A Novel Approach To Calibrate Mesopore Size from Nitrogen

Jul 18, 2013 - Conventional BJH mesopore determination from the adsorption branch is found to underestimate ... Maxim S. Mel'gunov , Artem B. Ayupov...
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A Novel Approach To Calibrate Mesopore Size from Nitrogen Adsorption Using X‑ray Diffraction: An SBA-15 Case Study Michael A. Smith,* Michael G. Ilasi, and Alexander Zoelle Department of Chemical Engineering, Villanova University, 800 Lancaster Avenue, Villanova, Pennsylvania 19085, United States ABSTRACT: The mesoporous silicate SBA-15 was prepared with a range of pore sizes by adjusting synthesis conditions, primarily tetraethyl orthosilicate to Pluronic P123 ratios and calcination temperature. Materials were characterized by SAXS to determine unit cell dimension and diffraction intensities, and by nitrogen adsorption to obtain mesopore diameters using the BJH and DFT methods. We modeled the electron density distribution and then used this to calculate the structure factor and theoretical diffraction intensities. Model results show points where the structure factor equals zero for the 2-D hexagonal system, corresponding to peak extinctions in the SAXS diffraction patterns. We compare model results to experimental data to establish biases in the mesopore diameters determined from nitrogen adsorption. Conventional BJH mesopore determination from the adsorption branch is found to underestimate pore diameters by 15% or ∼0.9 nm, BJH analysis from the desorption branch underestimates pore diameter by 40% or 1.9 nm, and DFT analysis is found to marginally overestimate pore diameter by 10% or 0.7 nm. Our results provide estimates for the biases in pore diameters from BJH analysis larger than bias estimates obtained by other means; more significantly, these results also reveal that DFT analysis overestimates mesopore diameters by 10%.

1. INTRODUCTION The effective utilization of porous materials used as adsorbents or catalyst supports depends upon an accurate assessment of pore structure. Gas adsorption using nitrogen (or less frequently argon) provides information on surface area, pore volume and pore diameters. For such materials the most widely used approach to evaluate mesopore size distribution from nitrogen isotherms employs the Barrett, Joyner and Halenda (BJH) methodology.1,2 Due to their regular and finely tunable pore geometry, mesoporous silicates such as SBA-15 and MCM-41 have become popular model systems for the study of adsorption3−5 and as supports in catalysis studies.6−10 Since their development, investigators have used the regular and relatively uniform nature of the cylindrical pore structure as a basis for the assessment and development of improved methods for pore size determination. It is now generally accepted that the BJH method applied to the desorption branch underestimates pore diameters;5,11−13 see also Thommes contribution.14 Notable among the improved methods are the Kruk, Jaroneic and Sayari (KJS) method, an empirical correction to the BJH procedure15−18 and methods based on nonlocal density functional theory (NLDFT or simply DFT).5,12,19 The BJH method is based upon the Kelvin equation which describes the equilibrium relationship between a curved liquid surface and vapor pressure. For a hemispherical meniscus in a closed pore, the relationship is given by: ⎛ p⎞ 2γVL 1 ln⎜⎜ ⎟⎟ = − p RT r ( ⎝ 0⎠ p − t)

liquid molar volume, rp the pore radius, and t the thickness of a liquid-like adsorbed film that accumulates prior to capillary condensation. The thickness of the adsorbed film t is typically given by the Frenkel Halsey Hill (FHH) equation: ⎛ p⎞ ⎛ n ⎞−1/ S ln⎜⎜ ⎟⎟ = −k ⎜ ⎟ ⎝ nm ⎠ ⎝ p0 ⎠

t = tm

(2b)

where k and s are taken as parameters fit to the multilayer adsorption on a non-microporous surface, n and nm are the molar coverage and molar coverage of a monolayer, and tm the monolayer thickness. While the surface tension and molar volume are typically left at default bulk-fluid values, alternative equations and different parameters for film thickness have been proposed. These selections can have an effect on reported results; however, rarely are the film thickness equations and parameters used in a given study reported. For an in-depth discussion of these considerations the reader is referred to Gregg and Sing;2 see also the discussion in Galarneau.11 Finally for MCM-41, isotherms typically have little to no hysteresis while SBA-15 exhibits a classic H1 hysteresis loop in the adsorption isotherm. Where hysteresis exists, the BJH method can been applied to either the adsorption or desorption branches of the isotherm.2

(1)

Received: January 9, 2013 Revised: July 18, 2013 Published: July 18, 2013

where p0 is the normal saturation pressure of the adsorbed gas, γ the surface tension of the liquid adsorbate, VL the adsorbate © 2013 American Chemical Society

n nm

(2a)

