ARTICLE pubs.acs.org/JPCC
Diffusion in Circularly Ordered Mesoporous Silica Fibers Hatem Alsyouri,†,‡ Oliver C. Gobin,‡ Andreas Jentys,* and Johannes A. Lercher Technische Univerisit€at M€unchen, Department of Chemistry and Catalysis Research Center, Lichtenbergstrasse 4, 85748 Garching, Germany
bS Supporting Information ABSTRACT: Transport processes of n-hexane and mesitylene in circularly ordered mesoporous silica fibers have been investigated. Combining theoretical calculations based on the complex hierarchical structure with experimental results from frequency response experiments, three well-defined pathways for the transport pathways of molecules in the ordered mesoporous silica fibers are identified, that is, diffusion in axial, tangential, and radial directions, all occurring in parallel. The radial diffusion is the dominant process and is governed by intrawall transport. Transport in the axial direction overlaps with the radial diffusion and can be governed by either intrawall or mesopore transport. Cracks or defects at the surface enable diffusion in the tangential direction, which is the third kinetic process experimentally observed.
1. INTRODUCTION Advances in self-assembly synthesis of inorganic materials and the understanding of the rule of defects in liquid crystal structures have led to the realization of ordered mesoporous materials with rich morphological features.14 Ordered mesoporous silica fibers, created by quiescent interfacial growth under acidic conditions,57 constitute one group of the novel products that possess a fibrous morphology and a unique circular pore structure.8 The cylindrical shape of the fibers is constituted by hexagonally packed tubular pores, similar to those of MCM-41, coiled around the central axis of the fiber.9 Typically, synthesis yields fibers with high aspect ratios (>30), with the fiber length and curvature influenced by the synthesis temperature.10 Silica fibers have attracted significant attention. They have been used as laser materials after doping with dyes,11 utilized as nanoreactors for extruding polymers through the extended nonconnected pores,12 used to prepare membranes with controlled pore orientation,13 and were also proposed for a variety of other applications in filtration and drug delivery. Most applications require a detailed understanding of the transport phenomena and diffusion characteristics through the complex structure. It appears logical on first sight that diffusion proceeds along the one-dimensional mesopore channels of the fibers. However, it was demonstrated that mesoporous materials exhibit diffusion anisotropy14 and that diffusion can also take place between the channels across the silica wall. The fact that the silica fibers are coiled adds even higher complexity. The anisotropic diffusion in combination with the circular pore architecture indicates multiple (parallel) pathways for diffusion in silica fibers, a feature that is not well addressed in the literature. r 2011 American Chemical Society
Only two groups have studied diffusion in coiled pores of mesoporous silica. Lin and co-workers15 used the gravimetric uptake method to study diffusion of gases in mesoporous silica fibers. The authors assumed that only the diffusion along the pores plays a role for the uptake and used a plane-sheet model (for 1D diffusion) to obtain a diffusion coefficient in the order of 106 cm2/s, which is consistent with Knudsen-type diffusion in mesopores. On the other hand, Marlow and co-workers16 utilized optical microscopy to trace the release of a dye tracer from coiled pores of SBA-3-like fibers. They reported that the release is dominated by radial diffusion across the walls and used a cylindrical model to calculate a diffusion coefficient in the range of 1011 cm2/s. Both groups controversially discussed their interpretations17,18 and agreed that, for a complete understanding of the diffusion processes, additional fiber-size-dependent experiments are required. The above discussion points to the presence of multiple pathways, that is, diffusion in the mesopores and diffusion across walls, in silica fibers with circular pores. We assume that the latter case is composed of two contributions, a radial and a longitudinal diffusion along the pore walls, and that the overall diffusion proceeds through these three pathways with different rates. However, it is experimentally challenging to separate these contributions, and in consequence, a comprehensive study of the individual diffusion pathways in silica fibers with circular pores has not been undertaken. The qualitative and quantitative knowledge about these transport pathways is critical for Received: October 27, 2010 Revised: February 16, 2011 Published: April 08, 2011 8602
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understanding the behavior of mesoporous silica fibers in practical applications. In this work, the diffusion of n-hexane and mesitylene in silica fibers with circular pores was investigated by means of the frequency response (FR) method. In the last decades, it has been adapted to measure diffusion in porous solids.1921 One conceptual advantage of this method is the possibility to simultaneously measure kinetic parameters of a system at various time scales. Multiple (kinetic) processes are additive in the frequency space, and thus, the individual contributions can be easier separated than by conventional diffusion measurement techniques. Therefore, this technique allowed us to identify multiple diffusive flows inside the mesoporous silica fibers and to quantify the contribution of these processes. Two fiber samples, with a different average fiber length, were investigated, in order to understand the dependence of the diffusional time constant on the length of the fibers and to identify the existing diffusional pathways.
2. THEORETICAL SECTION 2.1. Frequency Response Technique. In the typical frequency response experiment for batch adsorbers, a small amount of sorbate is equilibrated at constant pressure with the sorbent. Subsequently, the system is subjected to near-equilibrium pressure perturbations of varying frequencies achieved by small ((1%) sinusoidal or square-wave modulations of the system volume. The frequency response of such a system can be expressed as the in-phase and out-of-phase solutions of the mass balance of a closed volume given by
AB cosðj jB Þ 1 ¼ A AB sinðj jB Þ ¼ A
n
∑1 KFR;n 3 δinn
n
∑1 KFR;n 3 δout n
with KFR ¼ RT=V 3 ðDθ=DpÞ
ð1Þ
ð2Þ ð3Þ
Herein, KFR is related to the gradient of the adsorption isotherm (∂θ/∂p) at the temperature (T) and the total volume (V) of the experiment. R is the molar gas constant. The number of independent diffusion processes is defined by n. For instance, in the case of two diffusive flows inside the porous material, n equals 2. A blank experiment without sorbent is required for quantification of the time delays and of the nonideal behavior of the apparatus. The corresponding phase jjB and amplitude AB/A responses in eqs 1 and 2 are obtained for square-wave perturbations by a subsequent Fourier transformation of the pressure response or, in the case of a sinusoidal modulation, directly from the pressure waves. The in-phase and out-of-phase characteristic functions, δin and δout, are solutions to Fick’s second law. In the case of diffusion in a planar sheet with the thickness 2L, they are given by19 1 sinh η þ sin η δin ¼ δ1S ¼ ð4Þ η cosh η þ cos η δout ¼ δ1C
1 sinh η sin η ¼ η cosh η þ cos η
ð5Þ
Figure 1. Schematic representation of the three structural domains present in mesoporous silica fibers. Part (a) corresponds to the primary structure, (b) to the secondary structure, and the representation (c) to the ternary structure. The possible diffusion pathways in each structural domain are indicated by arrows.
