rheology of some polymeric systems

Ind. Eng. Chem. Prod. Res. Dev., Vol. 17. No. 2, 1978 95

TECHNICAL REVIEW

Rheology of Some Polymeric Systems

Hershel Markovitz’ and Guy C. Berry

Department of Chemistry, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213

Hershel Markovitz is Pro- fessor of Mechanics and Polymer Science in the De- partment of Chemistry, Carnegie-Mellon Uniuersity. He received his B.S. Chem. degree from the Uniuersity of Pittsburgh and his Ph.D. in Phvsical Chemistrv from Columbia Uniuersitv. He joinei Mellon Instituie’in 1949as a Fellow and became Senior Fellow in 1951. Wi th the creation of Carnegie- Mellon Uniuersity, he became Professor in the merged institution, initially, i n the Center for Special Studies. He was a Visiting Lecturer at the Johns Hopkins Uniuersity (1958-1959) and a Fulbright Lecturer (1964-1965) at Weizmann Institute, Israel. He has served as Chairman of the American Physical Society’s Diuision of High Polymer Physics and as President of the Society of Rheology (1969-1971) and was awarded that Society’s Bingham Medal (1967). He is Associate Editor of the Journal of Polymer Science. With B. D. Coleman and W . Noll, he co-authored “The Visco- metric Flow of Non-Newtonian Fluids”. He edited “Polymers in the Engineering Curriculum”, and (with E. F. Casassa) “Polymer Science: Achieuernents and Prospects”. He wrote and was principal i n the movie film “Rheological Behauior of Fluids”. He has about 50 publications in rheology and polymer physics.

Guy C. Berry is Professor of Chemistry and Polymer Science in the Department o f Chemistry, Carnegie- Mellon Uniuersity. He received his B.S. Chemical Engineering, M.S. Polymer Science, an.d Ph.D. decrees from the Uniuersity o i Michigan. He joined Mellon Institute in 1960 as a Fellow, becoming a Senior Fellow in 1964, and later, Professor in Carnegie-Mellon Uni- versity formed by merger of Mellon Institute with Carnegie Institute of Technology. He was a Visiting Professor at the Uniuersity of Tokyo (1973) under the auspices of the Japan Society for the Promotion of Science. His research interests include the physical chemistry of polymers in dilute solution and the rheology of polymers and their concentrated solutions. He has about 50 publications in these areas, including reuiews co-authored with T. G Fox and E. F. Casas- sa.

0019-7890/78/1217-0095$0.100/0 0 1978 American Chemical Society

96 Ind. Eng. Chem. Prod. Res. Dev., Vol. 17, No. 2, 1978

Introduction

The knowledge and control of rheological behavior of polymeric materials is required in many applications. Awareness of this behavior and how it changes may be important throughout the life cycle of the material from syn- thesis through processing and use until disposal. The rheological properties depend on a number of parameters, some of which may be controlled to achieve the desired results within the imposed constraihts; for example, choice of polymer (chemical species, molecular weight and its distribution, re- activity, degree of branching, components of mixtures, etc.), type and amount of additives (particulate solids, low molecular weight fluids, etc.), and environmental conditions (temperature, pressure, fluid surroundings, etc.). If it were known how the rheological properties varied with these parameters, it would be possible to do a better job of optimizing the se- lection or design of the material and the processes employed in handling it. Unfortunately, our knowledge is very incom- plete. One reason for this situation is the complexity required for the quantitative description of the rheological behavior. Unlike the common situation with nonpolymeric materials where the properties can be described in terms of constants (elastic constants for solids, viscosity for fluids), polymeric materials generally require sets of functions (viscosity as a function of rate of shear, the creep compliance, etc.) because of the time-dependent and nonlinear character of this behavior. In the current state of development, it is still not known what functions provide an adequately complete description. Furthermore, for the functions which have been determined experimentally, only a limited range of the independent variables has been covered and the variation of these functions with the material and environmental parameters has been incompletely studied.

The functions that have been investigated the most thoroughly are v ( K ) , the rate of shear dependence of the viscosity in steady shearing flow, and J ( t ) , the shear creep compliance, or equivalent linear viscoelastic property at low stress levels. Much less is known about phenomena such as the normal stress effect, transient behavior in the nonlinear region, and yield phenomena in multiphase systems. Rather than evalu- ating the needed functions it is a common practice to perform one-point measurements (such as 70, the zero-shear viscosity) or some standardized test (such as the melt index).

