Article pubs.acs.org/Langmuir
Validity of the t‑plot Method to Assess Microporosity in Hierarchical Micro/Mesoporous Materials Anne Galarneau,*,† François Villemot,†,‡ Jeremy Rodriguez,† François Fajula,† and Benoit Coasne*,†,‡,§ †
Institut Charles Gerhardt Montpellier, UMR 5253 CNRS-UM2-ENSCM-UM1, ENSCM, 8 rue de l’Ecole Normale, 34296 Montpellier Cedex 05, France ‡ MultiScale Material Science for Energy and Environment, CNRS/MIT (UMI 3466), 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States § Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States S Supporting Information *
ABSTRACT: The t-plot method is a well-known technique which allows determining the micro- and/or mesoporous volumes and the specific surface area of a sample by comparison with a reference adsorption isotherm of a nonporous material having the same surface chemistry. In this paper, the validity of the t-plot method is discussed in the case of hierarchical porous materials exhibiting both microand mesoporosities. Different hierarchical zeolites with MCM-41 type ordered mesoporosity are prepared using pseudomorphic transformation. For comparison, we also consider simple mechanical mixtures of microporous and mesoporous materials. We first show an intrinsic failure of the t-plot method; this method does not describe the fact that, for a given surface chemistry and pressure, the thickness of the film adsorbed in micropores or small mesopores (< 10σ, σ being the diameter of the adsorbate) increases with decreasing the pore size (curvature effect). We further show that such an effect, which arises from the fact that the surface area and, hence, the free energy of the curved gas/liquid interface decreases with increasing the film thickness, is captured using the simple thermodynamical model by Derjaguin. The effect of such a drawback on the ability of the t-plot method to estimate the micro- and mesoporous volumes of hierarchical samples is then discussed, and an abacus is given to correct the underestimated microporous volume by the t-plot method.
1. INTRODUCTION Zeolites present remarkable potential for catalysis, adsorption, and ion-exchange due to the unique properties of their networks of intracystalline micropores.1−10 Nevertheless, in most processes, and more particularly in catalysis, the nanometer size of the micropores leads to mass transfer limitations and reduced access to the active surface.4−7 To circumvent this limitation, postsynthesis treatments based on hydrothermal calcination and acid leaching, originally designed to stabilize the zeolite crystals, have been applied to generate a secondary network of mesopores allowing better transport of the reagents and products toward and from the intracrystalline active sites.11−13 Other routes to create ordered homogeneous distributions of mesopores inside zeolite crystals include dealumination followed by desilication in the presence of surfactants, addition of surfactants and/or long chain silanes or polymers during zeolite synthesis, use of hard templates such as carbon particles or carbon networks, and so forth. Several recent papers6,7,14,15 provide critical reviews of the different pathways to generate ordered mesoporous networks into zeolite crystals in order to increase accessibility to the zeolite active sites. Some of these methods lead to materials with © 2014 American Chemical Society
hierarchical micro/mesoporosity, which means that the mesopores are connected to the micropores and to the external surface of the particles. In catalysis, these systems are more efficient as they allow larger amounts of reagents to access the microporosity per unit of time.16 This interpretation is supported by our recent molecular simulation study on adsorption and transport in hierarchical porous materials built from faujasite zeolite (FAU) in which a regular cylindrical mesopore is carved out.17 Our molecular simulation showed that nitrogen adsorption isotherms in such hierarchical zeolites can be described as a linear combination of the weighed adsorption isotherms of the parent zeolite and of the mesoporous region. From a practical point of view, this theoretical work shows that hierarchical zeolites cannot be distinguished from a mechanical mixture of zeolite and mesoporous silica powders based on the sole knowledge of nitrogen adsorption isotherms. Received: July 12, 2014 Revised: August 22, 2014 Published: September 18, 2014 13266
dx.doi.org/10.1021/la5026679 | Langmuir 2014, 30, 13266−13274
Langmuir
Article
The existence of hierarchical microporous/mesoporous materials raises the question of their fine characterization using adsorption-based techniques.18−21 Among available techniques, the t-plot method22,23 is routinely used to determine the microporous and mesoporous volumes in porous materials including hierarchical zeolites. This method is based on the use of standard adsorption isotherms, that is, the so-called t-curve which relates the statistical film thickness t(P) on a flat surface as a function of pressure P for the same adsorbate and temperature (in fact, the reference system must possess a surface without any micro- or mesoporosity so that curvature effects such as capillary condensation can be neglected). We note that the use of standard adsorption isotherms for the characterization of porous materials from adsorption data has become a standard procedure.24 However, while the ability of the t-plot method to assess the surface area and microporosity of samples has been discussed in the literature for mesoporous materials, its validity in the case of hierarchical materials remains to be established. In order to address this issue, we prepared hierarchical micro/mesoporous materials with different amounts of ordered mesoporosity. Starting from faujasite crystals (FAU type), the pseudomorphic transformation (also referred to recrystallization synthesis) was used to create some mesoporosity similar to that found in templated mesoporous silica (MCM-41).6,25 To do so, faujasite crystals were put in contact with alkyltrimethylammonium bromide surfactants (CnTAB) in alkaline media (NaOH).6,25 The amount of sodium hydroxide in the pseudomorphic synthesis of FAU crystals was increased to increase the mesoporous volume until complete transformation into MCM-41-like materials. Different chain lengths of alkyltrimethylammonium bromide (CnTAB with n = 10−18) were used in the synthesis to study the influence of the mesopore size in the t-plot analysis of hierarchical micro/mesoporous materials. As a benchmark to assess the validity of the t-plot, we also considered the use of the t-plot method in the case of mechanical mixtures of known amounts of zeolite (FAU) and Al-MCM-41. These simple mixtures serve as reference systems, since their experimental micropore and mesopore volumes are known.
