Modeling of Surface Heterogeneity of Microporous Adsorbents with

Langmuir 2005, 21, 7243-7256

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Modeling of Surface Heterogeneity of Microporous Adsorbents with LBET Approach Jan T. Duda† and Janina Milewska-Duda*,‡ Faculty of Management and Faculty of Fuels and Energy, AGH - University of Science and Technology (AGH-UST), Al. Mickiewicza 30, 30-059 Krako´ w, Poland Received September 22, 2004. In Final Form: March 16, 2005 A package of new isotherm equations aimed at examination of pore structure of sub- and microporous materials with respect to a surface heterogeneity is proposed. One considers adsorption of small nearly spherical molecules in irregular pores of molecular size. A generalized BET theory is exploited together with a thermodynamic description of the process and handling restrictions for multilayer adsorption (LBET approach). Realistic multivariant relationships linking a surface energy distribution with the pore size are proposed, and exponential distribution of the pore size is accepted. As a result, the heterogeneous adsorption equations (uLBET formulas) are derived, dedicated for irregular microporous systems. It was shown that they may be well-approximated with analytical BET-like formulas (the models of the LBETtype), enabling fast multivariant examination of real microporous materials. Such an examination provides complementary information on adsorption mechanisms, thus, allowing more reliable evaluation of the system parameters. Results of the new model’s application to an empirical system (nitrogen adsorption on an activated carbon) are shown.

1. Introduction Evaluation of surface properties of porous materials based on adsorption measurements of small nearly spherical molecules (probing sorbates) from the volatile phase needs a mechanism of adsorption and its mathematical description to be presumed.1-3 It is especially important in the case of microporous materials, where the meaning of the surface itself is ambiguous2,3 and adsorption is affected by both a pore geometry and surface energy. There are a number of equations proposed to describe the adsorption equilibria of small molecules with different aspects of the process being stressed.1,3-6 For microporous materials the Dubinin-Radushkevitch (DR), Dubinin-Astachow, or BET equations are commonly recommended.1,7,8 They are useful to obtain a measure of the material surface area by using lower pressure data;1,2 however, results of surface structure and energy evaluation based on adsorption at higher pressures are doubtful. This is because effects of particular surface properties on the adsorption process are closely inter-correlated, so a specific shape of an isotherm may be formally explained with different adsorption mechanisms.3,4,9 Hence, in our opinion the surface structure and energy distribution in microporous material cannot be reliably determined but only evaluated in semiquantitative terms by multilateral verification of alternative models involving presumed geometric and energetic properties of the material. To * To whom correspondence should be addressed. Tel. (48 12) 617 2117; fax (48 12) 617 2066; e-mail [email protected]. † Faculty of Management, AGH-UST. ‡ Faculty of Fuels and Energy, AGH-UST. (1) Sing, K. S. W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska, T. Pure Appl. Chem. 1985, 57, 603. (2) Marsh, H. Carbon 1987, 25, 49. Mahajan, O. P. Carbon 1991, 29, 735. (3) Gauden, P. A.; Terzyk, A. P.; Rychlicki, G.; Kowalczyk, P.; C Ä wiertnia, M. S.; Garbacz, J. K. J. Colloid Interface Sci. 2004, 273, 39. (4) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (5) Dubinin, M. M. Prog. Surf. Membr. Sci. 1975, 9, 1. (6) Jaroniec, M. Adsorption 1997, 3, 187. (7) Clarkson, C. R.; Bustin, R. M.; Levy, J. H. Carbon 1997, 35, 1689. (8) Kats, B. M.; Kutarov, V. V. Adsorpt. Sci. Technol. 1998, 16, 257. (9) Milewska-Duda, J.; Duda, J. T.; Jodłowski, G.; Kwiatkowski, M. Langmuir 2000, 16, 7294.

make possible such an evaluation a mathematical description of the adsorption should be constructed in such a way to represent particular phenomenological effects with the parameters being of clear physical meaning. Such parameters can be defined to fulfill certain well-verifiable criteria. This paper presents an attempt to build the models satisfying the above demands. It follows the line of our earlier papers,9-14 where the BET theory4 was generalized and adapted to take into account specific properties of microporous adsorbents of irregular structure (LBET approach). In a recent paper13 a general mathematical model for constrained multilayer adsorption is derived and next simplified to the analytical form applicable for energetically homogeneous microporous surfaces (LcBET formula). In this paper the LBET approach is developed to handle an energetic heterogeneity of such surfaces. Links between structure of pores and adsorption energy distribution are stressed. A set of new equations describing the heterogeneous multilayer adsorption is proposed, and an example of its application is shown. 2. Generalized BET-like AdsorptionsTheoretical Basis and uniBET Formula The adsorption process in submicroporous and microporous materials may be considered as consisting of a number of sub-processes each of them involving mpa moles of adsorbate molecules with the same energy. At a temperature T and pressure P the sorption equilibrium of an ath subprocess can be generally described with the following formula:9-14,15

RT ln(π) )

∂∆H ∂∆S -T ∂mpa ∂mpa

(1)

where R denotes the gas constant, π ) f/fs stands for the (10) Milewska-Duda, J.; Duda, J. T. Langmuir 1997, 13, 1286. (11) Milewska-Duda, J.; Duda, J.; Nodzen˜ski, A.; Lakatos, J. Langmuir 2000, 16, 5458. (12) Milewska-Duda, J.; Duda, J. T. Langmuir 2001, 17, 4548. (13) Duda, J. T.; Milewska-Duda, J. Langmuir 2002, 18, 7503. (14) Milewska-Duda, J.; Duda, J. Colloids Surf., A 2002, 208, 303.

10.1021/la040117r CCC: $30.25 © 2005 American Chemical Society Published on Web 06/29/2005

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Langmuir, Vol. 21, No. 16, 2005

Figure 1. Illustration of adsorbate molecule clusterization mechanisms in micropores and types of clusters considered in the LBET approach: c1, a space of an individual cluster limited to four layers by the pore geometry (type II, the fifth layer needs an expansion of the pore) with the third layer branched due to rather compact shape of the space; c2, a space of a cluster limited to two layers by the competing cluster c3 (type I, c2 and c3 are separated arbitrarily); and c3, a space of a cluster limited to four layers by the pore geometry (the 5th layer energy is too large) and started at a flat surface enabling branching of the second layer. The values of βkn shown in the right panel were calculated for the set of clusters c1 and c3 and for single c2.

relative pressure (fugacity) of adsorptive at (P, T), and ∆H and ∆S are the total enthalpy and entropy changes, respectively, due to the process. In the case of small nearly spherical adsorbate molecules the formulas for ∆H and ∆S can be relatively easily derived by using a generalized BET approach.9,12-14 We consider creation of adsorbate clusters in highly dispersed space limited by a geometry of pores. The clusters are constructed by adding consecutive layers being in equilibrium with the volatile phase. The following fundamental assumptions have to be accepted:12,13 1. There are spaces of statistically presumed shape, preassigned to each cluster within the space of pores, such that creation and enlarging of a particular cluster (filling of its space) does not affect creation and enlarging of other ones. 2. Each cluster starts from only one site placed directly on the pore surface (primary site), and each molecule placed on a layer n > 1 may occupy only one site at the (n - 1)th layer. 3. The spaces preassigned to the clusters are divided into classes (counted with an index κ), such that within a class κ an adsorption energy profile along the layers n ) 1, 2, ... is identical. An adsorbate molecule may be placed at any site available at the (n - 1)th layer of the κth class with the same probability, irrespective the ocupancy of other sites in the system (intercluster interactions are neglected, but interlayer ones are taken into account). The pair (κ, n) points to an adsorption subsystem a considered in eq 1. The most essential is that the adsorbate clusters considered in this approach are assumed to be constructed in configurationally independent ways, each starting with a unique primary site (they can occupy only a space of a presumed structure and maximal size; see Figure 1). It means that possible multifoot clusters (like c2 and c3 shown in Figure 1) have to be viewed as a number of independent clusters sharing (in a presumed way) a common space within the pore. This will be called the competitive adsorption. The main feature discriminating the classes of clusters is the maximum number k of layers (it will be referred to as the cluster size, and clusters of the same k will be called the kth-type clusters). We distinguish between two different mechanisms restricting the cluster size. If a molecule placed at the layer k reaches a bound of the cluster space emerged formally from the larger space of a real multifoot cluster (competitive adsorption) we say that the top layer adsorption is of type

