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Ind. Eng. Chem. Res. 1997, 36, 1598-1602
Determination of the Surface Fractal Dimension for Porous Media by Capillary Condensation Fumin Wang and Shaofen Li* Department of Chemical Engineering, Tianjin University, Tianjin 300072, People’s Republic of China
We describe several methods of evaluating the surface fractal dimension of porous media. These include the thermodynamic method and the fractal version of Frenkel-Halsey-Hill theory. Neither method yields accurate estimates of the fractal dimensions of porous solids under the whole range of experimental scales. We propose a modified thermodynamic method that is relatively simple but is significantly more accurate than Neimark’s relation from the adsorption experiments. Then we use these methods to estimate the surface fractal dimension of several kinds of porous media. After a concrete analysis of the properties of topology and mercury porosimetry, N2 adsorption, and N2 desorption processes for porous media, we conclude that the real surface fractal dimension should be determined by Dabs (from the adsorption isotherm), Ddes (from the desorption isotherm), and Dm (from the mercury porosimetry) jointly as Dreal ) Dabs + (Dm - Ddes). Introduction Fractal geometry has been widely used in many areas of modern science. 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. The fractal dimension should be determined first before we can use the concept and knowledge of fractal geometry to characterize the structure of a given solid. The experimental methods used for the determination of the surface fractal dimension of porous solids have been reviewed by Avnir et al. (1992). The most common techniques are the adsorption and mercury porosimetry methods. In addition, electron microscopy image analysis and scattering methods (light, X-rays, neutrons) have also been used to demonstrate surface roughness for porous materials (Martin et al., 1986; Aubert et al., 1986; Freltoft et al., 1986). In this paper we restrict our attention to calculating the surface fractal dimension from adsorption measurements because they are the most commonly used methods to determine the fractal dimension D of solid materials. Perhaps one of the oldest methods of evaluating the surface fractal dimension is that based on the dependence of monolayer capacity on the adsorbate size, which was developed by Pfeifer and Avnir (1983). Although this method is simple, the fractal dimensions determined by this method are not always consistent, especially when the orientations of adsorbate molecules on the surface are different. Also, 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 adsorbateadsorbate interactions (Jaroniec, 1995). Moreover, in this method, one needs to evaluate the monolayer capacities of several adsorbates of different molecular sizes, which makes the experiment rather time-consuming. Because the molecules adsorbed play the role of the gauges, the range of scales available in this method is limited by the molecular sizes. These problems become particularly important for adsorption on some microporous solids that possess a high degree of surface irregularity (Jaroniec and Madey, 1988). * Author to whom correspondence should be addressed. S0888-5885(96)00555-6 CCC: $14.00
Pfeifer et al. (1991) developed an adsorption-based method for surface roughness determination in 1991. The principle is to measure the variation in surface area of the material coated with a series of presorbed films. In this method, they assume that every point of the film surface has the same shortest distance z to the solid, so the film area S(z) is related to the film volume V(z) by
S(z) ) dV/dz
(1)
When the surface of the solid is fractal, with fractal dimension D, V(z) is proportional to z3-D and, therefore, S(z) is proportional to z2-D, so one obtains
S ∝ V(2-D)/(3-D)
(2)
From eq 2, one can get the fractal dimension by measuring the surface area of the adsorbed film with varied film thicknesses of z. However, the equidistance assumption, on which this method is based, is open to doubt. The surface tension wants to make the filmvapor interface as flat as possible, so as to minimize the surface area of the interface. One cannot ignore the effect of the surface tension, especially when the film thickness is large (Pfeifer and Cole, 1990). From the simulation results of this method, we can see that the deviations are very large and the corresponding correlation coefficient is very small (Pfeifer et al., 1991). Another popular method for evaluating D is that given by the Frenkel-Halsey-Hill (FHH) equation, which in logarithmic form can be expressed as follows (Pfeifer and Cole, 1990):
ln(N) ) const - (3 - D) ln(µ)
(3)
where N is the amount adsorbed at the relative pressure P/P0 and absolute temperature T and µ is the so-called adsorption potential defined as
µ ) RT ln(P0/P)
(4)
Using eq 3, one can determine D from physical adsorption measurements on the fractal surface. On the other hand, Neimark et al. (1992, 1994) proposed the so-called thermodynamic method for calculating the fractal dimension D from the adsorption isotherm data. In this model, the fractal surface area © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1599
