Chain statistics characteristic ratio

The characteristic ratio changes from 1.3 to 2.8 with o changing from 0 to 1, when the virtual bond is used. On the other hand, when each bond of the phenylene group is taken into account individually, the two extreme values are 3.41 and 7.40. By assuming all the statistical weight factors to be unity, which corresponds to the freely-rotating chain, the characteristic ratio is 1.60 when the virtual bond is used, and 4.22 if it is not. 

The same conclusion has been drawn from the treatment of the unperturbed dimension of the PM chain. The characteristic ratio and its temperature coefficient din C o/dr were calculated using the parameter set given in equation (22). When the geometrical parameters lCCC = 112°, , = 0 and = 116.5° (cf. Table 1) were used, the following values were derived C = 7.65 and din Coo/dT= — 1.06x 10 at 140°C. Use of the statistical weights assembled in the second... 

Coo being the chain characteristic ratio. Consequently, the statistical weight of a closed loop comprising n chain bonds (i.e., r = 0) is given by... 

In a real chain segment-segment correlations extend beyond nearest neighbour distances. The standard model to treat the local statistics of a chain, which includes the local stiffness, would be the rotational isomeric state (RIS) [211] formalism. For a mode description as required for an evaluation of the chain motion it is more appropriate to consider the so-called all-rotational state (ARS) model [212], which describes the chain statistics in terms of orthogonal Rouse modes. It can be shown that both approaches are formally equivalent and only differ in the choice of the orthonormal basis for the representation of statistical weights. In the ARS approach the characteristic ratio of the RIS-model becomes mode dependent. 

The configurational-conformational characteristics of PP are discussed by considering every polymer chain as constituted by the periodic repetition of a sequence of monomeric units in a given configuration. Calculations are presented for the special case in which mesa and racemic diads are distributed according to Bemoullian statistics. Numerical results show that the characteristic ratio of atactic PP reaches an asymptotic value of 5.34 when the size of the periodic sequence corresponds to six monomeric units. 

Conformational features of meso and racemic diads of PVAc are examined using energy calculations. In contrast to other vinyl chains bearing planar substituents, the g conformation is not prohibited for this polymer. The shifts in the positions of the energy minima from perfect staggering are discussed in terms of the second order interactions. Calculated statistical weight parameters are used to reproduce the experimental data on NMR coupling constants and the characteristic ratios. 

A comparison is presented between the behavior of unperturbed stars of finite size whose configurational statistics are evaluated by R1S theory and the Kratky-Porod wormlike chain model. Emphasis Is placed on the initial slopes of the characteristic ratio, C, or g when plotted as a function of the reciprocal of the number of bonds, n. 

Expansion is considered for finite, regular polyethylene stars perturbed by the excluded volume effect. An RIS model is used for the chain statistics. The number of bonds in each branch ranges up to 10 240, and the functionality of the branch point ranges up to 20. The form of the calculation employed here provides a lower bound for the expansion. If the number, n, of bonds in the polymers is heid constant, expansion is found to decrease with increasing branch point functionality. Two factors dictate the manner in which finite stars approach the limiting behavior expected for very large stars, These two factors are the chain length dependence at small n of the characteristic ratio and of fa -a3) / n1/2. 

The characteristic ratio of polylA -ro-hydroxyethyl-L-glutamine) in water at 303 K is found to be 10 1, in agreement with results obtained by Brant and Flory [J. Am. Chem. Soc. 1965, 87, 2791) for four other polypeptides with -CH2—R side chains. The circular dichroism of polylA -m-hydroxyethyl-L-glutamine) under these conditions, where the polypeptide is in a statistical conformation, exhibits a positive band at 216 nm. 

The parameters a = l/rij5 the number of which equals m(m — IX are reciprocal reactivity ratios (2.8) of binary copolymers. Markov chain theory allows one, without any trouble, to calculate at any m, all the necessary statistical characteristics of the copolymers, which are formed at given composition x of the monomer feed mixture. For instance, the instantaneous composition of the multicomponent copolymer is still determined by means of formulae (3.7) and (3.8), the sums which now contain m items. In the general case the problems of the calculation of the instantaneous values of sequence distribution and composition distribution of the Markov multicomponent copolymers were also solved [53, 6]. The availability of the simple algebraic expressions puts in question the expediency of the application of the Monte-Carlo method, which was used in the case of terpolymerization [85,99-103], for the calculations of the above statistical characteristics. Actually, the probability of any sequence MjMjWk. .. Mrl 4s of consecutive monomer units, selected randomly from a polymer chain is calculated by means of the elementary formula ... 

Some important properties of polymer chains in dilute solutions [steric hindrance parameter, characteristic ratio, persistence length, radius of gyration, statistical chain segment length (introduced earlier, in Chapter 11), intrinsic viscosity, and viscosity at small but finite concentrations] will be discussed, and new correlations will be presented for the steric hindrance parameter and the molar stiffness function, in Chapter 12. 

For example, Lu and Jiang showed [100], based on a statistical thermodynamic derivation in which some of the parameters of the model were calibrated by using experimental data, that the approximation in Equation 6.20 can be made for polymers with a vinyl-type chain backbone, where Coo(Tg) is the value of the characteristic ratio at Tg. C, which will be discussed in Chapter 12, depends both on polymer chain stereoregularity and on the temperature. Direct applications of the method which will be presented in Chapter 12 for its prediction as a function... 

Several parameters, most of which are interrelated and can be estimated in terms of each other, are utilized to describe the conformational properties of polymer chains [1,2]. These quantities include the steric hindrance parameter a, the characteristic ratio Cx, the persistence length Ip, the statistical chain segment (or Kuhn segment) length lk, the root mean square radius of gyration Rg (often briefly referred to as simply the "radius of gyration"), and the molar... 

In its turn, the characteristic ratio C, which is polymer chain statistical flexibility characteristic [25], is determined according to the Eq. (97) [153] ... 

FIGURE 52 The dependence of conversion degree Q on polymer chain statistical flexibility, characterized by characteristic ratio C, for PUAr. 

One more parameter—the characteristic ratio Coo — is characteristic of polymer chain statistical flexibility and determined according to the Eq. (98). In Fig. 57 the relation between and o- for the considered polyarylates is adduced. Since these parameters characterize the same polymer chain property, then between them the correlation is expected [25] that confirms the plot of Fig. 57. Certain scatter of data for two from the considered polyarylates can reflect different values of angles between bonds in chain [25]. 

---