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Introduction
•Material transport by atomic motion
•Diffusion couple:
eg., Cu-Ni in close contact; hold at elevated temperature
for extended period and cool to room temperature.
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Introduction continue…
Before
After
•Interdiffusion or impurity diffusion
•Self diffusion: same type of atoms; no composition change
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Diffusion mechanisms
Vacancy diffusion
Interstitial diffusion
1.Vacancy Diffusion
Atom from normal lattice position changes position with
an adjacent vacancy (vacancy lattice site). So, the atoms
and vacancies travel in opposite directions. Both selfdiffusion and inter (impurity)-diffusion can occur thus
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Diffusion mechanisms continue …
Source: William Callister 7th edition, chapter 5, page 112, figure 5.3(a)
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Diffusion mechanisms continue …
2) Interstitial Diffusion
Atoms move from one interstitial site to another (vacant)
interstitial site
.
Source: William Callister 7th edition, chapter 5, page 112, figure 5.3(b)
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Steady state Diffusion
J=M/At
If J is constant, steady-state diffusion exists.
Where,
J= rate of mass transfer with time, kg/m2-sec or atoms/m2sec
A= Area across which diffusion is occurring
t= elapsed time, sec
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Steady state Diffusion continue….
Steady-state diffusion
Concentration profile
Source: William Callister 7th edition, chapter 5, page 113, figure 5.4
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Steady state Diffusion continue….
C: Concentration of diffusing species, kg/m3 or gm/cm3
x: Position
Hydrogen gas across palladium plate
Concentration Gradient: Slope at a particular point on the
curve
=dC/dx
 ΔC/ΔX CA C B
XA X B
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Steady state Diffusion continue….
Fick’s first law of diffusion
J  D d c
dx
D = Diffusion coefficient, m2/sec
dc/dx = Concentration gradient is the main driving force
for diffusion
(-) = Concentration gradient decrease
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Steady state Diffusion continue….
=-
Problem
Carbon diffusing through a plate of iron
Carbon
rich
Carbon
deficient
5mm
10mm
Concentration: 1.2 kg/m3 0.8 kg/m3
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- =
Steady state Diffusion continue….
Problem continue
Diffusion coefficient: 3 x 10-11 m2/sec
CA CB
J  D
XA  X B
11
 (3 10
(1.2  0.8)
m /sec)
(5 103 ) - (10 103 )
2
= 2.4 x 10-9 kg/m2-sec
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Non-steady state Diffusion
•Diffusion flux and concentration gradient vary with time;
net accumulation or depletion of diffusing species results
………… Fick’ second law
……… Modified Fick’s second law
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Semi-Infinite solid
Surface concentration at the other end is constant. eg, Bar
of length, l > 10Dt , i.e., none of the diffusing atoms reach
the bar end during the time-period of diffusion
Assumptions:
1. Co = Concentration before diffusion
2. x = Distance; at surface it is 0. It increases into the solid
3. t = Time; zero(0) at the instant diffusion starts
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Semi-Infinite solid continue….
We have, for t=0, C=Co at 0≤x≤
For t>0, C=CS (Constant surface concentration) at x = 0
Also, C= CO at x=
Error function
Chapter 5: Diffusion
This equation shows relationship
between concentration, position
and time
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Semi-Infinite solid continue….
erf
Where, CX=Concentration at depth x after time t.
erf
=erf2
•Concentration at distance x, CX is a function of
•If time (t) and position (x) are known and CO, CS and D
are given, CX can be determined
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Semi-Infinite solid continue….
If CX=C1 at a specific concentration of solute,
=constant
Therefore
=constant
=constant
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Semi-Infinite solid continue….
Surface concentration
Concentration C at
distance x
Concentration before
diffusion
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Problem
Carburization of steel using methane (CH4) at 950°C
(1750°F)
Steel: 0.25 wt% Carbon. Using CH4, carbon at surface is
suddenly brought to and maintained at 1.2 wt% carbon.
How long will it take to achieve a carbon content of
0.80% carbon at a position 0.5 mm below the surface?
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Problem continue…
D= 1.6 x 10-11 m2/sec
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Problem continue…
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Problem
The diffusion coefficients for copper in aluminum at 500
and 600 C are 4.8x10-14 and 5.3x10-13 m2/s, respectively.
Determine the approximate time at 500 C that will
produce the same diffusion result (in terms of
concentration of Cu at some specific point in Al) as a 10-h
heat treatment at 600°C.
To produce the same effect at 500°C, how long will it
take?
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Problem continue…
Dt= constant
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Factors in diffusion
•Temperature
•Time
D increases 5 orders of magnitude with temperature
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Factors in diffusion continue…
Q
D  Do exp( d )
RT
Do = temperature independent pre-exponential (m2/sec)
Qd = the activation energy for diffusion (J/mol,cal/mol
and ev/atom)
R = gas constant, 8.31 J/mol-K or 8.62 eV/atom-K
T = absolute temperature (K)
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Factors in diffusion continue…
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Summary
•Self Diffusion
•Inter-Diffusion
•Steady state
J=M/At
Fick’s First law
•Non-steady state Fick’s second law
•Temperature effect
•Activation energy
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