230 H. K. SCHACHMAN AND R. C. WILLIAMS 



from a theoretical point of view by Eiiistein (1906, 1911), who showed that 

 the viscosity increment caused by the addition of rigid, spherical particles 

 was simply proportional to the total volume fraction of the added material. 

 In some respects this was a surprising result, showing as it did that the size 

 of the spheres was unimportant, for many small spherical particles were 

 equivalent to a few large spheres having the same total volume. 



Einstein's treatment is restricted to solutions at concentrations sufficiently 

 low that the regions of disturbed flow from individual particles do not over- 

 lap. Subsequently more detailed investigations of this hydrodynamic prob- 

 lem have permitted the extension of the Einstein equation to solutions of 

 higher concentration by the addition of terms containing the second and 

 third power of the concentration. Although there is not as yet general 

 agreement on the values of the coefficients of these higher terms, uncer- 

 tainties here are obviated by the device of extrapolating measurements at 

 various concentrations to infinite dilution. In this way interaction effects are 

 avoided and data can be interpreted according to the Einstein equation or a 

 suitable modification of it. Implicit in the Einstein treatment was the 

 assumption that the particles are uncharged. Of course most macromolecules 

 of biological interest contain ionizable groups, and these cause electrostatic 

 forces leading to higher viscosities than would be observed for uncharged 

 molecules. To ehminate this, the electroviscous effect, it is general custom 

 to conduct experiments in solutions containing electrolytes, such as buffer 

 salts or sodium chloride. For most materials, 0.1 molar salt solutions sufiice 

 to damp out electrostatic forces which would otherwise render the Einstein 

 theory inapplicable. Until recently the vahdity of this fundamental theory 

 had been demonstrated only with particles of microscopic size, such as glass 

 spheres or yeast cells; but the availability of uniform, spherical, poly- 

 styrene latex particles has permitted a critical examination of the theory 

 with particles only slightly larger than many of the spherical viruses. As 

 with the suspensions of glass spheres, the results with the polystjTene latex 

 particles (Cheng and Schachman, 1955b) were completely in accord with the 

 Einstein equation, thereby lending confidence to the application of this 

 equation to solutions of spherical virus particles. 



With the development of colloid chemistry, viscosity measurements 

 achieved wide popularity and considerations were given to rigid particles of 

 other shapes. Most popular of these are the ellipsoids of revolution. A 

 prolate elhpsoid, resulting from the revolution of an ellipse about its long 

 axis, serves as a model for rodlike particles. Rotation of the ellipse about its 

 short axis gives an oblate ellipsoid which is a platelike object. The compli- 

 cated hydrodynamic problem of determining the viscosity increment caused 

 by such particles was solved in rigorous form by Simha (1940). For the cal- 

 culations of Simha to be applicable, it is mandatory that the particles be 



