1) Molecular dynamics simulations were performed on a binary mixture under oscillatory shear strain with varying amplitudes. 2) At small strains, particle motion was nearly reversible over thousands of cycles. At large strains, intermittent bursts of particle displacements led to faster structural relaxation. 3) Mobile particles were found to cluster together, with cluster sizes increasing at higher strains. This is unlike prior results for steadily sheared systems. 4) The fraction of mobile particles facilitated by neighboring particles also increased with strain amplitude.

1. Nikolai V. Priezjev
Department of Mechanical Engineering
Michigan State University
Movies, preprints @ http://www.egr.msu.edu/~priezjev
Acknowledgement:
NSF (CBET-1033662)
Heterogeneous Relaxation Dynamics in Amorphous Materials
under Cyclic Loading
N. V. Priezjev, “Heterogeneous relaxation dynamics in amorphous materials under cyclic
loading”, Phys. Rev. E 87, 052302 (2013). Preprint: http://xxx.lanl.gov/abs/1301.1666
March 22, 2013

2. Dynamical Heterogeneities in Granular Media and Supercooled Liquids
Candelier, Dauchot, and Biroli, PRL 102, 088001 (2009).
Spatial location of successive clusters of cage jumps
• Cyclic Shear Experiment on Dense 2D Granular Media
Power-law distribution of clusters sizes
• Fluidized Bed Experiment: Monolayer of Bidisperse Beads
• 2D Softly Repulsing Particle Molecular Dynamics Simulation (Supercooled Liquids at Eqm)
Candelier, Dauchot, and Biroli, EPL 92, 24003 (2010).
Candelier, Widmer-Cooper, Kummerfeld, Dauchot, Biroli, Harrowell, Reichman, PRL (2010).
• 3D metallic glass
under periodic strain?
Present study:
• Dynamical facilitation
of mobile particles
• Particle diffusion depends
on the strain amplitude
• Structural relaxation and
dynamical heterogeneities
• Particle hopping dynamics
clusters of mobile particles

3. -5
0
5
z
-5
0
5
x
-5
0
5
y σ
/
/
/
σ
σ
Details of molecular dynamics simulations and parameter values
Monomer density:  = A + B = 1.20  3
Temperature: T = 0.1 kB < Tg = 0.45 kB
System dimensions: 12.81σ  14.79σ  12.94σ
Lees-Edwards periodic boundary conditions
The SLLOD equations of motion: tMD= 0.005τ
Binary 3D Lennard-Jones Kob-Andersen mixture:


























6
12
4
)
(
r
r
r
VLJ






W. Kob and H. C. Andersen, Phys. Rev. E 51, 4626 (1995).
2940
,
,
5
.
0
1.5,
,
0
.
1 



 p
B
A
BB
AB
AA N
m
m



,
88
.
0
0.8,
,
0
.
1 

 BB
AB
AA 


Interaction parameters for   A and B particles:
Oscillatory shear strain: )
cos(
)
( 0 t
t 

 
 
Strain amplitude: ,
/
0
0 

 

Oscillation period:
1
02
.
0 
 



 16
.
314
/
2 

T
AA
A
AA m 

 
12,000 cycles (  7.5×108 MD steps)

4. 1 10 100 1000 10000
t /T
0.01
0.1
1
10
100
r
2
/
σ
2
Slope = 1.0
ωτ = 0.02
Mean square displacement as a function of time for different strain amplitudes
Oscillation period: 

 16
.
314
/
2 

T
= number of cycles
06
.
0
0 

07
.
0
0 

08
.
0
0 

05
.
0
0 

02
.
0
0 


2
cage
r
Reversible dynamics
Diffusive regime
mean
square
dispt
Strain amplitude:
03
.
0
0 


5. 1 10 100 1000 10000
t /T
0
0.2
0.4
0.6
0.8
Q
s
(a,t)
ωτ = 0.02
γ0
= 0.07
γ0
= 0.02
γ0
= 0.06
γ0
= 0.05
Self-overlap order parameter Qs(a,t) for different strain amplitudes









 


p
N
i
i
p
s
a
t
r
N
t
a
Q
1
2
2
2
)
(
exp
1
)
,
( 
12
.
0

a
= number of cycles
= probed length scale ~ max in
Department of Mechanical Engineering Michigan State University
)
,
(
4 t
a
X
)
,
( t
a
Qs describes
structural relaxation
of the material.
Measure of the spatial
overlap between partic-
les positions.
Strain
amplitude:
Reversible dynamics:
Qs(t)  constant
Diffusive regime:
Qs(t) vanishes at large t
?
)
,
(
4 
t
a
X
Susceptibility

6. is dynamical
susceptibility, which
is the variance of Qs.
Dynamical susceptibility as a function of time for different strain amplitudes
1 10 100 1000 10000
t /T
0.01
0.1
1
10
100
χ
4
(a,t)
0.02 0.06
γ0
1
7
ξ
4
ωτ = 0.02
Slope = 0.9
= number of cycles
 
