J. Cent. South Univ. (2016) 23: 1873-1882

DOI: 10.1007/s11771-016-3242-6

A revisit of strain-rate frequency superposition of dense colloidal suspensions under oscillatory shears

LI Jun-jie(Àî¿¡½Ü)^{1, 2}, CHENG Xuan(³Ìè¯)^{2, 3}, ZHANG Ying(ÕÅÓ±)^{2, 3}, SUN Wei-xiang(ËïξÏè)⁴

1. Department of Mathematics and Applied Mathematic, Xiamen University, Xiamen 361005, China;

2. Fujian Key Laboratory of Advanced Materials, Xiamen University, Xiamen 361005, China;

3. Department of Materials Science and Engineering, College of Materials, Xiamen University, Xiamen 361005, China;

4. Research Institute of Materials Science and State Key Laboratory of Luminescent Materials and Devices,

South China University of Technology, Guangzhou 510640, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract: Strain-rate frequency superposition (SRFS) is often employed to probe the low-frequency behavior of soft solids under oscillatory shear in anticipated linear response. However, physical interpretation of an apparently well-overlapped master curve generated by SRFS has to combine with nonlinear analysis techniques such as Fourier transform rheology and stress decomposition method. The benefit of SRFS is discarded when some inconsistencies of the shifted master curves with the canonical linear response are observed. In this work, instead of evaluating the SRFS in full master curves, two criteria were proposed to decompose the original SRFS data and to delete the bad experimental data. Application to Carabopol suspensions indicates that good master curves could be constructed based upon the modified data and the high-frequency deviations often observed in original SRFS master curves are eliminated. The modified SRFS data also enable a better quantitative description and the evaluation of the apparent structural relaxation time by the two-mode fractional Maxwell model.

Key words: strain-rate frequency superposition; medium amplitude oscillatory shear; linear viscoelasticity; fractional Maxwell model

1 Introduction

Ultraslow structural relaxation process in many soft materials is difficult to be directly probed experimentally due to its extremely time consuming in measurements and insufficient resolution of transducer, which limits the frequency range of the canonical method, small amplitude oscillatory shear (SAOS) [1], for investigating the linear viscoelastic responses of the materials [2]. A crude approximation of structural relaxation time is to directly calculate the reciprocal of the lowest frequency where storage moduli equals loss moduli, i.e. G'(w)= G"(w) [3-4]. Given that the frequency range of SAOS tests for typical soft materials is not low enough to observe this crossover, and the first harmonics moduli beyond SAOS tests does not mean the same physics as storage/loss moduli in linear viscoelasticity, WYSS et al [5] proposed the strain-rate frequency superposition (SRFS) principle to extend the range of frequency in linear viscoelasticity by shifting the inaccessible slow dynamics in soft glasses toward higher frequencies with the use of an applied shear rate amplitude. The essence of SRFS can be found similarly from time¨Ctemperature equivalent principle or time¨Ctemperature superposition principle [6-7]. Beyond the applied tests of SAOS, the measurement response is nonlinear, usually refered to large amplitude oscillatory shear(LAOS)[1]. However, one should note that hitherto all SRFS studies only consider the measurements with not very large amplitude, actually we call this not strongly nonlinear viscoelasticity as medium amplitude oscillatory shear(MAOS) [8]. In SRFS, the fast shear-driven relaxation in nonlinear viscoelasticity at relatively large frequency is believed to be equivalent to the slow structural relaxation in linear viscoelasticity at very low frequencies, and the relaxation time is supposed to be a function of strain-rate amplitude, which can be defined as in an oscillatory shear test. Consequently, the first harmonics moduli obtained from the measurements of constant strain-rate frequency sweeps might be superimposed into the master curves. The physical interpretation of the master curves by WYSS et al [5] has been only elucidated in terms of G' and G" [9]. However,with the application of Fourier transform rheology, KALELKAR et al [9] stated that with the same shift factors as forandhigher harmonic moduli (n>1) in SRFS tests could also be scaled to good overlapped master curves, and nonlinear responses are shown in low-frequency part of the master curve. HESS and AKSEL [10] used stress decomposition methods, following the extended version of ideas of CHO et al [11] introduced by EWOLDT et al [12], to provide physical interpretations of SRFS data with respect to strain softening/stiffening, and also shear thinning/ thickening. Studies about the physical interpretation of higher harmonics are being conducted. ERWIN et al [14] indicated that LAOS response of yield stress fluids could not be well interpreted as physically meaningful with the use of stress decomposition method and the Fourier/ Chebyshev description, but could be well elucidated in a sequence of physical processes. Instead of decomposing strain as apparent elastic and apparent plastic stresses from CHO¡¯s stress decomposition methods [11], DIMITRIOU et al [13] used a similar strain decomposition based on solid mechanics to determine elastic and plastic components of strain, which has a clearer physical background than CHO¡¯s method and is versatile in stress-controlled LAOS experiments. Advances in LAOS response of soft materials drive the need for well elucidating SRFS master curves in both linear and nonlinear analysis [9-10]. ERWIN et al [14] focused on inconsistencies between SRFS master curve and SAOS data at the low-frequency region and indicated the invalidity of Kramers¨CKronig (KK) relation between and which implies that the use of linear response theories to elucidate SRFS master curves is questionable. Careful investigations about conditions at which the SRFS master curves are physically meaningful are urgently needed.

