Diffusion

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Diffusion Chapter 8
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Selection of steel for gears
Wear resistance
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Fraction of cementitite by LEVER rule
More carbon
More cementite
More wear resistance
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Cementite is hard as well as brittle
Hardness or strength is desirable.
But brittleness is not.
Silica Glass is also hard and brittle
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Selection of steel for gears
High Carbon Steel Good wear resistance
but Btittle
Low Carbon steel Good ductility but poor
wear resistance
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Wear resistance is required only at the surface
High C steel on the surface
Mild steel inside
Q How do you achieve this?
Ans By case carburization
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Case carburization
Pack a mild steel gear in carbon and heat at a
high temperature in the austenite phase field for
some time.
Carbon will enter into the mild steel to give a
high-carbon wear resistant surface layer called
case.
How do carbon enter into solid steel?
At what temperature and how long should we do the
carburization?
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The process why which Carbon enters into solid
steel during Case carburization is an example of
DIFFUSION
Diffusion is relative movement of atoms inside a
solid
We can find appropriate time and tempearture for
case carburization by solution of Ficks second
law
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How do we create an n-p junction in silicon chip?
Ans by DIFFUSION
Deposit p type element
Deposit n type element
Heat
Si substrate
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Different kinds of flows in material
Heat flow of thermal energy
Electric current flow of electric charge
Diffusion flow of matter
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Heat flow of thermal energy
q heat flux (J m-2 s-1)
Gradient?
Temperature gradient
Fouriers law of heat conduction (1811)
Joseph Fourier (1768-1830)
Thermal conductivity
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Electric current flow of charge
j charge flux (C m-2 s-1), current density
Gradient?
Electric potential gradient, electric field E
Ohms law of electrical conduction (1827)
Georg Simon Ohm (1787-1854)
electrical conductivity
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Diffusion flow of mass
j mass flux (kg m-2 s-1, moles m-2 s-1)
Gradient?
concentration gradient, kg m-4
Ficks first law of diffusion1855
1829-1901
D Diffusivity, m2 s-1
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Mass in at x min A ?t j
Mass out at x? x mout A ?t (j ?j)
Mass accumulation between x and x ? x
?m min-mout
A ?t ( j j - ?j ) -A ?t ?j
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?m -A ?t ?j
Change in concentration in a volume ? V A ? x
and time interval ? t
Average rate of change of concentration between x
and x ? x in time interval ?t
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Instantaneous change in concentration at a time
t, at a point x
Ficks 2nd Law
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Using Ficks First Law
If D is independent of x
Ficks 2nd law
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Solution to Ficks 2nd law
Solution depends on the boundary condition.
A and B constants depending on the boundary
conditions
erf (z) Gaussian error function
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The Gaussian Error Function
Hatched area ? (2/??) erf (z)
z
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erf (0) 0, erf ( ?) 1, erf
(-z) - erf (z), erf (- ?) -1
erf (z)
z
TABLE 8.1
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Carburisation of steel
c (wt C)
cs
Surface concentration
Concentration profile after carburization for
time t at a temperature T
initial concentration
c0
x
Distance in steel from surface
Boundary conditions
1. cc0 at xgt0 , t0
2. ccs at x0 , tgt0
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Carburisation of steel
Boundary conditions
1. c(x,t) cs at x 0, t gt 0
2. c(x,t) c0 at x gt 0, t 0
B.C. 1 ? cs A B erf(0) A
A cs B cs c0
B.C. 2 ? c0 A B erf(?) A-B
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Case carburization of steel
Problem 8.4
Initial concentration c0 0.2 wt C Surface
concentration cs 1.4 wt C Temperature 900
ºC 1173 K Desired concentration c 1.0 wt C
at x 0.2 mm
At 900 ºC the equilibrium phase of steel is
austenite (?)
Diffusivity data for C in austenite D0 0.7 x
10-4 m2s-1 Q 157 kJ mol-1
7.13688 x 10-12 m2s-1
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Carburization of steels
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z erf(z) 0.30 0.3286 0.35 0.3794
0.305 0.3333
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t 15062 s 4 h 11 min
Ans
This is reasonable. If not, change D by changing T
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Temperature dependence of Diffusivity
D0 preexponential factor
Empirical constants
Q activation energy
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Self-Diffusion in Amorphous Se (Problem 8.3)
T (ºC) D (m2s-1) 35 7.7 x 10-1640 2.4 x
10-1546 3.2 x 10-1456 3.2 x 10-13
ln D
-28
-30
D0 2 x 1027 m2 s-1 Q 250 kJ mol-1
-32
-34
1/T
0.00305
0.00315
0.00325
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Atomic Mechanism of Diffusion
How does C enter into solid steel?
? is INTERSTITIAL solid solution of C in FCC Fe
C occupies octahedral voids in FCC Fe
Maximum solubility of C in austenite (?) is 2.14
wt at 1150 ºC.
Thus 9 out of 91 OH voids are occupied. 90 of
OH voids are empty
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C atoms can jump from one interstitial site to
another vacant interstitial site.
This is interstitial diffusion.
For OH voids the void size is 0.414 R but the
window through which C atoms can jump outside is
only 0.155 R.
Thus to jump out of an interstitial OH site the C
atoms will have to displace neighbouring Fe
atoms. This will increase the energy of the system
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A carbon atom can jump to a neighbouring site if
it has sufficient energy ?Hm.
It can gain this energy only through random
thermal vibration.
If thermal vibration frequency is ? then it makes
? attempts per second.
will have an energy ?Hm and will be successful.
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1
2
No. of successful jumps per second from plane 1
to plane 2, n1-gt2 A ? c1 ? exp(- ?Hm/RT) p
No. of successful jumps per second from plane 2
to plane 1, n2-gt1 A ? c2 ? exp(- ?Hm/RT) p
?
C1
C2
Net jumps per second from plane 1 to plane 2
?n n1-gt2 - n2-gt1
A ? (c1-c2) ? exp(- ?Hm/RT) p
Flux
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Ficks 1st Law
An atom making a successful jump may remain in
plane 1, go to the back plane or jump to forward
plane 2. Thus only a fraction p of successful
jumps are from plane 1 to plane 2. This factor
has been omitted in the textbook.
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Initially
After some time
Wt Ni
Ni
100
100
0
0
x
Adapted from Figs. 5.1 and 5.2, Callister 6e.
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Mechanism of substitutional diffusion
How is diffusion taking place in a substitutional
solid solution ?
Vacancy mechanism of substitutional diffusion
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However, only a very small fraction of the order
of 10-4 to 10-30 are vacant.
A jump can only be successful if the neighbouring
site is vacant.
Probability of finding a vacant site fraction
of vacant site
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Sbstitutional diffusion is usually slower than
interstitial diffusion due to difficulty of
finding a vacant site.
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Other Diffusion Paths
Lattice diffusion
Grain boundary diffusion
Surface diffusion
Experimentally Qsurface lt Qgrain boundary lt
Qlattice