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Pore Structure and Fluid Sorption in Ordered Mesoporous Silica. II. Modeling Dirk Mu¨ter,† Susanne Ja¨hnert,‡ John W. C. Dunlop,† Gerhard H. Findenegg,‡ and Oskar Paris*,†,§ Department of Biomaterials, Max Planck Institute of Colloids and Interfaces, D-14424 Potsdam, Germany, Institute of Chemistry, Stranski Laboratory, Technical UniVersity Berlin, D-10623 Berlin, Germany, and Institute of Physics, UniVersity of Leoben, A-8700 Leoben, Austria ReceiVed: NoVember 14, 2008; ReVised Manuscript ReceiVed: July 8, 2009
The nanostructure of SBA-15 ordered mesoporous silica is modeled by considering a full crystallite about 500 nm in size. The lattice sites of a 2D hexagonal lattice are occupied by circular mesopores that exhibit a size distribution and a rough pore wall. In addition, the walls of the ordered pores contain randomly distributed disordered pores. This model is introduced to quantitatively reproduce the experimental X-ray scattering profile from an evacuated SBA-15 sample considering both Bragg diffraction from the ordered pore lattice and the diffuse small-angle scattering from the disordered pores. In contrast to a previous form factor model which analyzed the Bragg diffraction peaks only, the present model reproduces correctly the total porosity of the material in good agreement with nitrogen sorption. The adsorption behavior of the model crystallite is studied by introducing simple filling rules for the ordered and the disordered porosity. The filling of the disordered pores as well as film formation and capillary condensation in the ordered pores are reproduced by the model. Good agreement between the calculated and measured scattering profiles along the adsorption isotherm of a contrast-matching fluid is found. In particular, the change of the integral diffuse scattering intensity with increasing vapor pressure of the fluid is quantitatively reproduced by the model. The integrated intensities from the Bragg reflections show however some quantitative discrepancies. Comparing the density profiles derived from the present lattice model with the one obtained from direct form factor fitting of the experimental data, we conclude that the experimental system is also subjected to fluctuations of the fluid density on the pore walls. Introduction Ordered mesoporous materials obtained from soft templating of self-assembled amphiphilic structures can exhibit high degrees of long-range order within their pore lattice. Experimentally, structural information on such systems can be deduced from small-angle X-ray scattering. Least-squares fitting of the intensities of the observed diffraction peaks permits the development of detailed structure models via Fourier transformation of trial single pore electron density distributions.1-6 However, it is also well-known that polymer-templated ordered mesoporous silica materials comprise additional pores, due to polymer chains protruding into the pore walls and due to processes occurring during the thermal post-treatment and template removal. SBA15 silicas for instance exhibit a significant amount of disordered pores within the mesopore walls,7 which may or may not exhibit the long-range order of the cylindrical mesopores. In a preceding paper,1 we reported on a method to estimate the contributions of ordered and disordered pores to the total pore volume of SBA-15. The porosity connected with the ordered cylindrical pores was determined from the X-ray diffraction peaks of the evacuated SBA-15 sample. The method is based on analytical models for the form factor of single mesopores and their porous corona and corrugated surface. The form factor models provide an excellent fit of the diffraction data and yield the pore size and specific pore volume of the * To whom correspondence should be addressed. E-mail: oskar.paris@ unileoben.ac.at. † Max Planck Institute of Colloids and Interfaces. ‡ Technical University Berlin. § University of Leoben.
