Surface Area and Microporosity of a Pillared Interlayered Clay (PILC

Dec 23, 2000 - Micromeritics Instrument Corp., Inc., Norcross, Georgia 30093, and MLO Consulting, 6105 Black Water Trail,. Atlanta, Georgia 30328...
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J. Phys. Chem. B 2001, 105, 623-629

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Surface Area and Microporosity of a Pillared Interlayered Clay (PILC) from a Hybrid Density Functional Theory (DFT) Method James P. Olivier† and Mario L. Occelli*,‡,§ Micromeritics Instrument Corp., Inc., Norcross, Georgia 30093, and MLO Consulting, 6105 Black Water Trail, Atlanta, Georgia 30328 ReceiVed: May 17, 2000; In Final Form: October 2, 2000

A hybrid density functional theory method previously developed for heterogeneous silica surfaces has been used to calculate model isotherms over a wide range of pore widths for the adsorption of nitrogen at 77 K within cylindrical pores having a clay-like surface. Using these models and the experimental isotherm data, the integral equation of adsorption was inverted by a regularization method to yield the micropore and mesopore size distribution of a pillared interlayered clay (PILC). The results obtained are compared with the results of more traditional data treatments.

Introduction It has long been recognized that gas adsorption isotherms can provide a great deal of information about the porous structure of solids. Virtually all adsorption isotherm equations stem from a common viewpoint that treats the adsorbed molecules as a separate homogeneous surface phase of the adsorptive in equilibrium with a homogeneous gas phase.1 The equation of state for the two-dimensional surface phase can, if known, be used to derive the isotherm equation, and vice versa.2 It can be argued that this is a valid viewpoint for classical thermodynamic treatment, but it prevents us from readily modeling the system on a molecular scale. To do the latter, it is better to consider the adsorptive as a single inhomogeneous phase having a density gradient near the surface of the adsorbent, produced by forces emanating from the solid. The quantity adsorbed is then given by the amount of adsorptive in the surface region in excess of the bulk concentration extended to the solid surface. The focus of study now becomes the problem of describing the density profile of the inhomogeneous phase, rather than the equation of state or the isotherm equation. Modern statistical thermodynamics provides alternative ways to model inhomogeneous systems. Although computationally expensive, computer simulation techniques, such as molecular dynamics and Monte Carlo processes, are very accurate ways to evaluate models of such systems. A reasonable alternative is available in density functional theory.3-6 None of these methods produce something that might be called an isotherm equation, but they do describe the particle density distribution. By defining a system geometry, a rule for the pairwise particle interaction potentials, the system temperature, and the chemical potential (bulk pressure), one can obtain the time average equilibrium distribution of particles near the surface boundary, hence an isotherm “point”. The use of density functional methods has grown rapidly over the past decade and is recognized as a powerful method for * Author to whom correspondence should be addressed. E-mail: [email protected]. † Micromeritics Instrument Corp., Inc. ‡ MLO Consulting. § E-mail: [email protected].

studying inhomogeneous classical fluids.7 Such systems are characterized by the spatial variation of the average one-body density of fluid molecules F(r), where r is the generalized coordinate vector. Density functional theory is therefore well suited to the study of physical adsorption, where the density of the adsorptive can change by several orders of magnitude over small distances near the surface of the adsorbent. The thermodynamics of homogeneous systems allows us to express the free energy of a fluid as a function of its bulk concentration or pressure. Similarly, density functional methods are based on the idea that the free energy of an inhomogeneous fluid can be expressed as a functional of the local concentration, F(r). Once this functional is known, all the relevant thermodynamic functions can be calculated. In addition, derivatives of this functional determine the equilibrium correlation functions that describe the microscopic structure of the inhomogeneous fluid. This allows the computation of tensions for interface problems, the determination of solvation forces for confined fluids, and the investigation of phase transitions for various types of inhomogeneity, such as fluids confined in a porous solid. The ability to model adsorption on a free surface having a given adsorptive potential, or in a confined pore space of slitlike or cylindrical geometry, has led to useful methods for extracting adsorptive potential distribution or pore size distribution information from experimental adsorption isotherms8-10 and has resulted in the development of practical numerical deconvolution methods that provide a best fit solution representing the adsorptive potential or pore distribution of the sample. In this paper we show how such a deconvolution method can be used for estimating the pore size distribution of a pillared interlayered clay (PILC). The PILC under study has been prepared by expanding a sample of Na-bentonite with the dodecameric [Al13O4(OH)24(H2O)12]7+ ion.11 After heating in air at 500 °C, this Keggin ion loses its water ligands forming Al13-blocks (1.09 nm × 0.98 nm) × 0.84 nm in size that become the structure supporting pillars.11 The determination of surface area (SA) and pore volumes (PV) in clays and expanded clays remain a subjects of controversy and debate.12,13 In PILCs, where the interlamellar distance is of the order of a few molecular diameters, only

