Corrosion-induced fracture of Cu–Al microelectronics interconnects

Consider a metal immersed in a solution, as shown in figure 2. The metal surface has a passivation layer that makes it resistant to corrosion. However, in the presence of a pit in the passivation layer, the corrosion can advance into the metal following the equilibrium between the dissolved metal atoms flux and the velocity of the pit boundary [17] as

$$\begin{equation}\left[ {D{\boldsymbol\nabla} c + \left( {c\left( {{\boldsymbol{x}},t} \right) - {c_{\text{s}}}} \right){\boldsymbol{v}}} \right] \cdot {\boldsymbol{n}} = 0{\,\text{ for }}{\boldsymbol{x}}{{\,\text{in}\,}\Gamma }\end{equation} \tag{ 3 }$$

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where $c\left( {{\boldsymbol{x}},t} \right)$ tracks the concentration of metal atoms, c_s is the concentration of atoms in the intact metal, $D$ is a diffusion coefficient, ${\boldsymbol{v}}$ is the velocity of the pit boundary, and n is the normal to the surface.

Initially, the process is activation controlled, and the pit velocity follows Tafel's equation [17]. When the concentration, $c\left( {{\boldsymbol{x}},t} \right)$, reaches the saturation concentration, c_sat, the process is controlled by diffusion, and the pit boundary normal velocity is given by

$$\begin{equation}{v_n} = { }\frac{{D{\boldsymbol\nabla} c \cdot {\boldsymbol{n}}}}{{\left( {{c_{\text{s}}} - {c_{{\text{sat}}}}} \right)}}.\end{equation} \tag{ 4 }$$

The solution of equations (3) and (4), known as the Stefan problem [21–24], requires imposing a boundary condition $c = {c_{{\text{sat}}}}$ in the moving boundary ${{\Gamma }}$. Here we use a phase field approach to follow the interface between the intact and corroded metal.

Following Mai et al [17] a phase field variable $\eta \left( {{\boldsymbol{x}},t} \right)$ is used to track the corroded material, with $\eta \left( {{\boldsymbol{x}},t} \right) = 1$ in the corroded or aqueous region and $\eta \left( {{\boldsymbol{x}},t} \right) = 0\,$ in the intact metal. To follow the concentration fraction of metal atoms, we define the concentration normalized by ${c_{\text{s}}}$

$$\begin{equation}\tilde c\left( {{\boldsymbol{x}},t} \right) = \frac{{c\left( {{\boldsymbol{x}},t} \right)}}{{{c_{\text{s}}}}}.\end{equation} \tag{ 5 }$$

The total free energy of the system is given by [18, 25]

$$\begin{equation}{\mathcal{F}_c}\left( {\tilde c,\eta } \right) = \mathop \int \nolimits \left( {{f_c}\left( {\tilde c,\eta } \right) + \frac{{{\varepsilon ^2}}}{2}{{\left( {\nabla \eta } \right)}^2} + w{\eta ^2}{{\left( {1 - \eta } \right)}^2}} \right){\text{d}}V.\end{equation} \tag{ 6 }$$

The first term in the integral in equation (6) is the local free energy density, the second is the interface energy, and the third is a double well potential. In equation (6) $\varepsilon$ and w are coefficients that control the thickness of the interface and are defined in terms of the interface energy density ${\gamma _{\text{s}}}$ and the interface thickness ${\delta _{\text{s}}}$ by [18]

$$\begin{equation}{\varepsilon ^2} = \frac{{8{\gamma _{\text{s}}}{\delta _{\text{s}}}}}{{{\pi ^2}}}\end{equation} \tag{ 7 }$$

$$\begin{equation}w = \frac{{4{\gamma _s}}}{{{\delta _s}}}.\end{equation} \tag{ 8 }$$

