Frank molecular theories

Recapitulating the foregoing discussion, it is clearly not our opinion that solution experiments appear to be inadequate for the purpose of comparison with molecular theories. However, we want to point out that due to the evident shortcomings of present theoretical and computational facilities a distinct scepticism is necessary in order to avoid the production of meaningless data. Of course, the solution experiments remain the main source of information, the data of which must be explained by theory. At the present stage of knowledge it is only possible to pick out selected properties of solutions which can be described satisfactorily on a molecular basis. For instance, referring to Frank and Wen s model of solvation shells W, the structure of the inner shell should not be modified too much by... 

Several molecular theories have been developed for correlating the Frank elastic constants with molecular constiments. The commonly employed one is mean-field theory [53,54]. In the mean-field theory, the three elastic constants are expressed as... 

The partially averaged potential (Eq. 29) can be used in the molecular theory of the nematic-isotropic transition (also being supplemented by the P4 term [19]). However, several other properties of nematics cannot be described in this way. For example, the full anisotropy of the Frank elastic constants can be accounted for only taking into account the explicit dependence of the interaction potential on the intermolecular vector [20]. In this case appropriate model potentials can be obtained using some more general expansion of the full potential y(a, Ty, a ). This potential can be expanded in terms of the spherical invariants... 

The literature on the molecular theory of liquid crystals is enormous and in this chapter we have been able to cover only a small part of it. We have mainly been interested in the models for the nematic-isotropic, nematic-smectic A and smectic A-smectic C phase transitions. The existing theory includes also extensive calculations of the various parameters of the liquid crystal phases Frank elastic constants, dielectric susceptibility, viscosity, flexoelectric coefficients and so... 

An alternative interpretation has been proposed by Barbero and co-workers [195, 218]. These authors expand the free energy to fourth order in the director derivatives. This increases the number of elastic terms by 35 and makes a general application of this so-called second-order elastic theory rather complicated. While this approach provides the correct torque balances, it results in large subsurface director deformations on a molecular length scale which are not consistent with the ideas of Frank s theory, which is based on the assumption of weak director gradients. 

There are three additional specific problems that have to be addressed in any molecular theory for polymer nematics. The first one is a possible effect of the Frank elasticity in these systems. It was shown [3] that the Frank and molecular polymer elasticity for LCEs have well-separated space scales, with their crossover, the characteristic scale I evaluated ash = s/K/G [3]. Here, G > 10 dyn/cm is a... 

Another mechanism for the role of cilia, in the crystallization from dilute solution, has been postulated by Toda and coworkers.(62) This theory is based on the works of Seto (47,63) and on Frank s theory,(64) which was discussed in Chapter 9. In essence, two steps traveling in opposite directions approach one another and are allowed to collide. As a consequence of the contact a pair of cilia is generated. Each can nucleate on the next layer and travel in both directions. The major conclusions from the analysis of this model are as follows. When the chain length is relatively small, nucleation by cilia can be neglected. For linear polyethylenes the cut-off molecular weight is approximately 3.0 x lO . Below this molecular weight cilia nucleation is not important. In Regime I, irrespective of whether or not cilia nucleation is taken into account... 

Whereas many scientists shared Mulliken s initial skepticism regarding the practical role of theory in solving problems in chemistry and physics, the work of London (6) on dispersion forces in 1930 and Hbckel s 7t-electron theory in 1931 (7) continued to attract the interest of many, including a young scientist named Frank Westheimer who, drawing on the physics of internal motions as detailed by Pitzer (8), first applied the basic concepts of what is now called molecular mechanics to compute the rates of the racemization of ortho-dibromobiphenyls. The 1946 publication (9) of these results would lay the foundation for Westheimer s own systematic conformational analysis studies (10) as well as for many others, eg, Hendrickson s (11) and Allinger s (12). These scientists would utilize basic Newtonian mechanics coupled with concepts from spectroscopy (13,14) to develop nonquantum mechanical models of structures, energies, and reactivity. 

Besides charge transfer interactions, dipolar coupling between ttk transitions of bases may lead to delocalization of the excited states. In order to obtain some guidelines for our experimental studies, we have undertaken the calculation of excited Frank-Condon states within the framework of the exci-ton theory [26]. These studies were enriched by combining data from quantum chemistry and molecular dynamics calculations in collaboration with Krystyna Zakrzewska and Richard Lavery [26,27,27-29]. The general formalism is described in the Chapter by E. Bittner and A. Czader in the present volume. 

Several interesting theoretical papers have appeared dealing with molecular dynamics and excimer formation in polymer systems. Frank and coworkers have developed a model to describe the transport of electronic excitation energy in polymer chains. The theory applies to an isolated chain with a small concentration of randomly placed chromophores, and a three-dimensional transport model was used to solve the problem which is based on a diagrammatic expansion of the transport Green function. (The Green function is related to time-dependent and photostationary depolarization and to transient and steady-state trap fluorescence.) The analysis is shown to be... 

At low enough shear rates, polymeric nematics ought to obey the same Leslie-Ericksen continuum theory that describes so well the behavior of small-molecule nematics. The main difference is that polymers have a much higher molecular aspect ratio than do small molecules, which leads to greater inequalities in the the numerical values of the various viscosities and Frank constants and to much higher viscosities. 

In microscale models the explicit chain nature has generally been integrated out completely. Polymers are often described by variants of models, which were primarily developed for small molecular weight materials. Examples include the Avrami model of crystallization,- and the director model for liquid crystal polymer texture. Polymeric characteristics appear via the values of certain constants, i.e. different Frank elastic constant for liquid crystal polymers rather than via explicit chain simulations. While models such as the liquid crystal director model are based on continuum theory, they typically capture spatiotemporal interactions, which demand modelling on a very fine scale to capture the essential effects. It is not always clearly defined over which range of scales this approach can be applied. 

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