Fractal Aggregates of Polydisperse Particles

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

205, 459 – 469 (1998)

CS985667

Fractal Aggregates of Polydisperse Particles Graeme Bushell and Rose Amal1 Centre for Particle and Catalyst Technologies, School of Chemical Engineering and Industrial Chemistry, University of New South Wales, Sydney 2052, Australia Received February 9, 1998; accepted May 18, 1998

In this work we examine structural effects of particle polydispersity on fractal aggregates by performing DLCA simulations with multiple primary particle sizes. We show that the fractal structure and the form of the cutoff function that describes the gross shape of the aggregates is unaffected by the details of the primary particle size distribution. The scattering behavior is evaluated in terms of partial structure factors, and depending on the details of primary particle size distribution and contrast, the scattering curve can deviate significantly from the typical q2Df behavior at the high q end of the spectrum. We also develop an expression for average primary particle size that allows the calculation of aggregate solid volume fraction for fractal aggregates with polydisperse primary particles. © 1998 Academic Press Key Words: fractal aggregates; polydispersity; solid volume fraction; RGD scattering; primary particles.

INTRODUCTION

Since the independent development of the cluster– cluster aggregation model by Meakin (1) and Kolb et al. (2), considerable work has been done in relating the fractal structures generated by the model to aggregates of colloidal particles. This has by and large been a very successful comparison. Aggregates of gold (3, 4), hematite (5, 6), silica (7, 8), polystyrene (9), carbon (10), and many other types of particles have been shown to possess a fractal structure, indicating that the form of these aggregates has little to do with the chemistry of the constituent particles and a great deal to do with the mechanism by which the particles meet and aggregate. Fractal structure in this context means mass fractal behavior, where the mass of the aggregates M are related to their linear size R and primary particle size r 0 by a power law, M}~R/r 0! Df,

[1]

where D f is the mass fractal dimension. The solid volume fraction of such aggregates, important for efforts to understand their hydrodynamic behavior (11, 12, 13), is simply derived from Eq. [1] and has the form 1

To whom correspondence should be addressed.

r }~R/r 0! Df23.

[2]

The original simulation consisted of identical occupied sites on a lattice which were allowed to diffuse into each other, sticking irreversibly on contact. Various improvements have since been made to the model. Moving the simulation off lattice, so that the constituent particles are free to move in any direction, and occupy any position in space rather than being restricted to hops between sites on a grid or lattice, had no effect on the long-range fractal structure but made the simulation somewhat more physically realistic in terms of the short-range order. The cluster– cluster aggregation model was originally of the diffusion-limited sort, whereby collision between particles always resulted in irreversible sticking. Such diffusion-limited cluster aggregation (DLCA) is known to produce structures having a mass fractal dimension of ;1.75–1.8 in three-dimensional space. Reaction-limited aggregation (RLCA) (14) assumes that the chance of sticking as a result of a collision event is vanishingly small, allowing colliding clusters to bounce off one another and explore all possible aggregation conformations. Such an aggregation mechanism allows the formation of much denser structures, having fractal dimensions of ;1.95– 2.1, depending on the aggregation kernel used and the resulting size distribution. The inclusion of chemically different monomer types (15) that can only react with each other but not with themselves produced fractal structures with mass fractal dimensions between the DLCA and RLCA values, depending on the concentrations of the individual species present. With an excess of one monomer or the other, aggregation stopped altogether as all active sites became occupied. Spontaneous restructuring during a collision event has also been modeled (16, 17) and was shown to increase the fractal dimension, in the case of RLCA with three-stage spontaneous restructuring, up to a mass fractal dimension of approximately 2.25. Primary particle polydispersity is important to understand from a practical perspective as virtually all aggregation processes apart from those in carefully controlled laboratory experiments involve primary particles of varying sizes. In particular, the treatment of water to remove suspended solids involves the addition of coagulants and flocculants such as ferric chloride and various polymers. These produce flocs constructed of relatively large particles of the suspended solid and very small particles of the hydrolyzed

459

0021-9797/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.

