ORIGINAL CONTRIBUTION

A molecular dynamics study of the stress–optical behaviorof a linear short-chain polyethylene melt under shear

Chunggi Baig & Brian J. Edwards & David J. Keffer

Received: 26 March 2006 /Accepted: 26 April 2007 / Published online: 22 May 2007# Springer-Verlag 2007

Abstract In this study, we present details of the stress–optical behavior of a linear polyethylene melt under shearusing a realistic potential model. We demonstrate theexistence of the critical shear stress, above which thestress–optical rule (SOR) begins to be invalid. The criticalshear stress of the SOR of this melt turns out to be 5.5 MPa,which is fairly higher than 3.2 MPa at which shear thinningstarts, indicating that the SOR is valid up to a point wellbeyond the incipient point of shear thinning. Furthermore,contrary to conventional wisdom, the breakdown of theSOR turns out not to be correlated with the saturation ofchain extension and orientation: It occurs at shear rates wellbefore maximum chain extension is obtained. In addition tothe stress and birefringence tensors, we also compare twoimportant coarse-grained second-rank tensors, the confor-mation and orientation tensors. The birefringence, confor-mation, and orientation tensors display nonlinearrelationships to each other at high values of the shearstress, and the deviation from linearity begins at approxi-mately the critical shear stress for breakdown of the SOR.

Keywords Birefringence . Stress–optical rule . Shear .

Nonequilibriummolecular dynamics .

Linear polyethylene melt

Introduction

Optical measurement of the anisotropy of materials hasproven to be very informative and useful in characterizingthe structure and stress of molecular crystals (Vuks 1966;

de Jong et al. 1991) and as a noninvasive tool in the studyof rheology of polymeric fluids (Morrison 2001; Janeschitz-Kriegl 1983). There are generally two types of birefrin-gence, form birefringence and flow birefringence. Theformer originates from the difference in the intrinsic(average) polarizability between the solvent and polymer,and the latter from anisotropic orientation of polymer chainbonds induced by the flow (Doi and Edwards 1986).Accordingly, form birefringence is expected to make asignificant contribution to the total birefringence in dilutepolymer solutions, providing us with information about thenonspherical shape of polymer aggregates in solutions.However, it can be ignored in concentrated polymersolutions or melts, for which flow birefringence is thedominant effect. Because flow birefringence is directlydetermined by a preferred orientation of molecules (bonds)in space under an externally imposed flow field, it would,in general, depend on specific polymers (and possibly theirmolecular weights), the detailed kinematics of the flow(type and strength), and system conditions (such astemperature and density).

An important application of birefringence to the rheol-ogy of polymer melts under flow comes from thesupposition that there exists a certain relationship betweenthe anisotropy of birefringence and that of stress, that is, thestress–optical behavior. A well-known fact is that thereexists, under a wide variety of conditions, a linearrelationship between the stress tensor σ and the real partof the refractive index tensor, or “birefringence tensor,” n,which is the so-called stress–optical rule (SOR), Δn=CΔσ.Here, C is called the “stress–optical coefficient” and thebirefringence Δn is the difference between the principalcomponents of the real part of the refractive index tensor.

To date, there are numerous experimental works exam-ining the SOR. One of the first detailed experimental

Rheol Acta (2007) 46:1171–1186DOI 10.1007/s00397-007-0199-2

C. Baig : B. J. Edwards (*) :D. J. KefferDepartment of Chemical Engineering, University of Tennessee,Knoxville, TN 37996-2200, USAe-mail: bje@utk.edu

studies of the stress–optical behavior of amorphouspolymers was carried out by Matsumoto and Bogue(1977) using polystyrene melts at temperatures above theglass transition temperature Tg. These experiments wereconducted under both isothermal and nonisothermal con-ditions. The polymer samples were uniaxially elongatedusing a mechanical–optical apparatus. They obtained amaster stress–birefringence curve on which all the datafrom both isothermal and nonisothermal conditions weresuperimposed. They reported that the SOR began to fail atstress values larger than about 1 MPa.

Since then, more extensive experimental studies of theSOR have been performed using more elaborate elonga-tional stress–optical apparati (Muller and Froelich 1985;Inoue et al. 1991; Kotaka et al. 1997; Okamoto et al. 1998;Venerus et al. 1999). Muller and Pesce (1994) observedsignificant deviations from the SOR at temperatures near Tgin both polystyrene and polycarbonate melts undergoinguniaxial elongation. They reported that at high temperatures(i.e., above Tg+20 K), the stress–optical behavior of thepolymer melts was independent of temperature and strainrate in the nonlinear regime as well as the linear regimewhere the SOR is valid. Inoue et al. (1996) investigated thestress–optical behavior of several vinyl polymers from theglassy to the rubbery regions through dynamic birefrin-gence measurements.

Recently, incorporating an optical measurement deviceinto an advanced elongational rheometer developed byMeissner and Hostettler (1994), Venerus et al. (1999)carried out a very careful stress–optical study and re-examined the stress–optical behavior of polystyrene meltsunder uniaxial elongational flow at sufficiently hightemperatures, above Tg+60 K. They reported that C of thepolystyrene melt was 4.8×10−9 Pa−1 and that the criticalstress for the breakdown of the SOR was roughly 1 MPa.These results are in good agreement with other previouslyreported values (Muller and Froelich 1985; Janeschitz-Kriegl 1983). They also observed that C decreases withincreasing extensional stress, consistent with many existingresults. Some time ago, Kotaka et al. (1997), however,reported the opposite behavior of polystyrene melts (but notpolyethylene melts) that C increases with increasingextensional stress. In this regard, Venerus et al. (1999)speculated that in the experiments of Kotaka et al. (1997)with polystyrene melts, a necking process might havedeveloped, causing the measured stress to decrease whilethe birefringence was not sufficiently relaxed.

More recently, also using polystyrene melts underelongation but with narrow molecular weight distributions,Luap et al. (2005) reported that the critical stress of theSOR was about 2.7 MPa, independent of temperature andstrain rate, and it corresponded to a Weissenberg numberWe ¼ lg

�� �approximately equal to 3. (The temperatures

employed in their study were above Tg+40 K.) They alsoshowed that the SOR was valid well into the strain-thinningregion of nonlinear flow behavior. By comparing theoret-ical predictions with existing experimental findings, VanMeerveld (2004) analyzed the effect of the molecularweight distribution on the critical stress value for break-down of the SOR, based on the idea that deviations to theSOR were effectively determined by the number ofstretched chains in the melt. In the meantime, Cormierand Callaghan (2002) applied another new experimentaltechnique, which combines rheometry with magneticresonance imaging spectroscopy, to measure segmentalalignment in polymer melts under shear flow. Because itis expected to become more and more advanced in thefuture, this technique seems to be very promising in therheo-optical study of polymer melts under various flows.On the theoretical side, Palierne (2004) demonstrated thatthe full Doi–Edwards reptation model predicts the failure ofthe SOR at large shear strains, although a restricted versionusing the so-called independent alignment approximationstill guarantees that the SOR is valid.

