Long-Term Atomistic Simulation of Hydrogen Diffusion in Metals

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1 Long-Term Atomistic Simulation of Hydrogen Diffusion in Metals Kevin Wang (1), Pilar Ariza (2), and Michael Ortiz (3) (1) Department of Aerospace and Ocean Engineering Virginia Polytechnic Institute and State University (2) Escuela Técnica Superior de Ingeniería Universidad de Sevilla, Spain (3) Division of Engineering and Applied Science California Institute of Technology

MOTIVATION Hydrogen energy in aerospace and ocean engineering

attractive attributes long endurance near-silent operation ease

hydrogen fuel cell vehicles would increase mission endurance and

2 MOTIVATION Hydrogen energy in aerospace and ocean engineering Type 212 submarine (Germany) UAVs submarines and UUVs - attractive attributes long endurance near-silent operation ease of control air-independent propulsion (for submarines and UUVs) - hydrogen fuel cell vehicles would increase mission endurance and stealth, while reducing logistical burdens and costs -- office of naval research

challenge: slow deformation-diffusion coupled process at atomic scale initial Barnoush, Hydrogen Embrittlement,

3 HYDROGEN STORAGE Metal hydride-based hydrogen storage (MgH 2, PdH, LiH, etc.) - advantages: high volume density, safety - topic of interest: design and fabrication of nanomaterials - challenge: slow deformation-diffusion coupled process at atomic scale initial Barnoush, Hydrogen Embrittlement, 2011 after 10 cycles Li et al, JACS (2007) Mg nanowires collapse after 10 cycles of charging/discharging; length scale: 1 nm; time scale: seconds to hours!

MULTISCALE FRAMEWORK Time-scale gap: from femtoseconds to hours experiments simulations molecular dynamics, APPLIED OPTICS (2012) A. Lakatos et al. Vacuum (2010) ab initio MD J.

4 MULTISCALE FRAMEWORK Time-scale gap: from femtoseconds to hours experiments simulations molecular dynamics, APPLIED OPTICS (2012) A. Lakatos et al. Vacuum (2010) ab initio MD J. Molecular Liquids (2012) - atomic level rate-limiting processes: thermal vibration, individual hops of atoms (fs) - macroscopic processes of interest: hydrogen diffusion, embrittlement, etc. (s h) - multiscale model for long-term atomistic simulation fine scale: non-equilibrium statistical thermodynamics coarse scale: discrete kinetic laws

5 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, otherwise - assumption: separation of time scales fine scale: thermal vibration (~fs) modeled p1 p2 coarse scale: global relaxation (>>fs) solved for probability: q, p, {n} ~ p q, p, {n}, - maximum entropy principle: max p with subject to: S p = k B log p 1 f = h 3N n O Γ NM (information entropy) f q, p, {n} p q, p, {n} dqdp H h i = Ee i i = 1,, N (atomic internal energy) n N ik = n ik x ik = Ni k = 1,, N (atomic molar fraction) i=1 pa qi pi p3 Position Momentum

6 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 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 γ ik = 1 k B S x ik (total entropy) (grand-canonical free entropy) particle chemical potential: μ ik = γ ik β i = k B T i γ ik - 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

7 NON-EQUILIBRIUM THERMODYNAMICS Meanfield approximations - we define trial Hamiltionians of the form h 0i q, p ; q, p, m, ω = 1 p 2 m i p 2 i + m 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 meanfield parameters: q i (mean atomic position), p i (mean atomic momentum), ω i (frequency of oscillation), m i (mean atomic mass) 2 q i q i 2 given atomic temperature {β} molar fraction {x}, the local equilibrium states { q i, p i, ω i, m i } can be uniquely determined q i q i 2

8 DISCRETE KINETIC LAWS Detailed balance of energy - balance of energy at a particle u i = w i + μ i T x i + q i v h ( w i = i α=1 A A α is the external power) α requires kinetic models for x i - assume x i can be divided into particle-to-particle fluxes of the form with - assume linear kinetics J ij = D ij (γ i γ j ) particle-to-particle mass fluxes D ij : pairwise diffusivity coefficient - enforce consistency with Fick s law for continuum D ij = D 0 2a 2 (x i + x j ) (for fcc and bcc lattices) Venturini, Wang, Romero, Ariza, Ortiz, JMPS (accepted)

9 APPLICATION Long-term atomistic simulation of mass and heat transport nanovoid growth in Cu Ariza, et al. IJ Fracture (2013) twin boundary migration in Mg Wang, Ramabathiran, Ortiz (in prep) heat conduction in SiNW Venturini, Wang, et al. JMPS (accepted)

10 APPLICATION 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

Computational model APPLICATION FCC Pd unit cell - 1D, constant temperature (300K) - meanfield positions of Pd atoms are fixed - linearization of Pd-H interatomic potential local

11 Computational model APPLICATION FCC Pd unit cell - 1D, constant temperature (300K) - meanfield positions of Pd atoms are fixed - linearization of Pd-H interatomic potential local equilibrium condition γ i = log x i 1 x i + j I H,j i C ij k B T x j + B i k B T - diffusion of hydrogen: discrete kinetic law x i = j Nei(i) D ij (γ j γ i ) x i 1D computational domain two phases!

12 MODEL VALIDATION Validation for H diffusion in α-phase PdH electolyte Dirichlet b.c. x = Pd film: x 0 = substrate Neumann b.c. flux = 0 - total simulation time: 1 sec. - time-step: ~10-6 sec. - fully atomistic Wang, Ortiz, Ariza, IJ Hydrogen Energy (submitted)

13 H molar fraction H molar fraction PHASE SEPARATION Propagation of α/β phase boundary 1 electolyte Dirichlet b.c. x = 0.99 classical model (Fick s 2 nd law) Pd film: x 0 = substrate Neumann b.c. flux = 0 new model Pd film (thickness: 46 nm) Pd film (thickness: 46 nm) Wang, Ortiz, Ariza, IJ Hydrogen Energy (submitted)

14 Velocity of phase boundary PHASE SEPARATION - velocity of phase boundary: ~100 nm/s - in many materials, phase boundaries move slowly. The prediction of phase boundary velocity is an important yet challenging task in multiscale analysis Wang, Ortiz, Ariza, IJ Hydrogen Energy (submitted)

15 SUMMARY AND FUTURE WORK Long-term atomistic simulation of mass transport - non-equilibrium statistical thermodynamics model maximum entropy principle meanfield approximation - mass transport: discrete kinetic law - coupling: a partitioned procedure - validations and applications Barnoush, (2011) Future work - large-scale, 3D simulations of hydrogen diffusion in metals - 3D validations A. Lakatos et al. Vacuum (2010) Acknowledgement: ARL, VT new faculty support (K.W.)

Long-Term Atomistic Simulation of Hydrogen Diffusion in Metals

Long-Term Atomistic Simulation of Hydrogen Diffusion in Metals K.G. Wang 1, M. and M. Ortiz 2 Universidad de Sevilla 1 Virginia Polytechnic Institute and State University 2 California Institute of Technology