Thermodynamic fluctuations in glass-forming liquids Ludovic - - PowerPoint PPT Presentation

Thermodynamic fluctuations in glass-forming liquids

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

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

title – p.1

The glass “transition”

• Many

materials become glasses (not crystals) at low temperature.

Temperature Energy Tk Tg Glass Ideal glass

DEC B2O3 PC SAL OTP GLY LJ BKS

Simulation Fragile Strong Tg/T τα/τ0 1 0.9 0.8 0.7 0.6 0.5 102 106 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

log10 ( η / poise)

• 4
• 2

2 4 6 8 10 12 14

T0 TK Tx TB Tc 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 highly

controversial (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, behaves

as 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

• 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. New

transport mechanisms revealed over the last decade: fluctuations matter.

10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90

• Spatial fluctuations grow (modestly)

near Tg.

• Clear indication that some kind of

phase 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 materials

Eds.: 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. New

transport mechanisms revealed over the last decade: fluctuations matter.

• Spatial fluctuations grow (modestly)

near Tg.

• Clear indication that some kind of

phase transition is not far – which?

• Structural origin not established:

point-to-set lengthscales and other structural indicators disappointing.

[Talks by Tanaka, Gradenigo...]

Dynamical heterogeneities in glasses, colloids and granular materials

Eds.: Berthier, Biroli, Bouchaud, Cipelletti, van Saarloos (Oxford Univ. Press, 2011).

title – p.6

Dynamical view: Large deviations

• Large deviations of fluctuations of

the (time integrated) local activity

mt = dx t

0 dt′m(x; t′, t′ + ∆t):

P(m) = δ(m − mt) ∼ e−tNψ(m).

• Exponential tail: direct signature of

phase 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

10

P(m)

tobs= 320 tobs= 640 tobs= 960 tobs= 1280 [Jack et al., JCP ’06]

• Equivalently, a field coupled to local dynamics induces a nonequilibrium

first-order phase transition in the “s-ensemble”.

[Garrahan et al., PRL ’07]

• Metastability controls this physics. Complex (RFOT) energy landscape

gives rise to same transition, but the transition exists without multiplicity of glassy states [cf Kurchan’s talk.]

[Jack & Garrahan, PRE ’10]

title – p.7

Thermodynamic view: RFOT

• Random First Order Transition (RFOT) theory is a theoretical framework

constructed over the last 30 years (Parisi, Wolynes, Götze...) using a large set 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 (apparition
• f metastable states), mode-coupling singularity (metastable states

dominate), 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 energy

cost (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 = 1

N

N

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

TK Simple liquid Tmct Tonset q V (q) 0.8 0.6 0.4 0.2 0.08 0.06 0.04 0.02

• V (Q) is a ‘large deviation’ func-

tion (in d dimensions), mainly studied in mean-field RFOT limit.

• P(Q)

= δ(Q − Qαβ) ∼ exp[−βNV (Q)]

• Overlap fluctuations reveal evolution of multiple metastable states. Finite

d 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 their
• verlap Q12, record the histogram of Q12.
• Problem: Two equilibrium configurations are typically uncorrelated, with

mutual overlap ≪ 1 and small (nearly Gaussian) fluctuations.

T = 9, N = 256 Q P(Q) 0.3 0.2 0.1 10−6 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 dynamics using Wi(Q) = ki(Q − Qi)2 to explore of Q ≈ Qi.

• Reconstruct

P(Q)

using reweighting techniques.

• Exponential

tail below

Tonset:

phase coexis- tence between multiple metastable states in bulk liquid.

• Static fluctuations control

non-trivial fluctuations in tra- jectory space, and phase transitions in s-ensemble.

T = 18, 15, 13, 11, 10, 9, 7 (a) Annealed Va(Q) 10 5 T = 13, 11, 10, 8, 7 (b) Quenched Q Vq(Q) 1 0.8 0.6 0.4 0.2 10 5

title – p.11

Equilibrium phase transitions

• Non-convex V (Q) implies that an equilibrium phase transition can be

induced 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 different

nature 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 region

ending at a critical point.

T = 13, · · · , 7 (a) N = 256 Q ǫ 1 0.8 0.6 0.4 0.2 10 20

6.43 6.33 6.17 5.98 5.09 4.17 ǫ = 0

(b) T = 8, N = 108 Q P(Q) 1 0.8 0.6 0.4 0.2 10 8 6 4 2

• P(Q) bimodal for finite N.
• Bimodality and static suscep-

tibility enhanced at larger N for

T Tc ≈ 9.8. → Equilibrium first-order phase

transition.

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 T Critical 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 the

presence of random pinning: parallel tempering. Limited (for now) to small system 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 order

parameter: The global overlap Q = Q12.

0.00 0.10 0.20 0.30 0.40 c 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

• verlap

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

N = 64

• Gradual increase at high T to abrupt emergence of amorphous order at

low 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 0.8 q 2 4 6 8 P(q) c=0 c=0.063 c=0.125 c=0.188 c=0.250 c=0.281 c=0.313 c=0.375 T=13 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q 2 4 6 8 P(q) c=0 c=0.031 c=0.078 c=0.109 c=0.125 c=0.156 c=0.188 T=4.8

N = 64

• Bimodal distributions appear at low enough T, indicative of phase

coexistence at first-order transition.

title – p.17

Thermodynamic limit?

• Phase transition can only be proven using finite-size scaling techniques

to extrapolate toward N → ∞.

• Limited data support enhanced bimodality and larger susceptibility for

larger 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 from

equilibrium measurements of microscopic order parameter on both sides.

0.04 0.08 0.12 0.16 0.2 0.24

c

2 4 6 8 10

T

Ton Tmct

TK from γ TK from χ

glass fluid

τ=10

7

τ=10

6

τ=10

5

τ=10

4

τ=10

3

(c)

• Glass formation induced by random pinning has clear thermodynamic

signatures 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 transition ending at simple critical point.

• Quenched coupling:

first-order transition ending 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