There have also been a few computer simulation studiesof the stress–optical behavior of polymer melts. Kröger etal. (1993, 1997) investigated the stress–optical behavior ofpolymer melts under shear (Kröger et al. 1993) and uniaxialelongation flow (Kröger et al. 1997) using a multi-beadanharmonic-spring model. They reported the breakdown ofthe SOR at high strain rates for both types of flows. Morerealistic potential models have been employed in simulationstudies of polyethylene melts under uniaxial elongation byGao and Weiner (1994) using molecular dynamics simu-lations, and later by Mavrantzas and Theodorou (2000a, b)using their advanced Monte Carlo technique (the so-calledend-bridging Monte Carlo simulation). Mavrantzas andTheodorou (2000b) observed the change of C with chainlength, with a plateau value obtained for chain lengthsgreater than C400H802.

Much of the previous work (both experiments andsimulations) on the stress–optical behavior has beenperformed with uniaxial elongational flow rather than shearflow. It seems to be mainly due to an experimental difficultyin achieving sufficiently high shear stress values to observethe failure of the SOR because of undesirable side effectssuch as melt fracture, nonuniform birefringence distributionbecause of frictional heat production, and parasitic birefrin-gence near the optical windows (Janeschitz-Kriegl 1983).This might have led people to think that the SOR would bevalid for shear flow, although Kröger et al. (1993) reportedthe failure of the SOR in the case of the multibead springmelts under shear. Therefore, a definitive statement con-cerning the breakdown of the SOR of polymer melts undershear flow seems to be necessary at this point, which hasmotivated the current work. To this end, computer

1172 Rheol Acta (2007) 46:1171–1186

simulations are thought to be the most appropriate tool atpresent because the shear stress (or equivalently shear rate)employed in simulations can be much higher than that inpractical experiments.

Another point worthy of investigation is the customaryview that the SOR is likely to break down at very highstresses because of the saturation of chain extension andorientation (Treloar 1975; Morrison 2001; Janeschitz-Kriegl1983). In fact, owing to its intrinsic kinematics, elonga-tional flow makes chains become much more extended andoriented than shear flow. That is, in shear flow, chains arenot likely to be stretched to such a high degree as inelongational flow, partly because of the rotational nature ofshear. Therefore, the breakdown of the SOR under shearwould not necessarily imply that the customary view iscorrect. This aspect is also considered in this study.

Herein, we further investigate the relationship betweenthe birefringence and two structural properties that havebeen regarded as very important in a coarse-grained level ofdescription (Beris and Edwards 1994): One is the confor-mation tensor c

�and the other the orientation tensor u. It is

easily seen from their definitions that the two are closelyrelated to each other:

c� ¼ 3 RRh i

R2h ieq; ð1Þ

u ¼ 3 br brh i; ð2Þwhere R and br represent the chain end-to-end vector andunit vector along the chain end-to-end direction, respec-tively. Notice that the two tensors become the unit second-rank tensor at equilibrium. It might be thought that thebirefringence tensor would be proportional to the confor-mation and orientational tensors at all strain rates. However,as we will see later in this article, this is not necessarilytrue.

Theoretical background for birefringence studies

To calculate the birefringence tensor of anisotropic materi-als, we first need to calculate the polarizability tensor of thesystem directly from the simulation and then the birefrin-gence tensor from the calculated polarizability tensor usingan appropriate theoretical formula. For isotropic materials,an exact formula exists, which is called the Clausius–Mossotti equation (see Eq. 9 below). Under quiescentconditions, this equation is adequate; however, under flow,the system of chain molecules will become anisotropic. Foranisotropic materials, to the authors’ knowledge, there doesnot seem to have appeared in the literature (before 2000) anexact generalized tensorial formula, although there have

appeared several derivations of certain approximate formulaapplied in the study of molecular crystals (Vuks 1966;Urano and Inoue 1976; de Jong et al. 1991). Moreover,Mavrantzas and Theodorou (2000a) presented an aniso-tropic version of the exact formula, but no derivation of itwas given in their article. In this study, we offer aderivation of this expression for anisotropic media, forcompleteness. The generalized Clausius–Mossotti tensorialformula for anisotropic materials reduces to the conven-tional Clausius–Mossotti equation for isotropic systems. Wewill also present in detail a procedure to calculate thepolarizability tensor of molecules making use of their bondpolarizability through a local coordinate transformation.

A derivation of the Clausius–Mossotti equationfor anisotropic media

For easier comprehension of the anisotropic derivationbelow, we first derive the Clausius–Mossotti equation forisotropic materials. Consider a linear dielectric mediumconsisting of spherically symmetric nonpolar atoms (suchas argon), or molecules (like methane). The polarizationvector (dipole moment per unit volume) P of a lineardielectric material is proportional to the local electric field,denoted by E:

P ¼ "0χe E: ð3ÞIn this expression, ɛ0 is the permittivity of a vacuum withthe value of 8:854� 10�12 C2 N m2ð Þ�1

and χe is theelectric susceptibility. In linear dielectric materials, χe isequal to "r � 1 ¼ "="0 � 1, where ɛ is the permittivity ofthe material and ɛr is called the relative permittivity or thedielectric constant of the material (Griffiths 1999). Theinduced dipole moment of molecule i, denoted by pi, iswritten as

pi ¼ aiEa; ð4Þ

where αi is the polarizability of molecule i and Ea

represents the applied electric field. Because we arecurrently dealing with identical and spherically symmetricmolecules, we can simply designate pi=p and αi=α (itshould be noted that the polarizability α should be, ingeneral, regarded as a second-rank tensor for nonsymmetricmolecules, as shown for anisotropic materials below). Notethat for these spherical molecules, p and Ea lie in the samedirection.

E will be, in general, different from Ea because at anypoint in space, there would be not only Ea but also aninduced electric field, denoted by Ed. This field results fromthe induced dipoles inside an infinitesimally small (but stillmacroscopic in the sense that it contains many molecules)volume of interest. Consequently, E=Ea+Ed. It should be

Rheol Acta (2007) 46:1171–1186 1173

also noted that in this material, Ed would lie in the oppositedirection to Ea.

