Engineering & Science
Darken's Equations: Alloy Interdiffusion Coefficient & Kirkendall Marker Velocity
Calculate the chemical interdiffusion coefficient D_tilde = (1 - X_A) * D_A + X_A * D_B and Kirkendall inert-marker velocity v_K = (D_A - D_B) * (dX_A / dx) in solid-state diffusion couples and solder joints.
Why Cu-Sn solder joints and Cu-Zn brass form Kirkendall voids!
In 1947, Ernest Kirkendall and Alice Smigelskas placed inert molybdenum wires at the interface between alpha-brass (70% Cu, 30% Zn) and pure copper at 785 °C—and stunned metallurgists by proving that the molybdenum wires physically migrated inward because zinc atoms diffuse faster out of brass via vacancies than copper atoms diffuse in! That net vacancy counter-flux condenses into "Kirkendall voids" that crack microchip Cu-Sn BGA solder bumps!
Lawrence Darken (1948) derived two landmark equations governing binary substitutional diffusion (ignoring thermodynamic non-ideality factor Phi = 1): Darken's Second Equation gives the effective chemical interdiffusion coefficient D_tilde = (1 - X_A) * D_A + X_A * D_B, and Darken's First Equation gives the lattice Kirkendall marker velocity v_K = (D_A - D_B) * grad_X_A! In 70Cu-30Zn brass (X_Zn = 0.30, D_Zn = 5.1e-13 m^2/s, D_Cu = 1.8e-13 m^2/s, composition gradient 250 m^-1), D_tilde = 4.11 x 10^-13 m^2/s and markers drift at 0.297 um/hour!
| Diffusion Couple State (785 °C Brass) | Mole Fraction X_Zn | Interdiffusivity D_tilde (10^-13 m^2/s) | Kirkendall Drift v_K (um/hr) |
|---|---|---|---|
| 30 at.% Zn Alpha-Brass (grad = 250 m^-1) | X_Zn = 0.30 | 4.110 (0.70*5.1 + 0.30*1.8) | 0.2970 um/hr |
| Dilute 10 at.% Zn Brass | X_Zn = 0.10 | 4.770 (Approaches fast D_Zn!) | 0.2970 um/hr |
| Equal Tracer Diffusivities (D_A = D_B = 1.8) | X_Zn = 0.30 | 1.800 | 0.0000 um/hr (Zero Kirkendall shift!) |
Darken Chemical Interdiffusion Coefficient D_tilde (10^-13 m^2/s)
Evaluates (1 - X_A) * D_A + X_A * D_B for tracer diffusivities B3 and B4 entered in units of 10^-13 m^2/s.
=IF(OR(B2 < 0, B2 > 1, B3 <= 0, B4 <= 0), "Invalid Diffusion Inputs", ROUND(((1 - B2) * B3) + (B2 * B4), 3))Kirkendall Inert-Marker Drift Velocity v_K (Micrometers per Hour, um/hr)
Converts (D_A - D_B) * 10^-13 m^2/s * grad_X_A (m^-1) into um/hr (* 10^6 um/m * 3,600 s/hr = * 3.6e-4).
=IF(OR(B3 <= 0, B4 <= 0), "Invalid Diffusion Inputs", ROUND((B3 - B4) * B5 * 0.00036, 4))Try the example
Use the original values or change them to test your understanding.