Rheol Acta (2012) 51:385–394DOI 10.1007/s00397-012-0622-1

ORIGINAL CONTRIBUTION

Stress and neutron scattering measurements on linearpolymer melts undergoing steady elongational flow

Ole Hassager · Kell Mortensen · Anders Bach ·Kristoffer Almdal · Henrik Koblitz Rasmussen ·Wim Pyckhout-Hintzen

Received: 25 August 2011 / Revised: 12 January 2012 / Accepted: 17 January 2012 / Published online: 26 February 2012© Springer-Verlag 2012

Abstract We use small-angle neutron scattering tomeasure the molecular stretching in polystyrene meltsundergoing steady elongational flow at large stretchrates. The radius of gyration of the central segmentof a partly deuterated polystyrene molecule is, in thestretching direction, increasing with the steady stretchrate to a power of about 0.25. This value is about half ofthe exponent observed for the increase in stress value σ ,in agreement with Gaussian behavior. Thus, finite chainextensibility does not seem to play an important rolein the strongly non-linear extensional stress behaviorexhibited by the linear polystyrene melt.

Keywords Scattering · Polymer melt ·Uniaxial extension · Polystyrene

O. Hassager (B) · A. BachDepartment of Chemical and Biochemical Engineering,Technical University of Denmark, 2800 Kgs. Lyngby,Denmarke-mail: OH@kt.dtu.dk

K. MortensenDepartment of Basic Sciences and Environment,Faculty of Life Sciences, University of Copenhagen,1871 Frederiksberg C, Denmark

K. AlmdalDepartment of Nanotechnology, Technical Universityof Denmark, 2800 Kgs. Lyngby, Denmark

H. K. RasmussenDepartment of Mechanical Engineering, TechnicalUniversity of Denmark, 2800 Kgs. Lyngby, Denmark

W. Pyckhout-HintzenJülich Centre for Neutron Science-1 & Institute for ComplexSystems, Forschungszentrum Jülich, Jülich 52425, Germany

Introduction

Elongational flow has long been and continues to bea topic of significant rheological interest (Petrie 1979;McKinley and Sridhar 2002). For dilute polymer so-lutions, it is known from visual observations (Perkinset al. 1997) that the polymer molecules may becomefully extended in strong elongational flows. Our un-derstanding of the fully extended state continues toimprove as a result of accurate theoretical modeling(Underhill and Doyle 2007). Moreover, the transitionfrom near equilibrium to the fully stretched state is arather sharp transition that occurs at a specific elonga-tion rate (De Gennes 1974).

For entangled polymer melts, our understanding ofthe molecular behavior in strong elongational flow isless satisfactory. Simple continuum models such as theOldroyd B model or the Lodge rubberlike liquid (Birdet al. 1987a) predict a stress singularity at a criticalelongation rate. This singularity may be suppressedby invoking finite extensibility in the associated struc-tural models (Bird et al. 1987b), but for most models,finite extensibility merely replaces the singularity witha sharp transition to an asymptotic behavior that maybe associated with a fully stretched state.

While one would not expect such simple models suchas the Oldroyd B model to provide more than just afirst-order approximation for non-linear dynamics, it ismore serious that the situation is not much better formost current structural models. This is true in particularfor the tube theory framework for modeling the dynam-ics of entangled polymer systems. Going back to thepioneering development by Doi and Edwards (1986),the basic model predicts a limiting stress in strongextensional flow. The molecular origin of the limiting

386 Rheol Acta (2012) 51:385–394

stress is the assumption of instantaneous chain retrac-tion so that the stress is due to orientation only. Perfectorientation of the tubes but with no chain stretchinggives a stress that is five times the plateau modulus.The corresponding steady elongational viscosity thendecreases with elongation rate ε̇ as ε̇−1. However, itwas recognized early that chain stretching will occur atdeformation rates faster than the inverse Rouse time(Marrucci and Grizzutti 1988; Pearson et al. 1989). Theconcept of chain stretching was consequently includedin several models based on the Doi and Edwards frame-work (Fang et al. 2000; Mead et al. 1998; Schieberet al. 2003). The result of including chain stretchingalone is to introduce a stress singularity at an elon-gation rate of the order the inverse Rouse time. Thissingulary may again be relieved by invoking finiteextensibility. The resulting models exhibit four flowregimes: a linear viscoelastic regime dominated by rep-tation wherein the elongational viscosity is constant, asecond regime wherein the elongational viscosity de-creases as ε̇−1, and a third regime with a sharp in-crease in the elongational viscosity to an asymptoticbehavior in the fourth regime. There was at the timeno data for entangled monodisperse polymer melts tocompare with (Mead et al. 1998). However, subse-quent data on steady elongational viscosity of narrowmolar mass distribution (NMMD) polystyrene (Bachet al. 2003a; Luap et al. 2005) demonstrated a ratherdifferent nonlinear behavior with an elongational vis-cosity scaling approximately as ε̇−0.5. The conflict wasrecognized immediately by Marrucci and coworkerswho introduced the concept of interchain pressure inpolymer dynamics (Marrucci and Ianniruberto 2004)whereby chain stretching is balanced by lateral forcesbetween the chain and the tube wall. In this way, theywere able to explain the ε̇−0.5 scaling. Moreover, theypredicted a maximum in the steady elongational vis-cosity for smaller molar masses that was subsequentlyobserved (Nielsen et al. 2006). Wagner and cowork-ers introduced the interchain pressure concept intotheir molecular stress function theory whereby theyobtained a constitutive equation with the correct ε̇−0.5

scaling (Wagner et al. 2005). Later refinements haveallowed quantitative predictions of available non-linearrheological measurements from linear viscoelastic dataalone (Wagner and Rolón-Garrido 2009, 2010). Theinterchain pressure concept has also been introducedinto the Rolie–Poly (Likhtman and Graham 2003)constitutive equation, whereby likewise a quantitativedescription of the elongational viscosity is obtained(Kabanemi and Hétu 2009). While there are currentlytwo tube-based models capable of describing the non-linear stress behavior in elongational flow, the situation

is not entirely satisfactory for two reasons. First thenumber of modifications needed on the basic modelis in itself worrying. Second the very fact that twodifferent tube-based models both describe the stressdata quantitatively does not leave a clear picture.