Following the KKS model [18] the concentration is interpolated by the concentration of the two phases

$$\begin{equation}\tilde c\left( {{\boldsymbol{x}},t} \right) = \left[ {1 - h\left( \eta \right)} \right]{\tilde c_1}\left( {{\boldsymbol{x}},t} \right) + h\left( \eta \right){\tilde c_2}\left( {{\boldsymbol{x}},t} \right)\end{equation} \tag{ 9 }$$

where ${\tilde c_1}$ is the concentration of metal atoms in the intact metal and ${\tilde c_2}$ is the concentration in the corroded material, and $h\left( \eta \right) = 3{\eta ^2} - 2{\eta ^3}$ is an interpolation function. Similarly, the local free energy density is written as an interpolation between the free energies densities of the two phases following

$$\begin{equation}{f_c}\left( {\tilde c,\eta } \right) = \,\left[ {1 - h\left( \eta \right)} \right]{f_1}\left( {{{\tilde c}_1}} \right) + h\left( \eta \right){f_2}\left( {{{\tilde c}_2}} \right).\end{equation} \tag{ 10 }$$

The free energy density functions for each phase can be approximated by a parabolic function with the minimum energies at the intact metal concentration ${\tilde c_{\text{s}}}$, and boundary condition ${\tilde c_{{\text{sat}}}}$.

$$\begin{equation}{f_1}\left( {{{\tilde c}_1}} \right) = \kappa {\left( {{{\tilde c}_1} - {{\tilde c}_s}} \right)^2}\end{equation} \tag{ 11 }$$

$$\begin{equation}{f_2}\left( {{{\tilde c}_2}} \right) = {{\kappa }}{\left( {{{\tilde c}_2} - {{\tilde c}_{{\text{sat}}}}} \right)^2}\end{equation} \tag{ 12 }$$

where $\kappa$ is a coefficient ${\tilde c_{\text{s}}} = 1$, and ${\tilde c_{{\text{sat}}}} = {c_{{\text{sat}}}}/{c_{\text{s}}}\,$ are the normalized equilibrium concentrations following the definition in equation (5). Finally, the free energy densities must satisfy the equilibrium of chemical potential

$$\begin{equation}\frac{{\partial {f_1}\left( {{{\tilde c}_1}} \right)}}{{\partial {{\tilde c}_1}}} = \frac{{\partial {f_2}\left( {{{\tilde c}_2}} \right)}}{{\partial {{\tilde c}_2}}}.\end{equation} \tag{ 13 }$$

The evolution of the phase fields follows the Allen–Cahn equation for the non-conserved field $\eta$ and the Cahn–Hilliard equation for the conserved variable $\tilde c$ [18]

$$\begin{equation}\frac{{\partial \eta }}{{\partial t}} = - {M_\eta }\left( {\frac{{\delta {\mathcal{F}_c}}}{{\delta \eta }}} \right)\end{equation} \tag{ 14 }$$

$$\begin{equation}\frac{{\partial \tilde c}}{{\partial t}} = \nabla { } \cdot {M_c}\nabla \left( {\frac{{\delta {\mathcal{F}_c}}}{{\delta \tilde c}}} \right)\end{equation} \tag{ 15 }$$

where ${M_\eta }$ is a kinetic parameter that controls the velocity of the interface, and ${M_c}$ is a diffusion coefficient. The functional derivatives $\frac{{\delta {\mathcal{F}_c}}}{{\delta \tilde c}}\,$ and $\frac{{\delta {\mathcal{F}_c}}}{{\delta \eta }}$ in equations (14) and (15) are derived in Kim et al [18].

The first chemical reaction in equation (1) is used to calculate c_sat which is determined by the metal atom concentration in ${\text{AlC}}{{\text{l}}_3}$ [2, 17]. Similarly, c_s is the metal atom concentration of ${\text{C}}{{\text{u}}_9}{\text{A}}{{\text{l}}_4}$. Following Scheiner and Hellmich [26] we write the mass density of ${\text{C}}{{\text{u}}_9}{\text{A}}{{\text{l}}_4}$ as:

$$\begin{equation}{\rho _s} = \sum\limits_{i = {\text{Cu}},{\kern 1pt} {\text{Al}}}^{} {{c_i}} {M_i} = {c_s}\sum\limits_{i = {\text{Cu}},{\kern 1pt} {\text{Al}}}^{} {\frac{{{c_i}}}{{{c_s}}}} {M_i} = {c_s}\sum\limits_{i = {\text{Cu}},{\text{Al}}}^{} {{a_i}} {M_i} = {c_s}{\bar M_s}\end{equation} \tag{ 16 }$$

where ${c_i}$ is the specific concentration of the species i, ${a_i}$ is the atomic % (note that it is equivalent to the mole fraction defined in reference [26]), ${M_i}$ is the molar mass, and ${{{\bar M}}_s} = \mathop \sum \nolimits_{i = {\text{Cu,Al}}}^{} {{{a}}_i}{ }{{{M}}_i}$.

