Coil molecules intrinsic viscosity

Branching reduces the hydrodynamic volume relative to the mass of the coil. The intrinsic viscosity of branched macromolecules is thus lower than that of their unbranched (linear) counterparts. The effect is particularly marked in the case of long-chain branching. If the number of branch points in a polymer homologous series increases with the molar mass, then the [rj] values also decrease relative to those of linear molecules. Thus, the slope of the log f (log 2) curve continuously decreases with increasing molar mass,... 

Universal SEC calibration reflects differences in the excluded volume of polymer molecules with identical molecular weight caused by varying coil conformation, coil geometry, and interactive propenies. Intrinsic viscosity, in the notation of Staudinger/ Mark/Houwink power law ([77]=fC.M ), summarizes these phenom-... 

According to the interpretation given above, the intrinsic viscosity is considered to be proportional to the ratio of the effective volume of the molecule in solution divided by its molecular weight. In particular (see Eq. 23), this effective volume is represented as being proportional to the cube of a linear dimension of the randomly coiled polymer chain,... 

The rheological behaviour of polymeric solutions is strongly influenced by the conformation of the polymer. In principle one has to deal with three different conformations, namely (1) random coil polymers (2) semi-flexible rod-like macromolecules and (2) rigid rods. It is easily understood that the hydrody-namically effective volume increases in the sequence mentioned, i.e. molecules with an equal degree of polymerisation exhibit drastically larger viscosities in a rod-like conformation than as statistical coil molecules. An experimental parameter, easily determined, for the conformation of a polymer is the exponent a of the Mark-Houwink relationship [25,26]. In the case of coiled polymers a is between 0.5 and 0.9,semi-flexible rods exhibit values between 1 and 1.3, whereas for an ideal rod the intrinsic viscosity is found to be proportional to M2. 

The solution properties of dendrigraft polybutadienes are, as in the previous cases discussed, consistent with a hard sphere morphology. The intrinsic viscosity of arborescent-poly(butadienes) levels off for the G1 and G2 polymers. Additionally, the ratio of the radius of gyration in solution (Rg) to the hydrodynamic radius (Rb) of the molecules decreases from RJRb = 1.4 to 0.8 from G1 to G2. For linear polymer chains with a coiled conformation in solution, a ratio RJRb = 1.48-1.50 is expected. For rigid spheres, in comparison, a limiting value RJRb = 0.775 is predicted. 

The effect of branching is to increase the segment density within the molecular coil. Thus a branched molecule occupies a smaller volume and has a lower intrinsic viscosity than a similar linear molecule of the same molecular weight. The degree of branching is often characterized in terms of the branching factor [1] in Eq. (14), where the subscripts B and L, respectively, refer to branched and linear polymers of the same molecular weight ... 

The fluid resistance experienced by a macromolecular solute moving in dilute solution depends on the shape and size of the molecule. A number of physical quantities have been introduced to express this. Typical ones are intrinsic viscosity [ry], limiting sedimentation coefficient s0, and limiting diffusion coefficient D0. The first is related to the rotation of the solute, while the last two are concerned with the translational motion of the solute. A wealth of theoretical and experimental information about these hydrodynamic quantities is already available for randomly coiled chains (40, 60). However, the corresponding information on non-randomly coiled polymers is as yet rather limited in number and in variety. 

Another interesting contribution to the study of viscosity behavior in the helix-coil Jransition region is the one due to Hayashi et al. (22) on a PBLA sample (Mw = 23.2 x 104) in m-cresol and a mixture of chloroform and DCA (5.7 voL-% DCA). As mentioned in Chapter B, PBLA undergoes an inverse transition in the chloroform-DCA mixture, while it undergoes a normal transition in m-cresol. Furthermore, its cooperativity parameter is distinctly smaller in the former solvent than in the latter. Thus we may expect that, when compared at the same helical fraction and chain length, the PBLA molecule in the chloroform-DCA mixture assumes a more extended shape and hence a larger intrinsic viscosity than in m-cresol, provided these two solvents have comparable solvent powers for the polymer. The experimental results shown in Fig. 32 are taken to substantiate this prediction, because the approximate agreement of the data points atfN=0 indicates that the two solvents have nearly equal solvent powers for the solute. 

Boss, et al., fitted Gq. (17) to their data vs. vdi enabling them to determine fp and D . At solvent concentration approaching vdiI = 0.95, the data revealed an enhancement above the value predicted by Eq. (17) as fitted to the lower-concentration data. The authors argued that under these circumstances macroscopic inhomogeneities in concentration (and hence in the free-volume distribution) should exist and enhance the diffusivity. Above v > 0.99 the polymer coils no longer overlapped substantially, depriving the solvent molecules of a set of obstacles fixed with respect to the laboratory, and solvent diffusion became related principally to intrinsic viscosity. 

