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