Tube/reptation model

The system of dynamic equations (3.37) for a chain of Brownian particles with local anisotropy of mobility appears to be rather complicated for direct analysis, and one ought to use numerical methods, described in the next Section, 

Following Doi and Edwards (1978), we shall consider a bead-spring model consisting of Z = M/Me subchains and assume that the distance between adjacent particles along the chain is constant and equal to a certain intermediate length , which is considered to be the radius of a tube , so that the number of particles is not arbitrary, but satisfies the condition 

The states of the macromolecule will be considered in points of time in a time interval At, so that the stochastic motion of Brownian particles of the chain can be described by the equation for the particle co-ordinates 

The set of equations (3.43) describes the stochastic motion of a chain along its contour. The head and the tail particles of the chain can choose random directions. Any other particle follows the neighbouring particles in front or behind. The smaller the time interval At is the quicker moves the chain. Clearly, the time interval cannot be an arbitrary quantity and is specified by the requirement that the squared displacement of the entire chain by diffusion for the interval At is equal to 2, so that 

The model described by equations (3.42)-(3.45) is valid for equilibrium situations. For chain in a flow, one ought to define displacements of the particles under flow and to consider the average values (3.44) to depend on the velocity gradient (Doi and Edwards 1986). McLeish and Milner (1999) considered mechanism of reptation motion of branched macromolecules of different architecture. 

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There were different generalisations of the reptation-tube model, aimed to soften the borders of the tube and to take into account the underlying stochastic dynamics. It seems that the correct expansion of the Doi-Edwards model, including the underlying stochastic motion and specific movement of the chain along its contour - the reptation mobility as a particular mode of motion, is presented by equations (3.37), (3.39) and (3.41). In any case, the introduction of local anisotropy of mobility of a particle of chain, as described by these equations, allows one to get the same effects on the relaxation times and mobility of macromolecule, which are determined by the Doi-Edwards model. 

Derived from linear approximation of the equations (3.37), the equilibrium correlation function (4.29), defines two conformation relaxation times r+ and r for every mode. The largest relaxation times have appeared to be unrealistically large for strongly entangled systems, which is connected with absence of effect of local anisotropy of mobility. To improve the situation, one can use the complete set of equations (3.37) with local anisotropy of mobility. It is convenient, first, to obtain asymptotic (for the systems of long macromolecules) estimates of relaxation times, using the reptation-tube model. 

It is not difficult to reproduce an expression for the correlation function Ma(t) and estimate times of relaxation due to the conventional reptation-tube model (see Section 3.5). Indeed, an equation for correlation function follows equation (3.48) and has the form... 

Each point is calculated as the asymptotic value of the rate of relaxation for large times (see examples of dependences in Fig. 6) for a macromolecule of length M = 25Me (x = 0.04, B = 429, ij) = 8.27) with the value of the coefficient of external local anisotropy ae = 0.3. The dashed lines reproduce the values of the relaxation times of the macromolecule due to the reptation-tube model. The labels of the modes are shown at the lines. Adapted from Pokrovskii (2006). 

The results of estimation of coefficient of self-diffusion due to simulation for macromolecules with different lengths are shown in Fig. 12. The introduction of local anisotropy practically does not affect the coefficient of diffusion below the transition point M, the position of which depends on the coefficient of local anisotropy. For strongly entangled systems (M > M ), the value of the index —2 in the reptation law is connected only with the fact of confinement of macromolecule, and does not depend on the value of the coefficient of local anisotropy. At the particular value ae = 0.3, the simulation reproduces the results of the conventional reptation-tube model (see equation (5.21)) and corresponds to the typical empirical situation (M = 10Me). 

The reptation-tube model, being used for interpretation of viscoelastic behaviour of the system, has allowed to obtain (Doi and Edwards 1986) the relation for terminal characteristics... 

Much literature involves extensions of the reptation/tube model of deGennes, Doi, cmd Edwards. The deGennes model [44] was originally proposed to describe a linear polymer chain diffusing in the presence of fixed obstacles, such as those presented by a covalently-crosslinked gel. In the original model, the chains of the gel are rigidly locked in plcice,... 

Work to further expand the reptation-tube model has been explored. Pokrovski (2008), for example, has shown that the underlying stochastic motion of a macromolecule leads to two modes of motion, namely, reptative and isotropically diffusive. There is a length of a macromolecule M = lOM where is the macromolecule length between adjacent entanglements above which macromolecules of a melt can be regarded as obstacles to motions of each other and the macromolecules reptate. The transition to the reptation mode of motion is determined by both topological restriction and the local anisotropy of the motion. 

That is why the numerous efforts to find the 3.4-index law for viscosity coefficient of linear polymers in frame of the reptation-tube model were doomed to fail and have failed during the last twenty years. 

