Elastic Fields of Vacancy Voids and Their Interaction with Radiation Defects in Body Centered Cubic Metals Fe and V: Calculation Methods

Abstract

The calculation of the sink strengths of vacancy voids for radiation defects, which are the parameters of phenomenological models of radiation damage of materials, requires knowledge of the energy of interaction of the radiation defects with the elastic fields created by vacancy voids in the bulk of the material. Direct calculation of the interaction energy by molecular statics demands enormous computational resources and therefore is not suitable for sink strength calculations. In this article, we propose a computationally efficient approach to calculating the interaction energy, which does not introduce a significant error in the calculations. This approach is based on the combined use of different methods: molecular statics is used to calculate the dipole tensors of radiation defects and the elastic strain fields created by vacancy voids, while the interaction of voids with radiation defects (elastic dipoles) is calculated using anisotropic linear elasticity theory. The validity of such an approach is demonstrated by directly comparing its results with the results obtained only by the method of molecular statics, which uses as a test problem the calculation of the interaction between spherical vacancy voids with diameters of 2 and 20 lattice parameters and self-point defects for the BCC metal Fe. Elastic strain fields of the spherical vacancy voids with diameters from 2 to 20 lattice parameters in the BCC metals Fe and V are calculated by molecular statics.

Appendices

APPENDIX A

DEPENDENCES OF ELASTIC FIELDS CREATED BY SPHERICAL VACANCY VOIDS ON THE DISTANCE TO THEIR SURFACE IN SELECTED DIRECTIONS

Fig. A1.

Dependence of the eigenvalues ε⁽¹⁾ (a, b) and ε⁽²⁾ = ε⁽³⁾ (c, d) of the VV elastic strain tensor on the distance to the VV surface in the 〈100〉 directions in Fe. The eigenvector corresponding to ε⁽¹⁾ is parallel to the particular 〈100〉 direction under consideration: () V₉; () V₁₅; () V₂₇; () V₅₉; () V₁₃₇; () V₂₂₉; () V₁₀₃₇; () V₂₂₇₇; () V₃₅₂₇; () V₅₀₆₅; () V₈₃₆₃.

Fig. A2.

Dependence of the eigenvalues ε⁽¹⁾ (a, b) and ε⁽²⁾ = ε⁽³⁾ (c, d) of the VV elastic strain tensor on the distance to the VV surface in the 〈111〉 directions in Fe. The eigenvector corresponding to ε⁽¹⁾ is parallel to the particular direction 〈111〉 under consideration: () V₉; () V₁₅; () V₂₇; () V₅₉; () V₁₃₇; () V₂₂₉; () V₁₀₃₇; () V₂₂₇₇; () V₃₅₂₇; () V₅₀₆₅; () V₈₃₆₃.

Fig. A3.

Dependence of the eigenvalues ε⁽¹⁾ (a, b) and ε⁽²⁾ = ε⁽³⁾ (c, d) of the VV elastic strain tensor on the distance to the VV surface in the 〈100〉 directions in V. The eigenvector corresponding to ε⁽¹⁾ is parallel to the particular direction 〈100〉 under consideration: () V₉; () V₁₅; () V₂₇; () V₅₉; () V₁₃₇; () V₂₂₉; () V₁₀₃₇; () V₂₂₇₇; () V₃₅₂₇; () V₅₀₆₅; () V₈₃₆₃.

Fig. A4.

Dependence of the eigenvalues ε⁽¹⁾ (a, b) and ε⁽²⁾ = ε⁽³⁾ (c, d) of the VV elastic strain tensor on the distance to the VV surface in the 〈111〉 directions in V. The eigenvector corresponding to ε⁽¹⁾ is parallel to the particular direction 〈111〉 under consideration: () V₉; () V₁₅; () V₂₇; () V₅₉; () V₁₃₇; () V₂₂₉; () V₁₀₃₇; () V₂₂₇₇; () V₃₅₂₇; () V₅₀₆₅; () V₈₃₆₃.

