Rotational Isomeric State Calculations

In a typical analysis of a polymer chain, the experimental values of configuration-dependent properties and their temperature coefficients are compared with the results of rotational isomeric state calculations. These comparisons yield values of the energies for the various rotational states about the backbone bonds, and these conformational preferences can then be used to predict other configuration-dependent properties of the chains. It is also possible to obtain such conformational information from potential energy calculations, using the methods of molecular mechanics.39,46 52... 

In addition to the above thermodynamic conditions for ring formation, the kinetics of the reactions must be considered. Thus, for a reaction to take place the two ends of the polymer chain must be in the correct conformation for sufficient time for the new bond to form. The kinetic factor for cyclization is proportional to Rg, so the net effect of the thermodynamic and kinetic factors is that rings are not favoured between n = 8 and k = 11. Suter (1989) has considered the theoretical approaches of Jacobsen and Stockmayer and compared theoretical and experimental values for macrocyclization equilibrium constants. This has also been performed for Monte Carlo as well as rotational-isomeric-state calculations for the statistical conformations of cyclic esters (decamethylene fiimarates and maleates) and agreement with experimental molar cyclization equilibrium constants found (Heath et al, 2000). 

Fig. 4.6. Establishment of the rectangular co-ordinate origin on the first bond for rotational isomeric state calculations bond 2 is trans to the hypothetical bond 0 the x-direction is along bond 1 the y-direction has a positive intercept on the projection of bond 0 the z-direction completes a right-handed Cartesian system (after Flory, 1969).

The consonance noted above between the predictions of scaling law theories and the results of experiments suggests that the assumptions underlying the theories are correct. Mattice (1981), however, has criticized the basic assumptions of the blob concept because they are not verified by rotational isomeric state calculations that incorporate the excluded volume term, admittedly in a somewhat empirical fashion. A major difficulty envisaged by Mattice in the simple application of the blob concept lies in its failure to account for the consequences of the repulsive interaction of the subchain with atoms elsewhere in the main chain. 

Hg. 21a. Schematic sketch of model used in calculation of scattering law for styrene-methyl methacrylate diblock copolymer in solution b Comparison of rotational isomeric state calculation (-----)... 

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. 

The objective of this section is to present a quantitative analysis of the photostationary state fluorescence of miscible and immiscible PS/PVME blends. In Section 4.1 we develop the onedimensional random walk model that is used in conjunction with rotational isomeric state calculations to analyze low concentration miscible blends. In Section 4.2 we treat miscible blends having high PS concentration using a spatially periodic three-dimensional random walk model. Finally, in Section 4.3 we present a simple two phase morphological model and demonstrate how it may be used to monitor phase separation kinetics. 

The conformational characteristics of PVF are the subject of several studies (53,65). The rotational isomeric state (RIS) model has been used to calculate mean square end-to-end distance, dipole moments, and conformational entropies. C-nmr chemical shifts are in agreement with these predictions (66). The stiffness parameter (5) has been calculated (67) using the relationship between chain stiffness and cross-sectional area (68). In comparison to polyethylene, PVF has greater chain stiffness which decreases melting entropy, ie, (AS ) = 8.58 J/(molK) [2.05 cal/(molK)] versus... 

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

In order to develop a molecular model of elongation from which M- s can be calculated, we have chosen the rotational isomeric state (RB)... 

From a stereochemical point of view the physical problem to be resolved (which antecedes that of calculation) consists in the identification of the rotational isomeric states to be considered and in the evaluation of their statistical weight, that is, their relative energy. 

A general theory is presented on the effect of elongation of a polymer chain on the apportionment of its bonds and bond sequences among various rotational Isomeric states. Numerical calculations are presented for PE chains and for syndiotactlc vinyl polymers. 

Bond length and bond angles are kept fixed at values used earlier (Suter, U, W. J. Am. Chem, Soc. 1979, 101, 6481). For the CH2-CH2 bonds three approximately staggered conformations are appropriate. At first, rotation about CO-CH2 bond is subdivided Into the same three isomeric states, but In the course at the calculations it becomes clear that the trees region cannot be satisfactorily modeled by a single conformation. The region is consequently split Into three equal domains, so that five rotational isomeric states result g" 0 to - 120°, t" - 120° to - 160°, t0 - 160° to 160°, t+ 160° to 120°, g+ 120° to 0°. 

Curro and Mark 38) have proposed a new non-Gaussian theory of rubber elasticity based on rotational isomeric state simulations of network chain configurations. Specifically, Monte Carlo calculations were used to determine the distribution functions for end-to-end dimensions of the network chains. The utilization of these distribution functions instead of the Gaussian function yields a large decreases in the entropy of the network chains. 

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