Lattice mismatch and twist partitioning at commensurate dichromatic pattern of two-phase interfaces

Abstract

Misfit dislocations are formed at a two-phase interface to reduce and even diminish coherency stress in the region far from the interface. Such semi-coherent interfaces are key structural features in a wide range of engineering materials. Burgers vectors of misfit dislocations are defined with the reference lattice named as commensurate dichromatic pattern (CDP) in the topological model. The CDP is not a geometrical average of boundary units as historically used in many examples. In this work, based on the Green’s function and theory of dislocations, both the mechanical effects of interfacial dislocation arrays and coherency strains due to geometry match at the interface are considered to get the CDP. We demonstrated that the mismatch and twist partitioning at a CDP are unequally partitioned in the two adjacent crystals, which only depends on elastic properties of the two crystals regardless of characters of misfit dislocations. Correspondingly, the method to determine the CDP of a two-phase interface in bi-crystal is developed.

Graphic abstract

Appendix: methodology for the strains due to interfacial dislocation arrays

For a bi-crystal with different elastic constants in two constitutes, the elastic fields associated with a dislocation array in the interface can be obtained by using the interfacial Green’s function [12] based on Stroh formalism [12,13,14]. Referring to the geometry in Fig. 1a, the displacement field due to a single dislocation with Burgers’ vector b at (x = X, z = 0) can be expressed as:

$${\mathbf{u}}_{i}^{\lambda } = \frac{1}{\pi }{\text{Im}} \left\{ {{\mathbf{A}}_{\lambda } \left\langle {\frac{1}{{z_{*}^{\lambda } - X}}\frac{{\partial z_{*}^{\lambda } }}{{\partial x_{i} }}} \right\rangle {\mathbf{q}}_{\lambda } } \right\}$$

(9)

with

$${\varvec{q}}_{1} = {\varvec{A}}_{1}^{ - 1} \left( {{\varvec{M}}_{1} + \overline{{\varvec{M}}}_{2} } \right)^{ - 1} \overline{{\varvec{M}}}_{2} {\varvec{b}}$$

$${\varvec{q}}_{2} = {\varvec{A}}_{2}^{ - 1} \left( {\overline{{\varvec{M}}}_{1} + {\varvec{M}}_{2} } \right)^{ - 1} \overline{{\varvec{M}}}_{1} {\varvec{b}}$$

$${\varvec{M}}_{1} = - i{\varvec{B}}_{1} {\varvec{A}}_{1}^{ - 1} ,\;{\varvec{M}}_{2} = - i{\varvec{B}}_{2} {\varvec{A}}_{2}^{ - 1}$$

$$\left\langle {f(x_{*} )} \right\rangle = {\text{diag}}\left[ {f(x_{1} ),f(x_{2} ),f(x_{3} )} \right]$$

$$z_{j}^{\lambda } = x + p_{j}^{\lambda } z\, \left( {j = {1},{ 2},{ 3}} \right)$$

where λ = 1, 2 represent the upper half-space material and the lower one, respectively. $p_{j}^{\lambda }$ , A_λ and B_λ denote the Stroh eigenvalues and the corresponding eigenmatrices for the material λ. Im{f} stands for the imaginary part of the variable or function f. According to Cottrell formula [15, 16], the elastic fields induced by dislocation array can be obtained through summing Eq. (9) over all the dislocations. The derivative of the displacements can be shown as [7, 8]

$${\mathbf{u}}_{i}^{\lambda } = \frac{1}{L}{\text{Im}} \left\{ {{\mathbf{A}}_{\lambda } \left\langle {\cot \left( {\frac{{z_{*}^{\lambda } - X}}{L}} \right)\left( {\delta_{i1} + p_{*}^{\lambda } \delta_{i3} } \right)} \right\rangle {\mathbf{q}}_{\lambda } } \right\}$$

(10)

where L is the period spacing of the dislocation array.