Rheology of hyperbranched polymer melts undergoing planar Couette flowT. C. Le,1 B. D. Todd,1,a� P. J. Daivis,2 and A. Uhlherr3

1Centre for Molecular Simulation, Swinburne University of Technology,P.O. Box 218, Hawthorn, Victoria 3122, Australia2School of Applied Sciences, RMIT University, GPO Box 2476V, Melbourne 3001, Victoria, Australia3CSIRO Information Management and Technology, Private Bag 33, Clayton South, Victoria 3169, Australia

�Received 1 April 2009; accepted 2 July 2009; published online 23 July 2009�

The melt rheology of four hyperbranched polymer structures with different molecular weights hasbeen studied using nonequilibrium molecular dynamics �NEMD�. Systems were simulated over awide range of strain rates to capture the crossover behavior from Newtonian to non-Newtonianregimes. Rheological properties including shear viscosity and first and second normal stresscoefficients were computed and the transition to shear thinning was observed at different strain ratesfor hyperbranched polymers of different sizes. The results were consistent with previous findingsfrom NEMD simulation of linear and dendritic polymers. Flow birefringence was characterized bytaking into account both form and intrinsic birefringences, which result from molecular and bondalignment, respectively. The stress optical rule was tested and shown to be valid only in theNewtonian regime and violated in the strong flow regime where the rule does not take into accountflow-induced changes of the microstructure. © 2009 American Institute of Physics.�DOI: 10.1063/1.3184799�

I. INTRODUCTION

Dendritic polymers including dendrimers and hyper-branched polymers are a class of highly branched polymersthat have unique properties due to their moleculararchitectures.1 Dendrimers with perfectly branched structuresare more difficult to synthesize due to the requirement ofchemical purity and stringent multiprocessing steps. In con-trast, hyperbranched polymers with imperfect branching canbe prepared easily using “one-pot” techniques that save timeand are suitable for large scale, low cost production. How-ever the one-pot techniques provide a polydisperse mixtureof randomly branched polymers with different size and to-pology. This leads to difficulties in experiments but it givessimulation a valuable opportunity to shed light on the struc-ture and rheology of monodisperse melts of hyperbranchedpolymers. In our previous report,2 the structural properties offour hyperbranched polymer topologies under shear flowwere analyzed. In this subsequent paper, the rheologicalproperties of those fluids will be discussed.

Possessing special structures, dendritic polymers includ-ing dendrimers and hyperbranched polymers have many in-teresting properties. One of the most interesting physicalproperties is the unusual melt rheology, which suggests thatthese macromolecules have potential application as rheologymodifiers and processing aids.3 In comparison with linearpolymers, dendrimers and hyperbranched polymers possessmuch lower viscosity and very high solubilities due to theirpacked structure with a large number of functional endgroups in one molecule. It has been demonstrated that den-dritic polyether and poly�amidoamine� melts exhibit aNewtonian behavior over the available measurement range in

experiments and the lack of entanglements in the systemsobserved suggests that as the dendritic topology leads toglobular conformations, molecules do not interpenetrate sig-nificantly in the melt and do not combine to form aggregatedstructures.4 In the case of dendritic polymers in solution, apeak is observed in the plot of intrinsic viscosity versus mo-lecular mass.5 This is in contrast with linear chains whichobey the Mark–Houwink equation6 and show a steady in-crease in the intrinsic viscosity as a function of molecularweight.

Although the number of experimental papers on hyper-branched polymers has increased considerably, only a few ofthem focus on rheology and especially melt rheology ofthese materials. One of the earliest studies on melt rheologyof hyperbranched polymers was reported by Farringtonet al.7 in 1998. The melt viscosities of dendritic poly�benzylether� including dendrimers and some end-substituted mono-and tridendrons were measured. Hsieh et al.8 then presentedrheological properties of processed commercial aliphatic hy-perbranched polyester melts. Shear thinning was observedfor the lower generation, whereas Newtonian behavior wasexhibited by the higher generation hyperbranched polyesters.It was also found that blends of these commercial polyestersshowed only Newtonian behavior under both steady shearand oscillatory shear if at least one of them is Newtonian.Another early study on melt rheology reported byKharchenko et al.9 characterized the behavior of hyper-branched polystyrenes. This was also the first publication onflow birefringence of this polymer. It was found that hyper-branched polystyrenes showed nonterminal behavior in thelow-frequency rheological response and the stress opticalrule �SOR� was only valid for polymers of high molecularweights. Moreover it was shown that when this rule holds,the stress optical coefficient for hyperbranched polystyrenesa�Electronic mail: btodd@swin.edu.au.

