Entangled model-branched polymers

Branched polymers may be classified into two categories from the point of view of rheology  

We will deal in this review article with monodisperse, model-branched polymers in order to describe the basic relaxation modes of branched polymers. The concepts described below are the source of current attempts to describe the viscoelastic properties of complex tree-like structures which are close to those fo md in low density polyethylene, for example. One may foimd interesting approaches of that problem in recent papers presented by Mac Leish et al [10]. 

If one considers that the reptation process is dominant for linear chains, one has to imagine additional processes of diffusion for polymers with long branches. The experimental data suggest strongly, however, that the basic kinetic unit of the chain (whatever it is) is the same as for linear chains the Rouse-like A and B processes are still there, which are still strong imprints of the tube . 

The exponential dependence of the relaxation time with arm length is a constant feature for all models describing the renewal of configuration of long branches, and the debate has focused on the non-exponential term fiNarm)- 

Doi and Kuzuu [6] have proposed a somewhat different approach based on the tube concept. They start with three basic assiunptions  

---

As an illustration of the Rouse model, consider the polydisperse mixture of polymers produced by random branching with short chains between branch points. The molar mass distribution and size of the branched polymers in this critical percolation limit were discussed in Section 6.5. Close to the gel point, some very large branched polymers (with M> 10 ) are formed and the intuitive expectation is that such large branched polymers would be entangled. However, recall that hyperscaling requires... 

The dynamics of highly entangled branched polymer motion in the presence of reversible cross-links is more complicated. In the case of entangled but unbranched polymers, it was recently shown how to include reversible scission effects within the framework of a tube model. With the results of Ref.6 to hand, a similar development is now possible in the branched case, and will be described in a future publication. ... 

Fig. 7 gives an example of such a comparison between a number of different polymer simulations and an experiment. The data contain a variety of Monte Carlo simulations employing different models, molecular dynamics simulations, as well as experimental results for polyethylene. Within the error bars this universal analysis of the diffusion constant is independent of the chemical species, be they simple computer models or real chemical materials. Thus, on this level, the simplified models are the most suitable models for investigating polymer materials. (For polymers with side branches or more complicated monomers, the situation is not that clear cut.) It also shows that the so-called entanglement length or entanglement molecular mass Mg is the universal scaling variable which allows one to compare different polymeric melts in order to interpret their viscoelastic behavior. 

The tube model gives a direct indication of why one might expect the strange observations on star melts described above. Because the branch points themselves in a high molecular weight star-polymer melt are extremely dilute, the physics of local entanglements is expected to be identical to the linear case each segment of polymer chain behaves as if it were in a tube of diameter a. However, in... 

A useful toy theoretical model which captures the essential features of self-entangled dendritic polymers is the monodisperse Cayley tree in which each chain segment branches with a fixed functionality z at each of its ends, except those at the extremity of the molecule (see Eig. 13). Smaller versions of these structures, too low in molecular weight to be entangled, have been synthesised and are usually referred-to as dendrimers [47]. 

It became clear in the early development of the tube model that it provided a means of calculating the response of entangled polymers to large deformations as well as small ones [2]. Some predictions, especially in steady shear flow, lead to strange anomaUes as we shall see, but others met with surprising success. In particular the same step-strain experiment used to determine G(t) directly in shear is straightforward to extend to large shear strains y. In many cases of such experiments on polymer melts both Hnear and branched, monodisperse and polydisperse,the experimental strain-dependent relaxation function G(t,Y) may be written... 

For concentrated solutions of amorphous polymers, Bueche s mathematical model shows the ratio of zero shear viscosities of branched and linear polymer above the critical molecular weight in the entanglement region to be (28) ... 

To understand this viscosity enhancement, it is easier to start with the theory for linear polymers. The behavior of linear polymers can be described by the reptation model.For a linear polymer of high molecular weight in the melt, chains can be modeled as a confined tube where the diffusion of the chain is restricted along the tube contour. Entanglements are formed between chains where the reptation of a chain along its contour becomes the dominant mode of movement. The addition of a branch point prevents reptation and other forms of movement must occur for the chain to change its configuration. In the case... 

---