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Rheologica Acta ISSN 0035-4511 Rheol ActaDOI 10.1007/s00397-011-0565-y

TTS in LAOS: validation of time-temperature superposition under largeamplitude oscillatory shear

Anja Vananroye, Pieter Leen, Peter VanPuyvelde & Christian Clasen

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Rheol ActaDOI 10.1007/s00397-011-0565-y

ORIGINAL CONTRIBUTION

TTS in LAOS: validation of time-temperaturesuperposition under large amplitude oscillatory shear

Anja Vananroye · Pieter Leen · Peter Van Puyvelde ·Christian Clasen

Received: 30 November 2010 / Revised: 29 April 2011 / Accepted: 5 June 2011© Springer-Verlag 2011

Abstract The validation of time-temperature super-position of non-linear parameters obtained from largeamplitude oscillatory shear is investigated for a model vis-coelastic fluid. Oscillatory time sweeps were performedon a 11 wt.% solution of high molecular weight poly-isobutylene in pristane as a function of temperature andfrequency and for a broad range of strain amplitudesvarying from the linear to the highly non-linear regime.Lissajous curves show that this reference material dis-plays strong non-linear behaviour when the strain am-plitude is exceeding a critical value. Elastic and viscousChebyshev coefficients and alternative non-linear pa-rameters were obtained based on the framework ofEwoldt et al. (J Rheol 52(6):1427–1458, 2008) as a func-tion of temperature, frequency and strain amplitude.For each strain amplitude, temperature shift factorsaT(T) were calculated for the first order elastic andviscous Chebyshev coefficients simultaneously, so thatmaster curves at a certain reference temperature Tref

were obtained. It is shown that the expected indepen-dency of these shift factors on strain amplitude holdseven in the non-linear regime. The shift factors aT(T)

can be used to also superpose the higher order elasticand viscous Chebyshev coefficients and the alternativemoduli and viscosities onto master curves. It was shownthat the Rutgers-Delaware rule also holds for a vis-coelastic solution at large strain amplitudes.

A. Vananroye · P. Leen · P. Van Puyvelde · C. Clasen (B)Department of Chemical Engineering, KatholiekeUniversiteit Leuven and Leuven Materials Research Centre,W. de Croylaan 46, 3001 Leuven, Belgiume-mail: christian.clasen@cit.kuleuven.be

Keywords Time-temperature superposition (TTS) ·LAOS · Lissajous · Non-linear parameters ·Viscoelastic solution · Rutgers-Delaware rule

Introduction

One of the most widely applied rheological techniquesto obtain viscoelastic properties of materials consists inperforming small amplitude oscillatory shear (SAOS)tests resulting in strain amplitude (γ 0) independentstorage moduli G′(ω) and loss moduli G′′(ω) as a func-tion of frequency ω (Ferry 1980; Macosko 1994). How-ever, the independency on strain amplitude is only validin the linear regime, where a scaled change in strainsignal γ (t) corresponds to a change in stress signal σ (t)with the same scaling factor.

The frequency range of SAOS experiments can beextended using time-temperature superposition (TTS).For a wide class of materials such as polymer melts andsolutions, changes in temperature T do not alter theshape of the linear moduli curves as a function of fre-quency. In a relaxation spectrum, the modulus G(λ) ofa relaxation time λ is directly proportional to the ther-mal energy and the number density of relaxing ele-ments and thus to the overall fluid density ρ, so thatG ∼ kTρ (and this will affect all moduli equally and in-dependent of the relaxation time). The relaxation timeitself depends on the ratio of internal friction and therelated internal spring constant. This spring constantfor an entropic spring is directly proportional to ther-mal energy. For a ‘thermo-rheologically simple’ fluid,it is assumed that the friction for each relaxation pro-cess is proportional to the same molecular friction coef-ficient ζ in the specific surrounding so that λ ∼ ζ /kT.

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Furthermore, this friction coefficient ζ is also a functionof the temperature T. With the assumption that all re-laxation times contributing to the relaxation spectrumdepend on the same internal friction coefficient ζ andthus have the same temperature dependency, a changein temperature will just shift G′(ω) and G′′(ω) along thetime or frequency axis. Also, the similar temperaturedependence of the different moduli at a given frequen-cy shifts the whole storage or loss modulus curve alongthe modulus axis (Larson 1998). Applying this TTSprinciple, moduli and viscosities measured at differenttemperatures can be shifted horizontally and verticallywith shift factors aT and b T , respectively, to superim-pose on master curves of reduced variables at an arbi-trary chosen reference temperature Tref (Ferry 1980):

G′ (ω, T) = b T G′r (aTω, Tref) ,

G′′ (ω, T) = b T G′′r (aTω, Tref) (1)

η′′ (ω, T) = b T

aTη′′

r (aTω, Tref) ,

η′ (ω, T) = b T

aTη′

r (aTω, Tref) (2)