where η = (2ωL2/D)1/2, ω is the angular frequency, and D is the intracrystalline transport diffusion coefficient. In practice, zeolites usually do not have a uniform particle size, and thus, the characteristic functions have to be modified by a normal particle size distribution19 _ ! Z ¥ _ 1 ðL L Þ2 dL ð11Þ δhðL Þ ¼ pffiffiffiffiffiffi δ 3 exp 2σ 2 σ 2π 0 _ where δh and Lh are the mean values of the characteristic function and of the crystal half thickness. 2.2. Mathematical Definition of Diffusion Pathways. Mesoporous silica fibers are a third-order hierarchical material. To understand and to quantify the diffusion in these materials, all three structural domains have to be considered. The three domains are shown in Figure 1. The primary structure is composed of cylindrical mesopores and secondary pores in a framework. The second structural domain is composed of the cylindrical mesopores arranged in an ordered hexagonal pore array, and the ternary structure is composed of rodlike particles with a length of up to several millimeters formed by a helix-like structure of the hexagonally aligned mesopores. Note that the helical mesopores are located up to the center of the fiber, and thus, no empty core is present, in contrast to the schematic representation in Figure 1. To analyze the frequency response data, the mathematical diffusion model presented by Marlow et al.9 was refined and extended. In the primary structural domain, the diffusion can be described by diffusion along the mesopore channels, Dp, and by diffusion through intrawall pores or larger cracks, Dw. From the dimension of the pores, it is reasonable to assume that the diffusion along the mesopores is in the Knudsen diffusion regime, while the molecules adsorbed at the pore wall may also be transported by surface diffusion.22 Such a system, in which surface and mesopore Knudsen diffusion coexist, can be represented by an effective diffusivity in the mesopores, Deff,p, by embedding both surface diffusion and pore diffusion coefficients Deff ;p ¼
εme Dp þ ð1 εme ÞKDS τ 3 ½εme þ ð1 εme ÞK
ð12Þ
where εme is the mesopore void fraction and (1 εme) the adsorbed liquid. The products εmeDp and (1 εme)KDs are the corresponding diffusion processes in the gas phase of the mesopores and in the adsorbed phase on the surface. K is the dimensionless Henry constant based on particle volume, and 8603
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)
perpendicular to the mesopores that is only determined by Dw. Figure 2 summarizes by a schematic representation the diffusivities of the primary and secondary domains. Following the suggestions of Marlow et al.,9 the ternary structure of the fibers is described by three effective diffusivities, that is, the tangential diffusion (Deff,t), radial diffusion (Deff,r), and axial diffusion direction (Deff,a), as shown in Figure 1c. In an ideal fiber, only the radial and the axial diffusivities can be followed by macroscopic measurements, whereas tangential diffusion is absent in an ideal fiber as open mesopores in the tangential direction do not exist (see Figure 1c). The diffusivities in this domain are related to the effective diffusivities Deff, and Deff,^ by the following equations
DS ¼ DW
ð14Þ
The second structural domain, as shown in Figure 1b, is composed of the primary cylindrical mesopores arranged in an ordered hexagonal pore array similar to the pore arrangement in MCM-41 materials. In this domain, the diffusion can be characterized by an effective diffusion parallel, Deff, , and perpendicular, Deff,^, to the mesopores. Assuming that the gas-phase capacity in the pore is much lower than the adsorbed capacity [εme , (1 εme)K], the effective diffusion coefficients are related to Dp and Dw by the following equations
Deff ;r ¼ Deff ;^
ð18Þ
Deff ;a Deff ;^ þ
Deff ; ¼ τfiber
Z
R
Lfiber dr D Lpore ðrÞ 3 eff ;
0
ð19Þ
ð15Þ
1 1 ½εme D1 K þ ð1 εme ÞðKDw Þ τ 3 ½εme þ ð1 εme ÞK DK . DW Dw τ
ð16Þ
A schematic representation of the orientation and dimensions of the helical mesopores inside the fiber is shown in Figure 3. The tortuosity in the axial direction of the fiber is given by τfiber(r) = Lpore(r)/Lfiber.
)
¼ Deff ;p εme DK Dw þ KH 3 τ τ
)
¼
ð17Þ
where Lpore(r) is the length of the primary mesopore as a function of the helix radius r. The macroscopic properties Lfiber and r are the (measurable) length and radius of the fiber. The mesopores are coiled to form helices with different radii, but identical pitches. If the pitch (h) of the helix is known, the length of the primary mesopore can be obtained by geometrical calculations as a function of the fiber length, Lfiber, and of the helix radius: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h Lfiber ð20Þ Lpore ðrÞ ¼ 2πr 3 1 þ 3 2πr h
Deff ;
Deff ;^
Deff ;t ¼ Deff ;
)
where rp = wd/2 is the average radius of the mesopores, M the molar mass of the molecule, R the molar gas constant, and T the temperature. If intrawall pores or cracks in the structure are present, the surface diffusion is identical to the intrawall diffusion in the adsorbed phase:
)
τ is the tortuosity of the mesopores. For MCM-41 materials, that is, cylindrical pores, τ = 1.12 was calculated by Salmas and Androutsopoulos.23 The term [εme þ (1 εme)K] represents the capacity of the system to accommodate adsorbed molecules in both phases. For the diffusion in the mesopores, the Knudsen diffusion flow can be described by rffiffiffiffiffiffiffiffiffi 4 rp 8RT ð13Þ Dp ¼ D K ¼ 3 2 πM
Figure 3. Schematic representation of an internal helical mesopore.
