Physics 557, Problem 15.45 "Vacancy mediated dynamics in binary alloys", Gould, Tobochnik & Christian
12/08/05
The applet can be viewed here
Spinodal decomposition is defined to be: a clustering reaction in a homogeneous, supersaturated solution (solid or liquid) which is unstable against infinitesimal fluctuations in density or composition. The solution therefore separates spontaneously into two phases, starting with small fluctuations and proceeding with a decrease in the Gibbs energy without a nucleation barrier.There are a lot of terms in this definition that require further explanation.
The system we simulated consists of a lattice of atoms of two types A and B. There are exactly the same number of A and B atoms. What we find, is that if the system is quenched instantaneously from high tempereature (when the system is homogeneous) to a temperature below the critical temperature a "clustering reaction" or domain formation occurs. This quenching could also be called "supersaturation". Atoms of type A cluster together, as do atoms of type B. This is truly are remarkable phenomenon, the reader should note however that it is not understood.
A description of the problem can be given, quite heuristically, as follows. The Gibbs free energy of a system is given by
where H is the enthalpy and S is the entropy.
It is well known that a "natural" process will occur spontaneously if dG less than zero and no spontanteous process will occur if dG greater than zero (dG is the change in Gibbs energy G(final) - G(initial). That is "systems want to minimize the Gibbs free energy".
A more detailed analysis gives the condition for spinodal decomposition. See this webpage.
The plot below shows the behaviour of dG for varying A and B atom concentrations (encoded in the w terms which we will not discuss here) and temperature. Note that the behaviour of dG seems quite normal for T greater than T_c. However when we reach the critical temperature, dG increases as X_B increases towards equal concentrations of B and A type atoms. This can be seen manisfestly with the inversion of the dG curve below.
Fig 1 : Plot of change in Gibbs energy with concentration
Let us look more closely at what Fig 1 actually means.
This curve is best understood by slices. That is, pick a concentration of X_0 and examine the behaviour of dG (see Fig 2)
Fig 2 : Picking a slice of the curve in Fig 1 for T lot less than T_c
We are therefore working on the premise that the total number of A type atoms and the total number of B type atoms does not change much, we are just "re-ordering" the atoms to minimize the energy by using small fluctuations in the concentrations of either type of atom. If we pick this concentration slice and examine the behaviour of dG an interesting observation is made. Note that if the slice we pick is along the part of the curve where the curvature is positive (Fig 3) a small fluctuation that results in the changing of the composition of A and B type atoms is unlikely since the result of such a change in composition is an increase in the Gibbs energy.
Fig 3 : Curve for T lot less than T_c and regime where nucleation occurs
Fig 4 : Spinodal decomposition condition
The endgame of this behaviour is a "nucleation" or "clustering" of each type of atom but not total separation of the two atom types. Note that if the slice we picked is along the part of the curve where the curvature (the second derivative of dG) is less than zero (Fig 4) we find that for a small fluctuation in the concentration of either A or B type atom the energy of the system will decrease. The system will therefore be unstable in this negative curvature part of the dG regime. This decreasing energy will therefore continue unabetted until complete decomposition occurs. This is known as "spinodal decomposition".The result is that the condition for spinodal decomposition is:
where X_B is the concentration of atoms of type B.
After all of this analysis we can see from the image below (Fig 5) that the end result is; if we quench the system with certain types of initial concentrations of A and B atoms we should see either nucleation or spinodal decomposition depending on where we are with respect to the inflection point of the dG curve. If the second derivative is less than zero we get instability and the system is spinodal. In contrast if we are at the part where the second derivative is greater than zero we are more stable to small fluctuations and only a small amount of clustering or nucleation occurs.
Fig 5 : Miscibility/phase space graph
In our simulation we studied a vacancy mediated binary alloy, using the Metropolis algorithm. The system is initiated with equal amounts of A and B (up and down spin) type atoms. We calculated the correlation function C(r) for this system. C(r) measure the average spin over a radius r. The first zero of C(r) is where there is no longer correlation. "r" is the only way we can measure this system, it is said to "characterize" the system. It is, in effect, a measure of the domain size or cluster size. The first set of images below (Fig 6) shows the system in its intial state when spins have been randomly assigned to the lattice points. The blue dot (cannot be seen on web image below) is the vacancy. The other dots are either A or B atoms. The second image (Fig 7) shows the system after a small number of steps and the final image (Fig 8) shows the complete spinodal decomposition.
Fig 6 : Initialization of lattice
Fig 7 : Situation after a short time
Fig 8 (a) : Complete spinodal decomposition
If we continue running the simulation we found that the regions of A and B atoms actually rotate. As shown in Fig 8 (b).
Fig 8 (b) : Rotation of spinodal decomposition after a long time
We also evolved the system for a temperature above the critical temperature and observed that no nucleation took place (Fig 9).
Fig 9 : For a temperature of T = 3*T_c we observed no nucleation or decomposition, as expected.
We also plotted the correlation (using two methods) below. The behaviour of C(r) is as expected (see Fig 10). After a long time C(r) is 50 for a 100*100 size lattice. R (the other measure of the cluster size) is roughly half of this (see Fig 11).
Fig 10 : Correlation function using covariance
Fig 11 : Correlation using domain growth kinetics
We also determined the scaling behaviour of R. We found that R scales with the 1/3 power of t (t = time), by plotting log R against log t. Note it converges to 0.33.
Fig 12 : Scaling exponent of 1/3
The final image below (Fig 13) shows what happends when you choose a temperature low enough and a composition sufficiently biased. No spinodal decomposition takes place but rather nucleation. This is the "nulceation" phase one can see in Fig 5 above. For Fig 13 below there is no zero for the correlation function C(r).
Fig 13 : Nucleation occurs at T = 0.2*T_c and with 9 times more B atoms than A atoms.