Nucleation Science 
4. Size and Density of Precipitates  
The Basics  
We have a supersaturation of point defects, we have some heterogeneous nucleation  finally we can make our precipitate. Two
(related) questions should now occur to you:
 
Looking at energy balances, as in the preceding modules, will not do the trick here. If you just minimize the energy, you will of course end up with exactly one spherical precipitate, with a size that is just right to accommodate all the point defects that were supersaturated in your crystal. You can't have a better surface to volume ratio that that.  
Equally
of course, that is not
what you will find in iron or steel at room temperature (or in most other crystals) if you
wait less than a sizeable fraction of the age of the known universe (about 14 billion years,
by the way). Remember: if nothing moves, nothing happens. The typical situation we have is:


So
how large can a precipitate be? There is an easy general answer: it can at best be so large
as to contain all the point defects that had a chance to reach it. In other words: All the point defects that were contained in a sphere with radius "total diffusion length" L_{to} determine the maximal size of a precipitate.  
Note
that the answer to the second question from above is also clear now. We have one precipitate
in a volume of about  
All that remains to do is to calculate L_{ to}. For that we need to know the temperature profile T(t), the way the specimen cools down with time.  
Then we calculate the diffusion coefficient at some temperature,
see how far a point defect could go in some small interval around that temperature for the
time the specimen stayed in that interval, repeat the procedure for other temperatures, and
add everything up. In other words; we integrate! It is easiest to do that for L^{2}_{to}; we can always take the root when we are done.  
Quite generally, the (square) of the average distance L or total diffusion length that a diffusing object covers during cooling down in three dimensions is given by  
 
The diffusion coefficient  
The most simple way of cooling is to put the specimen into some
cold environment. The temperature profile then follows proper beerfroth dynamics (get a beer and look it up!) or While the "decay parameter" l is most convenient for the equations, the temperature gradient Another convenient way for looking at coolingdown dynamics is to define a "cooling halftime" Combining the two, we can express the cooling halftime in terms of the temperature gradient as Here is the illustration for all that:  
 
Writing out the integral by inserting the basic equation for D(T) and the exponential decay function of the temperature from above, we get a little beauty:  
 
The integral
is a little math beauty because it looks neat, approachable and easy to conquer. It's like
female human beauties. Behind the comely surface often hides a strong mind, and conquering
is not all that easy either. Strangely enough, those humans who have no problem with the integral
(few and mostly male) typically have problems conquering that female beauty. Our little math beauty is not easy to do but needs to be wooed by suitable substitutions and approximations.  
What one gets for the total diffusion length L_{to} after a rather lengthy exercise in calculus is  
 
That
is an easy function to plot for some bandwidth of parameters. Shown below are curves for diffusion energies from 0.5 eV to 2 eV. Cooling rates from 1 K/s to 50 K/s and a  
 
Of course, diffusion lengths below 10^{–4} µm = 1 Å are meaningless for atoms and just shown for completeness.  
What we see quite generally is:
 
We knew that already to some extent. But now we have numbers for all kinds of circumstances.  
Let's look at a few numbers for possible precipitate sizes. For simplicities sake we assume that the volume is L^{3}_{to}, that all the impurity atoms contained in that volume end up in a "cube" precipitate, and that the size of one "particle" is 0.3 nm^{3}. That's what we get for the size of the precipitatecube in nanometer:  
 
Only in the two red cases the size of the precipitates surpass 1 µm, making them just about visible in a light microscope.  
Looking at carbon steel or "lowalloy" steel, containing fastdiffusing carbon in the 1 at% range, and slower diffusing stuff in the 0.01 at% to 1 at% range, we can expect to find carbon precipitates (Fe_{3}C or cementite) in the µm range and thus visible in a light microscope. Everything else should be far smaller, however, and thus had to escape the attention of early steel scientists, who could not yet command electron microscopes.  
The Great Simplification  
The total diffusion length L_{to} is quite important for many things related to iron and steel, and that's why I want to dwell on it a bit longer. I will get more specific for the atoms diffusing around in iron and steel, and more simplistic in how to look at the situation.  
For that it is beneficial to rewrite the equation
for L_{to} by introducing parameters that are more easy to relate to.
My choice is:


Inserting this in the equation above yields  
 
While this
may still look complicated, it makes estimations of L_{to} very easy
for all the atom where you have some Arrhenius diagram of their diffusion behavior. All we
need to do is to make a little table for typical values of the number 

Looking at typical migrations energies E_{D} for atoms moving
around in iron, we find
 
Looking at typical T_{0} temperatures for iron, we can take the melting point of pure iron at around 1800 K as an upper limit, and the "working " temperature off a smith at around 1000 ^{o}C (1832 ^{o}F) as a lower limit  in the form of 1300 K, of course. This gives us kT_{0} values of 0.16 eV or 0,11 eV, respectively.  
For cooling halftimes we might pick 36.000 s (10 hr) 3600 s (1 hr), 360 s (6 min) and 36 s, again covering most practical situations in between the two extreme values.  
Now we can come up with two tables for the values of the factor g(t_{half}, T_{0}, E_{D}) that (with interpolations) cover pretty much every coolingdown event.  
 
What the numbers mean is clear. An atom that moves a certain distance L_{to} within 1 second at one of the two temperatures given, will go for a distance that is larger by the number in the table for the conditions indicated.  
Let's look at an example. From the Arrhenius
plots in the "diffusion in iron" module
you can see that within 1 s carbon in iron covers roughly 50 µm at 1800
K, and 10 µm at 1300 K. It's migration
energy is close to 1 eV. Looking at the table you realize that carbon covers 53.300 µm = 53 mm for sluggish cooling from the melting temperature, and still 150 µm for extremely fast cooling. If the starting temperature is 1300 K, it moves 900 µm or 30 µm for the extremes in cooling, respectively. The same exercise for manganese, starting with 70 nm or 2 nm, respectively, and a migration energy of 1 eV, gives 5.25 µm or 140 nm for the extremes and 1800 K, and 128 nm or 4 nm for 1300 K cooling. 

Yes, I know it is not quite that easy. Cooling relatively pure iron from the melting point at 1800 K to room temperature involves a bcc fcc phase change at high temperatures and a fcc bcc at around 1.000 K, with the carbon moving much more sluggishly in the fcc austenite phase; this is clearly visible in the Arrhenius plot. So we overestimate the total diffusion length L_{to} somewhat  but who cares? We only get orders of magnitude anyway. The consequence of this little exercise are obvious and dramatic, no matter how you look at details:  
 
It doesn't matter how you cool down, and in most cases equilibration occurs over much larger distances. Provided, of course, the carbon is atomically dissolved and can move freely in all directions  
In other words: banging together thin sheets of low carbon / high carbon steel for damascene pattern welding won't do you much good. You don't get a compound material but a rather uniform material with an medium carbon content. I have stated that before but now we are one step more quantitative. I'll come back to this topic later again.  
Back to  
Nucleation Science Overview  
1. Global and local equilibrium for point defects  
2. Homogeneous nucleation  
3. Heterogeneous nucleation  
On to  
5. Precipitation and structures  
TTT Diagrams: 1. The Basic Idea
Phenomenological Modelling of Diffusion
Segregation at Room Temperature
Global and Local Equilibrium for Point Defects
© H. Föll (Iron, Steel and Swords script)