In this work, a decomposition criterion is proposed to define the experimental data range valid for constructing SRFS curves. The influence of the nonzero average shear strain in a cycle of shear strain at high frequencies is investigated and a data selecting criterion is proposed to eliminate those data with a large nonzero average shear strain. The SRFS master curves for a specific Carbopol suspension is then constructed based upon the decomposition criterion and the data selecting criterion. The application of two-mode fractional Maxwell model to quantitatively analyze the modified SRFS data is discussed.

2 Theoretical analysis

Under an externally applied oscillatory shear strain

                            (1)

the general stress response in a dense colloidal suspension in a steady state could be written as

            (2)

or

   (3)

whereand are Fourier coefficients and the phase angle is For stress response with a shear symmetry

                            (4)

Equations (2) and (3) degenerate to odd harmonic Fourier series [15-16]. A sufficiently small amplitude of the applied shear strain (¦Ã₀¡ú0) may lead to the domination of the first harmonic in Eq. (2) or Eq. (3), i.e., the magnitudes of and are much greater than those of and for all n>1. In this case, Fourier coefficientsandare normally considered as linear response moduli and and Eq. (2) may be rewritten as [17]

     (5)

with the coefficients in Eq. (2) and Eq. (5) related to each other by

          (6)

A similar relationship could be found in a multiple integral representation of Green¨CRivlin model [18]

              (7)

where memory kernels G₁, G₂, and G₃ are stress relaxation functions with fading memory behaviors, and ¦Ã is the time derivative of shear strain ¦Ã. One should bear in mind that the validity of Eqs. (6) and (7) indicating the region of measureable non-linear viscoelasticity is liminted in not very large strain amplitude, say ¦Ã₀<100%, which refers to MAOS [8]. Given the shear symmetry (4) and the applied shear strain (1), the second integral term in Eq. (7) would vanish, and the leading order nonlinearities in terms of ¦Ã₀ are the same as those in Eq. (6). It turns out that the stress relaxation functions and Fourier coefficients are inter-convertible [19].

The SRFS principle is [20]

 (8)

Equation (8) has been proposed to estimate the linearly viscoelastic behaviors in maximum range of frequencies with is the vertical scaling factor, and the horizontal shift factor. The significance of SRFS principle is to predict rheological behaviors of soft materials in the ranges of either low frequencies and/or high shear strain, in which the rheological measurement is limited because of the sensitivity of transducer and time-consuming tests. ERWIN et al [14] argued that the SRFS principle needs to be consistent with the KK relation for linear rheological responses. Since the rheological responses obtained from the oscillatory shear sweeps at low frequencies and high strain may be beyond the range of linear stress responses, it is interesting to find conditions under which the SRSF principle may hold away from the linear viscoelastic range is necessary. By substituting first two equations in Eqs.(6) and (8) and keeping the leading order ofone obtains

       (9)

Applying the KK relation in linear viscoelasticity [21], it is obtained

           (10)

                (11)

To the first harmonic coefficients and of Eq. (2) with the assumption:

  (12)

The modified SRFS principle becomes

           (13)

where K is the scaling factor between G¡ä(w) and Clearly, the SRFS principle may hold in certain regions outside of the classical linearly viscoelastic domain if the material has a response like Eq.(12). Although the assumption (12) is very strong in the sense that the effects of the strain magnitude and the oscillation frequency onare separable and the strain magnitude has the same effect on