2
2
4 )
,
(
)
,
(
)
,
( t
a
Q
t
a
Q
N
t
a
X s
s
p 
 
12
.
0

a = probed length scale ~ max in )
,
(
4 t
a
X
)
,
(
4 t
a
X
  3
/
1
max
4
4 )
(t
Χ


The dynamic correlation
length increases with in-
creasing strain amplitude
(in contrast to steadily
sheared supercooled
liquids and glasses).
02
.
0
0 

03
.
0
0 

06
.
0
0 

Tsamados, EPJE (2010)
Maximum
indicates the largest
spatial correlation
between localized
particles.
)
,
(
4 t
a
X
Berthier & Biroli (2011)
Mizuno & Yamamoto, JCP (2012);

7. 0 2000 4000 6000 8000 10000
t /T
0
400
800
N
c
0
40
80
N
c
0
40
80
N
c
(a)
(b)
(c)
γ0
= 0.02
γ0
= 0.04
γ0
= 0.06
= number of cycles
Power spectrum ~ frequency-2 = simple Brownian noise
Numerical algorithm for
detection of cage jumps:
Candelier, Dauchot,
Biroli, PRL (2009).
Strain amplitude:
Scale-invariant processes
or Pink noise = "1/f noise"
2940

p
N
Periodic deformation =
intermittent bursts of
large particle displace-
ments.
Number of particles undergoing cage jumps Nc as a function of time t/T

8. Typical clusters of mobile particles A (blue circles) and B (red circles)
-5
0
5
z
-5
0
5
x
-5
0
5
y
(a)
/ σ
/
σ
σ
/
-5
0
5
z
-5
0
5
x
-5
0
5
y
(b)
σ
σ
σ
/
/
/
-5
0
5
z
-5
0
5
x
-5
0
5
y
(c)
σ
σ
σ
/
/
/
-5
0
5
z
-5
0
5
x
-5
0
5
y
(d)
σ
σ
σ
/
/
/
Department of Mechanical Engineering Michigan State University
02
.
0
0 

03
.
0
0 

04
.
0
0 

05
.
0
0 

Strain amplitude:
Strain amplitude:
Single particle
reversible jumps:
Compact clusters;
Irreversible jumps:
9
.
0
0
4 
  The system is
fully relaxed over
about 104 cycles
06
.
0
0 


9. -5
0
5
z
-5
0
5
x
-5
0
5
y
(a)
/ σ
/
σ
σ
/
-5
0
5
z
-5
0
5
x
-5
0
5
y
(b)
σ
σ
σ
/
/
/
-5
0
5
z
-5
0
5
x
-5
0
5
y
(c)
σ
σ
σ
/
/
/
-5
0
5
z
-5
0
5
x
-5
0
5
y
(d)
σ
σ
σ
/
/
/
0 1000 2000 3000 4000 5000
Δt /T
0.2
0.4
0.6
0.8
1
N
f
/
N
tot
0 2000 4000 6000
Δt /T
0.2
0.4
0.6
0.8
1
N
f
/N
tot
ωτ = 0.02
02
.
0
0 

03
.
0
0 

04
.
0
0 

05
.
0
0 

06
.
0
0 

Strain amplitude:
= number of cycles
Oscillation period: 

 16
.
314
/
2 

T
Vogel and Glotzer, Phys. Rev. Lett. 92, 255901 (2004).
Cage jump
Mobile neighbor
t = time interval when a particles is immobile (inside the cage)
Fraction of dynamically facilitated particles increases with strain amplitude
Large surface area =
high probability to
have mobile neigh-
bors.

10. Important conclusions:
http://www.egr.msu.edu/~priezjev Michigan State University
• MD simulations of the binary 3D Lennard-Jones Kob-Andersen mixture at T = 0.1 kB
under spatially homogeneous, time-periodic shear strain.
• At small strain amplitudes, the mean square displacement exhibits a broad sub-diffusive
plateau and the system undergoes nearly reversible deformation over about 104 cycles.
• At larger strain amplitudes, the transition to the diffusive regime occurs at shorter time
intervals and the relaxation process involves intermittent bursts of large particle
displacements.
• The detailed analysis of particle hopping dynamics and the dynamic susceptibility
indicates that mobile particles aggregate into clusters whose sizes increase at larger
strain amplitudes. (In contrast to sheared supercooled liquids and glasses).
• Fraction of dynamically facilitated mobile particles increases at larger strain amplitudes.
9
.
0
0
4 
 
)
,
(
4 t
a
X
N. V. Priezjev, “Heterogeneous relaxation dynamics in amorphous materials under cyclic
loading”, Phys. Rev. E 87, 052302 (2013). Preprint: http://xxx.lanl.gov/abs/1301.1666