)
Figure 2. Schematic representation of the diffusivities of the primary and secondary structures. The porous silica framework is represented in blue color, the adsorbed liquid in gray color. The void in between the two framework walls represents the gas phase of a mesopore. Knudsen diffusion (orange) occurs in the gas phase; intrawall or surface diffusion (green) occurs in the liquid phase. The effective perpendicular and parallel diffusivities of the secondary structure are represented in red color.
where KH = εme þ (1 εme)K.24 Herein, it is assumed for Deff,^ that the intrawall diffusion Dw (the component of transport of molecules in the wall that is not normal to its boundaries) is much lower than DK, leading to an effective diffusion
3. EXPERIMENTAL SECTION 3.1. Materials. Mesoporous silica fibers were prepared by the self-assembly method as described in detail by Alsyouri and Lin.4,7 Tetrabutylorthosilicate (TBOS) silica precursor was added on the top of a premixed water solution containing 8604
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The Journal of Physical Chemistry C cetyltrimethylammonium bromide (CTAB) as a surfactant and HCl acid without mixing, creating a two-phase mixture. The silica fibers grow under quiescent conditions at the interface. After 2 weeks, the silica product was collected, dried, and calcined at 823 K. The silica product containing fibers of various lengths accompanied by particles with various shapes were separated by sieving. The fiber length was subsequently reduced by grinding the fraction containing fibers. 3.2. Physicochemical Characterization. Nitrogen physisorption isotherms were measured using a PMI automated sorptometer at liquid nitrogen temperature (77 K), after outgassing under vacuum at 473 K for at least 6 h. The apparent surface area was calculated by applying the BrunauerEmmettTeller (BET) theory to the adsorption isotherms over a relative pressure range from 0.10 to 0.30 p/p0. The pore volumes were evaluated using the Rs-comparative plot25 using nonporous hydroxylated silica26 as the reference adsorbent. In addition, calculations based on the nonlocal density functional theory (NLDFT) were performed using the Quantachrome Autosorb 1 software. As the DFT kernel, the nitrogen equilibrium model at 77 K on silica was used. Because of the limitations of the PMI instrument, the isotherms were measured at relative partial pressures higher than 105 p/p0. Scanning electron microscopy (SEM) images were recorded on a REM JEOL 5900 LV microscope operating at 25 kV with a resolution of 5 nm and a nominal magnification of 3.0 106. For SEM, the powdered samples were used without any pretreatment or coating. Transmission electron microscopy (TEM) was measured on a JEOL2011 electron microscope operating at 200 kV. Prior to the measurements, the samples were suspended in ethanol solution and dried on a coppercarbon grid. The X-ray powder diffraction (XRD) patterns were measured on a Philips X’pert Pro XRD instrument operating with the energy of CuKR1 radiation (λ = 1.54055 Å) at 40 kV using a Ni filter to remove the CuKβ line. Data points were recorded using a spinner system with a 1/4 in. slit mask between 2θ angles of 1.58 with a step size of 0.017 and a scan speed of 115 s per step. 3.3. Gravimetric Sorption Experiments. The gravimetric sorption capacities of the molecules were measured on a Setaram TG-DSC 111 thermoanalyzer. Activation was performed at 823 K for 1 h with a heating rate of 10 K min1 under vacuum (p < 107 mbar). The weight increase and the heat flux were measured during equilibration with the sorbate using small pressure steps up to 13 mbar. The enthalpies of adsorption were obtained by integration of the observed heat flux signal. Equilibrium constants were obtained by fitting the experimental data using either a Henry or a Langmuir model. The maximum loading of the corresponding sorbate was obtained from the liquid-phase density of the sorbates times the total pore volume obtained from the nitrogen physisorption analysis. 3.4. Frequency Response Experiments. A 30 mg portion of the powder sample was carefully dispersed on several layers of quartz wool at the bottom of a quartz tube in order to avoid artifacts from bed-depth effects. The sample tube was connected to a vacuum system, placed inside a heating oven, and pumped to 107 mbar at room temperature. The samples were heated to 823 K with a ramp of 10 K/min and activated under vacuum below 107 mbar for 1 h to remove adsorbed water. The sorbate gases were added with a partial pressure of 0.3 mbar into the system at temperatures between 333 and 423 K. After the sorption equilibrium was fully established, the volume of the system was modulated periodically by two magnetically driven plates sealed
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Figure 4. XRD pattern of the fiber sample.
Figure 5. TEM micrographs of the fiber edge.
with UHV bellows in the frequency range of 0.0012 Hz. The modulation resulted in a square-wave perturbation of the system volume with an amplitude of (1%. The pressure response of the system to the volume perturbation was recorded with an online Baratron pressure transducer (MKS 16A11 TCC). The amplitude and the phase lag of the frequency response were obtained by Fourier transformation of the pressure data. Nonlinear parameter fitting of the theoretical characteristic functions to the experimental frequency response was performed by using the CMA evolution strategy in Matlab.27 The root mean squared error normalized to the variance of the data (NRMS error) was used as the objective function to be minimized. To ensure that the globally optimal parameter set was found, each optimization run was repeated three times with varying parameter sets of the evolution strategy. The influence of heat or bed-depth effects was checked by variation of the sample amount and by the way the sample was placed in the sample tube, that is, with and without dispersing the sample on quartz wool. In all cases, the same frequency response 8605
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was observed, and therefore, we can exclude heat or bed-depth effects. The reproducibility of the resulting FR parameters was better than (5%.
4. RESULTS 4.1. Characterization Results. The internal structure of the fibers was analyzed by XRD, TEM, and nitrogen physisorption. The XRD pattern shown in Figure 4 corresponds to a hexagonal mesophase clearly showing the (100), (110), and (200) reflections. The internal hexagonal structure of the fibers was confirmed by TEM images, as shown in Figure 5. From the XRD pattern, a d(100) spacing of d = 3.72 nm and a unit cell parameter √ of a0 = 2/ 3d100 = 4.30 nm were calculated. The pore structure was further analyzed by nitrogen physisorption. The isotherm shown in Figure 6 is a typical type IV isotherm with high initial adsorption at low pressures and a well-defined capillary condensation step at 0.3 p/p0, indicating a very uniform and narrow pore structure. By applying the BET theory, a surface area of
Figure 6. N2 physisorption isotherm and the DFT pore size distribution (inset).
Table 1. Textural Properties of the Fibersa
a
SBET
Vmi
Vme
Vtot
(m2/g)
(cm3/g)
(cm3/g)
(cm3/g)
1008
0.01
0.63
0.64
wBJH
wDFT
wd
ε
(nm) (Nm) (nm)
(-)
2.35
0.58
3.53
3.44
Vmi, Vme, and Vtot are the micropore, mesopore, and total pore volumes as obtained by the Rs-plot. wBJH, wDFT, wd are the mesopore diameters as obtained by BJH, DFT, and by geometric considerations from XRD. ε is the mesoporosity.
