AnExaminationoftheShear-ThickeningBehaviorofHigh ... · havior manifests, the solutions begin to develop micron-size, optically isotropic particles composed of associated polymer

An Examination of the Shear-Thickening Behavior of HighMolecular Weight Polymers Dissolved in Low-ViscosityNewtonian Solvents

B. J. EDWARDS, D. J. KEFFER, C. W. RENEAU

Department of Chemical Engineering, University of Tennessee, Knoxville, Tennessee 37996

Received 1 June 2001; accepted 14 November 2001Published online 7 June 2002 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/app.10807

ABSTRACT: The anomaly of shear thickening at high shear rates can be observed undercertain conditions for high molecular weight polymers dissolved in low-viscosity New-tonian solvents despite the fact that shear-thinning behavior is considered the norm forthese fluids. The nature of the shear-thickening region of the flow curve is examinedherein through the application of a recent rheological model that has the capability ofquantifying not only the rheological properties of the material, but its internal micro-structural state as well. The results of this examination provide a self-consistentexplanation of the full flow characterization of this anomalous behavior, including bothrheological and optical experimental measurements. The results presented hereinsuggest that the shear-thickening behavior is actually caused by the destruction ofstructures formed during shear at lower shear rates, not by their formation, as previ-ously assumed. The linear birefringence and linear dichroism observed experimentallyin correlation with the shear-thickening behavior are well described by the rheologicalmodel and give predictions in line with experimental measurements. Furthermore,quantitative predictions are made for rheological characteristic functions, such as thefirst and second normal-stress coefficients, for which experimental measurements forthese solutions have not yet been made. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 85:1714–1735, 2002

Key words: dilute solutions; flow-induced phase separation; shear thickening; rheo-optical properties

INTRODUCTION

“. . . future work in this area should concentrate on theprediction of the relationship between shear-thicken-ing and structure formation from a first principlesmodel. The above explanation, while consistent, is anamalgamation of experimental observations and theo-ries or approximations. While all evidence supports theconclusions drawn from the rheo-optical study, a singletheory that could predict both the rheology and the

optical behavior simultaneously would characterize thenature of the structure formation more precisely. Asuccessful single theory would also lead to the possibleprediction of new phenomena to be investigated exper-imentally.”1

The passage quoted above is the final para-graph of the dissertation of A. J. Kishbaugh, Uni-versity of Illinois, 1992.1 This work, published byKishbaugh and McHugh,2,3 finally established aclear connection between the anomalous patternof shear thickening, which occurs under highshear conditions in several dilute polymer solu-tions, and the shear-induced structure formationin the same. Although this connection had been

Correspondence to: B. J. Edwards ([email protected]).Journal of Applied Polymer Science, Vol. 85, 1714–1735 (2002)© 2002 Wiley Periodicals, Inc.

1714

clearly established in refs. 1–3, the authors rec-ognized that no rheological theory had yet beendeveloped that could self-consistently describe allof the phenomena observed in the experimentalinvestigation, let alone be used to predict associ-ated phenomena that had not yet been observedexperimentally. In this article, we demonstratethat this statement is no longer true. The two-coupled Maxwell modes (TCMM) model, devel-oped in ref. 4 and analyzed extensively in ref. 5, isshown to give quantitative descriptions of all as-pects associated with the shear-thickening behav-ior of these solutions, as well as predictions forother viscometric properties that have not here-tofore been measured experimentally.

Before we continue, let us first explain the phe-nomenon under consideration. Dilute solutions ofhigh molecular weight polymers dissolved in low-viscosity, Newtonian solvents are expected to ex-hibit purely shear-thinning behavior at interme-diate and high shear rates (i.e., the viscosity ofthe solution decreases with increasing shearrate). However, an anomalous behavior was notedon some occasions, for many years now, which isstill not widely accepted. This anomalous behav-ior is termed shear-thickening, and it manifestsas the opposite effect to shear thinning: the vis-

cosity of the solution increases with increasingshear rate.6,7 Although the shear-thinning phe-nomenon is widely understood and accepted as anintramolecular effect occurring because of the ex-tension and orientation of the polymer chains insolution,8 prior to the past decade, no consensushad arisen to explain the origin of shear thicken-ing. Some minds even questioned whether it ex-isted at all in this kind of solution, preferring tobelieve that it was caused by some other well-known but not well-quantified phenomenon suchas viscous heating, polymer adsorption on walls,or capillary entrance effects. The work of Kish-baugh and McHugh,2,3 as mentioned above, hasfinally established the clear connection of thisunusual viscosity pattern with intermolecular ef-fects, not to mention establishing further docu-mentation as to its mere existence.

A typical flow curve for a dilute solution under-going steady shear flow is presented in Figure 1.This figure shows the experimental behavior ob-served with increasing shear rate, as observed byLayec-Raphalen and Wolff6 and Vrahopoulou andMcHugh.7 At low to intermediate shear rates (fol-lowing the common Newtonian plateau for verysmall shear rates), one observes the usual patternof shear thinning with increasing shear rate.

Figure 1 A typical plot of viscosity versus shear rate for a dilute polymer solution thatexhibits shear thickening. The increase in viscosity begins at the critical shear rate �c,and shear thinning resumes at �m.

HIGH MW POLYMER SHEAR THICKENING 1715

However, as the shear rate continues to increase,a critical shear rate is attained where the viscos-ity reaches a local minimum at �c and actuallybegins increasing with increasing shear ratethereafter. At very high shear rates, a local max-imum in the viscosity is achieved at �m, and aweak plateau and subsequent shear-thinning re-gion follow at extremely high shear rates.

