jamming at high densities
DESCRIPTION
marginally jammed. 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 - PowerPoint PPT PresentationTRANSCRIPT
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
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