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was much lower than that for linear analogs. After that,Suneel et al.10 reported their studies on melt rheology andmolecular weight distribution of short chain hyperbranchedaromatic polyesters. These papers confirmed that in hyper-branched polymer melts the chains are essentially unen-tangled. Another study on hyperbranched polyesters reportedby Gretton-Watson et al.11 showed that the viscosity of hy-perbranched poly�methylmethacrylate� melt was signifi-cantly lower than that of linear chains and exhibited shearthinning behavior.

As presented in our previous paper,2 a few molecularsimulation studies on hyperbranched polymer simulationhave been reported. However none of them focused on meltrheology. Therefore the purpose of this paper is to report thefirst study on the melt rheology of hyperbranched polymersusing a coarse-grained model and nonequilibrium moleculardynamics �NEMD� simulations.

A simple specific architecture of hyperbranched poly-mers has been studied. The hyperbranched polymers gener-ated have the same molecular weight �number of beads� asperfect trifunctional dendrimers but fewer branches at onebranching point and extra beads added in the outermost layerof the molecules. Beads along the chain interact viaWeeks–Chandler–Anderson12 �WCA� and finitely extensiblenonlinear elastic13 �FENE� potentials. More details of thehyperbranched polymer topologies and bead spring modelcan be found in our previous report.2

In the remainder of this paper, all quantities are ex-pressed in terms of site reduced units in which the reductionparameters are the Lennard-Jones interaction parameters �and � and the mass mi� of bead � in molecule i, which areset to the value of one. In terms of the corresponding quan-tities in real units, the reduced temperature is given byT�=kBT /�, the density by ��=��3, the pressure tensor byP�=P�3 /� and strain rate by ��= �mi��2 /��1/2�. For sim-plicity of notation, the asterisk will be omitted hereafter.

To simulate fluids under shear, the molecular version ofthe SLLOD algorithm14 was applied and the equations ofmotion are given as

ri� =pi�

mi�+ ri · �u , �1�

pi� = Fi� − �mi�/Mi�pi · �u − ��mi�/Mi�pi, �2�

where ri� and pi� represent the position and thermal momen-tum of an bead � in molecule i, �u is the strain rate tensor,ri=��=1

N� mi�ri� /Mi is the position of the molecular center ofmass of molecule i, Mi=��=1

N� mi� is the mass of molecule i,pi=��=1

N� pi� is the momentum of the molecular center ofmass of molecule i, and � is the thermostat constraint multi-plier which is given by

� =�i=1

N Fi · pi − ��i=1N pixpiy

�i=1N pi

2 . �3�

This expression for � is derived from Gauss’ principle ofleast constraint and used to keep the molecular center ofmass kinetic temperature constant. Throughout the simula-tion, there was only one single thermodynamic state point

defined by the reduced bead density of 0.84 and the reducedtemperature of 1.25. The equations of motion of atoms wereintegrated with time step �t=0.001 in reduced units using afifth-order Gear15 predictor corrector differential equationsolver. For hyperbranched polymers comprising 19 beads,NVT simulations were performed with 216 molecules,whereas for polymers comprising 43, 91, and 187 beads,NVT simulations were run with 125 molecules. Each simu-lation began with an equilibration period and 40 separateruns, which consist of 1�106 time steps each performed forevery system. The mean simulation data were evaluated fromresults of all the separate runs.

In Sec. II A of this paper, detailed analysis and discus-sion of rheological properties for four different hyper-branched polymers with molecular weight lying between 19and 187 are presented. Section II B focuses on the flow bi-refringence effect for hyperbranched polymers, which resultsfrom contributions from the form and intrinsic birefringence.In part C of Sec. II, comparisons with predictions of theSOR are shown. Finally, Sec. III presents a summary andconclusions.

