Thermodynamic fluctuations inglass-forming liquids

Ludovic Berthier

Laboratoire Charles CoulombUniversite de Montpellier 2 & CNRS

Physics of glassy and granular materials – Kyoto, July 18, 2013

title – p.1

The glass “transition”

• Many materials becomeglasses (not crystals) at lowtemperature.

Temperature

Energy

Tk Tg

Glass

Ideal glass

DEC

B2O3

PC

SAL

OTP

GLY

LJ

BKS

SimulationFragile

Strong

Tg/T

τ α/τ

0

10.90.80.70.60.5

102

106

1010

1014

• In practice, glass formation is a gradual process.

• What is the underlying “ideal” glass state?

• Existence of many metastable states: glasses are many-body “complex”systems, due to disorder and geometric frustration.

title – p.2

Temperature crossovers

• Glass formation characterized by several “accepted” crossovers. Onset,mode-coupling & glass temperatures: directly studied at equilibrium.

1 / T ( K-1)

0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050

log 10

( η

/ poi

se)

-4

-2

0

2

4

6

8

10

12

14

T0TK

TxTBTc

T*TA

Tg

Tb

Tm

[G. Tarjus] [Debenedetti & Stillinger]

• Extrapolated temperatures for dynamic and thermodynamic singularities:T0, TK . Ideal glass transition at the Kauzmann temperature is highlycontroversial (cf New York Times article in July 2008).

title – p.3

Molecular dynamics simulations

• Pair potential V (r < σ) = ǫ(1− r/σ)2: soft harmonic repulsion, behavesas hard spheres in limit ǫ/T → ∞.

• Constant density, decrease temperature. Dynamics slows down →computer glass transition. Tonset ≈ 10, Tmct ≈ 5.2. [Berthier & Witten ’09]

5

6

7

8

10

13

T = 30

Tmct = 5.2

t

Fs(q

,t)

10−1 101 103 105 107

1

0.8

0.6

0.4

0.2

0

• Fs(q, t) =1

N〈

N∑j=1

exp[iq · (rj(t)− rj(0))]〉title – p.4

Dynamic heterogeneity

• When density is large, particles must move in a correlated way. Newtransport mechanisms revealed over the last decade: fluctuations matter.

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90

• Spatial fluctuations grow (modestly)near Tg.

• Clear indication that some kind ofphase transition is not far – which?

• Structural origin not established:point-to-set lengthscales, other struc-tural indicators?[Talks by Tanaka, Gradenigo...]

Dynamical heterogeneities in glasses, colloids and granular materialsEds.: Berthier, Biroli, Bouchaud, Cipelletti, van Saarloos (Oxford Univ. Press, 2011).

title – p.5

Dynamic heterogeneity

• When density is large, particles must move in a correlated way. Newtransport mechanisms revealed over the last decade: fluctuations matter.

• Spatial fluctuations grow (modestly)near Tg.

• Clear indication that some kind ofphase transition is not far – which?

• Structural origin not established:point-to-set lengthscales and otherstructural indicators disappointing.[Talks by Tanaka, Gradenigo...]

Dynamical heterogeneities in glasses, colloids and granular materialsEds.: Berthier, Biroli, Bouchaud, Cipelletti, van Saarloos (Oxford Univ. Press, 2011).

title – p.6

Dynamical view: Large deviations

• Large deviations of fluctuations ofthe (time integrated) local activitymt =

∫dx

∫ t0dt′m(x; t′, t′ +∆t):

P (m) = 〈δ(m−mt)〉 ∼ e−tNψ(m).

• Exponential tail: direct signature ofphase coexistence in (d + 1) dimen-sions: High and low activity phases. 0.0 0.1 0.2 0.3

m

10-12

10-9

10-6

10-3

100

P(m

)

tobs

= 320tobs

= 640tobs

= 960tobs

= 1280

[Jack et al., JCP ’06]

• Equivalently, a field coupled to local dynamics induces a nonequilibriumfirst-order phase transition in the “s-ensemble”. [Garrahan et al., PRL ’07]

• Metastability controls this physics. Complex (RFOT) energy landscapegives rise to same transition, but the transition exists without multiplicity ofglassy states [cf Kurchan’s talk.] [Jack & Garrahan, PRE ’10]

title – p.7

Thermodynamic view: RFOT

• Random First Order Transition (RFOT) theory is a theoretical frameworkconstructed over the last 30 years (Parisi, Wolynes, Götze...) using a largeset of analytical techniques.[Structural glasses and supercooled liquids, Wolynes & Lubchenko, ’12]

• Some results become exact for simple “mean-field” models, such as the

fully connected p-spin glass model: H = −∑i1···ip

Ji1···ipsi1 · · · sip .

