EXPERIMENTAL EVALUATION OF VISCOELASTIC THEORIES J . 0 . D O U G H T Y ’ AND D. C. BOGUE Department of Chemical and Metallurgical Engineering, University of Tennessee, Knoxuille, T e n n .
Viscoelastic theories due to Oldroyd, Pao, Bogue, and Bernstein, Kearsley, and Zapas (BKZ) are examined for their ability to correlate linear dynamic data with nonlinear viscosity and normal stress data. Data are presented for solutions of 10% polyisobutylene in Decalin and 1 2y0 polystyrene in Aroclor. It is concluded that the three-constant Oldroyd and the Pao theories do not correlate the data quantitatively, the Pao theory producing a high shear rate viscosity plateau at too low a shear rate. The Bogue and BKZ theories, both of which contain an adlustable parameter in the forms used, are reasonably successful. essential experiments necessary to characterize viscoelastic fluids are made.
h STITUTIVE equations for viscoelastic fluids attempt to Corelate the stress response of these fluids to the strain history. I n the tensor equations from continuum mechanics one has material parameters (similar to viscosity) which bring in viscous, normal stress, and elastic effects. These parameters can be rearranged into parameters with dimensions of time and stress but in continuum mechanics no effort is made to relate them to molecular properties. Rather, the objective is to interrelate flow behavior from diverse instruments and geometries. O n the other hand, molecular theories deal with the relation of rheological behavior to molecular properties but the deformations studied are simple, linear ones. Despite these gaps in the theoretical structure, the material parameters in continuum mechanics do in some way reflect underlying molecular mechanisms and thus are not completely independent of each other or of the parameters from molecular theory. One expects materials which show normal stresses in steady shear also to show elasticity in a sinusoidal deformation. Materials that are highly elastic \\ill show strongly nonNewtonian viscosities. The dynamic viscosity as a function of frequency (in a linear sinusoidal deformation) is quantitatively similar to the viscosity as a function of shear rate (in a large shearing deformation). If these various effects could be brought together in a unifying phenomenological theory, one ksould be closer to having identified basic material parameters and the basic experiments to measure them. Thus considerable effort has recently been devoted to the development of tractable viscoelastic theories. Bird and coworkers have developed differential theories similar to the Oldroyd theory discussed below (Spriggs, 1965; Spriggs and Bird, 1965; Williams and Bird, 1962, 1964). These and other rheological theories are compared in a comprehensive review article by Spriggs, Huppler, and Bird (1966). An earlier review of viscoelastic theories was made by Markovitz (1957). As discussed in an earlier paper (Bogue and Doughty, 1966), the present work has been focused on several theories with a continuum mechanics origin. They are related to molecular theories in that they can be reduced to linear viscoelasticity and thus can be made to incorporate the results which molecular researchers display in this mathematical framework. I t cannot be said that the theories incorporate a truly mechanistic (molecular) explanation of non-Newtonian viscosities, since the arguments which bring in this effect are made on a macro-
scopic or a purely mathematical level. However, having established the shear-dependent viscosity, the theories make explicit predictions for the normal stresses. I n the present Ivork four theories are examined for quantitative correlation of experimental data from two polymeric solutions. In one sentence descriptions these theories may be classified as follows: (1) Oldroyd’s three-constant theory (1950, 1958) : a reduced form of Oldroyd’s eight-constant differential theory involving a generalization of linear viscoelasticity (relations among stress, strain, and first time derivatives of them) using a convected time derivative; (2) Pao’s theory (1957, 1962) : an integral theory involving superposition of a nonlinear strain tensor, a hfax\vell-type relaxation of stress, and stress-strain behavior tracked in a rotating COordinate system; (3) Bogue’s theory (1966) : an integral theory of the form of Coleman-No11 second-order theory but with a nonlinear memory function, constructed empirically to give reasonable shear rate-dependent viscosities; and (4) Bernstein, Kearsley, and Zapas’ (BKZ) theory (1963): a n integral theory of the general form of strain enrrgy elastic theory with time relaxation introduced. More details are given in the earlier paper (Bogue and Doughty, 1966). Below, the theories are examined for their ability to correlate linear dynamic data \sith viscosity and first normal stress data in viscometric flow. Only the directly pertinent theoretical results are presented for each theory. Predictions of Theories
Oldroyd’s Three-Constant Theory. Oldroyd’s theory differs from the others in that its predictions are not expressed in terms of the relaxation spectrum but rather directly in terms of the three material constants. Thus the frequencydependent dynamic viscosity is given by
In the present work Equation 1 was used to determine the three material constants: q o , XI, and X2. The viscometric predictions of Oldroyd’s model are (Bogue and Doughty, 1966) : 1f Ti2 -~ -
Present address, Department of .\erospace Engineering, University of Alabama, Tuscaloosa, Ala. 388
l&EC FUNDAMENTALS
Some comments on the
K
2
-
AiXzf?
