Simulations of the Kirkendall-Effect-Induced Deformation of Thermodynamically Ideal Binary Diffusion Couples with General Geometries

Abstract

The Kirkendall effect stems from the imbalance of atomic diffusion fluxes in a crystalline solid. Vacancy generation and annihilation, which compensate for the unbalanced fluxes, result in deformation that is experimentally observed as the motion of fiducial markers in a diffusion couple that can be approximated as a one-dimensional system. In multiple dimensions, such deformation occurs along both the directions parallel to and normal to the primary diffusion direction. In this article, we present a model that couples unbalanced interdiffusion and the resulting plastic deformation. One- and two-dimensional simulations are conducted with the analytically calculated diffusion coefficients of a thermodynamically ideal random alloy; the result shows that the ratio of the diffusion fluxes of the atomic species equals the ratio of atomic hop frequencies, which leads to the final volume ratio also given approximately by the hop frequency ratio if the initial volume ratio is equal. Moreover, the result also demonstrates that the conventional interdiffusion model fails to describe the Kirkendall void growth dynamics. For numerical implementation, we reformulate the diffusion equation to the smoothed boundary form and solve it within the deforming body governed by steady-state Navier-Stokes equation. This work demonstrates that the presented method can be a useful tool for studying Kirkendall-effect-induced deformation.

Appendices

Appendix: Diffusion Coefficients

According to the lattice-site conservation condition ($\mathbf{J}_A + \mathbf{J}_B + \mathbf{J}_V = 0$ and ∇(C _A  + C _B  + C _V ) = ∇ρ = 0), the flux expression, Eq. [2], can be rewritten in terms of the A and B fluxes:

$$ \left(\begin{array}{ll} {\mathbf{J}}_A \\ {\mathbf{J}}_B \end{array}\right)=-\left[ \begin{array}{ll} D_{AA} &D_{AB}\\ D_{BA} &D_{BB} \end{array} \right] \left( \begin{array}{l} \nabla C_A \\ \nabla C_B \end{array} \right) $$

(A1)

where the diffusion coefficients are related to those in Eq. [2] by D _VV  = D _AA  + D _BA , D _VB  = D _AA  − D _AB  + D _BA  − D _BB , D _BV  =  −D _BA , and D ^V _BB  = D _BB  − D _BA .[33] These diffusion coefficients can be determined by the kinetic transport coefficients and thermodynamic factors according to

$$ \left[ \begin{array}{ll} D_{AA} &D_{AB} \\ D_{BA} & D_{BB}\end{array} \right] = \left[ \begin{array}{cc} L_{AA} & L_{AB} \\ L_{BA} & L_{BB} \end{array} \right] \left[ \begin{array}{ll} \partial \tilde{\mu}_A/\partial C_A & \partial \tilde{\mu}_A/\partial C_B \\ \partial \tilde{\mu}_B/\partial C_A & \partial \tilde{\mu}_B/\partial C_B \end{array} \right] $$

(A2)

where L _ij are the kinetic transport coefficients, and $\tilde{\mu}_i = \mu_i - \mu_V$ is the chemical potential measured with respective to that of the vacancies.[30,33] According to the Onsager reciprocity theorem,[62,63] the matrix of kinetic transport coefficients is symmetric such that L _AB  = L _BA . In this article, we select a thermodynamically ideal random alloy system where the interaction between A, B, and V are assumed to be identical for simplicity. Analytical expressions of the kinetic transport coefficients for such a system have been derived in terms of the atomic hop frequencies of the diffusing species, lattice-structure-dependent parameters, and the composition of the alloy[33,59,64,65]:

$$ L_{AA} = \frac{\lambda a^2 \rho}{k_B T} X_V X_A \Upgamma_A \left( 1 - \frac{2 X_B \Upgamma_A}{\Uplambda} \right) $$

(A3a)

$$ L_{BB} = \frac{\lambda a^2 \rho}{k_B T} X_V X_B \Upgamma_B \left( 1 - \frac{2 X_A \Upgamma_B}{\Uplambda} \right) $$

