Chains with independent rotations

For present purposes only the form of Eq. (3.27) is required. Detailed formulation of Z can be found in Ref. (47) For chains with independent rotational potentials Z is equal to the bond rotational partition function. For high molecular weight, with int dependent rotational potential, Z is the largest eigenvalue of the statistical weight matrix describing this interdependence. 

The characteristic ratio of the poly(methylene) chain, which might be considered to be the simplest polymer molecule, as a function of chain length is shown in Fig. 4.4 for the simple models considered thus far. All save the simple freely jointed chain display end effects, manifest by an increase in Ci with chain length at small n, which will not be considered further. In what follows, only the asymptotic limit of the characteristic ratio for 00 (Coo) will be discussed. The values of for the freely jointed chain, the freely rotating chain and the chain with independent bond rotational potentials increase in value from 1 through 2 to ca 3-5. The latter value is, however, only ca one-half of the experimentally determined value of Coo=6-9 for poly(methylene). This serious discrepancy points to the fact that the bond rotational potentials are definitely not independent, i.e. the conformation of bond i depends upon the conformations of bonds (/—I) and (/-i-1). 

In order to calculate the mean square end-to-end vector or a radius of gyration we have to calculate averages (T,T,+i. .. Tj-i) (Eq. (5.24)). For bonds with independent rotational potentials this average is a product of averages (T) for single bonds. For chains with interactions between neighboring bonds we define for each bond i the supermatrix II T II of the order 9x9... 

By combining eqs (2.25), (2.27) and (2.28), an expression valid for a chain with hindered rotation with independent torsion angle potentials is obtained ... 

The diffo ence in the character of the nematic ordering in solutions of semiflexible macromolecules with diffaent mechanisms of flexibility is not only manifested in the thermodynamic characteristics of the phase transition itself, but also in the conformations of the polymer chains in the liquid-crystalline phase. For example, the dependences of the root-mean-square distance between chain ends (/ 2) on the concentration of polymer in the solution for semiflexible freely jointed and persistent chains calculated in [43,44] are shown in Fig. 1.4. Note that for the freely jointed model, the value of (jf) is almost independent of the concentration of the solution in the anisotropic phase (i.e., orientation of the segments but not uncoiling of the macromolecules takes place), while for a solution of persistent chains, the increase in (/ ) in the anisotropic phase with an increase in the concentration is very signiflcant (exponential). A solution of chains with the rotational-isomeric mechanism of flexibility (cf. Fig. 1.2c) behaves analogously in this case, as demonstrated in [35], in the... 

Each probability represents the fraction of the bonds with that rotation state. If the occurrence of bonds with such rotation states is independent of the states occupied by the neighboring bcmds, dien the probabilities may be used to generate chain sequences. It is possible to calculate various structural parameters directly from the probabilities. For example, in a homopolymer chain with two rotation states, the weighted average run of state 1 is givmi by... 

It is a familiar fact that the skeletal factor s defined in Eq. (2) can very often be decomposed into two independent parts, depending respectively on the restriction of bond angles and on the hindrance to internal rotations. In the simplest case of the polymethylene chain with an internal rotational potential that is symmetric about the irons conformation, we have... 

One explanation that can be offered to explain the two Tc values obtained for PMA at low values of a is that they represent the independent rotation of small clusters of the polymer chain. The larger value of (approximately 50 ns) can be associated with a rotating spherical cluster of radius 3.8 nm and of polymer molecular weight equal to 19000. Rotating units of similar size have been observed when the probes 9-methylanthracene and 9,10-DMA are solubilized in the PMA hypercoil structure (15,16) and when the more polar fluorescent probes Rhodamine B ( ) and 1,8-anilinonaphthalene sulfonic acid (1,8-ANS) (28) are bound to PMA for a value of a equal to 0. The smaller rotating unit present in PMA and PAA whose value of Tj, is approximately equal to 5 ns (which corresponds to particles whose radii are approximately equal to 2 nm) may arise from the rotation of a small section of the chain which is just sufficient to surround the 9,10-DMA probe and protect it from unfavourable entropic interactions with water. This shorter T, was... 

Simple chain with symmetric hindered rotation n, /, d,First-order interactions (independent bonds, symmetric torsion)... 

Chains with Fixed Bond Angles and Independent Potentials for Internal Bond Rotation 69... 

In the first part of this article the review of various theoretical models for polymer chains is given. The models of freely jointed chains, freely rotating chains (including wormlike chains), and chains with fixed bond angles and independent rotational potentials and with interdependent potentials, including rotatimial isomeric state approximation, are presented. 

CHAINS WITH FIXED BOND ANGLES AND INDEPENDENT POTENTIALS FOR INTERNAL BOND ROTATION... 

In the case of polar polymers the situation is more complex, since there are a large number of dipoles attached to one chain. These dipoles may either be attached to the main chain (as with poly(vinyl chloride), polyesters and polycarbonates) or the polar groups may not be directly attached to the main chain and the dipoles may, to some extent, rotate independently of it, e.g. as with poly(methyl methacrylate). 

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