Signatures of the full replica symmetry breaking in jamming systems - PowerPoint PPT Presentation

Aug. 11th, 2015 Japan-France Joint Seminar "New Frontiers in Non-equilibrium Physics of Glassy Materials" Signatures of the full replica symmetry breaking in jamming systems under shear Hajime Yoshino Cybermedia Center, Osaka University

Collaborators Francesco Zamponi (ENS, Paris) Corrado Raione (ENS, Paris & Sapienza-Univ. Rome) Pierefrancesco Urbani(CEA, Saclay) 中山大樹 Daijyu Nakayama(Osaka Univ.) 岡村論 Satoshi Okamura(Osaka Univ.) Financial Supports Synergy of Fluctuation and Structure : Quest for Universal Laws in Non-Equilibrium Systems 2013-2017 Grant-in-Aid for Scientific Research on Innovative Areas, MEXT, Japan JPS Core-to-Core program 2013-2015 Non-equilibrium dynamics of soft matter and information

温度効果のあるジャミング転移 � エマルション (emlusion; � 乳濁液， � 乳剤) � unjam � jam � 水と油など， 混ざり合わない液体が � ミセルを形成して � Emulsions, colloids,... 一方が液滴となって他方に分散している系 � 接触力 � hexagonal hexagonal lattice close packing volume ドデカン液滴 � 10 µ m fraction (in 水+グルコース) � “Jammed” Glass Glass エントロピー弾性 � Supercooled ϕ E. R. Weeks and � Liquid 温度効果なしや液体では0 � C. Holinger(2007) � 0 . 74 ϕ g ϕ RCP ϕ m ∼ 0 . 64 ∼ 0 . 58 ∼ 0 . 5 エマルションの圧力と剛性率の測定（室温） � Colloidal crystallization “random” close packing 身近では.) マヨネーズ， � 木工用ボンド， � など � (大きい○=圧力, 黒シンボル=剛性率) � E. R. Weeks, (s: 表面張力, R: � 粒径) � in "Statistical Physics of Complex Fluids", Eds. S Maruyama & M Tokuyama T. G. Mason et al. (1997) � k B T room / ✏ ∼ 10 − 5 (Tohoku University Press, Sendai, Japan, 2007). 圧力と剛性率の振る舞いがほぼ同じ � 液滴(粒子)間の相互作用の大きさで � → 温度効果のない数値計算 � 換算して温度 T ~ 10 -5 � では出てこない � Model cf. C. S. O’Hern et al. (2003) 等 � 1 � �� � U = v ( r ij ) r ij = | r i − r j | 0.8 α = 2 v ( r ) / � 0.6 � ij � 0.4 v ( r ) = � (1 − r/D ) α � (1 − r/D ) 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 r/D

Mean-Field Picture on Glass transitions F. Zamponi’s talk glass transition Gardner transition Jamming volume fraction ϕ G ϕ d < ϕ g < ϕ K ϕ Liquid Glass 1step replica symmetry Replica Symmetric (RS) “more” RSB breaking (1RSB) Kurchan-Parisi-Urbani-Zamponi, (2013). λ replicon = 0 Almeida-Thouless (AT) instability much like the MF models of spin-glasses

Okamura-Yoshino, unpublished (2013) Stress relaxation process Figure 1: This figure show snaphots before/after a plastic event trigered by thermal noises. Here we used a 2-dimensional version of the model (for the purpose of a demonstration) at volume fraction φ = 0 . 85 which is slightly above the jamming density φ J ∼ 0 . 84 (2-dim). The system is initially perturbed weakly by a shear-strain γ = 0 . 05 and let to relax at zero temperature by the conjugated gradient method which allows the system to relax using the harmonic modes. Then the thermal noise at (reduced) temperature T = 10 − 6 is switched on. The configuration of particles are represented by the circles and that of the contact forces f ij = − dv ij ( r ij ) /dr ij are represented by bonds whose thickness is chosen to be proportional to f ij . The panels a) and b) show the snapshots before/after a plastic event (which took about 10 4 t micro to complete). In panel c) the configuratoin of the particles before/after are overlaid : the one before the event is shown by the lighter color.

solids under shear mechanical properties liquid theory + replica of amorphous solids Yoshino-Mezard, PRL 105, 015504 (2010), Yoshino, JCP 136, 214108 (2012) Strain γ shear-stress (force/area) σ C. Rainone, P . Urbani, HY, F. Zamponi, Phys. Rev. Lett. 114, 015701 (2015). yield stress σ Y HY, F. Zamponi, PRE90, 015701(2014). σ = µ γ shear-strain γ 0

HY and F. Zamponi, Phys. Rev. E 90, 022302 (2014). Twisting replicated hardsphere liquid d → ∞ u a ... γ 1 γ 2 dx ρ ( x )[1 − log ρ ( x )] + 1 � � − β F ( { γ a } ) = dxdy ρ ( x ) ρ ( y ) f { γ a } ( x, y ) 2 Replicated Mayer function (under shear) m S ( γ ) µ ν = δ µ ν + γδ ν , 1 δ µ, 2 � e − β v ( | S ( γ a )( x a − y a ) | ) f { γ a } ( x, y ) = − 1 + a =1 − β F ( ˆ = 1 − log ρ + d log m + d 2 ( m − 1) log(2 π eD 2 /d 2 ) + d α m,m ) ∆ , { γ a } ) /N 2 log det(ˆ ⇣ 2 ( γ a − γ b ) 2 ⌘ R ∆ ab + λ 2 − d d λ 2 b 2 π F ϕ √

Corrado Rainone, Pierfrancesco Urbani, Hajime Yoshino, Francesco Zamponi, Phys. Rev. Lett. 114, 015701 (2015) dilatancy Yielding Following glassy states under shear/compression βσ /d γ

