^{1}, P. T. Callaghan

^{1}, G. Petekidis

^{2}and D. Vlassopoulos

^{2,a)}

### Abstract

Suspensions of multiarm star polymers are studied as models for soft colloidalglasses. Using an established pre-shearing protocol which ensures a reproducible initial state (the “rejuvenation” of the system), we report here the time evolution of the stress upon startup of simple shear flow for a range of shear rates. We show the existence of critical shear rates, which are functions of the concentration, . When the suspensions are sheared at rates below , the stress rises to a common value which is also a function of the concentration. The system thus develops a yield stress. This behavior manifests itself as an evolution from a monotonic slightly shear-thinning flow curve to a flow curve dominated by a stress plateau. We relate this bulk evolution to spatially resolved velocity profiles. Hence, yield stress is linked to shear banding in this class of soft colloids.

S.R. acknowledges the hospitality of FORTH, where the rheological component of this work was carried out. The financial assistance of the Royal Society of New Zealand Marsden fund, the Centres of Research Excellence fund and the EU NoE Softcomp (NMP3-CT2004-502235) and EU ToK Cosines (MTKD-CT-2005-029944) are gratefully acknowledged. The authors would like to thank Suzanne Fielding, Peter Olmsted, Mike Cates, and Brian Erwin for fruitful discussions. They are grateful to Jacques Roovers for providing the star polymer used in this work.

I. INTRODUCTION

II. EXPERIMENTAL

A. Materials

B. Rheology

III. RESULTS

A. Determining the conditions for rejuvenation: Establishing a protocol

B. Nonlinear strain response to oscillatory shear

C. The flow curves

D. Transient response to continuous steady shear

E. NMR velocimetry

1. Geometry

2. Evidence of shear banding

3. Evolution of banded structure

4. Characterizing the growth of the zero-shear band

F. Stress relaxation

IV. DISCUSSION: RECONCILING TRANSIENT RESPONSE

V. CONCLUSIONS

### Key Topics

- Materials aging
- 25.0
- Glasses
- 21.0
- Colloidal systems
- 19.0
- Stress relaxation
- 19.0
- Polymers
- 10.0

## Figures

Schematic representation of crowded systems. (a) Long linear polymers are confined to tubes at high concentrations and are said to be “entangled” and relax via reptation. (b) Colloidal hard spheres are kinetically arrested above a volume fraction . Neighbors form effective cages which must be broken for macroscopic flow. (c) Colloidal star polymers are kinetically arrested at higher volume fractions, , although the same ideas of cage escape are applied to their macroscopic behavior. Additional to the idea of local cages is the idea of arm engagement.

Schematic representation of crowded systems. (a) Long linear polymers are confined to tubes at high concentrations and are said to be “entangled” and relax via reptation. (b) Colloidal hard spheres are kinetically arrested above a volume fraction . Neighbors form effective cages which must be broken for macroscopic flow. (c) Colloidal star polymers are kinetically arrested at higher volume fractions, , although the same ideas of cage escape are applied to their macroscopic behavior. Additional to the idea of local cages is the idea of arm engagement.

(a) A log-log plot of the strain amplitude dependence of the storage and loss moduli, and , at for a sample of 12 880 stars in squalene at . At low amplitudes the material behaves linearly and solid-like. As the deformation increases, the moduli cross over and the material exhibits a shear-thinning liquid response at amplitudes above 10%. Solid lines represent asymptotic predictions for hard spheres. (b) Log-log plot of the frequency dependence of and at a fixed strain amplitude of 1% directly after rejuvenation. The storage modulus shows a weak power-law dependence and shows a minimum typical of glassy colloidal systems. This non-aged solid-like behavior is the benchmark for further investigations. Similar results are found for stars in squalene at .

