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Langmuir 1995,11, 2316-2317
Evaluation of the Fractal Dimension from a Single Adsorption Isotherm M. Jaroniec Department of Chemistry, Kent State University, Kent, Ohio 44242 Received February 21, 1995. In Final Form: March 30, 1995*
Fractal geometry, which has been widely used in many areas of modern science,l is also very popular in adsorption.2 Since 1983, when Pfeifer and Avni9 showed a successful application of this geometry to study adsorption on solid surfaces, a great number of papers have been published (e.g., see review by Avnir et al.4and references therein). In spite of some controversy in the literature5-I about assessing the fractal nature of solids, fractal geometry has played an important role in advancing the theoretical and experimental studies of adsorption on heterogeneous material^.^,^,^ The key quantity in fractal geometry is the fractal dimension D, which is an operative measure of the surface and structural irregularities of a given solid.2 Sources of surface heterogeneity are various irregularities from cracks, steps, flaws, impurities, and differing atomic species.8 The geometrical irregularities and roughness of the surface have an essential influence on the value of the fractal dimension D, which for solid surfaces can vary from 2 to 3. The lower limiting value of 2 corresponds to a perfectly regular smooth surface, whereas the upper limiting value of 3 relates to the maximum allowed complexity of the surface. In addition to the surface heterogeneity,the structural heterogeneity of a given solid, which is generated by the existence of pores of different sizes (especially fine pores), can contribute significantly to the fractal dimension, too.4,7,9-14 Several simple relationships have been proposed as a means of evaluating the fractal dimension from various types of experiments including adsorption, porosimetry, scanning electron microscopy, small-angle X-ray, and neutron scattering m e a ~ u r e m e n t s . ~ ,In ~ Jthe ~ current work the main relationships based on adsorption measurements will be discussed because they are most often used to determine the fractal dimension D of solid materials. Special emphasis will be given to comparison of the existing methods for evaluating the fractal dimension from a single adsorption isotherm. One of the most popular methods used to evaluate the fractal dimension is that based on the dependence of the (1)Fractal in Science; Bunde, A., Havlin, S., Eds.; Springer-Verlag: Berlin, 1994. (2) The Fractal Approach to Heterogeneous Chemistry;Avnir, D., Ed.; J. Wiley & Sons: New York, 1990. (3) Pfeifer, P.; Avnir, D. J . Chem. Phys. 1983,79,3558. (4)Avnir, D.; Farin, D.; Pfeifer, P. New. J . Chem. 1992,16,439. (5) Drake, J. M.; Levitz, P.; Klafter, J . New J . Chem. 1990,14,77. (6) Drake, J. M.; Yacullo, L. N.; Levitz. P.: Klafter. J . J . Phvs. Chem. 1994,98,380. (7) Conner, W. C.; Bennet. C. 0.J . Chem. Soc., Faradav Trans. 1983. 89,4109. ( 8 ) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (9)Avnir, D.; Pfeifer, P. J . Chem. Phys. 1989,80, 4573. (10)Jaroniec, M. Fuel 1990,69,1573. (11)Jaroniec, M.; Gilpin, R. K.; Choma, J. Carbon 1993,31,325. (12) Choma, J.; Burakiewicz-Mortka, W.; Jaroniec, M.; Gilpin, R. K. Langmuir 1995,9,2555. (13) Neimark, A. V. Physica A 1992,191,258. (14) Jaroniec, M.; Lu, X.; Madey, R.; Avnir, D. J . Chem. Phys. 1990, 92,7589. (15) Neimark, A. V. Russian J . Phys. Chem. 1990,64,1398.
