Measurement of the hydrodynamic fractal dimension ...

MOLECULARPHYSICS,1987, VOL. 62, No. 2, 411-418

Measurement of the hydrodynamic fractal dimension of aggregating polystyrene spheres by P. N. PUSEY and J. G. RARITY Royal Signals and Radar Establishment, St. Andrews Road, Malvern, Worcestershire, WR14 3PS, England

(Received 9 February 1987; accepted 3 April 1987) Slow (reaction-limited) aggregation of small polystyrene spheres in aqueous suspension was started by the addition of salt and was monitored by conventional and dynamic light scattering at small scattering angle. The average aggregation number and hydrodynamic radius obtained from these measurements were related through a 'hydrodynamic' fractal dimension dh. Once the average aggregate size became larger than about 1 #m we also measured, from the angular dependence of the scattered intensity, a ' structural' fractal dimension din. The two dimensions so obtained agreed within experimental error, implying a proportionality of structural and hydrodynamic radii; their values, about 2.08, are similar to those found by other workers for slow aggregation in three dimensions. 1.

Introduction

The aggregation of many small (seed) particles, suspended in a liquid, to form random clusters is a phenomenon of long-standing interest [13. There has been a recent upsurge of activity in this field involving theory, experiment and, in particular, computer simulation [23. Although a number of models of aggregation in suspension have been investigated, the most realistic appears to be that of clustercluster aggregation (CCA) [3, 4] first considered by Smoluchowski [1]. Here, at a given time during the growth process, new clusters are generated by the combination of clusters formed earlier. It has become apparent that several distinct types of scaling behaviour are found in this process. Firstly, on a spatial scale intermediate between the radius r o of a seed particle and the radius R of a cluster, the internal structure of a cluster shows the scale invariance characteristic of a fractal. Thus the number m(r) of seed particles contained inside a sphere of radius r within the aggregate is given by

m(r) oc ( r ~a', \ro/

(1)

where d,. is a fractal dimension and the constant of proportionality is of order one. A similar equation thus relates the total number of particles in a cluster to a measure R(m) of its radius

m ~ (R(m)~ din. \ ro /

(2)

Secondly the number N[m(t)]dm(t) of clusters with sizes (or masses) between re(t) and re(t) + dm(t) develops with time t during the aggregation process according to 9 Controller HMSO, London 1987

412

P . N . Pusey and J. G. Rarity

the scaling form [5-8]

m(t) m(t) N[m(t)]dm(t)= N ( t ) f [ - ~ ] d [ - ~ ] .

(3)

Here N(t) is the total number of clusters at time t and f(x) is a scaling function normalized by

f xf(x) dx Thus the total mass

=

1.

(4)

M x in the sample is given by M T = f m(t)N[m(t)]dm(t) = N(t)fn(t),

(5)

Fn(t) = MT/N(t )

(6)

which gives a definition

for the average cluster number (or mass) th(t). In this paper we describe experiments using conventional and dynamic light scattering to measure two fractal dimensions of aggregates of small polystyrene spheres in water. During the aggregation process we measured, as a function of time and at small scattering angle, both the average scattered intensity and the hydrodynamic radius obtained from the measured average translational diffusion constant. If the cluster size distribution scales as in equation (3), the fractal dimension dh relating cluster size to hydrodynamic radius (see w2) can be determined from these measurements (figure 1). Secondly, as others have done [9, 10], we measured the angular dependence of the average intensity scattered by large aggregates (figure 2). This yields the fractal dimension d,~ defined by equation (1). Our findings are discussed in w4.

