[PPT] - (Unifying?) rheology of soft glasses and jammed solids Ludovic PowerPoint Presentation

SLIDE 1

(Unifying?) rheology of soft glasses and jammed solids

Ludovic Berthier Laboratoire Charles Coulomb Universit´ e de Montpellier 2 & CNRS

Unifying concepts in materials, JAKS 2012 – Bangalore, February 7, 2012

title – p.1

SLIDE 2

Coworkers

On-going work with:
A. Ikeda (Montpellier)

P . Sollich (London)

Some previous work with:

P . Chaudhuri (Dusseldorf),

H. Jacquin (Paris),
S. Sastry (Bangalore),
T. Witten (Chicago),
F. Zamponi (Paris).

title – p.2

SLIDE 3

Disordered solid states

Atomic glasses (window glasses, plastics) are solid materials frozen in

an amorphous (non-crystalline, metastable) structure.

Dense granular materials are disordered solids.
Same/similar/(un-)related transitions? Similar properties in the ‘fluid’?

Similar mechanical response of the solid?

title – p.3

SLIDE 4

A geometric problem... really?

Athermal packing of soft repulsive spheres, e.g. V (r < σ) = ǫ(1 − r/σ)2.

c ϕ ϕ

Low ϕ: no overlap, fluid Large ϕ: overlaps, solid

Useful for non-Brownian suspensions (below), grains (at), foams and

emulsions (above). Many (oral) claims for glass-formers.

Aim: equilibrium statistical mechanics approach to jamming. See if and

how jamming emerges in the T → 0 limit of the (T, ϕ, σ) phase diagram.

title – p.4

SLIDE 5

Numerical observations

J-point from packing properties of soft

repulsive particles at T = 0.

[O’Hern et al. PRL ’02]

Scaling laws and structure of pack-

ings near jamming [vanHecke JPCM’10] Energy: E = 0 for ϕ < ϕJ; E ∼ (ϕ−ϕJ)α for ϕ > ϕJ. Contact number: z = 0 → z = zc +a(ϕ−

ϕJ)1/2 with zc = 2d (isostaticity).

A major numerical and experimental

effort over the last decade. A new nonequilibrium phase transition.

5
4
3
2

log (φ- φc)

3
2
1

log (Z-Z c)

6
4
2

log G

8
6
4
2

log p

α=2 α=5/2 α=2 α=5/2 3D 2D (a) (b) (c)

J

Unjammed Jammed

T 1/φ σ

title – p.5

SLIDE 6

Structure of soft colloids

Numerous experiments performed on soft colloidal particles (microgels,

emulsions) to probe the jamming transition.

[Zhang et al., Nature 2009]

Anomalous behavior of pair correlation function g(r) under compression.
Interpreted as a structural signature of the jamming transition. “Our results

conclusively demonstrate that length scales associated with the T = 0 jamming transition persist in thermal systems, not only in simulations but also in laboratory experiments.

title – p.6

SLIDE 7

Rheology of soft particles

Steady state rheology near jamming in overdamped (athermal)

numerical simulations of harmonic spheres. Diverging viscosity, emergence of yield stress.

10-3 10-2 10-1 10-5 10-4 10-3 10-2

etai etai etai etai etai etai etai etai etai etai etai etai etai etai etai fit2

inverse shear viscosity !"1 shear stress #

$=0.830 $=0.834 $=0.836 $=0.838 $=0.840 $=0.841 $=0.842 $=0.844 $=0.848 $=0.852 $=0.856 $=0.860 $=0.864 $=0.868 #=0.0012

10-2

[Olson, Teitel, PRL 07] [Norstrom et al., PRL 2010]

“Similar” behaviour (and scaling laws) observed experimentally.

“These results support the conclusion that jamming is similar to a critical phase transition.”

title – p.7

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Equilibrium fluid

Consider the fluid, V (r) = (1 − r)2, at equilibrium at (T > 0, ϕ, σ = 0).
Liquid state theory: solve structure, g(r), thus thermodynamics using

integral equations. We can use, e.g., HNC: g(r) = e−βV (r)+g(r)−1−c(r).

T = 1.20 · 10−3 Compress r g(r) 1.4 1.2 1 0.8 8 6 4 2

No glass or jamming transi-

tion is found.

Anomalous structural evo-

lution at all T! The system first ‘orders’, then ‘disorders’.

F = E − TS: Avoid overlap (reduce energy) at low ϕ. Difficult

(entropically disfavoured) at larger ϕ. Solution: increase overlap to gain entropy.

Softness matters (not jamming).

[Jacquin & Berthier, Soft Matter ’10]

title – p.8

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Why liquid state theory fails

Equilibrium phase diagram of soft harmonic spheres.

