Effective Temperatures in Driven Systems near Jamming

Transcription

1 Effective Temperatures in Driven Systems near Jamming Andrea J. Liu Department of Physics & Astronomy University of Pennsylvania Tom Haxton Yair Shokef Tal Danino Ian Ono Corey S. O Hern Douglas Durian Stephen Langer Sidney R. Nagel Physics, UPenn Physics, UPenn Chemistry, UCLA Chemistry, UCLA Mechanical Engineering, Yale Univ. Physics, UPenn. NIST James Franck Inst., U Chicago

2 Modern Challenge We understand a lot about the collective properties of many-particle systems in thermal equilibrium microscopic { systems near thermal equilibrium statistical mechanics { systems far from equilibrium }?? } macroscopic But flowing glassy systems are far from equilibrium

3 Driven Systems Near Jamming Unsheared foam (D. J. Durian) Sheared foam (M. Dennin) 95% gas 5% liquid energy barrier for bubble rearrangements "10 6 k B T Can fluctuations be described by effective temperature? If so, so what? Shearing drives system over energy barriers

4 Testing Effective Temperature We and others conducted numerical simulations of simple models under steady-state shear Calculate T eff in several independent ways In an equilibrium system, all calculations D. J. Durian, PRL 75, 4780 (1995). must yield same result Are they the same in a driven system?

5 Model: Equations of Motion m d 2 r i dt 2 Elastic forces Constant Shear-rate Viscous forces = r F i e + " r F i v # $% r v i + Fluid " v r i = v r i # $ y i x ˆ!& b.c. s thermostat L image image v x = " L v x = 0 v x = " # L Types of simulations: 1. Sheared; massless; athermal 2. Sheared; massive; athermal 3. Sheared; massive; thermal " > 0;m = 0;# = 0 " > 0;m # 0;$ = 0 " > 0;m # 0;$ = 0;T > 0 4. Equilibrium " = 0;m # 0;$ = 0;T > T g

6 Temperatures from Linear Response Thermal diffusion ("r(t)) 2 D. A. Weitz = 6Dt D = T " temperature drag 0.5µm Response to Force F Particles move at speed F /" where is drag " = 6#$a % ' T eff " & ( ' size of fluctuations (correlation) ( ) 2 D v /F, #V,... V$ T ) ' * + ' ease of creating fluctuation (response)

7 Other Linear Response Relations Adiabatic compressibility Pressure Fluctuations! S "1 = A T p " p ( ) 2 Langer and Liu, Europhys. Lett. 49, 68 (2000). Viscosity Shear Stress Fluctuations # $ 0! = A T dt " xy(t)" xy (0) c Heat Capacity! U!T = 1 T 2 Energy Fluctuations d U ( U " U ) 2 "!U = " dt T 2 ( ) 2

8 T Results Bidisperse T T eff T p T xy T U I. K. Ono, C. S. O Hern, D. J. Durian, S. A. Langer, A. J. Liu and S. R. Nagel, PRL 89, (2002). Berthier & Barrat, PRE, (2001); J Chem Phys, 116, 6228 (2002); PRL 89, (2002). Polydisperse Bidisperse !. 4 indep. def ns of T eff yield the same result Effective temperature appears to be a useful concept! AND there s more.

9 Textbook Definition of Temperature Monte-Carlo results for 1/T=dS/dU: 0-20 log 10 (!) N= I. K. Ono, C. S. O Hern, D. J. Durian, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. 89, (2002). T S T U from shearing runs T U H. Makse, J. Kurchan, Nature 415, 614 (2002) U/N Shear makes system nearly ergodic! BUT.

10 Two Time-Scale/Two Temperatures L. F. Cugliandolo, J. Kurchan, L. Peliti, PRE 55, 3898 (1997). L. Berthier, J.-L. Barrat, J. Kurchan, PRE 61, 5464 (2000). At short times, system sees T bath At long times, system sees T eff What happens if harmonic oscillator is inserted into system? T spr T eff T. Danino k B T spr = k spr < y 2 > T bath k spr /k

11 e.g. FD relation for density h ρ = perturbation Time-Dependent Linear Response C(t) = " ( k r,t)" # k r,0 R(t) = " r k,t ( ) ( ) # "( k r,0) h " ( ) = e i " r k,t N $ j=1 r k # r j (t ) correlation response R(t) = C(0) T # C(t) & % 1" ( $ C(0) ' R(t) = C(t) = C(t) C(0) R(t) = R(t) C(0) [1" C(t)] T Fluctuation- Dissipation Theorem

12 Intercept vs. Long-Time Slope Previous definitions based on static linear response R-intercept Definitions based on time-dependent linear response slope R(t) = R(t = ") =1/T intercept [1" C(t)] T slope System in thermal equilibrium

13 Sheared Glassy Systems The response-correlation curve is not a straight line! J. L. Barrat and L. Berthier, Phys. Rev. E 63, (2000). At long times, system can explore other minima At short times, system fluctuates within local minimum short-time slope: T S is bath temperature long-time slope: T L depends on shear rate Because curve is not straight, T L " T I Cugliandolo and Kurchan, PRL 71, 173 (1993)

14 Density fluctuations: Def. 4 c. f. Berthier & Barrat, PRL 89, (2002) " r k,t N $ ( ) = e i j=1 m>0;athermal " = 0.01 r k # r j (t ) /T - L (a) k=11 k=4.5 1/T - S C!(k) (t) T L from density fluctuations matches Defs. 1-3 based on static linear response! k=9

15 Pressure fluctuations P = N " j=1 P j k=0 quantity m>0;thermal " = /T " (b) --1/T Ss C P (t) Intercept, not slope, corresponds to consistent T eff

