A Quasicontinuum for Complex Crystals

A Quasicontinuum for Complex Crystals Ellad B. Tadmor Aerospace Engineering and Mechanics University of Minnesota Collaborators: Previous Work: Current Work: U. V. Waghmare, G. S. Smith, N. Bernstein, E. Kaxiras R. S. Elliott, M. Dobson, M. Luskin, R. D. James QC for Complex Crystals October 9, 2006 Minneapolis, MN

Materials with Multiple Functions Most materials with multiple functions such as Ferroelectrics, shape-memory materials, active thin films, intermetallics, ceramics are crystalline and share the property that their crystal structure is complex, i.e. they possess a lattice with a basis. unit cell contains more than one basis atom We are interested in using the quasicontinuum (QC) method to model the deformation of such materials at the macro scale while retaining atomistic resolution where necessary. E. B. Tadmor QC for Complex Crystals, October 9, 2006 2

Local QC for Complex Lattices u k k u j X 2 B 0T Ω e j Node i T u i Element e B 0 B 0U Tadmor et al., Phys. Rev. B, 59, 235 (1999). X 1 Varying displacement field Cauchy-Born in each element E. B. Tadmor QC for Complex Crystals, October 9, 2006 3

Cauchy-Born Rule The coordinates of the atoms in the reference configuration are: B 1 X " m k # = m i A i + B k Triplet of integers lattice Vectors sublattice positions k {1,...,M} According to the Cauchy-Born rule, the coordinates in the current configuration are: " # " # shift vectors m m x = FX + δ k k k = m i ( FA i )+ FB k + δ k δ = m i a i + b k + δ k E. B. Tadmor QC for Complex Crystals, October 9, 2006 4

Strain Energy Density (SED) F Point in Reference Configuration A 3 A2 A 1 FA 3 FA 1 FA 2 δ Point in Current Configuration strain energy density unit cell volume Minimize out shifts to obtain an effective continuum E. B. Tadmor QC for Complex Crystals, October 9, 2006 5

Effective SED Approach The internal shifts are minimized out locally Start with: W = W ( F, δ k ), k {1,...,M} Minimized SED with respect to shifts: W δ k F Define effective SED: = 0 δ k = δ k ( F) E. B. Tadmor QC for Complex Crystals, October 9, 2006 6

Importance of Shuffling Example: Simple shear of Si [111] [110] A [112] D B E C F w/o shuffling with shuffling E. B. Tadmor QC for Complex Crystals, October 9, 2006 7

Importance of Shuffling Example: Simple shear of Si 0.020 TB E 0.015 SW W [ev/a 3 ] 0.010 0.6 0.5 TB 0.005 0.000 D F 0 0.2 0.4 0.6 0.8 1 1.2 γ 0.4 W [ev/a 3 ] 0.3 0.2 C SW 0.1 0 A B 0 1 2 3 4 5 γ E. B. Tadmor QC for Complex Crystals, October 9, 2006 8

Local QC with Multilattices Local QC potential energy is: u k k u j X 2 B 0T Node i T u i Ω e j Element e Π h ({u}) = N e X e=1 Ω e W (F e ) N XR γ=1 f γ u γ B 0 where Fe = I + X j u j 0 N j (Xe) B 0U X 1 Equilibrium configurations are associated with minimizers of the potential energy subject to kinematic constraints: min u i Π h (u 1,...,u NR ) EQUILIBRIUM E. B. Tadmor QC for Complex Crystals, October 9, 2006 9

Nanoindentation in Si Local QC Examples metallic silicon Silicon (diamond) Smith, Tadmor and Kaxiras, Phys. Rev. Lett., 84, 1260 (2000). Smith, Tadmor, Bernstein and Kaxiras, Acta Mater., 49, 4089 (2001). E. B. Tadmor QC for Complex Crystals, October 9, 2006 10

Nanoindentation in Silicon QC EXPERIMENTAL Smith, Tadmor and Kaxiras, PRL., 84, 1260 (2000). Weppelmann et al., J. Mater. Res., 8, 830 (1993). E. B. Tadmor QC for Complex Crystals, October 9, 2006 12