Suppose that there are N molecules in a small volume Vof interest. If we take V as the local volume of the material,Ea in this volume will be the electric field including boththe externally applied field and the field induced by onlythe dipoles outside the volume. In other words, Ea is theelectric field due to everything except the dipoles insidethe volume. (It might also be possible to take Ea as a fielddue to everything except the molecule under consider-ation.) For the number density of molecules N

� ¼ N=V inthe volume under consideration, the polarization can bewritten as

P ¼ 1

V

XNi¼1

pi ¼ Np

V¼ N

�p ¼ N

�aEa: ð5Þ

In this equation, the second equality comes from theassumption that every molecule is identical and sphericallysymmetric and therefore has the same dipole moment. Toderive the relationship between α and χe or ɛr, we need toknow the induced electric field Ed because of the moleculesinside the volume under consideration. It is well known thatthe field created by the dipoles is (Griffiths 1999)

Ed ¼ � P

3"0¼ � N

�!

3"0Ea; ð6Þ

where Eq. 5 was used for the second equality. Therefore, thelocal electric field E is then

E ¼ Ea þ Ed ¼ 1� N�a

3"0

!Ea: ð7Þ

If we substitute Eq. 7 into Eq. 3, the polarization isexpressed in terms of Ea as

P ¼ "0χe 1� N!3"0

� �Ea : ð8Þ

By a direct comparison of Eqs. 5 and 8, after some algebraicoperations, we arrive at the well-known Clausius–Mossottiequation,

a ¼ 3"0N

"r � 1

"r þ 2

� �: ð9Þ

Furthermore, when the magnetic effect is neglected, ɛr isequal to n2, and Eq. 9 is written as

a ¼ 3"0

N�

n2 � 1

n2 þ 2

� �: ð10Þ

Equation 10 is called the Lorentz–Lorenz equation in optics.For anisotropic materials consisting of identical but

nonspherical molecules (such as carbon dioxide), the

polarizability and electric susceptibility are no longer scalarquantities but second-rank tensors, which we shall denoteby a and χe. Accordingly, the expression for P, corre-sponding to Eq. 3, is written as

P ¼ "0χe � E: ð11Þ

Furthermore, the equation corresponding to Eq. 4 isgiven by

pi ¼ αi � Ea: ð12Þ

Because we are dealing with identical but nonsphericalmolecules, the polarizability tensor of each molecule withrespect to a fixed frame of reference will be, in general,different than those of other molecules, depending on itsorientation in space. Moreover, in this material, pi would, ingeneral, not lie in the same direction as Ea. Because both piand ai are different for various molecules, it seems to bereasonable to use an averaged quantity for each of them.Taking a summation of Eq. 12 over all the molecules in Vand dividing by N, we obtain

p ¼ α � Ea; ð13Þwhere we have introduced the averaged quantities withp �PN

i¼1pi

�N and α �PN

i¼1αi

�N .

Using Eq. 13, the polarization is written as

P ¼PNi¼1

pi

V¼ Np

V¼ N

�p ¼ N

�α � Ea: ð14Þ

Therefore, the electric field induced by the dipoles is

Ed ¼ � P

3"0¼ � N

�

3"0α � Ea: ð15Þ

The local electric field E is then

E ¼ Ea þ Ed ¼ I � N�

3"0a

!� Ea; ð16Þ

where I is the second-rank unit tensor. Putting Eq. 16 intoEq. 11, the polarization is found to be

P ¼ "0χe: I � N�

3"0α

!� Ea: ð17Þ

Eliminating P from Eqs. 14 and 17, we arrive at

N�α� "0χe

� I � N�

3"0α

!" #� Ea ¼ 0: ð18Þ

Because Ea is arbitrary, the term within the square bracketsmust vanish. After some manipulation, we finally arrive at

1174 Rheol Acta (2007) 46:1171–1186

the following relationship between the polarizability andsusceptibility tensors for anisotropic materials:

α ¼ 3"0

N� χe þ 3I� ��1

� χe

¼ 3"0

N� "r þ 2I� ��1

� "r � I� �

:

ð19Þ

To obtain this expression, we used the tensorial equationχe ¼ "r � I . Equation 19 is the generalized Clausius–Mossotti (tensorial) formula of linear dielectric materials.If we put "r ¼ n �n in Eq. 19 in the absence of a magneticeffect, we obtain the generalized form of the Lorentz–Lorenz equation in optics,

α ¼ 3"0

N� n � nþ 2I� ��1

� n � n� I� �

: ð20Þ

Equation 20 is used in calculating the birefringencetensor in this study. Again, Eq. 19 was presented (without aderivation) in the article by Mavrantzas and Theodorou(2000a).

A local coordinate transformation used in calculatingthe polarizability tensor

To calculate the polarizability tensor of the molecules bysumming each contribution of the bonds, it is convenient tochoose a local Cartesian coordinate system (represented bylowercase letters) attached to each bond (Bower 2002). As

shown in Fig. 1, the X- and Y-axes in the laboratory frameof reference (represented by uppercase letters) are chosen tobe the flow and the velocity gradient directions, respective-ly, and the Z-axis is the neutral direction. In the localcoordinate system, the x-axis is located along the bonddirection, the y-axis is perpendicular to the x-axis and lyingin the X–Y plane, and the z-axis (orthogonal to both the x-and y-axes) is determined by taking the cross product of thex- and y-axes.

Using these reference frames, we need to find theorthogonal transformation matrix between the laboratoryand the local coordinate systems. First, the unit vector inthe x-direction, denoted by bx, is expressed in terms of theunit vectors ðbX ; bY ; bZ Þ in the laboratory coordinate system:

bx ¼ cos θbX þ sin θ cosφbY þ sin θ sinφbZ: ð21Þ

Next, because 2by is lying in the X–Y plane, it can beexpressed as

by ¼ abX þbbY : ð22Þ

Using its orthogonal relationship with bx and the unit lengthof its magnitude, a and b are

a ¼ � sinφ; b ¼ cosφ: ð23Þ

Let us choose a ¼ � sinφ and b ¼ cosφ. bz is thendetermined as

bz ¼ bx�by ¼ � cos θ cosφbX � cos θ sinφbY þ sin θbZ ð24Þ

X

Y

Z

x

z

y

X

Y

Z

X

Y

Z

x

z

y

θ

φ

X

Y

Z

x

z

y

X

Y

ZX

Y

Z

Fig. 1 Schematic diagram (left)of the laboratory coordinatesystem (represented by XYZ)and the local coordinate system(represented by xyz) attached toa bond. The x-direction is locat-ed along the bond. The arrowsin the picture (right) representthe velocity field, which is non-zero only in the X-direction. Thevelocity gradient and neutraldirections are the Y- and Z-directions, respectively

