Active microrheology: Brownian motion, strong forces and viscoelastic media

Transcription

1 OR1394 DFG Research Unit Nonlinear Response to Probe Vitrification Active microrheology: Brownian motion, strong forces and viscoelastic media Matthias Fuchs Fachbereich Physik, Universität Konstanz New Frontiers in Non-equilibrium Statistical Physics 2015

2 Outline Intro to dense colloidal dispersions Active microrheology: Theory Delocalization transition Length Scales & Populations Steady state motion Transient dynamics 2 / 33

3 to dense colloidal dispersions Intro to dense colloidal dispersions Intro 3 / 33

4 Brownian motion Stokes Einstein Sutherland: D 0 = k BT ζ 0 mean squared displacement (r(t) r(0)) 2 = 6D 0 t 4 / 33

5 Displacement fluctuations in glass Binary glass in d = 2 finite displacements h(ri (t) r i )2 i < with center of trajectory: R t 1 r i = t 0 dt ri (t) [Klix, Ebert, Weysser, MF, Maret, Keim, Phys. Rev. Lett.109, (2012) ] 5 / 33

6 microrheology: Theory Active microrheology: Theory Active 6 / 33

7 Active microrheology Hard-sphere suspension Fluid and glass states probe x z Probe displacement under strong forces Tails, heterogeneities etc. Parameters: φ and F d host volume fraction φ F >> kt/d 7 / 33

8 Microscopic (statistical) description Model: Interacting Brownian particles, no HI (D 0 = k BT ζ 0 ) Smoluchowski equation: Micro rheology t Ψ = Ω Ψ Ω = N D 0 i ( i βf i ) + D s s ( s βf s ) D s s F ex i ITT force-velocity relation (nonlinear, exact) ζ v t =F ex, ζ = ζ k B T 0 dt F s e Ω irr t F s (e) [ Gazuz, Puertas, Voigtmann, MF, Phys. Rev. Lett. 102, (2009)] 8 / 33

9 Microscopic (statistical) description (II) Mode coupling approximation: r, t F G s (r, t) Fourier transform Φ s q(t) t = 0 probe distribution function probe correlator Evolution equation: t Φ s q(t) + 1 τ q (F ) Φs q(t) + Parallel relaxation channels : t 0 m q (t t ) t Φ s q(t )dt = 0 m q (t) = F {Φ s q(t), Φ host q (t), F } Input (full MCT): Equilibrium structure of the host at φ [ Lang et al., PRL 105, (2010)] 9 / 33

10 transition Delocalization transition Delocalization 10 / 33

11 Tracer Nonergodicity Parameter in Real Space f s (r, t) = limt ρs (r, t) ϕ = / 33

12 Delocalization transition: phase diagram (MCT) glassy host delocalized probe fluid host glassy host localized probe 12 / 33

13 Delocalization transition φ = type B transition 13 / 33

14 Scales & Populations Length Scales & Populations Length 14 / 33

15 Exponential tails Glassy host F threshold localized probe MCT, φ = Marginal probability distribution parallel to the force [A. Puertas (U. Almeria), unpublished] 15 / 33

16 Probe distribution function G s (z, t ) Localized probe G tail s [ exp z ] ξ(f ) Growing correlation length ξ 16 / 33

17 state motion Steady state motion Steady 17 / 33

18 Probe motion in force direction: average drift δz(t) GLASSY HOST MCT φ = x F z δz(t) [d] 10 4 F = 10 [kt/d] 10 3 F = 20 delocalized probe F = F = 35 t 10 1 F = 40 F = 45 F 10 0 F = 50 localized probe t [d 2 /D 0 ] Delocalization transition 18 / 33

19 Probe motion in force direction: average drift δz(t) FLUID HOST MCT φ = x F z δz(t) [d] F [kt/d] = 1 F = 5 F = 10 F = 20 F = 30 F = 40 F = t [d 2 /D 0 ] t ---- linear response (FDT) Steady probe velocity for long times friction coefficient 19 / 33

