Langmuir 1997, 13, 1723-1728
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Diffusion Coefficients Calculated for Microporous Solids from Structural Parameters Evaluated by Fractal Geometry E. Schieferstein and P. Heinrich* Theoretical and Physical Chemistry, University Essen GHS, University Strasse 5, D-45117 Essen, Germany Received September 13, 1996. In Final Form: December 30, 1996X This paper presents the examination of 14 microporous active carbons. The Dubinin-Astakhov equation for adsorption processes in micropores modified by a function F(z) for gamma distributed pore sizes was used for obtaining the micropore size distribution based on experimental data of benzene adsorption isotherms. This distribution can be used to determine the fractal cut-off values xmin and xmax. It is shown by means of fractal geometry how to get the structural parameters porosity and tortuosity τ. Both values are needed for the further calculation of diffusion coefficients according to the Knudsen diffusion equation. A comparison with experimental data shows a very good agreement between calculated and measured values.
I. Introduction Many modern chemical production processes include the use of catalysts or adsorbents. These auxiliaries mostly consist of porous solids with large internal surfaces. Microporous solids fit especially well, because they offer the greatest ratio of surface to volume.1 Inside the pores the main parts of the reactions take place, and for these, adsorption processes are essential. The critical steps, namely, entering and leaving the pore system, are controlled by diffusion. Consequently a better understanding of the diffusion in microporous solids is of great interest. Especially the calculation of diffusion coefficients from structural data of individual porous solids was not possible up to now. The main reason for that is the lack of a realistic description of porous systems. In this paper it is shown by means of fractal geometry how to determine the structural parameters and τ that are essentially important for calculation of the diffusion coefficients. II. Characterization of Active Carbons Active carbons have a very heterogeneous structure.2,3 Micropores are responsible for the large internal surface, while macro- and mesopores enable diffusion into micropores. Electron microscopy experiments have shown that micropores in active carbons can be approximated as slits with half width x.4 Since every active carbon has pores of different sizes, it is necessary for its characterization to determine the distribution of the pore sizes. The latter is obtained from adsorption data. II.1. Interpretation of Adsorption Isotherms. We have analyzed the data of benzene adsorption on 14 different active carbons.5-7 Generally these active carbons can be classified as type I according to the BrunnauerDeming-Deming-Teller (BDDT) classification,8 where the adsorption isotherm of type I is shown by purely microporous active carbons (Figure 1). X
Figure 1. Adsorption isotherms (293 K) of the examined active carbons.5-7
In micropores the so-called “total volume filling” takes place. This adsorption process is well described by the Dubinin-Astakhov (DA) equation9,10
Abstract published in Advance ACS Abstracts, March 1, 1997.
(1) Bremer, H.; Wendlandt, K.-P. Heterogene Katalyse; Akademie Verlag: Berlin, 1978; Chapter 2.1. (2) Stoeckli, H. F. Carbon 1990, 28, 1. (3) Ju¨ntgen, H. Carbon 1977, 15, 273. (4) Dubinin, M. M. Carbon 1989, 27, 457. (5) Dubinin, M. M. Carbon 1987, 25, 593. (6) Heuchel, M.; Jaroniec, M. Private communication. (7) Vartapetian, R. Sh. Private communication. (8) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: New York, 1967; p 7.
S0743-7463(96)00889-X CCC: $14.00
Θ(A) )
[( ) ]
a A ) exp as βE0DA
nDA
(1)
with θ(A) being the degree of micropore filling, a the (9) Innes, R. W.; Fryer, J. R.; Stoeckli, H. F. Carbon 1989, 27, 71. (10) Stoeckli, H. F.; Kraehenbuehl, F.; Ballerini, L.; De Bernadini, S. Carbon 1989, 27, 125.