With ${\rho _s}$ = 6.85 g cm⁻³ for ${\text{C}}{{\text{u}}_9}{\text{A}}{{\text{l}}_4}$ and using the values in table 1 we obtain ${{{\bar M}}_s} = 52.3\frac{{{g}}}{{{\text{mol}}}}\,$and ${{{c}}_s} = \,{\rho _s}/\,{{{\bar M}}_s}$ = 0.131 mol cm⁻³. Repeating the procedure for AlCl₃ we obtain ${{{\bar M}}_{{\text{sat}}}} = 33.3$ g mol⁻¹. Using $\rho$ = 2.48 g cm⁻³ for AlCl₃, we obtain ${{{c}}_{{\text{sat}}}} = \,{\rho _{{\text{sat}}}}/\,{{{\bar M}}_{{\text{sat}}}}$ = 0.075 mol cm⁻³ and using equation (5), ${{{\tilde c}}_{{\text{sat}}}} = 0.57$.

Table 1. At% and molar mass Cu₉Al₄ and AlCl₃.

		# at	at% ${a_i}$	Molar mass (g mol−1) Mi
Cu9Al4	Cu	9	9/13	63.5
	Al	4	4/13	26.9
AlCl3	Cl	3	3/4	35.5
	Al	1	1/4	26.9

The mechanical responses of Cu and Al include plasticity, a fracture model is used in the IMC domain, and we incorporate the volumetric expansion of the corroded material [27]. The strain is decomposed into the elastic, ${{\boldsymbol{\varepsilon }}^e},$ and plastic, ${{\boldsymbol{\varepsilon }}^p},\,$ parts

$$\begin{equation}{\boldsymbol{\varepsilon }} = {{\boldsymbol{\varepsilon }}^e} + {{\boldsymbol{\varepsilon }}^p}.\end{equation} \tag{ 17 }$$

The plastic strain follows an isotropic flow rule

$$\begin{equation}{\dot \varepsilon ^p} = \frac{3}{2}\frac{{\boldsymbol{s}}}{\tau }\dot \gamma \end{equation} \tag{ 18 }$$

where the components of the deviatoric stress are ${s_{ij}} = {\sigma _{ij}} - \frac{{{\sigma _{kk}}}}{3}{\delta _{ij}}$, $\tau = \sqrt {\frac{3}{2}{s_{ij}}{s_{ij}}}$ is the von Misses stress (using Einstein's notation), and $\dot \gamma \,$ is the rate at which plastic strain occurs. We assume perfect plasticity with a flow rule [28]

$$\begin{equation}f\left( \tau \right) = \tau - {\sigma _y}\end{equation} \tag{ 19 }$$

where ${\sigma _y}$ is the yield stress. The irreversibility of the plastic deformation is obtained with the Kuhn/Tucker conditions:

$$\begin{equation}\dot \gamma \unicode{x2A7E} 0,\,f\left( \tau \right) \unicode{x2A7D} 0,{\text{ and}}\,\,f\left( \tau \right) \cdot \,\dot \gamma = 0.\end{equation} \tag{ 20 }$$

These conditions allow plastic deformation only when $\tau \unicode{x2A7E} {\sigma _y}$.

To track the evolution of cracks we use a phase field damage model [29–31] in which the field d(x) indicates undamaged material when d = 0 and damage when d = 1. The strain energy of the system is written as

$$\begin{equation}W\left[ {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}},d} \right] = {W_f} + \mathop \int \limits_V^{} {f_e}\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}},d} \right){\text{d}}V.\end{equation} \tag{ 21 }$$