Various explanations were advanced over theyears to rationalize this observed behavior. Someof these were intramolecular,9–12 and some wereintermolecular13–16; however, we omit a review ofthese, because all of them were demonstrated asinapplicable to the solutions under considerationby the experimental results of Kishbaugh andMcHugh2 (see below). Still, on the basis of thetrends observed in Figure 1, Vrahopoulou andMcHugh7 offered a rational intermolecular expla-nation for this behavior, which eventually evolvedinto the more sophisticated explanation of Kish-baugh and McHugh.2,3 In the explanation of theformer authors,7 experimental curves, such as theone displayed in Figure 1, arise as follows. Ini-tially, at low to moderate shear rates, the typicalflow behavior of a dilute polymer solution is ob-served, indicating the intramolecular chainstretching and orientation of the polymer mole-cules subject to hydrodynamic interactions only.However, at a critical shear rate, �c, intermolec-ular associations begin to develop which produceamorphous structures that can be much largerthan the average chain dimensions. These struc-tures grow in size with increasing shear rate,thus raising the solution viscosity. At some highercritical shear rate (�m), however, the magnitudeof the imposed shear becomes so severe that theformed structures are sheared apart, thus pro-ducing the second region of shear-thinning behav-ior at extremely high shear rates. Still, as long asonly rheological experiments were conducted, noconclusive evidence could be amassed to prove theexistence of a shear-thickening/structure-forma-tion region in the flow curves of these solutions: atthe high shear rates involved, a skeptic couldalways argue that the observed increase in vis-cosity was actually caused by some phenomenonunrelated to the structure formation, such aspolymer adsorption on the walls of the rheometer.

Definitive experimental evidence confirmingthe intermolecular nature of shear thickeningwas provided in 1992 by Kishbaugh andMcHugh.1,2 In these experiments, in situ opticalmeasurements of the linear dichroism and linearbirefringence, the imaginary and real parts of the

difference between principal indexes of the refrac-tive index tensor, were made simultaneously withmeasurements of the viscosity. The shear thick-ening behavior was observed to begin at a criticalshear rate that was typically slightly larger thanor equal to the shear rate where the linear dichro-ism displayed a global maximum, �d (see Fig. 2).At the same time, the linear birefringence in-creased monotonically with increasing shear rate.

On the basis of these experimental findings,Kishbaugh and McHugh3 advanced a sounderconceptual basis for the observed shear-thicken-ing behavior, which refines the explanation devel-oped by Vrahopoulou and McHugh.7 Before thecritical shear rate where the shear-thickening be-havior manifests, the solutions begin to developmicron-size, optically isotropic particles composedof associated polymer chains, which orient alongthe flow direction. These particles grow in size

Figure 2 Typical flow curves for viscosity, dichroism,and birefringence versus shear rate as observed in si-multaneous rheo-optical measurements. Note that �d

� �c.

1716 EDWARDS, KEFFER, AND RENEAU

with increasing shear rate, and, according to theanomalous diffraction approximation (ADA) lightscattering theory, this causes the maximum inthe linear dichroism curve.3 However, most poly-mer chains remain in the solution, not in theparticles, and it is the extension and orientationof these that cause the monotonic increase in thelinear birefringence with increasing shear rate.

In this article, we shall show that, according tothe TCMM model, the above explanation may notbe totally correct. In the above explanation, thereis really no reason the onset of shear thickeningmust always occur at an equal or slightly largervalue of the shear rate than the maximum indichroism curve. If amorphous particles are form-ing and enlarging in the polymer solution over arange of shear rates spanning the viscosity mini-mum, then it can be nothing more than coinci-dence that the maxima in the dichroic curvesalways correlate with the minima in the viscositycurves (see below). However, according to theTCMM model, as shown below, the particles (as-sociations of polymers) actually grow at shearrates well below the viscosity minimum. Thesestructures cause the rise in the dichroic curve upto the maximum values, and actually augmentthe shear-thinning behavior (i.e., they lower theeffective shear stress by relieving some of theforces acting on the anisotropic polymer chains inthe bulk flow). At the critical shear rate of thedichroic maximum, these structures begin to de-crease in size and aspect ratio because of thestrong hydrodynamic forces, thus raising theshear stress and increasing the effective viscosity,while causing the dichroic signal to begin retrac-ing its path backward as the effective particle sizedecreases. This provides a rational explanation asto why the viscosity minima and dichroic maximaalways occur at approximately the same shearrates, whereas the explanation in ref. 3 does not.

Another criticism of the Kishbaugh andMcHugh hypothesis is that the size of the parti-cles (on the order of microns) seems rather large.If such large amorphous structures are forming,then polymer molecules are being subtractedfrom the optically active bulk fluid to form them;however, the solution birefringence shows no signof this occurrence as it continues to increase ex-ponentially with increasing shear rate. Accordingto the TCMM model, the particles are actuallytwo orders of magnitude smaller, but more nu-merous than in the Kishbaugh and McHugh hy-pothesis. Consequently, their dichroic behaviorcan be described well by the Rayleigh scattering

theory, as demonstrated below. However, theabove-stated criticism is not overcome in thismanner. To overcome it, we must envision thesestructures not as separate entities composed ofamorphous polymer molecules, but rather as aninterpenetrating structured continuum imposedover the bulk chains in solution. Therefore, thestructures themselves may not be optically isotro-pic, as previously believed.2,3 Interestingly, theresults presented below can be used to under-stand the physics of the precursor state to flow-induced phase separation in polymer solutions ofhigher concentration. Furthermore, by fitting theparameters in the TCMM model against the ex-perimental data from the literature, predictionsare obtained for characteristic functions and op-tical properties that have yet to be measured ex-perimentally for these solutions.

THE TCMM MODEL

The TCMM model was developed by using thethermodynamic methodology put forth over thepast decade by Beris and Edwards4 and Grmelaand Ottinger.17,18 It first appeared in multiplemode form in ref. 4, and the two-mode version wasanalyzed extensively in ref. 5, wherein it wasshown that this model was able to reproduce thecomplex qualitative behavior that is typically as-cribed to polymeric liquids. This model is writtenin terms of two mode-conformation tensors, c1

and c2, one for each relaxation mode of the poly-meric liquid. Although most polymeric fluids pos-sess a whole spectrum of relaxation times, thequalitative behavior of both the linear and thenonlinear rheological properties can be ade-quately described with just two modes.5 For di-lute polymer solutions, two modes should allowample latitude to obtain reasonably good quanti-tative fits to the experimental data of interest.

The two relaxation modes give a direct quanti-fication of the microstructural state of the poly-meric fluid. The eigenvalues and eigenvectors ofthese second-rank tensors quantify the degree oforientation and its preferred direction, respec-tively, corresponding to the two relaxation modes.For the present case, c1(x, t) is taken as thesecond moment of the orientational distributionfunction, �(x, R, t)8:

c1 � � RR�d3R (1)


In this expression, R is the end-to-end vector of adissolved polymer chain, and c1 thus has units oflength squared.