II. RESULTS AND DISCUSSION

A. Rheology

The rheological properties of hyperbranched polymerfluids under shear flow, such as the shear viscosity and firstand second normal stress coefficients, can be calculated fromthe components of the molecular pressure tensor16 given by

PMV =��i=1

Nm pipi

Mi−

1

2�i=1

Nm

��=1

n

�j�i

Nm

�=1

n

rijFi�j� , �4�

where pi represents the total peculiar center of mass momen-tum of molecule i, as defined by the equations of motion,rij =r j −ri is the center of mass separation of molecule i andj, Fi�j is the intermolecular force on bead � in molecule idue to bead in molecule j and n is the total number ofinteracting beads in a molecule. The angular brackets denotean average over the nonequilibrium steady state. The non-Newtonian shear viscosity of hyperbranched polymer fluidssubject to planar shear flow, where the fluid flows in the xdirection with velocity gradient in the y direction, can becalculated from the components Pxy and Pyx of the molecularpressure tensor PM as =−�Pxy + Pyx /2�.

The viscosities at various shear rates computed for dif-ferent hyperbranched polymers using NEMD simulation arepresented in Fig. 1. As can be seen, at the same strain rate,larger hyperbranched polymers have higher viscosity values,indicating that under shear they have longer relaxation times.It can also be seen that the range of strain rates considered islarge enough to capture the shear thinning behavior of allsimulated hyperbranched polymer systems. At low strainrates, the viscosities remain constant whereas at high strainrates, these values decrease rapidly. This property of poly-meric fluids is very important for many engineering applica-tions. Shear viscosity data for different hyperbranched poly-mers in Fig. 1 were fitted using the Carreau–Yasuda model6

which is given by =0 / �1+ ����2�p, where 0 is the zero

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shear viscosity, � is a time constant and p is the power lawexponent. Table I presents the fitting parameters obtained forall simulated hyperbranched polymer systems. Apart fromusing the Carreau–Yasuda model, an alternative way to ob-tain the zero shear rate viscosity is by using the Crossequation,17 given as =�+ �0−�� / �1+ �K��m�, where �

is the infinite shear viscosity, K is the consistency index, andm is the power law index. The disadvantage of the Crossmodel is that in experiment, it is difficult to measure theinfinite shear rate viscosity �, therefore it is often set to bea very small value.18 Figure 2 presents the fitted lines ob-tained from the Cross equation in comparison with theCarreau–Yasuda equation for shear viscosity data of hyper-branched polymers comprising 187 monomers. It can be seenthat these two models agree very well. Fitted lines areslightly different at very high shear rates, in the transitionregion from Newtonian to non-Newtonian and at very lowshear rates. Cross equation fitting parameters for shear vis-cosity data are shown in Table II. The zero shear viscositiesfor hyperbranched polymers composed of 19, 43, and 91beads obtained from Cross equation fitting are quite similarto those from the Carreau–Yasuda model.

Figure 3 presents the dependence of the zero shear rateviscosity on the number of beads per molecule for differenthyperbranched polymers. The zero shear rate viscosity scalesas 0 N0.657�2� where N is the number of monomers/beadsper molecule, where by 0.657�2� we mean 0.657�0.002.The exponent of the power law for hyperbranched polymermelts is consistent with that for dendrimers which were

found to have 0 N0.646�2� �Ref. 19� due to similar molecularstructures. Furthermore, our results suggest that hyper-branched polymers are free of entanglements as the depen-dence of viscosities on the number of monomers does notbreak into two regions at low and high number of monomerswhile most of the linear analogs have viscosity dependenciesgiven as 0 Mw at low molecular weights and 0 N3.4 athigh molecular weights where the polymer chains entangle.20

However as WCA and FENE potentials were employed tosimulate hyperbranched polymers, beads along the polymerchain can vibrate and rotate freely. This results in a polymermodel that is more flexible than typical real materials andmodel molecules in shearing dense fluids may fold uponthemselves more than in reality. It has been found experi-mentally that the value of the power law exponent for den-drimers is 1.1 �Refs. 7 and 21� which is quite high com-pared to that from NEMD simulations, which was found tobe 0.646�2�. Therefore higher shear viscosity and higher val-ues of the power law exponent are expected in experimentson common hyperbranched polymers. However the compari-son between our model and real polymers should be made interms of the number of Kuhn steps per molecule, defined asNK= �Ns−1�2 /C�, where NS is the number of beads and C� isthe characteristic ratio. The value of NK makes it possible toestimate the molar mass of the real polymer with an equiva-lent NK to the polymer simulated. Our model and a closelyrelated model with constrained bond lengths22 have the char-acteristic ratio of 1.5–1.8, which can be compared withpolyethylene which has a characteristic ratio of about 7.2.