Rheol Acta (2007) 46:1171–1186 1175

The transformation between the two coordinate systems isthus expressed in matrix form as

x^

y^

z^

264375 ¼ M �

X^

Y^

Z^

26643775; ð25Þ

where

M ¼cos θ sin θ cosφ sin θ sinφ0 � sinφ cosφ

sinθ � cos θ cosφ � cos θ sinφ

24 35: ð26Þ

Denoting the longitudinal (x-direction) and transverse (y-and z-direction) polarizabilities of a bond by αp and αt,respectively, the polarizability tensor of each bond is indiagonal form in the local coordinate system:

αð Þxyz ¼αp 0 00 αt 00 0 αt

24 35: ð27Þ

Therefore, the polarizability tensor in the laboratorycoordinate system is

αð ÞXYZ ¼ MT � αð Þxyz � M

¼cos2 θð Þαp þ sin2 θ

� �αt sin θ cos θ cosφð Þ αp � αt

� �sin θ cos θ sinφð Þ αp � αt

� �sin θ cos θ cosφð Þ αp � αt

� �sin2 θ cos2 φ� �

αp þ sin2 φþ cos2 θ cos2 φ� �

αt sin2 θ sinφ cosφ� �

αp � αt

� �sin θ cos θ sinφð Þ αp � αt

� �sin2 θ sinφ cosφ� �

αp � αt

� �sin2 θ sin2 φ� �

αp þ cos2 φþ cos2 θ sin2 φ� �

αt

264375:ð28Þ

Before we discuss the simulation methodology, wewould like to point out that we have chosen to neglect theC−H bond altogether in calculating the birefringence tensorin this work. We did this because the potential model usedherein (described in the next section) to model the linearpolyethylene melt is a united-atom model, which treatsgroups of atoms (in this case, CH2 and CH3 groups).Although some scholars (Gao and Weiner 1994; Mavrantzasand Theodorou 2000a) have included the C−H bonds intheir calculation of the birefringence by assuming the trans-state of these bonds along the chain backbone, suchtreatments are still approximate and perhaps inconsistentwith the united-atom model. Therefore, to maintain theconsistency with the potential model employed in thesimulations, we have decided to ignore C−H bonds andonly consider C−C bonds in this work. This choice is notconsidered to be overly restrictive because all the propertiescan be still calculated without any corrections, except theabsolute value of the stress–optical coefficient C. In otherwords, the only weak point because of the neglect of C−Hbonds lies in obtaining a quantitative estimate of C. In thisregard, we should also note that there does not even seem toexist as yet universally accepted values of the experimentalbond polarizabilities for C−C and C−H bonds (Treloar1975). Therefore, our choice is reasonable, as it does notaffect the main purpose of this study and substantiallyreduces the computational cost of the simulation (by notincluding every atom explicitly). The bond polarizabilitiesof the C−C bond employed in this work were obtained from

Table 9.5 of Bower (2002): ap ¼ 10:8� 10�41 andαt ¼ 2:8� 10�41 Fm2.

Simulation methodology

Equations of motion

The melt studied in this work is C50H102, which is sufficientin length to see the flow effect on birefringence and, at thesame time, computationally feasible across a broad range ofshear rates. With this melt, we have performed NVTcanonical nonequilibrium molecular dynamics (NEMD)simulations using the SLLOD equations of motion for ahomogeneous shear flow (Evans and Morriss 1990). Tomaintain a constant temperature, we employed the Nosé–Hoover thermostat (Nosé 1984a, b; Hoover 1985). Theequations of motion are given by

q�ia ¼

piamia

þ qia � rν;

p�ia ¼ Fia � pia � rν � pς

Qpia;

ς� ¼ pς

Q;

p:

ς ¼Xi

Xa

p2iamia

� DNkBT ;

ð29Þ

where subscripts i and a are used as indices of molecularand atomic number, respectively. The quantities pia and qia,respectively, denote the momentum and position vectors of

1176 Rheol Acta (2007) 46:1171–1186

atom a in molecule i. Fia is the force on atom a in moleculei of mass mia. D represents dimensionality, V the systemvolume, N the total number of atoms, T the absolutetemperature, and kB the Boltzmann constant. ς and pς,respectively, are the coordinate- and momentum-like vari-ables of the Nosé–Hoover thermostat, and Q (=DNkBTτ

2)is the mass parameter of the thermostat. (τ is the char-acteristic frequency of the thermostat.) In shear flow,

rν ¼0 0 0γ�

0 00 0 0

24 35; ð30Þ

where g�is the applied shear rate. The equations of motion

were numerically integrated using a modified version(including the velocity field terms [Cui et al. 1996a]) ofthe reversible reference system propagator algorithm(r-RESPA), originally developed by Tuckerman et al. (1992).

Potential model

We employed the well-known Siepmann–Karaborni–Smitunited-atom model developed by Siepmann et al. (1993) forthe bond-bending, bond-torsional, and interatomic interac-tions, but replaced the rigid bond by a flexible one for thebond-stretching interaction. This potential model has beensuccessfully applied in NEMD simulations of various chainlengths of polyethylene melts under shear (Cui et al. 1996b)and planar elongation flow (Baig et al. 2005, 2006).

To avoid unnecessary repetition, we briefly present themain features of the potential model. The bond-stretching,bond-bending, bond-torsional, and interatomic Lennard–Jones (LJ) interactions, respectively, are given by

Vstr lð Þ ¼ 1

2kstr l � leq� �2

; ð31Þ

Vben qð Þ ¼ 1

2kben q � qeq� �2

; ð32Þ

Vtor φð Þ ¼X3m�0

am cosφð Þm; ð33Þ

and

VLJ rð Þ ¼ 4"absab

r

� �12� sab

r

� �6 : ð34Þ

The parameter ratios kstr/kB and kben/kB are equal to452,900 K/Å2 and 62,500 K/rad2, respectively, and a0/kB=1,010, a1/kB=2,019, a2/kB=136.4, and a3/kB=−3,165 K.The equilibrium bond distance is leq=1.54 Å, and the

equilibrium bond angle is θeq=114°. For the LJ interactionsbetween different atoms (say, a and b), the Lorentz–Berthelot mixing rule was employed as ɛab=(ɛaɛb)

1/2 andσab ¼ σa þ σbð Þ=2, with the energy and size parametersɛ/kB and σ being 47 K and 3.93 Å for the CH2 united-atomand 114 K and 3.93 Å for the CH3. The cutoff distance ofthe LJ interactions used in this work was 2.5σCH2.