Duda and Milewska-Duda

I that may be expected in more compact (holelike) or flat end large pores (slits). In turn, if the cluster size bound k is due to a pore shape, making the next layer energy large enough to stop the joining of further molecules,9,13 we say that the top layer adsorption is of type II, which is likely typical for small or narrow pores (small holes and channels). Despite serious simplifications and restrictions inherent in the above assumptions (see refs 13 and 14), they enable us to get a more realistic description of adsorption in microporous materials than the original BET theory, making it possible to express explicitly effects of pore geometry on the process. In particular, the assumptions 1-3 allow us to consider not only stacklike clusters (as assumed in BET theory), but also branched ones; see Figure 1. Such a non-BET clusterization is of essential influence on the configurational entropy of adsorption;12 hence, it can affect significantly adsorption isotherms.13,14 Its intensity depends on the number of adsorption sites offered by all molecules placed at the (n - 1)th layer of the κth class clusters for molecules of nth layer. It may be expressed as βκnmpκn-1, where βκn denotes the average number of sites offered by one molecule of the (n - 1)th layer in clusters of the same κ:

βκ1 ) 1, βκn > 0 for n ) 2, ..., k

(2)

The parameter βκn depends on a local pore width; thus, it may be viewed as a cluster layer shape factor. In the same way one can calculate a layer shape factor βkn, considering all classes κ of clusters limited to k g n layers. In real irregular porous structures the averaging is over a very large number of sites; hence, βκn ≈ βkn is very likely. The shape of such pores can be generally characterized by the overall factor β, being the arithmetic average of βkn for all considered layers (β ∈ (1, 1.3) seems to be typical for compact pores and β = 1 for narrow ones). The assumption 3 implies the following general formula for ∆H: K

∆H )

k

∑ ∑ ∑ mpκnQκn

(3)

k)1 κ n)1

where K means the maximum cluster size and Qκn is the molar energy contributed by placement of an adsorbate molecule at the (κ, n)th subsystem. By using in eq 1 formula 3 and a formula for ∆S derived in ref 12, one may arrive at the general model describing the local adsorption isotherm on the κth-type primary sites. It has the form of the set of following algebraic equations written for n ) 1, ..., k:

-Πκn* + (Πκn* + 1)θκn - θκnθκn+1 ) 0, θκk+1 ≡ 0 (4) where θκn denotes the coverage ratio of the (n - 1)th layer in clusters of the κth class def

θκn )

mpκn βκnmpκn-1

(5)

Πκn* is a transformed relative fugacity dependent on the relative pressure π, β, and θκn: def

Πκn* )

π (1 - θκn)1-βκn+1 Bκn

(6)

Bκn denotes the energetic parameter of the (κn)th subsystem:

Microporous Adsorbent Surface Heterogeneity

Langmuir, Vol. 21, No. 16, 2005 7245

( )

def

Bκn ) exp

Qκn RT

(7)

To simplify the model we neglect changes in internal entropy of adsorbate molecules. The formula 4 shows that the coverage ratios θκn do not depend on those of lower layers, and they can be calculated recursively, starting from the top (kth) layer with θκk+1 ≡ 0:

θκk )

Πκn* π and θκn ) for π + Bκk 1 + Πκn* - θκn+1 n ) k - 1, ..., 1 (8)

Next, the local adsorption mpκ of κth class can be calculated as k

mpκ ) mhAκθκ1[1 +

n

∑ ∏(βκjθκj)] n)2 j)2

(9)

where mhAκ denotes the amount (e.g., in mmol/g) of κ class primary sites. The eqs 4-9, referred to as the uniLBET model, makes it possible to consider any distribution of primary sites mhAκ, with an energy profile {Qκn; n ) 1, ...., k + 1} being specified for each κ (side interactions of low intensity on particular layers may be also taken into account; see ref 14). However, to make the model useful in practical examination of adsorption systems, it is necessary to assume additional relationships with a small number of parameters, linking the geometrical parameters and energetic properties with the adsorbent structure. 3. Pore GeometrysAdsorption Energy Relationships in Microporous Adsorbents of Irregular Structure and Geometry-Induced Energy Distribution Functions In our approach the energy Qκn is expressed in the following form:13,14 def

Qκn ) Up - ZcκnQcp - ZpκnQpp

The solubility parameter δc can be evaluated with the van Krevelen method.16 Relationships 10 and 11 are useful in validation of adsorption models, especially if applied to apolar adsorbates. They tighten an acceptability area for the magnitude of adsorption energies.11 Let QAκ denote the first layer adsorption energy (QAκ ) Qκ1). It may be expressed as

QAκ ) Up - ZAκ′Qcp

that is, Zcκ1 ) ZAκ′ and side adsorbate-adsorbate interactions are neglected (Zpκ1 ) 0). Formula 12 shows direct links between the pore surface shape and its adsorptivity. Primary adsorption sites appear at such places in a pore, where good adsorbatesurface contact is possible (e.g., in local niches and cavities), making ZAκ large enough to produce local minima of the energy QAκ, enabling the start of a cluster (creation of the first adsorption layer). In the case of natural microporous adsorbents, it is reasonable to assume that the values of ZAκ are almost uniformly distributed over a range (Zfk, ZAk) depending on the pore size k, and this range is wider for larger cluster spaces k. Let ZA denote the value of ZA1. The best contacts may be expected in the smallest pores (k ) 1), hence, ZA g ZAk and Zf1 e Zfk for k > 1. Let ζfk and ζAk denote the ratios def

ζAk )

Qcp) Cp2Vsδcδp ) Cp2δc(UpVs)1/2 and Qpp ) 2Vs(δp)2 ) 2Up (11) δc and δp stand for solubility parameters for the adsorbent and adsorbate, Cp is the polarity factor (Cp ) 1 for apolar adsorbates), and Vs is the molar volume of adsorbate in the adsorption system. (15) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.

ZAk def Zfk , ζfk ) , ζA1 ≡ 1 ZA ZA

(13)

The following relations are assumed to be held

ζAk g ζAk+1, ζfk g ζfk+1 0 e ζfk e ζAk, ζAmin < ζAk e 1, ζAmin ≈ 0.2 (14) By virtue of eq 12, the energy QAκ is also uniformly distributed within the limits (QAk, QAf):

QAk ) Up - ZAkQcp ) QA + ZA(1 - ζAk)Qcp, Qfk ) Up - ZfkQcp ) QA + ZA(1 - ζfk)Qcp (15)

(10)

where Up denotes the molar cohesion energy of adsorbate in its reference (bulk) state, Qpp is the molar adsorbateadsorbate interaction energy in the system, Qcp is the molar adhesion energy in ideal adsorbent-adsorbate contacts, and Zcκn and Zpκn are geometrical factors correcting interaction energies due to nonperfect adsorbentadsorbate and adsorbate-adsorbate contacts, respectively, specific for the considered system (they have to fulfill the relation 0 < Zcκn + Zpκn < 1). The quantities Qcp and Qpp are evaluated with the Berthelot rule (like in the FloryHuggins theory),15 by combining the cohesion energies of adsorbent and adsorbate:

(12)

def

QA ) Up - ZAQcp ) min(QAκ) k)1

(16)

and its distribution function over the primary sites of the kth type is

( )

fk

QAκ 1 RT RT ) ) RT (Qfk - QAk) ZAQcp ζAk - ζfk

(17)

The above assumptions imply a stepwise function of the surface energy distribution (with steps at consecutively growing QAk and Qfk), which depends also on a cluster size distribution. Distribution 17 may be related to the energetic parameter BAκ defined in eq 7:

1 1 ) ln(Bfk/BAk) BAκ 1 1 , BAκ ∈ (BAk, Bfk) (18) ZA(ζAk - ζfk) ln(Bcp) BAκ

fk(BAκ) )

where (16) Van Krevelen, D. W. Fuel 1965, 44, 229.