ordered pores. From this and the total pore volume determined by nitrogen adsorption, it was then possible to determine the pore volume of the disordered pores. In the experimental study of part I (DOI 10.1021/jp8100392), we also presented in situ X-ray scattering data for the adsorption of a density-matching fluid in the pores of SBA-15. The volume of fluid adsorbed in the ordered cylindrical pores was estimated with the same form factor models which were introduced for the determination of the ordered pore volume. In this way, it was possible to determine the adsorption isotherm of the fluid in the ordered pores. Indirect information about the adsorption in the disordered pores could be extracted from the diffuse scattering contribution and its decrease with progressing fluid adsorption. However, due to the ill-defined pore size and pore shape of these disordered pores, it was not possible to calculate the amount of fluid adsorbed in these pores in a similar way as for the ordered pores. It is well-known from the literature that for fully random porous systems stochastic reconstruction techniques can be applied to create structures with the same statistical characteristics as those extracted from X-ray or neutron scattering.8-10 However, these techniques have not been adapted for the description of ordered porous systems with walls containing disordered pores. In the present paper, we aim to quantitatively model in situ X-ray scattering data by considering a whole SBA15 crystallite. We adopt the structural parameters for the twodimensional hexagonal pore lattice of cylindrical pores with a porous corona and corrugated pore walls from part I (DOI 10.1021/jp8100392). By adding additional uncorrelated pores which do not exhibit the long-range order of the lattice, we
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intend to describe the diffuse scattering contribution as well as the disordered porosity in a quantitative way. Moreover, by applying simple rules for the filling of the ordered and disordered porosity within the model crystallite, we aim at a better understanding of the adsorption process of fluids in SBA-15. Models Lattice Model. Our approach is to construct a real-space lattice model for a typical SBA-15 crystallite which includes disordered pore regions superimposed on the ordered pores. SBA-15 consists essentially of parallel cylindrical mesopores arranged on a 2D hexagonal lattice. Since the pores are very long as compared to their diameter, it is sufficient to use a 2D model. By calculating the squared Fourier transform of this 2D model crystal, it is possible to derive scattering curves which can be directly compared to the experimentally measured scattering profiles. The numerical Fourier transform of the model crystallite described in more detail below was calculated using the C programming library “FFTW”.11 The 2D squared Fourier transform was then azimuthally averaged to produce the 1D scattering profiles. For comparison with the experimental data, we note that I3D(q) ∝ I2D(q)/q for long cylinders,12 where I(q) is the scattering intensity as a function of reciprocal space coordinate (length of the scattering vector) q. From experimental investigations, the typical size of SBA15 crystallites in the direction normal to the pore axes is known to be ≈0.6 µm.13 Recalling the size of the unit cell of the hexagonal lattice (a0 ) 11.52 nm1), the whole crystallite consists of about 2800 depicted ordered pores if they are placed on a square shaped crystallite. Accordingly, we describe the crystallite by a 2D matrix with the local electron density as matrix elements. The matrix consists of 7806 × 7806 pixels, where the pixels can have the values 1 (black) for empty space and 0 (white) for the silica matrix and the (matching) fluid. The pore centers are placed on the sites of a hexagonal lattice with a lattice parameter of 156 pixels. This leads to a resolution of 13.4 pixels/nm, which provides sufficient resolution to reproduce all of the essential features measurable in the X-ray scattering experiments. The crystallite is now constructed in two steps. In the first step, ordered pores exhibiting a porous corona and a corrugated surface are placed on the lattice sites. Some disorder is introduced already in this step via a pore size distribution of the ordered pores and a small random deviation of their centers from the lattice sites. In the second step, we implement additional disorder by randomly placing disordered pores into the mesopore walls. This step is motivated by the fact that SBA15 is strongly microporous, presumably even with interconnecting channels between the mesopores.7 To realize the first step, we use the results of the analytical density-gradient model for the single ordered pore developed in part I (DOI 10.1021/jp8100392).1 This model is characterized by a constant density (bulk silica) for r > R0, followed by two linearly changing density regions in the pore walls between R0 > r > R1 (porous corona) and R1 > r > R2 (corrugated pore surface), where r is the radial distance from the pore center. The fitting of this model to the experimentally measured Bragg reflections provided the average radial electron density profile F(r) for a single ordered pore. To implement this information into the numerical model, we convert the radial density distribution into a probability function which assigns each pixel a probability to be set empty ()black) based on the radial distance to the pore center, where F0 is the matrix density.