The time constant � in the Carreau–Yasuda model fitted

FIG. 1. Dependence of shear viscosity on strain rate for hyperbranchedpolymers of different molecular weights �solid lines representing fitting withthe Carreau–Yasuda model�.

TABLE I. Parameters of the Carreau–Yasuda model fitted to the shear vis-cosity vs strain rate dependence.

Number of beads 0 � p

19 10�2� 101�8� 0.138�1�43 18�3� 378�237� 0.143�5�91 31�11� 1069�299� 0.148�8�

187 48�6� 1422�615� 0.158�2�

FIG. 2. Carreau–Yasuda equation vs Cross equation fitted for shear viscositydata for hyperbranched polymers composed of 187 monomers.

TABLE II. Parameters of the Cross equation fitted to the shear viscosity vsstrain rate dependence.

Number of beads 0 � K m

19 10�2� 4�1� 24�9� 1.3�4�43 18.6�3� 3.2�9� 60�10� 0.78�9�91 35�1� 2�1� 178�20� 0.56�5�

187 53�2� 6�1� 310�41� 0.71�8�

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to the simulation data is also the longest relaxation time ofmolecules composing the fluids. Although the longest relax-ation time �0 can be the rotational relaxation time or thereptation relaxation time, �0 can be considered to be the ro-tational relaxation time as there is no entanglement in thesehyperbranched polymer systems and hence there is no repta-tion relaxation time. On the other hand, the inverse of thetime constant � is the strain rate �0 at which the onset ofshear thinning is observed. For hyperbranched polymerswith 19, 43, 91, and 187 beads, the values of �0 are 9.9�5��10−3, 2.6�4��10−3, 9.35�3��10−4, and 7.032�2��10−4,respectively. The shear thinning behavior occurs at lowerstrain rates for large hyperbranched molecules in comparisonwith small hyperbranched polymers. This can be explainedby relaxation times, the deformation and tendencies to alignwith the shear flow. Shear thinning occurs due to the defor-mation and increased alignment of the molecules to the flowfield, which is directly related to the relaxation time �0.23 Atshear rates higher than the inverse of �0, molecules arestressed and in order to reduce the stress, the molecules ex-tend, the deformation occurs and the alignment increases, thefluid moves into the non-Newtonian regime and the newstructural configuration results in lower viscosity. Becauselarger molecules have longer relaxation time �0, the value of1 /�0 is smaller than that of small molecules and the fluidscomposed of large polymers exhibit the crossover fromNewtonian to non-Newtonian behavior at lower strain rates.The stretching caused by the applied shear results in a moreellipsoidal configuration of hyperbranched polymers whichhas been discussed in detail in our previous paper.2

Figure 4 presents the dependence of longest relaxationtime �0 on the number of beads per molecule N for fourhyperbranched polymer systems. The logarithm of �0 for hy-perbranched polymers composed of 19, 43, and 91 beads hasa linear relationship with the logarithm of the number ofbeads N. The relaxation time for these systems scales as�0 N1.4�2�. The exponent value of 1.4�2� for hyperbranchedpolymers is consistent with previous NEMD simulation re-sults which found that the exponent value is 1.7 for linearpolymers and 1.3 for dendrimers.24 Our exponent result is

lower than those predicted by the Rouse model which gives�0 M2.25 On the other hand the value of �0 for the largesthyperbranched polymer system comprising 187 beads doesnot follow the same trend of the smaller polymers. The maincauses might be the increase in structural rigidity with thenumber of beads N and the flow-induced stretching behaviorof hyperbranched polymers which results in the more as-pherical shape of small molecules and more spherical shapeof large molecules. This is opposite to the behavior of linearmolecules. For long linear polymers above Ncritical, the long-est relaxation time, which is predicted to scale as N3 by thereptation theory, results from the entanglement of longchains.26 This is less likely to be the case for hyperbranchedpolymers with short branches and a large number of terminalgroups. Furthermore our computed values of the longest re-laxation time have large uncertainties due to the multistepprocedure used to obtain them.

