Chapter 3 Problems
1.
Estimate the surface energy of {111}, {200} and {220} surface planes in an fcc crystal.
Express your answer in J/surface atom and in J/m2.
3.
If a two-dimensional rectangular crystal is bounded by sides of length l1 and l2 show by
differentiation that the equilibrium shape is given by
l1 / l2   2 /  1
where 1 and 2 are the energies of the sides l1 and l2 respectively. (The area of the crystal l1l2 is
constant.)
6.
(a) Suppose a recrystallized, dislocation-free grain is growing into a deformed matrix
containing a dislocation density of 1016 m-2 (i.e. 1016 m/m3). If the dislocations have an energy
of b2/4 J/m calculate the pulling force acting on the recrystallized grain boundry. (Assume a
shear modulus  = 1010 N/m2 and a Burger vector b = 0.28 nm.)
(b) If the recrystallized grains grow from spherically shaped nuclei, what is the diameter of the
smallest nucleus that can expand into the surrounding matrix? (Assume a grain boundary
energy of 0.5 J/m2.)
9.
When a precipitate is surrounded by a spherical interface of radius r it is subjected to a
pressure above that of the matrix by 2g/r. Consider a faceted precipitate with an equilibrium
shape that of a square plate with a thickness of 2x1 and width 2x2. If the free energies of the
broad faces and edges are respectively 1 and 2 show that the broad faces exert a pressure on
the precipitate (P) given by
P  2 2 / x2
(Hint: consider the total face acting on the periphery of the broad faces.) Show that the same
result can be obtained by considering the pressure exerted by one of the edge faces of the plate.
11. Fe-rich GP zones can form in dilute Al-Fe alloys. Given that the atomic radii are 1.43 Å for Al
and 1.26 Å for Fe, would you expect the zones to be spherical or disc shaped?
12. Mg can dissolve in Al to form a substitutional solid solution. Mg atoms are, however, bigger
than Al atoms and each Mg atom therefore distorts the surrounding Al lattice, i.e. a coherency
strain field effectively exists around each Mg atom. Using Equation Gs  4  2  V estimate
the misfit strain energy. Express the answer in kJ/mol and eV/atom. (The shear modulus of Al
= 25 GPa, the radius of an Al atom = 1.43 Å, the radius of a Mg atom = 1.60 Å.) What
assumptions are implicit in this calculation?
14. Show that the passage of a Shockley partial dislocation over every one of a given set of
close-packed planes in fcc crystals produces a twin of the original crystal.
15. If the ledges on the planar semicoherent interface in the following figure move with a
transverse velocity u what will be the overall velocity of the interface perpendicular to CD.
Assume an infinite array of identical ledges of height (BC) = h and spacing (CD) = l.
18. By
using
a
similar
approach
to
the
derivation
of
Equation
  A2 n1 1Vm2 / N a RT  exp  G a / RT  G / Vm  for a high-angle grain boundary, show that
the net flux of B atoms across the / interface in the following figure is given by
J 
i
B
19. Derive Equation  Bi 
A n 
RT
  a  i
 B
exp  
RT


RT
 X i  X e  for an ideal or dilute solution.
Xe