Discrete models for defects and their motion in crystals

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Discrete models for defects and their motion in
crystals
A. Carpio, UCM, Spain
joint work with L.L.
Bonilla,UC3M, Spain
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Outline

• Defects in crystals.
• Models for defects.
• 2.1. The elasticity approach.
• 2.2. Molecular dynamics
  simulations.
• 2.3. Nearest neighbors discrete
  elasticity.
• 3. Mathematical setting.
• 3.1. Static dislocations
  singularities.
• 3.2. Moving dislocations discrete
  waves.
• 3.3. Analysis of a simple 2D
  model.
• Experiments.
• 5. Conclusions and perspectives.

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1. Defects in crystals
Fcc unit cell
Fcc perfect lattice
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Dislocations Defects supported by curves
Screw dislocation
Edge dislocation
Oscar Rodríguez de la Fuente, Ph.D. Thesis, UCM
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Goals

• predict thresholds for movement
• predict speeds as functions of the applied
  forces

Macroscopic theories mechanical properties,
growth
Control of defects
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2. Models for defects
2.1. The elasticity approach
u Displacement
div(?(u)) ??
Continuum limit
Navier equations Dirac sources
crystal dislocation

• ?u 1/r, r distance to the core of the
  dislocation
• Breakdown of linear elasticity at the
  core.
• u describes the arrangement of atoms far from
  the core
• (far field), the structure of the core is
  unknown.
• no information on motion.

Atomic scale
Motion along the principal crystallographic
directions, when the force surpasses a
threshold.
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2.2. Molecular dynamics simulations
m ui -?iltjV(ui- uj) -?i F( ?i?jui-uj )
glue potential

• Uncertainty about the potentials.
• Huge computational cost.
• Numerical artifacts numerical chaos, spurious
  oscillations
• - Time discretization long time
  computations
• - Boundary conditions reflected waves
• Qualitative information hard to extract from
  simulations.

Abraham (PRL 2000), Gao (Science 1999), Ortiz
(J. Comp. Aid. Mat. Des. 2002)
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2.3. Nonlinear discrete elasticity
Frank-Van der Merve, Proc. Roy. Soc., 1949
Crystal growth Suzuki, Phys. Rev B 1967
Dislocation motion Lomdahl, Srolovich Phys. Rev.
Lett. 1986, Dislocation generation Marder, PRL
1993, Pla et al, PRB 2000 Crack propagation
Ariza-Ortiz, Arch. Rat. Mech. Anal. to appear,
Dislocations Carpio-Bonilla, PRL 2003, PRB 2005
Dislocation interaction
- Linear nearest and next-nearest neighbour
models lattice structure
and bonds cubic, hexagonal
elasticity as continuum limit - Nonlinearity to
restore the periodicity of the crystal and allow
for glide motion
Which combination of neighbours yields
anisotropic elasticity? How to restore crystal
periodicity?
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Top down approach ? Simple cubic crystal
Displacement ui(x1,x2,x3,t), i1,2,3 Stress
?(x,t), Strain ?(x,t) ?ijcijkl?kl, ?kl 1
(?luk?kul)
2 Potential energy 1/2 ? cijkl ?kl ?ij
Navier equations ?ui- cijkl ?2 uk fi,
i1,2,3 ?xj ?xl
Displacement ui(l,m,n,t), i1,2,3 Discrete strain
?(l,m,n,t) ?kl 1 (g(Dluk)g(Dkul))
2 Potential energy 1/2 ? cijkl ?kl
?ij Discrete equations mui-D-j(cijkl
g(Dluk)g(Djui)) fi, i1,2,3
g periodic (periodlattice constant), normalized
by g(0)1 Dj, D-j forward and backward
differences in the direction j
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Can we extend the idea to fcc or bcc crystals?
Periodicity is expected in the three
primitive directions of the unit
cell ? Write the elastic energy in the (non
orthogonal) primitive coordinates
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Unit cell primitive vectors (ei ,e2,e3)
Elastic constants in this basis cijkl
Coordinates of the points of the crystal lattice
in this basis (l,m,n) Displacement
ui(l,m,n,t), i1,2,3 Discrete strain
?(l,m,n,t), ?kl 1 (g(Dluk)g(Dkul ))