The second mode tensor, c2(x, t), is associatedwith the intermolecular structures that form dur-ing shear. It, too, is correlated with the size of thestructures formed. If we define a size distributionfunction by f(x, a, t), where a is the vector span-ning the major axis of a spheroidal structure,then

c2 � � aafd3a (2)

The fundamental tenet of the TCMM model isthat these two mode-conformation tensors are notonly affected by the imposed deformation, butalso by each other.4 A typical differential rheo-logical model involves uncoupled modes, in whichthe individual mode tensors are affected by theimposed deformation only. Within the auspices ofthe generalized bracket approach to thermody-namic modeling, a general class of coupled relax-ation-mode models was derived.4 In the two-modelimit, the evolution equations for the mode-con-formation tensors are

�c��1

�t � ��c��1 c��

1 �� c��

1 �� 1�1

c��1

�kBT�1K1

��

2kBT �n2

n1

1

��1�2�K2�c��

1 c��

2 � c��2 c��

1 �

2kBTc��1 �

�c��2

�t � ��c��2 c��

2 �� c��

2 �� 1�2

c��2

�kBT�2K2

��

2kBT �n2

n1

1

��1�2� �K1�c��

1 c��

2 � c��2 c��

1 �

2kBTc��2 � (3)

In the present context, the polymer associationsquantified by c2 are assumed to possess the sametype of elasticity as the individual chains (i.e.,Maxwellian), but this elasticity acts over muchlarger length and time scales.

In the above expressions, five adjustable pa-rameters appear in addition to Boltzmann’s con-stant, kB, the absolute temperature, T, and theHookean spring constants, K1 and K2. These are�1, �2, n1, n2, and . The first two are the constant

relaxation times of the two modes, measured inunits of time. The second two are the effectiveconcentrations of the two modes, measured inunits of moles per volume. All four of these pa-rameters must be greater than or equal to zero forthe model to make sense physically. The last pa-rameter quantifies the degree of interaction be-tween the two modes. Thermodynamically, thisparameter should be within the range of �1 � � 1; however, as noted in ref. 5, is typically asmall positive fraction.

The extra stress tensor used for calculating therheological properties of the polymer solutions isa linear sum over the mode-conformation tensors:

�� i�1

2

�niNAKic��i niNAkBT�� (4)

where NA is Avogadro’s number. It has units offorce per length squared, and the shear viscosity,�, is calculated by dividing the shear stress com-ponent, �12, by the shear rate, �. Other rheologi-cal characteristic functions and optical propertieswill be defined where appropriate.

GENERAL BEHAVIOR OF THE TCMMMODEL IN STEADY SHEAR FLOW

The general behavior of the TCMM model insteady shear flow was already examined in ref. 5;however, no special attention was paid there tothe parameter region at high shear rates wherethe phenomenon of shear thickening may occur.Here, we examine more closely the high shearrate region of the flow curve, for varying param-eter ranges, to see what effects mode coupling hasupon not only the shear viscosity of the polymericsolutions, but also their other rheological charac-teristic functions. To do this more efficiently, wework in terms of dimensionless quantities.

The evolution equations, eq. (3), are made di-mensionless by defining the dimensionless shearrate as � � � �1�2, the dimensionless modetensors as c��

i � Kic��i /kBT, and dimensionless

time as t � t/�1�2. For a specified value of theshear rate, these coupled, nonlinear algebraicequations are solved via Newton’s method, andthe dimensionless shear stress is computed ac-cording to �12 � �12/n1n2 kBT. Hence, thedimensionless viscosity is given by � � �12/�.Furthermore, the first and second normal stress


coefficients, in dimensionless forms, are given by1 � (�11 � �22)/�2 and 2 � (�22 � �33)/�2. Whenexpressed in dimensionless form, the dependenceof the evolution equations, eq. (3), reduces tothree parameters, the ratio of the mode relaxationtimes, � � �2/�1, the ratio of the mode concentra-tions, n � n2/n1, and the coupling parameter, .

Figure 3 plots the results of calculations for thedimensionless viscosity versus dimensionlessshear rate for n � 0.05, � 0.1, and variousvalues of �. This demonstrates that the TCMMmodel is well capable of describing the qualitativebehavior of the experimental data for these solu-tions. For values of � less than approximately 3,the behavior is the expected shear thinning witha saturation at high values of the shear rate. For� values between approximately 3 and 100, onefinds a region of shear-thickening behavior athigh shear rates, followed by a maximum in theviscosity and a resumption of shear thinning.(Note that this latter effect is not noticeable inFig. 3, as it was drawn on a large scale to depicttrends in �. However, Fig. 1 was actually gener-ated by using � � 3 as a parameter, and it is thusevident that this shear-thinning region at highshear rates is well-described by the model.) As �continues to increase, above a value of about 100,the shear-thickening behavior is damped out and

replaced again by shear thinning and then satu-ration, as was the case for � � 2. This figure thusdemonstrates that the shear-thickening behaviorof the solutions only occurs when the relaxationtimes of the two modes are within two orders ofmagnitude of each other.

In Figure 4, we plot the dimensionless viscosityversus dimensionless shear rate for � � 3, � 0.1,and various values of n. This figure shows thatthere is also a concentration window within whichthe shear thickening appears. At values of n of0.05 and 0.075, we observe the pattern of Figure1: shear thinning, followed by shear thickening,and then shear thinning again. However, at thehighest concentration, only shear thinning is ev-ident. Also, although not shown on the graphbecause of scale, at lower concentrations, onlyshear thinning is evident as in the n � 0.10 caseshown. Interestingly, this general behavior wasobserved in the experiments of Kishbaugh andMcHugh,1,2 who reported that shear thickeningoccurred within a certain concentration windowfor a particular solution, and above or below thatwindow only shear thinning was observed. Evi-dently then, the appearance of a shear-thickeningregion of the flow curve is only possible undervery special circumstances. These circumstanceswill become clear later.

Figure 3 Dimensionless flow curves for n � 0.05, � 0.1, and various values of �.


Figure 5 depicts the dimensionless viscosityversus dimensionless shear rate for � � 3, n� 0.05, and various values of . When � 0.0,

one recovers the behavior of the two uncoupledMaxwell modes model (i.e., a viscosity that isindependent of shear rate). However, as shown in

Figure 4 Dimensionless flow curves for � � 3, � 0.1, and various values of n.

