Growth

The growth phase starts when a embryo exceeds r* (critical radius) and becomes a stable nucleus.

The way growth happens is through diffusion, a molecule will move through it's parent phase and across a phase boundary to join a nucleus. As the molecules need energy to move, and move faster the more energy they have, a high temperature causes a faster growth rate. From this, the equation

Where C is a constant, Q is the activation energy for long range diffusion, K is boltzmann's constant and T is temperature. If you look at the equation, you can see that as T grows larger, the exponential comes closer and closer to equaling 1, which will make C it's maximum possible value.

However, growth is also affected by the Nucleation rate. The colder a sample is, the more nuclei it will form. Think of homo and heterogenous nucleation, the colder a site is, the more sites will meet the energy requirements to be able to form a nucleus until eventually it will become homogenous nucleation, where the nuclei can form equally in the material. It's a bit like martensite forming, how you get lots of little needles of austenite and cementite.

As the graph above shows, the fastest transformation from liquid to solid occurs when the rate of Nuclei forming and the growth rate itself intersect, this is because the overall transformation rate is a product of the growth rate and nuclei formation rate. Lets explain this by considering some possibilities.

A bit below Melting temperature:

• • You have a very high growth rate as your molecules are easily able to move about through the parent phase and phase boundaries to find a nucleus. However, you need to actually have a nucleus for the molecule to find, and not many have formed as there aren't many sites with the right energy conditions to form. So overall you have a low transformation rate

A very low temperature:

• • You can form many, many different nuclei as almost everywhere in the material has the right conditions to form a nucleus. However, as the temperature is so low, it takes a long time for molecules to move and find a nucleus to grow the nucleus any more. So you also have a low overall transformation rate

When growth and nucleation rates are equal:

• • Now you have a good amount of sites which have favourable conditions for forming nuclei as it is not too hot, yet it is also hot enough that molecules have enough energy to diffuse through phases relatively easily. This results in an overall high transformation rate.

Now that you understand the basis of growth, let's talk about how this can affect a material and it's materials. There are typically two types of material grain structures, coarse and fine grains. Coarse grains form at temperatures just below the melting point and have a micro structure of large grains. Larger grains will be more ductile than coarse grains as there are less grain boundaries to prevent movement. If you think back to the previous example it makes sense, as few grains from but will end up being large. With fine grains, you have many more grains, but they are all very small and cause the material to be more brittle as there will be more grain boundaries preventing movement. Again, if we think of the last example, it explains it as many nuclei form but they won't grow much as the molecules can't diffuse.