Interdiffusion, Kirkendall displacement, Cu coating on nickel substrate

1. Introduction and physical geometry of the thin-film couple

In modern surface engineering, microelectronics, and high-temperature protective coatings, thin metallic films are frequently deposited on metallic substrates to impart oxidation resistance, electrical conductivity, or wear resistance. When such bi-material assemblies are exposed to elevated temperatures, interdiffusion between the film and substrate alters the interfacial chemistry and physical integrity of the coating.

The copper–nickel ($\text{Cu–Ni}$) binary system provides a model configuration for investigating these processes. Both copper and nickel crystallise in the face-centred cubic ($\text{fcc}$) lattice and exhibit complete, isomorphous mutual solubility across the entire compositional spectrum at typical processing temperatures ($700\ ^\circ\text{C}$ to $1050\ ^\circ\text{C}$).

We consider a physical geometry consisting of a flat, continuous thin film of pure copper of uniform initial thickness $h$ deposited on the planar surface of a semi-infinite nickel substrate bar. Let the spatial coordinate $x$ be oriented normal to the outer planar surface of the film and directed inward into the substrate:

• $x = 0$ corresponds to the free, unconstrained outer surface of the copper coating exposed to vacuum or an inert protective atmosphere. Mass cannot penetrate across this boundary into the ambient environment.
• $0 \le x \le h$ represents the initial spatial domain occupied by the as-deposited pure copper coating at time $t = 0$.
• $x = h$ defines the initial planar contact interface between the copper film and the underlying nickel bar.
• $x > h$ represents the semi-infinite bulk nickel substrate extending to $x \to +\infty$.

The initial concentration distribution of copper, expressed as an atomic (mole) fraction $C_{\text{Cu}}(x, 0)$, is described mathematically by a top-hat step profile:

Correspondingly, by site-conservation in a binary substitutional solid solution, the nickel concentration satisfies $C_{\text{Ni}}(x, t) = 1 - C_{\text{Cu}}(x, t)$ everywhere.

2. Governing boundary value problem for finite-source interdiffusion

Under isothermal conditions, one-dimensional atomic transport in the laboratory coordinate frame is governed by Fick's second law:

where $\tilde{D}$ is the chemical interdiffusion coefficient of the $\text{Cu–Ni}$ alloy. While $\tilde{D}$ exhibits mild composition dependence across the binary range, analytical treatment of the finite-source problem proceeds under the standard linearisation approximation $\tilde{D} \approx \text{constant}$ evaluated at the mean interfacial composition:

The physical constraints of the thin-film configuration impose two boundary conditions on the spatial domain $[0, +\infty)$:

1. Impermeable free surface boundary condition at $x = 0$: Because the ambient atmosphere is insoluble and non-volatile, the diffusive flux of copper across the outer surface is identically zero at all times $t \ge 0$: \begin{equation} J_{\text{Cu}}(0, t) = -\left. \tilde{D} \frac{\partial C_{\text{Cu}}}{\partial x} \right|_{x=0} = 0 \implies \left. \frac{\partial C_{\text{Cu}}}{\partial x} \right|_{x=0} = 0 \label{eq:bc_surface} \end{equation} This represents a homogeneous Neumann boundary condition.
2. Far-field semi-infinite substrate boundary condition as $x \to +\infty$: At infinite depth within the bulk nickel substrate, the copper concentration remains unperturbed: \begin{equation} \lim_{x \to +\infty} C_{\text{Cu}}(x, t) = 0, \quad \text{for all } t \ge 0 \label{eq:bc_farfield} \end{equation}

3. Derivation of the analytical concentration profile via the method of images

To solve the initial-boundary value problem defined by equations \eqref{eq:initial_cond}, \eqref{eq:linear_diff}, \eqref{eq:bc_surface}, and \eqref{eq:bc_farfield}, we employ the method of images (the reflection principle).

Infinite domain Green's function representation

Consider the one-dimensional diffusion equation on the unbounded real domain $x \in (-\infty, +\infty)$:

The fundamental solution (Green's function) corresponding to an instantaneous unit point source released at position $x = x'$ at $t = 0$ is the Gaussian kernel:

By linear superposition, the solution for any initial concentration distribution $C_0(x')$ on the infinite line is given by the convolution integral:

Construction of the symmetric extended initial condition

To satisfy the homogeneous Neumann condition $\left.\frac{\partial C}{\partial x}\right|_{x=0} = 0$ automatically for all $t > 0$, the concentration profile must remain an even function of $x$ across the origin ($C(-x, t) = C(x, t)$). Differentiating an even function with respect to $x$ yields an odd derivative, ensuring $\frac{\partial C}{\partial x}(0, t) = 0$.

