Monomer displacement in entangled linear melts

On time scales shorter than the relaxation time of an entanglement strand Te, the sections of a linear chain involved in coherent motion are 

For times shorter than the Rouse time of the chain (t TR), each monomer participates in coherent motion of a chain segment consisting of a/t/to neighbouring monomers. The time-dependent curvilinear coordinate of a monomer along the contour of the tube is s t) (Fig. 9.19). The mean-square monomer displacement along the tube is of the order of the mean-square size of this section in three-dimensional space [Eq. (8.58)]  

Curvilinear displacement of a monomer (labelled by a dark circle) along the contour of the tube between two conformations is A.v. Only a short section of the tube is shown. 

Since the tube itself is a random walk with step length a, the mean-square displacement of a monomer in three-dimensional space (Ar ) is the pro-duct of the step length a and the contour length displacement sj As )  

the mean-square monomer displacement in space exhibits a weak one-fourth power law in time when the chain is confined to a tube  

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