The Weissenberg number We, which is a dimensionlessshear rate, can be calculated from the longest relaxation timeby the definition We= ��0.6 Unlike the Deborah numberwhich is used to describe flows with a nonconstant stretchhistory, the Weissenberg number describes the flow with aconstant stretch history. Figure 5 presents the dependence ofthe ratio of shear viscosity and zero shear rate viscosity onthe Weissenberg number for different hyperbranched poly-mers. It can be seen that the ratios of shear viscosity to zeroshear rate viscosity obtained from NEMD simulations forfour hyperbranched polymer systems show a significant de-gree of consistency with a single master curve. At We�1,the value of /0 for all polymers starts to decrease and thefluids move from the Newtonian to non-Newtonian regime.The computed data were fitted with the Carreau–Yasudamodel and give a dependence of /0 to the Weissenbergnumber as /0=0.996�9� / ��1+ �1.1�1��We�2��0.143�6�. Us-ing the master curve established, the shear rate dependenceof the viscosity of other hyperbranched polymers in this se-ries could be predicted.

The first and second normal stress coefficients, whichcan be defined as �1= �Pyy − Pxx / �2 and �2= �Pzz− Pyy / �2,are presented in Fig. 6. These values describe the effect of

FIG. 3. Zero shear viscosity vs number of beads per molecule for hyper-branched polymers.

FIG. 4. Longest relaxation time vs number of beads per molecule for hy-perbranched polymers.

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the normal stress differences exhibited by polymeric fluids.Unlike Newtonian fluids which have the normal stress differ-ences exactly zero in shearing flow, polymeric fluids alwayshave positive first normal stress coefficient which is usuallymuch larger than the magnitude of the negative second nor-mal stress coefficient.6 An increase in these stress coeffi-

cients reflects a tendency of the fluid to deform in the normaldirections under shear. For all hyperbranched polymer sys-tems studied, the normal stress coefficients have a largepower-law region in which it might decrease by as much as afactor of 106 for a large molecular system. Furthermore, therate of decline of the first and second normal stress coeffi-cients with the strain rate is much greater than that of theshear viscosity. In the low shear rate region, the normal stresscoefficients reach a plateau as the ratios of Pyy − Pxx andPzz− Pyy are proportional to �2, whereas at higher shear rates,the values of the normal stress coefficients rapidly decrease.This again indicates that the hyperbranched polymer fluidhas a crossover from Newtonian to non-Newtonian behavior.At the same strain rate, larger hyperbranched polymers havehigher first and second normal stress coefficients as they tendto deform more in comparison with the small molecules. Thesecond normal stress coefficient, which has been studied lessextensively in experiments than the first coefficient, is nega-tive and much smaller in magnitude than the first normalstress coefficient. In comparison with other simulationresults,19 our normal stress coefficient values for hyper-branched polymers are larger than those for dendrimers butsmaller than those for linear polymers.

Data for the first normal stress coefficient for hyper-branched polymers consisting of 19 and 43 beads were fittedusing the Carreau–Yasuda model. The zero shear rate firstnormal stress coefficients �1,0 for these systems were foundto be 508�13� and 2430�45�, respectively. From these valuesand zero shear rate viscosities 0 obtained from the Carreau–Yasuda fit for shear viscosity, the viscous relaxation time �vcan be computed. The viscous relaxation time, which is de-fined by �v=�0

�tG�t�dt /�0�G�t�dt, where G�t� is the stress re-

laxation modulus from the theory of linear viscoelasticity,27

can be calculated by the expression �v=�1,0 /20. For hyper-branched polymers comprising 19 and 43 beads, the valuesof �v were found to be 25�3� and 67�7�, respectively. Thesevalues are in very good agreement with the time constants Kobtained from the Cross model which were found to be 24�9�and 60�10� for those hyperbranched polymer systems.

The normal stress coefficients for all hyperbranchedpolymers were fitted in the power-law region and the valuesof the exponents of the asymptotic dependences �1 �−� and �2 �− are presented in Table III. The values of these ex-ponents are within the range of experimental values for poly-mer melts and concentrated solutions.6 Similar to NEMDsimulation results for dendrimer melts,24 the values of the �and exponents do not vary systematically with the size ofthe hyperbranched polymer molecules. This is in contrastwith results for linear polymers24 which have the values of �and increasing with the chain lengths.

FIG. 5. Dependence of the ratio /0 on the Weissenberg number for dif-ferent hyperbranched polymers.

FIG. 6. First and second normal stress coefficients vs strain rate for differenthyperbranched polymers.

TABLE III. Estimated values of the exponents in the power-law regions forthe first and second normal stress coefficients.