460

BUSHELL AND AMAL

coagulant and/or strings of polymer. Understanding the structural effects of significant polydispersity of primary particles in aggregates is an essential step toward the application of the equations describing fractal aggregates to real-world situations. Most discussions on fractal aggregate structure in the literature have been confined to the case of monodisperse spherical primary particles, although the effects of primary particle size polydispersity have been given some attention (4, 18, 19, 20, 21). Farias et al. (18) assumed that polydispersity had no effect on the interparticle structure factor, although the work of Hasmy et al. (21) had already shown that it substantially lowers the value of q at which the fractal-to-Porod scattering transition occurs. Dimon et al. (4) also investigated the effect of polydispersity on the interparticle structure factor and found it to be unimportant, although the systems they considered had quite small polydispersities (;8– 13% rms width in particle diameter). In this paper we perform DLCA simulations with mono-, bi-, and tri-disperse primary particles. We develop models for partial distance distribution functions and partial structure factors in order to understand the scattering behavior of such systems, and demonstrate that the solid volume fraction of these aggregates can be determined using a fractal mean particle size. We also compare the cutoff functions, or Guinierto-fractal transition regions for the different primary particle polydispersity cases to determine whether or not the large scale features of fractal aggregates are affected by primary particle polydispersity. SIMULATIONS

The computer model used in this work is of the off-lattice three-dimensional DLCA type, with the modification that the primary spheres can have a discrete size distribution. The number of particles for a simulation is determined initially and then the particles are chosen, weighted according to their number fraction from the primary particle size distribution. Aggregation events happen sequentially. For each aggregation event a pair of particles (or as the simulation proceeds, aggregates) is chosen at random from the list of available particles, weighted according to a kernel that results in DLCA-type kinetics (7, 22). To perform each aggregation event, one of the chosen particles or aggregates is placed in a random direction from the other at a distance such that the centers of imaginary spheres just enclosing each aggregate are separated by a distance 10% greater than the sum of their radii. One of the aggregates is then held in a fixed position and the other allowed to diffuse according to a random walk with step size 20% of the smallest primary particle radius and a mean free path of three smallest primary particle radii. If the aggregates contact, they stick irreversibly with no restructuring to form a new aggregate, or if the distance between them becomes greater than a specified distance the event is restarted with the same aggregates at a newly chosen random starting position. The aggregation so

proceeds until only one aggregate, containing all of the initial particles, remains. The partial distance distribution functions of the simulated aggregates were determined in order to examine the aggregate structure and calculate the scattering behavior. The partial distance distribution function c ij (r) is simply the number of particles of type j found at a distance r from all particles of type i, c ij~uru! 5

O f ~r9!f ~r* 1 r!, i

j

[3]

r*

where f i (r*) is equal to one if a particle of type i is present at vector r* from an arbitrary origin, and zero otherwise. For a given magnitude, the vector r is taken over all orientations. Determination of the partial distance distribution functions using a computer is a simple matter of counting the number of different particle pairs for each pair type having separations within different discrete distance classes. For this work, each distance class had a width of 20% of the smallest primary particle radius. The distance distribution function of fractal aggregates (between length scales ;5r 0 , r ! R) manifests the mass fractal dimension in the manner c~r!}r Df21

[4]

in any space dimension as a direct consequence of differentiating Eq. [1]. The scattered intensity from the aggregates of monodisperse particles is often expressed as the so-called structure factor S(q) which is the space Fourier transform of the density– density correlation function f(r) of the scattering aggregate ( f(r) 5 c(r)/(4 p r 2 ) in 3-dimensional space). The variable q has units L21 and in the context of light scattering is the momentum transfer of scattered light. If we integrate over all orientations then we have the expression (23)

S~q! 5

E

`

c~r! z

0

sin~qr! dr. ~qr!

[5]

In the Rayleigh Gans Debye scattering limit (very low refractive indices and absorption), the scattered intensity I is equal to this structure factor multiplied by the square of the scattered amplitude function A(q) of the primary particles in the aggregate (23), I~q!}@A~q!# 2S~q!,

[6]

where for homogeneous particles, A~q!}

Oe k

2iq z OMk

.

[7]

461

FRACTAL AGGREGATES OF POLYDISPERSE PARTICLES

TABLE 1 Number of Primary Particles in the Simulations 1

2

3

5

seff

Simulations

1,200 1,157 980 360

0 0 0 720

0 43 212 0

0 0 8 120

0 0.35 0.59 0.55

80 86 211 30

Primary particle radius: Simulation Simulation Simulation Simulation

1 2 3 4

The OMk are vectors describing the positions of each of the elementary scattering units in the particle. The amplitude function squared is the same thing as the structure factor applied to a single particle and is sometimes called the form factor P(q). In the case that there are multiple particle sizes in the aggregate one can separate the overall aggregate structure into partial distance distribution functions, from which partial structure factors (24) can be computed. Each self-distance distribution function c ii or self-structure function S ii represents the structure of that particle type alone in the aggregate. The partial structure functions are then

S ij~q! 5

E

`

c ij~r! z

0

sin~qr! dr ~qr!

