Copper-nickel diffusion couple simulator

H. K. D. H. Bhadeshia and H. Yan, Queen Mary University of London

Isothermal solid-state interdiffusion across an initially flat contact interface between pure copper on the left and pure nickel on the right, showing exact equal-concentration crossing, dynamic vacancy supersaturation/subsaturation profiles, and Kirkendall marker displacement.

The Matano interface is the overall mass balance centre of the system. The full theoretical basis is available.

There is an image at the bottom of an actual Cu-Ni diffusion couple, with Kirkendall porosity in the copper after prolonged annealing.

Evaluates the divergence of the vacancy flux driven by the difference in intrinsic diffusivities: $J_{\text{v}} = -(J_{\text{Cu}} + J_{\text{Ni}}) = (D_{\text{Cu}} - D_{\text{Ni}}) \frac{\partial C_{\text{Cu}}}{\partial x}$
In the presence of dislocation sinks with relaxation time $\tau_{\text{v}}$, this produces: $\Delta C_{\text{v}} = -\tau_{\text{v}} \frac{\partial J_{\text{v}}}{\partial x} = -\tau_{\text{v}} (D_{\text{Cu}} - D_{\text{Ni}}) \frac{\partial^2 C_{\text{Cu}}}{\partial x^2}$
Copper side ($x < 0$): $\frac{\partial^2 C_{\text{Cu}}}{\partial x^2} < 0 \implies \Delta C_{\text{v}} > 0$. Generates a vacancy supersaturation peak ($C_{\text{v}} / C_{\text{v}}^{\text{eq}} > 1$), explaining Kirkendall void formation.
Nickel side ($x > 0$): $\frac{\partial^2 C_{\text{Cu}}}{\partial x^2} > 0 \implies \Delta C_{\text{v}} < 0$. Generates a vacancy subsaturation trough ($C_{\text{v}} / C_{\text{v}}^{\text{eq}} < 1$), driving dislocation climb.

Diffusion controls

Presets

Annealing duration 0.0 hours

Annealing temperature 1000 °C (1273 K)

Copper melting point is 1085 °C; nickel melting point is 1455 °C.

Dislocation sink relaxation factor (τ_v) 1.00 (well-annealed)

Higher values model low dislocation density where excess vacancies survive longer.

Display window width ± 120 µm

Display options

Copper concentration profile (C_Cu) Nickel concentration profile (C_Ni) Vacancy supersaturation ratio (C_v / C_v^eq) Matano interface position (x = 0) Kirkendall marker plane position (x_K)

Vacancy flux and Kirkendall voiding: Because copper diffuses faster than nickel (D_Cu > D_Ni), there is a net flux of vacancies from the nickel-rich side into the copper-rich side. With finite dislocation sink density, this creates a vacancy supersaturation peak (C_v / C_v^eq > 1) on the copper side (forming Frenkel/Kirkendall voids) and a vacancy subsaturation trough (C_v / C_v^eq < 1) on the nickel side.

Diffusion couple diagnostics

Parameter	Calculated value	Physical description
Annealing temperature	--	Isothermal furnace temperature
Annealing duration	--	Total elapsed diffusion time
Intrinsic copper diffusivity (D_Cu)	--	Mobility of copper atoms evaluated at interface
Intrinsic nickel diffusivity (D_Ni)	--	Mobility of nickel atoms evaluated at interface
Diffusivity ratio (D_Cu / D_Ni)	--	Confirms copper diffuses faster than nickel
Interdiffusion coefficient (D̃)	--	Darken average: D̃ = C_Ni D_Cu + C_Cu D_Ni
Characteristic diffusion distance (2√(D̃t))	--	Total penetration depth (zero at t = 0)
Kirkendall marker displacement (x_K)	--	Shift of inert markers into copper
Peak vacancy supersaturation (copper side)	--	Maximum C_v / C_v^eq ratio (void formation zone)
Peak vacancy subsaturation (nickel side)	--	Minimum C_v / C_v^eq ratio (lattice contraction zone)