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
Outline Intro to dense colloidal dispersions Active microrheology: Theory Delocalization transition Length Scales & Populations Steady state motion Transient dynamics 2 / 33
to dense colloidal dispersions Intro to dense colloidal dispersions Intro 3 / 33
Brownian motion Stokes Einstein Sutherland: D 0 = k BT ζ 0 mean squared displacement (r(t) r(0)) 2 = 6D 0 t 4 / 33
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, 178301 (2012) ] 5 / 33
microrheology: Theory Active microrheology: Theory Active 6 / 33
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
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 0 + 1 3k B T 0 dt F s e Ω irr t F s (e) [ Gazuz, Puertas, Voigtmann, MF, Phys. Rev. Lett. 102, 248302 (2009)] 8 / 33
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, 125701 (2010)] 9 / 33
transition Delocalization transition Delocalization 10 / 33
Tracer Nonergodicity Parameter in Real Space f s (r, t) = limt ρs (r, t) ϕ =.537 11 / 33
Delocalization transition: phase diagram (MCT) glassy host delocalized probe fluid host glassy host localized probe 12 / 33
Delocalization transition φ = 0.537 type B transition 13 / 33
Scales & Populations Length Scales & Populations Length 14 / 33
Exponential tails Glassy host F threshold localized probe MCT, φ = 0.516 Marginal probability distribution parallel to the force [A. Puertas (U. Almeria), unpublished] 15 / 33
Probe distribution function G s (z, t ) Localized probe G tail s [ exp z ] ξ(f ) Growing correlation length ξ 16 / 33
state motion Steady state motion Steady 17 / 33
Probe motion in force direction: average drift δz(t) GLASSY HOST MCT φ = 0.516 x F z δz(t) [d] 10 4 F = 10 [kt/d] 10 3 F = 20 delocalized probe F = 30 10 2 F = 35 t 10 1 F = 40 F = 45 F 10 0 F = 50 localized probe 10-1 10-2 10-3 10-4 10-3 10-2 10-1 10 0 10 1 10 2 10 3 t [d 2 /D 0 ] Delocalization transition 18 / 33
Probe motion in force direction: average drift δz(t) FLUID HOST MCT φ = 0.515 x F z δz(t) [d] 10 4 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10-4 F [kt/d] = 1 F = 5 F = 10 F = 20 F = 30 F = 40 F = 50 10-5 10-6 10-4 10-2 10 0 10 2 10 4 10 6 t [d 2 /D 0 ] t ---- linear response (FDT) Steady probe velocity for long times friction coefficient 19 / 33
Nonlinear friction coefficient: v = [ζ(f )] 1 F x z F ζ = ζ(f ) Force-thinning ζ/ζ 10 6 10 5 10 4 φ 10 3 10 2 φ = 0.500 (fluid) 10 1 0.510 (fluid) 0.515 (fluid) 10 0 0.516 (glass) 0.537 (glass) 10-1 0.62 (LD Sim.) 10-2 10 0 10 1 10 2 F [kt/d] F critical 20 / 33
Probe motion in perpendicular direction: δx 2 (t) GLASSY HOST x F z δx 2 (t) [d 2 ] 10 2 10 1 10 0 10-1 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 = 0.516 10-2 10-3 10-4 10-3 10-2 10-1 10 0 10 1 10 2 10 3 10 4 t [d 2 /D 0 ] Delocalization transition Diffusive motion for long times 21 / 33
Nonlinear diffusion coefficient - perpendicular direction δx 2 D orth t x z F D orth /D 0 10 2 φ = 0.500 (fluid) 10 1 0.510 (fluid) 0.515 (fluid) 10 0 0.516 (glassy) 0.537 (glassy) 10-1 0.62 (LD Sim.) 10-2 10-3 10-4 φ 10-5 F critical 10-6 10 0 10 1 10 2 F [kt/d] Orthogonal diffusion enhanced by increasing F 22 / 33
Stokes-Einstein Relation Equilibrium: D(ϕ)ζ(ϕ) kt Nonequilibrium: D orth (ϕ, F )ζ(ϕ, F ) kt = 1 = α(ϕ, F ) α-1 14 12 10 8 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 φ = 0.500 (fluid) 0.510 (fluid) 0.515 (fluid) 0.516 (glassy) 0.537 (glassy) 0.62 (LD sim.) 6 4 2 0 10 0 10 1 10 2 F [kt/d] 23 / 33
dynamics Transient dynamics Transient 24 / 33
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
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 ] 10 2 10 1 10 0 10-1 10-2 F = 40 [kt/d] F = 50 F = 60 F = 70 F = 80 F = 100 F = 120 F = 200 t 10-3 10-2 10-1 10 0 10 1 10 2 t [d 2 /D 0 ] [A. Puertas (U. Almeria), unpublished] 26 / 33
Probe motion in force direction: superdiffusion Brownian Dynamics simulations (2D) MSD δz 2 (t) [δz(t)] 2 x z F 27 / 33
Evolution of the probe distribution function G s (z, t) Delocalized probe G s (z,t) 10 0 10-1 t [d 2 /D 0 ] = 0.21 0.42 0.84 1.68 3.36 6.72 t 10-2 0 2 4 6 8 10 12 z [d] 28 / 33
Evolution of G s (z, t): Delocalized probe MCT, φ = 0.537, F = 50kT/d G s (z,t) 10 0 10-1 t [d 2 /D 0 ] = 0.21 0.42 0.84 1.68 3.36 6.72 t LD Simulations, φ = 0.62, F = 100kT/d G s (z,t) 10 0 10-1 t [d 2 /D 0 ] = 1.0 2.5 5.0 10.0 20.0 10-2 0 2 4 6 8 10 12 z [d] 10-2 0 2 4 6 8 z [d] [A. Puertas (U. Almeria), unpublished] 29 / 33
Trajectories Brownian dynamics simulations individual trajectories in the glassy host intermittent dynamics ζ s 10 6 10 4 10 2 ϕ= 0.79 ϕ = 0.81 10 0 10 0 10 1 10 2 10 3 F ex d [Harrer, unpublished] 30 / 33
Evolution of Gs (z, t): Delocalized probe 100 Gs(z,t) Probe trajectories Brownian dynamics simulations t [d2/d0] = 0.21 0.42 0.84 1.68 3.36 6.72 10-1 t 10-2 0 2 4 6 8 10 12 z [d] Active and inactive probes F 31 / 33
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
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