<table><caption>TABLE 3.3: Dislocation spacings $d_i$, angle between the two sets of dislocations $\phi$, and characters $\phi_i$ for three solutions, namely c1, c2, and c3, for which the fcc (here, Cu) and the proper Burgers vectors have been selected as the reference state in NW Cu/Nb interface.</caption><thead><tr><th colspan="6">Dislocation structures in NW Cu/Nb</th></tr><tr><th colspan="6">* solutions by selecting the fcc Burgers vectors</th></tr><tr><th>Cases</th><th>d<sub>1</sub> (nm)</th><th>d<sub>2</sub> (nm)</th><th>φ °</th><th>φ<sub>1</sub> °</th><th>φ<sub>2</sub> °</th></tr></thead><tbody><tr><td>c1 : {b<sub>1</sub><sup>fcc</sup>, b<sub>2</sub><sup>fcc</sup>}</td><td>1.1234</td><td>1.1234</td><td>15.03</td><td>37.51</td><td>37.51</td></tr><tr><td>c2 : {b<sub>1</sub><sup>fcc</sup>, b<sub>3</sub><sup>fcc</sup>}</td><td>4.2953</td><td>1.1234</td><td>82.49</td><td>60.00</td><td>82.49</td></tr><tr><td>c3 : {b<sub>2</sub><sup>fcc</sup>, b<sub>3</sub><sup>fcc</sup>}</td><td>4.2953</td><td>1.1234</td><td>82.49</td><td>60.00</td><td>82.49</td></tr></tbody></table><table><thead><tr><th colspan="6">* solutions by selecting the proper reference Burgers vectors</th></tr><tr><th>Cases</th><th>d<sub>1</sub> (nm)</th><th>d<sub>2</sub> (nm)</th><th>φ °</th><th>φ<sub>1</sub> °</th><th>φ<sub>2</sub> °</th></tr></thead><tbody><tr><td>c1 : {b&#770;<sub>1</sub><sup>ref</sup>, b&#770;<sub>2</sub><sup>ref</sup>}</td><td>1.1234</td><td>1.1234</td><td>15.03</td><td>39.62</td><td>39.62</td></tr><tr><td>c2 : {b&#770;<sub>1</sub><sup>ref</sup>, b&#770;<sub>3</sub><sup>ref</sup>}</td><td>4.2953</td><td>1.1234</td><td>82.49</td><td>57.89</td><td>82.49</td></tr><tr><td>c3 : {b&#770;<sub>2</sub><sup>ref</sup>, b&#770;<sub>3</sub><sup>ref</sup>}</td><td>4.2953</td><td>1.1234</td><td>82.49</td><td>57.89</td><td>82.49</td></tr></tbody></table>

Here and in the following, the superimposed hat will indicate quantities expressed in a frame with basis vectors, $\hat{x} = [100]$, $\hat{y} = [010]$, and $\hat{z} = [001]$. A schematic representation of a Cu/Nb interface in the NW OR is shown in Fig. (3.12a). Labeling of Burgers vectors for the other fcc/bcc systems of interest here follows the same pattern as shown for NW Cu/Nb in Figs. (3.12a) and (b).

If the fcc Cu material is used as the reference state, then three trial Burgers vectors may be selected in the interface plane:

$$b_1^{\text{fcc}} = \frac{a_{\text{Cu}}}{2} [\bar{1}01], \quad b_2^{\text{fcc}} = \frac{a_{\text{Cu}}}{2} [0\bar{1}1], \quad \text{and,} \quad b_3^{\text{fcc}} = \frac{a_{\text{Cu}}}{2} [\bar{1}10]. \qquad (3.45)$$

The transformation matrix $T_{\text{Nb}\to\text{Cu}}$ that represents the transformation of the bcc Nb material to the fcc Cu material may be written as

$$T_{\text{Nb}\to\text{Cu}} = I - F_{\text{Cu}\to\text{Nb}}^{-1}, \qquad (3.46)$$

where $I$ is the identity matrix and $F_{\text{Cu}\to\text{Nb}}$—the mapping that transforms the fcc Cu to the bcc Nb crystal—is written in the fcc reference system ($x_{\text{fcc}}, y_{\text{fcc}}, z_{\text{fcc}}$) as:

$$F_{\text{Cu}\to\text{Nb}} = \begin{bmatrix} 1.281998 & -0.009298 & 0.109180 \\ -0.009298 & 1.281998 & 0.109180 \\ -0.154404 & -0.154404 & 0.899935 \end{bmatrix}. \qquad (3.47)$$

For this interface, the Frank-Bilby equation has three different solutions, namely c1, which uses the pair {$b_1^{\text{fcc}}, b_2^{\text{fcc}}$}, c2 with {$b_1^{\text{fcc}}, b_3^{\text{fcc}}$}, and c3 with {$b_2^{\text{fcc}}, b_3^{\text{fcc}}$}. Due to the crystal symmetry along $\hat{z}$ in the NW OR, which exhibits the p2/m11 layer space group, two of the three solutions (c2 and c3) are mirror images. Analysis of dislocation structures for all three cases are given in Table 3.3, with $\phi$ the angle between the two sets of dislocations and $\phi_i$ their individual characters. The dislocation line directions and spacings are schematically depicted in Fig. (3.12c), where the filled circles represent the O-lattice points [32, 237].

If the bcc Nb lattice is used as the reference state, then corresponding expressions for $F_{\text{Nb}\to\text{Cu}}$ and $T_{\text{Cu}\to\text{Nb}}$ may also be obtained. In this case, Burgers vectors are equivalently expressed in the bcc crystal structure and the same dislocation geometries are found. Neither the fcc nor the bcc reference states satisfy the condition of vanishing far-field strains and stresses [123, 260] because neither accounts for the required partitioning of strains and rotations between the adjacent crystals [119].

There is a continuum of other possible reference states between these two extreme cases. To find the correct reference state, a dimensionless variable $\delta$ that interpolates linearly between the
<figure><img src="image_29.png" /><figcaption>FIGURE 3.12: (a) Representation of the NW OR between fcc {111} (blue atoms) and bcc {110} (red atoms) close packed planes in Cu/Nb interfaces. (b) The reference state is depicted by the dashed black polyhedron, within which the Burgers vectors (corresponding to the sides of each polyhedron) are defined. The difference between the positions of the fcc and bcc atoms have been exaggerated for clarity. (c) Schematic illustrations of two admissible dislocation structures (solutions c1 and c2) with O-lattice points (black circles) and the local elastic energy densities stored in a representative unit cell of the dislocation patterns. The colors of the dislocations are associated with the Burgers vectors that are colored in (b). Contour values (from the center of the patterns to the dislocation lines): {0, 0.2, 0.6, 1.2, 2.0, 3.2, 5.2} J.m⁻².</figcaption></figure>

pure Cu and Nb materials is introduced as follows

$$ \left\{ \begin{aligned} \mathrm{Cu}\mathbf{F} &= (1 - \delta)\mathbf{I} + \delta \mathbf{F}_{\mathrm{Nb} \to \mathrm{Cu}} \\ \mathrm{Nb}\mathbf{F} &= \delta \mathbf{I} + (1 - \delta) \mathbf{F}_{\mathrm{Cu} \to \mathrm{Nb}}. \end{aligned} \right. \qquad (3.48) $$

For $\delta = 0$, $T = T_{\mathrm{Nb} \to \mathrm{Cu}}$ and for $\delta = 1$, $T = T_{\mathrm{Cu} \to \mathrm{Nb}}$. Along the transformation pathway characterized by $\delta$, the elastic distortions (strain and rotation fields) in the NW ORs can also be computed.

### 3.4.2 Far-field strains and rotations

As shown in Refs. [126, 123, 260], and illustrated in Fig. (3.11b) the natural state of semi-infinite bicrystals is homogeneously transformed into a reference state by biaxial distortions parallel to the plane with normal $n \parallel \hat{y}$, so that the removal of the strains $\hat{e}_{2j \text{ tot}} = *$, with $j = 1, 2, 3$ [126, 123]. Thus, only six components (three for strains and three for rotations) of the distortion matrices are needed to meet the condition of vanishing total far-field strains and prescribed misorientations.

In the linear-elastic approximation, the distortion matrices $\hat{D}$ may also be separated into symmetric $\hat{E}$ and antisymmetric $\hat{\Omega}$ parts:

$$ \hat{D} = \underbrace{\begin{bmatrix} \hat{e}_{11} & * & \hat{e}_{13} \\ * & * & * \\ \hat{e}_{13} & * & \hat{e}_{33} \end{bmatrix}}_{\hat{E}} + \underbrace{\begin{bmatrix} 0 & -\hat{\omega}_{12} & \hat{\omega}_{13} \\ \hat{\omega}_{12} & 0 & -\hat{\omega}_{23} \\ -\hat{\omega}_{13} & \hat{\omega}_{23} & 0 \end{bmatrix}}_{\hat{\Omega}}. \qquad (3.49) $$

The coherency strain fields $\hat{E}_c$ on both sides of the interface are given by

$$ \mathrm{Cu}\hat{E}_c = \mathrm{sym}\,\mathrm{Cu}\hat{F}^{-1} - \mathbf{I}, \quad \text{and,} \quad \mathrm{Nb}\hat{E}_c = \mathrm{sym}\,\mathrm{Nb}\hat{F}^{-1} - \mathbf{I}, \qquad (3.50) $$
<figure><img src="image_30.png" /><figcaption>FIGURE 3.13: Dependence of the total far-field strain component $f_{cc} \hat{e}_{33}^\infty$ on $\delta$ in the fcc material for the Cu/Nb, Ag/V and Cu/Mo heterophase interfaces. The vertical dotted line shows the $\delta$ under the assumption that both materials at the interface have the same stiffness.</figcaption></figure>

where $c_{Cu} \hat{F}^{-1}$ and $c_{Nb} \hat{F}^{-1}$ are obtained from eqs. (3.48). Superposing the elastic strains produced by the interface dislocations in Cu and Nb, i.e. $c_{Cu} \hat{E}_{dis}^\infty$ and $c_{Nb} \hat{E}_{dis}^\infty$, the total far-field strain state in the entire bicrystal may be calculated [260].

Figure (3.13) shows the total strain component $c_{Cu} \hat{e}_{33}^\infty$ in Cu as a function of $\delta$ (black line). This strain vanishes, i.e. $c_{Cu} \hat{e}_{33}^\infty = 0$, for $\delta_{Cu/Nb} = 0.429103$. All other elastic components are consistent with the absence of strains in the far-field and the total far-field strain in Nb vanishes at the same $\delta$ as in Cu. Thus, the reference state is closer to Cu than to Nb, i.e. $\delta_{Cu/Nb} < 0.5$. This result cannot be easily predicted from inspection of the stiffness constants alone (see Table 3.2). Figure (3.13) also shows that $\delta_{Ag/V} = 0.623359$ and $\delta_{Cu/Mo} = 0.701109$, i.e. the reference state is closer to the bcc material (V and Mo) in both cases.

Knowing the $\delta$ value at which far-field stresses vanish, the crystal structure of the reference state is given by the uniform displacement gradients, obtained using eqs. (3.48) and (3.50):

$$ c_{Cu} \hat{E}_c = -c_{Cu} \hat{E}_{dis}^\infty = \begin{bmatrix} 0.022615 & 0.072664 & 0 \\ 0.072664 & 0.047550 & 0 \\ 0 & 0 & 0.107173 \end{bmatrix} \quad (3.51) \\ -c_{Nb} \hat{E}_c = c_{Nb} \hat{E}_{dis}^\infty = \begin{bmatrix} 0.030089 & 0.096675 & 0 \\ 0.096675 & 0.063262 & 0 \\ 0 & 0.154414 & 0.142588 \end{bmatrix}. $$

The Burgers vectors of the interfacial misfit dislocations are to be drawn from this reference state. The correct reference state of the NW OR is depicted by the dashed polyhedron in Fig. (3.12b), within which the Burgers vectors are defined by:

$$ \Sigma_{NW} \left\{ \begin{array}{l} \hat{b}_1^{\text{ref}} = -0.226379 \hat{x} - 0.141507 \hat{z} \ (\text{nm}) \\ \hat{b}_2^{\text{ref}} = -0.226379 \hat{x} + 0.141507 \hat{z} \ (\text{nm}) \\ \hat{b}_3^{\text{ref}} = \hat{b}_1^{\text{ref}} - \hat{b}_2^{\text{ref}} = -0.283015 \hat{z} \ (\text{nm}). \end{array} \right. \quad (3.52) $$
<table><caption>TABLE 3.4: Partitioning of strains and rotations for various fcc/bcc bicrystals.</caption><thead><tr><th rowspan="2">Systems</th><th rowspan="2">δ</th><th colspan="2">strains</th><th colspan="2">tilt rotations °</th></tr><tr><th>fcc ê<sub>33 c</sub></th><th>bcc ê<sub>33 c</sub></th><th>ϑ<sub>fcc</sub></th><th>ϑ<sub>bcc</sub></th></tr></thead><tbody><tr><td>Cu/Nb</td><td>0.429103</td><td>0.107173</td><td>-0.142588</td><td>-4.17</td><td>5.55</td></tr><tr><td>Ag/V</td><td>0.623359</td><td>0.031076</td><td>-0.018777</td><td>-6.03</td><td>3.68</td></tr><tr><td>Cu/Mo</td><td>0.701109</td><td>0.152295</td><td>-0.064925</td><td>-6.91</td><td>2.88</td></tr></tbody></table>

In addition to completely accommodating the coherency strains, interface dislocations also give rise to unequally partitioned rotation fields, given in the case of Cu/Nb in the NW OR by

$$ \begin{aligned} \text{Cu}\hat{\Omega}_{\text{dis}}^{\infty} &= -0.072664 (-\hat{x} \otimes \hat{y} + \hat{y} \otimes \hat{x}) \\ \text{Nb}\hat{\Omega}_{\text{dis}}^{\infty} &= -0.096675 (\hat{x} \otimes \hat{y} - \hat{y} \otimes \hat{x}), \end{aligned} \qquad (3.53) $$

yielding a net non-vanishing rotation vector, i.e.

$$ \hat{\omega} = \text{Cu}\hat{\omega}^{\infty} - \text{Nb}\hat{\omega}^{\infty} = (-0.072664 - 0.096675) \hat{z} = -0.169339 \hat{z}, \qquad (3.54) $$

about the $\hat{z}$ tilt axis. The unequal partition of far-field rotations given by eqs. (3.53) shows that, to achieve the NW OR, the upper material in the reference state must be rotated by a rigid-body rotation through a tilt angle $\theta_{Cu} \sim -4.17^\circ$ about the tilt axis $\hat{z} \parallel z_{fcc} = [1\bar{1}0]_{fcc}$ to the Cu material in the natural state. In addition, the lower material must be rotated through a tilt angle $\theta_{Nb} \sim 5.55^\circ$ about the tilt axis $\hat{z} \parallel z_{bcc} = [100]_{bcc}$ to form the Nb material. Thus, the net rotation angle is $\sim 9.72^\circ$ about $\hat{z}$, as discussed in Ref. [106]. This result can be shown by computing the polar decomposition of eq. (3.47) such that $F_{Cu \to Nb} = R(\sim 9.72^\circ, [1\bar{1}0]_{fcc}) \cdot B$, i.e.

$$ \mathbf{R} = \begin{bmatrix} 0.992799 & -0.007201 & 0.119573 \\ -0.007201 & 0.992799 & 0.119573 \\ -0.119573 & -0.119573 & 0.985599 \end{bmatrix}, \quad \text{and,} \quad \mathbf{B} = \begin{bmatrix} 1.291296 & 0 & 0 \\ 0 & 1.291296 & 0 \\ 0 & 0 & 0.913084 \end{bmatrix}, \qquad (3.55) $$

with $B_{11} = B_{22} = \sqrt{2}/\lambda$, $B_{33} = 1/\lambda$ and the lattice parameter ratio $\lambda = a_{Cu}/a_{Nb}$. In eqs. (3.55), the matrix $\mathbf{R}$ corresponds to a rigid-body rotation matrix of angle $\sim 9.72^\circ$ about $[1\bar{1}0]_{fcc}$ and $\mathbf{B}$ is the Bain strain matrix [16, 295]. The compression axis for the Bain strain is $[1\bar{1}0]_{fcc} \parallel \hat{z}$, because $B_{33} < 1$.

Table 3.4 summarizes the main results of unequal partitioning of elastic strains and tilt rotations between the adjacent materials of Cu/Nb, Ag/V and Cu/Mo systems in the NW OR.

### 3.4.3 Spurious fields from incorrect reference states

As indicated in Table 3.3, the correct dislocation Burgers vectors for the Cu/Nb interface in the NW OR differ from what they would have been had the fcc crystal (Cu) been selected as the coherent reference state. Their directions differ by $\sim 2.11^\circ$, which affect the character of the interface dislocations. The magnitudes of the Burgers vectors in the fcc crystal and the correct reference state also differ, with $|b_j^{fcc}| : |b_j^{ref}| = 0.90$. The consequences of these deviations in character and magnitude may be seen in Fig. (3.14): a residual stress state in Cu persists with $\hat{\sigma}_{33}^{\infty} = -20.01$ GPa, corresponding to a residual strain state $\hat{\sigma}_{33}^{\infty} = -0.10$, as shown in Fig. (3.13). A residual stress field exists in Nb as well, with $\hat{\sigma}_{33}^{\infty} = 16.67$ GPa. Figure (3.14) illustrates the variations of the spurious stress field component $\hat{\sigma}_{33}^{\infty}$ in the neighboring materials as a function of $\delta$. This elastic field arises when an incorrect reference state is selected.

To emphasize the need for accounting for the unequal partitioning of elastic distortions, the coherency strain matrices is recomputed under the assumption that both sides of the interface have the same stiffness (i.e. homogeneous elasticity problem), equal to that of Cu, but with their
<figure><img src="image_31.png" /><figcaption>FIGURE 3.14: Dependence of the total far-field stress component $\hat{\sigma}_{33}^{\infty}$ on $\delta$ in the fcc and bcc materials for the Cu/Nb, Ag/V and Cu/Mo interfaces.</figcaption></figure>

natural (unequal) lattice parameters, as in the original calculation for the Cu/Nb interface. The results are in agreement with the well-known approximate calculation for equally partitioned strains due to simple geometrical considerations [123], i.e.

$$ \mathrm{c}_{\mathrm{Cu}}\hat{\mathbf{E}}_{\mathrm{c}}^{\mathrm{iso}} = -\mathrm{n}_{\mathrm{Nb}}\hat{\mathbf{E}}_{\mathrm{c}}^{\mathrm{iso}} = \begin{bmatrix} 0.026451 & 0.085660 & 0 \\ 0.085660 & 0.055845 & 0 \\ 0 & 0 & 0.127132 \end{bmatrix}, \quad (3.56) $$

with $\mathrm{c}_{\mathrm{Cu}}\hat{\mathbf{e}}_{33}^{\mathrm{iso}} = \mathrm{n}_{\mathrm{Nb}}\hat{\mathbf{e}}_{33}^{\mathrm{iso}} = (a_{\mathrm{Nb}} - a_{\mathrm{Cu}}/\sqrt{2})/(a_{\mathrm{Nb}} + a_{\mathrm{Cu}}/\sqrt{2})$ and a net rotation vector $\hat{\mathbf{\omega}}^{\mathrm{iso}} = -2 \times 0.085660 \hat{\mathbf{z}}$, corresponding to equipartitioning of rotations with tilt angles $-\vartheta_{\mathrm{Cu}} = \vartheta_{\mathrm{Nb}} \sim 4.91^{\circ}$.

In the nomenclature given by eqs. (3.48), the homogeneous anisotropic (or isotropic) case is associated with $\delta = 0.5$, as depicted by the vertical dotted lines in Figs. (3.13) and (3.14). The vertical dotted line in Fig. (3.14) shows a (non-zero) excess far-field stress state with $\mathrm{c}_{\mathrm{Cu}}\hat{\mathbf{\sigma}}_{33}^{\infty} = 3.69$ GPa in Cu and $\mathrm{n}_{\mathrm{Nb}}\hat{\mathbf{\sigma}}_{33}^{\infty} = -3.09$ GPa in Nb in the NW Cu/Nb interface or $\mathrm{c}_{\mathrm{Cu}}\hat{\mathbf{\sigma}}_{33}^{\infty} = -7.36$ GPa and $\mathrm{n}_{\mathrm{Nb}}\hat{\mathbf{\sigma}}_{33}^{\infty} = 19.08$ GPa in the NW Cu/Mo interface. Thus, even if the choice of equipartitioning of strains and (tilt) rotations is better than selecting the fcc material as the reference state, a spurious far-field stress field still remains. As a consequence, the associated dislocation structures for the homogeneous anisotropic (or isotropic) elasticity case of the Cu/Nb bicrystal are designated as non-equilibrium structures.

### 3.4.4 Orientations differing from the Nishiyama-Wassermann relations

Another commonly studied misorientation of interfaces between close-packed planes of neighboring {111} fcc and {110} bcc solids is the KS OR [154]. In the KS OR, one of the <110> directions in a fcc {111} plane lies parallel to one of the <111> directions in a bcc {110} plane. A schematic representation of a Cu/Nb interface in the KS OR is shown in Fig. (3.15a), where the bcc atoms have been rotated by 5.26° from their positions in the NW OR. The geometrical characteristics (line directions and spacings) of dislocation structures in the KS OR for the three cases are given in Table 3.5 and depicted in Fig. (3.15c).

To treat the KS OR and other ORs related to the NW by an in-plane twist, the rigid-body rotation matrix $\mathbf{R}(\theta)$ that rotates all bcc atoms in the natural state is introduced with respect to the fixed fcc atoms by angle $\theta$ about the interface normal $\mathbf{n}$. The NW OR corresponds to $\theta = 0^{\circ}$. The
<figure><img src="image_32.png" /><figcaption>FIGURE 3.15: Similar illustration as in Fig. (3.12), but for a Cu/Nb interface in the KS OR. Contour values (from the center of the patterns to the dislocation lines): {0, 0.2, 0.4, 0.6, 1.0, 1.4, 2.8, 4.8} J.m⁻².</figcaption></figure>

KS OR differs from the original NW OR by a twist rotation of angle $\theta \sim 5.26^{\circ}$ about the interface normal axis $n$.

To describe the relation between the natural and reference states for fcc/bcc in the in-plane twisted ORs, $f_{\text{fcc}}\mathbf{F}^{-1}$ and $b_{\text{bcc}}\mathbf{F}^{-1}$ in eq. (3.2) are replaced by $f_{\text{fcc}}\mathbf{R}(\kappa\theta)_{\text{fcc}}\mathbf{F}_{\text{NW}}^{-1}$ and $b_{\text{bcc}}\mathbf{R}(\kappa\theta)_{\text{bcc}}\mathbf{F}_{\text{NW}}^{-1}$, where $\kappa$ is a dimensionless parameter that varies from 0 to 1, such that $\mathbf{R}(\kappa\theta)$ is the rotation matrix that continuously adjusts the reference state in the KS OR from the one determined in the NW OR. This rotation matrix is expressed in the fcc ($x_{\text{fcc}}, y_{\text{fcc}}, z_{\text{fcc}}$) and bcc ($x_{\text{bcc}}, y_{\text{bcc}}, z_{\text{bcc}}$) systems by $_{\text{Cu}}\mathbf{R}(\kappa\theta)$ and $_{\text{Nb}}\mathbf{R}(\kappa\theta)$ in the Cu/Nb bicrystal, respectively. Equipartitioning of twist between the adjacent crystals occurs when $\kappa = 0.5$ [193, 123].

The condition that determines $\kappa$ is that the far-field rotations produced by the interface dislocations must be in accordance with the prescribed twist misorientation. The $\kappa$ value that satisfies this condition for Cu/Nb in the KS OR is $\kappa = 0.570897$, yielding unequal partitioning of the twist rotations $\theta_{\text{Cu}} \sim 3.20^{\circ}$ and $\theta_{\text{Nb}} \sim -2.06^{\circ}$. The correct Burgers vectors associated with this reference state are illustrated in Fig. (3.15b). If the approximation of equipartitioning of distortions is considered, i.e. $\kappa = 0.5$, the partitioning of rotations gives rise to $\theta_{\text{Cu}} = \theta_{\text{Nb}} = 2.63^{\circ}$, such that the dislocation characters differ by $\sim 0.57^{\circ}$ from the results obtained with the unequally partitioned distortions. This difference is not large because $\theta \sim 5.26^{\circ}$ is small, but the elastic (short- and long-range) fields may be significantly affected by deviations associated with larger twist rotations [123].

### 3.4.5 Short-range elastic fields

Although the far-field strains vanish when the correct reference state for ORs differing from the NW by an in-plane twist is used, the dislocation structures depicted in Figs. (3.12b) and (3.15b) nevertheless generate non-zero short-range strains and stresses. For instance, Fig. (3.16) plots
<table><caption>TABLE 3.5: Dislocation structures associated with Cu/Nb in the KS OR. See the caption of Table 3.3 for definitions of notation.</caption><thead><tr><th colspan="6">Dislocation structures in KS Cu/Nb</th></tr><tr><th colspan="6">* solutions by selecting the proper reference Burgers vectors</th></tr><tr><th>Cases</th><th>d<sub>1</sub> (nm)</th><th>d<sub>2</sub> (nm)</th><th>φ °</th><th>φ<sub>1</sub> °</th><th>φ<sub>2</sub> °</th></tr></thead><tbody><tr><td>c1 : {b&#x0302;<sub>1</sub><sup>ref</sup>, b&#x0302;<sub>2</sub><sup>ref</sup>}</td><td>0.9073</td><td>1.2394</td><td>22.04</td><td>21.06</td><td>65.00</td></tr><tr><td>c2 : {b&#x0302;<sub>1</sub><sup>ref</sup>, b&#x0302;<sub>3</sub><sup>ref</sup>}</td><td>2.1457</td><td>1.2394</td><td>62.54</td><td>61.57</td><td>57.02</td></tr><tr><td>c3 : {b&#x0302;<sub>2</sub><sup>ref</sup>, b&#x0302;<sub>3</sub><sup>ref</sup>}</td><td>2.1457</td><td>0.9073</td><td>40.51</td><td>2.45</td><td>79.05</td></tr></tbody></table>

stress components $\sigma_{21}$ and $\sigma_{22}$ for set 1 only and for both sets of dislocations of c1 for the Cu/Nb interface in the NW OR, as a function of $x'$ ($x' \perp \xi_1$) and $y$ ($\hat{y} \parallel n$), with $z = 0$. Negative values (compression) are plotted in light grey and the positive values (extension) in dark grey. The thick black lines show the locations where the stresses are equal to zero. The fields are asymmetric due to the material elastic anisotropy and the characters of the dislocation arrays.

