Chain entanglements and the Edwards tube model

In the 1940s, it was recognized that the classical predictions of network modulus were bounded. A real network could certainly not be expected to have lower modulus than the phantom prediction, since it is based on unrestricted fluctuations of ideal strands that are allowed to pass through each other. At the other extreme, the classical models have no means to attain a higher modulus than the affine prediction, based on junctions that 

Each monomer is constrained to stay fairly close to the primitive path, but fluctuations driven by the thermal energy kT are allowed. Strand excursions in the quadratic potential are not likely to have free energies much more than kT above the minimum. Strand excursions that have free energy kT above the minimum at the primitive path define the width of the confining tube, called the tube diameter a (Fig. 7.10). In the classical affine -and phantom network models, the amplitude of the fluctuations of a 

The fact that two chains cannot pass through one another creates topological interactions known as entanglements that raise the network modulus. 

This tube diameter can be interpreted as the end-to-end distance of an entanglement strand of A e monomers  

A chain or network strand (thick curve) is topologically constrained to a tube-like region by surrounding chains. The primitive path is shown as the dashed curve. The roughly quadratic potential defining the tube is also sketched. 

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