Electrical Resistance Results E. B. Tadmor QC for Complex Crystals, October 9, 2006 13

Local QC Examples Ferroelectric (PbTiO 3 ) actuator (Shu and Bhattacharya, Phil. Mag. B, 81 2021-2054 (2001).) σ 0 σ 0 Ti Z Pb Δh E=0 h P P z x Δb E O 90 degrees Y P 180 degrees X 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 b 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 P P Tadmor, Waghmare, Smith and Kaxiras, Acta Mater., 50, 2989 (2002). E. B. Tadmor QC for Complex Crystals, October 9, 2006 14

Local QC Examples Ferroelectric (PbTiO 3 ) actuator: e zz 0.06 0.05 0.04 0.03 0.02 0.01 0-0.01 T(Z -) T(Z + ) T(X) -0.02-100 0 100 E(MV/m) e zz 0.06 0.05 0.04 0.03 0.02 0.01 0-0.01-0.02 T(Z -) T(Z + ) T(X) -100 0 100 E(MV/m) e zz 0.06 0.05 0.04 0.03 0.02 0.01 0-0.01-0.02 T(Z -) T(Z + ) T(X) -100 0 100 E(MV/m) 0.06 0.04 T(Z + ) 0.06 0.04 T(Z + ) 0.06 0.04 T(Z + ) P z (C/m 2 ) 0.02 0-0.02 T(X) P z (C/m 2 ) 0.02 0-0.02 T(X) P z (C/m 2 ) 0.02 0-0.02 T(X) -0.04-0.06 T(Z -) -100 0 100 E(MV/m) -0.04-0.06 T(Z -) -100 0 100 E(MV/m) -0.04-0.06 T(Z -) -100 0 100 E(MV/m) σ = 100 MPa σ = 300 MPa σ = 500 MPa E. B. Tadmor QC for Complex Crystals, October 9, 2006 15

Initial state Polarization Vectors stress = 400 MPa Local QC Examples Ferroelectric (PbTiO 3 ) actuator: Large applied E field E E. B. Tadmor QC for Complex Crystals, October 9, 2006 16

Limitations of Existing Methodology Lack of interfacial energy: leads to random structures in degenerate cases. leads to structures of arbitrary refinement. Failure of the Cauchy-Born rule due to changes in lattice periodicity. E. B. Tadmor QC for Complex Crystals, October 9, 2006 17

Degenerate Shifts Consider a 1D local QC mesh:... Strain can lead to dimerization. Two degenerate symmetry-related positions exist. The lack of interfacial energy in local QC means both can occur: If this region were refined down to the atomic scale and became nonlocal, this would become an anti-phase boundary (APB). E. B. Tadmor QC for Complex Crystals, October 9, 2006 18

Refinement lengthscale Si indentation Notice the fragmented structure with many different phases after unloading. Fully Loaded Fully Unloaded Fully connected Colors indicate degree of connectivity Minimally connected E. B. Tadmor QC for Complex Crystals, October 9, 2006 19

Interfacial Energy SOLUTION: Add on QC-style approximation to the interfacial energy: δ 1 1 F 1 a δ 1 1 F 1 a δ1 2 F 2 a δ1 2 F 2 a 0 interface element 1 element 2 element 3 00 11 00 11 00 11 δ 1 1 F 1 a δ 1 1 F 1 a δ1 2 F 2 a δ 2 1 F 2 a E. B. Tadmor QC for Complex Crystals, October 9, 2006 20

Local QC with Interfacial Energy Local QC formulation with interfacial energy is # of interfaces Shifts in elements adjacent to interface Note that the shifts are now global variables that are minimized along with the displacements. E. B. Tadmor QC for Complex Crystals, October 9, 2006 21

