Abstract Many metastable complex fluids, when subjectedto oscillatory shear flow of increasing strain amplitude atconstant frequency, are known to show a characteristic non-linear rheological response which consists of a monotonicdecrease in the elastic modulus and a nonmonotonic changein the loss modulus. In particular, the loss modulus increasesfrom its low strain value, crosses the elastic modulus, andthen decreases with further increase in the strain amplitude.Miyazaki et al. (Europhys Lett 75:915–921, 2006) proposeda qualitative argument to explain the origin of the nonmono-tonic nature of the loss modulus and suggested that in factthis response could be universal to all complex fluids ifthey are probed in a certain frequency window in whichthe fluid is dominantly elastic in the small strain limit. Inthis letter, we confirm their hypothesis by showing that awide variety of complex fluids, irrespective of their thermo-dynamic state under quiescent conditions, indeed show theaforementioned characteristic nonlinear response. We alsoshow that the maximum relative dissipation during yield-ing occurs when the imposed frequency resonates with thecharacteristic beta relaxation frequency of the fluid.
S. Kamble · A. Pandey · A. Lele (�)Polymer Science and Engineering Division,National Chemical Laboratory,Pune, Indiae-mail: [email protected]
S. KambleDepartment of Chemical Engineering, Indian Instituteof Technology Bombay (IITB), Mumbai, India
S. RastogiDepartment of Materials Loughborough,University Loughborough,England, UK
The nonlinear mechanical response of many materials,when subjected to large deformation or stress, changes frombeing predominantly elastic (solid-like) to predominantlyplastic (liquid-like) (Stokes et al. 2008). This transitionis called yielding. One of the experimental techniques toinvestigate the nonlinear response of soft materials involvessubjecting them to oscillatory shear flow in which the shearstrain γ = γ0 sin (ωt) is varied by ramping its amplitudeγ0 at a constant frequency. For an arbitrary strain ampli-tude, the measured stress σ(t) can be deconvoluted intoan in-phase response, characterized by the elastic modulusG′, and an out-of-phase response, characterized by the vis-cous modulus G′′. An accurate representation of the stressresponse would consist of Fourier harmonics of the elasticand viscous moduli (Ewoldt et al. 2008). However, har-monics higher than the first can be neglected for moderatestrain amplitudes at which yielding is most often seen insoft materials. Thus in an amplitude sweep test, the mechan-ical response of soft materials changes from being elastic(G′ > G′′) at small strain to viscous (G′ < G′′) at largestrain.
At intermediate strain, many metastable soft materi-als show a characteristic response in which the viscousmodulus increases from its linear value up to a max-imum value G′′
max before falling off, while the elasticmodulus decreases monotonically with strain and crossesbelow the viscous modulus at the yield point. This non-monotonic G′′ response, also called the type III LAOSbehavior (Hyun et al. 2002), has been reported earlier
for carbon composites of butyl rubber (Payne 1963),soft colloidal glasses (Brader et al. 2010), emulsions(Mason et al. 1995; Bower et al. 1999), gels (Altmannet al. 2004), electrorheological fluids (Parthasarathy andKlingenberg 1999; Sim et al. 2003), associating polymersolutions (Tirtaatmadja et al. 1997a, b), and weakly struc-tured materials such as xanthan gum solutions (Song et al.2006). On the other hand, polymeric fluids such as solu-tions and melts, which are ergodic, are typically known toexhibit strain-softening response (Doi and Edwards 1986)in which chain orientation causes both moduli to decreasemonotonically with increasing shear strain.
Recently, Miyazaki et al. (2006) proposed an elegantqualitative argument for explaining the origin of the non-monotonic G′′ response based on the reasoning that yield-ing involves a strain-induced decrease in the characteristicrelaxation time of the material. The authors proposed thatthe nonmonotonic G′′ behavior should be observable in allcomplex fluids. In this work, we validate this hypothesis bydemonstrating experimentally the universality of the yield-ing response. We also extend the argument further to inferan interesting dynamical feature.
