The reptation model

The mobility of the whole chain, free to move along the curvilinear axis of the tube is N/Nd smaller than the mobility of one bead, as the friction on the full necklace is the friction on one bead times the number of beads. The diffusion coefficient of the chain along the tube 

Tr is the longest relaxation time of the chain constrained by the obstacles. It is much 

The diffusion is much slower (by a factor N) than for a Rouse-type free chain. 

It is tempting to apply the ideas developed in section 2.1. to describe the dynamics of long linear polymer molecules in the melt state. The average radius of the chain is Rq = 

both the longest relaxation time Tr and the self diffusion coefficient can be estimated through eq. 5 and 6 respectively, replacing Nj by Ne- One gets  

In the de Gennes approach, the polymer chain is assumed to be contained in a hypothetical tube [Fig. 2.32(a)] which is placed initially in a three-dimensional network formed from other entangled chains. Although for simplicity these network knots are shown in Fig. 2.31 as fixed obstacles around which the chain under consideration must wriggle during translation, in practice these obstacles would also be in motion. The contours of the tube are then defined by the position of the entanglement points in the network. 

Mechanistically, reptation can be regarded as the movement of a kink in the chain along its length [see Fig. 2.32(b)] until it reaches the end of the chain and leaves it. Motion of this kind translates the chain through the tube and successive defects moving the chain in this way will eventually take it completely out of the tube. 

The time required for a chain to move completely out of the tube is the relaxation time T. Since the time necessary to move a certain distance 

Here Dt is the diffusion coefficient within the tube and can be expressed as the frictional coefficient for the chain within the tube confines, i.e., Dt = kT/ft. Since the reptation is assumed to occur by migration of a segmental kink along the chain, the force needed to do this is applied one segment at a time and so it is essentially proportional to the number of atoms Z) in the chain. Thus, 

Equation (2.71) shows that the relaxation time is proportional to the cube of the chain length. This is the fundamental result of the reptation model. 

Starting from the tube concept and considering the motions of the confined chains, Doi and Edwards devised a theory which became well-known as the 

Both the actual chain as well as the primitive path represent random coils. Since the end-to-end distances are equal, we have 

we have introduced the contour length of the primitive path, /pr, and an associated sequence length Upr- Upr characterizes the stiffness of the primitive path and is determined by the topology of the entanglement network. 

The process of disentangling, as it is envisaged in the reptation model, is sketched in Fig. 6.11. The motion of the primitive chain , the name given to the dynamic object associated with the primitive path, is described as a diffusion along its contour, that is to say, a reptation . The associated curvilinear diffusion coefficient can be derived from the Einstein relation, which holds generally, independent of the dimension or the topology. Denoting it D, we have 

Cp is the friction coefficient of the chain. As there are no entanglements within the tube, Cp equals the sum of the friction coefficients of all beads 

The reptation model, like the Rouse model, supposes that the friction involved in dragging the chain through its tube is proportional to the chain length, 5 = N i, Equation (33.33). The diffusion constant Dtubo for the chain moving through the tube is given by the Einstein-Smoluchowski relation, Equa- 

The reptation model also predicts a second type of diffusion constant. The diffusion constant Dtube cannot be readily measured because experiments cannot track how the chain moves along its tube axis. But you can measure a diffusion constant D that describes how the center of mass of the polymer chain moves in space over time. In this case, the average distance moved is (x-) oc Rj, where = Nb . The reptation model Equation (33.38) predicts that chain diffusion slows as the square of the chain length. 

Rouse-Zimm dynamics in a good solvent. Derive the dependence of the relaxation time T on the chain length N, for a dilute polymer solution in a good solvent. 

A tethered chain returns to the surface. A random-flight chain of length N is tethered at one end to an impenetrable surface. Write an expression for the probability P N) that the free end also happens to land on the surface. 

the polymer overlap concentration. Consider a solution of polymer molecules of length N. At concentrations c below the overlap concentration c a polymer molecule seldom collides with other polymers, c = c is the point at which the concentration of monomer units throughout the solution equals the concentration of monomers inside the circumscribed volume of the polymer chain. 

The snakelike motion along the tube is called reptation, from the Latin reptare, to crawl . The corresponding model of polymeric liquids is known as the reptation model. 

When two blocks of the same polymer are pressed together at a temperature above the Tg for a relatively short time t, interdiifusion of polymer chains takes place (by reptation) across the interface to produce a signi cant number of entanglements, thereby joining the blocks together (Sperling, 1986). The strength of the junction formed will, however, depend on time t. 

Problem 2.35 Mills et al. (1984) related the diffusion coef cient of polystyrene at 170°C to the weight-average molecular weight by the equation  

In the welding of polystyrene (M = 10 ) at a temperature of 170°C, calculate the ux of polymer molecules across the interface for diffusion over a 100 A distance. (Density of polystyrene = 1.0 g/cm. ) 

In one second, 48x10 molecules will diffuse through a 1-cm area of the interface. 

---

In order to draw some conclusions about viscosity from the reptation model, it is again necessary to anticipate some results from Chap. 9 on diffusion. The... 

In connection with a discussion of the Eyring theory, we remarked that Newtonian viscosity is proportional to the relaxation time [Eqs. (2.29) and (2.31)]. What is needed, therefore, is an examination of the nature of the proportionality between the two. At least the molecular weight dependence of that proportionality must be examined to reach a conclusion as to the prediction of the reptation model of the molecular weight dependence of viscosity. 

