1. 1 Discrete models for defects and their motion in crystals A. Carpio, UCM, Spain A. Carpio, UCM, Spain joint work with: L.L. Bonilla,UC3M, Spain L.L. Bonilla,UC3M, Spain

2. 2 1. Defects in crystals. 2. 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. 3.1. 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. Experiments. 5. Conclusions and perspectives. Outline

3. 3 1. Defects in crystals 1. Defects in crystals Fcc unit cell Fcc perfect lattice

4. 4 3500x3500 Å 2 Observaciones experimentales Oscar Rodríguez de la Fuente, Ph.D. Thesis, UCM Screw dislocation Edge dislocation Dislocations: Defects supported by curves

5. 5 Goals: Macroscopic theories: mechanical 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 defects

6. 6 u ~ 1/r, r = distance to the core of the dislocation u ~ 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 core u 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 approach 2.1. The elasticity approach crystal + dislocation Navier equations + Dirac sources + Dirac sources Continuum limit div ( (u)) = div ( (u)) = Motion along the principal crystallographic directions, when the force surpasses a threshold. Atomic scale scale u Displacement 2. Models for defects 2. Models for defects

7. 7 m u i ´´ = - i<j V´( |u i - u j | ) - i F( ij |u i - u j | ) m u i ´´ = - i<j V´( |u i - u j | ) - i F( ij |u i - u j | ) glue potential glue potential Uncertainty about the potentials. Uncertainty about the potentials. Huge computational cost. Huge computational cost. Numerical artifacts: numerical chaos, spurious oscillations Numerical 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), Ortiz (J. Comp. Aid. Mat. Des. 2002) 2.2. Molecular dynamics simulations

8. 8 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 · 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 for glide motion for glide motion Which combination of neighbours yields anisotropic elasticity? How to restore crystal periodicity?

9. 9 Displacement u i (x 1,x 2,x 3,t), i=1,2,3 Stress (x,t), Strain (x,t) ij =c ijkl kl, kl = 1 ( l u k + k u l ) ij =c ijkl kl, kl = 1 ( l u k + k u l ) 2 Potential energy 1/2 c ijkl kl ij 1/2 c ijkl kl ij Navier equations u i ´´ - c ijkl 2 u k = f i, i=1,2,3 u i ´´ - c ijkl 2 u k = f i, i=1,2,3 x j x l x j x l Displacement u i (l,m,n,t), i=1,2,3 Discrete strain (l,m,n,t) kl = 1 (g(D + l u k )+g(D + k u l )) kl = 1 (g(D + l u k )+g(D + k u l )) 2 Potential energy 1/2 c ijkl kl ij 1/2 c ijkl kl ij Discrete equations mu i ´´ -D - j (c ijkl g(D + l u k )g´(D + j u i ))= f i, mu i ´´ -D - j (c ijkl g(D + l u k )g´(D + j u i ))= f i, i=1,2,3 i=1,2,3 Top down approach Simple cubic crystal g periodic (period=lattice constant), normalized by g´(0)=1 D + j, D - j forward and backward differences in the direction j

10. 10 Can we extend the idea to fcc or bcc crystals? Periodicity is expected in the three primitive directions Periodicity is expected in the three primitive directions of the unit cell Write the elastic energy in the (non of the unit cell Write the elastic energy in the (non orthogonal) primitive coordinates orthogonal) primitive coordinates

11. 11 Unit cell primitive vectors: (e i ´,e 2 ´,e 3 ´) Elastic constants in this basis: c ijkl ´ Coordinates of the points of the crystal lattice in this basis: (l,m,n) Displacement: u i ´(l,m,n,t), i=1,2,3 Discrete strain: ´(l,m,n,t), kl ´= 1 ( g(D + l u k ´)+g(D + k u l ´ ) ) 2 g periodic, period=lattice constant, to be fitted g periodic, period=lattice constant, to be fitted Potential energy W= 1/2 c ijkl ´ kl ´ ij ´ Equations of motion W= 1/2 c ijkl ´ kl ´ ij ´ Equations of motion (Temperature and fluctuactions can be included following Landau) (Carpio-Bonilla, 2005)

12. 12 3.1. Static dislocations 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 setting Edge EdgedislocationScrewdislocation

13. 13 Class of functions S: sequences (u 1 ( l,m,n ), u 2 ( l,m,n ), u 3 ( l,m,n )) in Z 3 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 (c ijkl g(D + l u k )g´(D + j u i ))=0 in the class of functions S. Two options: a) Minimize the energy on S: a) Minimize the energy on S: 1/2 c ijkl ( g(D + l u k )+g(D + k u l ) ) /2 ( g(D + j u i )+g(D + i u j ) ) /2 1/2 c ijkl ( g(D + l u k )+g(D + k u l ) ) /2 ( g(D + j u i )+g(D + i u j ) ) /2 b) Compute the long time limit of the overdamped equations: b) Compute the long time limit of the overdamped equations: u i ´ -D - j (c ijkl g(D + l u k )g´(D + j u i ))= f i, i=1,2,3 u i ´ -D - j (c ijkl g(D + l u k )g´(D + j u i ))= f i, i=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: 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. 14 F F b glide 3.2. Moving dislocations: discrete waves (i,j,k+w ij ) Screw dislocations

15. 15 Traveling wave Traveling wave Displacement Deformed lattice u ij (t)=u(i-ct,j) u ij (t)=u(i-ct,j) (i+ u ij, j+v ij ) F F F b Edge dislocation Edge dislocation