SBET = 1008 m2/g was calculated. The pore volumes were obtained using the t- and Rs-comparative plot method. In both cases, an intercept close or below zero was observed, which is a clear confirmation that no or only a very small amount of micropores was present, very similar to the situation in MCM-41. A total pore volume of Vtot = 0.64 cm3/g and a mesopore volume of Vme = 0.63 cm3/g was obtained. A mesoporosity, εme = FVme/(1 þ FVtot), of 0.58 was calculated assuming a constant framework density of F = 2.2 g/cm3.28,29 The pore size can also be obtained by geometrical considerations by wpore = c 3 d 3 (FVme/(1 þ FVme))1/2, where c = (8/31/2π)1/2 = 1.213, d is from the XRD d(100) spacing, and the mesopore volume is Vme.29 Using this relation, a pore size of wpore = 3.44 nm was calculated. This pore size is in good agreement with the pore size of wDFT = 3.53 nm obtained from the DFT pore size distribution, as can be seen in the inset of Figure 6. Consistent with the results from the Rs-plot, only one pore population was identified from the DFT calculations with micropores being notably absent. It should be noted that, applying the BJH analysis, an average pore diameter of wBJH = 2.35 nm pores was obtained, but it is known that the BJH theory systematically underestimates the pore size. The pore wall thickness can be calculated from the difference between the pore size and the unit cell parameter. In the case of the DFT calculations, a wall thickness of ∼0.85 nm is obtained. All the structural properties of the sample are summarized in Table 1. The particle morphology was studied using SEM. It can be easily seen that the as-synthesized material contains fibers with a length up to a few millimeters and a thickness of 1525 μm and small spherical particles. According to Marlow et al.,9 the spherical particles can be described as small rotational symmetric particles. Their diameter is similar to the thickness of the fiber, and their internal structure is also composed of ordered hexagonally arranged mesopores belonging to the class of SBA-3 materials.9 A differentiation between fibers and small particles that have a similar internal structure than the fibers is only possible by XRD by performing several measurements at different sample orientations.30 In this case, the relative intensities of the XRD peaks change. For diffusion measurements, high uniformity of the particles drastically facilitates the interpretation. Therefore, the small particles were separated from the fibers by sieving in order to have a sample only composed of fibrous shapes. This sample is denoted as the parent fiber sample. Subsequently, the fiber length was reduced by mechanically breaking the fibers, resulting in a sample predominantly composed of small fiber pieces, denoted as the crushed fiber sample. In Figure 7, the SEM images of the sample containing the parent fibers (a), the nonuniform and small rotational symmetrical particles (b), and the crushed fibers
Figure 7. SEM images of the fiber samples: (a) the parent sample only containing fibers, (b) the fraction of nonfibrous particles, and (c) the crushed fiber sample. 8606
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The Journal of Physical Chemistry C (c) are shown. It can be easily seen that the length of the crushed fibers was drastically reduced. An average length of about 50 μm was determined by SEM, in contrast to an average length of 300 μm for the parent fibers. For the quantitative analysis of the transport experiments, the adsorption properties of hexane and mesitylene at the corresponding temperatures are required. The adsorption isotherms for mesitylene are shown in Figure 8. Up to 13 mbar, the isotherms are in all cases linear, and therefore, a Henry isotherm was used to fit the experimental data. Relatively low dimensionless weight-based Henry constants of 101, 32, and 10 at 70, 100, and 130 C, respectively, and a heat of adsorption of 44 kJ/mol were obtained for mesitylene. 4.2. Frequency Response Results. The frequency response results were performed using n-hexane and mesitylene as probe molecules on the parent and crushed fiber samples. The frequency responses of n-hexane for the parent and crushed fiber samples at 30 C are shown in Figure 9. In all cases, it can be clearly seen that the in-phase characteristic function is not approaching zero at high frequencies, and thus, the in- and out-of-phase characteristic functions are not crossing in the measured frequency range. This points to the existence of a second (fast) kinetic process located in the high-frequency range above 10 Hz, which cannot be directly followed due to the frequency limitations our system (maximum frequency 2 Hz) but can be deduced from the characteristic offset in the in-phase function. Note that the fast kinetic process is not visible in the
Figure 8. Sorption isotherms of mesitylene at 70, 100, and 130 C of the fiber sample. The fits are linear fits based on Henry’s law.
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out-of-phase function. Assuming two kinetic processes, the frequency responses for n-hexane can be fitted with a high accuracy. However, the exact time constant of the fast kinetic process cannot be unambiguously determined as the location is given by the out-of-phase characteristic function. Therefore, a time constant located in the frequency region above 10 Hz was assumed. In this case, only the amplitude of the response, which can be related to the relative number of molecules involved in this transport step (KFR value in eq 3), can be determined from the offset of the in-phase characteristic function. In Table 2, the kinetic data obtained from the fitting of the experimental frequency responses of n-hexane are given. A detailed discussion on the model selection can be found in the Supporting Information. Due to the fact that the characteristic length of the transport process is not known, only the time constants (i.e., L2/D), but not the diffusivities, are given in Table 2. The corresponding diffusivities can be calculated by assuming a certain characteristic length, L, of the diffusion pathway. From Table 2, it can be seen that the time constants of the slow kinetic process, that is, L21/D1 ≈ 20 s, are identical for the parent and crushed samples. The shape of the out-of-phase frequency response is very narrow and can be fitted by a one-dimensional diffusion model, as shown in Figure 9 and in Table 2. From the KFR values obtained by fitting the experimental frequency responses, the contribution of the very fast transport process at frequencies higher than 1 Hz in the frequency response analysis and of the slow process can be calculated. The relative contribution of the slow process to the overall transport, that is, KFR,1/(KFR,1 þ KFR,2), is given in Table 2. Herein, the KFR values were corrected by the corresponding sample mass. The dominant process with a contribution of ∼80% is the slow transport process for both samples, and a slight increase of the contribution of the fast process can be observed for the crushed sample. The transport of molecules being too large to diffuse through small micropores in the framework was studied using mesitylene. The transport of mesitylene was distinctly slower compared to nhexane; therefore, higher temperatures were used to perform the experiments. For n-hexane, the investigation of several temperatures was not possible, as the intensity of the frequency response was already very low at 30 C. The unusual type of frequency response31 observed for n-hexane was also seen for mesitylene. For both molecules, a kinetic process at high frequencies not directly visible in the investigated frequency range had to be assumed in order to fit the experimental data. In Figure 10 and Table 3, the corresponding measurements and parameters used for the fitting are summarized. Again, very similar results between the parent and the crushed sample were obtained with a slight
Figure 9. Frequency responses of n-hexane in the parent and the crushed sample at 30 C. 8607
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Table 2. Frequency Response Data for n-Hexanea sample
T (C)
L12/D1 (s)
L22/D2 (s)
KFR,1
KFR,2
KFR,1 þ KFR,2
KFR,1/(KFR,1 þ KFR,2)
σ (s)
parent
30
20.9
0.10
0.21
0.03
0.24
0.87
0
crushed
30
19.9
0.10
0.19
0.05
0.24
0.79
0
a
L12/D1 is the time constant of the kinetic process visible in the frequency response, L22/D2 the one of the kinetic process at high frequencies, which cannot be directly measured. The exact value of this process is not known. KFR,1 and KFR,2 are the contributions of the processes as given by eq 3. KFR,1/(KFR,1 þ KFR,2) is the relative contribution of the first kinetic process, and σ is the variance of the distribution as obtained by the fitting.