10.1021/jp001822l CCC: $20.00 © 2001 American Chemical Society Published on Web 12/23/2000

624 J. Phys. Chem. B, Vol. 105, No. 3, 2001 monolayer or restricted multilayer formation can occur and the PILCs surface area should be adequately approximated by the Langmuir equation.14 However, the Langmuir result includes adsorption on the PILC external surface and, as a result, SA values for the pores could be inflated.15 In PILCs supported by pillars 0.84 nm in height, multilayer adsorption of probe molecules required by the BET equation16 cannot occur and the BET method underestimates the PILC pore SA. More realistic PV values for PILCs have been obtained using the t-method of De Boer et al.15,17 The purpose of this paper is to study the PILC microstructure using a hybrid density functional theory (DFT) method based on a well-characterized model materials (MCM-41)18 and a cylindrical pore geometry.19 It is clear that any simple model chosen to represent a complex material such as a pillared clay must be only an approximation if it is to result in a useful method for characterizing an unknown sample of that material. The creation of an accurate model would require prior knowledge of the sample structure. While one might visualize a pillared clay as having rectangular pore openings, a channel structure of rectangular pores seems unlikely because such an ordered distribution of pillars would imply an equally ordered (homogeneous) distribution of charge density which in natural minerals is unlikely to occur. The adsorptive energy distribution of the material is a function of its geometry. The problem is compounded when the surface of the adsorbent is energetically heterogeneous as well, as are most surfaces. The adsorption isotherm reflects the net adsorptive energy distribution, but cannot differentiate the source of the variation. It is for these reasons that the hybrid method presented in this work was developed: the surface of the solid itself is used to describe the very low-pressure region of the model isotherms used to deconvolute the experimental data. It remained to choose a simple geometry with an appropriate surface area-to-pore volume ratio. A slit pore model was briefly investigated, but it was clear that it was inadequate, since it generated too little surface area for the observed pore volume. A pure slit pore model yields an area/volume ratio of 2/w, where w is the slit width. Introducing Keggin ions into this geometry adds net area and decreases the volume, or increases the area/volume ratio. A rectangular pore geometry, with square cross section, has an area/volume ratio of 4/w, as does a cylindrical geometry. The solution of the nonlocal density functional integral equations is currently not possible for other than a one-dimensional variation in wall potential. For the above reasons, a cylindrical model was chosen as the best practical compromise. Experimental Section The PILC material used in the present study was prepared with a sample of Na-bentonite (bentolite-H) obtained from the Southern Clay Products, Inc., Gonzales, Texas. The sample contains >90% of the clay mineral montmorillonite; quartz (∼3.9%) is the main impurity.1 Chemical analysis gave the following composition: 6.7% Al, 32.2% Si, 1.30% Mg, 0.65% Ca, 0.59% Fe, 2.60% Na, and 0.05% K. The pillaring reactions were performed with an excess of aluminum chlorhydrate (ACH) solutions (Reheis’s Chlorhydrol).The ACH/clay and the water/clay (wt/wt) ratio used was 1.0 and 100, respectively, and the slurry pH was ∼4.8. The reaction product was collected by centrifugation and extensively washed with distilled water at 60 °C. The washed samples were air dried at room temperature for 10 h and then oven-dried at 100 °C for an additional 4 h. Thermal stability was investigated by passing dry air in an oven heated at 500 oC/12 h.

