Rotational isomeric approximation

If 0 = 0, atoms that are covalently bonded to Ci and to C4 come into close contact consequently the potential energy of that shape or conformation is high. However, if 0 = 180°, atom C4 is at a maximum distance from Ci, and none of the atoms bonded to C4 comes close to an atom bonded to Ci. Hence, the potential energy has a minimum value. This is referred to as the trans t) conformation of the bond C2—C3. Further sub-minima at ( = -f 60° are referred to as gauche (g ) and gauche (g ) conformations. In the rotational isomeric approximation each C—C bond is assumed to be in a gauche or a trans conformation, and the shape of the polymer chain is defined by these conformations. 

This is the basis of the rotational isomeric approximation of Volkenstein [9] and others. The lowest energy conformation at 0 = 0° is the trans conformation, while the two minima at 0=120° and 240° are the gauche plus (g +) and gauche minus (g-) conformations (Figure 2.6). It is a good approximation in the calculation of molecular properties to assume that the molecule is an equilibriiun mixture of these conformers, so that... 

The conformational partition function, being the sum of all the terms appearing in the table, may be written for n-pentane, in the rotational isomeric approximation,... 

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. 

A very common approximation goes by the name of rotational isomeric state model (RIS). With such a model it is assumed that the molecular population is placed exclusively in a few energy minima, always according to the Boltzmann distribution. The conformations corresponding to the minima are called conformers, or rotational isomers, and are indicated by one or more letters (G, T) for butane three conformers are considered (G, T, G ), for pentane five (TT, G T, TG, G G, G G ). A more detailed examination of pentane reveals the existence of two further wells (fl, = 65 , 2 = 260° and 0, = 100°, 02 = 295°), sometimes indicated by G G and G G , respectively, close to the forbidden G G" conformation, but with a much tower energy (157, 158). 

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°. 

For random coils, is directly proportional to the contour length. If n is the number of main chain atoms in the chain, = an. The parameter a is relatively insensitive to environment (21), and has been calculated for a number of polymers from strictly intramolecular considerations using the rotational isomeric model (22). The root-mean-square distance of segments from the center of gravity of the coil is called the radius of gyration S. The quantity S3 is an approximate measure of the pervaded volume of the coil. For Gaussian coils,... 

W AXS-pattern of a polymer melt if these Boltzmann-weighted distance fluctuations are not accounted for. They characterize a relevant feature of polymer melt structures. We treat this in terms of the Rotational Isomeric State Approximation (RISA) discussed in the next section. 

C. If the isomeric form is not otherwise known, determine the optical rotation [see Optical (Specific) Rotation, Appendix IIB] of a 1 10 solution in chloroform. The optical rotation is approximately 0.05°. 

In the rotational isomeric state (RIS) approximation, each bond in the chain backbone is assumed to occur in any one of a small, discrete number of... 

More sophisticated approaches to the evaluation of wfr) exist. Some of them are the technique based on the Yoon-Flory expansion, variations of the Gaussian approximation, and an implementation of the Koyama distribution. Freely jointed chains and rotational isomeric chains have also been used as models of chains. Research is actively pursued on the subject of formulating more realistic closure relationships. In considering the choice of an intramolecular structure factor, a paper by McCoy et al. on model polyethylene melts will be useful. 

M. L. Mansfield, Macromolecules, 16, 1863 (1983). Effect of Fluctuating Internal Coordinates on the Rotational Isomeric States Approximation. 

Yoon and Flory > have compared the experimental data of Gawrisch et al. to the scattering of unperturbed chains calculated on the basis of the rotational isomeric state theory. They found good agreement up to the highest values of the scattering vector corresponding to distances of approximately one chain diameter... 

Now that the constraint arising from the steric energy has been removed, one may assume, as a first approximation, that all the ring formations (from x = 1 to oo) occur according to the known power law, (px OCX /2. Making use of the mean end-to-end distances computed from the rotational isomeric state models, one has... 

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