The vertical shift factor b T , introduced to account forthe (small) changes in density at different temperatures,is often close to unity and can be omitted. Originally, anempirical Arrhenius type equation, with Ea the activa-tion energy of flow and R the universal gas constant,was proposed to express the temperature dependenceof the shift factor aT :

aT = exp

[Ea

R

(1

T− 1

Tref

)](3)

This expression fits data quite good in the plateauand terminal zones for temperatures well above theglass transition temperature Tg. For polymer melts in atemperature range from Tg up to Tg + 100 K, the shiftfactor aT can be expressed by the Williams–Landel–Ferry (WLF) equation (Williams et al. 1955):

log aT = −c1(T − Tref)

c2 + T − Tref(4)

in which c1 and c2 are material parameters.The main advantage of TTS is the possibility to

extend the linear viscoelastic properties to a larger timeor frequency range. Examples and applications of TTSare therefore abundantly present in literature (Awasthiand Joshi 2009; Dealy and Plazek 2009; Han and Kim1993; O’Connell and McKenna 1997; Olsen et al. 2001;Palade et al. 1995; Tajvidi et al. 2005).

Although combined SAOS/TTS measurements offeran easy tool to characterize materials within their linear

regime, they often fail in describing the material prop-erties under real processing conditions where materi-als are frequently subjected to large deformations andrates beyond the linear limits (Osswald 1998). In ad-dition, a wide range of complex fluids already displaysnon-linear behaviour under mild processing conditions(Walker 2001; Yziquel et al. 1999). The non-linear re-sponse regime can in principle be probed by rheologicaltests that allow monitoring the stress response to a de-formation exceeding the linear viscoelastic limit of thefluid structure. A first category of non-linear measure-ments are (multi-)step shear strain experiments, wherein the case of time-deformation separability, a dampingfunction h(γ ) can be determined (Larson 1998; Rolon-Garrido and Wagner 2009). This damping function isgenerally experimentally obtained by vertically shiftingrelaxation moduli as a function of time at differentstrains onto a master curve. Another non-linear testmethod is the parallel or orthogonal superposition of asmall amplitude oscillation onto a non-linear shear de-formation to probe the linear response of a non-linearlydeformed structure (Vermant et al. 1998; Walker et al.2000). Parallel superposition has the disadvantage of astrong coupling between the basic shear field and thesuperimposed flow, which complicates the interpreta-tion of the stress response signals (Yamamoto 1971).With orthogonal superposition the coupling is lessstrong; nevertheless, difficulties in instrument develop-ment limit the use (Vermant et al. 1997). A third way tocharacterize the non-linear viscoelastic response is byperforming large amplitude oscillatory shear (LAOS)measurements with strain amplitudes beyond the linearlimit, resulting in both frequency and strain amplitude-dependent viscoelastic parameters. However, as higherorder harmonics start to play a role, analyzing theresponse using the linear viscoelasticity theory is notvalid anymore and a more sophisticated data treatmentbecomes necessary. Nevertheless, similar to a SAOSexperiment, a LAOS experiment has the advantageof reaching a cyclic steady state which allows probinghigher frequencies and faster relaxation processes thana step strain experiment. However, the transient regimebefore reaching a cyclic steady state in a LAOS ex-periment is inherently coupled with the induction ofa shear history, so that the non-linear regime probedwith LAOS will not be the same as with a step-strainexperiment.

In the past, several methods have been proposedto analyze the non-linear data obtained from LAOSexperiments. For instance, the raw stress signal σ (t) canbe graphically visualized as a function of strain γ (t)or strain rate γ̇ (t) resulting respectively in elastic orviscous Lissajous–Bowditch plots (Ewoldt et al. 2007a;

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Hyun et al. 2006; Philippoff 1966; Sim et al. 2003; Teeand Dealy 1975). Such curves qualitatively show evi-dence of non-linearities, as the shape of the closed loopsstarts to deviate from an ellipsoidal shape. One of themost frequently used quantitative techniques to analyzenon-linear data is Fourier Transform (FT) rheology(Wilhelm 2002) in which the stress response signal σ (t)is expressed as a Fourier series (Dealy and Wissbrun1990) capturing the higher harmonics (Carotenuto et al.2008; Fleury et al. 2004; Hyun et al. 2006; Sim et al. 2003):

σ(t) = γ0

∑n=odd

{G′

n(ω, γ0) sin nωt + G′′n(ω, γ0) cos nωt

}

(5)

in which G′n and G′′

n are the elastic and viscous Fouriercoefficients of order n. Recently, a generalization of lin-ear viscoelasticity from a geometrical point of view wasintroduced to analyze LAOS data, based on symmetryarguments (Cho et al. 2005). This method consists ofan orthogonal stress decomposition (SD) of the totalstress response signal σ (t) into an elastic contributionσ ′(x) with x = γ (t)

