Analytic molecular theory

We shall now consider what happens when the film thickness is of the order of the Debye length. In such a situation, no analytical expressions can be derived and numerical calculations should be used [125]. The real situation could be even more complicated, since an ill-defined film thickness can exist, like the example in Figure 2.6. We can use the molecular theory to obtain a self-consistently determined electrostatic potential profile across the interface as was shown in Figure 2.7 (see... 

Per-Olov Lowdin had a long and lasting interest in the analytical methods of quantum mechanics and my tribute to his legacy involves an application of the Wentzel-Kramers-Brillouin (WKB) asymptotic approximation method. It was the subject of a contribution(l) by Lowdin to the Solid State and Molecular Theory Group created by John C. Slater at the Massachusetts Institute of Technology. 

We should remark that no one of the mechanisms previously elaborated for water enables analytical description of the above fall of e"(v) for the case of ice. This phenomenon probably characterizes the transition from collective motions in a fluid typical for Debye relaxation, characterized by the relaxation time td, to vibration of individual molecules, characterized by much shorter lifetimes Tor, rq, Tjj. The latter times have the same order of magnitude in water and ice, but td drastically differs in these fluids. That is why the theory of far-IR spectra is rather similar in ice and water. On the contrary, it appears that the molecular theory pertaining to the low-frequency ice spectra (which still is not elaborated)... 

Although this chapter is concerned primarily with holonomic constraints, we comment on the role of nonholonomic constraints in the present context. Because the analytical dynamics theory of a system of particles subject to holonomic constraints is well established, the issues in (holonomic) constraint dynamics are mainly algorithmic in nature, as seen in this chapter. The situation is more complex for molecular dynamics with nonholonomic constraints, where theoretical difficulties exist. " ... 

In this section, we discuss the characteristics of an effective pair potential that can be used in a molecular theory of liquid water. As is the case for any liquid, neither theory nor experiment provides us with an analytical form of the entire pair potential as a function of six coordinates. Furthermore, the true pair potential is of no use for the study of liquid water. Therefore, one must resort to an effective pair potential. As we have discussed in Sec. 2.2 any effective pair potential must consist of essentially three terms one corresponding to the strong repulsive... 

C-C bonds in the actual molecules. The LCT is an analytical molecular-based theory for the statistical thermodynamics of molten polymers, associated with recognizing the degree to which the distinct chemical structures of the individual monomers are relevant. LCT also incorporates free volume and uses the nonrandom... 

Detailed analytical and numerical studies of the above questions are in progress, and a very rich and nonadditive dependence of the phase behavior on the precise nature of the attractive potentials, single chain architecture, and thermodynamic state is found [67, 72]. A full understanding of these issues would provide a scientific basis for the rational molecular design of polymeric alloys. The influence of asymmetries on the spinodal phase boundary of simple model polymer alloys using analytic PRISM theory with molecular closures has been derived by Schweizer [67]. In this section a few of these results are briefly discussed. 

In summary, the predictions of analytic PRISM theory [67] for the phase behavior of asymmetric thread polymer Uends display a ly rich dependence on the single chain structural asymn try variables, the interchain attractive potential asymmetries, the ratio of attractive and repulsi interaction potential length scales, a/d, and the thermodynamic state variaUes t) and < ). Moreover, these dependences are intimately coupled, which mathematically arises within the compressible PRISM theory from cross terms between the repulsive (athermal) and attractive potential contributions to the k = 0 direct correlations in the spinodal condition of Eq. (6.6). The nonuniversality and nonadditivity of the consequences of molecular structural and interaction potential asymmetries on phase stability can be viewed as a virtue in the sense that a great variety of phase behaviors are possible by rational chemical structure modification. Finally, the relationship between the analytic thread model predictions and numerical PRISM calculations for more realistic nonzero hard core diameter models remains to be fully established, but preliminary results suggest the thread model predictions are qualitatively reliable for thermal demixing [72,85]. 

In this section we present some of the characteristics of an effective pair potential that can be used in a molecular theory of liquid water. As is the case for any liquid, neither theory nor experiment provide us with an analytical form of the entire pair... 

---