Dilute-Solution Theories for Flexible Random Coils

Deduction from the Dilute-Solution Theories for Flexible Random Coils 

Although the use of reduced variables developed empirically in advance of the theories which support it, we shall introduce it as a logical consequence of some of the relations in the preceding chapters. 

In the theories for dilute solutions of flexible molecules based on the bead-spring model, the contribution of the solute to the storage shear modulus, loss modulus, or relaxation modulus is given by a series of terms the magnitude of each of which is proportional to nkT, i.e., to cRTjM, as in equation 18 of Chapter 9 alternatively, the definition of [C ]y as the zero-concentration limit of G M/cRT (equations 1 and 6 of Chapter 9) implies that all contributions are proportional to nkT. Each contribution is associated with a relaxation time which is proportional to [ri Ti)sM/RT-, the proportionality constant (= for r i) depends on which theory applies (Rouse, Zimm, etc.) but is independent of temperature, as is evident, for example, in equation 27 of Chapter 9. Thus the temperature dependence of viscoelastic properties enters in four variables [r ], t/j, T explicitly, and c (which decreases slightly with increasing temperature because of thermal expansion). 

It is implicit in equation 27 of Chapter 9, and the corresponding general equation for any degree of hydrodynamic interaction in which the factor b/ir is replaced by, that all the relaxation times have the same temperature dependence. This can be conveniently represented by the ratio of any specific relaxation time Tp, such as T], at temperature T to its value at an arbitrary reference temperature Tq  

From the relations among the viscoelastic functions, it follows that multiplication of any modulus function—(7 (w), G(l), E(t), etc.—by coTq/cT and plotting against war or will combine measurements at various temperatures to give a single composite curve which represents reduction of the data to Tq. Correspondingly, any compliance function multiplied by cT/cqTo can be reduced in a similar manner. Multiplication by the concentration-temperature ratio is often denoted by the subscript p. It should be emphasized that the ratio co/c differs from unity by only a very small amount associated with thermal expansion. 

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