sec-1) so that the diffusion of atoms can keep pace with the employed strain rate.
• The grains should easily slide and roll over one another under stress.
Examples for superplastic alloys are Al with 6 mass% Cu and 0.5 mass% Zr (Trade name SUPRAL), Bi-Sn, Al-Cu eutectic, Ti-6%Al-4%V, and Zn-23%Al.
Applications of superplasticity include the plastic deformation of certain consumer parts at modest stresses and by using inex-pensive die materials, making this technique suitable for small production runs.
It should be added that ultrafine-grained ceramics can also be superplastically deformed by the Nabarro–Herring mechanism.
Finally, as we will discuss in Chapter 15, glass may be drawn out rapidly in an appropriate temperature range into very long and fine fibers and thus, show likewise superplasticity.
Finally, a few words should be added about the diffusion of metal ions driven by strong electric fields. This mechanism is called elec-tromigration and is of great concern to the electronics industry.
It involves the forced motion of, for example, aluminum ions in aluminum thin film stripes that interconnect the individual tran-sistors in microelectronic circuits. As a consequence of the uni-directional motion of the Al ions from the cathode to the anode, voids near the cathode appear which lead to failure of the device.
The preferred paths for the electromigrating aluminum ions are the grain boundaries. This is understandable when it is recog-nized that the activation energy for diffusion in grain boundaries is only one-half of that for volume diffusion. Numerous attempts have been undertaken in the past to alleviate electromigration in thin films. Among them are the inoculation of grain boundaries by foreign atoms (such as copper) to block some of the grain boundary diffusion, orienting the grain boundaries perpendicu-lar to the electric field (creating a so-called bamboo structure), or utilizing better suited metallization materials, such as gold or cop-per which have a higher activation energy than aluminum.
The previous chapters have taught us that the arrangement of atoms or, better, their lack of regular arrangement, together with the motion of atoms and their mutual interactions, play a key role in understanding some fundamental mechanical properties of ma-terials. The motion of atoms is enhanced by thermal activation.
This thermal activation is often governed by an exponential law
6.3.9
Electro-migration
(c) During quenching, 70% of these vacancies anneal out; the rest are quenched-in and are present at 300 K. What is their concentration?
(d) What happens to these vacancies?
Do they anneal out at 300 K? De-termine the diffusion constant of the vacancies assuming Dvo 10 cm2/s. (Hint: Note that the diffusion constant for vacancies, Dv, equals Dvo exp[(Q Ef)/kBT].) Is there an observable vacancy diffusion?
(e) What happens to the atomic diffu-sion constant at 300 K during an-nealing out of these vacancies?
6.5. Calculate the approximate time that is needed for diffusing carbon into steel to a depth of 1 mm when the diffusion coefficient at the temperature of heat treatment is 2 107cm2/s.
6.6. Problem 6.5 above and the rule-of-thumb equation (6.13) do not specify the concentration Cx at the point X.
The following exercise alleviates this shortcoming for some specific cases:
Derive an equation similar to (6.13) from (6.11) by assuming (a) Cx C0
(“far field value”), and (b) Cx (Ci C0)/2 (“effective diffusion depth”
which is the “average value between surface and final concentrations”). (c) For what value of Cxdoes one obtain (6.13)?
Problems
6.1. Calculate Cxfor x ", x 0, and x " using Eq. (6.11) and compare the results with Figure 6.6.
6.2. Calculate the concentrations Cx for t씮 " and t 씮 0 using Eq. (6.11) and compare the results with Figure 6.6.
(Hint: For t씮 0, use positive x as well as negative x values.)
6.3. Calculate the ratio between the diffusion rates of grain boundary diffusion and volume diffusion assuming an activation energy for grain boundary diffusion to be one-half of that for volume diffusion.
Take T 500°C; Qv 2 eV; (f0)v 6 10151/s and (f0)g 1.5 10111/s.
6.4. In an aluminum alloy, the atomic dif-fusion constant D D0 exp (Q/kBT) is given by D0 10 cm2/s and Q 1.3 eV. The vacancies in this material are thought to have a formation energy, Ef, of 0.7 eV. For an observable mi-croscopic atomic diffusion, D should be larger than 1018cm2/s.
(a) Give the atomic diffusion constant for T 300 K (room temperature).
Is its value sufficient for observ-able diffusion?
(b) The alloy is homogenized at 750 K followed by a quench into water.
What is the vacancy concentration at this temperature? (Use the room temperature atomic mass and den-sity for Al.)
Suggestions for Further Study
R.T. DeHoff, Thermodynamics in Materials Science, McGraw-Hill, New York (1993).
that contains the absolute temperature and an activation energy as variables, as exemplified by a number of Arrhenius-type equations.
R. Haase, Thermodynamics of Irreversible Processes, Dover, New York (1990).
R.E. Hummel, “Electromigration and Related Failure Mecha-nisms in Integrated Circuit Interconnects,” International Ma-terials Reviews, 39 (1994), p. 97.
H. Mehrer, Diffusion in Metals and Binary Alloys, in: Diffusion in Condensed Matter, P. Heitjans and J. Kärger Eds., Springer-Verlag, Heidelberg, New York (2003).
P. Shewmon, Diffusion in Solids, McGraw-Hill, New York (1963).
7
Historians claim that the Iron Age began between 1500 and 1000
B.C. (at least in some parts of the world). This does not mean that iron was unknown to man before that time; quite the contrary is the case. Meteoric iron (which has a large nickel content) must have been used by prehistoric people as early as 4000 B.C. They made tools and weapons from it by shaping and hammering. It is thus quite understandable that in some ancient languages the word for iron meant “metal from the sky”. Naturally, the supply of meteoric iron was limited. Thus, stone, copper, and bronze were the materials of choice at least until the second millennium
B.C. There were, however, some important uses for iron ores dur-ing the Bronze Age and also durdur-ing the Chalcolithic period. As explained already in Chapter 1, copper needs a fluxing agent for the smelting process when using malachite. For this, iron oxide was utilized, which was known to react during smelting with the unwanted sand particles that are part of malachite. Eventually, a slag was formed which could be easily separated from the cop-per after the melt had cooled down.
It has been frequently debated and asked by scholars in which way early man might have produced iron, utilizing terrestrial sources in particular, since the melting point of iron is 1538°C.
This temperature was essentially unachievable during that pe-riod, at least in the western (or middle eastern) part of the world.
The answer can probably be found by considering the above-mentioned slag, large amounts of which have been found in ar-eas at which major copper smelting operations were conducted.
This slag was observed to contain some reduced iron, but in a porous condition, which is today known by the name of sponge iron or bloom. When bloom is repeatedly hammered at high temperatures, the slag can be eventually removed and the iron is compacted. In this way nearly pure iron is obtained. The end product is known among today’s metallurgists by the name of wrought iron. Therefore, it can be reasonably assumed that iron