(A3b)

$$ L_{AB} = L_{BA} = \frac{\lambda a^2 \rho}{k_B T} X_V \left( \frac{2 X_A \Upgamma_B X_B \Upgamma_A}{\Uplambda} \right) $$

(A3c)

where λ is the geometric factor of the crystal structure (λ = 1/6 for a face-centered-cubic [fcc] structure), a is the atomic hop distance, k _B is the Boltzmann constant, T is the absolute temperature, X _i  = C _i /ρ is the mole fraction of the ith diffusing species, $\Upgamma_i$ is the atomic hop frequency of the ith diffusing species,

$$ \begin{aligned} \Uplambda \,=\, & (F+2)(X_A \Upgamma_A + X_B \Upgamma_B)/2 - \Upgamma_A - \Upgamma_B + 2(X_A \Upgamma_B + X_B \Upgamma_A)\\ + & \sqrt{[(F+2)(X_A \Upgamma_A + X_B \Upgamma_B)/2 - \Upgamma_A - \Upgamma_B]^2 + 2 F \Upgamma_A \Upgamma_B} \end{aligned} $$

(A4)

F = 2 f ₀/(1 − f ₀), and f ₀ is the geometric correlation factor for a single component crystal (f ₀ = 0.7815 for fcc). The atomic hop frequency is estimated by $\Upgamma = \nu \exp(- \Updelta E_b /k_B T),$ where ν is the vibration prefactor and $\Updelta E_b$ is the activation barrier of an atomic hop. Under the ideal mixing condition, the free energy per lattice site of this system is determined by $g(X_A,X_B) = k_B T [X_A \ln{X_A} + X_B \ln{X_B} + X_V\ln{X_V}],$ where X _A  + X _B  + X _V  = 1 since each lattice site must be occupied by a diffuser atom or a vacancy. The chemical potentials are then obtained according to $\tilde{\mu}_i = \partial g/\partial X_i.$ Combining the kinetic transport coefficients and thermodynamic factors, one can analytically evaluate the diffusion coefficients in terms of the alloy composition and intrinsic atomic hop frequencies. For thermodynamically no-ideal alloys, both the kinetic transport coefficients and chemical potentials need to be evaluated by a combination of first principles calculation and kinetic Monte Carlo simulations.[33,66]

For a thermodynamically ideal random alloy, one can substitute the thermodynamic factors into Eq. [A2] to obtain

$$ D_{AA} = \frac{L_{AA}}{\rho} \left( \frac{1}{X_A} +\frac{1}{1-X_A-X_B} \right) + \frac{L_{AB}}{\rho} \left( \frac{1}{1-X_A - X_B} \right) $$

(A5a)

$$ D_{AB} = \frac{L_{AA}}{\rho} \left( \frac{1}{1-X_A-X_B} \right) + \frac{L_{AB}}{\rho} \left( \frac{1}{X_B} +\frac{1}{1-X_A - X_B} \right) $$

(A5b)

$$ D_{BA} = \frac{L_{AB}}{\rho} \left( \frac{1}{X_A} + \frac{1}{1-X_A-X_B} \right) + \frac{L_{BB}}{\rho} \left( \frac{1}{1-X_A - X_B} \right) $$

(A5c)

$$ D_{BB} = \frac{L_{AB}}{\rho} \left( \frac{1}{1-X_A-X_B} \right) + \frac{L_{BB}}{\rho} \left( \frac{1}{X_B} +\frac{1}{1-X_A - X_B} \right)$$

(A5d)

and then use the relations below Eq. [A1] to obtain Eq. [11].

