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MATLS 3T04 “Phase Transformations”
Test 2
Problem 1 (40 points)
Imagine that eutectic solidification
results in cylinders (rods) of
β
phase
surrounded by
α
phase. An arrangement of the
β
cylinders is shown in
Liquid
→α+β
Figure 1.
λ
0
90
β
α
Figure 1
If
f
is a known volume fraction of
β
phase, then what is the minimal
spacing
min
for a given
supercooling
?
T
Δ
Hint:
a
b
c
Problem 2 (40 points)
Consider a binary solid solution having a uniform composition. The mole fraction of the 2
nd
component in this solution is equal to
0
x
. Imagine that a very small
particle formed within this
solution. The particle has the same crystal structure as the solution (the particle, therefore, is a
cluster), but the mole fraction of the 2
nd
component in it is equal to
0
x
x
′ ≠
. Show that the change
of the Gibbs energy associated with the formation of one mole of such clusters is equal to
()
0
2
22
0
xx
Gx x
x x
=
′
∂∂
−
where
is a concentration dependency of the molar Gibbs energy of the solid solution.
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 Fall '06
 Malakov
 Thermodynamics, Gibbs, Condensed matter physics, Phase transition

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