(a) A log-log plot of the strain amplitude dependence of the storage and loss moduli, and , at for a sample of 12 880 stars in squalene at . At low amplitudes the material behaves linearly and solid-like. As the deformation increases, the moduli cross over and the material exhibits a shear-thinning liquid response at amplitudes above 10%. Solid lines represent asymptotic predictions for hard spheres. (b) Log-log plot of the frequency dependence of and at a fixed strain amplitude of 1% directly after rejuvenation. The storage modulus shows a weak power-law dependence and shows a minimum typical of glassy colloidal systems. This non-aged solid-like behavior is the benchmark for further investigations. Similar results are found for stars in squalene at .

Flow curve for a 12 880 star concentration of in the cone-and-plate geometry: log-log plots of the steady-state stress as a function of applied steady shear rate. Light gray: each datum (+) is acquired after shearing for a time following rejuvenation. Black: each datum (+) is acquired after shearing for time of to reach a steady state. The circles are taken from transient measurements (see below) with the unfilled corresponding to a time (youthful state) and the filled circles (aged). The plateau clearly visible in the long time limit is thus easily missed in a fast measurement. The solid lines are guides to the eye. The dotted line indicates the shear rate above which no aging occurs.

Flow curve for a 12 880 star concentration of in the cone-and-plate geometry: log-log plots of the steady-state stress as a function of applied steady shear rate. Light gray: each datum (+) is acquired after shearing for a time following rejuvenation. Black: each datum (+) is acquired after shearing for time of to reach a steady state. The circles are taken from transient measurements (see below) with the unfilled corresponding to a time (youthful state) and the filled circles (aged). The plateau clearly visible in the long time limit is thus easily missed in a fast measurement. The solid lines are guides to the eye. The dotted line indicates the shear rate above which no aging occurs.

Flow curve evolution for star concentration of in the cone-and-plate geometry: log-log plots of the stress as a function of applied steady shear rate for different ages of the sample. Each datum (+) is an average over acquired after shearing for a time indicated in the plot. The circles are taken from transient measurements (see text) with the unfilled symbols corresponding to a time (youthful state) and the filled circles (aged). Note the evolution of the plateau from lower shear rates toward higher rates. For clarity, a closeup of the critical region is displayed in the lower plot. The dotted line indicates the critical shear rate, above which no aging occurs.

Flow curve evolution for star concentration of in the cone-and-plate geometry: log-log plots of the stress as a function of applied steady shear rate for different ages of the sample. Each datum (+) is an average over acquired after shearing for a time indicated in the plot. The circles are taken from transient measurements (see text) with the unfilled symbols corresponding to a time (youthful state) and the filled circles (aged). Note the evolution of the plateau from lower shear rates toward higher rates. For clarity, a closeup of the critical region is displayed in the lower plot. The dotted line indicates the critical shear rate, above which no aging occurs.

Shear start-up for star concentration of and in the cone-and-plate geometry: lin-log plots of the stress response as reported by the analog output of the rheometer for waiting times , 5, 7, 10, 25, 50, 100, 500, 1000, and . The arrows indicate the shift in stress with increasing waiting time.

Shear start-up for star concentration of and in the cone-and-plate geometry: lin-log plots of the stress response as reported by the analog output of the rheometer for waiting times , 5, 7, 10, 25, 50, 100, 500, 1000, and . The arrows indicate the shift in stress with increasing waiting time.

(a) The stress residual from the preshear (points) decays with a power-law exponent of (solid line). (b) The maximum stress achieved at the plastic overshoot (points) increases as a power-law with an exponent of 0.096 (solid line).

(a) The stress residual from the preshear (points) decays with a power-law exponent of (solid line). (b) The maximum stress achieved at the plastic overshoot (points) increases as a power-law with an exponent of 0.096 (solid line).

Shear startup, , for star concentration of in the cone-and-plate geometry. Note the three types of shear rate-dependent response: at high shear rates there is an overshoot with no stress increase at long times (continual adolescence); at lower shear rates there is an overshoot followed by a steady adolescent response and then stress increase; at lower shear rates still, there is no overshoot with stress increase at long times. The strain of the stress overshoot shown in (b) is an increasing function of strain rate. The strain at which the small kink, visible just above a strain of at all rates, is also an increasing function of strain rate as shown in the inset. The dotted line is a logarithmic fit to the stress increase. In both (a) and (b) the arrows indicate the shift in stress due to increasing shear rate.