Failure of the Cauchy-Born Rule The Cauchy-Born rule is defined: " # m x = F ( m k i A i + B k ) + δ k, k {1,...,M} where A i are the primitive lattice vectors of the unit cell, and B k are the positions of the basis atoms. This is referred to as the essential description of a multilattice in contrast to a non-essential description employing a lattice supercell larger than necessary (Pitteri (1988), Ericksen (1998)). e.g. for a square lattice: essential non-essential The C-B rule can fail when the periodicity of the minimizing deformation exceeds that of the primitive lattice. E. B. Tadmor QC for Complex Crystals, October 9, 2006 23

Failure of the Cauchy-Born Rule 1D Chain 2-atom chain unit cell... Uniform stretch with shifts that can be described by C-B rule: unit cell periodic cell dimerization E. B. Tadmor QC for Complex Crystals, October 9, 2006 24

Failure of the Cauchy-Born Rule Period doubling (not accessible via C-B rule): unit cell unit cell periodic cell Period tripling (not accessible via C-B rule): unit cell unit cell unit cell periodic cell E. B. Tadmor QC for Complex Crystals, October 9, 2006 25

Failure of the Cauchy-Born Rule Si indentation In the QC simulations we were limited to 2-lattice transformations. In reality some phases are not 2-lattices: E. B. Tadmor QC for Complex Crystals, October 9, 2006 26

Weak Cauchy-Born Rule Failure of the C-B rule can be prevented by applying the rule to a non-essential description of the crystal. This is referred to as the Weak Cauchy-Born rule (Zanzotto, 1992): " # m x = F ³ m k i A c i + B k + δk, k {1,..., M} c cm = nm superlattice Vectors period extension Problem: How can we a priori know how large a supercell is needed? Cannot know this. SOLUTION: perform stability analysis on-the-fly within each element to determine required period Cascading Cauchy-Born Kinematics E. B. Tadmor QC for Complex Crystals, October 9, 2006 27

Stability Analysis u k k u j X 2 B 0T Ω e j Node i T u i Element e The element is in equilibrium with the rest of the structure. B 0 This means its potential energy is at a minimum. B 0U X 1 E. B. Tadmor QC for Complex Crystals, October 9, 2006 28

Stability Analysis u k k u j X 2 B 0T Ω e j Node i T u i Element e QUESTION: If we increase basis size do we remain at a minimum or does it become a saddle point? B 0 B 0U X 1 E. B. Tadmor QC for Complex Crystals, October 9, 2006 29

Cascading Cauchy-Born Kinematics 1. Perform load step 2. Minimize potential energy to find equilibrium positions 3. For all elements: a) Increase basis size M M +1 b) For all distinct non-essential unit cells of current basis: i. Construct Cauchy-Born Hessian K = 2 W/ δ k δ l ii. If any eigenvalue is non-positive adopt current basis and proceed to next element. c) If basis size is less then maximum to be checked goto (a) 4. If any elements had an increase in basis size goto 2 5. Goto 1 E. B. Tadmor QC for Complex Crystals, October 9, 2006 30

Efficient Method for Stability Analysis Consider a periodic supercell large enough to include all non-essential cells being considered. α 0 K " l l 0 α α 0 # = x Ã l α 2 W! x Ã l 0 α 0! 0 l l 0 α supercell l, l 0 {0, 1, 2,...,N} α, α 0 {0, 1, 2,...,M} N = number of essential unit cells in supercell M = number of basis atoms per unit cell E. B. Tadmor QC for Complex Crystals, October 9, 2006 31

Efficient Method for Stability Analysis Due to translational symmetry (with appropriate periodicity): K " l l 0 α α 0 # = K " l l 00 l 0 l 00 α α 0 # = K " 0 l 00 α α 0 # Select l 00 = l Consequently, the stiffness matrix is block circulant : K " l l 0 α α 0 # = l = 0 l 0 = 0 1 2 1 2 3 N A B C D... Z Z A B C D... Y. Z. A. B. C. D. (3M 3M) force constants Note: Due to potential cutoff radius there is a finite radius of interaction: A B C 0 0 E. B. Tadmor QC for Complex Crystals, October 9, 2006 32