was related to the average pore radius as (Neimark, 1992):
ln(S) ) const - (D - 2) ln(r)
(5)
and the surface area of the adsorbed film is calculated according to the Kiselev equation:
S(X) )
RT σ
N ∫N(X)
max
ln(X) dN(X)
(6) V(X) ) [Nmax - N(X)]VL
where X denotes the relative pressure and Nmax denotes the amount adsorbed at X tending to unity. The yardstick 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:
r)
-2σVL RT ln(X)
Theory Mandelbrot (1982) gave a correlation of the area for a fractal surface and the volume circumscribed by the surface:
(8)
In case the fractal surface is measured on a Euclidean area, the above relation can be changed into the subsequent form by dimensional analysis:
SE(δ) ) k0Dδ2-DVD/3
(9)
where k0 is a factor relating surface area with the corresponding volume. Just as illustrated by Neimark (1992), a given fractal surface can be approximated by its inscribed equicurvature surface (IES) of varying mean curvature radius (MCR). The decrease of the MCR, rc, causes penetration of the IES into surface indentations of smaller size. In other words, with a decrease of rc, the IES repeats all the peculiarities of the substratum relief. In the limit rc f 0, the IES is the fractal surface with the same dimension as the surface fractal dimension of the substratum. So, the mean curvature radius of the inscribed equicurvature surfaces can be chosen as a yardstick for measurement of the fractal surface. The
(10)
Substitution of SE and V(X) into eq 9 by eqs 6 and 10 gives the following expression: N ∫N(X)
max
-
ln(X) dN(X)
rc2(X)
(7)
The thermodynamic method proposed by Neimark (1992) and the method based on the FHH theory proposed by Pfeifer and Cole (1990) have been compared with one another by Jaroniec (1995). The theoretical analysis shows that both methods are essentially equivalent. However, the simulation results using Neimark’s relation show that the range of scale in which fractal exists is rather limited, which makes us suspect its validity and robustness. The objective of this work is to develop a more reliable scaling relation to determine surface fractal dimension of porous materials using the capillary condensation data. We also show that the new method, based on concrete analysis of the properties of topology and capillary condensation processes for porous media, is much more accurate than Neimark’s relation and covers a wide range of scales. Then the simulation results will be compared with those of the references.
S1/D ∼ V1/3
practical realization of this method can be provided by using the adsorption experiments. In the process of adsorption of nitrogen in porous media, the equilibrium interface between the liquid film and gas acts as the inscribed equicurvature surface. So, the surface area of the adsorbed film can be calculated according to the Kiselev equation (6). After assuming that the liquid cannot be compressed, one can get the following relation:
[
]
1/3 σ (Nmax - N(X)) ) k0DVLD/3 RT rc(X)
D
(11) Let N ∫N(X)
max
-
ln(X) dN(X)
rc2(X)
) A(X)
[Nmax - N(X)]1/3 rc(X)
) B(X) (12)
Equation 11 will be changed into the ensuring style
ln A(X) ) const + D ln B(X)
(13)
Accordingly, from the adsorption isotherm of nitrogen in the region of capillary condensation, we can compute a series of A(X) and B(X), where rc(X) can be predicted by the Kelvin equation (7) and then the surface fractal dimension can easily be achieved. Results and Discussion For a given porous solid, the desorption isotherm does not always retrace the adsorption isotherm but lies above it over a range pressures, forming a hysteresis loop, before eventually rejoining the adsorption isotherm. The fractal dimension determined by eq 13 can be calculated either from the adsorption isotherm or from the desorption isotherm. First, the simulation of surface fractal dimension for several samples has been made by eq 13 using the nitrogen adsorption and desorption data of the literature (Neimark et al., 1994), so that we can have a comparison with the simulation results by Neimark’s relation. The evaluation of the surface fractal dimension of SiO2(A) using eqs 5 and 13 is illustrated in Figures 1 and 2, respectively, as an example. The obtained results are reported in Table 1. This table also includes the range of scale from which the D was calculated. The above calculations and analyses show that the results simulated by means of eq 13 for several kinds of porous materials totally fall in the range of 2 < D < 3, which is predicted by fractal geometry, so this method is comparatively reasonable among the methods used to determine the surface fractal dimensions. From these results one can conclude that the method in this work is reliable over a sufficiently wide range of scales from 1 to 250 nm. Neimark’s relation, however, is seemingly reasonable only over a rather narrow range of scales. This is caused by the defects in the scaling relation (5). We know, from eq 9, that only when the
1600 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
Figure 1. Simulation results from eq 13 for SiO2 (A).
