Top Catal (2011) 54:121–134 DOI 10.1007/s11244-011-9631-z

ORIGINAL PAPER

Improvement in the Pore Size Distribution for Ordered Mesoporous Materials with Cylindrical and Spherical Pores Using the Kelvin Equation Jhonny Villarroel Rocha • Deicy Barrera Karim Sapag

•

Published online: 2 February 2011 Ó Springer Science+Business Media, LLC 2011

Abstract Ordered mesoporous materials such as MCM41 and SBA-15, which exhibit cylindrical pores open at both ends and SBA-16 with spherical pores, show a strong influence on adsorption and catalytic processes, basically due to their defined pore sizes. In general, the textural characteristics of these materials are obtained by N2 adsorption–desorption isotherms at 77 K where, for the calculus of the mesopores size, the ‘‘Kelvin equation’’ is used. Thus, several authors have conducted studies on the pore size distribution (PSD) for these materials, applying diverse methods such as: Barret, Joyner and Halenda (BJH); Dollimore and Heal (DH); and Kruk, Jaroniec and Sayari (BJH-KJS) methods. To obtain the PSD, the BJH and DH methods were proposed for cylindrical pores, using the desorption branch data of the isotherm, meanwhile the BJH-KJS method uses the adsorption branch data, but assumes the mechanism corresponding to the desorption branch for cylindrical pores. Due to the diversity of methods to evaluate the PSD, all of them with different considerations, it is difficult to determine the most suitable. In this work, with the aim to improve the analysis, the PSD was evaluated from the N2 adsorption–desorption isotherms at 77 K for a series of materials, MCM-41, SBA-15 and SBA-16 type, synthesized in our laboratory. By a modification in the Kelvin equation with the addition of a correction term (fc) and assuming appropriate mechanisms of capillary condensation and capillary evaporation, an improved method is proposed to be used for cylindrical as well as spherical pore geometries based on the BJH

J. Villarroel Rocha  D. Barrera  K. Sapag (&) Laboratorio de So´lidos Porosos, INFAP-CONICET, Universidad Nacional de San Luis, Chacabuco 917, CP 5700, San Luis, Argentina e-mail: [email protected]

algorithm. This term was obtained adjusting simulated isotherms with different values of fc to the experimental isotherm. The results were compared to those obtained by traditional methods and by the Non-Local Density Functional Theory (NLDFT) model. Keywords Ordered mesoporous materials  Mesopores size distribution  N2 adsorption–desorption isotherms at 77 K  Kelvin equation

1 Introduction The interest in the chemical and oil industries for the development of heterogeneous catalytic processes as substitutes of the homogeneous ones has progressively increased [1, 2]. Various solid catalysts have been studied, where porous materials play an important role both as catalysts and supports for diverse active phases [1]. Materials showing narrow pores (micropores, pores up to ˚ ), as the pillared clays [3], activated carbons [4] and 20 A zeolites [5], have been successfully applied to heterogeneous catalysts. However, diffusion complications have been found when using these materials with larger molecules than the pore size [1]. As a consequence, the development of materials with higher pore sizes, of the order of ˚ ) that the mesopores (pores ranging between 20 and 500 A improve the diffusion of reactants towards the catalytic sites, has become a focus of major interest [6]. Although several mesoporous materials have been successfully synthesized, the appearance of the Ordered Mesoporous Materials (OMM) has been one of the greatest achievements in material sciences. This represents a suitable alternative for the preparation of heterogeneous catalysts because of their porosity and specific surface, which

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have the ability to host active sites on its external and internal surface [7]. Due to their textural, morphological and structural properties, the most studied OMM are the MCM-41 [8], SBA-15 [9] and SBA-16 [10]. The pore sizes ˚ , while the SBAof the MCM-41 fall between 20 and 100 A ˚ 15 and SBA-16 between 20 and 300 A [11]. Structurally, the MCM-41 and SBA-15 materials are ordered in hexagonal arrangements of cylindrical channels and the SBA-16 materials show cage-like structures with spherical cavities [12]. As regards of the catalytic application, among other characteristics of the catalyst, the pore size distribution (PSD), which defines the different sizes and amounts of the pores, must be studied. Previous reports have shown the effect of the PSD of the catalyst on the selectivity and catalytic activity [13–15]. Consequently, diverse techniques and methods of characterization have been developed for the PSD determination. The most widely used experimental technique to determine textural characteristics (as the PSD) of porous materials is the N2 adsorption–desorption at 77 K [16, 17], with optimal results in the mesopores region [2, 11]. Among the extended researches and theories that have arisen from the data obtained by experimental isotherms are the methods based on the macroscopic theory of capillary condensation and the molecular microscopic theory [18]. Some of these latter are the molecular dynamics [19], the Monte Carlo simulation [20] and the Non-Local Density Functional Theory (NLDFT) model [21–23]. The Kelvin equation [16, 24], which is considered valid for the capillary condensation theory, is used in various methods proposed for the evaluation of the PSD (macroscopic methods). Among these macroscopic methods are: Barret, Joyner and Halenda (BJH) [25]; Pierce [26]; Dollimore and Heal (DH) [27]; Broekhoff and de Boer [28], among others. In addition, these methods require an equation to determine the statistical film thickness (t) of nitrogen adsorbed on the pore walls as a function of the equilibrium relative pressure. This corresponds to a semiempirical equation and the most acknowledged equation to predict thicknesses are proposed by Halsey [29]; Harkins and Jura [30]; and Broekhoff and de Boer [31]. The BJH and DH are the most used methods for mesoporous materials, where a cylindrical geometry of the pores is assumed and desorption branch data of the isotherm are used [25, 27]. Both, the BJH and DH methods, take into account the length and area of the pores walls in their initial considerations [16]. However, in the BJH method, due to simplification arguments, the effect of the pore lengths is not considered. As a result, the DH method provides an enhanced determination of the PSD. Several authors agree that the Kelvin equation is involved in the adsorption–desorption mechanism for