γ0and a viscous contribution σ ′′(y)

with y = γ̇ (t)γ̇0

, so that σ (t) = σ ′(t) + σ ′′(t). The sin-gle stress contributions σ ′(x) or σ ′′(y) represent singlevalued functions of strain and strain rate that can befitted by means of a polynomial regression (Cho et al.2005). Nevertheless, for the SD analysis, the non-linear material properties represented by the polyno-mial coefficients were not unique since they dependedon the highest order n chosen to reconstruct the signal.In addition, physical interpretation of the data based onthese polynomial coefficients or on the FT coefficientsis difficult. To overcome these problems, Ewoldt et al.(2008) extended the SD method of Cho et al. (2005)using a set of orthogonal polynomials over a finitedomain. They used Chebyshev polynomials of the firstkind Tn to fit the elastic stress σ ′(x) and viscous stressσ ′′(y) signals:

σ ′(x) = γ0

∑n,odd

en(ω, γ0)Tn(x) (6)

σ ′′(y) = γ̇0

∑n,odd

vn(ω, γ0)Tn(y) (7)

The Chebyshev polynomials Tn(x) and Tn(y) aresymmetric, bounded and orthogonal in [−1,1]. Thedefinition and the first few odd polynomials, used in thiswork, are given in Table 1.

The elastic and viscous Chebyshev coefficients, en(ω,γ 0)

and vn(ω,γ 0) appearing in the stress response signalscan easily be related to (or calculated from) the Fourier

Table 1 Chebyshev polynomials of the first kind

Tn + 1(x) = 2xTn(x)–Tn−1(x)

T0(x) = 1T1(x) = xT3(x) = 4x3 − 3xT5(x) = 16x5 − 20x3 + 5xT7(x) = 64x7 − 112x5 + 56x3 − 7x

coefficients G′n and G′′

n obtained from FT rheology(Ewoldt et al. 2008):

en = G′n(−1)(n−2)/2 n = odd, (8)

νn = G′′n

ωn = odd. (9)

In comparison to the polynomial coefficients of theSD analysis, the Chebyshev coefficients have the advan-tage that each coefficient represents a unique parameterquantifying non-linear behaviour, and is independentof the other coefficients and therefore, of the order ofthe fit. In the linear regime, the first order coefficientsrecover the linear viscoelastic limits (e1 = G′ and v1

= η′ = G′′/ω), whereas higher order contributions be-come vanishingly small. Ewoldt et al. (2008) also de-fined new physically meaningful elastic and viscousmoduli which describe material properties at charac-teristic points during an oscillatory deformation cycle.The minimum strain modulus G′

M and the large strainmodulus G′

L can be used to express the elastic responseat an instantaneous strain of zero and at maximumstrain, respectively:

G′M ≡ dσ

dγ

∣∣∣∣γ=0

≈ e1 − 3e3 + 5e5 − 7e7 + . . . (10)

G′L ≡ σ

γ

∣∣∣∣γ=±γ0

≈ e1 + e3 + e5 + e7 + . . . (11)

A graphical representation of these alternative elas-tic moduli is depicted in Fig. 1 in the form of aLissajous–Bowditch curve of the stress response as afunction of strain. G′

L can be obtained as the slope ofthe secant of the elastic stress response at maximumdeformation while G′

M can be found as the slope ofthe tangent of the elastic stress response at deformationγ = 0. Analogously, a minimum rate dynamic viscosityη′

M and a large rate dynamic viscosity η′L express the

viscous response at an instantaneous strain rate of zeroand at maximum strain rate respectively.

On rotational devices, the combination of largedeformations that are often needed to obtain non-linear behaviour, and high frequencies can cause flowanomalies as a distortion of the sample free interfaceand finally as the loss of sample. Therefore, from an

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Fig. 1 Geometric interpretation of the elastic alternative moduliG′

L and G′M obtained from a typical LAOS experiment

experimental point of view, the frequency regime iseven more limited in the case of LAOS than in the caseof linear SAOS experiments. Hence, in order to fullycharacterize non-linear parameters over a wide rangeof time scales, the applicability of the TTS principle alsoto the higher order harmonics is desirable. The questionthat arises is whether a non-linear deformation of thesystem is changing the internal friction coefficient ζ

on which the molecular interpretation of applicabilityof TTS is based. Moreover, it could be questionedwhether all parts of the relaxation time spectrum andthe related friction coefficients are affected similarly.However, one could expect that even for such partiallychanged internal friction coefficients the temperaturedependence is similar, and that TTS holds even in thenon-linear regime. Also from constitutive models, inparticular general memory integrals that include a time-deformation separability as of the Rivlin-Sawyer or K-BKZ type with factorizable kernel expressions for ex-ample of the form G(t–t′)h(γ ), it is expected to have allhigher order terms (which are expansions of the functionh(γ )) captured by the same shift factor (Bird et al. 1987).

