On the Yielding of Colloidal (and Other) Glass Formers Thomas - PowerPoint PPT Presentation

Transport Processes in Melts External Fields under Zukunftskolleg free • creative • connecting On the Yielding of Colloidal (and Other) Glass Formers Thomas Voigtmann Institute of Materials Physics in Space, German Aerospace Center, Cologne & Zukunftskolleg, Universität Konstanz, Germany J. A. Krumhansl Symposium, Bangalore, February 2012

Acknowledgments Ch. Harrer, S. Papenkort, A. Bhattacharjee A. Meyer (DLR), J. Horbach (Düsseldorf) M. Fuchs (Konstanz), J. Brader (Fribourg), M. E. Cates (Edinburgh) S. Egelhaaf (D’dorf), M. Ballauff, M. Siebenbürger (HZ Berlin) 2 / 29

Outline Dynamical Yield Stress Startup: Creep and Micro-Rheology Residual Stresses 3 / 29

Introduction 4 / 29

Rheology of Dense Fluids shear flow of dense fluids: external flow rate ˙ γ ∼ v/h [1 / s] velocity v large structural relaxation time τ [s] ⇒ large effect when ˙ γτ ≫ 1 height h (glassy) kinetic arrest: τ → ∞ apply perturbation (shear ˙ γ ) ⇒ measure response (stress σ ) F [ ˙ γ ] = σ constitutive equation F is a model of the material linear response, steady state : Newtonian liquid σ = ˙ γ × η 5 / 29

Visco-Elasticity: Maxwell’s Model Newtonian fluid: η = const. ⇒ σ ∝ ˙ γ shear stress σ Hookian elastic solid: σ ∝ γ viscous fluid dense fluids: ?? elastic solid Maxwell: combine σ ∼ γ and σ ∼ ˙ γ . t deformation (constant rate) γ = γ γ = ˙ ˙ σ/G ∞ + σ/η “spring-and-dashpot” model: Hookian spring constant G ∞ G ∞ η differential equation solved by � t γ ( t ′ ) G ∞ e − ( t − t ′ ) /τ dt ′ σ ( t ) = ˙ η = G ∞ τ −∞ output input model 6 / 29

Visco-Elasticity: Maxwell’s Model Newtonian fluid: η = const. ⇒ σ ∝ ˙ γ shear stress σ Hookian elastic solid: σ ∝ γ viscous fluid dense fluids: ?? elastic solid Maxwell: combine σ ∼ γ and σ ∼ ˙ γ . t deformation (constant rate) γ = γ γ = ˙ ˙ σ/G ∞ + σ/η “spring-and-dashpot” model: Hookian spring constant G ∞ G ∞ η Maxwell’s constitutive equation � t γ ( t ′ ) G ∞ e − ( t − t ′ ) /τ dt ′ σ ( t ) = ˙ η = G ∞ τ −∞ output input model 6 / 29

Nonlinear Rheology: Shear Thinning apply (steady) shear ⇒ dramatic decrease in apparent viscosity non-linear response linear response: η ∼ const. η → ∞ : glass η ∼ 1 / ˙ γ applications: painting, coating, lubrication, … “universal”: metallic melts, thermosensitive colloids geophysics, soft matter, … [Fuchs and Ballauff, J Chem Phys (2005)] 7 / 29

Nonlinear Rheology: Shear Thinning apply (steady) shear ⇒ dramatic decrease in apparent viscosity non-linear response linear response: η ∼ const. η → ∞ : glass η ∼ 1 / ˙ γ applications: painting, coating, lubrication, … “universal”: metallic melts, Pd 40 Ni 10 Cu 30 P 20 , various temperatures geophysics, soft matter, … [Kato et al. , JAP (1998)] 7 / 29