Olivier and Occelli Nitrogen sorption isotherms at liquid nitrogen temperature were obtained by the static-volumetric technique using a Micromeritics ASAP 2010 micropore unit. Prior to analysis, samples were outgassed in vacuo at 400 °C overnight. DFT Plus software, (from Micromeritics) was used to calculate pore size distributions, using the models and techniques described below. The Density Functional Theory (DFT) Method The density functional approach to understanding the structure of inhomogeneous fluids at a solid interface consists of constructing the grand potential functional, Ω[F(r)], of the average singlet density F(r) and minimizing Ω[F(r)] with respect to F(r) to obtain the equilibrium density profile and thermodynamic properties. One of the key results of density functional theory is that, for a given intermolecular potential energy Φ, the intrinsic Helmholtz free energy is a unique functional, F[F(r)], and does not depend on the form of the external potential V(r) that is the cause of the inhomogeneity. Because determining the exact free energy functional is not presently possible, a major goal of the theory is to find suitable approximations for a given type of fluid that are computationally tractable and sufficiently accurate for the problem being studied. The system model adopted above is described by the grand canonical ensemble, in which the chemical potential µ, system volume V, and temperature T are specified. An appropriate starting point, therefore, is to define the grand potential functional, Ω[F(r)], for the system:

Ω[F(r)] ) F[F(r)] +

∫ dr F(r)[Vext(r) - µb]

(1)

where µb is the bulk chemical potential imposed on the system and Vext(r) is the wall potential. At the equilibrium F(r), the value of Ω[F(r)] will be a minimum. Thus by differentiating eq 1 with respect to the density and equating to zero, it is possible, at least in principle, to solve the resulting equation for the equilibrium density profile F(r) in pores of a certain width by an iterative numerical method9 once expressions for F[F(r)] and Vext(r) are known. The net quantity adsorbed at a given pressure is then found by integrating F(r) from wall to wall and subtracting a quantity equivalent to the nonexcluded volume of the pore at the bulk gas density. The result is normalized to one square meter of wall area and expressed as cm3 STP for direct comparison with experimental data. Specification of V(r). As a first step in solving eq 1, we need to determine Vext(r) in the fluid region that consists of a given volume V bounded, on at least one side, by a wall that provides the external potential. The inhomogeneous fluid contained in the volume being considered is in equilibrium with a distant reservoir of the same fluid at a known chemical potential (hence fixed pressure). Being unaffected by the wall potential, the reservoir fluid is homogeneous. We know, therefore, that every point in the inhomogeneous system must have a net chemical potential equal to the bulk value despite large differences in local concentration. Considering a single, planar wall allows us to model adsorption at a free surface. Two parallel planar walls, as present in PILCs, confining the inhomogeneous fluid provides a model for a slit pore. Cylindrical and spherical walls can also be considered. We now define the particle-wall potential and the interparticle potential function. In the following discussions we model the wall as a semiinfinite solid with a topologically and energetically smooth, rigid surface of infinite lateral extent. Two parallel surfaces separated by a distance Wp between centers of surface atoms constitute a model slit-shaped pore. Assuming the Lennard-Jones pairwise

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interaction potential, Steele20 has described the interaction potential between a surface of this kind and a single fluid molecule by

Vw(z) ) 2πsfFsσsf2∆[2/5(σsf/z)10 - (σsf/z)4 σsf4/(3∆(z + 0.61∆)3)] (2) where z is the distance of the fluid particle from the wall, Fs is the molecular density of the solid, ∆ is a spacing parameter, and sf and σsf are the effective Lennard-Jones solid-fluid pairwise interaction parameters. We use the values Fs ) 0.114 A-3 and ∆ ) 3.35 A.21 Although eq 2 was derived specifically to model graphite, we will use it as a general form for the wall interaction, realizing that the value of sf should be viewed as a scaling factor. For the case of the model slit pore, the wall potential within the pore is the sum of the contributions from each wall, hence

Vext(z) ) Vw(z) + Vw(Wp - z) for 0 < z < Wp

(3)

The fluid-fluid pair interaction energy is modeled using the Lennard-Jones 6-12 potential

Φff(s) ) 4ff[(σff/s)12 - (σff/s)6]

