Kinetic Theory of Polymers

An examination of Figs. 7.1-7.4 indicate the similarity between the variation of relaxation modulus (creep compliance) with time and temperature. For this reason, the time and temperature variation of the moduli (or compliances) of a polymer are often said to be related or, in fact 

The theories of Rouse and Zimm were developed for dilute solutions of polymers above the Tg but Ferry and coworkers essentially extended these to bulk polymers in the rubbery state. In doing so, a number of assumptions were made among which were  

The Rouse equation for the relaxation time of an p (arbitrary) segment is, (see Nielsen (1965), Nielsen and Landel (1994) as well as an article by E. Passaglia and J.R. Knox in Baer (1964)), 

That is, the relaxation times of the bulk polymer at one temperature can be found from that at another temperature by multiplying each relaxation time by the shift factor or. 

the shifting of the data demonstrated in Fig. 7.3 should be represented by Eq. 7.4. The term thermorheologically simple refers to the key caveat that all relaxation times of the polymer must be affected by temperature in the same way. This assumption has been found to hold for a vast array of homogeneous polymer systems. Typically shift factors are found experimentally or by the WLF equation discussed in the next section. 

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From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

The reptation theory of viscoelasticity developed by Doi Edvards has paractically predetermined the appearance of Curtiss-Bird s reptation theory55). The latter is constructed on the basis of general kinetic theory of polymer fluids in phase space. According to the Curtiss-Bird theory, the longitudinal viscosity may increase only by a factor of two compared to the initial value. Note that a more significant increase in longitudinal viscosity was observed in experiments. 

Molecular theories, utilizing physically reasonable but approximate molecular models, can be used to specify the stress tensor expressions in nonlinear viscoelastic constitutive equations for polymer melts. These theories, called kinetic theories of polymers, are, of course, much more complex than, say, the kinetic theory of gases. Nevertheless, like the latter, they simplify the complicated physical realities of the substances involved, and we use approximate cartoon representations of macromolecular dynamics to describe the real response of these substances. Because of the relative simplicity of the models, a number of response parameters have to be chosen by trial and error to represent the real response. Unfortunately, such parameters are material specific, and we are unable to predict or specify from them the specific values of the corresponding parameters of other... 

In all kinetic theories of polymer crystallization [8] the crystal growth rate G... 

Once primary nuclei are formed the ensuing spherulites grow radially at a constant rate. Primary crystallization, which occurs initially on the surface of the primary nucleus and then on the surface of the growing lamellar, also involves a nucleation step, secondary nucleation. It is this step that largely governs the ultimate crystal thickness and which forms the focus of most kinetic theories of polymer crystallization. 

The whole discussion of polymer adsorption so far makes the fundamental assumption that the layer is at thermodynamic equilibrium. The relaxation times measured experimentally for polymer adsorption are very long and this equilibrium hypothesis is in many cases not satisfied [29]. The most striking example is the study of desorption if an adsorbed polymer layer is placed in contact with pure solvent, even after very long times (days) only a small fraction of the chains desorb (roughly 10%) polymer adsorption is thus mostly irreversible. A kinetic theory of polymer adsorption would thus be necessary. A few attempts have been made in this direction but the existing models remain rather rough [30,31]. 

The above reasoning has been used to eliminate the need for knowing the pair distribution function in the kinetic theory of polymer melts [9,14a]. 

We point out parenthetically that in the kinetic theory of dilute gases it is just the deviation from the Maxwellian velocity distribution that is of primary interest in the evaluation of the transport properties. In the kinetic theory of polymers, on the other hand, it has been assumed that the deviations from the Maxwellian distribution are of minor importance, and to date few calculations or estimations have been made of the errors introduced by this assumption [16-19], These exploratory efforts indicate that there may be a significant effect on the components of the complex viscosity in high-frequency oscillatory shearing flows. 

The current book is intended to be a concise introduction to polymer physics. As such, it will mainly focus on polymer structures as well as their relationships with properties (as elucidated by statistical thermodynamic and kinetic theories of polymers), and may not be able to provide an extensive survey on polymer properties and their wide applications. For a complementary knowledge about polymer properties, the readers are directed to other textbooks of polymer physics or specialized mmiographs about certain polymer properties. 

J.J. Point and M. Dosiere, Grystal growth rate as a function of molecular weight in polyethylene crystaUized from the melt an evaluation of the kinetic theory of polymer crystaUization, Polymer 30, 2292 2296 (1989). 

In the last fifteen years modern spectroscopical methods (ESR, IR) and conventional methods of structure research have permitted considerable progress in the investigation of deformation and fracture of polymeric materials. For the first time in western languages a unified view of the kinetic theory of polymer fracture is presented by one of the scientists contributing to its development. 

While data was collected from room temperature ( 25° C) to 130° C, only the data above 90° C is shown as it was not possible to shift data below this temperature to form a realistic extension to the data shown. Note that the TTSP method is an outgrowth of the kinetic theory of polymers which is only strictly valid above the Tg. While the TTSP is thought to be valid for temperatures below the Tg, the exact lower limit is not well defined. A guiding rule of thumb is that TTSP may be used below the Tg as long as data is shiftable to form a smooth master curve. 

The kinetic theory of polymers and the TTSP are only valid above the glass transition temperature. However, many feel that the procedure, in a modified form, is valid below the glass-transition temperature but exactly how far below is uncertain. The WLF equation, on the other hand is known to be only valid above the Tg because below this temperature the material can no longer be considered a super cooled liquid. In fact. Ferry, (1980) notes that the slope of the shift factor curve should be discontinuous at the Tg for the same reason that the coefficient of thermal expansion suffers a discontinuity at the Tg. 

Figure 8 Models and theories. The chart summarizes molecular models and outlines the development of kinetic theories of polymer liquids. The sketches show some simple physical pictures that form the basis of theoretical models describing...

This book on Polymer Fracture might as well have been called Kinetic Theory of Polymer Fracture . The term kinetic theory , however, needs some definition or, at least, some explanation. A kinetic theory deals with and particularly considers the effect of the existence and discrete size, of the motion and of the physical properties of molecules on the macroscopic behavior of an ensemble, gaseous or other. A kinetic theory of strength does have to consider additional aspects such as elastic and anelastic deformations, chemical and physical reactions, and the sequence and distribution of different disintegration steps. 

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