Efficient Method for Stability Analysis Discrete Fourier Transform of a block circulant matrix gives a blockdiagonal matrix with blocks of size (3M x 3M): x x x x x x x x x 0 0... 0 x x x x x x x x x 0 0...... 0 x x x x x x x x x Each block corresponds to a particular k-vector, that can be associated with a non-essential unit cell To detect instability, for each k-vector (=non-essential basis) compute the eigenvalues of the 3M x 3M matrix (instead of those of a 3MN x 3MN matrix!) E. B. Tadmor QC for Complex Crystals, October 9, 2006 33

Preliminary Results 1D Chain Model Interfacial Effects: Dimerization Model Fully-atomistic results (uniform strain) Local QC (uniform strain) Loca/Nonlocal QC (uniform strain) Cascading Cauchy-Born: Period Doubling Model Local QC (uniform strain) Local QC (uniform strain + external loads) E. B. Tadmor QC for Complex Crystals, October 9, 2006 34

1D Chain Model Two species of atoms A and B laid out in reference configuration equally spaced, and alternating: A B... They interact according to three Lennard-Jones (LJ) potentials: φ(r) =4²[(σ/r) 12 (σ/r) 6 ] where the parameters depend on the atom types involved: ² AA,² BB,² AB σ AA, σ BB, σ AB r cut = 3.5σ (3-neighbor interactions) E. B. Tadmor QC for Complex Crystals, October 9, 2006 35

LJ Models Two interaction models constructed: (1) LJ-D: dimerization Test interfacial energy correction (2) LJ-P: period-doubling Test cascading C-B kinematics Behavior explored using bifurcation and continuation approach of Elliott: LJ-D LJ-P 0.04 0.03 σ/e 0.02 0.01 0 unstable stable (2-lat) unstable stable (2-lat) 0 0.05 0.1 0.15 0.2 0.25 ² ɛ σ/e 0.01 0.0075 0.005 0.0025 0 unstable unstable stable (4-lat) stable (2-lat) 0 0.05 0.1 0.15 0.2 0.25 ² ɛ E. B. Tadmor QC for Complex Crystals, October 9, 2006 36

1D Dimerization Fully Atomistic [1] Increment uniform strain: [2] Apply square wave perturbation: Perturbation half wavelength This attempts to create dimerization domains [3] Relax, return to [1] until dimerization occurs. E. B. Tadmor QC for Complex Crystals, October 9, 2006 38

1D Dimerization Fully Atomistic Perturbation half-wavelength = 2 unit cells (fully-atomistic) Relaxed Atom Positions Applied Perturbation E. B. Tadmor QC for Complex Crystals, October 9, 2006 39

1D Dimerization Fully Atomistic Perturbation half-wavelength = 2 unit cells (fully-atomistic) Relaxed Atom Positions Applied Perturbation Atom positions less the uniform deformation E. B. Tadmor QC for Complex Crystals, October 9, 2006 40

Field Guide to Dobson Plots NO STRAIN UNIFORM STRAIN DIMERIZATION NO DIMER E. B. Tadmor QC for Complex Crystals, October 9, 2006 41

1D Dimerization Fully Atomistic Perturbation half-wavelength = 10 unit cells (fully-atomistic) Relaxed Atom Positions Applied Perturbation E. B. Tadmor QC for Complex Crystals, October 9, 2006 42

1D Dimerization Fully Atomistic Perturbation half-wavelength = 20 unit cells (fully-atomistic) Relaxed Atom Positions Applied Perturbation This is the critical length at which the dimerization domains are retained. E. B. Tadmor QC for Complex Crystals, October 9, 2006 43

1D Dimerization Study (LJ-D) RIGID Perturbation half-wavelength = 20 unit cells (fully-atomistic) Relaxed Atom Positions Applied Perturbation FLEXIBLE E. B. Tadmor QC for Complex Crystals, October 9, 2006 44

1D Dimerization Fully Atomistic Perturbation half-wavelength = 32 unit cells (fully-atomistic) Relaxed Atom Positions Applied Perturbation E. B. Tadmor QC for Complex Crystals, October 9, 2006 45