The first term on the right-hand side of equation (21) is the fracture energy and the second is the integral of elastic strain energy density:

$$\begin{equation}{f_e}\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}},d} \right) = {\left( {1 - d} \right)^2}{\psi ^ + }\,\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right) + {\psi ^ - }\,\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right)\end{equation} \tag{ 22 }$$

where ${\psi ^ + }\,\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right)$ is the part of the strain energy density that is degraded with damage and ${\psi ^ - }\,\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right)$ is not affected by damage and are defined by

$$\begin{equation}{\psi ^ + }{\mkern 1mu} \left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right) = \frac{{E\nu }}{{2\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)}}\langle tr{\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right)\rangle^2} + \frac{E}{{2\left( {1 + \nu } \right)}}{\mkern 1mu} {{\boldsymbol{\varepsilon }} {^{^{\prime}}}^{\boldsymbol{e}}}:{{\boldsymbol{\varepsilon }} {^{^{\prime}}}^{\boldsymbol{e}}}\end{equation} \tag{ 23 }$$

$$\begin{equation}{\psi ^ - }\,\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right) = \frac{{E\nu }}{{2\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)}}\left( {tr{{\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right)}^2} - \langle tr\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right)\rangle_{}^2} \right)\end{equation} \tag{ 24 }$$

$E$ and $\nu$ are Young's modulus and Poisson's ratio, 〈x〉 = x if $x > 0$, and 〈x〉 = 0 otherwise, and ${{\boldsymbol{\varepsilon }} {^{^{\prime}}}^{\boldsymbol{e}}} = {{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}} - \frac{{tr\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right)}}{3}{\boldsymbol{I}}$. Note that when d = 0, we recover the classical strain energy density as ${\psi ^ + }\,\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right) + {\psi ^ - }\,\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right)$. The fracture energy follows Griffith's criterion for brittle materials over the crack surface Γ [32]

$$\begin{equation}{W_f} = \,\mathop \int \limits_{{\Gamma }}^{} {G_c}d{{\Gamma }}\end{equation} \tag{ 25 }$$

where ${G_c}$ is the fracture surface energy density. The integral over the crack surface, ${{\Gamma }},$ can be replaced by an integral over the volume using a diffuse delta function in terms of the damage field [33–35]

$$\begin{equation}{W_f} = \mathop \int \limits_V^{} {G_c}\left( {\frac{{{d^2}}}{{2{l_0}}} + \frac{{{l_0}}}{2}{{\left| {\nabla d} \right|}^2}} \right){\text{d}}V\end{equation} \tag{ 26 }$$

where ${l_0}$ is the effective crack width [29–31, 33].

The evolution of the damage follows as

$$\begin{equation}\frac{{\partial d}}{{\partial t}}\left( {{\boldsymbol{x}},t} \right) = - L\frac{{\delta W}}{{\delta d}}.\end{equation} \tag{ 27 }$$

The rate of energy dissipation due to cracks should be positive, ${\dot W_f} \unicode{x2A7E} 0$, this leads to $\frac{{\partial d}}{{\partial t}} \unicode{x2A7E} 0$ i.e. the cracks can only grow with time. The mechanical response relaxes faster than the corrosion evolution, and therefore, we consider the steady state form of the conservation of linear momentum:

$$\begin{equation}\nabla \cdot {\boldsymbol{\sigma }} = 0\end{equation} \tag{ 28 }$$

where ${\boldsymbol{\sigma }}$ is the Cauchy stress tensor defined by

$$\begin{equation}{\boldsymbol{\sigma }} = \frac{{\partial {f_e}\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}},d} \right)}}{{\partial {{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}}} = {\left( {1 - d} \right)^2}\frac{{\partial {\psi ^ + }{ }\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right)}}{{\partial {{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}}} + \frac{{\partial {\psi ^ - }{ }\left( {{{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}} \right)}}{{\partial {{\boldsymbol{\varepsilon }}^{\boldsymbol{e}}}}}.\end{equation} \tag{ 29 }$$

