Large amplitude oscillatory shear of supramolecular materials

Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

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Journal of Non-Newtonian Fluid Mechanics journal homepage: http://www.elsevier.com/locate/jnnfm

Large amplitude oscillatory shear of supramolecular materials Alan Ranjit Jacob a,1, Abhijit P. Deshpande a,⇑, Laurent Bouteiller b a b

Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India Institut Parisien de Chimie Moleculaire (IPCM), UMR 8232, Chimie des Polymeres, Universite Paris 6-CNRS, 3, rue Galilee, Ivry sur Seine 94200, France

a r t i c l e

i n f o

Article history: Received 8 March 2013 Received in revised form 24 February 2014 Accepted 1 March 2014 Available online 12 March 2014 Keywords: Large amplitude oscillatory shear Supramolecular polymer Giesekus model Transient network model Secondary loops in Lissajous plots

a b s t r a c t Large amplitude oscillatory shear [LAOS] helps to investigate non-linear rheology and the dynamic behaviour of materials when subjected to large deformations. Qualitative characterization of non-linear rheology can be easily done but quantification and physical interpretation of non-linear rheological characteristics is one of the most challenging aspects. In this work, LAOS tests are done on different materials generally known as ‘living polymers’, with similar linear response, in order to study their non-linear response using both quantitative and qualitative analysis. The effect of living polymer networks on LAOS was examined using different concentrations of 2,4 bis(2-ethyl hexyl ureido)toluene [EHUT] gels. Cetyl trimethyl ammonium bromide [CTAB] wormlike micellar solution is prepared to compare and contrast with EHUT gels since both these materials show similar linear viscoelastic response. Elastic loading/ unloading and plastic deformations are identified as important processes that lead to the distinguishing behaviour for different materials. An equivalence is observed for Lissajous plots in the concentration frequency plane for EHUT. Giesekus model and Upper Convected Maxwell model coupled with transient network model [TNM], two well established models, are solved numerically for comparison with the experimental results. A modification of the Giesekus model to incorporate the network breakage/reformation as a linear function of strain amplitude, and nonlinear function of strain rate within an oscillation period is suggested. The proposed model is able to capture the competition between reptation, network breakage and reforming during LAOS. Moreover, the modified model suggests that the secondary loops during LAOS for EHUT gel occur soon after yielding due to competition between the network breaking and reforming at high shear rates and reptation at low shear rates within a cycle. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Supramolecular assembly of molecules is being increasingly used to obtain material systems with controlled properties. These assemblies can be from small molecules as well as from polymers [1–3], and are promising for diverse applications. Supramolecular polymer gels are assembled as networks of one-dimensional objects formed from interactions among small molecules. In a supramolecular polymer, repetitive arrangement of monomeric units is brought about by highly directional and reversible secondary interactions [4,5]. These secondary interactions can be hydrogen bonds, metal-coordination bonds, p–p interactions, van der Waals forces and dispersion forces. Unlike the covalently bonded polymers, the supramolecular polymers break and reform, and are also called living polymers. In general, the dynamic characteristics of networks and their role in rheological behaviour are studied ⇑ Corresponding author. Tel.: +91 44 22574169; fax: +91 44 22574152. 1

E-mail address: [email protected] (A.P. Deshpande). Current Address: FORTH-IESL, 71110 Heraklion, Crete, Greece.

http://dx.doi.org/10.1016/j.jnnfm.2014.03.001 0377-0257/Ó 2014 Elsevier B.V. All rights reserved.

for varied applications. The network formation in supramolecular polymer gels and their behaviour under different conditions have also been of wide interest. Therefore, rheology of supramolecular polymer gels is important not only for studying the deformation behaviour, but also to probe the network structure. 2,4 bis(2-ethyl hexyl ureido)toluene [EHUT], an organic gelator, has been shown to form tubular equilibrium aggregates in solvents such as dodecane [6]. The molecular structure of the EHUT monomer and the tubular aggregate are shown in Fig. 1 [7,8]. The hydrogen bonding between the amide groups and oxygen, represented by the dotted lines, is responsible for the aggregation. The tubular structures shown in the figure form entanglements/meshes which entrap the dodecane solvent [6,8]. It is also known that EHUT does not form these tubular structures in bulky solvents [9]. EHUT has been investigated for applications such as drag reducing agent and hardener for asphalt [10–12]. Supramolecular polymers, such as EHUT gels, have been characterized using rheological techniques like steady shear and dynamic oscillatory shear tests [6,13–15]. The linear viscoelastic response of the gels was shown to be similar to the Maxwell model. EHUT gels are similar to wormlike micelles

A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

41

1.1. Material models relevant for LAOS Many models have been used to examine the LAOS response, which include upper convected Maxwell, Giesekus [20,31], transient network model (TNM) [32], Leonov [33], molecular stress function (MSF) [34], corotational Maxwell [35] and Doi Edwards model [36]. A single relaxation mode as well as the relaxation spectrum has been used. The Giesekus model is written for polymer and solvent contributions to overall stress (sp þ ss ¼ s, respectively) [20,23,37–40].

!

r

k1 sp þ 1 þ

ak1 s s ¼ 2gp D; gp p p

ð1Þ

where D is the strain rate or stretching tensor, k1 is the characteristic relaxation time, gp is the viscosity coefficient, a is the non-linear r parameter or mobility factor, and sp is the upper convected derivative. The solvent contribution to stress tensor is Fig. 1. The molecular structure and the predicted microstructural thick filament structure (T) formed from each of the mono-meric units (S) where both of the systems are thermodynamically stable [7,8] The amide group is represented as a filled circle and the dotted lines represent the hydrogen bonds.

as they tend to undergo scission and recombination. Static and dynamic properties of the wormlike micellar systems have been exhaustively studied [16,17]. The similarities of EHUT and wormlike micellar systems are in linear rheology as well as non-linear rheology. At high shear rates, a non-monotonic rise of the stress with respect to shear rate has been observed. This behaviour and associated shear banding have been reported for both micellar solutions and EHUT [18,19]. While small amplitude oscillatory shear [SAOS] is used to probe the linear response of complex fluids, the non-linear behaviour of complex fluids can be investigated using large amplitude oscillatory shear [LAOS]. Large volume of research is going on to develop frameworks to quantify non-linear behaviour of materials and model such non-linear behaviour. LAOS has been investigated for many materials like entangled polymer systems, suspensions, emulsions, polymer blends, block copolymer melts and gels [20]. Fourier transform is one of the oldest techniques for analysis of LAOS data but it is difficult to provide physical interpretations of material behaviour with this method [21,22]. Lissajous plots is one of the methods to qualitatively analyse LAOS behaviour [23– 26]. One of the quantitative methods to analyse LAOS data was using G0L and G0M , derived from Chebyshev polynomials and can provide physical insights to viscoelastic material behaviour at LAOS [20,24]. Recently, analysis of data from strain, strain rate and stress space has been used to define new material functions to describe the physical processes during LAOS [27–30]. The appearances of the secondary loops in the Lissajous plots are due to specific changes in the microstructure of the material investigated [31] and have been investigated using constitutive models such as the Giesekus model. In the present work ‘living polymers’ based on EHUT and CTAB, which have similar linear rheological response are shown to be exhibit rich variety of LAOS response, due to different microstructural processes occuring during a cycle. An extensive characterisation of the non-linear rheological response of EHUT using LAOS is performed in order to gain insights into the development of ‘secondary loops’ appearing in the Lisajous plots. A concentration frequency equivalence is explored with the Lissajous plots. A microstructural hypothesis is offered with the help of these nonlinear experiments, which is further reinforced by modelling the EHUT response during LAOS with modifications to the Giesekus model. The results provide a platform and future directions for the understanding of non-linear rheology of ‘living polymers’.

ss ¼ 2gs D:

ð2Þ

This non-linear model has been used for LAOS extensively. The Giesekus model material functions for SAOS are given by, 0

2

G ¼ go x

G00 ¼ go x

k1  k2

!