2. Theory We consider light scattering by a dilute suspension of (non-interacting) aggregates at angles small enough that

KR

~ 1,

(7)

where K is the scattering vector and R the radius of a typical cluster. For this situation, independent aggregates scatter light in proportion to the square of their masses. The total scattered intensity, averaged over temporal fluctuations, is given by

(I) w_f mZ(t)N[m(t)]dm(t),

(8)

where we have assumed that the measurement can be made in a time over which m(t) does not change significantly. Taking the scaling form of equation (3) for the size distribution we get

(I) oc N(t)[Fn(t)]2 f x2f(x) dx. d

(9)

1000

ee•o

~ 1 7 69 9

esqe 9

,-" 100 OJ

9

10

I

I

t I I I I

I

[

I

I

I I I 11

100

1000

Hydrodynamic radius

hR /nm

Figure 1. Double-logarithmic plot of intensity (1) (in arbitrary units) against average hydrodynamic radius/~h, both measured at scattering angle 0 = 20 ~ (scattering vector K = 0.448 x 105 cm- 1). The solid line has slope 2.00. The results of two experiments are shown, which differ slightly in terms of initial seed particle concentration no and the setting of the (controlled) laser intensity. The slight drop of intensity observed in the saturation region is probably an experimental artefact e.g. a change of the laser intensity.

100 3~ 121.. ~

rt~

v

10

I

I

I

III

I

I

I

3x10s

0,5 1 Scaffering vecfor K/cm -1 plot of (I(K))/Po(K) (multiplied by

Figure 2. Double logarithmic sin 0, to allow for the changing size of the scattering volume) against scattering vector K (see equations (15) and (18)). The lower set of measurements (dots) were made when R h ~ 180nm and the upper set (crosses) when Rh m 550nm. The solid line has slope -2.08.

414


The last factor in equation (9) is independent of time; using equation (6) gives ( I ) oc MTrh(t).

(10)

Thus, as the aggregation proceeds, the scattered intensity (in the limit of equation (7)) increases with time in proportion to the average cluster size rh(t). From photon correlation spectroscopy one can obtain the initial slope F (or first cumulant) of the time correlation function of the scattered light field. For the situation under consideration this is given by

K z ~ N(m)m2D(m) dm r =

J

,

(11)

f N(m)m dm where D(m) is the translational diffusion coefficient of a cluster of size m. We define the hydrodynamic radius Rh(m) by

kT D(m) - 6ntlRh(m) '

(12)

where kT is the thermal energy and t/the viscosity of the liquid. We further assume that Rh(m) is related to m through an equation similar to equation (2) but with (hydrodynamic) fractal dimension dh

m oc (Rh(m)~ dh \ro/"

(2b)

Then, with use of equation (3), we get an expression for an average hydrodynamic radius

kT r~ Rh = 6ruI(F/K:) oc C

J

f f(x)x2 dx (13) f(X)X 2- l/dh dx

so that /~h OCth 1/e~.

(14)

Thus comparing equations (10) and (14) we see that a plot of In ( I ) versus In /~h, both quantities being measured during the aggregation process, should yield a straight line of slope dh. Note that, when the data are plotted in this way, dh is determined independently of the values of the constants of proportionality in equations (8) and (2 b). When an aggregate becomes large enough that equation (7) no longer holds the intensity it scatters can be written

(I(K)) oc Po(K)S(K).

(15)

Here Po(K) is the form factor of the seed particles. For a homogeneous sphere the Rayleigh-Gans-Debye approximation gives [11] 9

Po(K) = ,,-Z=---~,~(sin Kro - Kro cos Kro) 2. t/~roy

(16)

The hydrodynamic fractal dimension of aggregates

415

The structure factor S(K) is determined by the arrangement of seed particles within the cluster. For scattering vectors K in the range r o ~ K -1 ~ R

(17)

the structure factor of a fractal aggregate is given by [9, 10, 12-14,]

S(K) oc K -d"

(18)

where d,, is defined by equation (1) [15,]. Thus, for scattering vectors in the range (17), a plot of in [(I(K))/Po(K)] versus In K should yield a straight line of slope din.

