Primitive chain step length

Hie mesh size d is assumed to depend only on the effective spacing of obstacles in the medium and thus on the polymer concentration alone. We do not mean to imply that d would necessarily be of the order of the distance between chains (a few An troms in undiluted systems). Its effective value should be somewhat larger than this, since d must also reflect a certain amount of local freedom for mutual rearrangement of iKighbors in real systems. Nevertheless, like the primitive path step length a of Doi and Edwards, d should be independent of the large scale molecular structure. 

The Doi-Edwards theory treats monodisperse linear chain liquids by a model which suppresses fluctuations and assumes a topologically invariant medium. Two parameters are required, the monomeric friction coefficient which characterizes the local dynamics and the primitive path step length a which characterizes the topology of the medium. The step length is related to the entanglement molecular weight of earlier theories, = cRqT/Gn, by Eqs. 1 and 37 ... 

According to the Doi-Edwards theory, after time t = Teq following a step deformation at t = 0, the stress relaxation is described by Eqs. (8.52)-(8.56). In obtaining these equations, it is assumed that the primitive-chain contour length is fixed at its equilibrium value at all times. And the curvilinear diffusion of the primitive chain relaxes momentarily the orientational anisotropy (as expressed in terms of the unit vector u(s,t) = 5R(s,t)/9s), or the stress anisotropy, on the portion of the tube that is reached by either of the two chain ends. The theory based on these assumptions, namely, the Doi-Edwards theory, is called the pure reptational chain model. In reality, the primitive-chain contour length should not be fixed, but rather fluctuates (stretches and shrinks) because of thermal (Brownian) motions of the segments. 

That assumption simplifies the analysis of primitive path rearrangement. Local path jumps now correspond to random flips in the Orwoll-Stockmayer chain model, and we can apply these results directly. For our case the local jump distance is the path step length a, and the average time between jumps is 2r , where r is the mean waiting time for release of a constraint which allows a length preserving jump. The average number of such suitably situated constraints per cell is z(z < zo), and we assume for simplicity that all cells have z such constraints. 

Fig. 2A-C. Various representations of a polymer chain and its surroundings. The chain and segments of neighboring chains (A), the chain in a tube of uncrossable constraints provided by its neighbors (B), and the primitive path of a chain among the surrounding constraints provided by neighbors (a step length of the primitive path R = end-to-end vector of the chain) (C)...

Doi and Edwards considered the primitive chain as a freely jointed chain with step length a. The positions of the joints (or links) can be labeled as... 

Then R — Rn-i = o,. Assume that in a time interval At, the primitive chain jumps forward or backward with equal probability one step of length a. Then the curvilinear diffusion constant can be defined by... 

Let Sn be the point on the primitive-chain contour corresponding to the nth Rouse segment. Denote the positions of Sn in three-dimensional space before and after a step deformation E is applied as R°(S ) and R(S (t)), respectively. Then, before the application of E, the length vector /()v° along the primitive chain corresponding to the nth Rouse segment is given by... 

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

This shows explicitly that the step length a of the primitive chain is of the same order as the tube diameter Oq. The average of the fluctuation is calculated from eqn (6.71) as... 

Given the general agreement in the shape of the relaxation modulus, it is possible to determine the step length a of the primitive chain Though various ways are conceivable, a direct way is to use the plateau modulus... 

Ilg. 7J2. Explanation of the stress relaxation after large step strain, (a) Before deformation the conformatian of the fnimitive chain is in equilibrium (r = —0). (b) Immediately after deformation, the primitive chain is in the afiindy deformed conformation (t = -1-0). (c) After time Tj, the primitive chain contracts along the tube and recovers the eqi brium contour length (t Tj,). (d) After the time Xj, the primitive chain leaves the deformed tube by reptation (t Xa). The oblique lines indicates the deformed part of the tube. Reproduced from ref. 107. 

We consider the inextensible chain model. Figure 7.24 eiqilains the change of polymer conformation under the double step strain. Figure 7.24a shows the undeformed state just before the first deformation. Figure 7.24h represents the state immediately after the deformation the primitive chain is deformed by the shear Yi. Figure 7.24c indicates the state just before the second deformation the inner part AB still remains in the deformed tube, while the outer parts are in the undeformed tube. Now when the second deformation is applied, the inner part AB is deformed by the shear Yi + Yi from the equilibrium state, while the outer part is deformed by the shear Yi- It is important to note that the second shear stretches the contour length of the outer part by the factor < (72), but that of the inner part by the factor... 

Figure 4.32. The primitive chain is regarded as an ideal chain of a step length of and a contour length of L. The primitive chain and the test chain share the end-to-end distance.

The primitive chain shares some of the statistical properties with the parent test chain. Both are ideal. Their end-to-end distance is the same. The conformation of the primitive chain is a coarse-grained version of the conformation of the test chain. When we apply the random-walk model to the primitive chain, its step length is equal to the tube diameter ft, (Fig. 4.32). Because the tube eucases the test chain, bf > b. We can appreciate the coarse-grained nature of the primitive chain in this inequality. The contour length L of the primitive chain is shorter than that of the test chain. We can estimate L as follows. For the primitive chain to have a contour length of L, the random walk must have L/, steps. We equate the meau square end-to-end distance of the test chain and that of the primitive chain b (L/b = b N. Then,... 

Assume that the primitive path of the chain in the tube is a random-walk sequence of N steps with step length Then we have... 

These authors studied how tube models with reptation dynamics could be turned into a full theory of viscoelasticity. To do this one needs to describe the dynamics of the primitive path. Let the primitive chain make one step in time At. Define a random variable, (t) which is + 1( — 1) if the primitive chain moves backward (forward). Let p be a random vector of length a which is the new position of one of the ends of the primitive chain after one move (see Figure 23). Since the primitive chain is assumed to be made of N points R, . Rjy, connected by bonds of constant length a, the Langevin equation of the primitive chain is given by... 

Fig. 7. Illustration of a polymer chain (the tagged chain) confined in the fictitious tube of diameter d formed by the matrix. The contour line of the tube is called the primitive path having a random-walk conformation with a step length a=d. The four characteristic types of dynamic processes (dotted arrow lines) and their time constants Zs, Zg, Zr, and za defined in the frame of the Doi/Edwards tube/reptation model are indicated...

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