Langmuir 1990,6, 35-42 the exponent for flocculated suspensions is in the range 2.0-5.3. It should be mentioned, however, that the value of the exponent depends to some extent on the conditions to which a coagulated suspension has been subjected before the measurements were made. Thus, our result of an exponent of 6.0 seems reasonable. Recently, Ballz6used a novel method for calculating the elasticity (25) Russel, W. B. Powder Technol. 1987,51, 15.
35
of individual fractal clusters, and an exponent of 4.5 i 0.2 can be predicted for chemically limited aggregation in three dimensions and of 3.5 f 0.2 for diffusionlimited aggregation. Again, the power predicted by Ball's theory is lower than that found experimentally in our investigation. (26) Ball, R., paper presented a t the Society of Chemical Industry, to be published.
Motions of Particles in Concentrated Dispersions As Observed by Dynamic Light Scattering? W. van Megen* and S. M. UnderwoodS Department of Applied Physics, Royal Melbourne Institute of Technology, Melbourne, Victoria 3000, Australia Received October 4, 1988.I n Final Form: March 27, 1989 The motions of spherical particles in concentrated colloidal dispersions are measured by dynamic light scattering. Two complementary experimental systems are described. The first consists of sterically stabilized polymer particles dispersed in a mixture of organic liquids of a refractive index which is adjusted to match closely that of the particles. Correlation functions of the light scattered by the almost transparent dispersions give the coherent dynamic structure factor which reflects the collective particle motions. The second experimental system is a modification of the first, to which is added a trace number of inorganic particles identical in size and stabilized by the same steric barrier as the polymer (host) particles. Optical conditions are adjusted so that the incoherent dynamic structure factor can be measured, reflecting the motion of single (tracer) particles. Both experimental systems are studied as a function of concentration of polymer particles up to equivalent hard-sphere volume fractions approaching 0.5. The concentration and scattering vector dependence of the coherent and incoherent dynamic structure factors are compared and discussed. Finally, the results of the two experiments are combined to calculate the distinct dynamic structure factor. 1. Introduction
The most direct way to experimentally explore the structure and dynamics of a system of particles is with radiation of wavelength comparable to the particle size. Details regarding the particle motions and their relative positions are then impressed on the (elastically) scattered radiation. For dispersions of particles of colloidal dimensions, dynamic laser light scattering has proved to be a particularly useful technique, and several developments have occurred in recent years which significantly simplify the interpretation of the scattered light in terms of the particle dynamics.' Firstly, the availability of synthetic dispersions of uniform spherical particles with narrow size distributions means that complications due to variations in particle size and shape can be ignored. Examples of such dispersions include polystyrene, poly(methy1 methacrylate) (pmma), and silica spheres, in either electrostatically or
'
Presented a t the symposium on "Rheology of Concentrated Dispersions", Third Chemical Congress of North America, Toronto, June 5-10, 1988. Current address: IC1 Australia Research Group, Ascot Vale, Victoria 3032, Australia. (1) Dynamic Light Scattering; Pecora, R., Ed.; Plenum: New York, 1985.
*
0743-7463/90/2406-0035$02.50/0
sterically stabilized form, dispersed in a variety of liq~ids.~,~ Secondly, the strong multiple scattering, normally encountered in concentrated dispersions of near micronsized particles, can be eliminated by matching the refractive index of the supporting liquid to that of the part i c l e ~ . ~For ' ~ optically matched dispersions of particles, sterically stabilized by thin adsorbed layers, the interparticle force is steeply repulsive and may be approximated by the hard-sphere i n t e r a ~ t i o n . ~ This paper is concerned with measuring the dynamics of these essentially identical hard spherical particles in concentrated dispersions by dynamic light scattering (DLS). The complete characterization of the particle dynamics requires two experiments. Firstly, DLS is performed on dispersions of identical spheres. Here the fluctuations in the scattered light are a direct consequence of the fluctuations in the particle concentration. These are driven by diffusive motions to (2) Polymer Colloids II; Fitch, R. M., Ed.; Plenum: New York, 1980. Science and Technology of Polymer Colloids; Poehlein, G. W., Ottewill, R. H., Goodwin, J. W., Eds.; Martinus Nijhoff: The Hague, 1983. (3) van Helden, A. K.; Vrij, A. J. Colloid Interface Sci. 1980, 78, 312. (4) Pusey, P. N.; van Megen, W. J. Phys. (Les Ulis, Fr.) 1983, 44, 285. (5) Pusey, P. N.; van Megen, W. Nature 1986, 320, 340.
