A phase field computational procedure for electromigration with specified contact angle and diffusional anisotropy

Abstract

Electromigration in electrical interconnects with high current density causes voids to form and grow near the cathode. These voids can possibly grow large enough to cause open circuit failure. The simulation of electromigration void nucleation and growth is challenging because of the complex interaction of electrical, thermal and mechanical fields that lead to the observed voids. In this paper, a phase-field approach is developed to study void growth due to electromigration taking into account diffusional anisotropy and specified contact angle. Consideration of anisotropy and contact angle is critical in applications including electronic assemblies that most commonly use Sn-based solder interconnects. In such systems, Sn exhibits significant anisotropy in its self-diffusivity, which assumes importance in small solder joints that are known to contain only a few grains. Furthermore, the contact angle at the void-bonding pad-solder triple junction is known to influence the void size and shape. To the best of the authors’ knowledge, only anisotropy in surface diffusivity has been considered in phase-field electromigration literature, whereas the contact angle at a triple junction has not been considered. The developed phase-field model incorporates effects of both contact angle and anisotropy in self-diffusivity. The model is used to simulate systems with different contact angles and dominant directions of diffusion. It is observed that diffusional anisotropy plays a more dominant role in failure, while the contact angle dictates evolution of the shape of voids.

Appendix A: Formal asymptotic analysis

In general, it is necessary to show that the diffuse interface equations used in phase field modeling lead to the those given in Sect. 2 in the limit of the sharp interface. Such an exercise is carried out using the method of matched formal asymptotic analysis. A more detailed discussion of the technique of formal asymptotic analysis can be found in [54, 55].

1.1 Appendix A.1: Transformations to an interface coordinate system

The domain for the asymptotic analysis is shown in Fig. 13. In the outer region I, the Heaviside step function of Eq. (32), $H(\mathbf {x})= 1$, while in region II, $H(\mathbf {x})= 0$. In both region I and II, the Dirac $\delta$ function of Eq. (33), $\delta (\mathbf {x}) = 0$. At any instant, the interface $\varGamma$ moves with a normal velocity $v_n\mathbf {n}$, where $\mathbf {n}$ is the unit normal to the interface at the origin of the coordinate system. Following [46], a moving coordinate system attached to the interface $\varGamma$ based on the scaled distance $\rho$ from the interface is used (Fig. 14); $\rho$ is the normal signed distance from the interface, r, scaled by $\frac{1}{\epsilon }$:

$$\begin{aligned} \rho = \frac{r}{\epsilon } \end{aligned}$$

(53)

Fig. 13

Domain for the formal asymptotic analysis

In the moving inner coordinate system, any point $\mathbf {x}$  in space can be written as,

$$\begin{aligned} \mathbf {x}&= r\mathbf {n}+ \mathbf {P}\mathbf {x}\end{aligned}$$

(54)

where $\mathbf {P}$ is the projection tensor defined as:

$$\begin{aligned} \mathbf {P}=\mathbf {I}-\mathbf {n}\otimes \mathbf {n}\end{aligned}$$

(55)

with $\mathbf {I}$ being the identity tensor. Thus,

$$\begin{aligned} \frac{\partial \mathbf {x}}{\partial t}&= \frac{\partial r}{\partial t}\mathbf {n}+ \mathbf {P}\frac{\partial \mathbf {x}}{\partial t} \end{aligned}$$

(56)

Using the definitions given in [56] and described in detail in [57], the surface gradient of the scalar field $u(\mathbf {x},t)$ is:

$$\begin{aligned} \nabla _{\varGamma }u=\mathbf {P}\nabla u \end{aligned}$$

(57)

Similarly, the surface gradient and the surface divergence of any vector field $\mathbf {u}(\mathbf {x},t)$ is given by,

$$\begin{aligned}&\nabla _{\varGamma }\mathbf {u}=\mathbf {P}\nabla \mathbf {u} \end{aligned}$$

(58)

$$\begin{aligned}&\nabla _{\varGamma }\cdot \mathbf {u} =\mathrm {tr}\left( \nabla _{\varGamma }\mathbf {u}\right) =\mathbf {P}:\nabla \mathbf {u} \end{aligned}$$

(59)

Thus, using the definition of $\mathbf {P}$, it follows from Eqs. (57) to (59) that

$$\begin{aligned}&\nabla u = \frac{{u}_{\rho }}{\epsilon }\mathbf {n}+\nabla _{\varGamma }u \end{aligned}$$