Model
Following Miyazaki et al. (2006), we may represent anyviscoelastic fluid by a parallel combination of N Maxwellelements, each consists of linear springs1 in series withnonlinear dashpots so that the relaxation times (λi) of theMaxwell elements are given by some decreasing function ofthe strain amplitude such as
1
λi (γ0)= 1
λLVEi
+ k (ωγ0)m . (1)
In Eq. 1, λLVEi represents the characteristic relaxation time
for the ith mode in the linear regime, i.e., under smallimposed strain. The validity of Eq. 1 with m ≈ 1 fornonlinear deformations of metastable materials was demon-strated by Wyss et al. (2007), Yamamoto and Onuki (1998),Leonardo et al. (2005), and Kalelkar et al. (2010). Inciden-tally, Eq. 1 also describes the so-called convective constraintrelease mechanism of stress relaxation in entangled polymermelts subjected to high shear (Marrucci 1996). Thus, the useof Eq. 1 for describing the strain dependence of relaxationtimes of many soft materials in the nonlinear regime appearsjustified. Indeed, Eq. 1 is a simplified version of the model
1Nonlinear springs can be used without loss of generality. Strain-softening springs will cause a reduction in the prediction of themagnitude of G′′
max.
proposed by Derec et al. (2001) who, in addition to consid-ering the strain dependence of relaxation times, have alsotaken into account the influence of possible aging effects.
The elastic and viscous moduli for the Maxwell modelcan be written as
G′ (ω, γ0) =∑N
i
gi [ωλi (γ0)]2
1 + [ωλi (γ0)]2;
G′′ (ω, γ0) =∑N
i
gi [ωλi (γ0)]
1 + [ωλi (γ0)]2. (2)
Here, λi(γ0) is given by Eq. 1. Miyazaki et al. (2006)explained the strain dependence of G′′ by considering asingle mode (N = 1) in Eq. 2. The frequency regime ofinterest is one in which the material is predominantly elas-tic at small strains so that the Deborah number is given byωλLVE
C � 1, where λLVEC is a strain-independent character-
istic time of the material that is experimentally measured asthe inverse of the crossover frequency. At small strain ampli-tudes, G′ ∼ g and G′′∼g / ωλLVE
C so that G′ > G′′ and bothare independent of the applied strain. At moderate strainamplitudes, just after the linear regime, the Deborah numberis still ωλc (γ0) > 1 but the relaxation time decreases uponincreasing strain so that G′′ ∼ g / ωλc (γ0) is an increasingfunction of strain. For large strain amplitudes, the effectivestrain rate reduces the relaxation time to the extent whereωλC (γ0) � 1, so that the moduli in Eq. 2 can be approxi-mated as G′′ ∼ gi [ωλ (γ0)] ; G′ ∼ gi [ωλ (γ0)]2 indicatingboth G′ and G′′ to be decreasing functions of strain, withG′′ > G′. Further, the model also predicts the crossoverpoint, i.e., the macroscopic yield point at
ωλC
(γ0,y
) = 1, where G′ = G′′max = g / 2. (3)
In Eq. 3, γ0,y is the yield strain, which is unity whennormalized as γ̃0,y = kγ0,y
of soft materials nor their dynamical details, the Miyazakiargument presented above should be equally valid for allviscoelastic fluids as long as they are probed in an appropri-ate frequency window. In what follows, we demonstrate thisexperimentally for different complex fluids, which are cho-sen such that under near-quiescent conditions, some of themare in equilibrium state (polystyrene melt, surfactant lamel-lar phase) and some in metastable state (microgel densesuspension, hair gel, xanthan gum, and gelatin). The chosenmaterials also have very different microstructures.
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Experimental procedures
Sample preparation
Poly(N-isopropylacrylamide) microgels
Poly(N-isopropylacrylamide) (PNIPAm) microgels (Peltonand Chibante 1986) were synthesized by free radical precip-itation polymerization as prescribed by Senff and Richtering(2000). Sodium dodecyl sulfate (0.15 g) was used as astabilizer and potassium per sulfate (KPS 0.3 g) as an ini-tiator, and cross-linking was done by poly(ethylene glycol)diacrylate (Mw = 700 kg/mol, 0.157 g). Polymerizationwas carried out in a double-jacketed glass kettle reactorconnected to a temperature-controlled water circulator andan overhead stirrer. All reactants except the initiator weremixed in 480 ml of deionized water at 25 ◦C and stirred at300 rpm for 30 min under inert atmosphere. The reactionmixture was heated to 70 ◦C followed by addition of the ini-tiator (0.3 g of KPS in 20 ml deionized water). The reactionwas allowed to proceed for 4 h under nitrogen. The temper-ature was then reduced to 25 ◦C, and the reaction mixturewas stirred overnight at 100 rpm. Finally, the dispersion wasdialyzed (using dialysis bags having a molecular weight cut-off of 10,000 g/mol) against deionized water for 2 weeks.The dialyzed sample was lyophilized for 8 h and stored in adesiccator. Concentrated suspension (6 wt %) was then pre-pared by dispersing a known amount of polymer in deion-ized water. As these microgels are soft and compressible,they seldom crystallize at very high concentrations. Thehydrodynamic radius Rh = 137 nm at 25 ◦C was measuredfrom dynamic light scattering experiments (BrookhavenInstruments).
