J. CHEM. SOC. FARADAY TRANS., 1990, 86(4), 671-674 671

Determination of an Average Rate Constant for the Rapid Coagulation of Polydisperse Suspensions using Photon Correlation Spectroscopy

Thelma M. Herrington and Brian R. Midmore Department of Chemistry, University of Reading, Reading RG6 2AD

The rapid aggregation of various polydisperse mixtures of polystyrene latices and one titanium dioxide suspen- sion has been investigated using a photon-correlation technique. A method has been developed that gives reasonable values for the Smoluchowski rate constant of polydisperse suspensions without precise knowledge of t hei r part icl e-s ize d ist r i but i on.

Aggregation rate is an important measure of general colloid stability and also of stabilizing agent and aggregating agent efficiency. The measurement of this rate is not necessarily an easy task and it is perhaps for this reason that accurate studies have mainly been confined to monodisperse systems.' Most real colloidal systems, which might be of interest to industry, are of course polydisperse and development of a re- liable method for determining their rates of aggregation would be of significant value. Previous experimental and

divide g'(z) into two contributions, one from singlets and the other from aggregates. We then make the approximation that the hydrodynamic radii of the aggregates are the same as the hydrodynamic radii of aggregates formed from a mono- disperse suspension of spheres with a radius equal to the weighted mean of the radii of the polydisperse suspension, r' = 1 c, r, . The expression for g(')(z) is then given by

k g"'(z) =

theoretical work2-6 on the aggregation of polydisperse jmax

svstems has concentrated on size-distribution develoDment in r N 1 ( E ) 1 ck M,(e)exp(-rk Z, i- 1 N,(E)ii"(e)exp(-rj / L aerosol systems. In a previous publication7 we showed how

photon correlation spectroscopy could be used to determine the Smoluchowski rate constant of monodisperse polystyrene latices. In this paper we extend the technique to polydisperse

k j= 2 -I/

systems and show how an accurate rate constant can be determined for an aggregating powder suspension. The tech- nique should in principle be applicable to emulsion coagu- lation, which may prove of value in such areas as the food and cosmetic industries.

Theoretical The relationship between the empirical normalized intensity autocorrelation function g(2)(r) and the theoretical normal- ized field autocorrelation function g(l)(z) is given by the Siegert relation :

g'2'(Z) = 1 + c I g(')(z) l2 (1)

where C is a constant determined by the optics of the scat- tering apparatus. The field autocorrelation function g(')(z) for a polydisperse suspension of spheres is given in its discrete form as8

with

k and

r, = K ~ D , .

K = (4n/A)n0 sin(8/2) is the scattering wave vector where 2 is the wavelength of the incident light, no the refractive index of the dispersion medium and 8 the scattering angle. Mk(8) is the Mie scattering parameter, and c, is the number ratio of each particle with diffusion coefficient D, . In order to calculate g(')(z) for an aggregated polydisperse spherical system it is necessary to make some approximations. We choose to

with r; = K'D;. N , ( E ) is the number density of singlets, i;(8) is the intensity of light scattered from an average j-fold aggre- gate, NAE) and D; are, respectively, the number density and diffusion coefficient for an average j-fold aggregate. In order to construct a theoretical autocorrelation function we need values for N , , c,, M , , Nj, i;, and D ; . If Smoluchowski kinetics are assumed then the value of N,(E) is given by9

N ( E ) E'- ' A- - N o (l+EY'+' (4)

where N o is the initial number density of the suspension. E is defined as the ratio of the time t to particle halving time tl,2 and is related to the Smoluchowski second-order constant k2 by

E = k 2 Not. ( 5 )

Attributing a reasonable value to the Mie scattering param- eters proved less troublesome than anticipated. To a first approximation M , is proportional to the fourth power of the radius and we used this estimation in our analysis. Attempts to improve this estimate using more complicated algorithms had no effect on the generated results. If, on one hand, the particles are very different in size, then, because of the approximate fourth-power relationship, the smaller particles contribute very little to the total light scattered. Hence, even fairly large proportional errors in the ratio of the Mie param- eters result in small absolute errors in the total light scat- tered. If, however, the particles are of similar size, the suspension approximates to monodisperse form, in which case the Mie parameters cancel. It seems, therefore, that this analysis is very insensitive to errors in M , .