• Complex free energy landscape → sharp transitions: Onset (apparitionof metastable states), mode-coupling singularity (metastable statesdominate), and entropy crisis (metastable states become sub-extensive).

• Ideal glass = zero configurational entropy, replica symmetry breaking.

• Extension to finite dimensions (‘mosaic picture’) remains ambiguous.

title – p.8

A ‘Landau free energy’

• Complex free energy landscape → effective potential V (Q). Free energycost (configurational entropy) to have 2 configurations at fixed distance Q:

[Franz & Parisi, PRL ’97]

Vq(Q) = −(T/N)

∫dr2e

−βH(r2) log

∫dr1e

−βH(r1)δ(Q−Q12)

where: Q12 = 1N

∑N

i,j=1 θ(a− |r1,i − r2,j |). Quenched vs. annealed approx.

TK

Simple liquid

Tmct

Tonset

q

V(q

)

0.80.60.40.20

0.08

0.06

0.04

0.02

0

• V (Q) is a ‘large deviation’ func-tion (in d dimensions), mainlystudied in mean-field RFOT limit.

• P (Q) = 〈δ(Q−Qαβ)〉

∼ exp[−βNV (Q)]

• Overlap fluctuations reveal evolution of multiple metastable states. Finited requires ‘mosaic state’ because V (Q) must be convex: exponential tail.

title – p.9

Direct measurement?

• Principle: Take two equilibrated configurations 1 and 2, measure theiroverlap Q12, record the histogram of Q12.

• Problem: Two equilibrium configurations are typically uncorrelated, withmutual overlap ≪ 1 and small (nearly Gaussian) fluctuations.

T = 9, N = 256

Q

P(Q

)

0.30.20.1010−6

10−4

10−2

100

102

[see also Cammarota et al., PRL ’11]

• Solution: Seek large deviations using umbrella sampling techniques.[Berthier, arxiv.1306.0425]

title – p.10

Overlap fluctuations: Results

• Idea: bias the dynamicsusing Wi(Q) = ki(Q − Qi)

2

to explore of Q ≈ Qi.

• Reconstruct P (Q) usingreweighting techniques.

• Exponential tail belowTonset: phase coexis-tence between multiplemetastable states in bulkliquid.

• Static fluctuations controlnon-trivial fluctuations in tra-jectory space, and phasetransitions in s-ensemble.

T = 18, 15, 13, 11, 10, 9, 7

(a) Annealed

Va(Q

)

10

5

0

T = 13, 11, 10, 8, 7

(b) Quenched

Q

Vq(Q

)

10.80.60.40.20

10

5

0

title – p.11

Equilibrium phase transitions

• Non-convex V (Q) implies that an equilibrium phase transition can beinduced by a field conjugated to Q. [Kurchan, Franz, Mézard, Cammarota, Biroli...]

• Annealed: 2 coupled copies.

εa

H = H1 +H2 − ǫaQ12

• Quenched: copy 2 is frozen.

εq

H = H1 − ǫqQ12

TK

Tonset

T

ε

Critical point

• Within RFOT: Some differences be-tween quenched and annealed cases.

• First order transition emerges from TK ,ending at a critical point near Tonset.

• Direct consequence of, but differentnature from, ideal glass transition.

title – p.12

Numerical evidence in 3d liquid

• Investigate (T, ǫ) phase dia-gram using umbrella sampling.

• Sharp jump of the overlap be-low Tonset ≈ 10.

• Suggests coexistence regionending at a critical point.

