Nikolai V. PriezjevDepartment of Mechanical and Materials Engineering

Wright State University

Movies, preprints @http://www.wright.edu/~nikolai.priezjev/

Dynamical heterogeneity and structural relaxation in periodically deformed polymer glasses

N. V. Priezjev, “Dynamical heterogeneity in periodically deformed polymer glasses”, Physical Review E 89, 012601 (2014). N. V. Priezjev, “Heterogeneous relaxation dynamics in amorphous materials under cyclic loading”, Physical Review E 87, 052302 (2013).

Dynamical heterogeneity in quiescent and deformed polymer glasses

Gebremichael, Schroder, Starr, Glotzer, J. Chem. Phys. 64, 051503 (2001).

Riggleman, Lee, Ediger, and de Pablo, Soft Matter 6, 287 (2010).

Molecular dynamics simulations of a quiescent polymer glass near Tg

The most mobile monomers (top 5%) form transient clusters whose mean size increases upon cooling towards Tg

= constant strain rate

(b)

(a) 500-mer chains

10-mer chains

homogeneousheterogeneous

Dynamical heterogeneity 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 =0)

Candelier, Dauchot, and Biroli, EPL 92, 24003 (2010).

Candelier, Widmer-Cooper, Kummerfeld, Dauchot, Biroli, Harrowell, Reichman, PRL (2010).

• 3D polymer glasses under periodic strain?

Present study:

• Dynamical facilitation of mobile monomers

• Monomer diffusion dependson the strain amplitude

• Structural relaxation anddynamical heterogeneities

• Particle hopping dynamics clusters of mobile particles

50

0 m

m

500 cycles

-5

0

5

-5 0 5x-5

05

y σ/

/

/

σ

σz

Details of molecular dynamics simulations and parameter values

Monomer density: = 1.07 3, N = 3120

Temperature: T = 0.1 kB < Tg 0.3 kB

Cubic box: 14.29σ 14.29σ 14.29σ

Lees-Edwards periodic boundary conditions

SLLOD equations of motion: tMD= 0.005τ

Oscillatory shear strain: ) sin()( 0 tt

Strain amplitude: ,09.00

Oscillation period:

105.0

66.125/2 T15,000 cycles ( 3.8× 108 MD steps)

Lennard-Jones potential:

12 6

LJr rV (r) 4

FENE bead-spring model:

22

FENE o 2o

1 rV (r) kr ln 12 r

k = 30 2 and ro = 1.5

Kremer & Grest, J. Chem. Phys. 92, 5057 (1990)

1 10 100 1000 10000t /T

0.01

0.1

1

10

100

r2 /σ2

Slope = 0.87ωτ = 0.05

γ0 = 0.06

Mean-square-displacement of monomers for different strain amplitudes 0

Oscillation period: 66.125/2 T

number of cycles:

06.00

07.00

09.00

05.00

02.00 2cager

Reversible dynamics

Diffusive regime

1 10 100 1000 10000t /T

0

0.2

0.4

0.6

0.8

1

C9(

t)

0

0.2

0.4

0.6

0.8

1

C1(

t)

(a)

(b)

γ0 = 0.06

γ0 = 0.06

Time autocorrelation function of normal modes vs. strain amplitude 0

Oscillation period: 66.125/2 T

number of cycles

07.00

09.00 06.00

06.00

05.00

Relaxation of the whole chain

Segmental dynamics

Niptr

NtX

N

iip

)2/1(cos)(1)(1

p = mode number N = 10 chain length

p = 1

p = 9

)0()()( ppp XtXtC

Reversible dynamics

Chains are fully relaxed

05.00

07.00

The relaxation dynamics of polymer chains under-goes a transition at the strain amplitude 06.00

1 10 100 1000 10000

t /T

0

0.2

0.4

0.6

0.8

1

Qs(

a,t)

ωτ = 0.05

γ0 = 0.07

γ0 = 0.02

γ0 = 0.06

γ0 = 0.05

Self-overlap order parameter Qs(a,t) for different strain amplitudes 0

mN

i

i

ms a

trN

taQ1

2

2

2)(exp1),(

12.0a

= number of cycles

= probed length scale (about the cage size)

),( taQs describes structural relaxationof the material.Measure of the spatialoverlap between monomer positions.