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mesoporous materials. It has been found that the classic methods for the PSD analysis, based on the Kelvin equation, overestimate the capillary condensation/evaporation pressure. Subsequently, the average pore sizes for materials ˚ becomes underestimated [11, 18, 24, smaller than 100 A 32, 33]. A modification of the BJH method was then proposed by Kruk-Jaroniec-Sayari (BJH-KJS) to evaluate the PSD, but using the adsorption branch of the experimental isotherm of an OMM series of the MCM-41 type [33]. In the same work, an expression for the statistical film thickness of nitrogen adsorbed, t, was obtained for these authors by means of experimental data [33]. In the modified BJH method, they proposed to add a term to the Kelvin equation which was determined by a calibration procedure using pore sizes data obtained by X-ray diffraction (XRD) and the filling relative pressure of primary mesopores of several samples. The BJH-KJS method proposes that this term is constant for a determined range of pore sizes. It is noticeable that the additional term is obtained without considering the accepted filling mechanism for this isotherm branch. For these materials that exhibit cylindrical pores open at both ends [34–36], they assume the formation of hemispherical menisci in the capillary condensation, which has not physical meaning. In the present study were synthesized an OMM series with cylindrical (MCM-41, SBA-15) and spherical (SBA16) pore geometries in order to assess their PSD. An improvement in the DH method using a correction term (fc) in the Kelvin equation that adjusts to the experimental isotherm is presented. The proposed method (PM) produces a series of simulated isotherms with different values of fc to obtain a final correction term whose simulated isotherm adjusts to the experimental one. The results are compared to those obtained by methods traditionally used as DH and BJH-KJS and the NLDFT model, pointing out the differences among them in order to determine the most suitable.

2 Experimental 2.1 Materials Three materials were synthesized by the sol–gel method: MCM-41, SBA-15 and SBA-16. Variations in surfactant/ silica molar ratio were applied to each one and a total of nine samples were obtained. 2.1.1 MCM-41 Samples These materials were synthesized at room temperature and atmospheric pressure based on a modification to the synthesis process described by Gru¨n et al. [37]. Cetyl trimethyl

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Table 1 MCM-41, SBA-15 and SBA-16 samples: molar ratio surfactant:silica Ordered mesoporous materials (OMM)

Samples

Molar ratio

MCM-41

MCM-41_I

0.07

MCM-41_II MCM-41_III

0.12 0.27

CTAB:TEOS

P123:TEOS SBA-15

SBA-15_I

0.010

SBA-15_II

0.015

SBA-15_III

0.020 F127:TEOS

SBA-16

SBA-16_I

0.007

SBA-16_II

0.008

SBA-16_III

0.010

ammonium bromide (CTAB) was used as surfactant, Tetraethyl orthosilicate (TEOS) as silica source, NaOH as catalyst and water as solvent. Molar ratios used for the preparation of the MCM-41 samples were: xCTAB: 1TEOS:0.6NaOH:100H2O where x corresponds to the molar ratio CTAB/TEOS. CTAB suspended in deionized water was mixed with an aqueous solution of 1 M NaOH in vigorous stirring until a transparent solution was obtained. Subsequently, TEOS was added drop-wise to form an emulsion that was stirred for 24 h. The obtained solids were separated by filtration, washed with abundant deionized water up to a conductivity value smaller than 10 lS/cm, dried at 60 °C for 12 h and calcined at 550 °C for 6 h at a heating rate of 1 °C/min. The x ratio was varied, obtaining three samples described in Table 1.

ethanol/water mixture for 2 h to remove the surfactant. Finally, the samples were re-washed using deionized water, dried at 60 °C for 12 h and calcined at 550 °C for 6 h at 1 °C/min. The y was varied obtaining three samples described in Table 1. 2.1.3 SBA-16 Samples These materials were obtained by the synthesis method reported by Esparza et al. [39] with Pluronic F127 as surfactant and TEOS, HCl and ethanol (EtOH) with molar ratios of zF127:1TEOS:0.006HCl:9H2O:13.1EtOH. F127 was mixed with a solution of 0.037 M HCl at room temperature. EtOH was then added maintaining the mixture under stirring until the surfactant was completely dissolved. TEOS was added drop-wise to the resulting solution and once the incorporation was completed, the mixture was constantly stirred for 5 h. The solution was left for approximately 3 weeks at room temperature until it gets a gel state. The resulting material, showing transparent, monolithic and rigid appearance, was fragmented and washed with deionized water to be subsequently calcined at 550 °C for 6 h at a heating rate of 1 °C/min. The z was varied obtaining three samples, described in Table 1, which summarizes the surfactant/TEOS molar ratios as well as the designated nomenclature for each synthesized sample. 2.2 Nitrogen Adsorption Measurements Measurements of N2 (99.999%) adsorption–desorption at 77 K were carried out using a volumetric adsorption equipment (AUTOSORB-1MP, Quantachrome Instruments). Samples were previously degassed at a temperature of 150 °C for 12 h and 0.5 Pa.

2.1.2 SBA-15 Samples 2.2.1 Calculations from Isotherms Data These materials were obtained by the synthesis method reported by Esparza et al. [38]. The used reagents were the surfactant Pluronic P123, TEOS and HCl. The molar ratios used for the preparation of the SBA-15 were: yP123:1TEOS:6HCl:158H2O, where y corresponds to the molar ratio P123/TEOS. P123 was dissolved in an aqueous solution of 2 M HCl (pH 1) and kept under stirring at 50 °C until a transparent solution was obtained. The required quantity of TEOS was added drop-wise under vigorous stirring. Afterwards, the reaction mixture was aged 24 h at the same temperature without stirring. Subsequently, the temperature was raised to 80 °C and maintained at this value for 48 h. The solids were filtrated, washed with abundant deionized water until reaching a conductivity value smaller than 10 lS/cm and kept in an

The specific surface area (SBET) of the samples was estimated with the Brunauer, Emmet and Teller (BET) method [40], using the adsorption data in the range of relative pressures from 0.05 to 0.12, 0.05 to 0.18 and 0.05 to 0.23 for the samples MCM-41, SBA-15 and SBA-16 respectively, where conditions of linearity and considerations regarding the method were fulfilled [17]. The total pore volume (VTP) was obtained by the Gurvich’s rule at a relative pressure of 0.985 [17]. The a-plot method [16, 41] was used to calculate the micropores (VlP) and primary mesopores (VPMP) volumes, using the LiChrospher Si-1000 macroporous silica gel as the reference adsorbent [42]. The volume of secondary mesopores (VSMP) can be obtained from the difference between VTP and the sum of VlP and VPMP.