With respect to material parameters, perhaps most is known about the effect of molecular weight in the case of flexible coil, linear macromolecules. Much sparser is our un- derstanding of the variation of behavior with molecular weight distribution, chain stiffness, ionic charge, branching, strength of intermolecular bonds, nature and concentration of added low molecular weight fluids, and size, shape, and type of added particulate matter.

That so many lacunae remain is not surprising in view of the complexity of the rheological behavior, the many material and environmental parameters, the difficulty in obtaining well-characterized material, the limited range of most available instruments, the shortage of personnel with appropriate appreciation of both experiment and theory, and the limited funds devoted to this area of research.

Of course, it would hardly be justified to make a “complete” set of measurements for all materials. What is required is a study on selected materials, thorough enough to establish the basic patterns of behavior. This is then to be supplemented by a relatively few experiments which should be sufficient to deduce the desired properties of other materials of interest. In carrying out such a program, use is made of theoretically sound interrelations among various properties (for example, the calculation of stress relaxation modulus G ( t ) from the creep compliance J ( t ) or empirical deductions from previous

experimental results (such as the calculation of a material function at one temperature or concentration from that a t another by a superposition procedure).

In this brief review, we will discuss some basic experiments, with emphasis on both theoretical and practical (particularly with regard to sealants, adhesives, and coatings) interest. As to materials, we put special emphasis on two-component, two-phase systems, such as suspensions of hard particles in polymeric fluids. Many basic questions concerning the rheology of these classes of polymeric materials remain open. Some materials of these types exhibit solid-like behavior under some conditions while still maintaining their solubility and their ability to flow under other conditions. The rheological phenomena to be discussed are linear and nonlinear viscoelasticity in shear (creep and recovery) and nonlinear steady flow behavior (dependence of viscosity on rate of shear). Special attention is paid to the nature of yield phenomena which have been reported. The usefulness for these materials of various empirical interrelations such as time- temperature superposition, found valid for linear polymers and their solutions, will be discussed. Viscoelasticity studies on suspensions have been very rare, as pointed out in a recent review (19).

Because of the extensive literature, we shall restrict the area covered and the literature cited to those cases which we con- sider particularly pertinent to our topic. We divide this review into three categories: the rheology of (A) suspensions in Newtonian fluids, (B) polymeric fluids, and (C) suspensions in non-Newtonian fluids. The rheological properties and interrelations among them will be discussed as needed.

A. Suspensions in Newtonian Fluids There is a considerable amount of literature dealing with

the theory and experimental studies of the Newtonian viscosity of suspensions, much of it devoted to testing the Ein- stein relation (11) for dilute suspensions of rigid spherical particles and to extending it to higher concentrations and to particles of other shapes. The subject has been frequently reviewed ( I , 6, 13,17,18,48). A rigorous theory for nondilute suspensions is not yet available and the various approximate theories are not in agreement among themselves or with experiment (2,14,41,49).

Studies of the non-Newtonian flow behavior of suspensions of spheres have mostly been focused on the viscosity function v r defined as U / K , which reflects the nonlinear dependence of the shear stress u on the rate of shear K in steady simple shearing and in related steady viscometric flows.

Krieger and his co-workers (20-22,40) have made perhaps the most thorough investigation of ix for suspensions of non- interacting monodisperse spherical particles in Newtonian fluids. Using an argument based on dimensional analysis they introduced a time constant

7, = qma3/kT (1)

where pm is the viscosity of the medium, a is the radius of the particles, k is the Boltzmann constant, and T is the absolute temperature. Their data fell on a single curve when plotted as vK., the relative viscosity function ( q K ~ = q K / t m ) , against a reduced shear stress ur = ua3/kT (or equivalently a reduced shear rate K~ = K T ~ T O R ) . From a highly simplified model based on the formation and destruction of doublets, they obtained the formula (21)

(2)

where A is a numerical constant of the order of unity and, as the notation indicates, OR and v m ~ are the limiting values of v K ~ at low and high shear rates, respectively. This expression

