Jamming at High Densities

Jamming at High DensitiesJamming at High DensitiesNing Xu

Department of Physics & CAS Key Laboratory of Soft Matter ChemistryUniversity of Science and Technology of China

Hefei, Anhui 230026, P. R. Chinahttp://staff.ustc.edu.cn/~ningxu

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

Volume fraction

Point J (c)unjammed jammed

pressure, shear modulus > 0pressure, shear modulus = 0

marginally jammed

• 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

ijij

ijijijijij

r

rrrV

,0

,/)/1()(

Harmonic: =2; Hertzian: =5/2

• L-BFGS energy minimization (T = 0); constant pressure ensemble

• Molecular dynamics simulation at constant NPT (T > 0)

Potential Field

Low volume fraction High 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

pot

enti

al in

crea

ses

d

Critical Scalings

A crossover divides jamming into two regimes

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

Marginally Jammed

d

Critical Scalings

Potential )(~ cV

Bulk modulus 8.1)(~ cB

Pressure8.0)(~ cp

Shear modulus 2/3)(~ cG

Coordination number2/1)(~ cczz

zC=2d, isostatic value

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).

Marginally Jammed

Deeply Jammed

d

Critical Scalings

Potential ddVV ~)(

Bulk modulus 7.1)(~)( ddBB

Shear modulus 2.1)(~)( ddGG

Coordination number

ddzz ~)(

Deep jamming

Scalings do not rely on potential

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

Pressure7.1)(~)( ddpp

Structure Pair Distribution Function g(r)

What we have known for marginally jammed solids?

- c

g 1max

• First peak of g(r) diverges at Point J

• Second peak splits

• g(r) discontinuous at r = L, g(L+) < g(L

)

g1max

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).

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 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, 098301 (2005).C. Zhao, K. Tian, and N. Xu, Phys. Rev. Lett. 106, 125503 (2011).

d

D() ~ 2

increases

marginal

deep

Vibrational Properties Quasi-localization

Participation ratio

Define

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).

• 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

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, 56001 (2010).

Glass Transition and Glass Fragility

L. Berthier, A. J. Moreno, and G. Szamel, Phys. Rev. E 82, 060501(R) (2010).L. Wang and N. Xu, to be submitted (2011).

1/

1exp

00 TT

Vogel-Fulcher

Glass transition temperature and glass fragility index both reach maximum at d

d d

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).

t

a

1 < 2 < d

d < 3 < 4

Conclusions

• 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 USTCLijin Wang USTCKaiwen Tian will be at UPenn

Brought to you by National Natural Science Foundation of China No. 91027001

Thanks for your attentionThanks for your attention&&

Welcome to visit USTCWelcome to visit USTC