Table 1. Micropore and Mesopore Volumes Obtained from the t-plot Technique for Mechanical Mixtures of FAU CBV720 and MCM-41(6383) (Si/Al = 15) Prepared from Aerosil and C16TABa t-plot
experimental volumes FAU/(FAU+MCM), FAU wt %
Vmicro (mL/g)
Vmeso (mL/g)
Vmicro (mL/g)
Vmeso (mL/g)
100 (CBV720) 97.60 94.83 89.64 80.20 60.16 39.88 20.32 0 (MCM-41(C16))
0.357 0.348 0.338 0.320 0.286 0.215 0.142 0.072 0
0 0.015 0.032 0.064 0.122 0.245 0.369 0.489 0.614
0.211 0.205 0.201 0.188 0.178 0.144 0.107 0.067 0
0.144 0.157 0.171 0.205 0.289 0.374 0.456 0.614
a
The experimental volumes known from the amount of each sample added to the mixture are also shown.
Table 2. Micropore and Mesopore Volumes Obtained from the t-plot Technique for Mixtures of FAU CBV720 and MCM-41(6384) (Si/Al = 15) Prepared from Aerosil and C12TABa t-plot
experimental volumes FAU/(FAU+MCM), FAU (wt %)
Vmicro (mL/g)
Vmeso (mL/g)
Vmicro (mL/g)
Vmeso (mL/g)
100 (CBV720) 80.04 59.68 39.84 19.64 0 (MCM-41-C12)
0.357 0.286 0.213 0.142 0.070 0
0 0.088 0.153 0.200 0.237 0.532
0.211 0.179 0.144 0.108 0.067 0
0.196 0.270 0.336 0.414 0.531
a
The experimental volumes known from the amount of each sample added to the mixture are also shown.
this parent H-FAU-Y material, 0.36 mL/g, corresponds to that expected for pure crystalline faujasite. In order to prepare the hierarchichal microporous/mesoporous materials, an alkaline solution was prepared with CnTAB in sodium hydroxide solution at 50 °C under stirring. After complete dissolution of CnTAB in NaOH solution, the FAU zeolite powder was added and stirred for 1 h. The composition of the mixture in molar ratio corresponds to 1 Si/0.07 Al/ 0.1 CnTAB/0.05−0.25 NaOH/50 H2O. The slurry was then placed in a stainless autoclave at 115 °C (to test the effect of the NaOH amount) or 150 °C (to test the effect of the mesopore size) for 24 h. As will be discussed below, longer times were also considered. The resulting material was then filtered and washed until neutral pH, dried at 80 °C, and calcined at 550 °C for 8 h under air-flow. In a typical synthesis (example 6481, Table 3), 0.133 g of NaOH was stirred in 30 g of water until complete dissolution. Then 1.213 g of C16TAB was added and the mixture stirred until forming a homogeneous solution. Next, 2 g of FAU zeolite was added under stirring. The mixture was then transferred in an autoclave at 115 °C for 24 h in static condition. The autoclave was then cooled down in a fresh water bath, and the slurry filtered and washed until neutral pH, dried at 80 °C, and calcined at 550 °C for 8 h under air-flow. 2.2. Materials Characterization. X-ray diffraction (XRD) patterns of the materials were acquired using a Bruker D8 Advance diffractometer with a Bragg−Brentano geometry and equipped with a Bruker Lynx Eye detector. XRD patterns were recorded in the range 4−50° (2θ) to identify zeolite peaks and in the range 1−5° (2θ) to detect the low angle XRD peaks corresponding to the mesoporous part. The angular step size was of 0.0197° and the counting time of 0.2 s per step. The textural properties of the materials were determined
2. EXPERIMENTAL DETAILS 2.1. Materials Synthesis. Al-MCM-41 (Si/Al = 15) materials were synthesized by adding NaAlO2 (Carlo Erba) in an alkaline solution containing alkyltrimethylammonium bromide surfactant (CnTAB with n = 10-18) (Aldrich) (Tables 1 and 2 for n = 12 and 16). The mixture was stirred at 50 °C until complete dissolution before adding silica Aerosil 200 (Degussa) and stirring for 1 h. The composition of the mixture in molar ratio is 1 SiO2 / 0.07 NaAlO2 / 0.1 CnTAB/0.25 NaOH/50 H2O. The slurry was then placed in stainless autoclave at 115 °C for 24 h. The resulting material was then filtered and washed until neutral pH, dried at 80 °C, and calcined at 550 °C for 8 h under air-flow. Al-MCM-41 prepared with CnTAB will be referred as Al-MCM-41(Cn). Hierarchical micro/mesoporous FAU were prepared from dealuminated H-FAU-Y (Si/Al = 15) named CBV720, which was purchased from Zeolyst. This zeolite was chosen to keep the same Si/Al ratio as in Al-MCM-41 in order to ensure very close surface chemistries for both materials. This zeolite has been slightly dealuminated by the supplier to reach Si/Al = 15, which creates in the meantime some additional large mesopores (diameters > 30 nm) visible in the high-pressure region of the nitrogen adsorption isotherm (not shown). Nevertheless, this will not interfere with the present study as the mesoporosity created by pseudomorphism leads to much smaller mesopores (pore diameter < 5 nm). The micropore volume of 13267
