Chapter 5: Problem 4
Briefly explain the concept of steady state as it applies to diffusion.
Chapter 5: Problem 4
Briefly explain the concept of steady state as it applies to diffusion.
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Get started for freeAt approximately what temperature would a specimen of \(\gamma\)-iron have to be carburized for \(2 \mathrm{~h}\) to produce the same diffusion result as at \(900^{\circ} \mathrm{C}\) for \(15 \mathrm{~h}\) ?
(a) Briefly explain the concept of a driving force. (b) What is the driving force for steady-state diffusion?
An FCC iron-carbon alloy initially containing \(0.35 \mathrm{wt} \% \mathrm{C}\) is exposed to an oxygen-rich and virtually carbon-free atmosphere at \(1400 \mathrm{~K}\) (1127 \(\left.^{\circ} \mathrm{C}\right)\). Under these circumstances the carbon diffuses from the alloy and reacts at the surface, with the oxygen in the atmosphere; that is, the carbon concentration at the surface position is maintained essentially at \(0 \mathrm{wt} \%\) C. (This process of carbon depletion is termed decarburization.) At what position will the carbon concentration be \(0.15 \mathrm{wt} \%\) after a 10 -h treatment? The value of \(D\) at \(1400 \mathrm{~K}\) is \(6.9 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\).
Consider a diffusion couple composed of two semi-infinite solids of the same metal, and that each side of the diffusion couple has a different concentration of the same elemental impurity; furthermore, assume each impurity level is constant throughout its side of the diffusion couple. For this situation, the solution to Fick's second law (assuming that the diffusion coefficient for the impurity is independent of concentration) is as follows: $$ C_{x}=\left(\frac{C_{1}+C_{2}}{2}\right)-\left(\frac{C_{1}-C_{2}}{2}\right) \operatorname{erf}\left(\frac{x}{2 \sqrt{D t}}\right) $$ In this expression, when the \(x=0\) position is taken as the initial diffusion couple interface, then \(C_{1}\) is the impurity concentration for \(x<0\); likewise, \(C_{2}\) is the impurity content for \(x>0\). A diffusion couple composed of two silver-gold alloys is formed; these alloys have compositions of \(98 \mathrm{wt} \% \mathrm{Ag}-2 \mathrm{wt} \% \mathrm{Au}\) and \(95 \mathrm{wt} \% \mathrm{Ag}-5 \mathrm{wt} \%\) Au. Determine the time this diffusion couple must be heated at \(750^{\circ} \mathrm{C}\) (1023 K) in order for the composition to be \(2.5 \mathrm{wt} \% \mathrm{Au}\) at the \(50 \mu \mathrm{m}\) position into the 2 wt \(\%\) Au side of the diffusion couple. Preexponential and activation energy values for Au diffusion in \(\mathrm{Ag}\) are \(8.5 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\) and \(202,100 \mathrm{~J} / \mathrm{mol}\), respectively.
Aluminum atoms are to be diffused into a silicon wafer using both predeposition and drive-in heat treatments; the background concentration of \(\mathrm{Al}\) in this silicon material is known to be \(3 \times 10^{19}\) atoms/m \(^{3}\). The drive-in diffusion treatment is to be carried out at \(1050^{\circ} \mathrm{C}\) for a period of \(4.0 \mathrm{~h}\), which gives a junction depth \(x_{j}\) of \(3.0 \mu \mathrm{m}\). Compute the predeposition diffusion time at \(950^{\circ} \mathrm{C}\) if the surface concentration is maintained at a constant level of \(2 \times 10^{25}\) atoms \(/ \mathrm{m}^{3}\). For the diffusion of \(\mathrm{Al}\) in \(\mathrm{Si}\), values of \(Q_{d}\) and \(D_{0}\) are \(3.41\). \(\mathrm{eV} /\) atom and \(1.38 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}\), respectively.
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