First-principles localized cluster expansion study of the kinetics of hydrogen diffusion in homogeneous and heterogeneous Fe-Cr alloys

Metal alloys have a wide range of technological applications, from structural materials to catalysts. In many situations, the transport of hydrogen, whether intentionally for hydrogen storage and fuel cell applications or unintentionally in the case of tritium uptake in nuclear materials, is an important concern. Fe-Cr binary alloys, in particular, may be viewed as a simple model system to represent ferritic steels used in nuclear energy systems and, more generally, as a model binary alloy for examining the role of alloying elements on transport. In this work, we used density functional theory, cluster expansion, and kinetic Monte Carlo to study hydrogen kinetics in Fe-Cr alloys. In random homogeneous alloys, we observed that hydrogen diffusivity first decreased with increasing Cr content and then increased to its levels in pure Cr. Furthermore, the effects of heterogeneity, as might be induced by irradiation, were explored and it was concluded that local structural heterogeneities for the same overall Cr concentration may significantly affect the hydrogen diffusivity. The effect was attributed to the relative binding energy of hydrogen in different metals and this understanding was then utilized to predict hydrogen transport behavior in different element-segregated grain/grain boundary combinations and thus identify solute/solvent alloys where hydrogen transport might be either hindered or enhanced. Finally, we comment on the potential impact of radiation-induced segregation on hydrogen behavior in nuclear energy systems.

Figure 1

The cluster expansion fit to the DFT binding energy data. (a) A parity plot showing the agreement between the DFT recorded data and the cluster expansion (CE) model. The data points are clustered about the $y=x$ line. (b) The variation of the uncorrected binding energy as a function of the local environment up to the fourth-nearest neighbor shell inclusive. Both the calculated DFT results (red circles) and the cluster expansion results (blue + signs) are shown.

Figure 2

(a) A map of the CE-model-predicted zero-point-energy-corrected binding energy landscape of the system for a 2×2×2 supercell containing one Cr atom (0.0625 at.%). The location of the Cr atom is shown with a red (+) sign and periodic boundary conditions are utilized. (b) A histogram of the energy landscape for the given situation showing the number of T binding sites within the cell calculated to have an H binding energy within a given range.

Figure 3

Semi-logarithmic plots of histograms of the binding energy landscape for a 50×50×50 bcc supercell with differing Cr content. The Cr atoms were randomly distributed in each case: (a) 10 atomic% Cr, (b) 20 atomic% Cr, (c) 50 atomic% Cr, (d) 90 atomic % Cr.

Figure 4

Diffusivity as a function of temperature and the chromium content in the random Fe-Cr alloy. The markers indicate the data points obtained in this study and the lines are used to help guide the eyes. Error bars representing the uncertainty were also included from analyzing repeated simulations with different configurations.

Figure 5

Illustration of the cell used for exploring the effects of inhomogeneities. In our investigation, we selected blocks of size λ along the $z$ direction. λ was expressed as integers of the unit cell and we examined the following cases: $\lambda =1a$, $2a$, $5a$, and $10a$. For each of these cases, we investigated two situations: one in which the alternating layers had compositions of 0% Cr and 100% Cr for an overall composition of 50 atomic% Cr, and another in which the alternating layers had compositions of 0% Cr and 40% Cr for a total composition of 20 atomic% Cr.

Figure 6

The effects of heterogeneous alloy composition on calculated diffusivity. In each simulation, the simulation box was composed of (a) alternating layers in the $z$ direction containing 100% Cr and 0% Cr for an average of 50% Cr composition and (b) alternating layers of 40% Cr and 0% Cr for an average composition of 20% Cr. In each simulation, the thickness of the layer was denoted by λ (in units of the lattice constant). All simulations were performed in a 20×20×20 box. The diffusivity perpendicular to the direction of alternating layers $\left(D_z\right)$ is shown with red solid lines, the diffusivity parallel to the direction of the layering $\left(D_{xy}\right)$ is shown with green dotted lines, and the total diffusivity is shown with blue dashed lines. The dashed black lines represent the diffusivity in pure iron and have been included for reference. The dotted yellow lines represent the diffusivity in the random alloy corresponding to the overall composition of the given simulation and have also been included for reference. The error bars have been included for the collected data points and the lines between the data points were added to guide the eye.

Figure 7

The ZPE-corrected hydrogen binding energy landscape of the structures being investigated by introducing inhomogeneities averaged for all T-site positions in the same $xy$ plane. The figure outlines the variation in the average binding energy landscape along the $z$ direction.

Figure 8

Map of the energy of tetrahedral sites within multilayer configurations with (a) layers of 0% and 100% Cr and $\lambda =5a$, (b) layers of 0% and 40% Cr and $\lambda =5a$, and (c) layers of 0% and 40% Cr and $\lambda =10a$. Only sites with energies within 0.05 eV of the lowest-energy site (about 0.35 eV) are shown. The dark-blue layers are indicative of the pure Fe layers.

Figure 9

An illustration of the supercell considered for studying inhomogeneities which mimics the situation of a grain and a grain boundary. Structure (a) shows the entire supercell. Structure (b) illustrates the element that occupies the grain boundary; and structure (c) illustrates the element occupying the grain (the inside of the supercell). We considered two cases in our study: In case 1, the grain is composed of Fe and the grain boundary of Cr. In case 2, the grain is composed of Cr and the grain boundary of Fe.

Figure 10

Chart highlighting solute/solvent combinations that are expected to enhance (green) or retard (red) hydrogen transport as compared to a solid solution alloy. The shade of the color indicates the expected modification of transport as based on the solute segregation energies report in Ref. [69]. Those segregation energies are modulated by predicted binding energies (from Refs. [63, 64]) to the different component metals. If H prefers to go to the metal that is enriched at the GB, transport would be fast. Otherwise, it would be slow.

Figure 11

Schematic of how hydrogen transport through an Fe-Cr-based alloy might be modified as the clad is irradiated. Burning the fuel produces hydrogen. Initially, when the Fe-Cr alloy is more chemically homogeneous (randomly distributed), hydrogen has the highest rate of transport through the alloy. As radiation progresses and radiation-induced segregation of Cr to grain boundaries increases, the transport of hydrogen through the clad will be reduced and the leakage through the clad into the coolant will be likewise reduced.