Spatially periodic lattice model

In this section we examine the effect of PS concentration on the fluorescence of miscible PS/PVME blends cast from toluene. We then show how fluorescence data for miscible blends may be used to establish a "calibration curve for M, the probability of eventual non-radiative or radiative decay of the excitation by monomer fluorescence. Finally, we present a spatially periodic lattice model that leads to reasonable predictions for Wx for the three dimensional energy migration process. 

Spatially Periodic Lattice Model. The dependence of Id/% on PS volume fraction of lOy thick films of PS/PVME blends cast from toluene with PS molecular weights of 4000 and 100,000 is shown in Figure 7 [79]. Figure 7 also shows the results of phase-separated blends cast from tetrahydrofuran made with PS of molecular weight 100,000, to be considered in Section 4.3. 

ABSTRACT. Excimer fluorescence is developed as a quantitative probe of isolated chain statistics and intermolecular segment density for miscible and immiscible blends of polystyrene (PS) with poly(vinyl methyl ether) (PVME). Rotational isomeric state calculations combined with a one-dimensional random walk model are used to explain the dependence of the excimer to monomer intensity ratio on PS molecular weight for 5% PS/PVME blends. A model for a three-dimensional random walk on a spatially periodic lattice is presented to explain the fluorescence of miscible PS/PVME blends at high concentrations. Finally, a simple two-phase morphological model is employed to analyze the early stages of phase separation kinetics. 

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

Finally, the self-reproducibility in time of the lattice configuration (for two-dimensional flows) must be addressed. In the elliptic streamline region (A < 0), the lattice necessarily replicates itself periodically in time owing to closure of the streamlines. For hyperbolic flows (A > 0), the lattice is not generally reproduced however, in connection with research on spatially periodic models of foams (Aubert et al., 1986 Kraynik, 1988), Kraynik and Hansen (1986, 1987) found a finite set of reproducible hexagonal lattices for the extensional flow case A = 1. It is not clear how this unique discovery can be extended, if at all. 

The concept of the soliton has been introduced in the theoretical treatment by Su et al. (1979, 1980), in which they have employed a model Hamiltonian within the framework of the Hiickel approximation, including both cr-bond compressibility and the kinetic energy term of the CH units. The soliton is an elementary excitation and, in the case of the transient in Fig. 12a, is expected to satisfy a wave equation akin to the if/4 field theory (Krumhansl and Schrieffer, 1975). The estimated energy of the creation of a soliton is 0.4 eV and the periodic-lattice-induced activation energy for the soliton motion is 0.002 eV, the latter being consistent with the result of the ESR observation. The energy of a soliton is most stabilized when its tail (that is, the spatial halfwidth of the kink) extends over seven carbon sites. 

Durrett and Levin (4998) considered a simple lattice model occupied by three species in cyclic competition and observed that the behavior of the system in a spatially extended system with short range local interaction is different from the corresponding mean-field model. In general, cyclic competition in spatially extended systems produces a dynamical equilibrium in which all species coexist, while the mean-field model leads to either periodically oscillating total populations, or extinction of all except one of the species. 

Here the dielectric permittivity is spatially periodic, e(x + ti) = e(x), which means that the Kronig-Penney model is applicable. If we introduce the Bloch vector, i.e., the crystal momentum k, the periodicity of the electric field will be described by E x + d) = e E x)—Bloch or Floquet condition. The solution of the wave equation for an infinite ID lattice with periodically changing dielectric permittivity should have the form of a sum of a direct and a reflected wave... 

In conducting solids, the conduction electron density is spatially modulated, forming charge density waves (CDW) the periodic distortion accompanying the CDW (due to interaction between the conduction electron and the lattice) is responsible for the incommensurate phase (Overhauser, 1962 Di Salvo Rice, 1979 Riste, 1977). The occurrence of CDW and the periodic distortion can be understood in terms of the model proposed by Peierls and Frdhlich for one-dimensional metals. Let us consider a row of uniformly spaced chain of ions (spacing = a) associated with conduction electrons of energy E k) and a wave vector k. At 0 K, all the states are filled up to the Fermi energy, = E(kp). If the electron density is sinusoidally modulated as in Fig. 4.15 such that... 

It has been shown for molecules [189] that the expected l/R decay of distant correlation contributions is closely modeled by the atom-pair partitioning. The same type of decay is also observed in the periodic case and e " decays as 1/jo — rj . Of course, the extent of the lattice summations over prs in (5.78) greatly affects both the computational cost and the accuracy of the calculation. By taking advantage of the spatial decay properties mentioned above, the energy contraction can be carried out only for cells such that jo — rj < r-max, while the p and s lattice summation truncations must be controlled [185]. 

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