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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 230026, P. R. China

http://staff.ustc.edu.cn/~ningxu

Point J (c)

unjammed

jammed

Volume fraction

pressure, shear modulus = 0

pressure, shear modulus > 0

Will well-known properties of marginally jammed solids hold at high densities?

- 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)

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?

Critical Scalings

A crossover divides jamming into

two regimes

C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, 125503 (2011).

Pressure

Bulk modulus

Shear modulus

Coordination number

Marginally Jammed

zC=2d, isostatic value

d

Critical Scalings

Marginal jamming

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).

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).

g1max

- c

Structure Pair Distribution Function g(r)

What we have known for 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).

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).

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).

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).

d

d

Glass Transition and Glass Fragility

Glass transition temperature and glass fragility index both reach

maximum at d

L. Berthier, A. J. Moreno, and G. Szamel, Phys. Rev. E 82, 060501(R) (2010).

L. Wang and N. Xu, to be submitted (2011).

a

d < 3 < 4

t

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).

- 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. 91027001

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