1D Dimerization Study Critical Length Below a critical lengthscale dimerization domains are not retained. Why? Can we predict this lengthscale? To help answer consider the steapest-descent minimization sequence for a perturbation below the critical lengthscale: Perturbation half-wavelength = 2 unit cells (fully-atomistic) E. B. Tadmor QC for Complex Crystals, October 9, 2006 46

1D Dimerization Study Critical Length Mechanism for domain absorption: 1. Domain on right is in tension, placing adjacent domain in compression. 2. Compression increases until strain drops below critical value and chain undimerizes. 3. Undimerized chain dimerizes in opposite direction and is absorbed by right domain. Do the two different APB types play a role? E. B. Tadmor QC for Complex Crystals, October 9, 2006 47

1D Dimerization Local QC Local QC study of 1D dimerization Take a series of progressively finer elements: And ask: 4 unit cells = element size 1. Does the QC model reproduce the correct lengthscale? 2. Does the QC model converge to the correct structure with refinement? 2 1 (FULLY REFINED) E. B. Tadmor QC for Complex Crystals, October 9, 2006 48

1D Dimerization Local QC Perturbation half-wavelength = 32 unit cells (local QC) Coarsest possible model Fully Atomistic Local QC (size = 16 unit cell) Local QC reproduces the correct lengthscale. E. B. Tadmor QC for Complex Crystals, October 9, 2006 49

1D Dimerization Local QC Perturbation half-wavelength = 32 unit cells (local QC) Fully Atomistic Local QC (size = 8 unit cell) E. B. Tadmor QC for Complex Crystals, October 9, 2006 50

1D Dimerization Local QC Perturbation half-wavelength = 32 unit cells (local QC) Fully Atomistic Local QC (size = 4 unit cell) E. B. Tadmor QC for Complex Crystals, October 9, 2006 51

1D Dimerization Local QC Perturbation half-wavelength = 32 unit cells (local QC) Fully Atomistic Local QC (size = 2 unit cell) E. B. Tadmor QC for Complex Crystals, October 9, 2006 52

1D Dimerization Local QC Perturbation half-wavelength = 32 unit cells (local QC) Fully refined Fully Atomistic Local QC (size = 1 unit cell) Fully-refined local QC reproduces exact atomistic results! E. B. Tadmor QC for Complex Crystals, October 9, 2006 53

1D Dimerization Local/Nonlocal QC Local/Nonlocal QC study of 1D dimerization Local QC reproduces the correct periodicity and converges to the correct APB structure with refinement However, the APB strucutre is poor in coarse descriptions. Can accuracy be improved by incorporating a fully-atomistic region (=nonlocal QC) around APB? LOCAL NONLOCAL LOCAL E. B. Tadmor QC for Complex Crystals, October 9, 2006 54

1D Dimerization Local/Nonlocal QC Fully Atomistic Local/Nonlocal QC Cyan regions are local 1 element 1 element E. B. Tadmor QC for Complex Crystals, October 9, 2006 55

1D Dimerization Local/Nonlocal QC Fully Atomistic Local/Nonlocal QC 1 element 1 element E. B. Tadmor QC for Complex Crystals, October 9, 2006 56

1D Dimerization Local/Nonlocal QC Fully Atomistic Local/Nonlocal QC 1 element 1 element E. B. Tadmor QC for Complex Crystals, October 9, 2006 57

1D Dimerization Local/Nonlocal QC Fully Atomistic Local/Nonlocal QC 1 element 1 element E. B. Tadmor QC for Complex Crystals, October 9, 2006 58

1D Dimerization Local/Nonlocal QC Fully Atomistic Local/Nonlocal QC 1 element 1 element E. B. Tadmor QC for Complex Crystals, October 9, 2006 59

1D Dimerization Local/Nonlocal QC Fully Atomistic Local/Nonlocal QC 1 element 1 element E. B. Tadmor QC for Complex Crystals, October 9, 2006 60