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( )

def

Bcp ) exp


( )

Qcp QAk def , BAk ) exp ) RT RT

mp ) mhA1 def

πf1(BAκ)

∫BB

A1

BAκ + π

f1

BA(Bcp)ZA(1-ζAk), Bfk ) BA(Bcp)ZA(1-ζfk)

dBAκ +

K

∑

k)2

and

( )

QA BA ) exp RT def

def

( )

QC , QC ) Up(1 - 2Zpp) - ZC Qcp RT

(20)

In our work we assume that BC g 1; that is, QC g 0 and, thus, ZC e Up(1 - 2Zpp)/Qcp. The value Qκk for the top layer depends on the nature of the cluster boundary. If geometrically/energetically restricted adsorption is considered (type II), that is, Qκk+1 ≈ ∞, the value Qκk ) QC and so Bkk ) BC is justified. For competitive adsorption (type I) a lower value (but the same for all κ) should be used, that is, Qκk ) Qkk < QC (Bkk < BC). In this case it is convenient to presume that the value of Bκn (at each top layer) implies θκk ) θkk-1. Thus, according to eq 8 we have

Bκk ) Bkk )

{

(19)

The molar adsorption energy at layers n ) 2, ..., k may be directly expressed by eq 10. In this case the adsorption is mainly due to cohesive forces, so the factor Zpkn is of primary importance, although adhesive interactions may be significant (rather small but nonzero Zcκn seems to be typical). So far as the pore filling degree is moderately low, one may assume that each higher layer molecule is in contact with only one lower layer molecule. Thus, the parameter Zpκn depends only on the adsorbate molecule geometry; hence, it may be fixed at a value Zpκn ) Zpp ) constant ranging from 1/12 to 1/6. At a high filling degree of pores multiple contacts (i.e., greater Zpκn) may occur that cause rapid increase of adsorption. Such effects (discussed in ref 13) are beyond the scope of this paper. In turn, one may expect that in natural adsorbents the factor Zcκn is slightly varying from layer to layer (although it certainly depends on the geometry of the pores). It allows us to take Zcκn ) ZC ) constant. Accepting the above simplifications, we may assume that Qκn ) QC ) constant for each κ at layers n ) 2, ..., k - 1 (k ) 2, ..., K). Thus, the adsorption at layers n ) 2, ..., k - 1 may be calculated with a constant energetic parameter Bκn ) BC:

BC ) exp

∫BB

(βkjθkj)]

π(1 - θkk-1) , Bkk < BC, and θkk-1 8 BC (21) Bkk 9 πf0

As a consequence of the assumed homogeneity of layers n > 1, we may use the shape factor βkn averaged over all clusters of kth type (see Figure 1). It makes calculating the coverage ratio profiles θkn for n ) k, ..., 2 by using eq 8 possible, disregarding the surface heterogeneity and next θκ1 versus BAk (see eq 8). Then the total adsorption isotherm mp(π) may be calculated by integration of eq 9 with respect to the first layer adsorption energy distribution (eq 18), and next the obtained isotherms mpk(π) for consecutive k ) 1, ..., K are summed up:

mhA1 1 K

k

∑∏ ×

n)2 j)2

πfk(BAκ)

Ak

fk

n

k

mhAk[1 +

π + BAκ(1 - θk2)βk2 1

ln

ln(BA/Bf1) n

(βkjθkj)) ∑ mhAk(1 + n)2 ∑∏ k)2 j)2 ln

(

{

dBAκ )

( )} BA + π

Bf1 + π

+

1

1-

ln(BAk/Bfk)

)}

BAk(1 - θk2)βk2 + π Bfk(1 - θk2)βk2 + π

×

(22)

where mhAk denotes the total number of clusters limited to k layers (e.g., in mmol/g). In the above formula we assumed that the single molecules (clusters of k ) 1) are adsorbed only in the smallest pores; hence, the isotherm for k ) 1 is always of the Langmuir type. Examination of irregular microporous structures needs a large number K of the cluster types to be considered (K ≈ ∞). It makes the model 22 highly overestimated; thus, it cannot be reliably identified by fitting to any adsorption data (the numerical task is ill-conditioned).17 Hence, it is necessary to put additional relationships reducing radically the number of parameters and set arbitrarily such ones whose effect on the adsorption isotherm shape is weak. To reach considerable reduction of the model parameters number, for pores of random shape one may use an averaged shape parameter β ) βkn ) constant (see Figure 1), bearing in mind that it characterizes mainly the shape of small clusters and of a primary site neighborhood in larger ones.13 Hence, we take that the following relation should be held:

1 e β e βmax, βmax ≈ 1.5

(23)

For many microporous materials the primary site capacity distribution mhAk may be adequately expressed by the exponential function (see refs 3 and 9):

mhAk ) mhA‚(1 - R)Rk-1

(24)

where mhA denotes the total number of primary sites and R ∈ 〈0, 1) is an empirical parameter. The parameter R is of essential effect on the energy distribution function (see eq 18 and Figure 2). In microporous materials isotherm shape is usually slightly affected by a surface fraction of high energy QAκ, even if max(π) ≈ 1. It implies numerical obstacles for reliable evaluation of the right-hand-side profile of the energy distribution. In turn, the fraction of smallest clusters (k ) 1, 2, 3) is often highly dominant;9,11,13 hence, (17) Forsythe, G. E.; Malcolm, M. A.; Moler, C. B. Computer Methods for Mathematical Computations; Prentice Hall: Englewood Cliffs, NJ, 1977.


Langmuir, Vol. 21, No. 16, 2005 7247 Table 1. Settings of the Energy Distribution Function Parameters ζA2, ζAk, ζf1, ζf2, and ζfk for Particular Surface Heterogeneity Variants {h, d, η}

a

ζf∞ is a minimum value for ζfk determining max(QAκ). κ

Figure 2. Patterns of the primary site energy distribution functions applied in the variants 4-15 and 19-30 (see Table 3), calculated with parameters QA/RT ) -3.5 and Zf∞ ) 0; solid lines, R ) 0.4; dotted lines, R ) 0.8 (step points do not depend on R).

the parameters ZA, ζA2, ζA3, ζf1, and ζf2 may be of significant effect, mainly to a low-pressure section of isotherms. Bearing this in mind we propose avoiding the energy distribution overparametrization in the following ways: 1. We take ZA as the key parameter to be evaluated by the model fitting procedure (this gives the lower bound QA of the energy distribution and corresponding BA). 2. The step points of the distribution function (values for QAk and BAk) are fixed arbitrarily by setting the coefficients ζf1, ζAk, and ζfk for k ) 2, ..., K, with relations 14 being held: (a) Adsorption of single molecules (clusters of k ) 1) is always treated separately by using the Langmuir formula with the individual parameter ζf1 producing Bf1; (b) the isotherm for k ) 2 (doublets) is treated optionally either as a separate model with individul parameters ζA2 and ζf2 (option d ) 1) or together with remaining clusters k > 2 (option d ) 0) with the same ζAk ) ζA2 and ζfk calculated in the same manner as for k > 2; (c) for all cluster of k g 2 + d the distribution function starts at the same point ζAk ) ζA2+d; and (d) the right-hand-side limits ζfk of the distribution for k g 2 + d (assumed to be of less significance) are expressed as a simple function of k, fulfilling the relations 14 and enabling fast calculations of the model with large K (see the next section). 3. The identification of the model is proposed to be performed for a number of properly diversified variants of the energy distribution function, each of them having fixed values of ζf1, ζAk, and ζfk for k > 1.