I n the present work Zapas’ new function is employed with the modification of a n adjustable parameter, A , that has been introduced as follows: 722
Pao’s Theory. 1962) :
-
733
0
=
(4)
T h e viscometric predictions are (Pao,
(5)
If Zapas’ functions a ( s ) and c(s) are taken to be zero, his func1 tion p ( s ) is replaced by - m ( s ) , and the parameter A is set equal 2 to
1
+ K2XZ
-, 9
one obtains a reconciliation between Equation 11 and the
new Zapas function. Using Equation 11 and m ( s ) expressed as a series of exponentials (see Bogue and Doughty, 1966), the viscometric results are :
where XH(X) d In X
L
(7)
and
J-rn+ rn
p2 =
X*H(X)d In X 1 KZXZ
I n principle a n expre5sion for 7 2 2 - 7 3 3 can be obtained, but it is not a t present avai1,ible. Pao’s and Oldroyd’s predictions contain quantities that can be obtained from linear data alone. Recently Huseby and Blyler (1966) have suggested that a factor of 2 is needed before the second terms of Equations 5 and 6 to correct a n error in the original Pao formulation. This factor \iould have the effect of increasing the viscosity and first normal stress differmce a t high shear rates (the theoretical curves \\auld be higher by exactly a factor of 2 as K -+ a ) and would cause the disagreement shown in Figures 6 and 7 to be more pronounced. Details of the Huseby-Blyler modification have not been published. Bogue’s Theory. T h e viscometric predictions are (Bogue, 1966) : 712-
-- -
?(R)=
R 711
,f-- +
XH(X) d In A
1
K2
722
rn
alKIX
X2H(X)d In X
(1
+ alKIX12
(9)
(10)
An expression for the second normal stress difference is also available (Bogue, 1966). Bogue’s theory contains a material parameter, a, that does not appear in linear theory. If the second normal stress difference is considered, another material parameter appears. At present, no information is available concerning the importance or dependence of the second parameter. BKZ Theory. T h e incompressible, isothermal BKZ theory contains a n empirical function, W(1, 11, s), in which the time dependence is expressed by three material functions : m ( s ) , c(s), and a ( s ) . T h e function c(s) is associated with the high shear rate plateau on the viscosity curve, and a ( s ) is necessary for predicting shear-thickening behavior (Zapas, 1964). T h e final equations presented here assume both a ( s )and c(s) to be zero. The explicit results of the BKZ theory depend on the form of the empirical function, FV(1, 11, s). The first function presented by Zapas (1964) was discussed in the review article by Bogue and Doughty (1966). I n a recently published work a somewhat different function was presented (Zapas, 19G6).
An expression for Doughty, 1966).