The general form of shear stress in linear viscoelasticity with a continuous relaxation modulus can be represented as a Boltzmann causal integral equation [3, 22]

                   (14)

The principle of causality indicates that ¦Ò(t) at current time t depends on ¡¯s history [22]. The relaxation modulus G(t) is a completely monotonic function if the follow equation holds for every positive integer n [23]

                       (15)

Since the set of completely monotonic functions is closed under positive linear combinations and products, completely monotonic functions could be equivalently represented as [23]

                     (16)

for some positive measure on [0,¡Þ) according to Bernstein's theorem, the relaxation modulus G(t) may be expressed in a spectrum form [24-25]

                 (17)

or its discrete version [24-25]

                     (18)

where ¦Ó denotes the continuous relaxation time, H(¦Ó) denotes the non-negative relaxation spectrum, ¦Ó_n and g_n represent respectively the nth mode relaxation time and nth mode relaxation modulus/length. Equation(18) with a finite summation is widely known as the multi-mode Maxwell models when the terms in Eq. (18) are more than one. By introducing fractional Maxwell model (FMM) [26], it is obtained as

              (19)

where ¦Ó is a single relaxation time and g is elastic modulus, it is not difficult to show that when the applied shear strain ¦Ã(t)=¦Ã₀He(t) with He(t) the Heaviside function, the response of relaxation modulus of Eq. (19) for the single-mode model obtained by applying the Laplace transform is [26]

                (20)

where E_a,b(z) is the generalized Mittag¨CLeffler function [26-27], with its definition as

          (21)

For a small amplitude oscillatory shear (SAOS) as given by Eq. (1) with small shear strain amplitude ¦Ã₀, the responses of the Eq. (19) for the single-mode model become [26]

 (22)

and

 (23)

The requirement for completely monotonic relaxation modulus in linear viscoelasticity restricts 0¡Ü¦Â<¦Á¡Ü1. The linearity of Eq. (19) implies a natural extension of Eqs. (22) and (23) to the N-mode fractional Maxwell mode with

             (24)

and

  (25)

3 Experimental

The commercial Carbopol 980 (Hui You Chemical Industry, Beijing, China) was first dried in vacuum at 50¡ãC overnight, dispersed in deionized water (18.2 M¦¸¡¤cm, Millipore MilliQ) to a mass fraction of 0.2%, and stirred for 2h. Triethylamine (Enox) was then added into the dispersion under stirring drop by drop until the pH value of the mixture reached 7.5. Afterwards, the dispersion was stirred for another hour. Finally, the dispersion was put in vacuum at room temperature for 5min to remove bubbles before being stored at 4¡ãC.

Rheological tests of the samples were performed using an ARES-RFS (TA Instruments) at 25¡ãC with a cone-and-plate accessory in a geometry of 0.04 rad in cone angle and 50 mm in diameter. To avoid evaporation of solvents, a thin layer of silicon oil was supplied to smear the sample¡¯s age after loading [28]. The dynamic time sweep test procedure was employed, and an oscillatory shear strain was applied for a period of ten cycles for each pair of strain amplitudes ¦Ã₀ and frequencies w. Both strain and torque signals of the test were acquired from the analogue outputs through an analog-to-digital converter (ADC, National Instruments USB-9215A) with the use of a designed LabView program kindly supplied by Prof. Manfred Wilhelm¡¯s group [29]. Oversampled signals were acquired at 10⁵ s^-1 and decimated to 200 points per cycle to lower noise. A home-designed MATLAB code was used to analyze the harmonics and Lissajous curve of the data. The maximum signal-to-noise ratio was 45dB. The sweep tests were performed in the range of the strain rate amplitude from 0.01 s^¨C1 to 1.0 s^¨C1. A series of sweep tests was carried out for each fixed by selecting the pair of the shear strain ¦Ã₀ and the oscillatory frequency w according to with the oscillatory frequency w varying from 0.1 to 100 rad/s.

4 Results and discussion

In Fig. 1(a), the two components of linear moduli, G¡ä(w) and are presented for Carbopol suspensions in a frequency sweep test at ¦Ã₀=0.7 %. The suspension behaves like a typical soft solid [5, 30] in the sense of weakly frequency dependent storage modulus G' with the magnitude being generally one order greater than that of the loss modulus G".

