Reptation Tube

Diffusion coefficients are obtained by dividing the square of the length of distance covered by twice this time [Eq. (2.64)]. The length of the reptation tube is nlo therefore... 

Aside from the side chains, the movement of the backbone along the main reptation tube is still given by Eq. (2.67). With the side chains taken into account, the diffusion velocity must be decreased by multiplying by the probability of the side-chain relocation. Since the diffusion velocity is inversely proportional to r, Eq. (2.67) must be divided by Eq. (2.69) to give the relaxation time for a chain of degree of polymerization n carrying side chains of degree of polymerization n ... 

The term aK2v", derived from reptation theory, describes the velocity-dependent energy necessary to fracture the bulk adhesive. K2 is the consistency which relates the viscosity to the shear rate for a non-newtonian fluid. a = TtraL fh", with r being the chain radius, L the chain length, a the density of chains crossing over the fracture plane, and h is the distance between the chain and reptation tube. 

Motion within the tube is achieved by a random walk ( primitive path ) of unit steps of the order of the tube diameter, a. When a straight reptation tube is considered, for simplicity, reptation diffussional motion of the chain out of the tube is represented schematically in the steps depicted from Fig. 3.9(d)(i) to Fig. 3.9(d)(v). 

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 ... 

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... 

Pokrovskii VN (2006) A justification of the reptation-tube dynamics of a linear macromolecule in the mesoscopic approach. Physica A 366 88-106 Pokrovskii VN (2008) The reptation and diffusive modes of motion of linear macromolecules. J Exper Theor Phys 106(3) 604-607... 

Fig. 2.19 A model of the growth of a polyethylene crystal through multiple nucleation on a growth face of dimensions L and i, showing a sketch of a dynamic reptation tube delivering molecules from the melt to the growth face (from Hoffman (1983) and Chang and Lotz (2005) courtesy of Elsevier).

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,... 

The mobility of flexible chains in gels is well described by the biased reptation model [1], which is indicated schematically in Fig. 4. In the model, the fibers of the gel are coarse grained into a reptation tube that confines the chain. The chain thus slithers along the tube contour (the reptation part) under the influence of the electric field, which provides a tendency for the slithering motion to be in the direction of the electric field (the biased part). 

Electrophoresis, Fig. 4 When a long polyelectrolyte like DNA is electrophoresed through a gel, the fibers of the gel confine the DNA to a reptation tube. The force acting on each segment of the tube depends on the orientation vector Sx of that segment in the electric field. 

The molecular theory of extensional viscosity of polymer melts is again based oti the standard tube model. It considers the linear viscoelastic factors such as reptation, tube length fluctuations, and thermal constraint release, as well as the nonlinear viscoelastic factors such as segment orientations, elastic contractimi along the tube, and convective constraint release (Marrucci and lannirubertok 2004). Thus, it predicts the extensional stress-strain curve of monodispersed linear polymers, as illustrated in Fig. 7.12. At the first stage, the extensional viscosity of polymer melts exhibits the Newtonian-fluid behavior, following Trouton s ratio... 

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. 

To understand the dynamics of one chain in a melt, it is convenient to start from a slightly different problem. We consider one test chain of Ni monomers, embedded in a monodisperse melt of the same chemical species, with a number N of monomers per chain. We consider three types of motion for the test chain reptation, tube renewal, and Stokes-Einstein friction. We first describe tube renewal and show that this is probably negligible for most practical purposes. Then we discuss competition between reptation and Stokes-Einstein friction. 

Figure 6.31 Model tor regime II growth showing multiple nucleation. The quantity S represents the mean separation between the primary nuclei, and S denotes the mean distance between the associated niches. The primary nucleation rate is and the substrate completion rate is g. The overall observable growth rate is G//. Reptation tube contains molecule being reeled at rate ronto substrate (75).

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. 

Experiments give t oc iv (see Figure 33.10), which is slightly steeper than this prediction t oc of the reptation model. There is some evidence that this discrepency is due to the finite lengths of reptation tubes. This model accounts in a simple way for the steep chain-length dependence of the relaxation times of polymers in bulk melts. 

According to the reptation theory [74,75], the polymer chains are confined laterally to a tube-like region. The chains can only relax by sliding back and forth along the tube like a snake, and cannot cross the wall of the tube. Based on this physical picture of a polymer chain in liquid state, the volume of the reptation tube is presumably considered to be the free volume of a individual chain that can occupy. As illustrated in Figure 19, the volume of the tube, V l,b ,... 

---