[PPT] - On the Yielding of Colloidal (and Other) Glass Formers Thomas PowerPoint Presentation

SLIDE 1

under

Transport Processes Melts External Fields

in

Zukunftskolleg

free • creative • connecting

On the Yielding of Colloidal (and Other) Glass Formers

Thomas Voigtmann

Institute of Materials Physics in Space, German Aerospace Center, Cologne & Zukunftskolleg, Universität Konstanz, Germany

J. A. Krumhansl Symposium, Bangalore, February 2012

SLIDE 2

Acknowledgments

Ch. Harrer, S. Papenkort, A. Bhattacharjee
A. Meyer (DLR), J. Horbach (Düsseldorf)
M. Fuchs (Konstanz), J. Brader (Fribourg), M. E. Cates (Edinburgh)
S. Egelhaaf (D’dorf), M. Ballauff, M. Siebenbürger (HZ Berlin)

2 / 29

SLIDE 3

Outline

Dynamical Yield Stress Startup: Creep and Micro-Rheology Residual Stresses

3 / 29

SLIDE 4

Introduction

4 / 29

SLIDE 5

Rheology of Dense Fluids

shear flow of dense fluids:

velocity v height h

external flow rate ˙ γ ∼ v/h [1/s] large structural relaxation time τ [s] ⇒ large effect when ˙ γτ ≫ 1

(glassy) kinetic arrest: τ → ∞

apply perturbation (shear ˙ γ) ⇒ measure response (stress σ) F [ ˙ γ ] = σ constitutive equation F is a model of the material linear response, steady state: Newtonian liquid σ = ˙ γ × η

5 / 29

SLIDE 6

Visco-Elasticity: Maxwell’s Model

Newtonian fluid: η = const. ⇒ σ ∝ ˙ γ Hookian elastic solid: σ ∝ γ dense fluids: ?? Maxwell: combine σ ∼ γ and σ ∼ ˙ γ

deformation (constant rate) γ = γ . t shear stress σ elastic solid viscous fluid

˙ γ = ˙ σ/G∞ + σ/η “spring-and-dashpot” model: Hookian spring constant G∞ differential equation solved by

η G∞

σ(t) = t

−∞

˙ γ(t′) G∞e−(t−t′)/τ dt′ η = G∞τ

utput

input model 6 / 29

SLIDE 7

Visco-Elasticity: Maxwell’s Model

Newtonian fluid: η = const. ⇒ σ ∝ ˙ γ Hookian elastic solid: σ ∝ γ dense fluids: ?? Maxwell: combine σ ∼ γ and σ ∼ ˙ γ

deformation (constant rate) γ = γ . t shear stress σ elastic solid viscous fluid

˙ γ = ˙ σ/G∞ + σ/η “spring-and-dashpot” model: Hookian spring constant G∞ Maxwell’s constitutive equation

η G∞

σ(t) = t

−∞

˙ γ(t′) G∞e−(t−t′)/τ dt′ η = G∞τ

utput

input model 6 / 29

SLIDE 8

Nonlinear Rheology: Shear Thinning

apply (steady) shear ⇒ dramatic decrease in apparent viscosity

thermosensitive colloids

[Fuchs and Ballauff, J Chem Phys (2005)]

non-linear response

linear response: η ∼ const. η → ∞: glass

η ∼ 1/˙ γ applications: painting, coating, lubrication, … “universal”: metallic melts, geophysics, soft matter, …

7 / 29

SLIDE 9

Nonlinear Rheology: Shear Thinning

apply (steady) shear ⇒ dramatic decrease in apparent viscosity

Pd40Ni10Cu30P20, various temperatures

[Kato et al., JAP (1998)]

non-linear response

linear response: η ∼ const. η → ∞: glass

η ∼ 1/˙ γ applications: painting, coating, lubrication, … “universal”: metallic melts, geophysics, soft matter, …

7 / 29

SLIDE 10

Nonlinear Rheology: Shear Thinning

apply (steady) shear ⇒ dramatic decrease in apparent viscosity

10

8

10

7

10

6

10

5

10

4

10

3

γ .

