Large amplitude oscillatory shear rheology for nonlinear viscoelasticity in hectorite suspensions containing poly(ethylene glycol)

g
th
, C
ou 5
Hectorite suspension
Lissajous curve
gl
ted
nd
sho
ity
nite ca
30 nm
n will c
the ele
hindered by the growth of the aggregate structure following
a nonergodic process [6]. External shear reverses the system to
a fluid by partially destroying the structure during a process called
rejuvenation [7].
Addition of poly(ethylene glycol) (PEG) changes the dynamics
andphase behavior of the Laponite suspension. PEG adsorbedon the
which significantly limits the application of the materials. It is
therefore a common technique to add polymers to the suspensions,
which stabilizes the suspension by steric hindrance after adsorption
and prolongs the shelf live under a wider variety of pH and ionic
strength [14e16]. The effect of the adsorbed polymer on the linear
viscoelastic rheology, i.e. behaviors at the small deformation limit,
has been studied intensively [17e20]. However, knowledge of the
nonlinear rheology for these suspensions under large shear defor-
mation is desired to predict the flow property in the applications.
* Corresponding author. Tel.: þ86 20 87112886; fax: þ86 20 87110273.
Contents lists availab
ym
ls
Polymer 52 (2011) 1402e1409E-mail address: mcztong@scut.edu.cn (Z. Tong).the negative charge on the surface and positive charge at the rim.
The nature of this liquid to solid transition is still controversial [3].
Generally, this soft solid is considered, as a repulsive glass when pH
is high and/or salt concentration is low, for the dominant inter-
particle force is electrostatic repulsion of the surface charge [4].
When the salt concentration is increased to screen the surface
charge, the van der Waals attractive force affects the suspension
together with the electrostatic interaction and the soft solid is
formed through gelation of fractal aggregates of the particles [5]. In
both cases, mobility and diffusion of individual particles are
than 50k, the storagemodulusG decreasedwith increasingMw, and
then increased with further increasing Mw [12]. This reentrant
phenomenonwas believed to be resulted from the bridging effect of
the adsorbed polymer chains over the platelets [13]. All of the
progresses about the clay suspension structure with PEG have been
achieved on the data under small strain deformation.
The hectorite clay is widely applied as a rheology modifier in
surface coatings, paints, personal care products, adhesives etc. Due
to the charged nature of the clay particles, however, the product
property is sensitive to a subtle change in pH and ionic strength,1. Introduction
The synthetic hectorite clay Lapo
to monodisperse nanoplatelets of
thickness [1,2]. A Laponite suspensio
soft solid after aged over time due to0032-3861/$ e see front matter � 2011 Elsevier Ltd.
doi:10.1016/j.polymer.2011.01.048the Lissajous curve were found to be sensitive to the nonlinear viscoelasticity and the peak of hM and hL
appeared at lower g0 with higher maximum following the same dependency as I3/1 on PEG Mw and
concentration. The overall nonlinearity parameters NE and NV were proposed in this paper and
demonstrated to reflect the difference in the Laponite suspensions with PEG more clearly and more
effectively.
� 2011 Elsevier Ltd. All rights reserved.
n be exfoliated in water
diameter and 1 nm
hange into an isotropic
ctrostatic interaction of
Laponite surface in a very close-packed manner, acting as a steric
hindrancebetween theneighboringplatelets [8e10]. The PEGof low
molecular weight (Mw, but > 1k) delayed the aging process of
Laponite suspension more effectively [11]. The dynamic moduli in
the linear viscoelastic regime for an aged Laponite suspension
containing PEG depended on the PEG Mw. When the Mw was lower
0Keywords:
Large amplitude oscillatory shear (LAOS)Mw or lower concentration. I3/1 revealed the structure difference in the suspensions induced by
adsorbing PEG in the nonlinear regime. The minimum- and large-strain rate viscosities hM and hL fromnonlinear regime and leveled off at g0 � 100% with higher slope and constant value for the PEG of higherLarge amplitude oscillatory shear rheolo
hectorite suspensions containing poly(e
Weixiang Sun, Yanrui Yang, Tao Wang, Xinxing Liu
Research Institute of Materials Science, South China University of Technology, Guangzh
a r t i c l e i n f o
Article history:
Received 4 November 2010
Received in revised form
7 January 2011
Accepted 22 January 2011
Available online 1 February 2011
a b s t r a c t
The effect of poly(ethylene
taining NaCl was investiga
(Mw) of PEG was 4k, 10k a
The dynamic strain sweep
g0 ¼ 50e70%. The intens
Pol
journal homepage: www.eAll rights reserved.y for nonlinear viscoelasticity in
ylene glycol)
haoyang Wang, Zhen Tong*
10640, China
ycol) (PEG) on the nonlinear viscoelasticity of Laponite suspensions con-
with large amplitude oscillatory shear rheology. The molecular weight
35k, and the concentration of PEG was varied from 0.063 wt% to 2.4 wt%.