Nonlinear Rheology: Shear Thinning apply (steady) shear ⇒ dramatic decrease in apparent viscosity 4 non-linear response 10 linear response: η ∼ const. 3 10 η → ∞ : glass 2 η 10 η ∼ 1 / ˙ γ 1 10 applications: painting, coating, lubrication, … 0 10 -8 -7 -6 -5 -4 -3 10 10 10 10 10 10 . γ “universal”: metallic melts, granular simulation geophysics, soft matter, … [Olsson/Teitel, PRL (2007)] (?!?) 7 / 29

Rheo-Mode-Coupling Theory, Schematically � t nonlinear schematic model – strain history γ tt ′ = t ′ ˙ γ ( τ ) dτ � t � t MCT dt ′ ˙ dt ′ v σ ˙ γ ( t ′ ) φ 2 ( t, t ′ , [ γ ]) γ ( t ′ ) G ( t, t ′ , [˙ σ ( t ) ∼ γ ]) ≈ −∞ −∞ output input model � t t ′ m ( t, t ′′ , t ′ ) ∂ t ′′ φ ( t ′′ , t ′ ) dt ′ = 0 ∂ t φ ( t, t ′ ) + φ ( t, t ′ ) + v 1 φ ( t, t ′′ ) + v 2 φ ( t, t ′′ ) 2 � m ( t, t ′′ , t ′ ) = h [ γ tt ′ ] h [ γ tt ′′ ] � wave-vector advection cage effect γ ˙ 2 π/q y ( t, t ′ ) 2 π/q x 2 π/q x [Fuchs/Cates, PRL (2002); Brader et al. , PRL (2007); PRL (2008)] [Brader, ThV, Fuchs, Larson, Cates, PNAS (2009)] 8 / 29

Dynamical Yield Stress 9 / 29

Dynamical Yield Stress [Fuchs/Ballauff, JCP (2005)] thermosens. colloids: η ( ̺, ˙ γ ) flow curves in steady state σ (˙ γ → 0) = σ y > 0 in the (idealized) glass: dynamic yield stress � t γ → 0 is singular; σ = ˙ ˙ γ −∞ G ( t − t ′ , ˙ γ ) dt ′ 10 / 29

A Nonlinear Maxwell Model shear accelerates dynamics: relaxation time ∼ 1 / ˙ γ τ − 1 �→ τ − 1 + ˙ γ nonlinear Maxwell model G ∞ ˙ γ = ˙ σ + σ/τ + σ ˙ γ/γ c (plus a high-shear Newtonian viscosity…) γ ˙ σ = G ∞ τ 0 ˙ γ + G ∞ τ 1 + ˙ γτ/γ c ⇒ as τ → ∞ : critical dynamical yield stress σ y = G ∞ γ c > 0 [Fuchs/Cates, Faraday Discuss. (2003)] [ThV, EPJE (2011)] 11 / 29

Flow Curves granular material colloidal suspension -1 10 -3 10 −1 η -5 10 -7 10 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 σ [Olsson/Teitel, PRL (2007)] [data: M. Siebenbürger] scenarios for dynamical yielding: colloidal vs. granular remember: different protocols, k B T finite vs. zero 12 / 29

Flow Curves: Scaling Proposal granular material metallic melt α = 1.23 50K 0 10 β = 0.60 100K 300K T 0 = 860K 500K T = η -1 /|(T-T 0 )/T 0 | α 600K 700K -2 10 T>T 0 800K 840K 900K σ T α / β 940K -1 η -4 10 1000K 1040K T<T 0 1100K 0 1 2 3 10 10 10 10 σ T = σ /|(T-T 0 )/T 0 | β [Olsson/Teitel, PRL (2007)] [Guan/Chen/Egami, PRL (2010)] (granular) point J as a critical point γ x ⇒ hence: σ c scaling: suggests σ ( T → T c ) ∼ ˙ y = 0 13 / 29