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3 Methods for PSD Determinations 3.1 Traditional Methods Traditional methods, useful to obtain the mesopores size distribution with cylindrical geometry, following the BJH algorithm [25] were used in order to compare the results with the PM method. These methods were DH [16], BJH˚ [33] and KJS applicable to pore sizes between 20 and 65 A ˚ BJH-KJSimp applicable to pore sizes between 20 and 120 A [43]. For all the methods, the obtained pore radius (rp) was the sum of the corresponding Kelvin radius and the statistical film thickness of the N2 adsorbed layer (t). There are different proposals to obtain the t value, but in this work it was selected the one shown in the Eq. 1, derived by KJS [33], which gives optimal results for this kind of materials in the range of relative pressures (P/P0) from 0.1 to 0.95. ˚ ) was determined by: The t value (in A 2 30:3968 60:65  5 t¼4 ð1Þ 0:03071  log P=P0 According to the capillary condensation and capillary evaporation theory, the most accepted mechanisms for the filling or emptying of the mesopores are: (i)

Hemispherical menisci, for the filling and emptying of cylindrical pores open at one end [16]. (ii) Cylindrical menisci, for the filling of cylindrical pores open at both ends [44]. (iii) Hemispherical menisci, for the emptying of cylindrical pores open at both ends [44]. (iv) Hemispherical menisci, for the filling of spherical pores [45]. In addition, it is known that MCM-41 and SBA-15 materials present cylindrical pores open at both ends [34–36] and the SBA-16 materials show cage-like structures with spherical pores [12]. Therefore, all the considerations stated above should be taken into account in those traditional methods. 3.1.1 DH (Desorption Branch) The Kelvin radius used in the DH method [16] corresponding to a hemispherical meniscus is: 2cVL   rK ¼  RT ln P=P0

ð2Þ

˚ , c is the where, rk is the Kelvin radius expressed in A 2 surface tension of the liquid N2, 8.88E-3 (J/m ), VL is the

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˚ 3/mol), R is the molar volume of the liquid N2, 3.468E25 (A 2 ˚ ideal gas constant, 8.3143E20 (J A /(K mol m2)) and T is the absolute temperature of adsorption, 77 K [46]. The change in the volume of the pores emptying, applicable to cylindrical pores is: ! n1 n1 X X 2DVpi 2DVpi DVp ¼ DVn  Dtn þ Dtn tn  2 rpi pi i¼1 i¼1 r  2 rpn  ð3Þ rpn  tn At the n-th stage, DVn corresponds to the adsorbed N2 volume (as liquid), Dtn is the change in the thickness of the adsorbed layer on the pore walls, r pn is the average pore and tn is the average thickness of the adsorbed layer. As discussed above, this method can only be applied to the desorption branch of the isotherm data for materials with cylindrical pores like MCM-41 and SBA-15. Additional modifications should be made to extend its use to other cases. 3.1.2 BJH-KJS (Adsorption Branch) This method was proposed to be applied for the adsorption branch and cylindrical pores [33]. The Kelvin radius used in this method corresponding to a hemispherical meniscus, modified by a correction term, is: 2cVL  þ3 rK ¼  RT ln P=P0

ð4Þ

˚ and 3 is a where, rK is the Kelvin radius expressed in A term added by calibration with XRD data. The change in the volume of the pores filling, applicable for cylindrical pores is: ! !2   n1 X 2DVpi rpn  tn rpn DVp ¼ DVn  Dtn r Kn þ Dtn rpn rpi i¼1 ð5Þ where r Kn is the average Kelvin radius. It is noticeable that the Eq. 4 without the addition of the term 3, would corresponds to the original BJH method proposed for pores with cylindrical geometry used for the desorption branch data, where the accepted mechanism is the one ruled by the formation of hemispherical menisci. Thus, there is no physical justification to use this method with data of the adsorption branch. 3.1.3 BJH-KJSimp (Adsorption Branch) The empirical equation used to obtain the Kelvin radius with this method is [43]:

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0

rK

1 11:5   þ 2:7A ¼ 0:5@  log 0:875 P=P0

125

ð6Þ

˚ , which was where, rK is the Kelvin radius expressed in A again adjusted to XRD data, but with more parameters than in the case of the Eq. 4. Therefore it does not follow any filling mechanism. The change in the volume of the pores filling, applicable to cylindrical pores is: ! !2   n1 X r pn  tn 2DVpi r pn DVp ¼ DVn  Dtn r pn r pi r Kn þ Dtn i¼1 ð7Þ where, r  is the average Kelvin radius. Kn In all the methods, for a P/P0 variation in the adsorption/ desorption branch according to the chosen method, the filling or emptying of the pores (DVp) takes place along with a change in the pore sizes (Dwp) for an average pore size (wp ), where DVp/Dwp is the differential pore volume (denoted in PSD curves as dV/dw). Among the models based on the molecular microscopic theory, the Non-Local Density Functional Theory (NLDFT) model is one of the most reliable and precise models for the determination of the PSD [47]. Furthermore, as the profiles of equilibrium density are known for each pressure from an isotherm, suppositions regarding the mechanism for the filling of the pores are not required, contrasting with the macroscopic methods. As a result, a pore sizes analysis in the entire range of micro and mesopores can be performed by the NLDFT model [11]. The pore sizes obtained by this model agree with the results derived from experimental techniques based on X-ray diffraction (XRD) or transmission electronic microscopy (TEM) for OMM type MCM-41 [48, 49], SBA-15 [50] and SBA-16 [51, 52]. The pore size analysis for MCM-41 and SBA-15 materials can be obtained by the NLDFT model from the adsorption or desorption branch of the N2 isotherms at 77 K [9, 48]. For SBA-16 materials, the adsorption branch is commonly used for the PSD evaluation because the percolation phenomenon [39, 51], caused by the presence of ‘‘interconnected pores’’, takes place in the desorption branch [53]. The calculations by the NLDFT model for the PSD evaluations were made using an Autosorb 1 software (Quantachrome Instruments), where the kernels used for the materials with cylindrical pores (MCM-41 and SBA15) were ‘‘N2 at 77 K on silica, NLDFT adsorption branch model’’ (adsorption branch) [49] and ‘‘N2 at 77 K on silica, NLDFT equilibrium model’’ (desorption branch) [54], which have an application in the range of pore sizes from ˚ [55]. The kernel used for the materials with 3.5 to 1000 A

