The tube model

Though the tube model is successful, our present understanding of the dynamics in entangled systems is still incomplete. Agreement between theory and experiments is not yet complete as we shall discuss later. More seriously, the tube model does not describe all aspects of the dynamics it describes properties which depend on a single chain 

---

FIGURE 113 (a) The framework of bonds shown in the tube model of benzene are cr bonds (b) Each carbon is sp hybridized and has a 2p orbital perpendicular to the cr framework Overlap of the 2p orbitals generates a tt system encompass mg the entire ring (c) Electrostatic potential map of benzene The red area in the center corresponds to the region above and below the plane of the ring where the tt electrons are concentrated... 

The Ball and Wire model is identical to the Wire model, exeept that atom positions are represented by small spheres. This makes it possible to identify all atom locations in all molecules. The Tube model is identical to the Wire model, except that bonds, whether single, double or triple, are represented by single colored tubes. The tubes are useful because they better eonvey the three-dimensional shape of a molecule. The Ball and Spoke model is a variation on the Ibbe model atom positions are represented by colored spheres, making it possible to see all atom locations in all molecules. 

The most successful theoretical framework in which the accumulating data has been understood is the tube model of de Gennes, Doi and Edwards [2]. We visit the model in more detail in Sect. 2, but the fundamental assumption is simple to state the topological constraints by which contingent chains may not cross each other, which act in reality as complex many-body interactions, are assumed to be equivalent for each chain to a tube of width a surrounding and coarse-graining its own contour (Fig. 2). So, motions perpendicular to the tube contour are confined while those curvilinear to it are permitted. The theory then resembles a dynamic version of rubber elasticity with local dissipation, and with the additional assumption of the tube constraints. 

Fig. 2. The tube model replaces the many-chain system left) with an effective constraint on each single chain right). The tube permits diffusion of chains along their own contours only...

The fundamental example of the tube model s application is the simplest one of linear chains of identical molecular weight M or degree of polymerisation N. It will provide the starting point for more complex applications. 

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... 

This theory was able to account for both the molecular-weight scaling of the dynamic quantities Dg, r, and x as well as for the shape of the relaxation spectrum (see Fig. 5) apart from one important feature - the constant v in the leading exponential behaviour that multiplies the dimensionless arm molecular weight needed to be adjusted. This can be understood as follows. The prediction of the tube model for the plateau modulus from the stress Eq. (7) is... 

The approximate treatment described above accounts rather well for the linear rheology of star polymer melts. In fact it has been remarked that the case for the tube model draws its real strength from the results for star polymers rather than for linear chains, where the problems of constraint release and breathing modes are harder to account for (but see Sect. 3.2.4.). However, there are still some outstanding issues and questions ... 

Of course from a molecular point of view this is no longer surprising - we know that dynamic dilution is a highly cooperative process. However the quantitative prediction of the dynamic moduli of Fig. 14 is clearly a very demanding task for a theory with essentially no free parameters We outHne here how the tube model calculation is done in this case for details see [56]. 

Fig. 14. Data (points) for G (co) and G (co) for a range of compositions of a blend of two polyisoprene stars of molecular weights 28 and 144 kg mol The fractions of the bigger star are in order 0.0,0.2,0.5,0.8 and 1.0. Curves are theoretical predictions of the tube model with co-operative constraint release treated by dynamic dilution [56]. The choice of 2.0 rather than 7/3 for the dilution exponent p is compensated for by taking M = 5500 kg mol" ...

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... 

The challenge within our programme is to follow up the consequences of the tube model for the non-linear rheology of branched polymers - would such a theoretical framework lead to any understanding of the special behaviour of, for example, LDPE in complex flows We build up our tools as before in the context of linear polymers. 

When large non-linear deformations are made, we need to make additional assumptions within the tube model on how the tube itself deforms with the bulk strain. The simplest, and original assumptions are that ... 

Fig. 17. Damping functions in shear from the tube model for linear polymers (lowest curve) and various branched architectures. In the cases of comb and tree, the lower curves give the case of the structure with four levels of branching, the upper the limit of large structures hatched area covers published results on LDPE...

The step-strain experiments discussed above furnish the simplest example of a strong flow. Many other flows are of experimental importance transient and steady shear, transient extensional flow and reversing step strains, to give a few examples. Indeed the development of phenomenological constitutive equations to systematise the wealth of behaviour of polymeric liquids in general flows has been something of an industry over the past 40 years [9]. It is important to note that it is not possible to derive a constitutive equation from the tube model in... 

The tube models consider the oceanic mass of water as subdivided into columns. Mass transfer between columns takes place by advection and diffusion. Examples of tube models may be found in Munk (1966) and Bieri et al. (1966), to whom we refer readers for further clarification. 

Newer rubber elasticity theories based on the tube model (35) consider special constraint release mechanisms which allow a physi-... 

The distinct properties of liquid-crystalline polymer solutions arise mainly from extended conformations of the polymers. Thus it is reasonable to start theoretical considerations of liquid-crystalline polymers from those of straight rods. Long ago, Onsager [2] and Flory [3] worked out statistical thermodynamic theories for rodlike polymer solutions, which aimed at explaining the isotropic-liquid crystal phase behavior of liquid-crystalline polymer solutions. Dynamical properties of these systems have often been discussed by using the tube model theory for rodlike polymer solutions due originally to Doi and Edwards [4], This theory, the counterpart of Doi and Edward s tube model theory for flexible polymers, can intuitively explain the dynamic difference between rodlike and flexible polymers in concentrated systems [4]. 

---