Jamming and hard spheres Ludovic Berthier Laboratoire Charles Coulomb Universit´ e de Montpellier 2 & CNRS Spin glasses – Carg` ese, August 28, 2014 D4PARTICLES title – p.1
Outline 1 - Introduction: What is jamming? 2 - Early results 3 - Statistical mechanics approach 4 - Glassy phase diagram 5 - Microstructure of jammed packings 6 - Vibrational dynamics 7 - Rheology title – p.3
I-Introduction: What is jamming? title – p.4
A T = 0 geometric transition • Athermal packing of soft repulsive spheres, e.g. V ( r < σ ) = � (1 − r/ σ ) 2 . Fraction volumique ϕ j Low ϕ : no overlap, fluid Large ϕ : overlaps, solid • Clearly useful for: non-Brownian suspensions (below), hard grains (at), foams and emulsions of large droplets (above). • Since T = 0 , this is a purely geometric problem, possibly a nonequilibrium phase transition. A finite dimensional version of the ideal glass transition? title – p.5
I . 1 – Spin glass perspective title – p.6
Constraint satisfaction problem • Packing as a random constraint satisfaction problem → connection to spin glasses: Pack hard objects with no overlap. [Krzakala & Kurchan, PRE ’07] • A mean-field version of hard sphere fluid: each particle in- teract with z � ( N − 1) neigh- bors. • Interactions specified by quenched random graph. • Aim: Understanding jamming [Mari et al. , PRL ’08] within RFOT. • This model can then be solved as other random constraint satisfaction problems, e.g. cavity method. [Mézard et al. , JSTAT ’11] ( “This paper is meant to be read by specialists in the field, so we did not make much attempt to explain...” ) title – p.7
Connection to Giulio’s lecture • Evolution of free energy landscape in the context of q -coloring problem on random graphs. [Krzakala et al. , PNAS ’07] • “Clustering”: Mode-coupling transition. Equilibrium relaxation time diverges in mean-field. • “Condensation”: Ideal glass (Kauzmann) transition in finite d . • “Uncol”: No solution found which satisfies all constraints. This is a jamming transition. title – p.8
I . 2 – Broader perspective title – p.9
Disordered solid states • Dense granular materials are disordered solids. • Atomic glasses (window glasses, plastics) are solid materials frozen in an amorphous (non-crystalline, metastable) structure. • Two possible pictures: Force chains and geometry of contact network versus complex energy landscape characteristic of disordered materials. title – p.10
Jamming rheology • Observed by compressing soft or hard macroscopic particles. • Example: hard grain suspension. ‘Diverging’ athermal viscosity η 0 ( ϕ ) . 200µ m [Brown & Jaeger, PRL ’09] • Simple rheology: no time scale competes with shear rate γ ) = η 0 ( ϕ ) ∼ | ϕ J − ϕ | − ∆ . η ( ϕ , ˙ [Boyer & Pouliquen, PRL ’10] title – p.11
More ‘jamming’ transitions Γ ��� ��� τ � Air fluidized granular bed � τ [Daniels et al. , PRL ’12] � τ Γ Vibrated grains [Philippe & Bideau, EPL ’02] • Dense assemblies of grains, (large) colloids and bubbles stop flowing. Sheared foam [Langer, Liu, EPL ’00] title – p.12
Athermal rheology of soft particles • Overdamped ( T = 0 ) simulations of sheared harmonic spheres. Diverging viscosity, emergence of yield stress. T = 0 glass transition? $ =0.830 etai inverse shear viscosity ! " 1 $ =0.834 etai $ =0.836 etai $ =0.838 10 -1 etai $ =0.840 etai $ =0.841 etai $ =0.842 etai $ =0.844 etai $ =0.848 etai 10 -2 $ =0.852 etai $ =0.856 etai $ =0.860 etai $ =0.864 etai $ =0.868 etai # =0.0012 10 -3 etai 10 -5 10 -4 10 -3 10 -2 10 -2 fit2 shear stress # [Olsson & Teitel, PRL ’07] [Paredes et al. , PRL ’13] γ ) = | ϕ J − ϕ | − ∆ F (˙ γ | ϕ − ϕ J | β ) . Theoretical basis? • Scaling law: η ( ϕ , ˙ • Similar behaviour (and scaling laws?) observed in emulsion. title – p.13
Athermal rheology of soft particles • Overdamped ( T = 0 ) simulations of sheared harmonic spheres. Diverging viscosity, emergence of yield stress. T = 0 glass transition? [Olsson & Teitel, PRL ’07] [Paredes et al. , PRL ’13] γ ) = | ϕ J − ϕ | − ∆ F (˙ γ | ϕ − ϕ J | β ) . Theoretical basis? • Scaling law: η ( ϕ , ˙ • Similar behaviour (and scaling laws?) observed in emulsion. title – p.14
