Nonequilibrium transitions in glassy flows Peter Schall University of Amsterdam
Liquid or Solid?
Liquid or Solid? Example: Pitch Solid! 1 day 1 year Menkind 10-2 10 0 10 2 10 4 10 6 10 8 10 10 10 12 10 14 sec Time scale Liquid!
Glasses atomic vibrations + structural changes Atomic Volume atomic vibrations only Solidification Glass transition
Viscosity and Diffusion Macroscopic: Viscosity viscosity / Pa s 10 40 Glass transition 10 30 10 20 Viscosity10 η 10 10 12 glass ~ 1/D (diff.coeff.) ~ τ (relax.time) 1 day 1 year Menkind liquid temperature 10-2 10 0 10 2 10 4 10 6 10 8 10 10 10 12 10 14 sec Time scale P. Schall, Harvard University
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)
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
Glass Phenomenology Myth: Do cathedral glasses flow over centuries? Vogel-Fulcher-Tamman =
Colloidal Hard Spheres Hard-sphere Phase Diagram Fluid Quench 0.49 Fluid + Cryst 0.54 0.58 Crystal Glass 0.64 0.74 Volume Fraction (Alder, Wainwright 1957)
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
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 + 5.9 φ 2 )
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
Supercooled Liquids Dynamic Measurements... Weeks et al. Science (2000)
Dynamic Heterogeneity At the glass transition Packing fraction φ = 58 % Weeks et al. Science 2002
Glassy Flow - Basics i. Free volume ii. Correlations
Free Volume Theory Hard Spheres Bernal The structure of liquids et al. 1960s Canonical Holes
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 >)
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
Free Volume Theory V 0 Free volume from thermal expansion V i Big success of free volume theory! Viscosity =
Free Volume Theory and suspensions? (Chaikin, PRE 2002) φ max 1 / (Temperature) Volume fraction φ
Free Volume Theory Max. Packing Fraction φ m ~ 0.64 V 0 V i Free volume:??? Viscosity: =???
Free Volume Theory Max. Packing Fraction φ m ~ 0.64 V 0 V i Free volume: Viscosity: =
Free Volume Theory: Suspensions 10000 η / µ 1000 100 10 1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 φ φ max = (Cheng, Chaikin, PRE 2002)
Free volume Free energy Free Volume σ/2 Free energy =! Zargar, et al. Phys Rev Lett. (2013)
Free energy of glasses Glasses Crystals Glasses: free energy decrease over time Aging Zargar, et al. Phys Rev Lett. (2013)
Free energy of glasses Heterogeneity Relaxation ~200 particles Free energy barriers during relaxation N ~ F -α, α = 1.2 ~Gutenberg Richter
Glassy Flow - Basics i. Free volume ii. Correlations
Correlations in Traffic
Correlations in Traffic v i+1 v i v i-1
Correlations in Traffic v i v i+1 v i+2 v i+3 Velocity correlations " = # $ # $+! #($) ) Dynamic susceptibility * +",
Traffic simulations Dynamic susceptibility Max(χ 4 ) 1 - p De Wijn, Miedema, Nienhuis, P.S. PRL (2012)
Correlations in glasses r 4-point correlation function. - /,Δ2 = # 0,Δ2 # /,Δ2 Dynamic susceptibility * - = +. - /,Δ2,/
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
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 * 4 9 8 Correlation length
Glasses: Dynamic correlations Granular fluid of ball bearings Colloidal glass Computer simulation 2D repulsive discs Dynamic heterogeneity
Glass transition: critical phenomenon? Berthier, Biroli et al. 2011 No true divergence No dynamic criticality
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?