TY - JOUR

T1 - Kinetics of interaction of impurity interstitials with dislocations revisited

AU - Svoboda, Jiri

AU - Ecker, Werner

AU - Razumovskiy, V.I.

AU - Zickler, Gerald A.

AU - Fischer, Franz-Dieter

PY - 2018/10

Y1 - 2018/10

N2 - Interactions of impurity interstitial atoms with dislocations may play an important role in strengthening of structural materials as well as in other phenomena like hydrogen embrittlement. Impurity interstitial atoms occupy octahedral (typical for carbon) or tetrahedral (typical for hydrogen) sites in bcc metals, both showing solely tetragonal symmetry of three types distinguished by orientation with respect to the main crystallographic (1, 2, 3) directions. Placing an atom at one of such sites one of the three types of local multiaxial eigenstrain states is provoked. This eigenstrain state interacts with the stress field of the dislocation, and the corresponding interaction term provides a mechanical part of the generalized chemical potential of the interstitial component. For a proper description of the distribution of the interstitial component in the stressed lattice three site fractions X 1, X 2 and X 3, corresponding to the types of sites, are necessary. Interstitials like hydrogen and carbon are mobile even at room temperature and diffuse spontaneously to places with the lowest interaction energy. As the individual types of interstitial sites are interconnected by fast diffusion paths, one can assume local equilibrium among atoms occupying the three types of interstitial sites. This fact is expressed by a unique value of their generalized chemical potential. The gradient of this potential drives diffusion of interstitial atoms. At any time the atoms redistribute locally to the three types of sites according to the condition of local equilibrium. Thus, generally in the stressed bcc lattice the three site fractions X 1, X 2 and X 3 have different values and provoke a multiaxial eigenstrain state resulting in an eigenstress state. Hence, the stress field of the dislocation is superposed by the eigenstress state. This may cause a significant relaxation (i.e. reduction) of the total stress state depending on the amount of the interstitial atoms in the system and their diffusion kinetics. Very recently, the individual diffusion paths of interstitial atoms occupying octahedral or tetrahedral positions in a stressed bcc lattice have been identified. The respective model predicts a significant anisotropy of diffusion caused by the differences in local values of X 1, X 2 and X 3 in addition to the standard elastodiffusion due to deformation of the lattice. This leads to a significant modification of the standard diffusion equation, because the introduced “diffusion occupancy factors” are directly dependent on the values of X 1, X 2 and X 3. Assuming a straight dislocation (with its dislocation line coinciding with the z-coordinate axis and the slip plane as the x-z-plane of the local coordinate system) and its interaction with the interstitial component, it is advantageous to evaluate the eigenstrain tensor in the local coordinate system (x, y, z) and solve the problem in that system. Then the space around the dislocation can be divided into cylindrical representative volume elements with axes parallel to z-axis and small cross-sections in x- and y- directions. In each of the representative volume elements spatially constant values of X 1, X 2 and X 3 and of the eigenstrain tensor are assumed. As one can calculate the eigenstress state inside and outside of each cylindrical representative volume element, embedded in an eigenstrain-free infinite matrix, the total stress state is then given by linear superposition of the eigenstress states of all representative volume elements and the stress state induced by the dislocation. The goal of the proposed review paper is to comprise a collection of the knowledge of the last seven decades dealing with the following topics: • stress fields induced by edge and screw dislocation; a presentation of the most recent solution technique and discussion of previous solutions;• eigenstrains of carbon and hydrogen atoms placed in octahedral and tetrahedral sites; a selection of the values of reliable sources;• assembling of the interstitial atoms, acting as inclusions with a misfit eigenstrain state, in cylindrical representative volume elements;• eigenstrain states in cylindrical representative volume elements due to occupancy of interstitial atoms described by X 1, X 2 and X 3;• eigenstress fields generated by eigenstrains in cylindrical representative volume elements;• formulation of the generalized chemical potential of interstitial atoms in the stressed bcc lattice;• anisotropic diffusion equations for interstitials in a stressed bcc lattice, accounting for elastodiffusion and the effect of occupancy of different types of interstitial sites. A collection of equations provides a basis for acquiring a complex model of kinetics of interactions of interstitial atoms with (especially prioritizing) dislocations in the bcc lattice. This represents the “research part” of the review paper. Thus the paper focuses on kinetics of interactions of interstitials occupying tetrahedral and octahedral positions with edge and screw dislocations. As results of simulations the kinetics of distribution of the site fractions X 1, X 2 and X 3 around the dislocation as well as the relaxation (i.e. reduction) of the dislocation stress due to eigenstress are provided.

