Assignment 4

In our last homework we wrote a code to simulate the Ising model in two dimensions at constant temperature and with fixed magnetization, using Kawasaki dynamics. Using a mix of 30% up-spins and 70% down-spins, we brought the system from a random initial state toward equilibrium. We observed the formation and growth/coalescence of clusters.

In this assignment, we will explore how cooling rate affects microstructure during phase separation. Start your system at a relatively high temperature and try cooling it in a linear fashion, with temperature T=To-rt where To is the initial temp, t is time measured in Monte Carlo steps, and r is a cooling rate.

Note: Cool all three systems to the same final temperature of T=0. Thus the slower cooling rate system needs to run longer.

I tried To=2.0 and cooling rate r=0.00001 (slow), r=0.00005 (medium) and r=0.0001 (fast). The slowest run requires 200000 time steps. If this is beyond your capability, then use faster cooling rates.

Make a plot showing energy vs time for all three cooling rates. You will need to write a new routine for measuring the system’s total energy.

Count the number of clusters in the final state of each run. You may do this by eye if you like, or write an algorithm as we described in class. Does it depend in an interesting way on the cooling rate? If you write a cluster counting algorithm, try plotting the number of clusters vs time during one of your cooling runs.

Make pictures showing the final microstructure for all three cooling rates.