Chapter 5: Diffusion
HW: 2, 7, 12, 19, 22, 26, 30, D3 Due Wensday 20/10/2010
Atomic vibrations
 The atoms of a material become static only at 0oK.
 Under that condition the atoms settle down to their lowest
energy positions among their neighbors. As temp is
increased, the increased energy permits the atoms to vibrate
into greater and shorter inter-atomic distances.
 This produces a thermal expansion because the mean interatomic distance is increased.
 There is a spectrum of energy among the atoms which
extends from near 0 to very high values.
 The majority of the atoms have energies somewhere near the
mean value. Over a period of time, a specific atom will
experience a range of energies which extends from near 0 to
very high values. However, the energy of the atom is near the
mean value most of the time.
 Diffusion: probability of an atom having enough energy to
break its bond and jump to new location.
 Diffusion couple
Atomic Rearrangements
 As temp is increased and the atoms vibrate more
energetically, a small fraction of the atoms will relocate
themselves in the lattice.
 The fraction depends on: temp and how tightly the atoms are
bonded in position.
 For an atom to make such a move, two conditions must be
met:
o There must be an empty adjacent site




o The atom must have enough energy to break bonds with
its neighbor atoms and then cause some lattice
distorsion during the displacement
Activation energy: The energy req for an atom to change
position (E or Q).
A carbon atom is small and can sit interstitially among a
number of FCC Fe atoms. If it has enough energy, it can
squeeze between the Fe atoms to the next interstice when it
vibrates in that direction.
When all the atoms are the same size, or nearly so, the
vacancy mechanism becomes predominant
In most metal alloys, interstitial diffusion occurs much more
rapidly than vacancy diffusion
Steady state Diffusion
 When an atom moves into a vacancy, a new hole is opened.
In turn, this may receive an atom from any of the neighboring
sites. As a result, a vacancy makes a random walk through a
crystal.
 The same random walk mechanism may be described for a C
atom moving among Fe atoms from interstice to interstice.
 Diffusion is a time dependent process
 We encounter concentration gradients. As an example,
assume there is a C atom per 20 unit cells of FCC Fe at point
(1), and only 1 C atom per 30 unit cells at point (2), which is
one mm away.
 Since there are random movements of C atoms at each point,
we will find a net flux of C atoms from (1) to (2). This net
flow of atoms is called diffusion.
 The flux J of atoms (atoms/m2.s) is prop to the concentration
gradient, (Fick’s 1st law)
J  D
dC
dx
 D = diffusion coefficient, vary with the nature of the solute
atoms, nature of the solid structure, and change in temp.
(m2/s)
 Note the following:
1. Higher temp provide higher D, because the atoms have
higher thermal energies and therefore greater probability
of being activated over the energy barrier between atoms.
2. C has a higher D in Fe than does Ni in Fe because the C
atom is a small one.
3. Cu diffuses more readily in Al than in Cu because the CuCu bonds are stronger than the Al-Al bonds.
4. Atoms have higher D in BCC Fe than in FCC Fe because
the former has a lower atomic PF. FCC structure has
larger interstitial holes, however, the passageways
between the holes are smaller than in the BCC structure.
5. The diffusion proceeds more rapidly along the grain
boundaries.
 Example 5.1
Non-Steady State Diffusion
 Diffusion flux and concentration gradient at some particular
point in a solid vary aith time (F5.5)
 Fick’s 2nd Law:
C
 2C
D
t
x 2
 Boundary conditions (Figure 5.6):
1. For t = 0, C = Co @ 0 ≤ x ≤ ∞
2. For t > 0, C = Cs (constant surface concentration) @ x =
0, C = Co at x = ∞
C x C o
 x 
 1  erf 

C s Co
 2 Dt 
 Achieve some specific concentration of solute C1 in an alloy,
leads:
x2
 constant

Dt
 Example 5.2
 Example 5.3
Diffusion coefficients vs. temp
 Magnitude of diffusion coef D is indicative of the rate at
which atoms diffuse
 with diffusion, the activation energy for atom movements
corresponds to the energy E of Boltzmann’s eq:
D  Doe Q / RT
ln(D )  ln(Do ) 
Q
RT
log(D )  log( Do ) 
Q 1
 
2.3R  T 
 Do = temp indep preexponential, m2/s
 Q = activation energy for diffusion (J/mol, cal/mol, eV/atom)
 R = gas constant = 8.31 J/mol.K = 1.987 cal/mol.K = 8.62
eV/atom.K
 Example 5.4
 Example 5-5
 Design Example