Primitive path distance

Equation (16) defines the probability density of the fact that two concrete end points of AMink chain are connected by the primitive path of length /i. The probability density for an arbitrary end points having the distance equal to fi is defined by ... 

We may also obtain the mean-square tube radius by requiring that the mean-square end-to-end distance of the chain is given by a sum of independent contributions along the primitive path and between the chain ends and... 

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. 

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. 

The fluctuation forces also cause the curvilinear diffusion along the primitive path. However, because of the constraint effect of the tube, the positive and negative fluctuation forces perpendicular to the primitive path cancel each other out when averaged over a time period longer than required for a segment to travel over an entanglement distance, and thus make no net contribution to the curvihnear diffusion and the translational diffusion of the center of mass. Thus, in the study of the cmvilinear diffusion, Eq. (9.A.3 ) should be used, instead of Eq. (9.A.3). 

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

The time needed to achieve a complete disentangling can be estimated. In order to get disentangled, chains have to diffuse over a distance /pr, i.e. the original length of the primitive path, and this requires a time... 

How does this result change for an entangled melt The reptation model gives an answer. One has only to realize that the disentangling process is associated with a shift of the center of mass of a polymer molecule over a distance in the order of /pr along the primitive path and therefore leads to a mean-squared displacement... 

The width of the confining tube of a chain is determined by the spacing of the entanglement net, the distance between the primitive paths of the... 

Fig. 5. (a) Two molecules, A and B, are shown as represented by their primitive paths, (b) An end of B has reptated past A destroying a portion of B s tube. When that end moves back it can do so in any direction, loosing all memory of its original orientation. B was also part of the tube of A so that the illustrated motion represents a tube renewal process. A segment of A can make a limited step of distance Og until the next entanglement is encountered. The attachment of the released segment of A at both ends to the rest of the molecule allows only limited randomization of the orientation of this segment. 

A two dimensional picture of an entangled polymer is illustrated in Figure 8. The dashed line in the figure is called the primitive path, which is a shorter random walk and can be defined as the centre of the tube so that the distance between two non-crossable obstacles is on average the step length a of the primitive path and comparable to the width of the harmonic pipe potential. The polymer makes excursions between the entanglement points but returns to the tube le, always to the primitive path. Thus if we draw the centre of a tube, as in Figure 9, the polymer is a random walk confined in the tube. Such random walks are amenable to calculation and it is possible to evaluate the amount of polymer consumed by the excursions and hence the statistics of the primitive path itself. ... 

In the semidilute solution, the hydrodynamic interactions are shielded over the distance beyond the correlation length, just as the excluded volume is shielded. We can therefore approximate the dynamics of the test chain by a Rouse model, although the motion is constrained to the space within the tube. In the Rouse model, the chain as a whole receives the friction of N, where is the friction coefficient per bead. When the motion is limited to the curvilinear path of the primitive chain, the friction is the same. Because the test chain makes a Rouse motion within the tube, only the motion along the tube survives over time, leading to the translation of the primitive chain along its own contour. The one-dimensional diffusion coefficient for the motion of the primitive chain is called the curvilinear diffusion coefficient. It is therefore equal to Dq of the Rouse chain (Eq. 3.160) and given by... 

The mean square displacement in the three-dimensional space for a point on the primitive chain is proportional to Figure 4.37 illustrates the difference between the two distances. Recall that the mean square displacement is proportional to t in regular, unrestricted diffusion in solution. The exponent 1/2 in Eq. 4.53 is due to the one-dimensional random walk on a path created by another random walker in the three-dimensional space. 

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