Xanthan gum
A 2-wt % aqueous suspension of xanthan gum gives a softcolloidal glass (Song et al. 2006). Ninety-eight milliliters ofwater was taken in a beaker and stirred at ∼ 500 rpm withthe help of an overhead stirrer. Two grams of xanthan gumpowder was added very slowly to the continuously stirredwater. This ensured complete and homogenous mixing ofthe powder. Stirring was continued for another 30 min,and the suspension was stored at 5 ◦C in a screw-cappedcontainer.
Surfactant lamellar phase
A 90-wt % aqueous suspension of a nonionic surfactantC12E9 (Rylo) was prepared using deionized water. Thesuspension was heated above the isotropic temperature(∼ 40 ◦C) on a water bath. This warm solution was thenmixed vigorously using a vibrato meter. Bubbles trappedwere removed by sonication and multiple cycles of heating
and cooling above the isotropic melting temperature. Athermodynamically stable homogenous lamellar phase wasformed at room temperature (∼ 25 ◦C) as confirmed bysmall-angle X-ray scattering (Kulkarni et al. 2011).
Gelatin
A 14-wt % gelatin solution was prepared in deionized water.The gel was made in a petri dish. Heating the solution above40 ◦C helps to homogenize the gelation powder. On coolingthe solution to room temperature, the gel takes the shape ofthe container at room temperature (∼25 ◦C).
Hair gel
A commercial hair gel (Park Avenue) was used as received.
Polystyrene melt
Polystyrene (Mw ∼ 550, 000 g/mol, PDI ∼ 1.05) was pur-chased from Sigma-Aldrich (GPC grade), and disk samples(25 mm in diameter) were prepared by compression mol-ding at 170 ◦C.
Rheological measurements
Rheological experiments for all samples except polystyrenewere done on a MCR 301 (Anton Paar) rheometer. Thegeometry used was a cone–plate (cone angle 1◦, diame-ter 25 mm). Polystyrene disks were tested on an ARES-G2(TA Instruments) rheometer using 25-mm parallel plates.Oscillatory strain sweep tests were performed at constantfrequency and temperature. Experiments were conductedat various frequencies and at temperatures so chosen thatthe samples exhibited dominantly elastic response at smallstrain. Linear viscoelastic frequency response was measuredby performing frequency sweep tests at a constant strainchosen from the linear viscoelastic (LVE) regime.
Results and discussions
Figure 1(a–f) shows that all materials, including the entan-gled polystyrene melt and the lamellar surfactant phase,show a nonmonotonic G′′ response followed by a crossoverof the moduli, which we define as the macroscopicyielding event. Figure 1(g–l) shows that the LVE fre-quency response of all these materials, irrespective of theirmicrostructure and thermodynamic state, is qualitativelysimilar: G′
LVE > G′′LVE, suggesting that in the frequency–
temperature–density window of observation, the fluid is pre-dominantly elastic at small strain, G′′
LVE increases monoton-ically but weakly with frequency, whereas the G′′
LVE showsnonmonotonic frequency dependence with a minimum at
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Fig. 1 Amplitude sweep at frequency where maximum in normalizedG′′ is observed (a–f). Frequency sweep (in LVE) and normalized G′′ atdifferent frequencies for different materials (g–l). PNIPAm suspensionat 20 ◦C and frequency of 1 rad/s (a, g), lamellar phase of CnH2n+1surfactant at 25 ◦C and frequency of 1 rad/s (b, h), hair gel at 25 ◦Cand frequency of 0.5 rad/s (c, i), xanthan gum solution at 25 ◦C and
frequency of 1 rad/s (d, j), gelatin at 25 ◦C and frequency of 1.6 rad/s(e, k), and polystyrene at 170 ◦C and frequency of 16 rad/s (f, l). G′is represented by open circles, G′′ by filled circles, maximum normal-ized G′′ (G̃′′
max
)by filled stars, G′′ (G̃′′
max
)and dotted lines are guides
to the eye
intermediate frequency. Thus, the data corroborate the sim-ple arguments presented in the Section “Model” and under-line the similarity of patterns observed in the yieldingprocess in complex fluids.