Figure 5 Dimensionless flow curves for � � 3, n � 0.05, and various values of .


the inset of Figure 5, for finite values of the flowcurve develops a shear-thickening region at inter-mediate values of the shear rate. As continues toincrease, the relative magnitude of shear thicken-ing increases steadily, and the critical shear ratefor the onset of this behavior moves to lower val-ues of the dimensionless shear rate. The interac-tion parameter is thus shown to be critical for theobservation of not only the shear-thickening re-gion of the flow curve, but also for any shear-ratedependent rheological properties whatsoever.

Now let us examine the behavior of the twomode-conformation tensors for the case n � 0.05,� � 3, and � 0.1. This can give us importantclues as to the development of the anomalousshear-thickening behavior. Figure 6 shows thebehavior of the traces of the two mode-conforma-tion tensors versus dimensionless shear rate. Inessence, these are measures of the degree of ori-entation for the first mode, and the relative struc-ture size (or aspect ratio) for the second mode: trc1 � tr�RR�/tr�RR�0 and trc2 � tr�aa�/tr�aa�0,where the angular brackets denote the averagesdefined in eqs. (1) and (2), and the subscript de-notes that the averages are taken in the limit as�3 0. The first mode, representing the individualmolecular orientation and extension, increases

without limit with increasing shear rate. Thisimplies that the polymer molecules are furtherextended and oriented to a greater degree withincreasing applied shear. The second mode, how-ever, shows an unexpected behavior: the size ofthe structures increases only at low shear rates,reaches a maximum, and then decreases withincreasing shear rate thereafter. By comparingthe value of � where this maximum occurs withthe value at which the viscosity minimum occursfor the corresponding case in Figure 5, it is evi-dent that the maximum occurs at a slightly lowervalue of the shear rate than the minimum.

If the model is to be believed then, this im-plies that the shear-thickening behavior ob-served in these solutions is caused not by theformation of supermolecular structures as inthe Kishbaugh and McHugh hypothesis,3 butinstead by the degradation of that structure.The effect of this structure is, therefore, actu-ally to reduce the effective shear stress in ac-cordance with its intrinsic length scale: thelarger the structure (i.e., the more elongated itis), the more it reduces the apparent shearstress. (We shall return to this line of reasoninglater after comparing the model predictionswith experimental data.)

Figure 6 The traces of the dimensionless mode-conformation tensors versus dimen-sionless shear rate for � � 3, n � 0.05, and � 0.1.


In Figure 7, we plot the orientation angles ofthe two modes with respect to the direction offlow, in a counterclockwise progression. At lowshear rates, both angles decrease rapidly from thevalue of 45° as � 3 0. For intermediate and highvalues of the shear rate, the first mode orientsvery close to the flow direction. On the otherhand, the second mode orients along the flow di-rection to the greatest degree when the structuresize is relatively large. As the aspect ratio of thestructures decreases with increasing shear rate,the orientation angle increases marginally.

We can also use the model to predict the qual-itative behavior of the dimensionless first andsecond normal-stress coefficients for these solu-tions, which were never measured experimen-tally. These are plotted versus dimensionlessshear rate in Figure 8 for the case n � 0.05, �� 3, and � 0.1. Interestingly, the first normal-stress coefficient also exhibits a shear-thickeningeffect, whereas the second normal-stress coeffi-cient does not. Hence, the structure size affectsnot only the shear stress, but also the normalstress tensor components as well. Note that thesecond normal-stress coefficient is a small nega-tive fraction of the first normal-stress coefficient,in agreement with general trends in polymericfluid dynamics.

COMPARISON OF THE TCMM MODELWITH EXPERIMENTS

In this section, we continue our examination ofthis peculiar shear behavior by fitting the TCMMmodel to some of the experimental data of Vraho-poulou and McHugh7 and Kishbaugh andMcHugh.2 To compare the model with experimen-tal data, optimization of the five parameters iscarried out after evaluating eqs. (3) and (4) over arange of shear rates.

Vrahopoulou and McHugh7 present data forsolutions composed of high molecular weight poly-ethylene in xylene and polypropylene in tetralin.The data are presented in terms of relative vis-cosity versus shear rate, and reduced viscosityversus shear rate. The relative viscosity, �r, is theratio of the solution viscosity, �, to the solventviscosity, and is unitless. The reduced viscosity,�sp/c, is expressed in units of dl/g, and is definedas

�sp

c ��r 1

c (5)

where c is the solution concentration with units ofg/dl. In this article, we only examine a sampling of

Figure 7 The orientation angles (degrees) of the two modes versus dimensionlessshear rate.


the data available in the literature. This will al-low us to focus on the physics associated with theshear thickening without getting mired down inextensive data fitting. Future work will address afull parameterization of all available experimen-tal data, allowing us to achieve a more thoroughunderstanding of the variation of the mode pa-rameters with concentration, temperature, molec-ular weight, and atomistic polymer structure.

Before comparing the model calculations withthe experimental data, we must point out that the�sp/c versus � curves in ref. 7 were generated bytaking c with units of (wt %) instead of g/dl. Inconsequence, all of the reduced viscosity curves inref. 7 underestimate the actual values of �sp/c.Herein, we use the corrected values for this quan-tity.

In Figure 9, we show the optimized fit of themodel for the corrected polypropylene in tetralindata of Vrahopoulou and McHugh.7 Table I liststhe optimized parameter values for the three tem-peratures examined at a concentration of 0.01 wt%. (Note that we only examine the shear rateregions encompassing the initial shear-thinningand shear-thickening regions here because this iswhere all of the interesting physics takes place.)As evident from the figure, the model overpredicts

the viscosity at low values of the shear rate andunderpredicts it at high values. Nevertheless, theagreement is quite satisfactory with such a sim-ple model. Most likely, we could improve the fit byincorporating more complicated features, such asshear-rate dependent relaxation times or variablemode concentrations, into the model, but thesewould only obscure the essential information thatwe are trying to investigate. Therefore, we are, atpresent, satisfied with our model fits to the exper-imental data.

We find that the model faithfully representsthe general features of the experimental data.The viscosity decreases with increasing tempera-ture for any particular value of the shear rate,and the minima of the viscosity curves occur atincreasing � as the temperature increases.