3. Experimental details and results Samples were prepared from a stock suspension in water of colloidal polystyrene spheres of average radius about 22 nm. In pure water these charged particles are essentially stable due to strong electrostatic repulsions. Addition of salt to the suspension reduces the stabilizing maximum in the (DLVO) inter-particle potential, allowing the particles to get close enough together for the van der Waals attraction to form a permanent bond [16,]. The amplitude of this maximum, and hence the rate of aggregation, is controlled by the concentration of salt in the suspension. The stock suspension was filtered several times through a Millipore filter of pore size 0.22 #m to remove initial aggregates. It was then mixed with a roughly equal volume of sodium chloride solution and introduced quickly through a filter of 0.45 #m pore size into the light scattering cell (1 cm x 1 cm cross-section). The particle concentration was roughly 3 x 1012 cm -3 and the final salt concentration was varied from 0.1 M to 0.25 M. At an NaC1 concentration of 0.25 M large aggregates were formed within seconds and it was impossible to make meaningful light scattering measurements. At 0.1 M salt it took several weeks to form micron-sized aggregates. A reasonable aggregation rate (about one day to R = 1 #m) was achieved with 0.15M salt and this concentration was used for all measurements reported here. Standard light-scattering equipment was used with a Krypton ion laser operating at a wavelength in vacuo of 647 nm. At a scattering angle 0 of 20 ~ measurements were made automatically every few minutes of the average scattered intensity ( I ) and the average hydrodynamic radius defined by equation (13). A correlator sample time much smaller than the coherence time of the scattered light was used. The correlation functions were analysed by cumulant fits up to order four [17]. Little significant difference was found between the first cumulants obtained from the third and fourth order fits and the former are quoted here. The results of two such experiments are shown in figure 1 where In ( I ) is plotted against In/~h- Within experimental error, straight-line behaviour is observed up to/~h ~ 150nm. A line of slope 2.00 is shown in figure 1. On two occasions during the aggregation process, first when /~h ~ 180rim and second with /~h "~ 550 nm, measurements were made of the angular dependence of the average scattered intensity. As well as the usual sin 0 correction for the angular dependence of the scattering volume a small correction for background and dark count was made. The results are shown in figure 2 where In [I(K)/Po(K)] is plotted against In K. Po(K) was calculated from equation (16) taking ro = 22 nm; it deviated from one by less than 6 per cent for all values of scattering vector used. For the larger aggregates reasonable straight-line behaviour is found with slope 2.08 + 0.05.

416

P . N . Pusey and J. G. Rarity 4.

Discussion

The analysis of w2 showed that, with the assumptions of scaling behaviours for both the structure of the individual aggregates and for the time-dependent distribution of aggregate size, a double logarithmic plot of the intensity scattered by an aggregating suspension against the average hydrodynamic radius (both measured at small scattering vector) should yield a straight line. The slope of this line is the fractal dimension d h which relates the hydrodynamic radius of a cluster to its mass. The data of figure 1 shows such linearity up to a mean hydrodynamic radius of about 150 nm. At about this value of Rh, the condition K R ~ 1 (equation (7)) no longer holds (see below) and the plot starts to curve. Eventually, as is evident in figure 1, the intensity becomes independent of time though the cluster radius,/]h, continues to grow. This saturation of the intensity, found when K R >> 1, has been noted by others [9, 10, 12-14] and is a result of the ramified internal structure of the aggregates. As is seen in figure 2, the intensities scattered by aggregates of size Rh ~ 180 nm and R h ~ 550nm are nearly the same for K > 2 x 105 cm -1, implying that saturation has been reached. However an increasing difference in scattered intensity for the two cases is observed at smaller scattering vectors. With the possible exception of the point at K = 0.45 x 105 c m - 1, the data for the larger aggregates can be fitted well by a straight line. Thus, for R h ~ 550 nm, the condition KR >> 1 is fulfilled, equation (18) applies and the slope of the line yields d m . However, for Rh ~ 180 nm and K = 0.45 • 105cm -1 (0 = 20 ~ K R h : 0"81) the scattered intensity is still well below its saturation value (figure 1). The fractal dimensions d h ~ 2"00 and d m ~ 2"08, determined from the slopes of the lines in figures 1 and 2, are, within experimental error (~0.05), essentially equal. In three dimensional space, the lower and upper limits on the value of fractal dimension d are, of course, 1, corresponding to a linear aggregation, and 3, corresponding to the agglomeration of the seed particles into homogeneous spheres. Values of around 2 are consistent with a fairly open ramified but globular structure. It is now well established that for rapid, diffusion-limited aggregation, where clusters form a permanent bond on the first close encounter, d lies in the range 1.7 to 1.8 [-5, 10, 18, 19, 20]. However, for the slow, reaction-limited case, in which the probability is small that, in a brownian encounter, two clusters acquire enough thermal energy to surmount the maximum in the inter-particle potential, different clusters are, on average, able to interpenetrate each other significantly before sticking. This leads to a more compact structure with fractal dimension d around 2.1 [9, 10, 19, 20, 21], the value found in the present work. As indicated in w3 we chose the salt concentration in our experiment to be that corresponding to such a slow aggregation. An estimate of the degree of slowness can be obtained from Smoluchowski's result [-1] fn(t) = 2n ~ -k-T t

