DYNAMICAL CORRELATIONS NEAR DISLOCATION JAMMING Lasse Laurson ISI Foundation, Turin, Italy

Work done with: M.-Carmen Miguel (Barcelona) and Mikko Alava (Helsinki)

Phys. Rev. Lett. 105, 015501 (2010).

OUTLINE •Plastic deformation of crystalline solids - dislocations and their critical collective dynamics. •Dislocation jamming - earlier results. •Jamming in other systems (granular systems, foams, etc.) •Dynamical correlations near jamming: Tools and concepts such as the self-overlap order parameter, the four-point dynamical susceptibility, growing dynamical correlation length, etc. •Similar tools applied to the case of dislocation jamming - a divergent dynamical correlation length observed. •Summary.

PLASTIC DEFORMATION OF CRYSTALLINE SOLIDS - DISLOCATIONS Burgers vector:

�

d�u = �b

PLASTIC DEFORMATION OF CRYSTALLINE SOLIDS - DISLOCATIONS Dislocation glide:

Surface step (quantum of plastic strain):

γ˙ = bρm �v� Orowan relation Shear strain rate = Burgers vector * density of mobile dislocations * average velocity

RIGID DISLOCATIONS: INTERACTIONS Dislocations induce long-range stress fields in the host material:

The (Peach-Koehler) force on a dislocation in a stress field σ :

F� = (�b · σ) × ˆl

MICROSCALE PLASTICITY

M. D. Uchic et al., Science 305, 986 (2004).

PLASTIC STRAIN BURST DISTRIBUTION

P (s) ∼ s

−τ

, τ � 1.5

D. M. Dimiduk et al., Science 312, 1188 (2006).

LARGER SCALES: ACOUSTIC EMISSION •For larger systems, stressstrain curves are typically smooth. •Critical avalanche dynamics is still taking place - detectable via acoustic emission. •Such critical behavior hints that there might be an underlying non-equilibrium phase transition: the “yielding transition”. •How can we characterize this transition?

M.-C. Miguel et al., Nature 410, 667 (2001).

2D DISCRETE DISLOCATION DYNAMICS •Straight parallel edge dislocations, single slip. •Long-range stresses: N � σi = σe + sj σ(�ri − �rj )

Stress field modulus of a dislocation system:

j�=i

•Overdamped dynamics: −1 χd vi

b

= si bσi

•Annihilation of opposite sign M.-C. Miguel, L. Laurson, M.J. Alava, dislocations. Eur. Phys. J. B 64, 443 (2008). •Periodic boundary conditions.

JAMMING: PHASE DIAGRAM •A phase diagram spanned by variables such as density (volume fraction), applied load, and temperature. •In the jammed phase, the dynamics stops due to selfinduced constraints on the motion of the system constituents - the system develops a yield stress. •Here, look at the T = 0 plane.

A.J. Liu and R. Nagel, Nature 396, 21 (1998).

Focus on this part

DISLOCATION JAMMING •Study the response of the system to a constant external stress. •A critical stress σc separating moving and “jammed” steady states. −2/3 •For σe = σc , the critical relaxation of �dγ/dt� ∼ t � si vi the strain rate �dγ/dt� ∼ i corresponds to the Andrade creep law. •Non-linear stress-dependence of the steady state strain rate. 1.8 �dγ/dt� ∼ (σ − σ ) •The jammed state: dislocations form e c metastable configurations - a finite M.-C. Miguel et al., yield stress even in the absence of PRL 89, 165501 (2002). quenched disorder.

JAMMING: OTHER SYSTEMS “Dynamical heterogeneity” in a sheared granular material:

O. Dauchot et al., PRL 95, 265701 (2005).

Sheared wet foam:

J. Lauridsen et al., PRL 89, 098303 (2002).

JAMMING: DYNAMICAL CORRELATIONS A granular system of air-fluidized beads on approach to jamming: Self-overlap order parameter Q(l, τ ) and dynamical four-point susceptibility χ4 (l, τ ) : N � Q(l, τ ) = �Qt (l, τ )� 1 Qt (l, τ ) = wi N i=1 χ4 (l, τ ) = N [�Qt (l, τ )2 � − �Qt (l, τ )�2 ] A. S. Keys et al., Nature Phys. 3, 260 (2007).

