The lattice approach

Consider the mixing of Ri solvent molecules and ri2 polymer molecules, the subscripts 1 and 2 denoting solvent and polymer respectively. The total number of lattice sites no is given by 

Suppose molecules have been added to the lattice. We consider the number of different ways that the (i-i- l)th molecule can be sited on the lattice. The first segment can be placed on any one of (tto—xi) unoccupied sites. The next segment, which is connected to the first, can be placed in z(l —/) lattice sites, where z is the coordination number of the lattice and is the probability of a site being filled when at least one adjacent cell is empty. The third segment can be located in (z— 1X1 —fD different ways, since the second segment necessarily 

The total number of ways by which the (i+l)th molecule can be placed on the lattice is thus 

To find/-, it is necessary to assume that the solution is moderately concentrated so that fi can be replaced by its average value ( o-JcO/ o- Under this assumption. 

If all of the polymer molecules were distinguishable, the total number of ways Qfia which the polymer molecules could be mixed with the solvent would be 

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The lattice approach has also been used for the systematic description of inorganic crystal structures (Wells 1975, pp. 119-55 Hyde and Andersson 1989, pp. 6-49), but the method is not just geometric and descriptive. It has a sound physical basis and can therefore be used for structure modelling. 

Whereas in approach 1 lattice models are used, we will work in the continuum, making extensive use of interface thermodynamics. The advantage of such an approach, as it turns out, is that detailed properties such as the size distribution of microemulsion droplets and the interfacial tension of a flat monolayer separating a microemulsion and an excess phase can be predicted. On the other hand, the lattice approaches as summarized in item 1 predict global phase behavior, which is not (yet) possible with the thermodynamic formalism reviewed in the following section. The reason is that currently a realistic model for the middle phase is lacking. A more detailed discussion regarding this issue is presented in Sec. VIII. 

As mentioned earlier, studies of simple linear surfactants in a solvent (i.e, those without any third component) allow one to examine the sufficiency of coarse-grained lattice models for predicting the aggregation behavior of micelles and to examine the limits of applicability of analytical lattice approximations such as quasi-chemical theory or self-consistent field theory (in the case of polymers). The results available from the simulations for the structure and shapes of micelles, the polydispersity, and the cmc show that the lattice approach can be used reliably to obtain such information qualitatively as well as quantitatively. The results are generally consistent with what one would expect from mass-action models and other theoretical techniques as well as from experiments. For example. Desplat and Care [31] report micellization results (the cmc and micellar size) for the surfactant h ti (for a temperature of = ksT/tts = /(-ts = 1-18 and... 

The Bristow-Watson method is based on the Huggins equation deduced from a refinement of the lattice approach ... 

An introduction is given to the many and varied attempts to deal with the non-perturbative or confinement region of QCD, especially to the lattice approach and to the sum rule method, and to the exciting new ideas about baryon and lepton number violations. Necessarily the treatment of these is rather brief and not very comprehensive. We seek to convey the basic ideas and methods. 

The one problem of this approach appears in the case of two polymers with considerably different chemical structures - that is, with monomer units of unequal size. It is then necessary to choose repeating units such that these occupy the same volume for both polymers. A detailed discussion of the lattice approach and alternative derivations of entropy of mixing are provided in Ref. [11]. 

Equations (4.98) and (4.102) are the backbone of a method describing thermodynamic properties of macromolecular systems akin to the van der Waals approach to low molecular weight systems. The lattice approach outlined here was pioneered independently by Staverman and van Santen (Stavermann and van Santen 1941), Huggins (Huggins 1941,1942) and Flory (Paul John Flory, Nobel prize in chemistry for his work on the physical chemistry of macromolecules, 1974) (Flory 1941,1942 Koningsveld and Kleintjens 1988). 

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