TKR- l l m R - 1 - VOR - 4-R 1 4- A I

Ind. Eng. Chem. Prod. Res. Dev., Vol. 17, No. 2, 1978 97

The current status of some basic aspects of the rheology of polymeric materials, particularly dispersions, is reviewed. The rheological properties emphasized are the shear-rate dependent viscosity and viscoelasticity in shear, particularly as investigated by creep and recovery experiments in both the linear and nonlinear range. Ex- periments bearing on the existence of a yield value receive special attention. The time-temperature Superposi- tion procedure, originally proposed for linear viscoelasticity properties, and various extensions thersof to nonlinear rheological behavior, is useful for some polymeric materials but not for others.

is a good representation of the data obtained by Krieger and co-workers (20, 40) on dispersions of monodisperse, nonin- teracting polystyrene spherical particles in carefully chosen aqueous and organic Newtonian fluids for a range of solvents, particle concentrations, and sizes. No evidence for a yield value was found.

For other types of suspensions (e.g., those with appreciable coulombic interactions (22) ) , including some of commerical interest, deviations from Newtonian behavior are often found (32). Shear stress data in steady-state flow are often represented by the empirical Casson’s equation (7)

&2 = + qc’/2K’/2 ( K > 0) (3)

over the range of K covered (22 ,30) . Here qc and ay are empirical parameters. If the equation is assumed to hold over the entire range of shear rate, then cry is a yield stress, with no flow taking place a t shear stresses less than ay, and vc is the limiting viscosity q m at high rates of shear. When data are obtained at low enough rates of shear, it is frequently found (30, 4 7 ) that eq 3 is no longer satisfactory. In fact, the material flows at shear stresses lower than ay (or corresponding constant in some other empirical equation with an implied yield value), and indeed it flows at the lowest shear stresses tested. In spite of this, it is a very common practice to perform experiments over a limited range of shear rates, fit the data with some equation such as eq 3, and to refer to an empirical constant (such as uy) thus evaluated as “the yield stress”. Almost all “yield stresses” reported in the literature have been obtained by such extrapolation procedures and therefore have questionable physical significance. There are materials which appear to have a true yield stress; these will be discussed below.

We know of no viscoelastic measurements on colloidal suspensions in Newtonian media.

B. Polymer Fluids The rheological properties of polymers and their concen-

trated solutions have been more thoroughly investigated than any other class of materials.

With these materials, too, the most widely studied rheological property is qx. In addition, there is considerable literature on their viscoelastic properties, especially over the linear region of response but with growing attention to the nonlinear region. In our laboratories, the creep and recovery experiment is used to characterize this property.

In a creep experiment a shear stress a is suddenly applied to the material at rest in equilibrium and then held constant; one observes the shear strain y,(t) as a function of the time t after the stress is imposed. The property that characterizes the material in this experiment is the shear creep compliance J , ( t ) defined by the formula

J A t ) = r , ( t ) /u (4) If the applied stress u is small enough y,( t ) is proportional to a, J A t ) is independent of u, and the material obeys the laws of linear viscoelasticity. The limiting value of J,( t ) at low a is the linear shear creep compliance J ( t ) of linear viscoelasticity. If the material is a fluid, a steady state is reached in the

creep experiment with a constant rate of shear, K = dr,,(t)/dt, being attained. The steady-state viscosity q K is the ratio U / K .

The recovery experiment consists of removing the stress after a creep experiment has proceeded for a time S when the strain is r,(S) and noting the strain r,(S + 0) at a time 0 after the removal of the stress. The term recovered strain ~,,R(S, 0) is defined as

YoR(S9 0) = y u ( S ) - y u ( s + 0) (5) The material is characterized by the recovery function R,(S, 0 ) defined by the relation

Of particular interest is the value R ,,( a, a) of the recoverable compliance in an experiment where the creep has endured until a steady-state rate of shear K has been achieved and the recovery has been allowed to continue until equilibrium is restored. The name steady-state compliance and the symbol R , is given to flu(.., a), At low u, R , approaches the limiting value Ro (designated as Je0 in some of the literature of linear viscoelasticity), the linear steady-state compliance. It is a good simple measure of the elastic character of the fluid. I t is related to a number of other properties ( 1 5 ) , e.g., the normal stress function (8,29), the shift factor in superposition (271, and other linear viscoelastic functions ( 2 5 ) .

