Modified Cerf-Rouse Modes

One has to refer to dynamic equations (3.11) of the macromolecule to find independent modes of motion. The matrices A and G in these equations are defined by equations (1.8) and (3.10), and, in the normal co-ordinates (1.13), simultaneously have diagonal forms 

The zeroth eigenvalues of matrix A and G are zero, so that without any approximation, one can write an equation for diffusion mode 

The intermolecular forces are naturally absent from the equation for the zeroth mode, because this mode presents the motion of the mass centre of the coil. 

In accordance to approximate form (3.10), the other eigenvalues of the matrix G are constant and equal to unity, so that the set of equations for relaxation modes of the macromolecule now assumes the form 

Let us note, that the matrixes A and G are approximations of the real situation though, in any case, the zeroth eigenvalues of the matrixes must be zero and equation (4.1) for diffusive mode is valid, the other eigenvalues of matrix G depends, generally speaking, on the mode label. In fact, the written equations for the relaxation modes are implementation of the statements that the motion of a single macromolecule can be separated from others, and the motion of a single macromolecule can be expanded into an independent motion of modes. 

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One can see that the approximation of the theory, based on the linear dynamics of a macromolecule, is not adequate for strongly entangled systems. One has to introduce local anisotropy in the model of the modified Cerf-Rouse modes or use the model of reptating macromolecule (Doi and Edwards 1986) to get the necessary corrections (as we do in Chapters 4 and 5, considering relaxation and diffusion of macromolecules in entangled systems). The more consequent theory can be formulated on the base of non-linear dynamic equations (3.31), (3.34) and (3.35). 

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