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Critical Scaling of Jammed SystemsCritical Scaling of Jammed Systems
Ning Xu
Department of Physics, University of Science and Technology of ChinaCAS Key Laboratory of Soft Matter Chemistry
Hefei National Laboratory for Physical Sciences at the Microscale
1/Density
Temperature
Shear Stress
glasses
colloids
emulsionsfoams
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
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)
Part I. Marginal and deep jammingPart I. Marginal and deep jamming
Volume fraction
Point J (c)unjammed jammed
pressure, shear modulus > 0pressure, shear modulus = 0
marginally jammed
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 modulus2)(~ cB
Pressure12 )(~ 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 have we known about marginally jammed solids?
- c
g 1
• 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).
g1
11 )(~ cg
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
Normal modes of vibration
Dynamical (Hessian) matrix H (dN dN)
0
2
Rjiij
VH
,: Cartesian coordinatesi,j: particle index
zzijzyijzxij
yzijyyijyxij
xzijxyijxxij
HHH
HHH
HHH
Diagonalization of dynamical matrix
dNleeH lll ,...,2,1, 2ll m Eigenvalues : frequency of normal mode of vibration l
Eigenvectors : polarization vectors of mode lle
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() ~ 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).
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).
visc
osit
y
Tg/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
)/exp(0 TATg
Reentrant glass transition and glass fragility
1/
1exp
00 TTA
Vogel-Fulcher
Glass transition temperature and glass fragility index both reach maximum at Pd (d)
P < Pd
P > Pd
L. Wang, Y. Duan, and N. Xu, Soft Matter 8, 11831 (2012).
gTTg TTd
d |
)/(
)(ln
Reentrant dynamical heterogeneity
At constant temperature above glass transition, dynamical heterogeneity reaches maximum at Pd (d) Deep jamming at high density weakens dynamical heterogeneity
L. Wang, Y. Duan, and N. Xu, Soft Matter 8, 11831 (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 JPart II. Critical scaling near point J
g1
Z. Zhang, N. Xu, et al. Nature 459, 230 (2009).
/9.09.0max1 ~)(~ Tg c
)1/(2 )/(~)(~ pT c
Are the maxima merely thermal vestige of T = 0 jamming transition?
At maxima of g1
• 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 12 )(~/ cp
)(~)/(~ )1/(2cpV
L. Wang and N. Xu, Soft Matter 9, 2475 (2013).
= 2 = 5/2
Scaling collapse of multiple quantities
/)1(2/9.0
1 ),(T
pfTpTg g
/)1(2/1),(
T
pfTpT c
/)1(2
),(T
pTfpTV V
Critical at T = 0 and p = 0 (Point J)
L. Wang and N. Xu, Soft Matter 9, 2475 (2013).
/)1(2
2 ),(T
pfpTz V
= 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 = zc
L. Wang and N. Xu, Soft Matter 9, 2475 (2013).
)1/()1(2 )/(~ PTI
Phase diagram
2/~ pTg
)1/(2 )/(~ pTJ
)1/()1(2 )/(~ pTI
Glass transition (viscosity diverges)
Jamming-like transition (g1 is maximum)
Isostaticity (z = zc)
L. Wang and N. Xu, Soft Matter 9, 2475 (2013).
Glass transition
Jamming-like t
ransit
ion
Isos
tatic
ity
harmonic Hertzian
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 Wang Graduate student, USTCCang Zhao Graduate student, USTC
Grants:NSFC No. 11074228, 91027001 CAS 100-Talent ProgramFundamental Research Funds for the Central Universities No. 2340000034National Basic Research Program of China (973 Program) No. 2012CB821500
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