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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Kinetic Processes in Materials |
3.21 Spring 2002 |
Samuel M. Allen and W. Craig Carter |
Department of Materials Science and Engineering |
Massachusetts Institute of Technology |
77 Massachusetts Ave. |
Cambridge, MA 02139 |
Problem Set 7: Due Wed. April 17, 2002, Before 5PM in 4-049
Exercise 7.1
Use the symmetric molar regular free energy of mixing
for a binary alloy of and at fixed pressure.
|
(1) |
and do the following.
- Determine the critical temperature in terms of
and
.
- Plot the equilibrium compositions from to .
- Plot
for
.
- Plot
versus
for
.
Exercise 7.2
Consider the stagnation problem associated with the disappearance of
a nearly cylindrical grain in a thin sheet with thickness .
Figure 7-2-i:
Illustration of disappearing grain in a thin sheet.
The circular boundary groove, radius , which forms on each surface
creates a pinning force resisting boundary motion.
|
If a groove develops as shown, the grain boundary can become ``pinned.''
- Show that, for a pinned boundary,
is
the equilibrium shape of the grain boundary if all interfaces are isotropic.
- Calculate the net force on the groove due to the grain when the
radius of the groove . Note that
has two solutions when
,
and
.
-
ceases to have any solutions when
.
What happens to the grain when deceases to about 3/4 ?
Exercise 7.3
Calculating the fastest growing and smallest unstable wavelengths
for a cylinder
which is evolving due to surface diffusion.
Start with a uniform cylinder and perturb with an infinitessimal pertubation
. Use the small slope approximation
for the surface diffusion equation:
Find an expression for
and maximize with respect to .
Exercise 7.4
Determine the fastest growing and smallest unstable wavelengths
(if they exist) for:
- a nonconserved order parameter, with homogeneous free energy density:
- a conserved order parameter, with homogeneous free energy density:
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W. Craig Carter
2002-04-08