Jamming and the Glass Transition
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Jamming and the Glass Transition Andrea J. Liu Department of Physics & Astronomy University of Pennsylvania Michael Schmiedeberg UPenn Thomas K. Haxton UPenn Ning Xu USTC Sidney R. Nagel UChicago The Jamming Scenario C. S. O’Hern, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. 88, 075507 (2002). temperature C. S. O’Hern, L. E. Silbert, A. J. Liu, S. R. Nagel, Phys. Rev. E 68, 011306 (2003). jammed stress unjammed J 1/density • Focus on foams/emulsions or frictionless granular material – soft, repulsive, finite-range spherically-symmetric potentials • Such systems have T=0 1st/2nd-order phase transition The Jamming Scenario C. S. O’Hern, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. 88, 075507 (2002). temperature C. S. O’Hern, L. E. Silbert, A. J. Liu, S. R. Nagel, Phys. Rev. E 68, 011306 (2003). jammed ⎧ α ε ⎛ r ⎞ ⎪ ⎜1− ⎟ r ≤ σ V(r) = ⎨ α ⎝ σ ⎠ stress ⎪ ⎩ 0 r > σ unjammed J 1/density • Focus on foams/emulsions or frictionless granular material – soft, repulsive, finite-range spherically-symmetric potentials • Such systems have T=0 1st/2nd-order phase transition How we study Point J • Generate configurations near J ⎧ α ε ⎛ r ⎞ – Start with some initial config ⎪ ⎜1− ⎟ r ≤ σ V(r) = ⎨ α ⎝ σ ⎠ ⎪ ⎩ 0 r > σ – Conjugate gradient energy minimization (inherent structures, Stillinger & Weber) • Classify resulting configurations non-overlapped overlapped V=0 V>0 p=0 or p>0 Tf=0 Tf=0 Onset of Overlap is Onset of Jamming - 2 (a) D=2 - 4 α=2 D=3 •Pressures for different - 6 α=5/2 states collapse on a single α −1 8 - p ≈ p0(φ −φc) curve 0 (b) -2 α=2 •Shear & bulk moduli, G & B, =5/2 4 α vanish at same φc - α −1.5 G ≈ G0(φ − φc ) - 6 -4 -3 -2 0 (c) 3D φ φ γ≈1/2 log( - ) •G/B ~ ( - c) 1 φ φc 2D 2 3 D. J. Durian,5 PRL4 75, 47803 (1995); 2 log ( - ) C. S. O’Hern, S. A. Langer,φ φA.c J. Liu, S. R. Nagel, PRL 88, 075507 (2002). 2 (a) 4 Onsetα=2 of Overlap has 1st-Order Character 6 α=5/2 Just below Just above φ , no c 8 φc there are particles Zc overlap0 (b) overlapping 2 α=2 neighbors =5/2 4 α per particle 6 0 (c) 3D – Zc = 3.99 ± 0.01 (2D) -1 2D Zc = 5.97 ± 0.03 (3D) -2 0.5 Z − Zc ≈ Z0 (φ −φc) -3 -5 -4 -3 - 2 log(logφ -(φ φ-cφ)c) Verified experimentally: Durian, PRL 75, 4780 (1995). Majmudar, Sperl, Luding, Behringer, O’Hern, Langer, Liu, Nagel, PRL 88, 075507 (2002). PRL 98, 058001 (2007). Isostaticity • What is the minimum number of interparticle contacts needed for mechanical equilibrium? •No friction, spherical particles, D dimensions –Match unknowns (number of interparticle normal forces) equations (force balance for mechanical James Clerk Maxwell stability) –Number of unknowns per particle=Z/2 –Number of equations per particle = D Z = 2D Phillips, Thorpe, Boolchand, Edwards, Ball, Blumenfield Isostaticity and Diverging Length Scale M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05) •For system at φc, Z=2d •Removal of one bond makes entire system unstable by adding one soft mode + •This implies diverging length as φ-> φc For φ > φc, cut bonds at boundary of circle of size L Count number of soft modes within circle d −1 d Ns ≈ L − (Z − Zc )L Define length scale at which soft modes just