Reversible dynamics:Qs(t) constant

Diffusive regime:Qs(t) vanishes at large t

?),(4 taXSusceptibility

Department of Mechanical and Materials Engineering Wright State University

Nm = total number of monomers in the systemri(t) = the monomer displacement vector

Contour plots of dynamical susceptibility X4(a,t) and Qs(a,t) for 06.00

Measure of the spatialoverlap between monomer positions.

mN

i

i

ms a

trN

taQ1

2

2

2)(exp1),(

22

4

),(),(

),(

taQtaQ

NtaX

ss

m

cage

is dynamical susceptibility, which is the variance of Qs.

),(4 taX

Maximum indicates the largest number of monomers involved in correlated motion.

),(4 taX

Berthier & Biroli (2011)

1 10 100 1000 10000

t /T

0.01

1

100

χ 4(a

,t)

0.02 0.05 0.1

γ01

7

ξ4

ωτ = 0.05

γ0 = 0.09

γ0 = 0.07

γ0 = 0.06

Dynamical susceptibility as a function of time for different strain amplitudes 0

= number of cycles

224 ),(),(),( taQtaQNtaX ssm

probed length scale a = max in ),(4 taX

is dynamical susceptibility, which is the variance of Qs.

),(4 taX

3/1 max44 ),( taΧ

Transition at maximum correlation length

02.00

05.00

Tsamados, EPJE (2010)

Berthier & Biroli (2011)

Mizuno & Yamamoto, JCP (2012),(in contrast to steadily sheared supercooled liquids and glasses)

Nm = total number of monomers

06.00

3/1 max44 ),( taΧ

Maximum indicates the largest spatial correlation between localized monomers.

),(4 taX

0 2000 4000 6000 8000 10000t /T

0

800

1600

Nc

0

40

80

Nc

0

40

80

Nc

(a)

(b)

(c)

γ0 = 0.02

γ0 = 0.04

γ0 = 0.06

Number of monomers undergoing cage jumps Nc as a function of time t /T

= 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 processesor Pink noise = "1/f noise"

3120mN

Periodic deformation =intermittent bursts of large monomer displa-cements.

-5

0

5

z

-505x

-50

5y

(a)

/ σ

/σ

σ/

-5

0

5

z

-505x

-50

5y

(c)

σσ

σ

//

/

-5

0

5

z

-505x

-50

5y

(d)

σσ

σ

//

/

-5

0

5

z

-505x

-50

5y

(b)

σσ

σ

//

/

Typical clusters of mobile monomers for different strain amplitudes 0

Department of Mechanical and Materials Engineering Wright State University

03.00

04.00

05.00 06.00

Strain amplitude:

Single particlereversible jumps:

Compact clusters;Irreversible jumps:

The system is fully relaxed over about 100 cycles

07.00

Cluster spans thewhole system

0 1000 2000 3000 4000 5000

Δt /T

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nf/N

tot

ωτ = 0.05

Fraction of dynamically facilitated monomers increases with strain amplitude

02.00

03.00

04.00

05.00

06.00

Strain amplitude:

= number of cycles

Vogel and Glotzer, Phys. Rev. Lett. 92, 255901 (2004).

Cage jumpMobile neighbor

Large surface area =high probability to have mobile neighbors.

t = time interval when a particles is immobile (inside the cage)

Oscillation period: 66.125/2 T

Important conclusions:

http://www.wright.edu/~nikolai.priezjev Wright State University

• The coarse-grained bead-spring polymer glass at T = 0.1 kB < Tg under spatially homo-geneous, 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 the critical strain amplitude, the transition from slow to fast relaxation dynamics is associated with the largest number of dynamically correlated monomers as indicated by the peak value of the dynamical susceptibility.

• The detailed analysis of monomer hopping dynamics indicates that mobile monomers aggregate into clusters whose sizes increase at larger strain amplitudes. (This is in contrast to steadily sheared supercooled liquids and glasses).

• Fraction of dynamically facilitated mobile monomers increases at larger strain amplitudes.

3/1 max44 ),( taΧ

N. V. Priezjev, “Dynamical heterogeneity in periodically deformed polymer glasses”, Physical Review E 89, 012601 (2014).