Using these short-range fields at the interface, i.e. $y = 0$, the local self- and interaction energy densities are computed as a function of $x$ and $z$, as shown in Figs. (3.12c) and (3.15c) for all potential solutions predicted by the Frank-Bilby equation in the Cu/Nb NW and KS ORs, respectively. The unique solution of the Frank-Bilby equation is predicted by integrating the strain energy densities over each candidate solution and choosing the dislocation pattern with lowest elastic energy [258]. It is illustrated in the next section 3.4.6 that the present formalism predicts that c3 is in near perfect quantitative agreement with atomistic simulations for $\theta > 1°$. For instance, both approaches predict that Cu/Nb interface energy is minimized at $\theta = 2°$. The insets of Fig. (3.17) illustrates a qualitative comparison between the elasticity and atomistic calculations.

Using the minimum strain energy criterion for finding the likeliest dislocation structures, Fig. (3.17) plots the geometrical characteristics in terms of dislocation spacings, $d_i$ (in black), and characters, $\phi_i$ (light grey), for both sets of dislocations as a function of $\theta$ (between the NW and KS ORs). The geometry (i.e. dislocation spacing and character) of set 2 does not vary significantly as a function of $\theta$. In particular, the low spacing between misfit dislocations of set 2 is $d_2 \sim 1$ nm and is almost perfectly edge for $\theta = 2°$. On the other hand, the dislocation spacing and character of set 1 change markedly with $\theta$, e.g. from mixed dislocation character to almost perfectly screw character, and the dislocation spacing decreases almost by a factor 2. Set 1 is almost perfectly screw for $\theta = 4.75°$. The vertical line in Fig. (3.17) shows the lowest interface energy reported in Ref. [258] with the corresponding geometrical characteristics, i.e. dislocation spacings and characters. Surprisingly, this interface does not correspond to the interface with the largest dislocation spacings or nearly perfectly screw dislocation characters, contrary to what may be expected based on the theory of dislocations in uniform isotropic solids [7]. However, the approach predicts a dislocation structure with $d_1 = 3.5856$ nm, $\phi_1 = 24.37°$, $d_2 = 1.0426$ nm, $\phi_2 = 89.61°$, which is in agreement with the atomistic calculations [258].

### 3.4.6 Comparison with atomistic simulations

The present approach to interface design is to construct a mesoscale (as opposed to atomic-level) model that predicts misfit dislocation patterns with accuracy comparable to atomistic simulations, but at a fraction of the cost. The model is a reduced order model because it replaces the millions of variables associated with atomic positions with $\leq 15$ variables needed to describe misfit dislocations. The misfit dislocations are viewed as Volterra dislocations that have been inserted into the coherent reference state, suggesting that the total interface energies $\gamma$ be expressed as

$$ \gamma = \gamma_e(r_0) + \gamma_{\text{core}} + \gamma_{\text{relax}} + \dots \quad (3.57) $$
<figure><img src="image_33.png" /><figcaption>FIGURE 3.16: Contour plots of short-range stress component $\sigma_{21}$ and $\sigma_{22}$ for the Cu/Nb interface in the NW OR of c1, related to (a) the set 1, $\perp$, only and (b) both sets, $\perp$ and $\perp$, of interface dislocations. Contours with negative values (compression) are plotted in light gray while positive values (extension) are shown in dark gray. The thick black lines show the locations where stresses are zero.</figcaption></figure>

with $\gamma_e$ the elastic strain energy due to misfit dislocations from eq. (3.30), $\gamma_{\text{core}}$ the core energy, $\gamma_{\text{relax}}$ the energy part due to relaxations of the misfit dislocation network, and perhaps additional terms that have not yet been recognized. For the present purposes, it is not necessary to calculate the absolute value of $\gamma$, but rather only differences in $\gamma$ between the candidate solutions of the Frank-Bilby equation.

The outputs of the elasticity-based model are compared with atomistic calculations, which provide an opportunity for rigorous validation of the elasticity theory of dislocations. They are also convenient for atomistic simulations because embedded atom method potentials are available for several fcc/bcc binaries. The elasticity-based model is validated against the interface compositions: Cu/Nb [72], Ag/V [281], Cu/Fe [174], and Cu/Mo [103]. These choices fix the elastic constants, crystal structures, and lattice parameters of the adjoining constituents. Because attention is restricted to interfaces along fcc $\langle 111 \rangle$ and bcc $\langle 110 \rangle$ planes, only one crystallographic DoF remains to be specified: the twist angle $\theta$ describing the relative rotation of the crystals parallel to the interface plane. The $\theta$ is measured with respect to the NW OR, where a bcc $\langle 100 \rangle$ direction is parallel to a fcc $\langle 110 \rangle$ direction, such that $\theta = \pi/3 - \cos^{-1}(1/\sqrt{3}) \sim 5.26^\circ$ yields the KS OR. Due to the symmetry of the interface planes, all crystallographically distinct interfaces fall within $0^\circ \le \theta \le 15^\circ$. However, the analysis limited to $0^\circ \le \theta \le 10^\circ$ because for greater twists, misfit dislocations are too closely spaced to characterize reliably in atomic models.

For any composition and $\theta$, the Frank-Bilby equation has three distinct candidate solutions, as illustrated in Fig. (3.15b), which corresponds to one of three combinations of interfacial Burgers vectors, as described in the previous sections. The first candidate, termed "case 1" ($\equiv c1$), uses Burgers vectors $b_1$ and $b_2$. "Case 2" ($\equiv c2$) and "case 3" ($\equiv c3$) use Burgers vectors $b_1, b_3$, and $b_2, b_3$, respectively. Using the elasticity-based model, $\gamma_e$ of all three cases is computed for each composition and $\theta$ of interest. For all interfaces, the atomic-scale models are also constructed by joining cylindrical fcc and bcc blocks following the required interface crystallography. The models are large enough to contain a representative area of the misfit dislocation pattern and to avoid elastic images from free surfaces.

Figure (3.18a) compares $\gamma_e$ from the elasticity-based model with $\gamma$ from atomistic simulations for Cu/Nb interfaces. Because the relative energies of the three cases are the key quantities for
<figure><img src="image_34.png" /><figcaption>FIGURE 3.17: Dislocation spacings and characters predicted by the Frank-Bilby equation for both sets of dislocations in the Cu/Nb interface as a function of $\theta$ (from the NW, i.e. $\theta = 0^{\circ}$ to the KS, i.e. $\theta \sim 5.26^{\circ}$, ORs). The red line corresponds to the lowest energy interface for $\theta = 2^{\circ}$, reported in Ref. [258]. In insets: comparison of the dislocation geometries in the minimum energy state computed by the elasticity and atomistic approaches.</figcaption></figure>

comparison, both the elasticity-based model and atomistic data are shifted so that their energy minima occur at 0 J/m². The elasticity-based model predicts that case 3 has lowest $\gamma_e$ for all $\theta$. Furthermore, $\gamma_e$ for case 3 is in near perfect quantitative agreement with $\gamma$ for $\theta > 1^{\circ}$. Figure (3.18b) shows a similar comparison for Ag/V interfaces. Here, the elasticity-based model predicts that case 1 has lowest $\gamma_e$ for all $\theta$ outside $4.25^{\circ} < \theta < 5.25^{\circ}$, where $\gamma_e$ is lowest for case 2. $\gamma_e$ and $\gamma$ are in qualitative agreement over the entire twist angle range and in quantitative agreement for $\theta > 5^{\circ}$. As described in the Supplementary Note from Ref. [258], it is found comparable agreement between the elasticity-based model and atomistic interface energies for the remaining two compositions. Agreement between $\gamma_e$ and $\gamma$ is not sufficient to validate the present formalism. For that, it must be determined whether the lowest energy cases predicted by the elasticity-based model match the misfit dislocation patterns in atomistic simulations. Each of the three Frank-Bilby solutions predicts a different misfit dislocation pattern and therefore also a different disregistry. The present goal is to compare the disregistries of all three cases with that found in atomistic simulations. The model is validated if the case with lowest $\gamma_e$ has the best match with the atomistic disregistry. As shown in Figs. (3.18a) and (b), and detailed in Ref. [258], the disregistry analysis is in agreement with the elastic predictions for all Cu/Nb and Ag/V interfaces (circle filled with light grey) except Cu/Nb at $\theta = 0^{\circ}$. The disagreement is attributed to the reconstruction of the misfit dislocation network that is known to occur at that interface [272], which can be treated by further extensions from section 3.6. One further case of disagreement where dislocation network reconstruction occurs is found for Cu/Mo at $\theta = 0^{\circ}$ (see Supplementary Note). However, the agreement between the elasticity-based model and the atomistic models is excellent, overall.
<figure><img src="image_35.png" /><figcaption>FIGURE 3.18: Interface energies computed as a function of θ using the elasticity-based model (designated by ROMM as "Reduced Order Mesoscale Model") and atomistic modeling for (a) Cu/Nb and (b) Ag/V. Filled circles indicate atomic models whose disregistry was analyzed. The ringed numbers next to them state the case that best matches the atomic disregistry. Ad hoc parameters P, Q, and R for (c) Cu/Nb and (d) Ag/V.</figcaption></figure>

The general approach may be compared with several ad hoc parameters proposed previously to determine which of the cases predicted by the Frank-Bilby equation is likeliest. Bollmann suggested that the likeliest case minimizes [32]

$$P = \sum_{i} \frac{b_{i}^{2}}{d_{i}^{2}}, \qquad (3.58)$$

which is analogous to the Frank rule for predicting dislocation reactions [7]. Similarly, Ecob and Ralph propose two parameters [82] to distinguish between cases, defined by Q and R, as follows

$$Q = \sum_{i} \sum_{j} \frac{b_{i}b_{j}}{d_{i}d_{j}}, \quad \text{and,} \quad R = \sum_{i} \sum_{j} \sqrt{\frac{b_{i}b_{j}}{d_{i}d_{j}}}, \qquad (3.59)$$

using geometrical arguments for the energy of semicoherent interfaces. Figures (3.18c) and (d) plot these parameters for Cu/Nb and Ag/V interfaces. Comparing with Figs. (3.18a) and (b), none of them predicts the misfit dislocation patterns seen in atomistic models. For example, for Cu/Nb, all three parameters favor case 2, while the true interface structure is case 3. The elasticity-based model is therefore viewed as superior to these parameters and as validated for the purpose of
computational design of patterned interfaces.

## 3.5 Application to the sink strength of semicoherent interfaces

Clean, safe, and economical nuclear energy requires new materials capable of withstanding severe radiation damage. One way of removing radiation-induced defects is to provide a high density of sinks, such as GBs or heterophase interfaces [227] that continually absorb defects as they are created. This motivation underlies ongoing exploration of the radiation response of nanocomposite materials [74, 57], due to the large total interface area per unit volume they contain. These investigations have demonstrated wide variations in sink behavior of different interfaces. Some easily absorb defects, preventing damage in neighbouring material, but become damaged themselves [111]. Others are poor sinks for isolated defects, but excellent sinks for defect clusters [75]. The sink behavior of yet others changes with radiation dose [15, 14]. This wide variety of radiation responses prompts the physicists to ask:

* Are some specific interfaces best suited to mitigate radiation damage?

* Is it possible to identify them without resorting to resource-intensive irradiation experiments?

Here it is demonstrated that elastic interactions between point defects and semicoherent interfaces lead to a marked enhancement in interface sink strength. The conclusions stem from simulations that integrate first principles, object kinetic Monte Carlo, and anisotropic elasticity calculations. Surprisingly, the enhancement in sink strength is not due primarily to increased thermodynamic driving forces [144, 133], but rather to reduced defect migration barriers, which induce a preferential drift of defects towards interfaces. The sink strength enhancement is highly sensitive to the detailed character of interfacial stresses, suggesting that "super-sink" interfaces may be designed by optimizing interface stress fields. These findings motivate a computational search for "super-sink" interfaces: ones that optimally attract, absorb, and annihilate radiation-induced defects.

### 3.5.1 Computational multi-model strategy

To answer the aforementioned questions, an improved computational method for rapidly assessing the vacancy and interstitial sink strength of semicoherent interfaces is proposed. This method builds on the interfacial dislocation-based model for elastic fields of heterophase bicrystals, previously described. Such interfaces are of particular interest because many of them contain a high density of defect trapping sites [73, 220]. Moreover, semicoherent interfaces generate elastic fields that interact directly with radiation-induced defects [255]. These elastic fields have an unexpectedly large influence on interface sink strength, as quantified by the following computational multi-model approach.

#### Elastic dipole tensor calculation

Defect P-tensors are calculated using VASP [149], a plane wave-based, first principles density functional theory code. A fcc supercell containing 256 ± 1 atoms (+1 and −1 for interstitial and vacancy, respectively) is used. Calculations are also performed LAMMPS [207] classical potential simulations using embedded atom method potentials for Ag [88] and Cu [187] to study the convergence of the elastic dipole tensors up to supercell sizes of 2048 atoms. The discrepancy in the elastic P-tensor components between the 256-atom supercell and that of 2048-atom supercell is found lower than 4%. This supercell size ensures the convergence of defect formation energies to within few meV, as detailed in the Supplementary Note from Ref. [255]. The 256-atom density functional
theory simulations is therefore viewed as well converged with respect to model size. A 3 × 3 × 3 shifted Monkhorst-Pack K-point grid mesh, a Hermite-Gaussian broadening of 0.25 eV [184], and a plane wave cutoff energy of 400 eV are used. The change of the elastic dipole tensors is less than 0.5% compared to tighter settings. The Perdew-Burke-Ernzerhof [205] exchange-correlation functional is conveniently used within the projector-augmented-wave approach [150]. The structures are internally relaxed with a force convergence criterion of 10⁻³ eV/Å. The climbing image nudged elastic band method [117] is employed to find the saddle points for defect migration.

## Object kinetic Monte Carlo algorithm

The defect diffusion is investigated by using an object kinetic Monte Carlo code with a residence time algorithm to advance the simulation clock [38, 101]. At time t, the time step is chosen according to $\Delta t = -(\ln r_1)/w_{tot}$, where r₁ is a random number with $r_1 \in [0, 1]$ and $w_{tot}$ is the sum of frequencies of all events that may occur at t, i.e. $w_{tot} = \sum_{i}^{N} w_i$. The chosen event j is such that $\sum_{i}^{j-1} w_i < r_2 w_{tot} \le \sum_{i}^{j} w_i$, where r₂ is another random number with $r_2 \in [0, 1]$.

Three kinds of events are considered in the simulations: the jump of a point defect from one stable point to a neighbouring one, the absorption of a defect by an interface, and the creation of a new point defect through irradiation. Jump frequencies are given by $w_i = \nu \exp(-\Delta E_i / (kT))$, where $\nu$ is an attempt frequency and $\Delta E_i = E_i^{\text{sad}} - E_i^{\text{sta}}$ is the energy difference between the saddle position and the initial stable position of the jump considered. The stable point energy is

$$E_i^{\text{sta}} = - \sum_{k,l} P_{kl,i}^{\text{sta}} \varepsilon_{kl}^{\text{int}}(\mathbf{r}_i^{\text{sta}}), \qquad (3.60)$$

while the saddle point energy is

$$E_i^{\text{sad}} = E^{\text{m}} - \sum_{k,l} P_{kl,i}^{\text{sad}} \varepsilon_{kl}^{\text{int}}(\mathbf{r}_i^{\text{sad}}), \qquad (3.61)$$

with $E^{\text{m}}$ the migration energy in the absence of elastic interactions. Here, $\mathbf{P}^{\text{sta}}$ and $\mathbf{P}^{\text{sad}}$ are the defect P-tensors in the ground state and saddle point configurations, respectively. For simplicity, the position of the saddle point $\mathbf{r}_i^{\text{sad}}$ is taken mid-way between the two stable points explored by the jump [236].

The defect is considered to have been absorbed by an interface if it reaches the nearest atomic row to the interface. It is then simply removed from the simulation. This absorption condition is used to obtain a first estimate of sink strength, without taking into account the diffusion of point defects along interfaces or their possible reemission. The irradiation rate is fixed at the beginning of each simulation to keep the average number of point defects equal to 200 in the material where the measurements are performed, if no elastic interactions are considered. The actual number of point defects in the system, averaged over the simulation time when steady state is reached, constitutes the basis for the sink strength calculation.

The concentration of defects is recorded every 10⁴ iterations, after the concentration has become stationary. At the end of the simulation, an estimate of the average defect concentration $\bar{C}$ is computed by averaging over the values $C_j$, with $j = 1, ..., n$, as follows

$$\bar{C}_n = \frac{1}{n} \sum_{j=1}^{n} C_j. \qquad (3.62)$$

The final time is adjusted to obtain sufficient accuracy on $\bar{C}$ and thus on the associated sink strength $k^2$ in accordance with the mean field rate theory formalism [301]. For this purpose, the
<figure><img src="image_36.png" /><figcaption>FIGURE 3.19: Schematic illustration of the diffusion of radiation-induced point defects (illustrated by ovals) to interfaces under the influence of interface elastic fields. In general, materials A and B may be any two crystalline solids. In the present work, they are chosen to be either Cu or Ag.</figcaption></figure>

estimation of the error on the concentration is given by the standard error of the mean value, i.e.

$$ \delta \bar{C}_n = \frac{\sigma_n}{\sqrt{n}}, \qquad (3.63) $$

where

$$ \sigma_n^2 = \frac{1}{n-1} \sum_{j=1}^{n} (C_j - \bar{C}_n)^2 . \qquad (3.64) $$

The final time for each system is chosen so that the relative error on $\bar{C}$ and $k^2$ is less than 0.5%.

### 3.5.2 Kinetic Monte Carlo simulations with elastic interactions

Modelling the removal of radiation-induced point defects at sinks is a challenging task: on one hand, the variety and complexity of defect behaviors call for the flexibility of atomistic modelling. On the other, the relatively slow, thermally activated mechanisms of defect motion require longer simulation times than may be reached using conventional atomistic techniques, such as molecular dynamics. The object kinetic Monte Carlo (OKMC) method [38, 101, 56, 135] is employed, which is well suited to modeling long-time, thermally activated processes yet is also able to account for nuances of defect behavior uncovered through atomistic modeling.

Figure (3.19) illustrates the setup of the simulations containing two crystalline layers—A and B—separated by semicoherent interfaces. Periodic boundary conditions are applied in all directions, so each model contains two A-B interfaces. Due to their inherent internal structure, the interfaces create characteristic stress fields in the neighbouring crystalline layers. These stress fields interact with radiation-induced point defects, modifying their diffusion.

The interface stress fields is computed by the approach discussed in section 3.2. For illustration, two specific interfaces are treated in the present work: a low-angle twist GB on a (001) plane in Ag and a pure misfit (zero misorientation) heterophase interface between (001) planes of Ag and Cu. Figure (3.20a) shows a plan view of the Ag twist GB, where the adjacent GB planes have been rotated by $\pm\theta/2$ ($\theta$: twist angle). The boundary plane contains two sets of parallel, pure screw dislocations: one aligned with the $x = [110]$ direction and the other with the $y = [\bar{1}10]$ direction. For a relative twist angle of $\theta = 7.5^{\circ}$, the spacing between dislocations within each set is $\sim 2.2$ nm.
<figure><img src="image_37.png" /><figcaption>FIGURE 3.20: Planar semicoherent interfaces with identical misfit dislocation arrangements in (a) Ag twist GB with pure screw dislocations and (b) a Ag/Cu misfit interface with pure edge dislocations.</figcaption></figure>

Figure (3.20b) shows the interface plane of the Ag/Cu pure misfit interface. Similar to the twist boundary in Fig. (3.20a), this interface also contains two sets of parallel dislocations aligned with the $x = [110]$ and $y = [\bar{1}10]$ directions. Furthermore, the spacing between dislocations in the Ag/Cu interface is the same as in the twist boundary of Fig. (3.20a): ~ 2.2 nm. However, unlike in the twist boundary, both sets of dislocations in the misfit interface are of pure edge type.

The two interfaces in Fig. (3.20) have identical dislocation arrangements, but different dislocation characters. Thus, they contain identical dislocation densities, but have differing stress fields. For instance, all normal stress components for the twist GB are zero throughout the entire bicrystal. This stress field is therefore purely deviatoric. By contrast, due to symmetry, the shear stress $\sigma_{12}$ is everywhere zero for the Ag/Cu interface, but all of its other stress components are in general non-zero. In particular, this interface generates significant hydrostatic stresses. These differences have important implications for interface-defect interactions and defect migration pathways.

The force dipole moment approximation is used to compute elastic interaction energies between point defects and interfaces, $E^{\text{PD/int}}$ [140, 226, 71]:

$$E^{\text{PD/int}} = -P_{ij} \epsilon_{ij}^{\text{int}}(x,y,z). \qquad (3.65)$$

Here, $\epsilon_{ij}^{\text{int}}(x,y,z) = E_{ij}(x,y,z)$ are the short-range components of the previously calculated interface strain field, given by eq. (3.12a). On the other hand, $P_{ij}$ are the components of the elastic dipole tensor (the "P-tensor"), which describes the elastic fields generated by a point defect. $E^{\text{PD/int}}$ values are used to compute stress-dependent energy barriers for defect migration at each location in the simulation cell. A similar approach has been adopted in previous OKMC studies to describe point defect interactions with dislocations [228, 236].

The density functional theory is used to calculate P-tensors for two types of point defects in Ag and Cu: vacancies and self-interstitials of lowest formation energy, namely $\langle 100 \rangle$-split dumbbells [242]. The P-tensor values for these defects are obtained in their ground states as well as at their saddle point configurations during migration (found using the climbing image nudged elastic band method [117]). Starting from a simulation cell containing a perfect, stress-free crystal, the point defect of interest is inserted in the desired location and relax the atom positions while keeping the simulation cell shape fixed. The point defect induces stresses, $\sigma_{ij}$, in the simulation cell. They are related to the defect P-tensor through

$$P_{ij} = V \sigma_{ij} = P_{ij}^{\text{d}} + p^{\text{h}} \delta_{ij}, \qquad (3.66)$$
<table><caption>TABLE 3.6: Elastic dipole tensors P-tensors (in eV) of point defects from first principles for a (100)-split dumbbell self-interstitial and a vacancy in Ag and Cu at both the ground state and saddle point configurations. The ground state interstitial is oriented in the [100] direction. Its saddle point configuration is for a [100]-to-[010] migration path. The vacancy saddle point is for migration along the [110] direction.</caption><thead><tr><th rowspan="2">Element</th><th colspan="3">Interstitial</th><th colspan="3">Vacancy</th></tr><tr><th colspan="3">Ground state</th><th colspan="3">Ground state</th></tr><tr><th></th><th colspan="3">Saddle point</th><th colspan="3">Saddle point</th></tr></thead><tbody><tr><td rowspan="3">Ag</td><td>26.80</td><td>0</td><td>0</td><td>26.69</td><td>2.59</td><td>0</td><td>-3.04</td><td>0</td><td>0</td></tr><tr><td>0</td><td>26.86</td><td>0</td><td>2.59</td><td>26.69</td><td>0</td><td>0</td><td>-3.04</td><td>0</td></tr><tr><td>0</td><td>0</td><td>26.86</td><td>0</td><td>0</td><td>27.74</td><td>0</td><td>0</td><td>-3.04</td></tr><tr><td rowspan="3">Cu</td><td>17.46</td><td>0</td><td>0</td><td>18.01</td><td>1.78</td><td>0</td><td>-3.19</td><td>0</td><td>0</td></tr><tr><td>0</td><td>17.66</td><td>0</td><td>1.78</td><td>18.01</td><td>0</td><td>0</td><td>-3.19</td><td>0</td></tr><tr><td>0</td><td>0</td><td>17.66</td><td>0</td><td>0</td><td>18.46</td><td>0</td><td>0</td><td>-3.19</td></tr></tbody></table>

<figure><img src="image_38.png" /><figcaption>FIGURE 3.21: Elastic interaction energy between (a) an interstitial with the Ag twist GB ($E^{\text{PD/int}} < -0.002$ eV in the blue isovolume), and between the Ag/Cu misfit interface with (b) an interstitial and (c) a vacancy ($E^{\text{PD/int}} < -0.06$ eV in the blue isovolume; $E^{\text{PD/int}} > 0.06$ eV in the red; gray contours are locations with zero interaction energy).</figcaption></figure>

where V is the simulation cell volume. $P_{ij}^d$ and $p^h$ are the deviatoric and hydrostatic (isotropic) P-tensor components, respectively. The former is associated with a pure shear (no volume change) while the latter is related to isotropic tension (interstitials) or compression (vacancies), which leads to a volume change.

Table 3.6 lists the P-tensors used in the present study. All of them are expressed in the Nye frame, where the X-, Y-, and Z-axes are aligned with the [100], [010], and [001] Miller index directions, respectively. The form of the P-tensor reflects the symmetry of the corresponding defect. Thus, the P-tensor for a vacancy in its ground state is isotropic while that of an interstitial is tetragonal. P-tensors for defect orientations other than those given in Table 3.6 may be calculated using coordinate system rotations. The P-tensors for (100)-split dumbbell self-interstitials and vacancies in Cu agree with experimental data [114, 242, 285]. Furthermore, the present calculations of relaxation volumes of a vacancy in Ag and Cu are in very good agreement with recent ab-initio predictions [191].

Figure (3.21) shows the distribution of ground state interstitial and vacancy interaction energies with the Ag twist GB and the Ag/Cu misfit interface. A (100)-split dumbbell interstitial may take on three different orientations. Figure (3.21) uses the orientation with lowest $E^{\text{PD/int}}$. For the Ag twist GB, interstitial interaction energies are negative at all locations, as shown in Fig. (3.21a). Thus, all interstitials in the vicinity of this GB experience a thermodynamic driving force to migrate towards the boundary. The interstitials, however, have nearly isotropic P-tensors (see Table 3.6), so their interaction energies with the Ag twist GB are very small. The interaction
energy of vacancies with the Ag twist GB is everywhere zero due to the absence of hydrostatic
stresses near this interface. However, the anisotropy of the vacancy saddle point configuration
leads to non-zero interaction energies of migrating vacancies with the GB.

Interstitial interaction energies near the Ag/Cu misfit interface, shown in Fig. (3.21b), may be attractive or repulsive, depending on the location of the defect. Thus, interstitials in Ag are expected to migrate towards the center of the dislocation pattern while those in Cu are expected to migrate to dislocation cores. Figure (3.21c) shows the interaction energy between vacancies and the Ag/Cu misfit interface. The spatial variation of this interaction energy is similar to that of the interstitials, but with opposite sign.