Figure 10. Frequency responses of mesitylene in the parent and the crushed sample at 70, 100, and 130 C.
Table 3. Frequency Response Data for Mesitylenea T
L1 /
KFR,1 þ
L22/
sample (C) D1 (s) D2 (s) KFR,1 KFR,2
KFR,2
parent
crushed
a
2
σ
KFR,1/
(KFR,1 þ KFR,2) (s)
70
71.3
0.10
2.25 0.28
2.53
0.89
4.97
100
32.4
0.10
1.14 0.38
1.52
0.75
3.00
130
24.1
0.10
0.87 0.38
1.26
0.69
2.93
70
51.6
0.10
2.07 0.43
2.50
0.83
9.46
100
28.9
0.10
1.23 0.45
1.68
0.73
4.50
130
23.3
0.10
0.92 0.42
1.35
0.69
3.85
Parameters and variables are given in the description of Table 2.
trend to shorter time constants for the crushed sample. The slow transport process is again dominating the overall transport rate, with a contribution of 7090% depending on the temperature (see Table 3). In addition, for mesitylene, the out-of-phase function of the frequency response could not be fitted by an ideal diffusion model. In contrast to the results of n-hexane, a particle size distribution or, mathematically equivalent, a distribution of the diffusivities within the sample had to be assumed. The variance σ of this distribution was found to be temperaturedependent, as can be seen in Table 3. The apparent activation energy of the slow transport process obtained from the temperature dependence of L21/D1 (see Table 3) was calculated to be approximately 20 kJ/mol for the parent sample and about 16 kJ/mol for the crushed sample. The activation energy is lower for the crushed sample, mainly due to the difference in the time constants at 70 C. It should be noted,
however, that the experimental frequency response of mesitylene in the crushed fiber sample shows a second peak that may be related to an additional kinetic process. Also, the presence of a distribution of the time constants and the fact that this distribution is temperature-dependent lead to the conclusion that a third kinetic process at frequencies in the range of 10100 Hz must be present for mesitylene. Therefore, the time constant derived may not accurately represent the correct time constant of a single underlining process.
5. DISCUSSION To understand the complex situation present in these fibers, the experimental results are compared in the following with the results of the quantitative evaluations of the diffusivities according to eqs 1220. Due to the fact that the surface area at the end of the fibers is a very small fraction of the total surface area of the fibers, a low contribution of the effective axial diffusion process is expected, if diffusion in the radial direction is possible. Several studies by Marlow et al.9,16,17 showed that the radial diffusion pathway is the dominant one. This was concluded based on several facts: First, modification of the outer surface led to a distinct change in the uptake. Second, the concentration gradient of Rhodamine 6G (Rh6G), followed by light microscopy, did not show a gradient along the fiber, but a continuous depletion was observed. In addition, recent studies showed that it is possible to obtain stable carbon replicas of the fibers,28 which is only possible if such intrawall connections are present. However, these experiments do not unambiguously demonstrate that radial diffusion is possible, because the observations 8608
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Table 4. Calculated Effective Diffusivities and Equilibrium Constants KH (Henry) and KL (Langmuir) in the Fiber Samples for nHexane and Mesitylenea Deff,
compound
T (C)
KH
KL
DK
Deff,r
Deff,t
Deff,a(c,h1)
Deff,a(e,h1)
Deff,a(c,h2)
Deff,a(e,h2)
parent
mesitylene mesitylene
70 100
101 32
130 37
17 779 18 539
2.19 4.82
92.95 303.52
8.39 25.05
2.20 4.83
20.78 65.52
2.22 4.92
mesitylene crushed
)
Deff,^ sample
130
10
11
19 270
6.47
1009.01
73.74
6.51
208.27
6.81
n-hexane
30
13
15
19 734
7.49
790.12
60.16
7.51
165.51
7.75
mesitylene
70
101
130
17 779
3.03
93.78
9.28
3.03
21.78
3.06
mesitylene
100
32
37
18 539
5.41
304.11
25.69
5.42
66.24
5.51
mesitylene
130
10
11
19 270
6.70
1009.24
73.98
6.74
208.55
7.04
30
13
15
19 734
7.85
790.48
60.55
7.87
165.94
8.11
n-hexane
The effective diffusivities in the axial direction are given in the core “c” or at the edge “e” of the fiber calculated for a pitch of h1 = 4.30 nm and h2 = 43 nm. The units of all the diffusivities are 1012 m2/s. a
Table 5. Activation Energies of the Diffusivities Present in the Fibers, and the Heat of Adsorption of Mesitylene in the Parent Samplea EA
EA,
EA,^
EA,a
EA,a
EA,a
EA,a
(DK)
ΔHads
EA,t
EA,r
(c,h1)
(e,h1)
(c,h2)
(e,h2)
21.0
1.5
44.4
45.6
21.0
41.6
21.0
44.1
21.6
)
EA (exp) a
All values are given in kJ/mol. The nomenclature is identical to the one given in Table 4.