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Pempty(r) ) 1 -
F(r) F0
(1)
This results in density values from 1 for r < R2 to 0 for r > R0. Figure 1 depicts an exemplary mesopore constructed by this rule, demonstrating that this procedure creates a rough pore surface and a porous corona with decreasing density of small pores as the distance from the pore center increases. This process can lead to a small amount of isolated black or white pixels in a 2D image which would depict either inaccessible porosity (black pixels) or a mechanically impossible formation of isolated silica patches (white pixels). This problem is however easily resolved by considering that the real system is 3D with most of the disordered pores and silica patches connected via the third dimension. The nonideality of the pore lattice was taken into account in two ways. First, the centers of the ordered pores were allowed to deviate to some extent from their ideal positions, assuming a Gaussian distribution with a full width at half-maximum of 6% of the lattice parameter. This introduces essentially a static Debye-Waller factor into the model. Second, a size distribution of the ordered pores on the pore lattice was introduced. This size distribution was estimated from the pore condensation region of the adsorption isotherm of dibromomethane (DBM) in SBA-15 (Figure 1 in part I (DOI 10.1021/jp8100392)1), using the functional relation of the Kelvin equation in the form
P(R*) ) -
A ln(p/p0)
(2)
where P(R*) is the probability of finding a pore of radius R*, p/p0 is the relative pressure of DBM in the pore condensation region (0.67 e p/p0 e 0.78), and A is a normalization constant to make ∫P(R*) dR* ) 1. This leads to an approximately Gaussian pore size distribution with a full width at halfmaximum of roughly 10% of the mean pore diameter. To account for the additional diffuse scattering, randomly distributed disordered poresswhich we assume to be circularly shaped for simplicityswere placed into the ordered pore walls. We note that in SBA-15 these disordered pores may deviate from a circular shape, being rather wormlike channels connecting individual mesopores. Evidence for this comes from the synthesis process of SBA-15 and also from the fact that noncollapsing carbon structures can be fabricated from SBA15.7 However, this would require introducing at least two more freely adjustable parameters (pore shape and orientation) in our model. As we will show in the Results and Discussion section, circular disordered pores reproduce the measured scattering profiles well, and therefore, there is no need to introduce more complex pore geometries for our purpose. One might argue that this situation would lead to inaccessible disordered pores, since our system is below the percolation threshold for randomly placed pores. However, this is not the case, since the disordered pores need not be all interconnected but only be connected with an associated mesopore. We have calculated the fraction of disordered pores accessible from an adjacent mesopore via a simple algorithm. We found less than 15% of isolated disordered pores (i.e., disordered pores that are not directly connected to an ordered pore) corresponding to about 3% of inaccessible total porosity. As this portion will even be lower in 3D, fluid accessibility of the disordered pores is therefore not an issue to be considered as critical given the limits in experimental accuracy.
Nanostructure of SBA-15 Ordered Mesoporous Silica
Figure 1. Cross section of a model ordered pore with corona and corrugated pore wall (see text).
J. Phys. Chem. C, Vol. 113, No. 34, 2009 15213 function of the relative pressure of the fluid is accessible. In contrast, the X-ray scattering profiles from an in situ adsorption experiment of an ideally matching fluid contain in principle all of the details about the “pore structure” (i.e., the remaining empty space) for any filling fraction. For a contrast-matching fluid, adsorption in the lattice model can be treated simply by turning black pixels into white ones. The number of additional white pixels is proportional to the amount of adsorbed fluid in the experiment which is connected to the vapor pressure via the sorption isotherm. In order to compare the numerical results with the experiments where the control parameter was the relative vapor pressure, the adsorption isotherm of dibromomethane in SBA-15 (see part I, DOI 10.1021/jp81003921) was used to transform pore filling fractions into relative vapor pressures. To model the scattering profiles for different fluid filling fractions, we need to apply rules on how the pores are filled. We do this by introducing a local curvature driven filling mechanism for the disordered pores, and use the Kelvin equation for the ordered pores. During the whole filling process, the Kelvin equation is used to determine whether condensation should occur in a given pore by calculating the current radius of all ordered pores from their area. When an ordered pore radius falls below the critical radius R*, this pore will be filled, which means that all of the ordered pore pixels within a cutoff radius (corresponding to R1 in the analytical density-gradient model in part I, DOI 10.1021/jp8100392) will be set white. Since the Kelvin equation cannot be applied to the disordered pores due to their small size and irregular shape,15 we use a simple heuristic method to fill the disordered pores and the porous corona. A randomly chosen black pixel on the lattice point Ai,j is assigned a probability to be set white on the basis of the number of white pixels in its surrounding area: i+c
Figure 2. Cut-out of the SBA-15 model crystallite showing the 2Dhexagonal ordered pore lattice with additional random disordered pores.