It is further observed that the loss modulus G" of the suspensions varies more significantly with frequency and a valley-like curve with a shallow minimum around several rad/s. In Fig. 1(b), G' and G" of the suspensions are presented in the dynamic strain sweep test at a fixed oscillatory frequency w=6.28 rad/s. A similar behaviorcould be observed in the dynamic strain sweep test for suspensions in the range of strains smaller than the critical strain [31]. Beyond the critical strain, the loss modulus G" increases dramatically first as the magnitude of shear strain increases, and decays following a power law (v¡å=0.301) after reaching its peak value. However, the storage modulus follows the power law (v¡ä=0.601) when the magnitude of the applied shear strain is greater than the critical strain. The suspension demonstrates a typical scaling relationship v'/v"¡Ö2 reported in many experiments [32] for soft solids, which reminisces the standard terminal power law of linear viscoelastic behavior [25]. Also, the occurrence of the peak in G¡å(¦Ã₀) is normally regarded to be an indication of breakage of cage [2] formed in the quiescent concentrated suspensions. It is normally supposed that at very low frequencies, dense suspensions behave like a solid, and as frequency increases the suspensions behave increasingly like a liquid. The critical point of the dense suspension from a solid material behavior to a liquid material behavior would be signaled by the occurrence of an evident peak in the curve of G" against frequency, which indicates the structural relaxation in the suspension [5].

Fig. 1 Linear viscoelastic moduli with strain amplitude of 0.7% (a) and dynamic strain sweep test at frequency of 6.28 rad/s (b)

The plot of and against frequency for each are shown in Fig. 2. To construct their corresponding master curves, the least square method is employed and theandcurves for the smallest strain rate amplitude s^-1 are taken as referential curves. In order to include enough experimental data, a criterion based upon the intensity ratio of Eq. (3) is proposed to select the experimental data for constructing master curves.

I_n/I₁<¦Á, for all odd n, 1

The employment of the criterion (26) leads to a reconstruction of experimental stress data so that the data included are

     (27)

The master curves for the dense Carbopol suspension with ¦Á=0.03 and N=9 are presented in Fig. 3(a) with a unit vertical shift factorand horizontal shift factors presented in Fig. 3(b) as square dots for different (each color represents data from a single strain rate amplitude sweep). From Fig. 3(a), it is clear that good single master curves could be constructed from different curves of the storage and the loss moduli obtained in various strain rate amplitude sweep tests except at the high frequency region for the loss modulus. Also, a nice power law dependence of horizontal shift factor is given:

                     (28)

which may be obtained by fitting the calculated values of with 42.41 and v=0.5759.

Fig. 2 First harmonics against frequency in constant strain rate amplitude sweep tests:

Fig. 3 Master curves of SRFS data with for Carbopol suspension (Inset: modified linear SRFS data) (a) and horizontal shift factor (b)

Criterion (26) is a little more loose than that normally used to define linear viscoelastic region (I₃/I₁<0.5%) in FT-rheology [33]. However, it is safe to use an extended region of data though it might involve nonlinearity. For convenience, the data set defined by Eqs. (26) and (27) is called the linear SRFS data. In another attempt to isolate the linear viscoelastic response, WYSS et al [5] argued that the shear deformation at high frequency is independent of the microstructures of materials, and proposed that the master curve could be established by

             (29)

where denotes the complex modulus associated with the structural relaxation process (a) linear viscoelastic behavior), represents a high-frequency correction to the complex modulus with c being a proportionality factor determined from the high-frequency data of The existence of non- elliptical Lissajous¨CBowditch curves (¦Ò(t) versus ¦Ã(t)) in Pipkin space shown in Fig. 4 indicates that the suspension behaves non-linearly viscoelastically in various ranges of frequencies depending on the magnitude of Lissajous curves are plotted on the (¦Ã₀, ¦Ø) space rather than the (¦Ã₀, ¦Ø) space, which is commonly done in LAOS studies [34-35]. To understand the deviations in the master curve constructed by Eq. (8) from the linear response, the information about the input of shear strain ¦Ã(t) and the corresponding output of stress response ¦Ò(t) is investigated in detail. It turns out that for some small magnitudes of applied shear rate at high frequencies, the experimental data for shear strain exhibit a sinusoid form:

