Friction monomer 312--------------------------Rouse model

In addition to the Rouse model, the Hess theory contains two further parameters the critical monomer number Nc and the relative strength of the entanglement friction A (0)/ . Furthermore, the change in the monomeric friction coefficient with molecular mass has to be taken into account. Using results for (M) from viscosity data [47], Fig. 16 displays the results of the data fitting, varying only the two model parameters Nc and A (0)/ for the samples with the molecular masses Mw = 3600 and Mw = 6500 g/mol. 

In fact, the diffusion constant in solutions has the form of an Einstein diffusion of hard spheres with radius Re. For a diffusing chain the solvent within the coil is apparently also set in motion and does not contribute to the friction. Thus, the long-range hydrodynamic interactions lead, in comparison to the Rouse model, to qualitatively different results for both the center-of-mass diffusion—which is not proportional to the number of monomers exerting friction - as well as for the segment diffusion - which is considerably accelerated and follows a modified time law t2/3 instead of t1/2. 

Early-time motion, for segments s such that UgM(s)activated exploration of the original tube by the free end. In the absence of topological constraints along the contour, the end monomer moves by the classical non-Fickian diffusion of a Rouse chain, with spatial displacement f, but confined to the single dimension of the chain contour variable s. We therefore expect the early-time result for r(s) to scale as s. When all prefactors are calculated from the Rouse model [2] for Gaussian chains with local friction we find the form... 

Regardless of its complex architecture, any polymer relaxing with no topological constraints and no hydrodynamic interactions is well-represented by the Rouse model, with friction proportional to molar mass. To estimate the terminal response of randomly branched polymers, we apply this reasoning to the characteristic polymers, with size consisting of N monomers. The diffusion coefficient of these chains is given by the... 

The Rouse model is the simplest molecular model of polymer dynamics. The chain is mapped onto a system of beads connected by springs. There are no hydrodynamic interactions between beads. The surrounding medium only affects the motion of the chain through the friction coefficient of the beads. In polymer melts, hydrodynamic interactions are screened by the presence of other chains. Unentangled chains in a polymer melt relax by Rouse motion, with monomer friction coefficient C- The friction coefficient of the whole chain is NQ, making tha diffusion coefficient inversely proportional to chain length ... 

Rouse Model. In 1953 P. E. Rouse proposed a simple model to describe the dynamics of a polymer chain in dilute solution (21,59). The model considers the chain as a sequence of Brownian particles which are connected by harmonic springs. Being immersed in a (structureless) solvent the chain experiences a random force by the incessant collisions with the (infinitesimally small) solvent particles. The random force is assumed to act on each monomer separately and to create a monomeric friction coefficient. The model therefore contains chain connectivity, a local firiction and a local random force. All non-local interactions between monomers distant along the backbone of the chain, such as excluded-volume or hydrodynamic in-... 

What are the friction forces involved (in a Rouse model) The solvent forces are simple. The velocity for the n-th monomer is... 

FIG. 5 Monomer friction coefficient f vs. chain length N, obtained from mapping the atomistic MD data onto the Rouse model (squares) or the reptation model (circles). 

How does the relaxation time t depend on the chain length In the Rouse model, the friction coefficient is assumed to be the sum of frictional contributions, 5i, from each monomer unit... 

The result indicates that the ratio Cr/< r should be independent of the choice of the sequence. This is true if the friction coefficient Cr is proportional to the number of monomer units in the sequence. Strictly speaking, the latter property constitutes a basic requirement for the validity of the Rouse-model The friction coefficient of a sequence has to be proportional to the number of monomer units. In fact, this is not trivial and clear from the very beginning. It seems to be correct in a melt because, as we shall see, here the Rouse-model works quite satisfactorily, if compared with experimental results. On the other hand, the assumption is definitely wrong for isolated polymer chains in a solvent where hydrodynamic interactions strongly affect the motion we shall be concerned with this point in a subsequent section. 

The most important interaction present in a dilute suspension of a chain is the hydrodynamic interaction. In addition, the excluded volume interaction may be present depending on the nature of the solvent, polymer and temperature this interaction vanishes at the 6 temperature, so there is already an important problem when all interactions are ignored. However the interachain entanglement effects corresponding to the uncrossability of different portions of the same chain are always present. The simplest model which cannot actually be realized physically is to both ignore interactions and hydrodynamic effects and assume ad hoc that the solvent attributes a phenomenological friction coefficient for every monomer. This model is called the Rouse model. 

The Rouse model neglects hydrodynamic interactions and the monomer friction co-efQcients add up to give a total friction coefQcient that is linearly proportional to N. 

The simplest hydrodynamic model for a polymer chain in solution assumes that the solvent drains freely into the polymer chain. In this model, first studied by Rouse, the motions of different monomers are independent, i.e. the friction force on a given monomer depends solely on the velocity of that monomer and not on that of the other monomers. 

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