Ellipsoidal coils

In addition to an array of experimental methods, we also consider a more diverse assortment of polymeric systems than has been true in other chapters. Besides synthetic polymer solutions, we also consider aqueous protein solutions. The former polymers are well represented by the random coil model the latter are approximated by rigid ellipsoids or spheres. For random coils changes in the goodness of the solvent affects coil dimensions. For aqueous proteins the solvent-solute interaction results in various degrees of hydration, which also changes the size of the molecules. Hence the methods we discuss are all potential sources of information about these interactions between polymers and their solvent environments. 

Figure 4 demonstrates the results of several investigations. It can be seen that both methods lead to a linear dependence between c and Mw but differ by a factor of ten. The reason is seen in the fact that c ] depends on a model (Einstein s law), whereas c LS gives absolute results. In both cases the geometric shape of the polymer coils are assumed to be spherical but, in accordance with the findings of Kuhn, we know that the most probable form can be best represented as a bean-like (irregularly ellipsoidal) structure. 

As has been pointed out (63), this is a rather artificial model and, moreover, its application is quite unnecessary. In fact, (a> can be calculated from the refractive index increment (dnjdc), as has extensively been done in the field of light scattering. This procedure is applicable also to the form birefringence effect of coil molecules, as the mean excess polarizability of a coil molecule as a whole is not influenced by the form effect. It is still built up additively of the mean excess polarizabilities of the random links. This reasoning is justified by the low density of links within a coil. In fact, if the coil is replaced by an equivalent ellipsoid consisting of an isotropic material of a refractive index not very much different from that of the solvent, its mean excess polarizability is equal to that of a sphere of equal volume [cf. also Bullough (145)]. 

Fig. 2.37 Morphologies of PHIC-PS rod-coil block copolymers studied by Chen et al. (1996). 7bp TEM images from diblocks with different compositions.The dark regions correspond to PS, which have been preferentially stained with RuG4. (A) /,., K- = 0.42 (B) /PHIC = 0.73 (C) /pH,c = 0.89 (D) /Pmc = 0.96 (E) /PHIC = 0.98. The PHIC chain axis and lamellar normals are denoted by n and p. (F, G, H) schematic models showing the packing arrangement of the rod-coil chains. The PHIC block is represented by the white rod, and the PS block by the black ellipsoid. (F) Wavy lamellar morphology (G) zig-zag morphology (H) bilayer and interdigitated arrowhead morphologies.

In a deformed system, the average form of the macromolecular coil can be approximated by an ellipsoid. The effective volume of the macromolecular coil depends on the velocity gradients. The expansion of the effective volume as a series in powers of the velocity gradients does not contain the first-order term, so vu =0. This means that, at low velocity gradients, the coil does not change its volume (one says the coil is orientated by flow). At larger velocity gradients, the volume of the coil is increased. 

Fig. IS. Molecular-weight dependence of sedimentation constant (rc) and intrinsic viscosity ( ), for various degrees of draining and coil expansion. Full line is for coiled polymers without draining. Dotted curve is for rigid ellipsoids of revolntion at various axial ratios p. Experimental points a, cellulose nitrate in ethyl acetate 729) b, cellulose nitrate in acetone (181) c, cellulose acetate in acetone (125) d., ethyl cellulose in ethyl acetate 223 ) e, ethyl bydroxyethyl cellulose in water (772)...

Fig. 27. Viscosity-molecular weight relation lor paiy-y-benzyl-L-glutamate. Solid curve, theory lor random coil in DCA, Eqs. (58) and (86). Chain curve, theory lor rigid ellipsoids in DMF. Dashed curve, random coils in theta solvent, K — 58 10". Dotted curve, hypothetical curve lor random coils (or interrupted helices)...

The hydrodynamic properties of solutions of native double-stranded DNA have thus far eluded complete quantitative interpretation, in spite of very extensive investigation. A synthesis of experimental data has recently been furnished by Doty [84 ) some of the earlier experimental results may be found in the papers of Doty, Bunce-McGiix, and Rice (86) Doty, Makmur, Eignek, and Schildkraut (55) and Kawade and Watanabe (135 ). It is easy to see that the double helices are not perfectly inflexible, for the observed intrinsic viscosities are far lower than those of rigid rods or ellipsoids with the Watson-Crick dimensions, p = Af/4600. On the other hand, the customary flexible-coil treatments also do not apply to these data. For example, if the correlation plot of against g (a) M l [) ], / is attempted, it is found that... 

Macromolecules have large molecular weights and various random shapes that may be coil-like, rod-like, or globular (spheres or ellipsoids). They form true solutions. Their sizes and shapes affect their diffusion in solutions. Besides that, interactions of large molecules with the small solvent and/or solute molecules affect the diffusion of macromolecules and smaller molecules. Sometimes, reaction-diffusion systems may lead to facilitated and active transport of solutes and ions in biological systems. These types of transport will be discussed in Chapter 9. 

Polymers in Solution. Polyacrylamide is soluble in water at all concentrations, temperatures, and pH values. An extrapolated dieta temperature in water is approximately —40°C (17). Insoluble gel fractions are sometimes obtained owing to cross-link formation between chains or to die formation of imide groups along die polymer chains (18). In very dilute solution, polyacrylamide exists as unassociated coils which can have an ellipsoidal or beanlike structure (19). Large aggregates of polymer chains have been observed in hydrolyzed polyacrylamides (20) and in copolymers containing a small amount of hydrophobic groups (21). 

Recently, in a number of theoretical computational studies the shape of the Gaussian coil has been characterized by the components of radii of gyration Ri, Rj and R3 in the three main directions of the coil (fixed for each conformation). Calculations reveal that the average sh of the coil n be approximated by a three-axial ellipsoid with the ratio of squares of axes... 

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