To gain insight into the physics of strong elonga-tional flows from the experimental side, Luap et al.(2005) measured birefringence together with the stress.For their two NMMD polystyrene melts, they foundthe stress optical rule fulfilled up to a critical stress oforder 2.7 MPa or about ten times the plateau modulus.The fact that the ε̇−0.5 scaling sets in when the steadystress is of order the plateau modulus therefore suggeststhat the physics behind the ε̇−0.5 scaling is not relatedto finite extensibility of the chains. However, theirmeasurements also show that the ε̇−0.5 scaling extendsunchanged well beyond the onset of chain stretch in thenon-Gaussian regime (Luap et al. 2005) evidenced byfailure of the stress-optical rule. In other words, finiteextensibility does come into play, but it is not the primereason for the observed scaling nor does it lead to adrastic transition to an asymptotic behavior.

The purpose of this study is to gain further insightinto the physics of strong elongational flows by ap-plication of small-angle neutron scattering (SANS). Inpioneering work (Boué et al. 1982, 1983) on the appli-cation of SANS to understand dynamics in stretchedpolystyrene melts, the melts were rapidly extended bya stretch ratio of 3 at temperatures not far above theglass transition. The subsequent relaxation of the radiusof gyration in the direction transverse to the stretchingdirection was subsequently followed as function of timefrom a sequence of quenched samples. The early reduc-tion in the transverse direction predicted by the basictube theory was not observed possibly due to Rouserelaxation during the quenching process. The minimumwas, however, clearly observed later by Blanchard et al.(2005) for PI utilizing an in situ SANS device.

SANS measurements have also been performed onNMMD polystyrene samples quenched from steadyelongational flow (Muller et al. 1990). While the mea-surements were performed in the linear viscoelasticregime and therefore do not cast light on the ε̇−0.5 scal-ing, the study does give credence to a straightforwardanalysis in terms of radii of gyration in the paralleland transverse direction, respectively. More recently,neutron scattering has been reported on the steadyflow in a 4:1 planar contraction/expansion of NMMDpolystyrene samples (Bent et al. 2003). However, whilethe inhomogeneous flow is steady in a Eulerian sense,it is not steady in a Lagrangian sense so it is notstraightforward to compare with steady viscosity data.In fact while scattering from deformed polymer melts

Rheol Acta (2012) 51:385–394 387

has been analyzed in considerable detail both exper-imentally and theoretically (Read 2004), we do notknow of SANS measurements that may cast light on themolecular behavior behind the ε̇−0.5 scaling.

Experimental

Synthesis

To eliminate ambiguities due to polydispersity, weused anionic polymerization (Ndoni et al. 1995) toprepare polymer samples of narrow molar mass dis-tribution. The key polymer is a symmetric triblockcopolymer PS(H)–PS(D)–PS(H). The middle block(PS(D)) is polystyrene with hydrogen substituted bydeuterium, while the two identical end-blocks are ordi-nary polystyrene (PS(H)). This polymer is mixed into amatrix of ordinary polystyrene of the same molar mass.In this way, we obtain a solution of deuterium labeledmiddle segments in a sea of ordinary polystyrene. Wehave labeled only the middle block with deuterium fortwo reasons. Firstly, it is anticipated that the molecularrelaxation of stretch that inevitably will occur duringquenching primarily will take place near the ends ofthe chain. Secondly, if we labeled the entire chain,the molecular length scale in the direction parallel tothe macroscopic elongation would be too large to bemeasured accurately in the SANS facility used.

Chromatography

The molar masses of the polystyrene samples weredetermined by size exclusion chromatography (SEC)with toluene as the eluent and employing a columnset consisting of a 5pm guard column and two 300 ×8 mm2 columns (PLgel Mixed C and Mixed D). Thesystem is equipped with a triple detector system (acombined Viscotek model 200 differential refractiveindex (DRI) and differential viscosity detector plus aViscotek model LD 600 right angle laser light scatteringdetector (RALLS)). On the basis of calibration withnarrow molar mass polystyrene standards, the values ofMw and Mw/Mn were determined to be 145 kg/mol and1.13 for the triblock-copolymer and 150 kg/mol and 1.07for the homopolymer. Note that due to the calibrationtechnique, these numbers refer to molar masses fornon-deuterium substituted molecules. Thus, the molarvolume for the triblock and homopolymer are the samewithin experimental error. The molar mass numbersgiven in the main text are to be interpreted in thesame manner. Molar masses based on direct use of theDRI and RALLS signals were within error identical to

Table 1 Components in test material

Components Block copolymer HomopolymerPS(H)–PS(D)–PS(H) PS(H)

Mw 145 kg/mol 150 kg/molMw/Mn 1.13 1.07

The mass fraction of the deuterated middle block is 0.30 of theblock copolymer, while the mass fraction of the copolymer in themixture is 0.33 so the mixture contains 10% deuterium

the values obtained by calibration. The presence in theSEC setup of the RALLS detector enhances sensitivitytoward high molar mass impurities. No such impuritieswere detected. The final molecular characteristics aresummarized in Table 1.