Number of beads �

19 1.09�4� 0.96�2�43 1.09�2� 1.0476�9�91 1.05�3� 0.96�2�

187 1.00�2� 0.91�3�

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The ratios of −�2 /�1 are presented in Fig. 7, whichshows that the values of −�2 /�1 are 0.2 for all hyper-branched polymer systems. This is very similar to simulationresults for dendrimers as they all have compact, highlybranched architectures with globular shape and internalbond constraints which prevent pronounced stretching of themolecules.

Figure 8 shows the isotropic pressure of the hyper-branched polymer systems under shear which was calculatedas

p = 13Tr�PM� = 1

3 �Pxx + Pyy + Pzz� . �5�

At low strain rates, the pressure of smaller hyperbranchedpolymer fluids is higher than that of larger polymer fluids,while at high strain rates, the pressure of the large moleculefluids increases earlier and more rapidly than that of thesmall ones. This can be explained by the behavior of differ-ent molecules under shear as presented in our previouspaper,2 that larger molecules of hyperbranched polymers be-come aspherical more quickly under shear than for small

molecules. Fitting the pressure data using the Carreau–Yasuda model p= p0 / �1+ ��p��2�m which has been appliedfor shear viscosity data, the zero shear pressure of hyper-branched polymers can be investigated. The values of thezero shear pressure p0 are 5.540�3�, 5.414�3�, 5.349�6�, and5.299�8� for hyperbranched polymers composed of 19, 43,91, and 187 monomers, respectively. The critical strain rateat which the transition from Newtonian to non-Newtonianbehavior of the pressure occurs can be evaluated as the in-verse of �p obtained from the Carreau–Yasuda model. Thecritical strain rate was found to have the value of 0.217�2�,0.126�5�, and 0.06�2� for hyperbranched polymers compris-ing 43, 91, and 187 monomers, respectively. For hyper-branched molecules composed of 19 monomers, the value of�p obtained is very small and the value of the standard erroris large, hence the critical strain rate calculated is uncertain.In comparison to the critical strain rate for viscosity, thevalue of �critical for pressure is higher because a higher shearrate is required to distort the radial distribution function thanto distort the whole molecule. The trend of pressure changesdue to strain rate observed here for hyperbranched polymermelts is similar to that seen previously in dendrimer meltsand falls within the range between dendrimer and linearpolymer melts.19 This is because dendrimers have the mostcompact architecture, whereas hyperbranched polymers areless compact and linear polymers have the largest spatialseparation of monomers. This behavior can be found notonly for dendritic polymer melts but also for solutions. It hasbeen reported by Lue28 that dendritic polymers in low con-centration solution have lower pressure than linear polymers.In addition, it was found that concentrated solutions of low-generation dendrimers show similar behavior to linear poly-mers and the pressure increases more rapidly with concen-tration for high-generation dendrimers.

B. Flow birefringence

In the presence of a velocity gradient, the statistical dis-tribution of a flexible polymer is deformed from the equilib-rium isotropic state and the refractive index of the solutionbecomes anisotropic. This phenomenon is called flow bire-fringence or the Maxwell effect. The birefringence of a poly-mer system due to the alignment of the intramolecular bondsis called intrinsic birefringence whereas that caused by thealignment of the whole molecules is called formbirefringence.25 The contribution to the birefringence effectfrom the alignment of molecules is more important in thecase of solutions and arises due to the differences betweenthe polarizability of the molecules and the solvent.

1. Form birefringence

In order to characterize the flow induced molecularalignment of hyperbranched polymers, the molecular ordertensor Sm has been computed as

Sm = �i=1

N �uiui −1

3I� ,

where ui is the unit vector denoting the orientation of thesingle molecules and N is the total number of molecules in

FIG. 7. Ratio of the second and first normal stress coefficients for differenthyperbranched polymer systems.

FIG. 8. Dependence of the isotropic pressure on strain rate for differenthyperbranched polymers �solid lines representing fitting with the Carreau–Yasuda model�.

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the system. The direction in which molecules are aligned isindicated by the eigenvectors of the order tensor. Assumingthe eigenvector corresponding to the largest eigenvalue ofthe tensor of gyration denotes the orientation of the moleculeui, the birefringence extinction angle will be the angle be-tween the flow direction and the molecular alignmentdirection.

