Determination of colloid stability using photon correlation spectroscopy

Determination of Colloid Stability Using Photon Correlation Spectroscopy INTRODUCTION Coagulation rates for colloidal particles have been measured by a variety of methods, such as direct counting in the ultramicroscope (1-3) or in the Coulter counter (4, 5), light scattering (6-10), and turbidimetry (2, 11, 12). Detailed theoretical predictions for the first step of coagulation, doublet formation, are available which consider Brownian motion (13, 14), interparticle forces (1, 15-17) and hydrodynamic interactions (18, 19). This note describes a new technique to measure the rate of doublet formation in coagulating suspensions: photon correlation spectroscopy (PCS). PCS coagulation experiments were performed using monodisperse spherical Cr(OH)3 and SiO2 particles as a function of ionic strength. These results are compared to light scattering results available in the literature to validate this technique. The observed critical coagulation concentrations and appropriate ~"potentials were analyzed using DLVO theory to yield effective Hamaker constants, which were compared to literature values. PCS THEORY Photon correlation spectroscopy has been routinely used to measure the z-averaged translational diffusion coefficient of colloidal suspensions; excellent reviews exist in the literature (20, 21). In considering the initial stages of coagulation of monosized particles, it is found that singlets and doublets are present in the suspension. Thompson (22) and others (23-26) have studied bimodal distributions of polystyrene latex spheres in water and observed that PCS was sensitive to small numbers of larger particles in the presence of large numbers of small particles. With sufficiently precise data they were able to determine the two particle diameters and the ratio of scattering amplitudes. This approach is used to study the initial stage of coagulation. For a bimodal distribution, the field autocorrelation function, g'(r), is given by

g'(r)ot e x p ( - r l r ) + f e x p ( - r 2 r )

[ 1]

where r j is the decay constant where i equals 1 for the singlet and 2 for the doublet. The relative strength of ra with respect to l~l (20) is given by

f = N212(m, K) Nltl(m, K)

where X is the wavelength of light in the medium. The decay constant, Fi, is given by Pi = 1)i K2

[3]

where ~0i is the z-averaged diffusion coefficient. The diffusion coefficient can be related to the hydrodynamic diameter, di, by the Stokes-Einstein equation

~)i =

kT 3~r~d~

[4]

where k is the Boltzmann constant, T is the absolute temperature, and ~ is the viscosity of the medium. The hydrodynamic diameter of the doublet can be approximated by the surface diameter of the doublet for low Reynolds numbers (27). Thus, d2 = q~dl. Before coagulation begins, g'(r) will be given by the first term of Eq. [1]. A logarithmic plot of g'(r) gives Fl as the slope and N~Ii(m, K) as the intercept. As doublet formation proceeds, the second term of Eq. [1] increases from zero, since N2 increases and NI decreases. If Pl and r2 are not too different, g'(r) can be approximated by a single exponential curve g'(r)a exp(-I'r).

[5]

For a particular coagulation experiment, we know ]?j and 1"2.Noting Eq. [ 1] we find that any change in the measured value of I' is due to changes in f, assuming that the concentration of triplets and higher order species is negligible. From Eq. [2], we see that f i s proportional to N2. The coagulation of singlets to form doublets obeys the secondorder rate law

dN2 = koV2 dt

[61

where k¢ is the rate constant. In the initial stage of coagulation, we assume that N1 is known and is essentially constant. Thus the initial slope of f with time is proportional to dN2/dt, allowing the rate constant kc to be determined to within a proportionality constant. If the proportionality constant [12(m, K)/(NIII(m, K))] is known, then the absolute value of the rate constant, kc, can be determined; if it is not, the experimental stability ratio, IV, can be determined from the ratio of rate constants for rapid, kr, and slow, ks, coagulation conditions

[2]

W = k,/k,.

[71

All values needed for the proportionality constant [I2(m, K)/(N]Ii(m, K))] are available from experimental data except the scattering intensity per doublet Iz(m, K). The

where Ni is the number density of particles and I~(m, K) is the scattering intensity per particle of relative refractive index, m, and at scattering vector K = (4r/X) sin 0/2, and 584 0021-9797/84 $3.00 Copyright © 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.