© 1997 American Chemical Society
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Schieferstein and Heinrich
adsorbed amount at pressure p, as the saturation value for the adsorption in the micropores, A the adsorption potential ()differential molar work of adsorption5) [kJ/ mol], β the affinity coefficient, E0DA the adsorption energy of the reference adsorptive [kJ/mol], and nDA the homogeneity parameter. The parameter nDA rises with increasing homogeneity of the micropore system.11 Normally its value lies between 1 and 3 for active carbons.10 Experimental studies have shown that with nDA ) 3 the DA equation describes the adsorption in homogeneous microporous active carbons10,12 very well, but the technical important active carbons are heterogeneous. They have a broad micropore size distribution that must be taken into account. In micropores the adsorption energy increases with decreasing pore radius.13 Many equations exist for connecting the adsorption energy E0 and the half pore width x. Experimental studies lead to eq 2 evaluated by Stoeckli et al.,14 and it seems to be the most accurate relationship.15 It was used in the further considerations. 3
x ) f(z) ) 2852.5z + 15z + 0.014z
-1
- 0.75
with
z ) 1/E0 (mol/kJ)
(2)
Therefore, the micropore size distribution can be described by a distribution of adsorption energies. For that reason the DA equation is modified by a gamma distribution F(z) of the adsorption energies15-17
Θ(A) )
a ) as
∫F(z) exp[-(Az β)
nDA
] dz
(3)
with
F(z) )
nFν/n ν-1 z exp[-Fzn] ν Γ n
()
(4)
Figure 2. Isotherm data of examined active carbons (RIAA, RIB, RIC) fitted with eq 5. Table 1. Parameters ν, G, and n of Equation 5 and the Adsorption Energy E0DA of the DA Equation active carbon
n
F (kJ/mol)n
ν
E0DA (kJ/mol)
RIAA RIB RIC CAL GW F-200 ACT-K ACS ACZ AC-7 AC-10 AC-16 AC-66 FAC-3
3.59 3.47 3.47 3.25 2.80 3.02 4.05 2.88 2.20 2.44 2.98 2.32 3.04 2.16
5237.3 9848.1 14357.0 4641.9 3369.7 5574.4 1050.6 15217.0 168.1 4586.8 7234.0 3742.2 5679.2 441.0
2.11 2.28 2.35 2.36 2.79 2.65 1.78 2.63 1.71 4.43 3.58 4.12 3.85 3.56
16.3 20.0 22.2 18.1 21.6 21.7 8.7 31.1 14.0 27.5 21.0 29.9 17.9 15.1
heterogeneous active carbons is transformed into17
Θ(A) ) 18,19
ν, F, and n are parameters of the gamma distribution. F is the so-called heterogeneity parameter. With increasing value of F the distribution function F(z) becomes narrower thus indicating a more homogeneous structure of the solid. The parameters ν and n describe shape and width of the distribution function. Their influence is more complicated and shall not be discussed further. Choosing a gamma-distribution for describing the pore size distribution some simpler distribution functions are included,17,20 e.g., exponential, Weibull, Maxwell, and Rayleigh distribution functions. Since the DA equation may be regarded as a Weibull distribution8 close similarities are obtained for the n values of the gamma distribution and the nDA values. Providing that nDA and n are comparable, eq 3 for the benzene adsorption (β ) 1) on (11) Stoeckli, H. F. Carbon 1981, 19, 325. (12) Jaroniec, M.; Lu, X.; Madey, R.; Choma, J. Carbon 1989, 28, 243. (13) McEnaney, B. Carbon 1988, 26, 267. (14) Stoeckli, H. F.; Ballerini, L.; De Bernardini, S. Carbon 1989, 27, 501. (15) Jaroniec, M.; Gilpin, R. K.; Choma, J. Carbon 1993, 31, 325. (16) Choma, J.; Burakiewicz-Mortka, W.; Jaroniec, M.; Gilpin, R. K. Langmuir 1993, 9, 2555. (17) Jaroniec, M. Langmuir 1987, 3, 795. (18) Jaroniec, M.; Madey, R. J. Phys. Chem. 1989, 93, 5225. (19) Jaroniec, M.; Piotrowska J. Monatsh. Chem. 1986, 117, 7. (20) Mu¨ller, P. H. Wahrscheinlichkeitsrechnung und mathematische Statistik Lexikon der Stochastik; Akademie Verlag: Berlin, 1975; p 84. (21) Jaroniec, M.; Madey, R.; Choma, J. Mater. Chem. Phys. 1990, 26, 87. (22) Pfeifer, P.; Avnir, D. J. Chem. Phys. 1983, 79, 3558.