The predictions of the multimode Maxwell model (Eq. 2)are shown in Fig. 2 for the representative case of the PNI-PAm microgel suspension. For this calculation, an eight-mode relaxation spectrum was obtained by fitting the modelto the experimental linear viscoelastic frequency responseshown in Fig. 2a. The prediction of the multimode Maxwellmodel for an amplitude sweep experiment carried out at arepresentative frequency of 1 rad/s is compared with exper-imental data in Fig. 2b. Different values of the parameterm in Eq. 1 were tried, and it was found that m = 1 gavethe best fit to the experimental data. Previous experimentalinvestigations have also suggested the value of m = 1 (Wysset al. 2007; Kalelkar et al. 2010). The model provides only a
qualitative prediction of the nonmonotonic G′′ response.The poor agreement between model prediction and experi-mental data in the nonlinear region could be due to contri-butions from higher harmonics that have not been accountedfor in the model. However, in our earlier work (Kalelkaret al. 2010), we have shown that for a 14-wt % suspen-sion of PNIPAm microgels, the ratio of the third harmonicstress signal to the first harmonic stress signal is small (I3
/ I1 = 0.06). This suggests that the contribution of thehigher harmonics is likely to be small and may not be themain reason for the observed differences between the modeland experiments. In the present work, our interest is inseeking broader understanding of the phenomenon ratherthan aiming for quantitative predictions, for which moresophisticated models will be required. Hence, we presentbelow only qualitative trends predicted by the Maxwellmodel.
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Fig. 2 a Frequency sweep dataat low strain amplitude (linearviscoelastic response) ofPNIPAm microgel suspensionalong with fit of the multimodeMaxwell model. b Amplitudesweep data at 1 rad/s for thesame suspension together withpredictions of the multimodeMaxwell model, for m = 1
a b
While the multimode Maxwell model predicts similarnonlinear response as the single-mode version of Miyazakiet al. (2006), it allows us to interrogate what happens to thedissipation component when the strain sweep experimentsare done at different frequencies. To answer this, we definethe normalized viscous modulus G̃′′
max = G′′max/G
′′LVE (ω)
so that the dissipation can be compared for different fre-quencies relative to the linear limit. Figure 3a shows thatG′′
max has a weaker dependence on frequency relative toG′′
LVE (ω), which has a frequency dependence shown inFig. 2a. Therefore, the normalized viscous modulus G̃′′
maxhas a maximum at the characteristic frequency at whichthe G′′
LVE (ω) shows a minimum. To illustrate this, Fig. 3bshows the predictions of G̃′′
max at different frequencies ascalculated from the multimode Maxwell model for thecase of the PNIPAm microgel suspension. Also shown areexperimentally determined values of G̃′′
max and the linearfrequency response for this material. Indeed, the maximumin G̃′′
max is seen to occur at the frequency where G′′LVE (ω)
shows a minimum. That this is a common feature of yield-ing for all soft materials is seen in Fig. 1(g–l), which showsG̃′′
max at various frequencies for all materials investigatedhere. In each case, the maximum in G̃′′
max is seen at thefrequency where G′′
LVE (ω) shows a minimum. It may benoted that in a LAOS experiment, the energy dissipatedper cycle is Ø = ∫ 2π/ω
0 τ γ̇ dt = πG′′1γ
20 , where G′′
1 is thefirst harmonic of the loss modulus (Ganeriwala and Rotz1987). Thus, the maximum relative dissipation defined asØmax = πγ 2
0 G̃′′max (assuming that G′′ ∼= G′′
1) will havethe same frequency dependence as G̃′′
max. In other words,the maximum relative dissipation will be the highest at thefrequency where G′′
LVE (ω) shows a minimum.In order to better understand this phenomenon, we show
in Fig. 3c the predictions of Maxwell model for frequencydependence of viscoelastic moduli at various strain ampli-tudes starting from small strain (linear response) to largestrains (nonlinear response). The relaxation spectrum usedhere as an example is that for the PNIPAm suspension;however, similar features will be predicted for any othercomplex fluid. The calculations are extrapolated to low
frequencies where the Maxwell model predicts a crossoverof moduli corresponding with the structural relaxation timeλC . For increasing strain amplitude, several features areworth noticing in the figure: In the low-frequency regionω < ωLVE
C , the moduli decrease with increasing strain,suggesting a strain-softening behavior with G′′ > G′. Athigher frequencies ω > ωLVE
C , while G′ decreases withstrain, G′′ increases with strain. This corresponds with theupturn in G′′ seen in the amplitude sweep predictions. Itcan be seen that the ratio G̃′′ = G′′ (ω, γ0)/G
′′LVE (ω), i.e.,
the normalized loss modulus, is always the highest forthe frequency corresponding to the minimum in G′′
LVE (ω),indicating the maximum relative dissipation at this fre-quency. The crossover frequency ωC (γ0) increases withstrain amplitude, indicating a decrease in the structuralrelaxation time in accordance with Eq. 1. Thus, the slowstructural relaxation time scale approaches the fast timescale monotonically with increasing strain amplitude. At90 % strain for this material, the ωC approaches the fre-quency at which G′′
LVE (ω) shows a minimum. Above thisstrain, G′ decreases below G′′ (indicating yielding), andG′′ also decreases over the entire frequency range. Thus,the maximum relative dissipation is obtained at 90 % strainfor this fluid and at frequency close to ωβ , which is thefrequency corresponding to the minimum in G′′
LVE and isreferred to as the beta relaxation frequency of cage dynam-ics in the framework of the mode-coupling theory (Masonand Weitz 1995).