There are three basic time scales in the reptation model [49]. The first time Te Ml, describes the Rouse relaxation time between entanglements of molecular weight Me and is a local characteristic of the wriggling motion. The second time Tro M, describes the propagation of wriggle motions along the contour of the chain and is related to the Rouse relaxation time of the whole chain. The important... 

We present here a simple experiment, conceived to test both the reptation model and the minor chain model, by Welp et al. [50] and Agrawal et al. [51-53]. Consider the HDH/DHD interface formed with two layers of polystyrene with chain architectures shown in Fig. 5. In one of the layers, the central 50% of the chain is deuterated. This constitutes a triblock copolymer of labeled and normal polystyrene, which is, denoted HDH. In the second layer, the labeling has been reversed so that the two end fractions of the chain are deuterated, denoted by DHD. At temperatures above the glass transition temperature of the polystyrene ( 100°C), the polymer chains begin to interdiffuse across the... 

Although the mathematics of the reptation model lie beyond the scope of this book, we should notice two important predictions. They are ... 

The beauty of the reptation model is that it is able to make predictions about molecular flow both in solution and at fracture by assuming that the molecules undergo the same kind of motions in each case. For both self-diffusion in concentrated solutions and at fracture, the force to overcome in pulling the polymer molecule through the tube is assumed to be frictional. 

The reptation model thus predicts four dynamic regimes for segment diffusion. They are summarized in Fig. 18. 

A comparison with the tube diameter derived from plateau moduli measurements on PEB-7 underlines this assertion. The coincidence of tube diameters determined macroscopically by application of the reptation model and direct microscopic results is far better than what could have been expected and strongly underlines the basic validity of the reptation approach. 

The deformation of polymer chains in stretched and swollen networks can be investigated by SANS, A few such studies have been carried out, and some theoretical results based on Gaussian models of networks have been presented. The possible defects in network formation may invalidate an otherwise well planned experiment, and because of this uncertainty, conclusions based on current experiments must be viewed as tentative. It is also true that theoretical calculations have been restricted thus far to only a few simple models of an elastomeric network. An appropriate method of calculation for trapped entanglements has not been constructed, nor has any calculation of the SANS pattern of a network which is constrained according to the reptation models of de Gennes (24) or Doi-Edwards (25,26) appeared. 

The reptation model for polymer diffusion would predict that the thickness of the gel phase reflects the dynamics of disentanglement. The important factors here are chain length, solvent quality and temperature since they affect the dimensions of the polymer coils in the gel phase. The precursor phase, on the other hand, depends upon solvency and temperature only through the osmotic force it can generate in the system and the viscoelastic response of the system in the region of the front. These factors should be independent of the PMMA molecular weight. 

A schematic of a representation of the storage modulus is shown in Figure 5.27 for the reptation model. 

Figure 5.27 The storage modulus predicted by the reptation model...

The reptation model predicts that the viscosity of a melt scales with the chain length to the third power while the diffusion coefficient decreases with the second power of the chain length. 

We now consider the predictions of the reptation model for the mean-square displacement of the chain segments. Figure 3.13 gives an overview. 

Fig. 3.14 Mean-square displacement of a chain segment in the reptation model. The exponents of the different power law regimes are noted along the respective lines...

Fig. 3.16 Scaling presentation of the dynamic structure factor from a M =36,000 PE melt at 509 K as a function of the Rouse scaling variable. The solid lines are a fit with the reptation model (Eq. 3.39). The Q-values are from above Q=0.05,0.077,0.115,0.145 A The horizontal dashed lines display the prediction of the Debye-Waller factor estimate for the confinement size (see text)...

From Fig. 3.17 and Fig. 3.18 it is apparent that these data clearly favour the reptation model. The reptation model is the only model for which the dynamic structure factor has been calculated and which is in quantitative agreement with these results. We observe that the models of Ronca [63] (see Fig.3.17) and des Cloizeaux [66] (see Fig. 3.18) produce a plateau which is too flat. On the other hand, the model of Chaterjee and Loring relaxes too quickly and does not form a plateau (see Fig. 3 in [65]). 

Moreover, from Fig. 3.18 it is apparent that the model of des Cloizeaux also suffers from an incorrect Q-dependence of S(Q,f) in the plateau region, which is most apparent at the highest Q measured. It is important to note that the fits with the reptation model were done with only one free parameter, the entanglement distance d. The Rouse rate was determined earlier through NSE data taken for Kr. With this one free parameter, quantitative agreement over the whole range of Q and t using the reptation model with d=46.0 1.0 A was found. 

A (rheology) [7]. Obviously the local freedom for segmental motion is larger than anticipated so far from rheology. The result casts some doubts on the determination of the plateau modulus from rheological data in terms of the reptation model. 

Fig. 3. 27 NSE data from PE melts with a molecular weight of M =36,000 g/mol (left) and 190,000 g/mol (right) (M /M <1.02) for Q=0.03 (crosses), 0.05 (squares), 0,077 (circles), 0,096 (diamonds), 0.115 (triangles) and 0.145 A" (stars). For the common Q-values the data are identical. The lines represent a fit with the reptation model (using W =7xl0 AVns for the Rouse rate). (Reprinted with permission from [74]. Copyright 2000 EDP Sciences)...

---