FOR1394 DFG Research Unit FOR1394 Nonlinear Response to Probe Vitrification Higher harmonics in sheared colloids: Divergence of the nonlinear response Matthias Fuchs Fachbereich Physik, Universit¨ at Konstanz Japan-France Joint Seminar, YITP, Kyoto 2015
FOR1394 FOR1394 FOR1394 Maxwell Model of linear response Viscous flow Hookian elasticity Viscous flow Hookian elasticity σ xy = η ∂v x ∂u x σ xy = G ∞ ∂y ∂y stress, viscosity, velocity gradient stress, elastic constant, strain Visco-elasticity (J.C. Maxwell, 1867) Visco-elasticity (J.C. Maxwell, 1867) � t dt ′ G ( t − t ′ ) ∂v x ( t ′ ) σ xy ( t ) = ∂y −∞ G ( t ) = G ∞ e − t/τ Fluid: Fluid: G ( t ) rapid G ( t ) rapid Solid: Solid: G ( t ) slow G ( t ) slow G ( t ) = η δ ( t ) , η = G ∞ τ G ( t ) = G ∞ 2 / 26
FOR1394 FOR1394 FOR1394 Nonlinear response: FT Rheology G ( t, t ′ ) Non-time translational invariant G ( t, t ′ ) t � dt ′ ˙ γ ( t ′ ) G ( t, t ′ ) σ ( t ) = −∞ For the special case of oscillatory shear oscillatory shear: Input: γ ( t ) = γ 0 sin( ωt ) Output: � ∞ n =1 G ′ σ ( t ) = γ 0 n ( ω ) sin( nωt ) � ∞ n =1 G ′′ + γ 0 n ( ω ) cos( nωt ) [ fig: http://gain11.wordpress.com/2008/07/14/the-five-faces-of-distortion/ ] 3 / 26
FOR1394 FOR1394 FOR1394 3rd Harmonic & cooperativity Biroli-Bouchaud theory ∗ Biroli-Bouchaud theory ∗ 3rd harmonic χ 3 ( ω ) diverges at glass transition χ 3 ( ω ) ∝ ∂χ 1 (2 ω ) ∂T (using: T c ( E ) = T c (0) + κ E 2 , FDT) χ 3 ∝ N corr (number of correlated particles) Dielectric spectroscopy ∗∗ Dielectric spectroscopy ∗∗ χ 3 ( ω ) & N corr measured [ ⋆ Tarzia, Biroli, Lefevre & Bouchaud JCP 132 , 054501 (2010)]; also Biroli & Bouchaud, PRB 72 064204 (2005)] [ ⋆⋆ Bauer, Lunkenheimer & Loidl, PRL 111 , 225702 (2013); also Crauste-Thibierge, Brun, Ladieu, L’Hote, Biroli, Bouchaud, PRL 104 , 165703 (2010)] 4 / 26
FOR1394 FOR1394 FOR1394 Outline Nonlinear Dielectric Response Biroli-Bouchaud Theory Large Amplitude Oscillatory Shear (LAOS) strain Constitutive Equations in MCT-ITT Fourier Transform Rheology 3rd Harmonic Spectrum Scaling Laws Experiment Summary Nonlinear response of glass 5 / 26
FOR1394 FOR1394 FOR1394 Part II Large Amplitude Oscillatory Shear Constitutive Equations in MCT-ITT 6 / 26
FOR1394 FOR1394 FOR1394 Microscopic model Brownian particles in flow Brownian particles in flow Coupled random walks Coupled random walks � d y � dt r i − v solv ( r i ) ζ = F i + f i homogeneous flow v solv ( r ) = κ · r x e.g. simple shear F i interparticle force y f i random force ���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� x ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� � f α i ( t ) f β j ( t ′ ) � = 2 ζk B Tδ αβ δ ij δ ( t − t ′ ) ���� ���� z ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ��� ��� v solv ��� ��� = ˙ γ ( t ) y x Generalized Green Kubo relation (+ MCT approximation) Generalized Green Kubo relation � t � t t ′ ds Ω † ( s ) σ � ( e ) / ( k B TV ) dt ′ � Tr { κ ( t ) · σ } e − σ ( t ) = −∞ 7 / 26
FOR1394 FOR1394 FOR1394 Microscopic model Brownian particles in flow Brownian particles in flow Coupled random walks Coupled random walks � d y � dt r i − v solv ( r i ) ζ = F i + f i homogeneous flow v solv ( r ) = κ · r x e.g. simple shear F i interparticle force y f i random force ���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ��� ��� ���� ���� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� x ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� � f α i ( t ) f β j ( t ′ ) � = 2 ζk B Tδ αβ δ ij δ ( t − t ′ ) ���� ���� z ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ��� ��� ���� ���� ��� ��� ��� ��� v solv ��� ��� = ˙ γ ( t ) y x Generalized Green Kubo relation (+ MCT approximation) Generalized Green Kubo relation � t � t t ′ ds Ω † ( s ) σ � ( e ) / ( k B TV ) dt ′ � Tr { κ ( t ) · σ } e − σ ( t ) = −∞ 7 / 26
FOR1394 FOR1394 FOR1394 Linear rheology in colloids Linear response moduli Linear response moduli PNIPAM microgels η ∞ (HI) added G ∞ ——- radius R H ( T ) ηω ↑ 1 / τ ∝ ω φ eff = 4 π 3 R 3 N [Siebenb¨ urger, Ballauff (2009)] ≈ 0 . 63 H V stress magnitudes with 50% error stress magnitudes 8 / 26
FOR1394 FOR1394 FOR1394 Nonlinear rheology in colloids PNIPAM microgels Stress-strain relation in glass Stress-strain relation in glass 5 b) k B T /d 3 � 2 G c ∞ 1 Pe 0 = 10 -1 � radius R H ( T ) 10 -2 σ xy 10 -3 0.5 10 -4 10 -5 10 -6 0.1 1 γt ˙ scaling-law for ˙ γ → 0 (theo.) scaling-law 9 / 26
FOR1394 FOR1394 FOR1394 Nonlinear rheology in colloids Stress-strain relation in glass Stress-strain relation in glass PNIPAM microgels φ eff = 0.65 ε = 10 -3 5 σ xy [ k B T/d 3 ] 2 radius R H ( T ) Pe 0 = 10 -1 Pe eff /5.3 = 10 -3 10 -2 10 -4 10 -3 10 -5 1 10 -4 10 -6 10 -5 0.1 1 • t / γ res γ γ R 2 Pe 0 = ˙ H Siebenb¨ urger, Ballauff [JPCM 27 , 194121 (2015)] D 0 yield strain γ ∗ underestimated (factor 3) yield strain γ ∗ 9 / 26
FOR1394 FOR1394 FOR1394 Distorted structure MCT-ITT d = 3 MD metal melt ∗ d = 2 BD hard disks 6 4 0.2 2 0.1 y � Å � 0 0 � 2 � 0.1 � 0.2 � 4 � 6 � 2 � 4 0 2 4 6 � 6 � 6 � 4 � 2 0 2 4 6 x � Å � plastic deformation ( l = 4 ) plastic deformation MD metal melt confocal (MCT inset) 0.05 a) b) γ =0.01 γ =0.01 γ =0.35 γ =0.25 0 -0.05 δ g( θ ) -0.1 δ g( θ ) 0 -0.15 -0.4 -0.2 -0.8 0 π /4 3 π /4 π θ -0.25 0 π /4 π /2 3 π /4 0 π /4 π /2 3 π /4 π θ θ [ ∗ P. Kuhn, Th. Voigtmann; ∗∗ M. Laurati, S. Egelhaaf ; (unpublished, 2015) ] 10 / 26
FOR1394 FOR1394 FOR1394 Part III 3rd Harmonic Spectrum Scaling Laws Experiment schematic model used [J. Brader, T. Voigtmann, MF, R. Larson and M. Cates, PNAS, 106 , 15186 (2009)] 11 / 26
FOR1394 FOR1394 FOR1394 LAOS-model stress for applied shear rate γ ( t ) = γ 0 sin ωt ˙ stress for applied shear rate � t dt ′ G ( t, t ′ ) ˙ γ ( t ′ ) σ ( t ) = −∞ generalized shear modulus G ( t, t ′ ) = v σ Φ 2 ( t, t ′ ) � t γ ( t, t ′ ) = schematic F 12 model for strain schematic F 12 model for strain t ′ ds ˙ γ ( s ) � t � � ∂ t Φ( t, t ′ ) + Γ Φ( t, t ′ ) + t ′ d s m ( t, s, t ′ ) ∂ s Φ( s, t ′ ) = 0 memory kernel m ( t, s, t ′ ) = h ( γ ( t, s )) h ( γ ( t, t ′ )) ν 1 ( ε ) Φ( t, s ) + ν c 2 Φ 2 ( t, s ) � � strain decorrelation 1 h ( γ ) = 1 + ( γ/γ ∗ ) 2 12 / 26
FOR1394 FOR1394 FOR1394 Oscillatory shear – FT Rheology dimensionless parameters: γ R 2 shear rate: Pe 0 = ˙ (bare Peclet number) H D 0 shear rate: Pe = ˙ γτ (Peclet, Weissenberg number) Pe ω = ω R 2 frequency: H D 0 frequency: De = ωτ (Deborah number) σ × R 3 stress: H k B T γ = γ 0 strain: γ ∗ Input: ǫ = φ − φ c γ ( t ) = γ 0 sin( ωt ) , ( φ packing fraction) φ c Output: � ∞ � ∞ n =1 G ′ n =1 G ′′ σ ( t ) = γ 0 n ( ω ) sin( nωt ) + γ 0 n ( ω ) cos( nωt ) Parameters: v σ , Γ , γ ∗ & η ∞ 13 / 26
FOR1394 FOR1394 FOR1394 Motivation Object: ( ∝ γ 2 3rd Harmonic amplitude: I 3 = | G 3 ( ω ) | 0 ) Q 0 = 1 I 3 γ 2 I 1 0 Questions: Dependence on ω , ǫ ? I 3 related to N corr (number of correlated particles) ? Plastic decay ? Method: Taylor approximation of schematic MCT model for γ 0 → 0 14 / 26
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