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Analysis of pore structure and gas adsorption in periodic mesoporous solids by in situ small-angle X-ray scattering
Colloids and Surfaces A: Physicochem. Eng. Aspects 357 (2010) 3–10
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Analysis of pore structure and gas adsorption in periodic mesoporous solids by in situ small-angle X-ray scattering Gerhard H. Findenegg a,∗ , Susanne Jähnert a , Dirk Müter b , Oskar Paris b,c a b c
Institut für Chemie, Stranski Laboratory, Sekr. ER1, Technical University Berlin, D-10623 Berlin, Germany Department of Biomaterials, Max Planck Institute of Colloids and Interfaces, D-14424 Potsdam, Germany Institute of Physics, University of Leoben, A-8700 Leoben, Austria
a r t i c l e
i n f o
Article history: Received 31 July 2009 Received in revised form 23 September 2009 Accepted 25 September 2009 Available online 6 October 2009 Keywords: SBA-15 silica Ordered nanoporous materials In situ small-angle scattering Intrawall porosity Pore filling
a b s t r a c t This paper describes a new method for characterizing the pore structure and gas adsorption in periodic mesoporous materials based on small-angle X-ray diffraction and scattering. For SBA-15 as a case study it is shown that the effective pore radius and volume of the ordered cylindrical pores and their corona can be determined accurately from the intensities of the Bragg reflections. The contribution of the intrawall porosity to the pore volume is then obtained by comparing this ordered pore volume with the total pore volume from nitrogen adsorption. Adsorption of a fluid in the ordered and disordered pores of SBA-15 is studied by analyzing scattering data measured in situ as a function of the vapor pressure of the fluid. In this way the different stages of pore filling can be monitored, and separate filling isotherms of the ordered and intrawall pores can be obtained for an adsorbate (dibromomethane) having a similar electron density as the solid matrix. Our analysis shows that the filling of the intrawall porosity has a pronounced effect on the evaluation of the adsorbed amount in the ordered pores. These finding emphasize the importance of disordered intrawall porosity in polymer-templated silicas like SBA-15 and open up new ways for an in-depth characterization of these materials. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Periodic mesoporous silicas synthesized with amphiphilic block copolymers as structure-directing agent not only constitute regular arrangements of mesopores but also contain significant levels of disordered intrawall porosity, depending on the protocol of sample preparation [1–3]. Structural information about the ordered pore system can be obtained from X-ray small-angle diffraction data, either by adopting simple geometrical models for the density distribution which can be analytically Fourier transformed [4–6], or by numerical Fourier transformation of more sophisticated density distributions [7,8]. The former formalism can also be adapted to derive structural information about the adsorption of fluids in the ordered pores [5,6,9,10]. However, in the past most studies were focussed on the ordered pore system by analyzing the integral intensities of Bragg diffraction peaks arising from the mesopore lattice. Diffuse scattering caused by structural disorder (e.g., intrawall pores) was commonly treated as a background and not included in the modeling. Accordingly, no reliable quantitative estimates of the intrawall porosity exist. Recently we proposed a method for estimating this wall porosity by comparing the pore volume of ordered
∗ Corresponding author. E-mail address: fi[email protected] (G.H. Findenegg). 0927-7757/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2009.09.053
pores with the total pore volume as obtained by nitrogen adsorption [11]. It was also shown that information about the intrawall porosity can be derived from the diffuse small-angle scattering and its dependence on pore filling [12]. This paper gives an account of this new method for characterizing ordered and disordered porosity in periodic mesoporous materials, and its extension to the determination of adsorption isotherms of fluids in the ordered and disordered pores, based on in situ X-ray small-angle scattering. The formalism described below applies to materials exhibiting a two-dimensional (2D) hexagonal arrangement of cylindrical pores (plane group p6mm), such as MCM-41 and SBA-15, but may be extended to other periodic mesoporous solids, such as the cubic MCM-48 [7] and KIT-6 [13] (space group Ia3d) or the bcc structure of SBA-16 (space group Im3m) [14]. Dibromomethane (DBM, CH2 Br2 ) is chosen as the test fluid in the exemplary results presented in this paper, because DBM in its bulk liquid state has nearly the same electron density as silica, which facilitates the analysis of the scattering data [11]. The pore structure and pore filling of the ordered cylindrical pores are determined by analyzing the integrated intensities of up to 10 Bragg reflections in terms of analytical form factor models for cylindrical pores. In these models the corrugated pore walls and a microporous corona of the cylindrical pores are taken into account in a simple but physically meaningful way. In addition, randomly distributed, uncorrelated pores in the walls of the ordered mesopore lattice are considered
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as the source of diffuse scattering. Information about the gradual filling of these pores is extracted from the decreasing overall intensity of the diffuse scattering as a function of the relative pressure of the test fluid. Adsorption into the disordered pores in the matrix not only contributes to the overall adsorbed amount but also affects the determination of the amounts adsorbed in the ordered pores, due to the increase in electron density of the matrix when the disordered pores in the matrix are filled with the fluid. It is shown how this effect can be taken into account by a re-normalization of the density profiles in the form factor models, by which the amount of adsorbed fluid in the ordered pores can be converted to an absolute scale. The overall adsorption isotherms of the test fluid derived from the in situ X-ray study on the basis of this formalism is compared with the adsorption isotherm obtained by gravimetric adsorption technique. Finally, shortcomings and possible extensions of the new formalism are discussed briefly. 2. X-ray scattering and integrated scattering intensities A typical small-angle scattering profile I(q) for SBA-15 silica with the Bragg reflections resulting from the 2D hexagonal arrangement of the cylindrical mesopores is shown in Fig. 1a. The dashed area indicates diffuse small-angle scattering arising from disordered pores and other inhomogeneities of the matrix. Information about fluid adsorption in the ordered and disordered pores can be derived from diffraction and diffuse scattering, respectively, and their dependencies on the fluid pressure. Here we summarize the relations for the special case of a two-phase system [15], i.e., when the adsorbed fluid in its bulk liquid state has the same electron density as the solid matrix. As indicated in Fig. 1, the total scattered intensity can be separated into two parts: Bragg diffraction from the mesopore lattice (IBragg ), and diffuse small-angle scattering arising from disordered pores and other inhomogeneities of the matrix (Idiff ): I(q) = IBragg (q) + Idiff (q).
(1)
Integral intensities of the individual Bragg diffraction peaks (hk) can be determined by
˜IBragg (qhk ) =
IBragg (qhk )q2 dq,
(2)
where the diffuse scattering contribution is estimated from a double tangent at the base of the respective diffraction peak and subtracted separately for each peak. The total integral scattering intensity ˜I is given by
∞
I(q)q2 dq
˜I =
(3)
0
and can be determined by numerical integration of the scattering profiles within the appropriate q range (typically 0.3 nm−1 < q < 3.0 nm−1 ) and extrapolation of I(q) to zero and infinite q. Extrapolation to infinite q can be made using Porod’s law [15], since it is commonly found that the scattering curves are consistent with a q−4 behavior at large q. At the lower end of the experimental q range a constant scattering intensity may be assumed, i.e., I(q < qmin ) = I(qmin ) with qmin = 0.3 nm−1 . According to Eq. (1) the total integral intensity is given by the sum of the Bragg- and the diffuse scattering: ˜I = ˜IBragg + ˜Idiff ;
with ˜IBragg =
˜IBragg (qhk ).
(4)
hk
The contribution of diffuse scattering can be calculated from the total integral intensity ˜I (Eq. (3)) and ˜IBragg by Eq. (4). The effect of pores filling with the contrast-matching fluid DBM on the integral
Fig. 1. (a) X-ray scattering profile for SBA-15 silica with Bragg peaks resulting from the 2D hexagonal packing of cylindrical pores and a diffuse scattering background, indicated by the dashed area; (b) integral scattering intensities for DBM adsorption in SBA-15 as a function of relative pressure p/p0 of DBM: total integral scattering ˜I , integral intensity of the Bragg diffraction peaks ˜IBragg , and integral diffuse scattering intensity ˜Idiff . The pore condensation region is indicated by dotted vertical lines.
intensities ˜IBragg and ˜Idiff is shown in Fig. 1b. From these contributions one can derive more detailed information about the adsorption of the fluid in the ordered and disordered pores of the matrix, as explained in Sections 3 and 4. 3. Analysis of Bragg scattering The Bragg scattering intensity is proportional to the product of the structure factor S(q) of the mesopore lattice and the square of the scattering amplitude from a single mesopore (form factor) |F(q)|2 :
2
IBragg (qhk ) = KS(qhk )F(q) ,
(5)
where S(qhk ) represents a sum of ı-functions at the reciprocal lattice points (hk) and K is an instrumental constant. Changes in the intensity of the individual Bragg peaks as a function of the vapor pressure
G.H. Findenegg et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 357 (2010) 3–10
can be attributed to changes of the form factor as a consequence of fluid adsorption. The form factor is modeled by an appropriate analytical function and the parameters of this model function are obtained by fitting the set of integral intensities ˜IBragg (qhk ) to the experimental Bragg diffraction peaks.
