NON-STEADY STATE DIFFUSION

1
NON-STEADY STATE DIFFUSION
Non-steady state The concentration profile
changes with time. C C (x,t)
To conserve energy dJ/dX J(right) J(left)
/dx -?C/ ? t Ficks first law J
-D(?C/?x) dJ/ dx - ?/ ?x D(?C/?x) Substitute
for dJ/dX ?C/ ? t ?/ ?x D(?C/?x) Ficks
second law
If D is independent of concentration ?C/ ? t
D(?2C/?x2)
2
Solution of Ficks Second Law
"error function" Values calibrated in Table 5.1,
Callister 6e.
Cs concentration on the surface C0 initial
concentration
3
Application Case Hardening
Carburizing of Steels - Diffuse carbon atoms
into the surface of a gear made of a low carbon
steel (Steel is an alloy of iron and carbon). -
The concentration of carbon on the surface
increases and consequently the surface hardness
of the gear is increases. - The high hardness
reduces the wear and improves the resistance of
the gear to fracture under cyclic loading
(fatigue resistance).
Fig. 5.0, Callister 6e. (Fig. 5.0 is courtesy
of Surface Division, Midland-Ross.)
The carbon is introduced at high temperatures to
the surface by controlling the relative partial
pressure of Co and CO2 in the carburizing gas.
4
EFFECT OF TEMPERATURE
2
pre-exponential m
/s (see Table 5.2,
Callister 6e
)
activation energy
æ
ö
J/mol,eV/mol
Q
d
ç


D
D
exp
-
diffusivity
(see Table 5.2,
Callister 6e
)
o
è
ø
R
T
gas constant
8.31J/mol-K
Adapted from Fig. 5.7, Callister 6e. (Date for
Fig. 5.7 taken from E.A. Brandes and G.B. Brook
(Ed.) Smithells Metals Reference Book, 7th ed.,
Butterworth-Heinemann, Oxford, 1992.)
5
EFFECT OF CRYSTAL STRUCTURE AND MELTING POINT
Crystal Structure Diffusion is faster in more
open structures. Examples Self diffusion is
faster in a-Fe than it is in g-Fe. Carbon moves
faster in a-Fe than it does in g-Fe.
Melting Point Diffusivity decreases as the
melting point is increased. Example
DAlgtDCugtDg-Fe at a given temperature.