Limitation of Determination of Surface Fractal Dimension Using N2

Feb 27, 2003 - The surface fractal dimension, DS, can be determined by gas adsorption,1-7 ...... International Journal of Coal Geology 2016 154-155, 1...
0 downloads 0 Views 279KB Size
Download PDF
2632

Langmuir 2003, 19, 2632-2638

Limitation of Determination of Surface Fractal Dimension Using N2 Adsorption Isotherms and Modified Frenkel-Halsey-Hill Theory Patricia Tang,*,†,‡ Nora Y. K. Chew,† Hak-K. Chan,† and Judy A. Raper‡ Faculty of Pharmacy, University of Sydney, New South Wales 2006, Australia, and Department of Chemical Engineering, University of Sydney, New South Wales 2006, Australia Received August 7, 2002. In Final Form: December 16, 2002 Surface fractal dimensions, DS, of smooth and corrugated bovine serum albumin particles were obtained from N2 adsorption isotherms using modified Frenkel-Halsey-Hill (FHH) theory. It was found that for different particles, the correct DS values depended on the number of adsorbed layers, n. For corrugated particles, when 1 e n e 10, the value of DS is equal to 2.39, which agrees with the value obtained from light scattering (2.39 ( 0.05). Unlike the corrugated particles, the adsorption isotherm for the smooth particles generated the correct value of DS (2.12) only for 1.0 ( 0.5 e n e 2.0 ( 0.5 (i.e., around monolayer coverage). Determination of DS in the multilayer region (n > 2) produced a higher value than the one obtained from monolayer coverage. This was because the smooth particles were in closer contact with each other; at higher coverage the gas molecules probed the surface of the aggregates instead of the single particles. As there were fewer contact points between the corrugated particles compared to the smooth particles, this effect took place at higher coverage (pressure) causing deviation from the expected values. This finding is supported by the fact that for corrugated particles, the value of DS started to deviate at higher n and increased to 2.58 when n > 10. The use of modified FHH theory is thus limited by the number of adsorbed layers on the particles. The closer the particles come in contact, the thinner is the coverage region describing the correct DS. To ensure reliable determination of DS, it is therefore recommended to determine DS only around monolayer coverage.

I. Introduction A noninteger dimension called surface fractal dimension, DS, has been widely used to characterize particle surface roughness.1-7 Mandelbrot (1983) defines that a fractal object has a dimension D which is greater than the topological dimension but less than or equal to the dimension of the embedding Euclidean space.8,9 The value of DS varies from 2 for a perfectly smooth surface to 3 for a very rough surface. The most important property that distinguishes a fractal surface from a nonfractal one is self-similarity. This means that irrespective of length scales or magnification the repetition of disorder within the structure takes place. This concept is illustrated in Figure 1.10 As a first technique to determine the fractal dimension of a rugged boundary, a series of polygons of side length β are constructed. The perimeter of the polygon, P, then becomes the approximation of the perimeter at resolution β. The boundary fractal dimension, DL, can then be determined as follows:11,12

P ∝ β1-DL

(1)

* To whom correspondence should be addressed. E-mail: ptang@ pharm.usyd.edu.au. † Faculty of Pharmacy. ‡ Department of Chemical Engineering. (1) Ismail, I. M. K.; Pfeifer, P. Langmuir 1994, 10 (5), 1532-1538. (2) Suzuki, T.; Yano, T. Agric. Biol. Chem. 1991, 55 (4), 967-971. (3) Xu, W.; Zerda, T. W.; Yang, H.; Gerspacher, M. Carbon 1996, 34 (2), 165-171. (4) Weidler, P. G.; Stanjek, H. Clay Miner. 1998, 33, 277-284. (5) Wu, M. K. Aerosol Sci. Technol. 1996, 25, 392-398. (6) Avnir, D.; Jaroniec, M. Langmuir 1989, 5, 1431-1433. (7) Cellis, R.; Cornejo, J.; Hermosin, M. C. Clay Miner. 1996, 31, 355-363. (8) Sernkow, T. M. Environ. Int. 1996, 22 (Suppl.1), 567-574.

Figure 1. Self-similarity of a fractal surface [Reprinted from Powder Technol., 21, A. G. Flook, The Use of Dilation Logic on the Quantimet to Achieve Fractal Dimension Characterization of Textured and Structured Profiles, pp 295-298, Copyright (1978), with permission from Elsevier].