1 þ k21 x2 1 þ k 1 k 2 x2 1 þ k21 x2

ð3Þ

; ! :

ð4Þ

    where g0 ¼ gs þ gp and k2 ¼ k1 gs =gp is the retardation time [41]. Giesekus model reduces to the TNM model, when a ¼ 0 in Eq. (1), r

s þ k1 s ¼ 2g0 D:

ð5Þ

The SAOS Maxwell model material functions are given by substituting k2 ¼ 0 in Eqs. (3) and (4). The TNM model has been used to study LAOS, for example the start up flows in Couette geometry [42]. In order to capture the variation in the dynamic networks, formation and breakage of network points or bonds is hypothesized. A structural parameter quantifies the status of the network. This variation in networks is captured by writing an evolution equation for the structural parameter. Such a model was used for LAOS, along with the TNM model [32],

dx f1 ¼ ð1  xÞ  f2 x½IID ð1=2Þ : dt k1

ð6Þ

where x is the structural parameter. In this equation, f1 represents the rate of creation of network points, while f2 includes the breakage of these network points and the effect of shearing is included by c_ , (¼ IID , second invariant of D). x ¼ 0 implies that the network has completely broken down and x ¼ 1 implies the network in the undeformed state. Such an evolution equation is combined with TNM by assuming the parameters of TNM to be functions of x,

k1 ¼ ko1 x1:4 : ko1

ð7Þ

where is the relaxation time, when no shear is applied or the relaxation time of undeformed network. Hence the structural parameter which is governed by the breaking and reforming of the bonds when coupled with TNM model, is able to capture the temporary nature of networks in the non-linear regime [32]. Another important aspect in the non-linear modelling of material systems, during LAOS, is the effect of yielding of materials. Models incorporating yield stress and yield strain have also been used to analyze LAOS response [43]. Rigorous and generalized models incorporating the effect of maximum applied strain, on

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A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

the current state of material, have been used to describe the non-linear response of network systems such as rubber and cells [44–47]. In this work, the secondary loops were modeled using modifications of the Giesekus model, with aspects of TNM and yield models. In the modified one-dimensional model, the relaxation time is assumed to be a function of strain rate and strain difference between maximum applied strain and yield strain. 2. Materials and methods 2.1. Materials 2,4 Bis(2-ethyl hexyl ureido)toluene [EHUT] was synthesized as previously described [48]. Dodecane (>95% purity) was obtained from Spectrochem, India. Cetyl trimethyl ammonium bromide [CTAB] was bought from Sigma Aldrich (>99% purity). Sodium Salicylate [NaSal] was obtained from Merck Chemicals. 2.2. Sample preparation EHUT gel was prepared by stirring appropriate weight of EHUT in the required volume of dodecane at 80 °C in a water bath for 48 h. All the references to EHUT in the following discussions below refer to EHUT in dodecane. In order to prepare the wormlike 50mM CTAB and NaSal micellar solution, CTAB and NaSal were mixed in distilled water. 145.792g of CTAB and 64.04g of NaSal were separately mixed in 250 ml distilled water. Then both the solutions were poured into a single beaker and sonicated. The resulting solution was kept un-disturbed for 5 days, for attaining equilibrium structure. 2.3. Rheology Rheological tests were performed using Anton Paar MCR-301 rheometer. A cone and plate geometry with diameter 25 mm, the angle of inclination 1.01° and the truncation gap 0.047 mm was used. All tests were conducted at 25 °C. Data were collected using the rheometer software ‘Rheoplus’. All the tests were repeated three times to check for repeatability. The following types of experiments were done: 2.3.1. Linear visco-elastic regime The sample was made to undergo oscillatory shear by varying frequency from 0.01 rad/s to 100 rad/s at constant strain. This test was conducted to ensure the gels prepared were consistent [6]. Although all the tests were done from 0.01 rad/s to 100 rad/s, some of the data reported here are from 0.1 rad/s to 50 rad/s taking into consideration the instrument sensitivity, nominal stresses and system inertia. 2.3.2. Non-linear visco-elastic regime or LAOS In these tests, the strain on the sample was varied logarithmically from 1% to 1000%. All LAOS tests were carried out at 0.1, 1 and 5 rad/s with Direct Strain Oscillation (DSO) mode in MCR301. The actual strain applied on the material was examined/verified for all the LAOS experiments. For example, the applied strain was within ±1% strain for 250% and 1000%) and ±0.1% in the for 25% strain. The LAOS results were obtained as processed by Rheoplus also the torque signals collected were analyzed. Additional experiments were carried out in an ARES strain controlled rheometer. Cone and plates of different cone radius and materials of construction were used. Raw data during oscillatory shear were collected from the rheometer using a LABVIEW program.

2.3.3. Steady shear The strain rate was varied from 0:01 s1 to 100 s1 , and steady viscosity of EHUT gels was measured. The rheometer is connected to a MEphisto-Scope 1 oscilloscope, to acquire time series of electric torque and inertial torque signals. In order to get high signal to noise ratio appropriate amplification factors were used, while the data were being acquired in the oscilloscope. The time series data were collected at a sampling rate of 50 ls per data. A MATLAB code was used for the Fourier analysis. To improve signal to noise ratios, the torque signals were oversampled at a much higher sampling rate than that prescribed by Nyquist criteria and box car averaging was used for smoothening the data [49]. 2.4. Numerical methods The model equations in viscometric flows (simultaneous ordinary differential equations) were solved assuming viscometric flows. For a simple shear flow given by v 1 ð2Þ ¼ c_ ðtÞx2 . Therefore, four differential equations of stress components 11, 22, 33 and 12 for Giesekus equation from Eq. (1) are solved. Given an input of a particular oscillatory strain, the differential equations are solved simultaneously using ode45 in MATLAB. The results of interest for the present work, is the stress component 12 as a function of time. The material parameters of the model, such as g0 which is the zero shear viscosity and k1 which is the relaxation time, were obtained from the experimental data. The data extracted from experimental results were used for modelling (for all concentrations of EHUT and CTAB 50 mM), and are shown in Table 1. The non-linear parameter (a) of the Giesekus model was estimated from steady shear viscosity using least square fit regression analysis [40]. Similarly, differential equations of the TNM model in Eq. (5) coupled with Eq. (6) are solved using ode45 in MATLAB. Both the TNM model and structural parameter evolution are coupled together with the relaxation time using Eq. (7). The structure factor (x) is allowed to vary in between 0 (complete breakage of the network) and 1 (complete formation of the networks). f1 and f2 for the TNM were estimated by least square fit regression from Lissajous plots in the linear regime and these parameters used for different strains. 3. Results and discussions 3.1. SAOS The SAOS response of 5 g/L EHUT gel is shown in Fig. 2. It can be observed that the elastic and viscous modulus dominate at different frequency regimes. As expected, and similar to wormlike micellar solutions, there is a dominant relaxation. The relaxation time can be extracted from the crossover from the loss and storage moduli (G00 and G0 ). There are two relevant mechanisms that are

Table 1 Relaxation time, zero shear viscosities and plateau storage modulus of EHUT and CTAB. Material EHUT 1 g/L 3 g/L 5 g/L 7 g/L 9 g/L CTAB 50 mM

k1 (s)

g0 (Pas)

G (Pa)

De ¼ f k1

8.68 28.27 28.98 42.7 44.6

0.7 19.0 50.0 110.0 350.0

0.2 4.2 9.8 19.8 40.0

1.38 4.49 4.6 6.78 7.09

13.19

31.5

9.8

2.09

A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

3.2. LAOS

2

10

Viscous Modulus,G′′

Elastic Modulus,G′ [Pa] Viscous Modulus,G′′ [Pa]

Elastic Modulus,G′

1

10

0

10

−1

10

43

−2

10

−1

10

0

10

1

10

Angular Frequency [rad/sec] Fig. 2. Storage modulus G0 and loss modulus G00 for 5 g/L at c ¼ 10%.