dx.doi.org/10.1021/la5026679 | Langmuir 2014, 30, 13266−13274
Langmuir
Article
micropores, none of the linear regimes are getting through the origin of the t-plot, and the intercept of the linear fit in the lowpressure range is classically taken as the microporous volume of the solid. The validity of this technique will be discussed below. The key point of the t-plot method is hence to have an accurate description of the average thickness t(P) of the film adsorbed on the flat reference surface as a function of pressure P. Several theoretical models or empirical equations are available to describe this function. In case of weak adsorbate/ adsorbent interactions, t(P) is often considered independent of the nature of the solid surface.27 The Harkins−Jura equation, for instance, describes the value of the film thickness without accounting for adsorbate−adsorbent interactions.28 t(P) is also routinely described using the Frenkel−Halsey−Hill equation such as the Hasley t-plot.29 A detailed discussion on the use of the Frenkel−Halsey− Hill equation can be found in ref 30. For surfaces interacting weakly with the adsorbate, the disjoining pressure isotherm proposed by Churaev and Zorin31 gives accurate results. For different adsorbate−adsorbent interactions, Lecloux and Pirard32 demonstrated that the standard adsorption isotherm has to be chosen according to the intensity of the adsorbent−adsorbate interactions, expressed by the CBET constant. We have previously tested these t-curves27,29,32 for nitrogen adsorption at 77 K on purely mesoporous MCM-41 reference materials (CBET = 90−100). Unfortunately, none of these methods allow a linear fit going through the origin. Recently, a mixed reference isotherm has been developed exploiting the nitrogen isotherms of two silicas: the lowpressure part of the experimental reference curve of Aerosil and the high-pressure part of the reference curve of Lichrospher giving accurate results for MCM-41 reference materials (silica as well as polymers).33,34 Statistical analysis allows to describe the mixed isotherm by appropriate simple analytical functions for each portion of the relative pressure domain. These equations, initially developed for αs-plot analysis, have been converted into t-plot analysis by multiplying the equations by the thickness of the film adsorbed at P/P0 = 0.4 on Aerosil, t(P/ P0 = 0.4) = 0.538 nm. Such a t-plot can be described using the empirical equations given below in eq 1. Linear fits going through the origin for MCM-41 reference materials with mesopore diameters above 2 nm have been obtained. The corresponding t-curve, that is, the film thickness t(P/P0) as a function of the relative pressure P/P0 (P0 is the bulk saturating vapor pressure), obtained from the mixed reference can be described using the following empirical equation:
Table 3. Micropore and Mesopore Volumes Obtained from the t-plot Technique for Hierarchical FAU Prepared from FAU CBV720 (Si/Al = 15) with C16TAB at 115 °C with Different NaOH Molar Ratios and Synthesis Timesa linear combination
materials FAU 6480 6481 6482 6488 6455 6454 6453 MCM-41 6383
t-plot
NaOH/ Si
synthesis duration (h)
Vmicro (mL/g)
Vmeso (mL/g)
Vmicro (mL/g)
Vmeso (mL/g)
0 0.05 0.10 0.15 0.20 0.25 0.25 0.25 0.25
0 24 24 24 24 24 48 72 24
0.357 0.316 0.171 0.085 0.008 0.010 0 0 0
0 0.060 0.346 0.510 0.664 0.685 0.699 0.689 0.614
0.211 0.197 0.134 0.097 0.061 0.046 0.044 0.040 0
0.180 0.406 0.522 0.625 0.652 0.664 0.667 0.614
a
Comparison with Al-MCM-41 (Si/Al = 15) synthesized from Aerosil. The experimental volumes known from the linear combination method (see text) are also shown.
