134 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



and complements the other methods. Fig. 8 shows a photograph of the 

 equipment. 



III. MEASUREMENTS OF POLYMERS IN SOLUTION 



When such methods are apphed to a polymer solution, it is found 

 that the resistance and reactance components are no longer equal but 

 the resistance is invariably larger than the reactance. This indicates the 

 presence of a shear elasticity in the solution. If the molecules have a 

 single relaxation frequency, it has been found that the shear properties 

 of the liciuid can be represented by a stress-strain equation of the type 



T = r]A 



(8) 



where T is the shearing stress, S the shearing strain, t/^ the solvent 

 viscosity, i]b a molecular viscosity of some particular motion of the 

 chain which disappears when the reactance of the chain stiffness hb of 

 this motion is low enough so that the motion can follow the applied 

 shearing stress at the frequency of the measurement. When this type of 

 mechanism is present in the liquid, it has been shown^ that the impedance 

 the liquid presents to the crystals is 



PUbVb + j 

 Zo = Rm -{- jXm — 



ccpVaVb + {.Va -r Vb) 



CO 



(9) 



■I , -2. I 



Vb -r mb/w 



Fig. 9 shows a plot of the resistance and reactance components of an 

 assumed solution having a single relaxation frequency, and a viscosity 

 30 times the solvent viscosity. At very low freciuencies, the resistance 

 and reactance follow that of a solution, but for frequencies comparable 

 with the relaxation frequency, the resistance becomes larger than the 

 reactance while for very high frequencies the two come together on a 

 line determined by the solvent viscosity. If there is more than one 

 relaxation frequency, the resistance and reactance may coalesce for 

 several intermediate stages. A continuous distribution w^ould give a 

 definite relation between the frequency dependence of resistance and 

 reactance. 



The torsional crystal and the shear wave reflection method have 

 been applied to long chains of polyisobutylene dissolved in various sol- 



8 W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, 

 D. van Nostrand, 1950, p. 353. 



