I
E, W. MERRILL Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass.
Viscometric Classification of Polymer Solutions These polymer solutions illustrate the great diversity in kinds of non-Newtonian flow, the problems of theoretical attack, and hazards of extrapolating data beyond the experimental range
SOLUTIONS of high polymers constitute
a large fraction of the “pseudoplastic”
or “shearthinning” non-Newtonian fluids -characterized by absence of yield value and continuous decreaseof r / y with increase of r (shear stress) or y (shear rate). For molecular weight determination, polymer solutions are viscometrically examined a t low shear rate and high dilution-when each dissolved macromolecule (which occupies an enormous volume relative to its unsolvated size) approaches the ideal state of complete hydrodynamic independence of other macromolecules and negligible perturbation of its configuration in the solvent under zero flow. I n polymer solutions of industrial and engineering interest, the shear rate and level of concentration are, in general, large enough to preclude hydrodynamic independence and unperturbed configuration. The problem is further complicated by the fact that Newtonian flow is not limited to the ideal state of polymer solutions in which the macromolecules are hydrodynamically independent. Lodge (8) has shown that under conditions of moderate concentration (not dilute) in which network association of macromolecules predominates over the hydrodynamic effects of individual macromolecules, the solution should be Newtonian a t low values of shear stress. According to the theory of Bueche ( 3 ) > even in dilute solutions where the macromolecules are hydrodynamically independent, perturbation of the molecular configuration by moderate shear stress generally leads to pseudoplasticity. When the concentration is high enough to produce network aggregates and the shear stress is substantial (usual engineering conditions), pseudoplasticity results. No present theory combines the mechanics of perturbed network aggregates and individual macromolecules of perturbed configuration to predict adequately the behavior of moderately concentrated polymer solutions over a wide range of shear rate. Thus direct experimental determination is necessary. For comparison the empirical equation is used :
868
7
= cy‘
(1)
where b and s are constants at given temperature and concentration. Experimental Procedure
The data were collected on two coaxial cylinder viscometers. A modified Brookfield viscometer having a narrow gap between cup and bob gave readings at discrete points over the range of 1 to 20 sec.-l of shear rate, and a recording, wide-range viscometer gave a continuous plot of shear stress us. shear rate ai 200 to 20,000 sec.-l (9). The data refer to solutions of the following polymers in water (polyisobutylene in cyclohexane) : Ammonium alginate, a polyelectrolytic polyhexuronic acid (Superloid high viscosity ammonium alginate, Kelco Co.) Sodium carboxymethylcellulose (CMC), a polyelectrolytic polysaccharide (Hercules CMC-70 and CMC-70s) Amylopectin, a highly branched nonelectrolytic polysaccharide (obtained as waxy maize) Amylose, a linear nonelectrolytic polysaccharide (Bios Laboratories, New York) Polyisobutylene, a randomly coiling nonelectrolytic polymer (Enjay Co., Vistanex B 100) None of the polymer solutions studied was so concentrated that a yield value was visually or instrumentally detectable. Concentrated polymer solutions generally exhibit gross elastic behavior and cannot be described solely in terms of viscous response, which is the subject of this article.