(
~
-2 T~ ~ ~ ) is/ also K ~ available
(Bogue and
Experimanta I
Linear dynamic data and nonlinear steady shear data were taken for two polymer solutions: 10% polyisobutylene in Decalin and 12y0polystyrene in Aroclor. T h e approximate molecular weights were 90,000 and 170,000 for the polyisobutylene and polystyrene, respectively. All data were taken a t 25” C. with a Weissenberg Rheogoniometer [see Van Wazer, Lyons, Kim, and Colwell (1963) for details]. Viscous heating for the cone and plate flow was estimated by the results of Bird and Turian (1963), which predicted a maximum temperature difference in the sample of 0.4”C . for the most adverse condition. Data are presented without corrections for edge effects or secondary flow. Secondary effects were investigated for normal stress measurements by a method suggested by Ginn and Metzner (1 963)-that is, normal stress measurements were made on a NeLvtonian fluid (NBS oil M) under conditions similar to those for which viscoelastic data were taken with the assumption that any reading would be due to secondary effects; however, a normal stress difference of only 455 dynes per sq. cm. was observed for the most adverse condition. Since the viscoelastic fluids tested exhibited a difference of the order of 105 dynes per sq. cm. a t similar shear rates, the error was considered to be insignificant. The apparent viscosity and first normal stress difference were calculated directly from measurements of torque and thrust on the fluid in cone and plate flow. The linear dynamic functions were determined from measurements of the fluid in linear sinusoidal oscillation (see Figure 1). The relaxation spectra were approximated by an expression given by Staverman and Schwarzl (1956) and reported by Ferry (1961):
This first approximation was successively corrected until the relaxation spectrum would reproduce the G” curve. Corrections to the relaxation spectrum were taken as the difference between the reproduced G” curve and the actual G” curve. VOL. 6
NO. 3
AUGUST 1967
389
This technique is suitable because, to a rough approximation, one has (Ferry, 1961)
H(X) = G ” ( w )
Iw = x
a least squares fit was undertaken, the results being shown in Figure 3. The resulting constants are:
Thus, the process causes rapid convergence to a spectrum that will reproduce the G” curve (see Figure 2). Quantitative Evaluation of Theories
Of the theories being considered, Oldroyd’s is of the stressderivative form, whereas the others are of integral form. For that reason, it is convenient to consider Oldroyd’s theory apart from the others. Oldroyd’s Three-Constant Theory. I t was not possible to fit the three-constant form of Oldroyd’s theory, Equation 1, to the dynamic viscosity data-that is, the numerical values of the three constants were strongly dependent on which three points were selected for curve fitting. I n view of this difiiculty,
I 04
lo4
Io3
Io3
a
s .Fa
12y0Polystyrene
10y0 Polyisobutylene
(1 5)
1
to,poises
1083
XI, sec. Xz, sec.
0.0585
237 0.0108 0.00223
0.000914
The theoretical viscometric results are compared with data in Figures 4 and 5. The quantitative discrepancies are obvious from the comparison. It should be noted, however, that more general Oldroyd-type theories have been developed and tested successfully with viscosity and normal stress data (see Spriggs, Huppler, and Bird, 1966). Integral Theories. The three integral theories considered will accept any relaxation function that is single-valued in H ( X ) . Thus they are more flexible than Oldroyd’s threeconstant model for curve-fitting purposes. The viscometric predictions of Pao’s theory are shown in Figures 6 and 7. As was noted earlier by De Vries (1963), too-high viscosities are predicted a t moderate shear rates and the high shear rate Newtonian plateau occurs too quickly. The same behavior can be noted for the first normal stress difference predictions. Bogue’s theory and the BKZ theory were evaluated for values of the nonlinear material parameters, a and A , ranging between 0.1 and 1 .O. I t is not intended to suggest an absolute
102
IO1
IO‘
circle - 10% Polyisobutylene triangle - 12% Polystyrene IO0 open - 77’ Data closed - G” Data I 10-1
-
lo4
b
I o3
IO?
,p P
IO2
IO0 IO1 w (sec-1)
.-0
u 0)
IO*
IO‘
Figure 1. Linear dynamic data for 10% polyisobutylene and 12% polystyrene solutions
0
101.~.
io5
10%Polyisobutylene 12% Polystyrene
lo-’
IO0
IO’ w (sec-1)
IO2
IO
100
Figure 3. Least squares fit of Oldroyd’s three-constant theory to dynamic viscosity data