Figure 9 shows the molecular alignment angle for differ-ent hyperbranched polymers. It can be seen that all simulatedsystems reach the Newtonian region with the alignmentangle of 45° in the range of considered strain rates. The 45°angle is expected for systems in the Newtonian regime dueto the nonuniform spin angular velocity of molecular rotationin shear flow. Furthermore the alignment angle of large hy-perbranched polymers departs from 45° only at low strainrates whereas the alignment angle of small polymers remainsclose to 45° until higher strain rates are reached becausethese small polymers can rotate with the flow more easily. Incomparison with other NEMD simulation data, our align-ment angles in the non-Newtonian region are smaller thanthose for dendrimers and larger than those for linear poly-mers of the same molecular weight. This is because dendrim-ers have the most compact and constrained structure whilehyperbranched polymers have less rigid architecture and lin-ear polymers can stretch and align more easily with respectto the flow field, leading to anisotropic friction.

The order parameter Sm which describes the extent of themolecular alignment can be defined as 3/2 of the largest ei-genvalue of the order tensor, which is a measure of the an-isotropy of the average inertia tensor of a flexible moleculecaused by the shear field.25 This value equals zero in the caseof orientational disorder and reaches one for perfect align-ment. Figure 10 presents the molecular order parameter ofdifferent hyperbranched polymers. It remains constant at lowstrain rates and rapidly increases at high strain rate regions.This indicates that the orientational ordering increases andthe alignment of the polymeric chains is more pronounced athigh strain rates. It can also be seen that for any given strainrate larger N polymer systems have larger values of Sm incomparison to smaller N polymer systems. However when

the number of beads increases, the gap between the valuesof Sm for hyperbranched polymers decreases. For the twolargest systems of simulated hyperbranched polymers, theorder parameter curves almost overlap. Furthermore, at thehighest strain rate of 0.2, the order parameters for polymerscomprising 43, 91, and 187 beads reach the same value of0.73.

2. Intrinsic birefringence

In order to characterize the intrinsic birefringence of hy-perbranched polymer systems, the flow induced bond align-ment has been analyzed. The bond alignment tensor can becalculated as

Sbond = �i=1

Ns−1 �uiui −1

3I� ,

where Ns is the total number of beads in one molecule and ui

is the unit vector between neighboring beads which can bedefined as

ui =ri+1 − ri

ri+1 − ri .

The flow alignment angle and the extent of the bond align-ment can be calculated similarly to those of the molecularalignment in Sec. II B 1.

Figure 11 presents the bond alignment angle results fordifferent hyperbranched polymers and linear polymers ofequivalent molecular weight. As can be seen, the range ofconsidered strain rates is wide enough for all hyperbranchedpolymer systems to reach the Newtonian regime where thebond alignment angle � is 45°. In contrast, for large linearpolymers of 91 and 187 beads per molecule, the bond align-ment angle cannot reach 45° in the considered range of strainrate. In order to reach the alignment angle of 45°, the sys-tems would have to be simulated at lower strain rates. It canalso be seen that in the non-Newtonian region, the bondalignment angle decreases with increasing strain rate. At agiven strain rate, the bond alignment angle of larger mol-ecules is smaller than that of the smaller ones. Our data for

FIG. 9. Molecular alignment angle for hyperbranched polymers of differentmolecular sizes.

FIG. 10. Order parameter of the molecular alignment tensor for differenthyperbranched polymers.

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linear polymers are in good agreement with other NEMDsimulation results29 which indicated that the bond alignmentof systems comprising no more than 60 beads can reachthe Newtonian regime in the range of strain rate we haveinvestigated.

Alignment in the shear plane can be characterized by thevalues of Sbond, which are shown in Fig. 12. It can be clearlyseen that for both hyperbranched and linear polymer sys-tems, all these values increase with increasing strain rates,and at the same strain rate, values for larger polymers arealways higher than those for smaller polymers. This impliesthat for a given strain rate, the chain segments in large mol-ecules can more easily stretch and align with respect to theflow field. The Sbond function of shear rate is monotonicallyincreasing, but at high strain rates the values of Sbond becomethe same for all hyperbranched polymer systems while thosevalues are much higher for large linear polymers in compari-son with small ones. Furthermore, at a given strain rate, thealignment parameter for hyperbranched polymers is alwayslower than that for linear polymers. This is because it is moredifficult for hyperbranched chain segments to stretch andalign with respect to the flow field as they have a more com-pact and constrained architecture. Our alignment results forlinear polymers show good agreement with other NEMD

simulation results29 although our data show slightly strongeralignment due to the difference in temperature and chainlength of the systems.

