This document discusses thermally activated processes and diffusion in solids. It covers topics like rate processes, the probability of atoms acquiring activation energy, vacancy and interstitial diffusion mechanisms, and industrial applications of diffusion like surface hardening and integrated circuits. The key points are: - Reactions in solids require atoms to gain enough energy to overcome activation energy barriers. Higher temperatures provide more energy. - Diffusion occurs through vacancy or interstitial mechanisms as atoms move into vacant spaces in the crystal lattice. It is described by equations like Fick's laws. - Industrial uses of diffusion include carburizing steel surfaces to introduce carbon and dope silicon wafers with impurities to create integrated circuits. Diffusion

1. Thermally Activated Process and Diffusion in Solids.
Chapter 6

2. I. Rate Process in Solids
- Reactions occur in solid state resulting in more stable atomic arrangement.
- Reacting atoms must have sufficient energy to overcome activation energy barrier.
- At a given temperature, not all atoms have activation energy E*. It should be
supplied to them
∆𝐸 𝑃= ∆𝐸𝑟 + ∆𝐸∗
∆𝐸 𝑃 = Energy of Product
∆𝐸𝑟 = Energy of Reactants
∆𝐸∗
= Activation Energy

3. Type equation here. As temperature increases, more and more atoms acquire activation
energy level.
Probability if finding an atom/molecule with energy 𝐸∗
greater than average energy E of all
atom/ molecules is given by:
K = Boltzmann’s constant ( 1.38× 10−23
J/ atom.K )
T = Temperature in Kelvin
Probability = 𝑒− 𝐸∗−𝐸 /𝐾𝑇

4. The fraction of atoms having energies greater than E* in a system (when E* is greater than average
energy E) is given by:
𝑛
𝑁𝑡𝑜𝑡𝑎𝑙
= C𝑒 Τ−𝐸∗ 𝐾𝑇
n = Number of molecules greater than energy 𝐸∗
N = Total number of molecules
K = Boltzmann’s constant
C = constant
T = temperature in Kelvin
The number of vacancies at equilibrium at a particular temperature in a metallic crystal lattice is
given by:

5. 𝑛 𝑣
𝑁
= C𝑒 Τ−𝐸 𝑣 𝐾𝑇
𝑛 𝑣 = Number of vacancies per m3 of metal
𝐸𝑣= Activation Energy to form a vacancy
T = Absolute Temperature.
K = Boltzmann’s Constant.
C = Constant
The rate of chemical reaction is given by Arrhenius equation.
Rate of reaction = 𝐶𝑒−𝑄/𝑅𝑇
Q = Activation energy J/mol
R = Molar gas constant J/mol.K
T = Temperature in Kelvin
C = Rate constant ( Independent of temperature)

6. Rate of reaction depends upon number of reacting molecules.
Arrhenius equation can also be written as:
𝑙𝑛 𝑟𝑎𝑡𝑒 = ln 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 −
𝑄
𝑅𝑇
Or
𝐿𝑜𝑔10 𝑟𝑎𝑡𝑒 = 𝐿𝑜𝑔10 𝐶 −
𝑄
2.303𝑇
Which is equation of a straight line
With Y intercept as ‘b’ and slope ‘m’.
Y Log10(rate)
X (1/T)
b Log10(C)
m Q/2.303R
Y = b + m X
Which is similar to:
T

7. II. ATOMIC DIFFUSION IN SOLIDS
1. Diffusion
Diffusion is the mechanism or transportation of atom move through to another substance. The m
ovement of atom in the gas state is faster than liquid or solid-state. In solid-state atoms is hard to move. H
owever, vibration from the heat in a solid can made atoms movement. Diffusion in solids such as metals a
nd alloy is important because many reactions occur in solid-state is related to the movement of atoms such
as nuclei occur and grew grain on recrystallization processing by cold working.
Diffusion→ transport by atomic motion.

8. 2. Diffusion mechanism
Two mechanisms for diffusion of atoms:
1). Vacancy or Substitutional Mechanism
- Atoms diffuse in solids if Vacancies or other crystal
defects are present. There is enough activation energy
- Atoms move into the vacancies present
- More vacancies are created at higher temperatures.
- Diffusion rate is higher at high temperatures.
- As the melting point increases, activation energy also
increases

9. Self-diffusion or Substitutional mechanism Atom needs enough energy to break bonds and move
s into the vacancy. The energy that atom need Call the activation energy.
Activation Energy = Activation Energy + Activation Energy
of Self diffusion to forma Vacancy to move a vacancy

10. 2). Interstitial Diffusion mechanism
- Atoms move from one interstitial site to another
- The atoms that move must be much smaller than
the matrix atom.
- This mechanism is found for interdiffusion of
impurities such as hydrogen, carbon, nitrogen, and
oxygen, which have atoms that are small enough
to fit into the interstitial positions.
In most metal alloys, interstitial diffusion occurs much more rapidly than diffusion by the vacancy mode,
because the interstitial atoms are smaller and thus more mobile. The probability of interstitial atomic
movement is greater than for vacancy diffusion.