Figure 2. Simulation results from eq 5 for SiO2 (A).
volume encompassed by the fractal surface remains unchanged in the process can eq 3 be rewritten as
SE(δ) ∼ δ2-D from which eq 5 was obtained. But the capillary condensation process cannot satisfy this condition. With the increase of the relative pressure of nitrogen, the volume encompassed by the liquid-gas interface is decreased, so Neimark’s simulation results cannot characterize the real structure of the porous media. However, when the relative pressure of nitrogen X ) P/P0 is relatively small, the volume does not change remarkably, so Neimark’s relation shows seemingly reasonable results in this range. In the range of greater relative pressure of nitrogen, especially after the value
of X surpassed 0.8, the volume encompassed by the liquid-gas interface decreased remarkably with an increase of X, so the experimental data violated the simulated straight line. Neimark cannot find the defects in his scaling law and get to the wrong conclusion that the scaling interval for these porous solid is very narrow. So, it is necessary that the scaling relation developed to measure the surface fractal dimension using the capillary condensation data not only conform to the theory of fractal geometry itself but also consort with the concrete process of capillary condensation as well. Obviously, this idea has been considered when the scaling relation of this paper is being deduced. By contrast, the surface fractal dimensions determined by the adsorption data are smaller than those determined by the desorption data; this cannot be thought of as caused by the experimental probable error. The surface morphology is not the only factor that affects the desorption process; the topology of the porous space has a significant influence on this process. In the process of desorption, due to the “shielding” effect of the small pores on the large one, the calculating results of SE(X) are larger than those of SE in reality and thus make the discrepancy between Dabs and Ddes. The quantitative difference between the two indicates the degree of the influence of the shielding effect, so it reflects the topological feature of the porous solid in some degree. Jaroniec (1995) compared the fractal FHH equation (3) and Neimark’s relation (5) and concluded that the two methods are theoretically equivalent. So, the FHH equation meets the same difficulty when being used to evaluate the surface fractal dimension. Sahouli et al. (1996) reexamined the two methods and concluded that the FHH type equation is also sensitive to the microporous structures in contrast to Neimark’s relation, and this causes the disagreement of the results of the two methods for the solids with high surface area. In fact, the disagreement is due to the fact that the two methods were used in two different scale ranges. When being used in the same scale range, these two methods should yield the same results. The method in this work is based on the thermodynamic principle; another thermodynamic method to measure the surface fractal dimensions for the porous materials has been developed by Zhang and Li (1995). These two methods use different experimental data, with Zhang’s method using the mercury porosimetry data and the method in this work developed to use the capillary condensation data, but both are based on the same principle. In order to make a comparison between the two methods, we have done the experiments to get the mercury porosimetry and nitrogen adsorption isotherm of the same four porous materials. The experimental data of mercury porosimetry are measured by means of a J5-70 porosimeter made in Shanghai, People’s Republic of China. The range of pressure in
Table 1. Surface Fractal Dimensions Determined from Equations 13 and 5, Respectively, with Capillary Condensation Data in the Literature (Neimark et al., 1994) analyzed sample
Ddes
Dabs
porous glass Si300 MgO(A) cement MgO(B) SiO2(A) O2 SiO2(B)
2.52 2.48 2.41 2.68 2.51 2.53 2.67 2.55
2.35 2.34 2.33 2.63 2.33 2.41 2.65 2.50
eq 13 amin (nm) 1.0 1.5 1.3 1.8 1.2 1.5 1.1 1.1
amax (nm)
Ddes
Dabs
89.0 53.7 89.0 25.1 251.0 125.0 79.4 39.8
2.10 2.20 2.36 2.43 2.54 2.47 2.76 28.5
2.14 2.22 2.43 2.43 2.55 2.59 2.71 2.88
eq 5 amin (nm) 1.0 1.5 1.4 1.8 2.7 1.5 1.1 2.0
amax (nm) 16.0 10.0 22.4 7.2 13.2 12.5 5.5 8.0
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1601 Table 2. Surface Fractal Dimension Determined from Equation 13 and Zhang’s Relation analyzed sample
Dabs
γ-Al2O3(A) B106 γ-Al2O3(B) γ-Al2O3(C)
2.75 2.72 2.68 2.66
Ddes
2.75
eq 13 amin (nm) 1.0 1.0 1.0 1.0
