Density Functional Modeling of Nanocrystalline Materials
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1 Density Functional Modeling of Nanocrystalline Materials A new approach for modeling atomic scale properties in materials Peter Stefanovic Supervisor: Nikolas Provatas 70 / Part 1-7 February 007
2 Density Functional Modeling of Nanocrystalline Materials Classical density ( DFT ) as a new approach for modeling atomic scale properties in materials Incorporating dynamic elastic interactions Binary alloys & Extension to three dimension for direct coupling to molecular dynamic
3 Continuum Models of Materials FEM model of a polycrystalline microstructure (Bouaziz, lecture notes) Dendritic growth (Provatas) Phase field models Dislocations in phase separating binary alloys (Haataja, Mahon, Provatas, APL 86, 005) Common factors: Continuum representation of the matter Limited information about processes on the lowest scales Atomistic processes represented by effective parameters 3
4 Molecular Dynamics Simulations Computational 16 grains, 100,000 atoms Average grain diameter - 5. nm To simulate HPR - large size model required (up to 100 million of atoms sample in nm ) MD simulation High strain rate of 5*10 8 s -1. Time scale is set by atomic motion. Short timescales of nanoseconds are accessible Strain rates extremely high (>10 7 s -1 1% strain in 1ns) Schiøtz, Tolla, Jacobsen, Nature 391, (1998) 4
5 Models Comparison Computational MD simulations Only short time scales are accessible (pico-nano seconds) Sample size up to 100 million of atoms sample in nm Detailed information about atomistic processes Continuum models Limited information about processes on the lowest scales High length scales accessible Longer time scale accessible How to bridge two approaches? Our answer: Classical (DFT) 5
6 Project Objectives Project objectives To develop a new hybrid MD - phase field approach based on Density Functional Theory (DFT) To study elastic and plastic effects in phase transformations and microstructure growth To validate our approach against one or more experimental systems 6
7 Introduction to z Kinetic energy y atom position For each atom to calculate Newton s second law: d ri Fi = mr & i = m dt 1 1 E k = mr& = p m Atomistic Simulations x r i i th atom m - mass p momentum (mass x velocity) Total energy (Hamiltonian) k [U] ({ N p } { r N }) E = E + U = Η, Potential energy r r location of the atom simplest example of a hard sphere potential is the Lennard Jones potential 7
8 Introduction to Ergodic Hypothesis Time average (MD) Ensemble (configuration) average = Thermodynamic value is calculated as a time average over a number of simulation steps 1 A = t t 0 t + t 0 N ( τ ), p ( τ ) ( N ) A r dτ Helmholtz free energy: A Z cl Z cl cl = kt ln = 1 N! h N 3N N = 0 i= 1 e β ( H µ N ) Integral is difficult to evaluate for interacting systems dx dp i i 8
9 Introduction to Atomic Probability Density Hard-sphere representation of a fluid Hard-sphere representation of a solid Instantaneous density Density Instantaneous density Density Time averaged atomic density of liquid (constant) x t=1 t= t=3 t=1 t= t=3 (periodic) Time averaged atomic density of solid x 9
10 Introduction to Atomic Density Solid Liquid interface in TEM Solid Liquid interface - detail SOLID LIQUID SOLID LIQUID MRS Bulletin, Dec 004, pp Atomic density profile across the interface 10