appear 1 −0.5 ≈ ≈ (φ − φc ) Z Z − c Isostaticity and Diverging Length Scale M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05) •For system at φc, Z=2d •Removal of one bond makes entire system unstable by adding one soft mode + •This implies diverging length as φ-> φc For φ > φc, cut bonds at boundary of circle of size L Count number of soft modes within circle d −1 d Ns ≈ L − (Z − Zc )L Define length scale at which soft modes just appear 1 −0.5 ≈ ≈ (φ − φc ) Z Z − c Consequences of Diverging Length Scale • Solids are all alike at low T or ω: – density of vibrational states D(ω)~ω2 in d=3 – vibrational heat capacity C(T)~T3 – thermal conductivity κ(T)~Cvℓ~T3 • Why? Low-frequency excitations are sound modes. At long length scales all solids look elastic Consequences of Diverging Length Scale • Solids are all alike at low T or ω: – density of vibrational states D(ω)~ω2 in d=3 – vibrational heat capacity C(T)~T3 – thermal conductivity κ(T)~Cvℓ~T3 • Why? Low-frequency excitations are sound modes. At long length scales all solids look elastic BUT at Point J, there is a diverging length scale ℓ* So what happens? Vibrations in Marginally Jammed Solids L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (‘05) * / 1/2 D(ω) ω ω 0 Δφ ω * keff (α −2)/2 ω 0 ≡ m Δφ • More and more modes in excess of Debye prediction as φ->φc (boson peak) • New class of excitations distinct from sound modes originates from soft modes at Point J M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05) • Robust for systems near isostaticity Souslov, Liu, Lubensky, PRL 103, 205503 (2009); Mao, Xu, Lubensky arXiv: 09092616 Vibrational Modes Predict Soft Spots • Adams-Gibbs Cooperatively-Rearranging Regions? • Shear Transformation Zones? M. Lisa Manning’s talk next week Summary of Point J ⎧ α ε ⎛ r ⎞ ⎪ ⎜1− ⎟ r ≤ σ V(r) = ⎨ α ⎝ σ ⎠ ⎪ ⎩ 0 r > σ • Mixed first-order/second-order transition (RFOT) • Number of overlapping neighbors per particle ⎧ ⎪ 0 φ < φc Z = ⎨ β ≅1/2 Z + z φ − φ φ ≥ φ ⎩⎪ c 0 ( c ) c • Static shear modulus/bulk modulus γ ≅1/2 G / B φ − φ ( c ) • Two diverging length scales * −ν ≅−1/2 (φ − φc ) † † −ν ≅−1/4 (φ − φc ) Similarity to Other Models • This behavior has only been found in a few models, all in mean-field limit – 1-RSB p-spin interaction spin glass Kirkpatrick, Thirumalai, Wolynes – compressible frustrated Ising antiferromagnet Yin, Chakraborty – kinetically-constrained Ising models Sellitto, Toninelli, Biroli, Fisher – k-core percolation and variants Schwarz, Liu, Chayes, Toninelli, Biroli, Fisher,Harris, Jeng – Mode-coupling approximation of glasses Biroli, Bouchaud – 1-RSB solution of hard spheres Zamponi, Parisi • These other models all exhibit glassy dynamics!! First hint of quantitative connection between sphere packings and glass transition Point J and the Glass Transition N. Xu, T. K. Haxton, A. J. Liu and S. R. Nagel, PRL 103, 205503 (2009). temperature jammed stress unjammed J 1/density • How do ideal spheres behave at nonzero temperature? Dramatic Increase of Relaxation Time Colloidal glass transition Glass transition ⎛ ⎞ ⎛ A ⎞ A τ = τ 0 exp⎜ x ⎟ T T τ = τ 0 exp⎜ x ⎟ ⎝ ( − 0 ) ⎠ ⎝ (φ − φ0 ) ⎠ Brambilla, et al., arXiv/0809.3401 L.