Flow Curves: Finite Yield Stress Brownian hard spheres metallic melt 2 10 1 10 1 10 σ [ kT/R 2 ] 0 10 σ [GPa] 0 -1 10 σ y 10 -2 10 -1 10 -3 10 -6 -5 -4 -3 -2 -1 0 1 -6 -5 -4 -3 -2 -1 10 10 10 10 10 10 10 10 10 10 10 10 10 10 . . [10 γ 12 /s] γ [Guan/Chen/Egami, PRL (2010)] [data: Henrich et al. , Phil Trans Roy Soc (2009)] fits using schematic model of mode-coupling theory (MCT) prediction σ c y = O (0 . 1 k B T/R 3 ) – apparent power laws [ThV, EPJE 34 , 106 (2011)] 14 / 29

Different Scenarios for Flow Curves σ σ . . γ γ “discontinuous” vs. “continuous” yield-stress scenario different yielding mechanisms: local cages vs. avalanches energy densities k B T/R 3 vs. overlap energies 15 / 29

Different Scenarios for Flow Curves G ∞ log G ( t ) log G ( t ) log t log t “discontinuous” vs. “continuous” yield-stress scenario different yielding mechanisms: local cages vs. avalanches energy densities k B T/R 3 vs. overlap energies 15 / 29

Different Scenarios for Flow Curves η η η, σ η, σ σ σ 1/ϕ, T 1/ϕ, T “discontinuous” vs. “continuous” yield-stress scenario different yielding mechanisms: local cages vs. avalanches energy densities k B T/R 3 vs. overlap energies 15 / 29

Jamming Diagram iso-viscosity lines in the ( T, σ ) and (1 /̺, σ ) plane: 2000 0.4 1500 0.2 T � K � � � 1000 0.0 � 0.2 500 � 0.4 0 0 5 10 15 20 0 1 2 3 4 5 6 Σ � GPa � Σ (white: “jammed”) [ThV, EPJE 34 , 106 (2011)] 16 / 29

Startup: Creep and Micro-Rheology 17 / 29

Creep shear stress σ γ deformation γ ( t ) after sudden step stress time t σ 0 = 0.01 ∼ t σ = 0.9 kT / R 3 2 0 10 σ 0 = 0.05 10 σ = 0.5 kT / R 3 σ 0 = 0.10 σ = 0.1 kT / R 3 σ 0 = 0.20 1 10 -1 σ 0 = 0.30 10 σ 0 = 0.50 0 γ (t) γ ( t ) σ 0 = 1.00 10 -2 10 -1 10 -3 10 -2 10 experiment: M. Siebenbürger -4 -3 10 10 -2 -1 0 1 2 3 4 5 6 1 2 3 4 5 6 7 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 t t / τ 0 nonequilibrium transition: plastic deformation / flow static yield stress σ c anomalous flow behavior (creep)? [Cottrell, “The Time Laws of Creep”] [Siebenbürger, Ballauff, ThV (in preparation)] 18 / 29

Creep Continued creep laws (hard matter): 2 10 logarithmic, ˙ γ ( t ) t ∼ const. 1 10 t w = 60s 0 t w = 600s Andrade, ˙ γ ( t ) ∼ t − α , α ≈ 2 / 3 10 γ t w = 6000s -1 10 secondary, ˙ γ ( t ) ∼ const. -2 10 γ ( t ) ∼ t 1+ x “viscosity thinning”, ˙ -3 10 -3 10 related to stress overshoot? -4 10 aging-time dependent! -5 10 . γ -6 10 2.0 -7 10 -8 10 t w 1.5 -9 10 1 10 σ 1.0 0 10 -1 10 . t 0.5 γ -2 10 -3 10 0.0 -4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 10 γ -5 10 2 3 4 5 6 7 10 10 10 10 10 10 t / τ B [Siebenbürger, Ballauff, ThV (in preparation)] 19 / 29

Static Yielding: A Force Threshold steady external shear ⇒ glass molten (always) steady external force ⇒ yielding transition σ c microscopic analog? yielding of individual “cages” by local external force ⇒ microrheology delocalized applied force F liquid ex localized density [Gazuz, Puertas, ThV, Fuchs, PRL (2009)] 20 / 29