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measurements of microporosity in samples, and in the case of BJH, the specific equations and procedures employed to evaluate film thickness. Methods to evaluate these procedures that minimize such assumptions would be beneficial. Last, given the historical primacy of the BJH method, an accurate assessment of its inherent biases, especially in the range of typical SBA-15 pore diameters where there is greater uncertainty about pore size would be valuable. X-ray diffraction provides information on the distribution of electron density in the material, that is, structure and size of the unit cell, atomic distributions, etc. XRD reveals both SBA-15 and MCM-41 have pores arranged in a 2-D hexagonal pattern (space group p6mm) and thus show 10, 11, 20, 21, 30 and 31 peaks in the diffraction patterns. The intensity of higher order peaks, or the ratio of higher order peak intensity to the intensity of the 10 peak is commonly used to infer long-range order.20−23 A well-known outcome of X-ray scattering theory is that certain symmetry conditions in the atomic positions of crystalline materials lead to zeros in the structure factor and hence to specific reflection absences in XRD patterns characteristic of certain space groups.24 Zeros in the magnitude of the structure factor arise not only from symmetries in crystalline materials but can also arise from symmetries in electron density distribution in amorphous materials. Oster and Reilly investigated diffraction from arrays of infinite cylinders and showed that at certain defined ratios of diameter to spacing the structure factor dropped to zero.25 Beck et al. used Oster and Riley’s result to model the diffraction intensities in MCM-41 with reasonable success.26 Hammond et al. introduced changing wall thickness to this model and illustrated variation in peak intensity for discrete values of wall thickness and pore diameter.26,27 Imperor-Clerc et al. modeled the density distribution in the walls of SBA-15, fit this to synchrotron SAXS data, and concluded that there exists a microporous corona around the mesopore.23 A number of groups have studied adsorption in mesoporous materials using either neutron or X-ray scattering.28−35 Hofmann et al. obtained SAXS patterns for Kr adsorption on SBA-15 yielding detailed insight into the distribution of adsorbed Kr in and on the mesoporous walls of SBA-15.29 Zickler et al. similarly obtained synchrotron SAXS diffraction patterns for perfluoropentane adsorption on SBA-15.31 Both Hofmann and Zickler employed Imperor-Clerc’s corona model for micropore distribution; however, the corona model has been called into question in a recent study by Pollock et.al.33 In all these studies, analysis of diffraction patterns and intensities provides valuable structural information, where pore diameter from gas adsorption was generally employed as an input to the data analysis. In this paper, we minimize the information from adsorption into a model structure and consider whether the pattern of diffraction peak intensities provides information to test the consistency of adsorption models used to evaluate mesopore diameters. We model the density distribution of a solid with two-dimensional hexagonal pore system and show how the pore diameter relative to unit cell dimension affects XRD peak intensities. We prepare a wide range of SBA-15 materials with varying pore sizes, measure their pore diameters from nitrogen adsorption using BJH and DFT, and obtain high quality SAXS diffraction patterns. Thus, we have a set of experimental data for comparing the effect of the ratio of pore diameter (from nitrogen adsorption) to unit cell dimension on diffraction intensities. A comparison of the aggregate experimental data with theoretical diffraction intensities yields an independent

The current state -of-the-art for extracting pore structure information from isotherm data employs nonlocal density functional theory.5,12,19 In this procedure, DFT is used to model adsorptive intermolecular interactions within a homogeneous pore of defined shape. In practice DFT is used to create a kernel, a set of isotherms for model pores of fixed diameters, and the experimental pore size distribution is found from a weighted fit of the kernel isotherms to the experimental data. Ravikovitch and Neimark developed parameters for the DFT methodology based upon a set of MCM-41 adsorbents with pore diameters between 2 and 5 nm.12 To evaluate their method, these authors estimate pore diameter based on geometric arguments for the 2-D hexagonal structure, where the resulting equations for pore diameter are ⎛ 2 3 ⎞1/2 1/2 dp = ⎜ ⎟ aε ⎝ π ⎠ ε=

ρSiO2 (Vpore − Vmicropore) ρSiO2 Vpore + 1

(3)

where a is the unit cell dimension obtained from XRD, Vpore and Vmicropore are total and micropore pore volumes obtained from adsorption, and ρSiO2 is pore wall density assumed to be 2.2 g/cm3. Note that for MCM-41 materials the micropore volume, Vmicropore = 0. They showed that the method yields reasonable agreement with pore diameters obtained from geometric arguments but find that the BJH method underestimates pore diameters by 1.0 to 1.3 nm in the set of materials studied (see Figures 8 and 9 in ref 12). Ravikovitch, Neimark and co-workers also apply the DFT method to SBA-15 for pores between 5.0 and 9.0 nm.5 Two additional considerations are (1) one must evaluate wall microporosity that exists in SBA-15, and (2) because SBA-15 exhibits hysteresis the question of which branch of the isotherm is most appropriate to use must be considered. Ravikovitch and Neimark show that the adsorption branch of the isotherm represents a metastable state, the desorption branch represents the thermodynamic equilibrium. Given the appropriate kernel, the DFT method may be applied to either branch; however, this finding means that application of the BJH method to the adsorption branch is inappropriate, as it is inconsistent with the thermodynamic equilibrium assumption of the Kelvin equation.19 As they did with MCM-41, they estimate pore diameters from XRD and geometric arguments using eq 3 above, where microporosity is assessed from adsorption isotherms. They conclude that the DFT procedure gives results in good agreement with geometric estimates of pore diameter and find that the BJH method when applied to the appropriate desorption branch underestimates pore diameters by 1.4 to 1.9 nm, but they do not report the procedure used to estimate adsorbed film thickness.5 They also find that the BJH method when used with the adsorption branch yields results roughly in agreement with DFT. The authors clearly emphasize this is not justified on theoretical grounds and this result should be understood as accidental and not as confirmation of the BJH method. These conclusions are also generally consistent with Thommes’s statements that the BJH method underestimates pore diameters by up to 25% for pores 0.999 for the 5 points, in−2, in−1... in+2) was used to select points included in the surface area analysis; p/po values were between 0.01 and 0.20. Mesopore diameters were calculated using the BJH method on both the adsorption and desorption branches of the isotherm above a diameter of 2 nm, with the Kelvin equation for a hemispherical meniscus. The thickness of the adsorbed layer was estimated from the FHH equation with the typical default FHH parameters k = 5.0 and s = 3. We note that Gregg and Sing suggest parameters of k = 2.99 and s = 2.75, and these values model well nitrogen adsorption on the nonporous silica LiChrospher Si-1000;37 employing the Gregg and Sing values would give an adsorbed film thicknesses about 0.30 to 0.35 nm smaller and hence pore diameters about 0.7 nm smaller. Mesopore diameters were also calculated using Micromeritics DFT kernel for N2 in cylindrical pores for an oxide surface. The kernel operates on the adsorption branch of the isotherm. All calculations were performed using software Version 3.04 provided with the ASAP 2020 instrument. We report the mesopore peak maximum from the dV/dD pore size distribution curve. Small-angle X-ray scattering (SAXS) patterns were obtained using the University of Pennsylvania’s Laboratory for Research on the Structure of Matter (LRSM) multiangle X-ray diffractometer system (MAXS). This facility is equipped with a Bruker Nonius FR591 high brightness rotating-anode X-ray generator (Cu Kα), integral vacuum from generator to detector, focusing optics and a two-dimensional Bruker HiStar multiwire detector. Measurements were performed at an intermediate distance of 54 cm sample to detector which yields scattering angles from 0.3 to 5.0 2θ (scattering vector q = 0.2− 2.2 nm−1). Silver behenate was used as a calibration standard. Prior to loading, samples were stored for at least 48 h in a desiccator; samples were loaded under ambient conditions into