16. 16 kl = 1 ( g(D + l u k )+g(D + k u l ) ) kl = 1 ( g(D + l u k )+g(D + k u l ) ) 2 Variational formulation: Friesecke-Wattis (1994) m u n ´´=V´(u n+1 -u n )+V´(u n-1 -u n ) Properties of the energy? Restrictions on g? growth at infinity, convexity concentrated compactness Min 1/2 dx n,p c ijkl kl (x,n,p) ij (x,n,p) Min 1/2 dx n,p c ijkl kl (x,n,p) ij (x,n,p) 1= dx n,p |u x | 2 (x,n,p) Min 1/2 dx n,p |u x | 2 (x,n,p) Min 1/2 dx n,p |u x | 2 (x,n,p) 1= dx n,p c ijkl kl (x,n,p) ij (x,n,p)

17. 17 3D 2D vector scalar m u ij ´´ + u ij ´= (u i+1,j - 2u ij + u i-1,j ) + A [sin(u i.j+1 - u ij ) + sin(u i,j-1 - u ij )] u ij /2 : displacement of atom u ij /2 : displacement of atom (i,j) along the x axis. (i,j) along the x axis. A: stiffness ratio A: stiffness ratio m: inertia over damping ratio m: inertia over damping ratio Continuum limit: scalar elasticity u xx + A u yy = 0. Continuum limit: scalar elasticity u xx + A u yy = 0. Static edge dislocations are generated from the singular solution b (x,y/A)/2π (the angle function [0,2π)) Static edge dislocations are generated from the singular solution b (x,y/A)/2π (the angle function [0,2π)) 3.3. Analysis of a simplified model sin(x) x (Carpio-Bonilla, PRL, 2003)

18. 18 If we apply shear stress F directed along x: Two thresholds for the critical stress: Two thresholds for the critical stress: dynamic threshold F cd static threshold F cs dynamic threshold F cd static threshold F cs Below F cs, pinned dislocations (static Peierls stress). Below F cs, pinned dislocations (static Peierls stress). Above F cd, moving dislocations (dynamical Peierls stress). Above F cd, moving dislocations (dynamical Peierls stress). F cd =F cs, in the overdamped limit m=0. F cd =F cs, in the overdamped limit m=0. Moving dislocations identified with traveling wave fronts, Moving dislocations identified with traveling wave fronts, u ij = u(i-ct,j), its far field moves uniformly at the same u ij = u(i-ct,j), its far field moves uniformly at the same speed (x-ct,y/A) + F y speed (x-ct,y/A) + F y damped Analytical prediction overdamped

19. 19 Overdamped limit: Static critical stress and velocity Linear stability of the stationary solutions for |F| F cs negative eigenvalues, one vanishes at F=F cs. Linear stability of the stationary solutions for |F| F cs negative eigenvalues, one vanishes at F=F cs. Normal form of the bifurcationsnear F cs : = (F-F cs )+ 2 solutions blow up in finite time Normal form of the bifurcationsnear F cs : = (F-F cs )+ 2 solutions blow up in finite time Wave front profiles exhibiting steps above F cs Wave front profiles exhibiting steps above F cs at F cs profiles become discontinuous. at F cs profiles become discontinuous. Near F cs, wave velocity is the reciprocal of the width of blow up time interval : |c(F)|= (F-F cs )/π. Near F cs, wave velocity is the reciprocal of the width of blow up time interval : |c(F)|= (F-F cs )/π. Saddle-node bifurcation in the branch of traveling waves, |c(F)-c m |=k (F-F cd ), oscillatory front profiles. Saddle-node bifurcation in the branch of traveling waves, |c(F)-c m |=k (F-F cd ), oscillatory front profiles. Effects of inertia: Dynamic threshold

20. 20 Averaging densities N static edge dislocations at the points (x n,y n ) parallel to one dislocation at (x 0,y 0 ), separated from each other by distances of order L>>1 (in units of the burgers vector). Can the collective influence of N dislocations move that at (x 0,y 0 )? 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

21. 21 Inner model: discrete (atomic), Outer model: continuous (elasticity) Distortion tensor (to match the outer elastic description): w ij 1 = u i+1,j - u ij, w ij 2 = sin(u i.j+1 - u ij ), become u/x and u/y in the continuum limit 0, x-x 0 = i, y-y 0 = j finite. Distortion tensor seen by the dislocation at (x 0,y 0 ): w ij 1 = - A j /(Ai 2 +j 2 ) - 1 N A(y 0 - y n )/(A(x 0 - x n ) 2 +(x 0 - x n ) 2 ) w ij 1 = - A j /(Ai 2 +j 2 ) - 1 N A(y 0 - y n )/(A(x 0 - x n ) 2 +(x 0 - x n ) 2 ) w ij 2 = - A i /(Ai 2 +j 2 ) + 1 N A(x 0 - x n )/(A(x 0 - x n ) 2 +(x 0 - x n ) 2 ) w ij 2 = - A i /(Ai 2 +j 2 ) + 1 N A(x 0 - x n )/(A(x 0 - x n ) 2 +(x 0 - x n ) 2 ) The dislocation moves if F > F s (A). This is only possible as 0 when N=O(1/ ). Then, F becomes an integral: F= N dx dy A(x 0 - x) (x,y) /(A(x 0 - x) 2 +(x 0 - x) 2 ) F= N dx dy A(x 0 - x) (x,y) /(A(x 0 - x) 2 +(x 0 - x) 2 ) and we find a critical density for motion. F

22. 22 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 Cores: correct qualitative shape for fcc crystals Values of the static Peierls stress: correct order of magnitude Values of the static Peierls stress: correct order of magnitude Interaction of defects: attraction and repulsion, dipole and loop formation mechanisms are reproduced Interaction of defects: attraction and repulsion, dipole and loop formation mechanisms are reproduced Speeds? Speeds?

23. 23 5. 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 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 are discrete travelling waves. In a simplified 2D model for an edge dislocation we obtain 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, Open mathematical issues: existence of travelling waves, deriving macroscopic descriptions by averaging.