from the N2 adsorption/desorption isotherms at −196 °C measured on a Micromeritics Tristar 3000 apparatus. The samples were previously outgassed in vacuum at 250 °C for 12 h. The mesopore size was estimated using the Broekhoff−de Boer method as it is considered one of the most accurate methods for pore size determination in the case of mesoporous silica.26
3. RESULTS AND DISCUSSION 3.1. Application of the t-plot Method for Microporous and Mesoporous Adsorbents. Before discussing the validity of the t-plot analysis applied to hierarchical porous materials, it is important to recall the principles of this method and how it can be used to estimate microporous and mesoporous volumes. Although this part can be skipped by readers who are familiar with the t-plot method, we think that it is worth recalling its tenets and provide practical examples with pure microporous and mesoporous solids as this method, which is often used in a routine fashion, can lead to important errors in its interpretation. Let us consider the adsorption isotherm Nads0(P) (in mol/g) of a given gas on a flat surface having a specific surface area S (in m2/g). Assuming the adsorbed phase has an average density equal to the bulk liquid density ρ0 (in mol/m3), Nads0(P) can be converted into the average thickness of the film adsorbed on the surface t(P) = Nads0(P)/ρ0S. t(P) is known as the t-curve. Let us now consider adsorption on a solid of unknown geometry but having the same surface chemistry as the flat surface considered above. The t-plot is obtained by plotting for each pressure the adsorbed amount Nads(P) for this solid as a function of t(P). If the t-plot of a solid is a straight line, this solid has the same adsorption behavior as the flat surface so that it can be considered as a flat surface for the adsorbate, and the proportionality constant is simply the surface area of this solid. In other words, all linear regimes in a t-plot provide evidence that adsorption, for that specific pressure range and adsorbate, is taking place similarly as on a flat surface and the slope of the straight line gives the value of the surface area. In contrast, any departure from the linear regime indicates the presence of a porosity getting filled and corresponding to pore sizes given by the pressure at which this departure is observed. In the case of materials containing
t(P /P0) (in nm ) = ⎧ A1[1 − exp(A 2P /P0)] + A3[1 − exp(A4 P /P0)] (P /P0 < 0.03) ⎪ B B ⎪ B1(P /P0) 2 + B3(P /P0) 4 (0.03 ≤ P /P0 < 0.25) ⎪ C3 ⎤ C1 ⎪⎡ (0.25 ≤ P /P0 < 0.6) ⎨ ⎣⎢ C2 − log(P / P0) ⎦⎥ ⎪ ⎤D3 D1 ⎪ ⎡⎢ (0.6 ≤ P /P0 < 0.9) ⎪ ⎣ D2 − log(P / P0) ⎦⎥ ⎪ E (P /P )E2 + E (P /P )E4 (0.9 ≤ P /P ) ⎩ 1 0 3 0 0
(1)
where A1 = 0.1887299 nm, A2 = −481.3, A3 = 0.182099 nm, A4 = −23.78, B1 = 0.5675647 nm, B2 = 0.199735, B3 = 0.4116168 nm, B4 = 2.00834, C1 = 0.1423566, C2 = 0.1078, C3 = 0.4888, D1 = 0.08309076, D2 = 0.02995, D3 = 0.369, E1 = 1.268066 nm, E2 = 1.931, E3 = 0.76934 nm, and E4 = 51.09. If used as such, this equation provides the film thickness in nm. 13268
dx.doi.org/10.1021/la5026679 | Langmuir 2014, 30, 13266−13274
Langmuir
Article
We first discuss the applicability of the t-plot method for pure mesoporous (Al-MCM-41) and pure microporous (FAU zeolite) materials. Figure 1 shows the t-plot obtained for
specific surface area. As far as the mesoporous volume is concerned, there are two ways adopted in literature: one can take the intercept of the linear regime after condensation (red dashed line in Figure 1 for Al-MCM-41) or the pore volume corresponding to the first point which departs from this linear regime (shown by the black dashed line in Figure 1 for AlMCM-41). From the t-plot method applied to simulated mesoporous materials of known mesoporous volume, we have previously shown that the second estimate gives the more accurate results for mesopores of pore diameters larger than 2.4 nm.32 However, for the same simulated materials, the linear fit in the low pressure range does not go through the origin for materials with mesopore diameters