Ind. Eng. Chem. Res. 1995,34,1383-1386
1383
Determination of the Surface Fractal Dimension for Porous Media by Mercury Porosimetry Baoquan Zhang and Shaofen Li* Department of Chemical Engineering, Tianjin University, Tianjin 300072,People’s Republic of China
A scaling relation is presented after a concrete analysis of the properties of topology and mercury porosimetry processes for porous media. Then a new method and the corresponding procedure using the above scaling relation are set up in order to obtain the surface fractal dimensions of porous media from the experimental data of mercury intrusion. I n contrast with the existing methods of this kind exhibited in literature, the results simulated by means of the method here for several kinds of porous media conform to the prerequisite for being surface fractals, which means the surface fractal dimensions should be in the very range: 2 5 D < 3. At the same time, this method gives almost the same surface fractal dimensions as the adsorption method. The evidence deduced from the results and analyses of this paper shows that this method may be a promising one to decide or at least estimate the surface fractal dimension for porous media.
Introduction The fractal characterization of porous materials as an applicable and potential tool has been well documented (Avnir et al., 1992; Fairbridge et al., 1987). First, we have to determine the fractal dimension before the concept and knowledge of fractal geometry are applied to characterize a structure. Until now, a lot of researchers have published their results about the measurement of surface fractal dimension for porous media (Avnir et al., 1984, 1992; Friesen and Mikula, 1987; Neimark, 1990; Rothschild, 1991). By and large, there are two main experimental methods for determining the surface fractal dimensions of porous media-the adsorption method and mercury porosimetry method. In addition, small-angle scattering of X-rays or neutrons (SAXS, SANS) is also a practical way to demonstrate fractal roughness for porous materials. The theory, experimental results, and existent difficulties such as the peculiar surface behavior of porous silicas for the adsorption method have been thoroughly reviewed (Avnir et al., 1992; Drake et al., 1990; Pfeifer and Avnir, 1983). However, only a few articles have been devoted to the work that the surface fractal dimensions of porous media are reckoned by the method of mercury porosimetry. The procedure used in this respect is also immature and leaves much t o be desired. Two different scaling relationships have been proposed, respectively, so that the surface fractal dimensions of porous media can be obtained by mercury intrusion data. Friesen and Mikula (1987) proposed a relation in terms of the inherent relationship of an ideal fractal structure Menger Sponge: ln(dV/dP)
-
(D - 4)In P
(1)
So the surface fractal dimensions of porous media can be figured out in the light of the above equation by mercury intrusion data. On the other hand, Neimark (1990) gave the following equation by analogy with the relation used in the adsorption method determining the surface fractal dimensions: D=2-
However, in light of their scaling relations the surface fractal dimensions determined by mercury intrusion data are often different for different ranges of pressure, and some of them are more than 3, some even more than 5 . Even in the same regime of pressure, the deviations of simulation are rather large and the corresponding correlation coefficient is very small (Friesen and Mikula, 1987; Wan, 1993; Rothschild, 1991). According to the theory of fractal geometry, the fractal dimension of a surface must be in the range 2 ID < 3. D = 3 corresponds to volume-filling. So the values for D equal t o or more than 3 are nonphysical from a geometry point of view. Some of the results calculated by Friesen’s or Neimark’s relation are beyond this range. Thereupon, these methods themselves directly contradict the theory of fractal geometry, and there must be some defects in the scaling relations used in the methods. It is necessary to set up a scaling form which not only conforms to the theory of fractal geometry itself but also consorts with the concrete process of mercury intrusion porosimetry as well, so that the correct values of surface fractal dimension are able to be determined by means of mercury intrusion data. The aim of this contribution is just to develop a new scaling relation and the corresponding technique taking into account the preceding requirement and predict the surface fractal dimensions for several kinds of porous materials by means of this method, at length, compare the results of calculation with those of references.
Theory Thermodynamic Relation of Porous Media in the Process of Mercury Porosimetry. In the process of mercury porosimetry, with the increase of environmental pressure, the amount of mercury crushed into the pores of porous materials would have an increment so that the surface energy of the system is coincidentally elevated. As a hike of the surface energy in the system equals the work done by the surrounding, we can get the following equation for low-energy (nonwetting) surfaces (Rootare et al., 1967):
d[ln S(ri>l d Inki)
* To whom correspondence should be addressed.