(60)

$$\begin{aligned}&\nabla \cdot \mathbf {u} =\mathbf {n}\cdot \nabla \mathbf {u} \cdot \mathbf {n}+\nabla _{\varGamma }\cdot \mathbf {u} \nonumber \\&=\frac{1}{\epsilon }\mathbf {u}_\rho \cdot \mathbf {n}+ \nabla _{\varGamma }\cdot \mathbf {u} \end{aligned}$$

(61)

where the first derivative of the fields with respect to $\rho$ is denoted as ${u}_{\rho }=\frac{\partial u}{\partial \rho }$ and $\mathbf {u}_\rho =\frac{\partial \mathbf {u}}{\partial \rho }$. In the above expression, we have also used $\nabla \rho =\frac{1}{\epsilon }\nabla r = \frac{1}{\epsilon }\mathbf {n}$.

Fig. 14

Transformation of coordinate system from a global coordinate system to an interface attached coordinate system

The Laplacian of a scalar field $u(\mathbf {x})$ can next be derived as,

$$\begin{aligned} \nabla ^2u&= \nabla \cdot \nabla u = \nabla \cdot \left( \frac{{u}_{\rho }}{\epsilon }\mathbf {n}+\nabla _{\varGamma }u\right) \nonumber \\&= \nabla _{\varGamma }\cdot \left( \frac{{u}_{\rho }}{\epsilon }\mathbf {n}+\nabla _{\varGamma }u\right) +\frac{1}{\epsilon }\frac{\partial }{\partial \rho }\left( \frac{{u}_{\rho }}{\epsilon }\mathbf {n}+\nabla _{\varGamma }u\right) \cdot \mathbf {n} \nonumber \\ \Rightarrow \nabla ^2u&= \frac{1}{\epsilon ^2}{u}_{\rho \rho } + \frac{1}{\epsilon }{u}_{\rho } \nabla _{\varGamma }\cdot \mathbf {n} + \nabla _\varGamma ^2u \end{aligned}$$

(62)

where ${u}_{\rho \rho }=\frac{\partial ^2 u}{\partial \rho ^2}$. Using the curvature relation that $\nabla _{\varGamma }\cdot \mathbf {n}=-\kappa$, we get the relation for the Laplacian as,

$$\begin{aligned} \nabla ^2u = \frac{1}{\epsilon ^2}{u}_{\rho \rho } - \frac{1}{\epsilon }\kappa {u}_{\rho } + \nabla _\varGamma ^2u \end{aligned}$$

(63)

The signed distance field moves along with the interface and hence $\frac{d r}{dt}=0$. This gives,

$$\begin{aligned} \frac{d r}{dt}&= \frac{\partial r}{\partial t} + \mathbf {v}\cdot \nabla r = \frac{\partial r}{\partial t} + v_n = 0 \end{aligned}$$

(64)

$$\begin{aligned} \Rightarrow \frac{\partial r}{\partial t}&= -v_n \end{aligned}$$

(65)

Recognizing that, in the inner coordinate system, a scalar field is of the form $u(\mathbf {x}(r,\mathbf {P}),t)$, the temporal partial derivative of a scalar field is given by,

$$\begin{aligned} \frac{\partial }{\partial t}u(\mathbf {x}(r,\mathbf {P}),t)&= \frac{\partial }{\partial t}u(r,\mathbf {P},t) + \frac{\partial u}{\partial \mathbf {x}}\cdot \frac{\partial \mathbf {x}}{\partial t} \nonumber \\&= {u}_{t} + \left( \frac{\partial r}{\partial t}\mathbf {n}+ \mathbf {P}\frac{\partial \mathbf {x}}{\partial t}\right) \cdot \nabla u \nonumber \\&= {u}_{t} + \frac{1}{\epsilon }\frac{\partial r}{\partial t}{u}_{\rho } + \frac{\partial \mathbf {x}}{\partial t}\cdot \nabla _{\varGamma }u \nonumber \\ \Rightarrow \frac{\partial }{\partial t}u(\mathbf {x}(r,\mathbf {P}),t)&= {u}_{t} - \frac{v_n}{\epsilon }{u}_{\rho } + \frac{\partial \mathbf {x}}{\partial t}\cdot \nabla _{\varGamma }u \end{aligned}$$

(66)

1.2 Appendix A.2: Asymptotic analysis of the modified Cahn–Hiliard equations

The asymptotic analysis for Eq. (39) is based on the analysis in [46]. The objective of this asymptotic analysis is to determine the constants $c_1$-$c_3$ for the modified Cahn–Hiliard [Eq. (67)] equation given below.

