# Shear Thickening and Scaling of the Elastic Modulus in a Fractal Colloidal System with Attractive Interactions

Fall 2010 entry - Anna Wang

## Contents

## References

Shear Thickening and Scaling of the Elastic Modulus in a Fractal Colloidal System with Attractive Interactions. C. O. Osuji, C. Kim and D. A. Weitz. Phys. Rev. E. 77, 060402(R) (2008). cond-mat:0710.0042, Sep, 2007

## Background

Gels formed from colloidal particles possessing attractive potentials have properties determined by the critical volume fraction φ_{c} (above which gelation occurs), interaction energy U and fractal dimension d_{f}. They are also sensitive to the mechanical history of the system. Rheological studies of colloidal gels usually have a high rate pre-shear step to erase the mechanical history of the gel. The effectiveness of this is good and well understood for shear thinning systems, but not in shear thickening transitions for systems with sufficiently attractive interactive to form flocculated gels. In the latter case, the nature of the pre-shear flow would determine the underlying fluid microstructure and hence affect the gel elasticity after pre-shearing.

## Experiment

In this paper, shear thickening was observed for dilute dispersions (2-8 wt%) of fractal carbon black particles (~0.5μm diameter) with attractive van der Waals interactions. Tetradecane was chosen as the suspending medium to avoid the adsorption of solvent species onto the colloidal particles surfaces and complicating electrostatic interactions.

- Optical studies were made using a rheometer with a transparent base plate and CCD.
- Rheological measurements were made in strain-control mode using a rheometer with a cone plate and double-wall Couette.

The samples were pre-sheared at 100 s^{-1}, a rate known to provide consistent results.

## Results

The steady state viscosity was measured as a function of shear rate using a cone-plate geometry. Optical studies showed buildup of rolling, vorticity aligned cylindrical flocs at low shear rates (1-10 s^{-1}), a regime susceptible to wall-slip effects and hence requiring caution. At higher shear rates (>10 s^{-1}), homogenous flows are achieved. Figure 1 shows the shear thinning and Figure 2 shows the formation of densified clusters in this regime.

Shear thickening was observed at rate between 100 and 1000 s^{-1}, and the dense clusters from lower shear rates have broken up into a fine dispersion. This increases the effective volume fraction of the particles and hence viscosity. This is in constrast to hard sphere cases, which the shear thickening results from hydrocluster formation.

The dependence of elasticity on pre-shear stress was measured. First the samples were sheared to remove flow history, then pre-sheared at a rate of interest for 20mins and rested for 30mins. A cone-plate system was used to measure the viscoelastic storage modulus G' (Fig 3). A double-wall Couette cells was used to mitigate the effects of sedimentation at low shear rates or solutions too dilute for the cone-plate geometry.

The cone-plate data is rescaled and plotted in Figure 4. The results are interpreted in terms of the dependence of cluster size on shear stress; shear force and cohesive energy of the cluster balance at the optimal cluster size R. The elastic modulus can be written in terms of this cluster size : G' scales with number density n ~ σ/R^{df}) giving

G' ~ σφ^{df/3-df}

Using d_{f} = 1.8 as for diffusion limited aggregation systems, the experimental data is rescaled and is in good accord with the simple scaling law.