Average entanglements

The entanglement of formation of the mixed p is then defined as the average entanglement of the pure states of the decomposition [42], minimized over all decompositions of p ... 

This characteristic shear rate is equal to the reciprocal average entanglement time Tn at equilibrium. The solid curve is from Graessley s model, which reads... 

Exponential dependence of ij0 on Ma, as discussed earlier for star polymers, was also seen in the H-polymers. The crossover where rj0 of polystyrene H-polymers surpasses that of its linear homologues occurs at Mw 600000g/mol. This value is significantly lower than that observed for four-arm star polystyrenes [35]. Furthermore, the average entanglement of each of the five subchains of the H-polymers at this crossover is Ma 7Me, a value far lower than the level of arm entanglement required to reach the crossover in polystyrene star polymers. 

Polymer solutions are often characterized by their high viscosities compared to solutions of nonpolymeric solutes at similar mass concentrations. This is due to the mechanical entanglements formed between polymer chains. In fact, where entanglements dominate flow, the (zero-shear) viscosity of polymer melts and solutions varies with the 3.4 power of weight-average molecular weight. 

The viscous shear properties at any given shear rate are primarily determined by two factors, the free volume within the molten polymer mass and the amount of entanglement between the molecules. An increase in the former decreases the viscosity whilst an increase in the latter, i.e. the entanglement, increases viscosity. The effects of temperature, pressure, average molecular weight, branching and so on can largely be explained in the these terms. 

Cast material is stated to have a number average molecular weight of about 10. Whilst the Tg is about 104°C the molecular entanglements are so extensive that the material is incapable of flow below its decomposition temperature (approx. 170°C). There is thus a reasonably wide rubbery range and it is in this phase that such material is normally shaped. For injection moulding and extrusion much lower molecular weight materials are employed. Such polymers have a reasonable melt viscosity but marginally lower heat distortion temperatures and mechanical properties. 

Under quiescent conditions, polymer solutions are divided into four categories depending on the average distance separating the centers of mass of the molecular coils the dilute, the semi-dilute (or semi-concentrated), the concentrated and the entangled state. 

In the Eqs. (3.1) and (3.3), a second term can be introduced to account for the average molecular mass 1V1E between entanglements of chains [5, 29] ... 

The dashed line in Fig. 3.2 corresponds to a linear regression calculation yielding Me = 8 kg/mol for the average molecular mass between entanglements if no chemical crosslinks are present (MR - oo). This result agrees reasonably with values for various thermoplastics as determined from elasticity measurements on melts [30, 31, 32], Examples are given in Table 3.2. 

The diffusion coefficients of entangled polymers in solution will most certainly depend on the viscosity of the medium and vice versa. It is reasonable therefore to expect that the diffusion coefficient would correlate well with the weight average molecular weight of the polymer. M is therefore used with equation (lO) giving... 

According to Hess, the relative strength of the entanglement friction can be related to the more microscopic parameter q , describing the range of the true interchain interaction potential. A value of q 1 = 7 A, close to the average interchain distance of about 4.7 A, is obtained. 

Under -conditions the situation is more complex. On one side the excluded volume interactions are canceled and E,(c) is only related to the screening length of the hydrodynamic interactions. In addition, there is a finite probability for the occurrence of self-entanglements which are separated by the average distance E,i(c) = ( (c)/)1/2. As a consequence the single chain dynamics as typical for dilute -conditions will be restricted to length scales r < (c) [155,156],... 

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