix. Once convergence has been reached, the pressure is increased repeatedly: for each pressure P(n), the optimization of ri is done by starting from the local minimum obtained at P(n − 1). It may happen that one value of ri, say rb, drops to 0 during the optimization while the neighboring radii have finite values. This occurs if a bridge forms across the pore. Because the function r(z) is expected to be very steep close to zb, the node at zb is replaced by 11 nodes evenly positioned in the interval from zb−1 to zb+1. The optimization is then started again with these extra degrees of freedom, thereby enabling the position of the bridge to be determined more accurately. It has to be stressed that the conjugated gradient algorithm converges to a local minimum of Ω̅ , corresponding to the specific basin of the configuration used as the starting point of the optimization. The configuration reached by the procedure that we have just described may be metastable; it corresponds therefore to the adsorption branch of the isotherm. In order to explore the configuration space more thoroughly and to find the global minimum of Ω̅ , we proceed as follows. In addition to starting from the film configuration (see above) the minimization is repeated starting from a bridge configuration. In the latter case, the initial configuration is created by positioning a bridge at all the local minima of the pore radius R(z), and the optimization proceeds unchanged from there on. The two minima reached from the film and bridge

Clearly, the second optimization may overestimate the actual number of bridges in the global minimum of Ω̅ . The next step consists in considering each bridge individually and in removing those that lead to an increase of Ω̅ compared to the film configuration. The configuration reached after that step is illustrated in the third row of Figure 15. At this stage, it has to be remembered that Ω̅ is normalized to be 0 wherever r(z) = 0 (see eq 3). Therefore, the thermodynamic potential Ω̅ can be decomposed into a sum over all individual bubbles. The last step of the optimization therefore consists in removing those bubbles that contribute to positive values of Ω̅ . The final configuration, corresponding to the global minimum of Ω̅ , is shown at the bottom of Figure 15. From top to bottom, the configurations shown in Figure 15 have the values Ω̅ ≃ 0.004, 0.010, 0.003, and −0.002. The first three are metastable, and the last in indeed stable, compared to the completely filled pore, which has Ω̅ = 0 by construction. Surface Area of Gaussian Pores

The surface area of the Gaussian pore defined by eq 8 and eq 9 can be calculated by decomposing it into thin slices of thickness dz, ranging from za to zb = za + dz, and subsequently letting dz → 0. The lateral area of the infinitesimal frustum is dA = π(R a + R b) (R a − R b)2 + dz 2

(29)

with Ra = R(za) and Rb = R(zb). It has to be noted that Ra and Rb are random variables that depend notably on the particular values of za and zb. By construction, Ra and Rb obey bivariate Gaussian statistics and their joint distribution is therefore G(R a , R b) =

1

2πσ2R 1 − ρ2R ⎡ (R − R )2 + (R − R )2 − 2ρ (R − R )(R − R ) ⎤ a m b m m b m ⎥ R a exp⎢ − ⎢⎣ ⎥⎦ 2(1 − ρ2R )σ2R

(30)

where ρR is the correlation between Ra and Rb, defined as ρR (Δz) = 5112

⟨(R(z + Δz) − R m)(R(z) − R m)⟩ σ2R

(31)


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where the brackets stand for the average value calculated over all possible values of z. The average value of the elementary surface area dA is calculated by combining eq 29 with eq 30 and calculating a double integral over Ra and Rb. This leads to ⟨dA⟩ =

Rm σR

+∞

2π 1 − ρ2R

∫−∞

dη dz 2 + 2η2

⎡ ⎤ −η2 ⎥ × exp⎢ 2 ⎢⎣ 2σR (1 − ρ ) ⎥⎦ R

(32)

where ρR is evaluated for Δz = dz. The general relation between ρR(Δz) and the probability distribution of wave-vectors g(q) is obtained by by replacing eq 8 into eq 31 and integrating over z. The calculation leads to ρR (Δz) =

1 N

N

∑

cos[qnΔz] =

n=1

∫0

∞

Figure 16. Relation between σR2 ⟨q2⟩ and the roughness factor of the Gaussian pore, calculated through eq 36.

from that of dense silica (2 g/cm3) assuming that it is 30% microporous, a value that is supported by several independent estimations in SBA-15.64,65,83

cos[qΔz]g (q) dq

■

(33)

where the second equality holds in the limit N → ∞. For our present purpose, only infinitesimal values of Δz need to be considered. In this case, the cosine can be developed to the second order in Δz, which leads to 1 ρR ≃ 1 − ⟨q2⟩(Δz)2 (34) 2 2 2 where ⟨q ⟩ is the average value of q , or equivalently the second moment of g(q). Introducing the Taylor development eq 34 into eq 32, one finds ⟨dA⟩ = 2πR m dzF[σ2R ⟨q2⟩]

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research was initiated during a stay in the Department of Inorganic Chemistry and Catalysis of Utrecht University; it is a pleasure to acknowledge stimulating discussions with Pr. Krijn de Jong, Dr. Petra de Jongh, and Dr. Heiner Friedrich. This work is supported by the Fonds de la Recherche Scientifique (F.R.S.-FNRS, Belgium) through a research associate position.

(35)

where F[x] is the following function F[x] =

2 π

∫0

∞

dη 1 + x η2 exp[−η2 /2]

■

(36)

F[σR2 ⟨q2⟩]

Equation 35 shows that can be thought of as the roughness factor82 of the Gaussian pore, defined as the ratio of the surface areas of the corrugated and smooth pores with same average radius Rm. The function F[x] is plotted in Figure 16. In the case of γ-distributed values of q, with θ expressed in terms of the corrugation length as 2π/lC = (k − 1)θ, one has ⎛ σ ⎞2 k(k + 1) σ2R ⟨q2⟩ = (2π)2 ⎜ R ⎟ ⎝ lC ⎠ (k − 1)2

(37)

2πR mF[σ2R q2 ] 2 ( 3 a2 /2 − π(R m + σ2R ))ρsolid

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The relevant values for the realizations in the inset of Figure 14 are σR = 0.8 nm, lC = 4 nm, and k = 10, which leads to σR2 ⟨q2⟩ ≃ 2.1 This results in a roughness factor around 1.6. In order to convert this value to a specific surface area in the case of SBA15, it is necessary to normalize the surface area by the mass of silica. Calculating the surface of the hexagonal unit cell as √3a2/2, with a being the lattice parameter, and the average cross section area of the pore as π(Rm2 + σR2 ), the specific surface area is found to be Sm =

AUTHOR INFORMATION

(38)

where ρsolid is the density of the solid phase. Using the values a = 10.8 nm,39 Rm = 3.7 nm, σR = 0.8 nm, and ρsolid = 1.4 g/cm3 leads to Sm ≃ 470 m2/g. The value of the density is calculated 5113


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