I = Io

e 4 ⎛ 1 + cos2 2θ ⎞ 2 ⎟MF 2 ⎠ mcR ⎝

3

∏

2 2 2⎜

( πλ (⇀s − ⇀so )Nai i) π sin 2( λ (⇀ s −⇀ so )ai)

sin 2

i=1

(4)

where e and m are the charge and mass of an electron, c the speed of light, and R the distance to the observer. M is the multiplicity of the hk reflection: M = 6 for h0, k0, and when h and k are the same; M = 12 for mixed h and k. The quantity before M F2 is a function of instrument geometry (Io, R) and collection angle (θ), and there is no information here about the sample. The product quantity after F2 is over the three coordinate directions, Ni the number of repeat unit cells, and ai the dimension of the unit cell in the ith direction; it is essentially a spike at integral multiples of π, and zero everywhere else. Atomic positions, or more correctly the distribution of electron density within a unit cell appear only in the structure factor F. For a material where the electron density is described by a continuum, the structure factor F is given by F=

⎡

⎤

∫V ρ(⇀r )exp⎣⎢ 2λπi (⇀s − ⇀so )⇀r ⎦⎥dr

(5)

s − ⇀ s o the where λ is the wavelength of the radiation, ⇀ r the difference between incident and scattered beams, ⇀ r ) the density distribution position vector of scatterer, and ρ(⇀ of scatterers. The integration is over one unit cell. For a collection of cylindrical scatterers arranged in a 2-D hexagonal (p6mm) pattern, Oster and Reilly as well as Imperor-Clerc et al. give the solution to eq 5 as F = 2πr 2

J1(qrp) qrp

q=

4π 3

h2 + hk + k 2 a

(6)

where J1 represents the first-order Bessel function, rp is the pore radius, a the unit cell dimension, and h and k are the familiar Miller indices of a Bragg reflection. For 3-D crystalline materials 17495

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Table 1. Representative Synthesis Conditions and Analysis of SBA-15 Samplesa ID

TEOS/P-123 ratio (g/g)

synthesis gel mole ratios (H2O/ HCl/TEOS/P-123)

synth gel temp (°C)

calcine temp (°C)

unit cell, a (Å)

BET area (m2/g)

peak mesopore diameter, dp,BJH_a (Å)

A B C D E F Gb Hc Ic Jc

1.0 2.3 6.3 4.1 4.1 4.1 2.3 2.2 2.2 2.2

384/11/1/0.0343 173/4.9/1/0.0154 127/3.6/1/0.0056 192/5.5/1/0.0086 192/5.5/1/0.0086 192/5.5/1/0.0086 172/5.2/1/0.0154 185/5.6/1/0.0161 185/5.6/1/0.0161 185/5.6/1/0.0161

40 40 40 45 45 45 40 40 40 40

500 500 500 500 700 850 500 500 700 850

108.8 106.5 102.2 106.4 100.3 93.2 116.8 112.7 109.4 102.1

740 364 646 565 275 112 804 895 707 438

66.5 63.5 51 62 57.5 43 84 74 70 64

ratio ratio dp,BJH_a/a Vmicropore/Vtotal 0.61 0.60 0.50 0.58 0.57 0.46 0.72 0.66 0.64 0.63

0.20 0.22 0.45 0.27 0.10 0.01 0.16 0.25 0.21 0.12

a Samples A through F were all prepared with the same nominal batch size, ∼160 g of synthesis gel, and the same lot of raw materials. bSample G: different lots of TEOS and Pluronic P-123. cSamples H−J: 2.5× larger batch size and different lots of TEOS and P-123.

in which atomic positions occupy fixed positions in the cell, the structure factor becomes the familiar F=

