Tubes constraints

Completion Interval constraints Production tubing constraints... 

The main predictions of the scaling theory [40], concerning the dynamics behavior of polymer chains in tubes, deal with a number of characteristic times the smallest time rtube measures the interval of essentially Rouse relaxation before the monomers feel the tube constraints significantly, 1 < Wt < Wrtube = and diffusion of an inner monomer is... 

Of course, for time t = rtube the perpendicular displacement is of the order of D and, hence, the tube constraint is strongly felt. For t > rtube the perpendicular motions are essentially equilibrated and the dynamics is determined by individual motions along the tube ... 

Equations 22.3-22.14 represent the simplest formulation of filled phantom polymer networks. Clearly, specific features of the fractal filler structures of carbon black, etc., are totally neglected. However, the model uses chain variables R(i) directly. It assumes the chains are Gaussian the cross-links and filler particles are placed in position randomly and instantaneously and are thereafter permanent. Additionally, constraints arising from entanglements and packing effects can be introduced using the mean field approach of harmonic tube constraints [15]. 

We have expressed the relaxation behaviour in terms of the number of chains per unit volume. At this stage we are considering the polymer in an undiluted state. Suppose we now apply a step strain to the melt in the linear regime. Two different zones of behaviour can be seen relative to the time re. This is the time at which the tube constraints begin to affect the relaxation of the chain ... 

The most successful theoretical framework in which the accumulating data has been understood is the tube model of de Gennes, Doi and Edwards [2]. We visit the model in more detail in Sect. 2, but the fundamental assumption is simple to state the topological constraints by which contingent chains may not cross each other, which act in reality as complex many-body interactions, are assumed to be equivalent for each chain to a tube of width a surrounding and coarse-graining its own contour (Fig. 2). So, motions perpendicular to the tube contour are confined while those curvilinear to it are permitted. The theory then resembles a dynamic version of rubber elasticity with local dissipation, and with the additional assumption of the tube constraints. 

For times less than the Rouse time of an entanglement segment, Tg and short distances, the chain behaves as if it were free since no section has moved far enough to be strongly affected by the tube constraint. The characteristic decay-rate of the scattering function at wavevector k is dominated by the Rouse-time of chain segments whose size is the order of k % k. A detailed calculation gives for t % [2]... 

The chain tension arises in a physical way at timescales short enough for the tube constraints to be effectively permanent, each chain end is subject to random Brownian motion at the scale of an entanglement strand such that it may make a random choice of exploration of possible paths into the surrounding melt. One of these choices corresponds to retracing the chain back along its tube (thus shortening the primitive path), but far more choices correspond to extending the primitive path. The net effect is the chain tension sustained by the free ends. 

It is of interest to think about relaxing the assumptions (i) and particularly (ii) introduced at the beginning of Sect. 6.1, although hard experimental tests for specific assumptions of the deformation of the tube constraint itself can never be confined to rheology alone, but will involve at least careful analysis of neutronscattering experiments [46,63]. Not only might the tube diameter depend on the local strain, the localising field described by the tube may well take on an anisotropy consistent with the symmetry of the bulk strain. For discussions of how tube variables deform with strain see [67,68]... 

Now we turn to the single-chain dynamic structure factor, which is also strongly affected by the topological tube constraints. Qualitatively we would expect the following behaviour ... 

The trapping factor Te increases as the cross-link density increases, whereas ne and Ge—as terms that are specific to the polymer—are to a great extent independent of cross-link density. For the cross-link and tube constraint moduli, the following relations to molecular network parameters... 

It should be pointed out that the material parameter Ge can be determined in principle more precisely by means of equi-biaxial measurements than by uniaxial measurements. This is due to the fact that the first addend of the Ge-term in Eq. (45) increases linearly with X. This behavior results from the high lateral contraction on the equi-biaxial extension X2=X 2). It postulates a close dependency of the equi-biaxial stress on the tube constraint modu-... 

The important point lies in the determination of the topological tube constraint modulus Ge according to its physical value Ge l/2GN°, which was not realized in the first considerations. 

