When a binary alloy is rapidly quenched from high temperatures to a low temperature unstable state, a pattern of domain formation called spinodal decomposition takes place as the two metals within the alloy separate out. This process has been investigated in detail for bulk materials. Lifshitz and Slyozov1 predicted that the linear domain size scales with time as R ~ t1/3. This result is independent of the dimension for d >= 2 for binary alloys, and has been verified experimentally and in computer simulations. The behavior is modified for binary fluids due to hydrodynamic effects.
Most of the computer simulations of this growth process have been based on the Ising model with Kawasaki exchange dynamics. In this model there is an A or B atom at each site, where A and B represent labels for the two different metals. In the standard Ising model the energy of interaction between atoms on two neighboring sites is -J if the two atoms are the same type and +J if they are different. Monte Carlo moves are made by exchanging unlike atoms. A typical simulation begins with an equilibrated system at high temperatures. Then the temperature is changed instantaneously to a low temperature below the critical temperature Tc. If there are equal numbers of A and B atoms on the lattice, then spinodal decomposition occurs. If you watch a visualization of the evolution of the system, you would see wavy-like domains of each type of atom thickening with time.
Recently there has been some simulations performed with vacancy mediated dynamics,2 which have found that the scaling behavior begins at earlier times than with standard Kawasaki dynamics. Because of the large energy barriers which prevent real metallic atoms from exchanging position, it is likely that spinodal decomposition in real alloys also occurs with vacancy mediated dynamics. Because the number of vacancies in a real alloy is very small, we can do a realistic simulation by just including one vacancy. Then the only possible Monte Carlo move is to exchange one vacancy with one of its four neighboring atoms. To implement this algorithm, you simply need an array to keep track of which type of atom is on each lattice site and a variable to keep track of the location of the single vacancy. The simulation will run very fast because there is no complicated bookkeeping and all the possible trial moves are potentially good ones. In contrast, in standard Kawasaki dynamics it is necessary to either waste computer time checking for unlike nearest-neighbor atoms or keep track of where they are.
The major quantity of interest is the domain size R. One way to determine R is to measure the pair correlation function:
An alternative measure of the domain size is the quantity
The transition temperature for phase separation for the two dimensional square lattice is Tc = 2.269 ... J/k, where k is Boltzmann's constant. Using a lattice of size 50 x 50 or bigger run at least ten quenches. Begin with a random lattice and quench to Tc/2. Find the domain growth exponent and periodically show a visualization of the lattice so that you can see the growth pattern. A good idea is to take data at times t = 2n where n = 1, 2, 3 ...
At what value of time does the log (R) versus log (t) plot become linear? Does the behavior change for a different quench temperature? Try Tc/5 and 0.7Tc. Do both measures of the domain size give the same results? Repeat the measurements in three dimensions. Do you obtain the same exponent?
Please send suggestions and comments to Harvey Gould, hgould@clarku.edu, or Jan Tobochnik, jant@kzoo.edu.
Introduction to Computer Simulation Methods, Harvey Gould and Jan Tobochnik, Addison-Wesley (1996).Updated 26 October 1998.