Langmuir 1997, 13, 4391-4394
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Applicability of the Fractal FHH Equation Bendida Sahouli,*,† Silvia Blacher, and Franc¸ ois Brouers Physique des Mate´ riaux et Ge´ nie Chimique, Universite´ de Lie` ge, B5, Lie` ge 4000, Belgium Received December 19, 1996. In Final Form: May 20, 1997X The current work deals with the applicability of the fractal version of the Frenkel-Halsey-Hill (FHH) theory of multilayer adsorption. The concept of the standard isotherm is introduced to examine the weakness of this equation when the solid-gas potential controls the adsorption process of gas on a given solid. The results of the determination of the fractal dimension of typical samples of carbon blacks, activated carbon, and aerogels (mixed SiO2-ZrO2 and pure silica) are given. They show that the fractal dimension depends critically of the coverage exponent s for a flat surface of the same chemical nature.
θ ∝ [ln(P0/P)]-1/m
Introduction The concept of fractality introduced by Mandelbrot1 was applied for the first time by Avnir and Pfeifer2 to characterize the degree of complexity of a surface material from adsorption-desorption data. Since this pioneer work, many materials having a complex structure have been claimed as fractal. One way to define the fractal dimension D is to consider it as a measure of the ability of a solid to fill a three-dimensional volume. Different mathematical models are described in the literature3 to determine D from adsorption data. In this work, we are interested in one method for surface fractal analysis that is mostly4 used for determining the D value. This method is an adaptation of the Frenkel-HalseyHill (FHH)5 theory of multilayer adsorption to the fractal surfaces. The FHH theory5 is a model that describes the multilayer adsorption coverage. The classical FHH equation on a flat surface reads
θ ∝ [ln(P0/P)]-1/s
(1)
where θ is the adsorbed quantity and P/P0 is the relative pressure. The FHH exponent s can be considered as a parameter characterizing the shape of the isotherm in the multilayer region just as the CBET constant for the monolayer region. The generalization of the classical FHH equation (1) to fractal surfaces by Pfeifer et al.4 has been formulated as † Permanent address: L.O.C.M., Physique, Universite ´ d’Oran, Oran, Algeria. X Abstract published in Advance ACS Abstracts, July 1, 1997.
(1) Mandelbrot, B. Fractal Geometry of Nature; Freeman: San Francisco 1982. (2) Avnir, D.; Pfeifer, P. Nouv. J. Chim. 1983, 7, 71. (3) (a)Avnir, D.; Farin, D.; Pfeifer, P. Nature 1984, 33, 261. (b) de Gennes, P. G. Physics of Disordered Materials; Adler, D., Fritzsche, H., Ovhinsky, S. R., Eds.; Plenum, New York 1985. (c) Cole, M. W.; Holter,N. S; Pfeifer, P. Phys. Rev. B 1986, 33, 8806. (d) Fripiat, J. J.; Gatineau, L.; Van Damme, H. Langmuir 1986, 2, 562. (e) Pfeifer, P.; Obert, M.; Cole, M. W. Proc. R. Soc. London A 1989, 423,169. (f) Avnir, D.; Jaroniec, M. Langmuir 1989, 5, 1431. (g) Neimark, A. V. JETP Lett. 1990, 51, 607. (4) (a) Pfeifer, P.; Wu,Y. J.; Cole, M. W.; Krim, J. Phys. Rev. Lett. 1989, 62, 1997. (b) Pfeifer, P.; Cole, M. W. New J. Chem. 1990, 14, 221. (c) Pfeifer, P.; Kenntner, J.; Cole, M. W. In Fundamentals of Adsorption; Mersmann, A. B., Scholl, S. E., Eds.; Am. Inst. Chem. Eng.: New York, 1991; p 689. (d) Ismail, I. M. K.; Pfeifer, P. Langmuir 1994, 10, 153. (5) (a) Frenkel, J. Kinetic Theory of Liquids; Clarendon Press: Oxford, U.K., 1946. (b) Halsey, G. D. J. Chem. Phys. 1948, 16, 931. (c) Hill, T. L. Adv. Catal. 1952, 4, 211. (d) Pierce, C. J. Phys. Chem. 1959, 63, 1076. (e) Zettlemoyer, A. C. J. Colloid Interface Sci. 1968, 28, 343. (f) Steele, W. A. The interaction of gases with solid surfaces; Pergamon: Oxford, U.K., 1974; J. Colloid Interface Sci. 1980, 75, 13. (g) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic: London, 1982; p 90.