⎡ 2πi ⎤ (hxn + kyn + lzn)⎥ ⎦ λ

∑ fn exp⎢⎣ n

F=

(11)

The first exponential quantity in the expression is called the Debye−Waller factor, wherein un,hk is the projection of the scatter’s displacement normal to the hk diffraction plane. In the amorphous materials considered here, deviations of pore center within the unit cell may be thought of as random fluctuations of cylindrical micelle position frozen in the silica matrix; as such the Debye−Waller factor has also been applied to characterize the random but fixed fluctuations in mesopore position.23 Warren notes that the local fluctuations uav,hkl = ⟨uhkl2⟩1/2 may be different for hkl of different orientations.24 For this reason, our approach is to identify a uav,h0 that best fits the intensity ratios for the 10, 20 and 30 peaks, and as necessary, separate uav,hk that best fit the intensity ratios for the 21 and 11 peaks. There also exists a fundamental difference in the pore radius extracted from nitrogen adsorption and that used in our structural modeling. The pore radius found by adsorption corresponds to the void radius, from the edges of the electron cloud of opposing wall atoms. In contrast, the pore radius used in diffraction modeling corresponds to changes in electron density; because most of the electron density is found in core electrons held close to the nucleus, this pore radius corresponds to distance between atomic centers of opposing wall atoms. Therefore we expect there to be a systematic bias in pore radius found from adsorption versus that employed in the diffraction models, where the adsorption data are smaller by one atomic diameter, or ∼0.3 nm.

(7)

where λ is the wavelength of the radiation, f n the atomic scattering factor for all n atoms in the unit cell, and xn, yn, and zn the specific fractional coordinates of an atom.23,25 SBA-15 is amorphous at the atomic scale, so the approach here is to omit the z component and create a two-dimensional model of electron density within a unit cell by discretizing the cell into an m × m system, typically m > 200. In this study, diffraction intensities are normalized, so it suffices to substitute the atomic structure factor with an occupancy factor to represent electron density at each x, y, the fractional position in the discretized unit cell, where OCx,y = 1 when electron density is a maximum (in the pore wall), and OCx,y = 0 when not (center of the pore). For a discontinuous step-change in density at the pore wall this is expressed as L=

⎛1 ⎞2 ⎛ 1 ⎞⎛ 1 ⎞2 ⎛ 1 ⎞2 ⎜ − x⎟ − ⎜ − x⎟⎜ − y⎟ + ⎜ − y⎟ ⎝2 ⎠ ⎝2 ⎠⎝ 2 ⎠ ⎝2 ⎠

OCx , y

rp ⎫ ⎧ 1 if L > ⎪ ⎪ ⎪ a⎪ ⎬ =⎨ rp ⎪ ⎪ ⎪ 0 if L < ⎪ ⎩ a⎭

(8)

(9)

For a 2-D lattice (l = 0), the structure factor for eq 7 becomes 1 F= 2 m

m

3. RESULTS AND DISCUSSION Table 1 contains pertinent synthesis parameters for a representative sampling of prepared materials. Samples A−F in this set were all prepared with the same batch size, with the same initial quantity of water and HCl, and all with the same lots of TEOS and Pluronic P-123. Within this set at constant synthesis gel temperature, there exists a slight trend of decreasing unit cell size with increasing P-123/silica ratio and a more substantial trend of decreasing peak mesopore diameter with increasing P-123/silica ratio. Samples calcined to higher temperatures (700 °C, 850 °C) show a decrease in both unit cell dimension and a more marked drop in pore diameter, this too alters the pore diameter to unit cell ratio. It should also be noted that calcination to higher temperatures effectively anneals out micropores, and this is reflected in the significantly lower

m

∑ ∑ OCi ,j exp[2πi(hxi + kyj )] i=1 j=1

⎡ −8π 2⟨u 2⟩sin 2 θ ⎤ n , hk ⎥exp[2πi(hxn + ky )] OC exp ∑ n ⎢ n ⎢ ⎥⎦ λ2 ⎣ n

(10)

The summations are over indices i and j, so that the fractional positions are x = i/m and y = j/m. We have confirmed that the discretized result of eqs 8−10 yields the same result as the analytical solution of Oster and Riley, eq 6. The treatment above does not allow for displacements of electron density from a fixed unit cell position. In crystalline materials these deviations occur as a result of thermal motion. The effect is accounted for by allowing for a random component of displacement for any atom from its average position. When averaged over the sample, the result is a modification to the structure factor given by 17496

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Figure 1. (a) SAXS diffraction pattern and (b) nitrogen adsorption isotherms for two samples of SBA-15. The changing position of the hysteresis loop in the isotherm reflects different pore diameters. In the diffraction pattern, the 10 peaks are same position, indicating the same unit cell dimension, while the different intensities of higher order peaks is a consequence of the changing ratio of pore diameter to unit cell size.