DL, which varies from 1 to 2, describes the boundary line of a fractal object,13 while DS describes a surface morphology. The relationship between DL and DS is14

DS ) (1 + DL)

(2)

The surface fractal dimension, DS, can be determined by gas adsorption,1-7,15-22 light scattering,23-25 and image (9) Mandelbrot, B. B. The Fractal Geometry of Nature; W. H. Freeman: New York, 1983. (10) Flook, A. G. Powder Technol. 1978, 21, 295-298. (11) Kaye, B. H. Powder Technol. 1978, 21, 1-16. (12) Witten, T. A.; Sander, L. M. Phys. Rev. Lett. 1981, 47, 1400. (13) Fini, A.; Holgado, M. A.; Fernandez-Hervas, M. J.; Rabasco, A. M. Acta Technol. Legis Med. 1996, 7 (1), 41-56. (14) Farin, D.; Avnir, D. J. Pharm. Sci. 1992, 81, 54-57. (15) Suzuki, T.; Yano, T. Agric. Biol. Chem. 1990, 54 (12), 31313135. (16) Ludlow, D. K.; Vosen, W. M. Part. Part. Syst. Charact. 1993, 10, 313-320. (17) Vaimakis, T. C.; Skordilis, C. S.; Pomonis, P. J. J. Colloid Interface Sci. 1995, 172, 311-316. (18) Ng, S. H.; Fairbridge, C. Langmuir 1987, 3 (3), 340-345. (19) Fan, L. T.; Boateng, A. A.; Walawender, W. P. Can. J. Chem. Eng. 1992, 70, 387-390.

10.1021/la0263716 CCC: $25.00 © 2003 American Chemical Society Published on Web 02/27/2003

Determination of Surface Fractal Dimension

Langmuir, Vol. 19, No. 7, 2003 2633

analysis.13,26-33 The determination of surface fractal dimension by image analysis is very tedious. This method is only suitable when a sufficient number of particles can be collected and analyzed, the images have good contrast, and the particles are not overlapping. Due to these limitations, light scattering and gas adsorption have become more favored methods. There are three different methods usually employed to obtain fractal dimension by gas adsorption. In the first method, the particles are fractionated and the BrunauerEmmett-Teller (BET) surface area is measured for each size fraction.15,17,19,20 The DS can then be determined as34,35

S ∝ DPDS-3

(3)

where S is specific surface area (m2/g). DP is the mean particle diameter after fractionation into a narrow particle size distribution. In the second method, the particles do not have to be fractionated. However, several adsorbates, whose molecules have different cross-sectional areas, have to be used.2,16,18 The surface fractal dimension is then determined as34,35

S ∝ σ-(DS-2)/2

(4)

