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Physics of Thin Films

PES 449 / PHYS 549

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:

eutectic phase diagram and driving forces

  • 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:
  1. Nucleation of new phase
  2. 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

nucleation rate peaks below eutectic temp.

The maximum rate of nucleation is at some T < Te

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

transformation rate from nucleation and growth rates

Nucleation details

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

Two issues:

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

Homogeneous Nucleation

vapor --> liquid (solid) for a pure material with NO substrate

vapor to solid picture

Energy minimization involves two terms:

  1. volume transition
  2. surface formation

volume transition:

equation for free energy change per unit volume

where W is the atomic volume, PS is the pressure above the liquid (solid), and PV is the pressure in the vapor.

We want PV > PS so that ĘG is negative

=> supersaturation provides the driving force.

surface formation:

surface energy change

Change in surface energy is always positive when forming surfaces.

Total energy change:

total energy change

total free energy change and critical radius

note:

  • initial formation of nuclei has increase in 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

equations for critical radius and energy

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:

nucleation rate picture

expect nucleation rate to be given by equation of nucleation rate

N* = concentration of critical nuclei (nuclei/cm3)

A* = critical surface area of nuclei

w = flux of atom impingement (atoms / cm2sec)

Consider each of these three terms:

 nucleation rate terms - detail

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