)
mesitylene show a radial transport, as discussed previously. This leads to a situation in which the diameter of intrawall openings is larger than the thickness of the pore wall. Thus, the intrawall connections in the fibers are simply not visible in the nitrogen physisorption isotherm as the adsorption strength is identical to the one in the mesopores and the adsorption capacity is negligible, as can be seen in Table 1 (Vmi). Besides the effective diffusion in the radial direction, Deff,r, the other relevant diffusion process is the effective axial diffusion, Deff,a, as described in eq 19. It can be easily seen that this diffusivity is composed of both effective diffusivities perpendicular to the main mesopore channels, Deff,^, the intrawall diffusion, and of the effective diffusion in the mesopores itself divided by the tortuosity of the mesopores in the fiber, Deff, / τfiber. Therefore, molecules diffuse in the axial direction through two pathways in parallel, that is, the coiled mesopore channels and intrawall openings. As a consequence, the value of Deff,a ≈ Deff,^ þ Deff, /τfiber is close to Deff,r = Deff,^, as given in eq 19, which indicates that the axial process is determined by the intrawall diffusivity as )
Deff ;^
DK . DW
Dw
and not by the diffusion in the mesopore channels Deff, . It is necessary to analyze the dependency of the tortuosity of the fiber, τfiber, as a function of the radius and of the pitch of the mesopore helix, as given by eq 20. If a minimal pitch equal to the unit cell parameter a0 is assumed (hmin = 4.30 nm), the following tortuosities are obtained: τfiber = 15 for a helix radius very close to the center (r = 1 nm), and τfiber = 30 000 for a helix with a radius identical to the fiber radius (r = 20 μm). Larger pitches result in lower tortuosities, as for instance, a pitch of 10 hmin gives τfiber = 5 in the fiber center and τfiber = 3000 at the edge of the fiber. As an important conclusion, the diffusion time constant in the axial )
)
)
)
could also be explained by a strong surface barrier at the end of the fibers or by other effects leading to stable carbon replicas. Therefore, in this study, the average fiber length was shortened in order to analyze the diffusion as a function of the length of the fiber. If the axial diffusion pathway is significant, the diffusion time constants L2/D should differ by more than 1 order of magnitude for the parent and the crushed fiber sample. However, such dependence was not observed, as the main diffusion process was found to be independent of the fiber length. This confirms the assumption that the dominant transport process is the radial diffusion. An additional confirmation can be obtained from the effective diffusivities for n-hexane and mesitylene (see Table 4). For the calculations, the radial diffusion process Deff,r = Deff,^ was associated with the experimentally obtained L21/D1 (see Tables 2 and 3). According to eq 16, this diffusivity is identical to the intrawall diffusion Dw. The effective tangential diffusivity in the mesopores Deff, as given in eq 15 strongly depends on the temperature and may differ by several orders of magnitude compared to the Knudsen diffusion. This is the case, as the effective diffusion in the mesopores Deff, is mainly given by the amount adsorbed, that is, by surface diffusion, which itself depends on the adsorption strength. A similar situation was also recently found for the diffusion in large pore SBA-15 and SBA-16 materials.24,32,33 In Table 5, the activation energies of the several processes are given. It can be seen that the activation energy of the effective diffusion in mesopores, EA,eff, = 46 kJ/mol, is much larger than the one experimentally observed (1520 kJ/mol). This is another clear indication that the main diffusion process is not determined by the diffusion in the mesopore channels of the fiber material. The findings reported herein in combination with the results from Marlow et al. strongly suggest that perpendicular connections between the mesopore channels are present, leading to a radial transport in the fibers. The exact type of these connections is, however, still unclear. Interestingly, the analysis of the N2 physisorption isotherm does not indicate micropores below 2 nm. This can be explained by the fact that the thickness of the mesopore walls is only ∼0.85 nm as estimated by the DFT analysis of the nitrogen isotherm. Compared with the wall thickness of SBA-15 or -16 materials,24 the thickness of the studied materials is approximately 4 times smaller, subtly depending on the synthesis conditions. In addition, the minimum size of the intrawall connections must be larger than the critical diameter of mesitylene, that is, 0.84 nm,34 as both hexane and
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Table 6. Time Constants L2/D of the Radial and Axial Diffusion Processesa T (C)
τradial (s)
mesitylene
70
20.9
374.0
2994.7
135.9
2903.2
mesitylene
100
71.2
2681.9
10 244.9
1082.6
10 116.4
mesitylene
sample
compound
parent
crushed
τaxial(c,h1) (s)
τaxial(e,h1) (s)
τaxial(c,h2) (s)
τaxial(e,h2) (s)
130
32.4
898.1
4658.7
343.4
4572.5
n-hexane
30
24.1
305.1
3458.4
108.0
3304.6
mesitylene
70
19.9
10.3
79.4
3.8
77.1
mesitylene
100
51.6
67.4
206.4
28.7
204.5
mesitylene
130
28.9
24.3
115.2
9.4
113.3
30
23.3
8.4
92.8
3.0
88.8
n-hexane
For the radial process, τradial, a characteristic length of Lradial = 12.5 μm, i.e., the average half-diameter of the fibers, for both parent and crushed samples, is assumed. For the axial processes, τaxial, a characteristic length of Laxial,parent = 150 μm and Laxial,crushed = 25 μm are used for the parent and crushed fiber samples, respectively. The corresponding diffusivities D are reported in Table 4. The nomenclature is identical to the one given in Table 4. a
direction strongly depends on the radius of the corresponding mesopore helix in the fiber, thus leading to a distribution of the diffusion time constant and to a convolution of multiple axial diffusion processes. In Table 4, several effective axial diffusivities are given for mesopores located in the center of the fiber (“c”) and at the edge (“e”) for a pitch of the mesopore helix of p1 = 4.30 nm and p2 = 43 nm. It can be seen that the effective axial diffusivities obtained for mesopores located at the edge of the fibers are entirely determined by Deff,^, that is, by the intrawall diffusion. This is also confirmed by the analysis of the activation energies reported in Table 5. The activation energy of the effective axial diffusion in the center (EA,a(c) = 4244 kJ/mol) approaches the one of the effective mesopore (tangential) diffusion (EA,t = 45.6 kJ/mol), whereas the activation energy of the effective axial diffusion at the edge (EA,a(e) = 2122 kJ/mol) approaches the one of the radial diffusion (EA,r = 21 kJ/mol). This finding is remarkable, as it clearly shows that, even in the axial fiber direction, the diffusion is, to a large extent, given by the intrawall transport and not by the primary mesopore structure. Only the helical mesopores located in or close to the fiber center show an effective axial diffusion that is faster than the radial diffusion. By a macroscopic technique, such as the frequency response method, the diffusivity cannot be directly measured as, in principle, the diffusional time constant L2/D is observed. To assign all maxima in the experimental frequency response, it is necessary to transfer the diffusivities into diffusional time constants L2/D by knowledge of the corresponding characteristic length of the diffusion pathway. In the case of the radial effective diffusion, the average half-diameter, that is, Lradial = 12.5 μm, is the characteristic length for both fiber samples. The average halflength of the fiber, that is, the characteristic length of the axial diffusion process, is Laxial,parent = 150 μm for the parent and Laxial, crushed = 25 μm for the crushed sample. The time constant indicates the frequency at which the maximum of the out-ofphase characteristic function is expected in the experimental frequency response. The results given in Table 6 clearly show that, for the parent fiber sample, the time constants of the axial diffusion processes are larger in all cases than those of the radial processes, which indicates that the axial diffusion is not the dominant transport mechanism. For the crushed fiber sample, this situation is different. For the helical mesopores close to the center of the fiber, the time constants of the axial diffusion processes in the center of the fiber are lower, which relates to a faster transport than the one of the radial processes (see Table 6). This, however, is only valid for the helical mesopores close to the