We expect the disordered pores to have an irregular surface similar to the ordered pores. To account for this surface irregularity, each pixel at a distance r between the center of the disordered pore (position 0) and its outermost edge (position r0) is assumed to have a probability Pdis(r) to be black, using the following empirical relation:
Pdis,empty(r) ) 1 (r < r0 /3) ) 1 -
3r - r0 (r0 /3 < r < r0) 2r0 (3)
The area fraction and the size distribution of the disordered pores are used as control parameters for the modeling of the diffuse scattering. We estimate a lower bound for the size distribution by the requirement that the disordered pores must be accessible to dibromomethane (DBM). Using a volume of 0.116 nm3 (part I, DOI 10.1021/jp81003921) per molecule and assuming spherical geometry, this leads to a minimum radius of about 0.30 nm. Just summing up the atomic radii of two bromine atoms and one carbon atom14 yields essentially the same result. A part of the resulting lattice model can be seen in Figure 2. Pore Filling in the Lattice Model. From an adsorption isotherm, only the adsorbed amount (or filling fraction) as a
P(1f0)i,j ) 1 -
j+c
∑ ∑
a)i-c b)j-c
Aa,b (2 · c + 1)2
(4)
If a random number between 0 and 1 is smaller than the calculated probability of the chosen pixel, it will be set white. The control parameter c determines the size of the surrounding area to be taken into account: if c ) 1, only nearest neighbor pixels are considered, if c ) 2 next-nearest neighbors are also considered, and so on. With this definition, the probability of filling is high for small patches of black pixels, as in the corona and the disordered pores (see Figure 2), and is zero within the ordered pores as long as c is small. For c ) 1, this procedure works much like a simplified version of the KRT lattice gas model for a wetting fluid.16 However, since we expect the variation of the parameter c to have an influence on the filling process, we use it as a control parameter. Results and Discussion Dibromomethane (DBM), a fluid having the same electron density as SBA-15, was used as the adsorptive in the in situ sorption experiment reported in part I (DOI 10.1021/ jp8100392).1 Figure 3a shows four exemplary scattering profiles for different pressures of DBM (out of 67 measured in total along the adsorption isotherm). Up to 10 Bragg reflections from the 2D hexagonally ordered cylindrical pores can be recognized depending on the relative pressure. The nonmonotonic changes in the relative height of the Bragg reflections as well as in the diffuse scattering with pressure are obvious, and have been discussed in part I of this work (DOI 10.1021/jp8100392). We
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Figure 3. Experimental (a) and simulated (b) scattering curves for different relative vapor pressures (p/p0). Inset: Enlargement of the (10) peak on a linear scale.
Figure 4. Integrated intensities in dependence on the relative vapor pressure p/p0 from the pore filling model (b) and from the in situ X-ray scattering experiment (3) for the diffuse scattering (a) and the Bragg diffraction peaks (b).
refrain from a direct fitting of the scattering profiles but rather aim to reproduce the scattering curves qualitatively for the whole series of pressures by varying the few parameters of the model. As outlined in the previous section, the ordered pores with their porous corona and rough surface were constructed according to the results from the density-gradient model and their size distribution was estimated from the sorption isotherm. Therefore, the only free parameters in our model are the volume fraction of disordered pores and their size distribution (defined by r0, see eq 3), and the value of the parameter c in eq 4). An area fraction of randomly placed disordered pores in the twodimensional representation of the pore, φ2D dis ) 0.19, quantitatively reproduced the ratio of the integrated intensities from Bragg diffraction and diffuse scattering for the evacuated specimen. We found that a size distribution of the disordered pores in the range just above the minimum value, that is 4 to 6 pixels (0.30-0.45 nm), in combination with c ) 5 delivers the best agreement with the course of the diffuse scattering (see Figure 4), as will be discussed below. Figure 3b shows the calculated scattering profiles from the lattice model for the same relative pressures as Figure 3a. Due
to the different dimensionality of the experiment (3D) and the lattice model (2D), the spherical and radial sum, respectively, are displayed. It is seen that the Bragg reflections from the model crystal are much sharper than the measured ones. This is due to instrumental broadening which was not taken into account for the model crystal. Despite the width of the Bragg peaks, the experimental profiles in Figure 3a are quite well reproduced by the calculated profiles in Figure 3b. In particular, we note that the relative heights of all Bragg reflections are perfectly reproduced for the empty sample (p/p0 ) 0). During the course of pore filling, the following consistencies between measured and calculated scattering profiles are recognized: (1) The first-order Bragg reflection at q ) 0.623 nm-1 first increases at low relative pressures and then decreases steeply due to capillary condensation of the vapor above p/p0 ≈ 0.67. The behavior at low p/p0 can be attributed to the filling of disordered pores, leading to an increase in the density of the ordered pore walls. Since the prefactor to the intensity of the Bragg reflections is given by the electron density of the matrix, this increase is well understood.