$$\begin{aligned} \frac{\partial \phi }{\partial t}&= c_1 \left( \varOmega _\text {sol}\nabla \cdot \left[ \frac{M_\varGamma \delta (\phi ) }{\epsilon } \left( \nabla \mu _\text {sol}^\varGamma \right. \right. \right. \nonumber \\&\quad \left. \left. \left. +\, Z^*_\varGamma e \nabla \varphi _e +~Q^*_\varGamma \nabla T \right) \right] \right. \nonumber \\&\quad \left. +\, \frac{\delta (\phi )}{\epsilon } \lambda x_{\text {vac}} \left( \mu _\text {vac}- \mu _\text {sol}^\varGamma \right) \right) \end{aligned}$$

(67a)

$$\begin{aligned} \nonumber \\ \mu _\text {sol}^\varGamma&= c_2 \varOmega _\text {sol}\gamma \left( -\epsilon \nabla ^2 \phi + \frac{1}{\epsilon } f'(\phi ) \right) \nonumber \\&\quad +\, c_3 \varOmega _\text {sol}\delta (\phi ) W \end{aligned}$$

(67b)

The solutions $\phi ,\mu _\text {sol}^\varGamma$ to the above equation can be expanded in the outer regions as,

$$\begin{aligned} {\hat{\mu }}(r,\epsilon )&= {\hat{\mu }}^0 + \epsilon {\hat{\mu }}^1+ \epsilon ^2 {\hat{\mu }}^2 + \cdots \end{aligned}$$

(68a)

$$\begin{aligned} {\hat{\phi }}(r,\epsilon )&= {\hat{\phi }}^0 + \epsilon {\hat{\phi }}^1 + \epsilon ^2 {\hat{\phi }}^2 + \cdots . \end{aligned}$$

(68b)

The solution in the inner region can be written as,

$$\begin{aligned} \mu (\rho ,\epsilon )&= \mu ^0 + \epsilon \mu ^1 +\epsilon ^2\mu ^2 + \ldots , \end{aligned}$$

(69a)

$$\begin{aligned} \phi (\rho ,\epsilon )&= \phi ^0 + \epsilon \phi ^1 + \epsilon ^2 \phi ^2 + \ldots . \end{aligned}$$

(69b)

The analysis of outer region is performed first. Substituting Eq. (68) into Eq. (67b), the leading order in the outer expansion is the $\epsilon ^{-1}$ term. This leads to,

$$\begin{aligned} f'({\hat{\phi }}^0) = 0 \end{aligned}$$

(70)

Since the roots of $f'({\hat{\phi }}^0)$ are at $\pm 1$ (see Fig. 8), the solution to ${\hat{\phi }}$ in the outer regions converges to either $+ 1$ or $-1$. This allows the phases to be discriminated, with $+1$ indicating the presence of material and $-1$ the void. The only other term in the $\epsilon ^0$ order in Eq. (67b) is the $\varOmega _\text {sol}\delta ({\hat{\phi }}) W$ term, which is zero since $\delta ({\hat{\phi }})=0$ in the outer region. This leads to the leading order term ${\hat{\mu }}^0 = 0$ in the outer region, and furthermore, Eq. (67a) can be reduced to $0=0$ in all orders.

For the inner-expansion, Eq. (67) needs to be transformed into a co-ordinate system that is attached to the interface. Using Eqs. (63) and (66), the modified Cahn–Hilliard equation can be written as,

$$\begin{aligned} \begin{aligned}&{\phi }_{t} - \frac{1}{\epsilon } v_n{\phi }_{\rho } + \frac{\partial \mathbf {x}}{\partial t}\cdot \nabla _{\varGamma }\phi = \\&c_1\frac{\delta (\phi )}{\epsilon }\lambda x_\text {vac}\left( \mu _\text {vac}- \mu _\text {sol}^\varGamma \right) \\&\quad + c_1\varOmega _\text {sol}\left[ \frac{M \delta (\phi ) }{\epsilon }\left( \frac{1}{\epsilon ^2} {G}_{\rho \rho }- \frac{1}{\epsilon } \kappa { G}_{\rho } + \nabla _\varGamma ^2 G \right) \right. \\&\quad + \left. \frac{M\delta '(\phi )}{\epsilon } \left( \frac{{\phi }_{\rho }}{\epsilon } \mathbf {n}+\nabla _\varGamma \phi \right) . \left( \frac{ {G}_{\rho }}{\epsilon } \mathbf {n}+ \nabla _\varGamma G \right) \right] \end{aligned} \end{aligned}$$