The theory of Benoit et a1." gives an expression for the light scattered from a suspension of aggregated unequally sized spheres. Following Lips et al.," we make the assump- tion that this theory, which applies rigorously to Rayleigh

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672 J. CHEM. SOC. FARADAY TRANS., 1990, VOL. 86

scatterers, is also applicable to Mie scatterers. The light scat- tered from a single average j-fold aggregate, i; , is

ij” = j 1 M,(B)/k, k

2 ” + - 1 1 C ~ n ( e ) ~ m ( e ) I ” ’ (sin KrnmYKrnrn * (6)

Here I n , represents the centre-centre separation between spheres n and rn, the summation extending to all pairs of par- ticle centres in the aggregate. k, is number of sphere sizes and N is the number of j-fold aggregates over which iy is aver- aged.

As in the case of monodisperse aggregates, the second term in eqn (6) is best evaluated using computer-simulated aggre- gation, the principles of which have been described

The aggregates to be modelled were assumed to be entirely random in nature due to the absence of any potential barrier. In the case of a polydisperse suspension the additional parameter of the radius of the incoming sphere is included. The number of each type of sphere is governed by its concentration in the suspension to be modelled. As before, the values for M(6) were assumed to be proportional to r4.

In a previous p~bl ica t ion ,~ analysis of model glass aggre- gates falling through glycerine was used to determine the hydrodynamic radii of j-fold aggregates. These values are used in our analysis here assuming the value r’ for the aggre- gate monomer radius to calculate 09.

The effect of polydispersity on the Smoluchowski rate was first described by Muller.13.’4 If the effect of hydrodynamic interact on^'^ are ignored, k,, for unequal spheres of radii r , and r2 is given by

N 1 nm

(7)

where k, , is the rate constant of the equivalent monodisperse aggregation. This equation applies to singlet-singlet aggre- gation and the assumption that this rate constant is main- tained throughout the aggregation process is made. The rate equation for the disappearance of singlet particles is

dN dt - = -2k11 NZ

and for a polydisperse system we have

-- - -2 1 kijNiNj. dN

dt i j (9)

Thus, the effective overall rate constant for a polydisperse system, k , , is given by

1 kijNi Nj

Experimental All water used was MILLI-Q quality and had been filtered through a 0.4 pm filter. The latices used together with the monodisperse Smoluchowski rate constant in 1 mol dm-3 KCl are given in table 1. Latices 1-4 were obtained from the Interfacial Dynamics Corporation, latex 6 was prepared using a standard surfactant-free methodI6 and was cleaned by centrifugation.” The titanium dioxide was Fison’s S.L.R. grade and was dispersed in water at pH 10 without further treatment. AnalaR potassium chloride was used without further purification. The density of the titania particles was determined to be 3.67 g cm-3 by measuring the density of a

Table 1. Latex diameters and Smoluchowski rate constants in 1 mol dm-3 KCI

k, x lOl2/cm3 s - ’ latex dlnm

121 264 303 378 623

1.37 & 0.05 1.68 f 0.08 2.24 2 0.10 2.06 & 0.07 1.87 f 0.07

suspension of known mass fraction. The particles appeared approximately spherical under the electron microscope and the assumption that they approximate to spheres is made throughout. Using the Malvern 4700c apparatus, the z- average mean radius of the particles was determined to be 130 nm, with a polydispersity of 0.180. All suspensions used were ultrasonicated for 10 min immediately prior to use.

The photon correlation spectroscopy system used was the Malvern 470Oc apparatus. The system is homodyne and employs a Coherent Innova 90 laser. The scattering angle was 90” with a laser wavelength of 488 nm. The coagulation experiments were performed by first mixing equal portions of dilute suspensions and 2 mol dm-3 potassium chloride solu- tion. The mixture was then transferred to a precision silica optical cell which was then placed in the thermostatted cell head (25.0 0.1 C) and allowed to equilibrate for 5 min. Intensity autocorrelation functions of the coagulating suspen- sions were then collected at CQ. 2 min intervals allowing 30 s to collect the data for each.

Treatment of Data The principles governing the analysis of the data are essen- tially the same as for the monodisperse systems described in ref. (7). [Note that in ref. (7) the labelling of the axes of fig. 1 should be interchanged.] By means of a computer program the theoretical field autocorrelation g(’)(z) could be calculated from eqn (3) using aggregates up to j = 25. g(’)(z) was then converted to g(,)(z) using eqn (1). The parameter C in eqn (1) was calculated by assuming that the experimental g2(t) were linear at low z and extrapolating back to z = 0 where C = g(,)(O) - 1. The theoretical values of g(2)(z) were then fitted to the experimental values by means of a linear least- squares procedure using the single adjustable parameter E. The standard deviation of this fitting procedure was, on average, five times that of a cumulant fit.

