1

Discrete models for defects Discrete models for defects and their motion in and their motion in

crystalscrystals

A. Carpio, UCM, SpainA. Carpio, UCM, Spain

joint work with:joint work with: L.L. Bonilla,UC3M, SpainL.L. Bonilla,UC3M, Spain

2

1.1. Defects in crystals.Defects in crystals.2.2. Models for defects.Models for defects. 2.1. The elasticity approach.2.1. The elasticity approach. 2.2. Molecular dynamics simulations.2.2. Molecular dynamics simulations. 2.3. Nearest neighbors: discrete elasticity.2.3. Nearest neighbors: discrete elasticity.

3. Mathematical setting. 3. Mathematical setting. 3.1. Static dislocations: singularities.3.1. Static dislocations: singularities. 3.2. Moving dislocations: discrete waves.3.2. Moving dislocations: discrete waves. 3.3. Analysis of a simple 2D model.3.3. Analysis of a simple 2D model.

4.4. Experiments. Experiments. 5. Conclusions and perspectives.5. Conclusions and perspectives.

Outline

3

1. Defects in crystals1. Defects in crystals

Fcc unit cellFcc unit cell Fcc perfect latticeFcc perfect lattice

4

3500x3500 Å2

Observaciones experimentales

Oscar Rodríguez de la Fuente, Ph.D. Thesis, UCMOscar Rodríguez de la Fuente, Ph.D. Thesis, UCM

Screw dislocationScrew dislocation Edge dislocationEdge dislocation

Dislocations: Defects supported by curvesDislocations: Defects supported by curves

5

Goals:Goals:

Macroscopic theories: Macroscopic theories: mechanical properties, growthmechanical properties, growth

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

Control of defectsControl of defects

6

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

2.1. The elasticity approach2.1. The elasticity approach

crystal + dislocationcrystal + dislocation

Navier equationsNavier equations + Dirac sources+ Dirac sources

Continuum limitContinuum limitdivdiv(((u)) = (u)) =

Motion along the principal crystallographic directions, when the force surpasses a threshold.

AtomicAtomic scalescale

u Displacementu Displacement

2. Models for defects2. Models for defects

7

mm uuii´´ = -´´ = -i<ji<jV´(V´(|u|uii- - uujj||) -) -ii F( F( i≠ji≠j|u|uii--uujj| | )) glue potentialglue potential

• Uncertainty about the potentials.Uncertainty about the potentials.• Huge computational cost.Huge computational cost.• Numerical artifacts: numerical chaos, spurious oscillationsNumerical artifacts: numerical chaos, spurious oscillations - Time discretization: long time computations - Time discretization: long time computations - Boundary conditions: reflected waves- Boundary conditions: reflected waves

• Qualitative information hard to extract from simulations.Qualitative information hard to extract from simulations.

Abraham (PRL 2000), Gao (Science 1999), Abraham (PRL 2000), Gao (Science 1999),

Ortiz (J. Comp. Aid. Mat. Des. 2002)Ortiz (J. Comp. Aid. Mat. Des. 2002)

2.2. Molecular dynamics simulations2.2. Molecular dynamics simulations

8

2.3. Nonlinear discrete elasticity2.3. Nonlinear discrete elasticity

Frank-Van der Merve, Proc. Roy. Soc., 1949 Frank-Van der Merve, Proc. Roy. Soc., 1949 Crystal growthCrystal growthSuzuki, Phys. Rev B 1967 Suzuki, Phys. Rev B 1967 Dislocation motionDislocation motionLomdahl, Srolovich Phys. Rev. Lett. 1986Lomdahl, Srolovich Phys. Rev. Lett. 1986, Dislocation generation, Dislocation generationMarder, PRL 1993, Pla et al, PRB 2000 Marder, PRL 1993, Pla et al, PRB 2000 Crack propagationCrack propagation Ariza-Ortiz, Arch. Rat. Mech. Anal. to appear, Ariza-Ortiz, Arch. Rat. Mech. Anal. to appear, DislocationsDislocationsCarpio-Bonilla, PRL 2003, PRB 2005 Carpio-Bonilla, PRL 2003, PRB 2005 Dislocation interactionDislocation interaction