where σ is the adsorbate molecular cross-sectional area. The prediction of the molecular cross-sectional area is based on the correlations derived by McClellan and Harnsberger, which were developed based on the assumption that the same surface area was obtained by adsorption of different adsorbates.36 However, this is not true if the particle surface is rough. Adsorbate molecules having larger cross-sectional areas cannot access the finer structure of the surface. Therefore, the determination of DS from this method might not be accurate. Additionally, both of these methods require multiple measurements of adsorption isotherms. In the third method, however, only a single adsorption isotherm is required. The analysis of the single isotherm (20) Ladavos, A. K.; Trikalitis, P. N.; Pomonis, P. J. J. Mol. Catal. A: Chem. 1996, 106, 241-254. (21) Nagai, T.; Yano, T. Nippon Shokuhin Kogyo Gakkaishi 1991, 38 (4), 350-356. (22) Lee, C. K.; Tsay, C. S. J. Phys. Chem. B 1998, 102, 4123-4130. (23) Khatib, K.; Pons, C. H.; Bottero, J. Y.; Francois, M.; Baudin, I. J. Colloid Interface Sci. 1995, 172, 317-323. (24) Kriechbaum, M.; Degovics, G.; Trithart, J.; Laggner, P. Prog. Colloid Polym. Sci. 1989, 79, 101-105. (25) Martin, J. E.; Schaefer, D. W.; Hurd, A. J. Phys. Rev. A 1986, 33 (5), 3540-3543. (26) Barak, P.; Seybold, C. A.; McSweeney, K. Soil Sci. Soc. Am. J. 1996, 60, 72-76. (27) Podsiadlo, P.; Stachowiak, G. W. Wear 2000, 242, 180-188. (28) Stechemesser, H.; Partzscht, H.; Zobel, G. Miner. Process. 1996, 37 (9), 422-431. (29) Fini, A.; Fernandez-Hervas, M. J.; Holgado, M. A.; Rodriguez, L.; Cavallari, C.; Passerini, N.; Caputo, O. J. Pharm. Sci. 1997, 86 (11), 1303-1309. (30) Romeu, D.; Gomez, A.; Perez-Ramirez, J. G.; Silva, R.; Perez, O. L.; Gonzales, A. E.; Jose-Yacaman, M. Phys. Rev. Lett. 1986, 57 (20), 2552-2555. (31) Weidler, P. G.; Degovics, G.; Laggner, P. J. Colloid Interface Sci. 1998, 197, 1-8. (32) Dziuba, J.; Babuchowski, A.; Smoczynski, M.; Smietana, Z. Int. Dairy J. 1999, 9, 287-292. (33) Chesters, S.; Wang, H. C.; Kasper, G. Proceedings of the 36th Annual Technical Meeting, Institute of Environmental Sciences: New Orleans, Louisiana, April 23-27, 1990; Institute of Environmental Sciences: Mount Prospect, IL, 1990. (34) Pfeifer, P.; Avnir, D. J. Chem. Phys. 1983, 79, 3558-3565. (35) Avnir, D.; Farin, D.; Pfeifer, P. J. Chem. Phys. 1983, 79, 35663571. (36) McClellan, A. L.; Harnsberger, H. F. J. Colloid Interface Sci. 1967, 23, 577-599.

using a modified Frenkel-Halsey-Hill (FHH) theory allows the determination of surface fractal dimension. The original FHH theory was developed by Frenkel,37,38 Halsey,38 and Hill39 and later on was extended to fractal surfaces by Pfeifer et al.40 Smooth and corrugated particles have been shown to have different aerosol performance,41 and it will be useful to quantify the roughness of the particle surface and relate it with their performance. The aim of this paper is then to validate the determination of DS by gas adsorption using the modified FHH theory. We have found that the use of this theory is limited by the number of adsorbed layers on the particles. In this paper, the limitation of the modified FHH theory to determine surface fractal dimension, DS, from gas adsorption is discussed. DS was also determined by light scattering for comparison. II. Theory A. Determination of Surface Fractal Dimension by Laser Light Scattering. The determination of surface fractal dimension by light scattering utilizes the RayleighGans-Debye (RGD) scattering theory. There are two conditions to be met in order to utilize the RGD theory to predict DS accurately. They include the following: i. |m - 1| , 1, where m is the complex refractive index of the particle relative to that of the surrounding medium. This condition means that the incident beam is negligibly reflected at the particle-medium interface so that the direction of the incident light is the same everywhere in the scattering medium. ii. 2kR|m - 1| , 1, where R is the radius of the particle and k can be expressed as k ) 2π/λ (λ is the wavelength of the incident light). This condition means that the amplitude of the incident beam does not change significantly after it meets the particle. This theory neglects the multiple scattering effect since it assumes that the radiation illuminating each particle in the aggregate is the incident radiation and is totally unperturbed by the presence of the other particles in the aggregate. However, it has been shown experimentally and theoretically that even though multiple scattering does change the magnitude of the scattering intensity, it does not change the evaluated fractal dimension of the aggregates.42 During the scattering experiment, a laser light hits the particles suspended in liquid and it is scattered. The intensity of the scattered light, which is measured by a number of detectors positioned at different angles, can be used to generate DS. Scattering momentum, q, is used to relate λ, the scattering angle, θ, and the refractive index of suspending liquid, η:

q)

θ 4ηπ sin λ 2

()

(5)

In the case of a surface fractal, when q > DP-1,

I(q) ∝ q-6+DS

(6)

(37) Frenkel, J. Kinetic Theory of Liquids; Clarendon: Oxford, 1946. (38) Halsey, G. D. J. Chem. Phys. 1948, 16, 931. (39) Hill, T. L. Adv. Catal. 1952, 4, 211. (40) Pfeifer, P.; Wu, Y. J.; Cole, M.; Krim, J. Phys. Rev. Lett. 1989, 62, 1997. (41) Chew, N. Y. K.; Chan, H. K. Pharm. Res. 2001, 18 (11), 15701577. (42) Chen, Z.; Sheng, P.; Weitz, D. A.; Lindsay, H. M.; Lin, M. Y.; Meakin, P. Phys. Rev. B 1988, 37 (10), 5232-5235.

