Critical Scaling of Jammed Systems Ning Xu Department of Physics, University of Science and Technology of China CAS Key Laboratory of Soft Matter Chemistry.

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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