When the linear viscoelastic functions for some materials are graphed against time on logarithmic plots, it is found that the curves obtained a t different temperatures have similar shapes and can be superposed rather well by shifting along the coordinate axes. For the linear shear creep compliance, J ( t ) , for example, such a superposition procedure can be indicated as

where J ( t ; T ) is the function J ( t ) at temperature T, U T is the shift factor needed along the time axis to superpose the data at temperature T o n those at the reference temperature To, and bT is the vertical shift factor needed along the compliance axis. If the superposition procedure is valid over the whole time scale, the material is said to be thermorheologically simple and it can be shown that the shift factors can be expressed in terms of the viscoelastic constants (27)

bT = Ro(T)/Ro(To)

Here T ~ , defined by the equation

TC = 7 8 0 (10)

has the units of time and, as will be discussed below, is a useful time constant in describing time dependent properties of the material. The shift factor U T changes many orders of magnitude over the temperature range of interest; bT, on the other hand, is very insensitive to temperature, and frequently may be considered equal to unity.

I t should be noted that, while many papers and books give the contrary impression, thermorheological simplicity is not


1 1 I

I I I I

n Y T 1 A \-

r r

b

I I \

- 3 -2 - 1 0 1 L o g K (sec:')

Figure 1. Rheological data on a solution of polyisobutylene in cetane a t several temperatures: (a) R, vs. shear rate; temperature (in O C ) equal to 18.9, d; 23.8, C- ; 33.4, Q; 46.7, 9; 55.4,p; 63.1, 0; (b) qc vs. shear rate, symbols as defined above.

0

0 E \

a' 0 0 -I

-1

F

I I I 0

-1 1 -3 -2 - 1

L o g K / K ~

Figure 2. Reduced plots of the rheological data of Figure 1 on a solution of polyisobutylene in cetane at several temperatures. The symbols are identified in the caption of Figure 1: (a) reduced steady-state compliance vs. K / K O , where 3.62/~0 is equal to the time constant T~ defined in the text; (b) the reduced flow curve q, vs. K / K O = 0 . 2 7 6 ~ ~ ~ .

a universal law of mechanical behavior. While its failure with glassy and crystalline polymers with their secondary dispersions is widely recognized, deviations in the case of polymer fluids are not generally appreciated. Careful experimentation in our laboratories has demonstrated such departures from superposition in the case of polystyrenes (42) , particularly for low molecular weight samples (46 ) . In the former case a more complicated superposition procedure was found adequate ( 4 2 ) .

Although no material has been proved to be exactly thermorheologically simple (12a) , superposition remains a useful approximation in many cases. In fact, this idea has been greatly extended, with different degrees of success, in many directions (10, 26, 28, 39), for example, to the variation of properties with compositional and structural parameters such

as molecular weight, concentration and nature of solvent in solutions, particle size in suspensions ( 2 0 ) , and degree of cross-linking in vulcanized rubber ( 4 3 ) .

Superposition has also been found useful for other rheological functions, especially for ax. Curves obtained a t various temperatures for a given polymer or polymer solution can often be superposed by shifting along the horizontal axis in logarithmic plots of a,/oo against K (10, 26,391, that is

a,(T)/ao(T) = a , a , ( T o ) / ~ o ( ~ o ) (11)

where it can be shown under some circumstances (28 ) that UT is the same shift factor which is applicable to linear viscoelasticity, eq 8. It is sometimes convenient to rewrite eq 11 as (5 , 15)

7 = V O Q ( 7 C K ) (12)


1 0 . ~ ~ , , , , , , , , , , . , , , , , , , , , , , , , , , , , I , , , , ,

10-3 10-2 10-1 1 10

? o R o K

Figure 3. Rheological data on solutions of poly(n-methylstyrene) in chloronaphthalene. The circles give R , vs. ?&Oh, whereas the dashed curves give R o ( m , K - ~ ) vs. ?&OK.

where Q is a function independent of temperature with Q(0) = 1 and T~ is a shift factor ( 5 ) . On the basis of both molecular (15) and phenomenological (28) theory, it has been shown that the shift factor rc is determined by the viscoelastic constants Ro and 70 as indicated in eq 10. Examples of data and the corresponding reduced plot obtained in our laboratory are shown in Figures I b and 2b, respectively (50).

