Reptation

The summation is over all odd integers i. If reptation is the only mechanism of relaxation, then the relaxation modulus G(t) is proportional to P(t), i.e., G(t) = P t), and we have 

The distributions of relaxation modes Gj and relaxation times T are given by  

This time constant can be calculated from the monomeric friction coefficient g, the statistical segment length, b, the degree of polymerization N, and the tube diameter, a, as shown by Eq. 6.28 [1], 

The zero-shear viscosity, 77o can be obtained from the discrete spectrum of relaxation times given by (6.26) by using Eq. 4.16  

Using the expressions for G and T given in Eq. 6.27, and noting that the sum converges very rapidly, we find that the zero shear viscosity is given by Eq. 6.30. 

Although theories of the Rouse-Bueche-Zimm type have been very successful in rationalizing the behavior of polymeric systems from a molecular point of view, another class of theories is presently commanding the most attention. These theories treat the motion of polymer molecules in terms of reptation, a reptile-like diffusive motion of each polymer molecule through a matrix formed by its neighbors. To a considerable extent, this new approach has overcome some of the most important shortcomings of the normal-mode theories, which 

Exact calculations based on this model are complex. Nevertheless, it is relatively easy to develop certain scaling laws that relate how various macroscopic properties might depend on molecular properties. We will briefly sketch the development of such a scaling law for viscosity and chain length (or molecular weight) based on the reptation model.23,24 

According to De Gennes,23 this may be done as follows First, apply a steady force / to the chain and observe its velocity v in the tube. Under these circumstances the mobility, Aube, of the molecule in the tube is defined as 

Let n be the number of monomer units per molecule. To obtain the same velocity vc with molecules of various contour lengths, that is with various values of n, the force must be directly proportional to n. Thus, equation (3-109) may be rewritten  

The viscosity associated with this reptating motion may now be calculated as 

The restrictions of the motions by the presence of other chains are not effective on a monomer scale but rather permit lateral freedom on intermediate 

The viscosity relates to the longest relaxation time in a system. If we consider Rouse diffusion along the tube with a Rouse diffusion coefficient DJ l/ NQ) then an initial tube configuration is completely forgotten when the mean-square displacement along the tube fulfils (r (t))tube=(contour length ly. Thus, for the longest relaxation time, we obtain  

The diffusion coefficient is found by considering that during this time in real space the mean-square displacement just amounts to the end-to-end distance of the chain squared. Thus, we have  

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 will take a somewhat different but equivalent criterion in order to describe the crossover. As the crossover time r, we take the Rouse relaxation time of a polymer section, spanning the tube diameter  

The tube model, as the name suggests, assumes that the motion of a polymer chain 

FIGURE 13-67 Schematic diagram illustrating a snake-like motion among a set of obstacles. 

FIGURE 13-68 Schematic diagram illustrating the tube model. 

In order to follow the argument you need to recall a couple of things about Einstein s theory of diffusion. First, the diffusion coefficient depends upon a friction coefficient, (Equation 13-67)  

Second, the distance traveled, l, by a particle diffusing in a medium in a time t goes as tllz. This is, of course, the random walk problem. When we applied this to a polymer chain we were concerned with the distance between the ends in a walk of N steps here we are concerned with the distance traveled after a time t. Before we had R2 112 - N, here we have F 1/2 - t, or, more formally (Equation 13-68)  

The diameter, a, of the tube corresponds to the entanglement spacing, Mg. That is, a strand of polymer having molecular weight Mg spans a random walk end-to-end distance a (Fig. 3-24). Thus, = a M/Mg, and 

The tube itself is a random walk, each step of which has length a. This random walk is called the primitive path of the chain. The contour length of the tube, or the primitive path, is therefore Lj — aMfMg. For polymers of high molecular weight, the tube s contour length is much less than the contour length of the chain (see Fig. 3-24). Thus, the chain meanders about the primitive path. Some values for the tube diameter a for typical polymer melts are presented in Table 3-3. 