An experimental proof of the applicability of TTSfor non-linear measurements has so far only been ob-tained for step strain experiments (Einaga et al. 1973;Fukuda et al. 1975; Kapnistos et al. 2009). For LAOSexperiments, although they have a different non-lineardeformation history than the step strain experiments,one could expect from a molecular point of view thatTTS would also hold. However, there are so far noexperimental reports thereon and the aim of this paperis to close this gap and to investigate the validity of TTSapplied to LAOS measurements for a model viscoelas-tic material. The paper is structured as follows: firstly,the applicability of the TTS principle is shown for the

moduli obtained in the linear viscoelastic regime of aPIB/pristane solution. Then, the non-linear behaviourof this reference sample is measured over a range of fre-quencies and deformations and characterized in termsof Chebyshev coefficients at a reference temperatureTref. Finally, it is checked whether non-linear parame-ters as a function of temperature can be shifted ontomaster curves and whether the resulting shift factorscan still be fitted using the equations normally used inthe linear regime.

Materials and methods

Materials

The model material chosen is a shear thinning, elasticsolution of a high molecular weight polyisobutylene(PIB, Sigma–Aldrich) dissolved in 2,6,10,14-tetramethylpentadecane (pristane, Sigma–Aldrich) witha PIB mass fraction of 0.114. This fluid has similarcomponents as the NIST Standard Reference MaterialSRM 2490 that shows a characteristic viscoelastic be-haviour similar to that of polymeric melts with the advan-tage of being liquid at room temperature (Schultheiszand Leigh 2002).

The two components were mixed in an amber glassbottle and time (approximately 6 months) was givenfor the PIB to completely dissolve while performingonly a gentle agitation at room temperature to preventdegradation of the PIB (Khalil et al. 1994). The disso-lution process was monitored by performing repeti-tive rheological measurements on the sample until nochanges were observed and a stable solution was ob-tained. However, due to differences in PIB grade andin dissolution procedure, the absolute viscosity valuesand linear viscoelastic moduli differ from those of theNIST standard.

Methods

The linear viscoelastic response of the solution wasdetermined with dynamic oscillatory measurements ona ARES strain-controlled rheometer (TA Instruments)using a 25-mm cone geometry with a cone angle of 0.04radians and a bottom plate mounted on top of a fluidbath. The temperature was varied between 10◦C and40 ± 0.1◦C. Strain sweeps were performed to deter-mine the linear regime. For this reference material, acritical strain amplitude γ 0,crit of 20% at T = 25◦C wasobserved, determined as the strain amplitude at whicha change of more than 3% in moduli was seen. Subse-quently, frequency sweeps with ω from 0.1 to 100 rad/s

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were performed at different temperatures at a strainamplitude of 6%. Master curves for G′

r and G′′r, shown

in Fig 2a, were obtained by shifting the data sets atdifferent temperatures with respect to a reference tem-perature Tref of 25◦C using the TTS routines of the TAInstruments Orchestrator software. For the tempera-ture range under investigation, the data were shiftedonly using the horizontal shift factor aT . We verifiedthat it is possible to neglect the vertical shift factor b T

by plotting G′′r/G′

r as a function of aTω (Fig. 2a), whichresulted in a good master curve. For this system and atthe temperatures under investigation, the temperaturedependency of the shift factors aT(T), as shown inFig. 2b, could be described by the Arrhenius equation(Eq. 3) resulting in a material activation energy Ea

of 30.2 kJ/mole. The Arrhenius fit resulted in slightlyhigher coefficients of determination R2 compared tofits with the WLF equation. This is expected since themeasurement temperatures are strongly exceeding theglass transition temperature Tg of PIB (Tg = −70◦C)(Ferry 1980); therefore, the Arrhenius equation will

Fig. 2 a Linear dynamic master curves of the viscoelastic stan-dard at Tref = 25◦C. b Shift factor aT and Arrhenius fit as afunction of temperature T

further be used to fit the aT(T) data of the followingexperiments.

Non-linear LAOS measurements were obtained byperforming time sweep experiments at fixed ω, γ 0

and T. Strain and torque raw data waveforms werecollected using a National Instruments DAQ-Pad andTA Instruments Waveform & Fast Data Sampling soft-ware. For each set of frequency, strain amplitude andtemperature, at least 10 steady-state strain and torquecycles were collected for further processing.

