Chain statistics

Functional and end-functional polymers are precursors to block and graft copolymers and, in some cases, polymer networks. Copolymers with in-chain functionality may be simply prepared in copolymerizations by using a functional monomer. However, obtaining a desired distribution requires consideration of the chain statistics and, for low molecular weight polymers, the specificity of the initiation and termination processes, l hese issues are discussed in Section 7.5.6... 

The static properties of an isolated chain constitute a good starting point to study polymer dynamics many of the features of the chain in a quiescent fluid could be extrapolated to the kinetics theories of molecular coil deformation. As a matter of fact, it has been pointed out that the equations of chain statistics and chain dynamics are intimately related through the simplest notions of graph theory [16]. 

Hashimoto T., Shibayama M., Kawai H., and Meier D. J. Confined chain statistics of block polymers and estimation of optical anisotropy and domain size. Macromolecules, 18, 1855, 1985. 

Rg)cai2 is the calculated radius of gyration with the parameters for PIP, taken from P.J. Flory (Statistical Mechanics of Chain Molecules, Wiley Intersciences, New York 1969) and the assumption of Gaussian chain statistics. 

The molecular models of rubber elasticity relate chain statistics and chain deformation to the deformation of the macroscopic material. The thermodynamic changes, including stress are derived from chain deformation. In this sense, the measurement of geometric changes is fundamental to the theory, constitutes a direct check of the model, and is an unambiguous measure of the mutual consistency of theory and experiment. 

Szleifer, I., Ben-Shaul, A. and Gelbart, W. M. (1986). Chain statistics in micelles and bilayers effects of surface roughness and internal energy, J. Chem. Phys., 85, 5345-5358. 

In this model the linear chains become the rays of a star molecule. The rays, consisting of m repeating units, can be considered stiff rods, but in most cases they will be flexible and can be described in a first approximation by Gaussian chain statistics. A star molecule has only one branching unit among/xm units which belong to Hnear chains. Their properties can be expected to show a close... 

Up to this point we have considered only unperturbed chain statistics. However, even under -conditions when A2=0 unperturbed statistics cannot be... 

Assuming that the chain statistics is an effective Gaussian with the uniform expansion coefficient l, we obtain... 

Peckham, T. J., Schmeisser, J., Rodgers, M. and Holdcroft, S. 2007. Main-chain, statistically sulfonated proton exchange membranes The relationships of acid concentration and proton mobility to water content and their effect upon proton conductivity. Journal of Materials Chemistry 17 3255-3268. 

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. 

Fig. 5.1 Mode number dependence of the relaxation times Tj and T2 (solid lines). The dashed-dotted line shows the relaxation time ip in the Rouse model (Eq. 3.12). The horizontal dashed line displays the value of r. The dashed and the dotted lines represent the relaxation time when the influence of the chain stiffness is considered mode description of the chain statistics iq (dashed, Eq. 5.11) and bending force model tp (dotted, Eq. 5.7). The behaviour of the relaxation time used in the phenomenological description is also shown for the lowest modes (see text). (Reprinted with permission from [217]. Copyright 1999 American Institute of Physics)...

Probabilities of configurations conducive to the intramolecular back-biting abstraction of a hydrogen atom are evaluated for growing unperturbed PVAc chains. A realistic RIS model is used for the chain statistics, Probabilities are found to be smaller than those seen in an earlier treatment of the polyethylene chain. The smaller probabilities of PVAc contribute to the virtual absence of short branches. The present study therefore provides support for the validity of the Roedel mechanism for the formation of short branches in the free radical initiated polymerization of ethylene. 

The dynamic RIS formalism is used to calculate the rate of first passage from non-excimer-forming conformations to excimer-formlng conformations in seven aromatic polyesters with different flexible spacers between the aromatic rings. The equilibrium chain statistics provides a good description of the relative excimer population for these polyesters, even at times where the dynamic contribution is significant. 

Formation of complex branches, which might arise from two intramolecular rearrangements of the Roedel type, are investigated in polyethylene. An RIS model is used for the polyethylene chain statistics, with inclusion of the effect of a trifunctional branch point on weighting of all configurations. 

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. 

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