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For SBA-15, the seminal X-ray study of Impéror-Clerc et al. [4] indicated the existence of a corona of reduced density around the cylindrical pores. We have adopted two different geometrical models of the form factor |F(q)|2 to account for this corona and for the different stages of fluid adsorption in the mesopores: a step-density model [6], in which the corona and the liquid-like adsorbed film are assumed to have uniform densities, and a density-gradient model [11], which assumes layers of linearly varying density. By comparing the results obtained with the two models one can assess the reliability of the pore volume of ordered pores and of the adsorbed amount in the ordered pores that are extracted by such an approach.
pore with porous corona (see Section 3.2). Having fixed the values of R0 and R1 , the Bragg reflection data for the sample with adsorbed fluid are then analyzed by determining best-fit values of the reduced corona density ˛ and film thickness t. The parameter ˇ is either fixed as the bulk density of the liquid adsorbate or treated as a third adjustable parameter. An appealing property of the stepdensity model is that it separates adsorption into the matrix and growth of an adsorbed film, so that the adsorbed amount can be calculated from the parameters for fluids of any electron density. Explicit expressions for the scattering amplitude F(q) for this model in the different stages of pore filling are given in Ref. [6]. Results of an analysis of the Bragg diffraction data for the adsorption of DBM in SBA-15 in terms of the step-density profile are shown in Fig. 3. It is seen that the reduced corona density ˛ = 1 /0 and the thickness of the adsorbed film t are increasing with the relative pressure p/p0 up to the onset of pore condensation (at p/p0 = 0.67), while the amplitude factor (which is proportional to the constant K of Eq. (5)) decreases. Also shown is the fit variance 2 , which is generally higher for the step-profile model than for the densitygradient model [11].
3.1.1. Step-density model [6] This model (see Fig. 2a) assumes a corona of uniform (but variable) density 1 extending from an outer radius R0 to an inner radius R1 , and a liquid-like adsorbed film of constant density 2 extending from R1 to a variable radius R2 < R1 , with t = R1 − R2 the film thickness. In the presence of such an adsorbed film this model entails four density levels, corresponding to the empty core in the center of the pore ( = 0), the adsorbed liquid-like film (2 ), the corona (1 ) and the pore wall (0 ). In the analytical expression for the form factor only the reduced densities ˛ = 1 /0 (reduced corona density) and ˇ = 2 /0 (reduced film density) appear. The parameters R0 and R1 (outer and inner radius of the corona) and ˛* (density of the empty corona) are determined by fitting the Bragg reflections of the evacuated sample. These three parameters deter¯ defined as the radius mine the value of the equivalent pore radius R, of a cylinder having the same volume per unit length as the model
3.1.2. Density-gradient model [11] This form factor model (Fig. 2b) was adopted to take into account effects due to the non-ideality of the pore lattice and the roughness of the pore walls of SBA-15 materials. When the pore walls are strongly corrugated it becomes difficult to distinguish between adsorption into a corona and film formation. Accordingly, in this model the corona region is split into two parts: an outer corona region of gradually decreasing microporosity (extending from R0 to R1 ), and an inner region of corrugated wall (extending from R1 to R2 ). Fig. 2b shows a sketch of such a mesopore for which we assume linear density gradients in the outer corona region and the wall corrugation region. This model contains only one density parameter, = 1 /0 , which in the case of empty pores ( * ) represent the reduced mean electron density at the transition from the porous corona to the wall corrugation region (radius R1 ). The explicit analytical expression for the form factor |F(q)|2 of this density-gradient
3.1. Mesopore models
Fig. 2. Form factor models for the ordered cylindrical pores showing a sketch of the 2D unit cell and the respective electron density profile (r): (a) step-density profile and (b) density-gradient model (see text for explanations).
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Fig. 3. Fit parameters of the step-density model as a function of the relative pressure p/p0 for the adsorption of DBM in SBA-15: reduced density of the corona ˛ = 1 /0 , mean layer thickness of adsorbed film t = R2 − R1 , amplitude factor amp, and fit variance 2 .
model is given elsewhere [11]. Adsorption of a matching fluid (i.e., a fluid having the same electron density as the silica matrix in the adsorbed state) will cause changes of the parameters R1 , R2 and , which can be attributed to distinct adsorption processes, as explained in Ref. [11]. As in the step-density model, changes in the filling fraction of the ordered pore system caused by the adsorption of a matching fluid can be calculated from the changes in the model parameters, as explained in Section 4.
The pore volume of ordered pores can be obtained by analyzing the integrated intensities of the Bragg reflections of the evacuated sample in terms of the appropriate form factor models. For SBA-15 and other materials with 2D hexagonal arrays of cylindrical pores ¯ defined as it is convenient to introduce the equivalent pore radius R, the radius of a cylinder having the same volume per unit length as the model pore with porous corona. The equivalent pore radius for the step-density model and the density-gradient model is given by Eqs. (6) and (7):
R¯ 2 =
1 − ∗ 2 ∗ 2 (R0 + R0 R1 + R12 ) + (R + R1 R2 + R22 ). 3 3 1
ord =
Vord 2 = √ V 3
(6)
(7)
From the equivalent pore radius R¯ and the lattice parameter a0 one can calculate the volume fraction of the ordered cylindrical pores, ord = Vord /V, where V is the volume per unit length of the 2D √ unit cell, V = (a20 /2) 3, and Vord the respective volume occupied by the cylindrical mesopore, Vord = R¯ 2 . Hence the volume fraction of
R¯ a0
2 .
(8)
The overall volume fraction of pores (total porosity) tot , made up of the ordered cylindrical mesopores and disordered pores in the walls, can be determined from the total specific pore volume vp and the mass density of the matrix ˜ S as follows: tot = ord + dis =
3.2. Volume of ordered and disordered pores
R¯ 2 = R12 + (1 − ˛∗ )(R02 − R12 ),
ordered pore space (ordered porosity) is
vp ˜ S . 1 + vp ˜S
(9)
vp is commonly determined by nitrogen adsorption and ˜ S can be obtained by helium displacement measurements. Hence Eqs. (8) and (9) can be used to determine the volume fraction of disordered pore space, dis , from ord and tot . Alternatively, the volumes of ordered and disordered pores may be expressed by specific pore volumes vord and vdis = vp − vord , i.e., volumes per unit mass of silica matrix (see Appendix A). Assessment of the ordered and disordered porosity of periodic mesoporous materials by this formalism is straightforward [11]. For a high-quality SBA-15 material which had undergone hydrothermal aging at 100 ◦ C for 24 h, a total pore volume vp = 1.01 cm3 g−1 was found by nitrogen adsorption, corresponding to a total poros∗ = 0.686 (using ˜ S = 2.16 g cm−3 , from helium displacement ity tot measurements). Analysis of the Bragg reflection data by the two form factor models gave closely similar values of the equivalent pore radius R¯ and this value agrees with the pore radius from nitrogen adsorption based on the improved KJS prescription [16]. The (somewhat smaller) pore radius derived by the nonlocal density functional theory (NLDFT) method [17] agrees with the radius R1 of the step-density model [11]. For the given SBA-15 material, Eq. ∗ = 0.554 ± 0.003, and thus a dis(8) yields an ordered porosity ord
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∗ = ∗ − ∗ = 0.132. (Here and below the ordered porosity dis tot ord asterisk indicates properties of the evacuated sample.) Hence the disordered porosity makes up nearly 20% of the overall porosity of this sample.
4. Fluid adsorption in disordered and ordered pores When part of the pore space is occupied by an adsorbed fluid (F), Eq. (9) is replaced by ∗ = ord (p) + dis (p) + F (p) = 1 − S , tot
(10)
where ord (p) and dis (p) denote the remaining fractions of empty pore space and F (p) is the overall volume fraction occupied by the adsorbed fluid at pressure p. In this section it is shown that the volume fraction of fluid in the ordered and disordered pores can be estimated separately. These volume fractions are denoted as F,ord (p) and F,dis , and thus the total volume fraction of adsorbed fluid is F (p) = F,ord (p) + F,dis (p).