0 1990 American Chemical Society
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van Megen and Underwood
Langmurr, Vol. 6, No. 1 , 1990
which, in general, many particles contribute collectively.6 Modern detectors and signal processors (correlators) allow these intensity fluctuations to be monitored. The scattering angle (or scattering vector) determines the particular spatial Fourier component of the concentration fluctuations actually measured. The second experiment allows access to the motion of single particles in a concentrated dispersion. This requires a dispersion of two or more particle types, differing only in their scattering amplitudes (or refractive index) but identical in those aspects that determine their motion (i.e., their interaction^).^ Experimental conditions are selected so that the average particle scattering amplitude vanishes, thereby eliminating the light scattered by concentration fluctuations. The remaining light fluctuations can then be expressed in terms of single, uncorrelated, particle motion^.^^^ In these experiments, polymer and silica particles, of about 210-nm radius, stabilized with the same thin steric barrier, are dispersed in a liquid mixture whose refractive index is close to that of the particles. The interaction between the particles is a short-ranged repulsion. This is indicated by the observation that the phase behavior of these dispersions mimics that of a system of hard spheres5 In the following section of this paper, we outline the theory required for the interpretation of DLS data in terms of collective and single-particle motions in concentrated dispersions. Preparation of the dispersions and the interactions between the particles are discussed in section 3. In section 4, we present typical DLS data, firstly for the pmma dispersions, in which the collective motion of identical particles is studied. Secondly, for pmma dispersions containing a very small concentration of silica particles, optical conditions are selected to render the pmma particles essentially transparent, so that the average motion of single silica particles can be monitored. Thirdly, the results of the two experiments are combined to provide an indication of the correlated motion of pairs of particles. We conclude by comparing some of the properties of these dispersions with those of atomic liquids. 2. Theory The theory and experimental details of dynamic light scattering are discussed in several references.1,6,11,12He re we present the main aspects required for the interpretation of DLS applied to concentrated dispersions. The quantity measured in a DLS experiment is the normalized autocorrelation function of the scattered light field. For the present purpose, this is most conveniently expressed as g(')(q,t)= FM(q,t)/FM(q,O) (1) where we refer to FM(q,t)as the measured dynamic struc-
ture factor. For N spherical particles in the scattering (6) Pusey, P. N.; Tough, R. J. A. In Dynamic Light Scattering; Pecora, R., Ed.; Plenum Press: New York, 1985. (7) Pusey, P. N.; Fijnaut, H. M.; Vrij, A. J . Chem. Phys. 1982, 77, 4270. (8) van Megen, W.; Underwood, S.M.; Snook, 1. J. Chem. Phys. 1986, 8,5, 4065. (9) van Megen, W.; Underwood, S. M. J . Chem. Phys. 1988, 88, 784 1. (10) Copley, J. R. D.; Lovesey, S. W. Rep. Prog. Phys. 1975,38,461. (11) Chu, B. Laser Light ScatterinE; Academic Press: New York, 1974. (12) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1975.
volume, F'(q,t) is given by N
~ ~ ( q ,=t C(b,(q)bk(q) ) exp{ia.[r;(o) - ~
~ W I I (2) )
j,k=l
where rk(t) and b k ( q ) are, respectively, the position a t time t and the scattering amplitude of the kth particle; q is the scattering vector. The braces in eq 2 indicate an ensemble average over starting times t = 0. For an optically polydisperse system, in which the particles differ only in their scattering amplitudes but are identical with respect to those factors that determine their dynamics, eq 2 simplifies to7 -
+
F M ( q , t )= N(b2- b2)Fs(q,t) Nh2F(q,t)
(3)
where (4)
Fs(q,t) = (exp(iq[r(O)- r(t)lI) (5) is the incoherent (or self) dynamic structure factor and
is the coherent dynamic structure factor. The coherent dynamic structure factor is the correlation function of the qth spatial Fourier component of the concentration fluctuations p(q,t), given by
(7) In essence, different scattering vectors project different facets of the particle motions. If rm represents the most probable distance between a pair of particles in the system, then at small scattering vectors, such that 2 r / q >> r,, one observes large wavelength or macroscopic concentration fluctuations. A t the other extreme, for 2 r / q