A value of E was thereby determined at each time for the coagulating suspension. Plotting E against No t then yielded a straight line with slope k , which was calculated by a least- squares program. The value of No, the initial number density of the suspension, was determined in two ways. For the bidis- perse and polydisperse latex suspensions No was calculated from the known volume fraction and electron microscope radius of each component. This will be referred to as method (a). In the second method (b), volume fraction/radii data obtained from the 470Oc particle sizer were used. A com- mercial iterative exponential fitting program gave the particle-size distribution of the uncoagulated suspension. This method was employed on the titanium dioxide sol and for comparison with method (a), on the polydisperse latex suspension. The purpose of the comparison between the two methods was to establish a method for determining k, for systems where the number density of each size range was not known exactly. The commercial fitting procedure yielded seven size ranges and their volume ratios. From the centre- point of each size range an average radius was determined

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J. CHEM. SOC. FARADAY TRANS., 1990, VOL. 86 673

which was used to evaluate the number density of each radius from its volume ratio. This method has the advantage that it is partly self-adjusting. Overestimates in the radius, r, yield values of N o that are too low, but also the estimate of the rate is lower, as the fitting procedure calculates a smaller increase in the aggregation parameter, E, to account for a given increase in overall particle size. Using a range of r values for data from an aggregating monodisperse system7 showed that the percentage error in k , was approximately equal to the percentage error in r. Thus the rate constant is much less sensitive to errors in r than the inverse cube relationship between r and N o might suggest. The Malvern apparatus seems, however, to give estimates of the radii that are slightly too large. (This may be due to the near invisibility of very small particles or perhaps stray dust particles.) This was manifested by the negative intercepts in E / N o t plots (physically unrealistic). In order to account for this effect, all the radii measured by the photon correlation spectroscopy apparatus were adjusted downwards by the same percentage so that the E/Not plots passed through the origin. The radii were reduced by 4 and 13% in the cases of the polydisperse latex and titanium dioxide sol, respectively. It is to be remem- bered that unlike method (a) where E / N , t plots have a posi- tive intercept because of pre-aggregation, in method (b) the E / N o t should pass through the origin as here pre- aggregation is accounted for by an increase in the mean radius of the suspension with a corresponding decrease in N o -

Results and Discussion Aggregation experiments, performed in 1 mol dm-3 KC1 sol- ution, were carried out on 1 : 1 (by number) mixtures of latices 6 and 3 (mixture A) and 6 and 1 (mixture B). Experi- ments were also performed on a 1 : 2 : 3 : 2 : 1 polydisperse mixture of latices 1, 2, 3, 4 and 6 (mixture C) and also on the titanium dioxide powder suspension.

A comparison of the theoretical and experimental autocor- relation functions at two points in the coagulation of mixture 3 is shown in fig. 1. The E / N o t plots for mixtures A and B are given in fig. 2. The comparison between methods (a) and (b) for mixture C is shown in fig. 3 and the E / N o t plot for tita- nium dioxide powder, determined using method (b), is given in fig. 4. The theoretical rate constants for these systems were determined from eqn (10) and (7). The equivalent mono- disperse rate constant, k l l , to be used in eqn (7) was taken as the average of the rate constants of each monodisperse system given in ref. (7). In table 2 the experimental and theo-

7 0.4

0.2

0.1 1 0 1 2

T/ I 03 ps Fig. 1. Plot showing autocorrelation function [g2(7) - 1]1’2 at two points in the coagulation of mixture C analysed by method (a). +, E = 0.654; 0, E = 2.95; (--) theory.

3.0

2 . 5

2.0

E

1.5

1 .o

0.5

0 2 4 6 N,t/lO” ~ r n - ~ s

Fig. 2. Plot showing linear dependence of the degree of aggregation, E, with N o t for mixtures A and B. Mixture A: 0, N o = 6.10 x 10’ ~ m - ~ ; 0, N o = 3.05 x 10’ ~ r n - ~ . Mixture B: 0, N o = 4.00 x 10’

3.c

2.F

2.c

E 1.E

1 .c

0.5

0 I I 1 I I

2 4 6 8 10 Not/lO1’ ~ r n - ~ s

Fig. 3. Plot showing comparison between methods (a) and (b) for mixture C in the plot E and N o t . A, Method (a), N o = 4.84 x 10’ cm-3; A, method (b), N o = 3.79 x 10’ cmP3.