(11)
4.1. Adsorption in disordered pores Filling of the disordered pores in the matrix when a fluid is admitted can be assessed by analyzing the diffuse small-angle scattering from the sample. The integral intensity from diffuse scattering ˜Idiff is obtained from the total integrated intensity ˜I and the integrated intensity from Bragg scattering, ˜IBragg , as ˜Idiff (p) = ˜I (p) − ˜IBragg (p). Fig. 1b shows how the three quantities vary with relative pressure as the pore space is gradually filled with the contrast-matching fluid DBM. Specifically, the diffuse scattering ˜Idiff (p) decreases in a monotonic manner as the fluid pressure increases up to the onset of pore condensation, clearly indicating adsorption into disordered pores. As the adsorbed fluid has the same electron density as the matrix the filling of the disordered pores can be estimated from the dependence of ˜Idiff on the fluid pressure using the relation for a two-phase system: ˜Idiff (p) = Cϕm (1 − ϕm ) + B,
(12)
where ϕm is the intrawall porosity, i.e., the pressure-dependent volume fraction of empty pore space within the walls, ϕm (p) = ∗ + ), and B is a pressure-independent incoherent dis (p)/(dis S scattering background. When B is known, the constant C of Eq. (12) can be calculated from data of the empty sample, viz., ˜Idiff (p = 0) ∗ = ∗ /(∗ + ) = ∗ /(1 − ∗ ). A difficulty arises from and ϕm S dis dis dis ord the fact that the parameter B is not clearly defined experimentally. In the example presented in Fig. 1b, B was taken as the value of ˜Idiff (p) at a pressure at which pore condensation is just completed (p/p0 = 0.8). Based on this fixing the constant C of Eq. (12) was determined from the parameters for the evacuated sample ∗ = 0.296 and ˜ Idiff (0) − B = 0.394). The resulting dependence of (ϕm the wall porosity on the fluid pressure is shown in Fig. 4. The graph indicates that ca. 50% of the wall porosity is filled at relative pressures below 0.2, as expected for pore sizes below 2 nm. However, filling of the disordered pores continues up to and into the pore condensation region, indicating the existence of disordered pores of nearly the size of the cylindrical channels. Note, however, that this conclusion relies on the correct determination of the background scattering intensity B of Eq. (12). We return to this point in Section 5. The volume fraction of fluid in disordered pore space is obtained from ϕm (p) as follows: ∗ ∗ − dis (p) = dis (1 − ϕm (p)) − ϕm (p)S F,dis (p) = dis
(13)
According to Eq. (13) F,dis (p) increases as ϕm (p) decreases and ∗ when ϕ = 0, i.e., reaches its maximum value F,dis (max) = dis m when the intrawall porosity is completely filled.
Fig. 4. Matrix porosity ϕm as a function of the relative pressure p/p0 of DBM indicating the gradual filling of the intrawall porosity by the contrast-matching adsorbate. The inset shows a sketch of the porous matrix.
4.2. Adsorption in the ordered pores If the fluid has the same electron density as the matrix, the sample still conforms to a two-phase system. In this case Eq. (8) can be used to determine the volume fraction of empty space in the ordered ¯ which for the step pores, ord (p). Again one starts by calculating R, model is now given by R¯ 2 = R22 + (1 − ˇ)(R12 − R22 ) + (1 − ˛)(R02 − R12 ).
(14)
instead of Eq. (6), where ˇ = 1 for the case of a contrast-matching liquid. To calculate R¯ by the step model for a given pressure p of the adsorbed fluid one uses the values of R2 and ˛ derived for this pressure (Fig. 3) and the values of R0 and R1 obtained for the empty material. For the density-gradient model, Eq. (7) is applied with values of R2 , R1 and as derived for the given pressure, and the value of R0 as obtained for the empty material. The volume of adsorbed liquid in the ordered pores, expressed as volume fraction in a unit cell, F,ord , is then given by ∗ F,ord (p) = ord − ord (p),
(15)
∗ the volume fraction of ordered pores in the empty samwith ord ple. However, the values of F,ord (p) obtained from Eq. (15) may not represent the true (absolute) adsorbed amount, as is explained below. In the analysis of the Bragg scattering data in terms of the form factor models, the density profiles are expressed by the reduced densities ˛ = 1 /0 and ˇ = 2 /0 (step model), or = 2 /0 (density-gradient model). For materials without intrawall porosity, the wall density 0 is constant and thus adsorption-induced changes of these reduced densities can be attributed solely to changes in 1 and 2 (step model), or 1 (density-gradient model). SBA-15 and other polymer-templated ordered mesoporous materials, on the other hand, may have high levels of intrawall porosity [1–3]. Filling of these disordered pores with a contrast-matching fluid causes an increase of the wall density from 0∗ (evacuated sample) to S (skeleton density of silica), which is reached when the filling of the intrawall porosity has come to completion. This change in 0 can be taken into account by re-adjusting the density parameters ˛ and ˇ (step model) or (gradient model) to the new values ˛ , ˇ , and . Specifically, for the parameters ˛ and which
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refer to the density 1 : ˛, ≡
1 1 1 = 0 S 1 − ϕm
˛ , ≡
(16a)
1 1 1 = ∗ 0∗ S 1 − ϕm
(16b)
∗ ), with ϕ∗ the wall porosity of where we have used 0∗ = S (1 − ϕm m the evacuated sample. An analogous re-adjustment applies to the parameter ˇ which refers to the density level 2 of the step-density model. Eq. (16a) expresses densities in the ordered pores relative to the fluid density in the wall at the respective pressure. Accordingly, the adsorption calculated on this basis represents a relative adsorprel . These values are obtained when the reduced densities tion F,ord ˛(p) or (p) as obtained in the fitting of the form factor models are used (i.e., values from Fig. 3 in the present example). On the other hand, when adopting the adjusted reduced density ˛ (p) or (p) of Eq. (16b), the changes in these parameters are ascribed solely to density changes in the ordered pores. In this way the hidden dependence of densities in the ordered pores on the changing wall density is removed. Accordingly, the adsorption calculated on the basis of the density parameters of Eq. (16b) represents the absolute abs . From Eq. (16a,b) and the adsorption in the ordered pores, F,ord analogous relations for the reduced densities ˇ and ˇ one obtains:
ˇ 1 − ϕm ˛ = = = ∗ . ˛ 1 − ϕm ˇ
(17)
The ratio on the r.h.s. of Eq. (17) increases with the fluid pressure, as the intrawall porosity ϕm decreases when the disordered pores are gradually filled. Accordingly, the density of the fluid in the ordered pores, when calculated with the parameters ˛ and ˇ (or ), increases more strongly than when the original parameters ˛ and ˇ (or ) are used. This implies that the values of the absolute adsorption are higher than the values of the relative adsorption at any given pressure. Similar to the situation of high-pressure adsorption of fluids at flat solid surfaces [18], the relative adsorption may be zero or even negative, even though fluid has entered into the ordered pores. This effect can be seen most clearly at low pressures, when the fluid is accommodated only in the corona of the mesopores and in narrow pores within the walls (see density profiles in Fig. 2): If the density 1 increases in the same proportion as (or to a lesser extent than) the matrix density 0 , the relative rel adsorption F,ord will be zero (or negative), although the absolute
abs , is still positive. adsorption in the ordered pores, F,ord As an example, the isotherms of the relative and absolute rel (p) and abs (p), for DBM in the present SBA-15 adsorption, F,ord F,ord sample at pressures up to the onset of pore condensation are shown ∗ and in Fig. 5a. In this example the quantities ord ord (p) of Eq. (15) were calculated on the basis of the step-density profile, i.e., using Eqs. (14) and (8) with the pressure-dependent parameters ˛ and R2 of Fig. 3. As can be seen in Fig. 5a, the absolute adsorption exceeds the relative adsorption by nearly a factor of 2 in this case. The large rel (p) and abs (p) is a consequence of the difference between F,ord F,ord ∗ ≈ 0.3) as high intrawall porosity of the present SBA-15 sample (ϕm well as the high electron density of the present fluid in the adsorbed state. The overall adsorption of the fluid in the ordered and disordered pores, expressed as the filling fraction f(p), is calculated by the relations:
f (p) = 1 −
abs (p) + (p) ord dis ∗ tot
,
(18)
∗ + )ϕ (p). A comparison of f(p) with the fillwhere dis (p) = (dis m S ing fraction obtained by gravimetric adsorption measurements is shown in Fig. 5b. For the pressure range below pore condensation, to which this analysis applies, ftot (p) exceeds the gravimetric
Fig. 5. DBM adsorption in SBA-15: (a) absolute and relative values of the volume rel abs fraction of adsorbed liquid in the ordered pores, F,ord and F,ord , as a function of the relative pressure p/p0 for the step-density profile (full circles) and the densityabs indicated by smaller circles gradient model (open circles); the curve for F,ord
∗ illustrates the indirect effect of a lower interwall porosity ϕm on the volume fraction abs (see text) and (b) comparison of the of adsorbed liquid in the ordered pores, F,ord
filling fraction f derived from the in situ X-ray study (full symbols) with the filling fraction obtained by gravimetric adsorption measurements (open symbols).