                      (30)

with ¦Ä¡Ù0, a nonzero average shear strain in a cycle. Figure 5(a) plots the experiment data of shear strain (t) against time for shear strain amplitude ¦Ã₀=0.01 and w=10, and Fig. 5(b) is the corresponding stress responses. Figure 5(c) plots the stress responses ¦Ò(t; w, ¦Ã₀) against the modified shear strainwhich is merely a shift of shear strain by a nonzero value ¦Ä, and in Fig. 5(d), the stress response ¦Ò(t;w,¦Ã₀) is plotted against experimental input ¦Ã(t), i.e., the plot of the vertical value of Fig. 5(b) against that of Fig. 5(a) at the same time. The influence of nonzero ¦Ä is significant in the sense that the reconstructed stress response ¦Ò(t;w,¦Ã₀) based on the modified shear strain signal well fits the raw stress data from experiments.

Fig. 4 Pipkin diagram at different pairs of

The existence of nonzero strain shift in the sinusoid strain functions may be caused by the rheological testing system and the samples tested together. The reasons why and how this shift occurs are not our concerns and discussed in this work. However, the inevitability and unpredictability of strain shifts in the test data question the reliability of the response data. Since the strain shift may profoundly affect the construction of the master curves, considering the inevitability in, especially, the range of high frequencies, a modified data set {¦Ò_m(t), ¦Ã_m(t)} is chosen so that

      (31)

{¦Ò_m(t)} is the corresponding stress response of {¦Ã_m(t)} in {¦Ò_¦Á(t)}. Here in Eq. (31), e is a small and positive number, which is introduced to account for possible measuring errors. Clearly, the modified data set {¦Ò_m(t), ¦Ã_m(t)} is obtained by deleting those ¡°bad¡± points with |¦Ä|/¦Ã₀>¦Å from the linear SRFS data {¦Ò_¦Á(t), ¦Ã(t)}. In the inset of Fig. 3(a), the modified linear SRFS master curves are presented from the modified experimental data {¦Ò_m(t), ¦Ã_m(t)} with ¦Å=5%. The modified linear SRFS master curves overlap very well without high frequency correction.

Fig. 5 Experiment data of loaded shear strain ¦Ã(t;w,¦Ã₀) (a), corresponding stress response curve (b), ¦Ò(t;w,¦Ã₀)~¦Ã(t) curve (c), and ¦Ò(t;w,¦Ã₀)~curve (d)

It is noticed from Fig. 4 that non-elliptic Lissajous-Bowditch curves, which indicate a nonlinear viscoelastic behavior, exist for the strain rate amplitude as low as A careful observation of the Pipkin diagram reveals that at medium shear strain amplitudesthe suspension behaves like a Bingham fluid. Consequently, the relaxation time ¦Ó_ref associated with in Eq. (28) can not be regarded as the structural relaxation time of the suspension, in general. Since the relaxation time may be related to the horizontal shift factor by [5]

                              (32)

the structural relaxation time ¦Ó₀ of the Carbopol suspension may be found by defining which requires an extrapolation of the scaling factor to the zero shear strain rate according to Eq. (28). For the Carbopol suspension, the extrapolation of the fitted Eq. (28) results in b(0)=-2.27 and leads to a physically impossible negative structural relaxation time ¦Ó₀ by Eq. (32). WYSS et al [5] suggested that in the range of sufficiently low strain, the power law index v in Eq. (28) would asymptotically approach unity, and it might be reasonable to propose

       (33)

In this case, the scaling factor with its limiting value b(0)=0.5759. As an example, the SAOS data for the Carbopol suspension as shown in Fig. 1 are shifted with the scaling factor b(0)=0.5759, and compared with its SRFS master curve in Fig. 6 (Up and down triangle dots denote the SAOS data, the square and diamond dots denote data of the master curve) . Clearly from Fig. 6, the SAOS data could not be clasped well into the master curve by the scaling factor defined by b(0)=0.5759. Of course, one may argue that for those cases with a nonzero and small shear rate, the scaling factor may differ from the limiting value of b(0)=0.5759. However, for any nonzero shear rates, the corresponding scaling factoris greater than its limiting value 0.5759, the SAOS data are further shifted down to the low frequency range and can not match well with the master curve neither. Based upon the studies on thixotropic yield stress fluids [36-37], it has been argued that the construction of master curve by Eq. (8) might not be suitable for SAOS data since when the frequency and magnitude of shear strain are below some critical values, shear bandings and/or a heterogeneous and non-cyclic stress field appear. Since Carbopol suspension/gel has no evident thixotropy [2], it is questionable whether there exists a critical frequency below which the stress response could not be reconstructed well even in linear viscoelastic region crudely determined by dynamic strain sweep tests. However, the inconsistence in the rescaled SAOS data with the master curve might be due to the improper extrapolation of scaling curve Eqs. (28) to (33).

