Homework: MSE 335: Kinetics
Due date Tuesday March 23, 2017
A. As you probably know Pt is the catalyst of choice for many applications such
as acid fuel cells. The Pt is always used in nanoparticle form and the shape of
the nanoparticle is typically a cuboctahedron. In geometry, a cuboctahedron is
a polyhedron with eight triangular faces and six square faces.
All edges of a cuboctahedron have an equal length, L. On the left is a picture of
a cuboctahedron and on the right it has been “unfolded” showing the 6 square
faces and the 8 equilateral triangles. The area and volume of the cuboctahedron
is
(
)
A = 6 + 2 3 L2
V=
5 2 3
L
3
Consider the homogeneous nucleation of solid Pt from the liquid melt. When Pt
nanoparticles take a cubooctahedron shape the triangular faces have a (111)
orientation and the square faces have a (001) orientation.
Use the following data:
Heat of fusion; Lf = 2.52 x 109 Jm-­‐3. Tmp = 2041 K. kBoltzmann = 1.38 x 10-­‐23 J K-­‐1. The solid/liquid interfacial free energy for a (111) surface; γ(111)/liquid = 0.177 Jm-­‐2. The solid/liquid interfacial free energy for a (001) surface; γ(001)/liquid = 0.220 Jm-­‐2. (1) Assuming that the nuclei take the form of cuboctahedron particles,
calculate the size of the critical nucleus, L*, for an undercooling, ΔT = 200
K.
(2) Calculate the activation free energy, ΔG(L*) for nucleation.
B. Consider the homogeneous nucleation of solid silver in the form of a spherical
particle from a liquid silver melt. Determine the following:
(1.) Δg v as a function of the degree of undercooling ΔT ( = Tmp − T ) over a ΔT
range of 1-1000 K in 1 degree intervals. *Note Δg v is the free energy change
per atom in going from liquid to solid. Plot these results
(2.) the radius of the critical nucleus as a function of ΔT over the same range
in temperature as that in (1.) above. Plot these results
*Hint; Since Δg v is given on a per atom basis you need to be sure to convert this to a
free energy per (atomic) volume.
Parameter data base
parameter
Lf
1.88 x 10-20 J/atom
γ s/l
0.112 Jm-2
Tmp
4.08 x 10-10 m
1235 K
Io
1033 cm-3 s-1
lattice parameter
C. Consider now the heterogeneous nucleation of solid silver from a liquid silver
melt for which the nucleation process is catalyzed by contact of the liquid silver
with a graphite surface (the crucible or mold). The contact angle is 30º. Determine
the following:
(1.) Δg v as a function of the degree of undercooling ΔT ( = Tmp − T ) over a ΔT
range of 1-1000 K in 1 degree intervals. Note Δg v is the free energy
change per atom in going from liquid to solid. Plot these results.
(2.) the radius of the critical nucleus as a function of ΔT over the same range
in temperature as that in (1.) above. Plot these results.