Long-Term Atomistic Simulation of Heat and Mass Transport

Transcription

1 Long-Term Atomistic Simulation of Heat and Mass Transport Kevin G. Wang 1, Mauricio Ponga 2, Michael Ortiz 2 1 Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University 2 Division of Engineering and Applied Sciences, California Institute of Technology

2 Motivation Slow deformation-diffusion coupled processes hydrogen embrittlement in fuel cells initial after 10 cycles Li et al, JACS (2007) Mg nanowires collapse after 10 cycles of charging/discharging; temperature: 300K ~ 600K; time scale: seconds to hours!

3 Overview Long-term atomistic simulation of heat and mass transport - the essential difficulty: multiple disparate time scales atomic level rate-limiting processes: thermal vibrations, individual hops of atoms macroscopic processes of interest: hydrogen embrittlement, ductile fracture, fatigue, etc. time-scale gap: from molecular dynamics (femtoseconds) to to macroscopic (seconds, minutes, )! - modeling approach fine scale: non-equilibrium statistical thermodynamics coarse scale: discrete kinetic laws - previous work: Kulkarni et al ( HotQC ), JMPS (2008), Venturini et al, JMCE (2012), Ariza et al, IJF (2012)

4 Non-Equilibrium Thermodynamics Statistical thermodynamics for multi-species crystals - N particles, M species occupancy function 1, particle i is from species k for i = 1,, N, k = 1,, M, n ik = 0, oterwise - assumption: separation of time scales fine scale: thermal vibration (~fs) modeled coarse scale: global relaxation (>>fs) solved for probability: q, p, *n+ ~ p q, p, *n+, - maximum entropy principle: max p with S p = k B log p (information entropy) 1 f = 3N f q, p, *n+ p q, p, *n+ dqdp n O Γ NM subject to: H i = E e i i = 1,, N (atomic internal energy) nn ik = x ik = N k i = a 1,, N (atomic molar fraction) i<1 n ik

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6 Non-Equilibrium Thermodynamics Meanfield approximation (Yeomans 1992) - given a class of trial Hamiltonian parameterized by π : 0i ( p, q, *n+; π ), the trial probability density function can be defined as p 0 q, p, n; π ; β, *γ+ = 1 Ξ 0 exp β T 0 + γ T n - for fixed Lagrange multipliers β, *γ+, the optimal probability of the trial space *p 0 + is obtained by maximizing the constrained entropy with respect to π, i.e. p 0 = argmax *π+ L, p 0 π, β, *γ+ - - theorem (bounding principle for the free entropy): Φ *β+, *γ+ max Φ( 0, β, *γ+) h 0 the maximum free entropy of the trial space provides a lower bound of the actual free entropy p 0 converges to p as 0

7 Non-Equilibrium Thermodynamics Meanfield model: an example - in practice there is always a trade-off between accuracy and computability - we define the trial Hamiltionian as 0i q, p ; q, p, m, ω = 1 p 2m i p 2 i + m 2 iω i q i 2 i q 2 i *π+ uncoupled harmonic oscillators N p 0 ( q, p, *n+) = 1 1 exp β Ξ i p 0 2m i p 2 i + m 2 iω i q i 2 i q 2 i i<1 Gaussian distribution p 0 = argmax L,p 0, β, *γ+ - p i, q i, ω i,*m i + meanfield parameters: q i (mean atomic position), p i (mean atomic momentum), ω i (frequency of oscillation), m i (mean atomic mass) i = p p 0 2 2m + V i(*q+) explicit formulations of S, Φ, F,...

8 Modal Validation Thermal expansion coefficient: Mg single crystals - at fixed, uniform temperature, maximizes grand-canonical free entropy (Φ) with respect to atomic frequency (ω i ) and mean atomic position (q i ) q = argmin q,*ω+ Φ( q, ω, T) α V T = 1 V 0 V T 1 V 0 V 1 V 0 T 1 T 0 (volumetric thermal expansion coefficient) - comparison with experiments temperature simulation experimental rel. error (K) ( 10 ;6 ) ( 10 ;6 data source ) (%) Goens & Schmid (1936) Raynor & Hume-Rothery (1939) Goens & Schmid (1936) Hanawalt & Frevel (1938) Raynor & Hume-Rothery (1939) Raynor & Hume-Rothery (1939) 14.53

9 Modal Validation Nickel-Palladium lattice parameter - problem setup: FCC random solution of nickel (Ni) and palladium (Pd) 365 lattice sites; periodic boundary condition temperature and molar fraction are prescribed x Ni varies from 0 (pure Pd) to 1 (pure Ni) equilibrium lattice parameter a obtained by relaxing *q+ and *ω+ to maximize grand canonical free entropy uniform molar fraction (x Ni, x Pd ), uniform temperature T

10 Detailed balance of energy - balance of energy at a particle u i = w i + μ T i x i + q i Discrete Kinetic Laws (w i = v h i A A α α α<1 is the external power) requires kinetic models for x i and q i - assume x i and q i can be divided into particle-to-particle fluxes of the form with - assume linear kinetics Q ij = κ ij (β i β j ) J ij = D ij (γ i γ j ) κ ij : pairwise conductivity coefficient D ij : pairwise diffusivity coefficient - question: how to determine κ ij and D ij?