It is to be noted that this superposition procedure of dealing with the temperature dependence of qIK differs basically from the commonly used AE,,, activation energy for viscous flow, method. This latter method is a direct extension of a method used for dealing with the temperature dependence of New- tonian viscosity; it fails to take into account the fact that, with non-Newtonian behavior, one is dealing with a function rather than with a constant. As a result, there are frequent debates about the relative virtue of calculating AEv a t constant K or at constant u. If superposition is applicable, the temperature dependence is correctly expressed by the eq 11 or 12 with the appropriate shift factors. Actually, it can be shown that, due to the temperature insensitivity of b T , the constant u value of U, is a useful parameter if superposition is valid. However, if superposition is not valid (as is the case with some suspensions), a more complex analysis is required and use of either AEq can have very little predictive value.

Data obtained in our laboratory indicate that superposition can also be applied to R , (50). Logarithmic plots of the steady-state compliance R , vs. K may be superposed by shifting. This fact can be expressed mathematically as

R , = ROr(TCK)

where r is a function independent of temperature such that r ( 0 ) = 1 and rC is the same, eq 10, as it is for the viscosity function. See Figures l a and 2a.

Examination of Figures 2a and b indicates that nonlinearity becomes important for all the data for both the steady-state shear stress behavior and the steady-state compliance (i.e., Q and r deviate significantly from unity) at about the same value of T ~ K = I ~ & O K . Thus the combination of linear viscoelastic constants, 1f&0, can be used as a criterion for the value of K where significant deviations from linear behavior begin.

To some extent, data obtained on a set of high molecular weight, narrow distribution polymers which differ in molecular weight also can be superposed except for data a t high values of T,K. Thus the Q and r functions are independent of molecular weight for such polymers except a t high values of T ~ K . However, data obtained on polymers differing in molecular weight distribution cannot be superposed.

Recent results (4,15) indicate that, for narrow distribution

polymers and their concentrated solutions, the dependence of Ro on M and concentration of polymer c is represented by the relation

2 M 5 cRT

R o = - - ( E < k )

- - -- 2 p k M e ( E > k ) 5 c2RT

where p is the density of the material, M e is the molecular weight between entanglements, R is the gas constant, T i s the absolute temperature, k is a constant (= lo) , and E is the en- tanglement density cMlpM,. It is to be noted that as the temperature varies rc changes in proportion to 70/T as seen from eq 10, whereas T~ changes in proportion to qm/T.

Another empirical observation (4 ) is a correlation of R o ( m , 6') with R,. Here Ro(m, 6') represents the amount of recovery a t a time 6' after steady shearing flow has been stopped (in the linear viscoelasticity region) and R , represents the amount of recovery at an infinite time after a steady shearing flow with rate of shear K has ceased. For several concentrated polymer solutions, it has been found experimentally that Ro( m , K - ~ )

is close to R , for I~OROK < 10; Le., the total amount of recovery (after an infinite time) following a steady shearing a t a rate of shear K is related to the amount of (partial) recovery at a time equal to K - ~ if the stress during creep is low enough to be in the linear region of behavior. See Figure 3.

The concentrated polymer solutions of interest here do not exhibit a yield stress, but what might be considered an anal- ogous phenomenon has been seen. A total strain criterion has been found for the onset of appreciable nonlinear behavior in creep (50). The effect is seen in Figure 4, which shows J , ( t ) plotted as a function oft for various values of u. It is found that J , ( t ) is essentially independent of u up to a time tu* which decreases with increasing u. However, significant deviation from linearity in each case occurs when the strain attains the same critical value, y*.

A widely used empirical rheological rule is the Cox-Merz relation (9) between TJ* and 11f*(w)1, the magnitude of the complex dynamic viscosity of linear viscoelasticity

I f K = IIf*Ml (15)

which equates q K a t the rate of shear K with I ~ f * ( w ) I at an angular frequency w equal to K . This relation is found to hold for many polymers and their concentrated solutions.

Branched Polymers. For a long time, it had been eqpected that branched polymers (and their solutions) would have lower viscosities than the corresponding linear polymers of the same molecular weight. I t has been found, however, if the branches are long enough, that the zero shear viscosity can be higher than the linear polymer of the same molecular weight (3, 16,24,25). As a result, for such branched polymers, d log v/d log M can have a value considerably greater than the 3.4 found for linear polymers (16,24, 31 ). In a similar fashion, Ro is greatly enhanced in macromolecules with long branches (16).

Such branched polymers (cis-polybutadiene (23) and po- lydimethylsiloxanes (44,45)) also show nonlinear viscoelastic behavior at low stresses and give indications of the existence of a yield value. No permanent deformation is observed in creep experiments at low stresses but is found at high stresses. These materials thus have the potentially useful properties of being soluble but yet acting like solids if they are subjected only to low stresses while being capable of flow at high stresses.

