Nondraining coil

Such a coil is said to be nondraining, since the interior of its domain is unaffected by the flow. We anticipate using Eq. (1.58) to describe the molecular weight dependence of In view of this, we replace rg by (rg ) and attach a subscript 0 to the latter as a reminder that, under 0 conditions, solvent and excluded-volume effects cancel to give a true value. With these ideas in mind, the volume fraction of the nondraining coil is written... 

Equation (9.40) treats the nondraining coil as a rigid sphere and shows that in this limit [r ] (fg M-... 

The function f(X) approaches a constant value for nondraining coils to generate Eq. (9.40), and approaches some constant times nf/r o(r o ) for free-draining coils to generate Eq. (9.44). 

We remarked above that the Einstein equation might be pertinent in the case of a nondraining coil Equation (93) does not seem to bear this out. The Einstein theory in the form of Equation (41) predicts that (77/770 — 1) is proportional to the volume fraction of the dispersed particles. For the nondraining coils, the volume fraction is proportional to the volume of each coil domain times the number concentration of the particles. The first of these factors is proportional to Rg3 and the second to c/M. This leads to the prediction... 

Equation (9-148) describes the relationship between the intrinsic viscosity [97] of nondraining coils as a function of the molar mass and the radius of gyration. Two methods can be used to obtain [17] as a function of the molar mass alone ... 

Since e rarely takes on values above 0.23 in the case of nondraining coils, Or, values up to a maximum of 0.9 are obtained for nondraining coils. In the theta state, e = 0 and Equation (9-151) reduced to... 

In the case of a nondraining coil in the maximum good solvent (the self-avoiding limit), we have... 

In Eq. (27.24), a is the hydrodynamic interaction parameter. The parameter m accounts for the effects of collective motion it increases the term (a + tti) in the square bracket (correction for nonideal intermolecular friction) in the limit of infinitely dilute solution (Q —> 0 nondraining coils). Substituting the above equations into Eq. (27.22) and simplification leads to ... 

According to Eqs. (3,13) and (3.14) for nondraining coils and rods the product D,pp(q)I(q) should exhibit a slightly negative slope which should increase with chain stiffness. Experiments [121,123,137] using INS and DLS do... 

Equations (9.42) and (9.46) reveal that the range of a values in the Mark-Houwink equation is traceable to differences in the permeability of the coil to the flow streamlines. It is apparent that the extremes of the nondraining and free-draining polymer molecule bracket the range of intermediate permeabilities for the coil. In the next section we examine how these ideas can be refined still further. 

In the nondraining limit of Eq. (9.47), the coils are unperturbed in both senses of the word nondraining and 0 conditions. To emphasize the latter we attach the subscript 0 to [r ] when these conditions are met. Thus for high polymers under 0 conditions... 

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... 

Random coils. Equation (9.53) gives the Kirkwood-Riseman expression for the friction factor of a random coil. In the free-draining limit, the segmental friction factor can, in turn, be evaluated from f. In the nondraining limit the radius of gyration can be determined. We have already discussed f in Chap. 2 and (rg ) in this chapter and again in Chapter 10, so we shall not examine the information provided by D for the random coil any further. 

Kirkwood and Riseman have developed a theory that allows for variable degrees of solvent drainage through the coil domain. We shall not go into this theory in any detail, except to note that it should reduce to Equation (87) in the free-draining limit and to the Einstein equation in the nondraining limit. The Kirkwood-Riseman theory can be written in the form... 

For high molecular weight polymers in good solvents, fo] exceeds fo]0 because of coil expansion under nondraining conditions that is, as more solvent enters the coil domain than would be present under 0 conditions, Equation (92) continues to apply, with R replacing R2gfi. Using Equation (90) to quantify this expansion effect, we obtain... 

The lower transition temperature also indicates that the folding of the copolymer chains prepared at higher temperatures is much easier, or in a sense, these chains could memorize the parent collapsed globular state in which they were formed. As we discussed earlier, the conformational change can be better viewed in terms of the ratio of Rg)/ Rh). For a random coil and a uniform nondraining sphere, (Rg)/(Rh) 1.5 and 0.774, respec-... 

The dependence of the friction coefficient Cm of the macromolecule on its length M is affected by exclude-volume effects and effects of draining or nondraining (permeability of macromolecular coils). Taking into account equation (2.14), the coefficient of diffusion can be written as... 

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,... 

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