BET values for these samples.38 The remaining four samples listed illustrate the complexity in SBA-15 synthesis: Sample G was prepared using a lot of Pluronic P-123 different from that for A−F, while samples H, I, and J were all prepared in a larger (2.5×) batch size as well as different lots of P-123 and TEOS. Samples B and G have essentially identical recipes but differ by 10% in unit cell size and almost 20% in mesopore diameter. The point here is to illustrate the variation in synthesis parameters that resulted in a set of samples with moderately varying unit cell dimensions and a relatively broad range of pore diameters. Figure 1 depicts SAXS patterns and adsorption isotherms (inset) for two different samples of SBA-15, samples C and J from Table 1. The prominent 10 peak at 2θ ∼0.99 in the SAXS pattern indicates samples have very similar d-spacing and unit cell a. The diffraction pattern for sample J is typical of SBA-15 and clearly shows three prominent peaks for the 10, 11, and 20 reflections, but the 21 peak is nearly absent. Sample C shows the 10 and 20 peaks, but in contrast to sample J the 11 peak is nearly absent and the 21 peak is prominent. As shown below, this is due to the different ratio of mesopore diameter to unit cell dimension, dp/a, in these two samples. The inset in Figure 1 depicts the nitrogen isotherms for these same samples; here the position of the hysteresis loops and hence peak mesopore diameter is significantly different. Sample J BJH pore diameter based on the adsorption branch is 6.4 nm, and sample C pore diameter by the same measure is 5.1 nm. The ratio of BJH pore diameter to unit cell dimension, dp_BJHa/a, for samples J and C are 0.63 and 0.50, respectively. We use eq 6 to calculate the structure factor for the first seven peaks in a 2-D hexagonal structure (p6mm symmetry). Figure 2 shows the magnitude of the structure factor as a function of the ratio of pore diameter to unit cell dimension. Note that the 10 structure factor never equals zero, but this is not true for all higher order peaks. Thus, where the structure factor is zero will be dp/a ratios where these higher order peaks

Figure 2. Theoretical structure factors versus pore diameter/unit cell dimension ratio. Structures were modeled without variation in pore diameter (ideal pore).

are absent in the diffraction pattern. Ratios where the structure factor = 0 for higher order peaks are also tabulated in Table 2. For sample C, where the experimental dp_BJH/a = 0.50, we should expect to see the 11 peak much diminished, and the 21 peak to be present, quite in contrast to Figure 1. Similarly, sample J’s experimental ratio is 0.63, a point where one expects to observe the 11 peak. Diffraction data are routinely used to establish the crystallographic dimension within 0.01 nm; as well, it is broadly understood that the BJH method underestimates pore diameters. Both of these facts strongly suggest that the experimental dp/a ratios are not accurate. The global effect on diffraction patterns is illustrated in Figure 3, the simulated powder pattern of an ideal p6mm mesoporous oxide 17497

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Table 2. Calculated Ratios Where the Structure Factor for hkl Reflection = 0a peak hk 11 20 21 30 22 31 a

ideal pore/discontinuous density at wall 0.611 0.528 0.731 0.645 0.809 0.778

0.399 0.352 0.558 0.536

0.305 0.293

modeled density distribution, σ = 0.035 0.606 0.524 0.728 0.641 0.807 0.775

0.393 0.345 0.554 0.532

0.297 0.285

modeled density distribution, σ = 0.05 0.602 0.519 0.724 0.637 0.803 0.771

0.387 0.338 0.549 0.527

0.289 0.277

modeled density distribution, σ = 0.0.08 0.589 0.505 0.713 0.625 0.788 0.758

0.371 0.323 0.538 0.515

0.278 0.268

For all ratios, the third decimal place is not significant.

squares), dp_BJHa/a; the BJH method using desorption branch (yellow triangles), dp_BJHd/a; density functional theory (blue circles), dp_DFT/a. Also plotted is the ratio of the square of structure factors F112/F102 using an ideal smooth cylindrical pore given by eqs 8−10; the identical result is also obtained with eq 6. The magnitude of the theoretical intensity ratios is much larger than the experimental ratios so that theoretical values are plotted on the secondary y-axis. The similarity in the trend between the experiment and theory is obvious. The differences in the intercepts of the experimental data with the xaxis may be understood as a manifestation of generally accepted biases in the BJH values; the change in scaling between theory and experiment may be accounted for by incorporating a Debye−Waller factor. Note that the Debye−Waller factor has no impact on the intercept of the theoretical trend with the xaxis; it only effects the magnitude of Fhk2/F102. To account for the electron cloud of wall atoms and to make a more precise comparison between dp/a ratios determined from diffraction versus adsorption, 0.3 nm was added to the adsorption pore diameters. The structure model given in eq 9 is problematic for two reasons. First, it is widely accepted that there exist micropores in the wall of SBA-15, the result of interpenetration of Pluronic ethylene oxide fragments imbedded in the silica matrix. The possibility that the micropores are unevenly distributed in the silica wall must be considered. In their synchrotron SAXS study of SBA-15, Imperor-Clerc et al. concluded that the micropores are contained within a micropore corona around the mesopore; this corona was considered separate from a dense silica

with a unit cell dimension of a = 11.0 nm for a range of different pore diameter to unit cell ratios, dp/a.

Figure 3. Simulated diffraction patterns for a sample of p6mm mesoporous material with a = 110 Å and with a varying ratio of mesopore diameter.