monolayer capacity on the adsorbate ~ i z e : ~ , ~ J ~ n,
-012
(J
where n, is the monolayer capacity and (T is the area occupied by one adsorbate molecule. Although evaluation ofD on the basis of eq 1is simple, this procedure has some disadvantages related to evaluation of the monolayer capacity and selection of suitable adsorbates in order to avoid the effects associated with orientation of adsorbate molecules on the surface and with adsorbate-adsorbate interactions. Also, the range of u for available adsorbates is relatively narrow. These problems become particularly important for adsorption on microporous solids such as active carbons, which usually possess a high degree of surface i r r e g u l a r i t ~ . ~ J ~ . Another popular method for evaluating D is that utilizing the log-log plot of the pore size distribution. The slope of this plot is related to the fractal d i m e n ~ i o n , ~ , ~ log J ( r )= const - (D- 2) log r (2) where J ( r )is the pore size distribution and r is the average pore radius. Several authors (see references in the review by Avnir et al.4) have utilized the cumulative and differential pore size distributions in the mesopore range. Jaroniec et a1.1°-12 applied the method in the micropore range, which is essential to characterize heterogeneous active carbons. In addition to the relationships given by eqs 1 and 2, several isotherm equations have been derived for various models of physical adsorption on fractal surfaces.18-28 These equations contain the fractal dimension D is a parameter and describe the surface coverage as a function of the equilibrium pressure. One of simplest and most popular relationships is that given by the FrenkelHalsey-Hill (FHH) equation, which in logarithmic form can be expressed as follows:21 I n n = const - (3 - D)1nA (3) where n is the amount adsorbed at the relative pressure plpo and absolute temperature T and A is the so-called adsorption potential defined as A = RT In ( p d p )
(4)
The symbol po denotes the saturation vapor pressure. Equation 3 was first derived by Avnir and JaroniecZ1by extension of the Dubinin-Radushkevich isotherm equation to adsorption on structurally heterogeneous solids characterized by the fractal distribution of fine pores (see eq 2). Later Yin25obtained the same equation by assuming the sequential filling of pores from small size to large (i.e., condensation-type local adsorption isotherm) and integrating the fractal distribution of fine pores given by eq 2. He used the Kelvin equation to relate the average pore radius r t o the relative pressure plpo: (16) Avnir, D.; Farin, D.; Pfeifer, P. Nature 1984,308, 261. (17) Jaroniec, M.; Choma, J . Chem. Phys. Carbon 1989,22,197. (18)Cole, M. W.; Holter, N. S.; Pfeifer, P. Phys. Rev. B 1986,37, 8806. (19) Fripiat, J. J.; Gatineau, L.; Van Damme, H. Langmuir 1986,2, 162. (20) Levitz, P.;Van Damme, H.; Fripiat, J . J . Langmuir 1988,4,781. (21) Avnir, D.;Jaroniec, M. Langmuir 1989,5,1431. (22) Van Damme, H.; Fripiat, J. J. J . Chem. Phys. 1985,82,2785. (23) Nakaniski, K.; Soga, N. J . Non-Cryst. Solids 1988,4,781. (24) Pfeifer, P.; Cole, M. W. J . New Chem. 1990,14,221. (25) Yin, Y. Langmuir 1991,7,216. (26) Keller, J . U. Physica A 1990,166,180. (27) Urbakh, M.; Daikhin, L. Surf. Sci. 1993,2871288,847. (28) Albano, E. V.; Martin, 0. H. Phys. Reu. A 1989,39, 6003.
0743-746319512411-2316$09.00/0 0 1995 American Chemical Society
Langmuir, Vol. 11, No. 6, 1995 2317
Notes
or
A = 2yVmlr
(6) where y is the surface tension of the adsorbed liquid film and V, is the molar volume of the liquid adsorbate. An extensive theoretical study of the FHH exponent and its relation to the fractal dimension was made by They found that, for the adsorption Pfeifer et a1.24,29,30 model in which van der Waals attraction forces are dominating (i.e., when the liquidgas surface tension forces are negligible), the slope of the (In n vs In A) linear relationship is equal to (D - 3)/3. In this case the attractive van der Waals forces at the gaslsolid interface tend to form the adsorbed film, which replicates the surface r o ~ g h n e s s .However, ~~ at higher coverage the interface is controlled by the liquidgas surface tension (capillary condensation), because of greater thickness of the surface film. In this case the interface moves further away from the surface so that the interfacial area is reduced, giving the slope of the (In n vs In A) relationship equal to D 3. The ( D - 3) slope was first obtained by Avnir and J a r o n i e F and later confirmed b ~ Y i and n ~Pfeifer ~ et al.24 A common feature of the adsorption models discussed a b ~ ~ was e ~assumption ~ , ~ ~ of, the ~ volume ~ filling of fine pores, which in the case of mesopores occurs according to the capillary condensation mechanism. It is noteworthy that the FHH isotherm equation was successfully used to evaluate the fractal dimension of various ~ o l i d s . ~ l - ~ ~ Recently, Neimark et a1.13,35-38 proposed the so-called thermodynamic method for calculating the fractal dimension D from the adsorption isotherm data. The theoretical basis for this method is avery simple relationship between the surface area of the adsorbed liquid film, S , and the average pore radius:I3 In S = const - (D- 2) In r (7) In this method the surface area of the adsorbed film is calculated according to the Kiselev equation:
