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” 10 14 • Many materials become BKS glasses (not crystals) at low LJ Strong GLY temperature. OTP 10 10 SAL PC Energy B2O3 τ α /τ 0 DEC Fragile 10 6 Simulation Glass 10 2 Ideal glass Temperature 0.5 0.6 0.7 0.8 0.9 1 Tk Tg T g /T • 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. 14 12 T g 10 8 log 10 ( η / poise ) 6 T m 4 T b 2 0 T 0 T K T c -2 T x T B T A T* -4 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 1 / T ( K-1) [G. Tarjus] [Debenedetti & Stillinger] • Extrapolated temperatures for dynamic and thermodynamic singularities: T 0 , T K . 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. T onset ≈ 10 , T mct ≈ 5 . 2 . [Berthier & Witten ’09] 1 T = 30 13 10 0.8 8 7 6 0.6 F s ( q, t ) 5 0.4 T mct = 5 . 2 0.2 0 10 − 1 10 1 10 3 10 5 10 7 t N • F s ( q, t ) = 1 � N � exp[ i q · ( r j ( t ) − r j (0))] � j =1 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. 90 • Spatial fluctuations grow (modestly) 80 near T g . 70 60 • Clear indication that some kind of 50 phase transition is not far – which? 40 • Structural origin not established: 30 point-to-set lengthscales, other struc- 20 tural indicators? 10 [Talks by Tanaka, Gradenigo...] 0 0 10 20 30 40 50 60 70 80 90 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 T g . • 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 0 10 the (time integrated) local activity dx � t -3 0 dt ′ m ( x ; t ′ , t ′ + ∆ t ) : 10 m t = � P(m) -6 P ( m ) = � δ ( m − m t ) � ∼ e − tNψ ( m ) . 10 t obs = 320 t obs = 640 -9 10 t obs = 960 • Exponential tail: direct signature of t obs = 1280 -12 10 phase coexistence in ( d + 1) dimen- 0.0 0.1 0.2 0.3 sions: High and low activity phases. m [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 = − J i 1 ··· i p s i 1 · · · s i p . i 1 ··· i p • Complex free energy landscape → sharp transitions: Onset (apparition of 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] � � d r 2 e − βH ( r 2 ) log d r 1 e − βH ( r 1 ) δ ( Q − Q 12 ) V q ( Q ) = − ( T/N ) � N where: Q 12 = 1 i,j =1 θ ( a − | r 1 ,i − r 2 ,j | ) . Quenched vs. annealed approx. N 0.08 • V ( Q ) is a ‘large deviation’ func- Simple liquid 0.06 tion (in d dimensions), mainly T onset studied in mean-field RFOT limit. V ( q ) 0.04 T mct 0.02 • P ( Q ) = � δ ( Q − Q αβ ) � T K ∼ exp[ − βNV ( Q )] 0 0 0.2 0.4 0.6 0.8 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 overlap Q 12 , record the histogram of Q 12 . • Problem: Two equilibrium configurations are typically uncorrelated, with mutual overlap ≪ 1 and small (nearly Gaussian) fluctuations. 10 2 T = 9, N = 256 10 0 P ( Q ) 10 − 2 10 − 4 10 − 6 0 0.1 0.2 0.3 Q [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 T = 18 , 15 , 13 , 11 , 10 , 9 , 7 • Idea: bias the dynamics using W i ( Q ) = k i ( Q − Q i ) 2 10 (a) Annealed to explore of Q ≈ Q i . V a ( Q ) • Reconstruct P ( Q ) using 5 reweighting techniques. • Exponential tail below 0 T onset : phase coexis- tence between multiple T = 13 , 11 , 10 , 8 , 7 10 metastable states in bulk (b) Quenched liquid. V q ( Q ) 5 • Static fluctuations control non-trivial fluctuations in tra- jectory space, and phase 0 transitions in s -ensemble. 0 0.2 0.4 0.6 0.8 1 Q 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...] T Critical point • Annealed: 2 coupled copies. T onset T K ε a ε H = H 1 + H 2 − ǫ a Q 12 • Within RFOT: Some differences be- • Quenched: copy 2 is frozen. tween quenched and annealed cases. • First order transition emerges from T K , ε q ending at a critical point near T onset . • Direct consequence of, but different H = H 1 − ǫ q Q 12 nature from, ideal glass transition. title – p.12
Numerical evidence in 3 d liquid 20 (a) • Investigate ( T, ǫ ) phase dia- T = 13 , · · · , 7 gram using umbrella sampling. • Sharp jump of the overlap be- 10 ǫ low T onset ≈ 10 . • Suggests coexistence region N = 256 ending at a critical point. 0 0 0.2 0.4 0.6 0.8 1 Q 10 (b) • P ( Q ) bimodal for finite N . T = 8, N = 108 8 • Bimodality and static suscep- 6 P ( Q ) tibility enhanced at larger N for ǫ = 0 4.17 4 5.09 T � T c ≈ 9 . 8 . 5.98 6.17 6.33 2 6.43 → Equilibrium first-order phase 0 transition. 0 0.2 0.4 0.6 0.8 1 Q 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...] T Critical point • Within RFOT, ideal glass transition line ex- tends up to critical point. T onset [Cammarota & Biroli, PNAS ’12] T K • Pinning reduces multiplicity of states, i.e. de- creases configurational entropy: S conf ( c, T ) ≃ Pinning S conf (0 , T ) − cY ( T ) . Equivalent of T → T K . 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
Recommend
More recommend