lower than 3.2 nm, which could be misinterpreted as the presence of microporosity.32 While the t-plot method as described here provides accurate estimate of the specific surface area and mesoporous volume of mesopores with large diameters, it should be used with caution when dealing with small mesopores diameters (roughly less than 10 times the size σ of the adsorbate molecule, σ = 0.36 nm for nitrogen, σ = 0.34 nm for argon, and σ = 0.27 nm for water). Indeed, as will be discussed below, assuming that adsorption on the mesopore surface is similar to adsorption on a flat surface is only valid for mesopores with a large diameter. Figure 1 also shows the t-plot obtained for nitrogen adsorption at 77 K in a FAU zeolite (Si/Al = 15) with pore cavities of size 1.3 nm and pore windows of size 0.78 nm (the inset shows the nitrogen adsorption isotherm at 77 K which was used to obtain the t-plot). As in the case of a mesoporous solid, the adsorption isotherm in a microporous solid and the resulting t-plot can be divided in two regimes. At low pressure, adsorption first occurs in the micropores. Once the micropores are filled, adsorption only takes place on the external surface of the microporous particles. These two regimes clearly appear in the t-plot shown in Figure 1 for FAU zeolite. At low thicknesses (which correspond to low pressures), the adsorbed amount increases rapidly with increasing thickness. At larger thicknesses (which correspond to higher pressures), the t-plot is linear because adsorption only occurs on the external surface of the microporous particles in this pressure range. A linear regression in this pressure/thickness range provides an estimate of the external surface area of the particles (slope of the solid red line in Figure 1 for the FAU zeolite). The microporous volume represents the contribution to the adsorbed amount that does not come from the external surface and is deducted from the pore volume corresponding to the first point which departs from the linear regime of the external surface (shown by the black dashed line in Figure 1 for the FAU zeolite). A microporous volume of 0.36 mL/g is therefore obtained as expected for pure FAU crystals. However, the classical way to determine the microporous volume of a material by the t-plot method is to extrapolate the linear fit in the low-pressure range (blue line in Figure 1 for the FAU zeolite) and to take the intercept as the microporous volume. In the case of zeolite, it is clear that this intercept (pore volume V = 0.21 mL/g) does not provide the correct microporous volume and already shows the limitation of the t-plot method applied to pure microporous solids. This is due to the lack of validity of the t-plot for small pores as explained below. 3.2. Validity of the t-plot Method as a Function of the Range of Pore Sizes Considered. A major drawback of the tplot method, which has received only very little attention, is the fact that this method does not capture the effect of curvature on the thickness of the adsorbed film. While the thickness t(P) at a
Figure 1. (Top) t-plot for nitrogen adsorbed at 77 K in mesoporous Al-MCM-41(C16) (the inset shows the nitrogen adsorption isotherm at 77 K which was used to obtain the t-plot). The blue line, which goes though the origin, has a slope, which corresponds to the total specific surface area of the sample. The intercept of the dashed black line with the y-axis corresponds to the real mesoporous volume of the sample, while the intercept of the dashed red line with the y-axis corresponds to what is commonly used for determining the mesoporous volume. (Bottom) t-plot for nitrogen adsorbed at 77 K in FAU zeolite (the inset shows the nitrogen adsorption isotherm at 77 K which was used to obtain the t-plot). The blue and red lines, which correspond to linear regressions of the t-plot in different thickness ranges, provide different underestimation of the microporous volume of the sample, while the dashed black line gives the real microporous volume of the zeolite.