(4)

where s is the intermolecular separation and ff and σff are parameters of the bulk fluid. For nitrogen-nitrogen interactions, the values used for ff and σff were 1.2943 × 10-14 ergs/ molecule and 3.5746 × 10-8 cm, respectively. The Free Energy Density Functional. The free energy density functional, F[F(r)], to be used in eq 1, is not known exactly. Knowing it exactly would be equivalent to knowing the exact grand partition function of the inhomogeneous system. Following the prescription of Tarazona,22-24 we expand the functional in a perturbative fashion about a reference system of hard spheres:

∫∫ dr dr′F(2)(r,r′)Φ(r,r′)

F[F(r)] ) Fh[F(r)] + 1/2

(5)

where Fh[F(r)] is the free energy functional for a system of hard spheres, F(2)(r,r′) is the pair distribution function for particles at r and r′, and Φ(r,r′) is given by eq 4 using s ) |r - r′|. Clearly at this point the configurational part of the free energy has been separated into a short-ranged hard sphere repulsion contained in the term Fh[F(r)] and a longer range attractive potential represented by the last term. We follow previous authors22,25 in using the WCA prescription26 based on separation at the zero of force. The hard sphere functional can be divided into ideal and excess (configurational) components:

Fh[F(r)] ) Fid[F(r)] + Fex[F(r)]

(6)

The ideal component is local, i.e., given exactly by the density at that point:

Fid[F(r)] ) kT

∫ drF(r)[ln(Λ3F(r)) - 1]

(7)

where k is the Boltzmann constant and Λ is the de Broglie thermal wavelength for the adsorptive molecule: Λ ) [h2/ 2πmkT]1/2. The excess component is expressed as

Fex[F(r)] ) kT

∫ drF(r)fex[Fs(r)]

(8)

where fex is the excess Helmholtz free energy per molecule for a uniform hard sphere fluid and Fs(r) is the nonlocal density obtained by a weighted averaging of the density in the neighborhood of r:

Fs(r) )

∫ dr′F(r′)ω[|r - r′|; Fs(r)]

(9)

The hard sphere excess free energy is calculated from the Carnahan-Starling equation of state.27 The density-dependent weighting functional w(r;Fs) in eq 9 is chosen to give a good description of the hard sphere direct correlation function for the uniform fluid over a wide range of densities. The form used is that proposed by Tarazona.23 The Chemical Potential. Having completed the prescription for the grand potential, it now remains to calculate the equilibrium density profile for the system. Equation 5 can be put in a form to be solved for F(r) by substitution of eqs 3, 4, and 6-9. The minimization requirement is satisfied by differentiating the resulting equation with respect to density and equating to zero. The resulting Euler-Lagrange equation is

µb ) kT ln(Λ3F(r)) + fex[Fs(r)] +

∫ dr′F(r)f′ex[Fs(r)]∂Fs(r)/∂F(r) + ∫ dr′F(r′)Φ(|r - r′|) + Vext(r)

(10)

where the first term on the right is the ideal component of the chemical potential at r, the second and third terms are the contribution of the hard sphere repulsion determined for the nonlocal density, the fourth term is the contribution of the attractive forces now expressed in the mean field approximation, and the last term is the contribution of the wall potential. These five terms must sum to the imposed chemical potential of the bulk homogeneous phase at all system points. For the homogeneous system, eq 10 reduces to

µb ) kT ln(Λ3Fb) + fex(Fb) + Fb f′ex(Fb) 5.028πffσff3Fb (11) thus knowing experimental or calculated values of Fb for both the bulk gaseous and liquid adsorptive at the system temperature, we can solve eq 11 for numerical values of ff and σff. Equation 10 can be solved by iterative numeric methods22 and is applicable to any geometry. However, the problem simplifies considerably in the case of a slitlike pore where r ) z, allowing analytic integration over the x,y plane for which the density is assumed invariant, or more precisely, is assumed to have the same correlation as the bulk fluid. For cylindrical pores, the equations can be cast in cylindrical coordinates, allowing analytic integration over cylindrical shells of invariant density. When solving eq 10 for a confined fluid, one finds for a range of pore size and temperature that there are two profiles that provide solutions at a given bulk pressure, corresponding to two minima in the grand potential. These correspond to metastable gas and liquid phases, showing the possibility of pore capillary condensation. The equilibrium condensation pressure is that for which the grand potentials of the liquid and vapor metastable states are equal (eq 1). The DFT Calculation. The solution of eq 10 for a series of imposed pressures yields the equilibrium density profile of fluid molecules for each pressure. Figure 1 shows a representative profile for model nitrogen adsorbed in a model cylindrical pore at a relative pressure of 0.25. If we integrate a density profile

