Axial misfit stress relaxation in core–shell nanowires with polyhedral cores through the nucleation of misfit prismatic dislocation loops

Abstract

The theoretical model of axial misfit stress relaxation in polyhedral core–shell nanowires through the nucleation of prismatic dislocation loops is suggested. Different sites of dislocation nucleation in the nanowires with hexagonal, square and triangular shapes of the core cross section are considered. The energy change caused by the dislocation nucleation is calculated for every case under the assumption that the shell thickness is much smaller than the core size. The corresponding critical values of the misfit parameter for the dislocation nucleation are determined and compared with each other. According to this comparison, the most favorable sites in the core–shell nanowires and the optimal shapes of the dislocation loops are defined. Nanowires with round, hexagonal, square and triangle shapes of the core cross section are ranged with respect to their stability to dislocation loop nucleation.

Appendix 1

The axial stress in a core–shell NW with a cylindrical core (Fig. 1a) subjected to a 3D dilatation eigenstrain is

$$ \sigma_{zz}^{0} = 4\pi \,C\,\left[ {\frac{{R_{0}^{2} - R_{{}}^{2} }}{{R_{{}}^{2} }}{\text{H}}(R_{0}^{{}} - r) + \frac{{R_{0}^{2} }}{{R_{{}}^{2} }}{\text{H}}(r - R_{0}^{{}} )} \right] , $$

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where r is the polar radius, R₀ and R are the radii of the core and shell, respectively, $C = G\,f(1 + \nu )/[2\pi (1 - \nu )]$, f is the misfit parameter, and H(x) is the Heaviside function (H(x) = 1 for x > 0 and H(x) = 0 for x < 0).

The axial stress in a core–shell NW with a prismatic core of hexagonal cross section (Fig. 1b), subjected to a 3D dilatation eigenstrain, is

$$ \sigma_{zz}^{6} = 2C\,\left( {[{}_{{}}^{\infty } \varPsi_{zz}^{\text{hx}} + {}_{{}}^{*} \varPsi_{zz}^{\text{hx}} ]_{{k = \sqrt 3 /3,\,c = - R_{0} }}^{{k = - \sqrt 3 /3,\,c = R_{0} }} \left| {_{{y_{0} = \,0}}^{{y_{0} = \sqrt 3 R/2}} } \right. + [{}_{{}}^{\infty } \varPsi_{zz}^{\text{hx}} + {}_{{}}^{*} \varPsi_{zz}^{\text{hx}} ]_{{k = - \sqrt 3 /3,\,c = - R_{0} }}^{{k = \sqrt 3 /3,\,c = R_{0} }} \left| {_{{y_{0} = - \sqrt 3 R/2}}^{{y_{0} = 0}} } \right.} \right) $$

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with

$$ {}_{{}}^{\infty } \varPsi_{zz}^{\text{hx}} = - \frac{\pi }{2}\text{sgn} \frac{{y - y_{0} }}{x - k\,y - c} , $$

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$$ \begin{aligned} {}_{{}}^{ * } \varPsi_{zz}^{\text{hx}} &= - \cos \psi \,\left[ {q^{2} \cos (2\theta + \psi ) + \frac{4}{{p^{2} }}\left( {\sin (2\varphi + \psi )\ln \sqrt {p^{2} q^{2} - 2pq\cos (\varphi - \theta ) + 1} } \right.} \right. \\ &\quad \left. { + \cos (2\varphi + \psi )\,\,\tan^{-1} \left. {\frac{pq\sin (\varphi - \theta )}{1 - pq\cos (\varphi - \theta )}} \right)} \right], \\ \end{aligned} $$

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where p = r/R, q = ρ/R, $\rho = \sqrt {x_{0}^{2} + y_{0}^{2} }$, $\varphi = \tan^{-1} [y/x]$, $\theta = \tan^{-1} [y_{0} /x_{0} ]$, $\psi = \tan^{-1} [k]$, (x₀, y₀) and (ρ, θ) are the Cartesian and polar coordinates of a hexagonal corner, respectively, k and c are the slope and the y-intercept of the side.

The axial stress in a core–shell NW with a prismatic core of square cross section (Fig. 1c), subjected to a 3D dilatation eigenstrain, is

$$ \sigma_{zz}^{4} = 2C\;[{}_{{}}^{\infty } \varPsi_{zz}^{\text{sq}} + {}_{{}}^{*} \varPsi_{zz}^{\text{sq}} ]_{{x_{0} = -L/2 }}^{{x_{0} = L/2 }} \left| {_{{y_{0} = -L/2 }}^{{y_{0} = L/2 }} } \right. $$

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with

$$ {}_{{}}^{\infty } \varPsi_{zz}^{\text{sq}} = - \frac{\pi }{2}\text{sgn} \frac{{y - y_{0} }}{{x - x_{0} }} , $$

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$$ \begin{aligned} {}_{{}}^{*} \varPsi_{zz}^{\text{sq}} &= - q^{2} \sin 2\theta + \frac{4}{{p^{2} }}\left[ {\sin 2\varphi \ln \sqrt {p^{2} q^{2} - 2pq\cos (\varphi - \theta ) + 1} } \right. \\ & \quad \left. { - \cos 2\varphi \tan^{-1} \frac{pq\sin (\varphi - \theta )}{1 - pq\cos (\varphi - \theta )}} \right]. \\ \end{aligned} $$

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The axial stress in a core–shell NW with a prismatic core of triangular cross section (Fig. 1d), subjected to a 3D dilatation eigenstrain, is

$$ \sigma_{zz}^{3} = 2C\;[{}_{{}}^{\infty } \varPsi_{zz}^{\text{tr}} + {}_{{}}^{*} \varPsi_{zz}^{\text{tr}} ]\,_{{k = {{\sqrt 3 } \mathord{\left/ {\vphantom {{\sqrt 3 } 3}} \right. \kern-0pt} 3},\;c = {{ - \sqrt 3 R_{0} } \mathord{\left/ {\vphantom {{ - \sqrt 3 R_{0} } 3}} \right. \kern-0pt} 3}}}^{{k = {{ - \sqrt 3 } \mathord{\left/ {\vphantom {{ - \sqrt 3 } 3}} \right. \kern-0pt} 3},\;c = {{\sqrt 3 R_{0} } \mathord{\left/ {\vphantom {{\sqrt 3 R_{0} } 3}} \right. \kern-0pt} 3}}} \left| {_{{y_{0} = - R_{0} /2}}^{{y_{0} = R_{0} }} } \right. , $$

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$$ {}_{{}}^{\infty } \varPsi_{zz}^{\text{tr}} = - \frac{\pi }{2}\text{sgn} \frac{{y - y_{0} }}{x - k\,y - c} , $$

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$$ \begin{aligned} {}_{{}}^{ * } \varPsi_{zz}^{\text{tr}} & = - \cos \psi \left[ {q^{2} \cos (2\theta + \psi ) + \frac{4}{{p^{2} }}\left( {\sin (2\varphi + \psi )\ln \sqrt {p^{2} q^{2} - 2pq\cos (\varphi - \theta ) + 1} } \right.} \right. \\ & \quad \left. { + \cos (2\varphi + \psi )\tan^{-1} \left. {\frac{pq\sin (\varphi - \theta )}{1 - pq\cos (\varphi - \theta )}} \right)} \right]. \\ \end{aligned} $$

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