(71a)

$$\begin{aligned} \nonumber \\ \begin{aligned} \mu _\text {sol}^\varGamma = c_2 \varOmega _\text {sol}\gamma \left( \frac{-1}{\epsilon }{\phi }_{\rho \rho } + \kappa {\phi }_{\rho } +\frac{1}{\epsilon }f'(\phi ) - \epsilon \nabla _\varGamma ^2 \phi \right)&\\ + c_3\varOmega _\text {sol}\delta (\phi ) W&\end{aligned} \end{aligned}$$

(71b)

where

$$\begin{aligned} G = \mu _\text {sol}^\varGamma + Z^*e \varphi _e+ Q^*T \end{aligned}$$

(72)

Substituting Eq. (69) into Eq. (71), the leading order terms in Eq. (71b) are at the $\epsilon ^{-1}$ order and can be written as,

$$\begin{aligned} -\phi ^0_{\rho \rho } + f'(\phi ^0) = 0 \end{aligned}$$

(73)

The boundary conditions that Eq. (73) needs to satisfy the matching conditions with the outer regions I and II are, $\lim _{\rho \rightarrow -\infty }\phi ^0 \rightarrow -1$ and $\lim _{\rho \rightarrow \infty }\phi ^0 \rightarrow 1$. With these boundary conditions, the solution to Eq. (73) is found to be,

$$\begin{aligned} \phi ^0 = \tanh \frac{\rho }{\sqrt{2}} \end{aligned}$$

(74)

This implies that the equilibrium profile of $\phi$ is a hyperbolic tangent. Eq. (74) also explains the choice of the approximation to the Heaviside step function in Eq. (32), as that can be computed quite simply as $H(\mathbf {x})= \frac{1+\phi (\mathbf {x})}{2}$. Further, Eq. (74) also provides a relation for the normal and curvature in terms of $\phi ^0$ as,

$$\begin{aligned} \nabla \phi ^0&= \frac{1}{\sqrt{2}}\mathrm {sech}^2\frac{\rho }{\sqrt{2}}\nabla \rho = \frac{1}{\sqrt{2}\epsilon }\mathrm {sech}^2\frac{\rho }{\sqrt{2}}\mathbf {n}\end{aligned}$$

(75)

$$\begin{aligned} \nabla ^2\phi ^0&= \frac{-1}{\epsilon }\tanh \frac{\rho }{\sqrt{2}}\mathrm {sech}^2\frac{\rho }{\sqrt{2}}\nabla \cdot \mathbf {n} \nonumber \\&= \frac{-\kappa }{\epsilon }\tanh \frac{\rho }{\sqrt{2}}\mathrm {sech}^2\frac{\rho }{\sqrt{2}} \end{aligned}$$

(76)

where we have used the curvature-normal relation. Also, using the definition of surface gradient [Eq. (58)], we can now show that

$$\begin{aligned} \nabla _{\varGamma }\phi ^0&= \mathbf {P}\nabla \phi ^0 = \frac{1}{\sqrt{2}\epsilon }\mathrm {sech}^2\frac{\rho }{\sqrt{2}}\mathbf {P}\mathbf {n}= 0 \end{aligned}$$

(77)

Thus, at the $\epsilon ^0$ order in Eq. (71b), the terms that remain are,

$$\begin{aligned} \mu ^0 = c_2 \gamma \varOmega _\text {sol}\kappa {\phi ^0_\rho }_{\,}+ c_3\varOmega _\text {sol}\delta (\phi ^0)W \end{aligned}$$

(78)

To make the left hand side of Eq. (78) integrable, both sides are multiplied by $\phi ^0_\rho$. Integrating with respect to $\rho$ over $-\infty , \infty$, Eq. (78) can be shown to yield,

$$\begin{aligned} 2 \mu ^0 = \varOmega _\text {sol}\left( c_2 \frac{2\sqrt{2}}{3} \gamma \kappa + c_3 \frac{16}{15} W \right) \end{aligned}$$

(79)

Comparing with the sharp interface relation given in Eq. (27), we get

$$\begin{aligned} c_2=-\frac{3}{\sqrt{2}}, c_3=\frac{15}{8} \end{aligned}$$

(80)

In the above, the following integrals are used,

$$\begin{aligned} \int _{-\infty }^{\infty }&{\left( \tanh \frac{\rho }{\sqrt{2}}\right) }_{\rho }d\rho =2 \end{aligned}$$

(81a)

$$\begin{aligned} \int _{-\infty }^{\infty }&{\left( \tanh \frac{\rho }{\sqrt{2}}\right) }_{\rho }^2 d\rho = \frac{2\sqrt{2}}{3}\end{aligned}$$

(81b)

$$\begin{aligned} \int _{-\infty }^{\infty }&{\left( \tanh \frac{\rho }{\sqrt{2}}\right) }_{\rho }\left( 1- \phi ^2 \right) ^2 d\rho = \frac{16}{15} \end{aligned}$$

(81c)

An important assumption made in the above derivation is that the computed value of the strain energy is nearly constant over the interfacial region.

