1. CHAPTER 5: DIFFUSION IN SOLIDS
ISSUES TO ADDRESS...
• How does diffusion occur?
• Why is it an important part of processing?
• How can the rate of diffusion be predicted for
some simple cases?
• How does diffusion depend on structure
and temperature? 1

2. Chapter 5: DIFFUSION Why study Diffusion?
Heat-treated to improve their properties.
Heat-treatment almost always involve atomic diffusion.
desired results depends on diffusion rate
Heat-treatment temperature, time, and/or rate of heating/cooling can be predicted by the mathematics of diffusion
Steel gear  Case hardened to improve hardness and resistance to fatigue  diffusing excess carbon or nitrogen into outer surface layer.

3. 5.1 Introduction
Diffusion: The phenomenon of material transport by atomic motion.
Many reactions and processes that are important in the material treatment rely on the mass transfer:
Either with a specific solid ( at microscopic level )
Or from a liquid, a gas, or another solid phase.
This chapter covers:
Atomic mechanism
Mathematics of diffusion
Influence of temperature and diffusing species of the diffusion rate

4. 5.1 Introduction (Contd.) Phenomenon of diffusion
Explained using diffusion couple, formed by joining bars of two different materials having intimate contact
Copper and Nickel diffusion couple
Figure 5.1 shows as formed
Atom locations and concentration
Heated for an extended period at an elevated temperature ( but below melting temperature of both ) and cooled to room temperature.

5. DIFFUSION • Interdiffusion: In an alloy, atoms tend to migrate
from regions of large concentration.
After some time
Initially 3

6. 5.1 Introduction (Contd.) Chemical analysis reveals Alloy region
Variation of concentration
Atoms migrated or diffused into one another
Interdiffusion or impurity diffusion
Atoms of one metal diffuses into another
Net drift of atoms from high to lower concentration

7. DIFFUSION • Self-diffusion: In an elemental solid, atoms
also migrate. Self-diffusion
All atoms exchanging positions are of same type
No compositional Diffusion in pure metal
changes
Label some atoms
After some time 4

8. 5.2 Diffusion Mechanism
Atoms in solids are in constant motion rapidly changing positions.
Diffusion is just the stepwise migration of atoms from a lattice site to other lattice site.
Two conditions for movement:
1. There must be an empty adjacent site
2. Atom must have sufficient energy to break bonds with neighbor atoms
Atomic vibration (Section 4.7):
Every atom is vibrating very rapidly about its lattice position within the crystal
At any instant, not all vibrate with same frequency and amplitude.
Not all atoms have same energy
Same atom may have different level of energy at different time
Energy increases with temperature

9. 5.2 Diffusion Mechanism (Contd.)
Several different models for atomic motion
Two dominate for metallic diffusion
VACANCY DIFFUSION
Involves interchange of an atom from a normal lattice position to an adjacent vacant lattice site or vacancy
Necessitates presence of vacancies
Diffusing atoms and vacancies exchange positions  they move in opposite directions
Both self- and inter-diffusion occurs by this mechanism

10. DIFFUSION MECHANISMS Vacancy Diffusion:
• applies to substitutional impurities
• atoms exchange with vacancies
• rate depends on:
--number of vacancies
--activation energy to exchange. 5

11. 5.2 Diffusion Mechanism (Contd.)

12. 5.2 Diffusion Mechanism (Contd.)
INTERSTITIAL DIFFUSION
Atoms migrate from an interstitial position to a neighboring one that is empty
Found for interdiffusion of impuries such as hydrogen, carbon, nitrogen, and oxygen  atoms small enough to fit into interstitial positions.
Host or substitutional impurity atoms rarely have insterstitial diffusion
Interstitial atoms are smaller and thus more mobile  interstitial diffusion occurs much more rapidly then by vacancy mode
There are more empty interstitial positions than vacancies  interstitial atomic movement have greater probability

