Dynamic tube dilation

Comparison of the normalized relaxation modulus of P1308/PI21 blends at 40°C examined in Figure 3.5 with the prediction of full dynamic tube dilation (DTD) model (dotted curves) and partial-DTD model (solid curve). (Data taken, with permission, from Watanabe, H., S. Ishida, Y. Matsumiya, and T. Inoue. 2004b. Test of full and partial tube dilation pictures in entangled blends of hnear polyisoprenes. Macromolecides 37 6619-6631.)... 

For the global dynamics governing the rubbery/terminal relaxation, the thermorheological complexity of the components is one of the most prominent features. In PI/PVE blends associated with just a moderate dynamic asymmetry of the components, respective components exhibit very minor complexity and behave similarly to the components in chemically uniform blends such as PI/PI blends, as revealed from rheo-optical and dielectric studies. (PI chains have the type-A dipole so that their global motion is dielectrically detected.) The entanglement relaxation in the PI/PVE blends appears to occur through the mechanisms known for the chemically uniform blends, for example, through the reptation and constraint release (CR)/ dynamic tube dilation (DTD) mechanisms. 

F., and Marrucd, G. (2004) Highly entangled polymer primitive chain network simulations based on dynamic tube dilation. /. Chem. Phys., 121 (24), 12650-12654. 

Figure 12 Schematic illustration of (a) CLF and (b) CR. Accumulation of local CR-hopping results in dynamic tube dilation (DTD) see part b.

A feature of theories for tree-like polymers is the disentanglement transition , which occurs when the tube dilation becomes faster than the arm-retraction within it. In fact this will happen even for simple star polymers, but very close to the terminal time itself when very little orientation remains in the polymers. In tree-like polymers, it is possible that several levels of molecule near the core are not effectively entangled, and instead relax via renormalised Rouse dynamics (in other words the criterion for dynamic dilution of Sect. 3.2.5 occurs before the topology of the tree becomes trivial). In extreme cases the cores may relax by Zimm dynamics, when the surroundings fail to screen even the hydro-dynamic interactions between the slowest sections of the molecules. 

When the molecular weights of both components in a blend are greater than the entanglement molecular weight Mg, a new idea, tube dilation — a dynamic aspect of the tube associated with the high-molecular-weight component, needs to be introduced. 

As discussed in this section, the tube-dilation effect, i.e. M J/Me > 1, mainly occurs in the terminal-relaxation region of component two in a binary blend. This effect means that the basic mean-field assumption of the Doi-Edwards theory (Eq. (8.3)) has a dynamic aspect when the molecular-weight distribution of the polymer sample is not narrow. This additional dynamic effect causes the viscoelastic spectrum of a broadly polydisperse sample to be much more complicated to analyze in terms of the tube model, and is the main factor which prevents Eq. (9.19) from being applied... 

Entanglements of flexible polymer chains contribute to non-linear viscoelastic response. Motions hindered by entanglements are a contributor to dielectric and diffusion properties since they constrain chain dynamics. Macromolecular dynamics are theoretically described by the reptation model. Reptation includes fluctuations in chain contour length, entanglement release, tube dilation, and retraction of side chains as the molecules translate using segmental motions, through a theoretical tube. The reptation model shows favourable comparison with experimental data from viscoelastic and dielectric measurements. The model reveals much about chain dynamics, relaxation times and molecular structures of individual macromolecules. 

The constraint-release models discussed above have been tested by comparing their predictions to experimental data, as shown in Figures 7.9 and 7.10. For linear polymers for which the molecular weight distribution is unimodal, and not too broad, dynamic dilution is not very important, and theories that account for constraint release without assuming any tube dilation are adequate. Such is the case with the version of the Milner-McLeish theory for linear polymers used to make the predictions shown in Fig. 6.13. The double reptation theory also neglects tube dilation. The dual constraint theory mentioned in Chapter 6 does include dynamic dilution, although its effect is not very important for narrowly dispersed linear polymers. As described above, dynamic dilution becomes important for some bimodal blends, and is certainly extremely important for branched polymers, as discussed in Chapter 9. 