(19)

giving the time dependence of the mean cluster size for a fast, diffusion-limited aggregation; here no is the initial number density of seed particles. For our experiments n o ~ 3 x 1012cm -3 so that clusters of mean size 20, for example, should be formed in about 2.5 s under fast aggregation. In fact we found that it took well over one hour to achieve this condition, thus confirming that the aggregation in our system was indeed slow.

The hydrodynamic fractal dimension of aggregates

417

The similarity we find for the values of d~ and d h is not surprising on physical grounds. It implies that an aggregate behaves hydrodynamically like an effective solid sphere with a radius proportional to the aggregate's radius of gyration R h = 3R o,

(20)

where /~ is a constant. Evidently a considerable amount of liquid is entrapped hydrodynamically within the ramified structure of the aggregate. Such behaviour is found both theoretically [22] and experimentally [23] for polymer coils (a gaussian coil also has a fractal dimension of 2). Furthermore Meakin et al. [24] have also obtained similar results for aggregates formed in computer simulations of fast cluster-cluster aggregation, using the Oseen approximation for the hydrodynamic interactions between seed particles (as in the Kirkwood-Riseman theory of polymer friction [-22]). However, in this case, the fractal dimensions were found to be about 1.80, characteristic of the more open structure achieved in fast CCA. Two reservations concerning our data should be stated. The first concerns the degree to which the restriction (7) on scattering vector K is fulfilled in figure 1. For a monodisperse system the intensity scattered at small KR~ is given by Guinier's formula [12-14],

(I(K)) = (I(0))[1 - KZR~/3 +...].

(21)

Taking the value of/~ in equation (20) to be 0.665, that for a gaussian polymer coil [22, 23], and calculating K =(4~zn/2)sin (0/2) (refractive index n = 1-33, 2 = 647 nm, 0 = 20 ~ we find that (I(K)) is approximately 15 per cent smaller than (I(0)) when/~h = 100 nm. Applying such a correction to the data of figure 1 leads to a corrected dh ~ 2.09, now in excellent agreement with d,, = 2-08. When a distribution of cluster sizes is taken into account the difference between (I(K)) and (I(0)) becomes larger. However, with such polydispersity,/~h is also underestimated due to the reduced intensity scattered by the larger aggregates. Unpublished calculations, assuming a power-law form for f(x) (equation (3)) [15], show that these two effects more or less compensate leading to no significant change in the corrected value of dh given above. Secondly we note that, when /~h = 100nm, the average cluster size m(t),~ (R~/ro)ah is of order 20. Thus the linear portion of the plot in figure 1 spans only the early stages of aggregation where a scaling form for the size distribution (equation (3)) may not yet be fully applicable. In conclusion we have determined a hydrodynamic fractal dimension dh of small, slowly-aggregating polystyrene spheres from measurements of the scattered light intensity and mean hydrodynamic radius at small scattering angles. This hydrodynamic dimension is, within experimental error, the same as the mass fractal dimension d~, determined from the variation of scattered intensity with angle when the aggregates are large enough that the intensity has saturated. We find dh ~ d~ = 2.08 • 0.05 in agreement with other measurements on slowly aggregating systems. We thank Professor R. H. Ottewill for donating the polystyrene suspension and Drs R. C. Ball, J. E. Martin and D. A. Weitz for valuable discussions.