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(2) The intensities of the higher-order Bragg reflections change in a nonmonotonic manner within the interval 0 < p/p0 < 0.67, and start to decrease systematically at the onset of pore condensation. This behavior is attributed to the formation of a liquid-like film at the ordered pore walls, followed by the filling of the ordered pores via capillary condensation. Such changes of the Bragg reflections are well-known from the sorption of other fluids in SBA-15, and were analyzed by modeling the pore form factor already in previous work.1,2,4 It is particularly noteworthy that our simple mechanism for the filling of the disordered porosity in the numerical lattice model can reproduce the feature of film formation almost perfectly. (3) The higher-order Bragg reflections (q > 2 nm-1) are hardly seen for the empty sample, but they become clearly visible at higher pressures. Generally, the diffuse scattering at large q (q > 0.8 nm-1) decreases monotonically with increasing pressure. This must be attributed to the filling of pore space which is not associated with the longrange order of the ordered pore lattice and is therefore due to the filling of the additional disordered pores. (4) The diffuse scattering at very small q (q < 0.5 nm-1) increases strongly at the onset of pore condensation, reaches a maximum, and drops back again at completion of pore condensation. This behavior is related to transient disorder introduced on the pore lattice due to the selective filling of the ordered pores during pore condensation, as already discussed in part I (DOI 10.1021/jp8100392).1 In order to catch some of these features more quantitatively, we have calculated the integrated intensities from the experimental data and from the lattice model (∫Iq2 dq and ∫Iq dq representing the integrated intensities in three and two dimensions, respectively). The integrated intensities are shown separately for the diffuse scattering contributions (Figure 4a) and for the Bragg reflections (Figure 4b). The data for the empty sample (p/p0 ) 0) confirm that the implementation of an area fraction of φ2D dis ) 0.19 of additional disordered pores in the lattice model leads to nearly the same ratio of Bragg scattering to diffuse scattering of the integrated intensities (experimental, 0.720; simulated, 0.722). When relating the amount of disordered pores to the experimental system, we have again to consider the different dimensionality of the experiment and the simulation. Assuming the disordered pores to be spherical in 3D and isotropically arranged within the ordered pore walls, 2D ) we can deduce the connection between the area fraction (φdis 3D and the volume fraction (φdis ) of the disordered pores via the 3D 2D ≡ φord ): volume/area fraction of the ordered pores φord (φord
3D φdis )
2D 3/2 (φdis )
(1 - φord)1/2
(5)
2D ) 0.19 and the porosity arising from the ordered Using φdis pores, φord ) φord ) 0.554, as determined in part I (DOI 10.1021/ jp8100392),1 we obtain a volume fraction of disordered pores 3D ) 0.124 and a total pore volume fraction of φtot ) 0.68, of φdis in excellent agreement with the experimental values determined by X-ray diffraction and nitrogen adsorption, viz., φdis ) 0.132 and φtot ) 0.686 (see Table 3 of part I, DOI 10.1021/jp8100392). This demonstrates that our model is indeed capable of describing the total porosity on the basis of parameters derived from the experimental scattering curves alone.
Figure 5. Comparison between the total integrated intensity from the lattice model (b), from the X-ray scattering experiment (3), and the one derived from the sorption isotherm (s) in dependence on the relative vapor pressure.