spherical pores (SBA-16) was ‘‘N2 at 77 K on silica, NLDFT adsorption model’’ [51] with an application range ˚ [55]. in the pore sizes from 3.5 to 400 A 3.2 Proposed Method (PM) The novelty of the PM method is that it is self-consistent with the experimental isotherm data from which the PSD is obtained and takes into account the accepted filling or emptying mechanism for each type of pore geometry. 3.2.1 Basics and Equations 3.2.1.1 Cylindrical Pores The structure of the MCM-41 and SBA-15 materials show a high degree of arrangement and consist of cylindrical pores open at both ends [34–36]. For this reason, the PSD evaluations for this kind of materials can be accomplished by using either the adsorption or desorption branch, where the capillary condensation and capillary evaporation phenomena are ruled by the formation of cylindrical and hemispherical menisci, respectively [16, 44, 56]. A methodology for the calculation of the PSD is proposed as follows. Experimental data of relative pressure (P/P0) and adsorbed volume (Vads) are collected (N2 adsorption/desorption isotherm data at 77 K) for the chosen branch (adsorption or desorption) and points over the range between 0.10 and 0.95 of P/P0 are added through a nonlinear interpolation at intervals of 0.005. Along with the obtained data, the BJH algorithm using the DVp expression for cylindrical pores derived by DH was applied (Eq. 3) but using the modified Kelvin radius (for a cylindrical or hemispherical meniscus depending on the case) by the addition of a correction term, fc, as shown in Eqs. 8 and 9. The same equation of statistical film thickness, t, used in the traditional methods and proposed by KJS (Eq. 1), is also used. rKðcylÞ ¼ 

cVL   þ fc RT ln P=P0

ð8Þ

2cVL   þ fc RT ln P=P0

ð9Þ

rKðhemÞ ¼ 

˚ , for the Therefore, the pore radii, rp, expressed in A adsorption and desorption branches, respectively, are given by Eqs. 10 and 11. cVL   þ fc þ t rp ¼  RT ln P=P0

ð10Þ

2cVL   þ fc þ t rp ¼  RT ln P=P0

ð11Þ

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The Eqs. 8 and 10 or 9 and 11 are used to determine the average radii pore, r p , with their respective pore volumes, DVp, obtained from the Eq. 3 for a variation in the P/P0 increasing (adsorption) or decreasing (desorption), respectively. Using this information and the pore geometry, the equivalent lengths, Lp, were calculated with the Eq. 12. DVp Lp ¼  2 p rp

ð12Þ

Once the whole range of P/P0 is covered, the data base (r p , Lp) is obtained and from this, it is plotted the simulated adsorption or desorption isotherm for a given value of fc. Thus, a series of simulated isotherms with different values of fc are produced in order to obtain a final correction term whose simulated isotherm adjusts to the experimental isotherm, as it is explained afterwards (Sect. 3.2.2). 3.2.1.2 Spherical Pores The PSD evaluation of materials SBA-16 type, which exhibit spherical pores, was obtained using the adsorption branch (because of the presence of interconnected pores), where the capillary condensation phenomenon is ruled by the formation of hemispherical menisci [45, 57]. Using the experimental data of P/P0 and Vads obtained from the N2 adsorption isotherm at 77 K, the same procedure as described above was applied, employing the rK(hem) (Eq. 9) and rp (Eq. 11) expressions to obtain the filling volume of spherical pores (DVp). However, taking into account that the DH method is used for pores with cylindrical geometry, a new equation of DVp for spherical pores was obtained (Eq. 13). The development of Eq. 13 is shown in the Appendix. DVp ¼

n1 X 6DVpi  2 r pi i¼1 i¼1 r pi ! 3 n1 X 3DV r pn p i  Dtn ðtn Þ2  3 r pn  tn i¼1 r pi

DVn  Dtn

n1 X 3DVp

i

þ Dtn tn

ð13Þ

Once the r p are obtained with their corresponding DVp (calculated from Eq. 13), the value of the quantity of spherical pores with radii r p , Nðr p Þ, is obtained by Eq. 14. DVp Nðr p Þ ¼  3 4 3p r p

ð14Þ

Once the whole range of P/P0 is covered, the data base (r p , Nðr p Þ) is obtained. 3.2.2 Algorithm of Calculation Based on the previously mentioned considerations, phenomena of capillary condensation and capillary

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evaporation were modeled for cylindrical pores, assuming the formation of cylindrical and hemispherical menisci, respectively. For spherical pores, the phenomenon of capillary condensation was modeled assuming the hemispherical menisci formation. For the modeling, data bases of (r p , Lp) and (r p , Nðr p Þ) for cylindrical and spherical pores were respectively used. Starting with a P/P0 equal to 0.1, and increases of 0.001, it reaches a final relative pressure of 0.95. For each P/P0, the value for statistical film thickness of the adsorbed layer (t) (Eq. 1) and the pore radius (r p ) (Eqs. 10 or 11) are calculated. Whether this last value is equal to or higher than the pore sizes present in the data base of (r p , Lp) or (r p , Nðr p Þ) as appropriate, these pores would condensate/ evaporate, filling/emptying their volumes completely. On the other hand, the pores with sizes greater than this value (r p ) contribute to the adsorbed/desorbed volume only with the adsorbed layer on their walls. Therefore, the total adsorbed volume corresponds to the sum of these two contributions. For the OMM with the presence of micropores, the total adsorbed volume is given by the addition of this contribution, obtained by the a-plot method [41]. Repeating the same procedure for each value of P/P0 up to the last set value, the simulated adsorption or desorption isotherm is obtained. A set of simulated isotherms was obtained varying the fc ˚ . They were compared to the experivalue from 0 to 9 A mental isotherm to finally perform an adjustment (square minimum adjustment) for the determination of the final fc value that reproduces the experimental isotherm in the most suitable manner, validating the reliability and selfconsistency of the method. Finally, with the experimental isotherm data and the final fc value, the DVp/Dwp values for a wp were obtained. With these values, the PSD curves are plotted for cylindrical pores, using the adsorption or desorption branch, and for spherical pores, using the adsorption branch.