The Pilling–Bedworth (P–B) ratio [36, 37] is the ratio of the volume of the elementary cell of metal oxide, ${V_{{\text{ox}}}}$, to the unit volume of the metal, ${V_{\text{m}}}$, and it can be used to estimate the change in volume due corrosion [38]. This ratio is defined as

$$\begin{equation}{\text{PBR }} = { }\frac{{{V_{{\text{ox}}}}}}{{n{ }{V_{{\text{me}}}}}}{\text{ }}\end{equation} \tag{ 30 }$$

where n is the number of metal atoms in the oxide molecule. Note that equation (30) considers that the metal is a single component. Therefore, we will use here that ${V_{{\text{me}}}} = \frac{1}{{13}}99.12$ cm³ mol⁻¹ given the at% of 1 atom of Al in Cu₉Al₄. Using n = 2.67 and ${V_{ox}} = 37.07$, see table 2, we obtain PBR = 1.8, which represents an 80% volumetric expansion of the γAl₂O₃ due to corrosion. We assume that only Al in the Cu₉Al₄ phase is corroded as Cu precipitation is observed in the experiments [11]. Therefore, the volumetric expansion is proportional to the at% of Al in Cu₉Al₄, see table 1, which results in $\frac{4}{{13}}80\% = 24\% .$

Table 2. Mass density, molar mass, and volume of Cu₉Al₄ and $\gamma$-Al₂O₃.

	Mass density $\rho$ (g cm−3)	Molar mass Mi (g mol−1)	Molar volume Vi (cm3 mol−1)
Cu9Al4	6.85	679.84	99.12
$\gamma$-Al2O3 (Al2.67O4)	3.67	136.04	37.07

The field $\eta \left( {{\boldsymbol{x}},t} \right)$ used to indicate the corrosion in section 3.1 is used to track the volumetric expansion of the $\gamma$Al₂O₃,

$$\begin{equation}{\varepsilon ^v} = \beta \left( {\eta - {\eta _0}} \right)\end{equation} \tag{ 31 }$$

where ${\varepsilon ^v} = tr\left( {\boldsymbol{\varepsilon }} \right)$, ${\eta _0} = 0$ corresponds intact material, and $\beta = \,0.24$ is the volumetric expansion coefficient.

The KKS model presented in section 2.1 is implemented in the finite element solver MOOSE [39] to solve equations (14) and (15) using cylindrical coordinates with an element size 40 nm. A 50 μm radius Cu ball bond is considered in these simulations to compare with experimental results [9].

The mechanical part is not included in this section and we track only the evolution of the corrosion fields $\tilde c\left( {x,t} \right)\,$ and $\eta$ (x,t) in the Cu₉Al₄ region. Figure 3 shows the dimensions and initial conditions.

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The right boundary (outer surface) has a pit opening represented by a region with $\tilde c\left( {{\boldsymbol{x}},t} \right) = 0\,$ and $\eta \left( {{\boldsymbol{x}},t} \right) = 1$. Given that the initial corroded region will advance less than 10 μm from the outer surface during the simulation time we only simulate the region 40 μm ⩽ r ⩽ 50 μm Therefore, the region r ⩽40 μm contains intact material and we impose no flux of $\tilde c$ and $\eta$ in this boundary as well as the top and bottom boundaries.

The interface energy density, ${\gamma _s}$, is reported in literature for metal corrosion is in the range of 0.1–10 J m⁻² [2, 17, 40]. Therefore, we choose the interface energy density between the corroded material and the uncorroded material as ${\gamma _s} = 0.1$ J m⁻² and an interface thickness ${\delta _s} =$ 300 nm, which is seven times the element size. These values are used to calculate ${\varepsilon ^2}$ and $w$ in table 3. The experimental data in [9] is used to calibrate the other parameters in the simulation.

Table 3. Parameters used in the corrosion model.

${\varepsilon ^2}$ (${\text{J}}\,{{\text{m}}^{ - 1}}$)	$2.4 \times {10^{ - 8}}$
$w$ (${\text{J}}\,{{\text{m}}^{ - 3}}$)	$1.3 \times {10^6}$
$\kappa$ (${\text{J}}\,{{\text{m}}^{ - 3}}$)	$5 \times {10^7}$
${\tilde c_{{\text{sat}}}}$	0.57
${\tilde c_s}$	$1$
${M_c}$ (${{\text{m}}^{\text{5}}}\,{\text{J}}{{\text{s}}^{ - 1}}$)	$1 \times {10^{ - 25}}$
${M_\eta }$ (${{\text{m}}^{\text{3}}}\,{\text{J}}{{\text{s}}^{ - 1}}$)	2

