• Materials often heat treated to improve properties
• Atomic diffusion occurs during heat treatment
• Depending on situation higher or lower diffusion rates desired
• Heat treating temperatures and times, and heating or cooling rates can be determined using the mathematics/physics of diffusion
Example: steel gears are “case-hardened” by diffusing C or N to outer surface
Courtesy of P. Alpay

• Atomic mechanisms of diffusion
• Mathematics of diffusion
• Influence of temperature and diffusing species on
Diffusion rate
Courtesy of P. Alpay

Phenomenon of material transport by atomic or particle transport from region of high to low concentration
• What forces the particles to go from left to right?
• Does each particle “know” its local concentration?
• Every particle is equally likely to go left or right!
• At the interfaces in the above picture, there are more particles going right than left Î this causes an average “flux” of particles to the right!
• Largely determined by probability & statistics
Courtesy of P. Alpay

• Glass tube filled with water.
• At time t = 0, add some drops of ink to one end of the tube.
• Measure the diffusion distance, x, over some time.
t o t
1 t
2 t
3 x o x
1 x
2 x
3 x (mm) time (s)
Courtesy of P. Alpay

• Case Hardening :
-- Example of interstitial diffusion is a case hardened gear.
-- Diffuse carbon atoms into the host iron atoms at the surface.
Fig. 5.0,
Callister 6e .
(Fig. 5.0 is courtesy of
Surface
Division,
Midland-
Ross.)
• Result: The "Case" is
--hard to deform: C atoms
"lock" planes from shearing .
--hard to crack: C atoms put the surface in compression.
Courtesy of P. Alpay

• Doping Silicon with P for n-type semiconductors:
• Process
0.5mm
1. Deposit P rich layers on surface.
magnified image of a computer chip silicon
2. Heat it.
3. Result: Doped semiconductor regions.
light regions: Si atoms
Fig. 18.0,
Callister 6e .
silicon light regions: Al atoms
Courtesy of P. Alpay

Driving force of sintering process = reduction of energy associated with particles surfaces raw powder synthesis pressing or casting densification grain growth
“Green”
Two processes in competition: densification and coarsening coarsening
Porosity = 0
Transparent material
(if no refractive index mismatch between grains)
Porosity ≠ 0
Translucent material
Reducing scattering losses implies control of ceramic microstructure

Factors affecting solid-state sintering: . Temperature, pressure
. Green density
. Uniformity of green microstructure
. Atmosphere composition
. Impurities
. Particles size and size distribution
Sintering mechanisms

• Self-diffusion : In an elemental solid, atoms also migrate.
Label some atoms After some time
C
D
B
C
A
B*: self diffusion mobility
Self – diffusion coefficient in a simple cubic lattice
δ : interatomic distance v : jump frequency
A
B
D
Courtesy of P. Alpay

Tracer atom studies may not give the true self-diffusion coefficient, D i
*
(12.6) f: correlation coefficient
D i
*: true self-diffusion coefficient

• Interdiffusion : In an alloy or “diffusion couple”, atoms tend to migrate from regions of large to lower concentration.
Initially (diffusion couple) After some time
Adapted from
Figs. 5.1 and
5.2, Callister
6e .
100%
Cu Ni
100%
0
Concentration Profiles
0
Concentration Profiles
Courtesy of P. Alpay

If the concentration of component A is given in mass units, the rate equation for diffusion is:
(12.1)
W
Ax
: mass flux of A in the x-direction, kg (of A) m -2 s -1
ρ
A
: mass concentration of A, kg (of A) m -3 (of total material)
D
A
: diffusion coefficient, or diffusivity of A, m 2 s -1
(intrinsic diffusion coefficient)
In the materials field, the rate equation is usually written in terms of molar concentrations:
(12.2) j
Ax
: molar flux of A in the x-direction, mol (of A) m -2 s -1
C
A
: molar concentration of A, mol (of A) m -3 (of total material)

Fick’ s first law of diffusion states that species A diffuses in the direction of decreasing concentration of A, similarly as heat flows by conduction in the direction of decreasing temperature, and momentum is transferred in viscous flow in the direction of decreasing velocity.
Eq. (12.1) and (12.2) are convenient forms of the rate equation when the density of the total solution is uniform (in solid or liquid solutions, or in dilute gaseous mixture).
When the density is not uniform:
ρ : density of the entire solution, kg/m -3
ρ
A
*: mass fraction of A
(12.3)
C: local molar concentration in the solution at the point where the gradient is measured
(12.4)

Self – diffusion data apply to homogenous alloys in which there is no gradient in chemical composition.

