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Fractal Analysis of a Sol–Gel-Derived Silica by Adsorption Revisited
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
183, 289–290 (1996)
0546
NOTE Fractal Analysis of a Sol–Gel-Derived Silica by Adsorption Revisited METHODOLOGY The fractal analysis of microporous solids through the molecular tiling method is dependent on the geometry and orientation of molecules in the adsorbed phase as well as the mechanism of adsorption (monolayer-multilayer coverage or micropore volumefilling) occurring in the pores. The data of Sermon et al. (9) obtained with a series of n-alkanes on a microporous silica sol–gel have been reanalyzed by taking into account the micropore volume-filling process as well as the linear nature of the molecules and their parallel-lying, nearest neighbor, chain configuration on the surfaces of the sol–gel. q 1996 Academic Press, Inc. Key Words: adsorption; surface fractal dimension; silica sol–gel; molecular orientation; micropore volume-filling.
Linear, long-chain, cylindrically shaped molecules of n-alkanes are known to adsorb on fairly smooth, isotropic surfaces in such a manner that they form head-to-tail chains that lie parallel to one another (3). The fractal nature of such surfaces with this alignment of n-alkane molecules in nearest neighbor linear chains, in all the monolayers of the series as well as in bulk, can be discerned with the aid of the power-law expressions (3): for monolayer formation, ln(nm ) Å ln(k * ) 0 (dSF 0 1)ln( s ),
[2]
and for the micropore volume-filling process, ln(nm ) Å ln(k 9 ) 0 (dSF 0 1)ln( n ),
The surfaces of most materials are fractal at molecular length scales (1, 2). It is well known that geometric or structural heterogeneities on surfaces exhibit the property of self-similarity over certain length scales (2). An operative measure of the degree of surface irregularity is the surface fractal dimension, dSF , an intrinsic parameter taking on a value between 2.0 and 3.0 for most materials (3, 4). The surface fractality of porous materials has been frequently characterized by molecular tiling experiments wherein the monolayer uptake, nm , on a solid is measured with a series of adsorbates varying in their molecular dimensions (2, 3). The adsorbate series are typically the noble gases or homologous alcohols with approximately spherical molecules, or the linear molecules of the n-alkanes with a constant aspect ratio within the series (3). The value of dSF is determined from the equation ln(nm ) Å ln(k) 0 (dSF /2)ln( s ),
[1]
where s is the effective cross-sectional area of each adsorbate (usually retrieved from the density of the bulk-liquid phase) and k is a proportionality constant. By convention, nm is obtained from the BET equation (5) or the Langmuir equation (6) for each adsorbate. The procedure, however, is inappropriate for microporous solids since the phenomenon of volumefilling instead of layer-by-layer adsorption takes place in the constituent slit-shaped pores with dimensions less than approximately three times the molecular diameter of the adsorbate (6, 7). This is attributable to the enhancement of adsorptive potential of dispersive forces in the micropores due to the overlap of the potential from opposite pore walls (6). Under these conditions, the value of dSF recovered through Eq. [1] is invalid and does not reflect the true degree of irregularity of the surface probed (3, 8). The current work presents an example of the complexity involved in establishing a coherent picture of the surface morphology of microporous solids through the molecular tiling approach. Toward this end, the data obtained by Sermon et al. (9) for a microporous silica sol–gel have been reanalyzed by employing a more realistic representation of the geometry and orientation of molecules in the adsorbed phase. The sol–gel was generated through the acid-catalyzed hydrolysis of tetra-ethyl orthosilicate at 298 K, and its microstructure was probed with molecules of the homologous series of n-alkanes, namely, n-pentane, n-hexane, n-heptane, and n-octane (9).
where n is the molar volume of each hydrocarbon, and k * and k 9 are proportionality constants. As discussed previously, nm for a microporous solid is proportional to the volume (STP) of adsorbate in the micropores; therefore, Eq. [3a] for a microporous sol–gel can be expressed as ln( £mi ) Å ln(k - ) 0 (dSF 0 1)ln( n ),
JCIS 4409
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6g16$$$221
09-19-96 12:43:41
[3b]
where £mi is the effective volume adsorbed in the micropores, in units of volume (STP) per gram of adsorbent, and k - is a proportionality constant.
