Cluster Variation Method – 2-D Case – Configuration Variables, Entropy and Free Energy

Following the previous blog on the 1-D Cluster Variation Method, I illustrate here a micro-ensemble for the 2-D Cluster Variation Method, consisting of the original single zigzag chain of only ten units (see previous post), with three additional layers added, as shown in the following Figure 1.

In Figure 1, we again have an equilibrium distribution of fraction variables z(i). Note that, as with the 1-D case, the weighting coefficients for z(2) = z(5) = 2, whereas the weighting coefficients for the remaining fraction variables (z(1), z(3), z(4), and z(6)) are all 1.

In this illustration, we have an ensemble that wraps around both horizontally (as with the previous 1-D case) and vertically to create an envelope. The connections with the “other edge” – both vertically and horizontally – are shown as the textured units with a dashed border in Figure 1. There are 24 units in this micro-ensemble.

As with the previous 1-D case, this micro-system is at equilibrium, so that:
x(A) = x(B) = 0.5, where x(i) is the fraction of units in state A or state B, respectively.

Further, the pairwise interactions are also at their equilibrium values (for equal amounts of A and B, and for no interaction energy). These additional variables are shown in Figure 2 of the previous post, Visualizing Variables with the 1-D Cluster Variation Method.

The reduced free energy for the 2-D CVM ensemble is given as

where as before

Further, as with the 1-D free energy equation, we note that the last two terms, which are Lagrangians, of course are at zero in the equilibrium case. Also, the enthalpy term is at zero.

In this particular case (equilibrium at x(A)=x(B)=0.5), we can create a simpler equation, much as was done in the 1-D case, and solve for an analytic value of the reduced free energy.

Related Posts and Technical Reports

• Visualizing Configuration Variables with the 1-D Cluster Variation Method, July 9, 2014.
• Analytic Single-Point Solution for Cluster Variation Method Variables (at x! = x2 = 0.5), Dec. 8, 2011.
• The Cluster Variation Method II: 2-D Grid of Zigzag Chains: Basic Theory, Analytic Solution and Free Energy Variable Distributions at Midpoint (x1 = x2 = 0.5), THM TR2014-003 (ajm), July 2014
• The Cluster Variation Method I: 1-D Single Zigzag Chain: Basic Theory, Analytic Solution and Free Energy Variable Distributions at Midpoint (x1 = x2 = 0.5), THM TR2014-002 (ajm), June 2014