center of the fiber. At the edge of the fiber, the radial transport once again becomes faster than the axial. After calculation of the expected diffusion time constants present in the fibers, it is possible to explain the different processes visible in the experimental frequency response data as given in Table 3. Experimentally, three diffusion processes are observed, two diffusion processes at frequencies between 0.1 and 0.01 Hz and a fast process at frequencies higher than 1 Hz. The dominant transport process has to be composed of two processes, as it shows a relatively broad temperature-dependent frequency response. We conclude that, given the uniformity of the samples, this observation cannot be explained by a simple distribution of the fiber dimensions, but has to be related to a distribution of diffusivities due to the temperature dependency. This result is consistent with the finding obtained by the preceding calculations. The effective axial diffusion, as a second process, overlaps with the effective radial diffusion in the fibers and leads to a broadening, which is dependent on the temperature. This distribution is more pronounced for the crushed fiber due to the shorter diffusion pathway that shifts the time constant to a frequency region in which it is overlapping with the one of the radial diffusion process. The third process with characteristic frequencies above 1 Hz is difficult to understand. In Figure 11, three possible explanations for the fast process, which will be discussed in the following, are schematically shown. In the uptake rate analysis done by Stempniewicz et al.,16 such a fast process was also observed. Obviously, this shows that this fast process is not an experimental artifact of the present study, but it is related to a structural feature of the fibers or to a feature that is due to the synthesis method. Stempniewicz et al.16 related the fast process to cracks in the fibers and not to a diffusion process in the mesopores. To contribute to a fast diffusion process in macroscopic cracks, as shown in Figure 11a, a certain fraction of molecules would have to be adsorbed in these cracks. From the frequency response measurements, a contribution of ∼20% was observed; thus, this amount of molecules would have to be adsorbed in these cracks in order to contribute to the fast transport process. This situation seems very unlikely, as this pore fraction, which must be larger than the framework mesopores, is not seen in the N2 physisorption isotherm. If we assume in analyzing the presented arguments that macroscopic cracks are indeed the reason for the very fast transport process, the consequences of the existence of such a transport pathway would be dramatic. Because of the cracks in the fibers, the characteristic length of the axial diffusion pathway would be significantly reduced and the contribution of the 8610
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Figure 11. Schematic representation of the three possible explanations of the fast kinetic process. Part (a) corresponds to a macroscopic crack, (b) to the small rotational particles, and (c) to mesopore openings at the surface due to small cracks or surface nonidealities.
)
diffusion in the mesopores would be more important to the overall diffusion. This would lead to an activation energy of the observed transport process that is close to the one of EA, = 45 kJ/ mol. Thus, the observed activation energy would be expected to be higher than the one experimentally observed in the frequency response experiments (∼20 kJ/mol). This is due to the fact that the effective transport in the parallel direction to the mesopores is, to a large extend, given by the gas and liquid phase amount adsorbed in the mesopores, as given by eq 15. The amount adsorbed itself depends on the temperature dependence of the equilibrium in between these two phases. Thus, the temperature dependence of the effective diffusivity in the parallel direction is strongly depending on the temperature dependence of the adsorption constant as given in Table 4. This confirms the fact that macroscopic cracks in the fibers are not the reason for the fast transport process. Two types of shapes, fibers and small rotational symmetrical particles, grow during the synthesis at the phase boundary. Both have the same primary and secondary structures, but differ in their ternary structure. The particles are rather small, with diameters similar to the diameters of the fibers. In these particles, the effective diffusion in the mesopores would have time constants below 1s and, thus, be located in the frequency response at frequencies above 1 Hz. Therefore, it seems reasonable to assume that this second population of particles may be the reason for the fast transport process, as shown in Figure 11b. Also, a weight fraction of 1020% of the rotational symmetrical particles relative to the fiber fraction is reasonable from SEM images. However, the temperature dependence of the relative contribution of the processes cannot be explained by this model. As can be seen in Table 3, the relative contribution of the fast process is nearly 30% at 130 C of mesitylene, whereas it decreases to a contribution of ∼10% at 70 C. As the adsorption properties of the small rotational symmetrical particles and of the fibers are identical, this leads to the conclusion that the small particles cannot be the only explanation for the fast transport process. In Table 4, diffusivities of the effective tangential diffusion are given. This diffusivity is, in principle, identical to the effective diffusion in the mesopores. As this process is more than 1 order of magnitude faster than all other processes present in the fibers, it is a potential candidate for the very fast process experimentally observed. It was, however, stated that this diffusion pathway cannot be observed by a macroscopic method, as, in an ideal fiber, no open mesopore in the tangential direction exists. Assuming that indeed cracks or more precisely open mesopore walls at the edge of the fiber are present in the fibers, as shown in Figure 11c, this transport pathway is, in principle, possible and would lead to such a fast and also temperature-dependent transport process. In conclusion, the fast process is attributed to the effective tangential diffusion in the fibers (Figure 11c) in addition to a certain
fraction of fast desorbing rotational particles present in the samples (Figure 11b). The contribution of the assumed transport, that is, slow and fast transport processes, changes, as seen in Table 3. This is due to the fact that the transport in mesoporous silica fibers can be described by the transport pathways discussed above. However, one should be aware that the assumption of distinct transport processes is a simplification, and in reality, the transport in these fibers has to be understood as a population of diffusion processes. The distribution of this population is temperature-dependent, and therefore, a change in the contribution of slow and fast transport processes is observed as the temperature is varied.