The OKMC simulations assume a constant, uniform defect creation rate, G. Defects diffuse until they are absorbed by an interface. Only individual interstitials or vacancies are tracked in the simulations: defect reactions, such as clustering or recombination, are not considered. After a certain simulation time, defect distributions reach a steady state, whereupon the defect concentration is computed as a function of position along the z-direction (normal to the interface plane) based on the time spent by each defect on a given atomic site.

3.5.3 Effect of elastic interactions on interface sink strength

Figure (3.22) shows steady-state vacancy and interstitial concentrations for the two types of inter-faces described above for models with 10 nm-thick Ag and Cu layers. In the absence of elastic interactions between defects and interfaces, steady-state defect concentrations may be computed analytically, which are successfully compared with the simulation results.

Elastic interactions have a dramatic effect on defect concentration profiles. In all cases shown in Fig. (3.22) except vacancies near Ag/Cu interfaces, there are nearly no defects within ~ 2 nm-wide zones adjacent to the interfaces. By contrast, without elastic interactions, defect concentrations are zero only at the interfaces themselves. Moreover, even though defect-interface elastic interaction energies are negligible beyond ~ 2 nm, the zones depleted of defects near the interfaces have a pronounced effect on defect concentrations throughout the entire layer, markedly reducing the average defect concentration. For the simulations in Fig. (3.22), elastic interactions reduce defect concentrations by about a factor of two even in the middle of the layers. This effect is even more pronounced for thinner layers. For vacancies in Ag/Cu, local traps are responsible for the sharp increase in concentration near the interface.

The simulations account for numerous aspects of defect-interface elastic interactions, such as
defect anisotropy or differences in defect ground state and saddle point properties. To discover
which ones are primarily responsible for the defect concentrations shown in Fig. (3.22), some
of these characteristics are artificially "switched off" and repeated the OKMC simulations to see
whether doing so changes the steady-state defect concentrations. These calculations demonstrate
that the anisotropy of the **P**-tensor in the saddle point configurations is primarily responsible for
the reduced defect concentrations in Figs. (3.22a) and (3.22b).

The saddle point anisotropy is "switched off" by replacing the saddle point **P**-tensor with
**P**<sub>sad</sub> = *p*<sup>h</sup><sub>sad</sub> **I**, where **I** is the identity matrix and *p*<sup>h</sup><sub>sad</sub> is one third of the trace of the true saddle
point **P**-tensor. This assumption is tantamount to modelling defects at saddle points as misfitting
spherical inclusions in isotropic media. Concentration profiles obtained with this approximation
are markedly different from the anisotropic case, as shown in Fig. (3.22). In the case of the Ag
twist GB (Figs. (3.22c) and (3.22d)), isotropic saddle points yield the same defect concentrations as
when there are no defect-interface interactions at all. Indeed, since the twist interface generates no
hydrostatic strain field, only the deviatoric components of defect **P**-tensors may interact with these
interfaces. Ground state vacancies have zero deviatoric **P**-tensor components, so the interaction
energy with the Ag twist GB vanishes, similar to ground state interstitials with nearly isotropic
**P**-tensors (Table 3.6). The same conclusions hold at saddle positions if saddle point anisotropy
<figure><img src="image_39.png" /></figure>

FIGURE 3.22: Steady-state point defect concentrations as a function of location normal to interface planes. The black vertical lines represent the interface planes, while the continuous gray lines denote the reference case with no elastic interactions, computed analytically. OKMC results for both isotropic (orange) and anisotropic (blue) saddle point configurations are shown. (a) Vacancy and (b) interstitial profiles near Ag/Cu pure misfit interfaces. (c) Vacancy and (d) interstitial profiles near Ag twist GBs. Concentrations are normalized by the average concentration $\bar{C}$ obtained when no elastic interactions are taken into account.

is "switched off", as described above. Elastic interactions then do not affect migration energies, explaining why defect concentrations are identical to the case without elastic interactions.

For the Ag/Cu interface, concentration profiles computed without saddle point anisotropy lie between the non-interacting and fully anisotropic cases, as shown in Figs. (3.22a) and (3.22b). Vacancy concentrations are only marginally lower than the non-interacting case (Fig. (3.22a)), demonstrating the overriding importance of saddle point anisotropy in their behavior. Interstitial concentrations obtained without saddle anisotropy lie approximately mid-way between the fully anisotropic and non-interacting cases (Fig. (3.22b)), demonstrating that saddle point anisotropy is at least as important to their behavior as are $p\Delta V$ interactions, which are more commonly investigated.

Figure (3.23) gives a more detailed view of defect concentrations at different locations in the Ag layer of the Ag/Cu interface and in the Ag twist GB. Close to these interfaces, concentrations vary as a function of location parallel to the interface plane, following the strain field pattern created by the interfaces. Indeed, the strain field creates preferential paths for defect migration, as shown by the gray trajectories in Fig. (3.23). These paths are in general different for interstitials and vacancies. For both the Ag/Cu interface and Ag twist GB, vacancies preferentially migrate to the dislocation lines, while interstitials are mostly absorbed between dislocations. This preferential, non-random walk drift of point defects to specific locations is responsible for the enhanced interface sink strengths. Knowing the steady-state defect concentrations obtained by OKMC, sink strengths are derived for the two interfaces considered above. In the mean field rate theory formalism [41], "sink strengths" quantify the ability of sinks, such as interfaces, to absorb defects.
<figure><img src="image_40.png" /><figcaption>FIGURE 3.23: Preferential migration paths and local concentrations of (a) vacancies and (b) interstitials on the Ag side of the Ag/Cu interface and of (c) vacancies and (d) interstitials in the Ag twist GB. Migration paths are shown as gray lines originating from 1 nm away from the interface. The square grid of black lines represents interface dislocations. Concentrations are plotted in a plane located two atomic distances away from the interface. The concentrations are normalized by $\bar{C}$: the average concentration when no interactions are considered. Any normalized concentration values higher than 0.015 are shown as equal to 0.015.</figcaption></figure>

Within this formalism, the evolution equation for the average defect concentration, $\bar{C}$, follows

$$ \frac{d\bar{C}}{dt} = G - k^2 D \bar{C}, \qquad (3.67) $$

where G is the defect creation rate and D is bulk defect diffusivity. The second term on the right hand side is related to the loss of defects at sinks with associated sink strength, $k^2$. At steady state, the sink strength may be computed from the average concentration:

$$ k^2 = \frac{G}{D\bar{C}}. \qquad (3.68) $$

Using the average of the concentration profile computed for defect removal at interfaces in the absence of elastic interactions, the interface sink strength is analytically found to be $k^2 = 12/d^2$ [46]. When interactions between interfaces and defects are present, the sink strength is numerically determined through eq. (3.68), by using the average steady-state concentration obtained by OKMC simulations and the diffusion coefficient without elastic interactions. The resulting vacancy and interstitial sink strengths for both interfaces are shown in Fig. (3.24a–f) as a function of layer thickness.

In all cases, the sink strength increases significantly when elastic interactions are taken into
<figure><img src="image_41.png" /><figcaption>FIGURE 3.24: Enhancement in sink strength of Ag/Cu interfaces and Ag twist GBs for (a–c) vacancies ($k_v^2$) and (d–f) interstitials ($k_i^2$) in a given layer (Ag or Cu), as a function of layer thickness, $d$. (g–i) Bias factors of Ag/Cu interface and Ag twist GB. The gray line corresponds to the analytical solution when no interaction is present ($k^2 = 12/d^2$). Orange and blue lines correspond to OKMC calculations without saddle point anisotropy and with the fully anisotropic interaction model, respectively.</figcaption></figure>

account. This effect is especially pronounced for thinner layers, as defects undergo elastic interactions with interfaces over a larger fraction of the layer. It is particularly strong for interstitials, whatever the interface type, and for vacancies for the twist interface. These results also confirm the importance of saddle point anisotropy: by comparing with OKMC simulations that use isotropic saddle-point P-tensors, it yields order-of-magnitude increases in sink strength, in some cases.

Another quantity of interest for radiation response is the bias factor, $B$, which expresses the propensity of a given sink to absorb more interstitials than vacancies. It is defined as

$$B = \frac{k_i^2 - k_v^2}{k_i^2}, \qquad (3.69)$$

where $k_v^2$ and $k_i^2$ are the sink strengths for vacancies and interstitials, respectively. For example, small interstitial clusters and dislocations exhibit positive bias factors (typically between 0.01 and 0.3 [47, 116]) and thus absorb more interstitials than vacancies. The preferential absorption of interstitials by biased sinks leads to an excess of remaining vacancies, which cluster and eventually aggregate into voids [47, 178].

Bias factors for the semicoherent interfaces are shown in Fig. (3.24g–i). Values larger than 0.2
are obtained for the fully anisotropic interaction model in the case of the Ag/Cu interface. Such interfaces would compete for interstitials with dislocations. The presence of two sinks of differing bias magnitude has been given as a possible cause for void swelling suppression in ferritic steels [168]. Interestingly, for the Ag twist GB the bias factor is negative, meaning that these interfaces tend to absorb more vacancies than interstitials. Similar observations have been made in Ref. [229], where the bias factor for single screw dislocations is negative when using anisotropic elasticity theory and zero in the isotropic approximation. Such GBs may therefore deplete excess vacancy concentrations sufficiently to inhibit void nucleation.

## 3.6 Elastic strain relaxation in interfacial dislocation patterns

The interfacial dislocation-based model described in section 3.2 has been extended to investigate the equilibrium relaxed dislocation microstructures with specified constraints on semicoherent interfaces [249, 250]. The present parametric energy-based framework includes surface/interface stress and elasticity effects as additional constitutive relations, which are viewed as infinitely thin membranes in contact with each individual material, give rise to non-classical boundary conditions. The elastic field solutions are used to compute the corresponding strain energy landscapes for planar hexagonal-shaped configurations containing three sets of misfit dislocations with unextended three-fold nodes.

### 3.6.1 General considerations on hexagonal-shaped dislocation patterns

The mechanical dislocation-based problem for determining the elastic strain relaxation of interfacial patterns formed by joining two linear anisotropic elastic materials A and B is described by adopting specific notations and conventions in Fig. (3.25). In the global coordinate system (O, $x_1^{\text{or}}$, $x_2^{\text{or}}$, $x_3^{\text{or}}$), corresponding to the orientation relations along fixed crystal directions of the system of interest, the semicoherent interface is located at the coordinate $x_2^{\text{or}} = 0$, with $x_2^{\text{or}} > 0$ for material A, and $x_2^{\text{or}} < 0$ for material B. Such directions are not necessary related to high symmetry directions, so that the anisotropic elastic constants may be displayed in the most general form. In the present work, the unit vector normal to the interface is $n \parallel x_2^{\text{or}}$, and a coplanar free surface to the semicoherent interface is potentially introduced at $x_2^{\text{or}} = h_A$, whereas B is always a semi-infinite linear elastic crystal.

The crystallography of all interfaces is completely specified between close-packed planes of neighboring materials, so that both orientation relations of crystals A and B with relative misorientations (tilt and twist) and differing lattice parameters (misfit) are described using the previous concept of reference/natural states, as defined in section 3.2. As an example, the 2.5° Ta (tantalum) twist boundary is illustrated in Fig. (3.25a). In the reference state, the interface is coherent, but the interface is not coherent in the natural state, and the atomic structures of interfaces lead to the formation of periodic networks of misfit dislocations that may undergo local relaxations or reconstructions [94].

The closely related quantized Frank-Bilby equation [93, 30, 29] and the O-lattice theory [32] are crystallographic approaches used to describe intrinsic dislocation structures at semicoherent interfaces, which provide the interfacial dislocation geometries in terms of line directions and spacings for one, two, or three independent, planar, and uniformly spaced parallel sets of infinitely long straight dislocations. As illustrated in the previous sections, however, such purely geometrical approaches are not able to characterize local reactions of crossing dislocations to form dislocation segments with different Burgers vectors in mesh networks that are energetically favorable.

The extended formalism for predicting the interface dislocations arrays linking the quantized Frank-Bilby equation and anisotropic elasticity theory under the condition of vanishing the far-field stresses is used to identify the periodicity of the structures with two sets of dislocations from
<figure><img src="image_42.png" /><figcaption>FIGURE 3.25: Geometry of a hexagonal-shaped dislocation pattern containing three sets of interface dislocations with the associated individual Burgers vectors. (a) The orientation relationships between the adjacent linear materials are defined with respect to the global coordinate system (O, $x_1^{\text{or}}$, $x_2^{\text{or}}$, $x_3^{\text{or}}$), within which the semicoherent interface is located at $x_2^{\text{or}} = 0$. For illustration, the current intrinsic dislocation structure is associated with a planar $\{011\} \parallel n$ twist GB between two bcc crystals with a 2.5° rotation angle. (b) Anisotropic elasticity calculations are performed in the non-orthogonal (O, $x'_1 \parallel p_1^0$, $x_2 \parallel n$, $x'_3 \parallel p_2^0$) frame with fixed basis vectors, where $p_1^0$ and $p_2^0 \neq p_1^0$ are the O-lattice vectors that describe the periodicity of the dislocation structures. The fixed red and blue points characterize the initial lozenge-shaped unit cell and the pivot points for elastic strain relaxations, respectively. The grey points are related to the O-lattice points, separated by the networks of interfacial dislocations with three-fold dislocation junction nodes where the conservation law of Burgers vectors is satisfied, e.g. at the specific orange node $J_1$ that is parametrized by the dimensionless coordinates $(\eta_1, \eta_2)$. For convex hexagonal-shaped dislocation configurations, $J_1$ may move within the shaded triangular domain $T_{ABC}$ in dark grey.</figcaption></figure>

the pre-determined O-lattice vectors $p_1^0$ and $p_2^0 \neq p_1^0$, as illustrated in Fig. (3.25b). These two vectors characterize the initial lozenge-shaped unit cell of crossing dislocation sets (red points), for which the translations of the unit cell by the basis vectors $p_1^0$ and $p_2^0$ tessellate the entire interface plane. In the following, the superscript $\text{un}$ will be used to indicate quantities related to the unre-laxed dislocation configurations, e.g. $\xi_1^{\text{un}} \parallel p_2^0$ and $\xi_2^{\text{un}} \parallel p_1^0$ correspond to the initial dislocation directions of the two sets that consist of the lozenge-shaped patterns, with Burgers vectors $b_1$ and $b_2$, respectively, as stated in section 3.2. Planar energetically favorable interactions may lead to the formation of dislocation junctions with coplanar Burgers vector $b_3$, i.e.

$$b_1 + b_2 \rightarrow b_3, \qquad (3.70)$$

such that the current semicoherent interfaces contain infinite, planar, and periodic dislocation structures with three sets of misfit dislocations. As illustrated in Fig. (3.25b), the third newly formed set (in black) is associated with the junction formation due to the local rearrangements between two initial crossing dislocation arrays, shown by the blue and red dashed lines. The current directions of the three sets of misfit dislocations are denoted by $\xi_1$, $\xi_2$, and $\xi_3$ for which the latter is associated with the direction of the in-plane dislocation junctions.

The present reactions yield to hexagonal-shaped patterns with three-fold dislocation nodes, where the centers of the parent dislocation segments from the lozenge-shaped unit cells consist of pinning pivot points (blue points) for glissile planar dislocations. An useful triangular domain $T_{ABC}$ for performing parametric energy-based analyses, is represented by two blue pivot points (B and C) and the red intersection point A in which dislocation reactions occur, as shaded in dark grey in Fig. (3.25b). On the other hand, the newly formed representative hexagonal-shaped unit cell (light grey domain), which contains six vertices (dislocation nodes), indexed and ordered by
$J_1, J_2, J_3, J_4, J_5$, and $J_6$, is denoted by $\mathcal{H}_{J_1J_2J_3J_4J_5J_6}$. The determination of such infinitely repeated dislocation nodes with the type of rearrangement defined by eq. (3.70) produces neither orientation nor magnitude changes in the O-lattice vectors. Thus, the two-dimensional periodicity of the dislocation networks containing three families of straight parallel dislocation segments in the local Cartesian frame (O, $\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3$) with $\mathbf{x}_2 \parallel \mathbf{x}_2^{\text{or}} \parallel \mathbf{n}$, remains unchanged during the elastic strain relaxation processes.

In the previous non-orthogonal (oblique and fixed) frame with basis vectors (O, $\mathbf{x}'_1, \mathbf{x}_2, \mathbf{x}'_3$), where $\mathbf{x}'_1 \parallel \mathbf{p}_1^0 \parallel \xi_2^{\text{un}}$ and $\mathbf{x}'_3 \parallel \mathbf{x}_3 \parallel \mathbf{p}_2^0 \parallel \xi_1^{\text{un}}$, the oriented angle between $\xi_2^{\text{un}}$ and $\xi_1^{\text{un}}$ is denoted by $\phi^{\text{un}}$, so that $\mathbf{x}'_1 = \mathbf{x}_1 \csc \phi^{\text{un}}$ and $\mathbf{x}'_3 = \mathbf{x}_3 - \mathbf{x}_1 \operatorname{ctg} \phi^{\text{un}}$. Thus, any position vector in this non-orthogonal frame may be expressed as: $\mathbf{r} = \mathbf{x}'_1 \mathbf{p}_1^0 + \mathbf{x}'_3 \mathbf{p}_2^0 = (\mathbf{x}_1 \csc \phi^{\text{un}}) \mathbf{p}_1^0 + (\mathbf{x}_3 - \mathbf{x}_1 \operatorname{ctg} \phi^{\text{un}}) \mathbf{p}_2^0$. In particular, the mobile dislocation three-fold node of interest $J_1$, which is parametrized by the dimensionless coordinates $(\eta_1, \eta_2)$ in the first quadrant of the $(O, \mathbf{x}'_1, \mathbf{x}_2, \mathbf{x}'_3)$ frame, is also defined by: $\mathbf{j}_1 = \eta_1 \mathbf{p}_1^0 + \eta_2 \mathbf{p}_2^0$, with $(\eta_1, \eta_2) \in ]0, 1/2]^2$, excluding 0 and 1/2 to describe convex hexagonal-shaped patterns with six distinct dislocation edges. For example, the limiting case of equilibrium arrays with two sets of orthogonal misfit dislocations is given by: $\phi^{\text{eq}} = \pi/2$, $\eta_1^{\text{eq}} \to 1/2$, and $\eta_2^{\text{eq}} \to 1/2$, so that $J_2 \simeq J_3$ and $J_5 \simeq J_6$, as the (010) twist GBs in fcc materials. On the other hand, the regular equilibrium hexagonal network corresponds to the particular case where: $\phi^{\text{eq}} = \pi/3$, and $\eta_1^{\text{eq}} = \eta_2^{\text{eq}} = 1/3$, as the (111) twist GBs in fcc crystals.

### 3.6.2 Solution methodology for strain-relaxed rearrangements

During the non-random elastic strain relaxations without externally applied stresses, misfit dislocations are rearranged into hexagonal-shaped networks due to local reactions that lower the elastic strain energy at semicoherent interfaces [94, 7]. Such strain-relaxed rearrangements of interfacial dislocation patterns also involve the mechanical problem of finding the minimum-energy paths from a given initial non-equilibrium lozenge-shaped microstructure with two sets of parent misfit dislocations to new unique or multiple (with the same strain energy) stable equilibrium hexagonal-shaped dislocation patterns of lowest energies with possible metastable configurations.

Without changing the interface crystallographic characters upon the relaxation processes, the prescribed displacement jumps for each periodic hexagonal unit cell are also assumed to vary linearly with the (algebraic) directed distance between the O-lattice points (displayed by the grey points in Fig. (3.25b)) and the nearest neighbor interfacial dislocation segments. At the positions of the dislocation segments, the relative displacements are completely described by the directions and constant magnitudes of the associated individual Burgers vectors. Furthermore, the non-classical boundary conditions due to the free surface excess stress and the semicoherent interface excess stress contributions are therefore applied at: $\mathbf{x}_2^{\text{or}} = h_A$ and $\mathbf{x}_2^{\text{or}} = 0$, respectively. Thus, the minimum-energy paths are entirely obtained by measuring the removal of the short-range elastic strain energy with respect to the coordinates $(\eta_1, \eta_2)$ of $J_1$, along which the long-range elastic strain-free state is not altered by spurious non-zero far-field strains.

For a given crystallographic orientation relationship between materials A and B, the methodology for determining the equilibrium dislocation configurations for elastic strain relaxation processes along minimum-energy paths is described below. The two first items summarize the strategy procedure for computing the Burgers vectors of interface dislocations using anisotropic elasticity theory, which have been introduced in section 3.2.

1. The geometries in terms of dislocation spacings and line directions, i.e. $\xi_1^{\text{un}}$ and $\xi_2^{\text{un}}$, related to the initial lozenge-shaped patterns are found by using the quantized Frank-Bilby equation. For such networks containing two sets of straight, parallel, and infinite misfit dislocations, the periodicity of the dislocation structures is also obtained by mapping the O-lattice points at the interfaces. The corresponding computed O-lattice vectors $\mathbf{p}_1^0$ and $\mathbf{p}_2^0 \neq \mathbf{p}_1^0$ are conveniently associated with the fixed and non-orthogonal basis vectors of the $(O, \mathbf{x}'_1, \mathbf{x}_2, \mathbf{x}'_3)$ frame for elasticity analyses, where $\mathbf{x}'_1 \parallel \mathbf{p}_1^0 \parallel \xi_2^{\text{un}}$, $\mathbf{x}_2 \parallel \mathbf{n}$, and $\mathbf{x}'_3 \parallel \mathbf{p}_2^0 \parallel \xi_1^{\text{un}}$.
2. The reference state, within which the individual Burgers vectors of both dislocation sets are defined, i.e. $b_1$ and $b_2$, is determined by combining the Frank-Bilby equation with anisotropic elasticity theory that meets the constraints of interface crystallographic character and zero long-range strains (or stresses) for infinite bicrystals. Because the latter far-field condition is still fulfilled during the elastic strain relaxation processes, the third Burgers vector $b_3$ for the newly formed dislocation junctions is also obtained from the conservation eq. (3.70) of the Burgers vector content at the three-fold node $J_1$. In the limiting case where a coplanar free surface is located in material A, the reference state (and therefore also, the three Burgers vectors) is fully associated with material B, e.g. the case of a thin film on a semi-infinite substrate.

3. The specific triangular region $T_{ABC}$ in the representative lozenge-shaped unit cell, formed by the three fixed points A, B, and C in Fig. (3.25b), is discretized into four-node quadrilateral elements with respect to the $i^{th}$ nodal points with coordinates $(\eta_1^i, \eta_2^i)$, such that $\{\eta_1^i, \eta_2^i\} \in ]0, 1/2[^2$. This discretization allows to represent any convex hexagonal-shaped dislocation patterns in the non-orthogonal $(O, x'_1, x_2, x'_3)$ frame for mechanics-based calculations of elastic field solutions, e.g. displacements, stresses, traction forces, etc.

4. The elastic strain energy stored at semicoherent interfaces is computed at any mesh point $(\eta_1^i, \eta_2^i)$, by using the persistent short-range stress and strain field solutions for convex and irregular hexagonal-shaped dislocation configurations. Furthermore, the complete elastic energy landscape $\gamma_e(\eta_1, \eta_2)$ is interpolated for any $(\eta_1, \eta_2) \in ]0, 1/2]^2$ with the aid of standard finite element bilinear shape functions for four-node elements.

5. For energetically favorable reactions, the minimum-energy dislocation configurations are numerically obtained by using the conjugate gradient algorithm on the pre-computed energy landscapes with a given prescribed tolerance. Then, the nudged elastic band method [117, 222] is used to provide access to the minimum-energy paths between the initial (non-equilibrium) lozenge-shaped structures and the determined elastically relaxed dislocation patterns with the aid of the elastic forces: $f_e = -\nabla\gamma_e(\eta_1, \eta_2)$. In practice, all elastic field solutions are recomputed along the curvilinear reaction coordinates of the minimum-energy paths.

### 3.6.3 Parametric energy-based framework

This section is concerned with the complete expressions of elastic fields for hexagonal-shaped dislocation patterns located at heterophase interfaces between two dissimilar anisotropic materials. The Stroh sextic formalism of anisotropic linear elasticity combined with the surface/interface treatment in Ref. [108] and a Fourier series-based solution technique is therefore used to compute the elastic fields outside the cores of dislocations. In the general case, all surfaces of interest (i.e. semicoherent interfaces and free surfaces) are distinctly considered as infinitely thin membranes with different, separate, and appropriate constitutive equations than the relations for both (bulk) materials A and B.

Again, the pre-subscripts A and B in the elastic properties and also the field expressions will be omitted for clarity in the following if no distinction between materials is required.

