The transformation is abrupt as temperature is changed, but it happens
a bit at a time as you add energy. Water and ice coexist at the freezing
temperature. An ice-cube floating in a water glass will keep it at the
freezing point as heat seeps in from the room until it melts. Water when
heated at the boiling point forms bubbles of vapor. The amount of energy
per unit mass it takes to transform water into the new phase is called the
*latent heat*, **L**. The ice cube and the
vapor bubble have sharp boundaries separating them from the water, with
a surface tension (or free energy per unit area) s.
It's this surface tension which keeps bubbles and raindrops round.

One can often *supercool* or *superheat* at first-order phase
transitions. If you put very pure water cold, with no dust particles in it,
in a very smooth glass container, it can be supercooled by many
degrees
DT before freezing into ice.
Similarly, water vapor can be supercooled: when it's 110% humidity it means
that the water vapor in the air has been supercooled past the temperature
at which raindrops can first form.

The reason that you can supercool or superheat through first-order phase transitions is because there is a free energy barrier separating the two phases. Simply speaking, you need to make a large bubble of the new phase in the old one before the new phase can grow. The reason that a small bubble can't grow is the surface tension s. Small bubbles pay a large cost in surface tension for a small gain in volume of the new phase.

To see this quantitatively, we need a bit of knowledge from statistical
mechanics. The free energy of a droplet of radius R is the sum of the energy
cost due to the surface tension, 4psR^{2},
and the energy gain due to the volume of the new phase,
-(4/3)pR^{3}r Df,
where r is the density (mass per unit volume) and
the free energy difference per unit mass
Df =
**L**DT/T_{c}.
(This follows from the fact that the entropy difference is
**L**/T_{c},
and that the entropy of each phase is the derivative of the free energy with
respect to temperature). Plotting the free energy F(R) (left), we find
a radius R_{c} which gets bigger as the supercooling or superheating
DT gets smaller
(R_{c} ~1/DT), and a barrier height that
also diverges (B ~1/DT^{2}).

Now, the reason that you can supercool and superheat a container of
water is the barrier B. Because you need to nucleate a droplet of the
new phase of radius R_{c}, you need to pay an energy cost B. The
nucleation rate is a prefactor times the probability of being on top of
the barrier. The probability of being on top of the barrier is given by
the traditional Boltzmann (or Arrhenius) factor exp(-B/kT). When the
supercooling or superheating gets small, B gets large, and the
nucleation rate exp(-B/kT) gets really, really small. It gets so small
that, even though the experiment has lots of water and thus lots of places
for a new droplet to form, and even though water molecules rearrange very
quickly (giving a big prefactor), the probability of nucleating an ice
crystal or gas bubble can be negligable.

This is the theory of *homogeneous* nucleation. Usually in real
life the new phase nucleates on some kind of dust particle, or some
defect in the material, or some flaw on the surface of the container.
(That's why bubbles in soda pop or in boiling water usually form on the
sides and bottom of the container, and why people seed clouds to try to
get rain.) The atoms in the unstable phase will use any short-cut available,
and a dust particle around the size R_{c} which attracts ice or
water can form an easy bypass, allowing the phase transition to occur almost
where the equilibrium transition temperature says it should.

- Melting a Copper Cluster: Microcanonical Critical Droplets and Negative Specific Heat, where we find that holding the energy fixed can give a stable ``critical droplet'', and a negative specific heat.
- Nucleation and Surface Growth: Layer-by-Layer Forever, where we apply nucleation ideas to an out-of-equilibrium surface growth problem, where thermal fluctuations are replaced by shot noise from atoms falling from the sky.
- Nucleating Cracks: Elastic Theory has Zero Radius of Convergence, where cracks are critical droplets separating the solid phase from the broken phase.

Last modified: March 11, 2002James P. Sethna, sethna@lassp.cornell.edu

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).