Let us consider a polymer moving in a fixed network of obstacles. For the convenience of later discussion, we shall specify the problem in slightly 

To denote a point on the primitive chain, we use the contour length s measured from the chain end and call this the primitive chain segment s. If R(s, t) is its position at time t, the vector 

The dynamics of the primitive chain is characterized by the following assumptions. 

The first assumption corresponds to neglecting the fluctuations of the contour length. The second states that the motion of the primitive chain is reptation. The third guarantees that the conformation of the primitive 

The length a is called the step length of the primitive chain. 

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Figure C2.1.13. (a) Schematic representation of an entangled polymer melt, (b) Restriction of tire lateral motion of a particular chain by tire otlier chains. The entanglement points tliat restrict tire motion of a chain define a temporary tube along which tire chain reptates.

A reptation algorithm removes units from one end of the chain and adds them to the other end. 

Many simulations attempt to determine what motion of the polymer is possible. This can be done by modeling displacements of sections of the chain, Monte Carlo simulations, or reptation (a snakelike motion of the polymer chain as it threads past other chains). These motion studies ultimately attempt to determine a correlation between the molecular motion possible and the macroscopic flexibility, hardness, and so on. 

There are a number of important concepts which emerge in our discussion of viscosity. Most of these will come up again in subsequent chapters as we discuss other mechanical states of polymers. The important concepts include free volume, relaxation time, spectrum of relaxation times, entanglement, the friction factor, and reptation. Special attention should be paid to these terms as they are introduced. 

The tube is a construct which we might continue to sketch around an emerging chain as it diffuses out of the original sleeve. Instead, it is convenient to start with the tube initially in place and consider how long it takes for the molecule to escape. The initial entanglements which determine the contours of the tube comprise a set of constraints from which the molecule is relaxing, even if only to diffuse into another similar set. Accordingly, we identify this reptation time as a relaxation time r for the molecule. 

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... 

Figure 2.14 Reptation model for entanglements for (a) a linear molecule and (b) a branched molecule.

With these ideas in mind, let us consider how long it would take for a polymer chain to escape from the tube shown in Fig. 2.14 by reptation. 

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... 

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. 

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 ... 

Fig. 23.4. Each molecule in o linear polymer con be thought of as being contained in a tube made up by its surroundings. When the polymer is loaded at or above Tg, each molecule can move (reptate) in its tube, giving strain.

Polymers are a little more complicated. The drop in modulus (like the increase in creep rate) is caused by the increased ease with which molecules can slip past each other. In metals, which have a crystal structure, this reflects the increasing number of vacancies and the increased rate at which atoms jump into them. In polymers, which are amorphous, it reflects the increase in free volume which gives an increase in the rate of reptation. Then the shift factor is given, not by eqn. (23.11) but by... 

Figure 8.11. Reptation of a polymer chain. The chain moves snake-like through its confining...

The interdiffusion of polymer chains occurs by two basic processes. When the joint is first made chain loops between entanglements cross the interface but this motion is restricted by the entanglements and independent of molecular weight. Whole chains also start to cross the interface by reptation, but this is a rather slower process and requires that the diffusion of the chain across the interface is led by a chain end. The initial rate of this process is thus strongly influenced by the distribution of the chain ends close to the interface. Although these diffusion processes are fairly well understood, it is clear from the discussion above on immiscible polymers that the relationships between the failure stress of the interface and the interface structure are less understood. The most common assumptions used have been that the interface can bear a stress that is either proportional to the length of chain that has reptated across the interface or proportional to some measure of the density of cross interface entanglements or loops. Each of these criteria can be used with the micro-mechanical models but it is unclear which, if either, assumption is correct. 

The five time regions are based on the reptation theory proposed by De Gennes [46,47] and Doi and Edwards [48,49] for bulk dynamics of polymer melts and concentrated polymer solutions, and are discussed in detail in Chapter 3 of Ref. [1]. 

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