We observe now that Eq. (79) leads to ${\mu ^0_\rho } = 0$, and the results of Eqs. (74) and (80) can be used to analyze the solutions to Eq. (71a). Considering now the $\epsilon ^{-2}$ terms in the inner region and noting that $\mathbf {n}\cdot \nabla _{\varGamma }(\cdot )=0$ (surface gradient is orthogonal to normal $\mathbf {n}$) yields,

$$\begin{aligned}&{G^0_\rho }=0 \end{aligned}$$

(82)

$$\begin{aligned}&{G^0_{\rho \rho }}=0 \end{aligned}$$

(83)

The leading order in Eq. (71a) is thus $\epsilon ^{-1}$. Now, considering terms of order $\epsilon ^{-1}$, and using Eq. (77), we obtain the following,

$$\begin{aligned} \begin{aligned}&-v_n{\phi ^0_\rho } = c_1\varOmega _\text {sol}M\delta (\phi ^0) \nabla _\varGamma ^2 G \\&\quad +\, c_1 \delta (\phi ^0)\lambda x_\text {vac}(\mu _\text {vac}-\mu ^0) \end{aligned} \end{aligned}$$

(84)

Integrating the above from $-\infty$ to $\infty$ with respect to $\rho$, substituting the terms of G and writing $x_\text {vac}=\varOmega _\text {sol}c_\text {vac}$, the expression reduces to,

$$\begin{aligned} \begin{aligned} -2v_n= c_1\frac{4\sqrt{2}}{3}\varOmega _\text {sol}\Bigg [M \nabla _\varGamma ^2 \left( \mu ^0 +Z^*e\phi ^0 + Q^*T^0 \right) \Bigg .&\\ \Bigg .+\lambda c_\text {vac}(\mu _\text {vac}-\mu ^0) \Bigg ]&\end{aligned} \end{aligned}$$

(85)

On simplification, Eq. (85) therefore recovers the relation for the motion of the interface [Eq. (30)] if $c_1=\frac{3}{2\sqrt{2}}$. Thus, the constants in Eq. (39) are determined.

1.3 Appendix A.3: Asymptotic analysis of the vacancy diffusion equation

Finally, the phase field version of the vacancy diffusion equation is studied. A crucial assumption made in this derivation here is that the pressure variation in the interfacial region is negligible. This is not strictly true, as only the displacements are constant over the interfacial region. However, in this paper, the value of the pressure and other stress/strain related quantities is averaged over the interfacial region and hence ${p_{\text {in}}}_{\rho } =0$ is reasonable. The objective is to find a constant $c_4$ such that the vacancy diffusion [Eq. (86)] reduces to the sharp interface boundary condition for the vacancies on the moving interface Eq. (22).

$$\begin{aligned} \begin{aligned}&\frac{\partial }{\partial t}\left( H(\mathbf {x})x_\text {vac}\right) = \\&\nabla \cdot \left[ H(\mathbf {x})\frac{\mathbf {D}_\text {vac}}{RT}{x_\text {vac}} \left( RT \frac{\nabla x_\text {vac}}{x_\text {vac}} + \beta \nabla p + Z^*e \nabla \varphi _e\right. \right. \\&\quad \left. \left. ~~~+ Q^*\nabla T \right) \right] + c_4\frac{\delta (\phi )}{\epsilon } \lambda x_\text {vac}\left( \mu _\text {vac}- \mu _\text {sol}^\varGamma \right) \end{aligned} \end{aligned}$$

(86)

In the outer domain, the interfacial terms drop out as $\delta ({\hat{\phi }}) = 0$, and the standard bulk diffusion equation [Eq. (21)] is recovered. In the inner region, the vacancy mole fraction is expanded as

$$\begin{aligned} x_\text {vac}= x_\text {vac}^0 + \epsilon x_\text {vac}^1 + \cdots \end{aligned}$$