Small strain expansion 1 ,m m σ a γ a + 1 � � F ( { γ a } ) /N = F ( { 0 } ) /N + µ ab γ a γ b + · · · 2 a =1 a,b � � � β µ ab = d ∂ F − (1 − δ ab ) ∂ F translational � � δ ab � µ ab = 0 2 � ϕ invariance ∂ ∆ ac ∂ ∆ ab c ( � = c ) b Z 1 /m 1 ∆ ( y ) = γ ( y ) dz β ˆ µ ( y ) = z 2 γ ( z ) − m γ ( y ) y y Hierarchical RSB y = x/m µ 2 � µ 1 ∆ 2 0 1 µ 0 � ∆ 1 µ 2 µ 1 � ∆ 2 0 µ 2 µ 1 � 0 ∆ 2 µ 0 � ∆ 1 µ 1 µ 2 m � 0 ∆ 2 1RSB case : HY and M. Mezard (2010), HY (2012) m < x < 1

1 step RSB HY and F. Zamponi, Phys. Rev. E 90, 022302 (2014). ϕ d < � � ϕ < � ϕ Gardner � � δ ab − 1 H. Yoshino and M. Me´zard, PRL 105 , 015504 (2010). β � µ ab = β � µ EA H. Yoshino, The Journal of Chemical Physics 136 , 214108 (2012). m µ EA = � ∆ − 1 ∆ EA ∼ � � ϕ d ) 1 / 2 β ˆ ∆ d − C ( � ϕ − � EA in agreement with MCT G. Szamel and E. Flenner, PRL 107 , 105505 (2011). ˆ µ EA 1 / � ϕ 0 γ 1 / � ϕ d

1+continuous RSB ϕ Gardner < � � ϕ < � ϕ GCP ϕ → b b p ∝ 1 /m → ∞ ϕ − GCP γ ( y ) ∝ γ ∞ y − ( κ − 1) κ = 1 . 41575 HY and F. Zamponi, Phys. Rev. E 90, 022302 (2014). β µ EA = 1 / ∆ EA ∝ m − κ ∝ p κ consistent with scaling argument + effective medium computation E DeGiuli; E Lerner; C Brito; M Wyart, PNAS 111 (2014), 17054 “rigidity of inherent structures” 1 β b µ (1) = m γ (1) ∝ p γ “rigidity of metabasins”

Field Cooled/ Zero Field Cooled Susceptibilities in Spin-Glasses Full RSB solution of the Sherrington-Kirkpatrick (SK) model FC susceptibility (exact solution of the Edwards- Anderson spin-glass model ZFC in the limit ) d → ∞ susceptibility Z 1 χ FC = β [1 − dxQ ( x )] 0 χ ZFC = β [1 − Q (1)] (NOTE) spin-wave rigidity of spin-glass is also hierarchical reflecting RSB G. Kotliar, H. Sompolinsky, and A. Zippelius PRB 35, 311 (1987) H. Yoshino, JCP 136, 214108 (2012)

FC/ZFC shear response of glasses ? Nakayama-Yoshino-Zamponi, in progress Measure the remanent shear-stress shear strain µ = σ / γ Field Cooling γ initial conf. obtain via MD simulation at Zero Field Cooling T/ ✏ = 10 − 5 (1) temperature quench to T=0 (working at ) ϕ target (2) compression (working at T=0) Energy minimization : conjugated gradient method

Simulation of densely packed soft-spheres in 3 dim. 3 dim Harmonic-sphere(binary) O (10 3 ) − O (10 4 ) # of samples N = 320 , 1000 0.04 ZFC (compression) 0.035 ZFC (Temperature-quench) 0.03 0.025 µ 0.02 FC (compression) 0.015 FC (Temperature-quench) 0.01 γ = 10 − 4 0.005 0 γ = 10 − 3 0 0.001 0.002 0.003 0.004 0.005 p µ ZFC ∝ √ p Reminder: theory d → ∞ µ FC ∝ p

Divergence of non-linear susceptibility at spin-glass transition ◆ − γ ✓ T 1 − χ 2 ∝ T SG m = χ 0 h + χ 2 h 3 + . . . δ q EA = χ SG h 2 + . . . Edwards-Anderson Order parameter q EA = 1 X h S i i 2 N Spinglass susceptibility i χ SG = 1 [ h S i S j i 2 � h S i i 2 h S j i 2 ] X N ij Non-linear susceptibility and SG susceptibility χ 2 = − βχ SG ( T > T SG )

Non-linear shear-modulus H. Yoshino, in progress (2015) F ( ˆ F entropic ( ˆ ∆ ) + F interaction ( ˆ ∆ , { γ } ) = ∆ , { γ } ) fluctuation around the saddle point H ab,cd = ∂ 2 F ( ˆ ∆ ) , { γ = 0 } ∆ → ˆ ˆ ∆ + δ ˆ ∆ ∂ ∆ ab ∂ ∆ cd Z X N − 1 ln d ∆ ab e − β F ( ˆ ∆ ∗ + δ ˆ ∆ , { γ } ) − β F/N = a<b F (0) /N + γ 2 ✓ ◆ + 1 2 γ 4 Tr( cH − 1 c ) + . . . = 2 µ 0 + . . . − β ∂ 2 F int c ab = 1 X ∂ ∆ ab ∂ ∆ cd 2 c ( ∈ slave) ,d ( ∈ reference) σ = µ 0 γ + 1 3! µ 2 γ 3 + . . . shear stress ∂ 4 β F c ( λ ) 2 β µ 2 = 1 1 = − 1 non-linear X → −∞ shear modulus ∂γ 4 4! 2 λ N λ