prevalent in the experimental time scales i) Entanglement reptation and ii) Reversible Chain scission reactions. If the breaking of the tubes occurs faster than reptation then the material will have a single relaxation time as seen in Fig. 2. If the breaking of the tubes occurs at slower time scales than that of the reptation then we will never observe a single relaxation time. Thus at long time scales the material relaxes using reptation while at shorter time scales the material relaxes by scission of the tubes. It was proposed by Cates that entangled living polymers undergo a ‘breathing mechanism’ to relax [13] which is also seen for EHUT [6,14,50]. The variation of EHUT gel response with different concentrations is shown in Fig. 3. The response of the gel formed with 9 g/L EHUT is similar to that of 5 g/L, while the gel formed with 1 g/L shows a qualitatively different response. This can be attributed to the concentration of entanglements being very low, unlike at other concentrations. The SAOS material response is also compared with Maxwell and Giesekus models. It is observed that the Maxwell model describes the SAOS behaviour of 9 g/L EHUT gel, only below and around the crossover frequency. This is expected since the modes due to scission are important at higher frequency. The Giesekus model, therefore, describes the SAOS response of EHUT gels as shown in Fig. 3(b). As expected, the experimental results of 1 g/L EHUT are not described by Maxwell or Giesekus model. When the concentration of EHUT is increased, the length of the tubes is expected to increase [8]. This leads to smaller mesh size, giving rise to better gel behaviour as reported. A corresponding increase in the relaxation time is also observed, as given in Table 1. The similarity in behaviour of EHUT gels and CTAB solution is also apparent from Fig. 3(c) and Table 1. The dependence of plateau modulus and relaxation time on concentration for micellar solutions are Go / c2:25 and k1 / c1:77 , respectively [51]. The results with EHUT gels at different concentrations lead to Go / c2:25 and k1 / c0:77 , as observed earlier as well [6,14,50]. The theories for micellar solution seem to agree with EHUT gels, since both consist of flexible chains, and are living or equilibrium polymers. The parameters given in Table 1 are used in subsequent modelling of LAOS behaviour. The relationship between linear viscoelasticity and microstructure is well understood for these living polymers (such as scaling for the relaxation time with concentration). However, this is not the case in non-linear viscoelastic response of living polymers. This motivated us to investigate LAOS on EHUT at varying concentrations.

Non-linear rheology of EHUT gels was observed at medium and large amplitudes, and all these experiments were carried out at 1 rad/s. The variation of first harmonic response, or the linear moduli G0 and G00 , with strain amplitude is shown in Fig. 4(a). Both the moduli decrease monotonically with strain amplitude, except for G00 for 5 g/L. With higher concentration of EHUT, G0 becomes a function of strain at lower strain amplitudes, while G00 becomes a function strain at higher strain amplitudes. The limits of linearity, as observed from these data, are higher than 50% strain at all concentrations. The higher is the concentration of EHUT, the lower is the limit of non-linearity, as observed from Fig. 4(a). It should be noted that storage modulus is higher than the loss modulus (at low strains, or in the linear regime), implying the frequency of oscillation is greater than the crossover frequency (De > 1). Fig. 4(b) indicates that the same concentration of EHUT (5 g/L) shows varying behaviour at different Deborah numbers (De < 1 and De > 1). For De > 1, there seems to be a non-monotonic decrease in viscous moduli as seen for wide variety of soft materials recently [52]. This was attributed to the elastic behaviour of fluids in small strain limits. At higher concentrations greater than 5 g/L the number of entanglements is very high leading to stronger structures hence leading to late yielding and monotonic decrease of the viscous moduli as shown in Fig. 4(a). The effect of varying concentrations on the LAOS behaviour of EHUT gels is shown in Fig. 5 for 25%, 250% and 1000% strains. For comparison, LAOS behaviour of the CTAB solution is also shown. The non-linearity in material response seems to increase for increasing concentrations of EHUT at 400% strain, as is apparent from the shapes of Lissajous plots. Moreover, the secondary loops are also seen in Fig. 5(d2) and (e2). Such loops have been attributed to stress overshoot during a cycle and have been claimed to be useful in understanding response of different complex fluids such as polymer melts and solutions, soft glassy materials and structured fluids [31]. In the next section, preliminary models’ behaviour has also been used to highlight the observations. It can be observed that 50 mM CTAB exhibits qualitatively different behaviour, when compared to EHUT gels at different strain amplitudes. The SAOS behaviour of this micellar solution was shown to be similar to EHUT gel of 5 g/L. However, unlike the EHUT gels with different concentrations, no secondary loops are observed in CTAB upto 1000% strain because reptation dominates more than network breakage and reformation in CTAB. In Pipkin space, or at different values of De and strain, it is possible that different types of Lissajous plots can be obtained (for example, Fig. 9 from [24] by changing the frequency) whereas the theme explored in this work is to understand the material response variation at different De by systematically changing the concentration at a fixed frequency for EHUT gels. Usually, an order of magnitude increase in frequency leads to qualitatively different behaviour. Therefore, the comparison of CTAB with EHUT gels for qualitative rheological responses is reasonable given the similarity in their relaxation times, plateau modulus and zero shear viscosity as seen in Table 1. It should be noted that CTAB did exhibit secondary loops at strains higher than 1000% strain. However, due to uncertainties in analysis of extremely large amplitudes, the data is reported till 1000% strain. Slip at higher strain amplitudes should be ruled out, and this was ensured in the present work. This was based on the following observations; (a) gradual development of curves, or gradual changes in qualitative response of the material, is observed (b) no abrupt or random changes in the torque as detected by the transducer (c) no even harmonics observed in Fourier rheology, and (d) similar LAOS response observed with different cone and plate geometries of varying construction material/gaps. For

44

A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

10

10

10

1

1

(b) Scaled Storage Modulus,G′/ Go[1] Scaled Loss Modulus,G′′/ Go[1]

Scaled Storage Modulus,G′ / Go[1] Scaled Loss Modulus,G′′ / Go[1]

(a)

0

−1

G′′/Go 9 g/L G′/Go 9 g/L 10

−2

G′′/Go 1 g/L

10

0

10

−1

10

G′′/Go 9 g/L G′/Go 9 g/L G′′/Go 1 g/L

−2

10

G’/Go 1 g/L

G’/Go 1 g/L

G′ Giesekus Model

G′ Maxwell Model

G′′ Giesekus Model

G′′ Maxwell Model 10

−3

−3

10

−2

10

−1

10

0

10

1

10

2

10

−2

−1

10

0

10

Scaled Angular Frequency[1]

1

10

10

2

10

Scaled Angular Frequency[1]

1

Scaled Storage Modulus,G′/ Go[1] Scaled Loss Modulus,G′′/ Go[1]

(c)

10

0

10

−1

10

G′′ CTAB G′ CTAB

−2

10

G′ Giesekus Model G′′ Giesekus Model

−3

10

−2

10

−1

0

10

1

10

10

2

10

Scaled Angular Frequency [1] Fig. 3. (a) G0 =G ; G00 =G versus x=xc for 1 g/L and 9 g/L of EHUT and comparison with Maxwell model, (b) G0 =G ; G00 =G versus x=xc for 1 g/L and 9 g/L of EHUT and comparison with Giesekus model, (c) G0 =G ; G00 =G versus x=xc for 50 mM CTAB.

2

(a)

2

10

(b)

1

10

0

10

Storage Modulus G′ [Pa] Loss Modulus G′′ [Pa]

Storage Modulus G′ [Pa] Loss Modulus G′′ [Pa]

1

10

G′′ 3g/L G′ 3g/L G′′ 5g/L G′ 5g/L

−1

10

0

10

G′′ 0.1 rad/sec G′ 0.1 rad/sec G′′ 1 rad/sec −1

10

10

G′′ 7g/L

G′ 1 rad/sec

G′ 7g/L

G′′ 5 rad/sec

G′′ 9g/L

G′ 5 rad/sec

G′ 9g/L −2

10

−2

0

10

1

2

10

10

Strain [%]

3

10

10

10

0

10

1

10

2

3

10

Strain [%]

Fig. 4. The loss and storage modulus varying with strain for (a) different EHUT concentrations at constant frequency of 1 rad/s, and (b) different frequencies at constant concentration of 5 g/L.