Fig. 6 Comparison between scaled SAOS data and SRFS master curve

The lack of low frequencies data make the determination of relaxation spectrum either in discrete multi-mode Maxwell model Eq. (18) or continuous spectrum representation Eq. (17) ill-posed. It has been pointed out that the relaxation spectrum {¦Ó_n, g_n} in the range ofis unambiguous because of the oversampling principle [38-39] with w_min and w_max being the lowest and highest frequencies employed in the experiments. However, the relaxation spectrum outside of this reliable region is somewhat arbitrary. In order to overcome the difficulties in missing data in low frequency range, the two-mode fractional Maxwell model (FMM) (Eq. (24) and Eq. (25)) is employed to fit the SRFS data. The red solid and hollow dots in Fig. 7(a) are the modified linear SRSF data {¦Ò_m(t), ¦Ã_m(t)}, the pink solid and dashed lines in Fig. 7(a) are the corresponding fitted curves by two- mode FMM, respectively, with the fitted parameters listed in Table 1. As a comparison, the fitted curves (the black solid and dashed lines) by the two-mode FMM for the linear SRFS data {¦Ò¦Á(t), ¦Ã(t)} are also included in Fig. 7(a). Clearly, two-mode FMM generates a good fit to the modified linear SRFS data, and correctly predict the crossover between G'(w) and G"(w), which is about the same as that obtained from experimental data. Though the fit by the two-mode FMM for the linear SRFS data overlaps with the fit of the modified linear SRSF data in the range of high frequencies, it does not predict the crossover. Moreover, the fit by the two-mode FMM for the modified linear SRFS data predicts a much better terminal power law than that for the linear SRFS data. The apparent structural relaxation time predicted by the two-mode FMM is 4.05 s.

Fig. 7 Curves showing two-mode FMM fitted to linear SRFS data (a) and FMM fit and PM fit to modified linear SRFS data (b)

Table 1 Parameters estimation in two-mode FMM

It is interesting to compare the fits generated by the FMM and the parsimonious model (PM) [40] based on the modified linear SRFS data. The relaxation spectrums for the two-mode PM are {¦Ó₁=0.0243, g₁=1.121} and {¦Ó₂=3.648, g₂=239.2}. In Fig. 7(b), the linear portion of the fitted curves are plotted by FMM and PM as well as the linear SRFS data. In a crude estimation, the apparent structural relaxation time predicted by the two-mode PM is 3.648 s by neglecting the fast mode. Though PM is regarded as one of the most robust calculation of relaxation spectrum based upon a multi-mode Maxwell model, the two-mode FMM fits the data much better than those by the two-mode PM.

5 Conclusions

In this work, the principle of SRFS for predicting low frequency linear response is revisited in an attempt to extract meaningful material functions from well-overlapped SRFS master curves. By proposing a criterion according to the intensity ratios of stress modes to decompose the SRFS data and a data selecting criterion to eliminate the bad data with nonzero average shear rates, a method to construct reliable SRFS master curves from experimental data is established without involving a direct analysis of stress responses via Fourier-transform rheology and stress decomposition method or a prior examination of the validity of experimental data by the KK relationship. The application of the method to the Carbopol suspensions shows that good master curves with a prediction of the linear response region can be established even in the region of high frequencies, which presents a new insight to the deviation of from its linear portion of the master curve observed often in high frequencies. The success of the method to construct mater curves for the Carbopol suspensions implies that the valid KK relationship is not necessarily a prior indicator of linear viscoelasticity. Finally, the master curves constructed by the method can be fitted well with the two-mode FMM. Since MAOS experiments are normally performed at relatively not too large frequency, which belongs to intrinsically nonlinear oscillatory shears, the extraction of pure nonlinearity from SRFS master curve is a significant issue for further studies on the SRFS principle.

Acknowledgements

The authors would thank Prof. Henning Winter from University of Massachusetts, USA for providing calculation results of parsimonious spectrums.