Figure 45a-c shows an adaptation of the developed model to uniaxial stress-strain data of a pre-conditioned S-SBR-sample filled with 40 phr N220. The fits are obtained for the third stretching cycles at various prestrains by referring to Eqs. (38), (44), and (47) with different but constant strain amplification factors X=Xmax for every pre-strain. For illustrating the fitting procedure, the adaptation is performed in three steps. Since the evaluation of the nominal stress contribution of the strained filler clusters by the integral in Eq. (47) requires the nominal stress aR>1 of the rubber matrix, this quantity is developed in the first step shown in Fig. 45a. It is obtained by demanding an intersection of the simulated curves according to Eqs. (38) and (44) with the measured ones at maximum strain of each strain cycle, where all fragile filler clusters are broken and hence the stress contribution of the strained filler clusters vanishes. The adapted polymer parameters are Gc=0.176 MPa and neITe= 100, independent of pre-strain. According to the considerations at the end of Sect. 5.2.2, the tube constraint modulus is kept fixed at the value Ge=0.2 MPa, which is determined by the plateau modulus Gn° 0.4 MPa [174, 175] of the uncross-linked S-SBR-melt (Ge=l/2GN°). The adapted amplification factors Xmax for the different pre-strains ( max=l> 1-5, 2, 2.5, 3) are listed in the insert of Fig. 45a.

Fig. 14. The strain dependent part of the shear stress relaxation function for different values of the tube constraint parameter z...

Equation 4.2 presupposes [Pg.265]

Reptation assumes that the mobility of the matrix polymer plays no role in the relaxation of the tube constraint felt by a test molecule. However, if the matrix chains were very much more mobile than the test chain, additional lateral motion of the chain might be permitted by virtue of the constraining chains themselves moving away. This type of motion is called constraint release or tube renewal , and may be operative if some of the matrix chains are significantly shorter or intrinsically more mobile than the test chain (Green 1991, Composto etal. 1992). 

The time t denotes the onset of the effect of tube constraints for t < t , the chain behaves as a Rouse chain in free space, while for r > t the chain feels the constraints imposed by the tube,... 

Note that is proportional to f in the time region t, s r < xr. This specific diffusion behaviour, first predicted by de Gennes, is a consequence of the two effects, the Rouse-like diffusion equation (6.107), and the tube constraints equation (6.109). On the other hand, the behaviour... 

These results clearly indicate that the tube constraint for a polymer becomes weaker if it is made of shorter polymers. The weakening the tube can be expressed either by an increase in the step length, or by an increase in the constraint release process, or both. However, the interpretation seems to be still at a tentative level. 

Let us now study how the tube constraint affects the translational and the rotational Brownian motion. 

It is important to note that is not affected as seriously as D. This is because the viscous stress reflects very fast motions, for which the tube constraint is not effective. ... 

This force acts to constrain the lateral motion of the chains (tube constraints) and the average contour length of the tube coincides with the equilibrium tube length... 

Very high-frequency (short-time) Rouse modes where the tube constraints are not felt. 

Still other possibilities exist e.g., a concept revealing a weakness of the Doi-Ed-wards model the modelization of the melt by a temporary rubber leads to the use of the classical rubber theory this is known to encounter considerable discrepancies for actual rubbers (crosshnked melts). A slightly different description of a rubber would lead to greater motion of the crosslinks, and consequently of more extensive rearrangement of the tube constraints in the melt (see Sect. 13). 

At higher concentrations, where the molecules are substantially entangled, the non-Newtonian behavior is enormously amplified, and 77/770 may fall by several orders of magnitude. This phenomenon, as in undiluted polymers, is associated with topological restraints that can be represented by entanglement coupling or tube constraints, and will be discussed in Chapters 10,13, and 17. 

Removing the hypothesis of fixed obstacles the models In the context of the reptation model, the finite lifetime of the tube constraints leads to an extra relaxation process called tube-renewal or constraint-release. 

To summarize the first half of this paper, a relatively simple description of chain entanglement has provided considerable insight into an otherwise intractable topology problem. Our model is based on a series of conjectures regarding the nature of the tube constraints. In particular we conjectured the universality of chain entanglement via a coordination number N. At... 

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