## 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:- How large will those precipitates be (on average, of course) and what determines their
*size*? - How many will I find in a given volume or in other words: what kind of precipitate
*density*will I get?
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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. |
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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: - At high temperatures our point defects - vacancies, impurity atoms, whatever - can move vigorously and thus can go places. On the other hand, the supersaturation is low; definitely lower than if you have the same concentration at lower temperatures.
- The mobility of point defects is described by their diffusion
coefficient
; the average distance they covered after a time*D*(*T*) =*D*_{0}exp-(*E*_{D}/k*T*)is given by the diffusion length**t** .*L*= (*Dt*)^{½} - At low temperatures, the supersaturation is high but movement is sluggish.
- Nothing moves at room temperature.
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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" determine the
maximal size of a precipitate.L_{to} | ||||||||||||||||||||||||||||||||||||||||||||

Note that the answer to the second question from above is also clear now. We have one precipitate
in a volume of about (L^{3}_{to})1/(L^{3}_{to}) |
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All that remains to do is to calculate . For that
we need to know the temperature profile L_{ to}, the way the specimen cools down with time. T(t) |
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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 ; we can always take the root when we are done.
L^{2}_{to} | ||||||||||||||||||||||||||||||||||||||||||||

Quite generally, the (square) of the average distance or Ltotal
diffusion length that a diffusing object covers during cooling down in three dimensions is given by
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The diffusion coefficient D(T)D(T)
= D_{0} · exp–(E_{D} /kT) = migration energy
with typical values of E_{D}(0.5 - 2.5) eV. | ||||||||||||||||||||||||||||||||||||||||||||

The most simple way of cooling is to put the specimen into some cold environment. The temperature
profile then follows proper beer-froth
dynamics (get a beer and look it up!) or T( t) = T_{0} · exp–(lt ) is the starting temperature,
and T_{0}l
the "decay parameter".While the "decay parameter" l is
most convenient for the equations, the temperature gradient
[dT/dt]_{t = 0} = –lT_{0}1800 K with a gradient of –100 K/s, we have
a decay parameter l = 1/18 s^{–1} = 0.055 s^{–1}Another convenient way for looking at cooling-down dynamics is to define a " cooling
half-time" t_{half} = 1/l1/l actually takes you to 0.37
- but what the heck. If you are a stickler for correctness or just anal-retentive, you must go with T_{0}. t_{half}
= (1/l) · ln(0,5) » 0,69/l
Combining the two, we can express the cooling half-time in terms of the temperature gradient as t_{half}
= T_{0}/[dT/dt]Here is the illustration for all that: | ||||||||||||||||||||||||||||||||||||||||||||

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Writing out the integral by inserting the basic equation
for and the exponential decay function of the temperature from above, we get a little beauty:
D(T) | ||||||||||||||||||||||||||||||||||||||||||||

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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
after a rather lengthy exercise in calculus is L_{to} | ||||||||||||||||||||||||||||||||||||||||||||

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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 D_{0} = 10^{–5} cm^{2}s^{–1} |
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Of course, diffusion lengths below 10 are meaningless
for atoms and just shown for completeness.^{–4} µm = 1 Å | ||||||||||||||||||||||||||||||||||||||||||||

What we see quite generally is:- The cooling rate matters less than the migration energy.
- Most everything happens at high temperatures.
**1 µm**is already a large distance under many circumstances.
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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 , that all the impurity atoms contained in that volume
end up in a "cube" precipitate, and that the size of one "particle" is L^{3}_{to}0.3 nm. That's
what we get for the size of the precipitate-cube in nanometer:^{3} | ||||||||||||||||||||||||||||||||||||||||||||

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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 "low-alloy" steel, containing fast-diffusing
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 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._{3}C | ||||||||||||||||||||||||||||||||||||||||||||

The Great Simplification | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

The total diffusion length 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 L_{to} iron and steel, and more
simplistic in how to look at the situation. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

For that it is beneficial to rewrite the equation for
by introducing parameters that are more easy to relate to. My choice is:L_{to}- The "cooling half-time"
or*t*_{half}= 1/l .**l = 1/***t*_{half} - The diffusion length
for a time*L*_{0}**t**at the starting temperature_{1}= 1s. That is just the square root of the diffusion coefficient and directly obtainable from diffusion data. We then have*T*_{0} . This gives us*L*_{0}= (*D*(*T*_{0}) · t_{1})^{½}= (*D*_{0}· t_{1})^{ ½}· exp–(E_{D}/2k*T*_{0})**(***D*_{0})^{½}exp–(E_{D}/2k*T*_{0}) =*L*_{to}/(t_{1})^{½}
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Inserting this in the equation above yields | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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While this may still look complicated, it makes estimations of
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 L_{to}g(T_{ 0}, E_{D}, t_{half}). So let's look at the parameters. E_{D} | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Looking at typical migrations energies for atoms moving around
in iron, we findE_{D}- Values close to
**1 eV**for the interstitials carbon (C), nitrogen (N) and oxygen (O). - Values roughly around
**1.5 eV**for most substitutional atoms.
and E_{D} = 1 eV2 eV thus covers most everything. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Looking at typical temperatures for iron, we can take the melting
point of pure iron at around T_{0}1800 K as an upper limit, and the "working " temperature off a smith at around
1000 (^{o}C1832 ) as a lower limit - in the form of ^{o}F1300 K, of course. This gives
us k values of T_{0}0.16 eV or 0,11 eV, respectively. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

For cooling half-times 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.
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Now we can come up with two tables for the values of the factor
that (with interpolations) cover pretty
much every cooling-down event.g(t_{half}, T_{0}, E_{D}) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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What the numbers mean is clear. An atom that moves a certain distance 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. L_{to}
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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 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:L_{to} | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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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 |
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2. Homogeneous nucleation | ||

3. Heterogeneous nucleation |
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On to | ||

5. Precipitation and structures |
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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)