11 A simple example Discrete Kinetic Laws - consider only pairwise flux for only the nearest neighbors (based on a perfect crystal lattice) J ij = D ij γ i γ j, if i and j are nearest neigbors 0, oterwise - require consistency with Fick s law in continuum mechanics (in a finite-difference sense) Fick s second law: x = D 0 (x γ) discrete diffusion model: x i = D ij (γ i γ j ) j Nei(i) assume x, γ C 2 (R 3 ), x i = D 0 x i γ i + O(a 2 ) iff D ij = D 0 2a 2 (x i + x j ) (for fcc and bcc lattices) - can be extended to more than one layers of neighbors - the same procedure can be applied to heat conduction

12 Summary Thermo-chemo-mechanical coupled analysis given m n;1, q n;1, p n;1, ω n;1, β n, *x n + [1] assume trial Hamiltonian H 0 q, p ; π with π i = (m i, q i, p i, ω i ) [2] obtain optimal π n by maximizing free entropy: max m i, p i, q i, *ω i + L,ρ 0, β, γ - [5] solve kinetic equations and obtain β n:1, *x n:1 + [3] extract quantities of interest e.g. S n, Φ n, E n, [4]

13 Model Validation Heat conduction in Si nanowires - experiment (D. Li et al. 2003) specimen: single-crystal Si (111) nanowires R = 11, 18.5, 28, 57.5 nm measurement: thermal conductivity main discovery: significant size effect HRTEM image of a 22 nm SiNW

14 Model Validation Computational model and simulations - consider a slice of the SiNWs - create amorphous layer by random perturbation - pair-wise thermal conductivity: thermal conductivity amorphous layer atomic temperature

15 Model Validation Hydrogen diffusion in Pd nanofilms - experiment (Y. Li and Y.-T. Cheng, 1996) specimen: Pd (111) thin films with thickness L = 22 nm, 46 nm, 135 nm method: electrochemical stripping step 1: introduce H to Pd by cathodic polarization (set ΔV = 0.83V) Pd + H 2 O + e ; Pd H ads + OH ; step 2: remove H from Pd-H by anodic stripping (switch ΔV to 0.5V) Pd H ads + OH ; Pd + H 2 O + e ; l electrolyte solution substrate: Au-coated Ni Pd film measurement: time history of discharge current at the surface of Pd film C l, t I t = FS J H (l, t) = FSD H = FSD HZ x l, t l l<0 N A V cell l l<0

16 Models and simulations Model Validation - simplification of thermodynamic model 1D, constant temperature, fixed Pd lattice linearization of Pd-H interatomic potential equilibrium condition γ i = log x i 1 x i + - discrete Fick s law x i = D ij (γ j γ i ) j Nei(i) C ij k B T x j + B i k B T j I H,j i x i - highlights fully atomistic t max = 1.0 sec, dt ~ 10-6 sec L = 46 nm L = 135 nm

17 26nm 5nm Model Verification Nanovoid growth under tension (coupled with QC) - problem setup objective: study the evolution of nanovoid in a single FCC crystal of Cu representative volume: 26nm x 26nm x 26nm (~ 10 6 atoms) initial void diameter: 2nm initial atomic zone: 5nm x 5nm x 5nm (12a 0 x 12a 0 x 12a 0 ) interatomic potential: EAM-Mishin (Y. Mishin et al. 2001) uniaxial and triaxial loading with prescribed boundary displacement - strain rate (1/s): 10 5, 10 7, 10 8, 10 9, atomic zone void QC approximation cut-view of initial FE mesh

18 Nanovoid Growth Uniaxial loading, T = 300K, ε = /s 1/6,112-1/6,112-1/6,112-1/6,112- ε = 6.6% (emission of dislocation) ε = 6.8% (leading Shockley partials) atomic temperature ε = 7.0% (trailing Shockley partials) *111+ dislocation core identified by CSD* * centro-symmetric deviation: CSD = N 2 i<1 q i + q i: N 2 a 0 2 before dislocation emission after dislocation emission

19 virial stress (GPa) Nanovoid Growth Tri-axial loading, T = 300K, ε = 10 5, 10 8, 10 9, /s stress-strain response 10 dislocation core colored by CSD (ε = /s) - virial stress (Allen and Tildesley, 1989) ς ij = 1 V Ω m k q i k q i k q j k q j k q i l q i k f j kl k Ω ε = 10 5, ε = 10 8, ε = 10 9, ε = ε, = HotQC, 10 10, dynamic ε = ε, = MD 10 10, MD ref engineering strain

20 Summary and Future Work Long-term atomistic simulation of heat and mass transport - non-equilibrium statistical thermodynamics model maximum entropy principle meanfield approximation - discrete kinetic laws discrete Fick s law discrete Fourier s law - coupling: a partitioned procedure - validations and applications Future work - computational methods for thermo-chemo-mechanical coupling - design and validation of discrete kinetic laws - validations and applications Reference - Venturini, Wang, Romero, Ariza, and Ortiz, JMPS (submitted) Acknowledgement: Army Research Lab (ARL)