11 Introduction to Direct Correlation Functions Provide a description of a particle dynamics Can be readily found by molecular dynamics simulation Peaks of c(k) represent physical properties of solid (isothermal compressibility, bulk modulus and lattice constant) Particle Initial position at t 1 r 1 r Control volume- V Path of a particle Particle final position at t C(k) 15 c (k ) Correlation function for Copper Akusti Jaatinen, Master's Thesis, Helsinki University of Technology, Espoo, 006 k k 11
12 Introduction to F k T B = r dx ρ dx 1 dx 1 Free Energy Functional dx r ρ ln ρ dx ρ dx ( r ) r Non-interacting r ( r ) part δρ i.e. in ideal gas ( r ) Interacting part ( ) ( x c ) ( x, x, ρ, T ) ρ( x ) 3 l 1 ρ ( ) ( ) ( 3 x ρ x c ) ( x, x, x, ρ, T ) ρ( x ) 1 1 (Two particles interaction) r F r r ρ ( ) ( ) r = dx ln k T ρ δρ B ρl 1 dx1dxδρ dx1dxdx3δρ 1 6 (Three particles interaction) ( r ) ( ) ( x c ) ( x, x, ρ, T ) δρ( x ) L 1 ( ) ( ) ( 3 x ρ x c ) ( x, x, x, ρ, T ) δρ( x ) L R. Evans, Adv. Phys., Vol.8, (1979) L L 3 3 1
13 Free energy F k T B = where and r r dx ρ n = F = F ρk T B ( r ) Free Energy Functional r ρ ln ρ ( r ) ( ρ ρ ) F ρ liquid l liquid Fliquid r δρ 1 ( ) ( ) ( r dx dx δρ x c ) ( x, x, ρ T ) δρ( x ) c(k) [A 3 ] Simplified free energy L, Approximation by polynomial fit ˆ ˆ C = C0 + Ck ( ˆ ˆ + ˆ ) 4 C C C ˆ Cˆ k 4 4 k [1/A] 3 4 r 1 ρ 0 4 n n = dx n n Elder, Provatas, Berry, Stefanovic, Grant, PRB 75, (007) 13
14 Dynamics Dynamics with Elastic Interactions Fast propagating modes (early times) ρ t α - sound speed Slow propagating modes (late times) ρ β t + = β α D Equation is in the form of damped wave Contains two propagating density modes α Instantaneous elastic interactions (first term) Diffusive mode at late times (second term) Chemical potential δ [ ρ ] F ;T δρ - vacancy diffusion coefficient Stefanovic, Haataja, Provatas, Phys. Rev. Lett. 96, 5504 (006) 14
15 Dynamics Solidification of Copper ρ ρ + β = α t t Solid Liquid n ( ) 4 ˆ n n 1 C n + ρ l c(k) [A 3 ] Cu = 0.076A T = 1360K ρ l 3 Correlation fn. for Cu k [1/A] 15
16 Elastic Response in a Solid Bar Elastic interactions Load Displacement[x] D 1 D D N D - Displacements Deformed state Initial state α =15 = 0.9 β Linear response according to Hook's law (three different times/loads) x[a x ] 17 16
17 Elastic Response in a Solid Bar Elastic interactions Displacement[x] α = Ten fold increase in damping β = Non-Linear Visco-Elastic response (three different times/loads) x[a x ] 17 17
18 Elastic Interactions Dislocation Velocity Elastic interactions A portion of the sample used to examine dislocation glide velocity. 18
19 Elastic Interactions Dislocation Velocity Elastic interactions Dislocation velocity [ax/t] Experiment setup Dislocation glide velocity vs. applied strain rate. v Measured dislocation velocity Theoretical dislocation velocity (Orowan equation) D = & γ r ρ b D 0.05 b burgers vector (b= a x ) ρ D dislocation density (ρ D =.83x10-4 a - x ) 0 0.E+00.E-05 4.E-05 Applied Shear 6.E-05Rate [1/t] 8.E-05 γ& 19
20 Elastic interactions Elastic Interactions Dislocation Velocity Average strain in crystal [-] Experiment setup D is lo c a tio n v e lo c ity [a x /t] Measured dislocation velocity Theoretical dislocation velocity (Orowan equation) v D & γ = r ρ b D 0 0.E+00.E-05 4.E-05 Applied Shear 6.E-05Rate [1/t] 8.E-05 Dislocation glide velocity Average shear strain during deformation Applied strain rate = 8E-5 (Continuous dislocation glide) Applied strain rate = 3.3E-5 (Stick-slip dislocation glide) Applied strain Initial shear strain (single dislocation) Time [t] 5000 γ& 0
21 Elastic Interactions Atomistic Tensile Test Elastic interactions Strain concentration in a double notched sample under a uniaxial tension. Strain profile from the center of the sample into the root of the notch. Solid line is a guide to the eye. 1