45

A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56 1 0.8

(a2)

25%

1

0.4

0.4

0.4

0 −0.2 −0.4

Scaled Stress [1]

0.6

0.2 0 −0.2 −0.4

0 −0.2 −0.4 −0.6

−0.8

−0.8

−0.8

−1

−1

−1

−0.5

0

0.5

1

−1

1 0.8

−0.5

0

0.5

−1

1

−1

Scaled Strain Rate [1]

(b2)

25%

(b3)

250%

0.8

0.8

0.4

0.4

0.4

−0.2 −0.4

Scaled Stress [1]

0.6

0

0.2 0 −0.2 −0.4

0 −0.2 −0.4 −0.6

−0.8

−0.8

−0.8

−1

−1

0

0.5

1

−1

1 0.8

(c2)

25%

0.5

−1

1

(c3)

250%

0.8

0.4

0.4

−0.2 −0.4

Scaled Stress [1]

0.4

0

0.2 0 −0.2 −0.4

0

−0.4 −0.6

−0.8

−0.8

−0.8

−1

−1

0.5

1

−1

0

0.5

−1

1

−1

Scaled Strain Rate [1]

1 0.8

−0.5

(d2)

25%

(d3) 250%

0.4

0.4

−0.4

Scaled Stress [1]

0.4

Scaled Stress [1]

0.6

−0.2

0.2 0 −0.2 −0.4

0 −0.2 −0.4

−0.6

−0.6

−0.8

−0.8

−0.8

−1

−1

−0.5

0

0.5

1

−1

Scaled Strain Rate [1] 1 0.8

0

0.5

−1

1

(e2)

25%

(e3)

250%

0.8

0.4

−0.4

Scaled Stress [1]

0.6

−0.2

0.2 0 −0.2 −0.4

0.2 0 −0.2 −0.4

−0.6

−0.6

−0.6

−0.8

−0.8

−0.8

−1

−1

0

0.5

Scaled Strain Rate [1]

1

−1

−0.5

0

0.5

Scaled Strain Rate [1]

1

1000%

0.4

0

0.5

0.8

0.6

0.2

0

1

0.4

−0.5

−0.5

Scaled Strain Rate [1]

1

0.6

−1

−1

Scaled Strain Rate [1]

Scaled Stress [1]

(e1)

−0.5

1

0.2

−0.6

−1

0.5

1000%

0.8

0.6

0

0

1

0.6

0.2

−0.5

Scaled Strain Rate [1]

1 0.8

1

−0.2

−0.6

0

0.5

0.2

−0.6

Scaled Strain Rate [1]

0

1000%

0.8 0.6

−0.5

−0.5

1

0.6

−1

−1

Scaled Strain Rate [1]

1

0.6

0.2

(d1)

0

Scaled Strain Rate [1]

Scaled Stress [1]

Scaled Stress [1]

(c1)

−0.5

1

0.2

−0.6

−0.5

0.5

1000%

−0.6

−1

0

1

0.6

0.2

−0.5

Scaled Strain Rate [1]

1

0.6

Scaled Stress [1]

Scaled Stress [1]

0.2

−0.6

Scaled Strain Rate [1]

Scaled Stress [1]

1000%

−0.6

Scaled Strain Rate [1]

Scaled Stress [1]

0.8

0.6

0.2

(b1)

1

(a3)

250%

0.8

0.6

Scaled Stress [1]

Scaled Stress [1]

(a1)

1

−1

−1

−0.5

0

0.5

1

Scaled Strain Rate [1]

Fig. 5. The stress versus shear rate (viscous Lissajous) plots for (a) CTAB/NaSal at 50 mM, (b) EHUT at 3 g/L, (c) EHUT at 5 g/L, (d) EHUT at 7 g/L, (e) EHUT at 9 g/L for 25%, 250% and 1000% strains respectively.

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A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

(a)

1 0.8

(b)

1.1o 0.0254mm SST

1 0.8

o

0.6

0.6

0.4

0.4

Scaled Stress [1]

Scaled Stress [1]

2.2 0.0457mm Titanium

0.2 0 −0.2

0.2 o

1.1 0.0254mm SST

0 −0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1 −1

−0.5

0

0.5

1

Scaled Strain [1]

o

2.2 0.0457mm Titanium

−1 −1

−0.5

0

0.5

1

Scaled Strain [1]

Fig. 6. The elastic Lissajous plots obtained using a strain-controlled rheometer (a) linear regime 25% strain and 1 rad/s, and (b) non-linear regime 1000% strain and 1 rad/s for different cone angles, truncations and material of construction (5 g/L EHUT).

example, Fig. 6 shows that the LAOS response characterized in a strain controlled rheometer is similar within the limits of experimental repeatability. A good agreement among the data is observed with different geometries, both in the linear and the non-linear regime. At higher concentrations of EHUT, the onset of secondary loops occurs at lower strains. The relaxation time varies from 8.68 to 44.6, De increases for plots from (b) to (e) in Fig. 5. In order to complete the perspective, fixed concentration of EHUT was probed at different frequencies as shown in Fig. 7. It is observed that at lower frequency ((a) in Fig. 7), the secondary loops appear at lower strain amplitudes. From plots in Fig. 5 and in Fig. 7, it may be surmised that there is equivalence of LAOS behaviour in concentration-frequency plane. The concentration dependence of LAOS behaviour would be due to the increase in the length of the tubular chains, and the consequent increase in the entanglements at higher concentrations. The inverse frequency dependence is due to the fact it competes with the relaxation processes in the material. When the frequency is small, lesser than the relaxation time, the material has enough time to relax within the period of shear leading to secondary loops. While in the case of large frequencies, the material is not given enough time to relax before the next loading of strain takes place. Due to this inverse relation of concentration and frequency, the usual comparison across materials at same De leads to different behaviour. Conversely, qualitatively similar response is observed even if De is different. For example, Figs. 7(a3) and 5(e3) are similar responses at De ¼ 0:46 and De ¼ 7:1, respectively. Another example is of Figs. 7c3) and 5(a3) with De ¼ 23 and De ¼ 2:1, respectively. The LAOS behaviour of EHUT gels was also quantified by acquiring torque signals from the rheometer, as described in the Experimental section. At higher strains, odd harmonics upto the 7th harmonic were observed. At the same time, even harmonics were not observed for any of the samples. As an example, the ratio of 3rd to 1st harmonic is shown in Fig. 8 for EHUT gels, CTAB solution and for a standard viscosity fluid (silicon oil). The ratio increases rapidly for strains above 100%. The intensity ratio for silicone oil (60cp , Newtonian fluid standard) remains small even at very large strains as expected. It is seen that no significant difference can be spotted between EHUT gels at different concentrations, or between EHUT gel and CTAB solution. This holds true for all the concentrations of EHUT, though results for only two concentrations are shown in the figure. It can be noted that in spite of expected

differences in the microstructures of EHUT gels at different concentrations, and CTAB, there is no distinguishable difference in the way the intensity ratio varies with strain %. For CTAB, it was observed that G0L < G0M for high strain values which indicates that there is an intra cycle strain stiffening in the material at large strains. Additionally, G0M was higher at higher strains, indicating cycle strain hardening. G0M < 0 was observed for EHUT gels at many strains and concentrations, indicating that the material is unloading the elastic stress much faster than it is accumulating it. It can be observed from Table 3 that when G0M < 0 the secondary loops appear. At higher concentrations and larger strain, G0M is not only observed to negative, but is also close to zero. This implies a sustained plastic yielding with stress remaining relatively constant with changing strain. Therefore, secondary loops and plastic yielding are closely related to each other for these materials. EHUT gels, for most compositions and strains, show intracycle strain softening except for EHUT 7 and 9 g/L at 25% which shows intra cycle strain hardening. This might be due to the fact that, at higher concentration, larger number of entanglements is present. Therefore, at low strains these entanglements show a resistance to deformation. But as the strain increases it is seen that the difference between G0L and G0M changes from negative values to positive values which indicate that the entanglements have broken away and the material softens on applying more strain. For low concentration, even at low strains, the entanglements break away. It should be noted that secondary loops are only seen for higher concentrations of EHUT which indicates that both persistence length of the polymer and the number of entanglements might be contributing to this. The persistence length of EHUT was found to be greater than 100nm while for CTAB was found to be around 10 nm [6]. Since the persistence length of CTAB is an order of magnitude smaller than EHUT, the contribution of the relaxation is much more due to reptation rather than breaking/reformation in CTAB than in EHUT. Observation of the deformations within a period of cycle, similar to the sequence of physical process analysis [27], shows significant differences for CTAB and for EHUT gels. For example, in Fig. 9(a) EHUT 5 g/L at 1000% strain there exist sequential regimes for elastic loading, elastic unloading and deformation under almost constant stress (or plastic) deformation. When the strain increases from the point of strain reversal (during the period of elastic loading), stress would tend to accumulate. This is likely to be due to