C. Suspensions in Non-Newtonian Fluids Since only fragmentary work has been done on the rheolo-

gical properties of colloidal suspensions of monodisperse


n 2 X

c - b

3

4 u = 9 5 5 x 1 0 3 a = 6 0 6 X 1 0 3

* u = 3 l 6 X 1 O 3 0 u = 3 3 2 X 1 0 2

0 5 0 100 150 200 250 300 I I I I I

5 00 1000 1500 2000 2500 I (sec)

37

36

3 5

3 4

3 3

3 2

I 1

1

Figure 4. The creep as a function of time for a solution of poly(a-methylstyrene) in chloronaphthalene a t four stress levels.

l o g K

log 2!L 71

Figure 5. An example of possible rheological behavior of suspensions of solid particles dispersed in a polymeric fluid. The filler concentration increases for the curves - - -, - - -, and -.

spherical particles in non-Newtonian fluids, one is still not certain of the nature of the function v n over the entire range of K of interest, much less how v K depends on the size or concentration of particles, temperature, and the properties of the polymeric medium. An important area of uncertainty is the region of low shear stress. I t is common practice to report yield stresses on the basis of extrapolations of the type mentioned in our discussion of eq 3, but it is not often known whether or not the suspensions actually have yield values.

Some work (35,36) on suspensions suggests that 7, might have the appearance shown in Figure 5a for various values of 4, the volume fraction of suspended particles. The high K

section of the curves may be superposed by a translation of these logarithmic plots. Thus a reduced plot in the high K region can be obtained if vK/vf is plotted against voPRop where qoP and RoP are the values of vo and Ro for the polymer medium and where qf is the value of the viscosity in the plateau region. See Figure 5b. The low K section of the curves is mainly

conjectural since this region is not often observed because few instruments are capable of making the requisite measurements. I t is rarely known whether, as K approaches zero, vn approaches a limit (110) or whether there is a yield value with vr increasing, inversely proportional to K .

Onogi and his co-workers (30,37,38) have made the most extensive rheological study on suspensions in polymeric fluids. On their suspensions of 120-nm particles of cross-linked polystyrene in a polystyrene solution, they determined both qK and viscoelastic properties (37, 38). For the latter, they employed sinusoidal deformations y(t) = yo sinwt and expressed the properties in terms of the dependence of G’(w) and G ” ( w ) the shear storage and loss moduli, respectively, and related nonlinear functions on the angular frequency w and the shear amplitude yo. Although their v n data could be fit by the Casson equation (3) for much of the K range, they showed clearly that the extrapolated cry was not actually a yield stress (30). Steady flow did occur a t shear stresses below the so- called “yield stress” cry. In fact, a zero shear viscosity 70 was determined in some cases. However, they were not able to determine Ro. Their viscoelasticity measurements did not extend to low enough frequencies for the limiting value of G‘(w)/w2 to be determined. Thus the value of Ro could not be obtained from the relation (29)

Unlike the case with polymer solutions, plateaus were found at low K in logarithmic O-K plots for steady shearing flow and at low w in G”-w plots for nonlinear viscoelastic data at higher values of yo (30). Such plateaus are cited as evidence for yield stress phenomenon in other papers with less extensive data. They also found that the Cox-Merz relation equation (14), which seems to hold for many polymers and their solutions, does not hold for their suspensions (30,37).

It is not to be inferred that yield stresses are never observed. Indeed, they are. But here, too, discretion must be exercised in prescribing “the” yield stress. As we have observed in some recent experiments on solutions of stiff molecules (51) and on a proprietary suspension, where steady-state shearing experiments in which low rates of shear are imposed, essentially a constant shear stress ( U A ) is found over a wide range of K (vx


was inversely proportional to K ) . This is just the behavior ex- pected for a material with a yield value of UA. However, when creep experiments were performed at various low levels of stress, the material appeared to behave as a viscoelastic solid up to values of u which were higher than UA. Thus, a given constant value of stress can correspond to two different steady states (a constant strain or a constant rate of strain) depending on how that stress is reached.

