Non-draining

The same effect happens inside a random flight chain where the close proximity of the polymer segments offers mutual screening from the bulk flow field. The idea of a chain being non-drained was first considered by Debye Bueche who introduced the concept of a shielding length defined as [46] ... 

The simplest indicator of conformation comes not from but the sedimentation concentration dependence coefficient, ks. Wales and Van Holde [106] were the first to show that the ratio of fcs to the intrinsic viscosity, [/ ] was a measure of particle conformation. It was shown empirically by Creeth and Knight [107] that this has a value of 1.6 for compact spheres and non-draining coils, and adopted lower values for more extended structures. Rowe [36,37] subsequently provided a derivation for rigid particles, a derivation later supported by Lavrenko and coworkers [10]. The Rowe theory assumed there were no free-draining effects and also that the solvent had suf-... 

Flow properties of macroemulsions are different from those of non-emulsified phases 19,44). When water droplets are dispersed in a non-wetting oil phase, the relative permeability of the formation to the non-wetting phase decreases. Viscous energy must be expended to deform the emulsified water droplets so that they will pass through pore throats. If viscous forces are insufficient to overcome the capillary forces which hold the water droplet within the pore body, flow channels will become blocked with persistent, non-draining water droplets. As a result, the flow of oil to the wellbore will also be blocked. 

This result is appHcable to semi-dilute and concentrated solutions [21], and is also useful to check many simulations that do not include HI. For non-draining chains, introducing Gaussian statistics in Eq. (35), and transforming the summations over a large number of units in Eq. (34) into integrals, the translational friction coefficient can finally be written as [ 15]... 

A similar KR approach can be employed for the intrinsic viscosity. In this case it is necessary to assume the presence of a small amount of shear rate, which cancels out in the calculations. Now, the linear equations are used to evaluate an averaged crossed component of the stress tensor on unit i, .The final result for a long linear non-draining chain is... 

The calculation of g for Gaussian uniform star chains was carried out by Zimm and Kilb (ZK) [83]. They used a modified version of the dynamic Rouse theory including preaveraged HI (in the non-draining limit) that considers the particular connectivity of units consistently with the star architecture. This ap-... 

The preceding sections have demonstrated that dendrimers of lower generation are akin to branched polymeric structures. It is therefore to be expected that their flow behavior in dilute solution may be described in terms of the well-known concepts of dilute polymer solutions [14, 15]. Hence, dissolved dendrimers should behave like non-draining spheres. From an experimental comparison of and the immobilization of solvent inside the den-drimer can be compared directly since in this case the dendrimer may be approximated by a homogeneous sphere. Therefore R = 3/5 Rl where Ry, denotes the hydrodynamic radius of the dendrimer. This has been found experimentally [19]. 

Note A freely draining macromolecule and a non-draining macromolecule are two extremes of the concept of a partially draining macromolecule. 

In principle, it only remains to show that the non-draining case is actually prevailing in solutions of coil molecules. For this purpose, reference is made to the experiences gathered in the field of molecular weight determinations with the aid of intrinsic viscosity measurements. From eq. (3.37) one obtains, when eq. (3.58) is used for the relaxation... 

It is interesting to compare this parameter with the one obtained by Kirkwood and Riseman for the non-draining case, when these authors equated the centre of resistance with the centre of gravity. It reads... 

Fig. 3.1. Reduced stored free energy Fs vs. reduced shear rate py for three samples of anionic polystyrene in monobromo-benzene at 25° C 70). Data of these samples are gathered in Table 3.2. The following concentrations given in g/100 cm8, are used Taps. No. 5 ( -) 0.50, (- ) 0.35, (J) 0.20, (1) 0.10 Taps. No. 17 (o-) 1.00, (-O) 0.50, () 0.35, (i) 0.20 Taps. No. 15 (4) 1.00, (It) 0.633, (. ) 0.40. "F" indicates free-draining, N non-draining limit according to Zimm (SO). For the meaning of the broken lines see Section 5.3.2...

In this way an extrapolation of data to zero concentration seems avoidable. This is in accordance with an early expectation of Peterlin and Signer (114). A refinement of this view, however, will be given below and in Chapter 4. Furthermore, a fair reduction with respect to the molecular weight is noticed. All measuring points of Fig. 3.1, which were obtained at / values smaller than two fall within the limits of the theory indicated by the lines "N and F (non-draining and free-... 

The fact that the measured points of Fig. 3.1 lie more closely to the free-draining line, is in accordance with the experience obtained on anionic polystyrenes with other measuring techniques [dynamic oscillatory measurements (115), measurements of normal stresses (776)]. This result is quite surprising since for the description of intrinsic viscosity the non-draining case has clearly been shown to be valid (100). It will be shown below and in Chapter 4 that this inconsistency is in reality a consequence of the fact that the reduction with respect to concentration is less perfect than one would think at a first inspection of Fig. 3.1. 

Possible other reasons for a too high value of the reduced steady-state compliance, when compared with the expected non-draining value, are given by the facts that bromo-benzene is a very good solvent and that even the anionic polystyrenes have no completely uniform molecular weight. The second point will be treated in Section 3.8.3. As to the first point, it can be shown by experiment that the excluded volume... 

This value is very close to the theoretical values, given by eqs. (3.62a) and (3.62b) for the non-draining case. The reason why the just indicated extrapolation to zero concentration seemed unnecessary for the results obtained on solutions in bromo-benzene, lies in the lower measuring... 

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