Io4
io4
io3
** ** 103.
I
.-In Q
X I02
I
v
F 0
100
Io-‘
IO-^
IO-^
IO-’
too
10‘
lo2
X (sec-1) Figure 2. Relaxation spectra for 10% polyisobutylene and 1270 polystyrene solutions 390
I&EC FUNDAMENTALS
lO%PolyisobutyIene 12% Polystyrene
IO0 IO-’
lo2
10’
IOO
lo3
lo4
K (sec-1)
Figure 4. Comparison of theory with viscosity data
Oldroyd’s three-constant
limit on the range of these parameters, but rather, that this range was sufficient in this work. T h e viscometric predictions are shown and compared with data i n Figures 8 through 11. As can be seen from the figures, reasonable fits to the viscometric data are obtained into the non-Newtonian region by the proper choice of material parameters. Neither theory is presented in a form to predict a very high shear rate plateau. T o do so would involve including the material function a ( s ) i n the BKZ theory and a term with a very small relaxation time i n Bogue's theory (I 966). From the evidence avadable, Bogue's theory seems to provide a reasonable fit over a somewhat greater shear rate range, whereas the BKZ theory seems slightly less dependent on the material parameter-i.e., the BKZ theory would yield the better over-all fit if a single value of the material parameter were used for both materials. Of course, one is in fact mostly testing Zapas' empirical function W(I, 11, s), Equation 11, and a modification of it is certainly possible without affecting the remaining framework of the BKZ theory. Both theories produce viscosity curves that decrease too gently in the initial non-Newtonian region 01' the viscosity function. Of the two theories, numerical results are considerably easier to obtain from Bogue'i3 theory, since evaluation of single integrals is involved as opposed to the double integrals in the
o
BKZ theory. However, the BKZ theory is conceptually a more general theory which can include compressibility and energetic effects (Bernstein, Kearsley, and Zapas, 1964). T h e test of the theories was rather demanding, i n that correlation of very different types of data was attempted with, a t most, one experimental constant. I t seems particularly significant that Bogue's theory and the BKZ theory reasonably predicted normal stress data from linear data that in no direct way involved normal stresses. There is not sufficient evidence to determine clearly whether the material parameters, a and A , are constant for polymer solutions. I t does seem significant, however, that fairly good results for both solutions could be obtained from single values of the parameters-for instance, with a = 0.6 and A = 0.1. T h e significance of this is enhanced by the noticeable differences i n the behavior of the solutions-i.e., the different shapes of the relaxation spectra and the fact that the decrease in apparent viscosity is considerably more severe for the polyisobutylene solution. Characterization of Viscoelastic Fluids
One of the basic problems of viscoelasticity is that of specifying the essential experiments necessary to characterize rheological behavior. As with the various equations of state for
10% Polyisobutylene 12% Polystyrene-
10-1
-.o
IO%Polyisobutylene 12% Polystyrene
a 0
a
10-2 IO-I
io0
10' lo2 K (sec-')
lo3
10-2
11
Figure 5. Compariison of Oldroyd's three-constant theory with normal stress data
Figure 7. Comparison of Pao'stheory with normal stress data
io4
-.-x g
v
103
IO2
F
IO'
o
10%Polyisobutylene 12% Polystyrene
0
8
100
IO-' Figure 6. data
loo
Comparison of
IO' K( s e d)
lo2
Pao's theory
lo3
I"
lo4
to-'
roo
10'
lo2
io3
lo4
K (sec-I) with viscosity
Figure 8. data
Comparison of Bogue's theory with viscosity
VOL. 6
NO. 3 A U G U S T 1 9 6 7
391
K (sec-I) Figure
9. Comparison of Bogue's theory with normal stress data
IO-'
ioo
io1
lo2
lo3
104
K(sec-I) Figure 10. Comparison of Zapas' form of the BKZ theory with viscosity data
Io3
IO'
-
(u
5
IO2
'a 0
0) v)
1,
0)
IO'
a 'p Y
?INx
b-
loo Conclusions
Io-'
I o-2 ioo
io'
lo2
lo3
104
K ( sec-l ) Figure 1 1 . Comparison of Zapas' form of the BKZ theory with normal stress data 392