Simulation conditions

We employed 120 molecules of C50H102 in a rectangularbox, enlarged in the X-direction, with dimensions (X×Y×Z)of 93×45×45 Å3. The X-dimension was chosen to besufficiently large to avoid any undesirable system-sizeeffects at high shear rates where chains are quite extendedand oriented in the flow direction. The temperature anddensity were chosen as 450 K and 0.7438 g/cm3. We used18 different shear rates covering a large range of dimen-sionless shear rates, γ

�mσ2

�"

� �1=2 ¼ 0:0005� 1:0 in re-duced units (this corresponds to γ

� ¼ 2:1� 107 � 4:3�1011 s�1 in real units).

The lowest value of the shear rate is regarded as verysmall (although still high in practical experiments) in typicalNEMD simulations, considering that the longest relaxationtime (the so-called Rouse time) of the system λ was equal to500 ps (Baig et al. 2006). Consequently, the inverse of λ(regarded as an approximate critical shear rate for the onsetof shear-thinning behavior) is 0.0047 in reduced units,below which a large effect of Brownian motion of particlesmakes it difficult to measure the response of system to theflow field. (This will be discussed in more detail later in“Results and discussion”.) Furthermore, it usually takes amuch longer simulation time to reach a steady state as shearrate decreases. Despite these difficulties, we chose such lowstrain rates to observe in more detail the linear stress–opticalbehavior over a wide range of shear rates. Therefore, toobtain statistically reliable results for low shear rates, weperformed very long simulations, i.e., 70 ns at the lowestshear rate. In applying the r-RESPA integrator, two differenttime steps were used: the large time step of 2.35 fs wasemployed for intermolecular LJ interactions and the smalltime step of 0.235 fs for the bond-stretching, bond-bending,bond-torsional, and intramolecular LJ interactions.

Results and discussion

There are only four nonzero independent components of thestress tensor σ in simple shear flow, as represented by Eq. 30:these are σxx, σyy, σzz, and σxy (Morrison 2001). Thissituation also applies to the other tensors examined in this

Rheol Acta (2007) 46:1171–1186 1177

article, namely, the conformation tensor c�, the orientational

tensor u, and the birefringence tensor n. Furthermore,experimentally, we can only measure the differencebetween the normal (or diagonal) components of the stresstensor but not their absolute values (Bird et al. 1987).Therefore, in simple shear flow, we consider only threeindependent measurable physical quantities (or “character-

istic functions”) of the stress tensor (and the other tensors,c�, u, and n, as well): sxx � syy ; syy � szz, and sxy. The

corresponding three steady-state material functions insimple shear flow are defined as

η � σxy

.γ�; Ψ1 � � σxx � σyy

� �.γ� 2; Ψ2 � � σyy � σzz

� �.γ� 2:

ð35ÞFirst, we examine the three material functions, which are

displayed in Fig. 2. Consistent with many existingexperimental results (Morrison 2001; Bird et al. 1987), allthree properties show shear-thinning behavior with increas-ing shear rate, after a plateau in the linear regime at lowshear rates. The dotted line vertically drawn at the reducedshear rate of 0.004 in each part of the figure represents theboundary between the linear and nonlinear regimes. Thisvalue of the shear rate appears to agree well with thetheoretically predicted one γ

� ¼ 1=λ� �

at which the Weissenbergnumber We ¼ λγ

�� �is equal to unity and nonlinear

characteristics of material properties begin to appear. Thistrend has been observed for many polymeric systems—see,for example, Chapter 2 of Bird et al. (1987). One method tofind the characteristic time λ is to observe the timecorrelation function of the end-to-end chain vector, bymeans of which we have determined λ=500 ps (Baig et al.2006). Using the calculated value of λ, the critical shearrate for shear-thinning turns out to be 0.0047 in reducedunits γ

�mσ2

�"

� �1=2� �, which is very close to the boundary

value of γ�mσ2

�"Þ1=2 ¼ 0:004

�directly obtained from the

material functions. The other vertical line (represented bythe dashed line) drawn at γ

�mσ2

�"

� �1=2 ¼ 0:01 indicates anapproximate boundary between the linear and nonlinearstress–optical behavior between the stress and birefringencetensors. That is, it turns out that the linear SOR appears tobe valid up to γ

�mσ2

�"

� �1=2 ¼ 0:01, after which, it appearsto fail, as shown in Fig. 8. (This behavior will beconsidered in detail later.) Therefore, we inserted thesevertical lines in most of the figures presented in this

0.0001 0.001 0.01 0.1 1 100.0001

0.001

0.01

2/12 )/( ε γ m.

)sPa(η a

0.0001 0.001 0.01 0.1 1 101e-15

1e-14

1e-13

1e-12

1e-11

1e-10

)sPa( 2

1Ψ

b

0.0001 0.001 0.01 0.1 1 101e-15

1e-14

1e-13

1e-12

1e-11

1e-10

)sPa( 2

2Ψ−

c

σ

2/12 )/( ε γ m. σ

2/12 )/( ε γ m. σ

Fig. 2 Plots of the steady-state material functions vs shear rate: ashear viscosity η, b the first normal stress coefficient �1, and c thesecond normal stress coefficient �2

0.0001 0.001 0.01 0.1 1

<Rete2

> (

Å2)

600

800

1000

1200

1400

1600

<Rg

2>

(Å

2)

120

130

140

150

160

170

180

<Rete

2>

<Rg

2>

2/12 )/( ε γ m. σ

Fig. 3 Plots of the mean square chain end-to-end distance and the meansquare chain radius of gyration with respect to shear rate

1178 Rheol Acta (2007) 46:1171–1186

manuscript for a clear understanding where each data pointlies (i.e., whether it lies in the linear or nonlinear regimeswith respect to shear-thinning or the SOR).

In Fig. 3, we present two important structural quantities,the mean square chain end-to-end distance and the meansquare chain radius of gyration, denoted by R2

ete

� and

R2g

D E, respectively. As shear rate increases, the chains

become more and more aligned in the flow direction with

an anisotropically elongated shape. This results in theincrease in both R2

ete

� and R2

g

D Ewith shear rate, as shown

in the figure. However, at high shear rates, i.e.,γ�mσ2

�"

� �1=2> 0:1, both quantities appear to decrease with

shear rate. A similar behavior has also been observed byCui et al. (1996b) in their study using the short alkanechains C10H22, C16H34, and C24H50 under shear. Thisphenomenon can be explained in a dynamic sense, adoptinga mean-field concept of restricting the chain dimensions at

0.0001 0.001 0.01 0.1 1 100.01

0.1

1

10

100

1000

(MPa)xxyy σσ − a

0.0001 0.001 0.01 0.1 1 100.01

0.1

1

10

100

(MPa)zzyy σσ − b

0.0001 0.001 0.01 0.1 1 100.1

1

10

100

1000

)(MPaxyσ− c

2/12 )/( ε γ m. σ

2/12 )/( ε γ m. σ

2/12 )/( ε γ m. σ

Fig. 4 Plots of the three independent properties of the stress tensor σas functions of shear rate: a σxx–σyy, b σyy–σzz, and c σxy