(87)

This lets the chemical potential to be expanded as,

$$\begin{aligned} \begin{aligned} \mu&= \mu + \epsilon \mu ^1 + \epsilon ^2\mu ^2 + \cdots \\&= RT\ln \frac{x_\text {vac}^0}{x_\text {vac}^\text {eq}} + \epsilon RT\frac{x^1_\text {vac}}{x_\text {vac}^0} + \cdots&\\&+ \beta ( p^0 +\epsilon p^1 + \cdots ) \end{aligned} \end{aligned}$$

(88)

Thus, we get in the leading order

$$\begin{aligned} \mu ^0 = RT \ln \frac{x_\text {vac}^0}{x_\text {vac}^\text {eq}} + \beta p^0 \end{aligned}$$

(89)

Transforming Eq. (86) into the inner co-ordinate system, the diffusion equation in the inner region can be written using Eq. (60) as,

$$\begin{aligned} \begin{aligned}&\frac{\partial }{\partial t}(H(\phi )x_\text {vac}) - \frac{v_n}{\epsilon } {(H (\phi )x_\text {vac})}_{\rho } \\&\quad +\frac{\partial \mathbf {x}}{\partial t}\cdot \nabla _{\varGamma }(H(\phi )x_\text {vac}) = \\&\quad -\frac{1}{\epsilon } \mathbf {n}.{\mathbf {j}_\text {vac}}_{\rho } -\nabla _{\varGamma }\cdot \mathbf {j}_\text {vac} \\&\quad + c_4\frac{\delta (\phi )}{\epsilon } \lambda x_\text {vac}\left( \mu _\text {vac}- \mu _\text {sol}^\varGamma \right) \end{aligned} \end{aligned}$$

(90)

where $\mathbf {j}_\text {vac}$ is,

$$\begin{aligned} \begin{aligned} \mathbf {j}_\text {vac}=&-H (\phi )\frac{D_\text {vac}}{RT} x_\text {vac}\left( \nabla \mu _\text {vac}+ Z^*e \nabla \phi _e \right. \\&\left. + Q^*\nabla T \right) \end{aligned} \end{aligned}$$

(91)

Expanding Eq. (91) in the inner region and using Eq. (72), terms of order $\epsilon ^{-1}$ can be grouped to get,

$$\begin{aligned} H(\phi )D_\text {vac}x_\text {vac}\left( {\mu ^0_\text {vac}}_{\rho } + G^0_\rho - {\mu ^0_\text {sol}}_{\rho }\right) = 0 \end{aligned}$$

(92)

From Eq. (79), we know that $\mu _{\text {sol}_\rho }^0=0$. Next, using Eq. (82), and since $H(\phi )D_\text {vac}x_\text {vac}\ge 0$ in the inner region, the above relation implies that $\mu ^0_{\text {vac}_\rho }=0$ or $\mu ^0_{\text {vac}} = {constant}$ in the inner region. Therefore, from Eq. (17)

$$\begin{aligned} {x_\text {vac}^0}_{\rho } = -\frac{x_\text {vac}^0}{RT} \beta {p^0_\rho } \end{aligned}$$

(93)

With the stress/strain quantities smoothed over the interfacial region, ${p^0_\rho } \approx 0$ and this leads to,

$$\begin{aligned} {x_\text {vac}^0}_{\rho } = 0 \end{aligned}$$

(94)

Terms of order $\epsilon ^{-1}$ in Eq. (90) can now be grouped to get,

$$\begin{aligned} \begin{aligned}&-v_n{H(\phi )}_{\rho } x_\text {vac}- v_nH(\phi ){x_\text {vac}}_{\rho } = -\mathbf {n}.{\mathbf {j}_\text {vac}}_{\rho } \\&\quad + c_4 \delta (\phi ) \lambda x_\text {vac}\left( \mu _\text {vac}- \mu _\text {sol}^\varGamma \right) \end{aligned} \end{aligned}$$

(95)

Integrating the above equation from $-\infty$ to $\infty$, the relation can be rewritten as,

$$\begin{aligned} -v_nx_\text {vac}= -n.\mathbf {j}_\text {vac}+ c_4\frac{4\sqrt{2}}{3} \lambda x_\text {vac}\left( \mu _\text {vac}- \mu _\text {sol}^\varGamma \right) \end{aligned}$$

(96)

This recovers the boundary condition for the vacancies on the moving interface Eq. (22) for $c_4=-\frac{3}{4\sqrt{2}}$. This determines the required constant in Eq. (40).