Coil dimensions

To obtain isolated polymer chains, a solvent must be present. The solvent might be selectively excluded or imbibed by the coil, depending on the free energy of interaction, and thereby perturb the coil dimensions. 

At the beginning of this section we enumerated four ways in which actual polymer molecules deviate from the model for perfectly flexible chains. The three sources of deviation which we have discussed so far all lead to the prediction of larger coil dimensions than would be the case for perfect flexibility. The fourth source of discrepancy, solvent interaction, can have either an expansion or a contraction effect on the coil dimensions. To see how this comes about, we consider enclosing the spherical domain occupied by the polymer molecule by a hypothetical boundary as indicated by the broken line in Fig. 1.9. Only a portion of this domain is actually occupied by chain segments, and the remaining sites are occupied by solvent molecules which we have assumed to be totally indifferent as far as coil dimensions are concerned. The region enclosed by this hypothetical boundary may be viewed as a solution, an we next consider the tendency of solvent molecules to cross in or out of the domain of the polymer molecule. 

A good solvent is the technical as well as descriptive term used to identify a solvent which tends to increase coil dimensions. Since this is a consequence of thermodynamically favorable polymer-solvent interactions, good solvents also dissolve polymers more readily in the first place. 

By contrast, a poor solvent is the technical description of a solvent which tends to decrease coil dimensions. 

In summary, we see that the first two sources of deviation can be dealt with quantitatively, while the last two are dispatched by joining them in compensation for one another. (If you can t beat em, join em ) By convention, the coil dimensions under 0 conditions are given the subscript 0, so we write... 

In addition to and r nis ai other way of characterizing coil dimensions is to consider which end-to-end distance has the greatest probability of occurring for specified n and 1 values. Derive an expression for this most probable value of r, r, from Eq. (1.44). Compare the ratio r ms/ m the ratio from the kinetic molecular theory of gases (consult, say,... 

Strauss and Williamst have studied coil dimensions of derivatives of poly(4-vinylpyridine) by light-scattering and viscosity measurements. The derivatives studied were poly(pyridinium) ions quaternized y% with n-dodecyl groups and (1 - y)% with ethyl groups. Experimental coil dimensions extrapolated to 0 conditions and expressed relative to the length of a freely rotating repeat unit are presented here for the molecules in two different environments ... 

The subscript 0 on 1 implies 0 conditions, a state of affairs characterized in Chap. 1 by the compensation of chain-excluded volume and solvent effects on coil dimensions. In the present context we are applying this result to bulk polymer with no solvent present. We shall see in Chap. 9, however, that coil dimensions in bulk polymers and in solutions under 0 conditions are the same. 

Although the emphasis in these last chapters is certainly on the polymeric solute, the experimental methods described herein also measure the interactions of these solutes with various solvents. Such interactions include the hydration of proteins at one extreme and the exclusion of poor solvents from random coils at the other. In between, good solvents are imbibed into the polymer domain to various degrees to expand coil dimensions. Such quantities as the Flory-Huggins interaction parameter, the 0 temperature, and the coil expansion factor are among the ways such interactions are quantified in the following chapters. 

We saw in Sec. 1.11 that coil dimensions are affected by interactions between chain segments and solvent. Both the coil expansion factor a defined by Eq. (1.63) and the interaction parameter x are pertinent to describing this situation. 

Use of random flight statistics to derive rg for the coil assumes the individual segments exclude no volume from one another. While physically unrealistic, this assumption makes the derivation mathematically manageable. Neglecting this volume exclusion means that coil dimensions are underestimated by the random fight model, but this effect can be offset by applying the result to a solvent in which polymer-polymer contacts are somewhat favored over polymer-solvent contacts. 

Conditions in which the effects of item (2) exactly compensate are called 0 conditions. The expansion factor a gives the ratio of coil dimensions under non-0 conditions to those under 0 conditions. 

The geometrical problem. This involves evaluating the geometrical effect in item (2). It requires calculation of the volume of the overlapping regions as a function of d and the coil dimensions, say, r. The mathematics of this step are tedious and add little to the polymer aspects of the theory. 

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. 

This integral is a gamma function and is readily solved using a table of integrals. In writing the last result uIq has been replaced by rQ the mean-square coil dimensions under 0 conditions. Equation (9.49) involves rjj" not r so we note that rQ = nlQ and replace n by I i - j I, the number of units separating units i and j, to obtain... 

Next we consider the situation of a coil which is unperturbed in the hydro-dynamic sense of being effectively nondraining, yet having dimensions which are perturbed away from those under 0 conditions. As far as the hydrodynamics are concerned, a polymer coil can be expanded above its random flight dimensions and still be nondraining. In this case, what is needed is to correct the coil dimension parameters by multiplying with the coil expansion factor a, defined by Eq. (1.63). Under non-0 conditions (no subscript), = a(rg)Q therefore under these conditions we write... 

Since these forces work in opposition, the actual coil dimensions describe the point of balance between the two. 

K. R. Minard, R. A. Wind 2001, (Sole-noidal microcoil design. Part I Optimizing RF homogeneity and coil dimensions), Concepts Magn. Reson. 13(2), 128. 

Here c is the polymer concentration by weight. < the density of the polymer, a an effective bond length or measure of the coil dimensions, and to the monomeric friction factor. The subscript zero indicates the pure polymer. Since 2 (H), the mean-square end-to-end chain separation, the viscosity will be directly proportional to the polymer concentration unless the plasticizer modifies the coil swelling. At high molecular weight the monomeric friction factor is increased by the factor (MIMf)" and M, is increased relative to the undiluted polymer [equation (55)]. Thus... 

The exponent can vary from v=0.33 for hard spheres up to v=1.0 for rigid rods. For linear chains v=0.5 refers to unperturbed coil dimensions in -solvents and v=0.588 [6] to good solvent conditions. Equation (37) maybe re-writ-ten by expressing the molar mass as a function of the radius of gyration, i.e.. 

In some specific cases, dissolved macromolecules take up the shape predicted by the above theories of isolated chain molecules. In general, however, the interaction between solvent molecules and macromolecules has significant effects on the chain dimensions. In poor solvents, the interactions between polymer segments and solvent molecules are not that much different from those between different chain segments. Hence, the coil dimensions tend towards those of an unperturbed chain if the dimension of the unperturbed coil is identical to that in solution, the solution conditions are called conditions (ff solvent, temper-... 

Coil dimensions Static light scattering, sedimentation measurements, small angle X-ray, solution viscosity... 

AN. This very interesting result addresses the third concern expressed early in the Theoretical section and suggests that polymer-solvent interaction effects on coil dimensions as a function of copolymer composition are relatively abrupt, rather than continuous and gradual. This is necessary for the modified M-H-S equation expressed in eq.(ll) to be valid over any significantly wide polymer compositional range, and it is in contrast... 

For each coil, the peak magnetic field along with the average hoop stress has been calculated. In the table, (D, D ) provides the coil dimension at centre (r along to the radial if) and axial (z) coordinate directions. The hoop stress for each coil is reported in the last column. All 12 coils of the as3nnmetric magnet are provided. For S)nnmetric magnets, only... 

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. 

In Figure 2.14, it is implied that the terminal carbon atom can occupy any position on the rim of the cone that is, there is assumed to be perfectly free rotation around the penultimate carbon-carbon bond. This is equivalent to saying that all values of , the angle that describes the rotation (see Fig. 2.14), are equally probable. Any hindrance to free rotation will block certain configurations, expanding the coil dimensions still further. 

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