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Stochastic models for the space-time evolution of martensitic avalanches Pierluigi Cesana Institute of Mathematics for Industry Kyushu University, Japan Hysteresis, Avalanches and Interfaces in Solid Phase Transformations 20 th September, 2016

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Stochastic models for the space-time evolution of martensitic avalanches

Pierluigi Cesana

Institute of Mathematics for Industry Kyushu University, Japan Hysteresis, Avalanches and Interfaces in Solid Phase Transformations 20th September, 2016

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 1 / 52

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Overview

General Branching Random Walk model for martensitic avalanches Fragmentation model (SOC à-la-Bak) for the crystal variants of an elastic crystal

Niemann et al. APL Mater. 4, 064101 (2016)

Joint work with John Ball and Ben Hambly (Oxford)

  • J. Ball, P

.C., B. Hambly, Proceedings ESOMAT15 P .C., B. Hambly, in progress P .C., M. Porta, T. Lookman, JMPS 2014

  • S. Patching, P

.C, A. Rueland, in preparation

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 2 / 52

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Martensitic transformation

Aus-Mar interface, C.Chu

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 3 / 52

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Elasticity framework

F 2 R3×3 the deformation gradient ψ(F) the free-energy density

Figure : First-order phase transition, fixed temperature

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 4 / 52

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Effect of temperature

F 2 R3×3 the deformation gradient ψ(θ; F) the free-energy density

Courtesy Tim Duerig

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 5 / 52

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Self-similarity

Transformed sample of Cu-Zn-Al (after cooling). Optical microscope with polarized light 3mm x 2mm (Morin) LEFT-CENTER: SEM micrograph pictures, Ti-Ni; RIGHT: Ti-Ni-Cu (orthorombic martensite), self-acc.

  • M. Nishida et al. (2012) Self-accommodation of B19 martensite in Ti-Ni shape memory

alloys-Part 1. Morphological and crystallographic studies of the variant selection rule, Philosophical Magazine, 92:17, 2215-2233.

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 6 / 52

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Martensitic transformation

A martensitic transformation is a phase transition which involves a cooperative motion of a set of atoms across an interface causing a shape change and a sound. Philip C. Clapp, ICOMAT95

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 7 / 52

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Avalanches

intermittent evolution as a sequence of jerks (avalanches) athermal behavior jerky behavior is consequence of disorder

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 8 / 52

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Acoustic emissions

Avalanches detected by ultrasonic AEs

1) Polarized light optical micrograph of sample of martensitic NiMnGa at room temperature. 2) Emission hits per 0.01K temperature interval (acoustic activity). 3) Histogram of the number of hits vs the absolute energy. Niemann et al. (2014), PRB

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 9 / 52

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Universality

  • A. Planes et al. (2013) Acoustic emission in martensitic transformations, Journal of Alloys and

Compounds, 577S S699-S704

  • E. Salje et al. (2009) Jerky elasticity: Avalanches and the martensitic transition in

shape-memory alloys, APL 95

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 10 / 52

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Fragmentation

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 11 / 52

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Fragmentation

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Fragmentation

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Fragmentation

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 14 / 52

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Fragmentation

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Fragmentation

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n=100

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n=1000

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n=5000

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SEM micrograph (backscattered electron contrast) of an epitaxial Ni-Mn-Ga film in the martensitic state at room temperature. (b) A zoom-in shows two different microstructures. All contrast comes from mesoscopic twin boundaries. (c, d) TEM micrographs at cross-sections along the lines marked in (b).

  • R. Niemannet al. (2014) PRB

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 21 / 52

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3, 4 Phases

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 22 / 52

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Avalanche formation of a habit plane variant cluster with triangular morphology in TiNbAl [Kamioka, ...,T. Inamura, Proceedings of ESOMAT 2015]

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 23 / 52

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6 Phases

Self-accomodation structure in Ti-Ni-Cu Orthorombic Martensite, Watanabe et al., J. Japan Inst. Metals, 54, N.8 1990.