[
]
a An ) 1+ as F
-ν/n
(5)
This equation was fitted to the experimental isotherm data. The resulting values for ν, F, and n are summarized in Table 1. In the literature the DA equation is often used instead of eq 5. For comparison the adsorption energy E0DA is listed in Table 1 too. The accuracy of the fit for the active carbons RIAA, RIB, and RIC is shown in Figure 2. The distribution functions F(z) are calculated by inserting the obtained values of ν, F, and n into eq 4. Figure 3 shows the results for the examined active carbons. II.2. Evaluation of the Fractal Parameters. If F(z) is known, the pore size distribution j(x) can be calculated according to the following equation21
j(x) ) F(z) (dx/dz)-1
(6)
Figure 4 shows the pore size distributions j(x) of the tested active carbons. In Figure 4a RIC exhibits the highest amount of small micropores within the group of RIAA, RIB, and RIC. With increasing pore size the amount of pores falls steeply to zero. Therefore, RIC is the most homogeneous active carbon in this figure. RIAA turns out to be the most heterogeneous one showing a very broad range of different pore sizes. The active carbons F-200 and GW are very similar (Figure 4a). CAL displays less small pores than F-200 and GW.
Calculated Diffusion Coefficients for Microporous Solids
Langmuir, Vol. 13, No. 6, 1997 1725
Figure 3. Distribution functions F(z) of the examined active carbons.
The broadest range of micropore sizes is shown by ACTK; see Figure 4b. ACZ and FAC-3 have a higher amount of small micropores than ACT-K. In Figure 4c the homogeneity grows in the following sequence: AC-66, AC-10, AC-7, AC-16, and ACS. The last three carbons ACS, AC-16, and AC-7 have a very high amount of small pores. The curves steeply tend to zero with increasing radii of micropores. ACS and AC-16 have nearly equally distributed pore sizes. The pore size distribution j(x) is connected with the fractal dimension via22
j(x) ∝ x2-df
(7)
The double logarithmic plot of the half pore width x vs the pore size distribution j(x) is expected to be a straight line (slope m ) 2 - df). Figure 5 shows such a plot for the active carbons RIAA, RIB, and RIC as an example. Clearly one can see the linear regions in the range of xmin < x < xmax. These values are taken as the limits of the fractal region (cut-off values).23 We have determined each cut-off value explicitly out of the double logarithmic plot in following way: The equation jL(x) of the linear part in the double logarithmic plot was determined. Then the cut-off values were taken at the points where the deviation between the new calculated values jL(x) and the micropore size distribution j(x) exceeds (0.02. Figure 6 shows an example. From the double logarithmic plots of x and j(x) we received the fractal parameters df, xmin, and xmax (see Table 2). As expected the fractal dimension df commonly lies between 2 and 3. Only for AC-7, AC-16, and ACS were (23) Avnir, D. Fractal Approach to Heterogeneous Chemistry; J. Wiley & Sons: New York, 1990; p 16.