As one would expect, the mode relaxation timesdecrease with increasing temperature, as do themode concentrations. One can easily rationalizewhy the concentration of the second mode wouldshow this behavior: as the temperature increases,the individual molecules acquire a greater ther-mal kinetic energy, and the supermolecular struc-tures are harder to form. The concentration of thefirst mode, representing the individual polymerchains, decreases with increasing temperature,

Figure 8 The dimensionless first and second normal-stress coefficients as functions ofdimensionless shear rate for the case n � 0.05, � � 3, and � 0.1.


because the concentration in the experiments wasexpressed in weight percentage. Therefore, it var-ied with temperature because the density of thesolvent varied with temperature. By using thevalue for the weight-average molecular weight, 3 106 g/mol, for the polypropylene used in ref. 7,we calculate quiescent concentrations of the firstmode as 0.0306 mol/m3 at 90°C, 0.0302 mol/m3 at105°C, and 0.0298 mol/m3 at 120°C. The opti-mized concentrations for this mode are still twoorders of magnitude below these values. If wewere fitting data in the zero shear-rate limitwhere no (or very small) associations were form-ing, then we might have obtained higher values.At these high shear rates, however, the modeloptimization chose the best values for the param-eters to fit the data over the entire shear-raterange under investigation.

The parameter was practically independentof temperature. This suggests that the degree ofcoupling between the two modes is not affected bythe size of the structures formed under applica-tion of shear. (Note, though, that having a non-zero value of is absolutely required to obtain notonly shear thickening, but any shear-rate depen-dent rheological properties whatsoever.)

With these curves thus parameterized, we canmake predictions for the first and second normal-stress coefficients, which were unobtainable inthese experiments. These are plotted in Figure10. Figure 10(a) depicts the behavior of the firstnormal-stress coefficient plotted versus shearrate. This characteristic function decreases withincreasing temperature and shows both shear-thinning and shear-thickening behavior in thesame shear rate regimes as the shear viscosity

Figure 9 Model data fits for polypropylene in tetralin at the temperatures indicatedand at a concentration of 0.01 wt % for the parameter values listed in Table I.

Table I Parameter Values for 0.01 wt % Polypropylene in Tetralin

T (°C) �1 (s) �2 (s) n1 (mol/m3) n2 (mol/m3)

90 0.005 0.015 1.30 10�4 5.62 10�6 0.12105 0.004 0.012 1.18 10�4 5.20 10�6 0.12120 0.0033 0.010 1.17 10�4 5.00 10�6 0.12


curves. The magnitude of this quantity is rathersmall, leaving little hope at the present time ofobtaining any reliable measurements of its value.The second normal-stress coefficient predicted bythe model is displayed in Figure 10(b). It displaysonly shear-thinning behavior, is negative invalue, and its relative magnitude is two ordersless than that of 1.

More extensive data, containing both viscosityand optical measurements, can be found in thearticle of Kishbaugh and McHugh2 for solutions ofhigh molecular weight polystyrene in decalin.Here, we examine only one of the samples testedin refs. 1 and 2, and only two different concentra-tions—one in this section, and one in the penul-timate one. Again, a full parameterization of allthe available experimental data would only ob-scure the important physical information in abarrage of parameter values, so we shall considerthe parameterization of these additional data setsin future work.

The sample we shall consider has a weight-average molecular weight of 6.8 106 g/mol, andin this section we consider the solution of concen-tration 0.25 g/dl at a temperature of 25°C. At thisconcentration, one would expect that the fluidcontained roughly 0.00368 mol/m3 of polymer.

Figure 11 depicts the TCMM model fit to theshear viscosity, linear dichroism, and the dichroic

orientation angle for this solution with the pa-rameter values given in the figure caption andspecified in Table II. To obtain this figure, the fiveparameters were fit to the shear viscosity curveonly, and the remaining curves generated by us-ing those values, as given in Table II, in themanner explained below. Once again, note thatthe model overpredicts the shear viscosity at lowshear rates and underpredicts it at high ones.Most interesting, however, is the optical behavior.Note that the dichroism displays a maximum atabout the same shear rate as the minimum in theviscosity curve and decreases in magnitude there-after. At very high shear rates, it crosses theshear-rate axis and becomes negative. Also, theorientation angle decreases at low shear rates,levels off at intermediate ones, and then increasesgradually at high shear rates until an abrupt dropin value occurs at a very high shear rate. Themodel provides an accurate qualitative descrip-tion of the solution behavior, and a reasonablequantitative fit as well, all things considered.

The dichroism and its orientation angle werecomputed by using the same five parameter val-ues that were fitted to the viscosity curve. Thedichroism was assumed to arise from two sources:Rayleigh scattering from the structures and theinnate dichroism of the individual molecules.Thus, both modes contribute to the dichroic sig-nal, �n�, the expression for which is divided intotwo contributions, �n� � �n �1 � �n �2, where3

�n �1 �4�

5 k3cNAms

M ��12 �2

2�1�tr c1 3� (6)

and

�n �2 �8�

15 mpn2NAk3��12 �2

2�2

b�1 � 36/�2�

(7)

The first expression, eq. (6), is the innate di-chroism of the deformed polymer chains.3 Thequantities appearing therein are the wavenum-ber, k � 2�/(6.328 10�7 m); the polymerconcentration, c; the molecular weight, M; therefractive index of the solvent, ms � 1.474; andthe polarizability difference, (�1

2 � �22)1 � �1.25

10�42 cm6/molecule. Kishbaugh and McHugh1,3

computed the value of this last quantity by usingthe method of Gurnee.19 However, we have foundthat their rationalization is not particularly sat-isfying because the method is not exact and anorder of magnitude error in the value of this

Figure 10 Model predictions for the first and secondnormal-stress coefficients versus shear rate for polypro-pylene in tetralin at the temperatures indicated.


quantity produces unrealistic results. The twoquantities, �1 and �2, are so small that computingtheir squares is fraught with large errors.1,3 Thevalue of this parameter is then assumed to beindependent of concentration and used in the cal-

culations for the 0.30 dl/g solution below. It turnsout that �n �1 is only important at very low andvery high shear rates.

The second term in eq. (6) is the dichroismarising from the supermolecular structures ac-

Figure 11 The TCMM fit of the experimental viscosity, linear dichroism, and dichroicorientation angle for the 0.25 g/dl solution of polystyrene in decalin. Parameter valuesused to obtain the fit are �1 � 0.004 s, �2 � 0.012 s, n1 � 9.3 10�4 g/dl, n2 � 5.0 10�5 g/dl, and � 0.12.

