## Q. 5.4

Diffusion of Nickel in Magnesium Oxide
A 0.05 cm layer of magnesium oxide (MgO) is deposited between layers of nickel and tantalum to provide a diffusion barrier that prevents reactions between the two metals (Figure 5-11). At 1400 °C, nickel ions diffuse through the MgO ceramic to the tantalum. Determine the number of nickel ions that pass through the MgO per second. At 1400 °C, the diffusion coefficient of nickel ions in MgO is 9 × $10^{-12}$ cm²/s, and the lattice parameter of nickel at 1400 °C is 3.6 × $10^{-8}$ cm.

## Verified Solution

The composition of nickel at the Ni/MgO interface is 100% Ni, or

$c_{Ni/MgO}=\frac{4 \ Ni \ atoms}{(3.6 \times 10^{-8} \ cm)^3}=8.573 \times 10^{22} \ \frac{Ni \ atoms}{cm^3}$

The composition of nickel at the Ta/MgO interface is 0% Ni. Thus, the concentration gradient is

$\frac{\Delta c}{\Delta x}=\frac{0 \ – \ 8.573 \times 10^{22} \ \frac{Ni \ atoms}{cm^3}}{0.05 \ cm}=-1.715 \times 10^{24} \ \frac{Ni \ atoms}{cm^3 \cdot cm}$

The flux of nickel atoms through the MgO layer is

$J=-D\frac{\Delta c}{\Delta x}=-(9 \times 10^{-12} \ cm^2/s)(-1.715 \times 10^{24} \ \frac{Ni \ atoms}{cm^3 \cdot cm} )$
$J=1.543 \times 10^{13} \ \frac{Ni \ atoms}{cm^2 \cdot s}$

The total number of nickel atoms crossing the 2 cm × 2 cm interface per second is
Total Ni atoms per second = (J)(Area) = (1.543 × $10^{13} \frac{Ni \ atoms}{cm^2 \cdot s}$)(2 cm)(2 cm)
= 6.17 × $10^{13}$ Ni atoms/s

Although this may appear to be very rapid, in one second, the volume of nickel atoms removed from the Ni/MgO interface is

$\frac{6.17 × 10^{13} \frac{Ni \ atoms}{s}}{8.573 × 10^{22} \frac{Ni \ atoms}{cm^3}}= 7.2 × 10^{-10} \frac{cm^3}{s}$

The thickness by which the nickel layer is reduced each second is

$\frac{7.2 × 10^{-10} \frac{cm^3}{s}}{4 \ cm^2}= 1.8 × 10^{-10} \frac{cm}{s}$

For one micrometer ($10^{-4}$ cm) of nickel to be removed, the treatment requires

$\frac{10^{-4} \ cm}{1.8 × 10^{-10} \frac{cm}{s}}=556,000 \ s=154 \ h$