Question 5 of 7: Carbon Diffusion Flux in FCC Iron; Creep and Diffusion

Question 5: Carbon Diffusion Flux in FCC Iron; Creep and Diffusion (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Surface concentration: 1 C atom per 20 FCC unit cells; at $x=1$ mm depth: 1 C atom per 30 FCC unit cells; $D=3\times10^{-11}$ m$^2$/s at $1000^\circ$C; $a_0=0.365$ nm (FCC $\gamma$-iron unit cell edge).

Find. (a) Carbon atoms diffusing through one unit cell per minute. (b) Why creep is closely tied to diffusion. (c) General characteristics of creep-resistant materials.

Approach

(a) is a direct application of Fick’s first law, $J=-D\,dc/dx$, once the two given "atoms per unit cell" concentrations are converted to volumetric concentration (atoms/m$^3$) via the unit-cell volume $a_0^3$, and the resulting flux (atoms per unit area per second) is multiplied by the cross-sectional area of one unit cell face, $a_0^2$, to get atoms per unit cell per unit time.

1. Convert both concentrations to atoms/m$^3$. Unit cell volume $V_{cell}=a_0^3=(0.365\times10^{-9})^3=4.863\times10^{-29}$ m$^3$: $$c_1=\frac{1/20}{V_{cell}}=1.028\times10^{27}\ \text{atoms/m}^3\ \text{(surface)}$$ $$c_2=\frac{1/30}{V_{cell}}=6.855\times10^{26}\ \text{atoms/m}^3\ \text{(at 1 mm)}$$
2. Apply Fick’s first law over the 1 mm gap. $$J=D\left|\frac{\Delta c}{\Delta x}\right|=(3\times10^{-11})\times \frac{(1.028-0.6855)\times10^{27}}{1\times10^{-3}}\approx1.028\times10^{19}\ \text{atoms}/(\text{m}^2\cdot\text{s})$$
3. Scale the flux to one unit-cell cross-section. Area of one unit-cell face, $a_0^2=(0.365\times10^{-9})^2=1.332\times10^{-19}$ m$^2$: $$\dot{N}=J\times a_0^2\approx1.028\times10^{19}\times1.332\times10^{-19}\approx1.37\ \text{atoms/s}$$ $$\boxed{\dot{N}_{min}=1.37\times60\approx82\ \text{carbon atoms per unit cell per minute}}$$
4. (b) Why creep tracks diffusion. Creep — slow, time-dependent plastic strain under sustained stress, significant above $\approx0.4\,T_m$ — is rate-controlled at these temperatures by mechanisms that are themselves thermally-activated atomic transport: dislocation climb (a dislocation blocked by an obstacle can only bypass it by climbing out of its slip plane, which requires vacancies to diffuse to/from the dislocation core) and, at higher temperature/lower stress, diffusional (Nabarro–Herring/Coble) creep, where the strain itself is produced directly by stress-biased vacancy (or grain-boundary) diffusion from grain faces in tension to faces in compression. Both routes inherit an Arrhenius temperature dependence, and the measured creep activation energy is typically close to the activation energy for self-diffusion in the same material — direct evidence that atomic diffusion is the rate-limiting step.
5. (c) Characteristics of creep-resistant materials. High melting point (creep onset scales with $T/T_m$, so a higher $T_m$ pushes the onset to a higher service temperature); high elastic modulus; large grain size, or single-crystal/directionally-solidified microstructures that eliminate transverse grain boundaries (suppressing grain-boundary sliding and Coble creep, the dominant mechanism turbine-blade superalloys are designed against); stable, coarse precipitates or dispersoids that pin dislocations and resist coarsening/dissolution at service temperature; solid-solution strengthening by slow-diffusing, large-misfit solute atoms (which also slow the climb process itself); and good oxidation/corrosion resistance so the load-bearing section is not progressively thinned in service.