Molecular stress distribution function

IR spectroscopy can be used to study the effects of applied mechanical stress on highly oriented samples. The goal is to obtain the molecular stress distribution function, which is an important quantity for determining the stress relaxation moduli or creep compliances. Shifts in the peak frequencies are observed and an attempt is made to determine the molecular stress distribution by deconvolution [34]. The shifts as a function of stress for the 1168-cm band of oriented isotactic polypropylene [35] are shown in Fig. 4.37. 

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

If the crosslinking reaction is interrupted before the gel point, the molecular weight and functionality distributions of such functional precursors are wider but not basically different from that of polymers of BAf monomer. It was stressed recently that they resemble hyperbranched polymers [24] in a certain respects. The pre-gel polymers are generally not stable because the crosslinking reaction can occur during storage. Stable precursors, e.g. for RAf + R Ba, can be obtained in two ways ... 

The viscosity of some polymers at constant temperature is essentially Newtonian over a wide shear rate range. At low enough shear rates all polymers approach a Newtonian response that is, the shear stress is essentially proportional to the shear rate, and the linear slope is the viscosity. Generally, the deviation of the viscosity response to a pseudoplastic is a function of molecular weight, molecular weight distribution, polymer structure, and temperature. A model was developed by Adams and Campbell [18] that predicts the non-Newtonian shear viscosity behavior for linear polymers using four parameters. The Adams-Campbell model is as follows ... 

An empirical method to cope with the effect of molecular-weight distribution was proposed by Van der Vegt (1964). Fie determined viscosities of several grades of polypropylenes with different Mw and MMD as a function of the shear stress ash- A plot °f V/Vo vs. the product tvdv = t1wQ proved to give practically coinciding curves. This generalised curve has been reproduced in Fig. 15.21. 

From Eq. (3-13), polymer stresses in a flow or deformation can be calculated if the distribution function for that flow can be predicted. To make such predictions, molecular... 

Doi molecular theory adds a probability density function of molecular orientation to model rigid rodlike polymer molecules. This model is capable of describing the local molecular orientation distribution and nonlinear viscoelastic phenomena. Doi theory successfully predicts director tumbling in the linear regime and two sign changes in the first normal stress difference,as will be discussed later. However, because this theory assumes a uniform spatial structure, it is unable to describe textured LCPs. 

The starting point of a molecular constitutive theory is a simple mechanical model for the molecule that captures its most salient traits. Thus, flexible polymer molecules have been represented by elastic dumbbells and bead-spring chains, and rigid polymers by rigid dumbbells and rigid rods. For its simplicity, the evolution of the model molecule is easily described by a convection-diffusion equation. Then a Fokker-Planck equation is written for the probability distribution function of an ensemble of these molecules. Finally, the macroscopic stress tensor is derived in terms of the distribution function. This kinetic theory framework was pioneered by Kirkwood (see, for example, Ref. ). 

The requirements of theory both for solvation and transfer data of single ions are similar. A complete theory would require the knowledge of all molecular distribution functions and mean-force potentials between the ions and the solvent molecules. As already stressed in Section II such a theory is imavailable with the present state of knowledge. In the endeavour to represent solvation by models, the... 

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