Non-draining sphere

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]. 

Fig. 7. Molecular weight dependence of electrophoretic mobility of polyacrylate ion. Lines denoted by 1 and 2 are the calculated mobilities from Henry s theory assuming that the polyion sphere is a non-draining sphere having a surface charge of Equation (6). Degree of neutralization of polyacrylate ion is 1.0, 0.6, 0.4,0. 2, 0.1, 0.05 from top to bottom. NaCl concn. is 0.1 N. Reproduced from Reference [21].)...

Beside the effect of ionic atmosphere, there are various factors which may affect o Po. ThQ first may be the effect of drainage. If there is partial free drainage of solvent through the polymer coil, the powers of a in Equations (35) and (36) may be considerably different from 1 and 3 [18]. However, we expect that 0q Pq will be constant over a wide range of a, because the effect of partial free drainage on and [rj ] tend to compensate one another (e.g. as in Equation (37)). The second may be that, even in the limit of non-draining sphere, the sedimentation coefficient of Equation (35) and the intrinsic viscosity on Equation (36) may be proportional to lower powers of a than 1 and 3, respectively, as first pointed out by Kurata and Yamakawa [18]. Although there are various experimental works, the dependence of the sedimentation coefficient on a has not yet been completely solved [19, 19a]. 

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-... 

A nondraining polymer molecule, also referred to as the impermeable coil, can be represented by an equivalent impermeable hydrodynamic sphere of radius R. The frictional coefficient of this sphere which represents the frictional coefficient of the non-draining polymer coil can thus be written,... 

A non-draining polymer molecule can be represented by an equivalent impermeable hydrodynamic particle, i.e. one which has the same frictional coefficient as the polymer molecule. Thus a non-draining random coil can be represented by an equivalent impermeable hydrodynamic sphere of radius Rh- From Stokes Law... 

The predictions of the Zimm model for a Gaussian chain have a very simple interpretation the Zimm limit is a non-draining limit, the solvent flow does not penetrate into the polymer chain and, as far as hydrodynamic properties are concerned, a polymer chain can be considered as a hard sphere. The diffusion constant is then given by Stokes law, and the intrinsic viscosity by the Einstein equation for dilute solutions of hard spheres, the only difference being in the numerical factors. 

In the non-draining limit, polymer chains behave as hard spheres and the dynamic exponent is z = 3 leading to a variation of the characteristic time with the molecular mass... 

In the non-draining limit, which is a good approximation for dilute solutions, polymer coils behave as hard spheres. For hard spheres of radius R there are discrepancies between the values of the hydrodynamic interaction coefficients calculated by different authors. A critical comparison of most of the published approaches can be found in refs. 40 and 41. The most commonly assumed values are kD= 1.56[(47t/ )/3] and ks=6.55[(47rR )/3]. 

For the bead-spring model, a/b has to be smaller than 1/2 to avoid the interpenetration of the neighboring spheres. The value of ft described by Eq. (3.1) satisfies this criterion and so is consistent with the model on which the theory is based However, it should be noted that this favorable result is obtained within the framework of the Zimm theory. The value of ft at the non-free draining limit is 1/4 for Gaussian chains but it is different from 1/4 for chains of other distribution. Moreover,... 

The low intrinsic viscosities of the different lignins (dioxane, kraft, lignosul-fonate, alkali lignins) dissolved in various solvents and the Mark-Houwink exponent (a) ranged between 0 and 0.5 led what allow to conclude that the shape of lignin molecules is between an Einstein sphere and a non free-draining random... 

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