1D Dimerization Local/Nonlocal QC Fully Atomistic Local/Nonlocal QC 1 element 1 element E. B. Tadmor QC for Complex Crystals, October 9, 2006 61

1D Dimerization Local/Nonlocal QC Fully Atomistic Local/Nonlocal QC 1 element 1 element E. B. Tadmor QC for Complex Crystals, October 9, 2006 62

1D Dimerization Local/Nonlocal QC Fully Atomistic Local/Nonlocal QC 1 element E. B. Tadmor QC for Complex Crystals, October 9, 2006 63

1D Period Doubling Local QC Local QC study of 1D period doubling Take a series of progressively finer elements: And ask: 4 unit cells = element size 1. Does the QC model reproduce the correct lengthscale? 2. Does the QC model converge to the correct structure with refinement? 2 1 (FULLY REFINED) E. B. Tadmor QC for Complex Crystals, October 9, 2006 65

1D Period Doubling Local QC Perturbation half-wavelength = 32 unit cells (local QC) Local QC (size = 8 unit cell) Fully Atomistic Fully-refined local QC reproduces exact atomistic results! E. B. Tadmor QC for Complex Crystals, October 9, 2006 66

1D Period Doubling Local QC Perturbation half-wavelength = 32 unit cells (local QC) Local QC (size = 4 unit cell) Fully Atomistic E. B. Tadmor QC for Complex Crystals, October 9, 2006 67

1D Period Doubling Local QC Perturbation half-wavelength = 32 unit cells (local QC) Local QC (size = 2 unit cell) Fully Atomistic E. B. Tadmor QC for Complex Crystals, October 9, 2006 68

1D Period Doubling Local QC (forces) [1] Chain is subjected to two force pairs (not strong enough to cause period doubling) 256 atoms [2] Apply uniform strain increment The objective of the forces is to introduce some non-uniformity along the chain [3] Relax, return to [2] until period-doubling occurs. This is performed both fully atomistically and with local QC. Element size = 4 unit cells. E. B. Tadmor QC for Complex Crystals, October 9, 2006 69

1D Period Doubling Study Fully Atomistic STRAIN = 0 Local QC Negative slope = compression positive slope = tension E. B. Tadmor QC for Complex Crystals, October 9, 2006 70

1D Period Doubling Study Fully Atomistic STRAIN = 0.02 Local QC E. B. Tadmor QC for Complex Crystals, October 9, 2006 71

1D Period Doubling Study Fully Atomistic STRAIN = 0.04 Local QC E. B. Tadmor QC for Complex Crystals, October 9, 2006 72

1D Period Doubling Study Fully Atomistic STRAIN = 0.056 Local QC E. B. Tadmor QC for Complex Crystals, October 9, 2006 73

1D Period Doubling Study Fully Atomistic STRAIN = 0.058 Local QC Local QC reproduces period doubling observed in the exact system. E. B. Tadmor QC for Complex Crystals, October 9, 2006 74

Conclusions The standard local QC approach has limitations: Deformations involving extensions to the periodicity of the lattice are excluded by the standard Cauchy-Born rule. Interfacial energy is not considered (no penalty for multiple phases, no lengthscale) Note that although we have considered these limitations in the context of complex lattices they also exist in simple lattices. These limitations were addressed: Period extension is made possible through Cascading Cauchy-Born Kinematics. Atomistically-based estimate for Interfacial energy included in formulation. The results in 1D tests are promising: Fully-refined local model converges to exact solution. Method reproduces dimerization domain size and period doubling behavior of exact system. Accurate APB structures obtained by local-nonlocal approach. E. B. Tadmor QC for Complex Crystals, October 9, 2006 75

Remaining Issues and Future Directions APBs Analytical estimate for critical lengthscale Rigid vs. Flexible APB structure Mesh Refinement criterion Interplay with period doubling Localization Captured by cascading Cauchy-Born? Extension to higher dimensions Fully-refined local Fully-atomistic? Calculation of interfacial energy E. B. Tadmor QC for Complex Crystals, October 9, 2006 76