Elastic thread model

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

The last relation shows that a long macromolecule rolls up into a coil at high temperatures. The smaller the elasticity coefficient a is, the more it coils up. Another name for the model of flexible thread is the model of persistence length or the Kratky-Porod model. The quantity a/T is called the persistence length (Birshtein and Ptitsyn 1966). 

Pig. 12. Changes on shortening in the thermoelastic force (curves 1 and la), the potential-elastic force (curves 2 and 2a) and the mechanical force (curves 3 and 3a). Left, model fiber (A. Weber and H. H. Weber, 1951) right, actomyosin thread (Portzehl, 1950b). 

The above-mentioned complexity of the relationships between the structure and properties of textiles is further complicated by the non-linear mechanical properties of individual fibres caused by their visco-elastic behaviour, friction between fibres and threads, anisotropy, and statistical distribution of all properties. Modelling such complex materials requires application of a combination of experimental, analytical, and numerical methods, which will be considered in this chapter. 

Since fibers consist primarily of oriented crystallites, it is unfair to classify them as heterophase. However, the generalizations of time-temperature superposition that work so well with amorphous polymers do not apply to fibers. Fibers do exhibit viscoelasticity qualitatively like the amorphous polymers. It comes as a surprise to some that J. C. Maxwell, who is best known for his work in electricity and magnetism, should have contributed to the mathematics of viscoelasticity. The story goes that while using a silk thread as the restoring element in a charge-measuring device. Maxwell noticed that the material was not perfectly elastic and exhibited time-dependent effects. He noticed that the material was not perfectly elastic and showed time effects. The model that bears his name was propounded to correlate the real behavior of a fiber. 

In order to explain the observations made with natural rubber and other elastomers, it is necessary to understand the behavior of polymers at the microscopic level. This leads to a model that predicts the macroscopic behavior. It is surprising that in one of the earliest and most successful models, called the freely jointed chain [2,3], we can entirely disregard the chemical nature of the polymer and treat it as a long slender thread beset by Brownian motion forces. This simple picture of polymer molecules is developed and embellished in the sections that follow. Models can explain not only the basics of rabber elasticity but also the qualitative rheological behavior of polymers in dilute soluhon and as melts. The treatment herein is kept as simple as possible. More details are available in the literature [1-7]. 

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