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Have a read of the info below and then attempt the quizzes after each section. There is a pdf at the bottom of the page with extra information if needed.
What happens during Homogenous Nucleation?
Atoms cluster together to form spherical nuclei
These nuclei have radius r and form within the original phase (i.e solid nuclei forming within a liquid phase during solidification)
For the atoms to cluster together to form stable nuclei and nucleate to become stable, they need a negative value for Gibbs's free energy.
Why can nucleation take place?
During this solidification, there are two sources that contribute to Gibbs Free Energy. These sources ultimately affect if a nucleus can become stable.
Volume Free Energy = Free energy difference between solid and liquid phases
Surface Free Energy = Energy for the formation of the solid-liquid phase boundary of the nucleus
Remember dGv = H - TdS, so dGv will be negative if T is below solidification Temperature - as when dGv is negative the phase change (solidification) can spontaneously happen.
As solid atoms cluster to form a nucleus, the free energy of the nucleus (a combination of Volume/Surface Free Energy) increases.
At certain temperatures, the radius reaches a certain size (critical radius r* ) then its free energy surpasses the activation free energy (dG*) which means the nucleus becomes stable and can enter the growth stage.
If the nucleus does not reach this radius and doesn’t become stable, it loses energy and dissolves away into nothing again.
Key Definitions
Embryo - Cluster of atoms that haven’t yet reached the activation free energy (and thus don’t have a radius r* yet). This means they aren’t stable at this point.
Nucleus - Cluster of atoms that have reached r* and thus dG* and so are sable and can continue to begin to grow.
Activation-free energy (dG*) - Energy an embryo must have before it can become a stable nucleus and progress to the growth stage.
How do we find r* and dG*?
Through differentiating dG w.r.t r, finding stationary point (dG/dr = 0), and rearranging, we get an equation for critical radius r*:
Substitute r* equation above into original dG equation (with volume/surface free energy terms) to get an equation for free activation energy:
Undercooling?
Volume-free energy change dGv is the sole driving force that affects whether a phase transformation occurs. However, it is a function of Temperature.
When you decrease T further from Ts, dGv becomes more and more negative. The difference between Ts and this lower temperature value is called undercooling. Thus the greater the undercooling, the greater the driving force for nucleation, and the more readily nucleation can occur.
Now try part one of the quiz! : Click Here
Temperature dependant functions of r* and dG*
If we substitute the previous equation for dGv into the equations for r* and dG*, we are left with:
Both r* and dG* decrease as temperature decreases.
As temperature is lowered further from the solidification temp, nucleation occurs more readily.
Number of stable nuclei, n*
Number of stable nuclei, n* ( having radii > r*), is a function of temperature:
K1 is
a constant related to the total number of nuclei in solid phase
Frequency of attachment, vd
Temperature affects the rate at which atoms cluster together via short range diffusion during nucleation.
The frequency at which atoms from the liquid attach themselves to the solid nucleus, vd, is temperature dependent.
Nucleation rate, N
The nucleation rate (nuclei per unit volume per second), is proportional to the product of n* and vd.
Nucleation rate continued
At temperatures lower than Tm , the nucleation rate first increases, achieves a maximum, and subsequently diminishes.
At high temperatures, nucleation rate is small due to a low activation driving force.
At lower temperatures, a low atomic mobility suppresses the nucleation rate.
Amount of supercooling required for homogeneous nucleation
Now try part two of the quiz! : Click here
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