When vacancy concentration is assumed to be in equilibrium, as in the conventional treatment,[38] ∇μ _V  = 0 and the driving forces of diffusion are related by X _A ∇μ _A  + X _B ∇μ _B  = 0. Thus, the fluxes of A and B can be written by substituting Eq. [A2] to Eq. [A1] as

$$ {\mathbf{J}}_A = -\left(\frac{L_{AA}}{X_A} - \frac{L_{AB}}{X_B} \right) X_A \nabla \mu_A\quad \hbox{and}\quad{\mathbf{J}}_B = -\left(\frac{L_{BB}}{X_B} - \frac{L_{AB}}{X_A} \right) X_B \nabla \mu_B $$

(A6)

For a thermodynamically ideal random alloy, the thermodynamic factors are given by ∂μ _A /∂C _A  = k _B T/(X _A ρ) and ∂μ _B /∂C _B  = k _B T/(X _B ρ). Hence, the atomic fluxes can be directly related to their own concentration gradients and intrinsic diffusion coefficients by

$$ {\mathbf{J}}_A = - D_A \nabla C_A\quad \hbox{and}\quad{\mathbf{J}}_B = - D_B \nabla C_B $$

(A7)

where D _A  = (L _AA /X _A  − L _AB /X _B )/ρ and D _B  = (L _BB /X _B  − L _AB /X _A )/ρ.

Visco-Plasticity Model

Let us consider the formulation of the plastic deformation caused by the stress-free volumetric dilatational strain. In continuum mechanics, a strain is defined as the spatial derivative of displacement. Assuming the displacement is infinitesimal, one obtains the strain rate by taking the time derivative of the total strain in the laboratory frame, which can be expressed in terms of the velocity, v _i , as $\dot{\varepsilon}_{ij} = \partial \varepsilon_{ij} / \partial t= (\partial v_i /\partial x_j +\partial v_j /\partial x_i)/2,$ where v _i is the velocity component along the ith axis. This can be decomposed into a dilatational strain rate and a shear strain rate: $\dot{\varepsilon}_{ij} = \dot{\varepsilon}_U\delta_{ij}/d +\dot{\gamma}_{ij},$ where $\dot{\varepsilon}_U$ is the dilatational strain rate as defined in Section II–B, δ _ij is the Kronecker delta (δ _ij  = 1 for i = j; otherwise δ _ij  = 0), d is the dimensionality of the coordinate system, and $\dot{\gamma}_{ij}$ is the shear strain rate. The factor 1/d is adopted under the assumption that deformation is small and that any volume expansion/contraction due to generation/annihilation of lattice sites is isotropic. While the lattice structure, grain orientation, and grain boundary distribution can lead to anisotropy, we do not consider these effects in this macroscopic model since the entire solid is assumed to contain a high density of vacancy sources (crystalline defects) uniformly, which would lead to more isotropic behavior.

Similarly to References 27 and 29, we treat the plastically deformable solid as a very viscous fluid and employ a linear constitutive relation for a viscous Newtonian fluid to model the deformation process. This approach is commonly applied, for example, to model creep flow.[45,46] Since the solid is incompressible due to the assumption that ρ is constant, the considerations of elasticity effects are removed from the model. The shear strain rate is related to the stress components by $\dot{\gamma}_{ij} = ( \sigma_{ij} -\sigma_{kk} \delta_{ij}/d )/2\eta,$ where η is the viscosity of the material and σ _ij is the stress tensor. (The repeated indices denote summation over the indices.) The quantity (σ _ij  − σ _kk δ _ij /d) is the so-called deviatoric stress tensor in the plasticity. Substituting the shear and dilatational strain rates into the total strain rate and rearranging terms, we arrive at

$$ \sigma_{ij} = \frac{\sigma_{kk}}{d} \delta_{ij}+ \eta \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right) -\frac{2\eta}{d} \dot{\varepsilon}_U \delta_{ij} $$

(B1)

We assume the diffusion-induced deformation is very slow, such that the inertial force is neglected. This means that the deformation occurs via an instantaneous quasi-steady-state process where mechanical equilibrium (force balance) is maintained at all times: ∂σ _ij /∂x _j  = 0. Therefore, one obtains

$$ -\frac{\partial P}{\partial x_j} + \frac{\partial}{\partial x_j} \eta \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right) -\frac{\partial }{\partial x_j} \left( \frac{2 \eta}{d} \dot{\varepsilon}_U \right) =0 $$

(B2)

where P = −σ _kk /d is defined as an effective pressure. The first two terms correspond to the terms appearing in conventional Navier-Stokes equation for a quasi-steady flow, and the third term originates from the dilatational strain due to vacancy generation in the solid.