11. 3. Steady-state diffusion
when the conditions in diffusion do not change with time. The concentration remains the same,
as concentration is increased on one side, it decreases on the other (concentration input flows
from higher to lower) Fick’s first law.
The flux or flow of atoms is given by:
J = Flux or net flow of atoms.
D = Diffusion coefficient
𝑑𝑐
𝑑𝑥
= Concentration Gradient.
J = −D
𝑑𝑐
𝑑𝑥

12. For steady state diffusion condition, the net flow of atoms by atomic diffusion is equal to diffusion D
times the diffusion gradient dc/dx
𝐽
𝑎𝑡𝑜𝑚
𝑚2. 𝑠
= D
𝑚2
𝑠
𝑑𝐶
𝑑𝑥
𝑎𝑡𝑜𝑚
𝑚3 ×
1
𝑚
Diffusion coefficient (or Diffusivity) depends upon:
- Type of diffusion: Whether the diffusion is interstitial or substitutional.
- Temperature: As the temperature increases diffusivity increases.
- Type of crystal structure: BCC crystal has lower APF than FCC and hence has higher
diffusivity.
- Type of crystal imperfection: More open structures (grain boundaries) increases diffusion.
- The concentration of diffusing species: Higher concentrations of diffusing solute atoms will
affect diffusivity.

13. 4. Non-steady state diffusion
- Concentration of solute atoms at any point in metal changes with time in this case.
- Fick’s second law: Rate of compositional change is equal to diffusivitytimes the rate
of change of concentration gradient.






xd
Cd
D
xd
d
=
td
Cd xx

14. Table of the error function
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






Dt2
x
erf=
C-C
C-C
os
ox
Cs = Surface concentration of element in gas diffusing into the surface.
Co = Initial uniform concentration of element in solid.
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

15. Example Problem: Consider the gas carburizing of gear of 1020 steel at 927℃. Calculate the time in minutes
necessary to increase the carbon content to 0.40% at 0.05 mm below the surface. Assume that the carbon content
at the surface is 0.90% and that the steel has nominal carbon content of 0.20%. 𝐷927℃= 1.28 × 10−11 𝑚2/𝑠
Solution: 





Dt2
x
erf=
C-C
C-C
os
ox
CS = 0.90% x = 0.5 mm == 5.0 × 10−4 m
C0 = 0.20% 𝐷927℃= 1.28 × 10−11
𝑚2
/𝑠
Cx = 0.40% t = ?
Substituting these values gives:
( )( )









−
−
tsm2
m
erf=
-
-
/1028.1
100.5
20.090.0
40.090.0
211
4
7143.0
88.69
70.0
50.0
=





t
erf=
Let
t
=Z
88.69
then erf Z = 0.7143

16. We need a number for Z whose error function (erf) is 0.7143. From Table we find this
number by interpolation to be 0.75
erf Z Z
0.7112 0.75
0.7143 x
0.7421 0.80
75.080.0
7112.0
7112.07421.0
7112.07143.0
−
−
−
− x
=
x – 0.75 = (0.1003)(0.05)
x = 0.75 + 0.005= 0.755
thus,
755.0
88.69
=
t
=Z
6.92
755.0
88.69
==t
t = 8567 s = 143 min

17. III. Industrial Applications of Diffusion
1. Surface Hardening of steels by gas carburizing
- Sliding and rotating parts need to have hard surfaces.
- These parts are usually machined with low carbon steel as they are easy to machine.
- Their surface is then hardened by carburizing.
- Steel parts are placed at elevated temperature (927℃) in an atmosphere of hydrocarbon gas (CH4).
- Carbon diffuses into iron surface and fills interstitial space to make it harder.
Carbon Gradients In Carburized metals
C %
Low carbon
Steel part
Diffusing carbon atoms

19. 2. Integrated electronic circuits(IC) by doping Impurity Diffusion into Silicon wafer
- Impurities are made to diffuse into silicon wafer to change its electrical
characteristics.
- Used in integrated circuits.
- Silicon wafer is exposed to vapor of impurity at 1100℃ in a quartz tube
furnace.
- The concentration of impurity at any point depends on depth and time of
exposure

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