4. A variant is generated by taking a combination of three integer parameters, that is, the heterogeneity type h ) 0, ..., 9, the doublets treatment binary option d ) {0, 1}, and a binary option η ) {0, 1}, η e d (when η ) 1, the right-hand side of the distribution becomes steeper). 5. A couple of well-fitted variants (e.g., three ones) may be used to assess a reliability of the identification results. The formulas proposed to calculate the parameters ζf1, ζAk, and ζfk for particular sets {h, d, η} are gathered in Table 1. The parameter ζf∞ determines the smallest contact surface area enabling a place on a pore surface to be the primary adsorption site. The value of Zf∞ ) ZAζf∞ may vary from 0 or a small value ≈ 0.1 (if there are large flat areas in larger pores) to Zf∞ ≈ ZA2+d ≈ 1 (for almost homogeneous surfaces). Hence, in general, identification of real porous structures should be performed with Zf∞ being treated as the additional fitting parameter. Nevertheless, a contribution of the primary sites of very low ZAκ is mostly negligible. Thus, in practice, one may assume alternatively that the significant first layer adsorption sites fulfill the relation ZAκ < ZC, that is, Zf∞ ) ZC. It is justified when hypothetical sites of ZAκ > ZC are much more likely blocked by higher layers adsorption than are able to start new clusters (see Figure 1). Examples of more representative distribution functions of h > 2 and ζ0 ) ζA2+d are shown in Figure 2. Formula 22 with βkn ) β, assuming the exponential cluster size distribution (18), and used with fixed h, d, and η (energy distribution variant) will be referred to as the uLBET model. It involves five or six parameters (mhA, ZA, R, β, BC, and optionally Zf∞) to be adjusted by fitting the model to empirical adsorption data, with a chosen variant of the surface energy distribution function (Table 1). 4. Analytical Approximations of the uLBET FormulassModels of the LBET-type The multivariant identification of adsorption systems needs thousands calculations of theoretical isotherms. Hence, to make the calculations more effective we elaborated analytical formulas approximating the uLBET model 22 with relatively small errors. They will be referred to as the adsorption models of the LBETtype. Formula 22 with βkn ) β ) constant for n > 1 and the exponential distribution (24) of the primary site capacity (uLBET model) can be brought into an analytical form providing that (a) the coverage ratios θkn for k > 1 + d and

7248



n > 1 may be replaced by a constant (averaged) value θ and (b) the first layer coverage ratios θk1 (i.e., BAk, Bfk) do not depend on k for k > 1 + d, that is, θk1 ) θA, where θA stands for a constant coverage ratio of sites of k > 1 + d; d ) {0, 1} (d ) 1 means that the clusters of k ) 2 are to be considered separately with individual values θ21 and θ22). If the above conditions are fulfilled, the sum in eq 22 for k ) 2 + d, ..., K ) ∞ takes the form of the geometrical series denoted as Gd def

Gd )

∞

k

∑ k)2+d

Rk-1(1 +

(βθ)n-1) ∑ n)2

(25)

and the isotherm Id for clusters of k ) 2 + d, ..., ∞ may be written in the following concise form

Id ) mhA(1 - R) GdθA

(26)

For d ) {0, 1} we have

Gd )

d+1

R βθ 1 - (βθ)d + dβθ + (βθ)d 1 + ) 1-R 1 - Rβθ Rd+1 βθ d + (βθ)d 1 + (27) 1-R 1 - Rβθ

[

(

[

(

)] )]

It can be proved12,13 that application of eq 21 for Bkk (the 1st type of adsorption at top layers) in eqs 4-8, together with Bkn ) BC and βkn ) β for n ) 2, ..., k - 1, implies a constant coverage ratio θkn ) θ ) Π∞* for n > 1; that is, the adsorption at layer n > 1 fulfils the generalized Henry’s law:

mpκn ) mpκn-1βΠ∞* for n ) 2, 3, ..., k

(28)

where Π∞* denotes a transformed relative fugacity satisfying the following equation:

Π∞* ) (π/BC)(1 - Π∞*)1-β, 0 e π e min(1, πC) (29) πC ) BC(β - 1)β-1β-β and Π∞*(πC) ) β-1 (30) At π ) πC the theoretical isotherm becomes vertical13 (θ changes from 1/β to 1). This stepwise filling of pores is only due to configurational entropy changes;12,13 hence, we call it the geometrical condensation12 and a relative fugacity πC is referred to as the condensation pressure. The eq 28 with Bkn ) BC is strictly satisfied only for unlimited clusters, that is, for k ) ∞, thus, employment of eq 21 is fully legitimate, if the competing clusters are large enough to produce energetic effects representing a large (infinite) number of layers. Nevertheless, we proposed13 to use it for all clusters of k > 1 as a convenient approximation of a physical reality. If the limit k for cluster size is due to inconsistency of the cluster space and pore shape (2nd type of adsorption), the top layer isotherms are expressed by the Langmuir equation

θkk ) π/(BC + π)

(31)

and for large k the sequence θkn tends to Π∞* with n f 1, that is,

θkk-i f Π∞* kf∞ ifk+1

(see ref 13). Hence, we proposed13 to apply to eq 25

the averaged coverage ratio θ calculated with formula 32

θkn ) θ ) Π*

(

)

1 + wHΠ* for 1 + Π* n ) 2, ..., k, k ) 1 + d, ..., ∞ (32)

where wH is the coefficient calculated as the ratio of the number of layers n > 1, n < k - 1 (with adsorption satisfying eq 30 rather than eq 31) to the total number of layers n ) 2 + d, ... k

wH )

R(1 + R - R2) for d ) 0 and 2-R 2+R for d ) 1 (33) wH ) 3(2 - R)

where the symbol Π* denotes a transformed relative fugacity, which satisfies the following equation, similar to eq 29, but with eqs 32 and 33 being employed

Π* ) (π/BC)(1 - θ)1-β, 0 e π e πm

(34)

and πm is a maximal π producing an acceptable (maximal) θ (πm g πC). Let F(Π*) denote the polynomial defined below: def

F(Π*) ) wHβ(Π*)3 + wH(2β - 1)(Π*)2 + (β - 2)Π* - 1 (35) The following relations have to be fulfilled:13

F(Π*) e 0, F(Πm*) ) 0, πm ) BCΠm*(1 - θmax)β-1, θmax ) θ(Πm*) (36) The values for Π* can be easily calculated by direct iterations of eq 34 starting from Π* ) π/BC. If d ) 1, the value for θ22 in formula 22 is directly expressed by eq 31. Application of eq 22 and the model (eqs 25-27) to adsorbents with homogeneous surfaces (BAκ ) BA) leads to the following equation, referred to as the homogeneous LBET formula:13

mp π + ) (1 - R) mhA BA + π π + dR(1 - R)(1 + βθ22) BA(1 - θ22)β + π π βθ Rd+1 d + (βθ)d 1 + (37) 1 - Rβθ B (1 - θ)β + π A

[

(

)]

For heterogeneous adsorption formula 22 can be brought into an analytical form in the same way, providing that the energetic parameters BAk and Bfk do not depend on k for k > 1 + d (although they may be dependent on π). According to Table 1, for k > 1 + d we have BAk ) BA2+d. Let us assume that the parameter Bfk is of the same value Bfθ for k > 1 + d


Langmuir, Vol. 21, No. 16, 2005 7249

at given π. In such a case the heterogeneous adsorption isotherm can be expressed in the following form:

{

)}

(

mp BA + π 1 ) (1 - R) 1 ln mhA Bf1 + π ln(BA/Bf1)

{

[

ln(BA2/Bf2)

(

Bf2(1 - θ22)β + π

(

βθ 1 - Rβθ

ln

)}

BA2(1 - θ22)β + π

Rd+1 d + (βθ)d 1 +

ln

(

)

]{

1-

{

1+

+

1

dR(1 - R)(1 + βθ22) 1 -

∞

Idη ) mhA(1 - R)

ZA(ζA2+d - ζf∞) ln(Bcp) k - η - 1

[(

×

)}

BA2+d(1 - θ)β + π Bfθ(1 - θ)β + π

(38)

The model (38) is directly applicable to heterogeneous surfaces characterized by the energy distribution of type h ) 1 and h ) 2, in which Bfk ) Bfθ ) Bf∞ for each k (see Table 1). Nevertheless, to make them applicable to adsorbents with more diversified surface heterogeneity (h > 2), it is necessary to find appropriate formulas approximating the sequence Bfk for k > 1 + d with a value Bfθ implying a negligible errors. If the surface energy distribution is expressed in the form given in Table 1 for h > 2, the following approximation was found to be acceptable:

