# Stress Enhancement in the Delayed Yielding of Colloidal Gels

Review by Bryan Hassell: AP 255 Fall 11

**From:** Stress Enhancement in the Delayed Yielding of Colloidal Gels J. Sprakel, S. B. Lindström, T. E. Kodger, and D. A. Weitz. Phys. Rev. Lett. 106, (2011)

**Keywords:** colloids, viscosity, elasticity and creep

## Summary

Networks of weakly aggregated colloidal particles, or colloidal gels, exhibit a solidlike behavior: their elastic modulus G' is nearly independent of frequency <math>\omega</math> and significantly larger than their viscous modulus G". Interestingly, G" exhibits a feature common to many soft materials; it is largely frequency-independent but nevertheless displays a weak but noticeable increase at the lowest <math>\omega</math> typically measured. This rise suggests the presence of an ultraslow relaxation mechanism in the material. Such relaxation could reflect the ultimate fluid like behavior of the gel at frequencies so low they cannot be probed with oscillatory rheology. Alternatively it could result from aging, which leads to irreversible fluidization of the gel that cannot even be correctly measured with oscillatory rheological measurements. Instead, the mechanics of such soft solid materials can be probed with creep measurements. At low applied stress <math>\sigma</math>, the creep response of colloidal and polymeric gels typically displays the characteristics of a mechanically stable solid, with the deformation reaching a time-independent plateau that reflects the elasticity of the material. However, this solidlike stability persists only for a finite time, whereupon the gel suddenly and catastrophically yields. For polymer networks, the time between the application of a load and the time of yield, <math>\tau_d</math>, exhibits an exponential dependence on the applied stress; this results from stress enhancement of the thermal relaxation of individual bonds within the network. The yielding of colloidal gels also exhibits a strong dependence on stress, but the origin of this behavior has never been fully established.

## Experiment

Rheological experiments are carried out on a stress controlled rheometer (MCR501, Anton Paar) in a concentric cylinder geometry. We study a strong gel of carbon black particles (Cabot Vulcan XC72R) in tetradecane at 8 wt%. Its elastic modulus is <math>G_0</math> = 7690 Pa, measured before each experiment to confirm that the sample has not evolved due to evaporation or particle migration. To establish reproducible initial conditions, they are pre sheared for 60 s at a strain rate <math>\dot{\gamma}=500s^{-1}</math> and left to recover for 15 min before starting the measurement. Unless stated otherwise, all samples are subjected to a pre shear treatment prior to the creep experiments. For very low <math>\sigma</math>, the initial creep response is purely elastic: <math>\gamma</math> increases linearly with time, followed by an inertial ringing reflecting the elasticity of the sample after which <math>\gamma</math> becomes nearly independent of time [Fig. 1(a)] and <math>\dot{\gamma}</math> tends to zero [1(b)]. Virtually no creep is observed over the full length of the experiment [dashed line in Fig. 1(a)]. However, as <math>\sigma</math> is increased, the initial response remains the same, but the time-independent creep persists only for a finite time, whereupon the sample fails catastrophically and both <math>\gamma</math> and <math>\dot{\gamma}</math> increase sharply [Figs. 1(a) and 1(b)]. This occurs after a delay <math>\tau_d</math>, which we take to be the point where the increase in <math>\dot{\gamma}</math> is maximal, as shown by the arrows in Fig. 1(b). As <math>\sigma</math> is increased <math>\tau_d</math> decreases [Fig. 1(a)]. Ultimately, the range over which the creep is time independent becomes very small and failure is nearly instantaneous. Directly after yielding, the linear elastic modulus is essentially unchanged, but the gel yields at significantly shorter times, unless it is rejuvenated by a strong preshear treatment.