adsorption isotherm by 50% at low pressures and 10% at the highest pressures. The comparison of the two isotherms suggests that ∗ of the present we have overestimated the disordered porosity dis SBA-15 by ca. 15%. A reduction of dis affects the calculated adsorption in a dual way: directly, by lowering the adsorbed volume F,dis , and indirectly, via the difference between relative and absolute adsorption in the ordered pores, which depends on the gradual filling of the matrix porosity as the fluid pressure is increased (Fig. 4) [11]. 5. Discussion The new method for analyzing in situ small-angle X-ray diffraction and scattering data [11,12] is applicable to a wide class of ordered mesoporous materials with intrawall porosity. The method is based on a separation of the two constituents of the scattering profile I(q), viz., Bragg scattering IBragg (qhk ) from the mesopore lattice, and diffuse scattering Idiff (q) from the disordered pores in the matrix. Bragg scattering is analyzed in terms of geometric models of the cylindrical pores and their microporous corona. Two different prescriptions for the radial density profile of the pores (step-density model and density-gradient model) are adopted and found to give closely similar values of the equivalent pore radius R¯ in excellent agreement with the pore radius RKJS determined from nitrogen adsorption isotherms on the basis of the improved
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KJS method [16]. Hence reliable values of the volume fraction of ∗ are obtained from R¯ with Eq. (8). From this ordered pores ord ∗ of the sample the volume value and the known total porosity tot ∗ fraction of disordered pores dis can be derived. For the wellordered SBA-15 material (hydrothermal aging at 100 ◦ C for 24 h) used in the present case study, the disordered (intrawall) porosity amounts to nearly 20% of the total porosity. This result confirms earlier findings concerning the intrawall porosity of SBA-15 materials based solely on a NLDFT analysis of nitrogen adsorption isotherms [17]. The same formalism can be used to analyze in situ X-ray data for the adsorption of a fluid in the ordered and disordered pores of SBA-15. For the present case of a contrast-matching fluid the volume fraction of adsorbed fluid in the ordered pores, F,ord (p), can be calculated from the decrease in the volume fraction of empty pore space ord (p) with increasing fluid pressure p (Eq. (15)). The respective volume fraction of adsorbed fluid in the disordered pores, F,dis (p), cannot be determined directly because of the illdefined geometry of the intrawall pore system. However, in the case of a contrast-matching liquid, the volume fraction of empty intrawall pore space, dis (p), can be calculated from the integral diffuse scattering intensity ˜Idiff (p) on the basis of Eq. (12), which is applicable for a two-phase system. A difficulty with this procedure can be the correct determination of the incoherent scattering background B of Eq. (12). For a contrast-matching liquid Eq. (12) implies that ˜Idiff (p) reduces to this constant B when all intrawall pores are filled (ϕm = 0). However, we have found [11] that pore condensation in the ordered cylindrical pores also causes pronounced diffuse scattering which leads to a local maximum of ˜Idiff (p) in the pore condensation region (Fig. 1b). From a modeling study of the pore filling [12] it was concluded that the two contributions to ˜Idiff (p) are independent in good approximation and can thus be separated. Accordingly, ˜Idiff (p) should attain a constant value B at pressures beyond the pore condensation regime, when all pores are filled. Unfortunately, for the present fluid it was not possible to perform reliable in situ scattering measurements in the pressure range close to saturation pressure and thus the assumption underlying the determination of B could not be verified experimentally. Further work to resolve this problem is in progress. A quantitative determination of adsorption into the intrawall pores is important for the characterization of not only the disordered porosity, but also because of its effect on the electron density of the wall, 0 , by which it affects the determination of the adsorbed amount in the ordered pores. As we have seen, this coupling can be taken into account by a re-adjustment of the electron densities in Eq. (16). Fig. 5a shows that this re-adjustment has a pronounced effect on the calculated values of F,dis (p), as the adjusted (absolute) values are greater than the relative values by about a factor 2. This implies that the calculation of the adsorbed amount in the ∗ adopted ordered pores depends sensitively on the wall porosity ϕm in Eq. (16). To demonstrate this effect, the pore volume vp of the present sample was assumed to be 5% smaller than the experimental value (0.96 cm3 g−1 instead of 1.01 cm3 g−1 ) on account of the disordered pore space (i.e., keeping the volume of the ordered pore ∗ translates space constant). The resulting decrease in the value of ϕm abs (p) of about 8%, as indicated by the smaller into a decrease of F,ord symbols in Fig. 5a. Fig. 5b shows that in the pressure range up to pore condensation the values of the filling fraction f(p) derived from the X-ray data exceed the adsorption isotherm obtained by gravimetric measurements. This deviation is most pronounced at low pressures and amounts to about 10% at the highest pressures (close to pore condensation). As discussed above, this deviation may be attributed to an overestimate of the disordered porosity of the sample. However, the observed deviation might also reflect more fundamental shortcomings of the present treatment. Two possible arguments of this
9
kind are: (1) It may be argued that a separation of the scattered intensities in Bragg diffraction and diffuse scattering (Eqs. (1) and (4)) and the concomitant separation into ordered and disordered pore space is an unjustified simplification. However, this argument was refuted by our recent work [12] in which it was shown that a simple structural model for SBA-15 with hexagonally ordered mesopores and randomly distributed smaller interwall pores is able to reproduce quantitatively the experimental X-ray scattering curves. (2) The adsorbent (dibromomethane) chosen in this study may cause complications due to its high cohesive energy density and poor wetting behavior for silica. Contact angle measurements against a surface-oxidized silicon wafer indeed showed that DBM is not completely wetting the silica surface [11]. This may explain the observation that the filling of the disordered pores extends over a wide pressure range up to pore condensation (Fig. 4). An alternative interpretation of this finding is that the density of DBM in narrow pores is lower than the bulk density, so that it is not a contrast-matching liquid, and thus the underlying two-phase approximation is not correct. Studies with other adsorbates are in progress to test this point.
Acknowledgement Financial support from the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center SFB 448 (Projects B1 and B14) is gratefully acknowledged. Appendix A. Specific pore volumes. In Eqs. (8) and (9) the volumes of ordered and disordered pore space in the sample are expressed as volume fractions, i.e., pore volumes per unit length of the 2D unit cell of SBA15. The respective pore volumes per unit mass of the silica matrix (specific pore volumes) are then given by
vord =
Vord 1 ord = , VS ˜S ˜ S S
(A1)
vdis =
Vdis 1 dis = ˜S VS ˜ S S
(A2)
where VS is the matrix volume per unit length of the 2D hexagonal unit cell, VS = VS , and S = 1 − tot is the volume fraction of silica. In Eq. (15) the volume fraction of adsorbed liquid in the ordered pores, F,ord (p), is expressed by the volume fractions of empty pore space in the evacuated sample and at a pressure p. The respective volume of adsorbed liquid per unit mass of the matrix (specific adsorbed volume) is then given by
vF,ord (p) = v∗ord − vord (p), where v∗ord and vord (p) are calculated ∗ and ord ord (p), respectively.
(A3) by Eq. (A1) with the values
References [1] A. Galarneau, H. Cambon, F. Di Renzo, R. Ryoo, M. Choi, F. Fajula, New J. Chem. 27 (2003) 73–79. [2] O. Sel, A. Brandt, D. Wallacher, M. Thommes, B. Smarsly, Langmuir 23 (2007) 4724–4727. [3] O.C. Gobin, Y. Wan, D. Zhao, F. Kleitz, S. Kaliaguine, J. Phys. Chem. C 111 (2007) 3053–3058. [4] M. Impéror-Clerc, P. Davidson, A. Davidson, J. Am. Chem. Soc. 122 (2000) 11925–11933. [5] T. Hofmann, D. Wallacher, P. Huber, R. Birringer, K. Knorr, A. Schreiber, G.H. Findenegg, Phys. Rev. B 72 (2005) 064122.
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[6] G.A. Zickler, S. Jähnert, W. Wagermaier, S.S. Funari, G.H. Findenegg, O. Paris, Phys. Rev. B 73 (2006) 184109. [7] L.A. Solovyov, O.V. Belousov, R.E. Dinnebier, A.N. Shmakov, S.D. Kirik, J. Phys. Chem. B 109 (2005) 3233–3237. [8] F. Kleitz, T. Czuryszkiewicz, L.A. Solovyov, M. Lindén, Chem. Mater. 18 (2006) 5070–5079. [9] N. Muroyama, A. Yoshimura, Y. Kubota, K. Miyasaka, T. Ohsuna, R. Ryoo, P.I. Ravikovitch, A.V. Neimark, M. Takata, O. Terasaki, J. Phys. Chem. C 112 (2008) 10803–10813. [10] K. Miyasaka, A.V. Neimark, O. Terasaki, J. Phys. Chem. C 113 (2009) 791–794. [11] S. Jähnert, D. Müter, J. Prass, G.A. Zickler, O. Paris, G.H. Findenegg, J. Phys. Chem. C 113 (2009) 15201–15210.
[12] D. Müter, S. Jähnert, J.W.C. Dunlop, G.H. Findenegg, O. Paris, J. Phys. Chem. C 113 (2009) 15211–15217. [13] T.-W. Kim, F. Kleitz, B. Paul, R. Ryoo, J. Am. Chem. Soc. 127 (2005) 7601–7610. [14] F. Kleitz, T.-W. Kim, R. Ryoo, Langmuir 22 (2006) 440–445. [15] O. Glatter, O. Kratky, Small-angle X-ray scattering, Academic Press, New York, 1983. [16] M. Jaroniec, L.A. Solovyov, Langmuir 22 (2006) 6757–6760. [17] P.I. Ravikovitch, A.V. Neimark, J. Phys. Chem. B 105 (2001) 6817–6823. [18] G.H. Findenegg, M. Thommes, in: J. Fraissard, C.W. Conner (Eds.), Physical Adsorption: Experiment, Theory and Applications, NATO ASI Series C, Vol. 491, Kluwer, Dordrecht, 1997, pp. 151–179.