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... 

The following observations seem to lead to a key for the understanding of the quasistatic treatment of the stress-optical behaviour (777). With the aid of third eq. (5.4) and eq. (5.8), the ratio of Maxwell constant and intrinsic viscosity, as valid for Gaussian coil molecules in a matching solvent, can be formulated as follows ... 

Thus, the observations made at the beginning of this section [see eqs. (5.10) and (5.11a)] with respect to coil molecules and rigid rods, are confirmed for the behaviour of the dumb-bell models. In particular, a comparison of eqs. (5.18) and (5.22) shows that, other than for the intrinsic viscosity, the Maxwell constant can only be calculated when, besides (hi) also (hi) is known. This remark will be of importance for the next section, where theories for short chain molecules will be discussed. 

If a polymer molecule in solution behaves as a random coil, its average end-to-end distance is proportional to the square root of its extended chain length (see page 25) - i.e. proportional to Ai 5, where Mr is the relative molecular mass. The average solvated volume of the polymer molecule is, therefore, proportional to M 5 and, since the unsolvated volume is proportional to A/r, the average solvation factor is proportional to (i.e. Af 5). The intrinsic viscosity of... 

For a given polymer-solvent system, the intrinsic viscosity varies with the molecular weight of the polymer. According to Flory, the intrinsic viscosity is directly proportional to the hydrodynamic volume occupied by the random coil of the polymer molecule in a solution. In addition, the hydrodynamic volume is related to the cube of the typical linear dimension of the random coil (root mean square end-to-end distance). The intrinsic viscosity is expressed as ... 

However, to illustrate the effect of solvent on viscosity, we measured the intrinsic viscosities of three copolymers in a polar solvent—benzyl alcohol at 80°C. The two solvents are compared in Table B. Although the intrinsic viscosities of the hydroxyl-containing copolymers are similar in the two solvents, the value for pure polystyrene is much reduced in the polar solvent. This provides indirect evidence for the existence of a smaller coil volume for the copolymer molecules in the nonpolar solvent. 

Based on properties in solution such as intrinsic viscosity and sedimentation and diffusion rates, conclusions can be drawn concerning the polymer configuration. Like most of the synthetic polymers, such as polystyrene, cellulose in solution belongs to a group of linear, randomly coiling polymers. This means that the molecules have no preferred structure in solution in contrast to amylose and some protein molecules which can adopt helical conformations. Cellulose differs distinctly from synthetic polymers and from lignin in some of its polymer properties. Typical of its solutions are the comparatively high viscosities and low sedimentation and diffusion coefficients (Tables 3-2 and 3-3). 

In earlier studies on solutions of synthetic polymers (Ferry, 1980), the zero-shear viscosity was found to be related to the molecular weight of the polymers. Plots of log r] versus log M often resulted in two straight lines with the lower M section having a slope of about one and the upper M section having a slope of about 3.4. Because the apparent viscosity also increases with concentration of a specific polymer, the roles of both molecular size and concentration of polymer need to be understood. In polymer dispersions of moderate concentration, the viscosity is controlled primarily by the extent to which the polymer chains interpenetrate that is characterized by the coil overlap parameter c[r] (Graessley, 1980). Determination of intrinsic viscosity [r]] and its relation to molecular weight were discussed in Chapter 1. The product c[jj] is dimensionless and indicates the volume occupied by the polymer molecule in the solution. 

The GPC-viscometry with universal calibration provides the unique opportunity to measure the intrinsic viscosity as a function of molecular weight (viscosity law, log [17] (it versus log M) across the polymer distribution (curves 3 and 4 in Fig. 1). This dependence is an important source of information about the macromolecule architecture and conformations in a dilute solution. Thus, the Mark-Houwink equation usually describes this law for linear polymers log[i7] = ogK+ a log M (see the entry Mark-Houwink Relationship). The value of the exponent a is affected by the macromolecule conformations Flexible coils have the values between 0.5 and 0.8, the higher values are typical for stiff anisotropic ( rod -like) molecules, and much lower (even negative) values are associated with dense spherical conformations. 

The viscosity method makes use of the fact that the exponent, a, in the Mark-Houwink equation (see Frictional Properties of Polymer Molecules in Dilute Solution), rj = KM° , is equal to 0.5 for a random coil in a theta-solvent. A series of polymers of the same type with widely different known molecular weights is used to determine intrinsic viscosities [t ] at different temperatures and hence a at different temperatures. The theta-temperature can thus be determined either by direct experiment or, if it is not in the measurable range, by calculation. 

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