2634

Langmuir, Vol. 19, No. 7, 2003

Tang et al.

dominate, the relationship between A and DS can be expressed as follows:40

A)

DS - 3 3

(8)

However, later, surface tension (or capillary condensation) effects become more pronounced and the relationship between A and DS changes to44,45

A ) DS - 3

(9)

The van der Waals forces and the effect of surface tension represent limiting cases; however in general the adsorption forces are a mixture of the two.5 When surface coverage is low, Ismail and Pfeifer used δ to determine which effect dominates:1,5 Figure 2. Scattering curve for colloidal aggregates (DA, aggregate diameter; DP, primary particle diameter; DM, mass fractal dimension; DS, surface fractal dimension; b, bond length) (ref 43).

where I(q) is the intensity of scattered light. On the basis of eq 6, when I(q) versus q is plotted on a logarithmic scale, DS can be obtained from the slope. Basically, q can be described as a magnification scale. The higher the q values, the higher the magnification. To obtain DS, the magnification has to be high enough so that the particle surface structure can be resolved. The concept is illustrated in Figure 2.43 As can be seen from Figure 2, at q < DA-1 (limiting region) where DA is the diameter of the aggregate, the detailed structure of the colloidal aggregates cannot be resolved. For DA-1 < q < DP-1, the fractal region represents the packing density of the aggregates from which mass fractal dimension (1 e DM e 3) can be obtained. At q > DP-1 (Porod region), the detailed structure of the primary particle surface can be resolved. At this region, the surface fractal dimension, DS, can be obtained from the slope using eq 6. At the highest magnification (Bragg region), broad peaks that correspond to the smallest realspace features are observed and can be analyzed using Bragg’s law.16 B. Determination of Surface Fractal Dimension from the N2 Adsorption Isotherm Using Modified FHH Theory. Pfeifer et al. derived an expression for DS from an analysis of multilayer adsorption to a fractal surface:5,40

( )

ln

[ ( )]

P0 V ) C + A ln ln Vm P

(7)

where V is the volume of gas adsorbate at an equilibrium pressure P, Vm is the volume of gas in a monolayer, P is the adsorption equilibrium pressure of a gas, P0 is the saturation pressure of the gas at the given temperature, C is the preexponential factor, and A is a power law exponent dependent on DS and the mechanism of adsorption. At the early stages of adsorption (low P/P0), the effect of surface tension is negligible and the interaction between adsorbate molecules and the particles is mainly due to van der Waals forces.5 When the van der Waals forces (43) Schaefer, D. W.; Hurd, A. J. Aerosol Sci. Technol. 1990, 12, 876891.

δ ) 3(1 + A) - 2

(10)

Capillary condensation is negligible if δ g0 and significant if δ < 0. Extensive discussion on the effect of van der Waals forces and capillary condensation has been reported elsewhere.1,3 According to eq 7, a plot of log V versus log(log(P0/P)) should produce a straight line with a negative slope A, from which DS can be deduced. However, to the authors’ best knowledge, no one has previously ascertained the appropriate pressure range and hence range of thickness of the adsorbed layer coverage to accurately obtain DS. It is our objective then to show with justification the most appropriate region to determine DS from N2 adsorption isotherms using the modified FHH theory. From the BET equation,

(β - 1)P 1 P + ) VmβP0 V(P0 - P) Vmβ the volume of monolayer coverage, Vm, can be determined from the slope and intercept of a plot of P/[V(P0 - P)] versus P/P0,2 where β is a constant related to the energy of adsorption in the first adsorbed layer. The number of adsorbed layers, n, can be determined by the following relationship:1

n)

( ) V Vm

1/(3-DS)

(11)