wed that the nonlinearity appeared at g0 > 30% with a stress overshoot at
ratio I3/1 from Fourier-transform increased with g0 when entering the
le at ScienceDirect
er
evier .com/locate/polymer

Several methods have been proposed to study the viscoelasticity
under large shear. The large amplitude oscillatory shear (LAOS)
rheology was intensively accepted during recent years in studying
nonlinear behavior of complex fluids [21e23]. Furthermore, several
approaches for data interpretation proposed by different authors
have been known sensitive to the intricate and specific structures.
Fourier-transform rheology [24], for example, extracted a series of
high harmonic waves from the non-sinusoidal responsive stress.
The overtone intensity ratio provided the long chain branching [25]
or comb-like chain topology [26] in homopolymers, the droplet size
distribution in polymer blends [27], and the shear-induced phase
separation in block copolymers [28].
For such samples as colloids, emulsions, gels, however, multiple
high harmonics appeared depending on the experimental condi-
tions. In these cases, capture and comparison of the overall
nonlinear viscoelasticity only by the Fourier-transform rheology
becomes limited. Alternatively, four characteristic functions were
assigned to describe all the non-sinusoidal responsive stress for
these materials [29]. Although it was useful in understanding of
specified rheology behavior from the overall nonlinear response,
the arbitrariness in choosing the model functions hindered the
universality of the superposed function for various materials [30].
Recently, Ewoldt et al. proposed [22] several parameters to describe
the distortion of the Lissajous curve from ellipse, which were
powerful in interpreting structural-function relationship of
complex fluids [31].
In this work, hectorite clay suspensions containing PEG of
different Mw and NaCl have been investigated in nonlinear visco-
elastic regime using several methods based on the LAOS rheology.
What we focus on is which method is more sensitive to detect the
change in nonlinear viscoelasticity and microstructure of the clay
suspension induced by adsorbing PEG of different Mw at different
concentrations, which is hard to be distinguished by conventional
methods in the linear regime.
2. Experimental
Synthetic hectorite clay Laponite XLG ([Mg5.34Li0.66Si8O20(OH)4]
Na0.66, Rockwood Ltd.) was used after dried in vacuum at room
temperature overnight. Poly(ethylene glycol) (PEG, Uni-Chem)
samples of weight average molecular weight Mw 4000 (PEG-4 k),
10,000 (PEG-10 k) and 35,000 (PEG-35 k) with Mw/Mn ¼ 1.02e1.10
were used after dried in vacuum at 50 �C. Water was purified by
deionization and filtration with a Millipore purification apparatus
(18.2 M cm). Other chemicals were all analytic grade reagents.
Laponite suspension of 3 wt% and pH 9e10 was prepared by
mixing the clay powder with water under stirring for 15 min and
subsequently by ultrasonic radiation for another 15 min for
homogenization. PEG was dissolved in water with occasional
shakes for 1 day to obtain a 10 wt% solution. The solutions for
rheology measurements were prepared by mixing the Laponite
suspension and PEG solution with NaCl. The final concentration of
Laponite and NaCl in all the test solutions was fixed at 2 wt%, and
5 mM, respectively, while the concentration of PEG was varied. The
test solution was further ultrasonicated for 15 min and aged for
48 h prior to the measurement. The samples of Laponite suspen-
sions containing PEG of different Mw and concentrations cp were
denoted in the form of LeMwecp, where L meant the Laponite
W. Sun et al. / Polymer 52 (2011) 1402e1409 14030Fig. 1. Dynamic strain amplitude g0 sweep results of storage modulus G and loss modulus G
k at concentrations 0.063e2.4 wt%.00 for samples with indicated Mw of PEG at 0.63 wt% (A and B) and samples with PEG-35

of G0 and G00 at large g0 follows the power laws of G0wg�2v00 and
G00wg�v00 , where n0 z 0.9. The peak in G00 at the transition from
linear to nonlinear regime and the subsequent power law decay of
G0 and G00 were also observed from the LAOS results on Laponite
[1,35], hard or soft sphear colloids [36,37], block copolymer
micelles [38], foams [39,40] and emulsions [41] with the decay
exponential n0 varied among different systems.