We construct this symmetry by reflecting the initial copper slab of thickness $h$ across the $x = 0$ plane into the fictitious negative half-space $x \in [-h, 0]$. The resulting extended initial condition $C_{\text{ext}}(x, 0)$ on $x \in (-\infty, +\infty)$ is an even top-hat slab of total width $2h$ centred at the origin:

Evaluation of the convolution integral

Substituting equation \eqref{eq:extended_init} into the convolution integral \eqref{eq:convolution}:

We introduce the dimensionless change of variable:

The integration limits transform accordingly:

• Lower limit ($x' = -h$): $u_1 = \frac{-h - x}{2 \sqrt{\tilde{D} t}} = - \frac{h + x}{2 \sqrt{\tilde{D} t}}$
• Upper limit ($x' = +h$): $u_2 = \frac{h - x}{2 \sqrt{\tilde{D} t}}$

The integral transforms to:

Splitting the integral into two parts around $u = 0$:

Since $\exp(-u^2)$ is symmetric ($\exp(-(-u)^2) = \exp(-u^2)$), the second integral satisfies:

Recalling the standard definition of the Gaussian error function:

We obtain the exact analytical closed-form solution for the copper concentration profile:

Correspondingly, the complementary nickel concentration distribution is given by:

4. Verification of physical boundary invariants and asymptotic regimes

Strict verification of the surface Neumann condition

Differentiating equation \eqref{eq:thin_film_solution} with respect to $x$ using Leibniz's rule:

Letting $z_1(x) \equiv \frac{h - x}{2\sqrt{\tilde{D}t}}$ and $z_2(x) \equiv \frac{h + x}{2\sqrt{\tilde{D}t}}$, we evaluate the spatial derivatives:

The spatial gradient of the copper concentration profile is therefore:

Evaluating this gradient at the free outer surface ($x = 0$):

The zero-flux Neumann boundary condition is thus satisfied identically for all $t > 0$, guaranteeing zero atomic loss across the outer surface.

Global conservation of solute mass

The total amount of copper in the physical half-space $x \in [0, +\infty)$ must equal the initial mass of the deposited film ($h$ per unit area):

The mathematical model guarantees rigorous conservation of solute matter over all time.

Asymptotic regime 1: early-stage diffusion ($2\sqrt{\tilde{D}t} \ll h$)

When the characteristic diffusion penetration distance $2\sqrt{\tilde{D}t}$ is small compared to the film thickness $h$, the diffusion front has not yet felt the presence of the free surface. Under this condition, for positions near the interface ($x \approx h$):

Substituting this approximation into equation \eqref{eq:thin_film_solution}:

This recovers precisely the classical complementary error function solution for a semi-infinite diffusion couple centred at $x = h$. In this regime, the surface concentration remains pinned at pure copper:

Asymptotic regime 2: long-stage film depletion ($2\sqrt{\tilde{D}t} \gg h$)

When annealing times are long such that $2\sqrt{\tilde{D}t} \gg h$, solute has spread deeply into the nickel substrate, and the finite reservoir of the film becomes exhausted. Using the Taylor expansion of the error function for small arguments ($\operatorname{erf}(z) \approx \frac{2}{\sqrt{\pi}} z$ as $z \to 0$):

The concentration profile converges to a decaying Gaussian distribution reflecting off the impermeable wall:

In this regime, the copper concentration at the free surface decays algebraically as:

The thin copper film is progressively dissolved and diluted into the bulk nickel bar.

5. Intrinsic diffusivities, Darken's frame of reference, and lattice drift

In a binary substitutional metallic solution, atomic transport occurs predominantly via the vacancy exchange mechanism. Atoms of copper and nickel do not exchange positions symmetrically; instead, each species possesses its own characteristic jumping frequency and mobility.

Arrhenius kinetics of intrinsic diffusion

The intrinsic diffusion coefficients of copper ($D_{\text{Cu}}$) and nickel ($D_{\text{Ni}}$) obey Arrhenius temperature dependencies:

The kinetic parameters for tracer and intrinsic diffusion in the $\text{fcc}$ $\text{Cu–Ni}$ system are established experimentally:

• Copper: $D_{0, \text{Cu}} = 7.8 \times 10^{-5}\text{ m}^2/\text{s}$, $Q_{\text{Cu}} = 211.0\text{ kJ/mol}$ ($2.19\text{ eV}$).
• Nickel: $D_{0, \text{Ni}} = 1.9 \times 10^{-4}\text{ m}^2/\text{s}$, $Q_{\text{Ni}} = 280.0\text{ kJ/mol}$ ($2.90\text{ eV}$).