Journal of Colloid and InterfaceScience, Vol. 100, No. 2, August 1984

NOTES scattering intensity per doublet can be obtained by the integration over all possible spatial orientations (i.e., angles a and ~ ) of the doublet with respect to the scattering plane

12(m'K)= ~'/2 fo'/2Iil(°t' ~) + i2(°t'~)]

[8]

where it and/2 are the angular intensity functions where subscripts 1 and 2 denote incident radiation polarized perpendicular to and parallel to the scattering plane. The angular intensity functions ij and i2 for two identical (Rayleigh) spheres of any spatial orientation are given by Levine and Olaofe (39) (Eqs. [48] and [51]). The solution to Eq. [8] for various-sized spheres at various scattering angles will be addressed in a future note. EXPERIMENTAL All chemicals were analytical reagent grade, used without further purification. The solutions were made using deionized water (p ~> 16 × 106 ohm-cm) and were filtered through 0.22-~m Millipore filters. All experiments were performed at 25 _+0.1°C. Monodisperse Cr(OH)3 sol was prepared using the method of Demchak and Matijevi~ (28). The particle size distribution was determined using the polarization ratio method (29, 30), giving a geometrical mean diameter of 0.30 #m with a geometric standard deviation of 1.17. Monodisperse SiO2 sol was produced (31, 32) by the hydrolysis of tetraethylorthosilicate in ethanol solution containing ammonia. The particle size distribution was analyzed by image analysis of TEM micrographs, giving a geometrical mean diameter of 0.56 #m with a geometric standard deviation of 1.03 (32). The particle concentrations of the stock sols were determined by dry weight analysis using a density of 2.42 _+ 0.02 gin/ cm ~ (33) for Cr(OH)3 and 2.1 gm/cm 3 (32) for SiO2. The Malvern Ltd. K7025 correlator used in this study was similar to those described in the literature (21, 34, 35). The 120-channel correlator was controlled by a Commodore 3032 computer, which also analyzed the correlation functidn for F using the cumulant analysis method (21, 36). Selection of the parameters for the coagulation experiments required a compromise between accuracy and sampling (+ analysis speed). Speed was critical for unstable suspensions; data had to be obtained rapidly so that only a small change in the number of doublets occurred during the measurement. These requirements were satisfied using a sample rime of 200 #sec and 2 × 105 samples, which allowed data to be taken and analyzed at a rate of one per minute. Specific coagulation experiments were performed by diluting a stock sol into a solution having the desired salt concentration and pH, contained in a 3-ml glass cuvette. The cuvette was quickly capped, mixed, and transferred to the sample chamber of the correlator for analysis. The correlator gave F as a function of time. From the preceding analysis,f (and therefore N2) was determined as a function

585

of time, since Pl and 1?2 were known. In no case was the data used withflarger than 0.5, assuring that the fraction of triplets and higher was negligible. The coagulation rate constant, k~, was evaluated from the initial slope, and Nt using Eq. [6]. The stability ratio, W, was determined by Eq. [7] for rapid and slow coagulation conditions.

df/dtlt=o,

RESULTS AND DISCUSSION The results of the coagulation experiments for Cr(OH)3 in KC1 and K2SO4 solutions at a pH 3.5 are shown in Fig. 1. The data compare well with those obtained by Bleier and Matijevi~ (9) using low angle light scattering, also shown in Fig. 1. The critical coagulation concentration (CCC) values for Cr(OH)3 were 0.25 M for KC1 and 2.14 × l0 -3 M for K2SO4; the ratio of the CCC's was 0.009, which is about 60% of the value of 0.0156 (1/zt) required by DLVO theory (12). Effective Hamaker constants, A~2I, were calculated from the experimentally determined CCC's and the extrapolated ~'-poteutials (37) using the simple analysis set forth by Reerink and Overbeek (11). The corresponding values for were 3.3 × 10-21 J for KC1, using ~"= 13 mV (37). The average value was 9.6 × 10-21 J, which is larger than the value reported by Bleier and Marijevi6 (9) by a factor of 7. The stability ratios for SiO2 in KC1 and BaC12 solutions at pH 7.0 are given in Fig. 2. The ratio of CCC's for and BaClz was 0.047, which is three rimes larger than that predicted by DLVO theory. The Hamaker constants calculated from the CCC for KC1 and BaCI2 were 3.35 × 10-21 J, using ~"= - 1 6 mV (37), and 7.60 × 10-21 J, using ~"- - 1 3 mV (37), respectively. The average value of At2t was 5.48 X 10 -21 J, which lies well within the range of theoretically predicted and measured values of 2.0 to 17.0 × l 0 -2l J for vitrous SiO2 reported by Visser (38). Photon correlation spectroscopy has been shown to be a suitable method for measuring the rate of homoco-

Atzl

KC1

i

",

i

i

[

i

i

i

i

Mo,,,.i0,9

LOG MOLAR CONCENTRATION

FIG. 1. Cr(OH)2 coagulation. Coagulation rates, expressed as stability ratios, for monodisperse Cr(OH)3 at a pH = 3.5 in KCI (©) and K2504 (A) electrolyte solutions. The normalized curves reported by Bleier and Matijevi6 [9] (- - -) are included for comparison. Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984

586

NOTES i

4

i

,

i

~

5

2 £.9 O .d

j

O

1

I

-4

-5

i

15. 16. 17. 18.

o KCI

ABoC[ 2 JH = 7.0

o o n ~--.o~

/

-

LOG MOLAR CONCENTRATION

FIG. 2. SiO2 coagulation. Stability ratios for monodisperse SiO2 in KC1 (©) and BaC12(A) electrolyte solutions atpH 7.0.