S0743-7463(96)02119-1 CCC: $14.00
(2)
with m ) s/3 - D This equation, obtained by assuming s equal to the theoretical value 3 (nonretarded van der Waals interactions), is known as the fractal FHH equation and holds only at early stage of the multilayer formation. In the case of the capillary condensation (CC) regime, i.e., when the interface is controlled by the liquid-gas surface tension forces, the isotherm equation is given by
θ ∝ [ln(P0/P)]D-3
(3)
This equation was proposed for the first time by de Gennes3b and latter by Avnir et al.,3f Pfeifer et al.,4b,c Yin,6 and Neimark7a using different arguments. Recently, it has been shown8,9 that eq 3 and thermodynamic method proposed and developed by Neimark3g are equivalent in the region of capillary condensation in materials without microporosity. In the case when the dominating force is the substrate potential, the application of this method (eq 2) yields in most cases value of D smaller than 2 that is physically meaningless for a surface dimension. In addition to the argument of the crossover regime advanced by Pfeifer et al.,4b this can be linked to the choice of the coverage exponent (noted by s below) for a flat surface, i.e., s ) 3, a value which is almost never observed.5 This problem has been already mentioned by Neimark7a,b and very recently by us9 and Venkatraman et al.10 The main objective of this work is to suggest the use of the concept of the standard isotherm to determine and adjust the value of the s parameter. We have considered the two well-known different criteria used for the choice of a standard isotherm: (i) similarity in the chemical nature11 and (ii) the same CBET constant.12 We have considered a set of samples of very different materials and have calculated the fractal dimension using the nitrogen standard isotherms for silica,13 of Kaneko et al.14 (6) Yin, Y. Langmuir 1991, 7, 216. (7) (a) Neimark, A. V. Russ. J. Phys. Chem. 1990, 64, 1397. (b) Neimark, A. V. Adsorpt. Sci. Technol. 1990, 7, 210. (c) Neimark, A. V. Phys. Rev. B 1994, 50, 15435. (8) Jaroniec, M. Langmuir 1995, 11, 2316. (9) Sahouli, B.; Blacher, S.; Brouers, F. Langmuir 1996, 12, 2872. (10) Venkatrman, A.; Fan, L. T.; Walawender, W. P. J. Colloid Interface Sci. 1996, 182, 578. (11) Reference 5g, p 90. (12) Lecloux, A.; Pirard, J. P. J. Colloid Interface Sci. 1979, 70, 265. (13) Bhambhani, M. R.; Cutting, P. A.; Sing, K. S. W.; Turk, D. H. J. Colloid Interface Sci. 1972, 38, 109. (14) Kaneko, K.; Ishii, C.; Ruike, M.; Kuwabara, H. Carbon 1992, 30, 1075.
© 1997 American Chemical Society
4392 Langmuir, Vol. 13, No. 16, 1997
Sahouli et al.
and Rodriguez-Reinso et al.15 for carbon material, and of Lecloux et al.10 based on the CBET constant and the theoretical value s ) 3 corresponding to pure van der Waals forces. In addition, the s values of Pierce,5d Zettlemoyer5e and Dubinin et al.16 are considered. We note that this last value, s ) 2.24, has been already used by Neimark et al.17 in their comparative method for the discrimination of surface fractality. Materials and Adsorption Measurements We have considered two types of very different porous materials: (1) Two representative commercial carbon black samples (N115 and N550) and one commercial activated carbon (CAC) were considered in this investigation. The detailled studies of these carbon blacks have been published elsewhere.18 (2) The mixed SiO2-ZrO2 and pure silica (SiO2-P) aerogels were synthesized following the method reported elsewhere.19 Two different mixed samples were considered: one was prepared under acidic catalyst conditions (Mix-A) and the other one under under basic conditions (Mix-B). Nitrogen adsorption measurements were performed on a Sorpromatic Carlo-Erba Series 1800 appartus controlled by by an IBM Personal Computer. Prior to the nitrogen adsorption, all samples were outgassed in vacuum at 300 °C.
