Xiaoming Mao
Department of Physics and Astronomy, University of Pennsylvania
Collaborators: Tom Lubensky, Ning Xu, Anton Souslov, Andrea Liu
Feb. 22, 2009
What is isostaticity ?
• Isostatic systems are at the onset of mechanical rigidity
• Central force system of N particles in d dimensions
 total d.o.f.
dN
 # of constraints N C
 # of soft modes  dN  NC  d (d  1) / 2
• Large lattice, mean coordination




z
# of constraints NC  zN / 2
Isostaticity: z  2d at bulk
# of soft modes per particle O( N 1 )
Soft modes are associated with boundary
N  6, d  2,
N C  7, dN  N C  5
2 translations
1 rotation
2 soft modes
11 soft modes
•
•
•
•
J. C. Maxwell, Philosophical Magazine 27, 598 (1864).
S. Alexander, Physics reports 296, 65 (1998).
C. S. O’Hern, et al., Phys. Rev. E 68, 011306 (2003).
M. Wyart, et al., Phys. Rev. E 72, 051306 (2005).
What is interesting about isostaticity ?
• Zero-modes are present because of insufficient coordination,
rather than broken symmetry
• Jamming
– Granular packings
– Glasses and the Boson peak
– Emulsions, foams, colloids …
Courtesy of S. R. Nagel
• Rigidity percolation
Courtesy of D. J. Durian
Courtesy of D. A. Weitz
• Networks of semi-flexible polymers
• Applications in engineering
Courtesy of D. A. Weitz
General question:
How rigidity emerges in isostatic systems?
• Add more bonds
• Negative pressure: stretch
• Add angle-dependent force
• Thermal fluctuations
(+ excluded volume repulsion)
•
•
rubber
chemical gels
Current work: randomness and isostaticity
• Interplay of randomness and extra coordination
– Each NNN bond is present with
a given probability P
– For large system, threshold for
rigidity P ~ N 1  0
– How does rigidity scale with P ?
– What is new from randomness
• Nonaffine deformations
• Scattering of phonons: dissipation
Motivation: jamming
• Granular materials and point J
• Frictionless soft spheres
• One-sided repulsion
• At T=0, conjugate-gradient energy-minimization
T
unjammed

jammed
G=0
G>0

C
J
shear stress
1/ 
Packing fraction
• A. J. Liu and S. R. Nagel, Nature 396 N6706, 21 (1998).
• C. S. O’Hern, et al., Phys. Rev. E 68, 011306 (2003).
Point J is isostatic
• Jammed solids at point J are isostatic
G>0
J
C

Packing fraction
Z C  2d
• A. J. Liu and S. R. Nagel, Nature 396 N6706, 21 (1998).
• C. S. O’Hern, et al., Phys. Rev. E 68, 011306 (2003).
Jamming: scalings
G=0
G>0

•
•
•
•
•
Coordination number:
Shear modulus:
Bulk modulus:
Pressure:
Characteristic frequency:
z ~ (  C )1/ 2
Harmonic α=2
G ~ (  C ) 3 / 2  (z)1
B ~ (  C ) 2  (z )0
p ~ (  C ) 1  (z ) 2
  ~ (  C )( 1) / 2  (z)1
cL ~ B
l  L ~ (z ) 1
cT ~ G
l T ~ (z ) 1/ 2
• C. S. O’Hern, et al., Phys. Rev. E 68, 011306 (2003).
• L. E. Silbert, et al., Phys. Rev. Lett. 95, 098301 (2005).
• M. Wyart, et al., Phys. Rev. E 72, 051306 (2005).
The characteristic frequency scale 

• DOS for jammed solids
• Heuristic arguments by M. Wyart
l
•
•
•
•
•
•
Cut a region of size l
# of bonds severed: ~ l d 1
d
# of extra bonds in the region: ~ l z
The region has soft modes if l  l  ~ (z ) 1
Isostaticity length scale l 
Isostaticity frequency scale   ~ 1 / l  ~ (z )1
• M. Wyart, et al., Phys. Rev. E 72, 051306 (2005).
Dimensional crossover at 
• DOS for jammed solids

