Independent bond rotational potentials

It can be shown that the rms size of a polymer molecule of high molecular weight subject to independent bond rotational potentials is given by 

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). 

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Fig. 7a, k Second oidm- transition jn-obabilities a C,( Pa) for internal bo mis in polyethylene, obtmned from BD simulations, using independent bond rotational potentials, b C,( a T]P) for internal braids in polyethylene, obtained from BD simulations, using independent bond rotational potentiids... 

Fig. 4.4. Plots of the characteristic ratio of poly(methylene) at 140 °C as a function of the number of bonds using different theoretical models (1) the freely jointed chain (2) the rotating chain (3) independent bond rotational (4) interdependmt bond rotational potentials (after Flory, 1969).

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... 

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... 

Using Eqs. (5.23) and (5.27) we may calculate the mean-square end-to-end vector for fixed bond angles and independent potentials for internal bond rotation... 

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... 

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. 

Eg being the energy difference between the gauche and the trans states. Provided that the rotational potential of the bonds is independent of the actual torsion angles of the nearby bond, the average [Pg.28]

Thus the entropy of localized adsorption can range widely, depending on whether the site is viewed as equivalent to a strong adsorption bond of negligible entropy or as a potential box plus a weak bond (see Ref. 12). In addition, estimates of AS ds should include possible surface vibrational contributions in the case of mobile adsorption, and all calculations are faced with possible contributions from a loss in rotational entropy on adsorption as well as from change in the adsorbent structure following adsorption (see Section XVI-4B). These uncertainties make it virtually impossible to affirm what the state of an adsorbed film is from entropy measurements alone for this, additional independent information about surface mobility and vibrational surface states is needed. (However, see Ref. 15 for a somewhat more optimistic conclusion.)... 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

RiS theory is applied to investigate chain configuration of POLA. Independent conformations for each repeat monomer unit of the chain are assumed in the calculations of the unperturbed dimensions. Rotations about the oxygen-phenytene-carbon bonds are considered to be free with twofold symmetric potentials. The trans and cis conformations of the carbonyl-phenylene-carbon and the indan-carbonyl residues are assumed to have equal probability. The bond vectors for this model lie in a plane because every torsion angle is 0D or 180°. 

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