O O A ~q! A ~q!S ~q!. i

i

j

ij

RESULTS

DLCA simulations were performed with a number of different primary particle polydispersity conditions, shown in Table 1. The (dimensionless) effective standard deviation seff is as defined in (21):

[8]

s eff 5

and the total scattered intensity is the sum of partial scattered intensities I~q!}

over the size distribution, which was taken to be narrow and Gaussian. This has the effect of smearing the primary particle scattering over a wider range of q, suppressing the scattering behavior observed from monodisperse spheres whereby the intensity drops to zero at certain values of q.

[9]

j

Dimon et al. (4) approach the problem of primary particle polydispersity by averaging the primary particle form factors

FIG. 1. Distance distribution function for an aggregate of monodispersed primary particles.

1 2r 0

Î

1 N

O ~2r 2 2r ! . N

2

i

0

[10]

i51

In all simulations the aggregation mechanism was DLCA; the fractal dimension as determined from the slope of the partial distance distribution functions was independent of the primary particle size distribution and equal to 1.78 6 0.03. Figure 1 shows the distance distribution for simulation 1, the slope of the dotted line as we recall from Eq. [4] is equal to D f 2 1. In

FIG. 2. Partial distance distribution functions for an aggregate of bidispersed primary particles.

462

BUSHELL AND AMAL

FIG. 3.

Visualization of an aggregate from simulation 1, monodispersed primary particles.

Fig. 2 the partial distance distribution functions for simulation 2 are shown. Note that the slopes are all the same and all equal to 0.78, indicating that the fractal dimension is unaffected by the presence of multiple primary particle sizes, consistent with the work of Tence et al. (19). Shown in Figs. 3– 6 are images of the final aggregate taken from randomly chosen examples of simulations 1– 4, respectively. The structure in each of the four cases appears qualitatively similar, consistent with the conclusion that the mass fractal dimensions are the same. The individual particles that make up the aggregates can be seen to be more or less randomly dispersed throughout the aggregates with no particular connective preference, as is to be expected in a DLCA simulation where all collisions result in aggregation. The general effect on scattering of wide polydispersity in

primary particle size is a characteristic departure from the usual q 2D f dependence at the high q end of the spectrum. This can be qualitatively explained (6, 25) by the dilution of the larger particles within the fractal structure of the smaller ones. The magnitude of the fractal correlation between large particles is thereby reduced, whereas the self scattering is unchanged. In all primary particle polydisperse cases studied, the calculated scattered intensity function exhibited a characteristic departure from power law behavior at high q. In the extreme case where large particles are bound up in an aggregate of small particles with little or no contrast, the departure from q 2D f will be great, interfering with the determination of D f. A graphic example of this is shown in Fig. 7, which shows the partial and total scattering from aggregates in simulation 2, assuming the small

463

FRACTAL AGGREGATES OF POLYDISPERSE PARTICLES

established by Hasmy et al. (21) and the small dots their data. Beyond the conclusion that 2q c r 0 is certainly lower for our polydisperse cases than for the monodisperse simulation 1, agreement between our work and that of Hasmy et al. (21) is not good. This is almost certainly due to the fact that Hasmy et al. used Gaussian distributions for their primary particle sizes, whereas we used arbitrarily chosen combinations of monodisperse size classes. This is borne out to some extent by our own data: the bidisperse simulation 2 has seff 5 0.35 and 2q c r 0 5 2.1, whereas the tridisperse simulation 4 has 2q c r 0 almost the same at 2.2, but seff 5 0.55. Tridisperse simulation 3 has seff quite close to simulation 4 at 0.59, but 2q c r 0 5 2.7, opposite to the expected trend. MODEL AND DISCUSSION

Partial Distance Distribution Functions The form of Eq. [1] applies not only to the scaling of aggregate mass as a function of external size but also to the mass contained within an imaginary hypersphere of size r within the aggregate,

n5j

FIG. 4. Visualization of an aggregate from simulation 2, bidispersed primary particles.