626 J. Phys. Chem. B, Vol. 105, No. 3, 2001

Olivier and Occelli vector f(Hi) is most sensitive to imperfections in the data. For n > m, the solution is stabilized because of the additional data constraints. In this work we use a greatly overdetermined matrix for which n > 2m. There are additionally two other independent constraints on the solution that can be used to improve the stability of the process. One is the physical requirement that each fi be nonnegative, that is, only positive values of pore area or volume are allowed. The second regularization constraint is to require that for any real sample, the pore distribution should be smooth. As a measure of smoothness we use the size of the second derivative of f(Hi):

|

2

|

d f(H) 2 dH ∫ dH 2 Figure 1. Representative profile for model nitrogen adsorbed in a model cylindrical pore at a relative pressure of 0.25.

from the surface outward and subtract the quantity of adsorptive that would be present in the absence of wall forces, we obtain the quantity adsorbed per unit area of surface, Qads, which can be compared directly to experimental values. For a free surface,

Qads )

∫ dz(F(z) - F0(z))

(12)

To a high accuracy, we can let F0(z) ) 0 for z < σsf, F0(z) ) Fb for z > σsf. The upper limit of the integration for a free surface is chosen large enough that Qads is essentially constant upon further extension of the limit. For a slit or cylindrical pore, the integration limit is to the center of the pore. The integral equation of isothermal adsorption for the case of distributed pore size can be written as the convolution

Q(p) )

∫ dHq(p,H)f(H)

∑i q(p,Hi)f(Hi)

|Df|2 ) fTDTDf (15)

where D is the second derivative matrix, eq 16a. The problem is now reduced to finding the fi such that |Q qf|2 is small (a good fit to the data) and |Df|2 is small (a smooth pore size distribution), and f g 0 (no negative pore area). To do this we create the matrix q′ by augmenting λD to the bottom of the q matrix. We also create the vector Q′ by extending the Q vector with zeros (eq 16b).

-1 2 0 -1 · (a) D ) · · 0 · · · 0 · · ·

-1 0 0 2 -1 0 ·· 0 0

or in matrix notation Q ) qf (14)

where Q(p) is an experimental adsorption isotherm interpolated onto a vector p of pressure points; q(p,Hi) is a matrix of values for quantity adsorbed per square meter, each row calculated for a value of H at pressures p, and f(Hi) is the solution vector whose terms represent the area of surface in the sample characterized by each pore width Hi. The solution values desired are those that most nearly, in a least squares sense, solve eq 14. Since the data Q(p) contains some experimental error and the kernel models q(p,H) are not exact, we can expect the results, f(Hi), to be only approximate. Indeed it is a characteristic of deconvolution processes to be unstable with respect to small errors in the data. This problem can be somewhat mitigated by choice of matrix dimensions. If we consider m members of the set of H and a vector p of length n, it is clear that n g m must hold. If n ) m, the solution

0 0

0 · · · 0 0 · · · 0 · · · · 0 -1 2 -1 0 0 0 -1 2 -1 0 0

||

(13)

where Q(p) is the total quantity of adsorbate per gram of adsorbent at pressure p; q(p,H), the kernel function, describes the adsorption isotherm for an ideally homoporous material characterized by pore width H as quantity of adsorbate per square meter of pore surface, and f(H) is the desired pore surface area distribution function with respect to H. Equation 13 represents a Fredholm integral and its inversion is well-known to present an ill-posed problem. Since we are only interested in the numerical values of f(H), we can rewrite eq 1 as a summation:

