1. Introduction to Materials Science and Engineering
Chapter 5: Diffusion
Textbook: chapter 7

2. Content Introduction Diffusion Mechanisms Steady-state Diffusion
Nonsteady-state Diffusion
Factors that Influence Diffusion
Diffusion in Semiconducting Materials
Diffusion in Ionic and Polymer Materials

3. ISSUES TO ADDRESS... some simple cases? and temperature?
• 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?

4. Introduction
atoms and molecules can be quite mobile at Tset in both liquids and solids:
Drop of ink in a beaker of water spread water evenly colored.
Intermixing at molecular level diffusion
continuous motion of H2O molecules in water at R.T. exemplifies self-diffusion.
atomic-scale motion (diffusion) in liquids is relatively rapid and easy to visualize.
more difficult to visualize diffusion in rigid solids, but it does occur.

5. Introduction
Difference between liquid-state and solid-state diffusion is the slower diffusion rate in the solid.
Tight atomic structure of atoms has an impact on the diffusion of atoms or ions within the solid.
The energy requirements to squeeze most atoms or ions through a perfect crystal structure are so high that diffusion is nearly impossible.

6. Diffusion-Phenomenon (1)
Inter-diffusion in diffusion couple
atoms migrate from high- to low- concentration regions
Initially
after some time

7. Diffusion-Phenomenon (2)
Self diffusion in pure materials
all atoms exchanging positions are of the same type
label some atoms
after some time C C A D A D B B
* Labeling by isotope (tracer diffusion)

8. Content Introduction Diffusion Mechanisms Steady-state Diffusion
Nonsteady-state Diffusion
Factors that Influence Diffusion
Diffusion in Semiconducting Materials
Diffusion in Ionic and Polymer Materials

9. Diffusion Mechanism Vacancy diffusion
- atom interchange from a normal lattice position to an adjacent vacant lattice site.
- the extent of vacancy diffusion is controlled by the concentration of these defects.
- the direction of vacancy motion is opposite to direction of diffusing atoms.
- both self-diffusion and interdiffusion occur by this mechanism.

10. Diffusion Mechanism Interstitial diffusion
typical interstitial atoms: hydrogen, carbon.
- in most metals, interstitial diffusion occurs much more rapidly than vacancy diffusion.

11. Activation Energy for Diffusion
Conditions for atom migration:
- empty adjacent site.
- atom must have enough energy to break bonds and cause lattice distortion during displacement.
diffusive motion influenced by atom vibrational energies  f(T)

12. Content Introduction Diffusion Mechanisms Steady-state Diffusion
Nonsteady-state Diffusion
Factors that Influence Diffusion
Diffusion in Semiconducting Materials
Diffusion in Ionic and Polymer Materials

13. Steady-state Diffusion
Flux:

14. Steady-state Diffusion
Fick's First Law: dC = - J D x dx
the steeper the concentration profile, the greater the flux!

15. Steady-state Diffusion
Steady state: the concentration profile doesn't change with time.
Concentration, C, in the box doesn’t change w/time. J x
(right)
(left)
- apply Fick's First Law:
- if Jx)left = Jx)right , then
Result: the slope, dC/dx, must be constant
(i.e., slope doesn't vary with position)!

16. Steady-state Diffusion
Example
- steel plate at 700oCwith geometry shown:
- Q: how much carbon transfers from the rich
to the deficient side?

17. Content Introduction Diffusion Mechanisms Steady-state Diffusion
Nonsteady-state Diffusion
Factors that Influence Diffusion
Diffusion in Semiconducting Materials
Diffusion in Ionic and Polymer Materials

18. Nonsteady-state Diffusion
steady-state diffusion not commonly encountered in engineering materials.
in most cases the concentration of solute atoms at any point in the material changes with time  non-steady state diffusion. t2 t1 t0

19. Nonsteady-state Diffusion
Concentration profile,C(x),
changes with time.
to conserve matter:
Fick's First Law:
Governing Equation:
Fick’d Second Law

20. Nonsteady-state Diffusion
Example: copper diffuses into a bar of aluminum.
semi-infinite solid
General solution:
"error function"

21. Nonsteady-state Diffusion

22. Nonsteady-state Diffusion
Copper diffuses into a bar of aluminum.
10 hours at 600oC gives desired C(x).
How many hours would it take to get the same C(x) if we processed at 500oC?
key point 1: C(x,t500oC) = C(x,t600oC).
key point 2: both cases have the same Co and Cs.
Result: Dt should be held constant.
(Dt)
500oC
=(Dt)
600oC
Answer:
Note: values of D are
provided here.