T = 13, · · · , 7(a)

N = 256

Qǫ

10.80.60.40.200

10

20

6.43

6.33

6.17

5.98

5.09

4.17

ǫ = 0

(b)

T = 8, N = 108

Q

P(Q

)

10.80.60.40.20

10

8

6

4

2

0

• P (Q) bimodal for finite N .

• Bimodality and static suscep-tibility enhanced at larger N forT . Tc ≈ 9.8.

→ Equilibrium first-order phasetransition.

title – p.13

Ideal glass transition?

• ǫ perturbs the Hamiltonian: Affects the competition energy /configurational entropy (possibly) controlling the ideal glass transition.

• Random pinning of a fraction c of par-ticles: unperturbed Hamiltonian.

• Dynamical slowing down observed nu-merically. [Kim, Scheidler, Kuni...]

TK

Tonset

TCritical point

Pinning

• Within RFOT, ideal glass transition line ex-tends up to critical point.

[Cammarota & Biroli, PNAS ’12]

• Pinning reduces multiplicity of states, i.e. de-creases configurational entropy: Sconf(c, T ) ≃

Sconf(0, T )− cY (T ). Equivalent of T → TK .title – p.14

Random pinning: Simulations

• Challenge: fully exploring equilibrium configuration space in thepresence of random pinning: parallel tempering. Limited (for now) to smallsystem sizes: N = 64, 128. [Kob & Berthier, PRL ’13]

Low-c fluid High-c glass

• From liquid to equilibrium glass: freezing of amorphous density profile.

• We perform a detailed investigation of the nature of this phase change,in fully equilibrium conditions.

title – p.15

Microscopic order parameter

• No configurational entropy, no time scale, no extrapolation, no aging.

• We detect the glass formation using an equilibrium, microscopic orderparameter: The global overlap Q = 〈Q12〉.

0.00 0.10 0.20 0.30 0.40c

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

over

lap

T=4.8T=5.0T=5.5T=6.0T=7.0T=8.0T=13T=20

N = 64

• Gradual increase at high T to abrupt emergence of amorphous order atlow T at well-defined c value. Signature of first-order phase transition?

title – p.16

Fluctuations: Phase coexistence

• Probability distribution function of the overlap: P (Q) = 〈δ(Q−Qαβ)〉 .

• Numerical measurements using parallel tempering simulations to explore(c, T,N) phase diagram performing thermal and disorder averages.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8q

0

2

4

6

8

P(q

)

c=0c=0.063c=0.125c=0.188c=0.250c=0.281c=0.313c=0.375

T=13

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9q

0

2

4

6

8

P(q

)

c=0c=0.031c=0.078c=0.109c=0.125c=0.156c=0.188

T=4.8

N = 64

• Bimodal distributions appear at low enough T , indicative of phasecoexistence at first-order transition.

title – p.17

Thermodynamic limit?

• Phase transition can only be proven using finite-size scaling techniquesto extrapolate toward N → ∞.

• Limited data support enhanced bimodality and larger susceptibility forlarger N . Encouraging, but not quite good enough: More work needed.

title – p.18

Equilibrium phase diagram

• Location of the transition from liquid-to-glass determined fromequilibrium measurements of microscopic order parameter on both sides.

0 0.04 0.08 0.12 0.16 0.2 0.24c

0

2

4

6

8

10T

Ton

Tmct

TK from γTK from χ

glass

fluidτ=107

τ=106

τ=105τ=10

4τ=10

3

(c)

• Glass formation induced by random pinning has clear thermodynamicsignatures which can be studied directly.

• Results compatible with Kauzmann transition – this can now be decided.title – p.19

Summary

• Non-trivial static fluctuations of the overlap in bulk supercooled liquids:non-Gaussian V (Q) losing convexity below ≈ Tonset.

• Adding a thermodynamic field can induce equilibrium phase transitions.

TK

Tonset

T

ε

Critical point• Annealed coupling: first-order transitionending at simple critical point.

• Quenched coupling: first-order transitionending at random critical point.

• Random pinning: random first order transi-tion ending at random critical point.

• Direct probes of peculiar thermodynamic underpinnings of RFOT theory.

• A Kauzmann phase transition may exist, and its existence be decided.

title – p.20