A. R. Abate et al., Phys. Rev. E 76, 021306 (2007).

JAMMING: DYNAMICAL CORRELATIONS On approach to jamming, the dynamics∗slows down χ 4 ∗ ∗ n ≈ ( τ increases) and the size of the ∗ 1−Q dynamical heterogeneities increases:

A. R. Abate et al., Phys. Rev. E 76, 021306 (2007).

DISLOCATION JAMMING: THE FIRST PASSAGE TIME DISTRIBUTION •Look at the times tfp the dislocations first move a distance l in the steady state. •Their distributions can be collapsed, assuming that �tfp � ∼ l/�v� ∼ l(σe − σc )−β

and

δtfp ∼ �tfp �

•The best collapse for β ≈ 1.8

DISLOCATION JAMMING: THE DYNAMICAL SUSCEPTIBILITY

30 20

-_

m=0.06 m=0.055 m=0.05 m=0.045 m=0.04

r4(o)/(m-mc)

1 if dislocation n has not moved a distance l , 0 otherwise

1 0.8 0.6 0.4 0.2 0 40

r4(o!

N � 1 Qt (l, τ ) = wn N n=1

Q(o)

A “cumulative version” of P (tfp ) : the self-overlap order parameter:

10 0 1

Q(l, t) = �Qt (l, τ )�

χ4 (l, t) = N [�Qt (l, τ )2 � − �Qt (l, τ )�2 ]

0.2 0.1 0

0.01

1 0.1 -` o/(m-mc)

10

100

o

1000

DISLOCATION JAMMING: THE DYNAMICAL SUSCEPTIBILITY 60

r4*

40

100

0 0

r4*

The peak dynamical ∗ susceptibility χ4 diverges on approach to jamming:

20

10

∼ (σ − σc )

−α

m

L=200, l=1 L=300, l=1 L=200, l=2 L=200, l=4 L=200, l=16 _=1.14

0.01

∗ χ4

0.1

0.05

m-mc

, α � 1.1

0.1

DYNAMICAL HETEROGENEITIES: MEAN FIELD AVALANCHES χ∗4 ∼ (σ − σc )−α , α � 1.1 Can this be understood in terms of avalanches?

•Consider the avalanche size distribution with a cut-off: P (s) = s

−τ

f [s/(σa − σc )

−γ

]

•Assume that the number of dislocations

n swept to a distance l during an avalanche obeys �n� ∼ �s� , and that M �n� = N/2 : √ χ∗4 = N (δQ∗ )2 ∼ N ( M �n�/N )2 ∼ �n� ∼ (σa − σc )−γ(2−τ )

•Mean field avalanche size exponent τ

≈ 1.5 and α ≈ 1.0 would correspond to γ ≈ 2.0 , again consistent with the MF scenario.

DYNAMICAL AND STATIC CORRELATIONS Look at the dynamical correlations in space: ∗ ∗ P [w(l, τ , �r) = 0|w(l, τ , 0) = 0] Dynamical, σ = 0.06

Dynamical, σ = 0.04

Static, σ = 0.06

Static, σ = 0.04

The static structural correlations (calculated from snapshots of the dislocation configurations) show no significant dependence on σe.

DYNAMICAL AND STATIC CORRELATIONS Look at the dynamical correlations in space: ∗ ∗ P [w(l, τ , �r) = 0|w(l, τ , 0) = 0] Dynamical, σ = 0.06

Static, σ = 0.06

Dynamical, σ = 0.04

Static, σ = 0.04

Average over narrow strips estimate the correlation lengths.

The static structural correlations (calculated from snapshots of the dislocation configurations) show no significant dependence on σe.

DYNAMICAL CORRELATIONS: A DIVERGENT CORRELATION LENGTH 60

jx

100

40 20

jy

0.035

10

0.04

m

0.045

L=200b L=300b PL fit with iy=1.06

0.01

m-mc

0.1

•A divergent dynamical correlation length perpendicular to dislocation glide motion (single slip). • α ≈ νy : compact heterogeneities, the only divergent correlation length along y ? •“Static” structural correlations independent of σe .

SUMMARY •The dislocation jamming/yielding transition has similar features as jamming in many other systems (granular, etc.). •Divergent (and apparently anisotropic) dynamical correlations as jamming is approached, but no significant evolution of the dislocation structures. •The divergence of the dynamical susceptibility can be related to mean field -like avalanche dynamics.

Thank you!

L. Laurson, M.-C. Miguel, and M.J. Alava, “Dynamical Correlations near Dislocation Jamming”, Phys. Rev. Lett. 105, 015501 (2010).