(

1+

where the formulas for ζfk given in Table 1 combined with energy distribution functions (18) are employed, Wdη is an exponent minimizing the approximation error, and Bfθ is defined as

Bfθ )

Bcp-ZA(ζA2+d-ζf∞)(Wdη-(k-η-1)/(k-η))

def

(41)

∑

1 k-η-1

) {

Idη ) mhA(1 - R)

ZA(ζA2+d - ζf∞)Wdη ln(Bcp)

( )

where

1 i

∑

def

dη

)

∑

R

(

)}

BA2+d(1 - θ)β + π

(

i+η

i)1+d-η

Bfθ(1 - θ)β + π

{

1

1 - βθ

×

(44)

×

∑dη)WdηθA

(45)

)}

(46)

BA2+d(1 - θ)β + π Bfθ(1 - θ)β + π

(Rβθ)η ∞ (βθ)i - βθ ) 1 - βθi)1+d-η i 1 - βθi)1+d-η i Rη

∞

(

ln

ln(BA2+d/Bfθ)

)

1 - βθ(βθ)i+η

WdηθA ) mhA(1 - R)(Gd +

∑

Ri

∑

ln(1 - R) - (βθ)1+η ln(1 - Rβθ) + (d - η)R(1 - (βθ)2+η) 1 - βθ (47)

There is a freedom in selection of the exponent Wdη; however, it should be taken in such a way to minimize the last term in eq 39 with respect to contribution of particular clusters of size k > 1 + d depending on R. It can be proven that the following relations should be fulfilled: Rf0

Let Idη denote the isotherm for clusters of k > 1 + d obtained from formula 22 with θkn ) θ, βkn ) β for k > 1 + d, n ) 2, ..., k, and with approximation 39 being applied:

∑ (βθ)n-1) × 1

∞

lim Wdη ) (42)

(43)

n)2

Wdη 1 +

The corresponding maximal energy Qfk is

Qfθ ) Up - ZAζfθQcp ) QA2+d + ZA(ζA2+d - ζf∞)QcpWdη

]}

k

)

ζfθ ) ζA2 - (ζA2 - ζf∞)Wdη

ln(Bcp)

By employing in eq 44 the series Gd defined in eq 25 and using the new index i ) k - η - 1, we arrive at the following expression:

-Rη

BA2+d(Bcp)ZA(ζA2+d-ζf∞)Wdη ) BA(Bcp)ZA(1-ζfθ) (40)

)

k-η

Rk-1(1 +

ln

β

)

+

k-η-1

k)2+d

θA ) 1 -

Bfk

(

∞

Idη ) mhA(1 - R)

1+

) ln(Bfθ(1 - θ) + k-η-1 ln(Bcp) (39) π) - ZA(ζA2+d - ζf∞) Wdη k-η

def

Bfθ(1 - θ)β + π

×

If Wdη does not depend on k the formula for Idη can be written as follows:

ln(Bfk(1 - θ)β + π) = ln(Bfk(1 - θ)β + πBcp-ZA(ζA2+d-ζf∞)(Wdη-(k-η-1)/(k-η)))

)

BA2+d(1 - θ)β + π

ZA(ζA2+d - ζf∞) Wdη -

1 × ln(BA2+d/Bfθ)

(

k-η

1

ln

+

k

∑ Rk-1(1 + n)2 ∑ (βθ)n-1) × k)2+d

1+d-η , lim W ) 1 2 + d - η Rf1 dη

(48)

In particular, the most convenient expression for Wdη may be accepted, which brings formula 44 to the same form as in eq 26, that is,

Idη ) mhA(1 - R)(Gd +

∑dη)WdηθA ) mhA(1 - R)GdθA (49)

7250



Thus

∑

1 dη )1+ Wdη Gd

(50)

which leads to the following expression for Wdη, fulfilling the relations 48:

{

Wdη(R, βθ) ) 1 - [ln(1 - R) - (βθ)1+η ln(1 - Rβθ) + (d - η)R(1 - (βθ)2+η)]/ 1 - βθ βθ R1+d-η d + (βθ)d 1 + 1-R 1 - Rβθ

[

[

(

)]]}

-1

(51)

Approximation 39 was found to be also acceptable18 (with an additional error) to the energy distribution functions for h > 2 with ζ0 ) 1 (see Table 1):

ln(Bfk(1 - θ)β + π) = ln(Bfk(1 - θ)β + πBcp-ZA(1-ζf∞)(Wdη-(k-η-1)/(k-η)))

= ln(Bfθ(1 - θ) + π) k-η-1 ZA(ζA2+d - ζf∞) Wdη ln(Bcp) (52) k-η

(

β

)

In this case the parameter Bfθ and coefficient ζfθ (defined in eq 41) should be calculated as def

Bfθ ) BA(Bcp)ZA(1 - ζf∞)Wdη ) BA(Bcp)ZA(1-ζfθ), ζfθ ) 1 - (1 - ζf∞)‚Wdη (53) Thus, in general def

ζfθ ) ζ0 - (ζ0 - ζf∞)Wdη(R, βθ), ζ0 ) {1, ζA2+d}

(54)

As the result we obtain the general formula of the form of eq 38, referred to as the heterogeneous adsorption model of LBET type. Notice that for all k > 1 + d it uses the same uniform energy distribution function, but for h > 2 the effective energy range depends on the averaged surface coverage θ, with Wdη being the energy spectrum corrector. The function Wdη(R, βθ) plotted against βθ for selected R is shown in Figure 3 (see eq 51). Figure 4 illustrates typical magnitudes of relative discrepancies δmp/mp between isotherms mp produced by the “accurate” uLBET formula (eq 22) and those obtained with its analytical counterpart, that is, the LBET-type model. The “worst case” (R ) 0.4, d ) 0, η ) 0) is depicted in more detail in Figure 5. It can be seen that the relative approximation errors are mostly less than 1%. Notice that they are mainly due to simplifications 32-36, which are necessary to reach the analytical form of the local isotherm equation with the 2nd type of adsorption at top layers (dotted lines in Figure 4), and they do not occur if the 1st type of adsorption at the top layers is assumed. The energy distribution approximations 39, 51, and 52 proposed in this paper are of much smaller effect (see gaps between the solid and dotted lines in Figure 4). 5. Identifiability Assessment of Heterogeneous uLBET and LBET Equations The uLBET model may be viewed as an adequate phenomenological description of adsorption processes in (18) Kwiatkowski, M. Numerical description of microporous structures of carbonaceous materials. Ph.D. Dissertation, AGH-UST, Krako´w, Poland, 2004.

Figure 3. Coefficient Wdη versus βθ (eq 51) for selected values of R.

microporous materials of irregular structure. Thus, it may be used for checking (by computer simulations) effects of pore structure, adsorption energy, temperature, and pressure on adsorption isotherms. In such simulations the pore structure is represented indirectly by the uLBET model parameters affecting directly the adsorption energy and cluster size distributions. Semiquantitative relations between typical pore structures and the parameters of our model are outlined in Table 2. For pores of irregular (random) shape specification of more precise links seems to be unrealistic, because in such structures no single parameter (like pore size or pore width) is able to handle adequately mechanisms of adsorbent-adsorbate interactions. Hence, having an empirical system of parameters evaluated by a model identification, one can only infer the porous structure properties. The relations given in Table 2 may be used as a guide for such interpretations. To perform the identification we propose to apply the simplified LBET formulas. They make possible fast examination of real microporous adsorption systems by checking applicability of different model variants and, in this way, get information on their structural and energetic properties. It can be seen in Figure 4 that the uLBET-LBET model discrepancies, although being mostly of typical adsorption measurement error range, change to an extent the isotherm shape. The question arises if such errors are negligible; that is, does the LBET-model-based identification of real adsorption systems provide information of the same practical usefulness as a fitting of the uLBET formulas, when taking into account unavoidable processuLBET model mismatches and limited identifiability of the system due to numerical properties of the LBET formulas and optimization procedures? To answer this question we have carried out numerous identification calculations, based on adsorption isotherms generated by the LBET and uLBET formulas, assuming different values of the system parameters affecting the isotherm shape (QA/RT, BC, R, β). For each set of the parameters the “empirical” isotherms were generated in 30 variants characterized in Tables 3 and 4. Each data variant has been identified with 30 variants of the LBET model (see Table 3) with ζ0 ) ζA2+d, to find the most appropriate (the best fitted) variant. As it can be seen in Tables 1 and 3, the variants 1-3 and 16-18 are of a special kind: 1 and 16 assume homogeneous surface energy, 2 and 17 produce uniformly heterogeneous energy distribu-