0.0001 0.001 0.01 0.1 1 100.001

0.01

0.1

1

10

yyxx cc ~~ − a

0.0001 0.001 0.01 0.1 1 100.001

0.01

0.1

1

yyzz cc ~~ − b

0.0001 0.001 0.01 0.1 1 100.01

0.1

1

xyc~ c

2/12 )/( ε γ m. σ

2/12 )/( ε γ m. σ

2/12 )/( ε γ m. σ

Fig. 5 The same as Fig. 4 for the conformation tensor

Rheol Acta (2007) 46:1171–1186 1179

high strain rates because of a large number of molecularcollisions between atoms, which was originally proposedby Moore et al. (2000) and discussed by Baig et al. (2005).(Readers who are interested in the details on this issue arereferred to these papers.)

Another interesting point is to compare the values ofR2ete

� shown in Fig. 3 and that of the fully extended chain

with the equilibrium bond length and angle in the all-transstate. R2

ete

� of the fully stretched chain is calculated as

4,007 Å2. Thus, even the highest value, R2ete

� =1,338 Å2

observed at γ�mσ2

�"

� �1=2 ¼ 0:1, is found to be only aboutone third of that at the fully stretched state. Therefore,chains do not reach their maximum extensions in shearflow. This is interesting because (as mentioned earlier) theSOR has been customarily thought to become invalidbecause of the saturation of chain extension and orientation(Janeschitz-Kriegl 1983; Treloar 1975). However, as indi-cated by the dashed line in the figure and directly shown in

0.0001 0.001 0.01 0.1 1 100.001

0.01

0.1

1

10

yyxx uu − a

0.0001 0.001 0.01 0.1 1 100.001

0.01

0.1

1

yyzz uu − b

0.0001 0.001 0.01 0.1 1 100.01

0.1

1

xyu c

2/12 )/( ε γ m. σ

2/12 )/( ε γ m. σ

2/12 )/( ε γ m. σ

Fig. 6 The same as Fig. 4 for the orientational tensor u

0.0001 0.001 0.01 0.1 1 100.0001

0.001

0.01

0.1

yyxx nn − a

0.0001 0.001 0.01 0.1 1 101e-5

1e-4

1e-3

1e-2

yyzz nn − b

0.0001 0.001 0.01 0.1 1 100.0001

0.001

0.01

xyn c

2/12 )/( ε γ m. σ

2/12 )/( ε γ m. σ

2/12 )/( ε γ m. σ

Fig. 7 The same as Fig. 4 for the birefringence tensor n

1180 Rheol Acta (2007) 46:1171–1186

Fig. 8, the failure of the SOR begins at a fairly low strainrate without even being close to the maximum chainextension.

Figures 4, 5, 6, and 7 show the three independentproperties (xx–yy, yy–zz, and xy components) of the stress,conformation, orientation, and birefringence tensors, in thatorder, as functions of shear rate. In conjunction with thematerial functions shown in Fig. 2, the two normal stressdifferences and the shear stress shown in Fig. 4 appear toincrease monotonically with increasing shear rate, which isconsistent physically. However, the degree of increase foreach property is observed to vary with shear rate, asrevealed by its degree of shear-thinning behavior in Fig. 2.Notice that the yy–zz component of stress is shown to bestatistically less reliable, compared with the xx–yy and xyquantities. (In fact, as will be shown below, this is also truefor the other second-rank tensors.) Keeping this behavior inmind, we examine the variation of the conformation tensorwith respect to shear rate, which is shown in Fig. 5.Initially, all three quantities appear to increase withincreasing shear rate. At high shear rates, however, bothc�xx � c

�yy and c

�yy � c

�zz do not increase any further but

reach asymptotic values. Furthermore, c�xy is observed to

reach the maximum point around γ�mσ2

�"

� �1=2 ¼ 0:01�0:02 (which approximately coincides with the critical shearrate of the SOR) and then decreases with increasing shearrate. From the quantitative analysis of the results, thisseems to occur mainly because of the relatively largerdecrease in c

�yy than the increase in c

�xx above the critical

shear rate. (However, there seems to be other effectsoperating as well, such as a nonuniform distribution of theazimuthal angle in Y–Z plane.) Comparing with the resultsof the stress tensor shown in Fig. 4, it is evident that a linearrelationship between the stress and conformation tensorscan be realized only at low shear rates, and nonlinearbehavior occurs at high shear rates (see Fig. 10). Similarphenomena are also observed for the orientational tensor u,which is shown in Fig. 6. This result can be understoodthrough the close relationship between c

�and u under the

(plausible) assumption of a fairly narrow (approximatelyGaussian) distribution of the chain end-to-end distance.

We also examined the three components of the birefrin-gence tensor, which were calculated using Eq. 20 and arepresented in Fig. 7. All three quantities appear to increaserather quickly with shear rate, up to the critical shear ratearound γ

�mσ2

�"

� �1=2 ¼ 0:01. After that, all of them appearto increase relatively slowly and display asymptoticbehavior at high shear rates. This behavior is quite differentfrom that of the other tensors. Considering all the aboveresults for the four tensors, we would not expect a simplelinear relationship between any two of them.

The plots of the birefringence vs stress tensor arepresented in Fig. 8. Because the yy–zz quantity shown in

Fig. 8b experiences large fluctuations, we examine only thexx–yy and xy quantities. It is apparent from the figure thatthere is a clear, linear relationship between n and σ up toγ�mσ2

�"

� �1=2 ¼ 0:01, at which point, the shear stress andthe first and second normal stress differences are 5.5±0.2,6.5±0.3, and 1.4±0.2 MPa, respectively. (Also note that theonset of shear-thinning behavior [represented by dottedline] occurs at –σxy=3.2±0.1 MPa, σyy–σxx=2.2±0.2 MPa,and σyy–σzz=0.5±0.2 MPa.) These results appear to beconsistent with the experimental findings for the critical

0.01 0.1 1 10 100 10000.0001

0.001

0.01

0.1

yyxx nn −

(MPa)xxyy σσ −

a

0.01 0.1 1 10 1001e-5

1e-4

1e-3

1e-2

yyzz nn −

(MPa)zzyy σσ −

b

0.1 1 10 100 10000.0001

0.001

0.01

xyn

)(MPaxyσ−

c

Fig. 8 Comparison of the steady-state values between the birefrin-gence tensor n and the stress tensor σ

Rheol Acta (2007) 46:1171–1186 1181

tensile stress of the SOR under uniaxial elongation flowbeing of the order of megapascals (Janeschitz-Kriegl 1983;Luap et al. 2005). As mentioned earlier, considering anexperimentally known fact that it is very difficult to find the

critical stress of the SOR for polymeric materials for shearflow, the present result of the critical shear stress(≈5.5 MPa) of linear polyethylene melt of C50H102 is ofpotential value from both experimental and theoreticalviewpoints.