The freeware MATLAB-based MITlaos softwarepackage (Ewoldt et al. 2007b) was used to analyzethe time-dependent strain and stress signals (σ (ω,t),γ (ω,t)). The data are first trimmed to an integer num-ber of cycles. In the present analysis, 10 entire cycleswere selected for further analysis in order to reducethe effects of noise (Wilhelm et al. 1999). Secondly,a discrete Fourier Transform analysis is performed todetermine the individual harmonic contributions to thesystem response. Based on this analysis, the order ofthe highest significant harmonic (defined by a ratio ofharmonic amplitude to noise of greater than 1.4) wasdetermined and the harmonics up to this order wereused to reconstruct a noise-reduced stress signal forthe further analysis. In the present case, the highestharmonic taken into account was of order 7. Outgoingfrom this reconstructed stress, the viscoelastic parame-ters and Chebyshev coefficients en and vn are calculatedand Lissajous–Bowditch curves are reconstructed.

Results and discussion

Non-linear characterization at Tref = 25◦C

The non-linear oscillatory response of the referencematerial at a reference temperature Tref of 25◦C isshown in the form of Pipkin diagrams (Pipkin 1972).In Fig. 3, the effect of strain and frequency on themechanical behaviour of a material is visualized bymeans of normalized Lissajous–Bowditch curves. TheLissajous–Bowditch curves in Fig. 3a give for eachcombination of frequency ω and strain amplitude γ 0

the total stress response σ (t) as a function of strainγ (t) (full closed loops), and the elastic component σ ′(x)

(dotted line) from the orthogonal stress decomposition(Eq. 6). Both stress signals are normalized by the max-imum stress amplitude σ max of which the values areindicated on each Lissajous–Bowditch plot. Similarly,Fig. 3b shows the viscous response by plotting thenormalized total stress response σ (t)/σ max as a functionof the strain rate γ̇ (t) (full closed loops), and the nor-malized purely viscous component σ ′′(y)/σ max (dotted

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Fig. 3 Normalized Lissajous–Bowditch curves at T = 25◦C.a Normalized total stress response σ (t)/σmax (full line) and elasticstress response σ ′(t)/σmax (dotted line) as a function of nor-malized strain γ (t)/γ 0 and b normalized total stress response

σ (t)/σmax (full line) and viscous stress response σ ′′(t)/σmax (dot-ted line) as a function of normalized strain rate γ̇ (t)/γ̇0. The max-imum stresses σmax are added for each combination of frequencyand strain amplitude

line) from the orthogonal stress decomposition (Eq. 7).The total stresses σ (t) in Fig. 3 are reconstructed fromthe measured Fourier coefficients G′

n and G′′n up to a

highest harmonic nmax = 7 to remove noise from thedata (Fourier coefficients for harmonics higher thann = 7 are negligible in the present case even for thehighest amplitude and frequency). The higher harmon-ics determined for each measurement in Fig. 3 arecompiled in form of the elastic and viscous Chebyshevcoefficients en and vn as a function of frequency (for acompilation of all Chebyshev coefficients see Fig. 11 inthe Appendix that contains also the data from differenttemperatures shifted with TTS as shown in the follow-ing section).

At a strain amplitude of 6%, the Lissajous–Bowditchcurves can be approximated by ellipses, indicating thatthe material displays linear viscoelastic behaviour. Inthese cases, the elastic and viscous contributions (dot-ted lines) are straight lines. The large strain modulusG′

L and minimum strain modulus G′M reduce to e1 (or

the linear storage modulus G′) and the alternative dy-namic viscosities η′

L and η′M are equal to G′′ω = η′. Note

that at the lowest frequencies shown for the strain am-plitude of 6%, some overfitting of the stress responseis seen due to the high ratio of noise compared to the

G′n and G′′

n at the higher harmonics (for consistency, allharmonics up to n = 7 are used in Fig. 3 to construct thestrain curves even at the lowest strain amplitude).

With increasing strain amplitude γ 0, the total re-sponse starts to deviate from an ellipsoidal shape asnon-linearities are evolving. In these cases, the elas-tic and viscous contributions are not represented bystraight lines anymore and hence, a single elastic orviscous modulus is insufficient to describe the localelastic or viscous behaviour. In Fig. 3a for instance, itis seen at high strain amplitudes (γ 0 = 600%, γ 0 =1200%) that G′

L (slope of the secant of the elastic stressresponse at maximum deformation) is high comparedto G′

M (slope of the tangent of the elastic stress re-sponse at deformation γ 0 = 0), indicating that the PIBin pristane reference material behaves strain stiffeningat high strain amplitudes. Figure 4 shows the first orderChebyshev coefficients e1 (= G′

1) and the local elasticmoduli G′

M and G′L at a selected frequency of ω =

1 rad/s as a function of the imposed strain amplitudeγ 0. At low strain amplitudes (γ 0 = 6%) all three elasticmoduli converge to the linear storage modulus G′,indicating that linear behaviour prevails. With increas-ing strain amplitude, though all moduli simultaneouslydecrease, a clear divergence between the moduli is

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Fig. 4 Elastic moduli G′L, G′

