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## Nucleation and Growth

Ohring: Chapter 1, Section 1.7

Connection to Phase Diagrams

Can phase diagrams help us in understanding ** rates**
?

Consider cooling a liquid into a solid through a eutectic point:

- at point A: solid is not stable so will not form
- at point B: solid and liquid are both stable so no driving force to solid
- at point C: liquid is unstable - will form solid
- at point D: liquid is unstable - will form solid
- further from equilibrium => greater driving force to form solid

Transformation to solid phase

Requires:

- Nucleation of new phase
- growth of new phase

Nucleation:depends on:

- liquid phase instability

- driving force toward equilibrium (as above)
- increases as we move to lower temperatures
- diffusion of atoms into clusters

- increases at higher temperatures
combine these two terms (multiplication) to determine the total nucleation rate

The maximum rate of nucleation is at some T < T

_{e}

Growth:growth of the phase is diffusion controlled => increases with temperature

Transformation rate:total rate of forming solid is product of nucleation rate and growth rate

Nucleation details

When moving into a 2 phase region on phase diagram - how does the new phase form ?Two issues:

- Thermodynamics: Is nucleation possible ? (energy minimization)
- Kinetics: How fast does it happen ? (nucleation rate)

Homogeneous Nucleation

vapor --> liquid (solid) for a pure material with NO substrateEnergy minimization involves two terms:

- volume transition
- surface formation

volume transition:where W is the atomic volume, P

_{S}is the pressure above the liquid (solid), and P_{V}is the pressure in the vapor.We want P

_{V}> P_{S}so that ĘG is negative=> supersaturation provides the driving force.

surface formation:Change in surface energy is always

positivewhen forming surfaces.

Total energy change:

note:

- initial formation of nuclei has
increasein G => metastable- if r < r* then nuclei shrink to lower G
- if r > r* then nuclei grow to lower G
- r* is a critical radius for nuclei
Films will also have an interface term => heterogeneous nucleation (coming soon!)

Nucleation rate

How fast will the critical nucleus continue to grow ?Consider the rate at which atoms will join the critical nuclei:

expect nucleation rate to be given by

N* = concentration of critical nuclei (nuclei/cm^{3})A* = critical surface area of nuclei

w = flux of atom impingement (atoms / cm

^{2}sec)Consider each of these three terms: