of Long-Term Transport

Transcription

1 Atomistic Modeling and Simulation of Long-Term Transport Phenomena in Nanomaterials K.G. Wang, M. and M. Ortiz Virginia Polytechnic Institute and State University Universidad de Sevilla California Institute of Technology

2 Motivation - Diffusion hydrogen storage Slow deformation-diffusion coupled processes G.E. Froudakis, Materials Today, 2011 V. Tozzini and V. Pellegrini, Phys. Chem. Chem. Phys., 2013

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

4 Motivation Experiments, A. Lakatos et al. Vacuum (2010) Molecular Dynamics, APPLIED OPTICS (2012) NEED for time coarsening NEED for spatial coarsening Wanted: Cavitation pressure as a function void size, strain rate and temperature!

5 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 storage, hydrogen embrittlement, ductile fracture, fatigue, etc. time-scale gap: from molecular dynamics (femtoseconds) 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), Ponga et al, PRB (2014)

6 Non-Equilibrium Thermodynamics Statistical thermodynamics for multi-species crystals - N particles, M species occupancy function for i = 1,, N, k = 1,, M, n ik = - 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 S p = k B log p p 1 with f = h 3N n O Γ NM 1, particle i is from species k 0, otherwise (information entropy) f q, p, {n} p q, p, {n} dqdp subject to: h i = e i i = 1,, N (atomic internal energy) n ik = x ik i = 1,, N (atomic molar fraction) pa qi pi p1 p3 p2 Position Momentum

7 Non-Equilibrium Thermodynamics Thermodynamic potentials - solve the constrained optimization problem using Lagrange multipliers: L p; β, γ = S p k B β T h + k B γ T { n } ( constrained entropy ) p q, p, n ; β, {γ} = 1 Ξ exp β T h + γ T n with Ξ {β, {γ}) = n O NM e β T h γ T{n} dpdq S = k B log ρ = k B log Ξ + β T e + γ T {x} Φ β, γ = sup L ρ; β, γ = k B log Ξ( β, γ ) ρ β i = 1 S particle temperature: T k B e i = 1 i k B β i (total entropy) (grand-canonical free entropy) γ ik = 1 S particle chemical potential: μ ik = γ ik = k k B x ik β B T i γ ik i - however, p, Ξ, S, and Φ all depend on Hamiltonian h i = h i ( p, q, n ). in general, the closed-form formulation p, Ξ, S, and Φ are intractable.

8 Non-Equilibrium Thermodynamics Meanfield approximation (Yeomans 1992) - given a class of trial Hamiltonian: h 0i ( p, q, {n}), the trial probability density function can be defined as - for fixed Lagrange multipliers β, {γ}, the optimal probability of the trial space {p 0 } is obtained by maximizing the constrained entropy, i.e. - theorem (bounding principle for the free entropy): the maximum free entropy of the trial space provides a lower bound of the actual free entropy p 0 converges to p as h 0 h

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

10 Detailed balance of energy Discrete kinetic laws - balance of energy at a particle T u i = w i + μ i x i + v ( w i = α=1 requires kinetic models for x i and q i h i A α A α is the external power) - assume x i and q i can be divided into particle-to-particle fluxes of the form - assume linear kinetics Q ij = κ ij (β i β j ) J ij = D ij (γ i γ j ) q i D ij : pairwise diffusivity coefficient with κ ij : pairwise conductivity 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 neighbors 0, otherwise - require consistency with Fick s law in continuum mechanics (in a finite-difference sense) Fick s second law: discrete diffusion model: assume x, γ C 2 (R 3 ), x i = D 0 x i γ i + O(a 2 ) x = D 0 (x γ) x i = j Nei(i) D ij (γ i γ j ) 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 SMDS2014

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 SMDS2014

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 Pd film substrate: Au-coated Ni 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 Model Validation Models and simulations - simplification of thermodynamic model 1D, constant temperature, fixed Pd lattice linearization of Pd-H interatomic potential Zhou et al. J. Mater. Res., Vol. 23, 3 (2008) γ i = log x i 1 x i + - discrete Fick s law x i = - highlights equilibrium condition j Nei(i) j I H,j i D ij (γ j γ i ) x ij fully atomistic t max = 1.0 sec, dt ~ 10-6 sec C ij k B T x j + B i k B T mass transport restricted to nearest neighbors discretized in time by the mid-point rule L = 46 nm L = 135 nm

17 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 References - Venturini, Wang, Romero, Ariza, and Ortiz, JMPS (submitted) - Wang, Ortiz and Ariza, Int. J. Hydrogen Energy (submitted)

18 Atomistic Modeling and Simulation of Long-Term Transport Phenomena in Nanomaterials K.G. Wang, M. and M. Ortiz Virginia Polytechnic Institute and State University Universidad de Sevilla California Institute of Technology