Nucleation in the Ising Model

The Ising model can also be used to study the nucleation of one phase inside another. Supercooling water and waiting for an ice crystal nucleus to form can be shown to be quite analogous to changing the external field H from H>0 to H<0 at a temperature T<T_c. (The analogy with changing the temperature or pressure of a gas and waiting for a raindrop is even better!)

Magnetize the system at T = 1.5 in a field H = 2 (or just hit the M=1 button). Now set H = -0.4 and watch the magnetization. Flip the system a few times to get an idea of the nucleation rate. Notice that the system waits for a fluctuation: when a white region of sufficiently large size appears, it can grow and take over. Try lower fields H = 0.3 and 0.2): you'll surely get tired of waiting for the graphics, but you can set the "Sweep Increment" using the configure button higher (graphics is the speed bottleneck in this program) and use the zero-crossing of the M(t) graph to time the nucleation event. "reset" will zero out the old stuff on the graph.

The nucleation rate is often estimated using ``critical droplet theory''. Small droplets of the stable phase will shrink due to surface tension: large ones grow due to force from the external field. If the surface free energy is (2 pi R)*sigma, and the external field makes a energy difference of (pi R^2)*H*(2*M(T)), show that the energy as a function of R has a local maximum at R=sigma/(2 M(T) H). You can measure M(T) directly. We can guess that sigma will be roughly two for each "broken bond" (since we put the bond strength J equal to one) at low temperatures. Although sigma vanishes at the critical temperature T_c, it stays roughly near two except for temperatures quite near T_c. What do you estimate the critical radius is at T=1.5 and H=0.2?

We can test this critical radius idea by artificially creating droplets of various sizes. Starting at M=1, T=1.5, and M=-0.2, use the left mouse button (only one on Macs) to flip a region of radius four. (You'll want to change the Sweep Increment back to one.) Does it shrink or grow? Try a radius of ten. What is the critical radius, where the system shrinks roughly half of the time? How does it compare to our prediction above? Try this for larger and smaller fields. Is our analysis roughly right?

Before we get too excited about the powerful agreement between theory and experiment, we should try deducing the nucleation rate predicted by the critical droplet theory. What is the free energy barrier B we predict (surface energy plus external field) for this radius? Finding a particular critical droplet in our system (of a particular shape and position) will occur with probability exp(-B/kT). Multiply this by the number of positions that the droplet might start at (30x30). How many shapes for critical droplets do we need (related to the entropy at the saddle point) in order to get the nucleation rate you observed above? Does this seem sensible, given the kinds of fluctuations you saw in the droplet as it grew?


Other Ising Model Presets

Description of the Model.
Phase Diagram
Magnetization M(T)
Nucleation
Domain Coarsening

Last modified: June 1, 1997

James P. Sethna, sethna@lassp.cornell.edu, http://www.lassp.cornell.edu/sethna/sethna.html

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