M and e1 as a function of strainamplitude γ 0 for ω = 1 rad/s and T = 25◦C. Lines are added toguide the eye

developing. The large strain modulus G′L decreases less

than the minimum strain modulus G′M indicating that

the elastic response behaves strain stiffening at highstrain amplitudes. e1 is a weighted average of the elasticresponse in a strain cycle with e1

∼= 1/4 (3G′L + G′

M). Atstrain amplitudes of 600% and beyond, the minimumstrain modulus G′

M becomes even negative (dottedline at y = 0 in Fig. 3a), indicating that the materialis instantaneously releasing more elastic stress thanit is accumulating by the deformation (Ewoldt et al.2008). This relates to secondary loops in the total re-sponse signal as a function of strain rate (Fig. 3b),a phenomenon explained by Ewoldt and McKinley(2010) and corresponds to the stress overshoot in steadyshear start-up experiments on polymer solutions, whichis attributed to shear-enhanced disentanglements. Themost important distinction is that the LAOS flow fieldis periodically reversed, and that the changes in mi-crostructure must reform under the timescales of theoscillation to reversibly achieve negative G′

M values.Figure 5 shows the evolution of the elastic large strain

modulus G′L (Fig. 5a) and the elastic minimum strain

modulus G′M (Fig. 5b) as a function of frequency ω

and strain amplitude γ 0 at T = 25◦C. For all strainamplitudes applied, G′

L increases with increasing fre-quency, similar as in the linear regime. For G′

M, thissimilarity only holds for strain amplitudes up to 300%.The minimum strain modulus at high strain amplitudesshows specific behaviour as can be seen on the insetof Fig. 5b (linear-log scale). At a strain amplitude of600%, first a decrease of G′

M with frequency is observedfollowed by an increase for ω > 1 rad/s, whereas for astrain amplitude of 1200%, a decreasing trend is seen

up to ω = 6 rad/s. Furthermore, the decreasing G′M

eventually becomes negative.

TTS shift for Chebyshev coefficients en and vn

In the second part, the temperature dependent shift fac-tors for the first order Chebyshev coefficients e1 and v1

were determined for all strain amplitudes. Figure 6 showsan example of the effect of temperature on the elastic(Fig. 6a) and viscous (Fig. 6b) first order Chebyshevcoefficients e1 and v1 at a strain amplitude γ 0 of 1200%.

All curves in Fig. 6 display the same trend as a func-tion of frequency, irrespective of the temperature, andsimilar results were also seen at other strain amplitudes,indicating that master curves using shift factors couldbe obtained. For each strain amplitude, the shift fac-tors aT(T) were calculated with respect to a referencetemperature Tref of 25◦C based on a residual minimiza-tion algorithm for a joint shifting of the first orderviscous and elastic Chebyshev coefficients. For all in-

Fig. 5 Evolution of a the large strain modulus G′L and b the

minimum strain modulus G′M as a function of strain amplitude

γ 0 and frequency ω at T = 25◦C. Lines are added to guide the eye

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Fig. 6 Temperature dependence of a the elastic Chebyshevcoefficients e1 and b the viscous Chebyshev coefficients v1 at astrain amplitude γ 0 of 1200% as a function of frequency ω. Linesare added to guide the eye

vestigated strain amplitudes from 6% to 1200%, e1 andv1 coefficients could be simultaneously shifted for eachstrain amplitude, resulting in horizontal shift factorsaT(T) independent of the elastic or viscous characterof the Chebyshev coefficients. Again, it was verifiedthat vertical shift factor b T could be omitted. Thisresulted in an e1,r versus aTω master curve (as shownfor example for γ 0 = 1200% by the filled diamonds inFig. 7) and a v1,r/aT versus aTω master curve.

The obtained shift factors aT(T) for e1 and v1 at eachstrain amplitude could also be used to superpose allother en and vn coefficients up to n = 7. An exampleof the resulting master curves at Tref = 25◦C for thereduced elastic Chebyshev coefficients en,r is shownin Fig. 7 for a strain amplitude of 1200%. A com-plete overview of all master curves for both the elas-tic Chebyshev coefficients en,r and viscous Chebyshevcoefficients vn,r/aT for strain amplitudes from 100% to1200% is shown in Fig. 11 in the Appendix. For a strain

amplitude of 6% which is in the linear regime, all higherorder Chebyshev coefficients were negligible, as ex-pected. For strain amplitudes of 100% and 300% elasticand viscous contributions up to the third order, and fora strain amplitude of 600% Chebyshev coefficients upto the fifth order could be measured and were takeninto account for superposition. A general lower limitfor the determination of all Chebyshev coefficients withthe current setup was determined to be of O(0.01 Pa).Since results are presented on a logarithmic scale inFigs. 7 and 11, absolute values of the coefficients areshown and negative values are indicated as open symbols.