Because nickel has a substantially higher melting point ($T_{\text{m, Ni}} = 1455\ ^\circ\text{C} = 1728\text{ K}$) and greater cohesive bond strength than copper ($T_{\text{m, Cu}} = 1085\ ^\circ\text{C} = 1358\text{ K}$), the activation enthalpy for vacancy formation and migration in nickel is considerably higher ($Q_{\text{Ni}} - Q_{\text{Cu}} \approx 69\text{ kJ/mol}$). Consequently, across all operational annealing temperatures ($700\ ^\circ\text{C}\text{--}1050\ ^\circ\text{C}$):

Copper atoms diffuse significantly faster than nickel atoms.

Darken's frame transformation

In the crystal lattice frame (denoted with a prime $'$), atomic fluxes are driven by local concentration gradients:

Because $D_{\text{Cu}} > D_{\text{Ni}}$ and $\frac{\partial C_{\text{Cu}}}{\partial x'} < 0$ across the film–substrate interface, the outward intrinsic flux of copper leaving the film ($|J_{\text{Cu}}'|$) exceeds the inward intrinsic flux of nickel entering the film ($|J_{\text{Ni}}'|$). To prevent catastrophic local volumetric dilation and maintain constant atomic density, the crystal lattice planes must undergo a convective drift velocity $v$ relative to the laboratory frame:

The total flux of copper in the fixed laboratory coordinate system is:

where $\tilde{D}$ is Darken's interdiffusivity:

Here, $\Phi = 1 + \frac{\partial \ln \gamma_{\text{Cu}}}{\partial \ln C_{\text{Cu}}}$ is the thermodynamic factor representing non-ideality of mixing. In the $\text{Cu–Ni}$ solid solution, $\Phi \approx 1$ to a good approximation, yielding the weighted kinematic average $\tilde{D} = C_{\text{Ni}} D_{\text{Cu}} + C_{\text{Cu}} D_{\text{Ni}}$.

6. Kirkendall marker kinematics at the film–substrate boundary

Consider insoluble, chemically inert markers (such as sub-micron tungsten particles or molybdenum wires) placed precisely at the film–substrate contact interface ($x = h$) during deposition at $t = 0$.

Marker velocity equation

Because inert markers do not participate in substitutional lattice diffusion, they remain affixed to their local atomic plane. Their instantaneous velocity in the laboratory coordinate frame $v_{\text{K}}(t)$ matches the local Darken lattice drift velocity \eqref{eq:darken_velocity} evaluated at the marker plane position $x_{\text{K}}(t)$:

Substituting the analytical spatial gradient \eqref{eq:first_derivative} into \eqref{eq:marker_vel_exact}:

Direction of marker migration

Since the markers originate at $x_{\text{K}}(0) = h$, we evaluate the sign of the velocity for $x_{\text{K}} \approx h$:

Therefore:

Because $D_{\text{Cu}} > D_{\text{Ni}}$, this establishes that:

Integrated marker trajectory

During the early and intermediate stages of interdiffusion ($t \le h^2 / (4\tilde{D})$), the term $\exp(-(h+x_{\text{K}})^2 / (4\tilde{D}t))$ is negligible, and equation \eqref{eq:marker_ode} simplifies to:

Integrating with the initial condition $x_{\text{K}}(0) = h$:

The inward marker displacement relative to the initial film boundary $\Delta x_{\text{K}}(t) \equiv x_{\text{K}}(t) - h$ scales with the square root of time:

7. Non-equilibrium vacancy continuity and dislocation relaxation kinetics

Classical Darken theory assumes that the crystal lattice maintains thermodynamic equilibrium vacancy concentrations everywhere ($C_{\text{v}} \equiv C_{\text{v}}^{\text{eq}}$) via infinitely fast vacancy creation and annihilation at dislocation climb jogs. However, in real crystalline thin films, dislocation densities are finite, leading to significant local departures from equilibrium.

Net vacancy flux in the lattice frame

Because substitutional diffusion is mediated entirely by vacancy exchanges, each forward atomic hop requires a reverse vacancy hop. Site conservation requires:

Substituting the intrinsic atomic fluxes:

Because $\frac{\partial C_{\text{Cu}}}{\partial x} < 0$ and $D_{\text{Cu}} > D_{\text{Ni}}$, $J_{\text{v}}' < 0$. Vacancies are pumped across the interface from the nickel substrate into the copper coating.