nitrogen adsorption at 77 K in a mesoporous material AlMCM-41(C16) having regular mesopores of a diameter D = 3.6 nm (we also show in the inset the nitrogen adsorption isotherm at 77 K which was used to obtain the t-plot). The tplot applied to pure mesoporous materials exhibit two regimes. Prior to capillary condensation (corresponding to the sharp increase in the adsorbed amount in both the adsorption isotherm and t-plot), adsorption occurs both on the internal surface of the mesopores and on the external surface of the mesoporous particles. Above condensation, adsorption only occurs on the external surface. The first regime in the t-plot can be fitted with a straight line going through the origin (blue line in Figure 1 for Al-MCM-41). The slope of this line gives the total specific surface area that is the sum of the mesoporous and external surface areas. The slope of the linear regime after condensation gives the specific surface area of the external surface only (red line in Figure 1 for Al-MCM-41). From the tplot, it is also possible to obtain the mesoporous surface area by subtracting the external specific surface areas from the total 13269
dx.doi.org/10.1021/la5026679 | Langmuir 2014, 30, 13266−13274
Langmuir
Article
gases in porous silica). In agreement with the experimental and molecular simulation above, Derjaguin’s model predicts that, at a given pressure, t(P) increases with decreasing the pore diameter D. The fact that both molecular simulation and theory exhibit the same curvature effect on the thickness of the adsorbed film provides evidence that this effect is not an artifact arising from the treatment of the experimental data. This drawback affects the determination of the porous volumes using the intercept of the t-plot curve instead of using the pore volume corresponding to the first point (which departs from the linear regime after pore filling as explained in Figure 1). This effect can explain large underestimations of the mesopore volume observed for mesoporous materials such as MCM-41 porous silicas (see Figure 14 in ref 38 for instance). The physical origin of this curvature effect lies in the fact that, for cylindrical pores (or any curved geometries), the surface area and, hence, the free energy of the curved gas/liquid interface decreases with increasing the film thickness. As a result, for cylindrical pores of different diameters, the thermodynamics of the curved gas/liquid interface is different so that the thickness of the film adsorbed at a pressure P/P0 also depends on the pore diameter D. In order to provide a theoretical picture of the curvature effect, we consider the thermodynamical approach by Derjaguin.37,39,40 The Grand Potential of the system composed of the pore of length L and radius D and the film of a thickness t adsorbed at the pore surface writes:
given pressure can be assumed to be independent of the pore diameter D for large pores, it is known that for small pores t(P,D) also depends on D. For instance, in the specific case of argon adsorption at low temperature, molecular simulations have shown that t(P) is an increasing function of the pore diameter D for pores smaller than 10 times the size σ of the adsorbate molecule (σ = 0.34 nm for argon, and σ = 0.27 nm for water). 30,35 In order to illustrate this effect and demonstrates that it is not only observed in molecular simulations, we show in Figure 2 experimental data obtained
Figure 2. Thickness of the film adsorbed at 77 K and at P/P0 = 0.3 in mesopores MCM-41 of different diameters D (solid line) and on a flat surface (D → ∞) (dashed line), as predicted by Derjaguin’s model. The symbols are the film thicknesses obtained from the adsorption isotherms shown in Figure S1 of the Supporting Information.
Ω = −PGVG − PSVS − PLVL + γSLASL + γLGALG
by Kruk and Jaroniec36 for argon adsorption in MCM-41 samples with different diameters ranging from 2.4 and 6.5 nm (7σ and 20σ, respectively). Assuming that the adsorbed film has an average density equals to that of the bulk liquid ρ0 (note that the results below remain qualitatively valid if a different density is assumed), the experimental adsorbed amount were converted into a film thickness using the following equation: t=
D ⎡⎢ 1− 2 ⎢⎣
1−
Vads(P /P0) ⎤ ⎥ ⎥⎦ V0
+ ASL W (t )
(3)