Results and Discussion In Figure 1a,b we show the nitrogen adsorptiondesorption isotherms for the considered samples. The CAC adsorption isotherm is of type I in the BDDT classification, in which the initial part represents the micropore filling. For the other samples (N115, N550, SiO2-P, Mix-A, and Mix-B), the isotherms cannot be clearly identified with one of the five BDDT theoretical isotherms. For the carbon blacks, the isotherms are close to type IV according to the BDDT classification. This suggest that the adsorption occurs by the monolayer-multilayer mechanism at low and meduim relative pressures and by the capillary condensation at high relative pressures. The aerogels samples exhibit adsorption isotherms that can be considered as intermediate between type I and type IV which means that the adsorption process invloves the micropore filling, monolayer-multilayer, and the capillary condensation mechanism. Thus, these particular samples are chosen since, in our opinion, their adsorption process is characterized by the three fundamental mechanisms of the adsorption, namely, the pore filling, the multilayer buildup, and the capillary condensation. The fractal analysis using the method based on the FHH equation is illustrated in Figure 2 which shows the FHH plots of the studied samples. We note that in the case of the microporous solid CAC, the fractional filling was calculated using the micropore volume obtained from the Rs plot by considering the standard isotherm of Rodriguez-Reinso et al.15 The obtained results are reported in the Table 1a,b. This table includes the FHH exponent (m) of the sample under study, the FHH exponent (s) of the different isotherms standard, and the fractal dimension D calculated using relation 2 for each value of the parameter s and relation 3, respectively. (15) Rodriguez-Reinoso, F.; Martin-Martinez, J. M.; Prado-Burguete, C.; McEnaney, B. J. Phys. Chem. 1987, 91, 515. (16) Dubinin, M. M.; Kataeva, L. I.; Ulin, V. Proc. Acad. Sci. USSR 1977, 3, 510. (17) Neimark, A. V.; Unger, K. K. J. Colloid Interface Sci. 1993, 158, 412. (18) (a) Sahouli, B.; Blacher, S.; Brouers, F.; Sobry, R.; Van den Bossche, G.; Diez, B; Darmstadt, H.; Roy, C.; Kaliaguine, S. Carbon 1996, 34, 633. (b) Darmstadt, H.; Roy, C.; Kaliaguine, S.; Sahouli, B.; Blacher, S.; Pirard, R.; Brouers, F. Rubber Chem. Technol. 1995, 68, 330. (19) Blacher, S.; Pirard, R.; Pirard, J. P.; Sahouli, B.; Brouers, F. Langmuir 1997, 13, 1145.
Figure 1. Nitrogen adsorption-desorption isotherms for (a) the carbonaceous adsorbents (N115, N550, and CAC) and (b) the aerogel samples (SiO2-P, Mix-A, and Mix-B).
The results can be summarized as follows: (a) The range of relative pressure corresponding to the linear part, indicated by the arrows, of the curves (Figure 2) is not the same for all samples. (b) At first approximation, the m values seems to be depending on two factors: (i) the chemical nature of the adsorbent and (ii) the nature of porosity within a same class of solids. (c) The fractal dimension D depends on the value of the FHH exponent (s) which characterizes the standard isotherm in the multilayer range. The interval of length scales over which the fractal dimension is calculated can be estimated from lower and upper limits of the linear region in the FHH plots (Figure 2, a and b) by following the procedure of Ismail et al.4d It extends from a lower cutoff of about 0.35 nm (N2 monolayer) to a upper cutoff ranging from 2 nm to some 10 nm depending on the considered sample. (d) The values of the exponent m are in accordance with the known fact20 that the mesoporosity decreases the FHH exponent while the microporosity increases it. In other words, the FHH exponent can be related to the physical adsorption regime. In the light of these results, we can propose the following interpretation with regard to the theoretical boundary: Case 1: m . 3. When the system is essentially microporous having a horizontal plateau of a Type I isotherm, it is generally characterized by a very high value of the exponent m (such as the CAC sample). In that case the same D is obtained approximately for all the values of the parameter s (in the FHH regime). This is due to fact that the term (s/m) tends to zero in these conditions. (20) Carrot, P. J. M.; Sing, K. S. W. Pure Appl. Chem. 1989, 61, 1835.
Applicability of the Fractal FHH Equation
Langmuir, Vol. 13, No. 16, 1997 4393 Table 1. Main Results of the (a) Carbonaceous Adsorbents and (b) Aerogel Samples samples N115
N550
CAC
Figure 2. FHH plots for nitrogen adsorption on (a) the carbonaceous adsorbents (N115, N550, and CAC) and (b) the aerogel samples (SiO2-P, Mix-A, and Mix-B). θ refers to the adsorbed number of layers and the relative adsorption in the case of the CAC sample.