#### Elastic field equations and solutions in bulk materials

In the fixed Cartesian coordinate system $(O, x_1, x_2, x_3)$, the three-dimensional stress field $\sigma(x) = \sigma_{ij}(x_1, x_2, x_3)$ and the displacement field $u(x) = u_i(x_1, x_2, x_3)$ in both crystals A and B are related by the Hooke's law in index form from Eq. (3.12b), as follows

$$ \sigma_{ij}(x_1, x_2, x_3) = c_{ijkl} u_{k,l}(x_1, x_2, x_3), \qquad (3.71) $$
where a comma stands for differentiation, with repeated indices denoting summation convention ranging from 1 to 3, unless stipulated otherwise. The anisotropic elastic constants of the fourth-order stiffness tensor C are fully symmetric, i.e. $c_{ijkl} = c_{jikl} = c_{ijkl} = c_{klij}$, and the classical partial differential eq. (3.8) of mechanical equilibrium that is fulfilled in both crystals in terms of the displacement fields is given by

$$ \sigma_{ij,j}(x_1, x_2, x_3) = c_{ijkl} u_{k,jl}(x_1, x_2, x_3) = 0. \quad (3.72) $$

According to eq. (3.7), the complete displacement field is expressed as the superposition of the linear displacement contribution from the proper selection of reference states for constrained interfaces and the total displacement fields produced by the arrays of interfacial Volterra dislocations. The latter dislocation displacement fields are also given as a biperiodic Fourier series, i.e.

$$ u_k^{\text{dis}}(x_1, x_2, x_3) = \text{Re} \sum_{k \neq 0} e^{i2\pi k \cdot r} u_k^k(x_2) = 2 \text{Re} \sum_D e^{i2\pi k \cdot r} u_k^k(x_2), \quad (3.73) $$

where the Fourier series expansion involves the harmonics $(n, m)$ that belong to the upper two-dimensional half-plane domain, defined by $D = \{\{n \in \mathbb{N}^*\} \cup \{m \in \mathbb{Z}^*, n = 0\}\}$. For clarity, the subscript dis in eq. (3.7) has been changed to superscript in eq. (3.73). The components $k_1(n, m)$ and $k_3(m)$ of the wavevectors $k$ are given by eq. (3.6) as follows

$$ k \cdot r = \frac{n}{p_1^0} x_1' + \frac{m}{p_2^0} x_3' = \left( \frac{n \csc \phi^{\text{un}}}{p_1^0} - \frac{m \operatorname{ctg} \phi^{\text{un}}}{p_2^0} \right) x_1 + \frac{m}{p_2^0} x_3 = k_1(n, m) x_1 + k_3(m) x_3, \quad (3.74) $$

with $p_1^0 = |p_1^0|$ and $p_2^0 = |p_2^0|$. On the other hand, the far-field components are computed for two dislocation sets to determine the correct reference state [260], within which the Burgers vectors $b_1$ and $b_2$ (and also $b_3$, by virtue of eq. (3.70)) are defined. Because the elastic (short-range) strain relaxations do not alter the long-range strain state during the junction formation of the third dislocation sets, the removal of the far-field strains (or stresses) in the natural state is fulfilled by solving the tensorial far-field eqs. (3.1), exhibiting non-zero and heterogeneous short-range elastic fields for interfacial dislocation patterns, only. Thus, substituting the displacement field eq. (3.73) into eq. (3.71), the second-order differential equation applied to both materials is obtained in index form as follows

$$ -4\pi^2 W_{1_{ik}} \tilde{u}_k^k(x_2) + i2\pi (W_{2_{ik}} + W_{2_{ki}}) \tilde{u}_{k,2}^k(x_2) + W_{3_{ik}} \tilde{u}_{k,22}^k(x_2) = 0, \quad (3.75) $$

where $W_1, W_2$, and $W_3$ are $3 \times 3$ real matrices defined in eqs. (3.10). In eq. (3.75), the superimposed tilde to any quantities will be used to indicate that the corresponding field solutions are consistent with the Frank-Bilby equation under the condition of vanishing far-field strains (or stresses) for any dislocation patterns. For non-zero wavevectors $k$, the standard solutions satisfying eq. (3.75) can be written in the following form [64]

$$ \tilde{u}_k^k(x_2) = e^{i2\pi p^k x_2} a_k^k, \quad (3.76) $$

where $p^k = p$ and $a^k = a_k$ become the complex scalar and vectorial unknowns of the boundary value problems, respectively, for which the superscripts $k$ are omitted, for clarity. Introducing eq. (3.76) into eq. (3.75), the vector $a$ is found to satisfy the homogeneous linear system

$$ [W_{1_{ik}} + p(W_{2_{ik}} + W_{2_{ki}}) + p^2 W_{3_{ik}}] a_k = \Pi_{ik} a_k = 0, \quad (3.77) $$

which corresponds to the standard eigenvalue problem in anisotropic elasticity theory [234, 240]. A non-zero (non-trivial) solution can be found only if the determinant of $\Pi$ is zero, i.e.

$$ \det \Pi_{ik} = 0, \quad (3.78) $$
leading to a sextic equation for $p$. As mentioned in section 3.6.3, the solutions of eq. (3.78) have six imaginary roots, which are arranged such that the three first eigenvalue solutions $p^\alpha$ have positive imaginary parts, indexed by superscripts $\alpha = 1, 2, 3$. The remaining three solutions have negative imaginary parts, so that $p^{\alpha+3} = p_*^\alpha$. The corresponding eigenvectors $a^\alpha = a_k^\alpha$ are also complex conjugates with $a^{\alpha+3} = a_*^\alpha = a_{k_*}^\alpha$, so that the general solution may be rewritten as a linear combination of the three eigenfunctions, i.e.

$$ \tilde{u}_k^{\text{dis}}(x_1, x_2, x_3) = 2 \operatorname{Re} \sum_D e^{i2\pi k \cdot r} \sum_{\alpha=1}^3 \lambda^\alpha e^{i2\pi p^\alpha x_2} a_k^\alpha + \zeta^\alpha e^{i2\pi p_*^\alpha x_2} a_{k_*}^\alpha, \quad (3.79) $$

which differs from eq. (3.11) by a multiplicative $i2\pi$ term, without loss of generality. It also follows from eq. (3.71) that

$$ \tilde{\sigma}_{ij}^{\text{dis}}(x_1, x_2, x_3) = 4\pi \operatorname{Re} \sum_D i e^{i2\pi k \cdot r} \sum_{\alpha=1}^3 \lambda^{\alpha} e^{i2\pi p^{\alpha} x_2} H_{ij}^{\alpha} + \zeta^{\alpha} e^{i2\pi p_*^{\alpha} x_2} H_{ij_*}^{\alpha}, \quad (3.80) $$

where the $3 \times 3$ complex matrices $H^\alpha$ are related to the eigenvectors $a^\alpha$ by

$$ H_{ij}^{\alpha} = (k_1 c_{ijk1} + k_3 c_{ijk3} + p^{\alpha} c_{ijk2}) a_k^{\alpha}, \quad (3.81) $$

from selected elastic constants of materials A and B. In particular, the surface tractions at the semicoherent interfaces, i.e. $x_2 = 0$, are reduced to

$$ \tilde{t}_k^{\text{int}}(x_1, x_3) = \tilde{\sigma}_{ki}^{\text{dis}}(x_1, 0, x_3) n_i = 4\pi \operatorname{Re} \sum_D i e^{i2\pi k \cdot r} \sum_{\alpha=1}^3 \lambda^{\alpha} H_{k2}^{\alpha} + \zeta^{\alpha} H_{k2_*}^{\alpha}, \quad (3.82) $$

as well as the tractions at the free surface, i.e. $x_2 = h_A$, to

$$ \tilde{t}_k^{\text{fs}}(x_1, x_3) = \tilde{\sigma}_{ki}^{\text{dis}}(x_1, h_A, x_3) n_i = 4\pi \operatorname{Re} \sum_D i e^{i2\pi k \cdot r} \sum_{\alpha=1}^3 \lambda^{\alpha} e^{i2\pi p^{\alpha} h_A} H_{k2}^{\alpha} + \zeta^{\alpha} e^{i2\pi p_*^{\alpha} h_A} H_{k2_*}^{\alpha}. \quad (3.83) $$

## Free surface and semicoherent interface elasticity contributions

Combined with the surface tractions in eqs. (3.82) and (3.83), the additional surface/interface stress contributions, due to the work required by applying in-plane forces to elastically stretch the pre-existing free surfaces and interfaces neighboring both materials A and B into the correct reference states, are introduced as follows

$$ \tau_{\chi\phi}(x_1, x_3) = \gamma \delta_{\chi\phi} + \frac{\partial \gamma}{\partial \epsilon_{\chi\phi}^s(x_1, x_3)}, \quad (3.84) $$

where $\tau_{\chi\phi}(x_1, x_3)$ and $\epsilon_{\chi\phi}^s(x_1, x_3)$ are the $2 \times 2$ surface stress and strain tensors, and $\gamma$ is the surface free energy [224, 53]. Because eq. (3.84) is derived for the plane stresses acting in the surface area, the stress and strain fields have only in-plane components, and Greek indices take values 1 and 3, only. In order to solve the elasticity problems with appropriate constitutive relations between the surface stress and strain components, a linear constitutive equation analogous to eq. (3.71) is used [221], i.e.

$$ \tau_{\chi\phi}(x_1, x_3) = \tau_{\chi\phi}^0 + d_{\chi\phi\gamma\eta} \epsilon_{\gamma\eta}^s(x_1, x_3), \quad (3.85) $$

where $\tau_{\chi\phi}^0$ is the surface/interface residual stress tensor and $d_{\chi\phi\gamma\eta}$ is the fourth-order stiffness tensor of surface/interface elastic constants. When the surface/interface entities are considered as elastic isotropic media, the elasticity tensor contains also two independent constants, known as surface/interface Lamé constants [53].
In the case of realistic semicoherent interfaces, the atomic structures are not exactly like those generated by the linear mappings from a reference state, as idealized and illustrated in Fig. (3.25a) with no atomic relaxations. Indeed, electron microscopy and atomistic calculations have revealed that such boundaries consist of coherent patches separated by networks of interfacial dislocations. The coherency and bounding conditions between such boundaries and the adjacent bulk materials yield therefore to the expression for the surface stresses in terms of the derivatives of the bulk displacement fields, i.e.

$$2 \epsilon_{\gamma\eta}^{s}(x_1, x_3) = 2 \tilde{\epsilon}_{\gamma\eta}^{\text{dis}}(x_1, 0, x_3) = \tilde{u}_{\gamma,\eta}^{\text{dis}}(x_1, 0, x_3) + \tilde{u}_{\eta,\gamma}^{\text{dis}}(x_1, 0, x_3), \quad (3.86)$$

for all in-plane strain components, at $x_2 = 0$. Similarly to the model for interface stresses with application to misfit dislocations in Ref. [52], this interface/bulk conversion of strain fields in eq. (3.86) depends strongly on the presence of the misfit dislocations (and, therefore, on the non-arbitrary coherent reference states), which in the present case, gives rise to unequally partitioned distortion states, in terms of strains as well as tilt and twist rotations.

The elastic field solutions of the displacements, stresses and tractions of eqs. (3.79), (3.80), and (3.82–3.83) respectively, with respect to surface/interface effects defined by eqs. (3.85–3.86), are also written as linear combinations of the eigenfunctions, within which $\{\lambda^\alpha, \zeta^\alpha\}$ are complex unknown quantities that are to be determined by the boundary conditions. For hexagonal-shaped dislocation patterns, these specific required conditions are expressed in terms of the discontinuities of displacement and stress components across the semicoherent interfaces in bimaterials in presence (if any) of a free surface in the upper material.

### 3.6.4 Boundary conditions with surface/interface constitutive relations

In what follows in section 3.6.4, expressions of displacements, strains, and stresses, and also all related quantities that are needed to compute these field solutions (e.g., the elastic constants, Burgers vectors $b_1, b_2$, and $b_3$, eigenvectors $a^\alpha$, etc.), are expressed in the local oblique and fixed (O, $x'_1, x'_2, x'_3$) frame. In particular, the boundary conditions are written with respect to the geometry of the dislocation patterns, i.e. to the canonical coordinates ($\eta_1, \eta_2$) of the three-fold dislocation node $J_1$, as well as the magnitudes and directions of the individual Burgers vectors for three sets of dislocations.

#### Convergence of the elastic field solutions

For all constrained interfaces that are consistent with the Frank-Bilby equation, i.e. when eqs. (3.1) are fulfilled with respect to the correct reference state, the corresponding semi-infinite linear crystal (here, material B) is also necessary free of all far-field stress components. The elastic stress solution in eq. (3.80) is therefore required to converge to zero at long range, i.e. when $x_2 \to -\infty$. Hence, $B\lambda^\alpha = 0$, independently of interfacial boundary conditions. For infinite bicrystals of interest, the convergence conditions in both materials A and B yield to: $B\lambda^\alpha = A\zeta^\alpha = 0$, when $x_2 \to \pm\infty$, as already defined in section 3.2.5.

#### Relative displacement due to the interfacial dislocation patterns

In accordance with eq. (3.15), the prescribed relative displacement field $u^p(x_1, x_3)$ for any (irregular) hexagonal-shaped dislocation patterns at the interface, i.e. $x_2 = 0$, is obtained by superposing the contributions of both Burgers vectors $b_1$ and $b_2$, i.e.

$$u^p(x_1, x_3) = \frac{x_1 \csc \phi^{\text{un}}}{p_1^o} b_1 + \frac{x_3 - x_1 \operatorname{ctg} \phi^{\text{un}}}{p_2^o} b_2 = z_1(x_1) b_1 + z_2(x_1, x_3) b_2, \quad (3.87)$$
where $z_1 = z_1(x_1)$ and $z_2 = z_2(x_1, x_3)$ are dimensionless linear functions. Assuming that the displacement jumps are zero at all positions of the O-lattice points, e.g. at O in the representative unit cell in Fig. (3.25b), the prescribed displacement field is also an odd function with respect to $r$ in the oblique (O, $x'_1$, $x'_2$, $x'_3$) frame. According to the linear elasticity theory, these displacement jumps produced by each hexagonal-shaped dislocation cell can therefore be formally expressed as double Fourier series for any dislocation configurations with respect to $(\eta_1, \eta_2)$, i.e.

$$ \mathbf{u}^p(x_1, x_3) = \operatorname{Im} \sum_{k \neq 0} e^{i2\pi k \cdot r} \hat{\mathbf{u}}^p(\eta_1, \eta_2) = -\operatorname{Re} i \sum_{k \neq 0} e^{i2\pi k \cdot r} (\hat{\mathbf{u}}_1^p(\eta_1, \eta_2) + \hat{\mathbf{u}}_2^p(\eta_1, \eta_2)), \quad (3.88) $$

where all real-valued expansion coefficients $\hat{\mathbf{u}}^p(\eta_1, \eta_2)$ in eq. (3.88) are additionally decomposed into the individual contributions $\hat{\mathbf{u}}_1^p(\eta_1, \eta_2)$ and $\hat{\mathbf{u}}_2^p(\eta_1, \eta_2)$, associated with $b_1$ and $b_2$, respectively. In particular, the vector quantity $\hat{\mathbf{u}}_1^p(\eta_1, \eta_2)$ for $b_1$ is deduced by solving the double integral with respect to $z_1$ and $z_2$, as follows

$$ \hat{\mathbf{u}}_1^p(\eta_1, \eta_2) = \operatorname{Re} \left[ i \int_{\bar{z}_1(\eta_1)}^{\bar{z}_1(\eta_1)} \left( z_1 \int_{\bar{z}_2(z_1, \eta_1, \eta_2)}^{\bar{z}_2(z_1, \eta_1, \eta_2)} e^{-i2\pi(nz_1+mz_2)} dz_2 \right) dz_1 \right] b_1, \quad (3.89) $$

for any $(\eta_1, \eta_2) \in \mathcal{H}_{J_1J_2J_3J_4J_5J_6}$. Moreover, eq. (3.89) may be integrated over three separate unit domains, e.g. the parallelogram $\mathcal{P}_{J_1J_3J_4J_6}$ and both triangles $\mathcal{T}_{J_1J_2J_3}$ and $\mathcal{T}_{J_4J_5J_6}$, i.e.

$$ \hat{\mathbf{u}}_1^p(\eta_1, \eta_2) = \hat{\mathbf{u}}_1^p(\eta_1, \eta_2)|_{\mathcal{H}_{J_1J_2J_3J_4J_5J_6}} = \hat{\mathbf{u}}_1^p(\eta_1, \eta_2)|_{\mathcal{P}_{J_1J_3J_4J_6}} + \hat{\mathbf{u}}_1^p(\eta_1, \eta_2)|_{\mathcal{T}_{J_1J_2J_3}} + \hat{\mathbf{u}}_1^p(\eta_1, \eta_2)|_{\mathcal{T}_{J_4J_5J_6}}, \quad (3.90) $$

as illustrated by the different vertices in Fig. (3.25b). Because the boundaries of the hexagonal-shaped unit cells are composed of straight dislocation segments, the integral eq. (3.89) is necessarily bounded by affine functions with respect to the coordinates $\eta_1$ and $\eta_2$. The first quantity $\hat{\mathbf{u}}_1^p(\eta_1, \eta_2)|_{\mathcal{P}_{J_1J_3J_4J_6}}$ in the right-hand side of eq. (3.90) is also computed by using the following bounds, i.e.

$$ \forall \{\eta_1, \eta_2\} \in \mathcal{P}_{J_1J_3J_4J_6} : \left\{ \begin{array}{l} \bar{z}_1(\eta_1) = -\eta_1 \\ \bar{z}_1(\eta_1) = \eta_1 \end{array} \right. , \quad \text{and} \quad \left\{ \begin{array}{l} \bar{z}_2(z_1, \eta_1, \eta_2) = -\frac{1-2\eta_2}{2\eta_1}z_1 - \frac{1}{2} \\ \bar{z}_2(z_1, \eta_1, \eta_2) = -\frac{1-2\eta_2}{2\eta_1}z_1 + \frac{1}{2} \end{array} \right. \quad (3.91) $$

Similarly, the two quantities $\hat{\mathbf{u}}_1^p(\eta_1, \eta_2)|_{\mathcal{T}_{J_1J_2J_3}}$ and $\hat{\mathbf{u}}_1^p(\eta_1, \eta_2)|_{\mathcal{T}_{J_4J_5J_6}}$ in eq. (3.90) are determined by considering

$$ \forall \{\eta_1, \eta_2\} \in \mathcal{T}_{J_1J_2J_3} : \left\{ \begin{array}{l} \bar{z}_1(\eta_1) = \eta_1 \\ \bar{z}_1(\eta_1) = 1 - \eta_1 \end{array} \right. , \quad \text{and} \quad \left\{ \begin{array}{l} \bar{z}_2(z_1, \eta_1, \eta_2) = \frac{(1-2\eta_2)z_1 - 1 + \eta_2 + \eta_1}{1-2\eta_1} \\ \bar{z}_2(z_1, \eta_1, \eta_2) = \frac{2\eta_2 z_1 - \eta_2}{1+2\eta_1}, \end{array} \right. \quad (3.92) $$

$$ \forall \{\eta_1, \eta_2\} \in \mathcal{T}_{J_4J_5J_6} : \left\{ \begin{array}{l} \bar{z}_1(\eta_1) = \eta_1 - 1 \\ \bar{z}_1(\eta_1) = -\eta_1 \end{array} \right. , \quad \text{and} \quad \left\{ \begin{array}{l} \bar{z}_2(z_1, \eta_1, \eta_2) = \frac{2\eta_2 z_1 + \eta_2}{-1 + 2\eta_1} \\ \bar{z}_2(z_1, \eta_1, \eta_2) = \frac{(1 - 2\eta_2) z_1 + 1 - \eta_2 - \eta_1}{1 - 2\eta_1}, \end{array} \right. \quad (3.93) $$

respectively. Thus, after integrating eq. (3.90) analytically with respect to eqs. (3.91) and (3.92), it can also be found that

$$ \hat{\mathbf{u}}_1^p(\eta_1, \eta_2) = \sin(2\pi(m\eta_2 + n\eta_1)) \frac{-1 + 2\eta_1}{2\pi^2(m + n - 2(m\eta_2 + n\eta_1))(2m\eta_2 + n(-1 + 2\eta_1))} b_1, \quad (3.93) $$
for any given $(\eta_1, \eta_2)$. Analogously to eq. (3.89), the vector quantity $\hat{u}_2^p(\eta_1, \eta_2)$ for $b_2$ is written in the form

$$ \hat{u}_2^p(\eta_1, \eta_2) = \mathrm{Re} \left[ i \int_{\bar{z}_1(\eta_1)}^{\bar{z}_1(\eta_1)} \left( \int_{\bar{z}_2(z_1, \eta_1, \eta_2)}^{\bar{z}_2(z_1, \eta_1, \eta_2)} z_2 e^{-i2\pi(nz_1+mz_2)} dz_2 \right) dz_1 \right] b_2, \quad (3.94) $$

for which the same integral bounds defined by eqs. (3.91) and (3.92) are used to calculate eq. (3.94) over the hexagonal-shaped dislocation patterns. Hence, it follows

$$ \hat{u}_2^p(\eta_1, \eta_2) = \sin(2\pi(m\eta_2 + n\eta_1)) \frac{-1 + 2\eta_2}{2\pi^2(m + n - 2(m\eta_2 + n\eta_1))(m(-1 + 2\eta_2) + 2n\eta_1)} b_2, \quad (3.95) $$

for any $(\eta_1, \eta_2)$. Combining eq. (3.93) with eq. (3.95), the complete vectorial solution for $\hat{u}^p(\eta_1, \eta_2)$ is given by

$$ \hat{u}^p(\eta_1, \eta_2) = \frac{\sin(2\pi(m\eta_2 + n\eta_1))}{2\pi^2(m + n - 2(m\eta_2 + n\eta_1))} \left[ \frac{-1 + 2\eta_1}{2m\eta_2 + n(-1 + 2\eta_1)} b_1 + \frac{-1 + 2\eta_2}{m(-1 + 2\eta_2) + 2n\eta_1} b_2 \right], \quad (3.96) $$

which closely corresponds to the same expression given in Ref. [39], after minor corrections. It is worth noting that three singular values for $n$ and $m$ give rise to null denominators in eq. (3.96), so that three cases must be distinguished, i.e. c1: $m + n - 2(m\eta_2 + n\eta_1) \neq 0$, c2: $n - 2(m\eta_2 + n\eta_1) \neq 0$, and c3: $m - 2(m\eta_2 + n\eta_1) \neq 0$. By defining the function $z(n, m) = n z_1 + m z_2$ in the exponential terms of both eqs. (3.89) and (3.94), all corresponding real-valued expansion coefficients are also obtained by replacing $m$ with $m^*$ in $z(n, m)$ for all different cases, i.e.

$$ \text{c1: } m^* = -n \frac{1 - 2\eta_1}{1 - 2\eta_2}, \quad \text{c2: } m^* = n \frac{1 - 2\eta_1}{2\eta_2}, \quad \text{and} \quad \text{c3: } m^* = n \frac{2\eta_1}{1 - 2\eta_2}, \quad (3.97) $$

for which the expressions for these three cases are given in Appendix A from Ref. [249]. Finally, to exhibit the discontinuity condition in displacement, the prescribed jump in eq. (3.87) with the aid of the eqs. (3.96) may finally be related to the displacement fields generated by the interface dislocation patterns, i.e.

$$ u_k^p(x_1, x_3) = \left[ \tilde{u}_k^{\mathrm{dis}}(x_1, 0, x_3) \right]_{\mathrm{int}} = A \tilde{u}_k^{\mathrm{dis}}(x_1, 0, x_3) - B \tilde{u}_k^{\mathrm{dis}}(x_1, 0, x_3), \quad (3.98) $$

where the complete elastic field solutions in both materials A and B are given by eq. (3.79). The symbol $[y_k]_{\mathrm{int}} = \Delta y_k = A y_k - B y_k$ corresponds to the vectorial jump of the quantity $y$ across the interface at $x_2 = 0$. Although all physical displacement fields in eq. (3.98) are defined as the real quantities of complex Fourier series-based expressions, the real part designation in eqs. (3.79) and (3.88) are conveniently omitted to express the complex equality, as follows

$$ -i \hat{u}_k^p(\eta_1, \eta_2) = \sum_{\alpha=1}^{3} A \lambda^\alpha A a_k^\alpha + A \zeta^\alpha A a_{k_*}^\alpha - B \zeta^\alpha B a_{k_*}^\alpha, \quad (3.99) $$

so that both real and imaginary parts of eq. (3.99) lead to the equivalent homogeneous linear system $\Sigma_1$ of six real equations, i.e.

$$ (\Sigma_1) \quad \forall k \in \{1, 2, 3\} : \left\{ \begin{array}{l} 0 = \mathrm{Re} \sum_{\alpha=1}^{3} A \lambda^{\alpha} A a_{k}^{\alpha} + A \zeta^{\alpha} A a_{k_{*}}^{\alpha} - B \zeta^{\alpha} B a_{k_{*}}^{\alpha} \\ -\hat{u}_{k}^{p}(\eta_{1}, \eta_{2}) = \mathrm{Im} \sum_{\alpha=1}^{3} A \lambda^{\alpha} A a_{k}^{\alpha} + A \zeta^{\alpha} A a_{k_{*}}^{\alpha} - B \zeta^{\alpha} B a_{k_{*}}^{\alpha}, \end{array} \right. \quad (3.100) $$

where $\hat{u}^p(\eta_1, \eta_2)$ is defined in eq. (3.96), for any given $(\eta_1, \eta_2) \in ]0, 1/2]^2$ and for all $\{n, m\} \in D$.
Stress conditions at the semicoherent interfaces

Due to the presence of the interfacial excess energy close to grain and interphase boundaries, the discontinuity of the tangential stress components is introduced using the generalized Young-Laplace equation [108, 209, 80], as an equilibrium boundary condition to solve the present boundary-value problem with interface stress effects, i.e.

$$ \left[ \tilde{\sigma}_{\phi i}^{\text{dis}}(x_1, 0, x_3) n_i \right]_{\text{int}} + \tau_{\phi\chi,\chi} = 0, \qquad (3.101) $$

together with the stress discontinuity in normal direction of the boundaries, as follows

$$ \left[ \tilde{\sigma}_{ij}^{\text{dis}}(x_1, 0, x_3) n_j n_i \right]_{\text{int}} = \tau_{\chi\phi} \kappa_{\chi\phi}, \qquad (3.102) $$

with $\kappa_{\chi\phi}$ the curvature tensor of the solid-state interface of interest. Substituting the linear constitutive relation of eq. (3.85) into eqs. (3.101) and (3.102) respectively, the governing non-classical boundary equations lead to

$$ \begin{cases} 0 = \left[ \tilde{t}_{\phi}^{\text{int}}(x_1, x_3) \right]_{\text{int}} + d_{\phi\chi\gamma\eta} \tilde{u}_{\gamma,\eta\chi}^{\text{dis}}(x_1, 0, x_3) \\ 0 = \left[ \tilde{t}_{2}^{\text{int}}(x_1, x_3) \right]_{\text{int}} - (\tau_{\chi\phi}^{0} + d_{\chi\phi\gamma\eta} \tilde{u}_{\gamma,\eta}^{\text{dis}}(x_1, 0, x_3)) (\kappa_{\chi\phi}^{0} + \kappa_{\chi\phi}^{\Delta}), \end{cases} \qquad (3.103) $$

where $\kappa_{\chi\phi}^0$ and $\kappa_{\chi\phi}^{\Delta}$ are the deformation-independent curvature and curvature change tensors, respectively. In the classical theory of initially flat and infinitely thin membranes with small out-of-plane deflections [40], as the considered (and interpreted as surface stresses) elastically stretched membranes in Refs. [108, 109], the curvature change tensor may be approximated by

$$ \kappa_{\chi\phi}^{\Delta} = -\tilde{u}_{2,\chi\phi}^{\text{dis}}(x_1, 0, x_3), \qquad (3.104) $$

without internal moments. Under the treatment of such specific boundary conditions normal to the initially flat (but, stretched) membranes, the distortion response caused by the presence of interface dislocations may elastically warp the semicoherent interfaces with radii defined by $r_{\chi\phi} = 1/\kappa_{\chi\phi}^{\Delta}$. Thus, the right-hand side of the second equation in eqs. (3.103) is deduced by subsequently imposing no initial curvature and neglecting the second-order effects compared with unity, as follows

$$ (\tau_{\chi\phi}^{0} + d_{\chi\phi\gamma\eta} \tilde{u}_{\gamma,\eta}^{\text{dis}}(x_1, 0, x_3)) (\kappa_{\chi\phi}^{0} + \kappa_{\chi\phi}^{\Delta}) \approx -\tau_{\chi\phi}^{0} \tilde{u}_{2,\chi\phi}^{\text{dis}}(x_1, 0, x_3), \qquad (3.105) $$

thus, imposing $\kappa_{\chi\phi}^0 = 0$ and $\tilde{u}_{\gamma,\eta}^{\text{dis}}(x_1, 0, x_3) \tilde{u}_{2,\chi\phi}^{\text{dis}}(x_1, 0, x_3) \ll 1$. According to eq. (3.82), both discontinuous stress boundary conditions in eqs. (3.103) can also be recast in matrix form, i.e.

