I
D. W. DODGE
E. I. du Pont de Nemours & Co., Inc., Buffalo, N. Y.
Fluid Systems 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
FLUID
SYSTEMS are characterized by their laminar response to applied shearing stresses. T h e simplest type of fluid behavior, that commonly known as Newtonian, is defined by a direct proportionality between shear stress, 7, and induced rate of shear, (du/dy). 7 = P
(WdY)
(1)
The constant of proportionality, p , is the fluid viscosity. By exclusion, nonNewtonian systems embrace all fluids that do not conform to Equation 1. For these systems, the ratio of shear stress to shear rate becomes an apparent viscosity which varies with the imposed conditions of shear. Obviously, from a mathematical standpoint, the definition of non-Newtonian behavior encompasses a far wider realm of possible rheological performance than does that of Newtonian behavior. Non-Newtonian systems as recognized today may be grouped roughly into three major categories according to the distinguishing feature of their anomalous behavior: (1) viscous systems-fluids for which shear rate depends only on the imposed shearing stress; (2) timedependent systems-fluids for which shear rate depends on the duration of shear as well as the level of stress; and (3) viscoelastic systems-fluids that exhibit both viscous and elastic features. The main characteristics of these several types of behavior are reviewed briefly below as an introduction to the topic of non-Newtonian technology.
Viscous Systems The shear stress-shear rate relation-
ships for viscous non-Newtonian fluids traditionally have served as a criterion for further fluid classification. The figure compares the shear diagrams for Bingham plastic, psuedoplastic, and dilatant systems, representing the most widely used viscous subclassifications. The Bingham plastic model, like the Newtonian case, has a simple linear shear stress-shear rate relationship, but differs in that it sustains some finite shear stress before flow begins. The stress required to initiate flow is termed the “yield stress,” rY,and the slope of the shear diagram is designated as the “coefficient of rigidity,” 7. Pseudoplasticity, the most common type of nonNewtonian behavior, displays nonlinear shear diagrams (without yield stresses) wherein the apparent viscosity decreases with increasing shear stress. The slope of the tangent to a pseudoplastic shear diagram at the origin has been called the viscosity at zero rate of shear, and the slope of a straight line drawn asymptotically to the curve at large values of shear rate has been termed the limiting viscosity at infinite shear. Dilatant fluids also possess nonlinear shear diagrams with no yield stresses, but are opposite to pseudoplastics in that their apparent viscosities increase with increasing shear stress. Bingham plastic behavior has been ascribed to the presence of a threedimensional structure within the fluid while at rest which collapses completely when the yield stress is exceeded. Actually, the Bingham plastic model represents a very idealized behavior and it
is doubtful that any truly Bingham fluids exist. Some systems, however, particularly solid-liquid suspensions such as drilling muds or sewage sludges, approach this type of behavior sufficiently well to make the model useful. To promote pseudoplasticity the in, teractions between molecules and particles which give rise to the concept of viscosity must decrease with increasing stresses or shear rates. One plausible explanation lies in the idea that highly asymmetric molecules or particles, which are randomly interspersed and entangled when the fluid is at rest, tend to become disentangled and aligned in the direction of motion when placed in a shear field. The melts of most high polymers exhibit a tendency toward pseudoplastic behavior, due in part at least to this mechanism. A second possible explanation of pseudoplasticity may be found in the progressive shearing off of liquid carried by particles as envelopes of solvation, thereby decreasing their effective size with increasing shear rate. Examples of fluids which exhibit pseudoplasticity, possibly by this second mechanism, are polymer solutions and clay suspensions. Dilatancy occurs most frequently in suspensions of solids at high solids content. The explanation of dilatancy most often found in the literature pictures the dilatant system as one in which a dense packing of particles is graduallydestroyed with increasing shear rate.. The liquid phase which occupies a mini-. mum void space when the system is a t rest becomes insufficient in the expanded VOL. 51, NO. 7
JULY 1959
839
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.