Results Figure 1 gives 7 - y data for all solutions except amylose and amylopectin. The low shear rate viscometer shows substantially Newtonian behavior of the polyisobutylene and CMC solutions over its range; the high shear rate viscometer shows pronounced pseudoplasticity. I n contrast, the ammonium alginate is pseudoplastic over the range of both viscometers, and the whole curve can be reasonably ivell fitted with a single value of s and 6 (Equation 1 ) . The
INDUSTRIAL AND ENGINEERING CHEMISTRY
Newtonian behavior of CMC and polyisobutylene in the low shear range may correspond to the mechanics of network aggregates described by Lodge ( 8 ) , Philippoff and others (2) working on considerably more concentrated polyisobutylene solutions, in which network aggregates are even more likely, proved the existence of a lower Newtonian range below y values 10 to set.-' and appear to support the model proposed by Lodge ( 8 ) . I t is not to be inferred from Figure 1 that ammonium alginate does not have a Newtonian range at some value of y below 1 sec.?. In fact Eisenberg’s detailed work on deoxyribonucleic acid, a polyelectrolyte of great biological interest ( 5 ) ,showed a Sewtonian range at a somewhat lower shear rate than 1 sec.-l, followed by the marked pseudoplasticity observed here for alginate solutions. In the high shear range, rate of decrease of r / y with increase of y is considerably greater for CMC than for polyisobutylene solutions; alginate falls in between. This may reflect the known stiffness of the cellulose chain and probable stiffness of the polyhexuronic acid chain which leads to an extended, if not linear, rodlike form. Theoretically (4, 7) for extended particles the shear rate coefficient of T/y-i.e., d(~/y)/dyshould be much greater than for a symmetrical particle, such as the random coil characteristic of polyisobutylene. The comparison is necessarily incomplete because of the disparity in molecular weights and concentration. The behavior of CMC solutions is noteworthy because of the lower shear range over which Newtonian behavior is observed-suggesting network aggregate €ormation. Evidence for the association of sodium carboxymethyl cellulose molecules is reported by Allgen and Roswall ( I ) , who studied the dielectric properties of Na-CMC solutions. They report that “the shape of the (dielectric constant)concentration curve indicates strong interaction between the C M C molecules, especially a t concentrations above 0.01%, causing less polar aggregates to be
formed.’’ Results reported here concern CMC solutions or considerably higher concentration, measured at a lower rate of shear than those reported in precision experiments with capillary viscometers (6). Hence it is not surprising that the almost Newtonian behavior at low shear rates has not been noted before. In certain experiments the watersoluble dye Gentian Violet was added to the CMC solution. T h e dye presumably reacts directly with the carboxyl ion, forming the amine salt:
This voluminous but nonionized side group off carbon 6 must sterically prevent close alignment between chains. O n the other hand, it cannot prevent random association between chains by hydrogen bonding between hydroxyls.
Addition of Gentian Violet to CMC solutions causes the transition from Newtonian to pseudoplastic behavior to inasing titration, until
s. A typical curve ntaining a fraction of et is shown in Figure n from Newtonian to at 3000 sec.-I is
N(CH3)z
Genetian Violet does not substantially decrease viscosity. I t is inferred that the molecule is not significantly collapsed
from its fully ionized condition and that in aggregating the molecules must “tie up” about the same volume of solvent as they do individually. T h e distinctive shear stress-shear rate relations of amylose and amylopectin solutions are shown in Figure 2. I t appears fairly certain that the rheology is predominantly the result of network association of molecules, rather than the hydrodynamic behavior of individual molecules (70). Amylose is a linear, essentially rigid rod (collapsed helix), whereas amylopectin, because of its high degree of branching, is roughly spherical and probably rigid as the result of intramolecular hydrogen bonding. If these molecules did not attract others of their species, they should follow the mechanics of small rods subject to Brownian motion, which leads to a high shear dependence of viscosity o r show Newtonian behavior as Einsteinian spheres. Actually, because the molecules are nonelectrolytic and are literally bristling with hydroxyl groups, association into networks by hydrogen bonding is highly
Figure 1. Polymer solutions Newtonian in low shear range are non-Newtonian in high shear range Data from low-range viscometer shown as points, from wide-range viscometer as line. Shaded band represenir maximum and minlmum limits in several duplicate recorder plots. Value of I (Equation 1) Is slope of curve for reglon shown. Concentrations in weight percentages
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JULY 1959
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Figure 2. Timedependent curves for amylose and nontime- depend ent curves for amylopectin are distinctively d i f ferent from Figure 1 curves
-
Concentration, grams of polymer per 100 grams of water. Time, duration of shear in seconds a t each constant shear rate, measured from instant shear imposed. Data shown as points, computed from recorder traces of shear stress vs. time at different constant shear rates, obtained from widerange v i s c o m e t e r , 4 9 O c.
