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Description
A profound consequence of irradiation (neutron, proton, electron and heavy ion) of metal alloys is the spatial redistribution of alloy components. As a result, there are enrichment or depletion of the main, solute and impurity components of the alloy near the defect sinks. This phenomenon is called radiation-induced segregation (RIS) and leads to degradation of mechanical and physicochemical properties of materials.
The spatial and temporal evolution of the concentrations of alloy components $C_{k} $ $\left(k=\mathrm{Fe},\, \mathrm{Cr},\, \mathrm{Ni}\right)$ and point defects (PD) (vacancies $C_{\mathit{v}} $ and interstitials $C_{i} $) in the ternary concentrated Fe-Cr-Ni alloys under irradiation is described by the system of five coupled nonlinear partial differential equations [1-3]:
$\begin{equation}
\left\{\begin{array}{l} {\frac{\partial C_{k} }{\partial t} =-\mathbf{\nabla}\mathbf{J}_{k} ,} \\ {\frac{\partial C_{\mathit{v}} }{\partial t} =-\mathbf{\nabla}\mathbf{J}_{v} +K_{0} -R_{iv} C_{\mathit{v}} C_{i} -k_{v}^{2} D_{v} \left(C_{v} -C_{v}^{eq} \right),} \\ {\frac{\partial C_{i} }{\partial t} =-\mathbf{\nabla }\mathbf{J}_{i} +K_{0} -R_{iv} C_{\mathit{v}} C_{i} -k_{i}^{2} D_{i} \left(C_{i} -C_{i}^{eq} \right).} \end{array}\right.
\end{equation}$
where the fluxes of atoms species $k$ is $\mathbf{J}_k$, vacancies $\mathbf{J}_{\mathit{v}} $ and interstitial $\mathbf{J}_{i} $ defined as:
$\begin{equation}
\mathbf{J}_{k} =-\left(\sum _{d=v,i}d_{k,d} C_{d} \right)\mathbf{\nabla}C_{k} +C_{k} \left(d_{k,v} \mathbf{\nabla }C_{v} -d_{k,i} \mathbf{\nabla }C_{i} \right),
\end{equation}$
$\begin{equation}
\mathbf{J}_{v} =-\sum _{k=\mathrm{Fe},\, \mathrm{Cr},\, \mathrm{Ni}}d_{k,v} C_{k} \mathbf{\nabla }C_{v} +\alpha C_{v} \left(\sum _{k=\mathrm{Fe},\, \mathrm{Cr},\, \mathrm{Ni}}d_{k,v} \mathbf{\nabla}C_{k} \right),
\end{equation}$
$\begin{equation}
\mathbf{J}_{i} =-\sum _{k=\mathrm{Fe},\, \mathrm{Cr},\, \mathrm{Ni}}d_{k,i} C_{k} \mathbf{\nabla}C_{i} -\alpha C_{i} \left(\sum _{k=\mathrm{Fe},\, \mathrm{Cr},\, \mathrm{Ni}}d_{k,i} \mathbf{\nabla }C_{k} \right),
\end{equation}$
$K_{0} $ is the production rate of radiation PD, $R_{iv} $ is the recombination rate of PD, $k_{v}^{2} $ and $k_{i}^{2} $ are the sink strengths for vacancies and interstitials respectively, $C_{v}^{eq} $ and $C_{i}^{eq} $ are the equilibrium vacancy and interstitial concentrations, $D_{v} $ and $D_{i} $ are the diffusion coefficients of vacancies and interstitial, $d_{k,v} $ and $d_{k,i} $ are the diffusivity coefficients of vacancies and interstitial. The system with the corresponding initial and boundary conditions is solved numerically (a detailed solution algorithm is given in [2]).
The aim of the present paper is to calculate the governing quantitative characteristics of RIS for Fe-20%Cr-8%Ni alloy under the irradiation. That are: concentration profiles of atoms species $k$ $C_{k} \left(x\right)$ and PD $C_{v\left(i\right)} \left(x\right)\, $, surface concentration of atoms species $k$ $C_{k}^{Surf} $, the value of surface enrichment (depletion) of atoms species $k$ $\Delta C_{k} $, the full width of the concentration profile of atoms species $k$ at half maximum enrichment (depletion) $\mathrm{FWHM}_{k} $, segregation area of atoms species $k$ $S_{k} $ and discriminant of RIS of atoms species $k$ in a steady state $\mathit{{\mathfrak D}}_{k} $. For example, production rate dependence of surface depletion, $\mathrm{FWHM} $ and segregation area for Cr and Ni are shown in Fig. 1.
Fig. 1. Production rate dependence of surface Cr depletion $\Delta C_{\mathrm{Cr}} $ and Ni enrichment $\Delta C_{\mathrm{Ni}} $ (solid line), $\mathrm{FWHM}_{\mathrm{Cr}} $ and $\mathrm{FWHM}_{\mathrm{Ni}} $ (dashed line) and segregation area of Cr $S_{\mathrm{Cr}} $ and Ni $S_{\mathrm{Ni}} $ (dash-dotted line). Calculations were performed at temperature$T=400$~$\mathrm{{}^\circ}$C, dose $D=10$~dpa (displacement per atom).
The Authors acknowledge the support by the target research program of National Academy of Sciences of Ukraine “Nuclear and radiation technologies for the energy sector and social needs” for 2019-2023.
[1]. Was G.S. Fundamentals of Radiation Materials Science: Metals and Alloys (2nd ed.). New York: Springer-Verlag, 2017. 1002 p.
[2]. Skorokhod R.V., Koropov A.V. Modeling of Radiation-Induced Segregation in Fe–Cr–Ni Alloys. Physics of the Solid State. 2019. Vol. 61, P. 2269–2276.
[3]. Koropov O.V., Skorokhod R.V., in Proceedings of Ninth International Scientific-Practical Conference “Mathematics in Modern Technical University”, Kyiv, December, 28–29, 2020. Vinnytsia: Publisher FOP Kushnir Yu. V., 2021. P. 80-89. (in Ukrainian)
