(Unifying?) rheology of soft glasses and jammed solids Ludovic Berthier Laboratoire Charles Coulomb Universit´ e de Montpellier 2 & CNRS Unifying concepts in materials, JAKS 2012 – Bangalore, February 7, 2012 title – p.1
Coworkers • On-going work with: A. Ikeda (Montpellier) P . Sollich (London) • Some previous work with: P . Chaudhuri (Dusseldorf), H. Jacquin (Paris), S. Sastry (Bangalore), T. Witten (Chicago), F. Zamponi (Paris). title – p.2
Disordered solid states • Atomic glasses (window glasses, plastics) are solid materials frozen in an amorphous (non-crystalline, metastable) structure. • Dense granular materials are disordered solids. • Same/similar/(un-)related transitions? Similar properties in the ‘fluid’? Similar mechanical response of the solid? title – p.3
A geometric problem... really? • Athermal packing of soft repulsive spheres, e.g. V ( r < σ ) = ǫ (1 − r/σ ) 2 . ϕ ϕ c Low ϕ : no overlap, fluid Large ϕ : overlaps, solid • Useful for non-Brownian suspensions (below), grains (at), foams and emulsions (above). Many (oral) claims for glass-formers. • Aim: equilibrium statistical mechanics approach to jamming. See if and how jamming emerges in the T → 0 limit of the ( T, ϕ, σ ) phase diagram. title – p.4
Numerical observations -2 (a) • J -point from packing properties of soft repulsive particles at T = 0 . -4 α =2 [O’Hern et al. PRL ’02] T log p -6 α =5/2 Unjammed • Scaling laws and structure of pack- Jammed σ ings near jamming [vanHecke JPCM’10] -8 J 1/ φ Energy: E = 0 for ϕ < ϕ J ; E ∼ ( ϕ − ϕ J ) α 0 (b) for ϕ > ϕ J . α =2 -2 log G α =5/2 -4 Contact number: z = 0 → z = z c + a ( ϕ − ϕ J ) 1 / 2 with z c = 2 d (isostaticity). -6 0 (c) 3D log (Z-Z c ) -1 2D • A major numerical and experimental -2 effort over the last decade. A new nonequilibrium phase transition. -3 -5 -4 -3 -2 log ( φ - φ c ) title – p.5
Structure of soft colloids • Numerous experiments performed on soft colloidal particles (microgels, emulsions) to probe the jamming transition. [Zhang et al. , Nature 2009] • Anomalous behavior of pair correlation function g ( r ) under compression. • Interpreted as a structural signature of the jamming transition. “Our results conclusively demonstrate that length scales associated with the T = 0 jamming transition persist in thermal systems, not only in simulations but also in laboratory experiments. title – p.6
Rheology of soft particles • Steady state rheology near jamming in overdamped (athermal) numerical simulations of harmonic spheres. Diverging viscosity, emergence of yield stress. $ =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 -2 10 -5 10 -4 10 -3 10 -2 fit2 shear stress # [Olson, Teitel, PRL 07] [Norstrom et al., PRL 2010] • “Similar” behaviour (and scaling laws) observed experimentally. “These results support the conclusion that jamming is similar to a critical phase transition.” title – p.7
Equilibrium fluid • Consider the fluid, V ( r ) = (1 − r ) 2 , at equilibrium at ( T > 0 , ϕ, σ = 0) . • Liquid state theory: solve structure, g ( r ) , thus thermodynamics using integral equations. We can use, e.g., HNC: g ( r ) = e − βV ( r )+ g ( r ) − 1 − c ( r ) . 8 • No glass or jamming transi- 6 T = 1 . 20 · 10 − 3 tion is found. g ( r ) Compress 4 • Anomalous structural evo- lution at all T ! The system 2 first ‘orders’, then ‘disorders’. 0 0.8 1 1.2 1.4 r • F = E − TS : Avoid overlap (reduce energy) at low ϕ . Difficult (entropically disfavoured) at larger ϕ . Solution: increase overlap to gain entropy. • Softness matters (not jamming). [Jacquin & Berthier, Soft Matter ’10] title – p.8
Why liquid state theory fails • Equilibrium phase diagram of soft harmonic spheres. [Berthier & Witten, PRE ’09] 10 − 2 T 10 − 4 MCT VFT Scaling Super-Arrhenius 10 − 6 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 ϕ • The equilibrium fluid does not jam, but the glass structure does. • One cannot understand the jamming transition without dealing first with the glass phase. title – p.9
Metastable states & jamming Energy • Glasses depend on cooling history. Glass • Similarly, compressed fluids of hard spheres reach different glassy states. Ideal glass Temperature Tk Tg Equil. (hard) 1000 Equil. (soft) Glass (hard) • Jamming transition oc- Glass (soft) BMCSL curs along a range of den- Z ( ϕ ) sities [Chaudhuri et al. , PRL ’10] 100 • Theory must handle multiplicity of metastable ϕ onset ϕ MCT ϕ 0 10 states. 45 50 55 60 65 ϕ title – p.10