the experiment is from 0.1 to 300 MPa. The experimental data of the nitrogen adsorption isotherm are measured by means of a CHEMBET-3000 adsorption apparatus made by Quantachrome Co., Syosset, NY. The simulation results are demonstrated in Table 2. The simulation results show that the surface fractal dimensions determined by mercury porosimetry, Dm, are larger than both Dabs and Ddes. In the process of adsorption, before capillary condensation occurs, there has been a nitrogen film adsorbed on the surface of the porous solid. As a result, the interface between the film and the vapor, with which the film is in equilibrium, is no longer a simple replica of the film-solid interface. Due to the effect of the surface tension, the Dabs is smaller than the real dimension of the fractal surface. So, the discrepancy between Dabs and Dreal is determined by the competition between the attractive van der Waals gas-solid potential and the repulsive surface free energy of the nitrogen film. The potential wants to make the film-vapor interface follow the ups and downs of the surface as closely as possible, so that the adsorbed molecules get to set as close to the solid surface as possible. The surface tension, on the other hand, wants to make the film-vapor interface as flat as possible, so as to minimize the surface area of the interface; the two effects entangled together make the discrepancy between Dabs and Dreal. These effects, also, act on the determination of Ddes. On the other hand, the shielding effect makes the surface fractal dimension Dm determined by mercury porosimetry larger than the real dimension of the fractal surface, so it is not surprising that Dm of γ-Al2O3(C) is greater than 3. So, the shielding effect of the small pores on the large one and the effect of the surface tension result in Dm > Ddes > Dabs, which is proved by this work. From the analysis above, we can see that the discrepancies between Dm and Dreal, Ddes and Dabs show the shielding effect of the small pores on the large one, while the discrepancies between Dreal and Dabs, Dm and Ddes show the effect of the surface tension. So, we can easily get the conclusion that the following relation exists:
Dreal ) Dabs + (Dm - Ddes) ) Dm - (Ddes - Dabs) The real surface fractal dimension of porous media can be determined by Dabs, Ddes, and Dm jointly. Conclusion A new method is proposed to determine the surface fractal dimension for a porous solid. The method is based on the approximation of a given fractal surface by inscribed equicurvature surfaces of varying mean curvature radius. The practical realization of this method for determining the surface fractal dimension of real porous materials is provided by analyzing experimental data of capillary condensation. The simulations for several kinds of porous media are fulfilled by analyzing the nitrogen adsorption isotherm and mercury porosimetry from the literature and our
amax (nm)
Dm
24.0 24.0 24.0 24.0
2.81 2.80 2.72 3.07
Zhang’s relation amin (nm) 2.0 2.0 2.0 2.0
amax (nm) 25.7 25.7 25.7 26.3
own. The results show that Dabs, Ddes, and Dm are different and Dm > Ddes > Dabs. By analyzing the process of adsorption, desorption of nitrogen, and mercury incursion, the author gives a reasonable explanation of the results and achieves the conclusion that Dreal ) Dabs + (Dm - Ddes) ) Dm - (Ddes - Dabs). So, the real surface fractal dimension cannot be determined by Dabs, Ddes, and Dm, respectively, but jointly.
Acknowledgment Support from funds of Science and Technology of National Education Committee of People’s Republic of China is gratefully acknowledged.
Nomenclature amin ) outer cutoff of a finite scaling regime amax ) inner cutoff of a finite scaling regime D ) fractal dimension of pore surface k0 ) shape factor N ) physisorption quantity, mol P ) current pressure of nitrogen, Pa P0 ) saturation pressure of nitrogen, Pa rc ) mean curvature radius, m SE ) fractal area of pore surface in Euclidean space, m2 T ) absolute temperature of the adsorption, K V ) volume of the space encompassed by the fractal surface, m3 VL ) molar volume of liquid nitrogen, m3/mol Greek Symbols δ ) yardstick size of measurements, m µ ) chemical potential energy, J/mol σ ) surface tension between liquid and gas of nitrogen, J/m
Literature Cited Aubert, C.; Cannel, D. S. Restructuring of Colloidal Silica Aggregates. Phys. Rev. Lett. 1986, 56, 738. Avnir, D.; et al. A Discussion of Some Aspects of Surface Fractality and Its Determination. New J. Chem. 1992, 16 (4), 439. Freltoft, T.; et al. Power-law Correlations and Finite-size Effects in Silica Particle Aggregates Studied by Small-angle Neutron Scattering. Phys. Rev. B 1986, 33 (1), 269. Jaroniec, M. Evaluation of the Fractal Dimension From a Single Adsorption Isotherm. Langmuir 1995, 11, 2316. Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, The Netherlands, 1988. Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: New York, 1982; p 109. Martin, J. E.; et al. Fractal Geometry of Vapor-phase Aggregates. Phys. Rev. A 1986, 33 (5), 3540.
1602 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 Neimark, A. V. A New Approach to the Determination of the Surface Fractal Dimension of Porous Solids. Physica A 1992, 191, 258. Neimark, A. V.; et al. Determination of the Fractal Dimension for Porous Solids from Adsorption Isotherm of Nitrogen. Z. Phys. Chem. 1994, 187, 265. Pfeifer, P. Structure analysis of Porous Solids From Presorbed Films. Langmuir 1991, 7, 2833. Pfeifer, P.; Avnir, D. Chemistry in Noninteger Dimensions Between Two and Three. J. Chem. Phys. 1983, 79 (7), 3558. Pfeifer, P.; Cole, M. W. Fractals in Surface Science: Scattering and Thermodynamics of Adsorbed Films II. New J. Chem. 1990, 14, 221.
Zhang, B.; Li, S. Determination of the Surface Fractal Dimension for Porous Media by Mercury Porosimetry. Ind. Eng. Chem. Res. 1995, 34, 1383.
Received for review September 9, 1996 Revised manuscript received January 27, 1997 Accepted February 3, 1997X IE960555W
X Abstract published in Advance ACS Abstracts, March 15, 1997.