III. Method A. Determination of Surface Fractal Dimension by Light Scattering. The surface fractal dimension was measured using a Mastersizer S laser diffractometer (Malvern, Worcs, U.K.). A 2 mW He-Ne laser (λ ) 632.8 nm, beam width ) 18 mm) was used as the incident beam directed through the sample. The detector consisted of a 42 element composite array with two backscattering detectors. Powders of bovine serum albumin (BSA) were prepared by spray drying (Buchi 191, Flawil, Switzerland). Chloroform was used as the suspending medium. The concentration of BSA particles suspended in chloroform was approximately 0.1% (w/ w), which was suitable to give representative results and avoid the multiple scattering effect. The determination of surface fractal dimension was based on the refractive index (RI) of BSA (1.550), RIimaginary of BSA (1.200), and RI of chloroform (1.444). Each sample was measured 10 times to ensure reproducibility. B. Determination of Surface Fractal Dimension by Gas Adsorption. Nitrogen adsorption isotherms were generated by (44) Pfeifer, P.; Cole, M. W. New J. Chem. 1990, 14, 221-232. (45) Pfeifer, P.; Cole, M. W.; Krim, J. Phys. Rev. Lett. 1990, 65, 663.

Determination of Surface Fractal Dimension

Langmuir, Vol. 19, No. 7, 2003 2635

Figure 3. Iterative process to determine fractal dimension from N2 adsorption isotherms. an Autosorb (Quantachrome Corp., Boynton Beach, FL) gas adsorption unit. Approximately 0.5 g samples were outgassed at 45 °C for 24 h. Adsorption isotherms of N2 at 77 K were obtained from P/P0 ) 0.05 to P/P0 ) 0.99. To calculate the number of adsorbed layers as expressed in eq 11, an iterative process (illustrated in Figure 3) was required.

Figure 4. SEM photos of (a) smooth and (rough) BSA particles. Table 1. Summary of Equivalent Sphere Diameter Reported by the Malvern Mastersizer S for BSA Particles

IV. Results and Discussion A. BSA Particles. Scanning electron microscope (SEM) photos of the smooth and rough BSA particles are shown in Figure 4. Scattering curves for single particles (Figure 5a,b) have different shapes from that for colloidal aggregates (Figure 2). This is because the single particles suspended in chloroform during the measurement did not aggregate. Therefore, the fractal region describing the structure of aggregates did not exist. D(v,10), D(v,50), and D(v,90), which are the equivalent sphere diameters at 10%, 50%, and 90% cumulative volume for these particles, are shown in Table 1. Span, which is a measure of the width of the size distribution, was calculated as span ) [D(v,90) D(v,10)]/D(v,50). To ensure that the q values are high enough to resolve the detailed structure of the particle surface, the fractal region to obtain DS has to exist for q > DP-1. Using the D(v,50) values in Table 1, the corresponding q values to obtain DS for smooth and rough BSA particles are 4.63 × 10-4 and 5.65 × 10-4 nm-1, respectively. As can be seen from Figure 5a,b, the fractal regimes to obtain DS indeed exist for q > 4.63 × 10-4 nm-1 and q > 5.65 × 10-4 nm-1, respectively. Recalling that there are two conditions to be met in order to use the RGD scattering theory to accurately obtain DS, the first condition, that is, |m - 1| , 1, was satisfied. However, the second condition, 2kR|m - 1| , 1, was not

smooth BSA (Figure 4a) rough BSA (Figure 4b)

D(v,10) (µm)

D(v,50) (µm)

D(v,90) (µm)

span

1.08 0.99

2.16 1.77

3.96 3.26

1.33 1.28

satisfied due to the size of the particles. Therefore, it is not appropriate to call the dimension obtained from the slope of the scattering curve the surface fractal dimension. In this paper, this parameter will be called “apparent surface fractal dimension”. Although the absolute value of DS might not be correct, the light scattering technique can nevertheless be used as an additional tool to compare the roughness of particle surfaces in this work since the particles have similar sizes. Using eq 6, the values of apparent DS obtained from Figure 5a,b are 2.10 ( 0.04 and 2.39 ( 0.05, respectively. These values correspond to the expected trend since the smoother the particle surface, the closer the DS value is to 2. To obtain DS from N2 adsorption isotherms, the graphs in Figure 6 were plotted. For rough particles, the adsorption isotherm shows two different linear regions, while for smooth particles, it shows more than two linear regions. The question raised was then, which is the appropriate region to obtain DS? To answer this question, the iterative process to calculate the number of adsorbed layers, as illustrated in Figure 3, was done. Based on different ranges of n, the adsorption isotherms are constructed again for the rough and smooth particles (Figures 7 and 8).