• DOS for continuous elastic media
• 1D or right at isostaticity
D( ) ~
1
c
• 2D
D( ) ~

c2
2D
1D/isostaticity
• Below   the system behaves 2D, and above, like 1D/isostaticity.
• Mao, Xu and Lubensky, manuscript in preparation (2009).
• Souslov, Liu and Lubensky, manuscript in preparation (2009).
Our goal
• Can we systematically understand isostaticity using the tools
in condensed matter physics?
– Lattice models
– Naturally small parameter for perturbation P ~ z
• What is the role of disorder?
– Nonaffine deformations
– Damping of phonons
– Transport properties --- thermal conductivity of glasses
Our model
• Square lattice and kagome lattice
•
•
•
•
Spring constant of NN bonds k
Spring constant of NNN bonds 
Probability for each NNN bond to be present P
Map to effective medium periodic lattice with NNN
bonds all present with  m
• Mao, Xu and Lubensky, manuscript in preparation (2009).
Elastic expansion for lattice models
• Displacement field and potential energy

– Reference state r
  

– Displacement r  R  r  u
– Potential energy of a bond
 
 
 
force

r

R '
Rb

u
 
V ' (r )  2
1
|| 2
Vb ( R  R ' )  Vb " (rb ) ub  b b ub  O u 3
2
2rb
spring
constant

R

u '
rb
  
ub  u  u '

r '
Elasticity of the effective medium
• elastic free energy for square lattice
1


F   uq  Dq  u q
2 q
u qx
2 modes
u qy
– Dynamical matrix
 Dqxx
Dq   xy
 Dq
Dqxy 

yy 

Dq 
• NN bonds k
• NNN bonds  m
– Eigenvalues:
• Landau theory: continuous elasticity
strain
tensor
C11 ~ k   m
uij 
C12 ~  m
soft modes u xy
C44 ~  m
1
iu j   jui  iul  jul 
2
The characteristic frequency scale 

• Dispersion relation
F
1
  
u
  q  Dq  u q
2 q
x

1D/isostatic
D( x )
2D
qx  2
l  2
m
k
k
m
  2 m
qy
qx
• Phonon density of states
Van Hove
singularity
2D
1D/isostaticity
• NN bonds k
• NNN bonds  m
Longitudinal wave & characteristic length scale l
Isostaticity
(longitudinal waves)
Ising model
Free energy
Correlation function


F  q m 2  q 2 q

q
qq
Fx 
1
~ 2
m  q2

k
 2

u x q  qx  4 m sin 2 q y / 2u x q

2 q
k


uqx uxq ~
4
  1/ m
Characteristic length

l 
above
massive
2D
below
massless/critical
isostatic/1D
m
k
1
sin 2 q y / 2  q x
1 k
2 m
2
How to relate to the extra coordination z ?
• Scalings square lattice
  ~  m1/ 2
l  ~ 1/  ~  m
1/ 2
• Scalings in jamming
  ~ z
l  L ~ 1 /   ~ (z ) 1
• How to relate them?
 Simple guess
 m ~ z    ~ (z)1/ 2
inconsistent
• NN bonds k
• NNN bonds  m
Nearly isostatic random lattices
• Square lattice with random additional NNN bonds
k

• Spring constant of NN bonds k
• Spring constant of NNN bonds 
• Probability for each NNN bond to be present P
Nearly isostatic random lattices
• Square lattice with random additional NNN bonds
• Spring constant of NN bonds k
• Spring constant of NNN bonds 
• Probability for each NNN bond to be present P
k

