Rigid lattice model

A lattice model of uniaxial smectics, formed by molecules with flexible tails, was recently suggested by Dowell [29]. It was shown that differences in the steric (hard-repulsive) packing of rigid cores and flexible tails - as a function of tail chain flexibility - can stabilize different types of smectic A phases. These results explain the fact that virtually all molecules that form smectic phases (with only a few exceptions [la, 4]) have one or more flexible tail chains. Furthermore, as the chain tails are shortened, the smectic phase disappears, replaced by the nematic phase (Fig. 1). 

Usually it is assumed that tc is the only temperature-dependent variable in Eq. 9. This might be the case for an order-disorder type rigid lattice model, where the only motion is the intra-bond hopping of the protons, since the hopping distance is assumed to be constant and therefore also A and A2 are constant. This holds, however, only for symmetric bonds. Below Tc the hydrogen bonds become asymmetric and the mean square fluctuation amplitudes are reduced by the so-called depopulation factor (l - and become in this way temperature-dependent also. The temperature dependence of tc in this model is given by Eq. 8, i.e. r would be zero at Tc, proportional to (T - Tc) above Tc and proportional to (Tc - T) below Tc. 

Fig. 4.17 The total energy ol a cubic lattice of rigid anions and cations as a function of r+ with r fixed, for different coordination configurations. When the anions come into mutual contact as a result of decreasing r+ their repulsion determines the lattice constant and the cohesive energy becomes constant when expressed in terms or r. Thus near the values of r+/r at which anion-anion contact takes place, the radius ratio model predicta phase transitions to structures of successively lower coordination numbers. Note that the breaks" in the curves correspond to the values listed in Table 4.6. [From Treatise on Solid State Chemistry Hannay, N. B., Ed. Plenum New York, 1973.]...

Thermodynamic descriptions of polymer systems are usually based on a rigid-lattice model published in 1941 independently by Staverman and Van Santen, Huggins and Flory where the symbol x(T) is used to express the binary interaction function [16]. Once the interaction parameter is known we can calculate the liquid liquid phase behaviour. 

A partitioning function for a system of rigid rod-like particles with partial orientation around an axis is derived from the use of a modified lattice model. The free energy of mixing is shown as a function of the mole numbers, the axis ratio of the solute particles and a disorientation parameter this function passes through a minimum with increase in the disorientation parameter. The chemical potentials display discontinuities at the concentration at which the minimum appears and then separation into an isotropic phase and a somewhat more concentrated anisotropic phase arises. The critical concentration, v, is given in the form 13) ... 

A modification of the lattice model to account for semi-rigid polymers replaces the contour length, L, in the axial ratio with the Kuhn segmental length (34,35) i.e., x is replaced with xk = 2q/d... 

Higher volume fractions of polymer at phase separation are found in the Khokhlov-Semenov-Odijk theory for contour lengths greater than q. The virial results fall below that of the lattice model of completely rigid rods, however, when L < q. [This result is generally true for all q and d values.]... 

The Flory theory discussed in the next section is another important theory on rigid liquid crystalline polymers. Because of its clear picture of the lattice model and the incorporation of the Onsager theory, it has become a basic method for the theoretical study of liquid crystalline polymers. As a result of the constant efforts of Flory and his co-workers, the theory has been applied to binary and poly-disperse systems and also includes the soft interactions. 

Flory (1956, 1984) adopted the lattice model. The Flory theory starts with the partition function of systems consisting of rigid rods and solvent molecules. 

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