Lattice model molecule arrangement

In a lattice model, molecules are assumed to be arranged regularly on an array of sites or cells as shown in Figures 1 and 2, each polymer chain... 

The Flory-Huggins theory uses the lattice model to arrange the polymer chains and solvents. We have looked at the lattice chain model in Section 1.4 for an excluded-volume chain. Figure 2.1 shows a two-dimensional version of the lattice model. The system consists of si,e sites. Each site can be occupied by either a monomer of the polymer or a solvent molecule (the monomer and the solvent molecule occupies the same volume). Double occupancy and vacancy are not allowed. A hnear polymer chain occupies N sites on a string of N-l bonds. There is no preference in the direction the next bond takes when a polymer chain is laid onto the lattice sites (flexible). Polymer chains consisting of N monomers are laid onto empty sites one by one until there are a total tip chains. Then, the unoccupied sites are filled with solvent molecules. The volume fraction of the polymer is related to rip by... 

In lattice models each molecule (or segment of a molecule in the case of polymers) is assumed to occupy a cell in the lattice. The arrangement of the molecules or segments is assumed to depend upon only the composition and the size and shape of the molecules. In this case, the combinatorial (athermal) contribution is calculated from the number of arrangements statistically possible in the lattice. This contribution is also referred to as the entropic term. 

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.)...

By considering the number of arrangements of polymer and solvent molecules on a lattice, we can calculate the entropy of the mixture and from that the Gibbs energy (if we assume some value for the heat of mixing). A simplified two-dimensional model of a polymer molecule arranged on alatticeis shown in Fig. 35.5. We assume that a solvent molecule... 

A second way of dealing with the relationship between aj and the experimental concentration requires the use of a statistical model. We assume that the system consists of Nj molecules of type 1 and N2 molecules of type 2. In addition, it is assumed that the molecules, while distinguishable, are identical to one another in size and interaction energy. That is, we can replace a molecule of type 1 in the mixture by one of type 2 and both AV and AH are zero for the process. Now we consider the placement of these molecules in the Nj + N2 = N sites of a three-dimensional lattice. The total number of arrangements of the N molecules is given by N , but since interchanging any of the I s or 2 s makes no difference, we divide by the number of ways of doing the latter—Ni and N2 , respectively—to obtain the total number of different ways the system can come about. This is called the thermodynamic probabilty 2 of the system, and we saw in Sec. 3.3 that 2 is the basis for the statistical calculation of entropy. For this specific model... 

The exact cystal structure of quinacridone pigments has been published recently. So far, models indicated a planar arrangement of the molecules within the crystal lattice [21], In fact, it was now been proved recently by three-dimensional X-ray analysis that the pigment exists in two different crystal modifications [22] ... 

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