Mechanical spectroscopy and filament stretching

We performed small amplitude oscillatory shear todetermine the linear viscoelastic properties of thepolystyrene melt. We used a plate–plate geometry onan AR2000 rheometer from TA Instruments. The mea-surements were performed at 150◦C with the results asshown in Fig. 1.

To establish a well-defined steady elongational flow,we use a filament stretching rheometer (McKinley andSridhar 2002) equipped with a thermostat (Bach et al.2003b). Prior to flow, the sample is allowed to equi-librate for a time longer than the estimated longest

10−2 10−1 100 101 102100

101

102

103

Frequency ω (s−1)

Mo

du

li (k

Pa)

Fig. 1 G′ and G′′ for 150 kg/mol narrow molar mass distributionpolystyrene at 150◦C. Points represent data. Full red and greenlines represent BSW fit with parameters G0 = 220 kPa, ne = 0.23,ng = 0.67, λm = 19 s, λc = 0.013 s. Blue lines represent the con-tributions to G′′(ω) from the Likhtman model for the Rouse dy-namics (dashed-dotted line longitudinal mode relaxation, dashedline fast Rouse motion inside tube, full line sum of longitudinalmodes and fast Rouse motion)

388 Rheol Acta (2012) 51:385–394

relaxation time. Then at times t > 0, the plates areseparated from one another in such a fashion that theelongation rate at the plane of observation (the planeof symmetry) is kept constant at a given value ε̇. Theforce at the end-plate is measured online, as is thefilament diameter in the plane of observation. The platemotion is controlled to obtain a constant value of ε̇.At time t = 3/ε̇, the fluid in the plane of observationhas undergone locally a Hencky strain of ε = ε̇t = 3.This means that the fluid in the plane of observationhas been stretched by a factor e3 ≈ 20. The fluid in theplane of observation is the only part of the sample thatexperiences ideal elongational flow.

In Fig. 2, we show the development of the stress inthe plane of observation as function of the imposedHencky strain for elongation rates 0.001, 0.01, and0.1 s−1. It is seen that the stress increases monoton-ically reaching a constant value at a Hencky strainof about 2.5 for ε̇ = 0.1 s−1 and somewhat earlier forε̇ = 0.001 s−1. We assume therefore that a steady flowsituation has been established at Hencky strain 3. Alsoshown in Table 3 are the corresponding steady stressesdivided by the modulus G0 = 220 kPa. It is seen thatthe steady stresses at ε̇ = 0.001 s−1 are of order G0

while the steady stress at ε̇ = 0.1 s−1 are about 30G0

indicating that the experimental window is well into thenon-linear regime including stress levels for which Luapet al. found deviations from the stress optical rule.

Quenching and scattering

Assuming that steady flow conditions are establishedaround Hencky strain 2.5, a sequence of samples ob-tained by quenching the polystyrene bridge at Henckystrain 3 should contain molecular configurations rep-resentative of steady elongational flow. Thus, whenthe polymers in the plane of observation have reached

Fig. 2 Axial stress as function of continuum stretch ε = ε̇t in theplane of observation for elongational flow of polystyrene meltundergoing startup of elongational flow with constant elongationrates ε̇. The stress appears to reach a steady value at ε ≈ 2.5

Table 2 Physical properties of probe melt at experimental con-ditions (126.7◦C)

ρCp k τw η0 τR

2 106 J/m2 K 0.17 W/mK 970 s 70 MPa s 160 s

Density ρ, heat capacity Cp, longest relaxation time τw, zeroshear-rate viscosity η0, and Likhtman Rouse time τR

Hencky strain ε = 3, the liquid bridges were quenchedby a series of cooled nitrogen jets thereby providingsamples for the SANS experiments. Due to the atac-tic nature of the anionically synthesized polystyrene,this polymer does not crystallize but undergoes aglass transition around 100◦C whereby the molecu-lar configurations are frozen. All cooling equipmentwas made of stainless steel. Consequently, the coolingequipment acted as a heat sink in the oven even whenthe gas flow was turned off. The influence on filamenttemperature was investigated by placing a thermocou-ple in a filament and measuring the temperature. Whenthe oven temperature was set to 130.0◦C, the measuredfilament temperature was 126.7 ± 0.5◦C. We use thistemperature as the filament temperature for all elon-gational measurements, but the uncertainty in the tem-perature is a concern especially since the polymer timeconstants are highly dependent on the temperature.

The interpretation of the scattering results in termsof steady flow configurations hinges of course on theassumption that the polymer chains are frozen beforethey have time to relax. Hence, we need to know thetime needed to quench the entire filaments below theglass transition temperature. Because of the high flowrate of the cooling gas, we estimate that the coolingrate of the sample is limited primarily by temperaturegradients within the sample, whereas the surface of the

Fig. 3 Schematic illustration of the SANS setup, showing thecentral part of the elongated PS sample exposed to the neutronbeam

Rheol Acta (2012) 51:385–394 389

Fig. 4 Two-dimensional contour plots of neutron scattering dataand fits as obtained from sample of PS–PSd–PS/PS mixtureof polystyrene triblock copolymer mixed into a similar sizepolystyrene homopolymer. The data represent experiments un-der a equilibrium conditions and b–d steady-state elongationalflow at strain rates: b 0.001 s−1, c 0.01 s−1, and d 0.1 s−1.Left column is the measured scattering intensity, middle columnis a 2D Debye fit, and the right column is the difference be-tween the two first columns. The coordinates in each box arethe scattering vectors qx ∈ [−0.075 Å−1, +0.075 Å−1], and qz ∈[−0.075 Å−1, +0.075 Å−1]

filament obtains the temperature of the cooling gasimmediately. The internal gradient is governed by atime constant τk = ρCp R2/k where ρ, Cp, and k are

the density, heat capacity, and thermal conductivity,respectively, and R is the radius of the cylindricalshaped sample during quenching. Using R = 1 mm andthe physical constants in Table 2, we obtain τk = 3 s.With a gas temperature of 30◦C, we find that a cool-ing time of 1 s is sufficient to cool the sample belowthe glass transition. In reality, the molecular motion isslowed considerably well before the glass transition isreached. Since this is more than two orders of magni-tude smaller than the Rouse time, we feel confident thatthe quenched samples do indeed represent steady flowconditions.