$$ \left[ \tilde{t}_{k}^{\text{int}}(x_1, x_3) \right]_{\text{int}} - 4\pi^2 V_{ki} \tilde{u}_{i}^{\text{int}}(x_1, 0, x_3) = 0, \qquad (3.106) $$

where the 3 × 3 real matrix V is expressed as

$$ V_{ki} = V_{ik} = \begin{bmatrix} k_1^2 d_{11} + 2k_1 k_3 d_{15} + k_3^2 d_{55} & 0 & k_1^2 d_{15} + k_1 k_3 (d_{13} + d_{55}) + k_3^2 d_{35} \\ 0 & k_1^2 \tau_{11}^0 + 2k_1 k_3 \tau_{13}^0 + k_3^2 \tau_{33}^0 & 0 \\ k_1^2 d_{15} + k_1 k_3 (d_{13} + d_{55}) + k_3^2 d_{35} & 0 & k_1^2 d_{55} + 2k_1 k_3 d_{35} + k_3^2 d_{33} \end{bmatrix}, \qquad (3.107) $$

within which the surface/interface elastic constants are indexed using standard contracted notations. Mechanically balanced by the interface stress effects, eq. (3.106) shows that the infinitesimal in-plane strain fields in the membranes may influence the stresses in both bulk materials due to the elasticity contributions at the interphase boundaries. In contrast to the classical continuum elasticity, the tractions across the interface and the displacement fields are related to each other by the interface elasticity properties as well the interface geometries through the wavevector components.
Because the materials A and B are mapped separately from the reference state, the coherent regions at the interfaces (separated by the networks of interfacial dislocations) can also be viewed as infinitely thin membranes separately in contact with each individual bulk material. Furthermore, the determination of the reference states yielding (in general) to unequal partitioning of elastic distortions, the tractions that act on each individual upper and lower materials bonded by these coherent interfacial regions are consequently assumed to be different in both magnitude and direction. Using the concept of interface zone by in Ref. [146], the specific traction vector $\coth t^{\text{int}}(x_1, x_3)$, acting on both neighboring crystals with fictitious infinitely thin inter-layered coherent patches at $x_2 = 0$, is introduced to transfer traction forces from the upper material to the adjacent lower material. The equilibrium condition between the interface coherent regions and material A also reads

$$A \tilde{t}_k^{\text{int}}(x_1, x_3) - \coth t_k^{\text{int}}(x_1, x_3) - 4\pi^2 A V_{ki}^{\text{int}} A \tilde{u}_i^{\text{dis}}(x_1, 0, x_3) = 0, \quad (3.108)$$

by use of the boundary condition in eq. (3.106), while the equilibrium condition between the interface coherent regions and material B is given by

$$\coth t_k^{\text{int}}(x_1, x_3) - B \tilde{t}_k^{\text{int}}(x_1, x_3) - 4\pi^2 B V_{ki}^{\text{int}} B \tilde{u}_i^{\text{dis}}(x_1, 0, x_3) = 0, \quad (3.109)$$

where $A V^{\text{int}}$ and $B V^{\text{int}}$ depend on the elastic properties of the interfaces with respect to each material A and B, respectively. Summing both eqs. (3.108) and (3.109), it also follows that

$$(A \tilde{t}_k^{\text{int}}(x_1, x_3) - B \tilde{t}_k^{\text{int}}(x_1, x_3) - 4\pi^2 (A V_{ki}^{\text{int}} A \tilde{u}_i^{\text{dis}}(x_1, 0, x_3) + B V_{ki}^{\text{int}} B \tilde{u}_i^{\text{dis}}(x_1, 0, x_3))) = 0, \quad (3.110)$$

which yields to the non-classical stress discontinuity conditions at the mismatched interfaces. Using eqs. (3.79) and (3.82), eq. (3.110) gives rise to the additional linear system $\Sigma_2$ of six equations, i.e.

$$ (\Sigma_2): \left\{ \begin{array}{l} 0 = \text{Re} \left[ \sum_{\alpha=1}^{3} A \lambda^{\alpha} A h_1^{\alpha} + A \zeta^{\alpha} A h_{1_{*}}^{\alpha} + i2\pi (A V_{11}^{\text{int}} (A \lambda^{\alpha} A a_1^{\alpha} + A \zeta^{\alpha} A a_{1_{*}}^{\alpha}) + A V_{13}^{\text{int}} (A \lambda^{\alpha} A a_3^{\alpha} + A \zeta^{\alpha} A a_{3_{*}}^{\alpha})) \right. \\ \qquad \left. - B \zeta^{\alpha} (B h_{1_{*}}^{\alpha} - i2\pi (B V_{11}^{\text{int}} B a_{1_{*}}^{\alpha} + B V_{13}^{\text{int}} B a_{3_{*}}^{\alpha})) \right] = \text{Re} \sum_{\alpha=1}^{3} v_1^{\alpha} \\ 0 = \text{Re} \left[ \sum_{\alpha=1}^{3} A \lambda^{\alpha} A h_2^{\alpha} + A \zeta^{\alpha} A h_{2_{*}}^{\alpha} + i2\pi (A V_{22}^{\text{int}} (A \lambda^{\alpha} A a_2^{\alpha} + A \zeta^{\alpha} A a_{2_{*}}^{\alpha})) - B \zeta^{\alpha} (B h_{2_{*}}^{\alpha} - i2\pi B V_{22}^{\text{int}} B a_{2_{*}}^{\alpha}) \right] \\ = \text{Re} \sum_{\alpha=1}^{3} v_2^{\alpha} \\ 0 = \text{Re} \left[ \sum_{\alpha=1}^{3} A \lambda^{\alpha} A h_3^{\alpha} + A \zeta^{\alpha} A h_{3_{*}}^{\alpha} + i2\pi (A V_{13}^{\text{int}} (A \lambda^{\alpha} A a_1^{\alpha} + A \zeta^{\alpha} A a_{1_{*}}^{\alpha}) + A V_{33}^{\text{int}} (A \lambda^{\alpha} A a_3^{\alpha} + A \zeta^{\alpha} A a_{3_{*}}^{\alpha})) \right. \\ \qquad \left. - B \zeta^{\alpha} (B h_{3_{*}}^{\alpha} - i2\pi (B V_{13}^{\text{int}} B a_{1_{*}}^{\alpha} + B V_{33}^{\text{int}} B a_{3_{*}}^{\alpha})) \right] = \text{Re} \sum_{\alpha=1}^{3} v_3^{\alpha} \\ 0 = \text{Im} \left[ \sum_{\alpha=1}^{3} v_k^{\alpha} \right], \quad \forall k \in \{1, 2, 3\}, \end{array} \right. \quad (3.111) $$

with $h_k^\alpha = H_{k2}^\alpha$, for any given $(\eta_1, \eta_2) \in ]0, 1/2]^2$ and for all $\{n, m\} \in D$.

## Stress conditions at the free surfaces

Similarly to the semicoherent interface treatment, the free surfaces experience excess energy and excess energy due to different energy profiles close to such singular membrane-like boundaries. Thus, additional non-classical boundary conditions as eq. (3.110) are introduced on the outer free surface, at $x_2^{\text{or}} = h_A$, i.e.

$$A \tilde{t}_k^{\text{fs}}(x_1, x_3) + 4\pi^2 A V_{ki}^{\text{fs}} A \tilde{u}_i^{\text{dis}}(x_1, h_A, x_3) = 0, \quad (3.112)$$
where A**V**<sup>fs</sup> depends on the elastic constants of the free surfaces. It also yields to the following system Σ<sub>3</sub> of six other equations, i.e.

$$
(\Sigma_3) : \left\{
\begin{array}{l}
0 = \operatorname{Re} \left[ \sum_{\alpha=1}^{3} A \lambda^{\alpha} e^{i2\pi p^{\alpha} h_A} (A h_1^{\alpha} - i2\pi (A V_{11}^{\text{fs}} A a_1^{\alpha} + A V_{13}^{\text{fs}} A a_3^{\alpha})) \right. \\
\qquad \left. + A \zeta^{\alpha} e^{i2\pi p^{\alpha} h_A} (A h_{1*}^{\alpha} - i2\pi (A V_{11}^{\text{fs}} A a_{1*}^{\alpha} + A V_{13}^{\text{fs}} A a_{3*}^{\alpha})) \right] = \operatorname{Re} \sum_{\alpha=1}^{3} w_1^{\alpha} \\
0 = \operatorname{Re} \left[ \sum_{\alpha=1}^{3} A \lambda^{\alpha} e^{i2\pi p^{\alpha} h_A} (A h_2^{\alpha} - i2\pi A V_{22}^{\text{fs}} A a_2^{\alpha}) + A \zeta^{\alpha} e^{i2\pi p^{\alpha} h_A} (A h_{2*}^{\alpha} - i2\pi A V_{22}^{\text{fs}} A a_{2*}^{\alpha}) \right] \\
\qquad = \operatorname{Re} \sum_{\alpha=1}^{3} w_2^{\alpha} \\
0 = \operatorname{Re} \left[ \sum_{\alpha=1}^{3} A \lambda^{\alpha} e^{i2\pi p^{\alpha} h_A} (A h_3^{\alpha} - i2\pi (A V_{13}^{\text{fs}} A a_1^{\alpha} + A V_{33}^{\text{fs}} A a_3^{\alpha})) \right. \\
\qquad \left. + A \zeta^{\alpha} e^{i2\pi p^{\alpha} h_A} (A h_{3*}^{\alpha} - i2\pi (A V_{13}^{\text{fs}} A a_{1*}^{\alpha} + A V_{33}^{\text{fs}} A a_{3*}^{\alpha})) \right] = \operatorname{Re} \sum_{\alpha=1}^{3} w_3^{\alpha} \\
0 = \operatorname{Im} \left[ \sum_{\alpha=1}^{3} w_k^{\alpha} \right], \quad \forall k \in \{1, 2, 3\},
\end{array}
\right.
\qquad (3.113)
$$

for any given $(\eta_1, \eta_2) \in ]0, 1/2[^{2}$ and for all $\{n, m\} \in D$.

**Determination of the minimum-energy paths**

When the linear systems in eqs. (3.100) with (3.111) and (3.113) are combined, the set Ecst of all eighteen real unknowns (twelve and six for A and B, respectively) are also solved with respect to the prescribed boundary conditions, i.e.

$$
\mathrm{Ecst} = \sum_{\alpha=1}^{3} \{ \mathrm{Re}_{A} \lambda^{\alpha}, \mathrm{Im}_{A} \lambda^{\alpha}, \mathrm{Re}_{A} \zeta^{\alpha}, \mathrm{Im}_{A} \zeta^{\alpha}, \mathrm{Re}_{B} \zeta^{\alpha}, \mathrm{Im}_{B} \zeta^{\alpha} \}, \quad (3.114)
$$

completing the solutions of the elastic displacement and stress fields, given by eqs. (3.79) and (3.80), respectively. Following the procedure described in section 3.6.2, the upper triangular domain $\mathcal{T}_{\text{ABC}}$ in the representative unit dislocation cell, denoted by ABC in Fig. (3.25b), is discretized into four-node quadrilateral elements with respect to the $i^{\text{th}}$ nodal point coordinates $(\eta_1^i, \eta_2^i)$, such that $\{\eta_1^i, \eta_2^i\} \in ]0, 1/2[^2$ for convex hexagonal-shaped dislocation patterns. Thus, for any dislocation pattern that is geometrically characterized by the given coordinates $(\eta_1^i, \eta_2^i)$, the corresponding elastic strain energy can be computed as a volume integral over the heterostructure of interest, i.e.

$$
\gamma_e^i(\eta_1^i, \eta_2^i) = \frac{1}{2A} \iiint_V \tilde{\sigma}_{ij}^{\text{dis}}(x_1, x_2, x_3) \tilde{u}_{j,i}^{\text{dis}}(x_1, x_2, x_3) \, dV, \quad (3.115)
$$

where all persistent short-range field solutions of the integrand depend specifically on $(\eta_1^i, \eta_2^i)$ by
the treatment of boundary conditions, described in section 3.6.4. For far-field stress-free bicrystals
at equilibrium, the standard volume integral eq. (3.115) may be reduced to a surface integral by
the use of integration by parts, together with the divergence theorem without any body forces
[223, 260], as follows

$$
\gamma_e^i(\eta_1^i, \eta_2^i) = \frac{1}{2A} \iint_{A(r_0)} \tilde{t}_k^{\text{int}}(x_1, x_3) \left[ \tilde{u}_k^{\text{dis}}(x_1, 0, x_3) \right]_{\text{int}} \, dS, \quad (3.116)
$$

where A(r₀) is the hexagonal-shaped unit cell. In eqs. (3.115) and (3.116), the expressions of elas-
tic strain energy are conveniently expressed per unit area, for which A = A(r₀ = 0), and account
for several different contributions, i.e. interaction between Volterra-type dislocations against the
misfit strain state, self-energy induced by individual hexagonal-shaped dislocation configurations, as well as the interaction between the hexagonal-shaped unit cell with all infinitely repeated cells. Finally, the complete elastic strain energy landscape $\gamma_e(\eta_1, \eta_2)$ is interpolated for any $(\eta_1, \eta_2) \in ]0, 1/2[^{2}$, as follows

$$ \gamma_e(\eta_1, \eta_2) = \sum_{i=1}^{4} N_i(\eta_1, \eta_2) \gamma_e^i(\eta_1^i, \eta_2^i), \qquad (3.117) $$

where $N_i(\eta_1, \eta_2)$ are the standard finite element bilinear shape functions for four-node elements.

For elastic strain landscapes that favor the formation of dislocation junctions, the minimum-energy configurations are determined by computing the conjugate gradient algorithm, while the nudged elastic band method is used to find the corresponding minimum-energy paths. The nudged elastic band method is a chain-of-states method in which a string of images is used to describe the reaction pathways. These configurations are connected by spring forces to ensure equal spacing along the paths of interest. The ensemble of the configurations is then relax through a force projection scheme to converge to the most energetically favorable pathways [117, 222]. To identify the minimum-energy paths between the initial (non-equilibrium) lozenge-shaped pattern and the final elastically relaxed configurations (previously computed by the conjugate gradient algorithm), all images are simultaneously evolved to equilibrium under a nudged elastic band force (on image indexed by $s_{\eta}$) that contains two independent components on all images $s_{\eta}$, i.e.

$$ f_{s_{\eta}}^{\text{NEB}} = f_{s_{\eta}}^{\perp} + f_{s_{\eta}}^{\parallel}, \qquad (3.118) $$

where $f_{s_{\eta}}^{\perp}$ is the component of the elastic force acting normal to the tangent of the elastic landscape, as follows

$$ f_{s_{\eta}}^{\perp} = -\nabla\gamma_e(\eta_1, \eta_2) + (\nabla\gamma_e(\eta_1, \eta_2) \cdot \hat{\tau}_{s_{\eta}}) \hat{\tau}_{s_{\eta}}, \qquad (3.119) $$

with $\hat{\tau}_{s_{\eta}}$ the unit tangent to the elastic energy landscape. In addition, the spring force $f_{s_{\eta}}^{\parallel}$ in eq. (3.118), acting parallel to the energy landscape [117, 222] is defined by

$$ f_{s_{\eta}}^{\parallel} = k(|\eta_{s_{\eta}+1} - \eta_{s_{\eta}}| - |\eta_{s_{\eta}} - \eta_{s_{\eta}-1}|), \qquad (3.120) $$

where $\eta_{s_{\eta}} = \eta_{s_{\eta}}(\eta_1, \eta_2)$ is the position of the $s_{\eta}^{th}$ image and $k$ the spring constant. The spring interaction between adjacent images is added to ensure continuity of the chain.

The present numerical procedure is identical to nudged elastic band calculations recently performed to analyze the calculation of attempt frequency for a dislocation bypassing an obstacle [231] using a nodal dislocation dynamics simulation with non-singular treatments for isotropic elastic fields [51].

### 3.6.5 Application to Au/Cu heterosystems

The section gives applications to two examples of the general parametric energy-based framework. The first simple and limiting case is concerned with two dislocation sets in pure misfit (010) Au/Cu interfaces, for which the strain energy landscape for formation of dislocation junctions is unfavorable. The subsequent investigation of the effects of surface/interface stress and elasticity properties with different boundary conditions in (010) Au/Cu interfaces can be found in Ref. [249]. On the other hand, the second case deals with the minimum-energy reaction pathway of the pre-computed (111) Au/Cu elastic energy landscape, where the initial and unrelaxed dislocation pattern solution is described by the Frank-Bilby equation. The materials properties used in these examples are listed in Table 3.7.
<table><caption>TABLE 3.7: Lattice parameters *a* of Au and Cu crystals, material properties *c*<sub>ij</sub> of both bulk materials, surface stress τ<sub>χφ</sub> and surface modulus *d*<sub>χφ</sub> of the semicoherent Au/Cu heterophase interface and the (010) free surface in Au.</caption><thead><tr><th>Symbols</th><th>Au (material A)</th><th>Cu (Material B)</th><th>Units</th><th>References</th></tr></thead><tbody><tr><td colspan="5"><strong>Lattice parameters</strong></td></tr><tr><td><em>a</em></td><td>0.4078</td><td>0.3615</td><td>nm</td><td>[104]</td></tr><tr><td colspan="5"><strong>Elastic components (Voigt notation)</strong></td></tr><tr><td><em>c</em><sub>11</sub></td><td>187.0</td><td>168.4</td><td>GPa</td><td>[7]</td></tr><tr><td><em>c</em><sub>12</sub></td><td>157.0</td><td>121.4</td><td>GPa</td><td>[7]</td></tr><tr><td><em>c</em><sub>44</sub></td><td>43.6</td><td>75.4</td><td>GPa</td><td>[7]</td></tr><tr><td colspan="5"><strong>Elasticity properties for the semicoherent interfaces (Voigt notation)</strong></td></tr><tr><td colspan="5"><strong>* Interface stress</strong></td></tr><tr><td>τ<sub>11</sub></td><td>-0.0465</td><td>0.645</td><td>N/m</td><td>[146]</td></tr><tr><td>τ<sub>13</sub></td><td>0</td><td>0</td><td>N/m</td><td>[146]</td></tr><tr><td>τ<sub>33</sub></td><td>-0.0465</td><td>0.645</td><td>N/m</td><td>[146]</td></tr><tr><td colspan="5"><strong>* Interface modulus</strong></td></tr><tr><td><em>d</em><sub>11</sub></td><td>-6.84</td><td>-5.99</td><td>N/m</td><td>[146]</td></tr><tr><td><em>d</em><sub>13</sub></td><td>-3.47</td><td>0.6540</td><td>N/m</td><td>[146]</td></tr><tr><td><em>d</em><sub>33</sub></td><td>-6.84</td><td>-5.99</td><td>N/m</td><td>[146]</td></tr><tr><td><em>d</em><sub>15</sub></td><td>0.0042</td><td>0.0032</td><td>N/m</td><td>[146]</td></tr><tr><td><em>d</em><sub>35</sub></td><td>0.0042</td><td>0.0032</td><td>N/m</td><td>[146]</td></tr><tr><td><em>d</em><sub>55</sub></td><td>-1.91</td><td>-3.67</td><td>N/m</td><td>[146]</td></tr><tr><td colspan="5"><strong>Elasticity properties for the free surface (Voigt notation)</strong></td></tr><tr><td colspan="5"><strong>* Surface stress</strong></td></tr><tr><td>τ<sub>11</sub></td><td>1.49</td><td>—</td><td>N/m</td><td>[185]</td></tr><tr><td>τ<sub>13</sub></td><td>0</td><td>—</td><td>N/m</td><td>[185]</td></tr><tr><td>τ<sub>33</sub></td><td>1.49</td><td>—</td><td>N/m</td><td>[185]</td></tr><tr><td colspan="5"><strong>* Surface modulus</strong></td></tr><tr><td><em>d</em><sub>11</sub></td><td>-7.10</td><td>—</td><td>N/m</td><td>[185]</td></tr><tr><td><em>d</em><sub>13</sub></td><td>-5.67</td><td>—</td><td>N/m</td><td>[185]</td></tr><tr><td><em>d</em><sub>33</sub></td><td>-3.17</td><td>—</td><td>N/m</td><td>[185]</td></tr></tbody></table>
<figure><img src="image_43.png" /><figcaption>FIGURE 3.26: Dependence of the total far-field stress component $\sigma_{11}^c + \sigma_{11}^{\pm\infty}$ on $\delta$ in the Au and Cu materials for the (010) and (111) Au/Cu semicoherent interfaces.</figcaption></figure>

Case 1: The (010) Au/Cu interface with two sets of dislocations

As a limiting case, the atomically sharp (010) Au/Cu misfit interface contains two sets of orthogonal dislocations in cube-cube orientation relationship, i.e. $x_1^{\text{or}} = [10\bar{1}]$, $x_2^{\text{or}} = n = [010]$, and $x_3^{\text{or}} = [101]$. Similar to eq. (3.37), the net Burgers vectors are expressed by using the quantized Frank-Bilby equation [93, 30, 29], as follows

$$ \left( \frac{n \times \xi_1^{\text{un}}}{d_1^{\text{un}}} \cdot \mathbf{p} \right) b_1 + \left( \frac{n \times \xi_2^{\text{un}}}{d_2^{\text{un}}} \cdot \mathbf{p} \right) b_2 = n_1 \mathbf{b}_1 + n_2 \mathbf{b}_2 = (\mathbf{F}_{\text{Au}}^{-1} - \mathbf{F}_{\text{Cu}}^{-1}) \mathbf{p}, \quad (3.121) $$

where $d_1^{\text{un}}$ and $d_2^{\text{un}}$ are the regularly spaced inter-dislocation spacings, and the interface Burgers vectors $b_1 \parallel [10\bar{1}]$ and $b_2 \parallel [101]$ are both parallel to $x_1$- and $x_3$-axis, respectively. As a result of arbitrarily selecting the reference state identical to the Au (or Cu) natural state, for which the geometry of interface dislocations (line directions and spacings) is independent of the choice of reference state, the line directions are defined by $\xi_1^{\text{un}} \parallel [101]$ and $\xi_2^{\text{un}} \parallel [10\bar{1}]$, and the inter-dislocation spacings are given by $d_1^{\text{un}} = d_2^{\text{un}} = p_1^0 = p_2^0 = 2.25144$ nm. Thus, the Frank-Bilby equation predicts that an orthogonal network of straight parallel dislocations with pure edge characters is also needed to accommodate the pure misfit (010) Au/Cu interface.

The geometry of such orthogonal grid of dislocations can also be characterized by $\eta_1 \to 1/2$ and $\eta_2 \to 1/2$ in the general parametric framework, because $\phi^{\text{un}} = \pi/2$. According to the bilinear function $u^p(\eta_1 \to 1/2, \eta_2 \to 1/2)$ for the prescribed displacement field in eq. (3.87), the corresponding real-valued expansion functions in eq. (3.88) for the individual set 1 can be computed by imposing $m = 0$, as follows

$$ \left\{ \begin{array}{l} \lim_{\eta_1 \to 1/2} c_2 \hat{u}_1^p(\eta_1, \eta_2) = \lim_{\eta_2 \to 1/2} c_2 \hat{u}_1^p(\eta_1, \eta_2) = \frac{(-1)^{n+1}}{2\pi n} \mathbf{b}_1 \\ \\ \lim_{\eta_2 \to 1/2} \lim_{\eta_1 \to 1/2} c_2 \hat{u}_2^p(\eta_1, \eta_2) = \lim_{\eta_2 \to 1/2} \frac{(-1)^{n+1}(-1+2\eta_2)}{2\pi n} \mathbf{b}_2 = \mathbf{0}, \end{array} \right. \quad (3.122) $$

exhibiting that $\hat{u}^p(\eta_1 \to 1/2, \eta_2 \to 1/2) = \hat{u}_1^p(\eta_1 \to 1/2, \eta_2 \to 1/2)$ is evidently written as a function of $b_1$, for set 1. By superposing the similar contribution of set 2 with $n = 0$, the final prescribed displacement field produced by an orthogonal network of dislocations in eq. (3.88) is therefore written in the form of two distinct one-dimensional sawtooth-shaped functions with
<figure><img src="image_44.png" /><figcaption>FIGURE 3.27: Elastic strain energy landscapes $\gamma_e$ in J.m$^{-2}$ of the hexagonal-shaped patterns with three-fold dislocation nodes as a function of $\eta_1$ and $\eta_2$, for the (a) (010) and (b) (111) Au/Cu heterophase interface cases. The large points at $\eta_1 = \eta_2 = 1/2$ correspond to the initial lozenge-shaped patterns, for which the two crossing dislocation sets are related to equilibrium and non-equilibrium dislocation configurations for the (010) and (111) interface planes, respectively. The latter case gives rise to the presence of a minimum-energy path (in black) between the initial pattern and the fully elastically strain-relaxed dislocation structure (magenta point) at stable equilibrium state. An intermediate state is displayed by the orange point.</figcaption></figure>

Fourier sine series, as follows

$$ \mathbf{u}^p(x_1, x_3) = - \sum_{\substack{n \ge 1 \\ m=0}} \frac{(-1)^n}{\pi n} \sin(2\pi k_1(n, 0) x_1) \mathbf{b}_1 - \sum_{\substack{n=0 \\ m \ge 1}} \frac{(-1)^m}{\pi m} \sin(2\pi k_3(m) x_3) \mathbf{b}_2, \quad (3.123) $$

where $k_1(n, 0)$ and $k_3(m)$ are defined in eq. (3.74), with $\phi^{\text{un}} = \pi/2$. Here, the sawtooth-shaped functions in eq. (3.123) differ from eq. (3.19) by individual translations of magnitude $d_i^{\text{un}}/2$. Similarly to $\Sigma_1$ and $\Sigma_2$ in eqs. (3.20) and (3.25), the simplest limiting case of bicrystals without any surface/interface elasticity effects leads to a set of twelve real and linear equations, i.e.

$$ \left\{
\begin{array}{l}
\text{Re} \sum_{\alpha=1}^{3} A \lambda^{\alpha} A a^{\alpha} - B \zeta^{\alpha} B a_{*}^{\alpha} = \mathbf{0} \\
\\
\text{Im} \sum_{\alpha=1}^{3} A \lambda^{\alpha} A a^{\alpha} - B \zeta^{\alpha} B a_{*}^{\alpha} = \mathbf{\vartheta}, \quad \text{with: } \mathbf{\vartheta} = \begin{cases} -\frac{(-1)^n}{\pi n} \mathbf{b}_1 & \text{if } n=0 \quad n \in \mathbb{N}^* \\ -\frac{(-1)^m}{\pi m} \mathbf{b}_2 & \text{if } m=0 \quad m \in \mathbb{N}^* \\ \mathbf{0} & \text{if } nm \neq 0 \quad n \in \mathbb{N}^*, m \in \mathbb{N}^* \end{cases} \\
\\
\text{Re} \sum_{\alpha=1}^{3} A \lambda^{\alpha} A h^{\alpha} - B \zeta^{\alpha} B h_{*}^{\alpha} = \mathbf{0} \\
\\
\text{Im} \sum_{\alpha=1}^{3} A \lambda^{\alpha} A h^{\alpha} - B \zeta^{\alpha} B h_{*}^{\alpha} = \mathbf{0},
\end{array}
\right.
\qquad (3.124) $$

with respect to the six associated complex unknown quantities, i.e. $A\lambda^\alpha$ and $B\zeta^\alpha$.
Following the procedure described in section 3.3.3, the two deformation gradients $F_{Au}^{-1}$ and $F_{Cu}^{-1}$ in eq. (3.121) (also, the magnitudes of both $b_1$ and $b_2$) are determined by ensuring the condition of vanishing far-field stresses along a transformation pathway between both materials Au and Cu. For cube-cube orientation relation, this condition is met by continuously adjusting the reference lattice parameter $a_{ref}$ along a specified reaction pathway coordinate $\delta$, starting with the pure lattice parameter of Au to Cu, i.e.

$$a_{ref} = (1 - \delta) a_{Au} + \delta a_{Cu}, \qquad (3.125)$$

where $0 \le \delta \le 1$ is a dimensionless variable that interpolates linearly between $a_{Au}$ and $a_{Cu}$.

