Amorphous Polymers in the Bulk

Generally three stages are required to generate a realistic atomistic model 

Set up system in terms of atomic coordinates stored on a computer the physical state should resemble as closely as possible the state of interest (disordered liquid, amorphous solid, or crystalline solid). 

Perform simulation either employ MD to determine trajectories and, provided the simulation is long enough, calculate equilibrium properties, or employ molecular mechanics to probe the structure of low energy configurations and subsequently use MC sampling methods to compute equilibrium properties. 

Estimate pressure—volume—temperature (PVT) relations, cohesive energy densities, pair correlation functions, X-ray scattering curves, elastic constants, and other properties. 

One method for the simulation of dense polymer systems is based on Equation [67]  

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Fracture processes are associated with deformations. In glassy thermoplastics craze formation is the most frequent pre-failure deformation process. Just ahead of the crack tip, where the stresses are particularly concentrated, molecular chains of the polymer are drawn out of their amorphous arrangement in the bulk material into fibrils (see Fig. 1.1 and Chapter 1) under the action of the principal tensile stress... 

Several attempts to estimate the hole density from a comparison of the mean hole volume with the macroscopic volume are described in the literature. The drawback of such approaches is that assumptions must be made as to the value of or on the thermal expansion and compression of the volume that is not detected by o-Ps. Frequently, it is assumed that that this volume, denoted as occupied or bulk volume, expands Uke an amorphous polymer in the glassy state [Hristov et al., 1996 Dlubek et al., 1998c Band ch et al., 2000 Shantarovich et al., 2007]. Another assumption is that no variation with temperature or pressure is shown [Bohlen and Kirchheim, 2001]. Both assumptions are intuitive but physically not proved. The most successful attempt to estimate hole densities comes from a calculation of the hole free volume with... 

SANS is used in the determination of chain dimensions of polymers in the bulk amorphous state. Here the isotopic difference in neutron scattering cross-sections of deuterium and hydrogen are made use of. Jeffrey et al. studied the chain dimensions of rubber by using SANS technique. ... 

The fracture strength a of amorphous and semi-crystalline polymers in the bulk can be expressed according to the Net solution as. 

This would seem to indicate that the proportionality assumption has no physical basis. However, in many cases the bulk modulus is much larger than the shear modulus (except perhaps near / = 0) so that, for practical purposes, we can take V = Y- In other words, K(t) is so much larger than G(t) that it does not matter what shape we assume for K(t). This is the case for amorphous polymers at temperatures well above their glass transition temperatures. Also for many rigid plastics, for example, amorphous polymers in the glassy state, v is constant but less than j-, typically having values in the range 0.35-0.41 [Schapery (1974)]. In such cases, the proportionality assumption would seem to have approximate validity. In summary, therefore, this assumption, while motivated primarily by the need for mathematical simplicity, is a reasonable approximation for many materials. 

A suitable approach to the equilibration of an amorphous polymer system at bulk density becomes much more likely when the fully atomistic model in continuous space is replaced by an equivalent coarse-grained model on a lattice with sufficient conformational flexibility. Different strategies, which seek results at different levels of detail, can be employed to create an appropriate coarse-grained model. Section 4 (Doruker, Mattice) describes an approach which attempts to retain a connection with the covalent bonds in the polymer. The rotational isomeric state (RIS) [35,36] model for the chain is mapped into... 

Finally, we turn from solutions to the bulk state of amorphous polymers, specifically the thermoelastic properties of the rubbery state. The contrasting behavior of rubber, as compared with other solids, such as the temperature decrease upon adiabatic extension, the contraction upon heating under load, and the positive temperature coefficient of stress under constant elongation, had been observed in the nineteenth century by Gough and Joule. The latter was able to interpret these experiments in terms of the second law of thermodynamics, which revealed the connection between the different phenomena observed. One could conclude the primary effect to be a reduction of entropy... 

For all the cases cited above, which represent those data for which a comparison can be presently made, there is a direct connection between the critical molecular weight representing the influence of entanglements on the bulk viscosity and other properties, and the NMR linewidths, or spin-spin relaxation parameters of the amorphous polymers. Thus the entanglements must modulate the segmental motions so that even in the amorphous state they are a major reason for the incomplete motional narrowing, as has been postulated by Schaefer. ( ) This effect would then be further accentuated with crystallization. 

The close connection that exists between polymer morphology and mechanical properties stimulated extensive research in this field. In amorphous polymers, elastic neutron scattering led to important results. Using mixtures of conventional and deuterated macromolecules, the mean square radius of gyration < > of several amorphous polymers in bulk has been determined (237). This... 

As explained earlier (Sect. 1.3.1), macromolecules in a low-molecular-weight solvent prefer a coiled chain conformation (random coil). Under special conditions (theta state) the macromolecule finds itself in a force-free state and its coil assumes the unpertubed dimensions. This is also exactly the case for polymers in an amorphous melt or in the glassy state their segments cannot decide whether neighboring chain segments (which replace all the solvent molecules in the bulk phase) belong to its own chain or to another macromolecule (having an identical constitution, of course). Therefore, here too, it assumes the unperturbed ) dimensions. 

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