1. Jamming at High Densities
Ning Xu
Department of Physics & CAS Key Laboratory of Soft Matter Chemistry
University of Science and Technology of China
Hefei, Anhui , P. R. China
Point J (c)
unjammed
jammed
Volume fraction 
marginally jammed
pressure, shear modulus = 0
pressure, shear modulus > 0
Will well-known properties of marginally jammed solids hold at high densities?

2. Simulation Model 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
Harmonic: =2; Hertzian: =5/2
L-BFGS energy minimization (T = 0); constant pressure ensemble
Molecular dynamics simulation at constant NPT (T > 0)

3. Would it cause any new physics?
Potential Field
Low volume fraction
High volume fraction
potential increases
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?

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

5. Critical Scalings Marginal jamming Potential Pressure Bulk modulus
Marginally Jammed d
Marginal jamming
Potential
Pressure
Bulk modulus
Shear modulus
Coordination number
zC=2d, isostatic value
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).

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

7. Structure  Pair Distribution Function g(r)
What we have known for marginally jammed solids?
g1max
 - c
g1max
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).

8. Structure  Pair Distribution Function g(r)
What are new for deeply jammed solids? d
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).

9. Vibrational Properties  Density of States
 increases
marginal
deep d
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).

10. Vibrational Properties  Quasi-localizationd
Participation ratio
Define
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, (2011).
N. Xu, V. Vitelli, A. J. Liu, and S. R. Nagel, Europhys. Lett. 90, (2010).

11. 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 Vmax
Glass transition temperature may be maximal at d?
N. Xu, V. Vitelli, A. J. Liu, and S. R. Nagel, Europhys. Lett. 90, (2010).

12. Glass Transition and Glass Fragility
Vogel-Fulcher d d
Glass transition temperature and glass fragility index both reach
maximum at d
L. Berthier, A. J. Moreno, and G. Szamel, Phys. Rev. E 82, (R) (2010).
L. Wang and N. Xu, to be submitted (2011).

13. Dynamical Heterogeneity
At constant temperature above glass transition, dynamical heterogeneity
reaches maximum at d
 Deep jamming at high density weakens dynamical heterogeneity
L. Wang and N. Xu, to be submitted (2011).

14. Conclusions Acknowledgement
Critical scalings, structure, vibrational properties, and dynamics undergo
apparent changes at a crossover volume fraction d which thus separates
marginal jamming from deep jamming
Is the crossover critical?
Experimental realizations: charged colloids, star polymers
Acknowledgement
Cang Zhao USTC
Lijin Wang USTC
Kaiwen Tian will be at UPenn
Brought to you by National Natural Science Foundation of China No

15. Thanks for your attention
&
Welcome to visit USTC