Shear startup, , for star concentration of in the cone-and-plate geometry. Note the three types of shear rate-dependent response: at high shear rates there is an overshoot with no stress increase at long times (continual adolescence); at lower shear rates there is an overshoot followed by a steady adolescent response and then stress increase; at lower shear rates still, there is no overshoot with stress increase at long times. The strain of the stress overshoot shown in (b) is an increasing function of strain rate. The strain at which the small kink, visible just above a strain of at all rates, is also an increasing function of strain rate as shown in the inset. The dotted line is a logarithmic fit to the stress increase. In both (a) and (b) the arrows indicate the shift in stress due to increasing shear rate.

Shear startup for star concentration of in the cone-and-plate geometry: lin-log plots of the stress response as a function of time (a) and strain (b) to three steady shear rates after preshear. The same features as seen in the data of Fig. 7 are evident though the plateau occurs at a higher stress. Note the small kink visible at strains of as indicated by the inset in (b). This feature is clearer at the two lower rates than in the higher and, like the strain position of the stress overshoot, occurs at larger strains for higher rates.

Shear startup for star concentration of in the cone-and-plate geometry: lin-log plots of the stress response as a function of time (a) and strain (b) to three steady shear rates after preshear. The same features as seen in the data of Fig. 7 are evident though the plateau occurs at a higher stress. Note the small kink visible at strains of as indicated by the inset in (b). This feature is clearer at the two lower rates than in the higher and, like the strain position of the stress overshoot, occurs at larger strains for higher rates.

A one-dimensional velocity profile taken at times (a) and (c) for an imposed gap-average shear rate of at a temperature of with a waiting time . The purpose of the marker fluid is to allow for velocity extrapolation to determine the presence, or lack, of slip. No slip is seen during the entire experimental time frame. (b) and (d) show closeups of the velocity profile of the sample in the gap. A clear change can be seen from an initially liquid-everywhere response to a strongly banded state where one branch does not move. The dotted lines represent the limits used to track the evolution of the solid branch.

A one-dimensional velocity profile taken at times (a) and (c) for an imposed gap-average shear rate of at a temperature of with a waiting time . The purpose of the marker fluid is to allow for velocity extrapolation to determine the presence, or lack, of slip. No slip is seen during the entire experimental time frame. (b) and (d) show closeups of the velocity profile of the sample in the gap. A clear change can be seen from an initially liquid-everywhere response to a strongly banded state where one branch does not move. The dotted lines represent the limits used to track the evolution of the solid branch.

The evolution of the size of the solid branch for an imposed gap-average shear rate of at a temperature of with a waiting time for a single experiment as defined in the text. The ordinates correspond to the proportion of the sample that has a velocity within the values represented by the dotted lines in Fig. 9. The solid line corresponds to a Gompertz function and models the data well.

The evolution of the size of the solid branch for an imposed gap-average shear rate of at a temperature of with a waiting time for a single experiment as defined in the text. The ordinates correspond to the proportion of the sample that has a velocity within the values represented by the dotted lines in Fig. 9. The solid line corresponds to a Gompertz function and models the data well.

Stress relaxation as a function of time as response to a 1% strain following pre-shearing and varied waiting time . Waiting times are indicated. The arrow indicates the shift in stress with increased waiting time.

Stress relaxation as a function of time as response to a 1% strain following pre-shearing and varied waiting time . Waiting times are indicated. The arrow indicates the shift in stress with increased waiting time.

Definitions of the key stresses involved in a full description of the transient response to continuous strain for the colloidal star polymers.

Definitions of the key stresses involved in a full description of the transient response to continuous strain for the colloidal star polymers.

## Tables

Critical shear rates/stresses in stress response.

Critical shear rates/stresses in stress response.

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