2872
Langmuir 1996, 12, 2872-2874
Fractal Surface Analysis by Using Nitrogen Adsorption Data: The Case of the Capillary Condensation Regime 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 October 16, 1995. In Final Form: February 22, 1996
Introduction Recent calculations of the surface fractal dimension D of carbon blacks particles1 have shown that the values of D obtained using the fractal version of the Frenkel-HillHalsey (FHH) type equation in the case of capillary condensation (CC) regime and the “thermodynamic” method were in very good agreement. The FHH type equation was proposed for the first time by de Gennes2 and later by Avnir et al.,3 Pfeifer et al.4-7 and Yin.8 Here, only the last three approaches will be outlined with great attention to that developed by Pfeifer et al.4-7 The thermodynamic method was proposed and developed by Neimark et al.9-12 This experimental result led us to reexamine the theoretical basis of these two methods. Recently, Jaroniec13 found theoretically the equivalence of both methods in some conditions. In this paper, we want to examine this equivalence from an experimental point of view and provide additional arguments leading to the same conclusions. We take also the opportunity of this discussion to emphasize the practical difficulties met when applying these methods on real adsorbents by commenting on the results we have already reported for five grades of commercial carbon blacks.1 Classical FHH Theory The classical Frenkel-Halsey-Hill (FHH) theory14,15 describes the continuous growth of an adsorbate film with thickness zp according to the following assumptions: (1) the surface-adsorbate potential controls all layers, i.e., the source of potential is all the atoms in the solid which is considered as semi-infinite; (2) the multilayer formation is driven by the energy, i.e., the theory is based on the minimization of the energy resulting from a competition between solid-adsorbate and adsorbate-adsorbate interaction forces. This leads to a smooth film-vapor † Permanent address: Universite ´ d’Oran, Physique, Oran, Algeria.
(1) Darmstadt, H.; Roy, C.; Kaliaguine, S.; Sahouli, B.; Blacher, S.; Pirard, R.; Brouers, F. Rubber Chem. Technol. 1995, 68, 330. (2) de Gennes, P. G. In Physics of Disordered Materials; Adler, D., Fritzsche, H., Ovhinsky, S. R., Eds.; Plenum: New York, 1985. (3) Avnir, D.; Jaroniec, M. Langmuir 1989, 5, 1431. (4) Pfeifer, P.; Cole, M. W. New J. Chem. 1990, 14, 221. (5) Pfeifer, P.; Kenntner, J.; Cole, M. W. Fundamentals of Adsorption; Mersmann, A. B., Scholl, S. E., Eds.; American Institute of Chemical Engineering: New York, 1991; pp 689-700. (6) Pfeifer, P.; Obert, M.; Cole, M. W. Proc. R. Soc. London A 1989, 423, 169. (7) Pfeifer, P.; Wu, Y. J.; Cole, M. W.; Krim, J. Phys. Rev. Lett. 1989, 62, 1997. (8) Yin, Y. Langmuir 1991, 7, 216. (9) Neimark, A. V. JETP Lett. 1990, 51, 607. (10) Neimark, A. V.; Hanson, M.; Unger, K. K. J. Phys. Chem. 1993, 97, 6011. (11) Neimark, A. V.; Unger, K. K. J. Colloid Interface Sci. 1993, 158, 412. (12) Neimark, A. V. Adsorpt. Sci. Technol. 1990, 7 (4), 210. (13) Jaroniec, M. Langmuir 1995, 11, 2316. (14) Steele, W. A. The interaction of gases with solid surfaces; Pergamon: Oxford, 1974. (15) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic: London, 1982; p 90.
