The Single Phase Region

In the following sections, we shah demonstrate that the observed behavior of electro-optic activity with chromophore number density can be quantitatively explained in terms of intermolecular electrostatic interactions treated within a self-consistent framework. We shall consider such interactions at various levels to provide detailed insight into the role of both electronic and nuclear (molecular shape) interactions. Treatments at several levels of mathematical sophistication will be discussed and both analytical and numerical results will be presented. The theoretical approaches presented here also provide a bridge to the fast-developing area of ferro- and antiferroelectric liquid crystals [219-222]. Let us start with the simplest description of our system possible, namely, that of the Ising model [223,224]. This model is a simple two-state representation of the to- 

The average (poz)3 = cos30 , or order parameter, is calculated over the equilibrium density matrix, p=e H/T/Tr[e H/T], where Tr denotes the trace (sum over diagonal elements). The Ising Hamiltonian can be expressed as  

To simplify our calculations, let us define a reference lattice by assuming that the shape of a chromophore can be approximated as an ellipsoid (see Fig. 10). Taking the long axis of the ellipse as 10, we can define a closely packed cubic lattice of spheres of diameter 10. The number density associated with this lattice is N0=(l0) 3. For chromophore concentrations (loading in a host matrix) below this reference number density (i.e., N=N0x N0), the chromophores will on average be arranged in a cubic lattice with lattice constant, a=b=(N0) 1/3(x) l/3. For N N0, only the intermolecular distances in the plane perpendicular to the principle ellipsoid axis can decrease with increasing concentration, x. We have a simple tetrahedral lattice with lattice constants, a=(N0) 1/3(x) l/3 b=(N0) l/3. 

Simple models such as the Ising model have the advantage of permitting analytical results to be obtained for limiting cases. Let us consider these. 

However, we must ask how reasonable are the approximations employed in its derivation. This can be answered by comparing the above results with those of Monte Carlo (molecular dynamics) methods [240]. With Monte Carlo calculations, no approximations are made other than that of restricting considerations to a finite system (e.g., 1000 chromophores in a polymer lattice of finite dielectric constant). The full Hamiltonian can be used and calculations can be carried be- 

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The general thermodynamic treatment of binary systems which involve the incorporation of an electroactive species into a solid alloy electrode under the assumption of complete equilibrium was presented by Weppner and Huggins [19-21], Under these conditions the Gibbs Phase Rule specifies that the electrochemical potential varies with composition in the single-phase regions of a binary phase diagram, and is composition-independent in two-phase regions if the temperature and total pressure are kept constant. 

Because of the interest in its use in elevated-temperature molten salt electrolyte batteries, one of the first binary alloy systems studied in detail was the lithium-aluminium system. As shown in Fig. 1, the potential-composition behavior shows a long plateau between the lithium-saturated terminal solid solution and the intermediate P phase "LiAl", and a shorter one between the composition limits of the P and y phases, as well as composition-dependent values in the single-phase regions [35], This is as expected for a binary system with complete equilibrium. The potential of the first plateau varies linearly with temperature, as shown in Fig. 2. 

Figure 2.12a. Building blocks of binary phase diagrams examples of single-phase (two-variant) and two-phase (mono-variant) fields. In the figure the indication is given of the phases existing in the various fields and respectively of their number. The phase equilibrium composition in the two-phase fields is defined by the boundary (saturation) lines of the single-phase regions. (Pt), (Ag),...

Carbon is soluble to varying degrees in each of these allotropic forms of iron. The solid solutions of carbon in a-Fe, y-Fe, and <5-Fe are called, respectively, ferrite, austenite, and 8-ferrite. So, for example, the single-phase region labeled as y in... 

In three-phase systems the top phase. T. is an oleic phase. Ihe middle phase. M. is a inicroeuiulsion. and Ihe bottom phase. B. is an aqueous phase. Microcmulsions that occur in equilibrium with one or two other phases arc sometimes called "limiting mieroemulsions. because they occur at Ihe limits of the single-phase region... 

Figure 33 FT diagram showing the vapor-pressure curve for a pure substance and constant-volume lines in the single-phase regions.

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