Using the above results between n and σ, we can extractthe stress–optical coefficient C of the present system, forwhich only data at shear rates of γ

�mσ="ð Þ1=2 0:01 should

be used. The results are shown in Fig. 9. As mentioned

-2 0 2 4 6 8-0.002

0.000

0.002

0.004

0.006

0.008

0.010

yyxx nn −

C = 1.37*10-9 Pa-1

(MPa)xxyy σσ −

a

0 1 2 3 4 5 60.000

0.001

0.002

0.003

0.004

0.005

0.006

xyn

C = 0.89*10-9 Pa-1

)(MPaxyσ−

b

0.1 1 101e-5

1e-4

1e-3

1e-2

1e-1

yyxx nn −

slope = 1.82

c

)(MPaxyσ−

Fig. 9 Linear regression of data of n vs σ at low shear rates where theSOR is valid: a and b calculate the stress–optical coefficient C basedon nxx–nyy vs σxx–σyy and nxy vs σxy, respectively, and c shows theslope of nxx–nyy vs σxy. We present the stress optical coefficient as theaverage value between the two quantities from a and b. (The valuefrom the yy–zz quantity was not taken into account because of itsrather large statistical uncertainty, relative to the others.)

0.01 0.1 1 10 100 10000.01

0.1

1

10

yyxx cc ~~ −

(MPa)xxyy σσ −

a

0.01 0.1 1 10 1000.001

0.01

0.1

1

yyzz cc ~~ −

(MPa)zzyy σσ −

b

0.1 1 10 100 10000.01

0.1

1

xyc~

)(MPaxyσ−

c

Fig. 10 The same as Fig. 8 for the conformation tensor, c�, and the

stress tensor σ at various shear rates: a c�xx � c

�yy vs. σxx � σyy,

b c�xy � c

�zz vs. σyy � σzz, and c c

�xy vs. σxy

1182 Rheol Acta (2007) 46:1171–1186

above, because of the relatively large uncertainty in the yy–zzquantity, we have calculated C using only the xx–yy and xyquantities. As shown in Fig. 9a and b, the slope is found tobe (1.37±0.1)×10−9 Pa−1 for xx–yy and (0.89±0.3)×10−9 Pa−1 for xy. If we average these two values, thestress–optical coefficient C for C50H102 at T=450 K and ρ=0.7438 g/cm3 is estimated as (1.13±0.4)×10−9 Pa−1. Thisvalue is of the same order of magnitude as the experimen-tally reported value of 2.35×10−9 Pa−1 for a high molecularweight, high-density linear polyethylene melt at T=423 K

(Janeschitz-Kriegl 1983). As mentioned before, this quanti-tative difference might be largely due to the neglect of C−Hbonds. We also report the slope of nyy–nxx vs –σxy inFig. 9c. The slope of 1.82±0.2 agrees fairly well with thetheoretical value of 2 under the SOR.

Figure 10 shows the change of c�with respect to σ. As in

Fig. 8 for n vs σ, there appears to be a fairly linearrelationship between c

�and σ in both the xx–yy and xy

quantities up to a shear rate of γ�mσ2

�"

� �1=2 ¼ 0:01. With afurther increase in shear stress, c

�xx � c

�yy and c

�yy � c

�zz

eventually seem to reach certain asymptotic values, and c�xy

appears to even decrease after passing through a maximum.In Fig. 11, wemake a similar comparison between n and c

�.

It might be thought at first that n would be strictlyproportional to c

�, regardless of shear rate, because this

tensor represents the inherent microstructure of the melt.However, as seen in the figure, such a linear relationshipappears to be valid only up to γ

�mσ2

�"

� �1=2 ¼ 0:01,represented by the dashed line. This is particularly clear inFig. 11c for the xy quantity, where a sudden directionalchange occurs at γ

�mσ2

�"

� �1=2 ¼ 0:01. This result implies adefinite difference between the fine-grained property (n)and the coarse-grained property c

�� �in the highly nonlinear

regime. It is interesting to note the exact match between thecritical shear rate of the SOR and that of n vs c

�.

As another important physical property, the orientationangle χ is plotted as a function of shear rate in Fig. 12 foreach tensor. The orientation angle has been calculatedthrough the eigenvector analysis: the eigenvectorcorresponding to the largest eigenvalue of each tensor isthe director, and χ is the angle between the director and theflow direction (Allen and Tildesley 1987). As seen in thefigure, in general, χ appears to be different for each tensor,except for c

�and u (thus indicating the narrow Gaussian

distribution of the end-to-end distance). In other words, theprincipal frames of reference are observed to be different

0.01 0.1 1 100.0001

0.001

0.01

0.1

yyxx nn −

yyxx cc ~~ −

a

0.001 0.01 0.1 11e-5

1e-4

1e-3

1e-2

yyzz nn −

yyzz cc ~~ −

b

0.01 0.1 10.0001

0.001

0.01

xyn

xyc~

c

Fig. 11 The same as Fig. 8 for the birefringence tensor n and theconformation tensor c

�

0.0001 0.001 0.01 0.1 1 10

ori

enta

tion a

ngle

, χ

(o)

0

10

20

30

40

50

60conformation tensor,orientational tensor, ubirefringence, n

stress tensor, σ

c~

2/12 )/( ε γ m. σ

Fig. 12 Comparison of the orientation angle χ as a function of shearrate between , u, n, and σ

Rheol Acta (2007) 46:1171–1186 1183

between the various tensors. This difference becomes largerwith increasing shear rate, in particular, above the criticalshear rate of the SOR. While χ (also called the extinctionangle) of the birefringence continues to decrease withincreasing shear rate and eventually approaches an asymp-totic value of about 12° at high shear rates, those of c

�and u

asymptote to smaller values of about 5°. In contrast, χ ofthe stress tensor initially decreases with increasing shearrate, but appears to pass a minimum value and thenincreases slightly. Afterwards, for high shear rates, it seemsto approach an asymptotic value of 27°, which is quitedifferent than those of other tensors.