- Linear nearest and next-nearest neighbour models:- Linear nearest and next-nearest neighbour models: · lattice structure and bonds· lattice structure and bonds · cubic, hexagonal… elasticity as continuum limit · cubic, hexagonal… elasticity as continuum limit - Nonlinearity to restore the periodicity of the crystal and allow- Nonlinearity to restore the periodicity of the crystal and allow for glide motionfor glide motion

Which combination of neighbours yields anisotropic elasticity?Which combination of neighbours yields anisotropic elasticity?How to restore crystal periodicity?How to restore crystal periodicity?

9

Displacement uDisplacement uii(x(x11,x,x22,x,x33,t), ,t), i=1,2,3i=1,2,3

Stress Stress (x,t), Strain (x,t), Strain (x,t)(x,t)

ijij=c=cijklijklklkl, , klkl= = 11 (∂ (∂lluukk+∂+∂kkuull)) 22Potential energyPotential energy

1/2 ∫ 1/2 ∫ ccijkl ijkl klkl ijij

Navier equationsNavier equations

uuii´´´´-- ccijklijkl ∂∂

22 u ukk = f = fii, i=1,2,3, i=1,2,3

∂∂xxj j ∂x∂xll

Displacement uDisplacement uii(l,m,n,t), (l,m,n,t), i=1,2,3i=1,2,3

Discrete strain Discrete strain (l,m,n,t)(l,m,n,t)

klkl= = 11 (g(D (g(D++lluukk)+g(D)+g(D++

kkuull)))) 22Potential energyPotential energy

1/2 ∑ 1/2 ∑ ccijkl ijkl kl kl ijij

Discrete equationsDiscrete equations

mumuii´´´´-D-D--

jj(c(cijkl ijkl g(Dg(D++lluukk)g´(D)g´(D++

jjuuii))= f))= fii,,

i=1,2,3i=1,2,3

Top down approach Top down approach Simple cubic crystal Simple cubic crystal

g periodic (period=lattice constant), normalized by g´(0)=1g periodic (period=lattice constant), normalized by g´(0)=1

DD++jj,,

DD--jj forward andforward and backward differences in the direction jbackward differences in the direction j

10

Can we extend the idea to fcc or bcc crystals?Can we extend the idea to fcc or bcc crystals?

Periodicity is expected in the three primitive directionsPeriodicity is expected in the three primitive directions of the unit cell of the unit cell Write the elastic energy in the (non Write the elastic energy in the (non orthogonal) primitive coordinates orthogonal) primitive coordinates

11

Unit cell primitive vectors:Unit cell primitive vectors: (e (eii´́ ,e,e22´,e´,e33´) ´)

Elastic constants in this basis:Elastic constants in this basis: ccijklijkl´́

Coordinates of the points of the crystal lattice in this basis:Coordinates of the points of the crystal lattice in this basis: (l,m,n) (l,m,n)

Displacement:Displacement: u uii´(l,m,n,t), ´(l,m,n,t), i=1,2,3i=1,2,3

Discrete strain:Discrete strain: ´(l,m,n,t), ´(l,m,n,t), klkl´= ´= 11 ((g(Dg(D++lluukk´)+g(D´)+g(D++

kkuull´́ )))) 22 g periodic, period=lattice constant, to be fittedg periodic, period=lattice constant, to be fitted

Potential energyPotential energy

W= 1/2 ∑ W= 1/2 ∑ ccijklijkl´́ klkl´́ ijij´ ´ Equations of motionEquations of motion

(Temperature and fluctuactions can be included following Landau)

(Carpio-Bonilla, 2005)

12

3.1. 3.1. Static dislocationsStatic dislocations

Strategy: 1) Compute the adequate singular solution of the Navier Strategy: 1) Compute the adequate singular solution of the Navier equations (displacement far field).equations (displacement far field). 2) Use it as initial and boundary data in the damped discrete 2) Use it as initial and boundary data in the damped discrete model and let it relax to a static solution as time grows.model and let it relax to a static solution as time grows. 3) Rigorous existence results.3) Rigorous existence results.