[Berthier & Witten, PRE ’09]

Super-Arrhenius Scaling VFT MCT

ϕ T 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 10−2 10−4 10−6

The equilibrium fluid does not jam, but the glass structure does.
One cannot understand the jamming transition without dealing first with

the glass phase.

title – p.9

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Metastable states & jamming

Glasses depend on cooling history.
Similarly, compressed fluids of hard

spheres reach different glassy states.

Temperature Energy Tk Tg Glass Ideal glass BMCSL Glass (soft) Glass (hard)

Equil. (soft)
Equil. (hard)

ϕ0 ϕMCT ϕonset ϕ Z(ϕ) 65 60 55 50 45 1000 100 10

Jamming transition oc-

curs along a range of den- sities [Chaudhuri et al., PRL

’10]

Theory

must handle multiplicity of metastable states.

title – p.10

SLIDE 11

Statistical mechanics of glasses

Assume exponential number of metastable states exists:

f(T) = − T V log

d

f ′ exp

−Nf ′

T + Nsconf(f ′, T)

.
In practice, take m replica(s) and minimize the replicated free energy

[Monasson, PRL ’95, Mézard-Parisi PRL ’99]

f(m, T) = − T V log

d

f ′ exp

−Nf ′m

T + Nsconf(f ′, T)

.
New effective potential valid for both hard spheres (T → 0 small ϕ) and

soft glasses (T → 0 large ϕ), to treat analytically the glass & jamming transitions of harmonic spheres.

f(m, A, ϕ, T) = fharm(m, A) + fliquid(ϕ, T/m) − ρ 2

drg(r)[e−β(Veff(r)−mV (r)) − 1]

[Jacquin, Berthier & Zamponi PRL ’11]

title – p.11

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The ‘ideal’ glass transition

High T fluid: m = 1, sconf(T) > 0 and simple liquid theory is enough.
sconf(T) vanishes at TK(ϕ) > 0 for ϕ > ϕK ≡ hard sphere glass transition.

small cage new approximation

TK ∼ (ϕ − ϕK)2 ϕK ≈ 0.577 ϕ TK 0.75 0.7 0.65 0.6 0.55 10−8 10−6 10−4 10−2

FLUID GLASS

Super-Arrhenius Scaling VFT MCT

ϕ T 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 10−2 10−4 10−6

Low-T scaling: TK ∼ (ϕ − ϕK)2 (robust scaling) with ϕK ≈ 0.577 (value

depends on specific approx).

title – p.12

SLIDE 13

The ‘ideal’ jamming transition

Glass thermodynamics: energy, pressure, specific heat, fragility...
Jamming at T = 0 ⇔ Change in ground state glass properties.
ϕGCP = 0.633353... such that:

EGS = 0 below, EGS ≃ a(ϕ − ϕGCP )2 above.

Glass Close Packing: densest

T = 0 glass with no overlap.

[Zamponi & Parisi, RMP ’10]

PGS ∼ (ϕ − ϕGCP ).

JAMMED UNJAMMED GLASS FLUID

ϕ egs TK 3.10−3 2.10−3 1.10−3 0.72 0.68 0.64 0.6 0.56 1.10−4 5.10−5

Existence, location(s), and scaling laws of jamming from ‘first principles’.

title – p.13

SLIDE 14

Structure of jammed states

New predictions for g(r) near contact at all (T, ϕ). Isostaticity is derived.

0.62 0.625 0.63 0.635 0.64 0.645

ϕ

100 1000

gG

max

T=10

5

T=10

6

T=10

7

T=10

8

T=0 HS T=0 SS

10−10 10−9 10−8 10−7 10−6 T = 10−5 Uglass(T, ϕ) 10−6 10−8 10−10 10−12

[Berthier, Jacquin, Zamponi, PRE ’11]

Clear emergence of jamming singularity from finite temperatures

properties of glass phase–only if particles are not too soft, T/ǫ ≪ 10−6.

Still unconvinced G and J are different things? Rheology will do it.

title – p.14

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Rheology at finite temperatures

Aim: Study the harmonic sphere rheology at finite temperatures, then

approach T → 0.

‘SLLOD’: Newton eqs. + shear + thermostat (‘SLLOD’).

Problem: one cannot shear faster than thermal fluctuations, Pe = ˙

γτD < 1.

Here τD ∼ a/

kBT/m → ∞, cannot go athermal.
We use Langevin dynamics with shear and thermostat in d = 3:

ξ(dri dt − ˙ γyiex) = −

j

dV (|ri − rj|) dri + ηi,

with ηi(t)ηj(t′) = 2kBTξ1δ(t − t′).

Two important microscopic timescales:

(i) dissipation: τ0 = ξa2/ǫ = 1, our time unit. (ii) thermal time: τD = ξa2/(kBT) → ∞ when T → 0.