The fact that the same values of aT superpose allthe viscoelastic functions is one of the criteria for ap-plicability of the TTS procedure (Ferry 1980) to non-linear data also for LAOS experiments. In Fig. 8, theobtained shift factors aT are plotted as a function oftemperature for each of the strain amplitudes underinvestigation. As can be seen, results indicate that theshift factors are mainly independent of strain amplitudein the non-linear regime, at least in the strain amplituderange explored here.

Next, it was checked whether the aT(T) curves couldstill be fitted using the Arrhenius equation as expressedin Eq. 3. Results for the Arrhenius fit parameters Ea aresummarized in Table 2 as a function of strain amplitudetogether with the coefficients of determination R2.

As a function of strain amplitude in the non-linearregime, activation energy values slightly increase from25.4 to 29.3 kJ/mole, though the differences in Ea stillremain small. Hence, one could say that all Chebyshevcoefficients can still be reasonably shifted using the

Fig. 7 Master curves at Tref = 25◦C for the absolute valuesof the reduced elastic Chebyshev coefficients en,r at a strainamplitude γ 0 of 1200% as a function of reduced frequency aTω.Filled symbols represent positive values, open symbols representnegative values. The line is added to guide the eye

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Fig. 8 Shift factors aT as a function of temperature and strainamplitude for Tref = 25◦C

same aT factors independent of the applied strain am-plitude. The coefficients of determination R2 in the fitapproach were all above 0.96 and the fit parameter Ea

can still be referred to as the activation energy of flowof the system under investigation, since values in thelinear and non-linear regime are quite similar. Basedon the combined shifting of e1 and v1, an average valuefor the activation energy Ea of 28.1 kJ/mole can beassumed. The remark has to be made that this averagevalue is slightly below the value of 30.2 kJ/mole, ob-tained after simultaneously rescaling G′ and G′′ fromthe frequency sweep data in the linear regime (Fig. 2).

A closer look at the reduced Chebyshev coefficientsin Fig. 11 shows that increasing the strain amplitudegenerally has a stronger effect on the magnitude of theelastic coefficients en,r than on the viscous coefficientsvn,r for the investigated system. In particular the firstorder elastic coefficient e1,r shows a much stronger de-crease with strain amplitude than the viscous coefficientv1,r. This domination of the viscous contribution to thenon-linear response of the sample at high deforma-tions can also be seen when comparing the dissipatedenergy in the LAOS experiments to the viscosity ofa steady state shear rate experiment. In analogy tothe Rutgers-Delaware rule (Doraiswamy et al. 1991).Figure 9 shows v1,r/aT , which gives the total dissipatedenergy (Ewoldt et al. 2008; Ganeriwala and Rotz 1987;Giacomin and Oakley 1992) as a function of the re-

Table 2 Fit parameters Ea and coefficients of determinationR2

obtained by fitting the shift factors aT (T) of the first orderChebyshev coefficients e1 and v1 using Eq. 3

Strain amplitude (%) 6 100 300 600 1200

Ea (kJ/mol) 29.8 25.4 27.5 28.5 29.3R2 0.992 0.966 0.977 0.993 0.990

Fig. 9 Comparison of reduced first order Chebychev coefficientsv1,r/aT as a function of reduced shear rate amplitude aTωγ 0 withsteady state shear viscosity data as a function of shear rate atTref = 25◦C

duced shear rate amplitude aTωγ 0 at Tref = 25◦C.The Rutgers-Delaware rule was originally developedfor yield stress fluids that exhibit an inelastic, shearthinning behaviour in an oscillatory shear experimentonce the yield stress is overcome. For yield stress fluids,the rule holds only for strain amplitudes that are solarge that the shear stresses created in the yielded stateare large in comparison to the elastic stresses that arestored in the unyielded state of a deformation cycle.Similarly, the Rutgers-Delaware rule holds for a vis-coelastic solution only when the magnitude of viscousChebyshev coefficients is large in comparison to theelastic coefficients. As it can be seen in Fig. 9, a strainamplitude of 100% does not lead to an overlap ofv1,r/aT with the steady shear viscosity at Tref = 25◦C. Acomparison of the non-linear viscous moduli vn,rω withthe respective en,r from Fig. 11 for a strain amplitudeof 100% shows that both still have a similar order ofmagnitude. Only when applying higher strain ampli-tudes of 300% and larger, the viscous modulus v1,rω

starts to dominate and the dynamic values in Fig. 9 startto superpose onto the steady shear data. The sampleresponds to large deformation in LAOS mainly in ashear thinning manner. This observation can also berelated to the observed elastic response to high shearrates at large deformations. As shown in Figs. 5 and 12,the minimum strain modulus G′

M (Eq. 10) (that givesan elastic modulus at an instantaneous strain of zero,and therefore at the maximum strain rate) becomessmall with increasing strain amplitude and eventually,becomes negative. At the maximum strain rate whensubjected to large deformations during a deformationcycle, the entangled polymer solution is releasing moreelastic stress then it is accumulating.