The creep and recovery experiment is well suited for dis- tinguishing between a "true yield stress" from a so-called "yield stress" UY which is most often found by extrapolation of data obtained in steady state flow. In the creep experiment, a given shear stress is imposed. By doing a recovery experiment, it is possible to determine whether a permanent deformation (viscous flow) has occurred or not. Thus, in the ordinary routine of creep and recovery studies, it can be determined whether the material has a true yield stress, UT, that is a highest shear stress which can be applied to the unstressed material without causing an unrecoverable deformation. If there is a yield stress, the rheological character of the material for stresses below UT remains to be determined. Presumably it will be characterized by a shear modulus if elastic or by a shear creep compliance if it has the character of a viscoelastic solid as occurred in the case, mentioned above, of solutions of stiff molecules. Such measurements are rarely made.

In addition to uy and UT there is another stress that can be associated with a yield stress. As indicated above, it is found that the material flows at shear stresses that are less than the value, uy, found by extrapolation using Casson's (or some other) relation. However, there exists a shear stress UA (<uy) which is the lowest shear stress under which flow can take place during experiments where the shear rate is imposed. Experimentally UA can be determined from data obtained at low rates of shear as mentioned above. In the literature, it is customary to assume that UA = UT. For some materials, as indicated above it appears that UA < UT. Thus in the region of u, UA < u < UT, two different steady states (constant strain or constant rate of strain) are possible. Viscoelastic experiments need to be performed to better characterize these ob- servations quantitatively.

With regard to temperature dependence of rheological properties, the question of time-temperature superposition needs to be studied, e.g., to determine whether logarithmic plots such as those of qK/q0 and of RJRo vs. K (and similarly for J , ( t ) and R,(s, t ) vs. time) for various temperature can be superposed to give "master curves". Onogi et al. (37) found that they could superpose such plots of q,., G'(w), and G"(w) for their data on colloidal dispersions in a polystyrene solution. However, their data covered only a limited range of o and K ,

over which q'(w) for the solution was independent of w and V ( K ) varied only slightly with K . We suspect that, when a broader range of the K and time scale are covered, rheological simplicity will not occur since the long time response probably involves different processes from the short time response which is dominated by portions of the polymer molecules alone. If this is indeed the case, it may be possible to represent the various rheological functions as the sum of two components, each of which is thermorheologically simple. Such a scheme has proved useful previously in dealing with other thermorheologically complex materials (33, 3 4 , 4 3 ) .

Literature Cited (1) Batchelor, G. K., Ann. Rev. FluldMech., 6, 227 (1974). (2) Batchelor, G. K., Green, J. T., J. NuidMech., 56, 401 (1972). (3) Berry. G. C., Fox, T. G, Adv. Polym. Sci., 5, 261 (1968). (4) Berry, G. C., Hager, B. L., Wong, C.-P., Macromolecules, I O , 361 (1977). (5) Berry, G. C., Wong, C.-P., J. Polym. Scl., Polym. Phys. Ed., 13, 1761

(6) Brenner, H., "Progress in Heat and Mass Transfer," p 89, W. R. Schowalter,

(7) Casson, N.. in "Rheology of Disperse Systems," p 84, C. C. Mill, Ed., Per-

(8) Coleman, B. D., Markovitz, H., J. Appl. Phys., 35, l(1964). (9) Cox, W. P., Merz, E. H., J. Polym. Sci., 28, 619 (1958). (10) DeWitt, T. W., Markovitz, H., Padden, F. J., Jr., Zapas, L. J.. J. ColloidSci.,

(11) Einstein, A., Ann. Phys., [4] 17, 549 (1905); 19, 289, 371 (1906); 34, 591 (191 1); "Investigations on the Theory of the Brownian Movement," p 36, R. Furth, Ed., Dover, New York, N.Y., 1956.

(12) Ferry, J. D., "Viscoelastic Properties of Polymers," Wiley, New York, N.Y., 1970; (a) Chapter 1 1

(13) Frisch, H. L., Simha, R., in "Rheology," Vol. 1 , p 525, F. R. Eirich, Ed., Academic Press, New York, N.Y., 1956.

