1. Diffusion how atoms move in solids
Chapter Outline
Diffusion how atoms move in solids
Diffusion mechanisms
Vacancy diffusion
Interstitial diffusion
Impurities
Mathematics of diffusion
Steady-state diffusion (Fick’s first law)
Nonsteady-State Diffusion (Fick’s second law)
Factors that influence diffusion
Diffusing species
Host solid
Temperature
Microstructure
5.4 Nonsteady-State Diffusion – Not Covered / Not Tested

2. Diffusion transport by atomic motion.
What is diffusion?
Diffusion transport by atomic motion.
Inhomogeneous material can become homogeneous by diffusion. Temperature should be high enough to overcome energy barrier.

3. Interdiffusion vs. Self-diffusion
Concentration Gradient  Interdiffusion (or Impurity Diffusion).
Before
After
(Heat)
Self-diffusion: one-component material, all atoms that exchange positions are of same type.

4. Diffusion Mechanisms (I)
Vacancy diffusion
Atom migration
Vacancy migration
Before
After
To jump from lattice site to lattice site, atoms need energy to break bonds with neighbors, and to cause the necessary lattice distortions during jump. This energy comes from the thermal energy of atomic vibrations (Eav ~ kT)
Materials flow (the atom) is opposite the vacancy flow direction.

5. Diffusion Mechanisms (II)
Interstitial diffusion
Interstitial atom
before diffusion
Interstitial atom
after diffusion
Interstitial diffusion generally faster than vacancy diffusion because bonding of interstitials to surrounding atoms is normally weaker and there are more interstitial sites than vacancy sites to jump to.
Requires small impurity atoms (e.g. C, H, O) to fit into interstices in host.

6. Mass: J = M / (A t)  (1/A) (dM/dt)
Flux of diffusing atoms, J.
Number of atoms diffusing through unit area per unit time [atoms/(m2s)] or
Mass of atoms diffusing through unit area per unit time [kg/(m2 s)]
Mass: J = M / (A t)  (1/A) (dM/dt)
There is a difference between diffusion and net diffusion. In a homogeneous material, atoms also diffuse but this motion is hard to detect. This is
because atoms move randomly and there will be an equal number of atoms moving in one direction than in another. In inhomogeneous materials,
the effect of diffusion is readily seen by a change in concentration with time. In this case there is a net diffusion. Net diffusion occurs because,
although all atoms are moving randomly, there are more atoms moving in regions where their concentration is higher. J A

7. Steady-State Diffusion
Diffusion flux does not change with time
Concentration profile:
Concentration (kg/m3) vs. position
Concentration gradient: dC/dx (kg / m4)

8. Steady-State Diffusion
Fick’s first law: J proportion to dC/dx
D is the diffusion coefficient
Concentration gradient is the driving force (not a mechanistic force).
Minus sign means that diffusion is down the concentration gradient.

9. Nonsteady-State Diffusion
(not tested)
Concentration changing with time
Fick’s second law
Assumption: D does not depend on x (and C! )
Find C(x,t)

10. Diffusion – Thermally Activated Process (I)
Atom needs enough thermal energy to break bonds and squeeze through its neighbors. Energy needed  the activation energy
for vacancy motion. Em
Energy Em
Vacancy
Arrhenius equation was first formulated by Hood based on his experiments, but Arrhenius showed that it can be applied to any thermally activated process.
If you have stronger bonding, you have to spent more energy to create vacancy…
Atom
Distance
Diffusion – Thermally Activated Process (I)

11. Diffusion – Thermally Activated Process (II)
Average thermal energy of an atom at room temperature (kBT = eV)
Usually much smaller that activation energy Em (~ 1 eV/atom) (like Qv)
Therefore, a large fluctuation in energy is needed for a jump.
Probability of such fluctuations or frequency of jumps, Rj, depends exponentially on temperature. Swedish chemist Arrhenius :
R0 is an attempt frequency proportional to the frequency of atomic vibrations.

12. Diffusion – Thermally Activated Process
Probability of finding a vacancy in an adjacent lattice site (see Chapter 4):
Probability of thermal fluctuation needed to overcome energy barrier for vacancy motion
Assumption: D does not depend on x (and C! )
The diffusion coefficient = Multiply
Arrhenius dependence.

13. Diffusion – Temperature Dependence (I)
Diffusion coefficient is the measure of mobility of diffusing species.
D0 – temperature-independent (m2/s)
Qd – the activation energy (J/mol or eV/atom)
R – the gas constant (8.31 J/mol-K) or
k - Boltzman constant ( 8.6210-5 eV/atom-K)
T – absolute temperature (K) or
(lnD) vs. (1/T) or (logD) vs. (1/T)
Arrhenius Plots.

14. Diffusion – Temperature Dependence (II)
Graph of log D vs. 1/T has slop of –Qd/2.3R,
intercept of ln Do

15. Diffusion – Temperature Dependence (III)
Arrhenius plot of diffusivity data for some metallic systems

16. Diffusion of different species
Smaller atoms diffuse more readily than big ones, and diffusion is faster in open lattices or in open directions

17. Diffusion: Role of the microstructure (I)
Self-diffusion coefficients for Ag
Depends on diffusion path
Grain boundaries and surfaces less restrictive

18. Diffusion: Role of the microstructure (II)
Plots are from computer simulations
Initial positions are shown by the circles, paths are shown by lines. See difference between mobility in the bulk and in the grain boundary.

19. Factors that Influence Diffusion
Temperature - diffusion rate increases very rapidly with increasing temperature
Diffusion mechanism - interstitial is usually faster than vacancy
Diffusing and host species - Do, Qd is different for every solute, solvent pair
Microstructure - diffusion faster in polycrystalline vs. single crystal materials because of the rapid diffusion along grain boundaries and dislocation cores.

20. Make sure you understand
Summary
Make sure you understand
Activation energy
Concentration gradient
Diffusion
Diffusion coefficient
Diffusion flux
Driving force
Fick’s first and second laws
Interdiffusion
Interstitial diffusion
Self-diffusion
Steady-state diffusion
Vacancy diffusion

21. Reading for next class:
Chapter 6: Mechanical Properties of Metals
Stress and Strain
Tension
Compression
Shear
Torsion
Elastic deformation
Plastic Deformation
Yield Strength
Tensile Strength
Ductility
Resilience
Toughness
Hardness