2636

Langmuir, Vol. 19, No. 7, 2003

Figure 5. Scattering curves obtained from the light scattering technique for (a) smooth and (b) rough BSA particles.

Equation 9 was used instead of eq 8 to calculate DS because the values of δ, calculated by eq 10, are less than 0, which means that the effect of surface tension is not negligible and therefore the surface fractal dimension is more accurately determined by the use of eq 9. When DS was calculated using eq 8, the values obtained were lower than the minimum value of DS allowed (DS < 2). This observation has also been reported in the literature.3,6,40 It was concluded that this is an indication that the capillary condensation might already have taken place. Ismail and Pfeifer reported that at this low coverage the capillary condensation might take place at the points of contact between adjacent particles rather than from capillary condensation on the entire surface.1 From Figure 8a, when n ) 0.97-1.99, the DS value is 2.12, and from Figure 8b, when n ) 2.31-3.31, the DS value is 2.64 while for n ) 3.63-7.04, the DS value ) 2.80. The next question is, which of these DS values is correct since they do lie between 2 and 3? As can be seen from the SEM photo of these particles (Figure 4a), the particle surface is quite smooth and therefore the DS value should be close to 2. This value also agrees with the one obtained from the light scattering technique (Figure 5a). Although the length scale in light scattering is much higher than the length scale probed by nitrogen adsorption, the agreement between DS obtained by the two different methods is not surprising because fractal objects are self-similar and so DS should be the same irrespective of the length scales. It was shown earlier that the q values from light scattering are high enough to resolve the detailed structure of the particle surface and therefore the DS values would be expected to agree with nitrogen adsorption.

Tang et al.

Figure 6. N2 adsorption isotherms for (a) rough and (b) smooth BSA particles.

Figure 7. N2 adsorption isotherms for rough BSA particles for different numbers of adsorbed layers, n.

To substantiate this finding, standard spherical silica particles were used. The N2 adsorption isotherm was found

Determination of Surface Fractal Dimension

Langmuir, Vol. 19, No. 7, 2003 2637

Figure 9. Illustrations of multilayer adsorption on smooth spherical particles.

Figure 8. N2 adsorption isotherms for smooth BSA particles for different numbers of adsorbed layers, n.

to be of similar shape as the smooth BSA particles. The standard spherical silica particles with size 30.2 ( 1.0 µm produced a surface fractal dimension of 2.09 ( 0.03 for 1.0 ( 0.5 e n e 2.0 ( 0.5. This value agrees with the SEM photos of these particles which showed clearly that the surface was smooth, similar to the BSA smooth particles. At higher coverage, that is, n > 2, the value of DS increased to 2.67 ( 0.01. Therefore, it seems that when the adsorbed layer, n, varies from 1.0 ( 0.5 to 2.0 + 0.5, that is, around monolayer coverage, the correct value of DS was obtained. Interestingly, at a higher coverage of approximately 2.31-3.31 (Figure 8b), the value of DS increased. This means that when the adsorbed molecules form multilayers, the value of DS becomes incorrect. This can be explained by the schematic in Figure 9. Compared with rough particles, smooth particles have more contact area between the neighborhood particles. Because the N2 gas molecules are very small compared to the particle size (cross-sectional area of N2 molecule ) 16.2 Å, diameter ) 0.45 nm), the molecules can still be adsorbed even at the particle interface. It has been