In Figure 4a and 4b, we show the experimental diffraction intensity ratios (e.g., I11/I10) plotted versus dp/a for the 10 and 20 peaks. The three sets of experimental data points are calculated using three different techniques to determine pore diameter dp: the BJH method using adsorption branch (red

Figure 4. (a, b) Theoretical and experimental hk/10 peak intensity ratios versus ratio pore diameter to unit cell dimension. Yellow triangles: experimental data based on BJH desorption branch, dp,BJH_d; red squares: experimental ratio based on BJH adsorption branch, dp,BJH_a; blue circles: experimental ratio based on DFT pore diameter, dp,DFT. The solid line represents theoretical intensity ratios for an ideal mesoporous solid. 17498

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Figure 5. (a−d) Structure factor variation for (a) 11, (b) 20, (c) 21, and (d) 30 peaks, where the grey line is ideal pore, (− − −) σ = 0.035a, () σ = 0.05a, and (----) σ = 0.08a.

matrix.23 In their data analysis, however, they adopted the adsorption BJH value of pore diameter without considering the generally accepted biases of that method. A recent study by Pollock et al. used contrast matching SANS to conclude that the micropores are more evenly distributed through the silica wall.33 These authors find that the conclusion reached in the Imperor-Clerc study is due to an unrecognized, inherent bias in pore diameters obtained from BJH methodology. In addition, the cooperative self-assembly process for SBA synthesis, suggested to be (S0 H+)(X− I+), or (C2H5O−0H3O+)··· (xCl−···Si(OSi)4−x(OH2)x+x) by Zhao et al. indicates that condensed silica should be intimately associated with ethylene oxide fragments.36 Inverse templating studies with SBA-15 have also shown that the micropores interconnect the mesopore channels,39,40 again suggesting the silica micropores should be uniformly distributed in the walls of SBA-15. Finally, if we adopt an electron density distribution similar to that proposed by Imperor-Clerc for structure factor calculations, the result (not shown) shows that the extinction ratios markedly shift downscale. This shift is proportional to the assumed micropore distribution and hence degree of microporosity in the samples. The set of samples employed in this study has a large range of microporosity (see Table 1) and should be expected to exhibit significant variation in wall density distribution. If the ImperorClerc model is correct, the experimental data would be expected to show no discernible pattern in extinction ratios. This is not what we observe in Figure 4a and 4b. However, because our SAXS methodology made no special effort other than retaining samples in a desiccator before sample preparation, adsorbed water in SBA-15 may have minimized electron differences in the electron density distribution in wall. We therefore assume the electron density in our experimental SBA-15 samples is uniformly distributed and our model contains no correction for density variation within the wall. Second, SBA-15 pore size distributions from nitrogen isotherms typically show variation in mesopore diameter. In our sample set, the distribution of mesopore diameters typically

has a full-width at half-maximum of 0.5−1.3 nm. Assuming a normal (Gaussian) distribution, this corresponds to a standard deviation relative to typical unit cell dimension of σBJH/a = 0.02−0.05. We adopt values of σ/a = 0.035, 0.05 and 0.08 to illustrate the sensitivity to mesopore size variation on the scattering patterns and use the cumulative normal (Gaussian) probability distribution, CNF(x, μ, σ), to model this mesopore diameter variation, so that: ⎛ dp σ ⎞ OCn = CNF⎜L , , ⎟ a a⎠ ⎝

(12)

is used instead of eq 9. We use eqs 8, 10, and 12 to numerically calculate the structure factor. Figure 5a−d show this modified structure factor as a function of the ratio of pore diameter to unit cell ratio for the 11, 20, 21 and 30 reflections. In Figure 5a and 5b, there is a general decrease in structure factor magnitude, but the key observation is there are only modest shifts in the position of the zeros in the structure factor for the 11 and 20 reflections. For the 11 peak, the zero of observed diffraction intensity will shift from the ideal dp/a = 0.610, to a range of dp/a = 0.589−0.606. For typical SBA-15 with an 11.0 nm unit cell, the absence of the 11 peak in the diffraction pattern indicates a pore diameter of 6.5 to 6.7 nm. The decrease in the structure factor intensity and shifts in the zeros become somewhat more appreciable for higher order reflections (Figure 5c and 5d). Nevertheless, the absence of peaks in the diffraction pattern suggests important constraints in the true pore diameter. Table 2 also tabulates the zeros in the structure factor from these calculations from the model using eq 12. Figure 6a−d has the same format as Figure 4a and 4b; however, the theoretical intensity ratios here reflect the modified structure factor calculation using eq 12 and the effect of a Debye−Waller correction. The two theoretical lines in these plots represent the variation in theoretical intensity ratios based on variation in mesopore diameters σ/a = 0.035 and 0.08. The Debye−Waller correction was determined by fitting 17499

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a linear transformation between measured and best-fit pore diameter values; that is d p unbiased = αd p measured + β

where α and β are constants. To emphasize experimental data near the extinction ratio, the weighting function used in the least-squares procedure is inversely proportional to the intensity ratio. We evaluate a one-parameter addition to all adsorption-determined pore diameters in the data set (i.e., α = 1), a correspondingly simple percentage correction to all pore diameters (β = 0), and two-parameter correction to pore diameters. We find the two-parameter fit to experimental values is not significantly better than a simple percentage bias to the BJH values, and that the percentage bias works better than an absolute additive quantity to the BJH value. Results summarized in Table 3 indicate the BJH analysis based on Table 3. Estimated Biases in Adsorption-Based Pore Diameters from Three Isotherm Interpretation Models

Figure 6. (a−d) Theoretical and experimental hk/10 peak intensity ratios versus ratio pore diameter to unit cell dimension. Yellow triangles: experimental data based on BJH desorption branch; red squares: experimental data based on BJH adsorption branch; blue circles: experimental data based on DFT pore diameter. Solid line represents mesopore size variation of σ = 0.05a, dotted lines represent bracketing of σ = 0.035a and σ = 0.08a. See text for discussion of Debye −Waller factor used.

