Primitive with constraint release

The simple slip link algorithm described above applies most obviously to star polymers, which relax by primitive path fluctuations combined with constraint release. As mentioned above the equilibrium (or average) number of slip links is Nq = IMf, with the arm molecular... 

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. 

The center of gravity of a linear chain now moves by two uncorrelated processes, reptation and constraint release, so the diffusion coefficient is just the sum of the individual contributions. Equation 21 gives the reptation contribution. Equation 18 gives the constraint release contribution with

primitive path steps ... 

The case of star/linear blends is a challenging one, because the description of constraint release that works best for pure star polymers is dynamic dilution, while for pure linear polymers, double reptation , or some variant of it, seems to be the better description. However, Milner, McLeish and coworkers [23] have developed a rather successful theory for the case of star/ linear blends. In the Milner-McLeish theory, at early times after a step strain both the star branches and the ends of the linear chains relax by primitive-path fluctuations combined with dynamic dilution, the latter causing the effective tube diameter to slowly increase with time. Then, at a time corresponding to the reptation time of the linear chains, the tube surrounding the unrelaxed star arms increases rather quickly, because of the sudden reptation of the linear chains. The increase in the tube diameter would be very abrupt, if it were not slowed by inclusion of the constraint release-Rouse processes, which leads to a square-root-in-time decay in the modulus (see Section 7.3). With this formulation, the Milner-McLeish theory yields very favorable predictions of polybutadiene data for star/linear blends see Fig. 9.13, where the parameters have the same values as were used for pure linears and pure stars. 

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. 

Figure 11.7 compares the predictions of the MLD theory to steady-state shear data for a solution of nearly monodisperse polystyrene. In this comparison, the reptation time Tj and the plateau modulus have been taken as adjustable parameters. The theory used in this comparison is not the simplified toy model given by Eqs. 11.14 to 11.17, but a more complete theory with a contour variable, described in Mead et al [27]. The more complete version of the theory is able to include the effects of primitive path fluctuations as well as a more complete description of reptation and convective constraint release. Nevertheless, if the reptation time constant Tj is suitably adjusted to account phenomenologically for primitive path fluctuations (as discussed in Section 6.4.3 and 6.4-4.2), the predictions of Eqs. 11.14 to 11.17 are very similar to those of the full theory. 

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