Comparing the samples in Fig.1, one canfind that thePEGMwand
concentrationdependenceof theheightof theG00 peak is the sameas
G0 in the linear regime, i.e. the higher G0 in the linear regime, the
higher G00 peak appears. This peak is generally considered to come
from the dissipative energy during the oscillatory deformation for
destroy and rearrangement of the aggregated “house-of-cards”
microstructure in the Laponite suspension. The phenomenon is
understood as that when the gel structure is stronger the higher
dissipative energy arises under shear destruction. However, both
G0 and G00 become undistinguished among different samples at high
g0, so the method of dynamic strain sweep is not favorable for
detailed study of the nonlinear viscoelasticity.
3.2. Fourier-transform rheology and Lissajous curve
More detailed study on the nonlinear viscoelasticity is allowed
with the responsive stress in the LAOS experiment. Fig. 2 depicts
er 52 (2011) 1402e1409component, Mw the molecular weight of PEG, and cp the concen-
tration of PEG in wt%. For example, L denotes a sample without
addition of PEG, while L-35 k-0.63 denotes a sample with 0.63 wt%
of PEG-35 k.
Rheology measurements were carried out at 25 �C on a strain-
controlled rheometer (ARES-RFS) using a cone-plate fixture of
50 mm diameter and cone angle of 0.04 rad. Silicon oil was laid on
the rim of the fixture to prevent water evaporation. As a common
practice for characterizing thixotropic samples [32], the samplewas
pre-sheared at 200 s�1 (76.4 rpm) for 200 s and left at rest for 800 s
before each test in order to eliminate the deformation history and
establish a reproducible initial state. Dynamic strain sweep was at
an angular frequencyu of 5.0 rad/s. For the LAOS, sinusoidal strain g
(t) ¼ g0sin5t at several oscillatory shear strain amplitudes g0 was
applied and the responsive stress was recorded. 40 cycles were
repeated for each g0. For the Fourier-transform, the stress was
transferred from time domain to the frequency domain. The data
were processed with the MITLaos program [33] for the Lissajous
curve.
3. Results and discussion
3.1. Response to large amplitude oscillatory shear strain
Fig. 1 shows the storage modulus G0 and loss modulus G00 at
u¼ 5.0 rad/s as a function of shear strain amplitude g0 ranging from
1% to1000% for the Laponite suspensions containing0.63wt%of PEG
with different Mw (A and B) and PEG-35 k of different cp (C and D).
The linear viscoelasticity regime was observed at g0 < 10% for all
samples. In this regime, G0 decreases with decreasing PEG Mw
(Fig. 1A) or with increasing PEG concentration (Fig. 1C). By
comparing the results at the same concentration of 0.63wt% (Fig.1A
and B), G0 > G00 were observed, indicating that these samples
exhibited a gel-like behavior dominated by elasticity. For these
samples,G00 did not vary significantlywith PEGMwor concentration,
as seen fromFig.1B andD. In contrast, samples L-4 k-0.63 and L-35k-
2.4werefluidsmacroscopicallywithG0 <G00, and the values ofG00 for
these samples were lower than those of the gel-like samples.
Baghdadi et al. investigated the linear viscoelasticity behavior of
the Laponite-PEO system with PEO Mw ranging from 13k to 1070k
without addition of salts while allowed to stand for 80 days [12,34].
They found that with increasing Mw, the storage modulus G0 first
decreased and then increased until the value exceeding that of the
neat Laponite suspension. Therefore, a threshold PEO molecular
weight leading totheG0 minimumwassuggested toseparate the two
opposite effects of PEO Mw on the gelation and/or glass transition in
Laponite suspensions: the interparticle bridging by high-MW PEO
and reduction in the effective volume fraction by depletion force of
low-MW PEO. Nelson et al. [8,9] reported the adsorption of PEG on
the Laponite platelets in the suspension containing NaCl was
enhanced for the PEG with lower Mw to inhibit the gelation. The
present PEGMw rangewasmuchnarrower than that in the literature
and the samples contain NaCl, therefore only a part of reported
phenomena were observed. The present work will focus on the
rheology behavior of the Laponite suspension with PEG and NaCl
under the LAOS and does not go further to the linear regime.
All samples enter the nonlinear viscoelastic regime at g0 > 30%,
which is the lower boundary of LAOS measurements. From Fig. 1, it
is observed that G0 decreases monotonically with g0 when g0 > 30%
for all samples. For the gel samples whose G0 > G00 in the linear
regime, G00 starts to increase as g0 exceeds 30%, then decreases
monotonically, exhibiting a peak at about g0 ¼ 50e70% as the
yielding occurs in these samples. On the contrary, for the fluidic
samples with G0 < G00 in the linear regime, G00 directly decreases
W. Sun et al. / Polym1404with g0 when g0 > 30% without any peak. Furthermore, the decay
Fig. 2. Responsive stress in LAOS experiments: (A) effect of PEG Mw; (B) effect of PEG
concentration.