• Length scales in the nearly isostatic square lattice
• Compare with the case of jamming:
l2 ~ P 1/ 2
l1 ~ P 1
  ~ (z )1
cL ~ B
l  L ~ (z ) 1
cT ~ G
l T ~ (z ) 1/ 2
Method: Coherent Potential Approximation (CPA)
• Mapping to effective medium
map
 m   m ' i  m "
P
• Self-consistency equation
1  P 
P

0
 m   m ' i  m "
• P. Soven, Phys. Rev. 178, 1136 (1969).
• S. Feng, et al., PRB, 31, 276 (1985).
CPA: Green’s function and perturbation
• The effective medium
• Elastic free energy
F
• The perturbation
• Phonon Green’s function
1


uq  Dq  u q

2 q
D, '  D, '  V, '
G  Gm  Gm T  Gm
2
T  V  V  G m V  V  G m V  G m V   
V
1  Tr (G m V )
1
m   '
• The self-consistency equation
• Mao, Xu and Lubensky, manuscript in preparation (2009).
Static solution   0
• The self-consistency equation
At   0 , for P  1 asymptotic equation:
nonaffine
affine
 /k
nonaffine
affine
P
Comparison with simulation
• The self-consistency equation
At   0 , for P  1 asymptotic equation:
 /k
nonaffine
affine
P
 / k  102
 / k 1
Simulation with 100*100 lattice
C44   m
Numerical solution from the CPA
 / k  102
Asymptotic form (nonaffine)
Asymptotic form (affine)
Comparison with jamming
• The self-consistency equation:
 /k
nonaffine
affine
P
• Comparison with scaling at point J
 It is nonaffine near point J
 / k ~1
  m ~ P2
P ~ z  1
  ~  m1/ 2 ~ P
l ~ 1/  ~  m


1/ 2
~P
1
P ~ z
• CPA agrees with the scalings of jamming !
  ~ z
l  L ~ 1 /   ~ (z ) 1
Discussion of   0 solution
• How do we understand the scaling  m ~ P 2 ?
 Length scale in the effective medium
Fx 


k
 2

u x  q  q x  4 m sin 2 q y / 2u x q

2 q
k


qx  2
m
k
 Length scale in the random lattice
Length scale for

2
l ~P
1/ 2
l1 ~ P 1
ux
 l 
1
1 k


2 m
qx
waves propagating in
l1 ~ P 1
  m ~ P2
x
direction
CPA at finite frequency   0
• The self-consistency equation
At   0 , the solution gives a complex effective medium spring constant
 m   m ' i  m "
• Results
Rescaled
m"~ 
viscous elastic media
Density of states D( )
• Phonon DOS derived from Green’s function
DOS of jamming
DOS for various P
P  103
P  102
P  101
Density of states: simulation
Comparison with simulation:
P=0.1
simulation with 100*100 lattice
DOS from CPA (infinite vol)
DOS from CPA (10000 particles)

Response functions  (q,  )
• Phonon response function  ab (t , t ' )  
 , '
ua (t )
 f b' (t ' )

 xxm (q,  )  
1
 2  k qx2  4 m sin 2 (q y / 2)
1) q y  0 no damping
2) q y   strong damping
Im  xxm

qx

The kagome lattice…
 m  0.01
m  0
k1 0; p0
m
K
m"

k1 0.01 ; p0
M
K

K
K
  0.01
m  0
m '
M
m
k10.01;
p0
0.6
0.4

2
0.2
0
0
-2
0
-2
2
Summary and future work
• Conclusions
 We did a systematic study on random nearly isostatic lattices, in particular,
square lattices with random additional NNN bonds.
 Crossover between affine and nonaffine regime
m ~ P
m ~ P2
jamming
 Scaling of characteristic length and frequency scale in jamming can be
recovered by CPA calculation of random nearly isostatic lattices.
 Strong damping in phonon propagation smooth out Van Hove singularity,
similar DOS with jamming.
 Random nearly isostatic lattice models are able to capture some essential
physics in jammed solids
• Future work
 Transport properties
 Isostaticity in other systems