According to the far-field eq. (3.1), the dependence of the total large-range stress components $\sigma_{11}^c + \sigma_{11}^{\pm\infty}$ in Au (black line with symbols) and Cu (red line with symbols) on the transformation pathway coordinate $\delta$ is plotted in Fig. (3.26). For the (010) misfit case, both far-field stress components vanish for $\delta_{(010)} = 0.60392$, so that the corresponding reference state is closer to Cu than to Au, i.e. $\delta_{(010)} > 0.5$, where $Cu c_{11} < Au c_{11}$ and $Cu c_{12} < Au c_{12}$, but $Cu c_{44} > Au c_{44}$. All other elastic components are consistent with the absence of strains in the long range and no rotations are induced along the transformation path. Thus, it gives rise to the reference lattice parameter $a_{ref} = 0.37984$ nm, and also the magnitudes of correct Burgers vectors, i.e. $b_1 = b_2 = 0.26859$ nm, selected by the coherent reference state. When an incorrect reference state is arbitrary chosen, the corresponding Burgers vectors deviate in magnitude and non-zero spurious stress fields exist in the microstructure. For instance, a residual stress state in Au persists with $Au\sigma_{11}^c + Au\sigma_{11}^{\pm\infty} \simeq 6.29$ GPa and $\simeq -3.66$ GPa, for $\delta_{(010)} = 0$ and 1, respectively. A larger residual stress field exists in Cu as well, where $Cu\sigma_{11}^c + Cu\sigma_{11}^{\mp\infty} \simeq -8.77$ GPa, for $\delta_{(010)} = 0$, and $\simeq 5.09$ GPa, for $\delta_{(010)} = 1$.

For the following calculations in interfacial hexagonal-shaped dislocation patterns, the upper half-plane domain $D = \{\{0 \le n \le n_{max}\} \cup \{|m| \le n_{max}\} \setminus \{m \le 0, n = 0\}\}$ is defined by setting $n_{max} = 50$, which is large enough to ensure accurate solutions in truncated elastic stress fields with three sets of dislocations.

Figure (3.27a) shows the elastic strain energy landscape for the (010) Au/Cu misfit interface with classical boundary conditions between both neighboring semi-infinite Au and Cu crystals, for simplicity. To determine such energy landscape, the triangular domain $T_{ABC}$ is first discretized into 121 nodal points with coordinates $(\eta_1^i, \eta_2^i)$, such that $\{\eta_1^i, \eta_2^i\} \in ]0, 1/2]^2$, as depicted by the gray dots in Fig. (3.27a). Using the persistent short-range elastic fields, the finite (guaranteed by the zero far-field stresses) stored elastic energy per unit area is computed for any $(\eta_1^i, \eta_2^i)$ using eq. (3.116) with $r_0 = b_1/4$. Following the standard interpolation procedure of eq. (3.117), the elastic strain energy for any given $(\eta_1, \eta_2) \in ]0, 1/2]^2$ shows a smooth and symmetric landscape with respect to the median $(\eta_1 = \eta_2)$ of the triangular domain, within which the unique strain energy minima is obtained at $\eta_1 \to 1/2$ and $\eta_2 \to 1/2$, with $\gamma_e^{\min} = \gamma_e(\eta_1 \to 1/2, \eta_2 \to 1/2) \simeq 0.57344$ J.m$^{-2}$. Planar dislocation reactions and junctions for (010) misfit interfaces are also shown to be energetically unfavorable. It is therefore demonstrated that the initial orthogonal grid of uniformly spaced edge dislocations corresponds to the equilibrium structures for the (010)-type misfit interfaces, which satisfies the condition of vanishing far-field stresses as well as the minimum-energy criterion for predicting the most favorable dislocation structures.

Near the unreacted state of the (010) Au/Cu system, the present energy landscape shows concave slope profiles at $\eta_1 \simeq \eta_2 \simeq 1/2$. For calculations with other fcc/fcc heterosystems in the (010) cube-cube orientation relationship (not shown here), the corresponding unreacted state can exhibit convex energy profiles, which suggest different bound crossed states of dislocation reactions for the (010) twist GBs. Thus, the parent dislocations could also exhibit strong repulsive interactions or crossed states where local bend and twist of dislocations may locally occur at the short-range distances, as observed in non-coplanar dislocations [176].
<figure><img src="image_45.png" /></figure>

FIGURE 3.28: (a) Dependence on $\eta_1 = \eta_2$ of the elastic energy $\gamma_e$ in J.m$^{-2}$, i.e. along the bisecting lines of the admissible triangular domains $T_{ABC}$, as displayed in Figs. (3.27). The blue and the red curves correspond to the mismatched (010) and (111) Au/Cu interfaces, respectively. The latter exhibits black dots, indexed by $s_{\eta} = 1, \dots, 16$, which represent the minimum-energy path from Fig. (3.27b). The large points at $\eta_1 = \eta_2 = 1/2$ are related to the initial lozenge-shaped patterns with two crossing sets of dislocations, whereas the vertical arrow shows the minimum-energy configuration associated with the (111) semicoherent interface. (b) Dependence on $s_{\eta}$ of the dislocation characters $\phi_i$ for the three sets and the angle $\phi$ between the two parent dislocations for the corresponding (111) Au/Cu case. All these quantities are expressed in $^{\circ}$.

Case 2: The (111) Au/Cu interface with three sets of dislocations

In contrast to the (010) Au/Cu case, the (111)-oriented habit interface planes exhibit different arrangements of atoms, which yield to more complex interface dislocation patterns and also to general elastic states where both constituent strains and rotations are unequally partitioned between the crystals [123].

The present orientation relations associated with the (111) Au/Cu misfit case are defined by $\mathbf{x}_1^{\text{or}} = [11\bar{2}]$, $\mathbf{x}_2^{\text{or}} = [111] \parallel \mathbf{n}$, and $\mathbf{x}_3^{\text{or}} = [1\bar{1}0]$, within which the fcc {111} close-packed planes contain $a_{\text{ref}}/2\langle 110 \rangle$-type Burgers vectors. Similarly to the (010) case, such Burgers vectors must be defined in the proper reference state under the condition of vanishing far-field stresses in the (111) Au/Cu bicrystal. By arbitrarily choosing $\mathbf{b}_1 = a_{\text{Au}}/2[10\bar{1}]$ and $\mathbf{b}_2 = a_{\text{Au}}/2[0\bar{1}1]$ as the reference Burgers vectors, the quantized Frank-Bilby eq. (3.121) gives rise to the lozenge-shaped dislocation structure that is specifically comprised of two arrays of parallel dislocations (with no local reactions at nodes): the initial line directions are defined by $\xi_1^{\text{un}} \parallel [01\bar{1}]$ and $\xi_2^{\text{un}} \parallel [10\bar{1}]$, so that the individual characters are $\phi_1^{\text{un}} = \phi_2^{\text{un}} = 60^{\circ}$, and the angle between these two unrelaxed sets of dislocations is $\phi^{\text{un}} = 60^{\circ}$. In addition, $p_1^0 = p_2^0 = 2.25144$ nm, so that the inter-dislocation spacings are given here by $d_1^{\text{un}} = d_2^{\text{un}} = 1.94980$ nm.

As illustrated in Fig. (3.26), the dependence of the total far-field stress components in the (111) system, i.e. in both Au (black line) and Cu (red line) on $\delta$, yields to a predicted reference state for $\delta_{(111)} = 0.57962$, so that $a_{\text{ref}} = 0.38096$ nm, and also to the magnitudes of correct Burgers vectors are defined by $b_1 = b_2 = 0.26938$ nm. Moreover, Fig. (3.26) shows stronger spurious stress values for the (111) than (010) system cases, by a factor of 2.33 (2.15) in Au (Cu) when $\delta = 0$, i.e. when Au is improperly selected as the reference state. The same qualitative conclusion regarding the spurious stress state can be drawn for $\delta = 1$.

Using the aforementioned Frank-Bilby solution as the initial dislocation structure for possible elastic strain relaxation, Fig. (3.27b) shows the pre-computed elastic landscape as function of $\eta_1$ and $\eta_2$, associated with the (111) misfit interface case. The symmetric landscape has been computed using the same number of nodal points than in Fig. (3.27a), for which the orientations of both plots are different for clarity. The elastic energy per unit interface area for the unrelaxed
<figure><img src="image_46.png" /><figcaption>(b) Normal displacement field (nm)</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>(c) Displacement norm (nm)</figcaption></figure>

<figure><img src="image_48.png" /></figure>

FIGURE 3.29: Plots of initial, intermediate, and final states along the computed minimum-energy path for the (111) Au/Cu heterophase interface. The initial periodic network of lozenge-shaped misfit dislocations undergoes local relaxations and also leads to a final elastically relaxed hexagonal-shaped dislocation pattern with lowest short-range strain energy. (a) Dislocation structures. Distribution of (b) the normal displacement component $u_2$ and (c) the displacement norm $u$. See text for the displacement field expressions.
lozenge-shaped dislocation pattern is given by $\gamma_e(\eta_1 \rightarrow 1/2, \eta_2 \rightarrow 1/2) \approx 0.49568 \text{ J.m}^{-2}$, with $r_0 = b_1/4$, which is slightly lower than the stored energy for the (111) Au/Cu system for $\eta_1 \rightarrow 1/2$ and $\eta_2 \rightarrow 1/2$. Here, the landscape for the (111) system is qualitatively and quantitatively different than the (010) case, since the former gives rise to the existence of a unique minimum-energy dislocation configuration with three sets of dislocations resulting from junction reactions.

The energy minimization procedure that involves the conjugate gradient algorithm is performed by using a prescribed convergence criterion in the pre-computed energy landscape. The interface dislocation structures with the lowest elastic energy are considered to be found when the difference between the values of the stored elastic energy for two subsequent iterations is less than $10^{-4} \text{ J.m}^{-2}$. The corresponding minimum-energy path is determined by using the nudged elastic band method between the initial non-equilibrium and the minimum-energy states, for which the spring constant $k$ in eq. (3.120) has been varied over several orders of amplitude without noticeable effects on the computed path. The obtained minimum-energy path is displayed in Fig. (3.27b) by the black curved chain with equidistantly positioned images (i.e. intermediate states), where the final configuration state is designated as the final elastically strain-relaxed dislocation pattern. Here, the smooth path has no energy barrier (therefore also, no saddle point) and 15 intermediate states, which connect the initial and final states, are constructed. The minimum strain energy related to the relaxed dislocation pattern is given by $\gamma_e^{\min} = \gamma_e(\eta_1 \rightarrow 0.31981, \eta_2 \rightarrow 0.31981) = 0.44733 \text{ J.m}^{-2}$, which corresponds to a significant decrease in strain energy of 9.75%.

The variations of strain energy along the median ($\eta_1 = \eta_2$) of the two (010) and (111) Au/Cu landscapes, as displayed by the blue and red dotted lines in the insets of Fig. (3.28a), start from their initial corresponding lozenge-shaped dislocation structures at $\eta_1 = \eta_2 = 1/2$ with different stored energy values. The red (blue) line illustrates the (un)favorable elastic energy profile for junction formation that continuously decreases (increases) with decreasing both values of $\eta_1$ and $\eta_2$ from $1/2$ at the (111) ((010)) Au/Cu heterophase interface. The intermediate states between the lozenge-shaped and the relaxed hexagonal-shaped dislocation configurations for the (111) case are indexed by $s_\eta = 1, \dots, 15$. Such considerable saving in strain energy along $s_\eta$ is related to the change in dislocation structures, e.g. dislocation characters $\phi_i$ and the angle $\phi$ between $\xi_2$ and $\xi_1$, which can be examined along the determined minimum-energy path. Figure (3.28b) plots these geometrical characteristics in terms of $\phi$ (in green), $\phi_1$ (blue), $\phi_2$ (red), and $\phi_3$ (black, for the newly formed set of dislocation junction) as a function of $s_\eta$. It is also found that the geometrical equilibrium configuration of the minimum-energy dislocation pattern is characterized by $\phi^{\text{eq}} \approx 128.4^\circ$, $\phi_1^{\text{eq}} = \phi_2^{\text{eq}} \approx 85.8^\circ$, and $\phi_3^{\text{eq}} = 90^\circ$. Both sets 1 and 2 deviate by $4.2^\circ$ from pure edge characters, and the dislocation structure deviates by $8.4^\circ$ from regular hexagonal-shaped configuration. Such dislocation arrangement is in agreement with atomistic analysis in iron, where deviations from pure screw dislocations in (110) bcc twist GBs with comparable order of dislocation spacings have been reported using molecular statics simulations [290].

Figures (3.29) illustrate the strain-relaxed rearrangements of the interfacial dislocations from the lozenge-shaped configurations on the (111) heterophase interface using different elastic quantities, which can, for example, be used to analyze the likely regions for nucleating interface dislocations or absorbing and annihilating point defects (interstitials and vacancies). All contour plots are displayed at $x_2 = 3a_{\text{Au}}$ with respect to the three dislocation configurations shown in Figs. (3.29a), i.e. the "initial"<fnref n="1" /> at $s_\eta = 1$, intermediate ($s_\eta = 8$), and the final relaxed ($s_\eta = 16$) states, for which the specific intermediate case is located exactly halfway between both initial and final states, as depicted by the orange point along the computed minimum-energy path in Fig. (3.27b). A schematic representation of the atomically sharp (111) Au/Cu interface with current periodic dislocation lines is shown in Figs. (3.29a), where the Au (Cu) atoms are plotted by white (gray)

<footnote n="1">¹Here, "initial" means the first admissible configuration with three sets of dislocations, where an initially small dislocation segment for the junction has been introduced (in the direction of the steepest descent between the two parent sets) to solve the corresponding solutions for hexagonal-shaped dislocation patterns.</footnote>
dots. The three corresponding Burgers vectors on the (111) close-packed plane are represented as well.

Figures (3.29b) and (c) illustrate the normal displacement component $u_2 = \tilde{u}_2^{\text{dis}}(x_1, 3a_{\text{Au}}, x_3)$ and the displacement norm $u = |\tilde{u}^{\text{dis}}(x_1, 3a_{\text{Au}}, x_3)|$, respectively. Figures (3.29b) show that the minimum values of $u_2 = -0.01$ nm are located in the centers of the dislocation patterns, while the maximum values yield close to the dislocation junctions for the initial unrelaxed pattern. In the final relaxed dislocation configuration, the maximum values are unequally distributed at the three-fold dislocation nodes, e.g. $J_I = \{J_1, J_3, J_5\}$ versus $J_{II} = \{J_2, J_4, J_6\}$, for which the set of junction nodes $J_I$ gives rise to larger amplitudes of $u_2$ than $J_{II}$. Figures (3.29c) display the complex relief of displacement norm $u$ with the largest magnitudes at $J_I$, for illustration.

### 3.6.6 Comparison with atomistic simulations

The model interfaces for the present comparisons with atomistic simulations are selected according to the following criteria:

1. The structure of the interface is describable as a dislocation network. The present study is concerned with dislocation-based models of interface structure. Thus, interfaces to which these models do not apply are not suitable.

2. This dislocation network undergoes a relaxation through the dissociation of four-fold junctions into three-fold junctions. Some interfacial dislocation networks are not suitable for the study because they contain stable four-fold junctions that do not undergo any relaxation.

3. The interface dislocation network is initially periodic and remains so as it relaxes. Moreover, the dislocations in the network do not dissociate into partials. These choices are necessitated by current limitations in modeling capabilities [249, 250]. The requirement of periodicity is met by selecting special interfaces that may be modeled by two overlapping sets of misfit dislocations, whereas general interfaces involve three overlapping dislocation sets [1]. The requirement of no dissociation excludes from consideration GBs in low stacking fault energy materials.

4. The final structure of the relaxed interface is not the outcome of any inherent symmetry that the interface possesses. For example, while twist boundaries on {111} planes in aluminum meet all the foregoing conditions, they are excluded from consideration because the relaxed dislocation structure in these interfaces has the same $p6m$ symmetry as the underlying, unrelaxed dichromatic pattern [69, 70]. Such a symmetry-driven relaxation does not constitute a stringent test of the elasticity-based relaxation model.

5. Differences between the relaxed and unrelaxed dislocation network must be discernable in atomistic simulations. Thus, the dislocations should not be so closely spaced that they are difficult to distinguish yet not so far apart that they would require very large atomistic models. This criterion is met through judicious selection of the interface crystallographic character (misorientation, misfit, and plane orientation).

All of the foregoing criteria are met by the two classes of model interfaces selected for the present comparison: low-angle twist GBs on {110}-type planes in niobium (Nb t-GBs) as well as heterophase interfaces between {111}-type planes of silver and {110}-type planes of vanadium (Ag/V interfaces). For both interface types, a series of structures is considered by varying twist angle, $\theta$, i.e. $0^\circ \le \theta \le 10^\circ$ for both interface types. When $\theta = 0^\circ$, the Nb t-GB reduces to a perfect single crystal while the Ag/V interface is in the NW OR [279, 192], where $\langle 110 \rangle_{\text{fcc}}$ and $\langle 100 \rangle_{\text{bcc}}$ are parallel within the interface plane.

Ag/V interfaces formed in magnetron sputtered multilayers have been characterized extensively [280]. They are observed in a variety of ORs and with a wide range of interface planes.
<figure><img src="image_49.png" /><figcaption>FIGURE 3.30: Nb-t GB energies computed as a function of θ using the dislocation-based model and atom-istic modeling.</figcaption></figure>

Among the structures reported are Ag/V interfaces in the KS and NW ORs, both along Ag {111} and V {110} planes. They have been previously modeled using elasticity theory, albeit without accounting for network relaxations, as well as using classical potential [172]. Comparisons with atomistic simulations revealed discrepancies that were hypothesized to arise from nodal reconstructions of the kind investigated here. The dislocation-based model is presented in details in section 3.6, while the embedded atom method potentials are used to model atomic interactions in both Nb [294] and Ag/V [281].

No experimental investigations of Nb {110} t-GBs have been reported. Nevertheless, these interfaces were previously investigated by atomistic simulations [173], by anisotropic linear elasticity theory, and most recently using phase field models [211]. However, no quantitative comparison between structures predicted by the elasticity theory and atomistic modeling has been previously conducted.

### Nb {110} t-GBs

Figure (3.30) compares the energy of Nb t-GBs computed from atomistic models with values obtained using the dislocation-based model, the latter using two different core cutoff radii. Both atomistics and the elasticity theory reveal similar trends, with energies increasing monotonically as a function of θ within the range of twist angles investigated. Comparison of elastic results before and after relaxation of the dislocation network shows that this step in the calculation yields a relatively modest reduction in elastic energies. For example, for θ = 2°, the reduction is approximately 8% of the initial energy. Energies computed from atomistic models are higher than those obtained from the elasticity theory. This difference is due to dislocation core energies, which are inherently captured in the atomistic calculation, but are not accounted for in the dislocation approach. The larger the core cutoff, the lower the energy computed by the present calculations. Interestingly, regardless of the cutoff radius, the values are smaller than the atomistic ones by an apparently θ-independent factor, consistent with both the elastic and core energies scaling in proportion to the total length of dislocation segments in the network, to a first approximation.

Figure (3.31) compares the structure of Nb t-GB dislocation networks determined from atomistic modeling to ones found with the elasticity theory, using θ = 2° as an example. Other twist angles give rise to qualitatively similar structures. The atomistic structure in Fig. (3.31a) consists of a 2-D tiling of hexagonal regions separated by a connected network of misfit dislocation segments of predominantly screw character. Two types of segments are present: ones with 1/2 ⟨111⟩-type Burgers vectors as well as ones with ⟨100⟩-type Burgers vectors. As shown in Fig. (3.31a), the former are approximately twice as long as the latter. Consistent with previous studies in bcc Nb [173]
<figure><img src="image_50.png" /></figure>

FIGURE 3.31: Dislocation networks in Nb t-GBs obtained by (a) atomistic modeling and the dislocation-based model with core cutoff radii of (b) and one quarter of the Burgers vector magnitude and (c) two times the Burgers vector magnitude. Atoms in (a) are colored by their potential energy. The pattern is symmetric with respect to the dashed mirror lines in (a). In (b) and (c), dislocation line segments are superimposed on the unrelaxed dichromatic pattern of the GB.

and iron [288], both segment types have compact cores of atomic-scale dimensions. The hexagonal regions making up the t-GB are symmetric with respect to reflections about mirror lines parallel and perpendicular to the shorter segments (with 〈100〉-type Burgers vectors).

Similar to the atomistic structure, the network predicted by the elasticity theory—shown in Fig. (3.31b) and (3.31c)—also consists of predominantly screw character dislocation segments with 1/2 〈111〉- and 〈100〉-type Burgers vectors, the former with approximately twice the length of the latter. However, unlike in the atomistic structure, for a core cutoff radius of one quarter of the Burgers vector, the dislocation-based model network consists of slightly distorted hexagons with no lines of mirror symmetry, as evidenced by the unequal values of the angles $α_1$ and $α_2$ between the short and long dislocation segments in Fig. (3.31b). In addition to the geometry shown in Fig. (3.31b), the elasticity theory also predicts another stable dislocation network configuration with identical energy by reversing the circulation of the Burgers vectors and with the values of the two angles $α_1$ and $α_2$ reversed. Thus, the complete elastic energy landscape of the Nb t-GB dislocation network with this cutoff radius—expressed in terms of the nodal positions—shares the same symmetry as the GB dichromatic pattern itself, but the individual dislocation configurations corresponding to the minima in that landscape do not.

The discrepancy between the elasticity-based and atomistic structures has been analyzed in detail and confirmed that, for core cutoff radii of one quarter of the Burgers vector, it occurs systematically for all the twist angles and is not due to inadequate relaxation of either model. Its cause ultimately traces back to the character dependence of dislocation strain energies in bcc crystals [20, 11]. In elastically isotropic bcc crystals, screw dislocations have the lowest energy per unit length. By contrast, in elastically anisotropic materials, the dislocation energy per unit length is lowest for mixed dislocations. For instance, in Nb, dislocation arrays with [111]-type Burgers vectors exhibit a deviation of ~ 10° with respect to perfect screw character [250]. The asymmetry of the distorted hexagons in Fig. (3.31b) increases the edge component of the constituent dislocation segments, thereby reducing the elastic strain energy, as compared to the perfectly symmetric hexagons in Fig. (3.31a).

Interestingly, when the core cutoff radius is increased to two times the Burger vector, the elastic prediction of the relaxed dislocation network is symmetric, as shown in Fig. (3.31c). At first, it might be tempting to say that using a larger core cutoff changes the character dependence of the dislocation elastic energy, e.g. by lowering the energy of the pure screw relative to a mixed character. However, the form of the elastic field around an isolated dislocation has no characteristic length scale, so changing the core cutoff cannot lead to any change in the character dependence of dislocation properties [7]. Rather, the difference between the patterns in Fig. (3.31b) and (c) is
<figure><img src="image_51.png" /><figcaption>FIGURE 3.32: Ag/V interface energies computed as a function of $\theta$ using the dislocation-based approach and atomistic modeling.</figcaption></figure>

likely due to the length scale of the GB dislocation network itself, in particular to features of its elastic field within a distance of ~2b from the three-fold junctions that, when excluded from the elastic energy calculation, shift the elastic energy minimum to the symmetric state. Evidence for such near-node effects at twist GBs along {110}-type planes in bcc metals has been found in phase field simulations of dislocation networks, where dislocations are seen to acquire a slight curvature near three-fold junctions in some materials [270].