particles have a relative scattering power per unit volume ten times smaller than that of the large particles. It is clear that in general the departure from q 2D f scattering becomes more pronounced as a particular species of particle becomes more diluted within the aggregate. This increases the departure of its partial structure factor from q 2D f and, as its relative contrast increases, increases that particular species’ contribution to the overall scattering. The same departure from power law behavior can be seen in the work of Hasmy et al. (21). In their Fig. 2a, I(q) z q 4 was plotted against qa 0 (a 0 is the number weighted mean primary particle diameter—2r 0 ) in order to demonstrate the effect of primary particle polydispersity in moving the fractal-to-Porod scattering transition point. One can also see in that figure that as polydispersity increases the fractal part of the curve becomes less linear, especially when seff 5 0.34. For each of the simulations we have plotted I(q)zq 4 versus 2qr 0 in order to determine the position of the fractal-to-Porod scattering transition point 2q c r 0 . The results are shown in Fig. 8. The curves are labeled 1, 2, 3, and 4, corresponding to the monodisperse, bidisperse, and two tridisperse cases listed in Table 1. The obtained values of 2q c r 0 are shown as crosses in Fig. 9, with the solid line representing the linear behavior

SD r ro

Df

1 u,

[11]

where n is the number of particles having radius r 0 enclosed within a sphere of radius r. D f is the mass fractal dimension, j is a proportionality constant, and u is a scaling correction term in the spirit of that introduced by Ball and Jullien (26) and Warren (27). When there are multiple types of primary particles present within the aggregate, the equation is easily generalized with the introduction of a factor x j that represents the concentration of particle type j within the overall fractal structure of the aggregate:

SD

n j 5 jx j

r rj

Df

1 u j.

[12]

The partial distance distribution function for i–j particle pairs can be constructed from Eq. [12]. Equation [12] describes the number of type j particles within a hypersphere of radius r centered on an arbitrary particle. Differentiating it with respect to r gives the number of particles of type j at a distance r from the particle at the origin. Multiplying by the number of particles of type i, n i , gives c ij (r). The c ij (r) then represents the number of j-type particles located at distance r from all particles of type i,

c ij~r! 5 n i

dn j f z D f z r Df21, 5 n i z j z x j z r 2D j dr

[13]

464

BUSHELL AND AMAL

within the limits ;(ri 1 4rj) , r ! R (compare to Figs. 1 and 2). We checked this result by comparing the predictions of the above equation with simulation results. The particle types in the simulations are labeled consecutively in order of increasing size. Figure 10 shows three arbitrarily chosen partial distance distribution functions (c13, c22, and c33) from simulation series 4, corresponding to radius 1–radius 5, radius 2–radius 2, and radius 5–radius 5 distance distribution functions. The model (dotted lines) fits the simulation data (solid lines) quite well. Of course the distance distribution functions of the simulated aggregates decay to zero as the edge of the aggregate is reached (r approaches R). The model remains to be multiplied by a suitable cutoff function (convoluted with the cluster size distribution), though this is beyond the scope of the current discussion. The factor j was determined from the monomodal simulation to be equal to 1.2. The concentrations x i were determined by dividing the number of particles of type i in the aggregate by the number of particles of type i that would be required on their own to make up an aggregate of the same radius of gyration. Solid Volume Fraction We have seen that for diffusion-limited aggregation when the primary particles are polydisperse in size, aggregates are not mass fractals in the traditional sense because the mass contained within hyperspheres of increasing size is heavily

FIG. 5.

biased by the few large particles which contribute a lot of mass at r 5 0 (25) (i.e., the term u in Eq. [11] becomes large). The fractal nature of these aggregates is, however, preserved in the sense that the mass present at a distance r from a given occupied point (i.e., at the surface of the hyperspheres) still obeys the power law m~r!}r Df21,

[14]

for r greater than a few r 0 . This is essentially describing the same scaling as Eq. [13]. This feature allows us to derive an expression for the solid volume fraction of aggregates with multiple primary particle sizes. We solve the problem by considering an average primary particle size that when aggregated will produce a structure having the same mass and the same radius as the aggregate for which we are trying to determine the solid volume fraction. Consider a string of particles of different sizes r i lined up in a row like beads on a necklace. Such an aggregate has a mass fractal dimensionality of 1. In order to predict the solid volume fraction of such an aggregate, a single particle size r# must be chosen such that the linear aggregate of these particles has the same mass and the same length as the original structure. nr# 5

O nr

i i

i

Visualization of an aggregate from simulation 3, tridispersed primary particles.