APPENDIX B

ELASTIC-ISOTROPIC CONTINUUM MODEL OF A SPHERICAL VOID

The strain field created by a spherical void can be determined using the continuum models proposed by J. Eshelby [19, 20]. Let the elastic body be a ball with a radius R and volume V with a concentric spherical cavity. Let the forces normal to the surface of the cavity (the forces are directed to the center of the void) be applied uniformly to the surface such that the volume of the cavity decreases by δν (the radius of the spherical cavity thus becomes r₀). In this case, an elastic strain field will arise [5]:

$$\left. \begin{gathered} {{\varepsilon }_{{rr}}} = - \frac{{2C}}{{{{r}^{3}}}}\left( {1 - \frac{{1 - 2\nu }}{{1 + \nu }}\frac{{{{r}^{3}}}}{{{{R}^{3}}}}} \right), \hfill \\ {{\varepsilon }_{{00}}} = {{\varepsilon }_{{\varphi \varphi }}} = \frac{C}{{{{r}^{3}}}}\left( {1 + 2\frac{{1 - 2\nu }}{{1 + \nu }}\frac{{{{r}^{3}}}}{{{{R}^{3}}}}} \right), \hfill \\ \operatorname{Tr} \varepsilon = 6\frac{{1 - 2\nu }}{{1 + \nu }}\frac{C}{{{{R}^{3}}}}, \hfill \\ \end{gathered} \right\}$$

(B.1)

where

$$C = \frac{{\delta \nu }}{{4\pi \left( {1 + 2\frac{{1 - 2\nu }}{{1 + \nu }}\frac{{r_{0}^{3}}}{{{{R}^{3}}}}} \right)}}.$$

(B.2)

The change in the volume of the elastic body caused by the void is as follows [5]:

$${{V}^{{\text{R}}}} = 12\pi C\frac{{1 - \nu }}{{1 + \nu }}.$$

(B.3)

Since expression (B.3) establishes a relationship between the values of V^R and C, it is possible to take the value of V^R as a parameter determining the elastic field of the void. Let us rewrite expression (B.1) accordingly:

$$\left. \begin{gathered} {{\varepsilon }_{{rr}}} = - \frac{{{{V}^{{\text{R}}}}}}{{6\pi {{r}^{3}}}}\frac{{1 + \nu }}{{1 - \nu }}\left( {1 - \frac{{1 - 2\nu }}{{1 + \nu }}\frac{{{{r}^{3}}}}{{{{R}^{3}}}}} \right), \hfill \\ {{\varepsilon }_{{00}}} = {{\varepsilon }_{{\varphi \varphi }}} = \frac{{{{V}^{{\text{R}}}}}}{{12\pi {{r}^{3}}}}\frac{{1 + \nu }}{{1 - \nu }}\left( {1 + 2\frac{{1 - 2\nu }}{{1 + \nu }}\frac{{{{r}^{3}}}}{{{{R}^{3}}}}} \right), \hfill \\ \operatorname{Tr} \varepsilon = \frac{{1 - 2\nu }}{{1 - \nu }}\frac{{{{V}^{{\text{R}}}}}}{{2\pi {{r}^{3}}}} = \frac{2}{3}\frac{{1 - 2\nu }}{{1 - \nu }}\frac{{{{V}^{{\text{R}}}}}}{V}. \hfill \\ \end{gathered} \right\}$$

(B.4)

Proceeding from spherical to Cartesian coordinates, we obtain for the strain tensor

$$\left. \begin{gathered} {{\varepsilon }_{{ij}}} = \frac{{{{V}^{{\text{R}}}}}}{{4\pi {{r}^{3}}}}\frac{{1 + \nu }}{{1 - \nu }}\left( {\frac{{{{\delta }_{{ij}}}}}{3} - \frac{{{{x}_{i}}{{x}_{j}}}}{{{{r}^{2}}}}} \right) + \frac{2}{9}\frac{{{{V}^{{\text{R}}}}}}{V}\frac{{1 - 2\nu }}{{1 - \nu }}{{\delta }_{{ij}}}, \hfill \\ r = \sqrt {x_{1}^{2} + x_{2}^{2} + x_{3}^{2}.} \hfill \\ \end{gathered} \right\}$$

(B.5)

As can be seen from (B.1) or (B.4), the strain tensor trace Tr ε is the same at all points of the elastic body. For the combinations of values of V^R and V shown in Tables 1 and 3, Tr ε determined by (B.4) in absolute value does not exceed 4 × 10^–3%; therefore, the second summand in (B.5) has little effect on the appearance of isosurfaces ε_ij = ±0.02% shown in Figs. 1 and 2. In the calculations according to (B.5), the values of the Poisson ratio ν were assumed to be 0.30 for Fe and 0.36 for V. These values were obtained by the Vogt–Royce–Hill method [27] from the elastic constants c₁₁ (243.4 GPa for Fe and 227.5 GPa for V), c₁₂ (145.0 GPa for Fe and 119.3 GPa for V), and c₄₄ (116.0 GPa for Fe and 42.0 GPa for V) corresponding to the interatomic potentials [21, 22] for Fe and V.