the df values greater than 3. In the literature df values beyond 3 are found for active carbons with submicropores.24 Having a look at the very high adsorption energy values E0DA of about 30 kJ/mol of the mentioned active carbons (see Table 1) and thinking of the indirect proportionality between the adsorption energy E0 and the pore width x, one can conjecture that these active carbons have submicropores too. From the limits of the fractal region it becomes obvious that xmin is nearly constant. It corresponds to the size of the adsorbed benzene molecules. The upper cut-off value xmax varies more. This will be important for later calculations. All listed data depend on the choice of the saturation values of the adsorption as also given in Table 2. These as values were chosen at the beginning of the plateau of the isotherm.25,26 With a change of the as values for one active carbon, the range of the calculated df values varies only (0.05. Larger deviations occur if as values lying on the rapidly rising part at the beginning of the isotherm are taken. Analogous xmin values vary in a small range of (0.03 nm. The deviation in xmax usually is larger, about (0.1 nm. The parameters of the distribution function F(z) vary too when the as values are altered, but the maximum value zmax usually is the same for all these distributions. This is shown in Figure 7 for the active carbon RIAA. Therefore, (24) Hall, P. G.; Mu¨ller, S. A.; Williams, R. T. Carbon 1989, 27, 103. (25) Carrott, P. J. M.; Sing, K. S. W. In Characterisation of Porous Solids; Unger, K. K., et al., Eds.; Elsevier Science Publishers: Amsterdam 1988, p 77. (26) Kakei, K.; Ozeki, S.; Suzuki, T.; Kaneko, K. J. Chem. Soc., Faraday Trans. 1990, 86, 371. (27) Dubinin, M. M.; Vartapetian, R. Sh.; Voloshchuk, A. M.; Ka¨rger, J.; Pfeifer, H. Carbon 1988, 26, 515. (28) Vartapetian, R. Sh.; Voloshchuk, A. M.; Gur’yanov, V. V.; Ka¨rger, J.; Pfeifer, H. Russ. Chem. Bull. 1993, 42, 48.
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Figure 5. Double logarithmic plot of the micropore size distribution j(x) vs the half pore width x for the active carbons RIAA, RIB, and RIC.
Figure 6. Comparison of j(x) to jL(x) for the active carbon RIAA.
Figure 4. Pore size distributions j(x) of the examined active carbons.
zmax can be taken as a characteristic value of the specific active carbon. It is defined as18
zmax )
[ν nF- 1]
1/n
(8)
Taking z ) 1/E0, one can calculate E0z values (see Table 2). These values match the E0DA values within (5% (see Tables 1 and 2). II.3. The Mean Half-Pore-Width x j . For further calculations mean xj values are necessary. They should be determined by eq 915
xj )
∫x
xmax min
x j(x) dx
(9)
( )[( ) 3 - df 4 - df
xmax xmin
4-df
][( )
-1
xmax xmin
3-df
]
of the mean value of the gamma distribution function zj, given by18
zj )
The following approximate equation15 was used:
xj ) xmin
Figure 7. F(z) of RIAA calculated for different as values.
Γ((ν + 1)/n) -1/n F Γ(ν/n)
(11)
-1
-1
(10)
Another possibility for the determination of xj is the use
from which together with eq 2 xjz is calculated. Some calculated xj data are listed in Table 3 together with xjexp values from the literature.27,28 It must be stressed here that the experimental data xjexp show deviations too,
Calculated Diffusion Coefficients for Microporous Solids Table 2. Fractal Parameters df, xmin, and xmax Calculated by Equation 7, the Energy E0z Calculated by Equation 8, and the Saturation Values of Adsorption as active carbon
as (mmol/g)
xmin (nm)
xmax (nm)
df
E0z (kJ/mol)
RIAA RIB RIC CAL GW F-200 ACT-K ACS ACZ AC-7 AC-10 AC-16 AC-66 FAC-3
6.93 5.23 4.42 4.28 3.96 3.78 14.75 4.69 7.20 3.85 3.15 2.78 8.60 9.27
0.33 0.30 0.30 0.31 0.28 0.28 0.40 0.26 0.36 0.24 0.26 0.24 0.24 0.31
1.48 1.04 0.92 1.09 0.88 0.88 4.37 0.58 1.83 0.71 0.78 0.70 0.93 0.92
2.50 2.67 2.80 2.57 2.84 2.83 2.40 4.18 2.68 3.88 2.69 4.34 2.24 2.24
15.1 18.9 20.7 17.6 21.3 21.3 8.4 30.5 17.1 27.5 20.7 30.5 17.5 15.5
Table 3. Experimental and Calculated x j Values active carbon
xj (nm)
xjexp (nm)
xz (nm)
AC-7 (ref 27) AC-10 (ref 27) AC-16 (ref 27) AC-66 (ref 27) FAC-3 (ref 28)
0.41 0.49 0.38 0.57 0.60
0.49 0.66 0.36 0.67 0.87/0.72