4 Results and Discussion Figure 1 illustrates the experimental N2 adsorption– desorption isotherms at 77 K for the materials under study. The three kinds of materials, MCM-41, SBA-15 and SBA16 exhibit type IV isotherms according to the IUPAC classification [58], which are typical for mesoporous materials [32]. The isotherms for the MCM-41 materials are shown in Fig. 1a. As it may be observed, the adsorption–desorption process is reversible exclusively for the MCM-41_III sample. The MCM-41_I and MCM-41_II samples are reversible approximately up to 0.4 in P/P0 and at higher

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600

Table 2 Textural properties of the MCM-41, SBA-15 and SBA-16 samples

(a)

3

Vads (STP) [cm /g]

500

Samples

SBET (m2/g)

VlP (cm3/g)

VPMP (cm3/g)

VTP (cm3/g)

400

MCM-41 300

MCM-41_I

1075

–

0.57

0.74

200

MCM-41_II MCM-41_III

1290 1505

– –

0.71 0.86

0.86 0.91

MCM-41_I MCM-41_II MCM-41_III

100 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

SBA-15 SBA-15_I

455

0.03

0.30

0.46

SBA-15_II

670

0.05

0.37

0.78

SBA-15_III

645

0.05

0.25

0.61

SBA-16_I

470

–

0.27

0.28

SBA-16_II

490

0.02

0.31

0.36

SBA-16_III

465

–

0.30

0.32

SBA-16 525

(b)

3

Vads (STP) [cm /g]

420

315

210

SBA-15_I SBA-15_II SBA-15_III

105

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

250

(c)

3

Vads (STP) [cm /g]

200

150

100

SBA-16_I SBA-16_II SBA-16_III

50

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

P/P0 Fig. 1 Nitrogen adsorption–desorption experimental isotherms at 77 K for MCM-41, SBA-15 and SBA-16 samples

relative pressures show small type H4 hysteresis loops [58], typical for materials with interparticle pores, corresponding to secondary mesopores [32]. These isotherms present a step that corresponds to the filling of primary mesopores at a P/P0 interval between 0.1 and 0.35, where a pronounced inflexion indicates a narrow pore size distribution.

The isotherms of the SBA-15 materials (Fig. 1b) show a hybrid loop of hysteresis between the types H1 and H3 [58]. However, the hysteresis of the SBA-15_III sample is similar to a type H3 hysteresis loop. These isotherms are reversible up to a P/P0 of 0.45. The capillary condensation (adsorption branch) takes place on the primary mesopores at relative pressures higher than 0.45 up to approximately 0.8 where the secondary mesoporous filling is beginning. Meanwhile, three steps are observed in the desorption branch being the first related to the nitrogen desorption in the secondary mesopores, the second to the capillary evaporation of the primary mesopores and the third to the presence of blocked mesopores [59] or percolation effects. In the sample SBA-15_III these steps are not pronounced. In the hysteresis loops of the SBA-15_I and SBA-15_II samples, it can be observed that the adsorption and desorption branches are parallels, typical for materials with cylindrical pores of uniform size. On the other hand, the SBA-15_III sample exhibits an elongated hysteresis loop, indicating the presence of hexagonal arrays of distorted or low-quality cylindrical pores [38]. As a consequence, a wide pore size distribution was found (not shown). The isotherms of the SBA-16 materials (Fig. 1c) show notable hysteresis loops type H2 [58], typical of cage-like mesoporous materials [45]. The shape of this hysteresis is related to the percolation phenomenon occurring throughout the desorption process at a P/P0 of approximately 0.42 [53]. For all the samples, the capillary condensation on the primary mesopores takes places in one step, suggesting a defined pore size and a material with a high degree of arrangement [52]. From the N2 adsorption–desorption isotherms at 77 K, the textural properties of the materials under study were determined (Table 2). The values for specific surface areas

123

128 900

750

3

900

Cylindrical meniscus

MCM-41_I

V ads (STP) [cm /g]

Fig. 2 Adjustment using different correction terms (fc) in the Kelvin radius. Experimental and simulated adsorption/ desorption isotherms for MCM41, SBA-15 and SBA-16 samples

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Hemispherical meniscus

MCM-41_I

(Adsorption)

(Desorption)

750

600

600

450

450

300

300

Experimental 150

fc

Experimental

0

1

3

5

7

9

150

0 0.0

0.2

0.4

0.6

0.8

0

1.0

350

fc 0.0

3

7

9

0.4

0.6

0.8

1.0

SBA-15_I

(Adsorption) Cylindrical meniscus

250

300

(Desorption)

250

Hemispherical meniscus

3

V ads (STP) [cm /g]

1

5

350

SBA-15_I 300

0.2

0

200

200

150

150

100 50 0

100

Experimental

fc 0.0

0.2

0

1

3

5

7

9

0.4

0.6

Experimental

50

0.8

0

1.0

fc 0.0

0.2

0

1

5

7

0.4

0.6

3 9 0.8

1.0

P/P0

P/P0 320

Hemispherical meniscus

SBA-16_I 280 (Adsorption)

3

V ads (STP) [cm /g]