RESULTS AND DISCUSSION Sermon et al. (9) obtained two values of dSF for the microporous silica sol–gel with two sets of values of s for the n-alkanes. The first set for s was recovered from the liquid densities of the n-alkanes by assuming that the arrangement of these molecules on the surface of the sol–gel is identical to that observed in bulk liquid, i.e., the most dense, face-centered packing of spheres (10). On the other hand, the second set was retrieved from experimental values reported in the literature (11). For each set, a value
TABLE 1 Gurvitsch Rule Applied to a Microporous Silica Sol–Gel
n-alkane
Langmuir nma (mmol/g)
Liquid densityb (g/cm3)
Molar volume nb (cm3/mol)
Gurvitsch volumec (cm3/g)
n-pentane n-hexane n-heptane n-octane
1.0343 0.9390 0.8133 0.7423
0.6262 0.6603 0.6837 0.7025
115.21 130.57 146.78 162.60
0.119 0.122 0.119 0.120
a
Values at 295 K from Ref. (9). Values at 293 K from Ref. (14). c Product of nm and n. b
289
AID
[3a]
0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
coidal
AP: Colloid
290
NOTE It will be interesting to estimate the average dimensions of the micropores in this type of sol–gel through other techniques such as small-angle X-ray and neutron scattering (15) in order to discern the dominant mechanism of adsorption in the micropores. If the average width of the micropores is not greater than approximately two molecular diameters, the volume-filling process will be predominant; then, Eq. [3b] can facilitate the fractal analysis of the micropores with the advantage of reduced uncertainties in n’s compared to those of s in Eqs. [1] or [2].
CONCLUDING REMARKS The fractal analysis of microporous solids through the molecular tiling approach is dependent on the geometry and orientation assumed for molecules of adsorbate as well as on the primary mechanism of adsorption, i.e., monolayer-multilayer coverage or micropore volume-filling, taking place in the pores. Pertinent information on the fractal nature of such solids can be recovered from a power-law relationship between the effective micropore volume, £mi , and the molar volume, n, of the liquid adsorbate for the volumefilling process in the micropores.
ACKNOWLEDGMENTS This is contribution No. 96-335 J, Department of Chemical Engineering, Kansas Agricultural Experiment Station, Kansas State University, Manhattan, KS 66506-5102.
REFERENCES FIG. 1. Fractal analysis of microporous sol–gel derived silica.
of dSF was obtained by applying Eq. [1] to the nm’s estimated from the Langmuir model (6). The results are significantly different; they are 3.0 { 0.1 for the first set and 2.2 { 0.1 for the second set. The adsorption isotherms of the silica sol–gel measured with the nalkane series are all of Type I; furthermore, the molar uptakes of the nalkanes hardly increase for values of the relative pressure, p/p0 , exceeding approximately 0.1 (9). Hence, the value of £mi measured with each nalkane for the system of interest is approximately equal to the total pore or saturation volume, which is indicative of the extensive microporous nature of the adsorbent. Consequently, the value of 3.0 for dSF , measured with the first set of s’s reflects the volume-filling effects of adsorption in the microporous interfaces of the primary particles in the sol–gel and not the actual surface fractality of those interfaces. This fact has been further substantiated by applying Gurvitsch’s rule (6) to the Langmuir monolayer uptake of each n-alkane reported in (9); the results are shown in Table 1. It is evident from the table that the equivalent liquid volume uptakes of the n-alkanes by the micropores of the sol–gel are essentially identical to one another. Hence, the assumption of a spherical geometry and a face-centered packing for the n-alkanes on the sol–gel is invalid, and the value of dSF of 3.0 is of no significance in describing the actual morphology of the microporous interfaces of the silica sol–gel. On the other hand, a much more meaningful result for dSF is recovered when the values of £mi , or equivalently nm , for the molecules of n-alkanes which are assumed to lie sideways, are plotted against their respective n’s as shown in Fig. 1. The value of dSF extracted from the slope of the plot according to Eq. [3] is equal to 1.97 { 0.06, which is essentially 2. This value is in close agreement with those for similar silica sol–gels reported recently by other investigators (12). Hence, the interfaces of the micropores in the primary particles of the silica sol–gel, for all practical purposes, can be considered to be smooth and nonfractal. Alternatively, the micropores can also be construed as the spaces or voids between the primary particles in the closely packed aggregate. In either case, the result for dSF from applying Eq. [3b] to the adsorption data confirms the validity of the cylindrical, parallellying, head-to-tail chain model of the n-alkanes on the narrow micropores of the sol–gel. Note that the value of dSF equal to 2.2 from the second set for s (9) is coincidental since this set inherently assumes that the surfaces of the adsorbents probed are all smooth and nonfractal (13); therefore, it is obvious that this set is strictly limited to regular surfaces.