Statistical mechanics of glasses • Assume exponential number of metastable states exists: � � � f ( T ) = − T − Nf ′ f ′ exp + Ns conf ( f ′ , T ) . V log d T • In practice, take m replica(s) and minimize the replicated free energy [Monasson, PRL ’95, Mézard-Parisi PRL ’99] � � f ( m, T ) = − T � − Nf ′ m f ′ exp + Ns conf ( f ′ , T ) V log d . T • New effective potential valid for both hard spheres ( T → 0 small ϕ ) and soft glasses ( T → 0 large ϕ ), to treat analytically the glass & jamming transitions of harmonic spheres. f ( m, A, ϕ, T ) = f harm ( m, A ) + f liquid ( ϕ, T/m ) − ρ � drg ( r )[ e − β ( V eff ( r ) − mV ( r )) − 1] 2 [Jacquin, Berthier & Zamponi PRL ’11] title – p.11
The ‘ideal’ glass transition • High T fluid: m = 1 , s conf ( T ) > 0 and simple liquid theory is enough. • s conf ( T ) vanishes at T K ( ϕ ) > 0 for ϕ > ϕ K ≡ hard sphere glass transition. 10 − 2 FLUID T K ∼ ( ϕ − ϕ K ) 2 10 − 4 T K new approximation 10 − 2 GLASS small cage 10 − 6 T 10 − 4 ϕ K ≈ 0 . 577 MCT VFT Scaling Super-Arrhenius 10 − 8 0.55 0.6 0.65 0.7 0.75 10 − 6 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 ϕ ϕ • Low- T scaling: T K ∼ ( ϕ − ϕ K ) 2 (robust scaling) with ϕ K ≈ 0 . 577 (value depends on specific approx). title – p.12
The ‘ideal’ jamming transition • Glass thermodynamics: energy, pressure, specific heat, fragility... • Jamming at T = 0 ⇔ Change in ground state glass properties. 1 . 10 − 4 3 . 10 − 3 • ϕ GCP = 0 . 633353 ... such that: E GS = 0 below, FLUID E GS ≃ a ( ϕ − ϕ GCP ) 2 above. GLASS 2 . 10 − 3 T K • Glass Close Packing: densest e gs 5 . 10 − 5 T = 0 glass with no overlap. 1 . 10 − 3 [Zamponi & Parisi, RMP ’10] UNJAMMED JAMMED • P GS ∼ ( ϕ − ϕ GCP ) . 0 0 0.56 0.6 0.64 0.68 0.72 ϕ • Existence, location(s), and scaling laws of jamming from ‘first principles’. title – p.13
Structure of jammed states • New predictions for g ( r ) near contact at all ( T, ϕ ) . Isostaticity is derived. -5 T=10 1000 -6 T=10 10 − 6 -7 T=10 -8 T=10 U glass ( T, ϕ ) T=0 HS 10 − 8 T=0 SS max T = 10 − 5 g G 10 − 6 10 − 10 10 − 7 10 − 8 100 10 − 9 10 − 10 10 − 12 0 0.62 0.625 0.63 0.635 0.64 0.645 [Berthier, Jacquin, Zamponi, PRE ’11] ϕ • Clear emergence of jamming singularity from finite temperatures properties of glass phase–only if particles are not too soft, T/ǫ ≪ 10 − 6 . • Still unconvinced G and J are different things? Rheology will do it. title – p.14
Rheology at finite temperatures • Aim: Study the harmonic sphere rheology at finite temperatures, then approach T → 0 . • ‘SLLOD’: Newton eqs. + shear + thermostat (‘SLLOD’). Problem: one cannot shear faster than thermal fluctuations, Pe = ˙ γτ D < 1 . � Here τ D ∼ a/ k B T/m → ∞ , cannot go athermal. • We use Langevin dynamics with shear and thermostat in d = 3 : ξ ( d r i dV ( | r i − r j | ) � + η i , dt − ˙ γy i e x ) = − d r i j with � η i ( t ) η j ( t ′ ) � = 2 k B Tξ 1 δ ( t − t ′ ) . • Two important microscopic timescales: (i) dissipation: τ 0 = ξa 2 /ǫ = 1 , our time unit. (ii) thermal time: τ D = ξa 2 / ( k B T ) → ∞ when T → 0 . • We study both finite and zero temperatures, both thermal ( Pe < 1 ) and athermal ( Pe > 1 ) rheologies at once. [Ikeda, Berthier, Sollich, in preparation] title – p.15
Soft glassy rheology • Steady state rheology at T = 10 − 4 and increasing ϕ . Diverging viscosity, emerging yield stress. Here τ − 1 ∼ 10 − 4 . D 10 4 10 − 3 10 3 σ Y [ ǫ/a 3 ] η [ ξ/a ] 10 2 10 − 4 10 1 10 0 10 − 5 0.56 0.6 0.64 0.68 0.72 ϕ • This is a glass transition as seen in colloidal particles, star polymers, microgels, but also glassy liquids. • Theories of driven glasses capture competition between slow glassy dynamics and shear flow. title – p.16
From glass to jamming rheology • Same at T = 10 − 6 , here τ − 1 ∼ 10 − 6 . Glass physics shifts to lower shear D γ < 10 − 6 . rates: Pe < 1 → ˙ 10 2 10 − 3 10 − 4 10 1 σ Y [ ǫ/a 3 ] η [ ξ/a ] 10 − 5 10 0 10 − 6 10 − 1 10 − 7 0.56 0.6 0.64 0.68 0.72 ϕ • The athermal jamming physics emerges when τ − 1 γ ≪ τ − 1 0 . ≪ ˙ D • Two Newtonian regimes, two distinct viscosities, emergence of yield stress (when ˙ γ → 0 ), with funny density dependence. A real mess! title – p.17
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