Langmuir, Vol. 21, No. 16, 2005 7251

Figure 4. Relative differences δmp/mp between the adsorption mp produced by the uLBET model and that calculated by the approximated LBET-type formulas, versus relative fugacity π for systems with the 2nd type of adsorption at the top layers and QA/RT ) -4.5, Zf∞ ) 0, h ) 7, BC ) 1.5, and β ) 1.1. Solid lines, total errors of the model (32-36, 38, 39, 40, 51); point lines, only effect of using in eq 22 the averaged coverage ratio θ (see eqs 32-36) instead of θkj, without the simplifications proposed in eqs 39, 40, and 51 for the energy distribution functions. Table 2. Parametric Representation of Adsorbent Pore Structure in the uLBET Model pore structure type

Figure 5. Isotherm produced by the uLBET model (solid lines in the left subfigure) for a system with the primary site energy distribution shown in the upper right subfigure (dotted vertical line shows the energy corresponding to BC). The × marks and the dotted lines in the left panel show the isotherms calculated with the simplified formula of LBET-type (eqs 32-36, 38-40, 51), and the + marks show the isotherm produced by eq 22 with averaged θ. The lower right panel presents relative errors of the approximation (see Figure 4).

tion, and 3 and 18 are like 2 and 17 but with homogeneous primary sites of k ) 1. In the remaining variants (gathered into triples with the same h and different dη) the surface energy distribution becomes more symmetric with growing h and less slim along the triples (Figure 2). The isotherms have been generated for relative pressures π ranging to πmax ) 0.9 but not exceeding the condensation pressure (π < πC). In effect all they were smooth (like for typical microporous systems),4 with no steplike changes and with the initial slope depending on the energy distribution being assumed (the smallest slopes

1.

dominant fraction of small, and compact pores

2.

narrow channel-like pores of different lengths

3.

narrow tree-like pores of different lengths

4.

hole-like pores of moderately wide dimension spectrum

5.

large compact holes of irregular shape

6.

narrow slitlike pores of diversified volume/length

7.

slitlike pores of different and rather large widths and volume

uLBET parameters representing ffects of pore geometry R < 0.3; β ) 1; BC = 1; ZC < Zf∞ e ZA (narrow spectrum of the first layer adsorption energy QAκ or homogeneous system, i.e., QAκ ) QA); top layer adsorption of the 2nd type R ∈ (∼0.3, ∼0.6); β ) 1; BC ≈ 1; ZC < Zf∞ e ZA (narrow spectrum of QAκ or QAκ ) QA); top layer adsorption of the 2nd type R ∈ (∼0.3, ∼0.6); β ∈ (∼1.01, ∼1.3); BC ≈ 1; ZA < Zf∞ e ZC (narrow spectrum of QAκ); top layer adsorption of the 2nd type R ∈ (∼0.4, ∼0.7); β ∈ 〈1, ∼1.1〉; BC > 1; Zf∞ ≈ ZC (moderately wide spectrum of QAκ); top layer adsorption of the 2nd type R > 0.5; β > 1.05; BC > 1.5 (energetic bounds for cluster size); Zf∞ ≈ ZC; top layer adsorption of the 1st or 2nd type R > 0.3; β > 1.1; BC ≈ 1; Zf∞ < ZC (wide spectrum of QAκ); top layer adsorption of the 1st type (competitive adsorption) R ∈ (∼0.3, 1); β > 1.1; BC > 1.5 (larger for wider pores); Zf∞ < ZC (wide spectrum of QAκ); top layer adsorption of the 1st type (competitive adsorption)

are for the variants nos. 2, 15, 17, and 30). For such data the standard constrained optimization procedure “fmincon”, offered by the MATLAB software package19 for (19) Matlab-The Language of Technical Computing. Using Matlab, version 6; Mathworks, Inc., 2000.

7252



Table 3. Parameters of the Surface Energy Distribution Used in Variants of Calculations (see Table 1 and Figure 2) fitting LBET model variant no. option

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

h d

0 0

1 0

2 0

3 0

3 1

3 1

5 0

5 1

5 1

7 0

7 1

7 1

9 0

9 1

9 1

0 1

1 0

2 0

3 0

3 1

3 1

5 0

5 1

5 1

7 0

7 1

7 1

9 0

9 1

9 1

0

1

0

0

1

0

0

1

0

0

1

0

0

1

0

0

1

0

0

1

0

0

1

0

η type

top layer adsorption of type I

top layer adsorption of type II

Table 4. Differences (D) between the “Empirical” Data Generator and the Fitting LBET Model, Introduced To Study Effects of the Real Adsorption SystemsModel Mismatches perturbed parameters difference type D

data

A.5p B.5p B.6p C.5p D.6p

Zf∞ ) ZC ≈ 0.2 Zf∞ ) 0.16 Zf∞ ) 0.16 Zf∞ ) ZC ≈ 0.2 Zf∞ ) 0.16

parameter Zf∞ fitting LBET model Zf∞ is the same as the calculated ZC Zf∞ is the same as the calculated ZC Zf∞ calculated as the fitting parameter Zf∞ the same as the calculated ZC Zf∞ calculated as the fitting parameter

nonlinear tasks, was found to be effective enough to perform identification of the examined adsorption systems by minimization of the fitting error dispersion σe (nontypical isotherms need dedicated identification algorithms).13 The following quantities were selected11 to be adjusted during the optimization procedure: (a) VhA ) mhAVs, molar volume of the first layer region (Vs is the molar volume of adsorbate); (b) R and β, geometric parameters of the porous structure (see eqs 2 and 24); (c) ZA and ZC, correction factors used in eqs 16 and 20 to calculate QA and BC; and (d) Zf∞ (see Table 1) optionally (see Table 4), under the constraints

R∈ 〈0, 1〉; β∈ 〈1, 1.35〉; ZA∈ (0.2, 1); ZC∈ 〈1/12, Up(1 - 2Zpp)/Qcp〉; πm(β, BC) < πmax (55) Each time the identification has been started with R ) 0.01, β ) 1, ZC giving BC ) 1, VhA, and ZA, according to mhA and QA found by fitting the Langmuir equation to adsorption data of π < 0.1. Different perturbations have been put into the data generator (see Table 4), to obtain an insight into possible effects of model-process mismatches on the parameters evaluation errors. The calculations were performed for a hypothetical microporous adsorbent with the solubility parameter δc ) 28 (see eq 11; realistic value for natural carbonaceous adsorbents).11 Adsorption of two different vaporous substances has been taken into acount in the study, that is, (1) C6H6 at T ) 293 K (Up ) 31.17 kJ/mol, Vs ) 88.9 cm3/mol, Qcp/RT ) 38.29), assuming QA/RT ) -4.5 (typical for roomtemperature adsorption of small molecule vapors), and (2) N2 at T ) 77 K (Up ) 3.58 kJ/mol, Vs ) 34.7 cm3/mol, Qcp/RT ) 30.83), assuming QA/RT ) -12.0 (typical for low-temperature adsorption). All isotherms were generated with mhA ) 1.33 mmol/g, β ) 1.1, BC ) 1.5, and Zpp ) 1/6. More representative results obtained for poorly identifiable systems (R ) 0.4) are presented in Figures 6-10. The upper row diagrams show the correctness of recognition of the surface energy distribution and of the top layer

parameters h and ζ0 used to generate data variants h ζ0 hdata is the same as hmodel used in the fitting LBET model (see Table 3) for hmodel < 5: hdata is the same as in Table 3; for hmodel > 4: hdata ) hmodel -1, hmodel is from Table 3, hdata is h used in the model generating data