S0743-7463(95)00877-8 CCC: $12.00
interface. In this case, the classical FHH equation on a flat surface reads
-ln X ∝ R/zsp
(1)
where R depends on the solid-adsorbate interaction and X ) p/p0 is the relative pressure with the equilibrium pressure p on the sample and p0 the saturation pressure of nitrogen at 77 K. If one introduces the amount N adsorbed at the relative pressure p/p0, the eq 1 can be written in the following form:
N ∝ [-ln X]-1/s
(2)
Equation 2 is the well-known classical FHH isotherm where the exponent s may, according to Halsey,16 be taken as a rough guide to the strength of the adsorbate-solid interaction. In other words, s describes how fast the solidadsorbate interactions decrease with increasing distance from the solid surface. The value of s is determined by plotting the experimental isotherm (2). For example, Zettlemoyer et al.17 found values of s ) 2.2 to 2.7 for the adsorption of nitrogen on silica depending on its degree of hydroxylation. It is interesting to note that these values of s were obtained for nonporous solids. Fractal Version of the FHH Theory To generalize the classical FHH theory to fractal materials, Pfeifer et al.4-7 considered the amount adsorbed on a fractal surface. In that case, the volume of an adsorbed film is equal to the number n(z) of balls of radius z needed to cover the surface with a monolayer, multiplied by the ball volume (≈z3). The fractal dimension can be defined by assuming that n(z) is proportional to z-D. Then, the amount adsorbed as a function of the film thickness z on a fractal surface is given by
N ∝ z3-D
(3)
According to Pfeifer et al.,5 the film thickness z can be taken with a good approximation equal to the reference film thickness zp (eq 1). After minimization of the Helmholtz free energy, the equilibrium interface obeys to the equation
sub. potential (≈1/zs) + surf. tension (≈1/z0) ) chem. potential (≈-ln X) (4) In this equation, the approximation made by Pfeifer et al.5 for the mean radius of curvature of the interface of the condensed fluid (second term on the left-hand side) can be used. Two situations are then considered: (1) the polymolecular adsorption of FHH regime in which the dominating force is the substrate potential (first term in eq 4) and (2) the capillary condensation regime in which the dominating force is the surface tension (second term in eq 4). The crossover between the two regimes is given by a critical thickness zc which depends4 on the surface tension, σ, the solid-adsorbate interactions R, and the fractal dimension D. In the FHH regime, the film thickness is approached by (eq 4)
z ∝ [-ln X]-1/s
(5)
Combining eqs 5 and 3 with s equal to the theoretical value 3 (nonretarded van der Waals forces), Pfeifer et al. (16) Halsey, G. D. J. Chem. Phys. 1948, 16, 931. (17) Zettlemoyer, A. C. J. Colloid Interface Sci. 1968, 28, 343.
© 1996 American Chemical Society
Notes
Langmuir, Vol. 12, No. 11, 1996 2873
obtained the fractal FHH equation
N ∝ [-ln X]-1/m
(6)
where
m)
(s ) 3) 3-D
(7)
We will come back on the application of this equation at the end of the paper. In the CC regime, the mean radius curvature of the interface is approached by (eq 4)
z0 ∝ [-ln X]-1
(8)
Using eq 8, after insertion into eq 3, one obtains the practical expression of the capillary condensation prediction on a fractal surface (in log-log form):
ln(N) ) constant + (D - 3) ln(-ln X)
(9)
Using this equation one can determine D from adsorption measurements. It is worth noting that eq 8 has the same relative pressure dependence as the Kelvin equation which is defined below (eq 12). In the intermediate regime the multilayer formation is driven by the competion between the two mechanisms (FHH and CC). If the critical thickness zc is not large enough, the FHH regime can be masked by the crossover to the CC regime.4,18 In that case, the FHH exponent m is smaller than 3 and eq 9 must be used. Thermodynamic Method The thermodynamic method proposed and developed by Neimark10-13 is presented as an independent method. In this model, the fractal surface area19
S(r) ∝ r2-D
(10)
of the condensed adsorbate-gas interface is calculated, at a given pressure, from the Kiselev equation20
S)
∫NN (-ln X) dN
RT σ
h
l
(12)
R, T, Vmol, and σ are the gas constant, the temperature, the molar liquid nitrogen volume, and the surface tension, respectively. It must be pointed out that the Kiselev integral (eq 11) was initially proposed as an alternative method for the determination of the surface areas of porous (especially mesoporous) solids. The upper limit Nh is the coverage adsorption when the relative pressure X tends to 1 while the lower limit Nl corresponds usually to the beginning of the hysteresis loop. Taking eqs 10, 11, and 12 into account, the fractal surface dimension D is derived from the slope of the linear part of the following equation:
ln(S) ) constant + (D - 2) ln(-ln X)
Table 1. Fractal Dimension (D) of Commercial Carbon Blacks fractal dimensions D
(11)
The yardstick r is measured in terms of the average radius of curvature of the meniscus at the interface between condensed adsorbate and gas by the Kelvin equation
2σVmol r)RT ln X
Figure 1. FHH plot and Neimark plot (double-Y) of (a) N115 and (b) N774. The solid lines indicate the linear regression.
