Entanglement points

Figure C2.1.13. (a) Schematic representation of an entangled polymer melt, (b) Restriction of tire lateral motion of a particular chain by tire otlier chains. The entanglement points tliat restrict tire motion of a chain define a temporary tube along which tire chain reptates.

Apart from some numerical coefficients, the details of which we shall forgo, the required proportionality factor involves kT and the concentration of entanglement points. Multiplying Eq. (2.67) by these factors, we obtain... 

Fig. 23.6. A schematic of o linear-amorphous polymer, showing entanglement points (marked "E") which act like chemical cross-links.

At yet higher temperatures (>1.4T ) the secondary bonds melt completely and even the entanglement points slip. This is the regime in which thermoplastics are moulded linear polymers become viscous liquids. The viscosity is always defined (and usually measured) in shear if a shear stress o produces a rate of shear 7 then the viscosity (Chapter 19) is... 

Me(A) increases between entanglement points due to the retraction process at constant X. A more detailed treatment of disentanglement would account for the orientation function of the entanglements and lateral contraction, as discussed elsewhere [1]. Eq. 8.4 becomes... 

The range of semi-dilute network solutions is characterised by (1) polymer-polymer interactions which lead to a coil shrinkage (2) each blob acts as individual unit with both hydrodynamic and excluded volume effects and (3) for blobs in the same chain all interactions are screened out (the word blob denotes the portion of chain between two entanglements points). In this concentration range the flow characteristics and therefore also the relaxation time behaviour are not solely governed by the molar mass of the sample and its concentration, but also by the thermodynamic quality of the solvent. This leads to a shift factor, hm°d, is a function of the molar mass, concentration and solvent power. 

Recently, des Cloizeaux has conceived a rubber -like model for the transition regime to local reptation [56]. He considered infinite chains with spatially fixed entanglement points at intermediate times. In between these... 

Rubber-like models take entanglements as local stress points acting as temporary cross finks. De Cloizeaux [66] has proposed such a model, where he considers infinite chains with spatially fixed entanglement points at intermediate times. Under the condition of fixed entanglements, which are distributed according to a Poisson distribution, the chains perform Rouse motion. This rubber-like model is closest to the idea of a temporary network. The resulting dynamic structure factor has the form ... 

Selective bond rupture at entanglement points, or other such sites of stress concentration, could magnify the effect of a chain scission in the presence of an external stress, but it seems unlikely that this is occurring since the sol-gel data actually indicated a (slightly) lower ratio of scissions to crosslinks with an imposed stress. It also is difficult to visualize how the formation of free radicals, scissions, and crosslinks could directly cause the radiation expansion noted under no stress. Therefore, the mechanism of accelerated creep is probably not caused by the formation and reaction of macromolecular free radicals in the polymer specimens. 

Consider the random decoupling of a system of v primary Rouse molecules, each containing E entanglement points, with random assignment of the slow points. In the equivalent system, the concentration of strands between successive slow points which are made up of exactly i of the original vE network strands is simply (for E> 1) ... 

The first relaxation process (designated A hereafter) corresponds to a Rouse-like relaxation of chain segments between entanglement points. It is assumed that the entanglement points remain fixed during the time-scale of this relaxation and that no diffusion of monomers through the slip-links is allowed in such short times. The associated relaxation time,, is related to a monomeric friction coefficient, to thie number of monomers between... 

The first relaxation process (called the A relaxation process Fig. 8c) which occurs at the shortest times will be a local reequilibration of monomers without slippage through the slip-links. In other words, it is basically a Rouse relaxation process between entanglement points which are assumed to be fixed in that time scale. The characteristic relaxation time of this process is rather short and is independent of the overall chain length (see below). 

That relaxation process may be defined as a Rouse diffusion between entanglement points. The characteristic relaxation time of the (A) process is ... 

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