6. CONCLUSIONS Circularly ordered mesoporous silica fibers are complex, hierarchical materials for which all three structural domains of the fibers have to be considered as they contribute to the transport properties. The diffusion in the radial direction is the dominant transport process. It is related to intrawall pores present in the framework walls. Diffusion in the axial direction is also a potential transport pathway, which is essentially also governed by the intrawall diffusion and not by diffusion in the mesopores. This is due to the helical internal structure of the fibers that leads to high tortuosities of up to 104 at the edge of the fiber in the axial direction. In the fiber center, transport in the axial direction is caused by diffusion in the mesopore channels. In addition to these two transport pathways, also diffusion in the tangential direction in the mesopores is present. The process is very fast and is due to nonidealities of the fiber structure, leading to mesopore openings at the edge of the fiber. ’ ASSOCIATED CONTENT
bS
Supporting Information. A detailed discussion about the model selection including a table of the NRMSEs and figures showing the frequency responses of mesitylene in the parent and crushed samples for the different diffusion models. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Present Addresses †
University of Jordan, Department of Chemical Engineering, Amman 11942, Jordan.
Author Contributions ‡
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These authors contributed equally to this work. dx.doi.org/10.1021/jp1102912 |J. Phys. Chem. C 2011, 115, 8602–8612
The Journal of Physical Chemistry C
’ ACKNOWLEDGMENT The financial support from the Deutsche Forschungsgemeinschaft DFG under project JE260-8/1 is acknowledged. The authors are also grateful to Martin Neukamm for conducting the SEM measurements, to Xaver Hecht for the physisorption measurements, to R. Kolvenbach for the help in creating schematic representations of the fiber structure, and for the fruitful discussions in the framework of the center of excellence IDECAT. H.A. acknowledges the DFG and the Support to Research and Technological Development and Innovation Initiatives and Strategies in Jordan—an EU-funded program, for funding his research visits to Germany.
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(30) Marlow, F.; Kleitz, F.; Wilczok, U.; Schmidt, W.; Leike, I. ChemPhysChem 2005, 6, 1269. (31) Yang, Y. G.; Suzuki, M.; Fukui, H.; Shirai, H.; Hanabusa, K. Chem. Mater. 2006, 18, 1324. (32) Gobin, O. C.; Wan, Y.; Zhao, D. Y.; Kleitz, F.; Kaliaguine, S. J. Phys. Chem. C 2007, 111, 3053. (33) Hoang, V. T.; Huang, Q. L.; Eic, M.; Do, T. O.; Kaliaguine, S. Langmuir 2005, 21, 2051. (34) Satterfield, C. N.; Cheng, C. S. AIChE J. 1972, 18, 724.
’ REFERENCES (1) Yang, H.; Ozin, G. A.; Kresge, C. T. Adv. Mater. 1998, 10, 883. (2) Zhao, D. Y.; Yang, P. D.; Huo, Q. S.; Chmelka, B. F.; Stucky, G. D. Curr. Opin. Solid State Mater. Sci. 1998, 3, 111. (3) Yang, H.; Vovk, G.; Coombs, N.; Sokolov, I.; Ozin, G. A. J. Mater. Chem. 1998, 8, 743. (4) Naik, S. P.; Elangovan, S. P.; Okubo, T.; Sokolov, I. J. Phys. Chem. C 2007, 111, 11168. (5) Huo, Q. S.; Zhao, D. Y.; Feng, J. L.; Weston, K.; Buratto, S. K.; Stucky, G. D.; Schacht, S.; Sch€uth, F. Adv. Mater. 1997, 9, 974. (6) Kleitz, F.; Marlow, F.; Stucky, G. D.; Sch€uth, F. Chem. Mater. 2001, 13, 3587. (7) Alsyouri, H. M.; Lin, Y. S. Chem. Mater. 2003, 15, 2033. (8) Marlow, F.; Spliethoff, B.; Tesche, B.; Zhao, D. Y. Adv. Mater. 2000, 12, 961. (9) Marlow, F.; Khalil, A. S. G.; Stempniewicz, M. J. Mater. Chem. 2007, 17, 2168. (10) Kievsky, Y.; Sokolov, I. IEEE Trans. Nanotechnol. 2005, 4, 490. (11) Marlow, F.; McGehee, M. D.; Zhao, D. Y.; Chmelka, B. F.; Stucky, G. D. Adv. Mater. 1999, 11, 632. (12) Ye, Z. B.; Alsyouri, H.; Zhu, S. P.; Lin, Y. S. Polymer 2003, 44, 969. (13) Seshadri, S. K.; Alsyouri, H. M.; Lin, Y. S. Microporous Mesoporous Mater. 2010, 129, 228. (14) Stallmach, F.; Karger, J.; Krause, C.; Jeschke, M.; Oberhagemann, U. J. Am. Chem. Soc. 2000, 122, 9237. (15) Alsyouri, H. M.; Lin, J. Y. S. J. Phys. Chem. B 2005, 109, 13623. (16) Stempniewicz, M.; Rohwerder, M.; Marlow, F. ChemPhysChem 2007, 8, 188. (17) Marlow, F.; Stempniewicz, M. J. Phys. Chem. B 2006, 110, 11604. (18) Lin, Y. S.; Alsyouri, H. J. Phys. Chem. B 2006, 110, 11606. (19) Yasuda, Y. Heterog. Chem. Rev. 1994, 1, 103. (20) Shen, D. M.; Rees, L. V. C. Zeolites 1991, 11, 666. (21) Shen, D. M.; Rees, L. V. C. J. Chem. Soc., Faraday Trans. 1994, 90, 3011. (22) Ruthven, D. M.; DeSisto, W. J.; Higgins, S. Chem. Eng. Sci. 2009, 64, 3201. (23) Salmas, C. E.; Androutsopoulos, G. P. Ind. Eng. Chem. Res. 2001, 40, 721. (24) Gobin, O. C.; Huang, Q. L.; Vinh-Thang, H.; Kleitz, F.; Eic, M.; Kaliaguine, S. J. Phys. Chem. C 2007, 111, 3059. (25) Kruk, M.; Jaroniec, M.; Choma, J. Carbon 1998, 36, 1447. (26) Gregg, S. J.; Sing, S. K. W. Adsorption Surface Area and Porosity, 2nd ed.; Academic Press Inc.: New York, 1982. (27) Hansen, N. The CMA Evolution Strategy: A Comparing Review. In Towards a New Evolutionary Computation. Advances on Estimation of Distribution Algorithms; Lozano, J. A., Larranaga, P., Inza, I., Bengoetxea, E., Eds.; Springer, Berlin: 2006; p 75. (28) Chen, F. X.; Xu, X. J.; Shen, S. C.; Kawi, S.; Hidajat, K. Microporous Mesoporous Mater. 2004, 75, 231. (29) Sayari, A.; Liu, P.; Kruk, M.; Jaroniec, M. Chem. Mater. 1997, 9, 2499. 8612
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