Dynamics of a Colloidal Glass Dynamics of a Colloidal Glass During Stress-Mediated During Stress-Mediated

Structural ArrestStructural Arrest(“(“Relaxation in ReverseRelaxation in Reverse”)”)

Ajay Singh Negi and Chinedum OsujiDepartment of Chemical Engineering,

Yale University, New Haven, CT.

MotivationMotivation

Structural Glass

• At high temperatures (above Tg), system is liquid.

• Below Tg, the viscosity is very high.

• Below Tg, the system shows aging behavior.

Colloidal Glass

• Under high shear, the system flows.

• At low shear rates or rest, the system does not flow.

• Aging behavior is seen at rest or low shear.

Is shear temperature ?

MotivationMotivation

EXPERIMENTS: Ediger et al, Science, 323, 231 (2009)At higher creep stresses, the dynamics was faster and distribution narrower.

SIMULATIONS: Warren and Rottler, 2010: At higher stresses, dynamics is accelerated and distribution is narrowed.

How does stress influence the structural arrest of a colloidal glass?

MotivationMotivation

What is the role of stress on the arrest timescale?

How does stress affect the trajectory of the system during arrest? (peak-width in time-dependent viscosity)

System and MethodSystem and Method

• Laponite XLG

(a) Electrostatic screening length at

pH 10 ≈ 30 nm.

(b) Non-ergodic state at concentration

of 1 wt %.

• Bulk Rheology

(a) Constant stress measurements.

(b) Oscillation over a background steady flow.

time (s)Solid lines: Controlled variable Dashed lines: Measured variable

Schematic of Protocol Schematic of Protocol (constant stress)(constant stress)

t=0

Dynamical ArrestDynamical Arrest

0.1 1 10 100

1E-4

1E-3

0.01

0.1

1

10

100

3 Pa 4 Pa 6 Pa 9 Pa 13 Pa 18 Pa 24 Pa 31 Pa

shea

r ra

te (

1/s)

t (s)

Increasing Stress

Shear rate measured

as a function of time

for a constant applied

stress.

The time for arrest

increase with increase

in the applied stress.

Above a certain stress the system will NOT arrest.

Viscosity Bifurcation Coussot et al, PRL 2002

time (s)Solid lines: Controlled variable Dashed lines: Measured variable

Superposition Rheology ProtocolSuperposition Rheology Protocol

t=0

Linearity of Superposition RheologyLinearity of Superposition Rheology

10 100 1000

1

10

100

10 100 1000

1

10

100

G',

G"

(Pa)

time (s)

0.2 Pa, 5 Pa 0.4 Pa, 5 Pa 0.7 Pa, 5 Pa 1 Pa, 5 Pa 0.2 Pa, 10 Pa 0.4 Pa, 10 Pa 0.7 Pa, 10 Pa 1 Pa, 10 Pa

Same background

stress and

different probe

stresses give the

same result.

This ensured that

we are measuring

the linear

properties of the

system under flow.

WaveformsWaveforms

-1 0 1 2 3 4 5 6 7

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

G' > G"

Phase Angle (rad)

Stress Strain

Str

ess,

Str

ain

G' < G"

Waveforms are

NOT distorted

linear properties

are being

measured.

Finite phase lag for

a liquid sample.

The phase lag

vanishes when the

system solidifies.

Oscillation Over FlowOscillation Over Flow

10 100 10001

10

100

10 100 10001

10

100

filled symbol : G'open symbol : G"

G',

G"

(Pa)

19

t (s)

Dynamic

measurements on

the sample under

steady flow.

Probe stress = 1 Pa.

BentoniteOvarlez and Coussot, PRE, 76, 011406 (2007)

Shahin and Joshi, Langmuir, 26, 4219 (2010)

melting stress > yield stress

(yield stress)

Oscillation Over FlowOscillation Over Flow

10 100 10001

10

100

10 100 10001

10

100

filled symbol : G'open symbol : G"

G',

G"

(Pa)

13 19

t (s)

Dynamic

measurements on

the sample under

steady flow.

Probe stress = 1 Pa.

Oscillation Over FlowOscillation Over Flow

10 100 10001

10

100

10 100 10001

10

100

filled symbol : G'open symbol : G"

G',

G"

(Pa)

13 19 25

t (s)

Dynamic

measurements on

the sample under

steady flow.

Probe stress = 1 Pa.