3. Mathematical setting3. Mathematical setting

Edge Edge dislocationdislocation

Screw Screw dislocationdislocation

13

Class of functions S:Class of functions S: sequences (u sequences (u11((l,m,nl,m,n), u), u22((l,m,nl,m,n), u), u33((l,m,nl,m,n)) in Z)) in Z33

behaving at infinity like singular solutions of Navier equations,behaving at infinity like singular solutions of Navier equations,with a Dirac mass supported on the dislocation line as a source.with a Dirac mass supported on the dislocation line as a source.

Static dislocations: Static dislocations: solutions ofsolutions of D D--jj(c(cijkl ijkl g(Dg(D++

lluukk)g´(D)g´(D++jjuuii))=0 in the))=0 in the

class of functions S. Two options:class of functions S. Two options:

a) Minimize the energy on S: a) Minimize the energy on S:

1/2 ∑ 1/2 ∑ ccijkl ijkl ((g(Dg(D++lluukk)+g(D)+g(D++

kkuull))))/2/2 ((g(Dg(D++jjuuii)+g(D)+g(D++

iiuujj))))/2/2

b) Compute the long time limit of the overdamped equations:b) Compute the long time limit of the overdamped equations: uui i ´́ -D-D--

jj(c(cijkl ijkl g(Dg(D++lluukk)g´(D)g´(D++

jjuuii))= f))= fii, , i=1,2,3i=1,2,3

in S using the singular solution of Navier eqs. as initial datum.in S using the singular solution of Navier eqs. as initial datum.

The spatial operator is ‘elliptic’ near that solution.The spatial operator is ‘elliptic’ near that solution.

Outcome: Outcome: Shape of the dislocation core.Shape of the dislocation core. Threshold for motion: the spatial operator stops being Threshold for motion: the spatial operator stops being elliptic (change of type).elliptic (change of type).

14

FF

FF

bb

glideglide

3.2. 3.2. Moving dislocations: discrete wavesMoving dislocations: discrete waves

(i,j,k+w(i,j,k+wijij))

Screw dislocationsScrew dislocations

15

Traveling waveTraveling wave

DisplacementDisplacement Deformed latticeDeformed lattice

uuijij(t)=u(i-ct,j) (t)=u(i-ct,j)

(i+ u(i+ uijij, j+v, j+vijij)) F F

FF bb

Edge dislocation Edge dislocation

16

klkl= = 11 ((g(Dg(D++lluukk)+g(D)+g(D++

kkuull)))) 22

Variational formulation:

Friesecke-Wattis (1994)Friesecke-Wattis (1994) m um unn´´=V´(u´´=V´(un+1n+1-u-unn)+V´(u)+V´(un-1n-1-u-unn))

Properties of the energy? Restrictions on g?Properties of the energy? Restrictions on g?

growth at infinity, convexity concentrated compactness

Min 1/2 Min 1/2 dx dx n,pn,p ccijkl ijkl kl kl (x,n,p)(x,n,p) ij ij (x,n,p)(x,n,p)

1= 1= dx dx n,p n,p ||uuxx||22 (x,n,p)(x,n,p)

Min 1/2 Min 1/2 dx dx n,p n,p ||uuxx||22 (x,n,p)(x,n,p)

1= 1= dx dx n,pn,p ccijkl ijkl kl kl (x,n,p)(x,n,p) ij ij (x,n,p)(x,n,p)

17

3D 2D3D 2D vector scalarvector scalar

mm uuijij´´ +´´ + u uijij´= (u´= (ui+1,ji+1,j- 2u- 2uijij+ u+ ui-1,ji-1,j) + A [sin(u) + A [sin(ui.j+1i.j+1- u- uijij) + sin(u) + sin(ui,j-1i,j-1- u- uijij )] )]

uuijij/2/2 : displacement of atom : displacement of atom

(i,j) along the x axis.(i,j) along the x axis.