The maximal value of D (≈2.96) could reflect the volume filling process of the micropores rather than the surface fractal dimension of micropores wall. This is consistent with the recent work of Tsunoda et al.21 However, there is an alternative interpretation such that the exponent represents an imperfect sieve which may have a nonfractal power law distribution. Furthermore, one can consider rightfully that the use of eq 3 for the determination of fractal dimension for microporous solids22 in which the adsorption process is mainly the micropore filling and not layer-by layer coverage, and/or for mesoporous solids where the CC regime is predominating, as problematic. We recall that this equation was derived initially by Avnir and Jaroniec3f to investigate the texture of heteregeneous microporous solids. This apparent contradiction, has been clearly interpreted by Jaroniec8 when he wrote “... A common feature of the adsorption models discussed above was assumption of the volume filling of fine pores, which in the case of mesopores occurs according to CC mechanism ...”. In our opinion, eq 3 applies to microporous solids if the adsorbate-adsorbate interactions control the filling of the (21) Tsunoda, R.; Ando, J-I. J. Colloid Interface Sci. 1995, 171, 528. (22) (a) Jaroniec, M.; Lu, X.; Madey, R.; Avnir, D. J. Chem. Phys. 1990, 92, 7589. (b) Kaneko, K.; Sato, M.; Suzuki, T.; Fujiwara, Y.; Nishikawa, K.; Jaroniec, M. J. Chem. Soc., Faraday Trans. 1991, 87, 179.
m
s
D ) 3 - (s/m)
(a) Carbonaceous Adsorbents 1.76 2.43 3a 2.75b 1.86 c 2.12 2.12 2.63d 1.91 2.27e 2.06 2.24f 2.07 2.71g 1.88 2.22 3a 1.64 2.75b 1.76 2.12c 2.04 2.63d 1.81 2.27e 1.98 2.24f 1.99 2.71g 1.77 62.5 3a 2.95 b 2.75 2.95 2.12c 2.96 2.63d 2.95 2.27e 2.96 f 2.24 2.96 2.71g 2.95
Mix-A
2.50
Mix-B
4.48
SiO2-P
3.81
(b) Aerogel Samples 3a 1.80 2.75b 1.90 2.12c 2.15 2.64h 1.94 2.24f 2.10 2.71g 1.91 3a 2.33 2.75b 2.38 2.12c 2.52 2.64h 2.41 2.24f 2.50 2.71g 2.39 3a 2.21 2.75b 2.27 c 2.12 2.44 2.64h 2.30 2.24f 2.41 2.71g 2.28
D ) 3 - (1/m) 2.59
2.54
2.98
2.6
2.7
2.73
a Theory.4 b Pierce.5d c Zettlemoyer.5e d Rodriguez-Reinso et al.13 Kaneko et al.12 f Dubinin et al.14 g Lecloux et al.10 h Bhambani et al.13
e
wider micropores, i.e., secondary micropore filling23 which involves a quasi-multilayer formation. This case corresponds to the situation 2 < m < 3 discussed below. Case 2: m > ∼3. In this case, the multilayer formation is controlled by the competition between the two mechanisms (FHH and CC) but the former is the predominating one. In terms of the porosity, the solid is strongly microporous with a nonnegligible mesoporosity. The obtained D value with the FHH eq 2 seems less influenced by the value of the parameter s and it is smaller than that calculated by using eq 3. Only another method, such SAXS or SANS, of fractal analysis can confirm which value should be considered. This situation corresponds to the Mix-B and the SiO2-P samples. Case 3: 2 < m < 3. This case is the one we are mostly interested in since, as already mentionned above, the fractal FHH equation (2) gives a D < 2 that is unacceptable for a surface dimension. To explain the weakness of this equation one can consider three different physical behaviors: (a) The adsorption process is really either dominated by capillary condensation forces in the case of mesoporous solid or by quasi capillary condensation if the solid is microporous with the existence of large micropores. In a (23) Sing, K. S. W. Colloids Surf. 1989, 38, 113; J. Porous Mater. 1995, 2, 5.