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TTS shift for alternative elastic moduli and dynamicviscosities

Since all Chebyshev coefficients up to order n = 7 couldbe shifted using the same aT(T), it is expected basedon the definitions of the alternative non-linear moduliand viscosities in Eqs. 10 and 11, that also the elasticmoduli G′

L and G′M and the alternative dynamic vis-

cosities η′L and η′

M can be shifted onto master curves.In Fig. 10, an example of such TTS master curves ata strain amplitude of 1200% is shown for the reducedlarge strain modulus G′

L,r and for the reduced minimumstrain modulus G′

M,r as a function of reduced frequencyaTω (using the shift factors aT(T) obtained for thefirst order Chebyshev coefficients at the appropriatestrain amplitude (Fig. 8)). Results at other strain am-plitudes for the reduced alternative elastic moduli andfor the reduced alternative dynamic moduli are givenin Fig. 12 in the Appendix. In general, for all appliedstrain amplitudes, G′

L, η′L and η′

M data could be shiftedonto master curves simply using the shift factors aT(T)

from Fig. 8. One thing to recognize for G′M,r in Fig. 10

for a strain amplitude of 1200% (and also in Fig. 12for a strain amplitude of 600%), is that shifting theabsolute values for G′

M,r results in an apparently higherdegree of scatter on the master curves of more thanan order of magnitude above the lower determinationlimit of O(0.01 Pa) for the Chebyshev coefficients.The scatter on the G′

M,r data results from two issues:the contribution of higher order harmonics to G′

M,r isamplified since the higher order en,r enter multiplied

Fig. 10 Master curves at Tref = 25◦C for the reduced largestrain modulus G′

L,r and for the absolute values of the reducedminimum strain modulus G′

M,r at a strain amplitude γ 0 of 1200%as a function of reduced frequency aTω. Filled symbols representpositive values, open symbols represent negative values. The lineis added to guide the eye

with their order number n. Hence, the relatively smalldegree of scatter seen for the higher order harmonicsen,r (see Fig. 11 in the Appendix) contributes to a largerdegree to the scatter at high strain amplitudes for theG′

M,r data. Furthermore, the third order term entersEq. 10 as a negative value, which, for example for thecase of 600% strain amplitude in Fig. 11 cancels thecontribution from e1,r and leaves e5,r (that was deter-mined at the lower resolution limit) as the dominantcontribution.

Conclusions

The expected applicability of TTS also to non-linearLAOS measurements could be experimentally verifiedfor a thermo-rheologically simple fluid. This offers thepossibility to construct master curves over an extendedrange of frequencies and strain amplitudes and to coun-teract the inherently reduced frequency range for a sin-gle LAOS experiments due to the occurrence of flowanomalies at higher deformation rates in the non-linearregime. Such master curves could be used to charac-terize materials that show non-linear behaviour undertypical processing conditions which can be highly use-ful for processing industries. It was shown that hori-zontal shift factors aT(T) obtained for the first orderChebyshev coefficients can also be used to shift thehigher order Chebyshev coefficients as well as othernon-linear parameters at the same strain amplitude. Inaddition, it was seen that these shift factors aT(T) areindependent of strain amplitude, and hence also validin the non-linear response regime. Nevertheless, whenperforming TTS, whether it is in the linear or non-linear regime, one must be aware that shift factors validin specific regions of frequency and strain amplitude,might not apply to others and the data analysis alwayshas to be carefully monitored (Dealy and Plazek 2009).For the investigated model polymer solution it could beshown that the Rutgers-Delaware rule also holds for aviscoelastic solution if the deformation is high enough.This superposition of the first order viscous Chebyshevcoefficient v1 as a function of the shear rate amplitudeγ 0ω onto the steady-state flow curve only holds if themagnitude of the non-linear elastic moduli becomessmall compared to the viscous coefficients.

Acknowledgements The authors acknowledge the ResearchFoundation-Flanders (FWO Vlaanderen-post doctoral fellowship ofA. Vananroye) and Onderzoeksfonds K.U. Leuven (GOA 03/06)for financial support. CC would like to acknowledge financial sup-port from the ERC starting grant no. 203043-NANOFIB. Dr Ewoldtis gratefully acknowledged for providing the MITlaos software.

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Appendix

Fig. 11 Master curves after performing TTS shift for (E) the reduced elastic and (V) reduced viscous Chebyshev coefficients at a strainamplitude of a 100%, b 300%, c 600% and d 1200%. Open symbols represent negative values. Lines are added to guide the eye

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Fig. 12 Master curves after performing TTS shift for (E) the reduced elastic moduli and (V) reduced dynamic viscosities at a strainamplitude of a 100%, b 300%, c 600% and d 1200%. Open symbols represent negative values. Lines are added to guide the eye

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