Q(p) )

or in discretized form

q (b) q′ ) · · · λD

||

Q · · · Q′ ) 0 (16) · · · 0

With these definitions,

|Q′ - q′f|2 ) |Q - qf|2 + λ2|Df|2

(17)

The constant λ has been introduced to give an adjustment to the relative weight, or importance, of the smoothness constraint. The better the model and more error free the data, the smaller λ should be. The larger λ is, the smoother the result will be. It is useful to define λ ) λ′λ0 where λ20 ) Tr(q′q)/Tr(D′D). With this definition, λ′ is a dimensionless scaling factor for the relative weight of the smoothness constraint. When λ′ ) 1 there is about equal weight given to the smoothing and the fit to the data. Finding the vector f that minimizes |Q′ - q′f|2 subject to the constraint that all fi g 0 is a standard problem in pure linear algebra and can be solved to any desired precision. Creating the Model Matrix. To calculate the model isotherms using eq 10, we must first define a set of pore widths to be modeled and a set of pressure points at which to calculate the quantity adsorbed. The set of pore widths can be chosen somewhat arbitrarily, but the pressure vector should be specifically constructed to properly weight all pore widths. The algorithm for calculating an isotherm point in the matrix of model isotherms, q(p,Hi), proceeds similarly for all the models considered here and is described below.

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Figure 3. Nitrogen adsorption isotherms at 77 K for the original clay substrate (solid points), the PILC product (open points), and the rescaled substrate isotherm.

Figure 2. (A) Equivalent surface potential required to reproduce the observed critical porefilling pressure for 13X zeolite and MCM-41 materials.19 Solid points are data. The smooth line is the function used to interpolate to a given pore size class.19 (B) Critical relative pressure for pore filling as a function of pore width. Solid points are as predicted from DFT. Open points represent the correlation reported elsewhere.19

Choosing Model Pore Widths. In this work we model pore widths from about 0.4 nm to about 50 nm, covering the micropore and mesopore range. It is convenient to choose widths in a geometric progression with 30 to 60 size classes per decade. In addition, a free surface model is included by specifying an extremely large pore width, such that capillary condensation would not be experimentally observed. The smallest pore classes in the micropore range are spaced at intervals of 0.05 molecular diameters. Establishing the Pressure Vector. Experimental adsorption isotherms obtained with well-characterized materials18 have been used to correlate the critical pore condensation pressure, p*, with effective pore width. This is shown in Figure 2. The pressure vector should be such that no pair of adjacent pore size classes exhibit values of p* that fall between consecutive pressure points. To do this, a smooth, least squares interpolating spline routine was used to estimate the value of p* for each size class and also at the geometric mean of adjacent classes. In this way, a pressure vector with the desired properties and of twice the length of the pore size vector is generated. Once the pressure vector is established, the model matrix can be calculated. Results and Discussion The model isotherm predicted by nonlocal density functional theory for each pore size class was calculated by methods described above, based on a cylindrical pore geometry. The

Figure 4. The DFT calculated normalized isotherm for a 1.2 nm cylindrical pore in an energetically uniform solid (points), and the normalized isotherm for the clay substrate (line).

model isotherms calculated in this way do not reproduce the low-pressure region of the experimental isotherms because of the pronounced energetic heterogeneity of the clay surface. This portion of the model can, however, be described by the experimental isotherm for the parent clay. Figure 3 shows the nitrogen adsorption isotherms for the parent clay and the expanded PILC. Also shown is the parent clay isotherm scaled by a factor (2.2) that produces the best fit to the low-pressure region of the PILC. The scalability of the low coverage portion of the isotherm indicates that the nature of the surface of the clay is nearly unaffected by the pillaring reaction although very small differences at the lowest pressure may be due to the presence of alumina pillars. We can infer, therefore, that the desired model isotherm for each pore size class will follow the behavior of the parent clay prior to pore filling. This result also lends support to atomic force microscopy (AFM) images that represent the clay surface free from Al-species implying that all the Keggin ions added are inside the clay interlamellar space.28 In Figure 4, we show a model isotherm calculated by density functional theory for a 1.2 nm cylindrical pore together with the appropriately scaled isotherm for the parent clay. Using the algorithm previously described,19 a set of hybrid models can be created. An example, based on the curves in Figure 4, is shown in Figure 5. DFT calculations designed for slitlike pores are valid when w/D , 1, where w is the width and D the length of the pore. In PILCs, where w/D e 2.0, this inequality becomes invalid and a different pore geometry must be considered. Here we have approximated a rectangular pore opening by a cylindrical pore model.