418


Note added in proof--Since this paper was submitted Wiltzius (WILTZIUS,P., 1987, Phys. Rev. Lett., 58, 710) has also reported measurements of the hydrodynamic behaviour of fractal aggregates. References [1] VON SMOLUCHOWSKI,M., 1916, Phys. Z., 17, 585. [2] See, for example: (a) FAMILY,F., and LANDAU,D. P. (editors), 1984, Kinetics of Aggregation and Gelation (North Holland); (b) STANLEY,H. E., and OSTROWSKY,N. (editors), 1986, On Growth and Form (Martinus Nijhoff). [3] MEAKIN,P., 1983, Phys. Rev. Lett., 51, 1119. [4] KOL8, M., BOTET,R., and JULLIEN,R., 1983, Phys. Rev. Lett., 51, 1123. [5] KOL8, M., 1984, Phys. Rev. Lett., 53, 1654. [6] BOTET,R., and JULLIEN,R., 1984, J. Phys. A, 17, 2517. [7] VAN DONGEN,P. G. J., and ERNST,M. H., 1985, Phys. Rev. Lett., 54, 1396. [8] MEAKIN,P., VICSEK,T., FAMILY,F., 1985, Phys. Rev. B, 31, 564. [9] SCHAEFER,D. W., MARTIN, J. E., WILTZIUS,P., and CANNELL,D. S., 1984, Phys. Rev. Lett., 52, 2371. [10] WEITZ,D. A., HUANG,J. S., LIN, M. Y., and SUNG,J., 1985, Phys. Rev. Lett., 54, 1416. [11] KERKER,M., 1969, The Scattering of Light (Academic). [12] SINHA,S. K., FRELTOFT,T., and KJEMS,J., 1984, in [2(a)], p. 87. [13] TEIXERA,J., 1986, in [2(b)], p. 145. [14] BERRY,M. V., and PERCIVAL,I. C., 1986, Optica Acta, 33, 577. [15] MARTIN (MARTIN, J. E., 1986, J. appl. Crystallogr., 19, 25; also MARTIN, J. E., and LEYVRAZ, F., 1986, Phys. Rev. A, 34, 2346) has pointed out that, for a polydisperse suspension of aggregates, equation (18) applies only for certain forms of the size distribution N(m). However, he shows that, for the slow aggregation considered in this paper (w where N(m) oc m -~ with z = 1.5 + 0.3 [10], equation (18) should be valid. [16] VERWEY,E. J. W., and OWRBEEK,J. TrI. G., 1948, Theory of the Stability of Lyophobic Colloids (Elsevier). [17] KOPPEL,D. E., 1972, J. chem. Phys., 57, 4814. [18] WEITZ,D. A., and OLIWRIA,M., 1984, Phys. Rev. Lett., 52, 1433. [19] AUBERT,C., and CANNELL,D. S., 1986, Phys. Rev. Lett., 56, 738. [20] DIMON,P., SINHA,S. K., WEITZ,D. A., SAFINYA,C. R., SMITH,G. S., VARADY,W. A., and LINDSAY,H. M., 1986, Phys. Rev. Lett., 57, 595. [21] KOLa, M., and JULLIEN,R., 1984, J. Phys. Lett., Paris, 45, L977. [22] KIRKWOOD,J., and RISEMAN,J., 1948, J. chem. Phys., 16, 565. YAMAKAWA,H., 1971, Modern Theory of Polymer Solutions (Harper & Row). [23] SCHAEFER,D. W., and HAN, C. C., 1985, edited by R. Pecora, Dynamic Light Scattering (Plenum). [24] MEAKIN,P., CI-mY,Z-Y., and DEUTCH,J. M., 1985, J. chem. Phys., 82, 3786.