Rouse segment size

The force constant, 3kT/b, used in Eq. (9.A.2) originates from the entropy related to the number of microstructural configurations in a Rouse segment. The microstructural configurations have to be considered in three-dimensional space. Involving three-dimensional space is related to the idea that the size of the Rouse segment, b, is much smaller than the tube... 

For times less than the Rouse time of an entanglement segment, Tg and short distances, the chain behaves as if it were free since no section has moved far enough to be strongly affected by the tube constraint. The characteristic decay-rate of the scattering function at wavevector k is dominated by the Rouse-time of chain segments whose size is the order of k % k. A detailed calculation gives for t % [2]... 

For times shorter than the Rouse time of the chain (tcoherent 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)] ... 

Note In Ref. 3, saying that the Rouse and Kuhn segments are of the same order of magnitude in size is an incomplete description of their relatives sizes. 

The most studied relaxation processes from the point of view of molecular theories are those governing relaxation function, G,(t), in equation [7.2.4]. According to the Rouse theory, a macromolecule is modeled by a bead-spring chain. The beads are the centers of hydrodynamic interaction of a molecule with a solvent while the springs model elastic linkage between the beads. The polymer macromolecule is subdivided into a number of equal segments (submolecules or subchains) within which the equilibrium is supposed to be achieved thus the model does not permit to describe small-scale motions that are smaller in size than the statistical segment. Maximal relaxation time in a spectrum is expressed in terms of macroscopic parameters of the system, which can be easily measured ... 

The only parameter of the model (apart from the Rouse model parameters) is the lattice size g, which we expect to be analogous to the tube Kuhn segment a. Note that the model description is significantly shorter than that of the tube model, which means that there are much fewer assumptions and uncertainties in the model. The only important thing to note here is that we shall ignore the (implicit) forces from the constraints in the stress tensor calculation. More discussion about this can be found in the next sertion. 

The exact results of Equations 7.28, 7.31, and 7.32 for the Rouse model are in disagreement with the general expectations of Section 7.2.1 and are never observed experimentally in dilute solutions of polymers, where the model was intended. The reason for this discrepancy is the absence of hydrodynamic coupling between the segments in the Rouse model. Nevertheless, due to the general complexity of the problem of polymer dynamics, there have been considerable activities in simulating the dynamics of polymer chains in silico, usually without hydrodynamic interactions. For this artificial situation of a Rouse chain (i.e., without hydrodynamic interaction) with the size exponent v, the various theoretical results can be summarized as... 

The Zimm model is based on the Rouse model [6, 179], but includes long-range hydrodynamic interactions between the segments. Both models predict selfsimilarity, not only with respect to space, but also with respect to time. Therefore, the dynamics is conveniently described in terms of an exponent z, connecting the chain relaxation time Tk with the size of the coil R ... 

---