0.34 0.61 0.29 0.90 1.47
but no estimation of the error is found in the literature. Nevertheless, the qualitative agreement between the values discussed is good. Only for the active carbon FAC-3 xjz deviates stronger from xjexp and xj. A comparison of the obtained micropore size distributions and xj values with results of other theoretical or experimental investigations12,27-29 shows an excellent agreement. Therefore, a description of active carbons with fractal geometry offers a detailed and realistic picture of the pore structure. Since the porosity and the tortuosity τ are purely structural factors, it should now be possible to calculate them by means of the fractal parameters. III. Calculation of Porosity, Tortuosity, and Diffusion Coefficients The Knudsen diffusion coefficient Dkeff in micropores is defined as30,31
Dkeff )
1 d vj 3τ p
(12)
with the porosity, τ the tortuosity, dp the pore diameter (m), and vj the mean velocity of the considered gas (m/s). If the porosity and tortuosity τ are known, the diffusion coefficients can be deduced. In fractal theory32 the porosity is given by
)
[ ] xmin xmax
3-df
(13)
while the tortuosity τ is defined from eq 12 as30,31
τ)
vj dF 3Dkeff
(14)
Taking the pore diameter dp ) 2xj and eq 13 it follows (29) Heuchel, M.; Jaroniec, M. Langmuir 1995, 11, 1297. (30) Kast, W. Adsorption aus der Gasphase; Verlag Chemie: Weinheim, 1988. (31) Satterfield, C. N. In Heterogeneous Catal. Practice 1980, 170. (32) Katz, A. J.; Thompson, A. H. Phys. Rev. Lett. 1985, 54, 1325.
Langmuir, Vol. 13, No. 6, 1997 1727
τ)
[ ]
2xjvj xmin 3Dkeff xmax
3-df
(15)
The tortuosity τ usually is greater than 1 and describes the decrease of the velocity of the overall diffusion process in a porous system because of structural influences.33 Considering this we made following assumption:
τ)
[ ] xmax xmin
k
(16)
To estimate k, it is necessary to have diffusion systems with known structural parameters and diffusion-coefficients Dkeff. This applies to the active carbons: AC-7, AC-10, AC-16, AC-66, FAC-3. From their adsorption data27,28 we obtained the following result:
τ)
[ ] xmax xmin
1.5df+1.5
(17)
For every active carbon a characteristic structural factor could be defined as
[ ]
xmin ) τ xmax
0.5df+4.5
(18)
leading to an effective Knudsen diffusion coefficient of microporous solids:
Dkeff )
[ ]
xmin 2 xjvj 3 xmax
0.5df+4.5
(19)
The calculated diffusion coefficients Dkeff are listed in Table 4 together with the measured values Dkm.27,28 Moreover diffusion coefficients are reported which were calculated with other values of the pore width xj. The Dkexp values are based on the experimentally determined pore width xjexp (Table 3) while the Dkz values were calculated by means of xjz of Table 3. With regard to the errors in the determination of the fractal dimension df, the cut-off values xmin and xmax and the mean half pore width xj, the error in the calculation of the diffusion coefficient Dkeff is estimated to (10%. The experimentally determined diffusion coefficients show a strong dependence on the degree of micropore filling, and the values given in Table 4 are taken at the saturation of the micropores (if mentioned in the literature). A deviation of (1 × 10-10 m2/s can be taken as error in the determination of the experimental data. Within this range experimental and calculated values are in good agreement. A theoretical consideration can be done to confirm the validity of eq 19: The experiments14,34 show a lower portion of micropores and a larger amount of meso- and macropores with increasing activation of an active carbon. According to these findings the fractal dimension of the active carbon concerning its pore size distribution decreases. The pore diameter usually becomes greater, and therefore the diffusion coefficients should increase too. Under the condition that xmin and xmax are nearly constant, xj and the characteristic structural factor (/τ) are enlarged with decreasing df value (according eq 10) and thus Dkeff rises for higher activated carbons (see eq (33) Ka¨rger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; J. Wiley & Sons: New York, 1992; Chapter 11.1, p 342. (34) McEnaney, B. Carbon 1987, 25, 69.