16 T I (p)=t L (ρ k ) This result is robust. We have varied System (athermal with dissipation, thermal with no dissipation) Potential Density Shear rate Bath temperature C. S. O Hern, A. J. Liu, S. R. Nagel, PRL 93, (2004). Static and dynamic linear response give consistent T eff Cugliandolo-Kurchan picture is conceptually incomplete

17 For most observables, can choose consistent set But how to know which definition to use??? k=0 quantities: T I k>0 or local quantities: T L Others don t agree The Complications Quantity T I T L p(k=0) σ(k=0) p(k y ) ρ(k y ) X X p zz, (2p xx -p yy -p zz )/3, config. temps, force temp. (Shokef) X X

18 T T Summary Bidisperse L. Berthier & J.-L. Barrat, PRE (2001); J Chem Phys 116, 6228 (2002); PRL 89, (2002). I. K. Ono, C. S. O Hern, D. J. Durian, S. A. Langer, A. J. Liu and S. R. Nagel, PRL 89, (2002). 0 Polydisperse Bidisperse !. C. S. O Hern, A. J. Liu, S. R. Nagel, PRL 93, (2004). Altogether, 8 indep. def ns of T eff yield same value T p, T σ, T pk, T U, T ρ, T E, T spr,t K T S is consistent Effective temp. appears to be reasonable concept

19 Why should we care? If I tell you the temperature of a material, you don t get very excited. What does T eff tell us about the system (other than the relation between correlation and response)? Does T eff tell us anything about the dynamics of the system near jamming?

20 Jamming Phase Diagram A. J. Liu and S. R. Nagel, Nature 396 (N6706) 21 (1998). Temperature unjammed jammed Shear stress T eff (shear stress) 1/Density But this is a weird phase diagram Shear stress is nonequilibrium axis Can we replace shear stress with T eff?

21 Glass Transition log τ(s) C. A. Angell, J. Non-Cryst. Solids , 13 (1991). T g /T Super-Arrhenius dependence of relaxation time 1.0

22 Viscosity vs. T T. Haxton " = " 0 e A /(T #T 0 ) T 0 " equil 1/φ T σ Super-Arrhenius dependence of viscosity on T

23 Steady-state shear rheology " xy Solid: T<T 0 " Solid: T<T 0 Dashed: T>T 0 Dashed: T>T 0 " " xy System has apparent yield stress for T<T 0 Apparent yield stress increases with decreasing T Power-law rheology for T " T 0 (SGR)

24 Behavior of T eff T eff " Dashed: T>T 0 Solid: T<T 0 " " For T>T 0,T eff -> T as -> 0 " For T< T 0,T eff -> T eff,g as ->0 equil T 1/T eff η vs. T eff is super-arrhenius σ T eff,g < T 0 1/φ

25 What do shear-induced fluctuations do? 1/Teff Data collapse for T<<T 0 or large σ xy Limiting behavior: " xy # max V r $ %E 1/T eff " # 1 $E log % xy /% 0 [ ] " xy Let V r = typical size of rearrangement events Suppose that between rearrangements, local strain can build up to " max # exp ( $E /T ) eff due to to activated crossing of energy barriers ΔE with T eff. But "E # $ xy % max V r This yields " xy # " 0 exp($%e /T. eff )

26 Summary Steady shear appears to have two effects It lowers typical energy barriers But it also gives rise to fluctuations, which allow activated crossing of the barriers The typical energy barrier should depend on shear rate (Ashwin, Brumer, Reichman, Sastry, J Phys Chem B 108, (2004).) Still ahead: analysis of inherent structures

27 Second law of thermodynamics Do 2 systems in thermal contact equilibrate to same T eff? Recall earlier results: Small and large particles in binary sheared glass have same T eff (Berthier & Barrat, J Chem Phys, 116, 6228 (2002)) Massive particle has same T eff as small particles (Berthier & Barrat, PRL 89, (2002).) Harmonic oscillator has same T eff as system Requires matching of time scales Massive particle Low-frequency oscillator What about two large systems? How can we place them in contact?

28 Preliminary studies of 2nd law Separate by flexible chain with k chain This allows No particle exchange Propagation of rearrangement events Equilibration of shear stress if average horizontal position is not fixed Equilibration of pressure if average vertical position is not fixed "!& T. Danino!&

29 Pressure allowed to equilibrate? X X Preliminary results Shear stress allowed to equilibrate? X X T eff equilibrates? Final effective temps of both systems are equal if final pressures are equal. Does T eff depend only on pressure? (N. Xu & C. S. O Hern) X X

30 T. Haxton Does T eff does just depend on p? T eff p If T eff were just a function of p, then all data for different T, " would collapse on single curve No, T eff does not just depend on pressure. So why does equilibration of p allow equilibration of T eff???

31 Conclusions T eff seems to be reasonable for sheared glassy systems Experiments need to measure as many T eff as possible (Durian) Einstein relation Harmonic oscillator (optical or magnetic trap) T eff seems to control barrier hopping w/ Boltzmann prob. When is T eff useful? (2 time-scale/2 temp?) Second law? How is effective heat exchanged? Song, et al. (PNAS 05) Is configurational entropy conjugate to T eff even when particles have inertia?

32 Acknowledgements Tom Haxton Yair Shokef Tal Danino Ian Ono Corey S. O Hern Douglas Durian Stephen Langer Sidney R. Nagel Physics, UPenn Physics, UPenn Chemistry, UCLA Chemistry, UCLA Mechanical Engineering, Yale Univ. Physics, UPenn. NIST James Franck Inst., U Chicago Supported by NSF-DMR ,