10 10

1

10

2

10

3

10

4

η

granular simulation

[Olsson/Teitel, PRL (2007)]

non-linear response

linear response: η ∼ const. η → ∞: glass

η ∼ 1/˙ γ applications: painting, coating, lubrication, … “universal”: metallic melts, geophysics, soft matter, … (?!?)

7 / 29

SLIDE 11

[Fuchs/Cates, PRL (2002); Brader et al., PRL (2007); PRL (2008)] [Brader, ThV, Fuchs, Larson, Cates, PNAS (2009)]

Rheo-Mode-Coupling Theory, Schematically

nonlinear schematic model – strain history γtt′ = t

t′ ˙

γ(τ) dτ σ(t) ∼ t

−∞

dt′ ˙ γ(t′)G(t, t′, [˙ γ])

MCT

≈ t

−∞

dt′ vσ ˙ γ(t′) φ2(t, t′, [γ]) ∂tφ(t, t′) + φ(t, t′) + t

t′ m(t, t′′, t′)∂t′′φ(t′′, t′) dt′ = 0

m(t, t′′, t′) = h[γtt′]h[γtt′′]

v1φ(t, t′′) + v2φ(t, t′′)2

cage effect wave-vector advection

2π/qx 2π/qx 2π/qy(t, t′) ˙ γ

utput

input model 8 / 29

SLIDE 12

Dynamical Yield Stress

9 / 29

SLIDE 13

Dynamical Yield Stress

[Fuchs/Ballauff, JCP (2005)]

thermosens. colloids: η(̺, ˙

γ) flow curves in steady state σ(˙ γ → 0) = σy > 0 in the (idealized) glass: dynamic yield stress ˙ γ → 0 is singular; σ = ˙ γ t

−∞ G(t − t′, ˙

γ) dt′

10 / 29

SLIDE 14

[Fuchs/Cates, Faraday Discuss. (2003)] [ThV, EPJE (2011)]

A Nonlinear Maxwell Model

shear accelerates dynamics: relaxation time ∼ 1/˙ γ τ −1 → τ −1 + ˙ γ nonlinear Maxwell model G∞ ˙ γ = ˙ σ + σ/τ+σ ˙ γ/γc

(plus a high-shear Newtonian viscosity…)

σ = G∞τ0 ˙ γ + G∞τ ˙ γ 1 + ˙ γτ/γc ⇒ as τ → ∞: critical dynamical yield stress σy = G∞γc > 0

11 / 29

SLIDE 15

Flow Curves

granular material

[Olsson/Teitel, PRL (2007)]

colloidal suspension

10

4

10

3

10

2

10

1

10 10

1

10

2

σ

10

7

10

5

10

3

10

1

η

−1

[data: M. Siebenbürger]

scenarios for dynamical yielding: colloidal vs. granular remember: different protocols, kBT finite vs. zero

12 / 29

SLIDE 16

Flow Curves: Scaling Proposal

granular material

[Olsson/Teitel, PRL (2007)]

metallic melt

10 10

1

10

2

10

3

10

4

10

2

10

T>T0

50K 100K 300K 500K 600K 700K 800K 840K 900K 940K 1000K 1040K 1100K η

1

T = η-1/|(T-T0)/T0|α

σT= σ/|(T-T0)/T0|β

β = 0.60 α = 1.23

T0= 860K T<T0 σT

α/β

[Guan/Chen/Egami, PRL (2010)]

(granular) point J as a critical point scaling: suggests σ(T → Tc) ∼ ˙ γx ⇒ hence: σc

y = 0

13 / 29

SLIDE 17

[ThV, EPJE 34, 106 (2011)]

Flow Curves: Finite Yield Stress

Brownian hard spheres

10

6

10

5

10

4

10

3

10

2

10

1

10 10

1

γ .