Geometries featuring different pit sizes and varying numbers of pits are shown in figures 4(a)–(c). The boundary profiles at t = 0.5 h are shown in figures 4(d)–(f) and the evolution of the corrosion boundary location S(t) is plotted in figure 5. S(t) is the location of $\eta = 0.5$ and it is measured from the outer surface. S(t) is plotted as a function of the square root of time because the 1D analytical solution follows ${t^{1/2}}$. Due to the zero flux boundary conditions, once the corrosion reaches the top and bottom surfaces its profile is flat and the velocities are identical for the three cases following the form $S\left( t \right)\sim {t^{1/2}}$, see figure 5. Since there is no difference in the corroded material profile after 1 h, single 0.3 μm and 0.5 μm pits are chosen for the following simulations.

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Rongen et al [9] utilized scanning electron microscopy to measure the corrosion product and the intact metal radii in a Cu ball with a diameter of approximately 50 µm. The experiments were conducted with temperatures 110 °C and 130 °C, with RH levels of 75%, 85%, and 95%. Figure 6 shows the position of the corrosion pit as a function of time from Rongen et al [9] and our simulations. Lower temperatures reduce diffusivity, resulting in slower corrosion velocities. The parameters used in the simulation to obtain similar corrosion pit velocities are listed in table 3. It is important to notice the large uncertainty in the experiments, therefore, the mobility parameters in table 3 need to be considered an approximation.

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Figure 10 shows the evolution of corrosion field $\eta$, figure 11 shows the volumetric stress ${\sigma _v} = \frac{{tr\left( {\boldsymbol{\sigma }} \right)}}{3}$, and figure 12 shows the shear component in the stress tensor, ${\sigma _{rz}}$. The surrounding material inhibits the volumetric expansion of the corroded material, resulting in high compressive stress, see figure 11. Large shear stresses in the crack tip and in the Cu₉Al₄/Al are shown in figure 12. The shear stress in the Cu₉Al₄/Cu interface drive the growth of the crack as the corroded material expands.

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The evolution of the corrosion field in the 3D geometry is shown in figure 13. Corrosion starts at the three pits and the corroded regions merge forming a smooth front. The differences in time scales between the axisymmetric and 3D simulations are due to the different radii used in the geometries, 50 μm in the axisymmetric versus 3 μm in the 3D simulations. The time evolution of the position of the corrosion boundary is inversely proportional to the radius of the outer surface [47] and therefore, the rate of growth is higher in the 3D simulations.

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Figure 14 shows the damage field in the 3D geometry. The volumetric expansion due to the corrosion of the Cu₉Al₄ layer causes localized shear stresses between the pits in the surface of the Cu ball during the early stages (<1 h), see figure 15. Therefore, more pits could be nucleated in these regions and accelerate the corrosion process.

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Damage to the outer surface of the Cu ball in the Cu₉Al₄/Cu interface could induce the formation of new pits and therefore, accelerates the corrosion through faster diffusion of Cl⁻ . To take into account the coupling between damage and corrosion, we increase the diffusivity in the damaged regions with the following relationship:

$$\begin{equation}{M_c} = \left( {1 + K\mathcal{D}} \right)\,1 \times {10^{ - 25\,}}{{\text{m}}^5}\,{\text{J}}{{\text{s}}^{ - 1}}\end{equation} \tag{ 35 }$$

where $\mathcal{D}$ is a switch function that identifies the crack region, i.e. $\mathcal{D} = 0$ if $d \unicode{x2A7D} 0.9$, and $\mathcal{D} = 1$ otherwise, and $K$is a parameter. Note that in the regions without damage we recover the value of ${M_c}\,$listed in table 3. Figure 16 shows the corrosion boundary profile for K = 0, 30, 50, and 100 at 12 h. When K is not zero the mobility is increased in the crack region and the corrosion advances at a faster rate in the Cu₉Al₄/Cu interface forming a slanted profile. Due to this profile shape, a tensile stress appears in the Cu₉Al₄/Al in front of the corrosion boundary, see figure 17. That leads to the damage in the Cu₉Al₄/Al interface shown in figure 18.

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Figure 19 shows the position of the crack tip, C(t), and the corrosion boundary, S(t), measured from the Cu ball surface. As expected, enhancing the diffusion coefficient in the crack increases the rate of growth of the corroded region which in turn accelerates the damage.

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