22 Elastic Interactions Atomistic Tensile Test Strain concentration in a double notched sample under a uniaxial tension. Dynamics - Elastic interactions 3D Efficient solver &
23 Binary alloys Binary Alloys Free energy of a binary alloy made up of A and B atoms Correlation function now represents interactions between A and B atoms Dissipative dynamics driven by free energy minimization Elder, Provatas, Berry, Stefanovic, Grant, PRB 75, (007)
24 Spinodal Decomposition in a Single Crystal Binary alloys Dimensionless Temperature (T-T m )/T m Phase diagram A Liquid Solid C B Spinodal line 3
25 Spinodal Decomposition in a Single Crystal Concentration field - spinodal decomposition in a single crystal Binary alloys Amplitude Gaussian fit Characteristic wavelength log (wave length) slope = 1/3 x Wavelength D structure factor fitted with a Gaussian curve log (time) Evolution of the characteristic wavelength in time 4
26 Spinodal Decomposition and Dislocations Binary alloys t =48μs t =96μs t =40μs t =100μs Time sequences in the evolution of the concentration field Dislocations are labeled by a square on the dislocation core surrounded by a circle. The system size: 4nm x 4 nm 5
27 Spinodal Decomposition Dislocation is attracted to the phase boundary to relieve coherent strains Phase A Coherent Interface Phase B - Elastic interactions Binary alloys 3D Efficient solver & 7
28 Spinodal Decomposition and Dislocations Binary alloys Wavelength Inverse of the mean wave vector of the D structure factor of the concentration field vs. time Drag due to dislocations time 6
29 Rapid Solidification of Binary Alloys Binary alloys Size of the sample is 1000x1000 atoms (50nm) 7
30 Rapid Solidification of Binary Alloys Binary alloys c Microsegregation Microsegregation profile Side Branches Main Branch x 8
31 Extension to 3D Why go to 3D? More realistic representation of a material Some processes not manifested in two dimensions A single component crystal BCC structure The edge size is 64dx 3D Size of the sample was limited by a code performance Need for efficient PDE solver 9
32 - Elastic interactions 3D Efficient solver & Efficient Solver Iterative Methods Poisson equation: Discrete Poisson equation: Equilibrate low frequency modes on fine mesh inefficient (many iterations needed) Quickly equilibrate low frequency modes Then return to the fine mesh for only for few iteration h u h u = f Ω ( x, y) = fh ( x, y) ( x, y) Ωh 5 point Jacobi type iteration step 1 h uh h h h, h ( x, y) = [ 4u ( x, y) u ( x h, y) u ( x + h y) ( x, y h) u ( x y h)] u, h h + Fine mesh Coarse mesh 30
33 Multigrid in D Mesh Spacing h Four level multigrid grid Density function in a multigrid representation Efficient solver h 4h 8h Access to bigger samples Solve longer time scales faster 30
34 Summary Presented a new hybrid MD - phase field approach based on Density Functional Theory (DFT) Examined elastic effects using new model with elastic interactions Applied new DFT model to binary systems and compared with experimental and theoretical data Initiated work on efficient solver of PDE using a multigrid method To validate the approach against one or more experimental systems 31
35 Acknowledgements: Nikolas Provatas Mikko Haataja Ken Elder Materials Science Group at McMaster (Michael Greenwood, Juan Kong, Jianghua Li, Chaohui Tong, Alex Tetervak, Nana Ofori-Opoku, Tao Wu) More experimental results available at: 3
36 Structure factor in time
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