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Table 4. Measured and Calculated Diffusion Coefficients for Different Compounds on Active Carbonsa active carbon AC-7 (ref 27) AC-10 (ref 27) AC-16 (ref 27) AC-66 (ref 27) FAC-3 (ref 28)
diffusing compound
Dkeff (m2/s)
Dkm (m2/s)
Dkexp (m2/s)
Dkz (m2/s)
H2O C6H6 CH3OH H2O C6H6 CH3OH H2O H2O CH3OH H2O C6H6 C6F6 C2Cl3F3
1.5 × 0.7 × 10-10 1.1 × 10-10 3.1 × 10-10 1.5 × 10-10 2.4 × 10-10 1.2 × 10-10 1.1 × 10-10 0.8 × 10-10 5.3 × 10-10 2.5 × 10-10 1.6 × 10-10 1.6 × 10-10
1.6 × 0.3 × 10-10 0.6 × 10-10 3.0 × 10-10 0.8 × 10-10 1.6 × 10-10 1.0 × 10-10 2.5 × 10-10 3.0 × 10-10 8.0 × 10-10 3.5 × 10-10 1.4 × 10-10 1.8 × 10-10
1.8 × 0.9 × 10-10 1.3 × 10-10 4.2 × 10-10 2.0 × 10-10 3.0 × 10-10 1.1 × 10-10 1.3 × 10-10 1.0 × 10-10 7.6 × 10-10 3.7 × 10-10 2.4 × 10-10 2.4 × 10-10
1.3 × 10-10 0.6 × 10-10 0.9 × 10-10 3.9 × 10-10 1.9 × 10-10 2.9 × 10-10 0.9 × 10-10 1.8 × 10-10 1.3 × 10-10 12.8 × 10-10 6.2 × 10-10 4.0 × 10-10 4.0 × 10-10
10-10
10-10
10-10
a D m ) measured values, the other D values are calculated by eq 19 and different x j values. vj : H2O, 592.03 m/s; C6H6, 284.20 m/s; k k CH3OH, 443.73 m/s; C6F6, 184.15 m/s; C2Cl3F3 ) 183.50 m/s.
19). Unfortunately experimental data for a validation are not accessible. IV. Conclusion In this paper it is demonstrated that the calculation of diffusion coefficients in microporous solids from structural data is possible. As an example the structure of the microporous system of active carbons was described by means of fractal geometry. The basic assumption for such an application of this model is that the examined solid has slitlike micropores. For the determination of the structural parameters the benzene adsorption isotherms (293 K) of the active carbons5-7 were used. As a result the micropore size distribution and the mean half pore width xj of the active carbons were obtained. These results were in excellent agreement with those from other theoretical and experi-
mental investigations. Therefore, it is shown that the description of the pore system by means of fractal geometry is quite realistic. Under the assumption that the tortuosity is only a structural parameter and that the fractal model applies, an equation for a calculation of the tortuosity from structural data was developed. With this equation (19) the calculation of diffusion coefficients in microporous solids becomes possible. Acknowledgment. We thank R. Sh. Vartapetian for providing the isotherm data of FAC-3 and AC-7-AC-66 and the diffusion data and M. Heuchel and M. Jaroniec for the other isotherm data. We also thank J. Ka¨rger for helpful discussions. LA960889Q
