Nondraining limit

In the nondraining limit of Eq. (9.47), the coils are unperturbed in both senses of the word nondraining and 0 conditions. To emphasize the latter we attach the subscript 0 to [r ] when these conditions are met. Thus for high polymers under 0 conditions... 

Random coils. Equation (9.53) gives the Kirkwood-Riseman expression for the friction factor of a random coil. In the free-draining limit, the segmental friction factor can, in turn, be evaluated from f. In the nondraining limit the radius of gyration can be determined. We have already discussed f in Chap. 2 and (rg ) in this chapter and again in Chapter 10, so we shall not examine the information provided by D for the random coil any further. 

Kirkwood and Riseman have developed a theory that allows for variable degrees of solvent drainage through the coil domain. We shall not go into this theory in any detail, except to note that it should reduce to Equation (87) in the free-draining limit and to the Einstein equation in the nondraining limit. The Kirkwood-Riseman theory can be written in the form... 

Using the nondraining limit of the Kirkwood-Riseman theory gives... 

Both the scaling with co and the proportionality constant, V3, are confirmed by experimental data (see Fig. 3-13). The lines in Fig. 3-13 are proportional to G and G" — computed in the nondraining limit. The agreement with data for a polystyrene of high molecular weight (M = 860,000) in theta solvents is excellent. In addition to its agreement with experimental data, the predictions of Zimm theory are supported by molecular dynamics simulations (Pierleoni and Ryckaert 1991 DUnweg and Kremer 1991). 

Barrett [1984] has further proposed TP expressions for and an, formulated within the Kirkwood-Riseman hydrodynamic theory in the nondraining limit and using approximate formulas, based on numerical simulation, for the requisite statistical averages, (/f >and Rf/), where Ry refers to the distance between the ith andyth chain segments ... 

Attaining the nondraining limit is usually accomplished by using high-molecular-weight polymers. The cooperative effect of the large number of subchains entrains all the solvent within the coil. Measurements of the macromolecular friction coefficient for a random coil are then interpreted in... 

The intrinsic viscosity ( /] is obtained by extrapolation of reduced viscosity of the dilute polymer solution (q-qs)lcqs> to zero polymer concentration, c—>0 (here rj is the viscosity of the polymer solution and the viscosity of pure solvent). In the nondraining limit of large N, the coils behave in a shear flow as impermeable for the solvent particles of effertive radius J ,. In dilute-solution limit, the Einstein equation i/ = i/s[l+ (5/2) ] applies, where is the volume fraction of particles in the solution. Hence, the intrinsic viscosity [i/] measures the (inverse) average intramolecular concenttation of the monomer units assuming that they are confined within a sphere of radius J ,. 

In the nondraining limit with X large, the dependence of [>/] on Rq has been utilized to develop a number of approximate relations intended to deduce estimates of Rq.o and z/M from data on [ /] and M. These usually take the form of plots involving some combination of and For... 

Equation (9.40) treats the nondraining coil as a rigid sphere and shows that in this limit [r ] (fg M-... 

Aharoni and coworkers characterized 59 Denkewalter s cascade macromolecules 4 by employing classical polymer techniques viscosity determinations, photo correlation spectroscopy (PCS), and size exclusion chromatography (SEC). It was concluded that at each tier (2 through 10) these globular polymers were, in fact, monodisperse and behaved as nondraining spheres. The purity of these molecules was not ascertained and the dense packing limits were either not realized or simply not noted. 

Still another indication of the upper limit given earlier may. be found from another combination of sedimentation and viscosity data. By combining Eqs. (4), (53), (66), (68), and (69) we easily find for nondraining flexible chains ... 

Fixed points A and B are version. of free draining of the chain while C and D relate to a nondraining chain. The fixed points with u = 0 correspond to the 0 state, while u = 7T e/2 correspond to the self-avoiding limit (the maximum good solvent). 

In the case of a nondraining coil in the maximum good solvent (the self-avoiding limit), we have... 

Within this limit, called the nondraining chain, the hydrodynamic radius is proportional to the root mean-squared radius of gyration ... 

The translational diffusion coefficient D of a polymer coil can be found from the time-dependent correlation function of scattered intensity measured in dynamic light-scattering experiments. Using the Stokes-Einstein equation, the translational diffusion coefficient D can be related to the apparent hydro-dynamic radius (the radius of equivalent hard sphere). In the limit of nondraining for the solvent coil formed by infinitely long chain, the hydrodynamic radius is given by... 

In the limit of N—both l h and 1 , scale with the degree of polymerization in the same way as the gyration radius of the Gaussian or swollen coil N"a. Remarkably, for finite-length semiflexible chains, the nondraining condition does not strictly apply. As a result, deviation in the observable N dependence of the gyration radius and those of the hydrodynamic radii... 

In Eq. (27.24), a is the hydrodynamic interaction parameter. The parameter m accounts for the effects of collective motion it increases the term (a + tti) in the square bracket (correction for nonideal intermolecular friction) in the limit of infinitely dilute solution (Q —> 0 nondraining coils). Substituting the above equations into Eq. (27.22) and simplification leads to ... 

Surfaces. Corrosion is a surface phenomenon, and the effects of poorly prepared surfaces, rough textures, and con lex shapes and profiles can be expected to be deleterious. Figure 5 shows some exait les in which design specification could have considerably reduced the onset of corrosive damage. Design limitations include surfaces exposed to deposits, retained soluble salts (because of poor access for preparation before painting), nondraining assemblies, poorly handled components (distortion, scratches, and dents), and... 

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