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 24 / 52

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Other shapes

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 25 / 52

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The fragmentation model

Pick a point at random (nucleation) Choose a direction, V (with probability p) or H The general rectangle (a, b) splits into (a, b) ! 8 > > > > > > > > > < > > > > > > > > > : (aU, b), (a(1-U), b) with probability p (a, bU), (a, b(1-U)) with probability 1-p

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 26 / 52

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General Branching Random Walk

The whole structure can be captured in a tree as the evolution inside any rectangle does not affect what happens outside that rectangle. Thus in our tree each vertex will represent a rectangle . We use some results from Biggins1: super-critical GBRW have a shape theorem indicating the region where the number of particles will grow exponentially.

1How fast does a general branching random walk spread? IMA Vol. Math.

Appl., 84, Springer, New York, 1997.

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 27 / 52

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log transformation

Transformation: x = log a y = log b Each rectangle is an individual in the branching process and location is determined by its sides ! GBRW in R2

+.

Ancestor (0, 1) ⇥ (0, 1) ! (0, 0) The smaller the rectangles, the larger the coordinates

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 28 / 52

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Constructing the tree

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 29 / 52

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Branching Random Walk

In Crump-Mode-Jagers (General Branching Process) model an individual z:

is born at time σz 0, has a lifetime Lz 0, has offspring whose birth times are determined by a point process ξz on (0, 1).

For a General Branching Random Walk we include a point process for the birth position ηz (as well as birth times).

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 30 / 52

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General Branching Random Walk

Let the Bernoulli r.v. Bi = ⇢ 0 horizontal 1 p 1 vertical p Let Ui uniform r.v. in [0, 1] Outcomes of the general rectangle (a, b) are (a, b) ! 8 > > > > > > > > > < > > > > > > > > > : ⇣ BiUia, (1 Bi)Uib ⌘ ⇣ Bi(1 Ui)a, (1 Bi)(1 Ui)b ⌘

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 31 / 52

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General Branching Random Walk

Birth time: proportional to log(Area) of rectangle splitting (ηi, ξi) = space,time position of offspring of i

ηi, ξi = ⇢ (Bi log Ui, (1 Bi) log Ui) log Ui (Bi log(1 Ui), (1 Bi) log(1 Ui)) log(1 Ui)

The space-time point process:

(η(dx), ξ(dt)) = 8 > > > > < > > > > : (δ(− log u,0)(dx), δ− log t(dt))I{u=t}+ (δ(− log (1−u),0)(dx), δ− log (1−t)(dt))I{u=t} 1/2 (δ(0,− log u)(dx), δ− log t(dt))I{u=t}+ (δ(0,− log (1−u))(dx), δ− log (1−t)(dt))I{u=t} 1/2

birth time of σ∅ = 0 birth time of σij = σi + inf{t : ξi(t) j}

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 32 / 52

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Time

t = log(1 U1) log(1 U2) log(1 U3) Largest area is (1 U1)(1 U2)(1 U3) That is t = log(Area) at time t largest area is e−t

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 33 / 52

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Shape Theorem

  • We want to keep track of Nt(A) = # individuals in the set A at

time t. Take A ⇢ R2

+ closed, convex, non-empty interior.

1

If A \ {(x, y) : x + y = 1} = ; then t−1 log Nt(tA) ! 1 t ! 1, a.s.

2

If A \ {(x, y) : x + y = 1} 6= ; then t−1 log Nt(tA) ! β 0 t ! 1, a.s.

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 34 / 52

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where β = sup{α∗(a) : a 2 A} α∗(a) = infw{a · w + α(w)} α(a) = inf{φ : m(w, φ)  1} m(w, φ) = E Z e−w·x−φtη∅(dx)ξ∅(dt), w 2 R2, φ 2 R+. the moment-generating function for the position and birth times of offsprings.

