## Nucleation Science |

## 2. Homogeneous Nucleation | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

In the preceding module we have looked at a constant
but supersaturated concentration of vacancies or impurity atoms. That taught us a few important
things that we are going to need here - but the whole approach was unrealistic. The crystal,
after all, wants to get rid of the supersaturation and that involves that the concentration
comes down. We need to make a "big" precipitate with lots of point defects in it,
not just tiny clusters.As in the preceding module, we assume a crystal that is absolutely perfect except for some supersaturated point defects. The crystal has to do whatever needs to be done all by itself, in particular it must make the nuclei required to start precipitation without external or extrinsic "help". It would thus be logical to call that process "
intrinsic" nucleation. I won't, however, because it has always been called
"homogeneous nucleation" for historical reasons.
Since this is also a good term, let's go with the flow here. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Let's
assume that the precipitate is spherical, with a diameter of . The basic problem
in making precipitates is simple:r- Putting atoms
*inside*the precipitate leads to a*gain*in energy. This is good. - The interface precipitate - matrix has some interface energy. If the precipitate
grows because we keep putting atoms inside, the interface area increases. This
*costs*energy and that is bad.
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To get a feeling
of what is involved, let's look at some numbers. We have the following relations for the surface
area and the volume S of the spheres, the number V
of atoms or vacancies contained in the sphere, the number No. V of atoms or vacancies
sitting on the surface of the sphere, and the ratio between atoms in the bulk
and on the surface in percent. No. S | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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Of course, clusters smaller than 8, 9, or 12 atoms for bcc, prim. cubic or fcc crystals, respectively, have all of their atoms or vacancies at the "surface". | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

What we see very clearly is that the energy
gained by putting atomic defects into "prison" is pretty much lost again by having
most atoms or vacancies at the surface for very small clusters
or the nuclei of precipitates. Only for clusters larger
than 10 nm or so, the volume will win. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

We knew that all along. "You get more potato peel
from 20 small potatoes than from one big one!" is stated in the backbone right about here.True enough. But now we will calculate exactly how much more potato peel
we are talking here. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Essentially,
whenever we consider to produce a precipitate with radius by agglomerating
particles like impurity atoms, vacancies, interstitials or any mix thereof, we need to figure
out if the balance rD of the energies involved
makes the investment worthwhile. Only if the making of a precipitate offers a solid chance
for G(r)decreasing the free energy of a crystal
with a supersaturation of something, the investment would make sense. Note that I'm not saying
that there must be an Gimmediate or direct return on investment!
We have only two energy terms to consider: - The
**surface energy**that we have to invest for making a precipitate with a certain surface, or better interface area.*E*_{S}is quite simply given by the area*E*_{S}**4p**of the surface, times the specific interface energy*r*^{2}**g**of the interface in question. Its sign is always positive; we have to "pay". - The
(free) energy gained by imprisoning one surplus particle in the precipitate. This energy is
given by the volume
**4/3(p**of the precipitate times the energy*r*^{3})**D**gained per particle volume. Its sign is negative since we gain energy.*g*
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The energy balance thus quite simply writes | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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This is a simple function of the type y = ax^{2} – bx^{3}
a and b you enter, the general behavior
is always as shown below for the most simple case. In other words, our
curve looks exactly the same - just change the scales to G(r) and G
and provide the right numbers on the scales. With increasing radius r, starting
from r, the energy balance always increases, reaches a maximum, and then
decreases, eventually becoming negative as desired.r = 0 nm | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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What can we learn form this? Two things: |
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1. There is a critical radius for a viable nucleus, given by the r_{
crit}maximum
of the curve. A nucleus (or tiny precipitate) with this size will
G(r)not yet have provided a lower free energy for the crystal
but will produce a net return if it grows beyond that critical size. It then will tend to grow. Precipitates smaller than the critical radius would tend to
shrink. The critical size is easy to calculate from setting the differential quotient dG(r)/dr = 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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2.
I derived that result for the clustering of something like vacancies or impurity atoms inside
a solid crystal. Let's now look at the solidification of a liquid. Look, or even better, think. Did you get it? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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When a liquid solidifies, you need to start somewhere to put the atoms together in crystalline
form. There is some energy gain for temperatures below the melting point if atoms are put
into the system, and there is some solid - liquid interface with some interface energy g. Same thing as before. Just the numbers for g
and D are different.gYou do all the other examples. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Actually, looking
at freezing offers a little bonus: We know something about D
! We know that in order to melt a certain amount of a solid, you must invest a defined amount
of energy. When the liquid solidifies, you get exactly that energy back. It is called the "gheat of fusion" , and is rather
easy to determine. In contrast, the H_{F}D from above,
describing properties of particles like vacancies
or impurity atoms and their temperature dependence is not all that easy to get.g | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

For
the critical radius of a nuclei for solidification
we get: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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We have H_{F} · (T_{
M} – T)/T_{M}D.
gThe quantity , i.e. the difference
between the melting point temperature and the actual temperature at which the material is
still liquid, is called "T_{M} – Tsupercooling" or undercooling. Supercooling and supersaturation are closely related; they simply describe
how much a system deviates from equilibrium. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

The critical radius scales directly with the interface
energy g. The interface energy is pretty much a constant,
independent of temperature or supersaturation, and thus easy to assess.The critical radius scales inversely with the energy D that is to gain,
and we if we generalize a bit, with the supersaturation / supercooling. gSo the critical radius does decrease for small interface energies, large energy gains, and with increasing supersaturation. That makes sense. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

But let's not dawdle any more. Let's look at numbers for critical radii. There
is, of course, a large spread of possible g, and H_{F}
D values for whatever possible process,
supercooling or supersaturation you look at, but no matter:g | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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What does "far
too large" mean? Well, let's first consider what we could have in terms of
small nuclei. We did that already in the "Global
and Local Equilibrium for Point Defects" module. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

We might get a very few clusters with more
than 5 or 6 vacancies / impurity atoms in it, but that will be clusters with nominal radii
well below 1 nm. And we need much more for most
cases - 5 nm or 10 nm! Nuclei with such a critical radius carry far more energy
around than a typical vacancy or impurity atom, and the probability
for having clusters or nuclei with such a high energy is given by the ubiquitous Boltzmann factor E_{n}exp-
- and is zero for all practical concerns!E_{n}/kT | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

What that means is simple: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

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Well,
almost never. Solidification, for example, would require
undercoolings of up to 100 K before the critical radii required are small enough to initiate
freezing, and that's just not what we observe. Typically, undercoolings of at most a few degrees
will do the trick. Only ultra-clean liquids under tightly controlled laboratory conditions
can be supercooled to a large extent, i.e. stay liquid well below their freezing point. |
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Where does that leave us? Well - obviously
there is another, more efficient nucleation mechanisms. Yes, there is! It is called heterogeneous nucleation (surprise!), and it utilizes defects. |
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That will be the topic of the next module in this series. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Back to | ||

Nucleation Science Overview | ||

1. Global and local equilibrium for point defects | ||

On to | ||

3. Heterogeneous
nucleation | ||

4.
Size and density of precipitates | ||

5. Precipitation and structures | ||

8.2.2 It's a Long way to Nirvana

TTT Diagrams: 1. The Basic Idea

Experimental Techniques for Measuring Diffusion Parameters

Size and Density of Precipitates

Intrinsic Gettering in Silicon

Global and Local Equilibrium for Point Defects

Constitutional Supercooling and Interface Stability

TTT Diagrams 4. Experimental Construction of TTT and Phase Diagrams

© H. Föll (Iron, Steel and Swords script)