Statistical amorphous polymer chain

The experimental observation of the same Gaussian statistics of polymer chains in 0-solvent and condensed state is the main objection against local order availability in amorphous state polymers [105]. The equality of distances between macromolecules or subchains ends in the indicated states is considered as one of the pieces of evidence of this rule. Boyer [106] demonstrated schematically the possibility of local order existence at fulfilment of the indicated condition. However, strict confirmation of such a possibility was not obtained. Therefore the authors of paper [107] confirmed analytically Boyer s concept on the example of two series of epoxy polymers (EP-1 and EP-2). 

It is beyond our control how the cross-links are spaced along the polymer chains during the vulcanization process. This extraordinary important fact demands a generalization of the Gibbs formula in statistical mechanics for amorphous materials that have fixed constraints of which the exact topology is unknown. Details of a modified Gibbs formula of polymer networks can be found in the pioneering paper of Deam and Edwards [13]. 

Flory and Huggins developed an interaction parameter that may be used as a measure of the solvent power of solvents for amorphous polymers. Flory and Krigbaum introduced the idea of a theta temperature, which is the temperature at which an infinitely long polymer chain exists as a statistical coil in a solvent. 

Interactions between nonbonded atoms and groups in this chain are found to be either identical with or similar to interactions arising in PE, POM, and POE. Statistical weights obtained in the analysis of the dimensions and dipole moments of these chains are thus applicable to the present investigation of the polyltetramethylene oxide) chain, The calculated dipole moments are in good agreement with preliminary published results obtained on the undiluted, amorphous polymer. 

Two theoretical approaches for calculating NMR chemical shift of polymers and its application to structural characterization have been described. One is that model molecules such as dimer, trimer, etc., as a local structure of polymer chains, are in the calculation by combining quantum chemistry and statistical mechanics. This approach has been applied to polymer systems in the solution, amorphous and solid states. Another approach is to employ the tight-binding molecular orbital theory to describe the NMR chemical shift and electronic structure of infinite polymer chains with periodic structure. This approach has been applied to polymer systems in the solid state. These approaches have been successfully applied to structural characterization of polymers... 

According to the statistical-mechanical theory of rubber elasticity, it is possible to obtain the temperature coefficient of the unperturbed dimensions, d InsjdT, from measurements of elastic moduli as a function of temperature for lightly cross-linked amorphous networks [Volken-stein and Ptitsyn (258 ) Flory, Hoeve and Ciferri (103a)]. This possibility, which rests on the reasonable assumption that the chains in undiluted amorphous polymer have essentially their unperturbed mean dimensions [see Flory (5)j, has been realized experimentally for polyethylene, polyisobutylene, natural rubber and poly(dimethylsiloxane) [Ciferri, Hoeve and Flory (66") and Ciferri (66 )] and the results have been confirmed by observations of intrinsic viscosities in athermal (but not theta ) solvents for polyethylene and poly(dimethylsiloxane). In all these cases, the derivative d In sjdT is no greater than about 10-3 per degree, and is actually positive for natural rubber and for the siloxane polymer. 

As shown in Figure 23, PP is no longer able to crystallize when the stereoregularity of the chains is reduced below a threshold value (below about 70% m diad, or 40% mmmm pentad content for iPP, or below about 60% rrrr pentad content for sPP), and it becomes amorphous (amPP). When statistical randomness in the sequence of chirotopic methynes in the polymer chain is reached, the polymer is called atactic (aPP). In this case the pentad distribution is perfectly random Bcrnoullian mmmm mmmr rmmr mmrr (rmrr+ mrmm) mrmr rrrr rrrm mrrm 1 2 1 2 4 2 1 2 1. 

Presented polymer mixtures are composed of amorphous macromolecules with different molecular architecture homopolymers and random copolymers, with different segments distributed statistically along the chain, form partly miscible isotopic and isomeric model binary blends. The mixing of incompatible polymers is enforced by two different polymers covalently bonded forming diblock copolymers. Here only homopolymers admixed by copolymers are considered. The diblock copolymer melts have been described recently in a separate review by Krausch [17]. 

One criterion to distinguish the miscibility of blends is the glass transition temperature (Tg) that can be measured with different calorimetric methods [95]. Tg is the characteristic transition of the amorphous phase in polymers. Below Tg, polymer chains are fixed by intermolecular interactions, no diffusion is possible, and the polymer is rigid. At temperatures higher than Tg, kinetic forces are stronger than molecular interactions and polymer chain diffusion is likely. In binary or multi-component miscible one-phase systems, macromolecules are statistically distributed on a molecular level. Therefore, only one glass transition occurs, which normally lies between the glass transition temperatures of the pure components. 

The main asstrmptions used concern the Gaussian character of the chains and the absence of restrictions imposed by otber chains to tbe conformation of a given chain. They are based on the Flory theorem, which states that the statistical properties of polymer chains in a dense system are equivalent to those for single ideal chains. The reason is that in a imifoim, amorphous substance all the conformations of a certain chain are equally likely in a sense that they couespond to the same energy of interaction with other chains, because the surrotmdings of each emit are roughly the same. 

The long chains shown in Figure 1.3 also illustrate the coiling of polymer chains in the amorphous state. One of the most powerful theories in polymer science (2) states that the conformations of amorphous chains in space are random coils that is, the directions of the chain portions are statistically determined. 

However, many polymer scientists have been highly concerned with the density of polymers (332). For many common polymers the density of the amorphous phase is approximately 0.85 to 0.95 that of the crystalline phase (3,10). Returning to the spaghetti model, some scientists think that the polymer chains have to be organized more or less parallel over short distances, or the experimental densities cannot be attained. Others (32) have pointed out that different statistical methods of calculation lead, in fact, to satisfactory agreement between the experimental densities and a more random arrangement of the chains. 

Thus the conformational statistics of chain molecules are now increasingly able to provide a basis for estimating the rheological and viscoelastic behavior of linear amorphous polymers above their glass transition temperatures. 

---