C. Stress optical rule

In order to characterize the relationship between macro-scopic stress with the microscopic processes such as rota-tions and deformations of bonds,30 the SOR can be testedfrom the bond alignment tensor and stress tensor. It statesthat the mechanical and optical tensors are coaxial and pro-portional to each other.

Figure 13 presents the ratio of �Pxx− Pyy� and �Sxx−Syy�,as well as that of �Pyy − Pzz� and �Syy −Szz�, where Pxx, Pyy

and Pzz are components of the pressure tensor and Sxx, Syy,and Szz are components of the bond alignment tensor, forhyperbranched polymers of different molecular weight. Ascan be seen, the components of the stress and alignmenttensor are proportional at low strain rates. The proportional-ity constant, which is called the stress optical coefficient, isindependent of molecular weight. For all hyperbranchedpolymer systems, this coefficient has the value of 3.3�3�.The independence of the stress optical coefficient on molecu-lar weight for simulated hyperbranched polymers is in goodagreement with experimental results which showed that thestress optical coefficient is a function of the local condition

FIG. 11. Flow alignment angle for linear and hyperbranched polymers.FIG. 12. Bond order parameter for linear and hyperbranched polymers.

044902-8 Le et al. J. Chem. Phys. 131, 044902 �2009�

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such as the temperature, solvent or polymer concentrationbut does not depend on the features of molecular structure ona large length scale such as molecular weight, molecularweight distribution, branching and degree of cross-linking.25

At high strain rates, the SOR is invalid for all simulatedhyperbranched polymers. The reason is that the microstruc-ture of hyperbranched polymer melts is not included in thebond order tensor whereas the stress is determined by thealignment, deformation and microstructure of the systems.The SOR does not take into account the flow-inducedchanges of the radial distribution function which is distortedat high strain rates.

It has been discussed elsewhere that the failure of theSOR could be due to the influence of the molecular weightdistribution,31 the chain conformation,32 or the role of thenumber of entanglements in the fluid.33 There are also certaintypes of systems that do not follow the SOR at all such asrodlike polymers in the concentrated isotropic regime.34 Inexperiments, the experimental conditions can also lead to thefailure of the SOR, such as performing experiments near theglass transition temperature30,35 or at a rate higher than theinverse of the Rouse time of the chain in an elongationalexperiment.36 The SOR fails also because while birefrin-

gence saturates, the stress can continue to grow withoutlimit.37 These factors are not present in our study, hence wecan deduce that the SOR is violated as flow-induced changesof the microstructure of hyperbranched polymers were nottaken into account.

III. CONCLUSIONS

In this work, NEMD simulations have been performedfor four different hyperbranched polymer structures with mo-lecular weight varying from 19 to 187. Rheological proper-ties of these polymer melts were characterized and the cross-over from Newtonian to non-Newtonian behavior wascaptured in the considered range of strain rates. Computedshear viscosities are slightly higher than those for dendrimersand much lower than those for linear polymers of equivalentmolecular weight, which is in accordance with the moleculararchitecture being more open than that of dendrimers andmuch more compact than that of linear analogs. Shear thin-ning is observed for all hyperbranched polymer fluids al-though it occurs at different strain rates for different systems.Normal stress coefficients were also calculated and wereshown to have very large power-law regions. The magnitudeof the second normal stress coefficient is 20% of the firstnormal stress coefficients. In all cases, the obtained pressureof the polymeric fluid remains approximately constant at lowstrain rates and rapidly increases at very high strain rates.

The flow birefringence of hyperbranched polymer meltshas also been characterized by taking into account the calcu-lation of the form and intrinsic birefringence. For all sys-tems, the flow alignment angle converges to 45° at low strainrates and the order parameter results have shown that theintrinsic birefringence is fairly small in comparison with theform birefringence. Furthermore, the SOR has been shown tobe valid only in the Newtonian regime. The stress opticalcoefficient of 3.3�3� does not depend on the molecularweight of hyperbranched polymers. In our systems, the SORfails in the non-Newtonian region because it does not takeinto account the flow-induced changes of the microstructure.

ACKNOWLEDGMENTS

We would like to thank Swinburne University of Tech-nology Supercomputer Facility for a generous grant of com-puter time. One of the authors �T.C.L.� would like to thankDr. Jaroslaw Bosko for his valuable suggestions and discus-sions.

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