AID
JCIS 4409
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6g16$$$222
09-19-96 12:43:41
1. Farin, D., and Avnir, D., in ‘‘The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, and Polymers’’ (D. Avnir, Ed.), pp. 271–272. Wiley, Chichester, 1989. 2. Avnir, D., Farin, D., and Pfeifer, P., Nature (London) 308, 261 (1984). 3. Pfeifer, P., and Avnir, D., J. Chem. Phys. 79, 3558 (1983). 4. Fan, L. T., Neogi, D., and Yashima, M., ‘‘An Elementary Introduction to Spatial and Temporal Fractals,’’ pp. 30, 116–117. Springer-Verlag, Berlin, 1991. 5. Brunaeur, S., Emmett, P. H., and Teller, E., J. Am. Chem. Soc. 60, 309 (1938). 6. Gregg, S. J., and Sing, K. S. W., ‘‘Adsorption, Surface Area and Porosity,’’ 2nd ed., Chap. 4. Academic Press, London, 1982. 7. Dubinin, M. M., and Stoeckli, H. F., J. Colloid. Interface Sci. 75, 35 (1988). 8. Tsunoda, R., J. Colloid. Interface Sci. 152, 571 (1992). 9. Sermon, P. A., Wang, W., and Vong, M. S. W., J. Colloid. Interface Sci. 168, 327 (1994). 10. Emmett, P. H., and Brunaeur, S., J. Am. Chem. Soc. 59, 1553 (1937). 11. McCllelan, A. L., and Harnsberger, H. F., J. Colloid. Interface Sci. 23, 577 (1994). 12. Dolle, F. E., Holz, M., and Lahaye, J., Pure Appl. Chem. 65, 2223 (1993). 13. Stermer, D. L., Smith, D. M., and Hurd, A. J., J. Colloid. Interface Sci. 131, 592 (1989). 14. ‘‘CRC Handbook of Chemistry and Physics,’’ 66th ed., (R. C. Weast, Ed.), pp. C-295, C-303, C-381, C-393. CRC Press, Baco Raton, FL, 1985. 15. Venkatraman, A., Boateng, A. A., Fan, L. T., and Walawender, W. P., AIChE J. 42, 2014 (1996). A. VENK ATRAMAN L. T. FAN 1 W. P. WALAWENDER Department of Chemical Engineering 105 Durland Hall Kansas State University Manhattan, Kansas 66506-5102 Received March 5, 1996 1
To whom correspondence should be addressed.
coidal
AP: Colloid
183, 289–290 (1996)
0546
NOTE Fractal Analysis of a Sol–Gel-Derived Silica by Adsorption Revisited METHODOLOGY The fractal analysis of microporous solids through the molecular tiling method is dependent on the geometry and orientation of molecules in the adsorbed phase as well as the mechanism of adsorption (monolayer-multilayer coverage or micropore volumefilling) occurring in the pores. The data of Sermon et al. (9) obtained with a series of n-alkanes on a microporous silica sol–gel have been reanalyzed by taking into account the micropore volume-filling process as well as the linear nature of the molecules and their parallel-lying, nearest neighbor, chain configuration on the surfaces of the sol–gel. q 1996 Academic Press, Inc. Key Words: adsorption; surface fractal dimension; silica sol–gel; molecular orientation; micropore volume-filling.
Linear, long-chain, cylindrically shaped molecules of n-alkanes are known to adsorb on fairly smooth, isotropic surfaces in such a manner that they form head-to-tail chains that lie parallel to one another (3). The fractal nature of such surfaces with this alignment of n-alkane molecules in nearest neighbor linear chains, in all the monolayers of the series as well as in bulk, can be discerned with the aid of the power-law expressions (3): for monolayer formation, ln(nm ) Å ln(k * ) 0 (dSF 0 1)ln( s ),
[2]
and for the micropore volume-filling process, ln(nm ) Å ln(k 9 ) 0 (dSF 0 1)ln( n ),
The surfaces of most materials are fractal at molecular length scales (1, 2). It is well known that geometric or structural heterogeneities on surfaces exhibit the property of self-similarity over certain length scales (2). An operative measure of the degree of surface irregularity is the surface fractal dimension, dSF , an intrinsic parameter taking on a value between 2.0 and 3.0 for most materials (3, 4). The surface fractality of porous materials has been frequently characterized by molecular tiling experiments wherein the monolayer uptake, nm , on a solid is measured with a series of adsorbates varying in their molecular dimensions (2, 3). The adsorbate series are typically the noble gases or homologous alcohols with approximately spherical molecules, or the linear molecules of the n-alkanes with a constant aspect ratio within the series (3). The value of dSF is determined from the equation ln(nm ) Å ln(k) 0 (dSF /2)ln( s ),
[1]
where s is the effective cross-sectional area of each adsorbate (usually retrieved from the density of the bulk-liquid phase) and k is a proportionality constant. By convention, nm is obtained from the BET equation (5) or the Langmuir equation (6) for each adsorbate. The procedure, however, is inappropriate for microporous solids since the phenomenon of volumefilling instead of layer-by-layer adsorption takes place in the constituent slit-shaped pores with dimensions less than approximately three times the molecular diameter of the adsorbate (6, 7). This is attributable to the enhancement of adsorptive potential of dispersive forces in the micropores due to the overlap of the potential from opposite pore walls (6). Under these conditions, the value of dSF recovered through Eq. [1] is invalid and does not reflect the true degree of irregularity of the surface probed (3, 8). The current work presents an example of the complexity involved in establishing a coherent picture of the surface morphology of microporous solids through the molecular tiling approach. Toward this end, the data obtained by Sermon et al. (9) for a microporous silica sol–gel have been reanalyzed by employing a more realistic representation of the geometry and orientation of molecules in the adsorbed phase. The sol–gel was generated through the acid-catalyzed hydrolysis of tetra-ethyl orthosilicate at 298 K, and its microstructure was probed with molecules of the homologous series of n-alkanes, namely, n-pentane, n-hexane, n-heptane, and n-octane (9).
