Multiscale modeling of hydrogen diffusion in iron considering the effect of dislocations

TL;DR

This work develops a multiscale framework to predict hydrogen diffusion in BCC-Fe accounting for dislocations by integrating DFT-derived site energies, an object kinetic Monte Carlo (OkMC) lattice model, and a phase-field dislocation solver with FFT-based elasticity. Hydrogen diffusion is characterized through a tensor $\mathbf{D}$ that depends on temperature, external stress, and dislocation content, with rates governed by $\nu_{ij,t}=\nu_{0_{ij}}\exp(-\Delta E_{ij}/(k_B T))$ and a barrier decomposition $E=E_0+\Delta E_\mu+\Delta E_\sigma+\Delta E_{\nabla\sigma}$. The results show that shear stresses induce strong diffusivity anisotropy and that dislocations reduce overall diffusivity by about 30% at room temperature while breaking isotropy, with the maximum diffusion along $[0\,1\,\bar{1}]$ and the minimum along $[1\,1\,1]$. The framework yields a closed-form, stress- and dislocation-dependent diffusivity tensor suitable for upscaling to mesoscale models of hydrogen transport and embrittlement in iron.

Abstract

Modeling hydrogen diffusion and its absorption in traps is a fundamental first step towards the understanding and prediction of hydrogen embrittlement. In this study, a multiscale approach which includes DFT simulations, OkMC, and phase-field dislocations, is developed to study the movement of hydrogen atoms in alpha-iron crystals containing dislocations. At the nanoscale the interaction energies of hydrogen on different sites of the iron lattice are studied using DFT. At the microscale, this information is used to feed a lattice object kinetic Monte Carlo code (OKMC) which aims to evolve the arrangement of a large set of hydrogen atoms into the iron lattice considering point defects and the presence of dislocations. At the continuum level, an array of dislocations is introduced using a phase-field approach to accurately consider their elastic fields and core regions. The OKMC model includes both the chemical energies of H and vacancies and the elastic interactions between these point defects and the dislocations. The elastic interaction is obtained by an FFT-based approach which allows a very efficient computation of the elastic microfields created by the defects in an anisotropic medium. The framework has been used to obtain the diffusivity tensor of hydrogen as a function of the external stress state, temperature, and the presence of dislocations. It has been found that dislocations strongly affect the diffusivity tensor by breaking its isotropy and reducing its value by the effect of the microstresses around the dislocations.

Figures (6)

• Figure 1: Schematic representation of the simulation cell including a dislocation array. The sheared area between opposite dislocations ($\phi \approx 1$) is colored in red.
• Figure 2: Evolution over time of the mean and directional diffusion coefficient of hydrogen in iron at T =300 K. a) Under no external stress. b) Under shear external stress state $\boldsymbol{\sigma} = 200 (e_2 \otimes e_3)$ MPa.
• Figure 3: Stress induced anisotropy in the diffusion coefficient parameters: Diffusion coefficient at 300 K (D) (a,b), pre-exponential coefficient (D$^0$) (c,d) and effective energy barrier ($\Delta E$) (e,f). Under uniaxial stress (a,c,e) and shear stress (b,d,f). Note that in the uniaxial cases (a,c,e), the parallel ($\parallel$) and perpendicular ($\perp$) directions refer the direction parallel to the external load, and any direction perpendicular to it respectively. However, in the shear cases (b,d,f), the parallel and perpendicular attributes are related to the normal direction to the shear plane instead.
• Figure 4: Effect of stress in the diffusivity. Diffusion coefficient at 300 K (D) (a,b) pre-exponential coefficient (D$^0$) (c,d) and effective energy barrier ($\Delta E$) (e,f). Under uniaxial stress (a,c,e) and shear stress (b,d,f). Solid lines represent average parameters, shadows represent the region occupied by the directional counterparts.
• Figure 5: Probability distributions of residence time of mobile hydrogen with and without dislocations: a) volumetric distribution b) areal distribution