Polymers mathematical description

Parameter that provides a mathematical description of the fractal structure of a polymer network, an aggregated particulate sol, or of the particles that comprise them. 

Although many of the proposed applications for these gels requires that they operate under an applied pressure or generate some kind of mechanical force, a detailed understanding of these relationships does not currently exist. There is data available on the effect of the load on the rate of work and stroke, or generated force vs time, for example, but this is often presented on an empirical basis. Furthermore, much of the work has been carried out under conditions where the stimulus is rate-limiting, rather than the polymer network [66, 67], The development of a mathematical description of these phenomena using independently obtainable polymer parameters is needed. 

Section IIA summarizes the physical assumptions and the resulting mathematical descriptions of the "concentration-dependent (5) and "dual-mode" ( 13) sorption and transport models which describe the behavior of "non-ideal" penetrant-polymer systems, systems which exhibit nonlinear, pressure-dependent sorption and transport. In Section IIB we elucidate the mechanism of the "non-ideal" diffusion in glassy polymers by correlating the phenomenological diffusion coefficient of CO2 in PVC with the cooperative main-chain motions of the polymer in the presence of the penetrant. We report carbon-13 relaxation measurements which demonstrate that CO2 alters the cooperative main-chain motions of PVC. These changes correlate with changes in the diffusion coefficient of CO2 in the polymer, thus providing experimental evidence that the diffusion coefficient is concentration dependent. 

A number of attempts have been made to explain the nonlinear, pressure-dependent sorption and transport in polymers. These explanations may be classified as "concentration-dependent (5) and "dual-mode (13) sorption and transport models. These models differ in their physical assumptions and in their mathematical descriptions of the sorption and transport in penetrant-polymer systems. 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

Both the matrix-model and the dual-model represent the experimental data satisfactory (Fig. 1). After modeling sorption measurements in several gas-polymer systems we have observed no systematic differences between the mathematical descriptions of the two models. 

The rheokinetic phencanenology of curing is associated with the structure and covers a wide range of polymers which form cured materials. The universality of the mathematical description of rheokinetics of curing of different products indicates that the physico-chemical phenomena associated with the formation of crossHnked structures are genoaL... 

The importance of molecular weight distribution in studies of polymerization, polymer processing and the physical and mechanical properties of polymers creates a need for mathematical description of the distribution. Several models are commonly used (Flory [1], Schulz-Zimm... 

Symbols Presence or absence of events that determine the number of Mathematical description of individual polymer chains (N) scenarios... 

Matrix-type delivery systems are simple to make release is usually controlled by diffusion of drug through the polymer matrix. Mathematical descriptions of release are more complicated than are obtained for membrane-type devices, and it is difficult to produce devices that provide a constant rate of release. However, these materials are versatile and almost any compound can be formulated into a controlled-release matrix. 

The physical and mathematical description of the ribbon extrusion process was first given by Pearson [24], who simplified the conservation equations by using a onedimensional, isothermal, Newtonian fluid approach, and neglected the effects of polymer solidification. As in the case of blown film processes, several modifications and models have been proposed for the ribbon extrusion process (Table 24.2). 

The reason why the random flight model has proved so popular theoretically stems from its simplicity, which offers hope for the development of analytic solutions. The problem can usually be cast in the form of a diffusionlike or a Schrodinger-wave-equation-like differential equation, the solutions of which are reasonably well explored. A tendency has developed in recent times to apply extremely sophisticated mathematical procedures to what are really very primitive models for polymer chains (see, e.g. Levine et al., 1978). Whether the ends merit the means in such instances cannot yet be assessed objectively. A strategy that might be more productive in terms of the development of a practical theory for steric stabilization is to aim for a simpler mathematical description of more complex models of polymer chains. It should also be borne in mind in developing ab initio theories that a simple model that may well suffice in polymer solution thermodynamics may be quite inadequate for the simulation of the conformational properties of polymers. Polymer solution thermodynamics seem to be relatively insensitive to molecular architecture per se whereas the conformation of a polymer chain is extremely sensitive to it. 

Mathematical description of the process of polymer melting in the extrusion channel is complex when ultrasound is used. The description requires firstly, consideration of the mass flow of the polymer, knowledge of the flow characteristics of the melt, the temperature and pressure of extrusion, sizes of the channel and frequency of ultrasonic oscillations. Secondly, coefficient of swelling of the extrudate, effective viscosity of polymer, pressure of melt, and frequency of oscillations. 

The mathematical description of gas diffusion through a polymer is the same as that for heat diffusion considered in Section A.2 of Appendix A. Two material constants, diffusivity D and permeability P, are defined in terms of steady-state flow from a gas at a pressure pi, on one side of a polymer film of thickness L, to a pressure p2 on the other side (Fig. 11.3). The gas concentration in the polymer is constant at Ci and C2, respectively at the two surfaces. The flow rate Q through an area A of film is then given either by... 

In this chapter, we shall discuss different mathematical descriptions of the simplest model of a polymer, the ideal polymer coil (the reason for this name will become clear in Chapter 7). 

Indeed, all this can actually be observed, and is known as the gel effect during radical polymerization. The changes that occm are very dramatic. The rate of reaction jumps by a few orders of magnitude while the fraction of polymer increases only by a minute amount. This effect was noticed fairly long ago, well before the theory of reptation was proposed. However, it was the theory of reptation that enabled a proper mathematical description of the phenomenon. 

Thereby, by the results of different researches for the used experimental objects were determined introduced mathematical description of cross-linked polymers viscoelastic and electromagnetic properties parameters. Verification of prediction abilities of such kind mathematical descriptions is reahzed by experiments, conditions of which are different from conditions of the other experiments, where unknown model s parameters are determined. In our case verification was prediction of Thermomechanical and Thermooptical curves trend (Figure 2,)... 

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