The crystalline bridge model

In the absence of information regarding the arrangement of the crystalline bridges, it is assumed that they are randomly placed so that the probability of a crystalline sequence traversing the disordered regions to link adjacent crystalline blocks is given in terms of a single parameter p, defined as 

The first term in this expression, which corresponds to the crystalline bridge sequences, is next treated as an array of short fibres, so introducing the shear lag (efficiency) factor 

The aspect ratio of the fibre component, which is a measure of the width of the crystalline bridges, is not directly accessible but can be deduced from the value of the shear lag factor required to give the best match between the predicted and observed patterns of mechanical behaviour as a function of temperature. This exercise yields a radius of 1.5 nm for the crystalline bridge sequences, which suggests that each bridge is comprised of several extended polymer chains. Detailed considerations of the way in which the modulus increases at temperatures below —50°C suggest that the modulus of the matrix increases with increasing draw ratio due to an increase in E in Equation (9.24), which corresponds to an increase in the modulus of the non-crystalline material. 

Models of increasing sophistication have been developed to predict the elastic properties of composite materials from the properties of their constituent parts. These range from the simple rule-of-mixtures approach to the Halpin-Tsai and Mori-Tanaka analyses, where the geometry - essentially, the aspect ratio - of the reinforcing particles can be taken into account. This has the potential to model the effects of extreme aspect ratios that are seen in nanocomposites. Direct finite element simulation of the microstructure is an option that is becoming increasingly feasible at both the micro and nano levels. 

Even for block copolymers, in which the phase separation can be distinguished in electron micrographs, there are problems in matching parameters such as Poisson s ratios of the two components nevertheless the simple Takayanagi models, particularly when extended by a treatment to account for the finite length of the reinforcing component, can describe numerous features of static and dynamic elastic behaviour. 

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In POM a careful examination of WAXS and SAXS measurements on a wide range of drawn samples, showed that the average crystal length, even for the highest modulus material was only comparable with the long period Again it must be concluded that the crystalline bridge model is not applicable. 

Deformation Mechanism. The deformation mechanism has been studied in more detail for the case of semicrystalline polymers, polyethylene, in particular (1,2). There exist a number of models explaining the evolution of polymer structure at high draws. The models can be subdivided into two categories (2), namely, the microfiber plus the tie molecule model, as proposed by Peterlin and co-workers and the extended chain model, which is similar to Ward s crystalline bridge model. 

As a first stage, the contribution of the crystalline bridges can be considered as one element of a Takayanagi model (in Figure 8.9(b) this is the continuous phase) that is in parallel with the series combination of the remaining lamellar material and the amorphous component. Young s modulus would then be... 

The structure of high modulus polyethylene fibres obtained by optimized drawing of linear polyethylene is viewed as crystalline lamellae linked by intercrystalline bridges.Accordingly, the component B is then viewed as crystalline, and its content (1 — >1) corresponds to the volume fraction of the material incorporated in the crystalline bridges. A more complex model consisting of four components has been proposed for these fibres by Grubb. 

Another model assumes that gel zones are formed by hydrated lead dioxide (PbO(OH)2) and act as bridging elements between the crystallite particles. Electrons can move along the polymer chains of this gel and so cause electronic conductivity between the crystalline zones 137],... 

In view of this disagreement, as well as of evidence from polymer mesophases and MD simulations, we also propose an alternative model, based on the concept that the attractive interactions are so short-lived as to be effectively delocalized. As a consequence, bridges separating consecutive bundles are also taken into account in the evaluation of the average stem length of the growing crystal, in addition to the crystalline stems and to the loops... 

For the time being, the best model system to exemplify the particular type of alkynyl bridging may be the aluminium compound (CH3)2AIC2CH3 which is dimeric both in the crystalline and in the gaseous state (39). The Av(CEC)-values known for this main group metal alkynyl system match nicely those listed in Table VII. 

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