
2. Theory


Derivation
of the equation 

In this module I supposedly will show
how to calculate TTT diagrams. It is an easy module for me because I have a
message: forget
it! 


Calculating real TTT diagrams is beyond my
skills. Even for simple but real materials
it is not an easy task. That's also the reason why pretty much all real TTT or
CCT diagrams have been experimentally determined.
All I'm going to do is to derive the most simple approximation to the problem,
the celebrated Avrami equation, or to give it is's full name, the JohnsonMehlAvramiKolmogorov
(JMAK) equation. Those guys did not work together. Melvin Avrami
published a number of papers to the topic around 1940, but the others had
similar ideas around the same time or actually a bit earlier and credit must go
to all even so Mehl is the most prominent one of the
bunch^{1)} 


We look on some g ® a transformation and generally assume that the
transformation is finished after some time t. The JMAK equation
gives us the transformed fraction f(t) as a function of
time t and the relevant transformation parameters. It is based on
a number of (not very realistic) assumptions:
 Nucleation occurs randomly and homogeneously over the entire untransformed
(= g) portion of the material.
 The growth rate or the velocity of the precipitate interface does not
depend on the extent of the transformation.
 Growth occurs at the same rate in all directions; i.e. nuclei always grow
into spheres.


We start by assuming that the
concentration N of nuclei increases by the constant generation rate dN/dt =
G_{N}. They grow into spherical precipitates because the
interface between the a precipitate and the
g matrix moves outward with a
constant velocity v.
The precipitates only stop growing if they hit each other (the polite word is
"impinge"). We will neglect this for starters. In other words: we
first look at unrestricted growth 


We need to introduce two time scales. The first one is the regular time
t. The second one measures the time in which precipitates grow
and is denoted by t. It starts as soon as the
precipitate is nucleated. Since precipitates are nucleated all the time, they
all start growing at different times
t but always at their own t =
0. 


The (differential) number of nuclei that appear
during a time interval dt in a volume
V then is dN = G · V · dt; V is the volume of the
material 


The nuclei grow spherically and their interface
moves into the g matrix with a velocity
v that we also assume to be constant. After a nuclei appeared and
started to grow, its radius r(t) will be speed times
growing time or r = v · (t – t) 

The additional (differential) volume
dV_{ur} of the unrestrictedly growing phase a during a time interval dt then is given by: 




dV_{a, ur}^{ } 
= 
æ
ç
è 
4p
3 
v^{3} · (t – t)^{3}

ö
÷
ø 
· 
æ
ç
è 
G · V · dt 
ö
÷
ø 













Volume of
spheres 

Generation of
spheres 







Integration from t = 0 to t =
t will give the total volume of a
and thus the amount transformed after the time t has elapsed. We
get 





V_{a, ur}^{ } 
= 
p
3 
· G · V · v^{3} · t^{4}







This can't be correct, however. As
the fraction of a increases, more and more
precipitates can't grow anymore because they encounter other ones. You can
transform a g region only once, and if
another precipitate has already done that, growth must stop as soon as two
precipitates impinge on each other. 


If we look at some arbitrary time
t, the volume that has really
been transformed from g to a is just V_{a}, without the index "ur".
Expressed as a fraction of the total volume V it is
V_{a}/V. 


We now need to make the next assumption: At every
time increment the really transformed differential volume dV_{a} is directly proportional to the differential
volume dV_{a, ur} that we would get
without restrictions, and to the volume fraction of the not yet transformed
g phase, which is simply 1 –
(V_{a}/V). This
gives us 





dV_{a, }^{ } 
= 
dV_{a, ur}^{ }
· 
1 – (V_{a}/V) 
or
1
1 – (V_{a}/V) 
dV_{a,}^{ } 
= – 
dV_{a, ur}^{ } 







Integrating both sides gives 





ln {1 – (V_{a}/V) } 
= – 
V_{a, ur}^{ }
V 






Inserting the expression for
V_{a, ur} from above gives the
final result, the JMAK equation in its most simple form: 





V_{a}/V 
= 
fraction of
transformed
volume 
= 1 – exp– 
æ
ç
è 
p · G · v^{3} ·
t^{4}
3 
ö
÷
ø 










= 1 – exp– 
K ·
t ^{n} 






In the generalized version of the
JMAK equation we have two constants, K and n,
which we have calculated for a case where all the assumptions made would be
valid. Such a hypothetical case is not very realistic, however. For realistic
cases, we can expect that the general structure of the JMAK equation will stay
intact but that the values of K and n will differ from the
calculated values. 


That the general structure of the JMAK equation
will stay intact is clear from looking at the graph of that equation: 






Graph of the JohnsonMehlAvramiKolmogorov
equation 