Convergence of Neumann Boundary Condition Imposed by the Smoothed Boundary Method

To demonstrate that the smoothed-boundary-formulated diffusion equation satisfies the assigned Neumann boundary condition (specifying the boundary flux or normal gradient), we use the one-dimensional version of Eq. [13] for simplicity. Note the following analysis is also valid for higher dimensions by replacing dx with dn, where dn is the coordinate variable in the normal direction of the boundary. By reorganizing terms and integrating over the interfacial region, we obtain

$$ \int_{a_i-\xi/2}^{a_i+\xi/2}\psi \left[ \frac{\partial C_B}{\partial t} - v \frac{\partial C_B}{\partial x} - \frac{C_B}{\rho_a} \frac{\partial}{\partial x} \left( D_{VB} \frac{\partial C_B}{\partial x} \right) \right] dx = \left. \psi D_{BB}^V \frac{\partial C_B}{\partial x} \right|_{a_i - \xi/2}^{a_i + \xi/2} - \int_{a_i - \xi/2}^{a_i + \xi/2} \left| \frac{\partial \psi}{\partial x} \right| B_{N} dx, $$

(C1)

where a _i is the boundary position, ξ is the thickness of the smoothed boundary, and a _i  − ξ/2 < x < a _i  + ξ/2 is the region of the boundary. Following References 42, 43, 67, and 68, we shall apply the mean value theorem of integrals, which states that for a continuous function, f(x), there exists a constant value, h ₀, such that $\min{[f(x)]} < \int_{p}^{q} f(x) dx /(q-p) = h_0 < \max{[f(x)]},$ where p < x < q. By setting the second term on the right-hand side of Eq. [C1] to be zero (or B _N  = 0), the no-flux boundary condition can be imposed; the resulting equation is similar to those proposed in References 41 through 43, 67 and 68. However, we retain the term in order to maintain the generality of the method. By doing so, the analysis presented here leads to an extension of the original method in References 41 through 43 that greatly expands its applicability.

Since the function on the left-hand side of Eq. [C1] is continuous and finite within the boundary region, its value is proportional to the boundary thickness by h ₀ ξ, according to the mean value theorem of integrals. Using the conditions that ψ = 1 at x = a _i  + ξ/2 and ψ = 0 at x = a _i  − ξ/2, the first term on the right-hand side of Eq. [C1] is written as D ^V _BB (∂C _B /$\partial {x)_{a_{i}}}$. Since |∂ψ/∂x| = 0 for x < a _i  − ξ/2 or x > a _i  + ξ/2, the bounds of integral can be extended to $-\infty$ and $\infty.$ Thus, we can rewrite Eq. [C1] as $h_0 \xi = D_{BB}^V \partial C_B / \partial x |_{{a_{i}}+\xi/2} - \int_{-\infty}^{+ \infty} | \partial \psi / \partial x| B_{N} dx.$ By taking the limit $\xi \rightarrow 0,$ we obtain

$$ \begin{array}{c} D_{BB}^V \left|\frac{\partial C_B}{\partial x}\right|_{a_i} = \int_{-\infty}^{+ \infty} \delta(x-a_i) B_N dx =B_N \bigg|_{a_i}\end{array} $$

(C2)

where $\lim_{\xi \rightarrow 0} \partial C_B / \partial x|_{a_i+\xi/2} = \partial C_B / \partial x|_{a_i}, \lim_{\xi \rightarrow 0} |\partial \psi / \partial x | = \delta(x-a_i),$ and δ(x − a _i ) is the Dirac delta function, which has the property that $\int_{-\infty}^{+\infty} \delta(x-a_i) f(x) dx = f(a_i).$ One can show that these conditions are met when ψ takes the form of a hyperbolic tangent function. Therefore, Eq. [C2] clearly shows that the smoothed boundary method recovers the Neumann boundary condition at the boundary when the thickness of the smoothed boundary approaches zero.