Diffusion of Hydrogen in Materials Theory and Experiment

Diffusion of Hydrogen in Materials: Theory and Experiment Brent J. Heuser University of Illinois, Urbana, IL Outline • Diffusion—Fick’s 1 st and 2 nd laws; microscopic view. • Diffusion measurement techniques—some data on hydrogen diffusion. • Neutron scattering theory Part 1—coherent and incoherent. • Examples of scattering—Dispersion curves and vibrational density of states. • Neutron scattering theory Part 2—Incoherent quasi-elastic neutron scattering (QENS). • QENS—instruments, data related to the diffusion of hydrogen. 2007 LANSCE Neutron School

Useful books related to neutron scattering in general and QENS in particular: buy this book!!

Diffusion—Fick’s 1 st and 2 nd Laws—Macroscopic Point of View Diffusion acts to even out concentration gradients and determines kinetic response of system: C(x) J 1 Diffusion coefficient J 2 x 1 st Law J(x) J 1 J 2 x J 1 Dx J 2 2 nd Law

Solutions to Diffusion Equation Thin-film solution e-folding time: t = x 2/4 D Diffusion-couple solution C’ Separation of variables solution—diffusion out of a slab C’ h

Diffusion—Microscopic point of view Impurity interstitials—like hydrogen Diffusion processes at microscopic scale coupled to lattice defects in crystalline solid Vacancy self-diffusion hydrogen thermal vacancy Classical picture—transition state theory yields jump frequency over saddle point Interstitial self-diffusion-these arrangements are called crowdions

The Diffusion Coefficient Physics and temperature dependence come into problem via D Arrhenius behavior; Q is classical barrier height Classical Diffusion Do Q VSD vacancy VSD—vacancy self diffusion Diffusion of regular (non-impurity) atoms requires the presence of vacancies. Easier to form vacancies Easier for interstitials to move interstitial n is the vibrational or Debye frequency

Useful books related to hydrogen diffusion in metals

Effect of size Examples of Diffusion Data Small polaron theory of H diffusion in bcc metals Anomalous isotope effect in fcc metals Diffusion of hydrogen in metals: quantum effects such as tunneling and phonon-assisted processes are important.

Neutron Scattering Elastic (ħw=0) Diffraction SANS Inelastic (ħw≠ 0) Reflectometry Incoherent Coherent QENS Coherent Scattering Phonon Dispersion Vibrational Spectroscopy Coherent scattering—collective phenomena involving different nuclei that gives rise to interference effects. Incoherent scattering—scattering from the same nuclei at different times; no interference effects. Focus on this one for diffusion. Hydrogen has very large incoherent cross section— therefore incoherent QENS.

Double differential scattering cross section Coherent term—double summation over all nuclei that depends on correlation of the same nucleus at different times and different nuclei at different times. Incoherent term—single summation over all nuclei that depends on correlation of the same nucleus at different times.

Incoherent and coherent cross sections Two contributions to incoherent scattering: §non-zero nuclear spin §presents of more than one isotope

Dispersion curves (coherent) and vibrational density of states (incoherent) General Picture Local harmonic oscillator potential Hydrogen vibrational Do. S QENS Phonon dispersion curves in Pd. D 0. 6 optical modes due to hydrogen acoustic modes Phonon dispersion curves in Nb. D 0. 6 and Ta. D 0. 22

Mapping of physical processes and instrument coverage in Q-w space. NIST SNS Minimum energy resolution Dw determines maximum coherence time tcoh of the measurement via: where (sinx)/x is a “transmission” function in time and tcoh~1/Dw.

Basic Theory of Neutron Scattering Intermediate Scattering Functions Structure Factors Connection to Differential Cross Section This is measured in scattering exp.

van Hove Correlation Function Differential probability that given particle at origin at t=0, any particle will be at position r at time t. van Hove self-correlation function different particle same particle

Translation Jump Diffusion in Bravais Lattice Chudley-Elliott model Considering self diffusion QENS measures L

Limiting Cases of L(Q)

Hydrogen diffusion in simple metals Hydrogen diffusion via o-o jumps Polycrystalline Pd Hydrogen diffusion via o-o jumps Single crystal Pd Hydrogen diffusion in Nb—bcc metal with small polaron hopping

Instruments for QENS—NIST HFBS

DCS BS

Mapping of physical processes and instrument coverage in Q-w space. NIST SNS Minimum energy resolution Dw determines maximum coherence time tcoh of the measurement via: where (sinx)/x is a “transmission” function in time and tcoh~1/Dw.