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A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56 1

(a1)

(a2)

1 0.8

25%

250%

0.4

0.4

0.4

0.2 0 −0.2

Scaled Stress [1]

0.6

0.2 0 −0.2

0 −0.2

−0.4

−0.4

−0.6

−0.6

−0.6

−0.8

−0.8

−0.8

−0.5

0

0.5

−1 −1

1

−0.5

0.8

0.5

−1 −1

1

1

(b2)

0.8

25%

0.8

0.4

−0.2

Scaled Stress [1]

0.6

0.4

Scaled Stress [1]

0.6

0

0.2 0 −0.2

0

−0.4 −0.6

−0.6

−0.8

−0.8

−0.8

0

0.5

1

−0.5

Scaled Strain Rate [1]

(c1)

0.5

−0.4

−1 −1

1

(c2)

25%

1

0.8

(c3) 250%

0.8

0.4

0.4

0

Scaled Stress [1]

0.4

Scaled Stress [1]

0.6

−0.2

0.2 0 −0.2

0

−0.4 −0.6

−0.6

−0.8

−0.8

−0.8

0

0.5

Scaled Strain Rate [1]

1

−0.5

0

0.5

Scaled Strain Rate [1]

1

1000%

−0.2

−0.6

−0.5

1

0.2

−0.4

−1 −1

0.5

1

0.6

0.2

0

Scaled Strain Rate [1]

0.6

−1 −1

−0.5

Scaled Strain Rate [1]

1

0.8

0

1000%

−0.2

−0.6

−0.5

1

0.2

−0.4

−1 −1

0.5

1

(b3) 250%

0.4 0.2

0

Scaled Strain Rate [1]

0.6

−1 −1

−0.5

Scaled Strain Rate [1]

1

(b1)

0

1000%

0.2

−0.4

Scaled Strain Rate [1]

Scaled Stress [1]

0.8

0.6

−1 −1

Scaled Stress [1]

1

(a3)

0.6

Scaled Stress [1]

Scaled Stress [1]

0.8

−0.4

−1 −1

−0.5

0

0.5

1

Scaled Strain Rate [1]

Fig. 7. The stress versus shear rate (viscous Lissajous) plots for (a) EHUT at 5 g/L at 0.1 rad/s, (b) EHUT at 5 g/L at 1 rad/s, (c) EHUT at 5 g/L at 5 rad/s, for 25%, 250% and 1000% strains respectively.

entanglements, and would lead to a maximum in stress. As the strain decreases, or in other words strain rate increases, there is a sudden dip in the stress, which would be due to the yielding/ breakage of entanglements. This is in addition to the elastic unloading with decrease in strain. As the strain decreases even further, it is observed that the stress continues to change gradually, indicating possible microstructural steady state for high shear rates. In case of CTAB, as seen in Fig. 9(b), elastic loading and unloading seem to dominate. This plastic deformation is seen for all concentrations of EHUT above 3 g/L, and is an indicator of the secondary loops shown in Fig. 5. These results further show that although in the linear regime EHUT and CTAB show similar behaviour, there are significant differences in their LAOS behaviour. A closer look at the secondary loops of EHUT at different concentrations leads to further confirmation of the steady state-like

behaviour at higher strain rates during an LAOS cycle. However, this steady state is observed only at higher EHUT concentrations. For 9 g/L EHUT gel, as shown in Fig. 10(c), a region of stress-strain rate proportionality is observed. This region is not observed for 3 g/ L EHUT gel, as shown in Fig. 10(a). The corresponding elastic Lissajous plots are also shown in the same figure. As discussed earlier, region of plastic deformation is much less predominant in case of 3 g/L EHUT, when compared to 9 g/L EHUT. This also corresponds to higher negative G0M for 3 g/L EHUT, and close to zero G0M for 9 g/L EHUT (as shown in Table 3). The steady state of stress within a period can be observed for EHUT greater than 5 g/L. We believe that this has not been observed in detail before, although secondary loops have been observed [31]. This indicates further that at high shear rates some kind of microstructural steady state is reached within an LAOS cycle. Shear thinning response during

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A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

Ratio of third to first harmonic (i3/i1) [1]

0.35

5g/L EHUT 9g/L EHUT 50mM CTAB 60cp Silicone oil

0.3

0.25

0.2

0.15

0.1

0.05

0 0 10

1

2

10

3

10

10

Strain [%] Fig. 8. The ratio of intensity of 3rd harmonic to that of 1st harmonic from electrical torque for different materials.

steady shear can be compared with stress versus strain rate during an LAOS cycle (for example, as done in [29]). The results from this work suggest that this is not possible as with a given concentration of EHUT, a rich variety of stress-strain rate response is obtained during LAOS cycles depending on strain and frequency. This point is discussed in the next section as well. Fig. 11(a) shows that at low frequency, there exists a plastic regime for large range of strain. As described before there is stress accumulation at strain reversal which leads to yielding/breaking of entanglements giving rise to the plastic regime. Fig. 11(b) shows that at higher frequency, a different approach to the plastic regime. This is very interesting as it may be due to frequency being large enough (higher strain rates) to prevent any accumulation of stress in the material. As the materials are living polymers, at low rates there appears to be a process leading to stress accumulation. Figs. 9 and 11 also suggest that there is a concentration frequency equivalence. Another important point can be noted by comparing CTAB at 1 rad/s and 1000% as shown in Fig. 9(b) to EHUT at 5 rad/s and 1000% in Fig. 11(b). Even if there is an absence of secondary loops in viscous Lissajous plots, finer differences are seen in the elastic Lissajous plots. EHUT shows a plastic regime for a large range of

(a)

1

(b) 1000%

0.8

1 0.8

0.6

0.6

0.4

0.4

Scaled Stress [1]

Scaled Stress [1]

strains within a period, while there is no plastic regime observed for CTAB. This indicates a transient steady state within a period for EHUT, but for CTAB the loading and unloading mechanisms dominate. This emphasizes the fact that although similar response is seen in the linear regime, diffrent microstructural contributions lead to observations of large differences in LAOS. The secondary loops are known to form due to constitutive nature of complex fluids, or due to shear banding during viscometric flows [31]. An investigation using opto-rheology techniques, at large deformations, will give further insights into which of these aspects are important for EHUT gels. In the next section, the constitutive nature of LAOS response of EHUT gels is explored using different models. Based on these observations, especially the Lissajous plots, a hypothesis for what happens to EHUT at LAOS is speculated. The material relaxes due to reptation, if the time scale of oscillatory shear is large. During LAOS the entanglements are stretched to a large extent that there happens to be development of strain on the mesh. Since the tubes are formed from monomers due to directional reversible hydrogen bonding the polymer tends to break at these entanglements. Thus the material gives away to form shorter tubes, which would align in the flow direction. As soon as the deformation is stopped, the material comes back to its original microstructure as existed before the LAOS. This is because of the hydrogen bonding forming reversible bonds. Therefore, the transient networks of EHUT gels tend to break and regenerate during the oscillatory cycle. Since a terminal steady state is observed during subsequent LAOS cycles, the microstructural evolution processes such as breaking, reformation, orientation occur over smaller timescales when compared to the time scale of oscillation. These types of microstructural evolution processes are also precursors to phase separation and shear banding in complex fluids. In addition, they are practically utilized in applications such as hydrodynamic drag reduction using micellar solutions as well as EHUT gels [10,12]. The modelling of LAOS behaviour, assuming viscometric flows, is a preliminary investigation into the constitutive behaviour of EHUT gels, and the understanding obtained needs to be corroborated using further investigations. The experimental observations of LAOS behaviour of EHUT gels are compared with model predictions/fits with two types of models; Giesekus model and TNM model with structural change. Giesekus model, a non-linear rheological model, is expected to capture basic constitutive processes occurring in systems such as micellar solutions and EHUT gels. Breaking/reformation type thixotropic

0.2 0 −0.2

0.2 0 −0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1 −1

−0.5

0

Scaled Strain [1]

0.5

1

1000%

−1 −1

−0.5

0

0.5

Scaled Strain [1]

Fig. 9. (a) Elastic Lissajous plot for EHUT 5 g/L at 1000% strain (b)Elastic Lissajous plot for CTAB 50 mM at 1000% strain.