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Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Analysis of pore structure and gas adsorption in periodic mesoporous solids by in situ small-angle X-ray scattering Gerhard H. Findenegg a,∗ , Susanne Jähnert a , Dirk Müter b , Oskar Paris b,c a b c
Institut für Chemie, Stranski Laboratory, Sekr. ER1, Technical University Berlin, D-10623 Berlin, Germany Department of Biomaterials, Max Planck Institute of Colloids and Interfaces, D-14424 Potsdam, Germany Institute of Physics, University of Leoben, A-8700 Leoben, Austria
a r t i c l e
i n f o
Article history: Received 31 July 2009 Received in revised form 23 September 2009 Accepted 25 September 2009 Available online 6 October 2009 Keywords: SBA-15 silica Ordered nanoporous materials In situ small-angle scattering Intrawall porosity Pore filling
a b s t r a c t This paper describes a new method for characterizing the pore structure and gas adsorption in periodic mesoporous materials based on small-angle X-ray diffraction and scattering. For SBA-15 as a case study it is shown that the effective pore radius and volume of the ordered cylindrical pores and their corona can be determined accurately from the intensities of the Bragg reflections. The contribution of the intrawall porosity to the pore volume is then obtained by comparing this ordered pore volume with the total pore volume from nitrogen adsorption. Adsorption of a fluid in the ordered and disordered pores of SBA-15 is studied by analyzing scattering data measured in situ as a function of the vapor pressure of the fluid. In this way the different stages of pore filling can be monitored, and separate filling isotherms of the ordered and intrawall pores can be obtained for an adsorbate (dibromomethane) having a similar electron density as the solid matrix. Our analysis shows that the filling of the intrawall porosity has a pronounced effect on the evaluation of the adsorbed amount in the ordered pores. These finding emphasize the importance of disordered intrawall porosity in polymer-templated silicas like SBA-15 and open up new ways for an in-depth characterization of these materials. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Periodic mesoporous silicas synthesized with amphiphilic block copolymers as structure-directing agent not only constitute regular arrangements of mesopores but also contain significant levels of disordered intrawall porosity, depending on the protocol of sample preparation [1–3]. Structural information about the ordered pore system can be obtained from X-ray small-angle diffraction data, either by adopting simple geometrical models for the density distribution which can be analytically Fourier transformed [4–6], or by numerical Fourier transformation of more sophisticated density distributions [7,8]. The former formalism can also be adapted to derive structural information about the adsorption of fluids in the ordered pores [5,6,9,10]. However, in the past most studies were focussed on the ordered pore system by analyzing the integral intensities of Bragg diffraction peaks arising from the mesopore lattice. Diffuse scattering caused by structural disorder (e.g., intrawall pores) was commonly treated as a background and not included in the modeling. Accordingly, no reliable quantitative estimates of the intrawall porosity exist. Recently we proposed a method for estimating this wall porosity by comparing the pore volume of ordered
∗ Corresponding author. E-mail address: fi[email protected] (G.H. Findenegg). 0927-7757/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2009.09.053
pores with the total pore volume as obtained by nitrogen adsorption [11]. It was also shown that information about the intrawall porosity can be derived from the diffuse small-angle scattering and its dependence on pore filling [12]. This paper gives an account of this new method for characterizing ordered and disordered porosity in periodic mesoporous materials, and its extension to the determination of adsorption isotherms of fluids in the ordered and disordered pores, based on in situ X-ray small-angle scattering. The formalism described below applies to materials exhibiting a two-dimensional (2D) hexagonal arrangement of cylindrical pores (plane group p6mm), such as MCM-41 and SBA-15, but may be extended to other periodic mesoporous solids, such as the cubic MCM-48 [7] and KIT-6 [13] (space group Ia3d) or the bcc structure of SBA-16 (space group Im3m) [14]. Dibromomethane (DBM, CH2 Br2 ) is chosen as the test fluid in the exemplary results presented in this paper, because DBM in its bulk liquid state has nearly the same electron density as silica, which facilitates the analysis of the scattering data [11]. The pore structure and pore filling of the ordered cylindrical pores are determined by analyzing the integrated intensities of up to 10 Bragg reflections in terms of analytical form factor models for cylindrical pores. In these models the corrugated pore walls and a microporous corona of the cylindrical pores are taken into account in a simple but physically meaningful way. In addition, randomly distributed, uncorrelated pores in the walls of the ordered mesopore lattice are considered
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as the source of diffuse scattering. Information about the gradual filling of these pores is extracted from the decreasing overall intensity of the diffuse scattering as a function of the relative pressure of the test fluid. Adsorption into the disordered pores in the matrix not only contributes to the overall adsorbed amount but also affects the determination of the amounts adsorbed in the ordered pores, due to the increase in electron density of the matrix when the disordered pores in the matrix are filled with the fluid. It is shown how this effect can be taken into account by a re-normalization of the density profiles in the form factor models, by which the amount of adsorbed fluid in the ordered pores can be converted to an absolute scale. The overall adsorption isotherms of the test fluid derived from the in situ X-ray study on the basis of this formalism is compared with the adsorption isotherm obtained by gravimetric adsorption technique. Finally, shortcomings and possible extensions of the new formalism are discussed briefly. 2. X-ray scattering and integrated scattering intensities A typical small-angle scattering profile I(q) for SBA-15 silica with the Bragg reflections resulting from the 2D hexagonal arrangement of the cylindrical mesopores is shown in Fig. 1a. The dashed area indicates diffuse small-angle scattering arising from disordered pores and other inhomogeneities of the matrix. Information about fluid adsorption in the ordered and disordered pores can be derived from diffraction and diffuse scattering, respectively, and their dependencies on the fluid pressure. Here we summarize the relations for the special case of a two-phase system [15], i.e., when the adsorbed fluid in its bulk liquid state has the same electron density as the solid matrix. As indicated in Fig. 1, the total scattered intensity can be separated into two parts: Bragg diffraction from the mesopore lattice (IBragg ), and diffuse small-angle scattering arising from disordered pores and other inhomogeneities of the matrix (Idiff ): I(q) = IBragg (q) + Idiff (q).
(1)
Integral intensities of the individual Bragg diffraction peaks (hk) can be determined by
˜IBragg (qhk ) =
IBragg (qhk )q2 dq,
(2)
where the diffuse scattering contribution is estimated from a double tangent at the base of the respective diffraction peak and subtracted separately for each peak. The total integral scattering intensity ˜I is given by
∞
I(q)q2 dq
˜I =
(3)
0
and can be determined by numerical integration of the scattering profiles within the appropriate q range (typically 0.3 nm−1 < q < 3.0 nm−1 ) and extrapolation of I(q) to zero and infinite q. Extrapolation to infinite q can be made using Porod’s law [15], since it is commonly found that the scattering curves are consistent with a q−4 behavior at large q. At the lower end of the experimental q range a constant scattering intensity may be assumed, i.e., I(q < qmin ) = I(qmin ) with qmin = 0.3 nm−1 . According to Eq. (1) the total integral intensity is given by the sum of the Bragg- and the diffuse scattering: ˜I = ˜IBragg + ˜Idiff ;
with ˜IBragg =
˜IBragg (qhk ).
(4)
hk
The contribution of diffuse scattering can be calculated from the total integral intensity ˜I (Eq. (3)) and ˜IBragg by Eq. (4). The effect of pores filling with the contrast-matching fluid DBM on the integral
Fig. 1. (a) X-ray scattering profile for SBA-15 silica with Bragg peaks resulting from the 2D hexagonal packing of cylindrical pores and a diffuse scattering background, indicated by the dashed area; (b) integral scattering intensities for DBM adsorption in SBA-15 as a function of relative pressure p/p0 of DBM: total integral scattering ˜I , integral intensity of the Bragg diffraction peaks ˜IBragg , and integral diffuse scattering intensity ˜Idiff . The pore condensation region is indicated by dotted vertical lines.
intensities ˜IBragg and ˜Idiff is shown in Fig. 1b. From these contributions one can derive more detailed information about the adsorption of the fluid in the ordered and disordered pores of the matrix, as explained in Sections 3 and 4. 3. Analysis of Bragg scattering The Bragg scattering intensity is proportional to the product of the structure factor S(q) of the mesopore lattice and the square of the scattering amplitude from a single mesopore (form factor) |F(q)|2 :
2
IBragg (qhk ) = KS(qhk )F(q) ,
(5)
where S(qhk ) represents a sum of ı-functions at the reciprocal lattice points (hk) and K is an instrumental constant. Changes in the intensity of the individual Bragg peaks as a function of the vapor pressure
G.H. Findenegg et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 357 (2010) 3–10
can be attributed to changes of the form factor as a consequence of fluid adsorption. The form factor is modeled by an appropriate analytical function and the parameters of this model function are obtained by fitting the set of integral intensities ˜IBragg (qhk ) to the experimental Bragg diffraction peaks.