References

[1] HYUN K, WILHELM M, KLEIN C O, CHO K S, NAM J G, AHN K H, LEE S J, EWOLDT R H, MCKINLEY G H. A review of nonlinear oscillatory shear tests: Analysis and application of large amplitude oscillatory shear (LAOS) [J]. Progress in Polymer Science, 2011, 36(12): 1697-1753.

[2] KALELKAR C, LELE A, KAMBLE S. Strain-rate frequency superposition in large-amplitude oscillatory shear [J]. Physical Review E, 2010, 81: 031401.

[3] TSCHOEGL N W, TSCHOEGL N W. The phenomenological theory of linear viscoelastic behavior: An introduction [M]. New York: Springer-Verlag Berlin, 1989: 35-68.

[4] FERRY J D. Viscoelastic properties of polymers [M]. New York: John Wiley & Sons, 1980: 234-237.

[5] WYSS H M, MIYAZAKI K, MATTSSON J, HU Z, REICHMAN D R, WEITZ D A. Strain-rate frequency superposition: A rheological probe of structural relaxation in soft materials [J]. Physical Review Letters, 2007, 98: 238303.

[6] DEALY J, PLAZEK D. Time-temperature superposition¡ªA users guide [J]. Rheology Bulletin, 2009, 78(2): 16-31.

[7] ROULEAU L, DE J F, LEGAY A, LE L F. Application of Kramers¨CKronig relations to time¨Ctemperature superposition for viscoelastic materials [J]. Mechanics of Materials, 2013, 65: 66-75.

[8] EWOLDT R, BHARADWAJ N A. Low-dimensional intrinsic material functions for nonlinear viscoelasticity [J]. Rheologica Acta, 2013, 52(3): 201-219.

[9] KALELKAR C, LELE A, KAMBLE S. Strain-rate frequency superposition in large-amplitude oscillatory shear [J]. Physical Review E, 2010, 81(3): 031401.

[10] HESS A, AKSEL N. Yielding and structural relaxation in soft materials: Evaluation of strain-rate frequency superposition data by the stress decomposition method [J]. Physical Review E, 2011, 84(5): 051502.

[11] CHO K S, HYUN K, AHN K H, LEE S J. A geometrical interpretation of large amplitude oscillatory shear response [J]. Journal of Rheology, 2005, 49(3): 747-758.

[12] EWOLDT R H, HOSOI A E, MCKINLEY G H. New measures for characterizing nonlinear viscoelasticity in large amplitude oscillatory shear [J]. Journal of Rheology, 2008, 52(6): 1427-1458.

[13] DIMITRIOU C J, EWOLDT R H, MCKINLEY G H. Describing and prescribing the constitutive response of yield stress fluids using large amplitude oscillatory shear stress (LAOStress) [J]. Journal of Rheology, 2013, 57(1): 27-70.

[14] ERWIN B M, ROGERS S A, CLOITRE M, VLASSOPOULOS D. Examining the validity of strain-rate frequency superposition when measuring the linear viscoelastic properties of soft materials [J]. Journal of Rheology, 2010, 54(2): 187-195.

[15] BIRD R B, ARMSTRONG R C, HASSAGER O. Dynamics of polymeric liquids. Volume 1: fluid mechanics [M]. New York: John Wiley & Sons, 1987: 455-459.

[16] WILHELM M, MARING D, SPIESS H W. Fourier-transform rheology [J]. Rheologica Acta, 1998, 37(4): 399-405.

[17] PEARSON D S, ROCHEFORT W E. Behavior of concentrated polystyrene solutions in large-amplitude oscillating shear fields [J]. Journal of Polymer Science: Polymer Physics Edition, 1982, 20(1): 83-98.

[18] DRAPACA C S, SIVALOGANATHAN S, TENTI G. Nonlinear constitutive laws in viscoelasticity [J]. Mathematics and Mechanics of Solids, 2007, 12(5): 475-501.

[19] GANERIWALA S N, ROTZ C A. Fourier transform mechanical analysis for determining the nonlinear viscoelastic properties of polymers [J]. Polymer Engineering & Science, 1987, 27(2): 165-178.

[20] CHO K S, SONG K W, CHANG G S. Scaling relations in nonlinear viscoelastic behavior of aqueous PEO solutions under large amplitude oscillatory shear flow [J]. Journal of Rheology, 2010, 54(1): 27-63.