## Ag/V interfaces

Figure (3.32) plots energies of Ag/V interfaces as a function of twist angle, $\theta$. As discussed in section 3.4.6, the energies of heterophase interfaces, such as Ag/V, may be viewed as the sum of a chemical contribution, which is due to the difference in bonding between the two elements in the coherent reference state, and a contribution from the misfit dislocation network, which is associated with the relaxation of coherency. Only the latter depends on the twist angle while the former is a constant, independent of $\theta$. The elasticity-based model only computes the elastic contribution to the misfit dislocation network energy. Thus, to ease comparison of energies computed from atomistics to those computed using the elasticity theory, all plots in Fig. (3.32) have been shifted so that their minima are at an energy value of zero. For the atomistic calculations, a downward shift of 0.85 J/m$^2$ was imposed while all the elastic calculations were shifted downward by 0.24 J/m$^2$. The difference between these shift values, i.e. 0.61 J/m$^2$, is due to the chemical bonding contribution to the total interface energy. It is substantially larger than the elastic contribution. This conclusion is consistent with previous first-principles calculations, such as the one on Fe/VN interfaces reported in Ref. [134].

The Ag/V interface energies computed using the atomistic model exhibit local maxima at $\theta = 0^\circ$ (the NW OR) and $\theta = 5.25^\circ$ (the KS OR), two nearly degenerate minima at $\theta = 4.5^\circ$ and $\theta = 6^\circ$, and a monotonically increasing energy for twist angles greater than 6$^\circ$. Figure (3.32) plots the elastic energies for two unrelaxed dislocation network configurations, labeled "case 1" and "case 2". These cases correspond to two different solutions to the Frank-Bilby equation obtained by selecting two different combinations of misfit dislocation Burgers vectors, following the terminology introduced in section 3.4.6. Thus, case 1 is identified as the solution obtained using Burgers vectors $b_1$ and $b_2$ and case 2 as that obtained using $b_1$ and $b_3$, with $b_1 = a_{\text{ref}}/2[\bar{1}01]$, $b_2 = a_{\text{ref}}/2[0\bar{1}1]$, and $b_3 = a_{\text{ref}}/2[\bar{1}10]$ in the reference crystal. Case 1 has the lower energy for all twist angles, except in the interval ~4.25$^\circ$ < $\theta$ < ~5.25$^\circ$, where the energy of the latter is lower. One point of intersection between the case 1 and case 2 plots in Fig. (3.32) occurs near a local energy maximum, close the KS OR. Each of the unrelaxed configurations predicts one energy minimum, with twist angles and energies in reasonable agreement with one of the minima in the
<figure><img src="image_52.png" /><figcaption>FIGURE 3.33: The dislocation structures of the Ag/V interface obtained by the atomistic modeling (atoms coloured by their local potential energies using the same color bar as in Fig. (3.31) from -5.13 ev (blue) to -5.11 ev (red) and dislocation-based model (dislocation segments superimposed on dichromatic patterns). In (a), (b), and (c), the black line segment is the dislocation segment created upon dissociation of a four-fold node into two three-fold nodes. There is no such segment in (d) because the four-fold node in this network does not dissociate. The thin black lines in (a) illustrate the shape of the initial, unrelaxed dislocation network.</figcaption></figure>

atomistic model. Moreover, the energies of case 1 are in quantitative agreement with the atomistic model for $\theta > 6°$. However, at lower twist angles ($\theta \sim 6°$), case 1 systematically overpredicts the interface energy by approximately 20 mJ/m$^2$.

As demonstrated in Fig. (3.32), relaxation of the dislocation network structure in the elasticity-based model removes nearly all discrepancies in energy between the two models. Quantitatively accurate predictions of interface energies are achieved over the full range of twist angles, with the greatest differences being on the order of 10 mJ/m$^2$ and occurring within a relatively narrow range of twist angles centered approximately on $\theta = 5.5°$. Most notably, the match in the energy and twist angle of the minimum near $\theta = 4.5°$ improves and the discrepancy between the elasticity-based model and atomistic energies for $\theta \sim 4°$ is removed.

The dislocation network geometries predicted by atomistics and the elasticity theory are compared in Fig. (3.33) for several representative twist angles. All these examples exhibit good qualitative agreement between the two modeling methods, with the general shape of the relaxed patterns matching that of the atomistic models. Small quantitative discrepancies in the lengths and angles of individual dislocations are nevertheless apparent, e.g., in the orientation of the dislocation segments colored blue for $\theta = 2°$ in Fig. (3.33b). Interestingly, the elasticity-based model predicts no dissociation of four-fold dislocation nodes for $\theta = 8°$, thereby explaining why the unrelaxed network labeled "case 1" in Fig. (3.32) predicts the energy of this interface so accurately.

## Discussion

It is found that incorporating relaxation leads to improved predictions of interface energies for Ag/V interfaces, yielding nearly perfect quantitative agreement with atomistic models. In particular, it did not appear that these predictions might be improved by incorporating a dislocation core model into the elasticity theory. This apparent insensitivity to core structure may be due to the relatively large (1 nm-scale [258]) width of misfit dislocation cores in such interfaces as well as their confinement to the interface plane [73]. It indicates that the effectiveness of linear elastic,
dislocation-based models in predicting interface energies and structures is greater than anticipated
for some heterophase interfaces, such as the Ag/V interface studied here.

The investigation also shows that, while incorporating relaxations improved quantitative predictions of Ag/V interface energies, it did not alter the qualitative features of the energy versus twist angle plots obtained from unrelaxed dislocation models. In particular, it appears that qualitative features of this curve—such as the number of energy minima and maxima as well as their twist angles and relative energies—may be obtained via consideration of unrelaxed structures alone. Relaxation of the dislocation network is not essential for predicting those aspects of the interface energy dependence on twist angle. However, dislocation network relaxation is essential for correctly predicting the geometry of the dislocation networks in these interfaces.

For Nb t-GBs, marked differences between energies computed using the elasticity-based model and atomistic models are observed. These differences are attributed to the significant contribution of dislocation cores to the total energy of these interfaces: a contribution naturally accounted for in the atomistic model, but not in the elasticity theory. In particular, the incorporation of dislocation network relaxations in the elasticity theory does not resolve the observed discrepancies. Moreover, use of short (≈ b/4) core cutoff radii leads to asymmetric lowest energy network geometries, contrary to atomistic models. Thus, to better predict the energies of Nb t-GBs, the dislocation approach should be augmented with a core model. Overall, the introduction of core-spreading dislocations in continuum mechanics is a long-standing problem. One approach might be to include a Peierls-Nabarro type calculation with gamma-surfaces obtained from first principles calculations [70]. Some phase field models of dislocation behavior already use such techniques to model dislocation core spreading as well as the corresponding core energy [270, 211]. A second branch is based on generalized higher-order continuum dislocation mechanics [85, 155, 238, 208], which provides length-scale dependent field solutions. Besides these approaches, a recent core-spreading treatment to the present interfacial dislocation networks has been proposed, as described in section 3.8.

**3.7 Interaction with extrinsic dislocations in bimaterials**

In this section, lattice dislocation interactions with semicoherent interfaces are studied by means
of anisotropic field solutions in homo- and hetero-structures. The Stroh formalism cover different
shapes and dimensions of various extrinsic and intrinsic dislocations<fnref n="2" />. As illustrated in Fig. (3.34),
equi-spaced arrays of straight lattice dislocations and finite arrangements of piled-up dislocations
as well as polygonal and elliptical dislocation loops are considered using the superposition prin-
ciple in three dimensions. Interaction and driving forces are derived to compute the equilibrium
dislocation positions in pile-ups, including the internal structures and energetics of the interfacial
dislocations. For illustration, the effects due to the elastic and lattice mismatches are discussed
in the pure misfit Au/Cu and heterophase Cu/Nb systems, where the discrepancies from the
approximation of isotropic elasticity are shown.

**3.7.1 Extrinsic dislocation arrays and loops**

In classical dislocation dynamics calculations [246, 254], the material volumes of interest are usu-
ally regarded as a representative part of infinitely large crystals that are replicated by periodic
boundary conditions to preserve the translational invariance. It becomes also useful to derive ac-
curate field solutions for infinite dislocation arrays, for which the periodicity of lattice dislocations

<footnote n="2">²In accordance with the former derivations established by the Pan and workers, two explicit conventions have been changed with the foregoing formalism: the interface normal $n \parallel x_2$ with $n \parallel x_3$, as well as the positive with negative sign of the exponential in the Fourier transforms in eqs. (3.5) and (3.73), without loss of generality. These conventions will be adopted to the end of the manuscript.</footnote>
<figure><img src="image_53.png" /><figcaption>FIGURE 3.34: Linear superposition scheme of elastic field solutions in dissimilar anisotropic materials due to the presence of (a) misfit dislocation patterns at the heterophase interfaces, (b) infinite long straight lattice dislocations in the upper material A, (c) arbitrary polygonal and elliptical dislocation shear or prismatic loops in both crystals, (d) piled-up dislocation arrays onto specific glide planes in the lower material B. For each dislocation problem, the insets show particular features that are explained in the text. The three-dimensional applied stresses, internal stresses, image as well as self-stresses are superposed in (e) to compute some interaction force distributions on defects, as illustrated in the insets of (c) and (d).</figcaption></figure>

is consistent with infinitely periodic boundary conditions applied to the elementary representative volumes, without introducing truncation in replicating simulation cells. The elastic solutions for arrays of lattice dislocations and dislocation pile-ups located in bimaterials are analytically obtained using anisotropic elasticity, respectively. Without loss of generality, the following solutions are given for singularities located either in material A or B.

### Elastic fields for infinitely periodic dislocation arrays

The linear elasticity problem of the elastic field solutions in both materials A and B due to a Volterra-type lattice dislocation array with periodic inter-dislocation distances $h$ in bicrystals is solved for single dislocations [199, 198, 200] with infinite series [61]. For $n$ lattice dislocations from $-\infty$ to $\infty$, which are located in material A at $(x_1^{\text{lat}} + nh, x_3^{\text{lat}})$, the corresponding displacement field in material A, i.e. with $x_3 > 0$, is expressed therefore as

$$ A \mathbf{u}^{\text{lat array}}(\mathbf{x}) = \frac{1}{\pi} \sum_{n=-\infty}^{\infty} \text{Im} \left[ A \mathbf{A} \cdot \left\langle \ln(x_1 - (x_1^{\text{lat}} + nh) + A p^\dagger(x_3 - x_3^{\text{lat}})) \right\rangle_A \mathbf{q}^\infty \right] \\ + \frac{1}{\pi} \sum_{n=-\infty}^{\infty} \text{Im} \left[ \sum_{\alpha=1}^{3} A \mathbf{A} \cdot \left\langle \ln(x_1 - (x_1^{\text{lat}} + nh) + A p^\dagger x_3 - A p_*^\dagger x_3^{\text{lat}}) \right\rangle_A \mathbf{q}^\alpha \right], \quad (3.126) $$

where the diagonal complex matrices $\langle \_ \dagger \rangle$ are introduced as follows

$$ \langle \_ \dagger \rangle = \begin{bmatrix} 1 & & \\ & \ddots & \\ & & 1 \end{bmatrix}, \quad (3.127) $$

since the three eigenvalues $p^\dagger$ in eq. (3.126) do not allow an index representation. The first term in eq. (3.126) is the full-plane dislocation Green's functions in A, within which $A \mathbf{A} = [Aa^1, Aa^2, Aa^3]$ is the complex eigenmatrix associated with the corresponding Stroh eigenvalues $A p^\alpha$. Furthermore, the vector $A \mathbf{q}^\infty$ is defined by

$$ A \mathbf{q}^\infty = A \mathbf{B}^\dagger A \mathbf{b}^{\text{lat}}, \quad (3.128) $$
where the matrix A B is given by A B = [A h¹, A h², A h³], within which the subsidiary complex vectors hª are related to the vectors aª by

$$
p^{\alpha} h^{\alpha} = p^{\alpha} (\mathbf{R}^{\mathrm{t}} + p^{\alpha} \mathbf{T}) a^{\alpha} = -(\mathbf{Q} + p^{\alpha} \mathbf{R}) a^{\alpha}, \qquad (3.129)
$$

with Qik = cᵢ₁k₁, Rik = cᵢ₁k₃, and, Tᵢₖ = cᵢ₃k₃. On the other hand, the displacement field in material B is given by

$$
{}_B\boldsymbol{u}^{\text{lat array}}(\boldsymbol{x}) = \frac{1}{\pi} \sum_{n=-\infty}^{\infty} \text{Im} \sum_{\alpha=1}^{3} {}_B\boldsymbol{A} \cdot \left\langle \ln(x_1 - (x_1^{\text{lat}} + nh) + {}_B\boldsymbol{p}^{\dagger}x_3 - {}_A\boldsymbol{p}^{\alpha}x_3^{\text{lat}}) \right\rangle {}_B\boldsymbol{q}^{\alpha}, \quad (3.130)
$$

where the eigenmatrix B A is accordingly associated with the Stroh eigenvalues B pª. The second
term in eq. (3.126) as well as the single term in eq. (3.130) correspond to the dislocation Green’s
function solutions due to the presence of the elastic mismatches, for which the complementary
complex vectors A pª and B qª are determined using the standard continuity conditions along the
interface of the two semi-infinite materials. These unknown vectors can be obtained for perfect
inter interfaces [240, 200], as

$$
\begin{align}
{}_A \boldsymbol{q}^\alpha &= {}_A \boldsymbol{A}^{-1} \cdot ({}_A \boldsymbol{M} + {}_B \boldsymbol{M}_*)^{-1} \cdot ({}_B \boldsymbol{M}_* - {}_A \boldsymbol{M}_*) \cdot {}_A \boldsymbol{A}_* \cdot \boldsymbol{I}^\alpha {}_A \boldsymbol{q}_*^\infty \nonumber \\
{}_B \boldsymbol{q}^\alpha &= {}_B \boldsymbol{A}^{-1} \cdot ({}_A \boldsymbol{M}_* + {}_B \boldsymbol{M})^{-1} \cdot ({}_A \boldsymbol{M} + {}_A \boldsymbol{M}_*) \cdot {}_A \boldsymbol{A} \cdot \boldsymbol{I}^\alpha {}_A \boldsymbol{q}^\infty, \tag{3.131}
\end{align}
$$

where the diagonal matrices Iª are defined by

$$
\mathbf{I}^1 = \langle 1, 0, 0 \rangle, \quad \mathbf{I}^2 = \langle 0, 1, 0 \rangle, \quad \text{and,} \quad \mathbf{I}^3 = \langle 0, 0, 1 \rangle, \quad \text{so that:} \quad \sum_{\alpha=1}^{3} \mathbf{I}^{\alpha} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \mathbf{I}, \quad (3.132)
$$

and the impedance tensors **M** in eq. (3.131) by

$$
\mathbf{M} = -i \mathbf{B} \cdot \mathbf{A}^{-1}, \tag{3.133}
$$

for which both tensors **A** and **B** satisfy the classical orthogonality relations in both adjacent ma-
terials [240]. In index notation, by substituting the derivatives of the displacement fields from
eqs. (3.126) and (3.130) with respect to the space coordinates into the constitutive Hooke’s law in
eq. (3.71), and by virtue of the following explicit solution [63, 189] for the sum over *n* from −∞ to
∞, i.e.

$$
\sum_{n=-\infty}^{\infty} (z + nh)^{-1} = \frac{\pi}{h} \operatorname{ctg} \frac{\pi}{h} z, \qquad (3.134)
$$

both stress states in materials A and B can also be straightforwardly determined as follows

$$
\begin{equation}
\begin{split}
{}_A \sigma_{ij}^{\text{lat array}}(\mathbf{x}) ={}& \frac{1}{h} \operatorname{Im} \sum_{m=1}^{3} \left[ ({}_A c_{ijk1} + {}_A p^m {}_A c_{ijk3}) {}_A A_{km} {}_A q_m^\infty \operatorname{ctg}\left(\frac{\pi}{h}(x_1 - x_1^{\text{lat}} + {}_A p^m (x_3 - x_3^{\text{lat}}))\right) \right] \\
& + \frac{1}{h} \operatorname{Im} \sum_{m=1}^{3} \left[ \sum_{\alpha=1}^{3} ({}_A c_{ijk1} + {}_A p^\alpha {}_A c_{ijk3}) {}_A A_{km} {}_A q_m^\alpha \operatorname{ctg}\left(\frac{\pi}{h}(x_1 - x_1^{\text{lat}} + {}_A p^m x_3 - {}_A p_*^\alpha x_3^{\text{lat}})\right) \right],
\end{split}
\tag{3.135}
\end{equation}
$$

$$
{}_B \sigma_{ij}^{\text{lat array}}(\mathbf{x}) = \frac{1}{h} \text{Im} \sum_{m=1}^{3} \left[ \sum_{\alpha=1}^{3} ({}_B c_{ijk1} + {}_B p^{\alpha} {}_B c_{ijk3}) {}_B A_{km} {}_B q_{m}^{\alpha} \text{ctg}\left(\frac{\pi}{h}(x_1 - x_1^{\text{lat}} + {}_B p^{m} x_3 - {}_A p^{\alpha} x_3^{\text{lat}})\right) \right], \quad (3.136)
$$

which complete the stress fields in A and B produced by infinitely periodic arrays of lattice dislo-
cations with inter-dislocation spacings *h* located in the anisotropic upper material A.
Elastic fields for piled-up dislocation arrays

Since the single dislocation pile-up system can be viewed as a discrete set of identical parallel lattice dislocations lying in the same slip plane with combinations of attractive and repulsive forces on each dislocation until a barrier (here, the semicoherent interface) is encountered, the particular boundary-value problem consists of superposing the elastic stress fields produced by each dislocation from the entire pile-up. This summation over N dislocations that are individually located at $x^{\text{lats}} = (x_1^{\text{lats}}, x_3^{\text{lats}})$ for the s$^{th}$ piled-up lattice dislocation of interest (here, invariant along the $x_2$-axis), with the same (positive or negative) sign, is also carried out over the single stress field solutions, as follows

$$A \sigma_{ij}^{\text{lat pile-up}}(\mathbf{x}) = \sum_{s=1}^{N} A \sigma_{ij}^{\text{lat s}^{\text{th}}}(\mathbf{x}), \quad \text{and,} \quad B \sigma_{ij}^{\text{lat pile-up}}(\mathbf{x}) = \sum_{s=1}^{N} B \sigma_{ij}^{\text{lat s}^{\text{th}}}(\mathbf{x}), \qquad (3.137)$$

where the self-stress contribution should be omitted in the self-force calculations. Using the standard limit when $h \to \infty$, i.e.

$$\lim_{h \to \infty} \frac{1}{h} \operatorname{ctg} \frac{\pi}{h} z = (\pi z)^{-1}, \qquad (3.138)$$

in the expressions similar to eqs. (3.135) and (3.136), and replacing the pre-subscripts A with B (and contrariwise) for dislocations in the lower material, the classical field solutions based on the single-dislocation Green's function in material A for the s$^{th}$ individual lattice dislocation are therefore obtained, i.e.

$$A \sigma_{ij}^{\text{lats}^{\text{th}}}(\mathbf{x}) = \frac{1}{\pi} \operatorname{Im} \sum_{m=1}^{3} \left[ \sum_{\alpha=1}^{3} (A c_{ijk1} + A p^{\alpha} A c_{ijk3}) A A_{km} A q_{m}^{\alpha} (x_1 - x_1^{\text{lats}} + A p^{m} x_3 - B p^{\alpha} x_3^{\text{lats}})^{-1} \right], \qquad (3.139)$$

while the field solutions in material B are given by

$$B \sigma_{ij}^{\text{lats}^{\text{th}}}(\mathbf{x}) = \frac{1}{\pi} \operatorname{Im} \sum_{m=1}^{3} \left[ (B c_{ijk1} + B p^{m} B c_{ijk3}) B A_{km} B q_{m}^{\infty} (x_1 - x_1^{\text{lats}} + B p^{m} (x_3 - x_3^{\text{lats}}))^{-1} \right] \\ + \frac{1}{\pi} \operatorname{Im} \sum_{m=1}^{3} \left[ \sum_{\alpha=1}^{3} (B c_{ijk1} + B p^{\alpha} B c_{ijk3}) B A_{km} B q_{m}^{\alpha} (x_1 - x_1^{\text{lats}} + B p^{m} x_3 - B p_{*}^{\alpha} x_3^{\text{lats}})^{-1} \right], \qquad (3.140)$$

such that the piled-up dislocations field solutions in both materials A and B are determined by substituting eqs. (3.139) and (3.140) into eqs. (3.137), respectively.

Furthermore, in order to analyze the equilibrium conditions of discrete piled-up dislocation walls that consist of infinite and equi-spaced arrays of dislocation pile-ups [99, 218], the corresponding stress fields associated with such idealized dislocation arrangements can similarly be written as

$$A \sigma_{ij}^{\text{lat wall}}(\mathbf{x}) = \sum_{s=1}^{N} A \sigma_{ij}^{\text{lat array s}^{\text{th}}}(\mathbf{x}), \quad \text{and,} \quad B \sigma_{ij}^{\text{lat wall}}(\mathbf{x}) = \sum_{s=1}^{N} B \sigma_{ij}^{\text{lat array s}^{\text{th}}}(\mathbf{x}), \qquad (3.141)$$

where $A \sigma^{\text{lats}}$ and $B \sigma^{\text{lats}}$ are given by eqs. (3.135) and (3.136) respectively, within which all pre-subscripts A must be changed with B, and vice versa, and also $(x_1^{\text{lats}}, x_3^{\text{lats}})$ with $(x_1^{\text{lats}}, x_3^{\text{lats}})$ for the s$^{th}$ piled-up dislocation within the specific discrete wall of interest.

Extrinsic dislocation loops

Line-integral expressions for elastic stress fields due to planar polygonal and elliptical dislocation loops in the three-dimensional bimaterials are given by simple integrals. For simplicity, a
constant extended Burgers vector $A b^{\text{loop}}$ is defined over the planar cut-surfaces $S_A$ in material A, as depicted by the gray surfaces in Fig. (3.34c). The solutions for a corresponding displacement discontinuity in material B is found in Ref. [256]. Based on Betti's reciprocal theorem in three-dimensional space, the displacement components induced by the dislocation loops can be described as integral form [200], i.e.

$$u_k^{\text{loop}}(\mathbf{y}) = \int_S \sigma_{ij}^k(\mathbf{y}, \mathbf{x}) b_j^{\text{loop}}(\mathbf{x}) n_i^{\text{loop}}(\mathbf{x}) \, dS(\mathbf{x}) = \int_S c_{ijml}(\mathbf{x}) G_{mk,x_l}(\mathbf{y}, \mathbf{x}) b_j^{\text{loop}}(\mathbf{x}) n_i^{\text{loop}}(\mathbf{x}) \, dS(\mathbf{x}), \quad (3.142)$$

with $\sigma_{ij}^k(\mathbf{y}, \mathbf{x})$ the Green's stress at $\mathbf{x}$ induced by a unit point force at $\mathbf{y}$ in $k$-direction, and $G_{mk}(\mathbf{y}, \mathbf{x})$ the tensorial point-force Green's functions [112, 113, 201, 200]. The latter functions represent the elastic displacement in $m$-direction at location $\mathbf{x}$ caused by the unit force in $k$-direction applied at $\mathbf{y}$, while the displacement discontinuities are mathematically equivalent to the derivatives of the Green's tensors. In eq. (3.142), the vector $n^{\text{loop}}$ represents the unit normal to the planar surface S capping the dislocation loop.

Since the point force (called source point) acts at $\mathbf{y}$ in the upper half-space of a bimaterial, i.e. $y_3 > 0$, the general Green's function tensor at $\mathbf{x}$ (called field point) is separated into two parts, i.e.

$$\forall y_3 > 0: \mathbf{G}(\mathbf{y}, \mathbf{x}) = \begin{cases} A \mathbf{G}^{\dagger}(\mathbf{y}, \mathbf{x}) = A \mathbf{G}^{\dagger\infty}(\mathbf{y}, \mathbf{x}) + A \mathbf{G}^{\text{image}}(\mathbf{y}, \mathbf{x}), & x_3 > 0 \\ B \mathbf{G}^{\dagger}(\mathbf{y}, \mathbf{x}) = B \mathbf{G}^{\text{image}}(\mathbf{y}, \mathbf{x}), & x_3 < 0, \end{cases} \quad (3.143)$$

where $\mathbf{G}^{\dagger\infty}$ corresponds to the full-space part and $\mathbf{G}^{\text{image}}$ to the complementary image part, for which the latter is associated with the elastic mismatch in dissimilar materials. Here and in the following, the symbol $\uparrow$ ($\downarrow$) is introduced to unambiguously specify that the tensorial Green's functions are associated with a dislocation loop that is located in the upper (lower) material. For instance, $A \mathbf{G}^{\dagger\infty}(\mathbf{y}, \mathbf{x})$ represents the full-space Green's function tensor computed in the semi-infinite linear elastic crystal A at $\mathbf{x}$ when the point force $\mathbf{y}$ acts at the upper crystal $\uparrow$.