The small-angle neutron scattering experimentswere performed at the SANS-2 instrument, the PaulScherrer Institute in Switzerland, using standard datatreatment (Strunz et al. 2004). The solid PS sampleswere fixed to standard sample holders with tape andmounted in the beam so only the central part of thesample is exposed to neutrons (see Fig. 3). The SANSdata shown in this report are all obtained with a sample-to-detector distance equal 5.7- and 6-m pin-hole colli-mation with 16-mm-diameter entrance hole and 5-mmexit hole next to the sample. The neutron wavelengthwas 4.6 Å with 10% resolution. Instrumental settingwith 11.6-Å neutron wavelength was also applied, butdid not give reliable data due to very low signal-to-noiseratio, and is accordingly not included in this report.Correspondingly, we do not include results obtainedat smaller sample-to-detector distances, since thesedata do not contribute to the relevant length scalesstudied. The results are shown in Fig. 4 left columnand summarized in Table 3. The scattering statisticsreflects the sample volume scattered from: Samplesstretched at the highest rate were pre-stretched morethan the other samples and therefore ended up havingsmaller final diameter, lower volume and thus worsestatistics.

Table 3 Elongational stress and SANS data for hot-stretched and quenched samples

Sample ε̇ (s−1) Wi WiR Rg‖ (Å) Rg ⊥ (Å) Rm (Å) λ‖ λ⊥ σ (kPa) σ/G0

epsd 0001al3.0 0.001 1.0 0.16 66 41 48 1.4 0.89 210 0.95epsd0001al5.7 0.001 1.0 0.16 77 43 52 1.6 0.93 210 0.95epsd001bl3.0 0.01 10 1.6 147 32 53 3.2 0.69 1,380 6.3epsd001bl5.7 0.01 10 1.6 155 31 53 3.4 0.67 1,380 6.3epsd001cl3.0 0.01 10 1.6 149 33 54 3.3 0.72 1,370 6.2epsd001cl5.7 0.01 10 1.6 157 32 54 3.4 0.69 1,370 6.2epsd001d10l3.0 0.01 10 1.6 152 34 56 3.3 0.74 1,330 6.0epsd001d10l5.7 0.01 10 1.6 147 35 56 3.2 0.76 1,330 6.0epsd01cl3.0 0.1 100 16 228 23 49 4.9 0.50 5,700 26epsd01cl5.7 0.1 100 16 259 24 53 5.6 0.52 5,700 26epsd01a+b+cl5.7a 0.1 100 16 269 22 51 5.8 0.48 6,100 28

Here R3m = Rg‖ R2

g⊥ while λ‖ = Rg‖/Rg0 and λ⊥ = Rg⊥/Rg0 with the effective Rg0 = 46 Å. Also σ = σzz − σrr and G0 = 220 kPa. Allsamples were stretched to Hencky strain 3aSANS data obtained by measuring on three samples simultaneously

390 Rheol Acta (2012) 51:385–394

Analysis and results

Polymer relaxation times and stresses

The linear viscoelastic moduli in Fig. 1 are modeled bya continuous spectrum of relaxation times Baumgaertelet al. (1990).

H(λ) = neG0

[(λ

λm

)ne

+(

λ

λc

)−ng]

h(

1 − λ

λm

)(1)

G(t) =∫ ∞

0H(λ)

exp(−t/λ)

λdλ (2)

with parameters ne = 0.23, G0 = 220 kPa, ne = 0.23,ng = 0.67, λm = 19 s, and λc = 0.013 s (Jackson et al.1995) at 150◦C. The part associated with ne is associ-ated with the viscoelastic behavior, while the part withng is associated with the glassy behavior. G0 is theelastic modulus in fast deformation of the hypotheticalmaterial in which the glassy behavior is absent. Basi-cally G0 = G(0) when the glassy part of the relaxationspectrum is omitted. The parameter G0 as definedhere is denoted the plateau modulus by Jackson et al.(1995) although the physical interpretation is closer tothe entanglement modulus proposed by Likhtman andMcLeish (2002).

The longest relaxation time and the zero shear-rateviscosity are extracted from the viscoelastic part of therelaxation modulus,

η0 =∫ ∞

0G(s)ds

= neGλm/(1 + ne) ≈ 780 kPa s, (150◦C) (3)

τw =∫ ∞

0G(s)s ds/

∫ ∞

0G(s)ds

= λm(1 + ne)/(2 + ne) ≈ 10 s, (150◦C) (4)

The temperature dependence for polystyrene is de-scribed by a shift factor given by the WLF equation(Bach et al. 2003a)

log aT = −c01(T − T0)

c02 + (T − T0)

(5)

where c10 = 8.86, c2

0 = 101.6 K, T0 = 136.5◦C, and log isthe logarithm to base 10. We measured the tempera-ture in the filament stretching apparatus to be 126.7◦C.Consequently, the time constants at experimental con-ditions are about 97 longer than at 150◦C. Our estimatesof the longest relaxation time and zero shear-rate vis-cosity under the experimental conditions are shown inTable 2.

In addition to the longest relaxation time, we wouldlike to estimate a time scale for stretch relaxation. Sucha time scale is often taken to be represented by theRouse time τR. Unfortunately, the BSW spectrum is notuseful for extraction of a Rouse time from the presentdata. Basically Rouse scaling would require that G′(ω)

and G′′(ω) both scale as ω0.5 for large ω while the datahave a higher slope represented by the BSW parameterng = 0.67. Therefore, a Rouse time cannot be extractedfrom the BSW spectrum in a meaningful fashion. Toestimate a Rouse time, we use the (Likhtman andMcLeish 2002) relaxation modulus for the Rouse dy-namics of a linear chain in a tube

GLR(t)= Ge

Z

⎛⎝1

5

Z−1∑p=1

exp(

− p2tτR

)+

N∑p=Z

exp(

−2p2tτR

)⎞⎠

(6)

where Ge is the modulus, τR is the Rouse time, and N isthe total number of Kuhn links. The first term describeslongitudinal mode relaxation in the tube while the sec-ond term describes Rouse relaxation not constrained bythe tube. The high frequency part of the loss modulusmay be fitted with the parameters Ge = 220 kPa, Z =9, N = 200, and τR = 1.16 s as shown by the blue linesin Fig. 1. The lowest curve (dashed-dotted) representsthe longitudinal mode relaxation. The next curve frombelow represents the unconstrained Rouse dynamics,and the top curve is complete Rouse dynamics givenas the sum of the two individual contributions. It ap-pears from Fig. 1 that longitudinal mode relaxation isinsignificant compared to unconstrained Rouse motionfor this low value of Z . The complete Likhtman fitto the data including also escape from the tube andconstraint release is indistinguishable from the BSWfit and therefore not presented. The Likhtman para-meter τe = τR/Z 2 = 0.020 s could be compared to theBSW parameter λc = 0.013 s. Given the differences inapproach, this difference does not seem alarming. Webase therefore the analysis on the Likhtman Rousetime which when shifted to experimental conditions(126.7◦C) becomes 160 s.

The longest relaxation time is used to define anon-dimensional elongation rate in terms of theWeissenberg number Wi = ε̇τw. It appears fromTable 3 that the experiments cover the range fromWi = 1.0 to Wi = 100. For low values of Wi, one wouldestimate the steady stress to be σzz − σrr ≈ 3 η0ε̇. Indeedfor Wi = 1.0(ε̇ = 0.001 s−1), this gives a prediction of210 kPa in agreement with the observation in Table 3.However, for Wi = 10 and 100, the dynamics in steady

Rheol Acta (2012) 51:385–394 391

flow will be outside of linear dynamics. We heremake the stresses non-dimensional with the modulus,G0 = 220 kPa.

Scattering

For a-PS, it is known (Fetters et al. 2009) that the mean-square end-to-end distance is given by

〈R2〉0 = 0.437 M (7)

where M is the molar mass. The fully extended length isRmax = nl cos(θ/2) where n is the number of backbonebonds, l is the bond length, and θ the bond angle.The Kuhn length b = 〈R2〉0/Rmax = 18 Å. The deuter-ated middle block has M = 45 kg/mol whereby thecorresponding number of Kuhn links N = 60 and theequilibrium gyration of gyration Rg0 = 57 Å.

Principally we will limit our discussion to charac-teristics parallel and perpendicular to the stretchingdirection, respectively. Analyzing the data in terms ofnarrow intensity sectors of the two-dimensional data inthe directions perpendicular and parallel to the stretchdirections alone have, however, a very low signal tonoise ratio, as due to the small sample volume consti-tuted by the thin (1 mm diameter) filaments obtained atε = 3, and broadening the sectors will introduce errorswhen averaging in circular sectors, as is the commonprocedure. Hence, we opted for the simple assumptionof analyzing the full 2D data array using an ellipticalGaussian distribution whereby we could include alldata in the fitting procedure. Thus, we assume that thedistances between segment pairs n and m is Gaussianwith variance |n − m|b 2λ2

j in the j-direction so the pairdistribution function becomes:

ψ =3∏

j=1

(3

2π |n − m|b 2λ2j

)1/2

exp(

− 3(x j/λ j)2

2|n − m|b 2

)(8)

Here the λ j (=1 at equilibrium) are denoted the stretchfactor in the j-direction. The corresponding expressionfor the scattering function is a modified Debye functionof the form,

S(q) = 2x4 (exp(−x2) + x2 − 1) (9)

where in our situation,

x2 = (q‖ Rg‖)2 + (q⊥ Rg⊥)2 (10)

Here we have defined Rg‖ = λRg0 and Rg⊥ = λ⊥ Rg0.The iso-intensity contour lines are thus assumed el-liptical with constant eccentricity, reflecting that thelabeled middle part of the polymer chain is assumed to

be stretched uniformly at the length scales approached.The experimental data did not, within statistical errorbars, signify deviation from such behavior.

We extract the independent parameters Rg‖ and Rg⊥in Eqs. 9 and 10 from the 2D scattering data, I(q) byminimizing the function f (q) = I(q) − cS(q) in a leastsquares sense. Here c is an additional free parameterdetermined also in the minimization procedure, but ofno further significance. The assumption of simple ellip-tical Gaussian intensity distribution is a fair assumptionand fits very well the experimental data. Generally, theblock copolymer structure factor is characterized by apeak given by the correlation between the covalentlybonded PS(H)- and PS(D)-polymer blocks. This peakvanishes, however, due to the dilution with PS(H) poly-mers. The random phase approximation (RPA) resultfor such binary systems of dilute block copolymers inhomo and the structure factor of the free middle blockby itself can be distinguished (see Fig. 5), but the errorinduced using Gaussian coil statistics without chain in-teraction is limited to less than 20%, as discussed below.Strongly deformed polymer systems, like mixtures ofhigh molar-mass homopolymers or crosslinked poly-mer networks, lead typically to considerable lozengictype scattering patterns (Boué 1991; Straube et al.1995; Hayes et al. 1996; Read and McLeish 1997;Westermann et al. 1998). This is, however, not observedin our case. The main origin for their lozengic shapescattering pattern is polydispersity effects, but also dan-gling ends of the chains may contribute. The danglingends do not carry stress and have accordingly relaxed

0 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q [1/A]

S(q

)/S

(0)

Fig. 5 Comparison of normalized scattering intensities as mod-eled by the RPA (dotted line) and the Debye expression (fullline), respectively. The RPA is computed from Eqs. 15–20 withparameters N = 200, b = 18.4 Å, f = 0.35, and φ = 0.33

392 Rheol Acta (2012) 51:385–394

contributing isotropically to the 2D patterns and thusleading to the lemonlike lozengic structure. With thepresent triblock copolymer architecture, however, wehave successfully avoided this effect and are left with ascattering pattern that is well described by an ellipticalshape within a q-range which is easily accessible bythe SANS technique. In Fig. 4, the middle columnrepresents the resulting model fits, while the right col-umn represents the residuals. The three sets of data—experimental, simulation, and residuals—are all shownon the same intensity scale, making comparison easy.We see that the modified Debye function with the pa-rameters Rg‖ and Rg⊥ summarized in Table 3 describesthe scattering data to an approximation better than ex-perimental statistics. The largest discrepancy is locatednear the forward scattering (q � 0.02 Å

−1) where the

analytical calculation does not take the shadow of thebeam-stop into account. The top row in Fig. 4 repre-sents measurements for an un-stretched sample. Thecorresponding parameters determined by the modifiedDebye expression are Rg‖ = Rg⊥ = 46 Å which is about20% lower than the literature Rg◦ = 57 Å. This dis-crepancy may be explained in terms of the RPA to theblend of triblock (ABA) and homopolymer (A) systemat equilibrium (Maschke et al. 1993). For the presentsystem, the triblock has a total of N = 200 Kuhn links,with middle block 60 Kuhn links. The copolymer hasa volume fraction 0.33 in a homopolymer with 200Kuhn links. The corresponding normalized scatteringfunction described by the RPA and the simple Debyefunction are shown in Fig. 5. It appears that the appli-cation of the Debye expression does indeed lead to theradius of gyration at equilibrium being underestimatedby about 20%. Since we are only interested in the rela-tive deformation ratios, no extra correction is required.In Table 3, we also show values of a mean experimentalradius of gyration Rm defined from the experimentallydetermined parameters in various anisotropic samplesas R3

m = Rg‖ R2g⊥. In this way, the mistake in the deter-

mination of Rg,0 cancels and with Rg,0 ∼ Rm = (53 ±2) Å a good agreement with the estimated value isretrieved.

Discussion

Three distinct elongation rates have been consideredeach with multiple measurements:

• ε̇ = 0.001 s−1, Wi = 1, and WiR = 0.16: The stressis equal to 3η0ε̇ which is an indication that thedynamics is within the linear range or at least notfar into the non-linear range. This also agrees with

expectations at a Weissenberg number equal to 1,although ε̇ = 0.001 s−1 may represent the upperlimit for the application of linear viscoelasticity.The stretch factors indicate that significant devia-tion from isotropy occurs in the linear viscoelasticrange.

• ε̇ = 0.01 s−1, Wi = 10, and WiR = 1.6. The stressis significantly smaller than 3η0ε̇ which is an indi-cation that the dynamics is outside of the linearrange as would be expected at Wi = 10. Specificallythe stress is larger than five times the modulus,which is the upper limit that can be predicted bythe Doi–Edwards model without stretching. Conse-quently, according to the Doi–Edwards picture, thechains must be stretched relative to their equilib-rium length. This agrees also with the Rouse timeobtained from the Likhtman method. According toRouse theory, the coil stretch transition occurs atWiR = 0.5. Hence, some degree of chain stretchingat WiR = 1.6 is not unreasonable.

• ε̇ = 0.1 s−1, Wi = 100, and WiR = 16. These dataare in the fully non-linear region, with significantchain stretching taking place indicated by σ/G0 ≈27 as well as the stretch ratios. The value of λ‖ maybe questioned as discussed below.

In Fig. 6, we show the steady stress and stretchratios as function of the Weissenberg number Wi = ε̇τa.Given that there are only three data points, it is notpossible to rule out a coil-stretch transition completely.

100 101 10210−1

100

101

102

Weissenberg number, Wi

Fig. 6 Steady flow stress, σ/G0 (�), stretch ratios in flow direc-tion, λ‖ = Rg‖/Rg0 (�), and perpendicular to the flow direction,λ⊥ = Rg⊥/Rg0 ( ) as functions of non-dimensional elongationrate, De = ε̇τd for polystyrene of molar mass 150 kg/mol. Thedotted lines have slopes 0.6, 0.25, and −0.125, respectively. Thestretch ratios refer to the deuterated middle segment of thechain only

Rheol Acta (2012) 51:385–394 393

However, it does appear that the data are reasonablywell approximated by expressions of the form

σ/G0 ∼ Wiα (11)

λ‖ ∼ Wiβ (12)

λ⊥ ∼ Wi−γ (13)

with α ≈ 0.6, β ≈ 0.25, and γ ≈ 0.125. Keep in mindthat while the stresses refer to the entire molecules,the stretch factors refer just to the middle deuteratedblock. The power α ≈ 0.6 is in fact equal to the scalingobserved by Luap et al. (2005). It is within experimentaluncertainty equivalent to the ε̇−0.5 scaling reported byBach et al. (2003a). The measurements cover two or-ders of magnitude in the steady elongational rate withinthe non-linear range.

Although both λ‖ and λ⊥ are given separately inTable 3 (using Rg,0 = 46 Å), in view of the limitedq-range which was accessed in the experiment, theperpendicular data are to be trusted more than in theparallel direction. The max q-range considered in Fig. 4is between qRg = 0.01 × 46 and 0.07 × 46, i.e. 0.5Rg to4Rg. Hence, the parallel direction is fully out of range,shifted to un-measurable intensities in the beam stoparea, and tabulated values are strongly co-determinedby the perpendicular strain axis. For the latter axis,since the chain size is reduced due to incompressibility,the radius of gyration shifts to higher q and the simpleDebye equation which is then Guinier-like fulfills stillbetter the valid Guinier regime within the measuredq-range. The microscopic deformation ratios λ‖ arethen ∼ 1.5, 3.2, and 5.4. As an alternative to the directdetermination of λ‖, we may also estimate the effectivemicroscopic deformation ratios for the middle segmentof the triblock chain as λe

‖ = 1/λ2⊥. This would give

values ca. 1.3, 2, and 4 which are lower than thosein Table 3. The power law for the parallel directionis virtually unchanged, however. The data and line inFig. 6 are just shifted upward with a factor of about 1.2.

It remains to address the question whether the chainstretching approaches the fully stretched non-Gaussianlimit. For Gaussian behavior, one would expect σ ∼λ2

‖ − λ2⊥ which for large λ‖ would correspond to β = α/2

as observed within experimental accuracy. This may betaken as indirect evidence in support of the sugges-tion that finite extensibility is not the main mechanismbehind the scaling observed in non-linear elongationalflow (Luap et al. 2005). The maximum stretching onewould anticipate can be estimated from the number ofKuhn segments in the deuterated middle segment as√

60 = 7.7. Hence while the measurements are proba-

bly within the Gaussian range, the values at Wi = 90may be probing the upper limit Gaussian statistics.

Acknowledgments We are grateful to the Danish TechnicalResearch Council for financial support to the Danish PolymerCentre, to the Danish Natural Science Council for financial sup-port in the DANSCATT grant and to the Paul Scherrer Institutfor use of their neutron beam facility, and the SoftComp EU-Networks of Excellence.

Appendix

Consider a mixture of an ABA block-copolymer andpure A homopolymer both with a total of N segments.The ABA block co-polymer is symmetrical with twoend block of f N segments and a middle block of (1 −2 f )N segments. The mole fraction of ABA is called φ.The RPA (Doi 1992; Maschke et al. 1993), gives for thescattering function

I(q) = (ρA − ρB)2 S(0)

AA S(0)

BB − S(0)2AB

S(0)

AA + S(0)

BB + 2S(0)

AB

(14)

where ρA and ρB are the scattering length density of,respectively, A and B units (PSH and PSD in our case).S(0)

AA here includes contributions both from the triblockand from the homopolymer (S(0)

AA and S(0)

CC, respectively,in Maschke et al. (1993)). For equal size of homopoly-mer and triblock copolymers, we may normalize byreplacing the number of chains with the volume fractionof polymers. We let φ be the volume fraction of triblockso that (1 − φ) is the volume fraction of homopolymer.Apart from a numerical factor, the resulting structurefactor may then be written in the form

S(q) = S0AA S0

BB − (S0AB)2

Nh(1, x)(15)

where

S0BB = φNh(1 − 2 f, x) (16)

S0AA = φN(2h( f, x) + h(1, x) − 2h(1 − f, x)

+h(1 − 2 f, x)) + (1 − φ)Nh(1, x) (17)

and

S(0)

AB = φN(h(1 − f, x) − h( f, x) − h(1 − 2 f, x)) (18)

Here x and h( f, x) are defined by

x = q2 R2g = q2 Nb 2

6(19)

h( f, x) = 2x2 ( f x + exp(− f x) − 1) (20)

394 Rheol Acta (2012) 51:385–394

References

Bach A, Almdal K, Rasmussen H, Hassager O (2003a)Elongational viscosity of narrow molar mass distributionpolystyrene. Macromolecules 36:5174–5179

Bach A, Rasmussen H, Hassager O (2003b) Extensional viscos-ity for polymer melts measured in the filament stretchingrheometer. J Rheol 47:429–441

Baumgaertel M, Schausberger A, Winter H (1990) The relax-ation of polymers with linear flexible chains of uniformlength. Rheol Acta 29(5):400–408

Bent J, Hutchings L, Richards R, Gough T, Spares R, Coates P,Grillo I, Harlen O, Read D, Graham R, Likhtman A, GrovesD, Nicholson T, McLeish TCB (2003) Neutron-mappingpolymer flow: scattering, flow visualization, and moleculartheory. Science 301:1691–1695

Bird RB, Armstrong RC, Hassager O (1987a) Dynamics of poly-mer liquids: fluid mechanics, vol 1, 2 edn. Wiley, New York

Bird RB, Curtiss CF, Armstrong RC, Hassager O (1987b) Dy-namics of polymer liquids: kinetic theory, vol 2, 2 edn. Wiley,New York

Blanchard A, Graham RS, Heinrich M, Pyckhout-Hintzen W,Richter D, Likhtman AE, McLeish T, Read D, Straube E,Kohlbrecher J (2005) Small angle neutron scattering obser-vation of chain retraction after a large step deformation.Phys Rev Lett 95:166001

Boué F, Bastide J, Buzier M, Lapp A, Herz J, Vilgis TA (1991)Strain-induced large fluctuations during stress relaxation inpolymer melts observed by small-angle neutron scattering.Lozenges, butterflies, and related theory. Colloid Polym Sci269:195–216