where PG, PL, PS, VG = πL/4(D − 2t)2, VL = πL/4(D2 − (D − 2t)2), and VS are the pressure and volume of the gas, adsorbed, and solid phases, respectively. γLG, γSL, and ALG = πL(D − 2t), ASL = πLD are the gas-adsorbed phase and solid-adsorbed phase surface tensions and surface areas, respectively. The interface potential W(t) in eq 3 allows describing adsorption at the surface of the pore as it accounts for the interaction between the adsorbate molecule and solid surface. While different mathematical expressions are possible for W(t), we use in the present the simple exponential decaying function W(t) = S exp(−t/ξ), where S is the spreading coefficient as defined in ref 39 and ξ is the correlation length of the fluid (note that the reasoning below is valid regardless of the expression chosen for W(t) provided a decaying function of t is used). Using this expression and the fact that PL − PG = ρLkBT ln(PG/P0), one gets:
(2)
In this equation, which is consistent with the definition of the statistical film thickness t(P), one simply assumes that the adsorbate covers uniformly the surface of the cylindrical pore having a diameter D. V0 is simply the maximum adsorbed amount. Based on this equation and the data obtained by Kruk and Jaroniec (Figure S1 in the Supporting Information), we determined in Figure 2 the thickness of the adsorbed film at a pressure P/P0 = 0.3 in the MCM-41 mesopores of different diameters D. In agreement with the molecular simulations reported in refs 30 and 35, the experimental data show that, at a given pressure, the thickness of the film increases upon decreasing the pore diameter D. Interestingly, this curvature effect on the film thickness is also captured by simple thermodynamical approaches such as Derjaguin’s model.37 This simple model, which can be seen as an extension of the Kelvin equation accounting for adsorption of a film prior to capillary condensation, describes accurately adsorption and condensation of gases in mesoporous solids. In particular, as can be seen in the adsorption isotherms shown in Figure S1 in the Supporting Information, Derjaguin’s model provides a reasonable fit of the experimental adsorption isotherms for MCM-41 materials (see also ref 38 for a discussion on the validity of Derjaguin’s model to describe adsorption of various
⎛P ⎞ (D − 2t )2 Ω = ρL kBT ln⎜ G ⎟ + γLG(D − 2t ) 4 πL ⎝ P0 ⎠ + DS exp( −t /ξ)
(4)
where ρL is the liquid density of the adsorbate, kB Boltzmann’s constant, and T the temperature. At a given gas pressure PG and pore diameter D, the film thickness is given by the equilibrium condition (dΩ)/(dt) = 0, that is: ⎛P ⎞ DS (D − 2t )ρL kBT ln⎜ G ⎟ + 2γLG + exp( −t /ξ) = 0 ξ ⎝ P0 ⎠ (5) 13270
dx.doi.org/10.1021/la5026679 | Langmuir 2014, 30, 13266−13274
Langmuir
Article
For small film thicknesses with respect to the pore diameter, one can use a Taylor expansion of the exponential term to get the following solution: DS
(
D
from 20 to 80 wt %. In other words, the error in estimating the microporous volume of mechanical mixtures of microporous and mesoporous materials increases continuously with increasing the amount of microporosity (from 13 to 80% in volume ratio (Vmicro(t‑plot)/Vtotal(t‑plot) × 100)). Above 80 wt % microporosity (or 80% in volume), the error remains similar to Vmicro(t‑plot)/Vmicro(experimental) = 0.6. An abacus (eq 7) has been determined from Tables 1 and 2 to correct the microporous volume (Vmicro(t‑plot)/Vmicro(true)) in materials presenting both
)
− 1 − 2γLG ξ 2ξ D − t= P DS 2 2ρL kBT ln PG + 2
() 0
ξ
(6)
Equation 6 provides an estimate of the film thickness at a pressure P and its dependence on the pore diameter. Considering that S, γLG, and ξ are positive, t(P,D) is a decreasing function of D. This results demonstrates that, in agreement with the experimental and molecular simulation data above, t(P/P0) at a given pressure P/P0 increases upon decreasing D. 3.3. Application of the t-plot Method to Hierarchical Micro/Mesoporous Materials. To test the validity of the tplot method for samples containing different porosity scales, mechanical mixtures of different known amounts of FAU crystals and Al-MCM-41(C16) with pore diameter of 3.6 nm (Figure S2 in the Supporting Information) (or Al-MCM41(C12), pore diameter of 2.7 nm) were prepared. For each type of MCM-41 sample considered, the different mechanical mixtures and their known microporous and mesoporous volumes are shown in Tables 1 and 2. Given the fact that the microporous and mesoporous volumes are de facto known for a mechanical mixture, we can use these samples to assess the ability of the t-plot method to determine these different porous volumes. Figure 3 shows as an example the t-plot obtained for a
Figure 4. Abacus for t-plot analysis correction for hierarchical micro/ mesoporous materials featuring different microporous volume ratio determined by t-plot analysis. The plot shows the correction for two series of samples: (●) Hierarchical samples with MCM-41 (C16) mesoporosity and (○) hierarchical samples with MCM-41 (C12) mesoporosity. Solid line is the best fit (eq 7) of the data.