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 35 / 52

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Interpretation

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

Figure : n = 102, 103, 104

The line x + y = t Areas ⇠ e−t

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 36 / 52

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Application

At time t largest Area=e−t ! t = – log Area Shape theorem describes exponential growth of individuals N in a certain set A ⇢ R2

++

lim

t→∞ t−1 log(t Nt(A)) = β 0

for finite t, approximation formula

Nt(Ax,y) = et f( x

t , y t )

with:

f x

t , y t

  • =

8 > > < > > :

q 1 (2p 1)2 q 1 ( x

t y t )2+

+(1 2p)( x

t y t ) if x t + y t = 1

1 if x

t + y t 6= 1

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 37 / 52

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Numerical results (rectangles)

for p = 1

2, n = 2000, t ⇡ 6.9

Analytical solution f( x

t , 1 x t ) = 2

q

x t (1 x t )

holds on the line x

t + y t = 1

(ab = e−t)

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 38 / 52

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Numerical results (interfaces)

10

−1

10 10

−1

10 10

1

10

2

log(s)

2−direction 3−direction 4−direction 10

−1

10 10

−1

10 10

1

10

2

10

3

log(s)

2000 events final 1000 products final 500 products

Salje et al. (2009) Jerky elasticity: Avalanches and the martensitic transition in shape-memory alloy, APL 95

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 39 / 52

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Case with bias

1

  • 2
  • 1.5
  • 1

x/t 1

  • 0.5

0.5 0.8 0.5 y/t 1 0.6 0.4 0.2

Figure : Histograms, p = 0.1

  • 4
1
  • 3
0.8
  • 2
0.6
  • 1
Y 1 0.4 0.8 X 0.6 0.2 1 0.4 0.2

Figure : Histograms, p = 0.3

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 40 / 52

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Mondrian process

The Mondrian Process, D. M. Roy, Y. W. Teh, Advances in Neural Information Processing Systems 21 (NIPS 2008) Initial budget λ > 0 Nucleus picked at random Cut is made with probability proportional to its length Cost of cut / its length Arrest when the total length reaches the budget

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 41 / 52

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Fractal microstructure

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3D model

n = 10, 102, 103 events.

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 43 / 52

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3D model

  • 1.4
  • 1.2
  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2
  • 1
  • 0.5

0.5 1 1.5 2 2.5

Cloud and plate distribution.

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 44 / 52

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Random angles

Surface energy Unpinning strategy (joint w. A. Collevecchio, Monash)

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 45 / 52

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TIVP model

  • G. Torrents et al., Geometrical model for martensitic phase

transitions: understanding criticality and weak universality during microstructure growth, to appear discrete model

  • ur focus on new individuals

discrete features

10

−1

10 10

−1

10 10

1

10

2

log(s)

2−direction 3−direction 4−direction

statistics for interfaces and areas

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 46 / 52

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Self-similar nested microstructure

Figure : Star disclination in Pb3(VO4)2 (HREM), C. Manolikas, S. Amelinckx, Phys. Stat. Sol. 1980. Figure : The 2D version of the

hexagonal-to-orthorhombic transformation (Mg-Cd, Mg2Al4Si5O18) is the triangle-to-centered-rectangle (TR) transformation.

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 47 / 52

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Exact solutions

  • Ψ(F) min

F∈R3×3 + Kinematic Compatibility

E1 E2 E3

  • To find a deformation field y : Ω :! R3 s.t.

ry 2 {E1, E2, E3} and y is Hölder continuous.

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 48 / 52

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Kinematic compatibility (KC)

F1 = ry1 F2 = ry2

Conservation of tangential component of ry1, ry2 ! F1 F2 = a ⌦ n

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 49 / 52

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Kinematic compatibility

y : Ω ! R3 : ry 2 {E1, E2, E3} Parallelogram Martensite Microstructure

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 50 / 52

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Star-lisclination

! Rigidity: solution is unique

  • S. Patching, P

.C., A. Rueland P .C., M. Porta, T. Lookman JMPS14

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 51 / 52

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Acknowledgments

JSPS Grant in Aid for Young Scientists B 2016-19 European Research Council under the European Union’s Seventh Framework Programme (FP7/2007- 2013) - ERC grant agreement N. 291053 Department of Energy National Nuclear Security Administration under Award Number DE-FC52-08NA28613

  • Prof. A. Planes and E. Vives research group

LANL-work is unclassified p.cesana@latrobe.edu.au

Pierluigi Cesana (Kyushu/Melbourne) Martensitic avalanches HIA in SPT Oxford 52 / 52