5
For SBA-15, the seminal X-ray study of Impéror-Clerc et al. [4] indicated the existence of a corona of reduced density around the cylindrical pores. We have adopted two different geometrical models of the form factor |F(q)|2 to account for this corona and for the different stages of fluid adsorption in the mesopores: a step-density model [6], in which the corona and the liquid-like adsorbed film are assumed to have uniform densities, and a density-gradient model [11], which assumes layers of linearly varying density. By comparing the results obtained with the two models one can assess the reliability of the pore volume of ordered pores and of the adsorbed amount in the ordered pores that are extracted by such an approach.
pore with porous corona (see Section 3.2). Having fixed the values of R0 and R1 , the Bragg reflection data for the sample with adsorbed fluid are then analyzed by determining best-fit values of the reduced corona density ˛ and film thickness t. The parameter ˇ is either fixed as the bulk density of the liquid adsorbate or treated as a third adjustable parameter. An appealing property of the stepdensity model is that it separates adsorption into the matrix and growth of an adsorbed film, so that the adsorbed amount can be calculated from the parameters for fluids of any electron density. Explicit expressions for the scattering amplitude F(q) for this model in the different stages of pore filling are given in Ref. [6]. Results of an analysis of the Bragg diffraction data for the adsorption of DBM in SBA-15 in terms of the step-density profile are shown in Fig. 3. It is seen that the reduced corona density ˛ = 1 /0 and the thickness of the adsorbed film t are increasing with the relative pressure p/p0 up to the onset of pore condensation (at p/p0 = 0.67), while the amplitude factor (which is proportional to the constant K of Eq. (5)) decreases. Also shown is the fit variance 2 , which is generally higher for the step-profile model than for the densitygradient model [11].
3.1.1. Step-density model [6] This model (see Fig. 2a) assumes a corona of uniform (but variable) density 1 extending from an outer radius R0 to an inner radius R1 , and a liquid-like adsorbed film of constant density 2 extending from R1 to a variable radius R2 < R1 , with t = R1 − R2 the film thickness. In the presence of such an adsorbed film this model entails four density levels, corresponding to the empty core in the center of the pore ( = 0), the adsorbed liquid-like film (2 ), the corona (1 ) and the pore wall (0 ). In the analytical expression for the form factor only the reduced densities ˛ = 1 /0 (reduced corona density) and ˇ = 2 /0 (reduced film density) appear. The parameters R0 and R1 (outer and inner radius of the corona) and ˛* (density of the empty corona) are determined by fitting the Bragg reflections of the evacuated sample. These three parameters deter¯ defined as the radius mine the value of the equivalent pore radius R, of a cylinder having the same volume per unit length as the model
3.1.2. Density-gradient model [11] This form factor model (Fig. 2b) was adopted to take into account effects due to the non-ideality of the pore lattice and the roughness of the pore walls of SBA-15 materials. When the pore walls are strongly corrugated it becomes difficult to distinguish between adsorption into a corona and film formation. Accordingly, in this model the corona region is split into two parts: an outer corona region of gradually decreasing microporosity (extending from R0 to R1 ), and an inner region of corrugated wall (extending from R1 to R2 ). Fig. 2b shows a sketch of such a mesopore for which we assume linear density gradients in the outer corona region and the wall corrugation region. This model contains only one density parameter, = 1 /0 , which in the case of empty pores ( * ) represent the reduced mean electron density at the transition from the porous corona to the wall corrugation region (radius R1 ). The explicit analytical expression for the form factor |F(q)|2 of this density-gradient
3.1. Mesopore models
Fig. 2. Form factor models for the ordered cylindrical pores showing a sketch of the 2D unit cell and the respective electron density profile (r): (a) step-density profile and (b) density-gradient model (see text for explanations).
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Fig. 3. Fit parameters of the step-density model as a function of the relative pressure p/p0 for the adsorption of DBM in SBA-15: reduced density of the corona ˛ = 1 /0 , mean layer thickness of adsorbed film t = R2 − R1 , amplitude factor amp, and fit variance 2 .
model is given elsewhere [11]. Adsorption of a matching fluid (i.e., a fluid having the same electron density as the silica matrix in the adsorbed state) will cause changes of the parameters R1 , R2 and , which can be attributed to distinct adsorption processes, as explained in Ref. [11]. As in the step-density model, changes in the filling fraction of the ordered pore system caused by the adsorption of a matching fluid can be calculated from the changes in the model parameters, as explained in Section 4.
The pore volume of ordered pores can be obtained by analyzing the integrated intensities of the Bragg reflections of the evacuated sample in terms of the appropriate form factor models. For SBA-15 and other materials with 2D hexagonal arrays of cylindrical pores ¯ defined as it is convenient to introduce the equivalent pore radius R, the radius of a cylinder having the same volume per unit length as the model pore with porous corona. The equivalent pore radius for the step-density model and the density-gradient model is given by Eqs. (6) and (7):
R¯ 2 =
1 − ∗ 2 ∗ 2 (R0 + R0 R1 + R12 ) + (R + R1 R2 + R22 ). 3 3 1
ord =
Vord 2 = √ V 3
(6)
(7)
From the equivalent pore radius R¯ and the lattice parameter a0 one can calculate the volume fraction of the ordered cylindrical pores, ord = Vord /V, where V is the volume per unit length of the 2D √ unit cell, V = (a20 /2) 3, and Vord the respective volume occupied by the cylindrical mesopore, Vord = R¯ 2 . Hence the volume fraction of
R¯ a0
2 .
(8)
The overall volume fraction of pores (total porosity) tot , made up of the ordered cylindrical mesopores and disordered pores in the walls, can be determined from the total specific pore volume vp and the mass density of the matrix ˜ S as follows: tot = ord + dis =
3.2. Volume of ordered and disordered pores
R¯ 2 = R12 + (1 − ˛∗ )(R02 − R12 ),
ordered pore space (ordered porosity) is
vp ˜ S . 1 + vp ˜S
(9)
vp is commonly determined by nitrogen adsorption and ˜ S can be obtained by helium displacement measurements. Hence Eqs. (8) and (9) can be used to determine the volume fraction of disordered pore space, dis , from ord and tot . Alternatively, the volumes of ordered and disordered pores may be expressed by specific pore volumes vord and vdis = vp − vord , i.e., volumes per unit mass of silica matrix (see Appendix A). Assessment of the ordered and disordered porosity of periodic mesoporous materials by this formalism is straightforward [11]. For a high-quality SBA-15 material which had undergone hydrothermal aging at 100 ◦ C for 24 h, a total pore volume vp = 1.01 cm3 g−1 was found by nitrogen adsorption, corresponding to a total poros∗ = 0.686 (using ˜ S = 2.16 g cm−3 , from helium displacement ity tot measurements). Analysis of the Bragg reflection data by the two form factor models gave closely similar values of the equivalent pore radius R¯ and this value agrees with the pore radius from nitrogen adsorption based on the improved KJS prescription [16]. The (somewhat smaller) pore radius derived by the nonlocal density functional theory (NLDFT) method [17] agrees with the radius R1 of the step-density model [11]. For the given SBA-15 material, Eq. ∗ = 0.554 ± 0.003, and thus a dis(8) yields an ordered porosity ord
G.H. Findenegg et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 357 (2010) 3–10
7
∗ = ∗ − ∗ = 0.132. (Here and below the ordered porosity dis tot ord asterisk indicates properties of the evacuated sample.) Hence the disordered porosity makes up nearly 20% of the overall porosity of this sample.
4. Fluid adsorption in disordered and ordered pores When part of the pore space is occupied by an adsorbed fluid (F), Eq. (9) is replaced by ∗ = ord (p) + dis (p) + F (p) = 1 − S , tot
(10)
where ord (p) and dis (p) denote the remaining fractions of empty pore space and F (p) is the overall volume fraction occupied by the adsorbed fluid at pressure p. In this section it is shown that the volume fraction of fluid in the ordered and disordered pores can be estimated separately. These volume fractions are denoted as F,ord (p) and F,dis , and thus the total volume fraction of adsorbed fluid is F (p) = F,ord (p) + F,dis (p).