Vacancy continuity equation

The local time rate of change of vacancy concentration $C_{\text{v}}(x, t)$ is governed by the one-dimensional continuity equation:

where $\dot{\sigma}_{\text{sink}}$ represents the net rate of vacancy destruction at dislocation climb jogs and grain boundaries. According to standard chemical relaxation kinetics, the sink term is proportional to the local excess vacancy concentration $\Delta C_{\text{v}} \equiv C_{\text{v}} - C_{\text{v}}^{\text{eq}}$:

where $\tau_{\text{v}}$ is the vacancy relaxation lifetime. In physical metallurgy, $\tau_{\text{v}}$ is determined by the mean dislocation density $\rho_{\text{d}}$ and vacancy diffusivity $D_{\text{v}}$:

In well-annealed crystals with low dislocation density ($\rho_{\text{d}} \sim 10^{11}\text{ m}^{-2}$), $\tau_{\text{v}}$ is large, allowing excess vacancies to persist; in cold-worked or nanocrystalline films with dense dislocation networks ($\rho_{\text{d}} \sim 10^{15}\text{ m}^{-2}$), $\tau_{\text{v}}$ is small, forcing rapid relaxation toward equilibrium.

Quasi-steady-state approximation for excess vacancies

Because the vacancy migration energy ($E_{\text{m, v}} \approx 0.8\text{--}1.0\text{ eV}$) is far smaller than the total self-diffusion activation enthalpy ($Q \approx 2.2\text{--}2.9\text{ eV}$), vacancies redistribute much more rapidly than substitutional atoms. Over macroscopic interdiffusion time scales, the vacancy concentration field reaches a quasi-steady state ($\frac{\partial C_{\text{v}}}{\partial t} \approx 0$):

Rearranging yields the excess vacancy concentration:

Substituting the vacancy flux expression \eqref{eq:vacancy_flux}:

The non-equilibrium vacancy deviation is strictly proportional to the negative of the second spatial derivative (curvature) of the copper concentration profile.

8. Curvature analysis, vacancy supersaturation peaks, and interfacial delamination

Analytical evaluation of the second spatial derivative

To evaluate $\Delta C_{\text{v}}(x, t)$ explicitly, we differentiate the first derivative \eqref{eq:first_derivative} with respect to $x$:

Recalling $z_1 = \frac{h - x}{2\sqrt{\tilde{D}t}}$ and $z_2 = \frac{h + x}{2\sqrt{\tilde{D}t}}$:

Combining these expressions yields the exact analytical curvature of the thin-film profile:

Vacancy supersaturation ratio

Substituting \eqref{eq:second_derivative} into the master excess equation \eqref{eq:excess_vac_master}:

Normalising by the local thermal equilibrium concentration $C_{\text{v}}^{\text{eq}}$:

Spatial partitioning of supersaturation and subsaturation zones

The behaviour of equation \eqref{eq:vac_ratio_thin} across the physical domain is governed by the dominant term $z_1 = \frac{h - x}{2\sqrt{\tilde{D}t}}$:

• Inside the copper film ($0 \le x < h$): Here $x < h$, so $z_1 > 0$. The curvature $\frac{\partial^2 C_{\text{Cu}}}{\partial x^2} < 0$ is strictly negative. Consequently: \begin{equation} \Delta C_{\text{v}}(x, t) > 0 \implies \frac{C_{\text{v}}(x, t)}{C_{\text{v}}^{\text{eq}}} > 1 \quad \text{(Vacancy supersaturation)} \end{equation} Vacancies accumulate rapidly in the copper film near the original contact plane.
• At the inflection plane ($x \approx h$): At the interface, $z_1 = 0$, so $\frac{\partial^2 C_{\text{Cu}}}{\partial x^2} \approx 0$. The vacancy concentration crosses through its thermal equilibrium value ($C_{\text{v}} \approx C_{\text{v}}^{\text{eq}}$).
• Inside the nickel substrate ($x > h$): Here $x > h$, so $z_1 < 0$. The curvature $\frac{\partial^2 C_{\text{Cu}}}{\partial x^2} > 0$ is positive. Consequently: \begin{equation} \Delta C_{\text{v}}(x, t) < 0 \implies \frac{C_{\text{v}}(x, t)}{C_{\text{v}}^{\text{eq}}} < 1 \quad \text{(Vacancy subsaturation)} \end{equation} Vacancies are depleted below thermal equilibrium, driving dislocation climb and lattice plane insertion.

Thermodynamic driving force for void nucleation and coating delamination

The chemical potential of excess vacancies in the supersaturated copper film is:

When the supersaturation ratio $\frac{C_{\text{v}}}{C_{\text{v}}^{\text{eq}}}$ exceeds a critical barrier (typically $1.02\text{--}1.10$ in $\text{fcc}$ metals with low jog densities), excess vacancies undergo homogeneous or heterogeneous condensation into three-dimensional pores: Kirkendall voids (also termed Frenkel voids).

9. Comprehensive summary table of kinetic parameters and governing equations

The physical parameters, material constants, and mathematical equations governing interdiffusion and vacancy kinetics in the thin-film copper–nickel couple are compiled below: