NMR Measurement of the Alignment Tensor for a Polymer Melt underStrong Shearing Flow

M. L. Kilfoil and P. T. Callaghan*,

Institute of Fundamental Sciences-Physics, Massey University, Palmerston North, New Zealand, andDepartment of Physics and Physical Oceanography, Memorial University of Newfoundland,St. Johns, Newfoundland, Canada A1B 2X7

Received March 28, 2000

ABSTRACT: 2H NMR quadrupole interaction spectroscopy has been used to measure the deformation ofa 610 kD PDMS melt under shear in a Couette cell. The signals were acquired from a perdeuteratedbenzene probe molecule, which provides a motionally averaged sampling of the entire segmental ensemble.Our Rheo-NMR method involves the use of a selective storage precursor pulse, which enables one toobserve NMR spectra from preselected regions within the flow. We have measured the dependence onthe shear rate of the SXX (velocity) and SYY (velocity gradient) elements of the segmental alignment tensoras well as the angular dependence of the deuterium quadrupole splitting at a fixed shear rate. The resultsagree well with Doi-Edwards theory and return a value for the tube disengagement time of 310 ms.

IntroductionWhen a random coil polymer melt is subjected to

shear, the polymer chain suffers a biaxial deformationin which the principal axis of extension has a preferredorientation with respect to the hydrodynamic velocitydirection. The geometry is as indicated in Figure 1. Thedeformation may be usefully described by means of theaveraged segmental alignment tensor,

where ... represents the ensemble average and theintegral is taken over the curvilinear path of s chainsegments along the chain length L. In the Doi-Edwardsformulation of entangled polymer dynamics,1,2 the stresstensor R is shown to be directly proportional to theaverage alignment tensor SR. Where polymer segmentsare aligned so that SR is non-zero, other physicalproperties will be anisotropic as well. In particular, thedielectric properties that determine the material refrac-tive index will be affected, leading to the optical ani-sotropy known as birefringence. The correspondence ofthe stress tensor and the anisotropic part of the refrac-tive index tensor is known as the stress-optical lawand is expressed by the relation

C is known as the stress-optical coefficient.2,3Birefringence measurements may be used4 to inves-

tigate the segmental alignment tensor and so give someinsight regarding the deformation depicted in Figure 1.If incident plane polarized light has its polarization axisparallel to the extension axis, a unique refractive indexis observed. In this case extinction will occur betweenpolarizers crossed at some angle , generally called theextinction angle. Maximum birefringence effects will beapparent if the polarization axis is incident at 45 to

the extension axis and measurements of n and n| forsuch an arrangement may be used to investigate SR.

The measurement of polymer deformation by opticalmethods relies on some underlying assumptions, forexample, the identity of the macroscopic and microscopicpolarizability tensor and the dominance of intrinsicbirefringence over form birefringence. In the presentarticle we report on an entirely new method of measur-ing polymer deformation under shear in which theorientation of bond vectors is determined throughnuclear spin interactions. We use nuclear magneticresonance to determine the strength of the electricquadrupole interaction of deuterons, a quantity whichdepends in a very simple manner upon the relativeorientation of the electric field gradient axis (the bondaxis) and the polarizing magnetic field used to producethe nuclear Zeeman effect. In our experiments reportedhere, the deuteron is present in a small probe moleculethat rapidly samples the mean alignment of the hostpolymer. Because of the use of such an indirect probe,the measurement of alignment is not absolute and, like

* To whom correspondence should be addressed. Massey University. Memorial University of Newfoundland.

SR(t) ) s0L ds uR(s,t) u(s,t) - 13R (1)

nR ) CR (2)

Figure 1. Schematic diagram of a horizontal Couette cellin which the vorticity axis is aligned transverse to themagnetic field. The angle between the local velocity directionand the magnetic field is given by . The angle between theprincipal axis of the molecular (1,2,3) frame and the velocityaxis, X, is the so-called extinction angle, .

6828 Macromolecules 2000, 33, 6828-6833

10.1021/ma000554m CCC: $19.00 2000 American Chemical SocietyPublished on Web 08/17/2000

the birefringence method, relies on an equivalent to thestress-optical coefficient. However, the measurement ofextinction is absolute and we are able to investigatethe shear-rate dependence of the interaction, thustesting the predictions of Doi-Edwards theory.

TheoryMeasurement of Segmental Alignment by NMR.

A spin-one deuterium nucleus will experience an electricquadrupole interaction between the nuclear quadrupolemoment and the surrounding electric field gradient.This interaction is experienced in the presence of anoverwhelmingly larger Zeeman interaction between themagnetic dipole moment and the polarizing magneticfield B0 and, as a consequence, is observed as a firs-order perturbation projected along the spin quantizationaxis defined by the static magnetic field. This direction,normally labeled the z axis, is to be distinguished fromthe hydrodynamic vorticity direction, Z. For an axiallysymmetric electric field gradient, as is typically foundin a carbon-deuterium bond, the quadrupole interactiontakes the form

Here, is the angle between the field gradient axis andthe polarizing magnetic field, Vzz is the electric fieldgradient strength, Q is the nuclear quadrupole moment,and I is the nuclear spin quantum number. The effectof HQ is to split the deuterium resonance into a doubletseparated in frequency by (3/2eVzzQ/h)P2(cos ). Thequadrupole interaction strength 3/2eVzzQ/h is usually onthe order of 100 kHz in partially ordered systems. Theangular dependence arising from the second-rank Leg-endre polynomial, P2(cos ), makes it possible to use thequadrupole splitting to determine mean bond orienta-tion.

In simple isotropic liquids, molecular tumbling causes and hence HQ to fluctuate. Provided that this motionis more rapid than the quadrupolar interaction strength,a condition which is true for all but the most viscousliquids, the quadrupolar Hamiltonian is averaged tozero. In anisotropic liquids, the motional averaging isincomplete. Suppose there exists a local director, in-clined at angle R to the magnetic field B0, which is fixedin space or which fluctuates slowly by comparison withthe quadrupolar frequency. The orientation (,) of theelectric field gradient (bond) axis with respect to B0 canthen be decomposed into (R,R) describing the bond axiswith respect to the director, and R. Let us for themoment assume that distribution of (R,R) with respectto the axis labeled by R is independent of the particularorientation of that axis. The significance of this assump-tion will be discussed in the next section. Meanwhile,we note that the spherical harmonic addition theoremmay be used to factorize as follows,5

P2(cosR) thus defines a local order parameter thatscales the quadrupole interaction strength.

Here, we consider the case of a small benzene probemolecule that is placed as a dilute species near apolymer segment whose orientation u defines the localdirector. The tumbling probe molecule will undergosteric interactions with that segment and experience an

anisotropic mean orientation. The probe will thusexhibit a scaled down quadrupole splitting associatedwith that local site via a pseudo-nematic interaction.6This interaction is akin to the local anisotropic stericinteraction experienced by small probe molecules placedin a nematic liquid crystalline environment.7 On aver-age, the probe molecule samples an ensemble averagevalue for P2(cos R) as it diffuses over the moleculardimensions. For typical small molecules the diffusiontime to cross the molecular length scale is certainlysufficiently short to ensure that motional averagingoccurs in the sampling of the distribution of u vectors.The ensemble average of P2(cos R) is identically Szz )s0L ds uz(s) uz(s) - 1/3, and the quadrupole splittingmay be written

In the Doi-Edwards description of the alignment tensorSR, the relevant axis system is the hydrodynamic framedescribed respectively by the velocity (X), velocity gradi-ent (Y), and vorticity (Z) axes. Thus, to relate Doi-Edwards theory to the case of the NMR quadrupoleinteraction experiment, it is necessary to transform thistensor into the frame of the magnetic field, i.e.,

where the polar angle defines the direction of thevorticity axis Z relative to B0 and the azimuth definesthe orientations of X and Y. We are concerned with anexperiment in which the vorticity axis is situated normalto B0 and the projection along B0 is in the X-Y(velocity-velocity gradient) plane. We write

for the diagonal tensor element representing the inter-action strength measured in the NMR experiment. Notethat this expression reduces to P2(cos )SXX for SXY )0, SYY ) SZZ ) -1/2SXX, that is, when the deformationis uniaxial.

The Doi-Edwards Model for Segmental Align-ment under Steady Shear. The standard descriptionfor polymer deformation under shear is the tube-reptation model of Doi and Edwards,1 parametrized bythe tube disengagement time d and the tube diametera. d is the dominant relaxation time for the internaldynamics of the polymer; in this model it is the timefor the polymer chain to leave the deformed tube byreptation. By solving the diffusion equation in the tube,Doi and Edwards show that the equilibrium deformationis given by

(s,t) is the probability that the chain segment s is inthe deformed tube at time t and U is the independentalignment approximation for the tube orientation dis-tribution. In the independent alignment approximationthe segments deform affinely. The stress tensor iswritten

HQ )3eVzzQ

4I(2I - 1)P2(cos )(3Iz

2 - I2) (3)

P2(cos ) ) P2(cos R) P2(cos R) (4)

) (32 eVzzQh )P2(cos R)Szz (5)

R(,)(SXX SXY 0SXY SYY 00 0 SZZ )R-1(,) (6)

SZZ ) SXX cos2 - 2SXY sin cos + SYY sin

2 (7)

SR(s,t) ) s-t dt( @@t(s,t-t))URIA(E(t,t)) (8)

Macromolecules, Vol. 33, No. 18, 2000 Measurement of Alignment Tensor for a Polymer Melt 6829

where Ge ) 3kBTcb2/a2, b is the Kuhn segment size, andc is the segment concentration.

For the reptation model it is convenient to set (t) )1 for t < d and 0 for t > d. @/@t(t - t) reduces to (t- (t - d)) and the relevant deformation, E, is simplythat which occurs for shear rate over the duration, d.In the case of steady shearing at rate that deformationis d so that

However, here we have evaluated eq 8 using used theexact reptation expression for (t), namely, podd (8/2p2)exp(-p2t/d). Figure 2a shows the relevant elements ofthe SR tensor, where X, Y, and Z are the hydrodynamicvelocity, gradient, and vorticity directions. Figure 2bshows the shear-rate dependence of the extinction angle. In terms of the stress tensor, is the alignment anglethat the principal axes of R make with the velocitydirection X.

Nonlinear viscoelastic behavior becomes important atmoderate shear rates in polymers having large d values.Shear thinning occurs once the Deborah number dexceeds unity. In fact, the Doi Edwards model statesthat for shear rates in excess of 1/d, the rate of decreaseof viscosity is so sharp that the shear stress actuallydeclines, thus predicting unstable flow in shear regimeswhere, experimentally, only stable flow has been ob-served. Various refinements to the Doi-Edwards modelmodify the flow curve, and in particular the severity ofthe stress decline beyond d 1. These include theeffects of contour length fluctuations,8,9 nonaffine tubedeformation,10 and convected constraint release.11 In thepresent analysis we shall use the unrefined Doi-Edwards model to compare measured and calculatedsegmental alignment elements.

At this point we revisit the assumption of indepen-dence of the (R,R) distribution on R. For the entangledpolymer, consider shear deformation at a strain rate onthe order of or faster than the tube disengagement rated-1 but slower than the Rouse rate R-1. The orientationdistribution of tube segments becomes anisotropic butthe tube step length and the dynamics of the segmentalmotion at length scales smaller than a tube step areunperturbed. Consequently, we may take R to representthe tube step vector with the (R,R) distribution com-mon to each such vector, thus underpinning the as-sumptions behind eq 4.

Experimental SectionThe polymer sample studied in this work is high molecular

weight (Mw ) 610 kD) poly(dimethylsiloxane) (PDMS), Mw/Mn ) 2.0 (as measured by GPC), obtained from Polysciences(Warrington, PA). M0, the molecular weight per monomer, is74, and the molecular weight between chain entanglementsMe is 104 for this PDMS melt.12 In this was dissolved ap-proximately 10% w/w per-deuterated benzene (Aldrich, Stein-heim, Germany). Rheo-NMR measurements were carried outat a deuteron frequency of 42 MHz in an AMX300 spectrom-eter, using a specially constructed shear cell. Figure 3 showsour Rheo-NMR apparatus. It comprises a Couette cell madeof a machinable glass (MACOR) inner cylinder of outerdiameter 5 mm and a glass outer cylinder of inner diameter 6mm. The poly(dimethylsiloxane) is enclosed in the 0.5-mm gapbetween these cylinders. Surrounding the cell is an rf coiltuned for deuterium (2H). This assembly fits inside a set of

gradient coils and the entire probe is inserted in a 7 Tsuperconducting magnet such that the vorticity axis (thecylinder axis) is normal to the polarizing field. The innercylinder is rotated through a gear stage above the cell that isconnected via a drive shaft running up the bore of the magnetto a gearbox and stepper motor mounted above.

Standard NMR microimaging13 is used to view the PDMSin the Couette gap, to image the velocity distribution acrossthe gap (Figure 4a), or to excite a desired region of the samplefor spectroscopy experiments (Figure 4b) during steady-stateshear. This latter image shows that in one case the selectedregion has the velocity direction coincident with the polarizingfield B0, while in the other, the velocity gradient (shear axis)is parallel to B0. The selective excitation pulse sequence usedhas been specially devised to minimize exposure of selectednuclear spins to any relaxation, so that high-quality NMRspectroscopy can be performed in the desired region. Wedescribe this method in another article;14 however, the es-sential details are shown in Figure 5. The technique employsa selective precursor pulse sandwich that destroys magnetiza-tion outside the desired region but stores along the z-axis for

R(t) ) Ge1L s-L/2L/2 ds SR(s,t) (9)

R( ) GeSR( ) ) GeURIA( d) (10)

Figure 2. (a) Elements of the alignment tensor, SR, and (b)the extinction angle , as a function of the reduced shear rate, d, according to the Doi-Edwards model.

Figure 3. Mechanical arrangement of the horizontal Couettecell used in this work. The gear mechanism is driven by avertical shaft down the magnet bore.

6830 Kilfoil and Callaghan Macromolecules, Vol. 33, No. 18, 2000

later recall the magnetization from the region of interest.Using the pulse sequence of Figure 5a, this magnetization canbe used to obtain a confirmatory image (see Figure 4b), orusing the sequence shown in Figure 5b, it may be recalled forNMR spectroscopy. Our spectroscopic method involves a Hahnecho that refocuses all unwanted Zeeman interactions arisingfrom susceptibility inhomogeneity and leaves undisturbed thedesired but weak quadrupolar precession. Fourier transforma-tion of the echo amplitude with respect to the evolutiondimension t1 results in the quadrupolar spectrum. With thismethod we are able to resolve and measure quadrupolesplittings of a few Hz.

Results and Discussion

Figure 6 shows a series of 2H spectra obtained usingthe two-dimensional quadrupole spectroscopy sequenceshown in Figure 5b. These spectra were acquired for arange of shear rates as indicated, in the region of theCouette cell in which the velocity axis X is aligned withthe magnetic field direction. The shear rates werecalculated from the known inner cylinder rotation speed

and confirmed by velocity microimaging. Note theincreasing width of the peaks as the shear rate in-creases, an effect which we attribute to slight hetero-geneity in the shear field, thus leading to a distributionof splittings whose width is proportional to the average

Figure 4. (a) Speed map showing the shear across the Couettecell. This map was reconstructed from two NMR images thathad been respectively encoded for velocity components in thex and z direction. Details of the velocity encoding method canbe found in chapter 9 of ref 13. (b) NMR image obtainedfollowing selective storage of magnetization from a desiredregion of the Couette cell. The image shows the region ofsample that contributes to spectroscopy experiments when ) 0 (velocity direction parallel to B0).

Figure 5. Rf and gradient pulse sequence used to obtainimages and spectra from selected regions of the sample. Thesection of the pulse sequence before the dashed line is theselective storage segment that destroys all unwanted magne-tization and stores the desired magnetization along the z-axisfor later recall, in (a) for subsequent imaging and in (b) forsubsequent NMR spectroscopy. The spectroscopic pulse se-quence shown in (b) uses a spin-echo to refocus unwantedZeeman interactions while retaining the desired evolution, overthe t1 interval, under the quadrupole Hamiltonian.

Figure 6. Examples of deuterium NMR spectra obtained byFourier transforming the signal evolution in the t1 domain.The splitting arises from the electric quadrupole interactionand is a measure of the ensemble-averaged local order.

Macromolecules, Vol. 33, No. 18, 2000 Measurement of Alignment Tensor for a Polymer Melt 6831

shear rate. A corresponding set of data was acquiredfor the Couette cell region in which the velocity gradientdirection Y was coincident with the field axis. Figure 7shows the dependence of these splittings on shear rate.On the same graph we plot the corresponding Doi-Edwards alignment tensor curves (eq 10) in which theeffective quadrupole interaction strength is scaled tomatch the absolute alignment and the tube disengage-ment time is adjusted to match the reduced shear rate d. The fit to the two curves is quite good and anyremaining discrepancy from the Doi-Edwards theorymay be due to the chain length polydispersity or to thesmall nonuniformity in the shear rate. The best overallfit yields a tube disengagement time of 310 ms and apseudonematic order parameter, P2(cosR), of 4.74 10-4. We note that the tube disengagement time ob-tained by this method agrees well with that for un-cross-linked polymers of high molecular weight, calculatedaccording to15

Using the literature values15 for the tube diameter aand the monomeric frictional coefficient 0, we calculated ) 350 ms.

We have also carried out this experiment at a fixedshear rate ( d ) 3.9), varying the angle between thevelocity direction X and the magnetic field directionthrough a number of prescribed angles. This wasachieved by changing the orientation of the magneticfield gradient used in the precursor selective storagepulse. The angular dependence of the splitting is shownin Figure 8, along with the Doi-Edwards curve calcu-lated using eq 7. Again, the agreement is excellent. Atthis particular value of the reduced shear rate, the

extinction angle, , is predicted to be 14.6 from theresults in Figure 7. To gain some intuitive insightregarding the orientation of the polymer in the flow, wehave shown plotted in Figure 8 the theoretical curvefor P2(cos(-)) with adjusted to give the best fit () 17.7). Such a theoretical curve is naively based onthe assumption of an axially symmetric polymer defor-mation, whereas the Doi-Edwards formulation predictsa non-zero second normal stress difference, yy - zz.

Figure 7. Quadrupole splittings, , obtained from the spectra as shown in Figure 6, for a selected region of the horizontalCouette cell in which the velocity direction (solid circles) and gradient direction (open circles) are respectively parallel to themagnetic field. The lines are fits using the Doi-Edwards model in which the absolute splitting (vertical axis) is scaled to yieldthe pseudonematic order parameter and the horizontal axis is scaled to yield the tube disengagement time, d. The best compromisefit corresponds to d ) 310 ms.

d )a20MW

3

MeM022kBT

(11)

Figure 8. Quadrupole splittings, , versus orientation angle,, at a fixed Deborah number of d, )3.9.0 corresponds tothe velocity axis aligned with B0 and 90 to the gradient axisaligned with B0. The solid line shows the predictions of theDoi-Edwards model, with the angular dependence as givenby eq 7. The dashed line corresponds to a simple uniaxialtransformation (P2(cos(-))). is the corresponding extinc-tion angle in each case. The images show the selected regionsused to obtain ) 0, 45, and 90.

6832 Kilfoil and Callaghan Macromolecules, Vol. 33, No. 18, 2000

The poorer fit to the data given by this naive modelprovides independent confirmation of the biaxial natureof the deformation and further support for the Doi-Edwards description.

One defect of the present approach involves the needto use an indirect probe of the segmental order and thepossibility that the benzene diluent may plasticize thepolymer and significantly alter its dynamical behavior.To check this point, we have carried out flow curvemeasurements on 610 kD PDMS both with and withoutthe 10% benzene and find no significant difference inthe behavior. A different approach to direct measure-ment of the alignment tensor would be to use a per-deuterated polymer. Apart from the difficulty in obtain-ing high molecular mass PDMS, there exists anadditional complication in that the nature of the mo-tional averaging is quite different when signals areacquired from deuterons attached to the polymer chain.Now the characteristic time over which a given deuteron-labeled segment is able to sample the entire ensembleof possible orientations, u, is given by the tube disen-gagement time (J10 ms) rather than the time for asmall probe molecule to diffuse over the chain dimen-sions (j0.1 ms). Consequently, the averaged interactionwill arise from a subensemble of orientations and theoverall 2H NMR spectrum will reflect an inhomogeneousdistribution of such subensemble spectra. The distribu-tion of subensemble effective directors will lead tosomewhat different average quadrupolar splittings thanthose predicted by the Doi-Edwards curves of Figure2, although there should exist significant differences inthe nature of the quadrupole spectra observed whendifferent sample orientations are chosen.

In principle, it is possible to use the proton dipolarinteraction to gain similar information regarding chainsegment alignment, and such experiments avoid theneed to isotopically label. The proton spectrum may becomplicated by the influence of scalar spin-spin inter-actions (although these are absent in the case of PDMS),by the role of long-range dipolar couplings that aresuperposed on the desired, local two-spin couplings, andby the complexities of motional averaging discussed inthe previous paragraph. A recent study of weak dipolarspectral broadening in sheared PDMS has howeverrevealed a significant alignment effect and one whichis dependent on the selection angle .16

ConclusionWe have shown here that deuterium NMR spectros-

copy can provide useful insight regarding the deforma-tion of polymers under shear. Our results obtained from610 kD PDMS, up to a Deborah number in excess of 5,are consistent with the classical Doi-Edwards descrip-tion and yield a value for the tube disengagement time,which is consistent with other measurements for thisparticular polymer. The use of an isotopically labelledsmall-molecule probe has the advantage of experimentalsimplicity and the nature of the motional averagingachieved means the full segmental ensemble averageis returned in the NMR spectrum, albeit with anunknown scaling factor for the pseudonematic interac-tion. This scaling factor is however no different inprinciple from the stress-optical coefficient required inbirefringence studies. Our Rheo-NMR method has sig-nificant advantages in that it avoids the need for sampletransparency, is not affected by unwanted scattering

impurities, voids, or irregular surfaces, and provides alocal, microscopic description for which the known spinHamiltonian gives the basis for an exact theory.

By changing the nature of the labeled probe, it maybe possible to gain additional information. For example,site-specific deuteron labeling on the polymer couldreturn the alignment tensor, SR(s), corresponding todifferent points along the polymer chain. Where adiluent probe molecule is used, different motionalaveraging regimes could be accessed by tuning themolecular diffusion coefficient. It is possible, in principle,to study deformation at specific molecular sites withinblends, and by use of a probe molecule that can partitionbetween different regions of the sample, it may bepossible to study translational ordering. In a recentexperiment, anomalous alignment effects have beenobserved when the probe diluent is a small oligomer,an effect which was attributed to weak smectic order-ing.16

Other NMR isotopic labels are possible which accessorientational order. In particular, 13C has a largeanisotropic chemical shift whose angular dependencewhen measured in the B0 field can provide informationregarding the distribution of local director alignment.While 13C labeling is expensive, abundant sites do existin most polymers of interest and this spin probe couldprove effective in the study of model molecular systems.

Finally, we note that for d . 1 the Doi-Edwardspicture may hold longer. We hope to extend our Rheo-NMR observations further into this regime to investi-gate possible deviations.

Acknowledgment. M.L.K. acknowledges financialsupport from the National Research Council of Canadawhile P.T.C. is grateful to the New Zealand Foundationfor Research, Science and Technology and to the RoyalSociety of New Zealand respectively for funding supportunder the Public Good Science Fund and the MarsdenFund. The authors also acknowledge the assistance ofRyan Cormier in carrying out our flow curve measure-ments on PDMS samples.

References and Notes

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(2) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics;Oxford University Press: Oxford, England, 1986.

(3) Janeshitz-Kriegl, H. Polymer Melt Rheology and Flow Bire-fringence; Springer: New York, 1983.

(4) Fuller, G. G. Optical Rheometry of Complex Fluids; OxfordUniversity Press: New York, 1995.

(5) Abragam, A. The Principles of Nuclear Magnetism; OxfordUniversity Press: Oxford, England, 1961.

(6) Deloche, B.; Samulski, E. T. Macromolecules 1981, 14, 575.(7) Burnell, E. E.; de Lange, C. A. Chem. Rev. 1998, 98, 2359.(8) Doi, M. J. Polym. Sci. Lett. 1981, 19, 265.(9) Rubinstein, M.; Panyukov, S. Macromolecules 1997, 30, 8036.

(10) Marrucci, G. J. Polym. Sci. Phys. 1985, 23, 159.(11) Marrucci, G.; Ianniruberto, G. J. Non-Newt. Fluid Mech.

1999, 82, 275.(12) Callaghan, P. T.; Samulski, E. T. Macromolecules 1998, 31,

3693.(13) Callaghan, P. T. Principles of Nuclear Magnetic Resonance

Microscopy; Oxford University Press: New York, 1991.(14) Kilfoil, M. L.; Callaghan, P. T. (in preparation).(15) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.;

Wiley: New York, 1980.(16) Callaghan, P. T.; Kilfoil, M. L.; Samulski, E. T. Phys. Rev.

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MA000554M

Macromolecules, Vol. 33, No. 18, 2000 Measurement of Alignment Tensor for a Polymer Melt 6833