Entanglements coupling

This model, whilst not entirely satisfactory, can give a reasonable prediction of the elasticity in the plateau zone. The approach does not readily lend itself to a prediction of the dynamic behaviour. There have been numerous attempts to predict the low shear viscosity using entanglement models. For example Graessley obtained 

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En vs c [47] is usually associated with the establishment of entanglement couplings [48], Thus, the observed linearity of En vs c dependencies of dendrimer solutions seems to be another manifestation of dendrimer reluctance to interpenetrate, either with other dendrimers or with their parts. 

The symmetry properties of M alone are sufficient to prove that only the odd eigenvalues of A are affeded by subtraction of M. Thus, according to this view, only the relaxation times of odd order are affected by entanglement coupling. Furthermore, the shifts diminish rapidly as mode number increases, so only the longest odd relaxation times are affected. 

Each of the viscoelastic parameters G°, rj0, and Je° has associated with it a characteristic molecular weight which either measures an equivalent spacing of entanglement couples along the chain (Me, deduced from G with the kinetic theory of rubber elasticity), or marks the onset of behavior attributed to the presence of entanglements (Mc and AT, deduced from r/0 and Je° as functions of molecular weight). Table 5.2 lists Me, Mc, and M c for several polymers. Aside from certain difficulties in their evaluation, each is a rather direct and independent reflection of experimental fact. 

Kramer,0.,Ty,V, Ferry, J.D. Entanglement coupling in linear polymers demonstrated by networks crosslinked in states of strain. Proc. Natl. Acad. Sci. 69,2216-2218 (1972). 

Stratton,R. A. Non-newtonian flow in polymer systems with no entanglement coupling. Macromolecules 5,304-310 (1972). 

Many amorphous homopolymers and random copolymers show thermorheologically simple behavior within the usual experimental accuracy. Plazek (23,24), however, found that the steady-state viscosity and steady-state compliance of polystyrene cannot be described by the same WLF equation. The effect of temperature on entanglement couplings can also result in thermorheologically complex behavior. This has been shown on certain polymethacrylate polymers and their solutions (22, 23, 26, 31). The time-temperature superposition of thermorheologically simple materials is clearly not applicable to polymers with multiple transitions. The classical study in this area is that by Ferry and co-workers (5, 8) on polymethacrylates with relatively long side chains. In these the complex compliance is the sum of two contributions with different sets of relaxation mechanisms the compliance of the chain backbone and that of the side chains, respectively. 

Here Z is the average number of chain atoms between entanglement points, s is a factor to represent the strength of the entanglement couple s = 0 if the couple results in no constraint on the molecular motion and s = 1 if the couple acts as a permanent bond, and the remaining parameters are as defined previously. For typical systems, = 1 3, and is constant for a homologous series (except at very low Z). 

Barron, A.E., Blanch, H.W., and Soane, D.S., A transient entanglement coupling mechanism for DNA separation by capillary electrophoresis in ultradUutepolymer solutions. Electrophoresis, 15,597,1994. 

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