10

3

10

2

10

1

10 10

1

10

2

σ [kT/R2] [data: Henrich et al., Phil Trans Roy Soc (2009)]

metallic melt

10

6

10

5

10

4

10

3

10

2

10

1

γ . [10

12/s]

10

1

10 10

1

σ [GPa] σy [Guan/Chen/Egami, PRL (2010)]

fits using schematic model of mode-coupling theory (MCT) prediction σc

y = O(0.1 kBT/R3) – apparent power laws

14 / 29

SLIDE 18

Different Scenarios for Flow Curves

γ . σ γ . σ

“discontinuous” vs. “continuous” yield-stress scenario different yielding mechanisms: local cages vs. avalanches energy densities kBT/R3 vs. overlap energies

15 / 29

SLIDE 19

Different Scenarios for Flow Curves

log t log G(t) G∞ log t log G(t)

“discontinuous” vs. “continuous” yield-stress scenario different yielding mechanisms: local cages vs. avalanches energy densities kBT/R3 vs. overlap energies

15 / 29

SLIDE 20

Different Scenarios for Flow Curves

1/ϕ, T η, σ η σ 1/ϕ, T η, σ η σ

“discontinuous” vs. “continuous” yield-stress scenario different yielding mechanisms: local cages vs. avalanches energy densities kBT/R3 vs. overlap energies

15 / 29

SLIDE 21

[ThV, EPJE 34, 106 (2011)]

Jamming Diagram

iso-viscosity lines in the (T, σ) and (1/̺, σ) plane:

5 10 15 20 500 1000 1500 2000 Σ GPa T K 1 2 3 4 5 6 0.4 0.2 0.0 0.2 0.4 Σ

(white: “jammed”)

16 / 29

SLIDE 22

Startup: Creep and Micro-Rheology

17 / 29

SLIDE 23

[Siebenbürger, Ballauff, ThV (in preparation)]

Creep

deformation γ(t) after sudden step stress

time t shear stress σ γ 10

2

10

1

10 10

1

10

2

10

3

10

4

10

5

10

6

t 10

4

10

3

10

2

10

1

10 γ (t)

σ0 = 0.01 σ0 = 0.05 σ0 = 0.10 σ0 = 0.20 σ0 = 0.30 σ0 = 0.50 σ0 = 1.00

∼ t

10

1

10

2

10

3

10

4

10

5

10

6

10

7

t / τ0 10

3

10

2

10

1

10 10

1

10

2

γ (t) σ = 0.9 kT/R3 σ = 0.5 kT/R3 σ = 0.1 kT/R3 experiment: M. Siebenbürger

nonequilibrium transition: plastic deformation / flow static yield stress σc anomalous flow behavior (creep)?

[Cottrell, “The Time Laws of Creep”] 18 / 29

SLIDE 24

[Siebenbürger, Ballauff, ThV (in preparation)]

Creep Continued

10

3

10

2

10

1

10 10

1

10

2

γ tw = 60s tw = 600s tw = 6000s 10

9

10

8

10

7

10

6

10

5

10

4

10

3

γ . 10

2

10

3

10

4

10

5

10

6

10

7

t/τB 10

5

10

4

10

3

10

2

10

1

10 10

1

γ . t

creep laws (hard matter):

logarithmic, ˙ γ(t)t ∼ const. Andrade, ˙ γ(t) ∼ t−α, α ≈ 2/3 secondary, ˙ γ(t) ∼ const.

“viscosity thinning”, ˙ γ(t) ∼ t1+x

related to stress overshoot? aging-time dependent!