Boué F, Nierlich M, Jannink G, Ball R (1982) Polymer coil re-laxation in uniaxially strained polystyrene observed by smallangle neutron scattering. J Phys 43:137–148

Boué F, Nierlich M, Osaki K (1983) Dynamics of molten poly-mers on the sub-molecular scale—application of small-angle-neutron-scattering to transient relaxation. SympFaraday Soc 18:83–105

De Gennes P (1974) Coil-stretch transition of dilute flexiblepolymers under ultrahigh velocity gradients. J Chem Phys60:5030–5042

Doi M (1992) Introduction to polymer physics. Oxford UniversityPress, Oxford

Doi M, Edwards SF (1986) The theory of polymer dynamics.Clarendon, Oxford

Fang J, Kröger M, Öttinger HC (2000) A thermodynamicallyadmissible reptation model for fast flows of entangled poly-mers. II. Model predictions for shear and extensional flows.J Rheol 44:1293–1317

Fetters LJ, Lohse DJ, Colby RH (2009) Physical properties ofpolymers handbook. Springer, Berlin

Hayes C, Bokobza L, Boué F, Mendes E, Monneri L (1996)Relaxation dynamics in bimodal polystyrene melts: afourier-transform infrared dichroism and small-angle neu-tron scattering study. Macromolecules 29:5036–5041

Jackson JK, Winter HH (1995) Entanglement and flow behav-ior of bidisperse blends of polystyrene and polybutadiene.Macromolecules 28:3146–3155

Kabanemi K, Hétu J-F (2009) Dynamics of monodisperse linearentangled polymer melts in extensional flow: the effect ofexcluded-volume interactions. Polymer 50:5865–5870

Likhtman A, McLeish T, (2002) Quantitative theory for lin-ear dynamics of linear entangled polymers. Macromolecules35:6332–6343

Likhtman AE, Graham RS (2003) Simple constitutive equationfor linear polymer melts derived from molecular theory:Rolie–Poly equation. J Non-Newton Fluid Mech 114:1–12

Luap C, Muller C, Schweizer T, Venerus DC (2005) Simulta-neous stress and birefringence measurements during uniax-ial elongation of polystyrene melts with narrow molecularweight distribution. Rheol Acta 45:83–91

Marrucci G, Ianniruberto G (2004) Interchain pressure effect inextensional flows of entangled polymer melts. Macromole-cules 37:3934–3942

Marrucci G, Grizzutti N (1988) Fast flows of concentrated poly-mers: predictions of the tube model on chain stretching.Gazz Chim Ital 118:179–185

Maschke U, Ewen B, Benmouna M, Meier G, Benoit H (1993)Elastic coherent neutron scattering from mixtures of triblockcopolymers and homopolymers in the homogeneous bulkstate. Macromolecules 26:6197–6202

McKinley GH, Sridhar T (2002) Filament-stretching rheometryof complex fluids. Annu Rev Fluid Mech 34:375–415

Mead DW, Larson RG, Doi M (1998) A molecular theory for fastflows of entangled polymers. Macromolecules 31:7895–7914

Muller R, Picot C, Zang Y, Frolich D (1990) Polymer chainconformation in the melt during steady elongational flow asmeasured by sans. temporary network model. Macromole-cules 23:2577–2582

Ndoni S, Papadakis CM, Bates F, Almdal K (1995) Laboratory-scale setup for anionic-polymerization under inert at-mosphere. Rev Sci Instrum 66:1090–1095

Nielsen J, Rasmussen H, Hassager O, GH M (2006) Elongationalviscosity of monodisperse and bidisperse polystyrene melts.J Rheol 50:453–476

Pearson DS, Kiss Fetters LJ, Doi M (1989) Flow-induced bire-fringence of concentrated polyisoprene solutions. J Rheol33:517–535

Perkins TT, Smith D, Chu S (1997) Single polymer dynamics inan elongational flow. Science 276:2016–2021

Petrie CJS (1979) Elongational flows. Pitman, LondonRead D (2004) Calculation of scattering from stretched copoly-

mers using the tube model: incorporation of the effect ofelastic inhomogeneities. Macromolecules 37:5065–5092

Read D, McLeish T (1997) Microscopic theory for the “lozenge”contour plots in scattering from stretched polymer networks.Macromolecules 30:6376–6384

Schieber J, Neergaard J, Gupta S (2003) J Rheol 47:213–233Straube E, Urban V, Pyckhout-Hintzen W, Richter D, Glinka C

(1995) Small-angle neutron scattering investigation of topo-logical constraints and tube deformation in networks. PhysRev Lett 74:4464–4467

Strunz P, Mortensen K, Janssen S (2004) SANS-II at SINQ. In-stallation of the former Risø-SANS facility. Phys B CondensMatter 350:e783–e786

Underhill PT, Doyle PS (2007) Accuracy of bead-spring chains instrong flows. J non-Newton Fluid Mech 145:109–123

Wagner M, Kheirandish S, O H (2005) Quantitative predictionof transient and steady-state elongational viscosity of nearlymonodisperse polystyrene melts. J Rheol 49:1317–1327

Wagner MH, Rolón-Garrido VH (2009) Nonlinear rheology oflinear polymer melts: modeling chain stretch by interchaintube pressure and rouse time. Korea-Aust Rheol J 21:203–211

Wagner M, Rolón-Garrido VH (2010) The interchain pressureeffect in shear rheology. Rheol Acta 49:459–471

Westermann S, Urban V, Pyckhout-Hintzen W, Richter D,Straube E (1998) Comment on “lozenge” contour plots inscattering from polymer networks. Phys Rev Lett 80:5449