AA: stiffness ratio: stiffness ratio m: inertia over damping ratiom: inertia over damping ratio

• Continuum limit: scalar elasticity Continuum limit: scalar elasticity uuxxxx + A u + A uyyyy = 0 = 0 ..• Static edge dislocations are generated from the singular Static edge dislocations are generated from the singular solution solution bb(x,y/√A)/2π (the (x,y/√A)/2π (the angle function angle function [0,2π)) [0,2π))

3.3. Analysis of a simplified model

sin(x) xsin(x) x

(Carpio-Bonilla, PRL, 2003)

18

If we apply shear stress If we apply shear stress F F directed along x:directed along x:

• Two thresholds for the critical stress:Two thresholds for the critical stress: dynamic threshold dynamic threshold FFcdcd static threshold static threshold FFcscs

Below Below FFcscs, , pinned dislocations (pinned dislocations (static static Peierls stressPeierls stress).). Above Above FFcdcd, moving dislocations (, moving dislocations (dynamicaldynamical Peierls stressPeierls stress).).

FFcdcd=F=Fcscs, in the overdamped limit , in the overdamped limit m=0m=0..

• Moving dislocations identified with Moving dislocations identified with traveling wave fronts,traveling wave fronts, uuijij = u(i-ct,j), = u(i-ct,j), its far field moves uniformly at the same its far field moves uniformly at the same

speed speed (x-ct,y/√A) + F y(x-ct,y/√A) + F y

-5-4

-3-2

-10

-6-4

-20

24

6

2

4

6

y=i-ct

u(,j)

0.025 0.03 0.035 0.04 0.0450

0.005

0.01

0.015

F

c(F)

TextEnd damped

Analyticalprediction

overdamped

19

Overdamped limit: Static critical stress and velocityOverdamped limit: Static critical stress and velocity

• Linear stability of the stationary solutions for |F|Linear stability of the stationary solutions for |F|FFcscs negative eigenvalues, one vanishes at negative eigenvalues, one vanishes at F=FF=Fcscs..

Normal form of the bifurcationsnear Normal form of the bifurcationsnear FFcscs: : ’ = ’ = (F-F (F-Fcscs)+)+

22 solutions blow up in finite time solutions blow up in finite time

• Wave front profiles exhibiting steps above Wave front profiles exhibiting steps above FFcscs

at at FFcscs profiles become discontinuous. profiles become discontinuous.

• Near Near FFcscs, , wave velocity is the reciprocal of the width of blow wave velocity is the reciprocal of the width of blow

up time interval :up time interval : |c(F)|=√|c(F)|=√(F-F(F-Fcscs)/π.)/π.

• Saddle-node bifurcation in the branch of traveling Saddle-node bifurcation in the branch of traveling waves, waves, |c(F)-c|c(F)-cmm|=k √ (F-F|=k √ (F-Fcdcd)), oscillatory front profiles, oscillatory front profiles..

Effects of inertia: Dynamic thresholdEffects of inertia: Dynamic threshold

20

Averaging densitiesAveraging densities

N static edge dislocations at the points (xN static edge dislocations at the points (xnn,y,ynn) parallel to one dislocation) parallel to one dislocation

at (xat (x00,y,y00), separated from each other by distances of order L>>1 (in units), separated from each other by distances of order L>>1 (in units

of the burgers vector). of the burgers vector).

Can the collective influence of N dislocations move that at (xCan the collective influence of N dislocations move that at (x00,y,y00)?)?