ζ0 ) ζA2+d (the same as used in the fitting LBET model)

number of fitting parameters 5 5 6 5 6

ζ0 ) 1

adsorption type. The circles placed at the main diagonal mean the perfect recognition. In the case of incorrect recognition of the adsorption type, the circles appear at the lower-right or upper-left quarter. The index Wid is an identification certainty measure, being the ratio of the optimal fitting error dispersion to the mean value of error dispersion for the best, second, and third fitting quality variants, averaged over the set of 30 fitted data variants (the smaller Wid, the more reliable results found with the best fitted variants). Figure 6 illustrates accuracy of the system identification in an ideal case, when the set of the fitting model variants contains a pattern of the energy distribution, mapping exactly the distribution used in the data generator, and the LBET model can be fitted by independently adjusting five parameters: VhA, R, β, ZA, and ZC (D ) A.5p; Table 4). The left subfigures show that the LBET model itself is perfectly identifiable (when data are produced by the same LBET formula, all its adjusted parameters are determined with errors less than 1%; see lower row). In this ideal case the LBET model identification based on uLBET data is much worse (see the right subfigure), especially for the 2nd type of adsorption and for a flat energy distribution (see the variants nos. 17 and 25-30). It means that from a formal viewpoint, errors due to application of the averaged θ (eqs 32-36) in eq 22 are of significance. In turn, much smaller errors observed for the 1st type of adsorption give evidence that the approximation (eqs 39 and 51) of the high energy side of the energy distribution functions is well acceptable. Nevertheless, in a more realistic case, with perturbed energy distribution functions in the data generator (D ) C.5p), fitting of both the LBET and uLBET data leads to comparable estimation errors of the system parameters (see Figure 7). Much larger errors appear when Zf∞ differs from ZC (D ) B.5p), which may be expected in practice. Typical effects of such discrepancies on the system identification reliability are shown in Figure 8. They suggest that, in general, the parameter Zf∞ should be adjusted independently. Solving of such an extended identification task (involving six adjusted parameters) noticeably improves the identification results (compare


Figure 6. Identifiability characterization of isotherms produced by the LBET and uLBET formulas with typical values of QA/RT and R (β ) 1.1, BC ) 1.5) and fitted with the LBET models. The ideal case: the set of energy distribution patterns (see Figure 2) contains the functions applied for the data generation (D ) A.5p; see Table 4). For data and fitting model variants, see Table 3. The upper panels show recognition correctness of the top layer adsorption type and of the first layer adsorption energy distribution: O, the best fitted variants; *, second quality fittings; and +, third quality fittings. The lower panels show relative estimation errors 0 obtained for system parameters with the best fitted variants (0 ) |calculated value|/|actual value| - 1). The errors 0 are transformed to make better visible the errors of different magnitude. Dotted horizontal lines show the levels of (1%, (5%, (10%, (50%, and (100%. The pairs of numbers written in brackets show the absolute value of errors (in %) averaged over the set of 30 fitted data variants for the left and right subfigures.

Langmuir, Vol. 21, No. 16, 2005 7253

Figure 8. Identifiability of isotherms produced by the LBET and uLBET formulas with the same energy distribution functions as assumed in the fitting models, but with Zf∞ * ZC and with the five-parameter identification being applied (D ) B.5p; Table 4). For other explanations, see Figure 6.

Figure 9. Identifiability of isotherms produced by the LBET and uLBET formulas with the same energy distribution functions as assumed in the fitting models, with Zf∞ * ZC and with the six-parameter identification being applied (D ) B.6p; Table 4). For other explanations, see Figure 6.

Figure 7. Identifiability of isotherms produced by the LBET and uLBET formulas with energy distribution functions differing from those assumed in the fitting models (LBET) but allowing for application of the five parameter identification, that is, with Zf∞ ) ZC (D ) C.5p; see Table 4). For other explanations, see Figure 6.

Figures 9 and 8), although it takes about twice more computation time and gives a bit greater estimation errors than those shown in Figure 6 (where the system properties

allow for application of the five-parameter optimization). Nevertheless, the identifiability deterioration due to increasing the number of adjusted parameters is not significant (see the values for Wid at headings of the right subfigures in Figures 6 and 9). It is noteworthy that this time the system parameter estimation errors are of the same range for both the LBET and the uLBET data. The same was stated for other systems,18 with different R, QA/RT, and system-model mismatches (like those shown in Table 4). It means that uLBET-LBET model differences are of less significance for practical identifiability of real adsorption systems. Our study exhibits essential obstacles for the adsorption isotherm based examination of heterogeneous systems. Generally, the top layer adsorption type itself is mostly well recognized, which gives an insight into a nature of

7254



adsorbate clusterization constraints. Also QA/RT (the upper limit of the first layer energy) is well evaluated in each case (0 < 10%). However, the parameters characterizing the size and shape of clusters (BC, R, and β) are poorly determinable, especially for systems with a top layer adsorption of the 2nd type. Noticeably better identification reliability may be expected when the 1st type of adsorption is detected. The most uncertain are evaluations of the parameter BC, as its effect on isotherm shape is rather weak (especially for greater BC) and intercorrelated with the surface energy distribution and other factors (large evaluation errors of BC may be easily compensated by smaller deviations of R, β, and Zf∞ or by incorrect selection of the energy distribution variant). As an effect the parameter β often reaches its lower bound β ) 1 (0 ) -10%). Hence, a reasonable upper limit for BC should be taken, to improve the identification reliability (we assumed BC < 3.5). It is noteworthy that despite the large BC errors an evaluation accuracy of the corresponding parameter ZC is acceptable. In fact, formula 20 defining BC results in the relation

δBC = -QpcδZC ≈ -35δZC BC

(56)

Thus, overestimation of BC by 50% is produced by δZC ≈ -0.014. Generally, identifiability of systems with flat and wide energy distribution (i.e., with h ) 2 or h > 6, Zf∞ < ZC) is poor. For such systems the parameter estimation errors may reach 100%. It concerns also the primary sites capacity mhA (VhA), which is usually of main interest in examination of porous materials. Nevertheless, we stated that other adsorption models, including the commonly recommended DR and classical BET equations, give noticeably worse evaluations of this important parameter. It is shown in Figure 10 for two typical systems, where the uLBET isotherms are used as a representation of real isotherms, treated with different adsorption formulas. The heterogeneous LBET-type description provides not only better evaluation of the surface capacity (VhA) but also gives rough (although useful) quantitative information on the surface energy distribution, while the classical DR and BET equations show only an averaged adsorption energy. The lowest subfigures in Figure 10 illustrate effects of application of homogeneous LBET formulas to heterogeneous systems. It may be seen that the errors of R are close to those obtained with the optimal heterogeneous models, while BC and β take usually their lowest value BC ) 1 and β ) 1 (0 ) -33% and -10%, respectively). Thus, the parameters R, BC, and β evaluated by fitting of the heterogeneous LBET formulas give more information on the geometry of pores (see Table 2). Let us remember that Figures 6-9 present results obtained for poorly identifiable systems. In general, the parameter evaluation errors are smaller for larger R and much better identifiable are systems with narrower energy distribution18 (ZA2 > Zf∞ > ZC), which are likely more frequent than widely heterogeneous ones (Zf∞ < ZC). 6. Examination of an Empirical Adsorption IsothermsExample Because of high numerical efficiency of the LBET formulas (when using a typical PC Pentium1.5GHz, a single variant identification takes ca. 1-15 s), examination of empirical isotherms can be carried out with more variants of the energy distribution than listed in Table 4. We recommend to apply also h ) 4, 6, and 8 (see

Figure 10. Effect of QA/RT on the six-parameter identifiability of the uLBET-type isotherms with realistic data-fitting model discrepancies in energy distributions (D ) D.6p; Table 4) and comparison of the heterogeneous LBET-based parameter estimation errors with inaccuracies of quantitative uLBET systems characterization obtained by application of other adsorption models.