1

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A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

(a)

1

1000%

0.8

1000%

0.9

0.6

0.85

0.4

Scaled Stress [1]

Scaled Stress [1]

0.95

1

(b)

0.8 0.75 0.7

0.2 0 −0.2

0.65

−0.4

0.6

−0.6

0.55

−0.8

0.5 0.5

0.6

0.7

0.8

0.9

−1 −1

1

−0.5

(c)

1

1

1000%

0.8

0.9

0.6

1000%

0.85

0.4

Scaled Stress [1]

Scaled Stress [1]

0.5

1

(d)

0.95

0.8 0.75 0.7

0.2 0 −0.2

0.65

−0.4

0.6

−0.6

0.55

−0.8

0.5

0

Scaled Strain [1]

Scaled Strain Rate [1]

0.5

0.6

0.7

0.8

0.9

−1 −1

1

−0.5

Scaled Strain Rate [1]

0

0.5

1

Scaled Strain [1]

Fig. 10. The magnified view of the secondary loops in viscous Lissajous plots for EHUT at different concentrations (a) 3 g/L and (c) 9 g/L (overall plots shown in Fig. 5). Scaled elastic Lissajous plots for EHUT (b) 3 g/L and (d)9 g/L.

(a)

1 0.8

1

0.6

0.6

0.4

0.4

0.2 0 −0.2

0.2 0 −0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1 −1

−0.5

0

Scaled Strain [1]

0.5

1000%

0.8

Scaled Stress [1]

Scaled Stress [1]

(b)

1000%

1

−1 −1

−0.5

0

0.5

1

Scaled Strain [1]

Fig. 11. (a) Elastic Lissajous plot for EHUT 5 g/L at 1000% strain and 0.1 rad/s, and (b)Elastic Lissajous plot for EHUT 5 g/L at 1000% strain and 5 rad/s.

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A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

Table 2 Quantified model fit parameters used for Giesekus model, TNM model and Modified Giesekus model. Material

Strain (%)

a

f1

f2

25 250 1000

0.7 0.7 0.7

1 1 1

9 9 9

25 250 1000

0.33 0.33 0.33

– – –

– – –

EHUT 5 g/L CTAB 50 mM

Table 3 Quantification of the Lissajous plots of Fig. 5 using G0L and G0M . G0L and G0M are normalized as they are obtained from scaled elastic and viscous Lissajous plots respectively. Strain (%)

G0L (1)

25 250 1000

0.87 0.52 0.34

0.65 0.16 -0.43

5 g/L

25 250 1000

0.94 0.45 0.032

0.91 0.32 -0.14

7 g/L

25 250 1000

0.95 0.42 0.40

1.04 -0.22 -0.02

9 g/L

25 250 1000

0.97 0.43 0.35

1.22 0.05 0.04

25 250 1000

0.83 0.68 0.18

0.81 0.85 0.99

Material EHUT 3 g/L

CTAB 50 mM

G0M (1)

processes are specifically assumed to be part of TNM model with structural evolution. Based on the analysis with these two preliminary models and their simplistic modifications, we demonstrate additional modelling features required for capturing the LAOS behaviour of EHUT gels.

1

The model parameters for the Giesekus model were obtained experimentally from SAOS, as summarized in Tables 1 and 2. The non-linear parameter, a, was obtained by fitting the steady shear viscosity. The effect of a, called mobility factor, has been one of the reasons for the wide spread use of this model especially in the case of LAOS. This parameter defines the non-linearity of the system, which is physically explained as the drag force experienced by the polymer segments because of the solvent. Here a is fitted for each concentration of EHUT and 50 mM CTAB by least square regression analysis. It was found that the alpha is 0.7 and 0.33 respectively. Moreover, the variation of a in Giesekus model did not bring very significant qualitative difference in the model response during LAOS, as is shown in Fig. 12. Fig. 13(a) shows the stress responses predicted by the Giesekus model for EHUT 5 g/L and CTAB solution. It can be easily seen that qualitatively it is difficult to pin point the difference in the stresses at 25%, 250% and 1000% strains but quantitatively the stress is seen to be increasing with increasing strain. It can also be noticed from Fig. 13(b) that the stress evolution for CTAB appears to be similar to that of EHUT, 5 g/L. Furthermore, deviation of the stress sinusoid is slightly more in case of CTAB, than in EHUT. Given the qualitative trends observed in Fig. 5 for different materials, Giesekus model results are compared with the experimental observations in terms of Lissajous plots. Fig. 14 shows the LAOS response of 5 g/L EHUT gel at two different strains, 25% and 250%. Both the elastic (stress versus strain) and viscous (stress versus strain rate) Lissajous plots are shown. Given that the linear viscoelastic parameters have been obtained from SAOS and a is obtained from steady shear, no fitting parameters are used for these results of Giesekus model. The linear viscoelastic limits were greater than 25%, based on the moduli data presented in Fig. 4. Similarly, the appearance of third harmonic was at strains greater than 25% as shown in Fig. 8. Therefore, excellent fit of Giesekus model at 25% strain is expected. The fit of the steady shear data to obtain a is also shown in the inset of Fig. 14(a). The predictions using Giesekus model were observed to be valid at low strains for all concentrations of EHUT. The prediction of the Giesekus model is qualitatively different compared to experimental observations for 250% strain. Based on results of Figs. 4 and 8, this strain amplitude is higher than the linear limit.

(b)

1

0.8

0.8

0.6

0.6

0.4

0.4

Scaled Stress [1]

Scaled Stress [1]

(a)

3.3. Comparison with models

α=0.1

0.2

α=0.5

0

α=0.9

−0.2

−0.2

−0.6

−0.6

−0.8

−0.8

0

Scaled Strain [1]

0.5

1

α=0.9

0

−0.4

−0.5

α=0.5

0.2

−0.4

−1 −1

α=0.1

−1 −1

−0.5

0

0.5

1

Scaled Strain Rate[1]

Fig. 12. (a) The elastic Lissajous plot of 5 g/L EHUT for 0.1, 0.5 and 0.9 values of mobility factor(a), and (b) the viscous Lissajous plot of 5 g/L EHUT for 0.1, 0.5 and 0.9 values of mobility factor(a) for 1000% strain.

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A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

(a)

1

(b)

25%

0.8

1 25%

0.8

250%

250%

1000%

0.6

0.6 0.4

Stress [Pa]

Stress [Pa]

0.4

1000%

0.2 0

0.2 0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1

−1 1

2

3

4

5

6

7

8

9

1

Time [arb units]

2

3

4

5

6

7

8

9

Time [arb units]

Fig. 13. Stress response of the (a) EHUT at 5 g/L and (b) CTAB 50 mM for 25%, 250% and 1000% strain versus time.