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Figure 5. The resulting hybrid model isotherm for a 1.2 nm cylindrical pore.

Olivier and Occelli

Figure 7. The adsorption isotherm reconstructed by eq 2 from the distributions shown in Figure 4 (points) compared to the experimental isotherm (line), demonstrating the quality of the fit.

Figure 6. Differential pore volume distribution (A) and pore area distribution (B) obtained by fitting the hybrid models to the PILC adsorption isotherms.

Figure 8. Cumulative pore volume (A) and pore area (B) data obtained by fitting the hybrid models to the PILC adsorption isotherms.

Using the set of hybrid models constructed as described above as the function q(p,Hi) and the experimental adsorption isotherm for the function Q(p), eq 14 was solved for the distribution vector f(Hi). The resulting distributions of pore area and pore volume by pore width are shown in Figure 6A,B. In Figure 7 we show the original isotherm data together with the reconstructed isotherm calculated from the deconvolution result according to eq 14. The parent clay under study consists of crystallites formed by the long-range stacking of silicate layers. The distribution of micropores observed in Figure 6A for the Na-bentonite sample can be attributed to missing planes, and bent planes in

these crystallites. Stacking disorders of this type have been observed in high-resolution electron microscopy (HREM) images of bentonite samples.29 As a result, more than 40% of the total surface in the parent Na-bentonite is found in micropores, see Table 1 and Figure 8A. A major difference between simulated and measured results is the appearance of some pores in the meso-micro range; see Figure 6A. As the pore width increases, both the cumulative SA and PV increase due to nitrogen sorption in voids formed by crystallites agglomeration, Figure 8A,B. HREM images of expanded clay powders have shown that after drying, the clay platelet aggregates to form microporous crystallites irregular

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TABLE 1: Surface Area (SA, m2/g) and Pore Volume (PV, cm3/g) Results from Nitrogen Porosimetry Data for the Parent Na-bentonite method DFT Langmuir BET t-plot D-R

micro. SA

total SA

micro. PV

total PV

53

120 128 95 95

0.016

0.076

9 101

0.003 0.036

TABLE 2: Surface Area (SA, m2/g) and Pore Volume (PV, cm3/g) Results from Nitrogen Porosimetry Data for the Pillared Interlayered Clay (PILC) method DFT Langmuir BET t-plot D-R

micro. SA

total SA

micro. PV

total PV

324

341 315 275 275

0.106

0.129

238 37

0.092 0.133

in size and shape.30 These aggregates connect face-to-edge and edge-to-edge to create voids responsible for the expanded clay meso- and macroporosity. Thus, when a clay is pillared, micropores are created between its silicate layers in addition to the meso- and macropores already present; see Figure 6. In Figure 6B, micropores with an average pore width near 14 Å represent pores generated by the pillaring reaction whereas pores with an average pore width near 9.5 Å result from voids in crystallite agglomerates. In general, the contribution to the PILC overall SA and PV by the clay’s meso- and macroporosity strongly depends on the drying method used to separate the liquid from the solid phase.31 In Tables 1 and 2, the present results are compared to those of traditional methods. It can be seen that the BET method does underestimate the surface area of PILCs. The Langmuir and BET formalisms are based on the model of adsorption on a free surface. The Langmuir model assumes the surface saturates after the first adsorbed layer, while the BET model presumes that multilayers can form at higher pressures. Neither model allows for the filling of micropores. The surface area derived from these models will differ from the actual area in a way that depends on the solid microporous structure. The t-plot method is designed to determine micropore volume, and the data in Table 2 show results in reasonable agreement with the DFT approach. Some deviation would be expected, since the method depends on a generalized reference material. Like the t-plot method, the D-R technique32 is meant to determine total micropore volumes. While empirical, the results for total pore volume are in reasonable agreement with the DFT result, Table 2. To elucidate the microporous structure of materials such as PILCs, a method based on molecular scale statistical thermodynamics is needed. The DFT provides such a method, and allows us to extract from adsorption data a much more complete description of a porous material than traditional methods; see Table 2.