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I I I I I

0 2 4 6 8 1 0 1 2

N o t / l O 1 ’ ~ r n - ~ s Fig. 4. As fig 1 but for titanium dioxide suspension. ., N o = 7.40 x ~ O ~ c m - ~ .

retical rate constant of the polydisperse systems used are given. In each case the E/N,t plot is quite linear up to at least E = 2 and in most cases E = 3, indicating that Smolu- chowski kinetics are at least approximately obeyed. The agreement between the theoretical and measured rate con- stant is reasonably good, especially in the case of mixture A. Here the radius ratio is only 2 : 1 so it is more likely that the approximations are valid compared with the 5 : 1 radius ratio of mixture B. Even so the result for mixture B is reasonable, considering the gross nature of the approximations in this case.

The equivalence of the results for the two methods applied to mixtures is of importance. It implies that it is unnecessary to have exact particle-size distributions in order to determine

Table 2. Calculated and experimental polydisperse Smoluchowski rate constants in 1 mol dm-3 KCI

calculated experimental suspension k , x 10’2/cm3 s-’ k , x 10”/cm3 s - l

mixture A 2.20 mixture B 2.30 mixture C 2.10

titanium dioxide

2.18 & 0.16 2.77 0.06 2.47 & 0.06” 2.50 f 0.06’

2.61 f 0.09

Method (a), method (b): see text.

a meaningful estimate of the Smoluchowski rate constant. This is of significance for aggregation studies in polydisperse systems of spheres or approximate spheres such as emulsions or powder suspensions.

Conclusion The experimental method and results analysis described in this paper provide a reliable estimate for the rate constant of aggregation of polydisperse spherical or approximately spherical sols. This should prove of use in the study of emul- sion stability and also in the evaluation of the effectiveness of coagulants in such areas as water purification.

We are planning to extend the photon correlation spectros- copy technique to systems that do not obey Smoluchowski kinetics, by employing a suitable theoretical treatment. This should prove of use in studying the early stages of gelation.

We thank the S.E.R.C. for a grant to B.R.M.

References 1

2

3

4

5

6 7

8

9 10

11

12

13 14 15

16

17

H. Sonntag and K. Strenge, Coagulation Kinetics and Structure Formation (Plenum Press, New York, 1987), chapt. 3. S. K. Freidlander and C. S. Wang, J. Colloid Znterface Sci., 1966, 22, 126. C. S. Wang and S. K. Freidlander, J. Colloid Interface Sci., 1967, 24,170. E. R. Cohen and E. U. Vaughan, 3. Colloid Interface Sci., 1971, 35, 612. G. W. Mulholland, G. Lee and H. R. Baum, J. Colloid Interface Sci., 1977,62,406. K. W. Lee, J. Colloid Znterface Sci., 1983,92, 315. T. M. Herrington and B. R. Midmore, J. Chem. SOC., Faraday Trans. I , 1989,85,3529. H. Z. Cummins and P. N. Pusey, Photon Correlation Spectros- copy and Velocimetry, ed. H. Z . Cummins and E. R. Pike (NATO/Plenum Press, New York, 1977). B23, p. 164. M. von Smoluchowski, Physik Z., 1916,17, 557. H. Benoit, R. Ullman, A. J. De Vries and C. Wippler, J. Chim. Phys., 1962, 59, 88. A. Lips, C. Smart and E. Willis, Trans. Faraday SOC., 1971, 67, 2979. I. G. Clague and E. Dickinson, J. Chem. SOC., Farday Trans. 2, 1984,80, 1485. H. Muiier, Kolloid Z., 1926,38, 1. H. Muller, Kolloid Chem. Beih., 1928,26,257. J. A. Valioulis and E. J. List, Adu. Colloid Znterjiace Sci., 1984, 20, 95. J. W. Goodwin, J. Hearn, C. C. Ho and R. H. Ottewill, Colloid Polym. Sci., 1974, 252,464. I. H. Harding and T. W. Healy, J. Colloid Znterface Sci., 1982, 89, 185.

Paper 9/029721; Received 13th July, 1989

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