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

1/Density Temperature Shear Stress glasses colloids emulsions foams granular materials 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). Jamming phase diagram

Cubic box with periodic boundary conditions N/2 big and N/2 small frictionless spheres with mass m  L /  S = 1.4  avoid crystallization Purely repulsive interactions Simulation model Harmonic:  =2; Hertzian:  =5/2 L-BFGS energy minimization (T = 0); constant pressure ensemble Molecular dynamics simulation at constant NPT (T > 0)

Part I. Marginal and deep jamming Volume fraction  Point J (  c ) unjammedjammed pressure, shear modulus > 0 pressure, shear modulus = 0 marginally jammed

Potential field Low volume fractionHigh volume fraction At high volume fractions, interactions merge largely and inhomogeneously Would it cause any new physics? Interaction field on a slice of 3D packings of spheres potential increases

dd Critical scalings A crossover divides jamming into two regimes C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, (2011).

Marginally Jammed dd Critical scalings Potential Bulk modulus Pressure Shear modulus Coordination number z C =2d, isostatic value Marginal jamming Scalings rely on potential C. S. O’Hern et al., Phys. Rev. Lett. 88, (2002); Phys. Rev. E 68, (2003). C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, (2011).

Marginally Jammed Deeply Jammed dd Critical scalings Potential Bulk modulus Shear modulus Coordination number Deep jamming Scalings do not rely on potential C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, (2011). Pressure

Structure  Pair distribution function g(r) What have we known about marginally jammed solids?  -  c g1g1 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, (2006). g1g1

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, (2011). dd

Normal modes of vibration Dynamical (Hessian) matrix H (dN  dN) ,  : Cartesian coordinates i,j: particle index Diagonalization of dynamical matrix Eigenvalues : frequency of normal mode of vibration l Eigenvectors : polarization vectors of mode l

dd Vibrational properties  Density of states Plateau in density of states (DOS) for marginally jammed solids No Debye behavior, D(  ) ~  d  1, at low frequency If fitting low frequency part of DOS by D(  ) ~  ,  reaches maximum at  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, (2005). C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, (2011). D(  ) ~  2  increases marginal deep

Vibrational properties  Quasi-localization Participation ratio Define C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, (2011). N. Xu, V. Vitelli, A. J. Liu, and S. R. Nagel, Europhys. Lett. 90, (2010). 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 dd

What we learned from jamming at T = 0? A crossover at  d separates deep jamming from marginal jamming Many changes concur at  d States at  d have least localized low frequency modes Implication: States at  d are most stable, i.e. low frequency modes there have highest energy barrier V max Glass transition temperature may be maximal at  d ? N. Xu, V. Vitelli, A. J. Liu, and S. R. Nagel, Europhys. Lett. 90, (2010).

What is glass transition? P. G. Debenedetti and F. H. Stillinger, Nature 410, 259 (2001). L.-M. Martinez and C. A. Angell, Nature 410, 663 (2001). viscosity Tg/TTg/T 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

Reentrant glass transition and glass fragility Vogel-Fulcher Glass transition temperature and glass fragility index both reach maximum at P d (  d ) P < P d P > P d L. Wang, Y. Duan, and N. Xu, Soft Matter 8, (2012).

Reentrant dynamical heterogeneity At constant temperature above glass transition, dynamical heterogeneity reaches maximum at P d (  d )  Deep jamming at high density weakens dynamical heterogeneity L. Wang, Y. Duan, and N. Xu, Soft Matter 8, (2012). N

Maxima only happen when volume fraction (pressure) varies under constant temperature (along with colloidal glass transition) At the maxima Part II. Critical scaling near point J g1g1 Z. Zhang, N. Xu, et al. Nature 459, 230 (2009).

Are the maxima merely thermal vestige of T = 0 jamming transition? At maxima of g 1 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).  = 2  = 5/2

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 = z c L. Wang and N. Xu, Soft Matter 9, 2475 (2013).

Phase diagram Glass transition (viscosity diverges) Jamming-like transition (g 1 is maximum) Isostaticity (z = z c ) L. Wang and N. Xu, Soft Matter 9, 2475 (2013). Glass transition Jamming-like transition Isostaticity harmonicHertzian

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 WangGraduate student, USTC Cang ZhaoGraduate student, USTC Grants: NSFC No , CAS 100-Talent Program Fundamental Research Funds for the Central Universities No National Basic Research Program of China (973 Program) No. 2012CB Thanks for your attention!