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such case, relation 3 must be used and the sample N115 can be considered as an illustrative example9 for the former case. Also, the thermodynamic method proposed by Neimark3g would be used as a further check and a test of the mesoporosity of the studied solid. (b) According to Pfeifer et al.,4b this corresponds to the intermediate regime of multilayer formation; i.e., both FHH and CC adsorption mechanisms operate simultaneously. Consequently, as pointed out by Ismail et al.,4d the D value is necessarily qualitative due to the lack of an unique relation between the exponent m and D in this case. (c) As noted previously, this can be linked to the fact that this theory has been derived by assuming s ) 3 for a smooth surfaces, a value which is almost never observed.5 In this case, the nonclassical FHH exponent, m, of the experimental isotherm might be due to the energetic inhomogeneities of the surface (e.g., the effects of adsorbate-adsorbent interactions constant as well as the presence or not of the polar groups on the surface) rather than to any geometric roughness. However, it must be pointed out that in a real system, it is extremely difficult to decouple24 both effects particularly from adsorption measurements. Therefore, in order to find a remedy and to be more or less sure that the departures of the experimental FHH exponent m are due exclusively to the existence of the geometric irregularities, we propose the use of the experimental value of the s parameter chosen on the basis of the appropriate standard istherm and not the theoretical value 3. In principle, since the aim is texture studies, the reference material and the tested sample must have the same chemical nature. Here, we have considered the referenced standard isotherms12-15 to determine the FHH exponent s. The obtained values are repoted in Table 1 which includes also the theoretical value s ) 3 and that of Pierce’s,5d Zettlemoyer’s,5e and Dubinin et al.16 The dependence of D on the values of s is shown in Table 1. For example, the commercial carbon blacks samples (N115, N550) indicate that the use of the exponent s corresponding to nonporous carbon material and to that of Zettlemoyer s ) 2.12 (which describes adsorption on low-energy surfaces) gives a D in the range between 2 and 3. These results suggest that the carbon black surface can be regarded as flat on atomic length scale. The value D ) 2.04 for N550 is closest to that found by Ismail et al.4d from the molecular tiling technique.3a For the case of the mixed aerogel synthesized under acidic conditions (Mix-A), the problem is more complicated (24) Sokolowska, Z. Z. Phys. Chem. 1989, 270, 1113.
Sahouli et al.
since for all values of the parameter s (except that of Zettlemoyer s ) 2.12 which cannot be considered because the aerogels samples have not low-energy surfaces) D is always less than 2. This means that the adsorption process is dominated by the CC regime and hence relation 3 must be used. From the FHH plot of these samples (Figure 2b), one observes a negative deviation at high relative pressure and this behavior can be related to the known fact25 that the nitrogen condensation could disrupt the aerogel structure. Conclusions In this work, the applicability of the fractal version of the Frenkel-Halsey-Hill theory of multilayer adsorption has been assessed. We have confirmed that the FHH exponent depends on the dominating forces which govern the adsorption process. When the solid-gas potential controls the adsorption process, eq 2 must be used with an experimental value of the exponent s instead of the theoretical value 3. To this end, the concept of the standard isotherm is introduced to determine the s value and to examine the weakness of this equation in these circumstances. When the adsorption process is dominated by the capillary condensation, one shoud use eq 3 for the materials without microporosity or in general the thermodynamic method.3g The obtained results for typical samples of carbon blacks, activated carbon, and aerogels (mixed SiO2-ZrO2 and pure silica) indicate that the choice of the parameter s characteristic of the reference flat surface is critical for determining the fractal dimension by methods based on the FHH theory. However, the choice of the parameter s, as we have noted above, depends on the one hand CBET and the associated standard isotherms or on the other hand the chemical nature of the reference isotherm. The result of our study leads us to stress that it is more reasonable to base the choice of s on the second criteria. Acknowledgment. The authors thank the Communaute´ Franc¸ aise de Belgique, the Region Wallonne, and the Administration Ge´ne´rale de la Coope´ration au De´veloppement (AGCD, Brussels, Belgium), and they gratefully acknowledge R. Pirard for valuable discussion and adsorption measurements. We express our sincere thanks to the referees for their constructive comments and for the proposed alternative interpretation when the fractal dimension D is close to 3. LA962119K (25) Scherer, G. W.; Stein, J.; Smith, D. M. J. Non-Cryst. Solids 1995, 486, 309.