0.0 0.1 0.2 0.3 0.4 0.5 0.6

γ

0.0 0.5 1.0 1.5 2.0

σ tw 19 / 29

SLIDE 25

[Gazuz, Puertas, ThV, Fuchs, PRL (2009)]

Static Yielding: A Force Threshold

steady external shear ⇒ glass molten (always) steady external force ⇒ yielding transition σc microscopic analog?

yielding of individual “cages” by local external force

⇒ microrheology

F

ex

localized delocalized liquid density applied force

20 / 29

SLIDE 26

[Gazuz, Puertas, ThV, Fuchs, PRL (2009)]

Local Melting of the Glass

F ex < F ex

c : localized probe

distorted probe probability density φs( r, t → ∞)

1 2 4 6 8

Fex 0

1 0.5 0.5 1 0.2 0.4 1 2 3 4 5 6

Fex 10

1 0.5 0.5 1 0.2 0.4 0.2 0.4 0.6 0.8 1 1.2 1.4

Fex 20

1 0.5 0.5 1 0.2 0.4 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Fex 30

1 0.5 0.5 1 0.2 0.4

F ex

F ex > F ex

c : delocalized probe

F ex

c

≫ kBT/σ: cages, not thermal forces

21 / 29

SLIDE 27

[Gnann, Gazuz, Puertas, Fuchs, ThV, Soft Matter (2011)]

Microscopic Yielding

sim.: A M Puertas / exp.: Habdas et al. [Europhys Lett (2004)]

10 10

1

10

2

10

3

Fex [kT/a]

10 10

1

10

2

10

3

ζ / ζ0

ϕ = 0.62 ϕ = 0.57 ϕ = 0.55 ϕ = 0.50 ϕ = 0.40 ϕ = 0.45

10 10

1

10

2

10

3

10

4

10

5

10

6

ζ / ζ0

ϕ = 0.55 ϕ = 0.53 ϕ = 0.52 ϕ = 0.50 ϕ = 0.45

10

2

10

1

10 10

1

Fex [pN]

depinning signature at F ≈ Fc: measures typical cage strength fits: schematic model (MCT)

modes F ex, ⊥ F ex high-force plateau: fluctuations ⊥ force strong influence of hydrodynamic interactions

MCT power laws v∞ ∼ (F − Fc)1/a−1

F

ex

22 / 29

SLIDE 28

[Gnann, Gazuz, Puertas, Fuchs, ThV, Soft Matter (2011)]

Microscopic Yielding

sim.: A M Puertas / exp.: Habdas et al. [Europhys Lett (2004)]

10 10

1

10

2

10

3

Fex [kT/a]

10 10

1

10

2

10

3

ζ / ζ0

ϕ = 0.62 ϕ = 0.57 ϕ = 0.55 ϕ = 0.50 ϕ = 0.40 ϕ = 0.45

10 10

1

10

2

10

3

10

4

10

5

10

6

ζ / ζ0

ϕ = 0.55 ϕ = 0.53 ϕ = 0.52 ϕ = 0.50 ϕ = 0.45

10

2

10

1

10 10

1

Fex [pN]

depinning signature at F ≈ Fc: measures typical cage strength fits: schematic model (MCT)

modes F ex, ⊥ F ex high-force plateau: fluctuations ⊥ force strong influence of hydrodynamic interactions

MCT power laws v∞ ∼ (F − Fc)1/a−1

F

ex

22 / 29

SLIDE 29

Residual Stresses

23 / 29

SLIDE 30

Constitutive Equations

common Ansatz: nonlinear Boltzmann superposition principle σ(t) = t

−∞

γtt′ ψ(t − t′, γtt′) dt′ (BKZ) ⇒ prediction: single-step-strain response contains it all ⇒ σ(∞) = 0 whenever

i γi = 0 in n-step-strain

test: double step strain

note: MCT does not reduce to BKZ form ⇒ prediction of residual stresses

24 / 29

SLIDE 31

[ThV, Brader, Fuchs, Cates, Soft Matter (submitted)]

Stress Recovery After Reversing Strain

time t deformation γ

t = 0 ∆t measure stress relaxation σ(t)