Other theories which are not based on the reptation/tube model have also been developed. While some make an ansatz about how a polymer moves in a melt, others are more microscopic. Schweizer uses a modecoupling approach. His theory predicts the emergence of a plateau shear... 

The DE Constitutive Equations. The DE model (52-56) made a major breakthrough in polymer viscoelasticity in that it provided an important new molecular physics based constitutive relation (between the stress and the applied deformation history). This section outlines the DE approach that built on the reptation-tube model developed above and gave a nonlinear constitutive equation, which in one simplified form gives the K-BKZ equation (70,71). The model also inspired a significant amount of experimental work. One should begin by... 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

Figure 16 Tube model for reptation of a branched polymer molecule from the work of Blackwell et al. [124]. Reproduced with permission from Blackwell et al. [124]. Copyright 2000, The Society of Rheology, Inc.

A second appealing feature of tube model theories is that they provide a natural hierarchy of effects which one can incorporate or ignore at will in a calculation, depending on the accuracy desired. We will see how, in the case of linear polymers, bare reptation in a fixed tube provides a first-order calculation more accurate levels of the theory may incorporate the co-operative effects of constraint release and further refinements such as path-length fluctuation via the Rouse modes of the chains. 

In a celebrated series of papers, Doi and Edwards [80] used the reptation, or tube model, to develop a theory for the rheological properties of entangled chains. In this model, the dynamics of a chain is described by the tube of entanglements. The configuration of the... 

The Doi-Edwards, reptation based model makes specific predictions for the relaxation dynamics of different portions of a polymer chain. Specifically, the relaxation of the chain ends is predicted to be substantially faster than the relaxation of the center. This is a result of the reptation dynamics, which have the ends first leaving the confines of the tube. Using polymer chains that were selectively deuterated either at the ends or at the middle, Ylitalo and coworkers [135] examined this problem and found that the Doi-Edwards model was able to successfully predict the observed behavior once the effects of orientational coupling was included. The same group further explored the phenomena of orientational coupling in papers that focused on its molecular weight [136] and temperature [137]... 

The pom-pom polymer reptation model was developed by McLeish and Larson (60) to represent long chain-branched LDPE chains, which exhibit pronounced strain hardening in elongational flows. This idealized pom-pom molecule has a single backbone confined in a reptation tube, with multiple arms and branches protruding from each tube end, as shown in Fig. 3.12(a). Mb is the molecular weight of the backbone and Ma, that of the arms. 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

The tube model avoids dealing with specification of an average molecular weight between entanglement points (Mg), but the diameter of the tube d) is an equivalent parameter. Further, the ratio of the contour length to diameter L/d) is taken to be approximately the magnitude of the entanglements per molecule (M/Me). Some of the important relationships predicted from the reptation model include (Tirrell, 1994) ... 

Despite these complications, there are now numerous evidences that the tube model is basically con-ect. The signatory mark that the chain is trapped in a tube is that the chain ends relax first, and the center of the chain remains unrelaxed until relaxation is almost over. Evidence that this occurs has been obtained in experiments with chains whose ends are labeled, either chemically or isotopically (Ylitalo et al. 1990 Russell et al. 1993). These studies show that the rate of relaxation of the chain ends is distinctively faster than the middle of the chain, in quantitative agreement with reptation theory. The special role of chain ends is also shown indirectly in studies of the relaxation of star polymers. Stars are polymers in which several branches radiate from a single branch point. The arms of the star cannot reptate because they are anchored at the branch point (de Gennes 1975). Relaxation must thus occur by the slower process of primitive-path fluctuations, which is found to slow down exponentially with increasing arm molecular weight, in agreement with predictions (Pearson and Helfand 1984). 

In interpreting the relaxation behavior of polydisperse systems by means of the tube model, one must consider that renewal of the tube occurs because the chain inside it moves thermally, either by reptation mode, by fluctuation of the tube length in time (breathing motion), or in both ways (13,14). Moreover, the tube wall can be renewed independently of the motion of the chain inside the tube because the segments of the chains of the wall are themselves moving. The relaxation mechanism associated with the renewal of the tube is called constraint release. 

Figure 11.7 Reptation of star chains in the tube model. (From Ref. 3.)...

Figure 1 (a) Reptation of a linear polymer molecule in a tube, (b) Arm retraction mechanism in the tube model for a star polymer... 

In general the terminology follows that of Ferry for viscoelastic functions and molecular variables and Doi and Edwards for reptation and tube model variables. 

Two theories of viscoelasticity in reptating chain systems have appeared since the original Doi-Edwards publications, the theory of Marrucci and coworkers and the theory of Curtiss and Bird ° They differ in various ways from the Doi-Edwards tube model and from the model suggested in Part II. It is difficult and probably premature to provide detmled criticisms at the present time, but it is perhaps worthwhile to point out at least a few of the differences. 

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