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Critical Scaling of Jammed Systems. Ning Xu Department of Physics, University of Science and Technology of China CAS Key Laboratory of Soft Matter Chemistry Hefei National Laboratory for Physical Sciences at the Microscale. glasses. colloids. emulsions. foams. granular materials.

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## Critical Scaling of Jammed Systems

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**Critical Scaling of Jammed Systems**Ning Xu Department of Physics, University of Science and Technology of China CAS Key Laboratory of Soft Matter Chemistry Hefei National Laboratory for Physical Sciences at the Microscale**glasses**colloids emulsions foams granular materials Jamming phase diagram Temperature Shear Stress 1/Density A.J. Liu and S.R. Nagel, Nature 396, 21 (1998). V. Trappe et al., Nature 411, 772 (2001). Z. Zhang, N. Xu, et al. Nature 459, 230 (2009).**Simulation model**• Cubic box with periodic boundary conditions • N/2 big andN/2 small frictionlessspheres with mass m • L/ S= 1.4 avoid crystallization • Purely repulsive interactions Harmonic: =2; Hertzian: =5/2 • L-BFGS energy minimization (T = 0); constant pressure ensemble • Molecular dynamics simulation at constant NPT (T > 0)**marginally jammed**Part I. Marginal and deep jamming Point J (c) unjammed jammed Volume fraction pressure, shear modulus = 0 pressure, shear modulus > 0**Low volume fraction**High volume fraction potential increases Potential field Interaction field on a slice of 3D packings of spheres At high volume fractions, interactions merge largely and inhomogeneously Would it cause any new physics?**d**Critical scalings A crossover divides jamming into two regimes C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, 125503 (2011).**Coordination number**Marginally Jammed zC=2d, isostatic value d Critical scalings Marginal jamming Potential Pressure Bulk modulus Shear modulus Scalings rely on potential C. S. O’Hern et al., Phys. Rev. Lett. 88, 075507 (2002); Phys. Rev. E 68, 011306 (2003). C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, 125503 (2011).**Potential**Pressure Bulk modulus Shear modulus Coordination number Marginally Jammed Deeply Jammed d Critical scalings Deep jamming Scalings do not rely on potential C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, 125503 (2011).**g1**g1 - c Structure Pair distribution function g(r) What have we known about marginally jammed solids? • First peak of g(r) diverges at Point J • Second peak splits • g(r) discontinuous at r = L, g(L+) < g(L) L. E. Silbert, A. J. Liu, and S. R. Nagel, Phys. Rev. E 73, 041304 (2006).**d**Structure pair distribution function g(r) What are new for deeply jammed solids? • Second peak emerges below r = L • First peak stops decay with increasing volume fraction • g(L+) reaches minimum approximately at d C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, 125503 (2011).**Eigenvalues : frequency of normal mode of**vibration l Eigenvectors : polarization vectors of mode l Normal modes of vibration Dynamical (Hessian) matrix H (dN dN) ,: Cartesian coordinates i,j: particle index Diagonalization of dynamical matrix**marginal** increases deep D() ~ 2 d Vibrational properties Density of states • Plateau in density of states (DOS) for marginally jammed solids • No Debye behavior, D() ~ d1, at low frequency • If fitting low frequency part of DOS by D() ~ , reaches maximumat d • Double peak structure in DOS for deeply jammed solids • Maximum frequency increases with volume fraction for deeply jammed • solids (harmonic interaction) change of effective interaction L. E. Silbert, A. J. Liu, and S. R. Nagel, Phys. Rev. Lett. 95, 098301 (2005). C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, 125503 (2011).**d**Participation ratio Define Vibrational properties Quasi-localization • Low frequency modes are quasi-localized • Localization at low frequency is the least at d • High frequency modes are less localized for deeply jammed solids C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, 125503 (2011). N. Xu, V. Vitelli, A. J. Liu, and S. R. Nagel, Europhys. Lett. 90, 56001 (2010).**What we learned from jamming at T = 0?**• A crossover at dseparates deep jamming from marginal jamming • Many changes concur at d • States at d have least localized low frequency modes • Implication: States at dare most stable, i.e. low frequency modes there have • highest energy barrier Vmax Glass transition temperature may be maximal at d? N. Xu, V. Vitelli, A. J. Liu, and S. R. Nagel, Europhys. Lett. 90, 56001 (2010).**viscosity**Tg/T What is glass transition? • Viscosity (relation time) increases by orders of magnitude with small drop • of temperature or small compression • A glass is more fragile if the Angell plot deviates more from Arrhenius behavior P. G. Debenedetti and F. H. Stillinger, Nature 410, 259 (2001). L.-M. Martinez and C. A. Angell, Nature 410, 663 (2001).**Vogel-Fulcher**Reentrant glass transition and glass fragility P < Pd P > Pd Glass transition temperature and glass fragility index both reach maximum at Pd (d) L. Wang, Y. Duan, and N. Xu, Soft Matter 8, 11831 (2012).**N**Reentrant dynamical heterogeneity At constant temperature above glass transition, dynamical heterogeneity reaches maximum at Pd (d) Deep jamming at high density weakens dynamical heterogeneity L. Wang, Y. Duan, and N. Xu, Soft Matter 8, 11831 (2012).**g1**Part II. Critical scaling near point J • Maxima only happen when volume fraction (pressure) varies under constant • temperature (along with colloidal glass transition) • At the maxima Z. Zhang, N. Xu, et al. Nature 459, 230 (2009).**Are the maxima merely thermal vestige of**T = 0 jamming transition? = 5/2 = 2 • At maxima of g1 • Equation of state and potential energy • change form • Kinetic energy approximately equals to • potential energy • Fluctuation of coordination number is • maximum • Scaling laws at T = 0 are recovered • above maxima L. Wang and N. Xu, Soft Matter 9, 2475 (2013).** = 2** = 5/2 Scaling collapse of multiple quantities Critical at T = 0 and p = 0 (Point J) L. Wang and N. Xu, Soft Matter 9, 2475 (2013).**Isostaticity and plateau in density of states**• Isostatic temperature at which z=zc is scaled well with temperature • Plateau of density of states still happen when z = zc L. Wang and N. Xu, Soft Matter 9, 2475 (2013).**Glass transition (viscosity diverges)**Jamming-like transition (g1 is maximum) Isostaticity (z = zc) Phase diagram harmonic Hertzian Glass transition Jamming-like transition Isostaticity L. Wang and N. Xu, Soft Matter 9, 2475 (2013).**Conclusions**• A crossover volume fraction divides the zero temperature jamming into • marginal and deep jamming, which have distinct scalings, structure, and • vibrational properties. • Reentrant glass transition is understandable from marginal-deep jamming • transition • Jamming in thermal systems is signified by the maximum first peak of • the pair distribution function • Zero temperature jamming transition is critical**Acknowledgement**Collaborators: Lijin Wang Graduate student, USTC Cang Zhao Graduate student, USTC Grants: NSFC No. 11074228, 91027001 CAS 100-Talent Program Fundamental Research Funds for the Central Universities No. 2340000034 National Basic Research Program of China (973 Program) No. 2012CB821500 Thanks for your attention!

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