Primitive path orientation

In the above, X is the chain stretch, which is greater than unity when the flow is fast enough (i.e., y T, > 1) that the retraction process is not complete, and the chain s primitive path therefore becomes stretched. This magnifies the stress, as shown by the multiplier X in the equation for the stress tensor a, Eq. (3-78d). The tensor Q is defined as Q/5, where Q is defined by Eq. (3-70). Convective constraint release is responsible for the last terms in Eqns. (A3-29a) and (A3-29c) these cause the orientation relaxation time r to be shorter than the reptation time Zti and reduce the chain stretch X. Derive the predicted dependence of the dimensionless shear stress On/G and the first normal stress difference M/G on the dimensionless shear rate y for rd/r, = 50 and compare your results with those plotted in Fig. 3-35. 

Doi and Edwan argue for a particular relationship betwera stress and orientation They assume that the primitive path stepi deform affinely (Fig. 4) and that the chain segments contained in those steps respond initially like indepemlent Gau an strands. 

Doi and Edwards then derive expressions for the changing contour length and change in the orientation of the primitive path. The development is beyond the scope of the current article. The important expression is that for the chain orientation tensor Q as a function of the macroscopic deformation gradient tensor ... 

Eniture efforts will address other systems such as entangled (linear and branched) polymers where, inspired by the corresponding GENERIC formalism, one should resort to a description in terms of the orientational distribution function of an entanglement segment along the primitive path of the chain. 

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. 

The earliest tube models included only the simplest nonlinearities, that is, convective constraint release was neglected (since its importance was not clearly recognized), and the retraction was assumed to occur so fast relative to the rate of flow that the chains were assumed to remain imstretched. The linear relaxation processes of constraint release and primitive path fluctuations were also ignored, so that the model contained only one linear relaxation mechanism, namely reptation, and only the nonlinearity associated with large orientation of tube segments, but no stretch. Subsequent models added the omitted relaxation phenomena, one at a time, and in what follows we present the most important constitutive models that included these effects, starting with models for monodisperse linear polymers. 

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