240 200 160 120

Experimental

80 40

fc

0 0.0

0.2

1 7

0 5 0.4

0.6

3 9 0.8

1.0

P/P0

(SBET) and the primary mesopores volumes (VPMP) reported in this work are typical for this kind of materials. In the MCM-41 and SBA-16 materials the micropores are absent, except for the SBA-16_II sample. The primary mesopores predominates for all the materials. Therefore, from the isotherms analysis of all the samples it is possible to conclude that the obtained materials belong to typical OMM. In Fig. 2, the set of simulated isotherms obtained from the proposed algorithm (PM method) for some samples is shown. These isotherms were simulated using values of the ˚ , for the adsorption correction terms, fc, between 0 and 9 A and desorption branches in the case of the MCM-41_I and

123

SBA-15_I samples and only for the adsorption branch in the case of the SBA-16_I sample. As it may be seen, for fc values equal to zero (unmodified Kelvin equation), the adsorbed volume is overestimated, in accordance with the known fact that the pore size is underestimated in the traditional methods. As the correction term, fc, increases, this overestimation decreases. The fc value, in which the simulated isotherm agrees the experimental one, is considered as the final correction term. The final fc values obtained for each material are shown in Table 3. In general, as shown in Table 3, the final correction terms are different for each sample and branch (adsorption

Top Catal (2011) 54:121–134

129

Table 3 Final correction terms (fc) for the MCM-41, SBA-15 and SBA-16 samples ˚) fc (A

Samples Adsorption

Desorption

MCM-41-I

8.5

5.6

MCM-41-II

8.5

5.6

MCM-41-III SBA-15-I

7.8 6.4

5.0 4.1

SBA-15-II

9.7

6.0

SBA-15-III

7.4

4.6

SBA-16-I

6.8

–

SBA-16-II

9.5

–

SBA-16-III

9.3

–

or desorption) under study. It is noticeable that, for the studied materials, for a fixed fc value, the overestimating of the adsorbed volume is higher in the adsorption branch than in the desorption one. This may be related to the fact that the desorption occurs closer to the equilibrium than the adsorption process [54]. The PSD evaluations by adsorption and desorption branches performed by using the PM method obtained for the MCM-41_I and SBA-15_I samples, are shown in Fig. 3a and b, respectively. As Fig. 3a illustrates, the PSD obtained from the adsorption as well as the desorption branch, do not show considerable differences in the estimation of the primary mesopores size because in the experimental isotherm the filling and emptying stages of these pores are reversible (see Fig. 1a). In Fig. 3a, the PSD ˚, from desorption branch shows a second peak near to 52 A which could be related to the percolation effect of the interconnected secondary mesopores. The MCM-41_II sample shows a similar behavior but in the MCM-41_III sample, which exhibits a completely reversible isotherm, the second peak is absent. According to the behavior shown

Fig. 3 Comparison of the PSD obtained by the PM method for both branches, adsorption and desorption

0.12

by these samples, it can be concluded that if the isotherms are reversible at the filling and emptying stages of the primary mesopores, the PSD of these samples can be obtained either by the adsorption or the desorption branch. The primary mesopores sizes in the PSD obtained from the adsorption and desorption branches of the SBA-15_I ˚ sample, present a difference of approximately 10 A (Fig. 3b), being higher for the desorption branch. The additional peak shown by the PSD assessed by using the ˚ , could be related desorption branch at approximately 50 A to the blocked primary mesopores. This behavior was observed in the other two samples, where the pores sizes values obtained by the adsorption and desorption branches ˚ for the SBA-15_II and SBA-15_III differ in 8 and 14 A samples, respectively. The difference in the pore sizes obtained for each branch, when the materials exhibit a hysteresis similar to those of the SBA-15, makes critical the selection of the isotherm branch to evaluate the correct pore size. The selection of the adequate branch in the PSD evaluation is subject of major discussions for various authors. Some of them justify the selection of the adsorption [33] or the desorption branch [54], being this latter the most accepted because, according to them, the desorption branch reflects the transition of the equilibrium phase, which supports the macroscopic methods that uses the Kelvin equation. Figure 4 illustrates the comparison of the PSD obtained for some samples by the PM method with the traditional methods and the NLDFT model. For the MCM-41_II and SBA-15_II samples, the PSD obtained by the PM method for cylindrical pores were compared with the BJH-KJS method and the NLDFT(ads) model by using the adsorption branch and with the DH method and the NLDFT(des) model, by using the desorption branch. As it may be observed from Fig. 4a and b, for the MCM41_II sample, the PSD obtained by the PM method show a

(a)

0.10

(b)

SBA-15_I (Adsorption)

0.025

MCM-41_I (Desorption)

0.08

0.020

0.06

0.015

0.04

0.010

0.02

0.005

SBA-15_I (Desorption)

3

dV/dw [cm /Å/g]

0.030

MCM-41_I (Adsorption)

0.00 20

30

40

Pore size [Å]

50

60

0.000 20

40

60

80

100

120

Pore size [Å]

123

130

(b)

(a) 0.15

0.15

MCM-41_II

PM(cyl. pore)

(Desorption)

0.12

3

MCM-41_II

PM(cyl. pore)

(Adsorption)

dV/dw [cm /Å/g]

Fig. 4 Comparison of the PSD obtained by the PM and traditional methods and the NLDFT model for adsorption/ desorption branches

Top Catal (2011) 54:121–134

0.12

NLDFT

NLDFT

BJH-KJS

DH

0.09

0.09

0.06

0.06

0.03

0.03

0.00 20

30

40

50

0.00 20

60

(c) 0.04

30

40

50

(d) 0.04 PM(cyl. pore)

SBA-15_II

PM(cyl. pore)

SBA-15_II (Desorption)

(Adsorption)

NLDFT

NLDFT

0.03

0.03

DH

BJH-KJSimp

3

dV/dw [cm /Å/g]

60

0.02

0.02

0.01

0.01

0.00 20

40

60

80

100

120

0.00 20

40

60

Pore size [Å]

(e) 1.8

SBA-16_III

3

100

120

PM(spher. pore)

(Adsorption)

1.5

dV/dlog(w) [cm /g]