13. INTERSTITIAL DIFFUSION
• Applies to interstitial
impurities.
• More rapid than
vacancy diffusion.
• Simulation:
--shows the jumping of a
smaller atom (gray) from
one interstitial site to
another in a BCC
structure. The
interstitial sites
considered here are
at midpoints along the
unit cell edges.
(Courtesy P.M. Anderson) 7

14. 5.3 Steady-State Diffusion
The quantity of an element that is transported within another is a function of time  diffusion is a time-dependent process.
Diffusion flux (J)
Rate of diffusion or mass transfer
Defined as “mass or number of atoms (M) diffusing through and perpendicular to a unit cross-sectional area of solid per unit time.
Mathematically, J = M / (At)
In differential form: J = (1/A)(dM/dt)
A: area across which diffusion is occuring
t: elapsed diffusion time

15. 5.3 Steady-State Diffusion (Contd.)
If the diffusion flux does not change with time  steady-state diffusion
Example:
Diffusion of a gas through a plate of metal
Concentration (or pressure) of diffusing species on both side are held constant
Concentration profile: Concentration versus position
Assumed linear concentration profile as shown in figure (b)

16. 5.3 Steady-State Diffusion (Contd.)
Concentration gradient
Slope at a particular point on the concentration profile curve
Concentration gradient = dC / dx
For linear concentration shown in figure 5.4b:
Conc. Gradient = DC/Dx = (CA – CB) / (xA – xB)
Fick’s first law: For steady-state diffusion, the flux is proportional to the concentration gradient
J = -D(dC/dx)
D: diffusion coefficient (sq. m per second )
-ve sign: direction of diffusion from a high to a low concentration

17. 5.3 Steady-State Diffusion (Contd.)

18. 5.4 Nonsteady-State Diffusion
Most practical diffusion situations are non-steady
Non-steady
Diffusion flux and the concentration flux at some particular point of solid vary with time
Net accumulation or depletion of the diffusing species
Figure shown concentration profile at three different times

19. NON STEADY STATE DIFFUSION
• Concentration profile,
C(x), changes
w/ time.
• To conserve matter:
• Fick's First Law:
• Governing Eqn.: 14

20. Solution for Semi-infinite Solid with constant surface concentration
Assumptions
Initial concentration C0
X = 0 at the surface and increases with distance into the solid
Initial time = 0
Boundary conditions
For t = 0, C = Co at 0  x  
For t > 0, C = Cs (Constant surface concentration) at x=0
C = C0 at x = 
Solution
erf ( ) : Gaussian error function
Values given in Table 5.1

21. NON STEADY STATE DIFFUSION
• Copper diffuses into a bar of aluminum.
• General solution:
"error function"
Values calibrated in Table 5.1, Callister 6e. 15

24. EXAMPLE PROBLEM • Dt should be held constant. • Answer:
• Copper diffuses into a bar of aluminum.
• 10 hours at 600C gives desired C(x).
• How many hours would it take to get the same C(x)
if we processed at 500C?
Key point 1: C(x,t500C) = C(x,t600C).
Key point 2: Both cases have the same Co and Cs.
• Dt should be held constant.
Note: values
of D are
Given here.
• Answer: 16

25. Factors That Influence Diffusion

26. Factors That Influence Diffusion (Contd.)
DIFFUSING SPECIES
Magnitude of diffusion coefficient (D)  indicative of the rate at which atoms diffuse
D depends on both the diffusing species as well as the host atomic structure
Self-diffusion Fe in a-Fe 3.0E(-21) m2/s Vacancy Diffusion
Inter-diffusion C in a-Fe 2.4E(-12) m2/s Interstitial Diffusion
Interstitial is faster than vacancy diffusion

27. Factors That Influence Diffusion (Contd.)
TEMPERATURE
Temperature has a most profound influence on the coefficients and diffusion rate
Example: Fe in a-Fe (Table 5.2)
500oC D=3.0E(-21) m2/s
900oC D=1.8E(-15) m2/s  approximately six orders

28. DIFFUSION AND TEMPERATURE
• Diffusivity increases with T.
• Experimental Data:
D has exp. dependence on T
Recall: Vacancy does also! 19