As seen in Figure 4a, the development of the diffuse scattering during the pore filling process (p/p0 > 0) agrees well with the experimental data over the whole pressure range. This indicates that our attempt to model the filling of the disordered pores is quite successful. The model also correctly predicts the pronounced maximum due to “Laue scattering” from the ordered pore lattice at capillary condensation. As mentioned above and in part I (DOI 10.1021/jp8100392), the maximum in the integrated intensity is caused by a strong increase of the diffuse scattering at small q (q < 0.5 nm-1). This behavior suggests that the ordered pores are indeed filled randomly according to their size. In contrast to the diffuse scattering, the pressure dependence of the integrated intensity from the Bragg reflections in Figure 4b is not so well reproduced by the model. Only for low pressures (p/p0 < 0.1), the integrated intensity in the numerical lattice model agrees with the experiment. At higher pressures, the experimental values of the integrated intensity of the Bragg reflections stay practically constant up to the onset of capillary condensation, but the respective values resulting from the model continue to increase in a pronounced manner. Above the onset of capillary condensation (p/p0 > 0.67), both the experimental and calculated integrated intensities of the Bragg reflections drop to zero as expected. The increasing integrated intensity of the Bragg peaks found in the lattice model suggests that the crystalline order in the system increases by fluid sorption. Such a behavior is indeed expected if we consider that the filling of the disordered pores and the formation of a liquid film should lead to a better defined circular cross section of the ordered pores, thus enhancing the crystalline order of the system. However, the experimental data rather indicate that the crystalline order reflected by the integrated intensity of the Bragg reflections remains nearly unchanged during the filling of the disordered porosity and film formation. Figure 5 compares the total integrated intensities from the present model with the experiment and with the integrated intensity calculated from the adsorption isotherm.1 For a matching fluid, the connection between the volume fraction of silica plus adsorbed fluid φ(p/p0) at a relative pressure p/p0 is
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Figure 6. Density distribution from the form factor model and the lattice model with different values for the parameter c for the empty pore (p/p0 ) 0.00) (a) and for a pressure value just below the pore condensation pressure (p/p0 ) 0.67) (b).
related to the integrated intensity I˜(p/p0) by I˜ ) K1φ(1 - φ) + K2, where K1 and K2 are constants. It is obvious from Figure 5 that there is quite good agreement between the intensity from the lattice model and the intensity derived from the sorption isotherm on the basis of this relation. However, both differ considerably from the measured integrated intensities. From Figure 4, we conclude that this discrepancy is mainly due to the behavior of the Bragg scattering, as the experimental diffuse scattering is well reproduced by the model. The difference in pressure dependence between experimental and calculated values of the integrated Bragg intensities could not be improved noticeably by changing the model parameters within physically meaningful limits. It is noteworthy in this context to recall one major result of part I (DOI 10.1021/ jp8100392), i.e., the radial density distribution of a single ordered pore obtained from fitting the density-gradient model to the Bragg reflections.1 Figure 6 shows the density profiles for two selected pressures from these fits, together with the normalized density profiles extracted from the numerical lattice model. As expected from the construction of the crystallite, the density profiles are practically identical for the empty sample (Figure 6a). The width of the experimental density profile remains almost unchanged for higher pressures (Figure 6b), which suggests that the surface roughness in the corona is not markedly smoothed by the adsorbing film. This fact was already noted in the first part of this work (DOI 10.1021/jp8100392),1 and was tentatively attributed to incomplete wetting of dibromomethane in SBA-15 silica. In contrast to the experimental density profile, the profile obtained from the lattice model becomes considerably sharper for higher fluid pressures, suggesting that the filling rule for the disordered porosity favors the smoothing of the pore walls. It should be noted that the degree of smoothing depends on the parameter c in eq 4. Figure 6 illustrates this dependence by showing the mean density profile for two values of the parameter c. The smoothing of the pore walls is even more pronounced for c ) 1 as compared to c ) 5. However, for no realistic values of c, satisfactory agreement between the experimental and calculated density profiles could be obtained without sacrificing the agreement of the pressure dependence of the diffuse scattering. We conclude that our model is not able to reproduce the pressure dependence of the integrated Bragg peak intensity