micropores and mesopores (Figure 4). This abacus can be described using the equation: y = A + BxC
(7)
where A = 0.34, B = 2.70, C = −0.59, y = Vmicro(t-plot)/ Vmicro(true), and x = Vmicro/Vtotal(t-plot) × 100. In the case of FAU, Vmicro/Vtotal (t-plot) = 59% and y = 0.59, therefore for purely microporous materials the error obtained by t-plot is constant y = 0.59. In this abacus, the percentage of microporosity has been determined by the t-plot method by using as microporous volume (V micro(t‑plot) ) the value determined classically at the intercept of low-pressure linear part (blue line in Figure 3) and as total volume (Vtotal(t‑plot)) the pore volume corresponding to the first point, which departs from the linear regime after pore filling (intercept with the dashed black line in Figure 3). In order to evaluate the true microporosity in “real” hierarchical microporous/mesoporous materials, we considered hierarchical materials obtained by pseudomorphic synthesis (also referred as recrystallization synthesis) of MCM-41 type porosity in FAU zeolite materials (in this context, “real” refers to materials with entangled micro and mesoporosity by opposition to simple mechanical mixtures of microporous and mesoporous solids considered above). Different hierarchical micro/mesoporous materials have been synthesized in order to assess the microporous and mesoporous volumes in such materials. Starting from FAU crystals and adding CnTAB in alkaline media with increasing amounts of NaOH, it was possible to increase the mesoporosity of hierarchical materials until NaOH/Si = 0.20 (Figures S3 and S4 of the Supporting Information file). At higher NaOH amount (NaOH/Si > 0.25), the transformation into pure MCM-41 mesoporous materials occurs. Hierarchical materials with different mesopore diame-
Figure 3. t-plot for nitrogen adsorbed at 77 K in a mechanical mixture of 20 wt % FAU and 80 wt % Al-MCM-41(C16) (the inset shows the nitrogen adsorption isotherm at 77 K which was used to obtain the tplot). The intercept of the dashed black line with the y-axis provides an estimate of the total porous volume, while the intercept of the blue line provides an estimate of the microporous volume.
mechanical mixture consisting of 20 wt % FAU and 80 wt % AlMCM-41(C16). We also show in the inset of Figure 3 the nitrogen adsorption isotherm at 77 K, which was used to obtain the t-plot. As described in subsection 3.1, the intercept of the dashed black line with the y-axis provides an estimate of the total porous volume while the intercept of the blue line provides an estimate of the microporous volume (the mesoporous volume can be estimated as the difference of these two volumes). Tables 1 and 2 compare the known microporous and mesoporous volumes of the different mechanical mixtures considered in this work with those obtained from the t-plot method. As evidenced in this comparison, the t-plot method significantly underestimates the micropore volumes of the mixtures from 0.9 < Vmicro(t‑plot)/ Vmicro(experimental or true) < 0.6 when the amount of FAU increases 13271
dx.doi.org/10.1021/la5026679 | Langmuir 2014, 30, 13266−13274
Langmuir
Article
the t-plot. In the t-plot method applied to a micro/mesoporous material, extrapolation to a zero film thickness of the linear regime prior to capillary condensation (the y-intercept of the blue line) provides an estimate of the microporous volume. On the other hand, from the t-plot data in the pressure above mesopore filling, the intercept of the black dashed line represents the total porous volume. The mesoporous volume is simply given by the difference between the total pore volume and the microporous volume. Tables 3 and 4 compare the microporous and mesoporous volumes obtained from the t-plot method and the values obtained using the linear combination technique. At a low level of FAU transformation (0.05 < NaOH/Si < 0.10), micropore volumes obtained from the t-plot analysis are underestimated in comparison to the linear combination. To analyze the influence of the mesopore diameter on the validity of the t-plot method, hierarchical FAU were also prepared at NaOH/Si = 0.125 using surfactants with different chain lengths: CnTAB with n = 10−18 (Table 4). For hierarchical micro/mesoporous materials with the smallest mesopore diameters (synthesized with alkyl chain of C10TAB and C12TAB), microporous volumes obtained from the t-plot analysis are underestimated in comparison with the linear combination (Table 4). In fact, the underestimation by the tplot method does not depend on the mesopore size but is related to the amount of microporosity in the solid (Table 5). Indeed, the transformation of FAU using shorter alkyl chains of surfactants (C10TAB and C12TAB) is slower and the remaining microporosity is therefore higher for a similar time of pseudomorphic synthesis. The micropore (and the mesopore) of the “true” hierarchical materials have been recalculated (Table 5) from the t-plot analysis using the abacus (Figure 4). Corrected microporous volumes are in good agreement with the microporous volumes determined by the linear combination for the hierarchical materials featuring a microporous volume ratio in the solid (determined by t-plot) higher than 20%. In contrast, the t-plot analysis underestimates the microporous volume. As an example (sample 6480, Table 5), the t-plot method gives a microporous volume of 0.20 mL/g whereas 0.33 and 0.32 mL/g have been obtained for the corrected microporous volume and the microporous volume determined by the linear combination, respectively. For hierarchical materials with a lower amount of microporosity (
