Lecture on DIFFUSION IN SOLIDS

Applications of Diffusion in Solids(besides nucleation and growth)

  Hard Facing (Carburizing of Steels)  Tough Tools and Parts.

Wear Facing of Gears, Wheels and Rails

Chemical Tempering of Glass and Ceramics Toughened Ceramics (Corel Ware)Shard resistant safety glass

Figure 4.41

  Thin Film Electronics (CMOS and Bipolar Transistors)Doping of Semiconductors

   Diffusion Bonding -- (Adhesives and cements for ceramic, metallic and polymer materials)

  Portland Cement as Bonding for ConstructionSolvents Cements for PVC Polymeric Piping Solders and Welds for Thermocouple Junctions

   Corrosion Protection

  Galvanizing, Electroplating, Anodizing, Inhibiting    Gas (Chemical) Separation Processes –   Diffusion membranes

K)*J/(atom 101.38x = constant sBoltzmann = k

K e,temperatur absolute = T

e y Probabilit

23-

kT

E - E-*

'

Diffusion is a RATE PROCESS   Probability of Finding an atom with energy E*

Fraction of atoms or molecules having energies greater than E* which is itself much greater than the average energy E.

constant = C

eV/K 108.62x = constant sBoltzmann = k

K e,temperatur absolute = T

systemin molecules or atoms of number total = N E than greaterenergy withatoms of number = n where

e C = N

n

5-

total

*

kTE-

total

*

'

Arrhenius' equation for the rate of many chemical reactions

constant rate = C

K) cal/(mol 1.986 or K)*J/(mol 8.314 = constant gas molar = R

K e,temperatur absolute = T

cal/mol or J/mol energy, activation = Q where,

e C = Reaction of Rate T R

Q-

T R 2.303

Q - C = rate

T R

Q - C = rate

1010 loglog

lnln

Rewritten as linear functions of the reciprocal of the absolute temperature.

ATOMIC DIFFUSION IN SOLIDS Diffusion can be defined as the mechanism by which matter is transported into or through matter.  Two mechanisms for diffusion of atoms in a crystalline lattice:  1. Vacancy or Substitutional Mechanism.  2. Interstitial mechanism.

Vacancy Mechanism

Atoms can move from one site to another if there is sufficient energy present for the atoms to overcome a local activation energy barrier and if there are vacancies present for the atoms to move into.

The activation energy for diffusion is the sum of the energy required to form a vacancy and the energy to move the vacancy.

Interstitial Mechanism

Interstitial atoms like hydrogen, helium, carbon, nitrogen, etc) must squeeze through openings between interstitial sites to diffuse around in a crystal.

The activation energy for diffusion is the energy required for these atoms to squeeze through the small openings between the host lattice atoms.

Steady-State Diffusion: Fick's First Law of Diffusion. For steady state conditions, the net flux of atoms is equal to the diffusivity times the concentration gradient.

4

2

2

d CJ = - D

d x

atomswhere J = flux or net flow of atoms

* sm

mD = diffusivity or diffusion coefficient s

d C atoms = concentrationgradient

d x m

Diffusivity -- the proportionality constant between flux and concentration gradient depends on: 1. Diffusion mechanism. Substitutional vs interstitial.2. Temperature. 3. Type of crystal structure of the host lattice. Interstitial diffusion easier in BCC than in FCC.4. Type of crystal imperfections.

(a) Diffusion takes place faster along grain boundaries than elsewhere in a crystal.

(b) Diffusion is faster along dislocation lines than through bulk crystal. (c) Excess vacancies will enhance diffusion.

5. Concentration of diffusing species.

Temperature Dependence of the Diffusion Coefficient

 D is the Diffusivity or Diffusion Coefficient ( m2 / sec )

Do is the prexponential factor ( m2 / sec )

Qd is the activation energy for diffusion ( joules / mole )

R is the gas constant ( joules / (mole deg) )T is the absolute temperature ( K )

T R

Q - D = D d

o exp

T R

Q - D = D d

olnln

Temperature Dependence of Diffusivity

Non-Steady-State Diffusion:

Fick's Second Law of Diffusion

  In words, The rate of change of composition at position x with time, t, is equal to the rate of change of the product of the diffusivity, D, times the rate of change of the concentration gradient, dCx/dx, with respect to distance, x.

x dC d

D x d

d =

t dC d xx

 Second order partial differential equations are nontrivial and difficult to solve.  Consider diffusion in from a surface where the concentration of diffusing species is always constant. This solution applies to gas diffusion into a solid as in carburization of steels or doping of semiconductors.

Boundary Conditions For t = 0, C = Co at 0 x

For t > 0 C = Cs at x = 0 and

C = Co at x =oo

where Cs = surface concentration

Co = initial uniform bulk concentration

Cx = concentration of element at distance x from

surface at time t.x = distance from surfaceD = diffusivity of diffusing species in host latticet = timeerf = error function

Dt2

x erf - 1 =

C - C

C - C

os

ox

Carburizing or Surface Modifying System:  Species A achieves a surface concentration of Cs and at time

zero the initial uniform concentration of species A in the solid is Co . Then the solution to Fick's second law for the

relationship between the concentration Cx at a distance x

below the surface at time t is given as  

where Cs = surface concentration, Co = initial uniform bulk

concentrationCx = concentration of element at distance x from surface at time t.

x = distance from surfaceD = diffusivity of diffusing species in host latticet = time

Dt2

x erf - 1 =

C - C

C - C

os

ox

Carbon diffusion into Steel – Hard Facing

Temperature Dependence of Diffusivity

N-type and P-type Dopant diffusion into Silicon. The making of devices.

Interdiffusion with Interface motion