Answer to Question #260475 in Material Science Engineering for IST

Question

A diffusion couple is made from nickel and iron. After 30 hours of heat treatment at 1400 K, 

the concentration of nickel is 3.2 wt % at a distance of 1.5 mm from interface. If an identical 

diffusion couple is heated at 1200 K for 30 hours, at what depth from the interface will the 

concentration of nickel be 3.2 wt %? The pre-exponential factor (Do) and activation energy for 

diffusion is 1.8 x 10-5 m2

/s and 152 kJ/mol respectively, and the surface concentration of nickel

is 100 wt%. (Note that this is a non-steady state diffusion process).

Answer 1

We will use the following solution for Fick's second law for a diffusion couple for this problem:

C_x = C₂ + (C₁ - C₂)/2 x [1 - erf(x/2√Dt)]

C₁ is impurity at x < 0 and C₂ is impurity content for x > 0. Let us consider Ni as impurity and it is kept at x < 0. Then C₁ = 100 wt % of Ni, C₂ = 0.

When concentration is constant

C₂ + (C₁ - C₂)/2 x [1 - erf(x₁/2√D₁t₁)] - C₂ + (C₁ - C₂)/2 x [1 - erf(x₂/2√D₂t₂)]

x₁/√D₁t₁ = x₂/√D₂t₂

x₁ = 1.5 mm, x₂ = x mm, t₁ = 30 h, t₂ = 30 h.

2.25 / D₁ = x² / D₂

D₂ = x²D₁/2.25

D₀ x exp(-Q_d/RT₂) = x²D₀/2.25 x exp(-Q_d/RT₁)

-Q_d/RT₂ = ln(x²) - ln 2.25 - Q_d/RT₁

- 152 000 J/mol / (8.31 J/mol K х 1200 K) =

ln(x²) - 0.81 - 152 000 J/mol / (8.31 J/mol K х 1400 K)

x = 2.10 mm