Nonequilibrium transitions in glassy flows. Peter Schall University of Amsterdam
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1 Nonequilibrium transitions in glassy flows Peter Schall University of Amsterdam
2 Liquid or Solid?
3 Liquid or Solid? Example: Pitch Solid! 1 day 1 year Menkind sec Time scale Liquid!
4 Glasses atomic vibrations + structural changes Atomic Volume atomic vibrations only Solidification Glass transition
5 Viscosity and Diffusion Macroscopic: Viscosity viscosity / Pa s Glass transition Viscosity10 η glass ~ 1/D (diff.coeff.) ~ τ (relax.time) 1 day 1 year Menkind liquid temperature sec Time scale P. Schall, Harvard University
6 Viscosity and Diffusion Simple Liquids: Arrhenius E act Log(viscosity) simple liquids Diffusion coefficient D ~ D 0 e ( -E act/k B T) Viscosity η ~ η 0 e ( E act/k B T) many glasses (1/T g ) (1/T)
7 Strong and Fragile Glasses Angel plot Arrhenius η = η 0 exp(e/k B T) Vogel-Fulcher- Tamman η = η 0 exp( ) B T-T 0
8 Glass Phenomenology Myth: Do cathedral glasses flow over centuries? Vogel-Fulcher-Tamman =
9 Colloidal Hard Spheres Hard-sphere Phase Diagram Fluid Quench 0.49 Fluid + Cryst Crystal Glass Volume Fraction (Alder, Wainwright 1957)
10 Single Particle Dynamics Diffusion in liquids <r(t) 2 > 0 0 t Mean square displacement < x(t) 2 > = 2Dt Fluctuation-Dissipation ξd = k B T
11 Single Particle Dynamics Dilute suspensions <r(t) 2 > 0 t Einstein (1906): η(φ) = η 0 (1 + 5/2 φ) Batchelor (1977): η(φ) = η 0 (1 + 5/2 φ φ 2 )
12 Single Particle Dynamics Diffusion (Molecules or small particles in a supercooled liquid) Mean-square Displacement R Liquid r 2 ~ t Supercooled liquid Arrest in plateau time 0 t D Diffusion time τ Glass relaxation time
13 Supercooled Liquids Dynamic Measurements... Weeks et al. Science (2000)
14 Dynamic Heterogeneity At the glass transition Packing fraction φ = 58 % Weeks et al. Science 2002
15 Glassy Flow - Basics i. Free volume ii. Correlations
16 Free Volume Theory Hard Spheres Bernal The structure of liquids et al. 1960s Canonical Holes
17 Free Volume Theory V 0 V i Free Volume V f ~ (V i V 0 ) Free Volume Theory: P(V f ) ~ exp(-v f / <V f >)
18 Free Volume Theory V 0 V i Rearrangements occur if V f ~ V 0 Viscosity η ~ P(V f ~V 0 ) -1 ~ exp(+δv 0 /<V f >) ~1
19 Free Volume Theory V 0 Free volume from thermal expansion V i Big success of free volume theory! Viscosity =
20 Free Volume Theory and suspensions? (Chaikin, PRE 2002) φ max 1 / (Temperature) Volume fraction φ
21 Free Volume Theory Max. Packing Fraction φ m ~ 0.64 V 0 V i Free volume:??? Viscosity: =???
22 Free Volume Theory Max. Packing Fraction φ m ~ 0.64 V 0 V i Free volume: Viscosity: =
23 Free Volume Theory: Suspensions η / µ ,0 0,1 0,2 0,3 0,4 0,5 0,6 φ φ max = (Cheng, Chaikin, PRE 2002)
24 Free volume Free energy Free Volume σ/2 Free energy =! Zargar, et al. Phys Rev Lett. (2013)
25 Free energy of glasses Glasses Crystals Glasses: free energy decrease over time Aging Zargar, et al. Phys Rev Lett. (2013)
26 Free energy of glasses Heterogeneity Relaxation ~200 particles Free energy barriers during relaxation N ~ F -α, α = 1.2 ~Gutenberg Richter
27 Glassy Flow - Basics i. Free volume ii. Correlations
28 Correlations in Traffic
29 Correlations in Traffic v i+1 v i v i-1
30 Correlations in Traffic v i v i+1 v i+2 v i+3 Velocity correlations " = # $ # $+! #($) ) Dynamic susceptibility * +",
31 Traffic simulations Dynamic susceptibility Max(χ 4 ) 1 - p De Wijn, Miedema, Nienhuis, P.S. PRL (2012)
32 Correlations in glasses r 4-point correlation function. - /,Δ2 = # 0,Δ2 # /,Δ2 Dynamic susceptibility * - = +. - /,Δ2,/
33 Analogy: Magnetic Coupling Magnetic spins in external field m(r) r m(r+ r) H Correlation function " 4 Δ/ = 6 / 6 /+Δ/ 7 Susceptibility * 4 = +" 4 /,5
34 Analogy: Magnetic Coupling 2nd Order Phase Transitions m(r) r m(r+ r) H Critical Scaling close to T c " 4 / / / : Divergence of Correlation length Susceptibility : < 8 * Correlation length
35 Glasses: Dynamic correlations Granular fluid of ball bearings Colloidal glass Computer simulation 2D repulsive discs Dynamic heterogeneity
36 Glass transition: critical phenomenon? Berthier, Biroli et al No true divergence No dynamic criticality
37 Summary Glasses, Suspensions Liquid and Solid, depending on time scale Free volume theory Glasses: temperature - viscosity (Vogel-Fulcher) Suspensions: Divergence of viscosity Correlations Coupling Self-Organization?
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