80

Pore size [Å]

NLDFT BJH-KJS

1.2 0.9 0.6 0.3 0.0 20

40

60

80

100

120

Pore size [Å]

higher mesopores size respect to the other distributions. Moreover, in all the distributions shown in Fig. 4b, peaks related to the presence of secondary mesopores are observed. Regarding the PSD of the SBA-15_II sample (Fig. 4c), it can be seen that the PM method underestimates the primary mesopores size with respect to the NLDFT(ads) model and the BJH-KJSimp method. In Fig. 4d, the PSD obtained by the PM method is similar to the obtained by the NLDFT(des) model and presents higher pore sizes than the

123

DH method. All the PSD are bimodal, where the pores with the lower size could correspond to the blocked pores. Regarding the SBA-16_III sample (Fig. 4e), using the adsorption branch of the isotherm, the comparison of the PSD obtained by the PM method for spherical pores with the BJH-KJS method and the NLDFT(ads) model was made. This Figure shows that the PSD obtained by using the PM method for the SBA-16_III sample, approximates to a great extent to the PSD evaluated by the NLDFT(ads) model, while the BJH-KJS method underestimate the mesopores

Top Catal (2011) 54:121–134

131

˚ ) obtained by the PM and Table 4 Size of primary mesopores (in A traditional methods and the NLDFT model Samples

PM

NLDFT

BJH-KJS

DH

Ads

Des

Ads

Des

MCM-41_I

35.6

36.5

31.8

31.8

31.4

25.2

MCM-41_II

34.9

35.4

31.8

31.8

30.1

24.2

MCM-41_III SBA-15_I

33.4 63.7

34.0 73.9

31.8 75.9

31.8 81.4

30.0 83.5*

23.9 65.7

SBA-15_II

69.6

77.7

73.1

81.4

78.6*

65.7

SBA-15_III

61.4

75.9

64.0

79.7

78.7*

66.7

SBA-16_I

42.6D

–

40.6D

–

34.3

–

SBA-16_II

62.4D

–

58.8D

–

49.4

–

SBA-16_III

54.1D

–

51.3D

–

41.0

–

( )

* Pore sizes obtained by the BJH-KJSimp method

(D) Method and model for spherical pores

size respect to the first ones. It is noticeable that, at difference with macroscopic methods, the NLDFT(ads) model (Quantachrome software) does not show a defined pore size (histogram representation). Table 4 summarizes the primary mesopores size for the materials under study calculated by various methods (traditional and proposed) and by the NLDFT model. It can be observed several values for primary mesopores sizes, depending on the sample, the chosen branch (adsorption or desorption) and the selected method. The results summarized in Table 4 for the MCM-41 and SBA-15 samples show that the DH method underestimates the pore sizes between 16 and 25% respect to the pore sizes obtained by the NLDFT model, agreeing determinations previously reported by other authors [9]. It should be noted that the PSD obtained by the BJH-KJS method for the adsorption branch are in concordance with the NLDFT model for the desorption branch, as these authors suggested. For the SBA-16 samples the BJH-KJS method underestimates the mesopores size between 16 and 20% respect to the NLDFT model. This underestimation was previously reported elsewhere [60]. The PM method shows diverse behaviors in the results of the PSD evaluation for the three studied materials. If the results with this method are compared to these obtained by the NLDFT model, the following can be seen: (i) For the MCM-41 materials, a difference (overestimation) between 5 and 15% is observed; (ii) The MCM-41 do not exhibit a significant difference in the obtained results by the adsorption and desorption branches due to the filling and emptying of the primary mesopores are reversible; (iii) Regarding the SBA-15, differences in the PSD are detected when adsorption or desorption branches are used. Thus, using the desorption branch, there is a smaller underestimation in comparison to the adsorption branch; (iv) For the

materials with spherical pores, SBA-16, the PM method presents a small overestimation (about 5%) in the pore size. From these results and the ones discussed by different authors [22] it can be suggested that for samples with cylindrical pores if their isotherms show hysteresis, the PM method must be used for the desorption branch. However, it is important to take into account that, if the hysteresis loop closes at near 0.42 of relative pressure, the percolation effects (due to interconnected pores) could be present. In general, it was found that the PM method gives suitable approximations for the PSD evaluation. Among the advantages of this method, it is the versatility shown for three types of materials with different geometries. Regards to the application range, the proposed method can be used within a range of relative pressures between 0.1 and 0.95 of the N2 experimental isotherm at 77 K, over the whole mesopores range. This method could be used with other gases than N2 at various temperatures, taking into account the adequate parameters of these gases and the precise estimation of the adsorbed layer thickness, t, of the chosen adsorbate. Finally, the proposed method, contrasting with the traditional methods, can accurately reproduce the experimental isotherm from the PSD data. These comparisons are shown in Fig. 5, using the adsorption branch for the MCM41_II and SBA-16_III samples and the desorption branch for the SBA-15_II sample. For materials with cylindrical geometry, it can be seen that the BJH-KJS (Fig. 5a) and DH (Fig. 5b) methods overestimate the adsorbed volume compared to the experimental isotherm. For the material with spherical geometry (Fig. 5c), it was found that the BJH-KJS method overestimates the adsorbed volume. In Fig. 5a, b and c it can be noted that only the PM method (macroscopic method, using the Kelvin equation) and the NLDFT model (microscopic model) coincide with the experimental isotherms.

5 Conclusions Nine OMM materials were synthesized, based on MCM41, SBA-15 and SBA-16, with cylindrical and spherical pores. For all the samples, the N2 adsorption–desorption isotherms at 77 K were carried out to study their textural characteristics. From adsorption–desorption data, the PSD of the samples were analyzed using traditional (or macroscopic) methods (where the Kelvin equation is valid) and a microscopic model (NLDFT model). Based on the Kelvin equation, an improved method was proposed to evaluate the PSD, denominated PM method, which takes into account the correct filling/emptying mechanism for each type of pores, cylindrical or spherical.