where n is the molar volume of each hydrocarbon, and k * and k 9 are proportionality constants. As discussed previously, nm for a microporous solid is proportional to the volume (STP) of adsorbate in the micropores; therefore, Eq. [3a] for a microporous sol–gel can be expressed as ln( £mi ) Å ln(k - ) 0 (dSF 0 1)ln( n ),
JCIS 4409
/
6g16$$$221
09-19-96 12:43:41
[3b]
where £mi is the effective volume adsorbed in the micropores, in units of volume (STP) per gram of adsorbent, and k - is a proportionality constant.
RESULTS AND DISCUSSION Sermon et al. (9) obtained two values of dSF for the microporous silica sol–gel with two sets of values of s for the n-alkanes. The first set for s was recovered from the liquid densities of the n-alkanes by assuming that the arrangement of these molecules on the surface of the sol–gel is identical to that observed in bulk liquid, i.e., the most dense, face-centered packing of spheres (10). On the other hand, the second set was retrieved from experimental values reported in the literature (11). For each set, a value
TABLE 1 Gurvitsch Rule Applied to a Microporous Silica Sol–Gel
n-alkane
Langmuir nma (mmol/g)
Liquid densityb (g/cm3)
Molar volume nb (cm3/mol)
Gurvitsch volumec (cm3/g)
n-pentane n-hexane n-heptane n-octane
1.0343 0.9390 0.8133 0.7423
0.6262 0.6603 0.6837 0.7025
115.21 130.57 146.78 162.60
0.119 0.122 0.119 0.120
a
Values at 295 K from Ref. (9). Values at 293 K from Ref. (14). c Product of nm and n. b
289
AID
[3a]
0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
coidal
AP: Colloid
290
NOTE It will be interesting to estimate the average dimensions of the micropores in this type of sol–gel through other techniques such as small-angle X-ray and neutron scattering (15) in order to discern the dominant mechanism of adsorption in the micropores. If the average width of the micropores is not greater than approximately two molecular diameters, the volume-filling process will be predominant; then, Eq. [3b] can facilitate the fractal analysis of the micropores with the advantage of reduced uncertainties in n’s compared to those of s in Eqs. [1] or [2].
CONCLUDING REMARKS The fractal analysis of microporous solids through the molecular tiling approach is dependent on the geometry and orientation assumed for molecules of adsorbate as well as on the primary mechanism of adsorption, i.e., monolayer-multilayer coverage or micropore volume-filling, taking place in the pores. Pertinent information on the fractal nature of such solids can be recovered from a power-law relationship between the effective micropore volume, £mi , and the molar volume, n, of the liquid adsorbate for the volumefilling process in the micropores.
ACKNOWLEDGMENTS This is contribution No. 96-335 J, Department of Chemical Engineering, Kansas Agricultural Experiment Station, Kansas State University, Manhattan, KS 66506-5102.