Conclusions A formalism for constructing models describing adsorption in cylindrical pores, originally developed for mesoporous silica,19 has been shown to be useful for extracting surface area and micropore size distribution information for PILC materials. It is believed that this formalism can also adequately describe the porous properties of other heterogeneous catalysts such as the fluid cracking catalyst microspheres used in gas oil cracking. Acknowledgment. This work has been supported in part by NATO collaborative grant CRG-971497 to M.L.O. References and Notes (1) Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience: New York, 1964; Chapter 1. (2) Ross, S.; Olivier, J. P., loc. sit., p 18. (3) Evans, R. AdV. Phys. 1979, 28, 143. (4) Sullivan, D. E.; Telo de Gama, M. M. Fluid Interfacial Phenomena; Croxton, C. A., Ed.; Wiley: New York, 1987. (5) Evans, R.; Tarazona, P. Phys. ReV. Lett. 1984, 53, 557. (6) Walton, J. P. R. B.; Quirke, N. Chem. Phys. Lett. 1986, 129, 382. (7) Evans, R. Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Marcel Dekker Inc.: New York, 1992; pp 85-175. (8) Seaton, N. A. Carbon 1989, 27, 853. (9) Olivier, J. P.; Conklin, W. B. Presented at First International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, Kazimierz Dolny, Poland, 1992. (10) Lastoski, C. M.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (11) Occelli, M. L.; Bertrand, A.; Gould, S.; Dominguez, J. M. Micro. Meso. Mater. 2000, 34, 195-206. (12) Bergaya, F. J. Porous Mater. 1995, 2, 91. (13) Remy, M. J.; Vieira Coelho, A.C.; Poncelet, G. Micro. Meso. Mater. 1996, 7, 287-297. (14) Langmuir, I. J. Am. Chem. Soc. 1916, 38, 2267. (15) Zhu, H.; Vansant, E. J. Porous Mater. 1985, 2, 107. (16) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309. (17) Osinga, T. J. J. Colloid Interface Sci. 1966, 21, 405-414. (18) Kruk, M.; Jarionec, M.; Sayari, A. Langmuir 1997, 13, 6267. (19) Olivier; J. P.; Koch, S.; Jaroniec, M.; Kruk, M. 1999 Studies in Surface Science, Vol. 128; Characterization of porous solids V; Unger, K. K., et al., Eds.; Elsevier: Amsterdam, 2000; p 71. (20) Steele, W. A. Surf. Sci. 1973, 36, 317. (21) Olivier, J. P. In Carbon ’94 Extended Abstracts; The Spanish Carbon Group, Granada, Spain, 1994; p 248. (22) Tarazona, P. Phys. ReV. A 1985, 31, 2672. (23) Tarazona, P. Phys. ReV. A 1985, 32, 3148. (24) Tarazona, P.; Marconi, U. M. B.; Evans, R. Mol. Phys. 1987, 60, 573. (25) Lastoski, C. M.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (26) Weeks, J. D.; Chandler, D.; Anderson, H. C. J. Chem. Phys. 1971, 54, 5237. (27) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (28) Occelli, M. L.; Drake, B.; Gould, S. A. C. J. Catal. 1993, 142, 337-348. (29) Dominguez, J. M.; Occelli, M. L. Synthesis of Microporous Materials, Vol. II; Expanded Clays and Other Microporous Solids; Occelli, M. L., Robson, H., Eds.; Van Nostrand and Reinhold: New York, 1992; p 81. (30) Occelli, M. L.; Lynch, J.; Sanders, J. V. J. Catal. 1987, 197 (2), 557. (31) Occelli. M. L.; Peaden, P. A.; Ritz, G. P.; Iyer, P. S.; Yokoyama, M. Microporous Mater. 1993, 1, 99-113. (32) Dubinin, M. M.; Radushkevich, L. V. Dokl. Akad. Nauk. SSSR 1947, 55, 331.