If the pore surface of a porous material is fractally rough, the area S in eq 3 ought to be fractal. Now let
0888-5885/95/2634-1383$09.00/00 1995 American Chemical Society
1384 Ind. Eng. Chem. Res., Vol. 34, No. 4,1995
us measure the pore surface on Euclidean area, eq 3 can be rewritten as
P d V = -yL
COS
8 bsE
To characterize the structure of porous media, the above equation will be adjusted into the following equality to the pores of radius ri:
(4)
Integrating the above equation for the different stages of pressure in the process of mercury porosimetry, we can obtain
After assuming that 8 keeps constant in the whole process of experiment, the above equation can be W h e r transformed into n
Although the fractal area SEcovered by mercury in different pressure intervals cannot be measured directly in Euclidean space, it can be assessed in terms of eq 6. Inasmuch as the scale form itself is much more important than the absolute value of surface area for determining the surface fractal dimensions, the estimates from eq 6 might be anticipated accurate enough.
Scaling Relationship of the Fractal Surface Area and the Corresponding Volume Encompassed by the Surface in Porous Media. Mandelbrot (1982) gave a correlation of the area for a fractal
By comparing the relation concerning the length of a fractal curve, its shadow, and yardstick sizes, the shadow length of pore circumference is k,&r, (Zhang et al., 1994);therefore, the yardstick size of measuring the circumference of the pores with radius equal to or smaller than r i should be
(dJi = kr&ric
(13)
Only one substance, mercury, is used in the experiment of mercury porosimetry. If porous media are considered t o be isotropic, both k , and E can be regarded as constants, and there is enough evidence for us to choose the same yardstick size over all the directions for the measurement of surface area, that is
di = (d,)i
(14)
Equation 9 can be modified as
where
surface and the volume circumscribed by the surface: sl!D
~1.13
(7)
Combining the above equality with eq 6, we can attain the final scaling relationship:
In case the fractal surface is measured on a Euclidean area, the above relation can be changed into the subsequent form by dimensional analysis:
S E = kD 62 - D p l 3
(8)
where k is a factor relating surface area with the corresponding volume. Applying the former equation to calculate and analyze the experimental data of mercury intrusion, the relationship between the volume squeezed into the pores and the corresponding coverage on the pore surface can be elicited:
On the other hand, the perimeter of pore cross section for a porous material with surface fractality also have a fractal structure. If different yardstick sizes are used to measure the perimeter, the results would be Merent. Mandelbrot (1982) proposed a proportional relation connecting the circumference which is considered as a fractal with the cross-section area surrounded by it: ~1lD'. ~ 1 ! 2
= k,d(l-Dr)/D, A 112
(16)
where
c' = --K(D,+,
COS
e
The fractal dimension D of the internal surface for porous media, therefore, is able to be simulated in terms of eq 16 by mercury intrusion data.
Calculating Procedure and Results of Surface Fractal Dimension The surface fractal dimensions can be simulated in terms of eq 16. First, let
n
W, = ZFiAVi i=l
So eq 16 will be changed into the ensuing style: (10)
Providing the fractal circumference is measured on Euclidean length, the below equality would be created after dimensional analysis: p+l!Dr
~P,AV =, i=l
(11)
where k , is a factor relating the circumference and the area of cross section.
ln(Wn)= C
+ In(&,)
(17)
where C = In C'. Accordingly, we can compute a pair of Qn and Wn for a natural number n equal t o or greater than 1 using mercury intrusion data Vi Pi of a certain sample, where ri can be predicted by the Washburn equation (also known as the Laplace equation), after assuming a surface fractal dimension D between 2 and 3. A series of Qn and W, can be procured further for n = 1 , 2 , 3 , ....
-
Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 1385 Table 1. Surface Fractal Dimensions Determined from Ea 17 and Other Methods eq 17 sample
D
R
Y-AlzO3 2.86 0.982 silica gelA 2.90 0.994 silicagelB 2.91 0.979
adsorption method“ Neimark’s relationb alcohol 2.72 2.98 2.99
alkane 2.85 2.74 3.00
Di 3.58 2.72 4.07
D2 2.40 3.06 2.82
Table 2. Surface Fractal Dimensions Determined by Eq 17 and Friesen’s Relation for Mercury Intrusion Data in Literature (Friesen and Mikula, 1987) eq 17
D3
2.40 4.70 5.55
a Alcohol and alkane mean the measurements are performed using a series of alcohol and n-alkane molecules, respectively. D1 is measured from 0.1 to 15 MPa, DZfrom 15 to 100 MPa, and D3 from 100 to 250 MPa.