gases, the answer will depend on the situation. For instance, if a fluid were adequately described by Oldroyd's three-constant model, only three material constants would be needed; or if Pao's theory were used, only the relaxation spectrum would be required. O n the other hand, unless universal values could be assigned to the material parameters, a and A , appearing in Bogue's theory and the BKZ theory, some nonlinear data would be necessary. The latter case seems reasonable, since one might expect effects in nonlinear behavior that were latent in linear behavior. The experiments considered in the present framework are those involving the three viscometric functions (the viscosity function, the first normal stress function, and the second normal stress function) and any one of the linear functions (the relaxation spectrum, the dynamic viscosity, etc.). Using either the BKZ theory or Bogue's theory, one finds that the viscosity function and first normal stress function are specified by the relaxation spectrum and one point in the nonlinear region of either of the viscometric functions. A less direct but equally valid characterization \vould result from either nonlinear function with one point on the relaxation spectrum or some other nonlinear function. I n fact, the relaxation spectrum could be retrieved from Equation 9 or 10 by methods very similar to those used n i t h linear data. I n practice, it would be more difficult to retrieve from Equation 12 or 13 because of the double integral. Treatment of the second normal stress difference \I ith the theories above would involve no more than one additional material constant. In the matter of deciding on the ranges and accuracy of the basic data, several comments can be made. The fluids tested exhibit a relaxation function that decreases sharply with increasing relaxation time. \Vhen used \vith the integral theories, the most significant portion of such a function associated with a particular shear rate is a range that is centered in a relaxation time equal to the reciprocal of the shear rate. Dynamic data are needed in a region centered about w = 1, X to produce a point on the relaxation spectrum a t A. Therefore, if one is taking dynamic data to make predictions over a certain shear rate range, the dynamic data should cover a somewhat greater range in w and be centered about the shear rate range Using the integral theories the relaxation spectrum need not be represented \vith absolute precision at each point, since an integration is involved. This allo\\s some liberty in representing and extrapolating the function and suggests the possibility of empirically representing the function \vith a small number of constants (an attracrive feature for dimensional analysis)for instance, the spectrum for the polyisobutylene solution can be represented by a four-constant equation (Doughty, 1966). These remarks apply to characterization in the realm of viscometric and linear deformations. I t is certainly the intent of the theories that they not be so limited and the form is such that they can, in principle, be extended to any flow, however, complex, \vithout additional parameters. The suitability of the theories for such f l o ~ has s not been tested, however.
I&EC FUNDAMENTALS
Four nonlinear viscoelastic theories \vere quantitatively examined for their ability to correlate linear dynamic behavior and nonlinear viscometric data (nonlinear viscosity and first normal stress data) \vith the follo\ving conclusions: Oldroyd's three-constant theory and Pao's theory produce viscometric stress functions that are qualitatively but not quantitatively correct.
Bogue’s theory and Zapas’ form of the BKZ theory correlate data u p to moderate shear rates. One adjustable parameter is involved i n both theories; however, it is possible that these parameters will prove to be constant for classes of materials. T h e latter theories predict nonlinear stress behavior from linear data (the relaxation spectrum) assuming prior knowledge of a material constant. Possibly another constant is required to predict the second normal stress difference. Acknowledgment
T h e authors express appreciation to the National Science Foundation for its support under NSF GP2005. T h e terminal work of one of the authors (JOD) was supported by the National Aeronautics and Space Administration under NASA NsG 671. T h e support of this agency is also gratefully acknowledged. Nomenclature = dimensionless material constant
= = = = = =
a decay function of time shear loss modulus, F/L2 continuous relaxation spectrum, F/L2 first invariant o f the Finger strain tensor second invariant of the Finger strain tensor shear rate, T-‘ decay function function defined by Equation 7 function defined by Equation 8 a backward-running time index, T a n arbitrary scalar in the BKZ theory, F / L 2
