Discrete models for defects and their motion in crystals

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

2. 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.

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

4. Observaciones experimentales 3500x3500 Å2 Dislocations: Defects supported by curves Screw dislocation Edge dislocation Oscar Rodríguez de la Fuente, Ph.D. Thesis, UCM

5. Goals: • predict thresholds for movement • predict speeds as functions of the applied forces Control of defects Macroscopic theories: mechanical properties, growth

6. 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.

7. 2.2. Molecular dynamics simulations mui´´ = -i<jV´(|ui- uj|) -iF( i≠j|ui-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)

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 · 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?

9. Top down approach  Simple cubic crystal Displacement ui(x1,x2,x3,t), i=1,2,3 Stress (x,t), Strain (x,t) ij=cijklkl, kl= 1 (∂luk+∂kul) 2 Potential energy 1/2 ∫ cijkl klij Navier equations ui´´-cijkl∂2 uk = fi, i=1,2,3 ∂xj ∂xl Displacement ui(l,m,n,t), i=1,2,3 Discrete strain (l,m,n,t) kl= 1 (g(D+luk)+g(D+kul)) 2 Potential energy 1/2 ∑ cijkl kl ij Discrete equations mui´´-D-j(cijkl g(D+luk)g´(D+jui))= fi, i=1,2,3 g periodic (period=lattice constant), normalized by g´(0)=1 D+j,D-jforward andbackward differences in the direction j

10. 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

11. 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), i=1,2,3 Discrete strain: ´(l,m,n,t), kl´= 1(g(D+luk´)+g(D+kul´)) 2 g periodic, period=lattice 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)

12. 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

13. 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 ofD-j(cijkl g(D+luk)g´(D+jui))=0 in the class of functions S. Two options: a) Minimize the energy on S: 1/2 ∑ cijkl (g(D+luk)+g(D+kul))/2(g(D+jui)+g(D+iuj))/2 b) Compute the long time limit of the overdamped equations: ui ´-D-j(cijkl g(D+luk)g´(D+jui))= fi, i=1,2,3 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).

14. 3.2. Moving dislocations: discrete waves Screw dislocations b F (i,j,k+wij) F glide

15. Traveling wave Edge dislocation  Displacement Deformed lattice (i+ uij, j+vij) uij(t)=u(i-ct,j) F   F b

16. Variational formulation: Min 1/2 dx n,pcijkl kl (x,n,p)ij (x,n,p) 1= dx n,p |ux|2(x,n,p) kl= 1(g(D+luk)+g(D+kul)) 2 Min 1/2 dx n,p |ux|2(x,n,p) 1= dx n,pcijkl kl (x,n,p)ij (x,n,p) Properties of the energy? Restrictions on g? m un´´=V´(un+1-un)+V´(un-1-un) Friesecke-Wattis (1994) growth at infinity, convexity concentrated compactness

17. 3.3. Analysis of a simplified model vector scalar sin(x) x 3D 2D muij´´ + uij´= (ui+1,j- 2uij+ ui-1,j) + A [sin(ui.j+1- 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/√A)/2π (the angle function   [0,2π))

18. 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 (dynamicalPeierls stress). • Fcd=Fcs, in the overdamped limit m=0. • Moving dislocations identified with traveling wave fronts, • uij = u(i-ct,j),its far field moves uniformly at the same • speed (x-ct,y/√A) + F y overdamped damped Analytical prediction

19. Overdamped limit: Static critical stress and velocity • Linear stability of the stationary solutions for |F|Fcs negative eigenvalues, one vanishes at F=Fcs. • 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)|=√(F-Fcs)/π. Effects of inertia: Dynamic threshold • Saddle-node bifurcation in the branch of traveling waves, |c(F)-cm|=k √ (F-Fcd), oscillatory front profiles.

20. 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 L>>1 (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 createdby the others reduced field dynamics, not particle dynamics

21. Inner model: discrete (atomic), Outer model: continuous (elasticity) Distortion tensor (to match the outer elastic description): wij1 =ui+1,j- uij, wij2 = sin(ui.j+1- 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 /(Ai2+j2) -  1N A(y0- yn)/(A(x0- xn)2+(x0- xn)2) wij2= - A i /(Ai2+j2) +  1N A(x0- xn)/(A(x0- xn)2+(x0- xn)2) The dislocation moves if F > Fs(A). This is only possible as 0 when N=O(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 

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 • Values of the static Peierls stress: correct order of magnitude • Interaction of defects: attraction and repulsion, dipole and loop formation mechanisms are reproduced • Speeds?

23. 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.