agulation for monodisperse sols. The stability data obtained are similar to those obtained by low-angle light scattering, and effective Hamaker constants agree with those found in the literature. REFERENCES 1. Overbeek, J. Th. G., in "Colloid Science" (H. R. Kruyt, Ed.), Vol. I, pp. 292-295, 282-283. Elsevier, Amsterdam, 1952. 2. Wartillon, A., Romerowski, M., and van Grunderbeck, F., Bull. Soc. Chim. Belg. 68, 450 (1959). 3. Smith, A. L., Spec. Discuss. Faraday Soc. 1, 32 (1970). 4. Higuchi, W. I., Okada, R., Stelter, G. A,, and Lembuerger, A. P., J. Pharm. Sci. 52, 49 (1963). 5. Matthews, B. A., and Rhodes, C. T., J. Colloid Interface ScL 32, 332 (1970). 6. Lips, A., Smart, C., and Willis, E., Trans. Faraday Soc. 67, 2979 (1971). 7. Lips, A., and Willis, E., J. Chem. Soc, Faraday 169, 1226 (1973). 8. Lichtenbelt, J. W. Th., Pathmamanoharan, C., and Wiersema, P. H., £ Colloid Interface Sci. 49, 281 (1974). 9. Bleier, A., and Matijevi6, E., £ Colloid Interface Sci. 55, 510 (1976). I0. Sasaki, H., Matijevi6, E., and Barouch, E., J. Colloid Interface Sci. 76, 319 (1980). I 1. Reerink, H., and Overbeek, J. Th. G., Discuss. Faraday Soc. 18, 74 (1954). 12. Verwey, E. J. W., and Overbeek, J. Th. G., "Theory of the Stability of Lyophobic Colloids." Elsevier, Amsterdam, 1948. 13. Tambour, Y., and Seinfeld, J. H., J. Colloid Interface Sci. 74, 260 (1980). 14. von Smoluchowski, M., Phys. Z. 17, 557, 585 (1916); Z. Phys. Chem. 92, 129 (1917).

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1984

Fuchs, N., Z. Phys. 89, 736 (1934). Cooper, W. D., Kolloid-Z. Z. Polym. 250, 38 (1972). Marmur, A., J. Colloidlnterface Sci. 72, 41 (1979). Spielman, L. A., J. Colloid Interface Sci. 33, 562 (1970). 19. Honig, E. P., Roebersen, G. J., and Wiersema, P. H., £ Colloid Interface Sci. 36, 97 (1971). 20. Chu, B., "Laser Light Scattering." Academic Press, New York, 1974. 21. Cummings, H. Z., and Pusey, P. N:, in "Photon Correlation SpectrosCopy and Velocimetry" (H. Z. Cummings and E. R. Pike, Eds.), pp. 164-199. Plenum, New York, 1977. 22. Thompson, D. S., J. Chem. Phys. 54, 1411 (1971). 23. Bargeron, C. B., Appl. Phys. Lett. 23, 379 (1973). 24. Bargeron, C. B., J. Chem. Phys. 60, 2516 (1974). 25. Lee, S. P., and Chu, B., Appl. Phys. Lett. 24, 201 (1974). 26. Chen, F. C., Tscharnoth, W., Schmidt, D., and Chu, B., J. Chem. Phys. 60, 1675 (1974). 27. Allen, T., "Particle Size Measurement" 3rd ed., p. 104. Chapman & Hall, London, 1981. 28. Demchak, R., and Matijevi6, E., J. Colloid Interface Sci. 31, 256 (1969). 29. Kerker, M., Matijevi6, E., Espenscheid, W. F., Farone, W. A., and Kitani, S., £ Colloid Interface Sci. 19, 223 (1964). 30. Jacobsen, R. J., Kerker, M,, and Matijevi6, E., J. Phys. Chem. 71, 514 (1967). 31. Stober, W., Fink, A., and Bohn, E., J. Colloid Interface Sci. 26, 62 (1968). 32. Huynh, T. C., Ph.D. thesis, M.I.T., Mass., 1984. 33. Bell, A., and Matijevi6, E., J. Phys. Chem. 78, 2621 (1974). 34. Chen, F. C., Yeh, A., and Chu, B.yJ. Chem. Phys. 66, 1290 (1977). 35. Gulari, E., Bedwell, B., and Alkhafaji, S., J. Colloid Interface Sci. 77, 202 (1980). 36. Koppel, D. E., J. Chem. Phys. 57, 4814 (1972). 37. Barringer, E., Ph.D. thesis, M.I.T., Mass., 1983. 38. Visser, J., Advan. Colloid Interface Sci. 3, 331 (1972). 39. Levine, S., and O!aofe, J. Colloid Interface Sci. 27, 442 (1968). E. A. BARRINGER B. E. NOVZCH T. A. RING

Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Received September 26, 1983