Tube segment

Tj stretch relaxation time of a pom-pom cross-bar t(s) relaxation time of a tube segment with arc co-ordinate s... 

Fig.4. By curvilinear diffusion a chain evacuates its original tube and creates new tube segments from an isotropic distribution. Forgotten portions of the original tube are shown shaded...

The observations above can be rapidly turned into a semi-quantitative theory for star-polymer stress-relaxation [24] which is amenable to more quantitative refinement [25]. The key observation is that the diffusion equation for stress-re-lease, which arises in linear polymers via the passage of free ends out of deformed tube segment, is now modified in star polymers by the potential of Eq. (16). Apart from small displacements of the end, the diffusion to any position s along the arm will now need to be activated and so is exponentially suppressed. Each position along the arm, s, will possess its own characteristic stress relaxation time T(s) given approximately by... 

Finally we require an expression for the relaxation modulus consistent with the dilution hypothesis in which each tube segment relaxing its stress typically at a time r(x) does so in a background whose effective density of entangled strands is 0 The appropriate general form is... 

The probabihty distribution/(u,s,f) for the local segment distribution is related to the contribution from the ensemble of segments at s to the total stress. It is calculated self-consistently from the survival probabihty function by letting the surviving tube segments be deformed by the total deformation tensor over their lifetimes, Eftt k... 

In order to calculate the effects of CLF we have to ask how the fraction of monomers that is released through CLF at the chain ends grows with time. It has been recently shown that for Ktr the effect of reptation on escaping from the tube is negligible in comparison to CLF [90]. It is the first passage of a chain end that is assumed to relax the constraint of a tube segment on a chain. From the scale invariance of the Rouse equation (Eq. 3.7) an exact asymptotic result... 

With this, all relevant lengths scale with Therefore, for early times the fraction of still-confined tube segments must behave as ... 

A Two Phase Fluid having Uqutd and Gas Phases NAME OF LINE SEGMENT I Tube Segment 2/3 Point C to D ... 

The function F(t — t ) is related, as with the temporary network model of Green and Tobolsky (48) discussed earlier, to the survival probability of a tube segment for a time interval (f — t ) of the strain history (58,59). Finally, this Doi-Edwards model (Eq. 3.4-5) is for monodispersed polymers, and is capable of moderate predictive success in the non linear viscoelastic range. However, it is not capable of predicting strain hardening in elongational flows (Figs. 3.6 and 3.7). 

A field-scale simulator based on this approach would probably require simpler equations that captured the relevant phenomena without explicitly addressing many of them. Chapter 15, by Prieditis and Flumerfelt, models two-phase flow in a network of interconnected channels that consist of constricted tube segments. Work on the creation of a model that contains capillary snap-off in a network similar to that of Figure 6 has very recently been started at the University of Texas (R. Schechter, personal communication, October 26, 1987). 

This complex two phase flow is now modeled as a network of interconnected channels consisting of constricted tube segments which contain the flowing and non-flowing fractions of the gas and liquid phases. Although overly simplistic in many respects, this representation provides a basis for obtaining qualitative and semi-quantitative permeability results which are useful in... 

Viscous Pressure Drop. For continuous bubble trains with perfectly mobile interfaces moving through a given Dm channel, the dynamic pressure drop in the gas (foam) phase over a single tube segment D follows from Equation 16 ... 

The total number of bubbles N in the D tube segment can be given in terms of the average bubble volume Vba in the Dm channel by... 

Summing up the pressure drops across each of the D tube segments, the total pressure drop across the porous medium is obtained, i.e.. 

In obtaining Equation 45 we have let the bubble contact length Lf in each tube segment be Kil, where i is the tube length. 

Figure 17 The tube renewal concept an internal tube segment is lost when the attached entanglement disappears.

The relaxation modulus is a single exponential decay because a typical tube segment relaxes only after it has been passed off from one chain to another many times. Hence, before relaxing, each tube segment samples chains of many lengths as well as many different distances from a chain end this produces motional averaging, which results in nearly monoexponential relaxation. 

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