[15]

465

FRACTAL AGGREGATES OF POLYDISPERSE PARTICLES

FIG. 6.

n~r# ! 3 5

Visualization of an aggregate from simulation 4, tridispersed primary particles.

O n ~r ! i

3

i

[16]

O n ~r !

n~r# ! Dw 5

i

i

O n ~r ! n~r# ! 5 O n ~r ! i

2

i

[17]

i

3

i

3

i

[18]

i

In the general D f dimensional case we can say that the model aggregate of monodisperse elements must have the same D w dimensional weight and the same D f space filling characteristic as the distribution of spheres present in the actual aggregate, n~r# ! Df 5

O n ~r ! i

i

i

Dw

,

[20]

i

In the case of a planar aggregate of various particle sizes, the aggregate modeled with a single particle size must have (neglecting the complication of close packing of the spheres) the same area and the same mass as the original aggregate, n~r# ! 2 5

i

Df

[19]

giving us an expression for an average size of the constituent particles which when used alone to create an aggregate of the same linear size and same D f dimensionality will exhibit the same amount of D w measure:

O n ~r ! r# 5 S O n ~r ! i

i

i

i

i

i

D

Dw 1/Dw2Df Df

.

[21]

The solid volume fraction for such aggregates can then be calculated as

r5e

SD Re r#

Df23

1 u 9,

[22]

where R e is the external radius of the aggregate and u9 is an error term analogous to u in Eq. [11]. This result is easily

466

BUSHELL AND AMAL

FIG. 7. Calculated partial and total scattered intensity functions from an aggregate of bidisperse primary particles, large particles with relatively high scattering power.

checked against the simulation data: the actual solid volume fraction is simply the volume of the constituent particles divided by the volume of the smallest sphere enclosing the aggregate. In Table 2, we compare the actual and modeled solid volume fraction for the 4 simulations. The value of e, although theoretically related to j through the cutoff function, was determined empirically from the monomodal simulation 1 to be 0.459 in this case, assuming that the error term u9 is negligible for aggregates of 1200 particles. In real applications e and j are expected to be significantly different from the values reported here as they are quite dependent on the shortrange structure of the aggregates, which may not be well modeled by DLCA.

FIG. 8. Scattered intensity curves plotted to show the fractal-to-Porod scattering transition point 2q c r 0 , indicated by small vertical bars.

FIG. 9. Fractal-to-Porod scattering transition point as a function of the primary particle number weighted primary particle radius standard deviation, seff. The solid line is the linear behavior proposed by Hasmy et al. (21); the dots are their data and the crosses are the results of this work.

Equation [21] has consequences for the contention of Hasmy et al. (21) that primary particle polydispersity has no effect on the overall size of aggregates. We have calculated the fractal mean particle size r# corresponding to D w 5 0 and D f 5 2 for the Gaussian distributions used by Hasmy et al. with seff 5 0.04, 0.08, 0.16, and 0.34. This is the monodisperse primary particle size that is required to produce an aggregate of the same size and same number of particles as the polydisperse case. We find that r# equals 1.0008, 1.0032, 1.0127, and 1.0543, respectively. Thus the more polydisperse of the aggregates generated by Hasmy et al. should be of the order of a few percentage points larger than the monodisperse ones. This is

FIG. 10. Selected partial distance distribution functions from a trimodal case (solid lines) and the corresponding model distance distribution functions (Eq. [13], dotted lines).

467

FRACTAL AGGREGATES OF POLYDISPERSE PARTICLES

TABLE 2 Actual and Predicted Solid Volume Fractions

Simulation Simulation Simulation Simulation

1 2 3 4

r#

Re

( i n i r 3i

ractual

rpredicted (Eq. [22])

1 1.46 2.42 3.29

83.1 92.6 128.2 184.4

1,200 2,361 7,704 21,120

0.00209 0.00297 0.00366 0.00334

— 0.00291 0.00363 0.00337

only a small difference which would be difficult to detect graphically, impossible on a log scaled plot. Cutoff Function Real aggregates are of course finite in size (unless they gel), whereas fractals are infinite in extent. To account for the finite extent of real aggregates, a cutoff function h(r/R) is introduced, where R represents a measure of the size of the aggregate. c~r! 5 r Df21 z h~r/R!