AB - Interactions of impurity interstitial atoms with dislocations may play an important role in strengthening of structural materials as well as in other phenomena like hydrogen embrittlement. Impurity interstitial atoms occupy octahedral (typical for carbon) or tetrahedral (typical for hydrogen) sites in bcc metals, both showing solely tetragonal symmetry of three types distinguished by orientation with respect to the main crystallographic (1, 2, 3) directions. Placing an atom at one of such sites one of the three types of local multiaxial eigenstrain states is provoked. This eigenstrain state interacts with the stress field of the dislocation, and the corresponding interaction term provides a mechanical part of the generalized chemical potential of the interstitial component. For a proper description of the distribution of the interstitial component in the stressed lattice three site fractions X 1, X 2 and X 3, corresponding to the types of sites, are necessary. Interstitials like hydrogen and carbon are mobile even at room temperature and diffuse spontaneously to places with the lowest interaction energy. As the individual types of interstitial sites are interconnected by fast diffusion paths, one can assume local equilibrium among atoms occupying the three types of interstitial sites. This fact is expressed by a unique value of their generalized chemical potential. The gradient of this potential drives diffusion of interstitial atoms. At any time the atoms redistribute locally to the three types of sites according to the condition of local equilibrium. Thus, generally in the stressed bcc lattice the three site fractions X 1, X 2 and X 3 have different values and provoke a multiaxial eigenstrain state resulting in an eigenstress state. Hence, the stress field of the dislocation is superposed by the eigenstress state. This may cause a significant relaxation (i.e. reduction) of the total stress state depending on the amount of the interstitial atoms in the system and their diffusion kinetics. Very recently, the individual diffusion paths of interstitial atoms occupying octahedral or tetrahedral positions in a stressed bcc lattice have been identified. The respective model predicts a significant anisotropy of diffusion caused by the differences in local values of X 1, X 2 and X 3 in addition to the standard elastodiffusion due to deformation of the lattice. This leads to a significant modification of the standard diffusion equation, because the introduced “diffusion occupancy factors” are directly dependent on the values of X 1, X 2 and X 3. Assuming a straight dislocation (with its dislocation line coinciding with the z-coordinate axis and the slip plane as the x-z-plane of the local coordinate system) and its interaction with the interstitial component, it is advantageous to evaluate the eigenstrain tensor in the local coordinate system (x, y, z) and solve the problem in that system. Then the space around the dislocation can be divided into cylindrical representative volume elements with axes parallel to z-axis and small cross-sections in x- and y- directions. In each of the representative volume elements spatially constant values of X 1, X 2 and X 3 and of the eigenstrain tensor are assumed. As one can calculate the eigenstress state inside and outside of each cylindrical representative volume element, embedded in an eigenstrain-free infinite matrix, the total stress state is then given by linear superposition of the eigenstress states of all representative volume elements and the stress state induced by the dislocation. The goal of the proposed review paper is to comprise a collection of the knowledge of the last seven decades dealing with the following topics: • stress fields induced by edge and screw dislocation; a presentation of the most recent solution technique and discussion of previous solutions;• eigenstrains of carbon and hydrogen atoms placed in octahedral and tetrahedral sites; a selection of the values of reliable sources;• assembling of the interstitial atoms, acting as inclusions with a misfit eigenstrain state, in cylindrical representative volume elements;• eigenstrain states in cylindrical representative volume elements due to occupancy of interstitial atoms described by X 1, X 2 and X 3;• eigenstress fields generated by eigenstrains in cylindrical representative volume elements;• formulation of the generalized chemical potential of interstitial atoms in the stressed bcc lattice;• anisotropic diffusion equations for interstitials in a stressed bcc lattice, accounting for elastodiffusion and the effect of occupancy of different types of interstitial sites. A collection of equations provides a basis for acquiring a complex model of kinetics of interactions of interstitial atoms with (especially prioritizing) dislocations in the bcc lattice. This represents the “research part” of the review paper. Thus the paper focuses on kinetics of interactions of interstitials occupying tetrahedral and octahedral positions with edge and screw dislocations. As results of simulations the kinetics of distribution of the site fractions X 1, X 2 and X 3 around the dislocation as well as the relaxation (i.e. reduction) of the dislocation stress due to eigenstress are provided.

UR - http://www.scopus.com/inward/record.url?scp=85058229143&partnerID=8YFLogxK

U2 - 10.1016/j.pmatsci.2018.10.001

DO - 10.1016/j.pmatsci.2018.10.001

M3 - Article

VL - 101.2019

SP - 172

EP - 206

JO - Progress in materials science

JF - Progress in materials science

SN - 0079-6425

IS - April

ER -