Table II Parameter Values for Polystyrene in Decalin at 25°C

c (g/dl) �1 (s) �2 (s) n1 (mol/m3) n2 (mol/m3) �aa�0 (m2)

0.25 0.004000 0.012 9.3 10�4 5.0 10�5 0.12 3.06 10�16

0.30 0.004333 0.013 1.0 10�3 1.0 10�6 0.10 4.20 10�16


cording to the Rayleigh scattering theory.3 Thequantities appearing in this expression are therefractive index of the polymer, mp � 1.59, andseveral other functions. The first is an anisotropyfunction that depends on the sphericity or shapeof the structure, p:

b �p2 1p2 � 1 (8)

with p taken as

p � �1 �32 �tr c2 3��3/4

(9)

The quantity � is a dimensionless measure of theshear rate relative to the size and shape of theassumed structures. It is given by3

� ��SVp�p�

kBT � (10)

where VP is the volume of the structure

VP �4�

3p2 a3 �4�

3p2 ��tr c2 3��aa�0�3/2 (11)

and

1�p�

�p2

p4 � 1 ��1 �2p2 1

2p�p2 1ln�p � �p2 1

p �p2 1��(12)

Note that �aa�0 is a parameter that is fit to thedichroism curve by matching the maximum valueof �n� at the appropriate shear rate. After fitting,a can be used to determine the effective structuresize at any shear rate by taking the square root ofthe primary eigenvalue of c2 multiplied by thefactor �aa�0: a � (�p�1)�aa�0.

The last quantity appearing in eq. (7) is thepolarizability difference of the structures, (�1

2

� �22)2. It is given by

16�2

VP2 ��1

2 �22�2 � � 1

L1 � 1/�ms2 1��

2

� 1L2 � 1/�ms

2 1��2

(13)

where

L1 �1 e2

e2 ��1 �12e ln�1 � e

1 e�� (14)

L2 �1 L1

2 (15)

e2 � �1 1p2� (16)

The orientation angle of the dichroism is cal-culated by determining which mode is dominatingthe dichroism. During the shear-thickening re-gion of the flow curve, the structures dominatethe dichroism, and the orientation angle of thedichroism is that of the second mode. At very highshear rates, after the structures were reduced toisotropic spheroids, the individual chains domi-nate the dichroism and the orientation angleswitches to that of the first mode.

The effective structure size a is plotted versusshear rate in Figure 12. This curve was calculatedby fitting the parameter �aa�0 � 3.06 10�16 m2

to the dichroic maximum value at the criticalshear rate. This demonstrates that the size of thestructures is increasing with shear rate at lowshear rates, until the size becomes so large andextended that the shear forces tend to reducetheir aspect ratio. This causes the maximum inthe linear dichroism curve. These structures arequite extended and range in size from about ahundredth to a tenth of a micron over the shear-rate range examined. After the structure is com-pletely spheroidal at very high shear rates, theinnate dichroism of the individual molecules is allthat remains, and this is inherently negative.This causes the crossover to negative values of thedichroism at very high shear rates. Furthermore,at low shear rates, the orientation of the struc-tures decreases toward a saturation value veryclose to the flow direction. As the structures beginto be reduced in size at higher shear rates, theorientation angle increases with increasing shearrate, until the structures are spherical and theorientation angle drops precipitously to that ofthe individual molecules.

The TCMM model descriptions of the linearbirefringence, �n�, and its orientation angle aredepicted in Figure 13. The birefringence is calcu-lated according to

�n� � 2�n1NA

50 ms��tr c1 3��i

� �tr c1 3��fs � 4�e1�f� (17)


where it is assumed that only the first mode hasany appreciable effect on the birefringence. (Thisassumption was tested by Kishbaugh andMcHugh,3 and also by the present authors, bycalculating the birefringence arising from the sec-ond mode in Rayleigh scattering theory. We willnot describe this calculation here because the re-sult did not affect the value of the total birefrin-gence.) A factor of 50 occurs in the denominator ofeq. (17) to bring the prediction of (17) in line withthe experimental data. In other words, it allowsthe result of calculation with eq. (17) to be on thesame order of magnitude with the experimentalbirefringence. It should thus be viewed as an ad-ditional fitting parameter; however, once its valueis set for any one of the polystyrene/decalin solu-tions at a given concentration, it should notchange as concentration, temperature, or polymermolecular weight is varied.

In the above expression, three types of phe-nomena contribute to �n�. The first of these is theintrinsic optical anisotropy of the polymer chains.This effect is represented through �i, which isgiven by20

�i �35 ��1 �2�1 (18)

The polarizability difference, (�1 � �2)1 � �5.49 10�24 cm3/molecule, is assumed to have thevalue estimated by Kishbaugh and McHugh,3 ac-cording to the method of Gurnee.19 [As alluded toabove, there may be up to two orders of magni-tude of uncertainty in this quantity, and this ex-plains the magnitude corrector (50) in Eq. (17).]The intrinsic optical anisotropy of polystyrene isthus a negative quantity.

The second contributor to the overall birefrin-gence is the macroform anisotropy,21 which iscaused by the difference between the refractiveindices of the polymer and solvent. In eq. (17),

�f � �ms2 � 23 � 2 � mp

2 ms2

4�ms�NA�2 M2

(19)

e12 � �1

1p1

2� (20)

where � � 1.065 g/ml, and M2/ � 1.543 1028

g/mol for polystyrene at the specified value of M(6.8 106 g/mol). Note, however, that p1 appear-ing in eq. (20) is now a function of the trace of thefirst mode-conformation tensor, instead of the sec-ond, p1 � (1 � 3/ 2 [tr c1 � 3])3/4.

Figure 12 Model prediction for the average size of structures versus shear rate for the0.25 g/dl polystyrene in decalin solution under consideration.


The third contribution to the birefringence isthe microform anisotropy,22 which increases withthe extension of the polymer chain. It is taken as

�fs �35 �ms

2 � 23 � 2 �mp

2 ms2

4�ms� 2 4�M0�es

�NA(21)

where � � 5.4 is the number of monomeric unitsper statistical chain segment, M0 � 104 g/mol isthe monomer molecular weight, and es � 0.1 isthe optical shape factor for the segment.3 Notethat in the above expressions, no fitting parame-ters appear beyond the five that were used to fitthe TCMM model to the shear viscosity versusshear rate curve and the magnitude corrector ineq. (17).