2 g periodic,
periodlattice constant, to be fitted Potential
energy W 1/2 ? cijkl ?kl ?ij ?
Equations of motion
(Temperature and fluctuactions can be included
following Landau)
(Carpio-Bonilla, 2005)
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3. Mathematical setting
3.1. Static dislocations
Strategy 1) Compute the adequate singular
solution of the Navier
equations (displacement far field).
2) Use it as initial and boundary data in the
damped discrete model and
let it relax to a static solution as time grows.
3) Rigorous existence results.
Edge dislocation
Screw dislocation
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Class of functions S sequences (u1(l,m,n),
u2(l,m,n), u3(l,m,n)) in Z3 behaving at
infinity like singular solutions of Navier
equations, with a Dirac mass supported on the
dislocation line as a source. Static
dislocations solutions of D-j(cijkl
g(Dluk)g(Djui))0 in the class of functions S.
Two options a) Minimize the energy on S
1/2 ? cijkl (g(Dluk)g(Dkul))/2
(g(Djui)g(Diuj))/2 b) Compute the long time
limit of the overdamped equations ui
-D-j(cijkl g(Dluk)g(Djui)) fi, i1,2,3
in S using the singular solution of Navier eqs.
as initial datum. The spatial operator is
elliptic near that solution. Outcome Shape
of the dislocation core.
Threshold for motion the spatial operator stops
being elliptic (change of
type).
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3.2. Moving dislocations discrete waves
Screw dislocations
b
F
F
(i,j,kwij)
glide
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Traveling wave
Edge dislocation ?
Displacement
Deformed lattice
(i uij, jvij)
uij(t)u(i-ct,j)
F ?
? F
b
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Variational formulation
Min 1/2 ?dx ?n,p
cijkl ?kl (x,n,p) ?ij (x,n,p) 1 ?dx ?n,p ux2
(x,n,p)
?kl 1 (g(Dluk)g(Dkul)) 2
Min
1/2 ?dx ?n,p ux2 (x,n,p) 1 ?dx ?n,p
cijkl ?kl (x,n,p) ?ij (x,n,p)
Properties of the energy? Restrictions on g?
m unV(un1-un)V(un-1-un)
Friesecke-Wattis (1994)
growth at infinity, convexity concentrated
compactness
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3.3. Analysis of a simplified model
vector scalar
sin(x) x
3D 2D
m uij ? uij (ui1,j- 2uij ui-1,j) A
sin(ui.j1- uij) sin(ui,j-1- uij )
(Carpio-Bonilla, PRL, 2003)
uij/2? displacement of atom
(i,j) along the x axis.
A stiffness ratio m inertia over damping
ratio

• Continuum limit scalar elasticity uxx A uyy
  0 .
• Static edge dislocations are generated from the
  singular solution b?(x,y/vA)/2p (the angle
  function ? ? 0,2p))

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• If we apply shear stress F directed along x
• Two thresholds for the critical stress
• dynamic threshold Fcd ? static threshold
  Fcs
• Below Fcs, pinned dislocations (static Peierls
  stress).
• Above Fcd, moving dislocations (dynamical
  Peierls stress).
• FcdFcs, in the overdamped limit m0.
• Moving dislocations identified with traveling
  wave fronts,
• uij u(i-ct,j), its far field moves
  uniformly at the same

overdamped
damped
Analytical prediction
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Overdamped limit Static critical stress and
velocity

• Linear stability of the stationary solutions for
  F?Fcs ? negative eigenvalues, one vanishes at
  FFcs.
• Normal form of the bifurcationsnear Fcs ? ?
  (F-Fcs)? ?2 ? solutions blow up in finite
  time
• Wave front profiles exhibiting steps above Fcs
• ? at Fcs profiles become discontinuous.
• Near Fcs, wave velocity is the reciprocal of the
  width of blow up time interval
  c(F)v??(F-Fcs)/p.

Effects of inertia Dynamic threshold

• Saddle-node bifurcation in the branch of
  traveling waves, c(F)-cmk v (F-Fcd),
  oscillatory front profiles.

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Averaging densities
N static edge dislocations at the points (xn,yn)
parallel to one dislocation at (x0,y0), separated
from each other by distances of order Lgtgt1 (in
units of the burgers vector). Can the
collective influence of N dislocations move that
at (x0,y0)? Reminds problem of finding the
reduced dynamics for the centers of
2D interacting Ginzburg-Landau vortices (Neu
1990, Chapman 1996) Big difference the
existence of a pinning threshold implies that
the reduced dynamics is that of a single
dislocation subject to the mean field created by
the others ? reduced field dynamics, not
particle dynamics
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Inner model discrete (atomic), Outer model
continuous (elasticity) Distortion tensor (to
match the outer elastic description) wij1
ui1,j- uij, wij2 sin(ui.j1- uij), become
??u/?x and ??u/?y in the continuum limit ??0,
x-x0 ?i, y-y0 ?j finite. Distortion tensor
seen by the dislocation at (x0,y0) wij1 - ?A
j /(Ai2j2) - ? ?1N ?A(y0- yn)/(A(x0- xn)2(x0-
xn)2) wij2 - ?A i /(Ai2j2) ? ?1N ?A(x0-
xn)/(A(x0- xn)2(x0- xn)2) The dislocation
moves if F gt Fs(A). This is only possible as
??0 when NO(1/?). Then, F becomes an integral
F ?N ?? dx dy ?A(x0- x) ?(x,y) /(A(x0-
x)2(x0- x)2) and we find a critical density
for motion.
F
?
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4. Experiments

• To asses the validity of the model we compare
  with available
• quantitative and qualitative experimental
  information
• Cores correct qualitative shape for fcc
  crystals
• Values of the static Peierls stress correct
  order of magnitude
• Interaction of defects attraction and
  repulsion, dipole and loop formation mechanisms
  are reproduced
• Speeds?

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5. Conclusions and open problems

• We have introduced a class of nonlinear discrete
  models for
• defects the simplest correction to elasticity
  theory that accounts
• for crystal defects and their motion.
• We have constructed solutions that can be
  identified with
• static, moving and interacting dislocations.
  Moving dislocations
• are discrete travelling waves.
• In a simplified 2D model for an edge dislocation
  we obtain
• an analytical theory for depinning transitions
  that explains the
• role of static and dynamic Peierls stresses and
  predicts scaling
• laws for the speed of the dislocations. This
  information may be
• used to find homogeneized descriptions.
• Open mathematical issues existence of
  travelling waves,
• deriving macroscopic descriptions by averaging.