Oscillation Over FlowOscillation Over Flow

10 100 10001

10

100

10 100 10001

10

100

filled symbol : G'open symbol : G"

G',

G"

(Pa)

10 13 16 19 21 23 25 28

t (s)

Dynamic

measurements on

the sample under

steady flow.

Varying background

stresses, σm.

Probe stress = 1 Pa.

The cross-over between G’ and G” is delayed as the background

stress was increased.

ττ vsvs σσmm

5 10 15 20 25 30

10

100

1000

2.7 3 3.3 3.610

100

0 (P

a), E

(wt %)

0

E

2.75% 3% 3.5%

(s)

m (Pa)

System will not arrest

above a stress σ0.

VFT equation

ττ vsvs σσmm

5 10 15 20 25 30

10

100

1000

2.7 3 3.3 3.610

100

0 (P

a), E

(wt %)

0

E

2.75% 3% 3.5%

(s)

m (Pa)

System will not arrest

above a stress σ0.

VFT equation

Evolution of the arrest time with applied stress? VFT dependenceWidth of the loss mode peak with applied stress?

Width of Loss PeakWidth of Loss Peak

0.1 1 10 100 10001

10

100

G"

a( m

)

t b(m

)

10 13 16 19 21 23 25 28

G” peaks broaden

as stress is

decreased,

precluding time-

stress

superposition.

It is similar to broadening of loss peak on approaching glass transition temperature.

Peak Width of Loss ModulusPeak Width of Loss Modulus

1.0 1.5 2.0 2.5 3.00.4

0.8

1.2

1.6

2.0

2.4

1.0 1.5 2.0 2.5 3.00.4

0.8

1.2

1.6

2.0

2.4

13 16 19 21 23 25 28

log(t) (s)

log

(G'')

(P

a)

Lines are fit to Lorentzian function

12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

Pe

ak

Wid

th

stress (Pa)

Loss mode peak narrows with increasing stress.

Cole Davidson ExponentCole Davidson Exponent

~ 1/peak width12 14 16 18 20 22 24 26 28 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1/(p

eak

wid

th)

stress (Pa)

Stickel et al (1993): decreases on approaching glass transition for PDE, an organic glass former.

SummarySummary

• Stress delays the onset of structural arrest.• Dynamic measurement gives a temporal response

similar to frequency dependence observed in non-aging systems.

• The arrest time τ has an exponential dependence on inverse stress.

• Above a critical stress σ0, the arrest time diverges.

• Loss peak narrows with increasing stress.

Negi and Osuji, EPL, 90, 28003 (2010)

Effect of FrequencyEffect of Frequency

The frequency of the probe stress was varied.

Frequency DependenceFrequency Dependence

At higher frequency system arrests much faster.

3 10 90

0

100

200

300

400

500

600

700

800

900

t_cr

oss

ing

(s)

frequency (rad/s)

1 pa 2 Pa 4 Pa 6 Pa 8 Pa 9 Pa

1

10

100

1000

1

10

100

1000

1

10

100

1

10

100

10 100 1000

0.01

0.1

1

10

100

10 100 1000

0.01

0.1

1

10

100

70 40 20 10 7 4

1 Pa

G',

G"

(Pa)

4 Pa

9 Pa

t (s)

Width of Loss ModulusWidth of Loss Modulus

The lines are fitted to logistic power peak function to estimate the width.

3 10 90

1

2

3

4

5

6

7

pea

k w

idth

freq (rad/s)

1 Pa 2 Pa 4 Pa 6 Pa 8 Pa 9 Pa

1.1

1.2

1.3

1.4

1.5

1.6

1.1

1.2

1.3

1.4

1.5

1.6

0.5 1.0 1.5 2.0 2.5 3.00.0

0.4

0.8

1.2

1.6

2.0

0.5 1.0 1.5 2.0 2.5 3.00.0

0.4

0.8

1.2

1.6

2.0

0.8

1.0

1.2

1.4

1.6

0.8

1.0

1.2

1.4

1.6

1 Pa

70 40 20 10 7 4

log(t) (s)

9 Pa

4 Palog

(G")

(P

a)

SummarySummary

• The response of the system at higher frequency is qualitatively similar to its response at the lower stresses.

• The system seem to be more arrested at higher frequencies.

POSSIBLE REASON• At higher frequency or at short times, the system seems

more solid like.

AcknowledgementsAcknowledgements• Funding.

• Osujilab group.http://www.eng.yale.edu/polymers/index.html

Thank you for your attention!!!

Questions?