The effect of the macroform and microformanisotropies is always positive and is dominant atlow shear rates where the polymer chains are notextended or oriented to any large degree. How-ever, at intermediate shear rates, the intrinsicbirefringence of the polymer chains begins todominate the response as the chains become moreand more extended and oriented along the flowdirection. This causes the sign change seen inFigure 13. Considering that no additional param-eters were used for this calculation, other than

the magnitude corrector, the agreement betweenthe model prediction and the experimental dataare quite reasonable. If we had used the correctoras a true fitting parameter, we could have gottenmuch better agreement; however, there seemed tobe no point in doing so.

The orientation angle of the birefringence isalso plotted in Figure 13. This angle was calcu-lated directly from the eigenvectors of c1. Thetheoretical prediction follows very closely the be-havior of the experimental data: it decreases rap-idly from its zero shear rate limit of 45° withincreasing shear rate and saturates very near tothe direction of flow.

Predictions for the first and second normal-stress coefficients for the 0.25 g/dl solution areventured in Figure 14. The same qualitative be-havior is seen here as for the polypropylene solu-tions examined earlier.

A DISCUSSION OF THE DIFFERENCESBETWEEN THE MODEL BEHAVIOR ANDTHE KISHBAUGH AND MCHUGHHYPOTHESIS

Kishbaugh and McHugh3 hypothesized that theshear thickening is manifested in the viscosity

Figure 13 The birefringence and its orientation angle versus shear rate for the 0.25g/dl solution of polystyrene in decalin.


curve because of the growth of optically isotropicparticles at high shear rates. Before the criticalshear rate where the shear-thickening behaviormanifests, the solutions begin to develop micron-size particles composed of amorphous polymerchains that orient along the flow direction. Theseparticles grow in size with increasing shear rate,and, according to the ADA light scattering theory,this causes the maximum in the linear dichroismcurve.3 However, most polymer chains remain insolution, not in the particles, and it is the exten-sion and orientation of these that cause the mono-tonic increase in the linear birefringence withincreasing shear rate.

Kishbaugh and McHugh were led to this hy-pothesis by two notions, one experimental andone theoretical. The experimental one is thatmore concentrated solutions of these polymersthan those discussed above can experience a flow-induced phase separation at large values of theshear rate.23–27 Hence, it seemed reasonable thatthe precursor stage to this phenomenon was caus-ing the shear thickening. Once the phase separa-tion occurred, the separated phase was opticallyisotropic and plainly visible to the naked eye.25–27

Thus, it seemed reasonable to rationalize that theshear thickening arose from the growth of opti-

cally isotropic particles and that the continuedgrowth of these particles with increasing shearrate would eventually lead to phase separation athigher concentrations. Rather than pursue thisidea here, however, we postpone this discussionuntil the next section of the article.

The theoretical notion leading to the Kish-baugh and McHugh hypothesis derived from theADA scattering theory of Meeten,28–31 and it isthe subject of the present section. Under the aus-pices of this theory, the linear dichroism of thescattering centers is essentially a function ofthree variables: the concentration of particles, n2;the shape of the particles, p; and the size of theparticles, a. The theory is valid for particle sizesthat are large relative to the wavelength of theincident light. In the present case, the wave-length is �i � 6.328 10�7 m, so the theory isvalid when 2�a/�i is large, which implies that thescattering particles must be micron-size or largerfor the theory to be valid.

The results of the ADA show that for fixed n2and p, the dichroism varies sinusoidally with in-creasing particle size a. Consequently, Kish-baugh and McHugh3 fit the ADA during the firsthalf-period of its sinusoidal cycle in a to theirlinear dichroism data for the 0.25 g/dl polystyrene

Figure 14 The first and second normal-stress coefficients for the 0.25 g/dl solution ofpolystyrene in decalin.


solution by using n2, p, and a as fitting parame-ters. For fixed values of the first two parameters,they then could plot �n� versus particle size a.They found that the dichroism data versus shearrate could be fit well if a varied between 1.5 and 4�m, with n2 � 9.46 10�14 mol/m3 and p� 1.45. Comparing this with the experimentaldata, they could then correlate increasing particlesize with increasing shear rate.

Interestingly, the experimental data revealthat the maxima in the dichroic curves alwayscorrelate closely with the minima in the viscositycurves. However, there is nothing in the abovehypothesis that explains why this must be thecase: the particle size increases throughout theshear rate range under investigation. In thisviewpoint, therefore, this correlation must be con-sidered as coincidental.

In the TCMM model, one does not have asmuch freedom with the ADA, because two of thefitting parameters used above, n2 and p, are de-termined by the model. The first is given by theoriginal fit of the model to the viscosity data, andthe second is determined by the second mode-conformation tensor [see eq. (9)]. Therefore, theonly fitting parameter in the TCMM ADA calcu-lation is the parameter �aa�0. For micron-size par-ticles, we found that the calculation produced di-chroic behavior with many periods over the ap-propriate shear-rate range. This is so because pbecomes larger, with increasing shear rate, thanthe constant value assumed in ref. 3, and becausethe value of n2 that is required to fit the viscositydata is much larger than the value assumed inref. 3 (see Table II). With such a large value forthe concentration of structures, to get sensibleresults one must have much smaller structures,for which the ADA is no longer valid. Thus, whenusing the TCMM model, we were forced to use theRayleigh scattering theory to explain the behav-ior of the linear dichroism. This theory is validwhen the particle size is approximately �i/ 20,which is exactly the order of magnitude for a thatfit the experimental dichroic data.

Because the particle size actually begins to de-crease, and does not continue to increase, at theminima on the viscosity curves, the TCMM modelprovides a rational explanation as to why theviscosity minima and dichroic maxima always oc-cur at approximately the same shear rates,whereas the ADA does not. Furthermore, the ori-entation angle in the ADA associated with theKishbaugh and McHugh dichroism saturatesalong the flow direction,3 whereas the experimen-

tal data clearly does not (see Fig. 11). Again, oneis forced to the TCMM model to obtain the properqualitative behavior for this quantity.

