Tulin 



where Xj are the thermal forces, £ is the Lagrangian or, in this case, the neg- 

 ative strain energy, s is the entropy, a. is the principal strain (x. /relaxed mo- 

 lecular radius), and p is the principal strain probability distribution. 



Almost all rheological effects exhibited by dilute polymer solutions are due 

 to the existence of this strain energy. So is the elasticity of rubber, which is of 

 precisely the same fundamental nature as the elasticity of dilute polymer solu- 

 tions (5). Incidentally, the strain energy stored in stretched rubber, being 

 thermal in nature, may be felt by the hand. 



NORMAL STRESSES AND MOHR'S CIRCLE 



Their own inertia being negligible, the thermal and viscous forces acting on 

 the molecules are in balance. The reaction of the viscous forces upon the fluid 

 transmit the molecular thermal forces to the bulk medium. Therefore, in reac- 

 tion to the internal thermal forces, boundary stresses must arise on an infini- 

 tesimal fluid element; both shear stresses and nonisotropic pressures result. 

 In order to calculate these boundary stresses (a tensor) in flow coordinates it 

 becomes useful to introduce that most ubiquitous construction from the mechan- 

 ics of deformable media — Mohr's circle (for simplicity we take a two- 

 dimensional viewpoint here). Now quite clearly the boundary stresses in flow 

 coordinates depend upon the orientation of the particles' principal strain axes 

 relative to the flow as well as upon the magnitude of the principal strains them- 

 selves. As indicated at the top of Fig. 2, the orientation of the strain axes de- 

 pends upon the product of the molecular relaxation time and the strain rate (note 

 that the rotation time in shear is the inverse of the latter, i.e. Cdu/dyy 0- 

 Mohr's circle may easily be used to show how both normal and shearing 

 stresses arise due to the presence of strained molecules, and how their magni- 

 tudes depend in general upon the molecular strains and on the molecular orien- 

 tation. In the case of low shears (left side of Fig. 2), and therefore with |aj J 

 slightly greater than \cr^^\, the normal stresses a^^ and a^ are positive, al- 

 most equal (a^^ > cTyy), and, we may show, proportional to the square of the 

 strain rate —all in accord with the contentions of Marcus Reiner (Ref. 6, page 

 42). For high enough shears, however, the normal stresses may become of dif- 

 ferent sign, with cr^^ positive and much larger than a^ in magnitude —as has 

 sometimes been contended by others. Perhaps some of these old controversies 

 can be squared away within the confines of Mohr's circle. 



THE DYNAMICS OF MACROMOLECULES 



It turns out to be somewhat easier to predict the effect upon the fluid 

 stresses of molecular strains and orientations than it is to predict these latter 

 quantities themselves — as a generation of rheologists can testify. Here I must 

 particularly mention such physical chemists as Kirkwood and Riseman (7), 

 Rouse (8), Bueche (9), Peterlin (10), Sadron (11), and Hermanns (12). Although 

 they have endowed us with a great legacy, a really satisfactory treatment of 

 flexible molecules in even simple flows cannot be said to exist, that is, free 

 from restrictions on small strain, etc. Then, too, as hydrodynamicists we must 

 be concerned with rather complicated flows — in comparison to simple shearing. 



