Unentangled melt

Usually the model is solved for a ring with no free ends. If the chain ends are free, as for all linear chains, the first and the last monomer have to be treated differently. For i = 1, the first term on the right hand side is —A (ri - F2) and similarly for i = N. The distribution of random forces f, is Gaussian with zero mean and second moment. 

Note that this model does not contain any specific interactions between monomers except those due to the chain connectivity. Since the model neglects both hydrodynamic effects and excluded volume, it does not describe the dynamics of an isolated chain in solution. However in a dense melt, the long-range hydrodynamic interactions are screened out, much like the excluded volume interactions. For this reason, it was suggested that this model could describe the motion of chains in a melt, except that C arises from other chains in the melt and is larger than in a dilute solution. 

The Rouse model can be solved by transforming to the normal coordinates Xp t) of the chain. For a discrete monomer chain these are given 

For chains in a melt, the Rouse modes are expected to be eigemnodes of the chains. This has been verified by for a melt of short chains by 

The case of = 1 corresponds to a coupling constant kf = 2k for the end-bonds. This correction for free chains is in most cases negligible, since this results in a 1/iV correction term. All the MD simulations presented in this review used = 1. For walks within a network = l since there the last monomer within an arbitrary walk through the net is not only coupled to the previous monomer, but also to the following one. This leads to an effectively stronger coupling, which for calculating motion quantities is not important. However for a correct estimate of the modulus this extra term is crucial. For more details see Refs 13, 162 and Section 4.5 on networks. 

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Compare this Zimm diffusion coefficient Dz with the Rouse diffusion coefficient Dr of part (ii). Hint. The viscosity of an unentangled melt of shorter /Vg-chains is predicted by the Rouse model [Eq. (8.53)]. 

Calculate the stress relaxation modulus G(t) for a polydisperse unentangled melt of linear polymers with a power law distribution of chain lengths... 

The viscosity of a polymer melt is predicted to be proportional to molar mass for unentangled melts (the Rouse model) and proportional to the cube of molar mass for entangled melts (the reptation model). 

In an unentangled melt, the number density of modes relaxing with rate e corresponds to the number of chain sections containing K Kuhn segments such that... 

The Rouse theory is clearly not applicable to polymer melts of a molar mass greater than (M ) for which chain entanglement plays an important role. This is obvious from a comparison of eqs (6.40)-(6.42) and experimental data (Figs 6.13 and 6.14) and from the basic assumptions made. However, for unentangled melts, i.e. melts of a molar mass less than (M ), both the zero-shear-rate viscosity and recoverable shear compliance have the same molar mass dependence as was found experimentally (Figs 6.13 and 6.14). The Rouse model does not predict any shear-rate dependence of the shear viscosity, in contradiction to experimental data. 

CENERIC Coarse-Craining Applied to Unentangled Melts Foundations... 

Chapter 6 treats mean-field theories of melt behavior. We begin with the Rouse model for molecules in dilute solution and its modification by Bueche to deal with unentangled melts. The longest Rouse relaxation time emerges from this treatment and plays an important role in all molecular models. The tube model is introduced, in which the basic relaxation... 

Figure 5.2 Relaxation moduli of three samples of a linear polymer A) an unentangled molten sample, B) an entangled,monodisperse molten sample,C) an entangled, polydisperse molten sample, and D) acrosslinked sample. At short times,all the samples relax first by a glassy mechanism and then by Rouse relaxation involving only very short segments of the chain (log scales). The unentangled melt then flows in the terminal zone.The entangled, monodisperse melt has a plateau modulus followed by terminal relaxation, while in the polydisperse melt the plateau zone of the longest molecules overlaps with the terminal zones of the shorter molecules.

For multi-armed stars, where is less than M, Ham s model for unentangled melts [85] indicates that the ratio of /° values for /-armed stars and linear molecules having the same molecular weight, is given by ... 

While the Rouse model was originally intended to describe dilute polymer solutions, Bueche [6] noted that the freely-jointed chain model should be able to describe the behavior of an unentangled melt. It has been found experimentally that the static interactions between a polymer molecule and its surroundings are normally the same in the melt as in a solution in its theta state, although Krishnamoorti et al. [7] have noted a few cases where chain dimensions are different in the melt and at the theta state. They attribute this to the ability of some theta solvents to induce a conformer population different from what is favored in the melt state . 

Then we further assume that this equation describes the hypothetical, unentangled melt with M > Mq, and we call the zero-shear viscosity of this hypothetical material... 

Inserting this expression for %(M) into Eq. 6.12 in place of the tIq of the unentangled melt, we obtain ... 

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