Flow of Glasses 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!
Fundamental Transition Elastic solid Elastic Modulus µ Viscous Liquid diffusive, Viscosity η Symmetry change Temporal symmetry Elastic F F(-t) = F(t) F(-t) = - F(t) Plastic F Energy storage Energy loss
Glass Formation Cooling from Liquid Solidification Glass transition
Viscosity and Diffusion Macroscopic: Viscosity viscosity / Pa s 10 40 10 30 Glass transition 10 20 10 12 glass 10 10 liquid 1 day 1 year Menkind 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? NO! Vogel-Fulcher-Tamman = Relax. Time > 10 18 s
Hard-Sphere Suspensions 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 (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)
Insight into Flow Phenomena
Food Personal care products and Applications Paint Processing Construction
Suspension Flows γ. Shear thickening 10-1 10-2 10-3 Diffusion rate t D -1 Shear thinning Driven Flow 10-4 Free volume V f /V p?? Viscous Flow Fluid 0.49 Supercooled Fluid 0.58 Glass 0.64 φ
Glassy Flow - Basics i. Free volume
Free Volume Theory Hard Spheres Bernal The structure of liquids et al. 1960s Canonical Holes
Model systems: Hard spheres Voronoi Volume
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 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)
Flow : Constitutive description Fraction f of flow spot Strain per STZ = 1. Free volume Theory (! =" # $#= %& & 2. Strain per STZ ~ 1 3. Jump Rate = & Applied Stress σ Jump Rate (Spaepen Acta Met. 1977) )*/, -)*/, (forward) (backward)
Flow : Constitutive description Fraction f of flow spot Strain per STZ = Jump Rate (Spaepen Acta Met. 1977) =!. /012 3*, =!. 45 9: 678 78 sinh(x) ~ x?
Flow : Constitutive description Models Creep of Bulk Metallic Glass Pd 41 Ni 10 Cu 29 P 20 Heggen et al. 2003 T = 550K
Local Correlation Shear Modulus From thermal fluctuations <ε ij2 > t Free Volume From Voronoi volume 1 R k B T = µ <ε ij2 > 2 t V 0 Rahmani et al. Phys Rev. E (2014)
Local Correlation Shear Modulus µ [k B T/R 3 ] V f [10-3 V 0 ] Free Volume Rahmani et al. Phys Rev. E (2014)
Glassy Flow - Basics i. Correlations T T g Increasing cooperativity Adam & Gibbs (1965)
Elastic Field Elastic continuum
Elastic Field Displacements Strain Field (Hutchinson 2006) Strain Field. @A B C long-range Correlations between flow spots?
Elastic Correlations? Internal coupling in external field
Analogy: Magnetic Coupling Magnetic spins in external field m(r) r m(r+ r) H Correlation function F E ΔG = J G J G+ΔG B Susceptibility D E = "F E G $H
Analogy: Magnetic Coupling 2nd Order Phase Transitions m(r) r m(r+ r) H Critical Scaling close to T c F E G G O G P Divergence of Correlation length Susceptibility P L L R M D E L L N M Correlation length
Glasses: Dynamic correlations Dynamic correlation function v(0, t) r v(r, t) 4-point correlation function T S G,ΔV = # 0,ΔV # G,ΔV Dynamic susceptibility D S = "T S G,ΔV $G
Glasses: Dynamic correlations Granular fluid of ball bearings Colloidal glass Computer simulation 2D repulsive discs Dynamical criticality? T S G O B X Y
Glasses: Dynamic correlations Berthier et al. PRL 2003 Dynamical criticality? T S G O B X Y
Glass transition: critical phenomenon? Berthier, Biroli et al. 2011 No true divergence for quiescent glass
Summary Glasses Liquid and Solid, depending on time scale Flows Liquid/Glassy, Flow rate ~ t D -1, τ D -1 Structural ingredient Free volume local modulus Correlations Elastic field Coupling Self-Organization?