Using In(&,) as the abscissa and ln(Wn) as the ordinate, we can linearly simulate the series of In(&,) and MW,) and get a slope f. If fis approximately equal to 1,the assumed surface fractal dimension D satisfies eq 17, which shows that D is the surface fractal dimension of the sample. Otherwise, we have to assume a new D and do the above calculations again until f approximately equals 1. The surface fractal dimensions of y - A l 2 0 3 and two silica gel samples are simulated in the light of the above method, the results of simulation and the corresponding correlation coefficients are shown in Table 1. The two silica gel samples: silica gel A and silica gel B have different pore-size distributions, and their average pore radii are 30 and 50 A, respectively. The above experimental data of mercury porosimetry are measured by means of a 55-70 porosimeter made in Shanghai, P.R. China. The range of pressure in the experiment is from 0.1 to 300 MPa. The results gained by the adsorption method and Neimark’s relation (Wan, 1993) are also shown in Table 1 so as to have a clear comparison among the methods. Obviously, the results simulated in terms of eq 17 and those procured through the adsorption method are consistent with each other very well, and most of them are almost in the very range 2 ID -= 3. However, the method of Neimark does not have the same consequence as the former two methods, and its results, as mentioned in the Introduction, violate the prerequisite for being fractals and have the problem of division in different ranges of pressure. We also calculate the surface fractal dimensions by Friesen’s relation utilizing the preceding experimental data. However, the correlation coefficients in this case are so small that they cannot be regarded as straight lines. Furthermore, the simulations of surface fractal dimension for coal samples have also been made through eq 17 using the mercury intrusion data of literature so that we can have a comparison with the simulation results by Friesen’s relation exhibited in Friesen and Mikula (1987). The results are given in Table 2. The correlation coefficients for all the samples are also more than 0.99. To check the fitting degree of eq 17 and experimental data obviously, we can make an illustration shown in Figure 1 as an example. The straight line simulated fits in with the experimental data of silica gel B very well with a slope f = 1.001. The same result can also be procured for other samples except for the different ranges of quantities Qn and W,. This is further evidence that the methodology here is dependable. In the past decade a number of studies have been performed to decide the surface fractal dimension for a diversity of porous materials including alumina, silica gels, and coals through the adsorption method. The analysis of published data has led to the conclusion that the surfaces of most porous materials are fractals. Avnir et al. (1984) derived D % 2.80 for an alumina sample using polystyrene molecules and nitrogen as
Friesen’s relationb
sample
D
R
Di
Dz
D3
coal A coal A treated coal B coal B treated coal A old char(oxidized) coal A new char(oxidized) coal A old char coal A old char(uncrushed) coal A new char(uncrushed)
2.73 2.72 2.81 2.80 2.68 2.67 2.68 2.66 2.67
0.993 0.993 0.992 0.992 0.993 0.993 0.993 0.993 0.993
1.84 1.54 1.70 1.82 1.73 1.93 1.68 2.84 2.80
2.84 2.92 2.64 2.62 2.95 2.97 3.00 2.94 2.99
3.37 3.33 3.84 3.67 2.95 2.97 3.00 2.94 2.99
a D is measured from 1.0 to 200 MPa. D1 is measured from 0.0 to 0.1 MPa, DZfrom 1.0 to 10.0 MPa, and D3 from 10.0 to 200 MPa. (The size range of the crushed samples was 0.5 x 0.25 mm, while that of the uncrushed samples was approximately 5 x 10 mm.)
D=2.9 1
10
-$/,
,,,,,,,, ’
to-’
I
I , i l l , ,I
, , ,,,,,,,
1
, , ,,,,,,, , , LLLys-
10
10
Figure 1. Linear correlation between Q,,and W,, under D = 2.91 for silica gel B.
adsorbates. For the silica gels having approximately similar properties to the above silica gels A and B the values of D are around 3.0; however, the surface fractal dimensions of a number of coal particles vary from 2.20 to 2.60 (Avnir et al., 1992). Except for being a little bit higher for alumina and coal samples, the surface fractal dimensions obtained from eq 17 here are compatible with those of the counterparts compiled in the literature (Avnir et al., 1984, 1992).