The symbols in Fig. 9.18 are experimental data of Daniels etal. [25]. The solid and dotted lines are predictions of the hierarchical model with monodisperse and polydisperse arms and backbone molecular weights, respectively. The parameters are given in the caption of Fig. 10.6 with a= 4/3 the parameter value = 1/12 is used in Eq. 9.9 for the branch-point mobility, as suggested by Daniels et al. [25]. Once the arms relax, the backbone is assumed to reptate in a tube dilated by the dynamic dilution due to relaxation of the star arms. 

So the criterion that the effective constraint-release must be fast enough to allow local pieces of umelaxed chain to explore any dilated tube fully confirms the assumption of dynamic dilution for nearly the whole range of relaxation timescales exhibited by star polymers. 

Here, the overall flow q is dependent on variables such as the pressure applied across the tube (AP), the radius (r), length (L) of the tube, and the viscosity ( u.) of the solution. Unfortunately, this equation must be modified somewhat because blood is a non-Newtonian fluid having some pseudoelasticity. Therefore, at similar applied pressures, blood will flow faster than a typical Newtonian fluid. Furthermore, in vivo, the vessel diameter (due to dilation and constriction of the resistance vessels) is dynamic and constantly changing. Moreover, the viscosity of blood is very difficult to measure because the hematocrit of blood changes depending on the diameter of the vessel in which it flows. Because of this, blood viscosity in vivo is often reported as an apparent viscosity. Finally, in the bloodstream, blood flow is somewhat pulsatile, especially in the venous side of the circulation. In sum, it is difficult to know the exact linear rate of the blood flow or a pressure drop at any one point in the microcirculation. 

Finally, we remark that the idea of self-consistent dynamic dilution was applied first by Marrucci [20] to the case of monodisperse linear polymers, and was then adapted by BaU and McLeish [11] to monodisperse stars. We also note that theories combining reptation, primitive path fluctuations, and constraint release by dynamic dilution have been applied successfully by Milner and McLeish and coworkers to monodisperse linear polymers [21], monodisperse stars [13], bimodal star/star blends [22], and star/linear blends [23], as well as H-branched polymers [24], and combs [25]. The approach taken for all these cases is similar at early times after a small step strain, the star arms and the tips of linear molecules relax by primitive path fluctuations and dynamic dilution. At some later time, if there are linear chains that reach their reptation time, there is a rapid relaxation of these linear chains. This produces a dilation of the effective tubes that surround any remaining unrelaxed star arms by constraint-release Rouse motion (see Section 7.3). Finally, after dilation has finished, the primitive path fluctuations of remaining portions of star arms begin again, in the dilated tube. We refer to this set of theories for stars, linears, and mixtures thereof as the Milner-McLeish theory . The details of the Milner-McLeish theory are beyond the scope of this work, but the interested reader can learn more from the original articles as well as from McLeish and Milner [26], McLeish [14], Park and Larson [27], and by Watanabe [19]. 

For the multibranched polymers, an additional mechanism not considered for the linear/star-branched polymers needs to be introduced. For the simplest multibranched polymer, the pom-pom polymer schematically illustrated in Figure 16, McLdsh and Larson combined the equilibrium mechanisms (arm retraction/trunk reptation in the dynamically dilated tube) and the nonequilibrium mechanisms (arm/trunk stretch, CCR, and the arm withdrawal) to formulate a constitutive equation. The arm withdrawal mechanism, leading to partial contraction of the trunk up to a point of tension balance with the arms, is the mechanism not considered for the linear/ star-branched chains. The resulting constitutive equation for the pom-pom chains cannot be cast in a Bemstein-Zapas-Kearsley (BKZ)-type convolution form. This pom-pom con-stimtive equation reproduces the hierarchical relaxation (from... 

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