Lastly, we plot the ratios among the three independentquantities for each tensor in Fig. 13. As shown in Fig. 13a,the qualitative behavior of the ratio between the xx–yy andyy–zz quantities of each tensor with varying shear rateseems to be similar. However, the quantitative resultsappear to vary significantly. In Fig. 13b, Je ¼ xx� yyð Þ=2 xyð Þ2h i

, which is called the shear compliance for the stresstensor, is plotted as a function of shear rate. Je appears to beconstant at relatively low shear rates but can changedramatically at higher shear rates. More specifically, while

Je of the conformation and orientation tensors appear toincrease with shear rate and seem to approach asymptoticvalues at very high shear rates, Je of the stress tensorcontinues to decrease with increasing shear rate. In contrast,Je of the birefringence appears to pass through a maximumat an intermediate shear rate. The quantitative value of Je ofthe stress for this melt is roughly equal to 1.1×10−7 Pa−1.

There have recently appeared in the literature somedetailed discussions of the effect of the glassy stress on thebreakdown of the conventional SOR (Osaki and Inoue1996; Inoue et al. 1999, 2000). This can be readily un-derstood from a modified equation for the SOR, asdescribed in the above references,

Δn tð Þ ¼ CRΔsR þ CGΔsG: ð36Þ

Herein, the subscripts R and G represent the rubbery andthe glassy contributions to the stress tensor, respectively.This glassy contribution is, however, expected to be verysmall in the present polyethylene system for severalreasons, as follows.

First, Inoue et al. (1999) investigated four relativelyshort-chain polystyrene melts in the range of Mw=1,050–10,500g/mol, and in the later dynamic birefringence study(Inoue et al. 2000), only the shortest polystyrene samplewith Mw=1,050g/mol was employed. This chain lengthcorresponds to only one Kuhn segment of a polystyrenechain. In contrast, based on the existing experimentalresults of the mean-squared end-to-end distance, the Kuhnsegment of linear polyethylene melts is estimated to containapproximately ten methylene groups (see, for example,Flory 1969). Therefore, each molecule of C50H102

employed in the present study would consist of five Kuhnsegments, which correspond to polystyrene melts of Mw=5,250g/mol. In addition, it was reported by Inoue et al.(1999) that polystyrene melts with Mw>5,000g/mol can bewell fitted by the multibead-spring (Rouse) model. Further-more, one must take into account that linear polyethylenechains are much more flexible than polystyrene chains,which supports the notion that the glassy contribution in thepresent system would be much weaker. Second, as shownby Inoue et al. (1996), for polystyrene melts, the (planar)phenyl side group attached to the main chain backboneapparently magnifies the glassy contribution. Such a sidegroup is entirely absent in linear polyethylene chains.Third, whereas the highest temperature used for the shortestpolystyrene melt (Mw=1,050g/mol) in Inoue et al. (2000)was only Tg+30 °C, the temperature imposed in our systemwas 177 °C (450 K), which is even higher than the meltingpoint of HDPE, Tm=135 °C (≈Tg+300 °C, assumingTg=−120 °C, as reported in some of the literature). At thisvery high temperature, the glassy contribution is surelyeven more negligible.

0.0001 0.001 0.01 0.1 1 100.001

0.01

0.1

1

10conformation tensor, orientational tensor, birefringence tensor, n

stress tensor, σ

c~( )( )yyxx

yyzz

−− a

0.0001 0.001 0.01 0.1 1 100.001

0.01

0.1

1

10

100

1000

conformation tensor, orientational tensor, ubirefringence tensor, n

stress tensor, σ (MPa-1)

( )( )22 yx

yyxxJe

−=

c~

b

2/12 )/( ε γ m. σ

2/12 )/( ε γ m. σ

Fig. 13 Plots of a (zz–yy)/(xx–yy) and b the equilibrium shearcompliance as functions of shear rate between c

�, u, n, and σ

1184 Rheol Acta (2007) 46:1171–1186

As a consequence of the above statements, the break-down of the SOR in the present study should not beinterpreted as the effect of the glassy-stress contribution,but as the true deviation from the linear relationshipbetween structural and rheological responses to strong flowfields. In a sense, we might even consider the deviation ofthe SOR with increasing shear rate as caused by theincrease in a ‘glassy-like’ contribution coming from aneffective ‘freezing’ behavior of relatively longer relaxation-time modes in response to the applied flow field. Aquantitative description of this consideration might be quiteinformative to this subject. On the other hand, it would bevery interesting in a future study to investigate the generaltrend of the glassy contribution to the birefringence usinglinear/nonlinear polyethylene melts of various chain lengths(and the side-chain length in the case of nonlinearpolymers) over a wide range of temperatures.

Conclusions

In this article, we presented in detail the results of asimulation-based examination of the stress–optical behaviorof a linear, short-chain polyethylene melt, C50H102, undershear. We have presented a derivation of the generalizedClausius–Mossotti formula for anisotropic media. Fourimportant second-rank tensors, the stress tensor, birefrin-gence tensor, conformation tensor, and orientation tensor,were calculated directly from simulations at various shearrates and compared with each other. We summarize belowseveral main conclusions drawn from the present work.

A linear relationship between the stress and birefrin-gence (SOR) appears to be valid up to a certain shear ratewell beyond the incipient point of shear thinning. In thepresent system, the critical shear stress for shear thinningand the breakdown of the SOR were found to be 3.2 and5.5 MPa, respectively.

The slopes of the birefringence vs stress curves appeared todecrease with increasing stress for all three quantities (xx–yy,yy–zz, and xy), consistent with many existing experimentalresults (Matsumoto and Bogue 1977; Janeschitz-Kriegl1983; Venerus et al. 1999).

The orientation angles obtained from each of the fourtensors (σ, n, c

�, and u) were shown to be close to each

other at low strain rates but became more and more distinctas shear rate increased. This implies that the principal frameof reference of each tensor does not coincide with that ofthe other tensors, in general (except for c

�and u), thus

indicating a narrow Gaussian distribution of the chain end-to-end distance.

Rather surprisingly at first, even n and c�(also u as well)

were shown to be nonlinear at high shear stress values. Thecritical stress value for the onset of nonlinearity was

approximately the same as that at which breakdown of theSOR occurred.

The customary view that the SOR breaks down becauseof the saturation of chain extension and orientation wasdemonstrated to be incorrect under shear because the failureof the SOR was observed to occur at a much earlier stage inboth chain extension and orientation. Specifically, the chainextension at the point of breakdown of the SOR was about27% of the full extension (see Fig. 3) and the orientationangle of the birefringence was 23°.

Acknowledgments This research used resources of the Center forComputational Sciences at Oak Ridge National Laboratory, throughthe University of Tennessee Computational Sciences Initiative.Additional support for CB was provided by the University ofTennessee Computational Sciences Initiative. ORNL is operated forthe DOE by UT-Battelle, LLC, under contract number DE-AC0500OR22725.

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