# Lectures on DIFFUSION IN SOLIDS Applications of Diffusion in by 1hAsMAwq

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```									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 Construction
Solvents Cements for PVC Polymeric Piping
Solders and Welds for Thermocouple Junctions

 Corrosion Protection

Galvanizing, Electroplating, Anodizing, Inhibiting

 Gas (Chemical) Separation Processes –

Diffusion membranes
Diffusion is a RATE PROCESS

Probability of Finding an atom with energy E*

-

 E* - E   

Probability  e   
  kT       


T = absolute temperature, K
k = Boltzmann' s constant = 1.38x10- 23 J/(atom* K)
Fraction of atoms or molecules having energies greater
than E* which is itself much greater than the average
energy E.

n               E* 
= Ce   - 
 kT 
N total

where n = number of atoms with energy greater than E*
N total = total number of atoms or molecules in system
T = absolute temperature, K
k = Boltzmann' s constant = 8.62x10-5 eV/K
C = constant
Arrhenius' equation for the rate of many
chemical reactions
 Q 
-   
Rate of Reaction = C e     RT 

where, Q = activationenergy, J/mol or cal/mol
T = absolute temperature, K
R = molar gas constant = 8.314 J/(mol* K) or 1.986 cal/(mol K)
C = rate constant
Rewritten as linear functions of the reciprocal of the
absolute temperature.

Q
ln rate = ln C -
RT

Q
log 10 rate = log 10 C -
2.303 R T
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

dC
J = -D
dx

 atoms 
where J = flux or net flow of atoms  2     
  m *s 
 m2 
D = diffusivity or diffusion coefficient  
 s 
dC                                atoms 
dx                                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

 Qd 
D = Do exp  -  
 RT 

Qd
ln D = ln Do -
RT
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 )
Temperature Dependence of Diffusivity

Fick's Second Law of Diffusion

d Cx    d  d Cx 
=    D     
dt    d x d x 
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.
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
C x - Co                     x 
= 1 - erf                
C s - Co                     2 Dt 
where Cs = surface concentration
Co = initial uniform bulk concentration
Cx = concentration of element at distance x from
surface at time t.
x = distance from surface
D = diffusivity of diffusing species in host lattice
t = time
erf = error function
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

C x - Co                     x 
= 1 - erf                
C s - Co                     2 Dt 
where Cs = surface concentration, Co = initial uniform bulk
concentration
Cx = concentration of element at distance x from surface at time t.
x = distance from surface
D = diffusivity of diffusing species in host lattice
t = time
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

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