conditions, the phenomenon of viscoelasticity is of concern only when the shear stress acting on a fluid element is changing or during transient flow situations. Under turbulent conditions, however, the formation and dissipation of turbulent perturbations constitute a perpetual unsteady state, and hence it is to be expected that elastic properties will contribute markedly to turbulent behavior. ( If steady-state laminar motion of a viscoelastic fluid is suddenly stopped, the Maxwell model predicts that the shear stress will decay exponentially according to the relationship
structures to lubricate passage of particles past one another. Many expressions, theoretical and empirical, have been proposed over the years for relating shear rate to shear stress for pseudoplastic and dilatant systems. The simplest and most widely used of these equations is an empirical power function of the form 7
= K(du/dy)”
(2)
where n and K are constants. Equation 2, which defines the behavior of SOcalled “power-law” fluids, closely approximates the behavior of many real fluids over rather wide ranges of shear rate. In accordance with the specified viscosity characteristics, n < 1 defines the domain of pseudoplasticity, whereas n > 1 defines the domain of dilatancy. When n equals unity the power law specifies Kewtonian behavior, K becoming the Newtonian viscosity. The exponent n. termed the flow behavior index, is a measure of the departure from Newtonian behavior in that the further n is removed from unity, either above or below, the more pronounced become the non-Newtonian characteristics.
Time-Dependent Systems Time-dependent non-Newtonian fluids exhibit a reversible change in viscosity at constant temperature with duration of shear. This definition excludes irreversible changes due to permanent alteration of particles or molecules within the fluid, and is limited to steady-state effects in contrast to the unsteady-state time effects associated with viscoelasticity. The classification of “thixotropic” is commonly used to denote those timedependent materials for which the viscosity decreases with the duration of shear. “Rheopectic” fluids display the opposite type of time-dependent behavior-i.e., rheir viscosities increase with duration of shear. The physical mechanisms responsible for thixotropy and rheopexy probably are very similar to those causing the analogous cases of pseudoplasticity and dilatancy, with the exception that a significant time period is required to reach equilibrium. The severity of the time-dependency characteristic reflects the extent to which the system is displaced from the equilibrium state. Typical examples of thixotropic systems can be found among paints and printing inks. Rheopexy has been observed primarily in suspension systems such as gypsum in water.
Viscoelastic Systems With a purely viscous fluid, energy imparted to the system as work during
840
Pseudoplastic
Newtonian 7/70
Shear Rate,d/y/
dy
Arithmetic shear diagram nowNewtonian fluids
for classical
These relationships are used as a criterion for fluid classification
shear becomes dissipated as heat. In contrast, energy imparted to an ideally elastic material during strain becomes stored as potential energy, and may be recovered upon removal of the stress. Viscoelastic fluids distribute imparted energy according to both mechanisms. Viscoelasticity, characteristic of longchain molecules, is fostered by elastic elements linked together by the frictional restraints of molecular entanglements and strong intermolecular forces. Polymer chains that possess coiled configurations at conditions of rest are placed under tension and extended when subjected to shear, thereby storing potential energy. The strain caused by the extension of these elastic elements becomes superimposed upon the motion of viscous shear. Polyethylene melts and Napalm incendiary gels offer typical examples of viscoelasticity. The simplest rheological relationship that has been used to describe the behavior of viscoelastic systems is based on Newton’s viscosity law for the viscous component of behavior and Hooke’s law for the elastic component.
+ x(d7/de)
= P
(d~dd~)
(3)
Such behavior, defining a model known as a Maxwell body, may be likened mechanically to the response of a spring in series with a dashpot. For a situation of steady and uniform flow where the time derivative of shear stress (&/de) is zero, the elastic component of Equation 3 disappears and the Maxwell model reduces to the Newtonian case, demonstrating a basic characteristic of viscoelastic behavior. Under laminar flow
INDUSTRIAL AND ENGINEERING CHEMISTRY
= e@/&
(4)
The parameter A, called the “relaxation time,” represents a characteristic time constant for the stress decay, and as such is indicative of the degree of elasticity of the system. By the analogy to a spring in series with a dashpot, it can be shown that X is equal to the ratio of the viscosity to the elastic shear modulus. The initial division of fluids into the categories of viscous, time-dependent, and viscoelastic represents an oversimplification of fact. Nature has not been so kind as to keep these effects separate, and many fluids may be found that manifest several of the distinguishing features. It is not intended to imply that the physical phenomena discussed in this brief review are complete. Other mechanisms have been and will continue to be recognized. Similarly, the rheological models mentioned here, such as the power law and the Maxwell element, are not the only models available to today’s workers, but rather they represent some of the simpler models which illustrate the peculiarities of non-Newtonian systems. RECEIVED for review January 12, 1959 ACCEPTEDMarch 12, 1959
CORRECTION
Vapor-Liquid Equilibria. Microsampling Technique Applied to a New Variable Volume Cell I n the article on “Vapor-Liquid Equilibria. Microsampling Technique Applied to a New Variable Volume Cell” [T. J. Rigas, D. F. Mason, and George Thodos, IND. ENG. CHEM. 50, 1297 (1958)], all pressures concerned with the experimental investigation should be increased by 15 p.s.i.a. This change brings the data of this study and those of Sage, Hicks, and Lacey within the reported experimental error.