For a specific surface $S_A$ bounded by a dislocation loop in the upper material A with constant elastic stiffness and uniform Burgers vector, the corresponding displacement gradients can straightforwardly be separated into two parts as follows

$$A u_{k,q}^{\text{loop} \uparrow}(\mathbf{y}) = A c_{ijml} A b_j^{\text{loop}} A n_i^{\text{loop}} \int_{S_A} \left[ A \mathbf{G}_{mk,x_l y_q}^{\dagger\infty}(\mathbf{y}, \mathbf{x}) + A \mathbf{G}_{mk,x_l y_q}^{\text{image}}(\mathbf{y}, \mathbf{x}) \right] dS(\mathbf{x}), \quad (3.144)$$

where differentiation on the left-hand side is with respect to $\mathbf{y}$. Only the corresponding derivatives of the point-force Green's function tensors are therefore needed to determine the elastic distortion (also, the elastic stress) fields, which is discussed and detailed in Appendix A from Ref. [256]. Thus, the complete point-force Green's displacement tensor in real space is given by

$$A \mathbf{G}^{\dagger\infty}(\mathbf{y}, \mathbf{x}) = \begin{cases} -\frac{1}{2\pi^2} \int_0^\pi A \mathbf{A}_* \cdot \mathbf{F}(A p_*^\dagger) \cdot A \mathbf{A}_*^\text{t} \, d\theta, & x_3 > y_3 \\ \frac{1}{2\pi^2} \int_0^\pi A \mathbf{A} \cdot \mathbf{F}(A p^\dagger) \cdot A \mathbf{A}^\text{t} \, d\theta, & 0 \le x_3 < y_3, \end{cases} \quad \text{and, } A \mathbf{G}^{\text{image}}(\mathbf{y}, \mathbf{x}) = \frac{1}{2\pi^2} \int_0^\pi A \mathbf{A}_* \cdot A \mathbf{C}^{\dagger} \cdot A \mathbf{A}^\text{t} \, d\theta, \quad (3.145)$$

where both matrices $F(Ap^\dagger)$, or $F(Ap_*^\dagger)$ by substitution, and $A C^\dagger$ for the full-space and complementary image parts, respectively, are defined by

$$\begin{aligned} \mathbf{F}(A p^\dagger) &= ((x_1 - y_1) \cos \theta + (x_2 - y_2) \sin \theta + A p^k (x_3 - y_3))^{-1} \mathbf{I} \\ A \mathbf{C}^\dagger &= ((x_1 - y_1) \cos \theta + (x_2 - y_2) \sin \theta + A p_*^k x_3 - A p^j y_3)^{-1} \left( -A \mathbf{A}_*^{-1} \cdot (A \mathbf{M}_* + B \mathbf{M})^{-1} \cdot (A \mathbf{M} - B \mathbf{M}) \cdot A \mathbf{A} \right). \end{aligned} \quad (3.146)$$
By virtue of eq. (3.71) and eq. (3.144) with eqs. (3.145) and (3.146), the stress fields produced in A due to dislocation loops located at the upper material A are expressed as follows

$$ A \sigma_{vw}^{\text{loop} \uparrow}(\mathbf{y}) = \begin{cases} (2\pi)^{-1} A_{vwkq} A_{ijml} A_{j}^{\text{loop}} A_{i}^{\text{loop}} \int_{0}^{\pi} \left[ -A_{*} \cdot \frac{\int_{S_{A}} \mathbf{F}_{,x_{p}y_{q}} (A p_{*}^{\dagger}) \, dS(\mathbf{x}) \cdot A_{*} \mathbf{A}^{\dagger} + A_{*} \cdot \int_{S_{A}} A_{*} \cdot \mathbf{C}_{,x_{p}y_{q}}^{\dagger} \, dS(\mathbf{x}) \cdot A_{*} \mathbf{A}^{\dagger}}{\underbrace{(1)}_{(1)}} \right]_{mk} \, d\theta, & x_3 > y_3 \\ (2\pi)^{-1} A_{vwkq} A_{ijml} A_{j}^{\text{loop}} A_{i}^{\text{loop}} \int_{0}^{\pi} \left[ A_{*} \cdot \frac{\int_{S_{A}} \mathbf{F}_{,x_{p}y_{q}} (A p^{\dagger}) \, dS(\mathbf{x}) \cdot A_{*} \mathbf{A}^{\dagger} + A_{*} \cdot \int_{S_{A}} A_{*} \cdot \mathbf{C}_{,x_{p}y_{q}}^{\dagger} \, dS(\mathbf{x}) \cdot A_{*} \mathbf{A}^{\dagger}}{\underbrace{(2)}_{(2)}} \right]_{mk} \, d\theta, & 0 \le x_3 < y_3 \end{cases} \quad (3.147) $$

in terms of two surface integrals (1) and (2) over a given arbitrary, uniform, and flat polygonal surface $S_A$. The analytical solutions of these surface integrals are given in Appendix B from Ref. [256], so that eqs. (3.147) contains convenient line-integral expressions for both the full-part and the image parts over $[0, \pi]$, only.

On the other hand, the stress field solutions that are generated in the lower material B (i.e. $x_3 < 0$) are needed to complete the entire stress fields in anisotropic bimaterials. Following the same derivation as in the upper half-space, the analogous displacement gradients in B are expressed as

$$ B u_{k,q}^{\text{loop} \uparrow}(\mathbf{y}) = A c_{ijml} A b_{j}^{\text{loop}} A n_{i}^{\text{loop}} \int_{S_A} B G_{mk,x_l y_q}^{\uparrow}(\mathbf{y}, \mathbf{x}) \, dS(\mathbf{x}), \quad (3.148) $$

where the complementary (i.e. image only, without full-space contribution) tensorial point-force Green's functions from eq. (3.143) are defined by

$$ B \mathbf{G}^{\uparrow}(\mathbf{y}, \mathbf{x}) = B \mathbf{G}^{\uparrow \text{image}}(\mathbf{y}, \mathbf{x}) = \frac{1}{2\pi^2} \int_{0}^{\pi} A \mathbf{A}_{*} \cdot B \mathbf{C}^{\uparrow} \cdot B \mathbf{A}_{*}^{\text{t}} \, d\theta, \quad (3.149) $$

for which the matrix $B\mathbf{C}^{\uparrow}$ is given by

$$ B \mathbf{C}^{\uparrow} = ((x_1 - y_1) \cos \theta + (x_2 - y_2) \sin \theta + A p_{*}^{k} x_3 - B p_{*}^{j} y_3)^{-1} (A \mathbf{A}_{*}^{-1} \cdot (A \mathbf{M}_{*} + B \mathbf{M})^{-1} \cdot (B \mathbf{M}_{*} + B \mathbf{M}) \cdot B \mathbf{A}_{*}), \quad (3.150) $$

so that the completing stress field solutions in the lower materials B are finally given by

$$ B \sigma_{vw}^{\text{loop} \uparrow}(\mathbf{y}) = (2\pi)^{-1} B c_{vwkq} A_{ijml} A_{j}^{\text{loop}} A_{i}^{\text{loop}} \int_{0}^{\pi} \left[ A_{*} \cdot \frac{\int_{S_{A}} \mathbf{C}_{,x_{p}y_{q}}^{\dagger} \, dS(\mathbf{x}) \cdot \mathbf{B} \mathbf{A}_{*}^{\dagger}}{(3)} \right]_{mk} \, d\theta, \quad (3.151) $$

where the expression for the integral (3) is analytically determined as well.

Equations (3.147) and (3.151) yield the final inhomogeneous stresses in both neighboring crystals A and B with different anisotropic elastic constants, respectively, which are induced by arbitrary polygonal-shaped as well as elliptical dislocation loops in the upper material. These expressions are suitable for numerical treatments (e.g., using the weighted Chebyshev-Gauss quadrature method) since line integrals over $[0, \pi]$ are needed for both full-part and image parts in stresses.

### 3.7.2 Internal forces on intrinsic and extrinsic dislocations

A general concept in classical dislocations dynamics simulations is based on the assumption of equilibrium of forces at each time increments that act along the dislocation lines or loops (e.g., polygonally discretized into segments). These forces arise externally from the dislocation of interest (as a long-range force) and from the dislocation itself (as a self-force). In the following, these two contributions are described.
## The long-range Peach-Koehler formula

Many features of crystalline solids can be explained based on the conservative driving forces to dislocation lines and loops. In the present work, to determine the driving force exerted on dislocations by the total stress field $\sigma^{\text{tot}}(\mathbf{x}^P)$ applied at the coordinate $\mathbf{x}^P$ of point P, the long-range Peach-Koehler force $f^{\text{PK}}(\mathbf{x}^P)$ per unit length is used [203], i.e.

$$f_l^{\text{PK}}(\mathbf{x}^P) = \epsilon_{ikl} b_j(\mathbf{x}^P) \bar{\sigma}_{ij}^{\text{tot}}(\mathbf{x}^P) \zeta_k(\mathbf{x}^P) = \epsilon_{ikl} b_j(\mathbf{x}^P) (\sigma_{ij}^{\text{app}} + \bar{\sigma}_{ij}^{\text{int}}(\mathbf{x}^P) + \bar{\sigma}_{ij}^{\text{lat}}(\mathbf{x}^P) + \bar{\sigma}_{ij}^{\text{loop}}(\mathbf{x}^P)) \zeta_k(\mathbf{x}^P), \quad (3.152)$$

where $\epsilon_{ikl}$ is the permutation tensor, and the superimposed bars to any stress quantities are used to indicate that the stress fields exclude the singular self-stress field components which would yield unrealistic divergent components. Thus, the local stress fields in eq. (3.152) are originated from external applied (here, uniform) stresses $\sigma^{\text{app}}$, and the internal stresses produced by the other dislocations, i.e. including the misfit dislocation stresses $\bar{\sigma}^{\text{int}}$, lattice dislocation structures (arrays and pile-ups) $\bar{\sigma}^{\text{lat}}$, and dislocation loops $\bar{\sigma}^{\text{loop}}$, where all field solutions include the complementary image stresses arising from the presence of dissimilar interfaces.

## Self-force on planar dislocation loops

Without considering any nonlinear interatomic interactions due the dislocation cores, the standard self-force is thought of as resulting from the elastic self-energy of the dislocation loops. From Ref. [97], the self-energy of a planar dislocation loop with arbitrary shape is defined to be the strain energy exterior to a tube of cut-off radius $\epsilon$ surrounding the dislocation loop L, such that the corresponding stored energy value is formally finite. The total self-energy $W^{\text{self}}$ for a given dislocation loop is also separated into two contributions [45, 97, 13], i.e.

$$W^{\text{self}} = W_{T_{\epsilon}}^{\text{self}} + W_{S_{\epsilon}}^{\text{self}} = \frac{1}{2} \int_{T_{\epsilon}} \sigma_{ij}^{\text{loop}}(\mathbf{x}) u_{i}^{\text{loop}}(\mathbf{x}) n_{j}^{\text{tube}} \, dS + \frac{1}{2} \int_{S_{\epsilon}} \sigma_{ij}^{\text{loop}}(\mathbf{x}) b_{i}^{\text{loop}} n_{j}^{\text{tube}} \, dS, \quad (3.153)$$

where $n^{\text{tube}}$ is the inner normal on the tube $T_{\epsilon}$ surrounding L, referred as the tube self-energy contribution, and $S_{\epsilon}$ is the portion of an open surface S bounded by L not enclosed by $T_{\epsilon}$, i.e. the cut self-energy part. After substantial manipulation [98, 97], the tube contribution to the self-energy is given by

$$W_{T_{\epsilon}}^{\text{self}} = \frac{1}{2} \int_{L} d\ell \int_{0}^{2\pi} \sigma_{ij}^{\text{loop}}(\omega) u_{i}^{\text{loop}}(\omega) n_{j}^{\text{tube}} \epsilon \, d\omega = H(\alpha) \int_{L} d\ell, \quad (3.154)$$

where

$$H(\alpha) = -\frac{1}{2} \int_{0}^{2\pi} \frac{\partial \phi_{i}}{\partial \omega} d\omega = \frac{1}{2} \int_{0}^{2\pi} \phi_{i} \frac{\partial u_{i}^{\text{loop}}}{\partial \omega} d\omega, \quad (3.155)$$

with $\omega$ the polar angle about the dislocation in any plane cross-section of the tube, and $\phi_i$ the Airy stress function vector. The rigorous derivation in Ref. [97] by varying the self-energy in eq. (3.153) with respect to an arbitrary in-plane virtual normal displacement of the dislocation loop gives rise to the self-consistent treatment of the total distributed (and also signed) self-force on the dislocation loop at point P of interest. The complete line-tension self-force expression is also defined as follows

$$f^{\Gamma}(\mathbf{x}^P) = \underbrace{-\frac{1}{2} b_i n_j (\sigma_{ij}^{\text{loop}}(\mathbf{x}^P + \epsilon \mathbf{m}) + \sigma_{ij}^{\text{loop}}(\mathbf{x}^P - \epsilon \mathbf{m}))}_{\text{cut self-force}} + \kappa E(\phi) - \underbrace{-\kappa \left[ H(\alpha) + \frac{\partial^2 H}{\partial \alpha^2} \right]}_{\text{tube self-force}}, \quad (3.156)$$

where $\mathbf{m} = \mathbf{n} \times \boldsymbol{\tau}$ points inward for convex planar dislocation loops (i.e. toward the centers) with $\boldsymbol{\tau}$ the unit local tangent vector to the planar loop, $\kappa$ is the local curvature at point P, E is the
standard pre-logarithmic energy factor of an infinite straight dislocation with tangent $\xi$, and $H$ is a tube integral around the same latter dislocation. The term $E$ in eq. (3.156) is expressed as function of the polar angle $\phi$, which is measured between the Burgers vector and the line direction $\xi$, i.e. $\phi$ denotes a given character angle, where $\phi = 0^{\circ}$ corresponds to a pure screw dislocation [219], while $H$ depends on the angle $\alpha$ between $\tau$ and an arbitrary datum [97, 13]. The datum is conveniently chosen here along the fixed Burgers vector of the corresponding planar dislocation loop. Hence, $\alpha$ is also the local character angle, measured counter-clockwise from the Burgers vector to the local tangent vector $\tau$ at every point on the loop. In particular, $\alpha = \phi + 90^{\circ}$ for circular shear dislocation loops, where more complex geometrical relations arise for arbitrarily-shaped dislocation loops.

Importantly, because these energy contributions vary with dislocation characters (e.g., edge and screw dislocations have different energies), the line-tension self-forces can exert a torque on the line portion of the curved dislocation of interest in order to rotate it into its orientation of lowest energy. It is worth noting that the first part in the cut self-force contribution in eq. (3.156) is given in Ref. [43], where the singularity is removed by defining the stress as an average of stress evaluated at two points on the opposite side of and at a short distance $\epsilon$ away from the dislocation line. The correction term $\kappa E(\phi)$, proportional to the curvature $\kappa$, has been consistently obtained in Ref. [97]. Here, if the force calculated from eq. (3.156) is positive, then it acts along $-m$ and vice versa, so that

$$f^{\Gamma}(\mathbf{x}^P) = -f^{\Gamma}(\mathbf{x}^P) m. \qquad (3.157)$$

For large values of $\kappa^{-1}$ and $(\kappa\epsilon)^{-1}$, the dominant cut self-force contribution in eq. (3.156) is $-\kappa\Gamma$, where $\Gamma$ represents the classical local line-tension approximation [13], i.e.

$$\Gamma = \left( E(\phi) + \frac{\partial^2 E}{\partial \phi^2} \right) \ln((\kappa\epsilon)^{-1}), \qquad (3.158)$$

for which $-\kappa\Gamma$ vanishes for any $P$ along infinitely long straight dislocations, i.e. when $\kappa \to 0$. In the following, $\epsilon$ is chosen as the pre-determined cutoff distance $r_0$ that excludes the dislocation singularities, i.e. $\epsilon = r_0$.

### 3.7.3 On the piled-up dislocations in the (111)Cu/(011)Nb bimaterial

The considered interaction problems are related to the force equilibrium of piled-up systems with infinitely long straight dislocations under the action of an externally applied shear stress that maintains the lattice dislocations toward the impenetrable interfaces (i.e. without slip transmission across the boundaries) in (111)-type glide planes. Without loss of generality, the dislocations with Burgers vectors $b_{B}^{\text{lat}}$ are embedded in the material B. Several cases, from the simple single-crystal case of equilibrium pile-ups to the piled-up dislocations between the bimetallic semicircular Cu/Nb interface with relaxed interfacial dislocation arrangements and a shear dislocation loop, are presented. The materials properties used in the calculations are listed in Table 3.1.

#### Computational procedures

The present calculations of equilibrium positions of $N$ piled-up dislocations embedded in the lower materials are carried out by using a numerical iterative relaxation scheme under constant applied stresses. As commonly used in dislocation dynamics simulations, linear mobility law for all (except for the leading dislocations) piled-up dislocations, individually located at $\mathbf{x}^{\text{lats}}$ in single glide planes, is phenomenologically introduced, i.e.

$$v_{\text{glide}}(\mathbf{x}^{\text{lats}}) = B^{-1} f_{\text{glide}}^{\text{PK}}(\mathbf{x}^{\text{lats}}), \qquad (3.159)$$
<figure><img src="image_54.png" /><figcaption>FIGURE 3.35: Specific internal stress components produced by the equilibrium (a) 6-mixed, (b) 9-mixed, and (c) 12-screw piled-up dislocations against the interface in Cu using anisotropic elasticity. The same component as in (c) is shown in (d) using the approximation of isotropic elasticity (obtained by the Voigt averages). The horizontal lines denote the impenetrable interfaces.</figcaption></figure>

where B is an isotropic viscous coefficient and is ~ $10^{-5}$ Pa.s for fcc solids [153]. In the following calculations, $B = 5.10^{-5}$ Pa.s. In eq. (3.159), the Peach-Koehler force is also used to compute the velocity $v_{\text{glide}}$ for dislocation glide, which in turn, is used to predict the newly positions of the dislocation by adopting a standard time integration scheme (the explicit forward Euler time discretization) accordingly, such that $x^{\text{lat}s}(t + \Delta t) = x^{\text{lat}s}(t) + \Delta t v_{\text{glide}}(x^{\text{lat}s})$.

The initial distributions of the piled-up dislocations are aligned and arbitrary equi-spaced on the same glide plane until the net resolved force on each dislocation (except acting on the first leading dislocations at the interfaces) is less than $10^{-5}$ N/m. For the specific leading dislocations, strictly located at the interfaces, i.e. $x^{\text{lat}1\text{st}} = 0$, a zero velocity is therefore imposed, while the corresponding Peach-Koehler force $f^{\text{PK}}(x^{\text{lat}1\text{st}} = 0)$ is not necessary equal to zero. These forces acting on the leading dislocations at interfaces are therefore separated into both resolved glide and climb components, i.e.

$$f_{\text{glide}}^{\text{PK, 1st}} = f^{\text{PK}}(x^{\text{lat}1\text{st}} = 0) \cdot v_{\text{glide}}, \quad \text{and,} \quad f_{\text{climb}}^{\text{PK, 1st}} = f^{\text{PK}}(x^{\text{lat}1\text{st}} = 0) \cdot v_{\text{climb}}, \qquad (3.160)$$

where $v_{\text{glide}} = v_{\text{climb}} \times \zeta$, while $v_{\text{climb}}$ are the directions of the dislocation pile-ups and the normal of the slip planes, respectively. Here, the piled-up dislocation line directions $\zeta$ are chosen such that $v_{\text{glide}}$ points away from the interfaces, so that a positive value of the glide force in eq. (3.160) indicates a repulsive force from the interface.

In order to illustrate the numerical procedure for dislocation pile-ups, single-crystal Cu systems (which consist of a particular case where both materials A and B are identical, i.e. without lattice misfit nor image forces) are considered with the following orientation, i.e. $x_1 = [1\bar{1}0]$, $x_2 = [110]$, and $x_3 = n = [001]$. Two piled-up systems with different dislocation characters are examined: the piled-up dislocations with 60° mixed and pure screw characters. The line directions $\zeta \parallel [110]$ are defined along the planar glide plane normal to $v_{\text{climb}} \parallel [1\bar{1}1]$, which is not orthogonal to the impenetrable interface since an angle of 54.7° is defined with the pile-up plane. The calculations are performed in Cu, with the moderately high anisotropy ratio $A_{\text{Cu}} = 3.21$, and the associated Burgers vectors are defined by $b^{\text{lat}} = a_{\text{Cu}}[101]$ and $b^{\text{lat}} = a_{\text{Cu}}[110] \parallel x_2$, for 6-, 9-, or 12-mixed and screw piled-up dislocations, respectively. As quantified in the foregoing sections, because the corresponding elastic coherency stress states that characterize the semicoherent interfaces can be very large in the far-field stress-free materials, simple shear stresses are applied to the nanostructured elastic problems with mixed and pure screw piled-up dislocations, respectively.

Figures (3.35a), (b) and (c) illustrate the stress field components $\sigma_{22}^{\text{lat}}$ or $\sigma_{12}^{\text{lat}}$ produced by the equilibrium piled-up arrangements of 6, 9 or 12 mixed and screw dislocations, respectively. For
<table><caption>TABLE 3.8: Equilibrium positions (in nm) of 60° mixed and pure screw piled-up dislocations measured obliquely along the piled-up slip direction from the interface in Cu.</caption><thead><tr><th># Mixed dislocation</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>f<sup>PK, 1st</sup><sub>glide</sub></th><th>f<sup>PK, 1st</sup><sub>climb</sub></th></tr></thead><tbody><tr><td>Isotropic</td><td>0</td><td>0.52</td><td>1.80</td><td>3.91</td><td>7.08</td><td>11.97</td><td></td><td></td><td></td><td></td><td></td><td></td><td>2.67</td><td>-0.47</td></tr><tr><td>Anisotropic</td><td>0</td><td>0.49</td><td>1.67</td><td>3.62</td><td>6.55</td><td>11.06</td><td></td><td></td><td></td><td></td><td></td><td></td><td>2.67</td><td>0.13</td></tr><tr><td>Isotropic</td><td>0</td><td>0.35</td><td>1.20</td><td>2.54</td><td>4.42</td><td>6.92</td><td>10.18</td><td>14.48</td><td>20.51</td><td></td><td></td><td></td><td>4.01</td><td>-0.61</td></tr><tr><td>Anisotropic</td><td>0</td><td>0.33</td><td>1.11</td><td>2.35</td><td>4.09</td><td>6.41</td><td>9.43</td><td>13.41</td><td>19.01</td><td></td><td></td><td></td><td>4.01</td><td>0.61</td></tr><tr><td>Isotropic</td><td>0</td><td>0.27</td><td>0.90</td><td>1.89</td><td>3.25</td><td>5.02</td><td>7.22</td><td>9.91</td><td>13.18</td><td>17.18</td><td>22.18</td><td>28.89</td><td>5.38</td><td>-0.55</td></tr><tr><td>Anisotropic</td><td>0</td><td>0.24</td><td>0.82</td><td>1.74</td><td>3.01</td><td>4.65</td><td>6.69</td><td>9.19</td><td>12.23</td><td>15.95</td><td>20.60</td><td>26.86</td><td>5.36</td><td>1.44</td></tr><tr><td># Screw dislocation</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td><td>12</td><td>f<sup>PK, 1st</sup><sub>glide</sub></td><td>f<sup>PK, 1st</sup><sub>climb</sub></td></tr><tr><td>Isotropic</td><td>0</td><td>1.09</td><td>3.72</td><td>8.14</td><td>14.83</td><td>25.11</td><td></td><td></td><td></td><td></td><td></td><td></td><td>14.70</td><td>-0.19</td></tr><tr><td>Anisotropic</td><td>0</td><td>0.76</td><td>2.66</td><td>5.78</td><td>10.35</td><td>17.64</td><td></td><td></td><td></td><td></td><td></td><td></td><td>14.70</td><td>-7.45</td></tr><tr><td>Isotropic</td><td>0</td><td>0.62</td><td>2.33</td><td>5.12</td><td>9.05</td><td>14.31</td><td>21.20</td><td>30.29</td><td>43.21</td><td></td><td></td><td></td><td>22.05</td><td>-7.78</td></tr><tr><td>Anisotropic</td><td>0</td><td>0.59</td><td>1.67</td><td>3.72</td><td>6.40</td><td>10.01</td><td>14.92</td><td>21.11</td><td>30.28</td><td></td><td></td><td></td><td>22.04</td><td>-10.26</td></tr><tr><td>Isotropic</td><td>0</td><td>0.55</td><td>1.79</td><td>3.86</td><td>6.67</td><td>10.38</td><td>15.04</td><td>20.75</td><td>27.73</td><td>36.38</td><td>47.23</td><td>62.17</td><td>29.40</td><td>0.92</td></tr><tr><td>Anisotropic</td><td>0</td><td>0.46</td><td>1.23</td><td>2.67</td><td>4.68</td><td>7.33</td><td>10.54</td><td>14.45</td><td>19.36</td><td>25.42</td><td>33.03</td><td>43.22</td><td>29.39</td><td>-13.54</td></tr></tbody></table>

comparison with Fig. (3.35c), Fig. (3.35d) shows the equilibrium 12-screw piled-up dislocations and the corresponding stress component $\sigma_{12}^{\text{lat}}$ that are obtained by using the isotropic elastic approximation based on the Voigt averages of the elastic constants. Considered as the exhaustion of Frank-Read sources, these plots illustrate the back stress concentrations generated by the pile-ups, which are considerably affected by the number of piled-up dislocations, the individual dislocation characters, as well as the anisotropic elasticity.

Table (3.8) reports the corresponding dislocation positions in the different equilibrium pile-ups and forces on the fixed leading dislocations in anisotropic and isotropic media. As found in the earliest studies of discrete edge or screw dislocation pile-ups on simple single glide planes by use of the Laguerre polynomials as routine procedures [86, 115], the present results essentially illustrate a $\sim x^{-1/2}$ dependence of the dislocation density on the distance $x$ to the impenetrable interfaces. For the mixed piled-up dislocations, the results from anisotropic and isotropic elastic calculations are practically indistinguishable with 6 and 9 dislocations. For screw piled-up dislocations, however, the results exhibit the discrepancies in dislocation spacings and forces (in magnitude in N/m and sign, especially for the climb components) resulting from the considered approximation of isotropic elasticity, with relative errors that vary between 16.4% and 31.3% in dislocation positions.

## Dislocation geometries and orientations

An arbitrary microstructure is chosen here to demonstrate the ability of the present elastic super-position scheme in complex piled-up dislocation problems. The short-range stress fields generated by the infinitely periodic misfit dislocation pattern and interacted with various types of lattice defects are investigated for the semi-infinite Cu/Nb system in the NW OR. As defined in eq. (3.44), the following specific NW relations are used, i.e. $x_1 \parallel [1\bar{1}0]_{\text{fcc}} \parallel [100]_{\text{bcc}}$, $x_2 = [11\bar{2}]_{\text{fcc}} \parallel [01\bar{1}]_{\text{bcc}}$, and, $x_3 \parallel n \parallel [111]_{\text{fcc}} \parallel [011]_{\text{bcc}}$.

In the upper dislocated Cu material, two sets of the infinitely periodic mixed dislocations with identical line directions $\xi \parallel [11\bar{2}]_{\text{fcc}} \parallel x_2$, but different Burgers vectors $A\mathbf{b}^{\text{lat (1)}} \parallel [101]$ (i.e. almost edge) and $A\mathbf{b}^{\text{lat (2)}} \parallel [01\bar{1}]$ (i.e. 30° mixed) are randomly introduced near the Cu/Nb interface, respectively. The inter-dislocation distance $h$, which consists of the translationally periodic boundary conditions with respect to the $x_1$ direction, and each set of the infinitely long dislocations has the same number of positive and negative signed dislocation characters, such that these dislocations are viewed as statistically-stored dislocations [9], without producing long-range stress effects. On the other hand, the misfit dislocations that are characterized by a non-zero Burgers vector content (to necessarily realize the compatibility at the NW Cu/Nb interface) are, by definition, geometrically-necessary dislocations with zero far-field stresses as well. Following the procedure from section 3.6, these geometrically-necessary dislocations are analyzed in Fig. (3.36).