(11)
4.1. Adsorption in disordered pores Filling of the disordered pores in the matrix when a fluid is admitted can be assessed by analyzing the diffuse small-angle scattering from the sample. The integral intensity from diffuse scattering ˜Idiff is obtained from the total integrated intensity ˜I and the integrated intensity from Bragg scattering, ˜IBragg , as ˜Idiff (p) = ˜I (p) − ˜IBragg (p). Fig. 1b shows how the three quantities vary with relative pressure as the pore space is gradually filled with the contrast-matching fluid DBM. Specifically, the diffuse scattering ˜Idiff (p) decreases in a monotonic manner as the fluid pressure increases up to the onset of pore condensation, clearly indicating adsorption into disordered pores. As the adsorbed fluid has the same electron density as the matrix the filling of the disordered pores can be estimated from the dependence of ˜Idiff on the fluid pressure using the relation for a two-phase system: ˜Idiff (p) = Cϕm (1 − ϕm ) + B,
(12)
where ϕm is the intrawall porosity, i.e., the pressure-dependent volume fraction of empty pore space within the walls, ϕm (p) = ∗ + ), and B is a pressure-independent incoherent dis (p)/(dis S scattering background. When B is known, the constant C of Eq. (12) can be calculated from data of the empty sample, viz., ˜Idiff (p = 0) ∗ = ∗ /(∗ + ) = ∗ /(1 − ∗ ). A difficulty arises from and ϕm S dis dis dis ord the fact that the parameter B is not clearly defined experimentally. In the example presented in Fig. 1b, B was taken as the value of ˜Idiff (p) at a pressure at which pore condensation is just completed (p/p0 = 0.8). Based on this fixing the constant C of Eq. (12) was determined from the parameters for the evacuated sample ∗ = 0.296 and ˜ Idiff (0) − B = 0.394). The resulting dependence of (ϕm the wall porosity on the fluid pressure is shown in Fig. 4. The graph indicates that ca. 50% of the wall porosity is filled at relative pressures below 0.2, as expected for pore sizes below 2 nm. However, filling of the disordered pores continues up to and into the pore condensation region, indicating the existence of disordered pores of nearly the size of the cylindrical channels. Note, however, that this conclusion relies on the correct determination of the background scattering intensity B of Eq. (12). We return to this point in Section 5. The volume fraction of fluid in disordered pore space is obtained from ϕm (p) as follows: ∗ ∗ − dis (p) = dis (1 − ϕm (p)) − ϕm (p)S F,dis (p) = dis
(13)
According to Eq. (13) F,dis (p) increases as ϕm (p) decreases and ∗ when ϕ = 0, i.e., reaches its maximum value F,dis (max) = dis m when the intrawall porosity is completely filled.
Fig. 4. Matrix porosity ϕm as a function of the relative pressure p/p0 of DBM indicating the gradual filling of the intrawall porosity by the contrast-matching adsorbate. The inset shows a sketch of the porous matrix.
4.2. Adsorption in the ordered pores If the fluid has the same electron density as the matrix, the sample still conforms to a two-phase system. In this case Eq. (8) can be used to determine the volume fraction of empty space in the ordered ¯ which for the step pores, ord (p). Again one starts by calculating R, model is now given by R¯ 2 = R22 + (1 − ˇ)(R12 − R22 ) + (1 − ˛)(R02 − R12 ).
(14)
instead of Eq. (6), where ˇ = 1 for the case of a contrast-matching liquid. To calculate R¯ by the step model for a given pressure p of the adsorbed fluid one uses the values of R2 and ˛ derived for this pressure (Fig. 3) and the values of R0 and R1 obtained for the empty material. For the density-gradient model, Eq. (7) is applied with values of R2 , R1 and as derived for the given pressure, and the value of R0 as obtained for the empty material. The volume of adsorbed liquid in the ordered pores, expressed as volume fraction in a unit cell, F,ord , is then given by ∗ F,ord (p) = ord − ord (p),
(15)
∗ the volume fraction of ordered pores in the empty samwith ord ple. However, the values of F,ord (p) obtained from Eq. (15) may not represent the true (absolute) adsorbed amount, as is explained below. In the analysis of the Bragg scattering data in terms of the form factor models, the density profiles are expressed by the reduced densities ˛ = 1 /0 and ˇ = 2 /0 (step model), or = 2 /0 (density-gradient model). For materials without intrawall porosity, the wall density 0 is constant and thus adsorption-induced changes of these reduced densities can be attributed solely to changes in 1 and 2 (step model), or 1 (density-gradient model). SBA-15 and other polymer-templated ordered mesoporous materials, on the other hand, may have high levels of intrawall porosity [1–3]. Filling of these disordered pores with a contrast-matching fluid causes an increase of the wall density from 0∗ (evacuated sample) to S (skeleton density of silica), which is reached when the filling of the intrawall porosity has come to completion. This change in 0 can be taken into account by re-adjusting the density parameters ˛ and ˇ (step model) or (gradient model) to the new values ˛ , ˇ , and . Specifically, for the parameters ˛ and which
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G.H. Findenegg et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 357 (2010) 3–10
refer to the density 1 : ˛, ≡
1 1 1 = 0 S 1 − ϕm
˛ , ≡
(16a)
1 1 1 = ∗ 0∗ S 1 − ϕm
(16b)
∗ ), with ϕ∗ the wall porosity of where we have used 0∗ = S (1 − ϕm m the evacuated sample. An analogous re-adjustment applies to the parameter ˇ which refers to the density level 2 of the step-density model. Eq. (16a) expresses densities in the ordered pores relative to the fluid density in the wall at the respective pressure. Accordingly, the adsorption calculated on this basis represents a relative adsorprel . These values are obtained when the reduced densities tion F,ord ˛(p) or (p) as obtained in the fitting of the form factor models are used (i.e., values from Fig. 3 in the present example). On the other hand, when adopting the adjusted reduced density ˛ (p) or (p) of Eq. (16b), the changes in these parameters are ascribed solely to density changes in the ordered pores. In this way the hidden dependence of densities in the ordered pores on the changing wall density is removed. Accordingly, the adsorption calculated on the basis of the density parameters of Eq. (16b) represents the absolute abs . From Eq. (16a,b) and the adsorption in the ordered pores, F,ord analogous relations for the reduced densities ˇ and ˇ one obtains:
ˇ 1 − ϕm ˛ = = = ∗ . ˛ 1 − ϕm ˇ
(17)
The ratio on the r.h.s. of Eq. (17) increases with the fluid pressure, as the intrawall porosity ϕm decreases when the disordered pores are gradually filled. Accordingly, the density of the fluid in the ordered pores, when calculated with the parameters ˛ and ˇ (or ), increases more strongly than when the original parameters ˛ and ˇ (or ) are used. This implies that the values of the absolute adsorption are higher than the values of the relative adsorption at any given pressure. Similar to the situation of high-pressure adsorption of fluids at flat solid surfaces [18], the relative adsorption may be zero or even negative, even though fluid has entered into the ordered pores. This effect can be seen most clearly at low pressures, when the fluid is accommodated only in the corona of the mesopores and in narrow pores within the walls (see density profiles in Fig. 2): If the density 1 increases in the same proportion as (or to a lesser extent than) the matrix density 0 , the relative rel adsorption F,ord will be zero (or negative), although the absolute
abs , is still positive. adsorption in the ordered pores, F,ord As an example, the isotherms of the relative and absolute rel (p) and abs (p), for DBM in the present SBA-15 adsorption, F,ord F,ord sample at pressures up to the onset of pore condensation are shown ∗ and in Fig. 5a. In this example the quantities ord ord (p) of Eq. (15) were calculated on the basis of the step-density profile, i.e., using Eqs. (14) and (8) with the pressure-dependent parameters ˛ and R2 of Fig. 3. As can be seen in Fig. 5a, the absolute adsorption exceeds the relative adsorption by nearly a factor of 2 in this case. The large rel (p) and abs (p) is a consequence of the difference between F,ord F,ord ∗ ≈ 0.3) as high intrawall porosity of the present SBA-15 sample (ϕm well as the high electron density of the present fluid in the adsorbed state. The overall adsorption of the fluid in the ordered and disordered pores, expressed as the filling fraction f(p), is calculated by the relations:
f (p) = 1 −
abs (p) + (p) ord dis ∗ tot
,
(18)
∗ + )ϕ (p). A comparison of f(p) with the fillwhere dis (p) = (dis m S ing fraction obtained by gravimetric adsorption measurements is shown in Fig. 5b. For the pressure range below pore condensation, to which this analysis applies, ftot (p) exceeds the gravimetric
Fig. 5. DBM adsorption in SBA-15: (a) absolute and relative values of the volume rel abs fraction of adsorbed liquid in the ordered pores, F,ord and F,ord , as a function of the relative pressure p/p0 for the step-density profile (full circles) and the densityabs indicated by smaller circles gradient model (open circles); the curve for F,ord
∗ illustrates the indirect effect of a lower interwall porosity ϕm on the volume fraction abs (see text) and (b) comparison of the of adsorbed liquid in the ordered pores, F,ord
filling fraction f derived from the in situ X-ray study (full symbols) with the filling fraction obtained by gravimetric adsorption measurements (open symbols).
