Number of Entanglements

The above variables are not independent of each other. Solution viscosity is a function of both concentration and polymer molecular mass. Also, these two parameters affect the number of entanglements. 

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For concentrated polystyrene solutions (c>c ) in n-butylbenzene, Graessley and co-workers [36] observed that the shift factor hconc depends on the number of entanglements per macromolecule E. 

With the number of entanglement strands ne/volume, the desired number of binary contacts per entanglement strand becomes... 

Equation (29) shows that the modulus is proportional to the cycle rank , and that no other structural parameters contribute to the modulus. The number of entanglements trapped in the network structure does not change the cycle rank. Possible contributions of these trapped entanglements to the modulus therefore cannot originate from the topology of the phantom network. 

The time ie represents the entanglement formation time at the current shear rate. The second term g is the average number of entanglement sites for a chain of fixed length in steady flow relative to the number in the zero shear rate limit ... 

Already Flory predicted that the number of entanglements would be lower for polymers based on A B monomers, with subsequent reduction in mechanical strength [1]. 

Rouse normal mode Dynamic response matrix Longitudinal susceptibility Number of entanglements/chain... 

FIGURE 16.9 Idealized relationship between chain length and number of entanglements and polymer strength. 

The De Gennes method can be applied directly to Bueche s model without the defect diffusion formalism. Consider a chain with n main chain atoms, mean square end-to-end distance , and a large number of entanglement obstacles along its contour. For simplicity, suppose the number of main chain atoms between successive obstacles and the scalar distance between obstacles are... 

Initially, the elasticity of concentrated polymer systems was ascribed to the existence of a network in the system formed by long macromolecules with junction sites (Ferry 1980). The sites were assumed to exist for an appreciable time, so that, for observable times which are less than the lifetime of the site, the entangled system appears to be elastic. Equation (1.44) was used to estimate the number density of sites in the system. The number of entanglements for a single macromolecule Z = M/Me can be calculated according to the modified formula... 

Slow crystallization promotes disentangling since the process of reeling-in of chains from the entangled melt onto the crystal surface is promoted and this in turn enhances the maximum drawability, despite the fact that this increases the crystallinity. However, there is a critical lower limit for the number of entanglements between crystals in order to achieve high drawability and in the extreme limit of very slow isothermal crystallization, linear standard polyethylenes (not the UHMW-PE grades) can lose their drawability completely and become brittle materials. 

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