Bernal’s insight “ This theory treats liquids as homoge- neous, coherent and irregular assem- blages of molecules containing no crys- talline regions or holes. ” • Theory of liquids as a random packing problem. • Experiments with grains, com- puter simulation. [J. D. Bernal, “ A geometrical approach to the structure of liquids ”, Nature (1959)] title – p.15
Dynamics in colloidal hard spheres • Glass ‘transition’: Dramatic increase in viscosity when ϕ decreases. • Viscosity measurements diffi- cult → light scattering. • τ T = σ 2 /D 0 ∼ 1 ms . • T value irrelevant, but T > 0 for thermal equilibrium ̸ = jamming? • Mode-Coupling Theory fit? τ α ∼ ( ϕ c − ϕ ) − γ , ϕ c ≈ 0 . 58 . • ‘Free volume’ → 0 at ϕ J . τ α ≈ exp( P/T ) ∼ exp( A/ | ϕ J − ϕ | ) . Jamming’s back! [van Megen et al. , PRE ’98] • No neat experimental answer from early work (’86 - ’05). title – p.16
Activated colloidal dynamics • Most recent set of light scat- tering data to date. • Mode-coupling prediction fails, no algebraic divergence at ϕ c . • Best fit to ‘activated’ dynam- ics: 4 4 A log( τ α / τ 0 ) τ α ≈ exp( ( ϕ 0 − ϕ ) δ ) , δ ≈ 2 . 2 2 log( τ α / τ 0 ) 0 • Simulations suggest ϕ 0 dis- 0 tinct from jamming point, but 0.55 ϕ 0.60 relie on extrapolations. -2 -4 [Brambilla et al. , PRL ’09] 0.0 0.2 0.4 0.6 ϕ title – p.17
Colloidal rheology • Non-linear rheological study: Transition from viscous fluid to yield stress amorphous solid, in the presence of thermal fluctuations, non-linear rheology: η = η ( ϕ , ˙ γ ) . 10 3 0.620 10 2 0.603 σ a 3 /k B T 10 1 0.586 10 0 0.560 10 -1 0.472 0.517 10 -2 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 . [Petekidis et al. , JPCM ’04] γτ T • Linear viscosity: η T ( ϕ ) = η (˙ γ → 0 , ϕ ) . Similar to η o ( ϕ ) for non-Brownian suspensions, or activated dynamics as found for τ α ?? title – p.18
The glass ‘transition’ 10 14 • Many molecular materials BKS become glasses at low tem- LJ Strong GLY perature. OTP 10 10 SAL PC Energy B2O3 τ α / τ 0 DEC Fragile 10 6 Simulation Glass 10 2 Ideal glass Temperature 0.5 0.6 0.7 0.8 0.9 1 T g /T Tk Tg • Glass ≡ liquid “too viscous” to flow. Glass formation is a gradual process with activated dynamics (not algebraic) in thermal equilibrium. • ‘Ideal’ glass transition at equilibrium? Jamming point for molecules? • Existence of many metastable states: glasses are many-body “complex” systems, due to disorder and geometric frustration. title – p.19
Molecular glassy liquids rheology • Flow curves at finite shear rate ˙ γ in simple shear flow: σ = σ (˙ γ , γ ) = η (˙ γ )˙ in binary LJ mixture. • Transition from viscous fluid to solid material (finite yield stress), with non-linear behaviour (shear-thinning). [Berthier & Barrat JCP ’01] • Rheological behaviour (again) very similar to colloidal hard spheres and non-Brownian soft particles. Viscosity shows activated dynamics. title – p.20
Jamming phase diagram • Suggested by similarity of rheological behaviour observed in non-Brownian & Brownian suspensions, and molecular liquids. J [Liu & Nagel, Nature ’98 - cited 846] [Trappe et al. , Nature ’01 - cited 398] • Has given rise to a whole field of ‘jamming’ studies, many experimental measurements in connection to jamming phase diagram. title – p.21
II-Early results title – p.22
Harmonic spheres • “Bubble model” introduced by Durian in ’95 to study wet foams: V ( r < σ ) = � (1 − r/ σ ) 2 . Can be used to explore the complete ( T, ϕ , σ ) “jamming phase diagram” at once. • Two ways of going athermal, � /T → ∞ . (i) T = const. and � → ∞ : Brownian hard spheres (e.g. colloids). (ii) � = const. , T → 0 : athermal soft suspensions (e.g. foams), equivalent to hard spheres below jamming. • For thermal systems, T/ � quantifies the particle softness: large T/ � = soft particles. Useful for emulsions, microgels, simple liquids. title – p.23
Durian’s bubble model: ’95 • Nice early work: [Bolton & Weaire, PRL 65, 3449 (1990)] • Durian: Rheological study at T = 0 us- ing computer simulations. • Transition from fluid to solid behaviour at ϕ J with scaling properties. • Scaling laws for emergence of shear modulus G , packing pressure P and number of contacts per particle z at crit- ical density ϕ J . Φ [Durian, PRL ’95] title – p.24
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