123

132

(a)

Top Catal (2011) 54:121–134 b Fig. 5 Comparison of adjustments between the simulated isotherms

675

MCM-41_II (Adsorption)

and experimental one, obtained by the PM and traditional methods and the NLDFT model

3

Vads (STP) [cm /g]

560

445

330

Experimental PM

215

-cyl. pore-

BJH-KJS NLDFT 100 0.0

(b)

550

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

SBA-15_II (Desorption)

400

3

Vads (STP) [cm /g]

475

325

250

Experimental PM-cyl. pore-

175

NLDFT DH 100 0.0

(c)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

220

SBA-16_III (Adsorption)

3

Vads (STP) [cm /g]

190

160

Experimental 130

For the studied samples, it was found that the correction term, fc, for the adsorption branch of the isotherm in materials with cylindrical pore geometry were 1.5 times higher than those from the desorption branch. Using the PM method, for materials with cylindrical pores MCM-41 type where the primary mesopores filling (adsorption) and emptying (desorption) coincide (up to 0.4 of P/P0) it was found that any isotherm branch can be used to evaluate the primary mesopores size, which is physically acceptable and does not occur using the traditional methods. It was obtained an expression for the calculation of DVp for materials with spherical pore geometry (SBA-16), which is not referred in the bibliography. Therefore with this new expression and applying the PM method it is possible to obtain a more reliable PSD for this kind of materials. The PSD obtained for SBA-15 and SBA-16 materials using the PM method, are similar to the PSD evaluated by the NLDFT model when the most commonly used isotherm branch for these materials is selected (desorption branch for SBA-15 samples and adsorption branch for SBA-16 samples). The PM method for the three kinds of materials under study can be applied in the P/P0 ranges between 0.1 and 0.95, covering the whole range of mesopores. In addition, the PM method could be applicable to different gases knowing their adequate parameters. From the PSD data obtained by the PM method, the experimental isotherm for each studied sample can be reconstructed, which demonstrates the self-consistence of the method. The pore geometry and the filling/emptying mechanism for the materials under study are critical factors for considerations regarding the PSD evaluation. Finally, we can conclude that the proposed method (PM method) is suitable to obtain a ‘‘more realistic’’ PSD for mesoporous materials, in accordance with the experimental data (adsorption–desorption isotherm).

Appendix: Determination of the Filling Volume of Spherical Pores (DVp)

PM -spher. poreBJH-KJS NLDFT

100 0.0

0.1

0.2

0.3

0.4

0.5

P/P0

123

0.6

0.7

0.8

0.9

1.0

At the stage i of the adsorption process, there is an increase on the multilayer by capillary condensation in the pore radii (rpi ). The multilayer volume of the pores with radius rp, when rp [ rpi , is established by the difference between

Top Catal (2011) 54:121–134

133

its total pore volume and its inner capillary volume (Vci ), as described in the Eq. A1.  4 3 4 3 pr  pðrp  ti Þ Nðrp Þdrp 3 p 3 4 ¼ p 4rp2 ti  4rp ti2 þ ti3 Nðrp Þdrp ðA1Þ 3 where ti is the statistical film thickness of the adsorbed layer at the stage i and N(rp) is the number of spherical pores of radius rp. Considering the spherical geometry of the pore, the total area of pore filling at the adsorption stage (Ap) is given by the Eq. A2. Ap ¼ 3

DVp rp

ðA2Þ

The multilayer volume on the walls of the all pores, Vm, with radii higher than rpi is given by the Eq. A3. Z1  4 Vm ¼ p 4rp2 ti  4rp ti2 þ ti3 Nðrp Þdrp ðA3Þ 3 rpi

Rearranging the Eq. A3 and assuming that ti is constant: Z1 Z1 2 2 4prp Nðrp Þdrp  ti 4prp Nðrp Þdrp Vm ¼ ti r pi

1 þ ti3 3

Z1

rpi

4pNðrp Þdrp

ðA4Þ

rpi

ðA9Þ

For a small and finite step i, the multilayer increases by capillary condensation (DVm) in pores with radii higher than rpi , is given by the Eq. A10.   Ap DVm ¼ Ap ½ [ rpi Dti  2 ½ [ rpi ti Dti rp ! Ap þ ðA10Þ ½ [ rpi ti2 Dti rp2 By defining DVi as the total adsorbed volume during the stage i, the filled volume within the internal capillaries (cores) (DVc) is given by the Eq. A11. DVc ¼ DVi  DVm

ðA11Þ

Considering the geometry of the pore, the Eq. A12 shows the existing relation between DVc and DVp.  3  3 r pi r pi DVp ¼ DVc ¼ DVc ðA12Þ r Ki r pi  ti Finally, the expression for the calculation of DVp is obtained by the Eq. A13, as shown. DVp ¼

DVn  Dtn

n1 X 3DVp i¼1

r pi

i

þ Dtn tn

n1 X 6DVpi  2 i¼1 r pi 3

! n1 X 3DVpi r pn  Dtn ðtn Þ  3 r pn  tn i¼1 r pi

where: Z1

Differencing the Eq. A8 as a function of ti:   Ap dVm ¼ Ap ½ [ rpi dti  2 ½ [ rpi ti dti rp ! Ap þ ½ [ rpi ti2 dti rp2

ðA13Þ

2

 4prp2 Nðrp Þdrp ¼ Ap [ rpi

ðA5Þ

r pi

Z1

 4prp Nðrp Þdrp ¼

r pi

Z1 r pi

  Ap

[ rpi rp

ðA6Þ

!

4pNðrp Þdrp ¼

 Ap

[ rpi rp2

ðA7Þ

where Ap ½ [ rpi  is the sum of the areas of pores exhibiting     A A radii higher than rpi , rpp ½ [ rpi  and r2p ½ [ rpi  are the p

sums of the quotients

Ap rp

and

Ap rp2

of all the radii that are

higher than rpi . Substituting the Eqs. A5, A6 and A7 in the Eq. A4: !   1 3 Ap 2 Ap Vm ¼ ti Ap ½ [ rpi   ti ½ [ rpi  þ ti ½ [ rp i  3 rp rp2 ðA8Þ

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