REFERENCES FIG. 1. Fractal analysis of microporous sol–gel derived silica.
of dSF was obtained by applying Eq. [1] to the nm’s estimated from the Langmuir model (6). The results are significantly different; they are 3.0 { 0.1 for the first set and 2.2 { 0.1 for the second set. The adsorption isotherms of the silica sol–gel measured with the nalkane series are all of Type I; furthermore, the molar uptakes of the nalkanes hardly increase for values of the relative pressure, p/p0 , exceeding approximately 0.1 (9). Hence, the value of £mi measured with each nalkane for the system of interest is approximately equal to the total pore or saturation volume, which is indicative of the extensive microporous nature of the adsorbent. Consequently, the value of 3.0 for dSF , measured with the first set of s’s reflects the volume-filling effects of adsorption in the microporous interfaces of the primary particles in the sol–gel and not the actual surface fractality of those interfaces. This fact has been further substantiated by applying Gurvitsch’s rule (6) to the Langmuir monolayer uptake of each n-alkane reported in (9); the results are shown in Table 1. It is evident from the table that the equivalent liquid volume uptakes of the n-alkanes by the micropores of the sol–gel are essentially identical to one another. Hence, the assumption of a spherical geometry and a face-centered packing for the n-alkanes on the sol–gel is invalid, and the value of dSF of 3.0 is of no significance in describing the actual morphology of the microporous interfaces of the silica sol–gel. On the other hand, a much more meaningful result for dSF is recovered when the values of £mi , or equivalently nm , for the molecules of n-alkanes which are assumed to lie sideways, are plotted against their respective n’s as shown in Fig. 1. The value of dSF extracted from the slope of the plot according to Eq. [3] is equal to 1.97 { 0.06, which is essentially 2. This value is in close agreement with those for similar silica sol–gels reported recently by other investigators (12). Hence, the interfaces of the micropores in the primary particles of the silica sol–gel, for all practical purposes, can be considered to be smooth and nonfractal. Alternatively, the micropores can also be construed as the spaces or voids between the primary particles in the closely packed aggregate. In either case, the result for dSF from applying Eq. [3b] to the adsorption data confirms the validity of the cylindrical, parallellying, head-to-tail chain model of the n-alkanes on the narrow micropores of the sol–gel. Note that the value of dSF equal to 2.2 from the second set for s (9) is coincidental since this set inherently assumes that the surfaces of the adsorbents probed are all smooth and nonfractal (13); therefore, it is obvious that this set is strictly limited to regular surfaces.
AID
JCIS 4409
/
6g16$$$222
09-19-96 12:43:41
1. Farin, D., and Avnir, D., in ‘‘The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, and Polymers’’ (D. Avnir, Ed.), pp. 271–272. Wiley, Chichester, 1989. 2. Avnir, D., Farin, D., and Pfeifer, P., Nature (London) 308, 261 (1984). 3. Pfeifer, P., and Avnir, D., J. Chem. Phys. 79, 3558 (1983). 4. Fan, L. T., Neogi, D., and Yashima, M., ‘‘An Elementary Introduction to Spatial and Temporal Fractals,’’ pp. 30, 116–117. Springer-Verlag, Berlin, 1991. 5. Brunaeur, S., Emmett, P. H., and Teller, E., J. Am. Chem. Soc. 60, 309 (1938). 6. Gregg, S. J., and Sing, K. S. W., ‘‘Adsorption, Surface Area and Porosity,’’ 2nd ed., Chap. 4. Academic Press, London, 1982. 7. Dubinin, M. M., and Stoeckli, H. F., J. Colloid. Interface Sci. 75, 35 (1988). 8. Tsunoda, R., J. Colloid. Interface Sci. 152, 571 (1992). 9. Sermon, P. A., Wang, W., and Vong, M. S. W., J. Colloid. Interface Sci. 168, 327 (1994). 10. Emmett, P. H., and Brunaeur, S., J. Am. Chem. Soc. 59, 1553 (1937). 11. McCllelan, A. L., and Harnsberger, H. F., J. Colloid. Interface Sci. 23, 577 (1994). 12. Dolle, F. E., Holz, M., and Lahaye, J., Pure Appl. Chem. 65, 2223 (1993). 13. Stermer, D. L., Smith, D. M., and Hurd, A. J., J. Colloid. Interface Sci. 131, 592 (1989). 14. ‘‘CRC Handbook of Chemistry and Physics,’’ 66th ed., (R. C. Weast, Ed.), pp. C-295, C-303, C-381, C-393. CRC Press, Baco Raton, FL, 1985. 15. Venkatraman, A., Boateng, A. A., Fan, L. T., and Walawender, W. P., AIChE J. 42, 2014 (1996). A. VENK ATRAMAN L. T. FAN 1 W. P. WALAWENDER Department of Chemical Engineering 105 Durland Hall Kansas State University Manhattan, Kansas 66506-5102 Received March 5, 1996 1
To whom correspondence should be addressed.
coidal
AP: Colloid