This would be due to EHUT undergoing complex mechanisms of relaxation in addition to dis-entanglement and scission. The Giesekus model predictions for LAOS response of CTAB are shown in Fig. 15. Unlike EHUT gel, Giesekus model predicts the response very well. Giesekus model has been used earlier for CTAB solutions (with zero shear viscosity 10 Pas and relaxation times 0.01 s), and has been shown to describe SAOS as well as LAOS response for nonbanding and shear banding systems [26,53,54]. It can be observed from Table 1 that both the 5 g/L EHUT and CTAB solution have similar viscosity, but different relaxation time. The LAOS behaviour, according to the Giesekus model, does not change significantly with change in zero shear viscosity. However, model response is significantly affected by changes in the relaxation time. It is interesting to note that the material response of EHUT 3 g/L shows more non-linearity for a fixed strain than 9 g/L according to the Giesekus model. This is contrary to the response observed shown in Fig. 5. Moreover, the secondary loops for EHUT are observed at extremely high strains (3000%) based on Giesekus model predictions, irrespective of the concentration. On the other hand, experimentally, the EHUT gels exhibit secondary loops when the strain is more than 200%. EHUT are tube like structures as shown in Fig. 1, and during LAOS they are expected to evolve dynamically. Therefore, dependence of tube orientations, and conformation dependent drag based models may be better choices for capturing LAOS of EHUT gel. Given the hypotheses about structural changes during an LAOS cycle in complex fluids such as EHUT gels and micellar solutions, a structural evolution equation combined with TNM was used for comparison with experimental data. The rate constants of formation and breakage are found out through fitting by least square regression at 1000% strain was 1 and 9 for 5 g/L EHUT respectively. It should be kept in mind that the rate constant of breakage was greater than the rate constant of reformation for large strains. It is observed from the TNM model that as the strain increases the Lissajous curves change significantly. It can be observed from Fig. 16(a) and (b) that the model response is similar to the experimental data for 25% strain. It can be seen that at 250% strain the material shows very significant changes in stress for small changes in the strain or shear rate than for 25% strain. At 250% strain and beyond there is no match between experimental and model data as seen from Fig. 16(c) and (d). Thus this model is not sufficient to predict how the EHUT gels behave during LAOS. The two models, compared with the experimental data, do not predict the LAOS response observed for EHUT gels under the

specifed experimental conditions. The Giesekus model seems to fail because the model underpredicts the significant changes in microstructure which occur at the large strain amplitudes. The TNM, on the other hand, does not capture the essential elements of structural changes and therefore, fails qualitatively as well quantitatively at large strains. In the Giesekus model we found out that the Lissajous plots, the response (both elastic and viscous) of the model was a strong function of relaxation time. This motivated us to modify the relaxation time according to the strain applied. Modification of the Giesekus model such that the relaxation time is a function of the strain, relaxation time (relaxation time an exponential function of strain [55]) has already been reported . In the present work, we used the relaxation time to be a linear and decreasing function of strain. Additionally, the network model was combined with the Giesekus model. Another important point to note is that the structure factor in this case was allowed to vary between 1 and a non-zero lower limit. In a physical sense this implies that we allow less than complete breakage, by not allowing the structural parameter to reduce to 0. As earlier, the rate constants of breakage and formation were obtained by fitting the data at low strains with TNM. Therefore, the modified model equations are as follows:

!

ak1 ðc; c_ Þ k1 sp þ 1 þ sp sp ¼ 2gp D gp r

ð8Þ

where k1 evolves as function of strain amplitude, and instantaneous strain rate. As mentioned earlier, the relaxation time depends on strain amplitude as,

  k1 ¼ ko1  ðkÞ co  cy

ð9Þ

where ko1 is the relaxation time in the linear viscoelastic regime obtained from the dynamic frequency sweep, co is the maximum applied strain, cy is the strain at which non-linearily appears or the yield strain, k1 , is the modified relaxation time at LAOS and k is the slope that indicates the magnitude of decrease of relaxation time. The higher the non-linear strain, the lower is the value of the relaxation time k1 . cy is taken as 100% from Figs. 4 and 8. Thus the relaxation time is a function of strains at LAOS according to Eq. (9). In the following discussion, results are shown with different k to show the effectiveness of this modification. The limitations of Eq. (9) are that it is a one dimensional equation, and can be used only for oscillatory flows with known

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A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56 1

(b)

1

0.8

0.8

0.6

0.6

0.4

0.4

Scaled Stress[1]

Scaled Stress[1]

(a)

0.2

Giesekus model

0

EHUT 5g/L −0.2

EHUT 5g/L 0 −0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1 −1

−0.5

0

0.5

Giesekus model

0.2

−1 −1

1

−0.5

Scaled Strain Rate[1]

(c)

1 0.8

0

0.5

1

0.5

1

Scaled Strain Rate[1]

(d)

Giesekus model

1 0.8

0.6

0.6

0.4

0.4

Scaled Stress[1]

Scaled Stress[1]

EHUT 5g/L

0.2 0 −0.2

0.2

Giesekus model 0 −0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1 −1

−0.5

0

0.5

EHUT 5g/L

−1 −1

1

−0.5

Scaled Strain[1]

(e)

10

0

Scaled Strain Rate[1] 0

exp data for 5g/L EHUT

Scaled Viscosity [1]

fit data with alpha = 0.7

10

−1

10

−2

10

−3

10

−1

10

0

10

1

10

2

Shear rate [1/sec] Fig. 14. LAOS response of 5 g/L EHUT; elastic Lissajous plots at (a) 25% , (c) 250% and viscous Lissajous plots of the experiment and the model at (b) 25%, (d) 250%. The steady shear fit for EHUT 5 g/L gel is shown in (e), a = 0.7.

amplitude of oscillation, co . Therefore, it is not valid for flows of arbitrary histories in the stated form. However, we believe that borrowing from the rigourous and 3-dimensional models incorporating ideas related to the effect of maximum applied strain in the past and yielding, further modelling effort can lead to improved description of LAOS response.

Since EHUT gels are living polymers, the networks break and reform continuously. Within a single period of oscillation the number of networks points are minimum at maximum rate/minimum strain and maximum at minimum rate/maximum strain. This is the motivation to modify the relaxation time as a function of network points within a cycle of oscillation as shown in Eqs. (10) and (11)

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A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

(a)

1 0.8

(b)

Giesekus model

1 0.8

0.6

0.6

0.4

0.4

Scaled Stress [1]

Scaled Stress [1]

CTAB 50mM

0.2 0 −0.2

0.2 Giesekus model

0

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1 −1

−0.5

0

0.5

CTAB 50mM

−0.2

−1 −1

1

−0.5

Scaled Strain [1] 1

(d)

0.8

0.6

0.6

0.4

0.4 Giesekus model

0.2

CTAB 50mM

0 −0.2

1

−0.2

−0.6

−0.6

−0.8

−0.8

0

0.5

CTAB 50mM

0

−0.4

−0.5

1

0.5

1

Scaled Strain [1]

Giesekus model

0.2

−0.4

−1 −1

0.5

1

0.8

Scaled Stress [1]

Scaled Stress [1]

(c)

0

Scaled Strain Rate [1]

−1 −1

−0.5

0

Scaled Strain Rate [1]

Fig. 15. The comparison of elastic Lissajous plots of the experiment and Giesekus model at (a) 25%, (c) 250% and viscous Lissajous plots of the experiment and the model at (b) 25%, (d) 250% where the material parameters used are that of 50 mM CTAB where a ¼ 0:33.