20 Nonlinear friction coefficient: v = [ζ(f )] 1 F x z F ζ = ζ(f ) Force-thinning ζ/ζ φ φ = (fluid) (fluid) (fluid) (glass) (glass) (LD Sim.) F [kt/d] F critical 20 / 33

21 Probe motion in perpendicular direction: δx 2 (t) GLASSY HOST x F z δx 2 (t) [d 2 ] F = 10 [kt/d] F = 20 F = 30 F = 35 F = 40 F = 45 F = 50 F D orth t delocalized probe localized probe MCT φ = t [d 2 /D 0 ] Delocalization transition Diffusive motion for long times 21 / 33

22 Nonlinear diffusion coefficient - perpendicular direction δx 2 D orth t x z F D orth /D φ = (fluid) (fluid) (fluid) (glassy) (glassy) (LD Sim.) φ 10-5 F critical F [kt/d] Orthogonal diffusion enhanced by increasing F 22 / 33

23 Stokes-Einstein Relation Equilibrium: D(ϕ)ζ(ϕ) kt Nonequilibrium: D orth (ϕ, F )ζ(ϕ, F ) kt = 1 = α(ϕ, F ) α φ = (fluid) (fluid) (fluid) (glassy) (glassy) 0.62 (LD sim.) F [kt/d] 23 / 33

24 dynamics Transient dynamics Transient 24 / 33

25 Probe motion in force direction: superdiffusion MD Simulations - Yukawa mixture Winter et al., PRL 108 (2012) MSD δz 2 (t) [δz(t)] 2 x F z MSD (parallel) 25 / 33

26 Probe motion in force direction: superdiffusion Langevin Dynamics simulations φ = 0.62 MSD δz 2 (t) [δz(t)] 2 x F z δz 2 (t) - [δz(t)] 2 [d 2 ] F = 40 [kt/d] F = 50 F = 60 F = 70 F = 80 F = 100 F = 120 F = 200 t t [d 2 /D 0 ] [A. Puertas (U. Almeria), unpublished] 26 / 33

27 Probe motion in force direction: superdiffusion Brownian Dynamics simulations (2D) MSD δz 2 (t) [δz(t)] 2 x z F 27 / 33

28 Evolution of the probe distribution function G s (z, t) Delocalized probe G s (z,t) t [d 2 /D 0 ] = t z [d] 28 / 33

29 Evolution of G s (z, t): Delocalized probe MCT, φ = 0.537, F = 50kT/d G s (z,t) t [d 2 /D 0 ] = t LD Simulations, φ = 0.62, F = 100kT/d G s (z,t) t [d 2 /D 0 ] = z [d] z [d] [A. Puertas (U. Almeria), unpublished] 29 / 33

30 Trajectories Brownian dynamics simulations individual trajectories in the glassy host intermittent dynamics ζ s ϕ= 0.79 ϕ = F ex d [Harrer, unpublished] 30 / 33

31 Evolution of Gs (z, t): Delocalized probe 100 Gs(z,t) Probe trajectories Brownian dynamics simulations t [d2/d0] = t z [d] Active and inactive probes F 31 / 33

32 Conclusions micro-rheology delocalization threshold Fc ex force-induced diffusion force two population behavior close to depinning rare large excursions on diverging length scale k B T/σ 32 / 33

33 Acknowledgements Gustavo Abade, Markus Gruber (Konstanz) Antonio Puertas (Almeria) Thomas Voigtmann (DLR, Düsseldorf) Igor Gazuz, Christian Harrer, Manuel Gnann (Konstanz) David Winter, Jürgen Horbach (Düsseldorf) movies/discussion: Peter Keim (Konstanz), Stefan Egelhaaf (Düsseldorf), Rut Besseling (Edinburgh), Eric Weeks (Emory) DFG Research Unit Nonlinear Response to Probe Vitrification Thank you for your attention 33 / 33