Random coiled Polymer

An interesting outgrowth of these considerations is the idea that In r versus K or Vj should describe a universal calibration curve in a particular column for random coil polymers. This conclusion is justified by examining Eq. (9.55), in which the product [i ]M is seen to be proportional to (rg ), with r = a(rg 0 ) - This suggests that In rg in the theoretical calibration curve can be replaced by ln[r ]M. The product [r ]M is called the hydrodynamic volume, and Fig. 9.17 shows that the calibration curves for a variety of polymer types merge into a single curve when the product [r ]M, rather than M alone, is used as the basis for the cafibration. 

Although R2 is the easiest quantity to be obtained theoretically, there is no straigthforward experimental method for its determination. For this reason, two other quantities are widely in use to characterize the dimensions of a randomly coiled polymer molecule ... 

The lowermost curve in Fig. 45 represents P(0) plotted against according to Eq. (31) for random coil molecules. The results of similar calculations for spherical and for rod-shaped particles of uniform density are shown also. The curve for the former of these is not very different from that for randomly coiled polymers at corresponding values of the abscissas the factor P(0) for rods differs appreciably, however. 

Fig. 46.—Dissymmetry ratio for light scattered at 45° and 135° as a function of -x/r /X for random coil polymer chains. ...

Apart from their utility in determining the correction factor 1/P( ), light-scattering dissymmetry measurements afford a measure of the dimensions of the randomly coiled polymer molecule in dilute solution. Thus the above analysis of measurements made at different angles yields the important ratio from which the root-mean-square... 

Thus both the numerator and denominator terms in Eq. (41), or in Eq. (44), depend on the concentration. Because of this situation empirical extrapolation of D is particularly hazardous (for random coiling polymers). If F2 is known from osmotic or light-scattering measurements at a series of concentrations, extrapolation according to Eq. (44) will be facilitated. (If such measurements have been carried out, however, the molecular weight also will have been determined.)... 

It is essential that the solution be sufficiently dilute to behave ideally, a condition which is difficult to meet in practice. Ordinarily the dilutions required are beyond those at which the concentration gradient measurement by the refractive index method may be applied with accuracy. Corrections for nonideality are particularly difficult to introduce in a satisfactory manner owing to the fact that nonideality terms depend on the molecular weight distribution, and the molecular weight distribution (as well as the concentration) varies over the length of the cell. Largely as a consequence of this circumstance, the sedimentation equilibrium method has been far less successful in application to random-coil polymers than to the comparatively compact proteins, for which deviations from ideality are much less severe. 

Thus we may retain the root-mean-square end-to-end distance as a measure of the size of the random-coiling polymer chain, and the parameter jS required to characterize the spatial distribution of polymer segments (not to be confused with the end-to-end distribution) can be calculated from It should be noted that the r used here... 

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 MW dependence of R for random-coil polymers is related to the molecular exp nsion ( )... 

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. 

For instance, one would like to know the types of structures actually present in the native and denatured proteins.. .. The denatured protein in a good solvent such as urea is probably somewhat like a randomly coiled polymer, though the large optical rotation of denatured proteins in urea indicates that much local rigidity must be present in the chain (pg. 4). 

The geometric properties of highly denatured states appear to be consistent with those expected for a random-coil polymer. For example, proteins unfolded at high temperatures or in high concentrations of denaturant invariably produce Kratky scattering profiles exhibiting the monotonic increase indicative of an expanded, coil-like conformation (Fig. 1) (Hagihara et al., 1998 see also Doniach et al., 1995). Consistent... 

Since little chain entanglement would be expected from these structures, one would expect poor bulk properties compared to traditional linear random coil polymers. 

Here a0 is a constant called the effective bond length of the chain, and as(z) is a dimensionless quantity called the linear expansion factor of the chain. The latter depends on long-range interactions between pairs of monomer units and chain length through the so-called excluded-volume parameter z. For details of these quantities characterizing the dimensions of random-coil polymers, the reader is referred to a recently published book by Yamakawa (40). At this place we simply note that as tends to unity in the absence of excluded-volume effect. 

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. 

The viscosity-molecular weight relations of polypeptides in helix-breaking solvents such as DCA and TFA are, as would be expected, very similar to those of randomly coiled polymers this can be seen from the v values summarized in Table 4. 

It is seen that

characteristic behavior suggests that the molecular shape of PBLG in the mixed solvent studied does not differ very much from swollen spheres of randomly coiled polymers at stages where the helical fraction is less than about 0.6. In this connection, it is worth recalling from Chapter C, Section 2.b that the dimensional features of a polypeptide remain close to Gaussian at such stages of helix-coil transition, provided the chain is sufficiently long. 

The dimensions of a randomly coiled polymer molecule are a topic that appears to bear no relationship to diffusion however, both the coil dimensions and diffusion can be analyzed in terms of random walk statistics. Therefore we may take advantage of the statistical argument we have developed to consider this problem. 

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