BJH adsorption BJH desorption DFT

range of pore diameters predicted by diffraction analysis

range of pore diameters from nitrogen isotherm analysis

50−90

45−85

15

9.3

50−90

35−65

40

18.7

50−90

55−100

−10

−7.4

average average percentage absolute difference, % difference

the adsorption branch yields pore diameters that are 15% too small, and BJH values based on the desorption branch are 40% too small. Over the range of samples examined in this study, this amounts to a ∼0.9 nm underestimation of pore size using BJH from the adsorption branch and 2.0 nm underestimation if the thermodynamically correct desorption branch is used. These observations are consistent with previous observations that the BJH method underestimates pore diameter; however, the magnitude of the biases found in this study are somewhat larger than reported in earlier investigations.11−13 Ravikovitch et al. reported a rough equivalence of BJH based on the adsorption branch with true pore diameter; in contrast, we find this method yields results ∼15% too low. The same authors also reported a 1.4−1.9 nm discrepancy based on the desorption branch, our results are at the very upper end of this range.12 Similarly, the percentage underestimation in BJH based on the desorption branch found here is larger than that cited by Thommes (40% versus 25%).13 For ratios determined from DFT pore diameters, dp_DFT/a, the diffraction intensity ratio appears to intersect the x axis marginally greater than the theoretical value, see especially Figure 6b for 20/10 data. As shown in Table 3 the experimental DFT pore diameters are shifted upscale by about 10%, or ∼0.7 nm for smaller pores. We also observed two samples with DFT pore diameter/unit cell dimension ratio exceeding 1. If this were true, the mesopores would overlap and there would be on average no wall separating them, and we reach a conclusion inconsistent with the structural integrity of SBA-15. This experimental dp_DFT/a ratio greater than 1 leads to the conclusion that the DFT results overestimate pore diameters by at least 10% or 1−2 nm for larger pores above 10 nm. The bias estimates in pore diameters suggested by this analysis were developed on an aggregate collection of samples.

the theoretical curve to a geometrical assessment of pore diameter using eq 3; we find a value of uav,h0 = 0.46 nm for the 10, 20 and 30 peaks fits best. This value also works well to model the 21 peak, but for the 11, we found a uav,11 = 0.59 nm. These values are somewhat smaller than the value found by Imperor-Clerc et al. who identified a single uav = 0.95 nm. While we have no conclusive explanation for the large value for u in the 11 direction, this is toward the nearest-neighbor micelles. A large value for uav,11 in this direction suggests that the random micropore and complementary mesoporous interconnections between micelles effect the structure factor in ways our model cannot capture. Alternatively, these interconnecting channels could perturb mesopore position in the 11 diffraction direction, which would show as a larger Debye−Waller factor in that direction. The existence of interconnecting micropores and complementary mesopores in SBA-15 is well established,40,41 and inverse templating studies have revealed that the SBA-15 structure is potentially more complex than a simple hexagonal arrangement of cylindrical mesopores.42,43 In all parts of Figure 6, the experimental data trends with similar slopes to the theoretical curves, suggesting that there are simple scaling corrections to estimate biases in pore diameters obtained from gas adsorption studies. In Figure 6a for the 11/ 10 peak ratio, the intersection of the trend in experimental diffraction intensity ratios with the x axis (dp/a point of peak extinction = 0.60) clearly shows that the ratios based on BJH approaches are shifted downscale. To determine the best estimate of bias to BJH data, we employ a weighted leastsquares minimization between theoretical and experimental intensities, where the theoretical intensities are calculated from 17500