2. Diffusion under the influence of a composition gradient
ƒ Vacancy mechanism
ƒ Ring mechanism
ƒ Interstitialcy mechanism

Flux of gold atoms (according to an observer sitting on a plane moving with the solid’s velocity):
Total flux of gold atoms (according to an observer sitting on an unattached plane in space):
(12.7)
The accumulation of gold as a function of time within a unit volume:
(12.8)

Accumulation of nickel in the same unit volume:
(12.9)
(12.10)
(12.11)

If the vacancy concentration within the unit volume is constant, then the volume is approximately constant and:
(12.12)
(12.13)
Only in terms of the gold concentration gradient:
(12.14)

Accumulation of gold solely in terms of diffusion coefficients and concentration gradients:
(12.15)
Fick’s first law in its simplest form, for gold:
(12.16) side-steps the issue of bulk flow.
The accumulation of gold within the unit volume:
(12.17)

The unsteady-state equation for unidirectional diffusion in solids ( Fick’s second law ):
(12.18)
Equation (12.18) is used to obtain analytical solutions for diffusion problems
(12.19)
D
Au and D
Ni
: Intrinsic diffusion coefficients

3. Darken’s equation
(12.20) n
Ax
: flux of A atoms passing through an unit area in the x-direction n
A
: the number of A atoms per unit volume
B
A
: mobility of A atoms in the presence of the energy gradient
_
: Avogadro’s number
G
A
: partial molar free energy of A (chemical potential of A)

a x
: activity of A
K
B
: Boltzmann’s constant = R/N
0
X
A
: Mole fraction of A a
A
= X
A
(for a thermodynamically ideal solution)
NERNST-EINSTEIN EQUATION
(12.21)
(12.22)
(12.23)

Gibbs – Duhem equation:
(12.24)
And only if B
A
= B
A
* and B
B
= B
B
*, then Eq. (12.19) can be written as:
(12.25)
If the system is ideal:
(12.26)

4. Temperature dependence of diffusion in solids
(12.27)
(Arrhenius equation)
According to Zener, if the jump process is an activated one, jump frequency
ν can be described by:
(12.28)
ν
0
= vibrational frequency of the atom in the lattice
Z = coordination number
∆ G
:
= free energy of activation required for the atom to jump from one site into the next

(12.29)

Sherby and Simnad developed a correlation equation for predicting selfdiffusion data in pure metals:
(12.30)
K
0 depends only on the crystal structure
V = normal valence of the metal
T
M
= absolute melting point
D
0 is approximated as 1 × 10 -4 m2 s -1 for estimation purposes
(12.31)

The general equation for the transport of the species in the x-direction under the influence of an electric field and a concentration gradient in the x-direction:
(12.32)
In the absence of a concentration gradient, the flux is:
(12.33)

The current density:
(12.34)
(12.35)
The electrical conductivity σ:
(12.36)
(12.37)
This equation is known as the “extended” Nernst-Einstein equation

t i
: transference number of species i
For NaCl:
(12.38)
(12.39)
(12.40)

For CoO:

Thermodynamic factor can be calculated by:
P °
O 2
: standard oxygen pressure, 1 atm

Hydrodynamical theory:
The force on a sphere moving at steady-state in laminar flow: and since B = V
∞
/F,
(12.41)
Stokes – Einstein equation

Sutherland – Einstein equation
(12.42)

(12.43)
A is a constant
M = atomic weight or molecular weight
T
M
= melting point
D* = self-diffusion coefficient
V
M
= molar (or atomic) volume
If it is assumed that the atoms in the liquid are in a cubic array, then for materials generally the simple theory predicts that:
(12.44)

Hole theory : This theory presumes the existence of holes or vacancies randomly distributed throughout the liquid and providing ready diffusion paths for atoms and ions. The concentration of these holes would have to be very great in order to account for the volume increase upon melting, thus resulting in much higher diffusion rates in liquids than in solids just below the melting point.
Eyring theory :
If the liquid is considered quasi-crystalline, and the atoms are in a cubic configuration, then an expression relating D* and η results:
(12.45)

Fluctuation theory :
In this theory, the ‘extra’ volume (over that of the solid) in the liquid is distributed evenly throughout the liquid, making the average nearest neighbour distance increase.
Reynik’s fluctuation model:
(12.46)
(12.47)

Diffusion in liquid metals :

Diffusion in molten salts and silicates :

Diffusion in common liquids :

Based on the kinetic theory of gases, self-diffusivity of spherical A atoms diffusing in pure A can be predicted by:
(12.48)
For the diffusivity of two unequal size spherical atoms A and B, kinetic theory predicts:
(12.49)
K
B
: Boltzmann’s constant, 1,38 × 10 -16 ergs molecule -1 K -1

Chapman – Enskog theory:
For monoatomic gases
(12.50)

The collision integral can be calculated by:
(12.51) where A = 1.06036, B = 0.15610, C = 0.19300, D = 0.47635, E = 1.03587,
F = 1.52996, G = 1.76474, and H = 3.89411.
(12.52)
(12.53)

(12.54)

(12.55) void fraction effective diffusivity tortuosity
Tortuosity is a number greater than one and it ranges from values of 1.5 -2.0 for unconsolidated particles, to as high as 7 or 8 for compacted particles .
An analysis of the flow of molecules through a cylindirical pore radius r under a concentration gradient yields

(12.56)
Knudsen diffusion coefficient mean free path of the gas concentration of the molecules, cm collision diameter of the molecule, cm
-3
If r and λ are of the same order of magnitude, or if the pore diameter is only one order larger, then Knudsen diffusion is presumably important.

If the ratio D
AB is small, then the ordinary diffusion is the rate determinig step. Conversely, if D
AB
/D
K is large, it is presumed that Knudsen diffusion controls.
/D
K
(12.57)