reported by Krupp (1967) that the interparticle distance of particles ranging from 1 to 1000 µm in air, determined by Born’s repulsion force, is approximately 0.4 nm46,47 and therefore there is only enough room for monolayer coverage. As shown in Figure 9, when the particles are covered by a monolayer of gas molecules, the adsorption isotherm would describe the surface roughness correctly. As the relative pressure increases, more gas molecules are adsorbed onto the particle surface to form a multilayer. However, at the contact area between particles there is not enough space for more gas molecules to reside. As a result, the adsorption isotherm from the multilayer region would describe the aggregate as a whole. The fractal dimension obtained from this region would no longer describe the roughness of the primary particle surface but would describe the outline of the aggregate. Therefore, the DS values obtained from this region (Figure 8b,c) are higher than the DS value obtained from monolayer coverage (Figure 8a). The value of DS for n ) 3.63-7.04 (Figure 8c) is higher than the one obtained for n ) 2.313.31 (Figure 8b). This finding agrees with the illustration shown in Figure 9 that when more gas molecules were adsorbed, more molecules were available to cover the aggregate outline and therefore the value of surface fractal dimension would be more likely to increase. It was also expected that as more layers are built up, the interface of the adsorbate with the adsorbed molecules becomes smooth, as reported by Wu,5 and therefore the DS should decrease. However, this smoothing effect was not observed in our work. This means that the molecules must have been distributed evenly instead of accumulating on the contact points between particles. For rough particles, there was less contact area and larger average space between a particle and its neighborhood particles. Therefore, at higher coverage the DS value still represents the particle surface roughness, and (46) Powder Technology Handbook, 2nd ed.; Gotoh, K., Masuda, H., Higashitani, K., Eds.; Marcel Dekker: New York, 1997. (47) Krupp, H. Adv. Colloid Interface Sci. 1967, 1, 111.

2638

Langmuir, Vol. 19, No. 7, 2003

Tang et al.

their SEM photos, is 0.39 ( 0.08 µm. It is then proved that 43 layers of adsorbed gas molecules (thickness of 0.019 µm) were not enough to show the smoothing effect. A decade length scale (number of adsorbed layers × diameter of gas molecule) is usually required for a welldefined fractal. However, it has been shown here that for smooth particles a linear region in the adsorption isotherms covering a decade length scale did not exist. Therefore, on the basis of the findings reported in this work, it is recommended to determine the surface fractal dimension around monolayer coverage for smooth particles when the modified Frenkel-Halsey-Hill theory is used. V. Conclusion

Figure 10. Illustrations of multilayer adsorption on corrugated particles.

deviation from linearity was observed at a higher pressure than that for smooth particles. As shown in Figure 7a, for rough particles, when the number of adsorbed molecule layers, n, varied from 1.09 to 10.28, the value of DS obtained from eq 9 is 2.39, which agrees with the value obtained from the light scattering technique. For n ) 12.63-43.13, the value of DS increased to 2.58 (Figure 7b). From Figure 10, multilayer adsorption would additionally describe the internal area within the aggregate (e.g., regions A and B) as corrugated surfaces. This might additionally account for the increase in DS at higher coverage. Similarly as with the smooth particles, the smoothing effect resulting in the decrease of DS was not observed with the rough particles although the calculated maximum number of layers adsorbed is 43. As can be seen from Figure 4b, the concavity of the particle surface is quite steep and so it will take many layers to show the smoothing effect. The concavity, measured from the distance between the particle surface and the lowest point of the depression, of 105 particles, obtained from

The modified Frenkel-Halsey-Hill theory has been applied to obtain surface fractal dimensions of BSA particles from N2 adsorption isotherms. It was found that for smooth particles this method generates the correct value of DS only around monolayer coverage (1.0 ( 0.5 e n e 2.0 ( 0.5). When monolayer coverage was achieved, the DS value of the smooth BSA particles was 2.12 ( 0.04, which agrees with the apparent DS obtained by light scattering. SEM photos of these particles further support this finding. However, for high surface coverage, the DS value increased. This is because the gas molecules adsorbed no longer probed the single particle surface but probed the outline of the particle clumps instead, indicating an apparent increase of surface roughness. For the rough particles, this effect started to show at much higher coverage (pressure). Up to n ) 10.28, the value of DS still agrees with the value obtained from light scattering. This is because the particle surface is rough and so there was less contact area between particles. Therefore the adsorbed gas molecules did not probe the particles as clumps until the multilayer was built up significantly. These findings show that accurate determination of the surface fractal dimension from adsorption isotherms using the modified FHH theory depends on choosing an appropriate coverage region. For smooth BSA particles, this region is 1.0 ( 0.5 e n e 2.0 ( 0.5, while for the corrugated particles it is 1.09 e n e 10.28. To ensure that the determination of DS is reliable, it is therefore recommended to determine DS only around monolayer coverage. Acknowledgment. The authors sincerely thank Professor Peter Pfeifer for the valuable information and discussion made for this work. LA0263716