-M. Martinez and C. A. Angell, Similar fitting functions. Nature 410, 663 (2001). WHY?? Very Different Underlying Pictures Colloidal glass transition Map of Lake O'Hara - YellowMaps TopoGlass Guide transition 8/5/09 4:25 PM Copyright © 2004 YellowMaps. All rights reserved. Terms of Use | Privacy Policy | Home | Close free volume complex energy landscape Pressure is most important Temperature is most important It governs amount of free It allows system to overcome volume energy barriers http://www.yellowmaps.com/topo/_lakeohara_bc/index2.htm Page 1 of 2 Very Different Underlying Pictures Colloidal glass transition Map of Lake O'Hara - YellowMaps TopoGlass Guide transition 8/5/09 4:25 PM Copyright © 2004 YellowMaps. All rights reserved. Terms of Use | Privacy Policy | Home | Close free volume complex energy landscape Pressure is most important Temperature is most important It governs amount of free It allows system to overcome volume energy barriers There are systems for which these two transitions are the same phenomenon http://www.yellowmaps.com/topo/_lakeohara_bc/index2.htm Page 1 of 2 Relaxation Time temperature • Look at relaxation time along different jammed trajectories – Fix p, lower T stress – Fix T, raise p unjammed J 1/density Relaxation Time temperature • Look at relaxation time along different jammed trajectories – Fix p, lower T stress – Fix T, raise p unjammed J 1/density 6 10 5 (a) 10 4 10 ! 3 10 2 10 1 10 -8 -7 -6 -5 -4 10 10 10 10 10 T 6 10 (b) 5 10 4 ! 10 3 10 2 10 3 4 5 6 7 8 10 10 10 10 10 10 1/p Relaxation Time temperature • Look at relaxation time along different jammed 6 trajectories 10 5 (a) – Fix p, lower T 10 stress 4 10 – Fix T, raise p ! 3 10 2 unjammed J 10 1 10 -8 -7 -6 -5 -4 1/density 10 10 10 10 10 T 6 6 10 10 5 (a) (b) 5 10 10 4 10 4 ! 3 ! 10 10 2 3 10 10 1 2 10 -8 -7 -6 -5 -4 10 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 10 10 T 6 1/p 10 (b) 5 10 4 ! 10 3 10 2 10 3 4 5 6 7 8 10 10 10 10 10 10 1/p Data Collapse! 3 10 -7 6 p=2x10 (a) 10 -6 (a) ⎛ T ⎞ p=2x10 5 -5 10 2 τ p = f p=2x10 -4 1/2 ⎜ ⎟ 4 10 ⎝ p ⎠ p=2x10 10 τ p -8 ! T=10 3 -7 10 /m) T=10 2 -6 ! 1 T=10 10 -5 10 T=10 1 (p 10 -8 -7 -6 -5 -4 10 10 10 10 10 " T 6 0 10 (b) 10 5 10 0 0.1 0.2 0.3 0.4 0.5 0.6 4 ! 10 3 3 T/pT/p! 10 2 10 3 4 5 6 7 8 (b) 10 10 10 10 10 10 Data100 for different (p, T) collapse 1/p 1/2 on to single scaling curve! /m) 10 ! (p " 1 -5 -4 -3 -2 -1 10 10 103 10 10 p! /# Dimensional Analysis • Recall interaction potential ⎧ 2 ε ⎛ r ⎞ ⎪ ⎜1− ⎟ r ≤ σ V(r) = ⎨ 2 ⎝ σ ⎠ particle mass ⎪ ⎩ 0 r > σ • We have 3 dimensional parameters in model: ε, m, σ interaction energy ε ⎛ T pσ 3 ⎞ Relaxation h , particle diameter τ 2 = ⎜ ⎟ time mσ ⎝ ε ε ⎠ or equivalently pσ ⎛ T pσ 3 ⎞ τ = g , m ⎝⎜ pσ 3 ε ⎠⎟ Dimensional Analysis • Recall interaction potential ⎧ 2 ε ⎛ r ⎞ ⎪ ⎜1− ⎟ r ≤ σ V(r) = ⎨ 2 ⎝ σ ⎠ particle mass ⎪ ⎩ 0 r > σ • We have 3 dimensional parameters in model: ε, m, σ interaction energy ε ⎛ T pσ 3 ⎞ Relaxation h , particle diameter τ 2 = ⎜ ⎟ time mσ ⎝ ε ε ⎠ or equivalently 0 pσ ⎛ T pσ 3 ⎞ τ = g , m ⎝⎜ pσ 3 ε ⎠⎟ 3 10 -7 p=2x10 (a) -6 p=2x10 -5 2 p=2x10 -4 1/2 10 p=2x10 -8 T=10 -7 /m) T=10 -6 ! 1 T=10 -5 10 T=10 (p " 0 10 0 0.1 0.2 0.3 0.4 0.5 0.6 Dimensional Analysis3 T/p! (b) 100 1/2 /m) 10 ! (p " 1 -5 -4 -3 -2 -1 10 10