where nmaxdenotes the amount adsorbed at plpo tending to unity. However, Kelvin eq 5 is used to convert the equilibrium pressure, p , to the average pore radius, r. Application of Neimark's method for evaluating the fractal dimension D from nitrogen adsorption isotherms is demonstrated in refs 37 and 38. The main aim of the current paper is comparison of two methods, which are often used to evaluate the fractal dimension from a single adsorption isotherm: (i) the thermodynamic method proposed by Neimark13,35,36 and (ii) a method based on the FHH eq 3 proposed by Avnir and Jaroniec (AJ).21 Since Neimark used general relationships of interfacial thermodynamics, his method has (29) Pfeifer, P.; Cole, M. W.; Krim, J. Phys. Rev. Lett. 1990,65,663. (30) Pfeifer, P.; Kenntner, J.; Cole, M. W. In Fundamental of Adsorption; Mersmann, A. B., Scholl, S. F., Eds.; American Institute of Chemical Engineers: New York, 1991; p 689. (31) Ismail, I. M.K.; Pfeifer, P.Langmuir 1994, 10, 1532. (32) Kaneko, K.; Sato, N.; Suzuki, T.; Fajiwara, Y.; Nishikawa, K.; Jaroniec, M. J . Chem. SOC., Faraday Trans. 1991,87, 179. (33) Lefehvre, Y.; Lacelle, S.; Jolicoeur, C. J . Mater. Res. 1992, 7, 1 RXA (34) Lefehvre, Y.;Jolicoeur, C. Colloids Surf. 1992, 63, 67. (35)Neimark, A.V.JETP Lett. 1990, 51, 607. (36)Neimark, A. V.Adsorpt. Sci. Technol. 1990, 7, 210. (37) Neimark, A.V.;Unger, K. K. J . ColloidZnterfaceSei. 1993,158, 412. (38) Neimark, A.V.;Hanson, M.; Unger, K. K. J . Phys. Chem. 1993, 97,6011.
no connection with a specific isotherm equation and can be claimed as a model-independent method. In contrast, the AJ method is based on the FHH model of multilayer adsorption and seems to be model-dependent. It will be shown that both methods are essentially equivalent in the range of applicability of the FHH equation. Let us start with the FHH isotherm equation derived by Yin:25 n = K(D)(3- D)-1(2yVm)3-DAD-3
(9)
or
drz = - K ( D ) ( ~ Y V , ) ~ - ~ dA A~-~
(10) where K(D)is the normalization constant of the pore size andA distribution given by eq 2 in the range xrmn-xmax1l is the adsorption potential defined by eq 4. Similarly to the case of Neimark's Yin's consideration^^^ are based on Kelvin eq 5. Neimark35-3a used eq 7 to calculate the surface area of the adsorbed film. This equation rewritten in terms of the adsorption potentialA has the following form:
s = y-l Sn"""
A dn
(11)
Substitution of dn in the integral in eq 11 by eq 10 gives the following expression:
s = K(D)(2yvm)3-Dy-1hAP-3 dx
(12)
Integration of eq 12 gives the analytical relationship between the surface area of the adsorbed film, S , and the adsorption potential A for the FHH eq 9:
S = K(D)(D- 2 ) - ' ( 2 ~ V , ) ~ - ~ y - ~ A ~ - '(13) The relationship between S and r can be obtained after substitutingA in eq 13 by eq 6:
s = ~v,J(D)(D
-2)-VD
(14) The logarithmic form of eq 14 is identical with eq 7, which is a key expression in Neimark's m e t h ~ d . ~ ~ - ~ ~ Note that eq 7, which relates the surface area of the adsorbed film to the pore radius, is the fundamental dependence of N e i m a r k ' ~ ~method. ~ - ~ ~ To plot this dependence on the basis of a single adsorption isotherm one needs to convert (i)the adsorbed amount to the surface area of the adsorbed film using Kiselev eq 8 and (ii)the relative pressure to the pore radius using Kelvin eq 5. Theoretical considerations presented above demonstrate that Kiselev eq 8 combined with FHH eq 9 [with the (D - 3) exponent] and Kelvin eq 5 gives the same relationship between S and r as that used by Neimark.35-3a Thus, from a thermodynamic viewpoint both methods are identical in the range of applicability of the FHH equation. The equivalence of both methods in the above mentioned range and the generality ofNeimark's method (which does not require an analytical isotherm equation) indicate also that the FHH isotherm equation is a general thermodynamic relationship that represents the pore-filling adsorption process in the range of the Kelvin equation. Since the equivalence of both methods is obtained for the FHH equation with the ( D - 3) exponent, this analytical form of the FHH exponent seems to be characteristic for the pressure range in which the Kelvin equation is valid. LA950132J @
Abstract published in Advance ACS Abstracts, June 1, 1995.