23. Content Introduction Diffusion Mechanisms Steady-state Diffusion
Nonsteady-state Diffusion
Factors that Influence Diffusion
Diffusion in Semiconducting Materials
Diffusion in Ionic and Polymer Materials

24. Factors That Influence Diffusion
Diffusing Species
- Magnitude of diffusion coefficient
: indicative of the rate at which atoms diffuse
: depends on diffusing species and host materials
- example
: Diffusing species in a-Fe
Fe (self diffusion) << C (interdiffusin)
3.0x10-21 m2/s x10-12 m2/s at 500oC
vacancy interstitial mechanism
: Host material (crystal structure)
a-Fe (BCC) > g-Fe (FCC)
1.8x m2/s x10-17 m2/s at 900oC
( lower packing factor- open structure
low coordination number- fewer bonds broken)

25. Factors That Influence Diffusion
Diffusing Species

26. Factors That Influence Diffusion
Temperature
- diffusion- thermally activated process

27. Factors That Influence Diffusion
Temperature
lnDo
ln D
-Qd/R

28. Factors That Influence Diffusion
Determination of activation energy

29. Factors That Influence Diffusion
참고자료
Factors That Influence Diffusion
Diffusion paths
- short circuit diffusion path- dislocation, grain boundary,
external surface (open structure)
- diffusing area should be considered for the comparison
of relative contribution of each path

30. Factors That Influence Diffusion
참고자료
Factors That Influence Diffusion
Miscellaneous
- melting temperature concentration dependence
The activation energy for self-diffusion
increases as the melting point of the
metal increases
The dependence of diffusion
Coefficient of Au on concentration

31. Factors That Influence Diffusion
Summary
Diffusion FASTER for...
• open crystal structures
• lower melting T materials
• materials w/secondary
bonding
• smaller diffusing atoms
• cations
• lower density materials
Diffusion SLOWER for...
• close-packed structures
• higher melting T materials
• materials w/covalent
bonding
• larger diffusing atoms
• anions
• higher density materials

32. Content Introduction Diffusion Mechanisms Steady-state Diffusion
Nonsteady-state Diffusion
Factors that Influence Diffusion
Diffusion in Semiconducting Materials
Diffusion in Ionic and Polymer Materials

33. Diffusion in Semiconducting Materials
1. Electrical properties of semiconducting materials can be tuned by incorporating substitutional impurities into the semiconductor – which we call doping.
N-type doping with P and As
P-type doping with B
For instance,

34. Diffusion in Semiconducting Materials
How you incorporate doping elements into Si?
Using two-step process:
Pre-deposition (Furnace Diffusion or Ion Implantation) – surface concentration fixed
Drive-in Diffusion – total amount of impurity is fixed

35. Content Introduction Diffusion Mechanisms Steady-state Diffusion
Nonsteady-state Diffusion
Factors that Influence Diffusion
Diffusion in Semiconducting Materials
Diffusion in Ionic and Polymer Materials

36. Diffusion in Ionic and Polymeric Materials
Ionic Materials:
Diffusion mechanism is much more complicated than for metals – should consider the charge nature:
Ion vacancies occur in pairs
They form in non-stoichiometric compounds
They are created by substitutional impurity ions having different charge states
Ions can move with electric field applied to materials – field driven transport
2. Polymer Materials:
Steady-state diffusion through a polymer membrane is,
where, PM=DS (D: diffusivity, S: solubility)
where, S=C/P