We study both finite and zero temperatures, both thermal (Pe < 1) and

athermal (Pe > 1) rheologies at once.

[Ikeda, Berthier, Sollich, in preparation]

title – p.15

SLIDE 16

Soft glassy rheology

Steady state rheology at T = 10−4 and increasing ϕ. Diverging viscosity,

emerging yield stress. Here τ −1

D

∼ 10−4.

ϕ σY [ǫ/a3] η[ξ/a] 10−3 10−4 10−5 0.72 0.68 0.64 0.6 0.56 104 103 102 101 100

This is a glass transition as seen in colloidal particles, star polymers,

microgels, but also glassy liquids.

Theories of driven glasses capture competition between slow glassy

dynamics and shear flow.

title – p.16

SLIDE 17

From glass to jamming rheology

Same at T = 10−6, here τ −1

D

∼ 10−6. Glass physics shifts to lower shear

rates: Pe < 1 → ˙

γ < 10−6.

ϕ σY [ǫ/a3] η[ξ/a] 10−6 10−7 10−3 10−4 10−5 0.72 0.68 0.64 0.6 0.56 102 101 100 10−1

The athermal jamming physics emerges when τ −1

D

≪ ˙ γ ≪ τ −1

0 .

Two Newtonian regimes, two distinct viscosities, emergence of yield

stress (when ˙

γ → 0), with funny density dependence. A real mess!

title – p.17

SLIDE 18

Pure jamming rheology

Rheology at T = 0. Here τ −1

D

= 0, i.e. Pe = ∞.

ϕ σY [ǫ/a3] η[ξ/a] 10−3 10−4 10−5 0.72 0.68 0.64 0.6 0.56 103 102 101 100

Glass rheology has gone, T = 0 jamming transition remains: diverging

viscosity, emergence of yield stress.

No microscopic theory. Driven dynamics at T = 0 is difficult to attack

from first principles (driven glass theories fail badly).

title – p.18

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Rheology: summary

T = 10−6 T = 10−4

T = 0 ϕ η[ξ/a] 0.7 0.68 0.66 0.64 0.62 0.6 0.58 0.56 0.54 10000 1000 100 10 1 0.1

T = 0 T = 10−7 T = 10−6 T = 10−5 T = 10−4

ϕ σY [ǫ/a3] 0.72 0.68 0.64 0.6 0.56 10−3 10−4 10−5 10−6 10−7 10−8

We observed two types of Newtonian regimes, depending on the Peclet

number and particle softness.

Clearly, η(T > 0) does not converge to η(T = 0) when T → 0. These are

distinct divergences at distinct densities with distinct physics.

Solidity emerges at the glass transition at any T > 0, and “transition to

jammed solid” only exists at T = 0. cf. “Melting by freezing”.

To see jamming: Pe ≫ 1 (kinetics) and kBT/a3 ≪ ǫ/a2 (hardness).

title – p.19

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Microscopic dynamics

Widely different mean-squared displacements.

t2 t

2

t

T = 10−4: Glass physics because particles explore their ‘cage’ due to

thermal fluctuations.

T = 10−6: crossover towards athermal dynamics when thermal

fluctuations do not allow sufficient exploration of the cage.

At T = 0, ballistic (non-affine) short-time dynamics due to shear flow,

then diffusive behaviour.

title – p.20

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Experiments?

Micron-sized microemulsions seem ideal to observe the

athermal/thermal crossover.

[Mason, Bibette, Weiz, 1996]

Star polymers are too soft, grains are too large. What about microgel

’pnipam’ particles? Both small (100 nm) and large (1 micron) colloids available, but quite they are also quite soft...

title – p.21

SLIDE 22

How confusing!

title – p.22

SLIDE 23

How confusing!

title – p.23

SLIDE 24

Confusion?

Rheology of soft microgels of about 1 micron [Nordstrom et al., PRL 2010]
Scaling analysis: τ/τ0 ∼ 107 − 1011, σY /ǫ ∼ 10−6 − 10−4.
Thermal units: ˙

γ = 1s−1 → Pe ≈ 5. → τ/τD ∼ 1 − 104.

Rheological “transition” occurs at Pe < 1, jamming rheology outside the

shear rate window, diverging viscosity and emergence of yield stress not controlled by T = 0 physics.

title – p.24

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Summary

Why not use thermal fluctuations and statistical mechanics approaches

that we know to describe the emergence of interesting physics at T = 0?

Useful to understand relationship with glass transition, glass theories

provide first principles approaches and new predictions for structural quantities and correlation functions.

Also seems promising to understand steady state rheology.
Need some theory for to study better athermal rheology, where glass

theories fail.

title – p.25

SLIDE 26

Even more confusing

title – p.26