Quasicrystalline lattice model

To relate heats of mixing to energies of interaction we make use of the quasicrystalline lattice model of a liquid. In this the molecules are pictured as occupying points on a lattice whose co-ordination number (or number of nearest neighbours to a given molecule) is 2 . In the case of liquids 2 must be regarded as a statistical average. 

We see that the quasicrystalline lattice model employed above must be used with prudence, especially when the molecules are of complicated shapes. 

Figure 3.2 Schematic representation of quasicrystalline lattice model for solutions, (a) Simple solution mixture of molecules of equal size, white circles representing the solvent molecules and filled circles the solute molecules. It is assumed that solvent molecules can exchange sites with solute molecules. This results in an increase in the number of ways they can be arranged, and hence in an increase in entropy Jb) Polymer solution mixture of solvent molecules (unfilled circles) With a polymer molecule composed of chain segments (each segment represented by a filled circle) tied with chemical bonds. It is assumed that solvent molecules can exchange sites with polymer chain segments. This results in an increase in entropy. (After Flory, 1953.)...

Roe (1975) developed a quasicrystalline lattice model for conditions where Xi2 Xcr (where Xcr is the critical value of the interaction parameter at the phase separation) and for Xn Xcr Xcr- Under the first conditions (high immiscibility), the theory predicted a proportionality between Vj2 and Xn whereas under the secmid (near the phase separation), a proportionality between Vi2 and xn was predicted. By contrast with the previously summarized Helfand and Tagami predictions. Roe s theory indicates that the product V12A/ should be proportional to... 

These treatments suppose the surface molar areas of A and B atoms to be equal. However, for alloys with A 0, the low molar volume much greater than that of A (up to 100%). Typical examples are the Cu-Pb, Cu-Bi and Al-Pb systems, for which the experimental decreases in rxLV caused by adding small quantities of B are significantly more than those predicted by the regular solution model. These differences have been attributed to the fact that not all the sites of the quasicrystalline lattice on which A and B atoms are distributed are equivalent (Goumiri et al. 1979). 

The molecules in the pure liquids and in their solution are considered to be solid-like, in a quasicrystalline state, where the molecules do not translate fully in a chaotic manner as in a gas, but where each molecule tends to stay in a small region, a more or less fixed position in space about which it vibrated back and forth. The quasicrystalline picture of the liquid state supposes molecules to sit in a regular array in space, called a lattice, and therefore liquid and liquid mixture models based on this simplified picture are called lattice models. 

---