Table 1), with ζ0 ) 1 (see eq 54) producing a bit narrower distribution functions than those shown in Figure 2 (variants nos. 31-39 and 40-48). The data should be treated both with the five-parameter identification (assuming Zf∞ ) ZC) and with Zf∞ being adjusted independently. Moreover, identification based on data of different pressure ranges can be carried out. Each time a number of well-fitted variants may be taken under study (we used three of them) to assess the identification reliability and possible properties of the system. Evaluations obtained with Zf∞ ) ZC are preferable, providing that the fitting


Langmuir, Vol. 21, No. 16, 2005 7255

Table 5. Results of Multivariant Examination of the Empirical Isotherm of N2 on Active Carbon AMBERSORB572 at 77.4 K (ref 20): N2 Cohesion Energy, 3577 J/mol; Molar Volume, 34.7 cm3/mol options πmax 6par.a

0.8

5par.b

0.8

6par.a

0.006

3pt13 c 3pt13 c

0.983 0.983

evaluation of the system parameters

σe

var.

type

h

d

η

mhA

QA/RT

ZA

BC

ZC

Zf∞

R

β

0.308 0.309 0.313 0.311 0.314 0.320 0.449 0.449 0.449

6 18 33 19 40 7 18 3 6 1 1

I II I II II I II I I I II

3 2 4 3 4 5 2 2 3 0 0

1 1 1 0 0 0 1 1 1 -

0 0 0 0 0 0 0 0 0

13.38 13.37 13.37 14.09 14.01 13.92 15.17 15.17 12.92 11.77 11.77

-11.4 -11.4 -11.8 -11.4 -11.7 -12.0 -11.2 -11.2 -11.2 -6.79 -6.79

0.55 0.55 0.57 0.55 0.56 0.57 0.55 0.55 0.55

3.18 3.18 3.18 3.18 3.18 3.18 1.00 1.00 1.00 3.04 3.03

0.083 0.083 0.083 0.083 0.083 0.083 0.121 0.121 0.121

0.149 0.260 0.145 0.083 0.083 0.083 0.100 0.100 0.100

0.52 0.52 0.52 0.53 0.54 0.55 0.52 0.52 0.44 0.57 0.62

1.00 1.10 1.00 1.12 1.11 1.00 1.07 1.00 1.00 1.09 1.07

a 6par.: the three best fitted variants found by the six-parameter identification (Z b f∞ adjusted). 5par.: the three best fitted variants found by the five parameter identification (Zf∞ ) ZC). c 3pt13: the parameters evaluated by the “three point” fitting of the homogeneous LBET model with πC ) πmax (see ref 13).

Figure 11. Best results of application of the LBET and uLBET models to examination of the N2 adsorption isotherm on the active carbon AMBERSORB-572 at 77 K (Choma and Jaroniec, ref 20), by the six-parameter identification. Larger pictures, data with πmax ) 0.8 (see the best triple in Table 5); insets, data with πmax ) 0.006. The columns of subfigures show the three best fitted LBET model variants (the optimal is in the right column); upper panels: O, empirical data; solid lines, the theoretical LBET and uLBET isotherms (indistinguishable); dotted lines, the theoretical adsorption at the first layer.

quality and the value for Zf∞ are close to those reached by the six-parameter identification. Examination results of an empirical isotherm20 are presented in Table 5. The system parameter evaluations obtained in our earlier paper13 by a special identification procedure with the homogeneous LBET model are also presented (3pt).13 The procedure exploited in ref 13 was focused on explanation of a rapid growth of adsorption observed for π > 0.8, and data of π < 0.006 (exhibiting evident heterogeneity of the first layer energy) were omitted. It provided reliable evaluations of β and BC. The multivariant fitting of the LBET-type formulas was aimed at analysis of surface heterogeneity effects; hence, the isotherm limited to π < 0.8 has been taken under examination. Moreover, to get more information on the energy distribution function, the fitting of the very lowpressure data (π < 0.006) was also carried out. The six- and five- parameter identification, based on data of π < 0.8, give comparable fitting quality (σe), and resultant values of the adjusted parameters are close (except for Zf∞). In each case the parameter ZC meets its (20) Choma, J.; Jaroniec, M. Langmuir 1997, 13, 1026.

lower bound that yields the same, high value for BC. It shows that energetic constraints for multilayer adsorption are significant (the same was stated in ref 13; see 3pt13 results in Table 5). The best variants are slightly distinctive; hence, the results shown in Table 5 (in the rows 6 par. and 5 par.) should be viewed as a complementary quantitative characterization of the system. In particular, they suggest that there is no dominant mechanisms limiting the number of layers, as the variants assuming 1st- and 2nd-type adsorption at top layers are of comparable reliability. The results obtained with the six-parameter identification seem to be more reliable; hence, they are presented in more detail in Figure 11. The small pictures placed within the larger ones show the fittings of the initial isotherm section (π < 0.006) obtained individually (see Table 5 for the corresponding parameters). It may be seen that energy distributions found with the three best variants are of different shapes, which confirms our opinion that the surface energy distribution of microporous materials is hardly determinable. Nevertheless, they all give evidence that there is a large fraction of small, highly adsorptive

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micropores with an almost heterogeneous surface. The evaluated system parameters and energy distributions corresponding to the best fitted variants suggest that the material contains a large number of very small micropores or cavities, capable of keeping only single molecules, providing about 50% of the primary sites of almost the same adsorption energy (the single molecule adsorption is homogeneous). Larger clusters are created in rather wide channel-like or compact pores with smooth walls (small ZC), in which branching of second and third layers is likely (variants with β > 1 are acceptable) and competitive clusterization is possible. The ends of channels and local niches in compact pores make primary adsorption sites of moderately diversified energy (energy distributions of the best fitted variants are rather narrow). Noticeable model-data discrepancies observed in Figure 11 for π ∈〈0.005, 0.15) suggest that there is a larger fraction of highly adsorptive primary sites than predicted by our model assuming the uniform distribution of ZAκ for k > 1 (see eq 12). It is noteworthy that evaluations of the parameter R obtained by fitting of the heterogeneous LBET formulas to the very initial section of the isotherm (πmax ) 0.006) are very close to those based on wide pressure range data (including 3pt results). Also the surface capacity (mhA) and the energy distributions are adequately determined. It is due to the relationships linking a primary site capacity k with its adsorption energy distribution (see Table 1). They point to essential correlations of the lowest pressure isotherm shape (strongly affected by the surface energy distribution) with the structure of pores, which makes possible gaining adequate information on the main properties of adsorbents, by fitting a lowest pressure section of isotherms.

that the pore size distribution may be expressed in an exponential form, and the first layer adsorption energy distribution is closely linked with a pore geometry. The simplified LBET formulas are accurate enough to be effectively used in identification of real adsorption systems of irregular microporous structure, instead of their rigorous counterparts, uLBET models (it reduces radically the identification computing time). In particular, a rough approximation of the high energy side of the surface energy distribution proposed in the paper is of negligible effect on the identification results. The multivariant identification of adsorption systems, with a surface energy distribution structure being presumed in each variant, makes it possible to avoid numerical problems caused by a large number of the system parameters to be evaluated. Final analysis of a number of the best fitted variants (e.g., of three of them) provides complementary information on adsorption mechanisms, thus, enabling more reliable evaluation of an adsorbent pore structure. The most suitable solution may be selected with respect to additional (nonformal) criteria. An advantage of the uLBET/LBET formulas is that they may be reliably identified by using only low pressure data. The multivariant identification method, based on fitting of the LBET-type formula, is easily applicable to examination of real adsorption isotherms. It gives an insight into the nature of restrictions for multilayer adsorption and into geometrical properties of pores and outlines a first layer energy distribution. The LBET-type formulas are accurate enough to make possible drawing more information on heterogeneous adsorption systems than provided by classical adsorption equations.

7. Conclusions

Acknowledgment. The research is led within the AGH-UST Grant 11.11.210.62.

The uLBET model may be treated as a formal tool for studying properties of the adsorption system, assuming

LA040117R

Modeling of Surface Heterogeneity of Microporous Adsorbents with

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