[21] BOOIJ H C, THOONE G P. Generalization of Kramers-Kronig transforms and some approximations of relations between viscoelastic quantities [J]. Rheologica Acta, 1982, 21(1): 15-24.

[22] DROZDOV A D. Constitutive models in linear viscoelasticity [M]. London: Academic Presss City, 1998: 25-106.

[23] ANDERSSEN R S, LOY R J. Completely monotone fading memory relaxation modulii [J]. Bulletin of the Australian Mathematical Society, 2002, 65(3): 449-460.

[24] ROBERT B. B, ROBERT C A, OLE H. Dynamics of polymeric liquids, kinetic theory [M]. New York: Wiley-Interscience, 1987: 112-120.

[25] JOHN M D, RONALD G. Structure and rheology of molten polymers: from structure to flow behavior and back again [M]. Munich: Hanser Gardner Publications, 2006: 91-99.

[26] JAISHANKAR A, MCKINLEY G H. Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations [J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 2013, 469: 2149-2166.

[27] NIKIFOROV A F, UVAROV V B. Special functions of mathematical physics [M]. Boston: Springer, 1988: 2149-2166.

[28] WANG Bao, WANG Li-jun, LI Dong, WEI Qing, ADHIKARI B. The rheological behavior of native and high-pressure homogenized waxy maize starch pastes [J]. Carbohydrate Polymers, 2012, 88(2): 481-489.

[29] SHU Rui-wen,SUN Wei-xiang, WANG Tao, WANG Chao-yang, LIU Xin-xing, TONG Zhen. Linear and nonlinear viscoelasticity of water-in-oil emulsions: Effect of droplet elasticity [J]. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 2013, 434: 220-228.

[30] van HORN, B L, WINTER H H. Dynamics of shear aligning of nematic liquid crystal monodomains [J]. Rheologica Acta, 2000, 39(3): 294-300.

[31] HOWARD A B, HUTTON J F. An introduction to rheology [M]. Elsevier, 1989: 46-50.

[32] BRADER J M, SIEBENBRGER M, BALLAUFF M, REINHEIMER K, WILHELM M, FREY S J, WEYSSER F, FUCHS M. Nonlinear response of dense colloidal suspensions under oscillatory shear: Mode-coupling theory and Fourier transform rheology experiments [J]. Physical Review E, 2010, 82(6): 061401.

[33] HYUN K, WILHELM M. Establishing a new mechanical nonlinear coefficient Q from FT-Rheology: First investigation of entangled linear and comb polymer model systems [J]. Macromolecules, 2008, 42(1): 411-422.

[34] PIPKIN A C. Lectures on viscoelasticity theory [M]. Springer-Verlag, 1986: 132-135.

[35] EWOLDT R H, HOSOI A E, MCKINLEY G H. Rheological fingerprinting of complex fluids using large amplitude oscillatory shear (laos) flow [J]. Annual Transactions-Nordic Rheology Society, 2007, 15: 3-8.

[36] MOLLER P C F, MEWIS J, BONN D. Yield stress and thixotropy: On the difficulty of measuring yield stresses in practice [J]. Soft Matter, 2006, 2(4): 274-283.

[37] P C F, RODTS S, MICHELS M A J, BONN D. Shear banding and yield stress in soft glassy materials [J]. Physical Review E, 2008, 77(4): 041507.

[38] DAVIES A, ANDERSSEN R S. Sampling localization in determining the relaxation spectrum [J]. Journal of Non-Newtonian Fluid Mechanics, 1997, 73(1): 163-179.

[39] MUSTAPHA S S, PHILLIPS T. A dynamic nonlinear regression method for the determination of the discrete relaxation spectrum [J]. Journal of Physics D: Applied Physics, 2000, 33(10): 1219.

[40] BAUMGAERTEL M, WINTER H H. Determination of discrete relaxation and retardation time spectra from dynamic mechanical data [J]. Rheologica Acta, 1989, 28(6): 511-519.

(Edited by FANG Jing-hua)

Foundation item: Project(11372263) supported by the National Natural Science Foundation of China

Received date: 2015-07-29; Accepted date: 2015-12-09

Corresponding author: ZHANG Ying, Professor, PhD; Tel/Fax: +86-592-2185959; E-mail: yzh@xmu.edu.cn