A DISCUSSION OF THE RELATIONSHIPBETWEEN SHEAR THICKENING ANDPHASE SEPARATION IN POLYMERSOLUTIONS

As mentioned earlier, as one increases the con-centration of the polymer in solution, a thresholdwill be crossed where no further shear thickeningis observed: purely shear thinning is evident overthe entire flow curve. If we increase the concen-tration of the solution under consideration to 0.30g/dl, we see the experimental manifestation ofthis phenomenon in Figure 15.1,2

For these higher concentrations, where thepurely shear-thinning behavior resumes, Kish-baugh and McHugh3 argue that the structures donot form because they would be too large to besupported by the shear field. This is hard to ac-cept because at even higher concentrations, aflow-induced phase separation can occur.23–27 Ob-viously then, some associations are forming,which eventually phase separate from the bulksolution. Furthermore, if the structures causeshear thickening and none are forming, then whyis the viscosity of the 0.30 g/dl solution higherthan that of the 0.25 g/dl solution (compare Figs.11 and 15) in the shear-rate region where thelatter solution exhibits shear thickening? Let usexamine what the TCMM model has to say aboutthis phenomenon.

As evident in Figure 15, the TCMM model alsofits the experimental viscosity data reasonablywell for this purely shear-thinning solution withthe parameter values listed in Table II. The di-chroism saturates at high shear rates, as does theorientation angle. However, the interesting thingabout the model predictions is that, instead ofpredicting a total lack of structures or very smallones, the model actually predicts that structuresare formed. These structures are much larger, butless numerous, than in the 0.25 g/dl solution (seeFig. 16). Furthermore, these structures are notreduced in size by the increasing shear rate, aswas the case for the 0.25 g/dl solution. Hence, noshear thickening is observed for the 0.30 g/dl so-lution.

The above analysis provides a rational expla-nation for flow-induced phase separation in poly-mer solutions. At high enough concentrations,


structures form under shear that become so largethat the shear forces cannot affect them. There-fore, at some critical size and shear rate, phaseseparation is instigated. At lower concentrations,the structures never grow to a size appreciableenough to start phase separation because theyare sheared down by the applied deformation.

For completeness, we also show the birefrin-gence data and model fit for the 0.30 g/dl solutionin Figure 17. As was the case for the 0.25 g/dlsolution, the agreement is remarkably good con-sidering that no additional parameters, otherthan the magnitude corrector, are used to fit thedata beyond those needed to fit the shear viscos-ity. Predictions for the first and second normal-

stress coefficients are given also in Figure 18.Note that there is no apparent shear thickeningin the first normal-stress coefficient for this solu-tion.

CONCLUSIONS

The TCMM model suggests a consistent, rationalexplanation for every facet and nuance of therheological and optical behavior associated withshear thickening in dilute polymer solutions. Fur-thermore, it offers sensible predictions for rheo-logical properties that cannot be measured exper-

Figure 15 The TCMM fit of the experimental viscosity, linear dichroism, and dichroicorientation angle for the 0.30 g/dl solution of polystyrene in decalin. Parameter valuesused to obtain the fit are �1 � 0.004333 s, �2 � 0.013 s, n1 � 1.0 10�3 g/dl, n2 � 1.0 10�6 g/dl, and � 0.10.


Figure 16 Structure size versus shear rate for the 0.30 g/dl polystyrene solution.

Figure 17 Linear birefringence and its orientation angle versus shear rate for the0.30 g/dl polystyrene solution.


imentally because of apparatus limitations. Themain conclusions of this article are the following.

Shear thickening is only observed in a givenconcentration window, dependent upon the mo-lecular weight of the polymer in solution. Belowthis window, only shear thinning is observed, asany structures formed are too small to affect ei-ther the rheological or the optical response of thesolution. Within this window, structures form atlow shear rates, increasing in size with increasingshear rate, and then begin to shrink near thecritical shear rate where shear thickening begins.These structures are anisotropic and are one totwo orders of magnitude smaller than micron sizeat maximum dimensions. The higher the shearrate, the more elongated they become, until ulti-mately they are reduced in size and aspect ratioby the increasing shear forces. For concentrationsabove this window, the structures continue to in-crease in size with increasing shear rate, eventu-ally culminating in phase separation.

Perhaps the most interesting point, however, isthat the model indicates that shear thickening isnot at all related to phase separation, as assertedin the past, but is actually the antithesis of thesame: shear thickening only appears when theassociations prevalent in the solution are de-

stroyed, not formed. One must first raise the so-lution concentration beyond the region whereshear thickening occurs to see any phase separa-tion whatsoever.

In conclusion, it seems fair to say that thehopeful prophecy of Kishbaugh, quoted at the be-ginning of the introduction, may have come topass. Future work should focus on an understand-ing of the TCMM model in terms of more detailedmolecular- and atomistic-based formulations, sothat the internal physics of this structure forma-tion may be evaluated.

REFERENCES

1. Kishbaugh, A. J. Ph.D. Dissertation, University ofIllinois, 1992.

2. Kishbaugh, A. J.; McHugh, A. J. Rheol Acta 1993,32, 9.

3. Kishbaugh, A. J.; McHugh, A. J. Rheol Acta 1993,32, 115.

4. Beris, A. N.; Edwards, B. J. Thermodynamics ofFlowing Systems; Oxford Univ. Press, New York,1994.

5. Edwards, B. J.; Beris, A. N.; Mavrantzas, V. G.J Rheol 1996, 40, 917.

Figure 18 The first and second normal-stress coefficients versus shear rate for the0.30 g/dl polystyrene solution.


6. Layec-Raphalen, M.; Wolff, C. J Non-NewtonianFluid Mech 1976, 1, 159.

7. Vrahopoulou, E. P.; McHugh, A. J. J Non-New-tonian Fluid Mech 1987, 25, 157.

8. Bird, R. B.; Curtiss, C. F.; Armstrong, R. C.; Has-sager, O. Dynamics of Polymeric Fluids, 2nd ed.;John Wiley and Sons: New York, 1987; Vol. 2.

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M. N. Polymer 1979, 20, 281.14. Wolff, C. Adv Colloid Interface Sci 1982, 17, 263.15. Vrahopoulou, E. P.; McHugh, A. J. J Rheol 1987,

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19. Gurnee, E. F. J Appl Phys 1954, 25, 1232.20. Kuhn, W.; Grun, F. Kolloid-Z 1942, 101, 248.21. Copic, M. J Chem Phys 1957, 26, 1382.22. Tsvetkov, V. N. Sov Phys Uspekhi 1964, 6, 639.23. Ver Strate, G.; Philippoff, W. J Polym Sci, Polym

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AnExaminationoftheShear-ThickeningBehaviorofHigh ... · havior manifests, the solutions begin to develop micron-size, optically isotropic particles composed of associated polymer