(Figure 4). Furthermore, there is a discrepancy between the measured total integrated intensity as compared to the one calculated from the measured sorption isotherm (Figure 5). This suggests that some features of the experimental Bragg scattering are neither covered by our (static) model nor covered by the gravimetric adsorption isotherm. We propose that the intensity of the experimentally measured Bragg reflections is strongly reduced by dynamic fluctuations of the fluid film adsorbed on the pore walls, and that these fluctuations increase as the fluid pressure increases. Such fluctuations can explain qualitatively both the missing sharpening of the experimental density profile as described above and the diminishing of the intensity from the Bragg peaks due to dynamic disorder. Since such disorder leads to thermal diffuse scattering (which is subtracted from the experimental data as incoherent scattering before the integrated intensity is calculated), a systematically lower integrated intensity would be expected as compared to the static case, in agreement with Figure 4. Finally, we note that “nonsharp” density profiles of the liquid adsorbed film have already been reported by other groups4 but have not been attributed to dynamic film fluctuations so far. Conclusion We have developed a structural model for an SBA-15 crystallite consisting of hexagonally ordered pores with a porous corona and a corrugated surface as well as randomly distributed smaller pores within the ordered pore walls. This simple geometrical model is able to quantitatively reproduce the experimental X-ray scattering profile from an evacuated SBA15 sample, and delivers the correct total porosity measured independently by nitrogen sorption. Although the model does certainly not take the full compexity of the SBA-15 nanostructure into account, the agreement with the experimental data is very good. By introducing simple filling rules for disordered pores and ordered pores, we were able to describe the filling of the disordered porosity, film formation, and capillary condensation in the ordered pores. The simulated scattering profiles as a function of fluid vapor pressure along the adsorption isotherm reproduce all of the important experimental features, and thus provide a better understanding of the details of the adsorption
Nanostructure of SBA-15 Ordered Mesoporous Silica process. In particular, it is demonstrated that the filling of the disordered porosity and film formation are concurrent processes in SBA-15, and that the ordered pores are successively filled according to their size in agreement with the classical Kelvin equation. The model predicts a sharpening of the density distribution at the ordered pore walls with increasing pressure, which would correspond to a smoothing of the rough walls by the adsorbed film. Since such a smoothing is not found for the experimental system, it is concluded that fluctuations of the fluid film produce a similar type of roughness as the corrugated surface of the empty pores. This interpretation is corroborated by the discrepancy of the experimental integrated intensity as compared to the one calculated from an independently measured adsorption isotherm. It is possible that this behavior is caused by the fact that the fluid studied in this work is at the border from complete to partial wetting of the silica surface. In order to check this supposition, it would be of interest to perform a similar study with a complete wetting fluid that is forming laterally uniform (multilayer) films at the pore walls. Acknowledgment. The authors acknowledge financial support from the “Deutsche Forschungsgemeinschaft” (DFG) within the Collaborative Research Center Sfb 448, projects B1 and B14, and from the Max Planck Society.
J. Phys. Chem. C, Vol. 113, No. 34, 2009 15217 References and Notes (1) Ja¨hnert, S.; Mu¨ter, D.; Prass, J.; Zickler, G.; Paris, O.; Findenegg, G. H. J. Phys. Chem. C. DOI: 10.1021/jp8100392. (2) Zickler, G. A.; Ja¨hnert, S.; Wagermaier, W.; Funari, S. S.; Findenegg, G. H.; Paris, O. Phys. ReV. B 2006, 73, 184109. (3) Albouy, P. A.; Ayral, A. Chem. Mater. 2002, 14 (8), 3391–3397. (4) Hofmann, T.; Wallacher, D.; Huber, P.; Birringer, R.; Knorr, K. Phys. ReV. B 2005, 72, 064122. (5) Impe´ror-Clerc, M.; Davidson, P.; Davidson, A. J. Am. Chem. Soc. 2000, 122, 11925–11933. (6) Solovyov, L. A.; Kirik, S. D.; et al. Microporous Mesoporous Mater. 2001, 44, 17–23. (7) Ryoo, R.; Ko, C. H.; Kruk, M.; Antochshuk, V.; Jaroniec, M. J. Phys. Chem. B 2000, 104, 11465–11471. (8) Kikkinides, E.; Stefanopoulos, K.; Steriotis, T.; Mitropoulos, A.; Kanellopoulos, N.; Treimer, W. Appl. Phys. A 2002, 74, 954–956. (9) Salazar, R.; Gelb, L. Langmuir 2007, 23, 530–541. (10) Eschricht, N.; Hoinkis, E.; Madler, F.; Schubert-Bischoff, P.; RohlKuhn, B. J. Colloid Interface Sci. 2005, 291, 201–213. (11) Frigo, M.; Johnson, S. G. FFTW 3.0.1, www.fftw.org. (12) Fratzl, P. J. Stat. Phys. 1994, 77, 125–143. (13) Schreiber, A.; Ketelsen, I.; Findenegg, G. Phys. Chem. Chem. Phys. 2001, 3, 1185–1195. (14) Slater, J. C. J. Chem. Phys. 1964, 41, 3199–3204. (15) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: 1982; p 113. (16) Kierlik, E.; Rosinberg, M.; Tarjus, G.; Viot, P. Phys. Chem. Chem. Phys. 2001, 3, 1201–1206.
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