GREEK LETTERS = decay function
a(s)
of time
q
= material function (the viscosity), F T / L 2
q,,
= material constant in Oldroyd’s theory,
XI, X u
FT/L2
= dynamic viscosity, F T / L 2
q’
~
Bernstein, B., Kearsley, E. A., Zapas, L. J., J. Res. Natl. Bur. Stds. B68B, 103 (1964). Bernstein, B., Kearlsey, E. A., Zapas, L. J., Trans. SOG.Rheol. 5 , 391 (1963). Bird, R. B., Turian, R. M., Chem. Eng. Sci. 17, 331 (1963). Bogue, D. c.,IND. ENG.CHEM. FUNDAMENTALS 5,253 (1966). Bogue, D. C., Doughty, J. O., IND.ENG. CHEM.FUNDAMENTALS 5,243 (1966). De Vries, A. J., “Proceedings of 4th International Congress on Rheology,” Part 3, pp. 321-44, Interscience, New York, 1963. Doughty, J. O., Ph.D. dissertation, University of Tennessee, Knoxville, Tenn., 1966. Ferry, J. D., “Viscoelastic Properties of Polymers,” Wiley, New York, 1961. Ginn, R. F., Metzner, A. B., “Proceedings of 5th International Congress on Rheology,” Part 2, pp. 583-601, Interscience, New York, 1963. Huseby, T. \V., Blyler, L. L., Jr., “Steady Flow and Dynamic Oscillatory Experiments,” Society of Rheology, Atlantic City, N.J., October 1966; Trans.Sot. Rheol., in press. Markovitz, H., Trans.SOG.Rheol. 1, 37 (1957). Oldroyd, J. G., Proc. Roy. SOC.A200, 523 (1950). Oldrovd. J. G.. Proc. Rov. SOC. A245. 278 (1958). Pao, Y . ,J. A/$. Phys. 28, 591 (1959). Pao, Y . , J . Polymer Sci. 61, 413 (1962). Spriggs, T. W., Chem. Eng. Sci.20, 931 (1965). Spriggs, T. \V., Bird, R. B., IND.ENG. CHEM.FUNDAMENTALS 4. 182 11965). Spriggs, T. \V.; Huppler, J. D., Bird, R. B., Trans. SOG.Rheol. 10 ( l ) , 191 (1966). Staverman, A. J., Schwarzl, F., in Stuart, H. A,, “Die Physik der Hochpolymeren,” Vol. IV,Springer-Verlag, Berlin, 1956. Van Wazer, J. R., Lyons, J. I V . , Kim, K. Y., Colwell, R. E., “Viscosity and Flow Measurement,” p. 113, Wiley, New York, 1963. Williams, M. C., Bird, R. B., IND.ENG.CHEM.FUNDAMENTALS 3, 42 (1964). Williams, M. C., Bird, R. B., Phys. Fluids 5 , 1126 (1962). Zapas, L. J., “Correlations of an Elastic Fluid with Experiments,” Society of Rheology, Pittsburgh, Pa., 1964, and personal communication. Zapas, L. J., J . Res. Nat. Bur. Stds. 70A, 525 (1966). .
= dimensionless material constant = = = = =
Literature Cited
X2
= material constants i n Oldroyd’s theory,
RECEIVED for review September 22, 1966 ACCEPTED March 29, 1967
T
= relaxation t h e , T i
j
= components of the deviatoric stress tensor,
I
F/L2 in
Cartesian coordinates = angular frequency, T-l
Symposium on Mechanics on Rheologically Complex Fluids, Society of Petroleum Engineers, AIME, Houston, Tex., December 1966.
FLOW OF VISCOELASTIC FLUIDS THROUGiH POROUS MEDIA R. J.
M A R S H A L L A N D A. 6 . M E T Z N E R
University of Delaware, iVewark, Del. 19711 HILE the pragmatic: significance of studies of flows through porous media requires no discussion, there is a very strong motivation for such studies from a strictly theoretical point of view: Flows in this geometry provide a n excellent opportunity for studying the behavior of viscoelastic fluids a t high Deborah number levels (Metzner et al., 1966b). This dimensionless group, representing a ratio of time scales of the material and the flow process, may be defined as (Astarita, 1967; Metzner et al., 1966a, b) ;
in which denotes the relaxation time of the fluid under the conditions of interest in the problem under consideration
and IId represents the second invariant of the deformation rate tensor (Bird et al., 1960; Fredrickson, 1964). This latter term depicts the intensity or magnitude of the deformation rate process, and the dimensionless group defined by Equation 1 may be considered to represent the ratio of the time interval required for the fluid to respond to a change in imposed conditions of deformation rate to the time interval between such changes. I t is thus an index of the extent to which the velocity field is unsteady from the viewpoint of an observer moving with a given fluid element as it proceeds along its course or trajectory in a process, using the relaxation time of the fluid as a unit of time. For perfectly steady flows-e.g., under laminar flow conditions in a very long tube-the Deborah number is identically zero; for highly unsteady processes it may be large. VOL. 6
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