[23]

The form of h(r/R) must be such that it satisfies the conditions h~r/R! , 1

r!R

[24]

h~r/R! 3 0

r . R.

[25]

Additionally, in order that the fractal scaling in the distance distribution function c(r/R) is recovered, h(r/R) must decay to zero faster than any power law. The use of the correct form of the cutoff function is essential to have any hope of recovering the correct mean aggregate size and aggregate size distribution from the light scattering data (28). Several forms for the cutoff

FIG. 11. Comparison of normalized scattering curves for aggregates of monomodal, bimodal, and two cases of trimodal particles.

function have been proposed, including exponential (8), stretched exponential (29), overlapping spheres (30), and Gaussian (31). The cutoff function has been discussed by Lin et al. (32), and a good review is also given by Sorensen (33). The most appropriate form for the cutoff function is a complex, and we believe as yet unresolved, issue that will be treated separately in a future publication. Whatever the correct form for the cutoff function, it is of interest to know whether or not it is affected by the primary particle size distribution. We can check this by comparing the structure factors for the four different simulations considered so far. Each of the equal mass bimodal and two arbitrary trimodal cases were normalized by S(0) and plotted as functions of qRg and compared with the monomodal case. Figure 11 shows that there is no apparent difference between the structure factors. At each q value, Student’s T-test was performed to determine whether the means of the structure factors were different. As we can see from Fig. 12, there is no significant difference between the different polydisperse cases and the monodisperse case at the 70% confidence level. Indeed, one would not expect the primary particle size distribution to have any effect on the form of the cutoff function as the cutoff function is concerned with structural fea-

FIG. 12. Student’s T-test shows no significant difference between scattering curves (and hence cutoff functions) for the different simulations at the 70% confidence level.

468

BUSHELL AND AMAL

tures on the scale of the aggregate, very much larger than the influence of primary particles. SUMMARY

We have performed diffusion-limited cluster– cluster aggregation simulations with polydisperse primary particles. The resulting structures had mass fractal dimensions of 1.78 6 0.03, independent of the size distribution chosen for the primary particles (consistent with the findings of Tence et al. (19)). Scattering and structural effects associated with polydispersity of primary particles in fractal aggregates can be modeled quantitatively using partial distance distribution (or equivalently, density correlation) functions and structure factors. The deviations from power law scattering produced by these effects can be severe, particularly when there are very diluted (small x i ) particles with high contrast present in the aggregate. The form of the cutoff function is, however, not affected by primary particle size polydispersity. We find a variation in the position of the fractal-to-Porod

scattering transition point broadly similar with that found by Hasmy et al. (21). The exact position of the transition point is, however, somewhat dependent on the details of the primary particle size distribution rather than being encapsulated by a simple measure of polydispersity such as standard deviation of the radius. The linear behavior found by Hasmy et al. may only be accurate for Gaussian-type size distributions. An expression for a fractal mean primary particle size is developed, which is of general use in determining the way macroscopic features of fractal aggregates are affected by primary particle size polydispersity. In particular, the volume weighted D f mean particle size allows the extension of the calculation of aggregate solid volume fraction to the polydisperse case. This has significant implications to industrial and natural processes which involve packing or sedimentation of aggregates of polydisperse particles. It will be particularly useful when there is a wide bimodal distribution of primary particle sizes, such as the coagulant and contaminant particles in water and waste water treatment.

APPENDIX: NOMENCLATURE

Variable

Units

i, j

—

M m R r0 Df Dw seff r

M M L L — — —

c ij r r fi(r) q qc S I A u e ni j xi r# h

— L L — L21 L21 — ML2T22 M1/2LT21 — — — — L —

Description Particle type index. Numerical particle indices start at the smallest particle type and increase toward the largest. Total aggregate mass Mass within a differential shell A linear measure of the overall size of an aggregate Number mean primary particle radius Mass fractal dimension Weighting dimensions Effective standard deviation in primary particle polydispersity Solid volume fraction, equal to the volume of the solid constituent particles divided by the volume of the smallest sphere that fully encloses the aggregate. The i–j distance distribution function Distance Displacement A function equal to 1 if a particle of type i is present at r, 0 otherwise Momentum transfer Critical q value at which fractal-to-Porod scattering transition occurs Structure factor Scattered intensity function Scattered amplitude function Error term Proportionality constant Number of particles of type i in the aggregate Proportionality constant Concentration of particle type i in the fractal structure Fractal mean primary particle radius Cutoff function

FRACTAL AGGREGATES OF POLYDISPERSE PARTICLES

ACKNOWLEDGMENT The authors would like to thank the Australian Research Council for the funding which supported this work.