10

2

10

1

10 10

1

10

2

10

3

t

8
4

4 8

σ(t) ∆t

γ1+ γ2 = 0

0.0 0.2 0.4 0.6 0.8 1.0

strain magnitude γ0

0.75 0.80 0.85 0.90 0.95 1.00

recovered stress: 1 − |σ(∞)/γ0G∞|

v2 = 4.1 v2 = 5.0 ∆t=0.1 ∆t=100 ∆t=10 ∆t=1000 ∆t=1

glass: finite σ(∞) even for reversed strain recovered stress: < 100% due to memory effects

25 / 29

SLIDE 32

[ThV, Brader, Fuchs, Cates, Soft Matter (submitted)]

Double Step Strain: Protocol Dependence

0.0 0.5 1.0 Γ0 0.0 0.5 1.0 Γ1 0.5 0.0 0.5 1.0 Σ 0.0 0.5 1.0 Γ0 0.0 0.5 1.0 Γ1 0.5 0.0 0.5 Σ

protocol dependence: sudden “affine” deformation vs. strain-rate-ramp response: “echo” vs. “wipe-out”

26 / 29

SLIDE 33

[Siebenbürger/Ballauff, ThV, SFB-TR6 A6/A7 collab. (unpublished results)]

Residual Stresses: Switching off Steady Shear

time t shear rate γ .

ss eq tw = 0 tw1 tw2

measure decay of shear stress σ(t)

10

2

10 10

2

10

4

10

6

10

8

time since cessation 0.0 0.2 0.4 0.6 0.8 1.0 σ(t)/σy γ . = 10

−5

γ . = 10

−4

γ . = 10

−3

γ . = 10

−2

γ . = 10

−1

γ . = 10

glass can sustain finite residual stress σ(∞) strong shear “forgotten” quicker

27 / 29

SLIDE 34

[Siebenbürger/Ballauff, ThV, SFB-TR6 A6/A7 collab. (unpublished results)]

Residual Stresses: Switching off Steady Shear

time t shear rate γ .

ss eq tw = 0 tw1 tw2

measure decay of shear stress σ(t)

10

3

10

2

10

1

10 10

1

10

2

10

3

10

4

scaled time γ . t 10

2

10

1

10 σ(t)/σy γ . = 10

−3

γ . = 10

−2

γ . = 5×10

2

theory 10

3

10

2

10

1

10 10

1

10

2

10

3

10

4

10

5

scaled time γ . t 10

3

10

2

10

1

10 σ(t)/σy

T = 15°C, γ .= 10

−4 ... 10 2

T = 18°C, γ .= 10

−4 ... 10 2

experiment (Siebenbürger)

glass can sustain finite residual stress σ(∞) – transition strong shear “forgotten” quicker

27 / 29

SLIDE 35

Frozen-In Stresses: History Dependence

glass re-freezes after shear melting

⇒ frozen in stresses: memory of flow history σ(t) = ˙ γ

−∞

dt′ G(t>0, t′ <0; ˙ γ)

10

8

10

6

10

4

10

2

10

shear rate γ .

10

1

10

shear stress σ(0), σ(∞)

σy residual σ∞ after cessation σ steady state (flow curve)

relevant for applications

residual stress in a worn railway rail

[Webster et al., MRS Forum (2002)]

stress birefringence

[image: wikipedia.org] 28 / 29

SLIDE 36

Summary

dynamical yield stress σy

goes to zero continuously as liquid is approached? ⇒ “granular case” goes to constant? ⇒ “Maxwell case” (G∞ = 0 in liquid)

creep

analogy colloidal systems – metallic systems “failure time” – overshoot in startup flow

residual stresses

signature of energy dissipation due to relaxation processes not obtained in standard BKZ-type constitutive equations

FOR1394 DFG Research Unit

Nonlinear Response to Probe Vitrification

FOR1394 Zukunftskolleg

free • creative • connecting

29 / 29