adsorption isotherm by 50% at low pressures and 10% at the highest pressures. The comparison of the two isotherms suggests that ∗ of the present we have overestimated the disordered porosity dis SBA-15 by ca. 15%. A reduction of dis affects the calculated adsorption in a dual way: directly, by lowering the adsorbed volume F,dis , and indirectly, via the difference between relative and absolute adsorption in the ordered pores, which depends on the gradual filling of the matrix porosity as the fluid pressure is increased (Fig. 4) [11]. 5. Discussion The new method for analyzing in situ small-angle X-ray diffraction and scattering data [11,12] is applicable to a wide class of ordered mesoporous materials with intrawall porosity. The method is based on a separation of the two constituents of the scattering profile I(q), viz., Bragg scattering IBragg (qhk ) from the mesopore lattice, and diffuse scattering Idiff (q) from the disordered pores in the matrix. Bragg scattering is analyzed in terms of geometric models of the cylindrical pores and their microporous corona. Two different prescriptions for the radial density profile of the pores (step-density model and density-gradient model) are adopted and found to give closely similar values of the equivalent pore radius R¯ in excellent agreement with the pore radius RKJS determined from nitrogen adsorption isotherms on the basis of the improved
G.H. Findenegg et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 357 (2010) 3–10
KJS method [16]. Hence reliable values of the volume fraction of ∗ are obtained from R¯ with Eq. (8). From this ordered pores ord ∗ of the sample the volume value and the known total porosity tot ∗ fraction of disordered pores dis can be derived. For the wellordered SBA-15 material (hydrothermal aging at 100 ◦ C for 24 h) used in the present case study, the disordered (intrawall) porosity amounts to nearly 20% of the total porosity. This result confirms earlier findings concerning the intrawall porosity of SBA-15 materials based solely on a NLDFT analysis of nitrogen adsorption isotherms [17]. The same formalism can be used to analyze in situ X-ray data for the adsorption of a fluid in the ordered and disordered pores of SBA-15. For the present case of a contrast-matching fluid the volume fraction of adsorbed fluid in the ordered pores, F,ord (p), can be calculated from the decrease in the volume fraction of empty pore space ord (p) with increasing fluid pressure p (Eq. (15)). The respective volume fraction of adsorbed fluid in the disordered pores, F,dis (p), cannot be determined directly because of the illdefined geometry of the intrawall pore system. However, in the case of a contrast-matching liquid, the volume fraction of empty intrawall pore space, dis (p), can be calculated from the integral diffuse scattering intensity ˜Idiff (p) on the basis of Eq. (12), which is applicable for a two-phase system. A difficulty with this procedure can be the correct determination of the incoherent scattering background B of Eq. (12). For a contrast-matching liquid Eq. (12) implies that ˜Idiff (p) reduces to this constant B when all intrawall pores are filled (ϕm = 0). However, we have found [11] that pore condensation in the ordered cylindrical pores also causes pronounced diffuse scattering which leads to a local maximum of ˜Idiff (p) in the pore condensation region (Fig. 1b). From a modeling study of the pore filling [12] it was concluded that the two contributions to ˜Idiff (p) are independent in good approximation and can thus be separated. Accordingly, ˜Idiff (p) should attain a constant value B at pressures beyond the pore condensation regime, when all pores are filled. Unfortunately, for the present fluid it was not possible to perform reliable in situ scattering measurements in the pressure range close to saturation pressure and thus the assumption underlying the determination of B could not be verified experimentally. Further work to resolve this problem is in progress. A quantitative determination of adsorption into the intrawall pores is important for the characterization of not only the disordered porosity, but also because of its effect on the electron density of the wall, 0 , by which it affects the determination of the adsorbed amount in the ordered pores. As we have seen, this coupling can be taken into account by a re-adjustment of the electron densities in Eq. (16). Fig. 5a shows that this re-adjustment has a pronounced effect on the calculated values of F,dis (p), as the adjusted (absolute) values are greater than the relative values by about a factor 2. This implies that the calculation of the adsorbed amount in the ∗ adopted ordered pores depends sensitively on the wall porosity ϕm in Eq. (16). To demonstrate this effect, the pore volume vp of the present sample was assumed to be 5% smaller than the experimental value (0.96 cm3 g−1 instead of 1.01 cm3 g−1 ) on account of the disordered pore space (i.e., keeping the volume of the ordered pore ∗ translates space constant). The resulting decrease in the value of ϕm abs (p) of about 8%, as indicated by the smaller into a decrease of F,ord symbols in Fig. 5a. Fig. 5b shows that in the pressure range up to pore condensation the values of the filling fraction f(p) derived from the X-ray data exceed the adsorption isotherm obtained by gravimetric measurements. This deviation is most pronounced at low pressures and amounts to about 10% at the highest pressures (close to pore condensation). As discussed above, this deviation may be attributed to an overestimate of the disordered porosity of the sample. However, the observed deviation might also reflect more fundamental shortcomings of the present treatment. Two possible arguments of this
9
kind are: (1) It may be argued that a separation of the scattered intensities in Bragg diffraction and diffuse scattering (Eqs. (1) and (4)) and the concomitant separation into ordered and disordered pore space is an unjustified simplification. However, this argument was refuted by our recent work [12] in which it was shown that a simple structural model for SBA-15 with hexagonally ordered mesopores and randomly distributed smaller interwall pores is able to reproduce quantitatively the experimental X-ray scattering curves. (2) The adsorbent (dibromomethane) chosen in this study may cause complications due to its high cohesive energy density and poor wetting behavior for silica. Contact angle measurements against a surface-oxidized silicon wafer indeed showed that DBM is not completely wetting the silica surface [11]. This may explain the observation that the filling of the disordered pores extends over a wide pressure range up to pore condensation (Fig. 4). An alternative interpretation of this finding is that the density of DBM in narrow pores is lower than the bulk density, so that it is not a contrast-matching liquid, and thus the underlying two-phase approximation is not correct. Studies with other adsorbates are in progress to test this point.
Acknowledgement Financial support from the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center SFB 448 (Projects B1 and B14) is gratefully acknowledged. Appendix A. Specific pore volumes. In Eqs. (8) and (9) the volumes of ordered and disordered pore space in the sample are expressed as volume fractions, i.e., pore volumes per unit length of the 2D unit cell of SBA15. The respective pore volumes per unit mass of the silica matrix (specific pore volumes) are then given by
vord =
Vord 1 ord = , VS ˜S ˜ S S
(A1)
vdis =
Vdis 1 dis = ˜S VS ˜ S S
(A2)
where VS is the matrix volume per unit length of the 2D hexagonal unit cell, VS = VS , and S = 1 − tot is the volume fraction of silica. In Eq. (15) the volume fraction of adsorbed liquid in the ordered pores, F,ord (p), is expressed by the volume fractions of empty pore space in the evacuated sample and at a pressure p. The respective volume of adsorbed liquid per unit mass of the matrix (specific adsorbed volume) is then given by
vF,ord (p) = v∗ord − vord (p), where v∗ord and vord (p) are calculated ∗ and ord ord (p), respectively.
(A3) by Eq. (A1) with the values
References [1] A. Galarneau, H. Cambon, F. Di Renzo, R. Ryoo, M. Choi, F. Fajula, New J. Chem. 27 (2003) 73–79. [2] O. Sel, A. Brandt, D. Wallacher, M. Thommes, B. Smarsly, Langmuir 23 (2007) 4724–4727. [3] O.C. Gobin, Y. Wan, D. Zhao, F. Kleitz, S. Kaliaguine, J. Phys. Chem. C 111 (2007) 3053–3058. [4] M. Impéror-Clerc, P. Davidson, A. Davidson, J. Am. Chem. Soc. 122 (2000) 11925–11933. [5] T. Hofmann, D. Wallacher, P. Huber, R. Birringer, K. Knorr, A. Schreiber, G.H. Findenegg, Phys. Rev. B 72 (2005) 064122.
10
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[6] G.A. Zickler, S. Jähnert, W. Wagermaier, S.S. Funari, G.H. Findenegg, O. Paris, Phys. Rev. B 73 (2006) 184109. [7] L.A. Solovyov, O.V. Belousov, R.E. Dinnebier, A.N. Shmakov, S.D. Kirik, J. Phys. Chem. B 109 (2005) 3233–3237. [8] F. Kleitz, T. Czuryszkiewicz, L.A. Solovyov, M. Lindén, Chem. Mater. 18 (2006) 5070–5079. [9] N. Muroyama, A. Yoshimura, Y. Kubota, K. Miyasaka, T. Ohsuna, R. Ryoo, P.I. Ravikovitch, A.V. Neimark, M. Takata, O. Terasaki, J. Phys. Chem. C 112 (2008) 10803–10813. [10] K. Miyasaka, A.V. Neimark, O. Terasaki, J. Phys. Chem. C 113 (2009) 791–794. [11] S. Jähnert, D. Müter, J. Prass, G.A. Zickler, O. Paris, G.H. Findenegg, J. Phys. Chem. C 113 (2009) 15201–15210.
[12] D. Müter, S. Jähnert, J.W.C. Dunlop, G.H. Findenegg, O. Paris, J. Phys. Chem. C 113 (2009) 15211–15217. [13] T.-W. Kim, F. Kleitz, B. Paul, R. Ryoo, J. Am. Chem. Soc. 127 (2005) 7601–7610. [14] F. Kleitz, T.-W. Kim, R. Ryoo, Langmuir 22 (2006) 440–445. [15] O. Glatter, O. Kratky, Small-angle X-ray scattering, Academic Press, New York, 1983. [16] M. Jaroniec, L.A. Solovyov, Langmuir 22 (2006) 6757–6760. [17] P.I. Ravikovitch, A.V. Neimark, J. Phys. Chem. B 105 (2001) 6817–6823. [18] G.H. Findenegg, M. Thommes, in: J. Fraissard, C.W. Conner (Eds.), Physical Adsorption: Experiment, Theory and Applications, NATO ASI Series C, Vol. 491, Kluwer, Dordrecht, 1997, pp. 151–179.