15. 16. 17. 18.

REFERENCES

19.

1. Meakin, P., Phys. Rev. Lett. 51, 1119 (1983). 2. Kolb, M., Botet, R., and Jullien, R., Phys. Rev. Lett. 51, 1123 (1983). 3. Weitz, D. A., Huang, J. S., Lin, M. Y., and Sung, J., Phys. Rev. Lett. 54, 1416 (1985). 4. Dimon, P., Sinha, S. K., Weitz, D. A., Safinya, C. R., Smith, G. S., Varady, W. A., and Lindsay, H. M., Phys. Rev. Lett. 57, 595 (1986). 5. Amal, R., Raper, J. A., and Waite, T. D., J. Colloid Interface Sci. 140, 158 (1990). 6. Bushell, G. C., Amal, R., and Raper, J. A., Physica A 233, 859 (1996). 7. Martin, J. E., Wilcoxon, J. P., Schaefer, D., and Odinek, J., Phys. Rev. A 41, 4379 (1990). 8. Sinha, S. K., Freltoft, T., and Kjems, J., in “Kinetics of Aggregation and Gelation” (F. Family and D. P. Landau, Eds.). Elsevier, Amsterdam/New York, 1984. 9. Zhou, Z., and Chu, B., J. Colloid Interface Sci. 143, 356 (1991). 10. Sorenson, C. M., Cai, J., and Lu, N., Appl. Opt. 31, 6547 (1992). 11. Meakin, P., Chen, Z-Y., and Deutch, J. M., J. Chem. Phys. 82, 3768 (1985). 12. Hess, W., Frisch, H. L., and Klein, R., Z. Physik B 64, 65 (1986). 13. Veerapaneni, S., and Wiesner, M. R., J. Colloid Interface Sci. 117, 45 (1996). 14. Jullien, R., and Kolb, M., J. Phys. A. 17, L776 (1984).

20. 21. 22. 23. 24. 25.

26. 27. 28. 29. 30. 31. 32. 33.

469

Meakin, P., and Miyazima, S., J. Phys. Soc. Japan 57, 4439 (1988). Meakin, P., and Jullien, R., J. Chem. Phys. 89, 246 (1988). Jullien, R., and Meakin, P., J. Colloid Interface Sci. 127, 265 (1989). Farias, T. L., Koylu, U. O., and Carvalho, M. G., J. Quant. Spectrosc. Radiat. Transfer 55, 357 (1996). Tence, M., Chevalier, J. P., and Jullien, R., J. Physique 47, 1989 (1986). Hasmy, A., Foret, M., Pelous, J., and Jullien, R., Phys. Rev. B 48, 9345 (1993). Hasmy, A., Vacher, R., and Jullien, R., Phys. Rev. B 50, 1305 (1994). Botet, R., J. Phys. A Math. Gen. 18, 847 (1985). Guinier, A., and Fournet, G., “Small Angle Scattering of X-Rays.” Wiley, New York, 1955. Salgi, P., and Rajagopalan, R., Adv. Colloid Interface Sci. 43, 169 (1993). Bushell, G. C., Amal, R., and Raper, J. A., in “Fractals and Chaos in Chemical Engineering—Proceedings of the International CFIC 96 Conference” (M. Giona and G. Biardi, Eds.), p. 225. World Scientific, Singapore, 1997. Ball, R. C., and Jullien, R., J. Phys. Lett. Fr. 45, L1031 (1984). Warren, P. B., J. Physique I 3, 1509 (1993). Sorensen, C. M., Lu, N., and Cai, J., J. Colloid Interface Sci. 174, 456 (1995). Mountain, R. D., and Mulholland, G. W., Langmuir 4, 1321 (1988). Hurd, A. J., and Flower, W. L., J. Colloid Interface Sci. 122, 178 (1988). Jullien, R., J. Physique I 2, 759 (1992). Lin, M. V., Klein, R., Lindsay, H. M., Weitz, D. A., Ball, R. C., and Meakin, P., J. Colloid Interface Sci. 137, 263 (1990). Sorenson, C. M., in “Handbook of Surface and Colloid Chemistry” (K. S. Birdi, Ed.), p. 533. CRC Press, New York, 1997.