VOLUME 68, NUMBER 22 PH YSICAL REVIEW LETTERS 1 JUNE 1992 Spinodal-Type Dynamics in Fractal Aggregation of Colloidal Clusters Marina Carpineti and Marzio Giglio Dipartimento di Fisica, Universita di Milano, via Celoria 16, Milano 20133, Italy (Received 25 February 1992) The aggregation of dense colloidal solutions has been investigated by means of low-angle static light scattering. We show that the scattered pattern exhibits a finite-q-vector peak, whose intensity and posi- tion q change with time. We find that the intensity distributions scale according to S(q/q, t) q (t) 4F(q/q ), in agreement with the scaling law for spinodal decomposition. While d=3 for spi- nodal decomposition, here scaling requires that d =df, the fractal dimension of the clusters. PACS numbers: 64.75.+g, 05. 40.+j, 64.60. Ht, 82.70. Dd The spinodal decomposition (SD) is a phase-separation process that has been investigated in a large number of quite dissimilar systems like small-molecule liquid mix- tures [1-4], metallic alloys [5, 6], polymer blends [7,8], inorganic glasses [9], and thermodynamically unstable colloidal systems [10]. The peculiar feature of SD is that the long-wavelength diffusion coefficient becomes nega- tive so that fluctuations grow instead of decaying. The fastest-growing fluctuation occurs at a finite wave vector and this gives rise to the well-known ring in the pattern of scattered radiation. The intensity and radius of the ring change in time as the thermodynamically stable state is approached. In spite of the diversity of the physical sys- tems, universal features in the dynamics are observed. It should also be pointed out that other phase-separating systems, although not exactly falling in the class of SD, exhibit the same behavior [11, 12]. Colloidal aggregation is another area where a substan- tial amount of work has been produced in recent years [13]. In this case also diffusion plays an essential role. Monomers diffuse to form fractal clusters, and the clus- ters themselves difl'use to coalesce into even larger clus- ters and so on. In this paper we will show that colloidal aggregation exhibits the same features of SD in spite of the fact that nothing anomalous occurs to the diffusion coefficient of the monomers and clusters. By using very-low-angle stat- ic light scattering and high monomer concentration, we present for the first time clear evidence of a ring in the scattered intensity pattern. We will show that during the later stages the dynamics is in agreement with the scaling predictions of Marro, Lebowitz, and Kalos [14], also put forward by Furukawa [15] and by Binder and Stauffer [16]. The position of the scattered peak q and the scaled structure factor S(q/q, t) are related by the equation S(q/q, t) =q (t) F(q/q ), where F(q/q ) is a time-independent scaling function. For ordinary spinodal decomposition, d=3, while here we find that the relation holds if we take d =df, where df is the fractal dimension of the clusters. The surprising simi- larities between these results and those related to the spi- nodal decomposition are likely to suggest some underly- ing common mechanism in the dynamics of these irrever- sible processes. The measurements have been performed on a solution of polystyrene spheres 0. 0190 pm in diameter in a water-heavy-water mixture. The mixture was adjusted so as to make the solution isopycnic, in order to avoid differential sedimentation problems. The monomer con- centration is c=8. 25x 10' cm (volume fraction 4 2.96&10 4), more than 2 orders of magnitude larger than in previous works [17-20]. The aggregation was in- duced by adding a divalent salt to the solution. Runs were taken at various MgC12 concentrations slightly above the critical flocculation value (4-8 mM). By so doing, the reaction proceeds on manageable time scales in spite of the high value of c. The low-angle light scattering setup has already been described elsewhere [19] and only a brief account will be given here. A parallel beam of He-Ne laser light im- pinges on a cuvette. The scattered light and the transmit- ted beam are collected by a lens. A multielement sensor array is placed in the back focal plane. Each element has an annular shape, and is centered around a tiny hole which allows the focused transmitted beam to pass clear of the sensing elements. Consequently, each sensor col- lects light scattered around a given angle 8. The thick- ness and the average radius of the elements are geometri- cally spaced and cover scattering angles 0. 18 & 0 &12. 1' corresponding to the wave-vector range 4. 15 x10 & q & 2.78&&10 cm '. Here q =(4tr/A, ) sin(8/2). We show in Fig. 1 a sequence of intensity distributions at various times after the start of the aggregation process. The run was at 8-mM salt concentration, and is quite representative of all the runs taken at various salt concen- trations, the only differences being dilation of the time scale and slight changes of df. At variance with all the previous observations via static light scattering from ag- gregating colloids, no maximum is observed at q-0. In- stead the curves show a peak at a finite value of q. The peak height increases in time, and the peak position q shifts to smaller and smaller values in a way which is strongly reminiscent of the spinodal decomposition dy- namics. It should also be pointed out that at large q all the distributions collapse onto the same asymptotic curve. This behavior is well known, and always occurs in the ag- 1992 The American Physical Society 3327