kt ¼ k1 x1:4 d^x f1 ¼ ð1  ^xÞ  f2 ^x½IID ð1=2Þ dt kt x  xt ^x ¼ xi  xt

ð10Þ ð11Þ ð12Þ

where ^ x is the modified structure factor. xt is the terminal structure factor and xi is the initial structure factor, always taken as 1, assuming un-disturbed networked structure in the beginning. With a particular strain amplitude, the relaxation time evolves during an LAOS cycle based on Eqs. (6), (7) and (12). In Eqs. (6), (7), ^ x is used instead of x. In Eq. (12), different values of xt can be used to understand how the terminal structure factor modification affects the LAOS behaviour. In Fig. 17, it can been seen that by the modification of the relaxation time in the Giesekus model, there is a good fit for describing the Lissajous plot at very large strain amplitude as well. In this figure, xt ¼ 0:2, while results with three different values of k are shown to demonstrate the effect. As can be observed, for the slope of 0.4 for the relaxation time versus strain curve, good fit is obtained. The slopes of 0.1 and 0.5 for relaxation times show a much larger deviation from the experimental data. Similarly, we fixed the slope of the relaxation time versus strain and varied the terminal

structure factor to observe how the fit varies. It can be seen that in Fig. 18, the model shows very close behaviour to the actual material when xt ¼ 0:2. For these results, k is kept constant at 0.4. With xt of 0 and 0.9, significant departure from the actual material behaviour is observed. All this evidence points out to the fact that LAOS of EHUT leads to a point where the time-strain separability breaks down, and a model with this feature can capture the LAOS in a realistic manner. It is observed from Figs. 17 and 18 that relaxation time being a function of strain alone is not sufficient, and the relaxation time should also be a function of shear rate. Based on the closer observation of the loops given earlier, the first one corresponds to structural organization due to large deformations leading to different loading-unloading behaviour. On the other hand, the second one corresponds to steady state-like response during an LAOS cycle, and the plastic deformations. Therefore, the Giesekus model, which does not incorporate these features, does not capture the experimental material behaviour for EHUT. These results point towards drastic changes to the material microstructure during LAOS of EHUT gels. The modifications to the Giesekus model which lead to reasonable agreement with experimental observations suggest that steady shear response cannot be compared with stress-strain rate response during an LAOS cycle. The loading, unloading and

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A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

(a)

1 0.8 0.6

(b)

0.8

UCM model EHUT 5g/L

0.6 0.4

Scaled Stress [1]

0.4

Scaled Stress [1]

1

0.2 0 −0.2

0.2 UCM model

0

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1 −1

−0.5

0

0.5

EHUT 5g/L

−0.2

−1 −1

1

−0.5

Scaled Strain [1] 1

(d)

0.8

0.6

0.6

0.4

0.4

0.2 UCM model

0

EHUT 5g/L

−0.2

−0.6

−0.8

−0.8

0.5

1

EHUT 5g/L

−0.2

−0.6

0

0.5

UCM model

0

−0.4

−0.5

1

0.2

−0.4

−1 −1

0.5

1

0.8

Scaled Stress [1]

Scaled Stress [1]

(c)

0

Scaled Strain Rate [1]

−1 −1

1

−0.5

Scaled Strain [1]

0

Scaled Strain Rate [1]

Fig. 16. The comparison of elastic Lissajous plots of the experiment and TNM at (a) 25%, (c) 250% and viscous Lissajous plots of the experiment and the model at (b) 25%, (d) 250% where the material parameters used are that of 5 g/L EHUT.

(a)

1

(b)

0.8

0.8

0.6

0.6

0.4

Scaled Stress [1]

0.4

Scaled Stress [1]

1

Relax time strain slope 0.1

0.2

Relax time strain slope 0.4 Relax time strain slope 0.5

0

EHUT 5g/L

−0.2

0.2 0 −0.2 −0.4

−0.4

−0.6

−0.6

Relax time strain slope 0.1 Relax time strain slope 0.4

−0.8

Relax time strain slope 0.5

−0.8 −1 −1

−1

−0.5

0

Scaled Strain [1]

0.5

1

−1

EHUT 5g/L

−0.5

0

0.5

1

Scaled Strain Rate [1]

Fig. 17. The elastic (a) and viscous (b) Lissajous plots at different slopes when the Giesekus model relaxation time is made a linear function of strain at fixed terminal structure factor of 0.2. Experimental Lissajous plot at 1000% strain for 5 g/L EHUT is also shown here for comparison. Parameters used are for modelling are of 5 g/L EHUT.

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A.R. Jacob et al. / Journal of Non-Newtonian Fluid Mechanics 206 (2014) 40–56

(a)

1

(b)

0.8

0.8

0.6

0.6

0.4

Scaled Stress [1]

Scaled Stress [1]

0.4 0.2 Terminal structure factor 0

0

Terminal structure factor 0.2 Terminal structure factor 0.9

−0.2

EHUT 5g/L

0.2 0 −0.2 −0.4

−0.4

−0.6

−0.6

Terminal structure factor 0 Terminal structure factor 0.2

−0.8

Terminal structure factor 0.9

−0.8 −1 −1

1

−1

−0.5

0

0.5

1

−1

EHUT 5g/L

−0.5

Scaled Strain [1]

0

0.5

1

Scaled Strain Rate [1]

Fig. 18. The elastic (a) and viscous (b) Lissajous plots at different terminal structure factors at fixed slope of relaxation time versus strain of 0.4. Experimental Lissajous plot at 1000% strain for 5 g/L EHUT is also shown here for comparison. Parameters used are for modelling are of 5 g/L EHUT.

plastic processes during a cycle are different depending on strain and frequency, and therefore cannot be mapped on the stready shear response which only depends on strain rate. Fig. 18 on closer observation gives insight on how the sequence of physical processes is being captured at the 1000% strain soon after yielding. When the terminal structure factor goes to zero, a pure plastic deformation regime is observed just after elastic unloading as seen in Fig. 10 9 g/L of EHUT gel. However, at lower concentrations of 3 g/L and 5 g/L, similar behaviour is observed when the terminal structure factor is 0.2. This gives an indication that at lower concentrations not all the network points break as the tube length of the EHUT gels are small favouring more reptation. It is interesting to note that at high shear rates within a LAOS cycle the network breakage and reformation are more dominant than reptation and this leads to secondary loops. Observing Fig. 5 as the concentration/De increases for 1000% strain, it can be seen that the point of origin of the secondary loop occurs at lower shear rates. This indicates that for higher concentrations and for larger number of network points the breakage and reformation starts appearing at lower shear rates. The modified Giesekus model as seen in Fig. 18(b) at terminal structure factor zero is able to capture the shoulder seen to develop at 1000% for increasing concentration of EHUT gels just before the secondary loops start. Hence CTAB whose persistence length are much smaller than EHUT gels shows secondary loops only after 1000% strain while in EHUT gels of higher concentartions form secondary loops start 100% strain. Although a phenomenological approach to modelling of this system is suggested in the current work, we establish a direction for rigorous modelling efforts for supramolecular systems. Similar to the model proposed for self associating telechelic systems [56], supramolecular polymers will also need to be modeled for a better understanding of these interesting systems.

non-linearity as well as non-linear features such as secondary loops in Lissajous plots, were observed at lower strains at higher concentrations of EHUT which is due to the increase in network points breakage and reformation. Though similar in SAOS response, EHUT gels at different concentrations and the micellar solutions different behaviour in LAOS due to different microstructural contributions. Similar to the Pipkin space of strain amplitude and frequency, qualitative variation of EHUT gels suggested similar behaviour at different strain amplitude and EHUT concentration was observed. All the experimental results were compared with Giesekus and upper convected Maxwell model coupled with transient network model [TNM]. With parameters estimated from steady shear and SAOS the Giesekus and TNM models showed large deviations in LAOS response for EHUT gels, but showed a good prediction for CTAB. By making the relaxation time a function of strain and structural parameter in the modified Giesekus model, it was shown that good agreement between model response and experimental observations could be obtained. The model response demonstrates that network breakage and reformation as well as reptation processes play an important role in the formation of secondary loops. The modified Giesekus model is able to capture the network breakage and reformation and therefore is in agreement with the experimental LAOS behaviour of these specific supra molecular material systems. Acknowledgements The authors would like to thank Mutharasu C. for the initial help with the modelling. We thank Dimitris Vlassopoulos and Jan Mewis for their insightful suggestions and stimulating discussions. We acknowledge George Petekidis for the permission to use the rheometers in IESL, FORTH to conduct experiments. The authors also would like to acknowledge the Department of Science and Technology, India for the funding of this project.

4. Conclusions References Large amplitude oscillatory shear [LAOS] behaviour of a supramolecular polymer 2,4 bis(2-ethyl hexyl ureido) toluene [EHUT] gels at varying concentrations and cetyl trimethyl ammonium bromide [CTAB] wormlike micellar solution at a specific concentration was examined. It was observed that the Lissajous plots could be used to discern the qualitative response of different materials, which show similar response in the linear regime. The onset of

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