The above observations together with Eq. 3 imply thatthe maximum relative dissipation
(G̃′′
max
)just before macro-
scopic yielding will occur in an amplitude sweep experi-ment when the imposed frequency satisfies ω = 1/λc =1/λβ ; here, λC(γ0) is the structural relaxation time (i.e.,the so-called alpha relaxation time) and λβ = 1/ωβ is thebeta relaxation time. For soft glassy materials consisting ofparticles trapped in a local “cage,” λC represents the timescale over which a trapped particle would escape its cage,while λβ corresponds to the cooperative motion of the par-ticles within the cage (Roldan-Vargas et al. 2010). Thus, themaximum relative dissipation prior to macroscopic yielding
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Fig. 3 a Maxwell modelpredictions of the straindependence of normalized G′′for various frequencies. bComparison of experimentallydetermined normalizedG̃′′
max(filled circles) as a functionof frequency with predictions ofthe Maxwell model (dotted line)for the normalized G̃′′
max. Thedashed–dotted line through theexperimental data only serves asguide to the eye. The figure alsoshows the experimental linearviscoelastic frequency response(triangles) and model fit for thesame (bold and dashed lines). cPrediction of the Maxwell modelfor the frequency dependence ofG′ and G′′ at different strainsγ0 = 1 % (LVE), 5 %, 20 %,40 %, 60 %, and 90 % (arrowsindicate increasing strainamplitudes). The closed andopen symbols represent theexperimental G′ and G′′,respectively, for γLVE = 0.6 %
is seen to occur when the dynamics of alpha relaxationare accelerated by the imposed shear to an extent wherethey become equal to the beta relaxation dynamics so that,effectively, a particle in the fluid does not feel the presenceof topological constraints.
For entangled polymers, whose dynamics may be under-stood using the tube model, Marrucci (1996) argued that thetube renewal time scale λC will decrease upon imposition oflarge and rapid deformation by the so-called convective con-straint release process in which neigboring entangled chainsare convected away from the test chain, releasing entangle-ments locally along its contour. Under sufficiently strongflows, if the rate of CCR is of the same order as the relax-ation of a disentangled polymer, then the polymer chainwould not experience the presence of topological constraintstube in a dynamical sense. The latter is approximately givenby the Rouse reorientation time λR of half the chain in itstube. Hence, when the imposed frequency of a large ampli-tude oscillatory flow approaches λR , a complete destructionof the tube is possible. This is akin to the destruction ofthe cage structure in a soft glassy material. It is well knownthat the Rouse frequency ωR = 1/λR is slightly lower thanthe frequency at which G′′
LVE shows a minimum (Rubinsteinand Colby 2003; Doi and Edwards 1986). If the imposedfrequency equals the Rouse frequency, a polymer chainwould not feel the presence of topological constraints of theentanglements. Therefore, the criterion ω = 1/λC = 1/λβ
results in the maximum relative dissipation in the case ofentangled polymers as well.
In summary, we have ascertained that soft materialsexhibit universal features of yielding in an oscillatoryshear test when conducted in an appropriate frequency–temperature–concentration window. Specifically, the linearfrequency response and the nonlinear strain response arerelated such that the maximum relative dissipation prior tomacroscopic yielding is obtained when the imposed fre-quency resonates with the microscopic time scale of thematerial. Under this condition of high shear, the micro-scopic structural entities that make up the material do notfeel topologically constrained any more.
Acknowledgments We are grateful to the Council of Scientific andIndustrial Research, India for funding this research.
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