Discussion The above calculations and analyses show that the results simulated by means of eq 17 are comparatively reasonable among the methods used to determine the surface fractal dimensions from the experimental data of mercury intrusion, and also the deviations of simulation are rather small. As a matter of fact, if one is going to decide the surface fractal dimensions of porous media utilizing mercury intrusion data, he has t o analyze the inherent characteristics of the experiment process so as to propose a suitable scaling relation which not only observes the theory of fractal geometry but also reflects the properties of the corresponding process of experiment. Obviously, this idea has been considered when the scaling relation of this paper is being deduced. Therefore, the results attained through simulating mercury intrusion data according to the method here are rational and in line with those of the adsorption method.
1386 Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995
Equation 1 was derived from the scaling relation between the inner surface area and the pore-size distribution of an ideal fractal model-Menger Sponge. Commonly, Menger Sponge has a different value of surface fractal dimension from a real porous medium; further, there is a large difference in pore-size distributions for them. The pore-size distribution is one descriptor of the pore structure. It can provide the roughness information of a surface at different resolutions statistially (Avnir et al., 1983;Friesen and Mikula, 1987). As implemented by Friesen and Mikula (1987), it is a straightforward way t o create a scaling relationship for determining surface fractality of porous structures through research into the pore-size distribution information. Hence, the method which simulates the surface fractal dimensions of real porous media making use of the pore information of Menger Sponge deserves doubting its correctness. The results attained from eq 1 listed in Table 2 actually have answered the doubt that preceding. On the other hand, relation 2 was proposed through carrying out an analogy with the corresponding relation of the adsorption method. The two processes of measurement for pore surface by the adsorption and mercury porosimetry method are very different undoubtedly. For the adsorption method the pore surface is measured in terms of varying the yardstick size-the size of molecules used in the experiment. If the molecules used are not too large, commonly, they can diffuse into any pores in the range of measurement. That is to say, the whole area of the pore surface to be considered can be measured after prescribing a yardstick size. But the process of mercury porosimetry is not the case. With the increase of pressure exerted by the environment, mercury forcedly goes into smaller pores and the measured area of the surface grows correspondingly for a porous medium. In different stages of mercury porosimetry, S(rJ calculated by eq 6 is only part of the surface area to be wanted. S ( r J is not the whole area of the measured surface until the process of mercury porosimetry ends. Thus, applying the same relation to the two processes having different principles and backgrounds is not reasonable, and the results, of course, cannot characterize the real structure of porous media. Finally, the precision of calculation needs to be noticed in utilizing the method here. If each variation of pressure in the process of mercury intrusion is decreased, the experimental points of the same integral interval of pressure are enhanced, so a higher precision of calculation can be reached in figuring out the surface area by eq 6 . Conclusions
A fitting scale relation was established through combining the feature of mercury intrusion process with the rationale of fractal geometry for porous media. Then a method to obtain the surface fractal dimensions of porous media by means of simulating mercury intrusion data was generated. The simulations for several kinds of porous media are implemented using the experimental data of mercury intrusion porosimetry from the literature and our own. The results represent that this method is better than the other methods of this kind, which are not only rational but approximately equal to the values of surface fractal dimension by the adsorption method as well.
Acknowledgment The authors are grateful for the financial support of the National Natural Science Foundation of P.R.China.
Nomenclature A = cross-sectional area of pores, m2 D = fractal dimension of pore surface D,= fractal dimension of the circumferenceof cross section G = circumference of the cross section in a pore, m P = pressure, Pa P = average pressure, Pa r = pore radius, m R = correlation coefficient SE= fractal area of pore surface in Euclidean space, m2 V = volume of the mercury crushed into porous media, m3 W = work done by the surrounding, J
Greek symbols
surface tension between mercury and pore surface, Jlm 6 = yardstick size of measurement, m 6 = scale 0 = contact angle between mercury and pore surface y~ =
Subscripts
i = ith interval of pressure in the process of mercury porosimetry n = pressure interval number
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Received for review J u n e 1, 1994 Revised manuscript received October 28, 1994 Accepted November 25, 1994@ IE940350S Abstract published in Advance A C S Abstracts, February 15, 1995. @