Reminds problem of finding the reduced dynamics for the centers of 2DReminds problem of finding the reduced dynamics for the centers of 2Dinteracting Ginzburg-Landau vortices interacting Ginzburg-Landau vortices (Neu 1990, Chapman 1996)(Neu 1990, Chapman 1996)

Big difference: the existence of a pinning threshold implies that theBig difference: the existence of a pinning threshold implies that thereduced dynamics is that of a single dislocation subject to the meanreduced dynamics is that of a single dislocation subject to the meanfield createdfield created by the othersby the others reduced field dynamics, not particle reduced field dynamics, not particledynamicsdynamics

21

Inner model: discrete (atomic), Outer model: continuous (elasticity)Inner model: discrete (atomic), Outer model: continuous (elasticity)

Distortion tensor (to match the outer elastic description):Distortion tensor (to match the outer elastic description):wwijij

1 1 == uui+1,ji+1,j- u- uijij, w, wijij22 = sin(u = sin(ui.j+1i.j+1- u- uijij),), become become ∂u/∂x∂u/∂x and and ∂u/∂y∂u/∂y

in the continuum limit in the continuum limit 0, x-x0, x-x00= = i, y-yi, y-y00= = j finite.j finite.

Distortion tensor seen by the dislocation at Distortion tensor seen by the dislocation at (x(x00,y,y00):):

wwijij11= - = - A j /(AiA j /(Ai22+j+j22) - ) - 11

N N A(yA(y00- y- ynn)/(A(x)/(A(x00- x- xnn))22+(x+(x00- x- xnn))22) )

wwijij22= - = - A i /(AiA i /(Ai22+j+j22) + ) + 11

N N A(xA(x00- x- xnn)/(A(x)/(A(x00- x- xnn))22+(x+(x00- x- xnn))22) )

The dislocation moves if F > FThe dislocation moves if F > Fss(A). This is only possible as (A). This is only possible as 00

when N=O(1/when N=O(1/). Then, F becomes an integral:). Then, F becomes an integral:

F= F= NN ∫∫∫∫ dx dy dx dy A(xA(x00- x) - x) (x,y) /(A(x(x,y) /(A(x00- x)- x)22+(x+(x00- x)- x)22))

and we find a critical density for motion.and we find a critical density for motion.

FF

22

4.4. ExperimentsExperiments

To asses the validity of the model we compare with available To asses the validity of the model we compare with available quantitative and qualitative experimental information:quantitative and qualitative experimental information:

• Cores:Cores: correct qualitative shape for fcc crystals correct qualitative shape for fcc crystals• Values of the static Peierls stress:Values of the static Peierls stress: correct order of magnitude correct order of magnitude• Interaction of defects:Interaction of defects: attraction and repulsion, dipole and attraction and repulsion, dipole and loop formation mechanisms are reproducedloop formation mechanisms are reproduced

• Speeds?Speeds?

23

5. Conclusions and open problems5. Conclusions and open problems

• We have introduced a class of nonlinear discrete models for We have introduced a class of nonlinear discrete models for defects: the simplest correction to elasticity theory that accounts defects: the simplest correction to elasticity theory that accounts for crystal defects and their motion.for crystal defects and their motion.

• We have constructed solutions that can be identified with We have constructed solutions that can be identified with static, moving and interacting dislocations. Moving dislocations static, moving and interacting dislocations. Moving dislocations are discrete travelling waves.are discrete travelling waves.

• In a simplified 2D model for an edge dislocation we obtainIn a simplified 2D model for an edge dislocation we obtainan analytical theory for depinning transitions that explains thean analytical theory for depinning transitions that explains therole of static and dynamic Peierls stresses and predicts scalingrole of static and dynamic Peierls stresses and predicts scalinglaws for the speed of the dislocations. This information may belaws for the speed of the dislocations. This information may beused to find homogeneized descriptions.used to find homogeneized descriptions.

• Open mathematical issues: existence of travelling waves, Open mathematical issues: existence of travelling waves, deriving macroscopic descriptions by averaging.deriving macroscopic descriptions by averaging.