2
0
4
8
6
/O
SHEAR R A T E , S f C . - ’ X IO-’
probable. The rheology of these two molecular species in aqueous solution was treated according to a network association theory, similar to that of Goodeve ( 7 ) : modified to suit these distinctive molecular features (70). When the rate of destruction of association links is at all times equal to the rate of reformation -i.e., when imposition of shear instantly causes equilibrium between breakdown and build-up-the following equation is obtained. 7 =-
cvNo
+
70
1 +,iY
(2)
k l , and k2 have the same signifi01, cance as in Equation 2. At long times of shear t: the terms in Equation 3 containing t vanish, so that Equation 3 in the limit t + w is identical with Equation 2. Equation 2 fits accurately the experimental data shown in Figure 2 on amylopectin solutions over the range observed. (Data at very low shear rates are not available.) The instantaneous equilibrium in rates of destruction and reformation of links. Dostulated in the arqument leading to Equation a consequence of the supposition that the effectiveness of a collision of amylopectin molecules with a network aggregate not depend O n whether the fluid is under shear or at rest (all orientations of this spherical molecule are equally effective), With amylopectin solutions no time dependency of the - y relation was observed. In marked contrast. solutions of amylose showed a marked time dependency. The curves in Figure 2 show shear stress T after 1.0 and 45 seconds measured from the beginning of shear (from rest) at constant shear rate. The original complete data from which these curves for amylose (Figure 2) were constructed are correlated by Equation 3 Lvith fair accuracy.
-
where
2j
apparent viscosity, r / y a constant number of association links per unit volume under zero shear ‘Onstant
for destruction
Of
links rate constant for reformation of links limiting viscositv for no association u links When the rate of destruction of network links is initially greater than the rate of reformation until rates balance ultimately, the theory predicts an exponential dependence of viscosity on time, t , as well as a complicated function of shear rate, y :
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Because amyiose molecules at res: are randomly oriented but under shear are oriented in the direction of flow, viscosity a t constant shear rate should decrease initially with time measured from imposition of shear, as networks of randomly oriented molecules in the solution at rest are destroyed, while networks or more oriented molecules developed under shear. Except over meaninglessly limited ranges, the empirical exponential Equation l is inapplicable to the rheology of amylose and amylopectin solutions. Conclusions
In general, for engineering purposes, the actual shear stress-.shear rate rclation of polymer solutions over the range corresponding to the engineering problem of interest must be obtained viscometrically, because of molecular interference and (in some cases) molecular association, and the lack of theory that unites the dynamics of network aggregates under moderate to high stress with the hydrodynamics of individual macromolecules under the same levels of stress (whether the macromolecules be mutually interfering or not). Any such theory would have to take into account the chemical individuality of each kind of macromolecule for which the r - y relation of the solution was desired-e.g., stiffness of the main chain, possibility of intra- or intermolecular hydrogen bonding, and the kind of solvent-polymer interaction. This would require detailed information to be worked into the general theory before useful predictions could be made. The data on polymer solutions presented in Figure 1 show it to be impossible to extrapolate viscometric data taken from a low range of shear rate to a considerably higher range, or vice versa, with any assurance of being correct even in order of magnitude. Literature Cited (1) Allgen, L., Roswall, S., J . Polymer Sci. 12, 229 (1954). (2) Brodnyan, J. G., Gaskins, F. H-.,
Philippoff, Trans. Rheol. 1* 109 (1957). (3) Bueche, F., J. Chem. PIzys. 22, 1570 (1954). (4) Burgers, J. M., “Second Report on ViscositY and Plasticity,” PP. 114-84, Academy of Sciences, Amsterdam, 1938. (5) Eisenberg, H., J . Polymer Sci. 23,
579 (1957).
(6) Fujita, H., Homma, T., Ibid., 15,
277 (1954).
(7) Goodeve, C. F., Traiis. Faraday Sac. 35, 342 (1935). (8) Lodge, A. S., Zbid.,52, 120 (1956). ( 9 ) Merrill, E. W., ZSA Journal 3, 124 (1956). (10) Storey, B. T., Merrill, E. W., J. sei, 33, 361 (1958),
RECEIVED for review November 21, 1958 ACCEPTEDMay 1, 1959
