# Limits to Gelation in Colloidal Aggregation

Original entry: Hsin-I Lu, APPHY 225, Fall 2009

'Limits to Gelation in Colloidal Aggregation'

S. Manley, L. Cipelletti, V. Trappe, A. E. Bailey, R. J. Christianson, U. Gasser, V. Prasad, P. N. Segre,
M. P. Doherty, S. Sankaran, A. L. Jankovsky, B. Shiley, J. Bowen, J. Eggers, C. Kurta, T. Lorik, and D. A.Weitz, PRL **93**, 108302 (2004)

## Summary

This paper studies the dynamics of large fractal aggregates in polystyrene colloids. Two experiments were conducted in microgravity and on Earth with dynamic light scattering to determine the aggregate size and internal elasticity. The authors can describe the clustering dynamics well with a combination of translational and rotational diffusion and internal elastic fluctuations. They found cluster growth is limited by gravity-induced restructuring in the presence of gravity and thermal fluctuations ultimately inhibit fractal growth in microgravity experiment.

## Soft Matter Keywords

Diffusion-limited cluster aggregation (DLCA), fractal, gelation, colloid

## Soft Matter

- Experiment:

The authors study the structure and dynamics of clusters using both static light scattering (SLS) and dynamic light scattering (DLS). A narrow beam with a wavelength 532 nm, passes through the sample. Scattered light is collected from a volume of approximately <math>10^{-5} cm^3</math> with a single-mode optical fiber, and detected with a photon-counting detector; this fiber can be rotated and probes scattering vectors <math>3 \mu m^{-1} < q < 31 \mu m^{-1}</math>. Additionally, a broad beam from a second laser illuminates the sample in a direction perpendicular to the first beam. Scattered light from a larger volume is imaged through a spherical lens onto a CCD camera to measure small-angle SLS or DLS at many wave vectors simultaneously.

- Results:

Fig. 1(a) shows the static scattering intensity, <math>I(q)</math> data in microgravity experiemnt 16 days after aggregation was initiated. Fractal clusters are formed, as evidenced by the powerlaw decay of <math>I(q)</math>. The fractal dimension is <math>d_f=1.9</math>, in agreement with the value expected for DLCA.

The dynamic structure factor, <math>f(q,\tau)</math>, exhibits complete relaxation at all q and at all times, for both ground and space experiments. The authors fit data to a stretched exponential, <math>f(q,\tau) ~ exp(-[q^2 D_{\alpha} \tau]^{\alpha}) </math>, where <math>D_{\alpha}</math> has the form of a diffusion coefficient, and <math>\alpha</math> is the stretching exponent. The solid lines in Fig. 2. shows the fitted curve.

Fig. 1(b) shows a distinct crossover between the high- and low-q regimes for fitting parameters <math>D_{\alpha}</math> and <math>\alpha</math>. For all aggregation times, <math>D_{\alpha}</math> is nearly <math>q</math> independent and <math>\alpha \approx 1</math> at low <math>q</math>, reflecting diffusive motion of clusters.

To estimate the cluster size, the authors determine the first cumulant of <math>f(q,\tau)</math>, <math>\Gamma_1</math>, for the small-angle scattering data and calculate the effective diffusion coefficient, <math>D_{eff} =q^2</math> For contributions of rotational and translational diffusion alone, the authors expect <math>D_{eff}</math> to plateau at very large <math>q</math>. However, data at large <math>q</math> of Fig. 3(a) is not consistent with this argument. The authors point out the deviaiton should come from the internal fluctuations within the cluster. Such internal dynamics has been observed in colloidal gels, where the fractal clusters are constrained by the network and thus can neither translate nor rotate.

The authors compare the aggregation kinetics measured in space with those measured on Earth in Fig. 4. In microgravity the clusters grow at a rate in excellent qualitative agreement with that expected for DLCA, <math>R_c \approx t^{1/d_f}</math>. Gelation is not observed because the duration is less than the gelation time. By contrast, the time evolution of Rc on Earth shows significant deviation from the behavior anticipated for DLCA. The initial growth of the clusters is identical to that in space, DLCA; however, after 10 h, the growth deviates from the expected behavior and reaches a plateau, with no further growth observed. Thus, this system will not gel as the maximum cluster size <math>R_c \approx 20 \mu m</math>is more than 1 order of magnitude lower than that needed to form a gel.

- Discussion:

This cluster growth observed on Earth suggests that gravity can ultimately limit gelation. As an aggregate sediment, its buoyant weight is balanced by hydrodynamic drag from the fluid. Hydrodynamic resistance acts primarily at the surface of the cluster, but is screened in the interior, whereas gravity is a body force distributed throughout the cluster. This results in an internal shear stress across the cluster and can tear the aggregates apart.