(14) Goodwin, J. W., ColloidSci., 2, 246 (1976). (15) Graessley, W. W., Adv. Polym. Sci., 16, l(1974). (16) Graessley, W. W., Masuda, T., Roovers, J. E. L., Hadjichristides, N., Mac-

(17) Happel, J., Brenner, H., "Low Reynolds Number Hydrodynamics," Chapter

(18) Hermans, J. J., Ed., "Flow Properties of Disperse Systems," Chapters 1

(19) Jeffrey, D. J., Acrivos, A,, AlChEJ., 22, 417 (1976). (20) Krieger, I. M., Adv. Colloidlnterface Sci., 3, 1 1 1 (1972). (21) Krieger, I. M., Dougherty, T. J., Trans. SOC. Rheol., 3, 137 (1959). (22) Krieger, I. M., Eguiluz. M., Trans. SOC. Rheol., 20, 29 (1976). (23) Kraus, G., Gruver, J. T., J. Appl. Polym. Sci., 9, 739 (1965). (24) Kraus, G., Gruver, J. T., J. Polym. Sci. A, 3, 105 (1965). (25) Long, V. C., Berry, G. C., Hobbs, L. M., Polymer, 5, 517 (1964). (26) Markovitz, H., Trans. SOC. Rheol., 1, 37 (1957). (27) Markovitz, H.. J. Phys. Chem., 69, 671 (1965). (28) Markovitz, H., J. Polym. Sci., Polym. Symp., 50, 431 (1975). (29) Markovitz, H., Coleman, B. D., Adv. Appl. Mech., 8, 69 (1964). (30) Matsumoto. T., Hitomi, C., Onogi, S., Trans. SOC. Rheol., 19, 541

(31) Mendelson, R. A., Bowles, W. A., Finger, F. L., J. Polym. Sci. PartA2, 8,

(32) Mewis, J., Spaull. A. J. B., Adv. Colloid lnterface Sci., 6, 173 (1976). (33) Murray, A. D., Markovitz. H., J. Polym. Sci., Polym. Phys. Ed., 12, 587

(34) Nakayasu, H., Markovitz, H., Plazek, D. J., Trans. SOC. Rheol., 5, 261

(35) Nicodemo, L., Nicoiais, L., J. Appi. Polym. Sci., 18, 2809 (1974). (36) Nicodemo, L., Nicolais. L., Landel. R. F., Chem. Eng. Sci., 29, 729

(37) Onogi, S., Matsumoto, T., Warashina, Y., Trans. SOC. Rheol., 17, 175

(38) Onogi, S., Masuda, T., Matsumoto, T., Trans. SOC. Rheol., 14, 275

(39) Padden, F. J., DeWitt, T. W., J. Appl. Phys., 25, 1086 (1954). (40) Papir, Y. S., Krieger, I. M., J. Colloid lnterface Sci., 34, 126 (1970). (41) Peterson, J. M., Fixman, M., J. Chem. Phys. 39, 2516 (1963). (42) Plazek. D. J., J. Phys. Chem., 69, 3480 (1965). (43) Plazek, D. J., J. Polym. Sci., Part A2, 4, 745 (1 966). (44) Plazek, D. J., personal communication. (45) Plazek, D. J., Dannhauser, W., Ferry, J. D., J. Colloid Sci., 16, 101

(46) Plazek, D. J., O'Rourke, V. M., J. Polym. Sci., Part A2, 9, 209 (1 97 1). (47) Rosen. S. L., Rodriguez, F., J. Appl. Polym. Sci., 9, 1601 (1961). (46) Rutgers, R., Rheol. Acta, 2, 202, 305 (1962). (49) Vand. V., J. Phys. ColloidChem., 52, 277 (1948). (50) Wong, C.-P., Berry, G. C., Polym. Prepr., Am. Chem. SOC. Div. Polym.

Chem., 15 (2), 126 (1974). (51) Wong, C.-P., Berry, G. C., in "Structure-Solubility Relationships in Poly-

mers," p 71, F. W. Harris and R. B. Seymour, Ed., Academic Press, New York,

(1975).

Ed., Pergamon. New York, N.Y., 1972.

gamon, London, 1959.

10, 174 (1955).

romolecules, 9, 127 (1976).

9, Prentice-Hall, New York, N.Y., 1965.

and 4, Interscience, New York, N.Y., 1953.

(1975).

105, 127 (1970).

(1974).

(1961).

(1974).

(1973).

(1970).

(1961).

N.Y., 1977.

Receiued for reuieu, September 10,1977 Accepted M a r c h 6, 1978

T h i s work was supported in p a r t b y Gran t AFOSR 77-3404 from t he Air Force Off ice of Scienti f ic Research.

rheology of some polymeric systems

Documents