(13)
In this method, the only adsorption process taken into account is the capillary condensation phenomenon which can be described by the Kelvin equation. (18) Ismail, I. M. K.; Pfeifer, P. Langmuir 1994, 10, 1532. (19) Mandelbrot, B. Fractal Geometry of Nature; Freeman: San Francisco, CA, 1982. (20) Reference 14, p 171.
BET (m2/g)
p/p0 (FHH)
FHH eq 9
Neimark (eq 13)
p/p0 (Neimark)
N 115
145
0.17-0.91
2.59
N 375 N 539 N 660 N 774
90 43 36 29
0.2-0.92 0.38-0.94 0.4-0.96 0.54-0.98
2.55 2.56 2.58 2.57
2.41 2.43a 2.38 2.58 2.55 2.59
0.35-0.94 0.17-0.91a 0.3-0.93 0.38-0.94 0.4-0.96 0.54-0.98
samples
a
This value is obtained with a correlation coefficient R ) 0.997.
Results and Discussion Obviously, the FHH-like equation for the CC regime (eq 9) and the thermodynamic model (eq 13) are based on two similar relations (8) and (12). In fact, both methods are derived by using the Kelvin equation in a fractal context: (i) Neimark uses the fractal definition of the surface area of the adsorbed film while (ii) in the FHHlike equation one introduces the fractal definition of the film volume or of the pore-size distribution. As a consequence, the use of the thermodynamic method and of the fractal FHH equation with exponent (3 - D) will lead to the same results as this has been recently shown theoretically by Jaroniec.13 We have re-examined our experimental results on five commercial rubber grade carbon blacks: N115, N375, N539, N660, and N774 in the light of the Jaroniec13 conclusion. The experimental details have been given elsewhere.1 The determination of the fractal dimension D using eqs 9 and 13 is illustrated in Figure 1 for two
2874
Langmuir, Vol. 12, No. 11, 1996
representative samples. The obtained results are reported in Table 1. This table also includes the BET surface areas of the samples and the relative pressure range over which the fractal dimension has been calculated. From these results one can conclude that in the three last cases (low-surface area grades) when the microporosity is negligible and the main adsorption process is capillary condensation, there is a total equivalence in the range of experimental errors. For the two high surface area grades (N115 and N375) the results are obviously different. This can be understood as follows: (a) With large BET surface area, the microporosity cannot be neglected and the adsorption process is a mixture of polymolecular and capillary condensation. (b) In contrast to Neimark’s method, the FHH type equation is also sensitive to the microporous structures3,21 and this most probably explains the disagreement in the case of these two samples. Finally, we want to make a last remark for the case when the dominating force is the substrate potential. (21) Jaroniec, M.; Liu, X.; Madey, R.; Avnir, D. J. Chem. Phys. 1990, 92, 7589.
Notes
Sometimes, eqs 6 and 7 yield a value of D smaller than 2 that is physically meaningless for a surface dimension. In addition to the above arguments (crossover to CC regime), this can be linked to the choice of s ) 3 in eq 7, a value which is almost never observed even for smooth surfaces.15 Conclusion We have shown how experimental arguments lead to the conclusion that the equivalence between a FHH type equation and the thermodynamic method holds only in the case where the adsorption process is dominated by capillary condensation forces and that it depends on the nature of the porosity. Acknowledgment. The authors thank the Communaute´ Franc¸ aise de Belgique, the Region Wallonne, the Administration Ge´ne´rale de la Coope´ration au De´veloppement (AGCD, Brussels, Belgium) and they gratefully acknowledge Professor C. Roy and collaborators for providing the commercial carbon blacks samples. LA950877P