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(6) Hess, C. Nanostructured Vanadium Oxide Model Catalysts for Selective Oxidation Reactions. ChemPhysChem 2009, 10, 319−326. (7) Rioux, R. M.; Song, H.; Hoefelmeyer, J. D.; Yang, P.; Somorjai, G. A. High-Surface-Area Catalyst Design: Synthesis, Characterization, and Reaction Studies of Platinum Nanoparticles in Mesoporous SBA-15 Silica. J. Phys. Chem. B 2005, 109, 2192−2202. (8) Song, H.; Rioux, R. M.; Hoefelmeyer, J. D.; Komor, R.; Niesz, K.; Grass, M.; Yang, P. D.; Somorjai, G. A. Hydrothermal Growth of Mesoporous SBA-15 Silica in the Presence of PVP-Stabilized Pt Nanoparticles: Synthesis, Characterization, and Catalytic Properties. J. Am. Chem. Soc. 2006, 128, 3027−3037. (9) Zeidan, R. K.; Dufaud, V.; Davis, M. E. Enhanced Cooperative, Catalytic Behavior of Organic Functional Groups by Immobilization. J. Catal. 2006, 239, 299−306. (10) Zeidan, R. K.; Hwang, S. J.; Davis, M. E. Multifunctional Heterogeneous Catalysts: SBA-15-Containing Primary Amines and Sulfonic Acids. Angew. Chem., Int. Ed. 2006, 45, 6332−6335. (11) Galarneau, A.; Desplantier, D.; Dutartre, R.; Di Renzo, F. Micelle-templated silicates as a test bed for methods of mesopore size evaluation. Microporous Mesoporous Mater. 1999, 27, 297−308. (12) Ravikovitch, P. I.; Wei, D.; Chueh, W. T.; Haller, G. L.; Neimark, A. V. Evaluation of Pore Structure Parameters of MCM-41 Catalyst Supports and Catalysts by Means of Nitrogen and Argon Adsorption. J. Phys. Chem. B 1997, 101, 3671−3679. (13) Thommes, M. Physical Adsorption Characterization of Nanoporous Materials. Chem. Ing. Tech. 2010, 82, 1059−1073. (14) Lu, G. Q.; Zhao, X. S., Eds. Nanoporous Materials: Science and Engineering; Imperial College Press: London, 2004; Vol. 4, pp 317− 364. (15) Kruk, M.; Jaroniec, M.; Sayari, A. Application of Large Pore MCM-41 Molecular Sieves To Improve Pore Size Analysis Using Nitrogen Adsorption Measurements. Langmuir 1997, 13, 6267−6273. (16) Broekhoff, J. C. P.; de Boer, J. H. Studies on Pore Systems in Catalysts: IX. Calculation of Pore Distributions from the Adsorption Branch of Nitrogen Sorption Isotherms in the Case of Open Cylindrical Pores A. Fundamental Equations. J. Catal. 1967, 9, 8−14. (17) Broekhoff, J. C. P.; de Boer, J. H. Studies on Pore Systems in Catalysts: X. Calculations of Pore Distributions from the Adsorption Branch of Nitrogen Sorption Isotherms in the Case of Open Cylindrical Pores B. Applications. J. Catal. 1967, 9, 15−27. (18) Lukens, W. W.; Schmidt-Winkel, P.; Zhao, D. Y.; Feng, J. L.; Stucky, G. D. Evaluating Pore Sizes in Mesoporous Materials: A Simplified Standard Adsorption Method and a Simplified Broekhoff-de Boer Method. Langmuir 1999, 15, 5403−5409. (19) Ravikovitch, P. I.; Odomhnaill, S. C.; Neimark, A. V.; Schuth, F.; Unger, K. K. Capillary Hysteresis in Nanopores: Theoretical and Experimental Studies of Nitrogen Adsorption on MCM-41. Langmuir 1995, 11, 4765−4772. (20) Huang, Z.; Bensch, W.; Sigle, W.; van Aken, P. A.; Kienle, L.; Vitoya, T.; Modrow, H.; Ressler, T. The Modification of MoO(3) Nanoparticles Supported on Mesoporous SBA-15: Characterization Using X-ray Scattering, N(2) Physisorption, Transmission Electron Microscopy, High-Angle Annular Darkfield Technique, Raman and XAFS Spectroscopy. J. Mater. Sci. 2008, 43, 244−253. (21) Brieler, F. J.; Grundmann, P.; Froba, M.; Chen, L. M.; Klar, P. J.; Heimbrodt, W.; von Nidda, H. A. K.; Kurz, T.; Loidl, A. Formation of Zn(1−x)Mn(x)S Nanowires within Mesoporous Silica of Different Pore Sizes. J. Am. Chem. Soc. 2004, 126, 797−807. (22) Froba, M.; Kohn, R.; Bouffaud, G.; Richard, O.; van Tendeloo, G. Fe2O3 Nanoparticles within Mesoporous MCM-48 Silica: In Situ Formation and Characterization. Chem. Mater. 1999, 11, 2858−2865. (23) Imperor-Clerc, M.; Davidson, P.; Davidson, A. Existence of a Microporous Corona around the Mesopores of Silica-Based SBA-15 Materials Templated by Triblock Copolymers. J. Am. Chem. Soc. 2000, 122, 11925−11933. (24) Warren, B. E. X-ray Diffraction; Dover Publications: New York, 1990. (25) Oster, G.; Riley, D. P. Scattering from Cylindrically Symmetric Systems. Acta Crystallogr. 1952, 5, 272−276.

In principle one could develop a methodology to estimate pore diameters for a single sample and XRD pattern. For example, a set of simulated XRD patterns similar to those shown in Figure 3 could be used as kernel for a fit of theoretical intensities to the experimental pattern, much as DFT is executed for determining pore size distributions from nitrogen isotherms today. However, one would have to make a priori assumptions about mesopore position fluctuations, distribution of electron density in walls, and variation in mesopore sizes within each sample. As such, we do not believe that such an effort is warranted.

4. CONCLUSION A well-known outcome of X-ray scattering theory is that various symmetry conditions in purely crystalline materials lead to reflection absences characteristic of certain space groups. In the same way, certain ratios of pore diameters to unit cell dimension in ordered mesoporous silicates leads to extinctions of peak intensities. Thus, SBA-15 samples that exhibit demonstrable absences of certain peaks must have certain ratios of pore diameter to unit cell size. More generally, the ratio of peak intensities is a function of the dp/a ratio. For a collection of SBA-15 samples with variation in this ratio, we use trends in peak intensities to establish an unbiased and essentially assumption-free estimate of the accuracy of various methods to extract pore diameter from adsorption techniques. We find that traditional BJH method based on the adsorption branch underestimates pore diameters by ∼15%, BJH applied to the thermodynamically appropriate desorption branch underestimates pore diameters by 40%, and somewhat surprisingly the DFT method employed in this study overestimates pore diameters by ∼10%.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; tel: 610 519-5968; fax: 610 519-7354. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the support from Villanova University.

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REFERENCES

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