the stress waveform responsive to a sinusoidal strain g at the
amplitude g0 of 100%. The Lissajous curves were shown in the left
column and the corresponding stress waveforms were shown in
the right column. Nonlinear viscoelasticity was clearly indicated by
nonsinusodal stress response and non-ellipse shape of Lissajous
curves. As shown in Fig. 2, addition of PEG weakens the nonlinear
behavior of the suspension comparedwith that of the neat Laponite
suspension L. Increasing PEG Mw or decreasing PEG concentration
was found to enhance the nonlinearity.
The viscoelastic nonlinearity can be interpreted with by Four-
ier-transform of the responsive stress, which results in a series of
odd high harmonics. Fig. 3A demonstrates the Fourier-transform
of the stress in the frequency domain at the amplitude g0 of 500%
for the samples at a constant PEG concentration. The number and
intensity of the resolved high harmonic peaks decrease the
following sample sequence of L, L-35 k-0.63, L-10 k-0.63, and L-4
k-0.63, suggesting the decrease in viscoelastic nonlinearity with
Fig. 3. (A) Fourier-transform spectra of samples L, L-35 k-0.63, L-10 k-0.63 and L-4 k-
0.63 with the primary frequency of u/2p ¼ 0.8 Hz and g0 ¼ 500%, the curves were
vertically shifted to avoid overlapping; (B) g0 dependence of the relative intensity In/1
for the nth harmonics for the sample L-35 k-0.63.
Fig. 4. Relative intensity I3/1 of the 3rd harmonic to the primary one varying with amptitude
dependence of the I3/1 plateau value; (B) suspensions with indicated PEG concentration cp
W. Sun et al. / Polymer 52 (2011) 1402e1409 1405decreasing PEG Mw.
The intensity of the nth harmonics related to the primary one In/1
is plotted as a function of oscillatory strain amptitude g0 in Fig. 3B
for the suspenstion L-35 k-0.63 as an example. With increasing
g0, the intensity of high harmonics becomes more evident as
g0 < 200%, and then levels off at higher g0. At the same time, the
intensity of higher n harmonics becomes undistinguishable.
Therefore, the relative intensity of the amplitude of the 3rd
harmony to the primary one, I3/1 is widely used to characterize the
nonlinear viscoelasticity. In Fig. 4, I3/1 is plotted against g0. For all
samples, I3/1 increases dramatically with g0 at small g0 (30e100%),
while is almost a constant at large g0 (>100%). The I3/1 plateau level
here is higher than those reported in literature [24e28], mani-
festing that the present system has more evident viscoelastic
nonlinearity as an ideal object for using LAOS. The increase of I3/1
with g0 becomes steeper with higher plateau value for the sample
with higher Mw or lower cp of PEG. The range of g0 (30e100%)
where I3/1 increases steeply in Fig. 4 coincides with the g0 range
where G0 begins to decrease and G00 displays a peak in Fig. 1.
Therefore, this increase in I3/1 would be related to the yielding in
the samples. The yielding transition is the sharpest for the neat
Laponite suspension L because the interparticle interaction has no
any segregation due to absent of PEG.
The above results indicate that the Laponite/PEG suspensions
are complex yield stress fluids (YSFs) whose yielded flow state
shows nonlinear viscoelasticity rather than simple plasticity
[42,43], The value of I3/1 depends on PEG Mw and concentration,
similar to the G0 dependence on Mw and cp in the linear regime, so
that the viscoelastic nonlinearity in the yielding flowmay be linkedg0: (A) suspensions with indicated PEG Mw at 0.63 wt% and the inset presents the Mw
and the inset is the cp dependence of the I3/1 plateau value.

to the elasticity before yielding, i.e., in the linear regime. Lower PEG
Mw and higher cp lead to a restricted gelation by PEG adsorption on
the platelets and separation of the particle aggregation. Therefore,
if G0 is lower in the linear regime, the corresponding viscoelastic
nonlinearity is also lower in the yielded flow under the LAOS.
3.3. Descriptive parameters GM, GL, hM, and hL
The I3/1 parameter cannot represent the overall viscoelastic
nonlinearity of the present Laponite suspensions containing PEG
and NaCl because the harmonic higher than the 3rd is ignored.
Alternatively, the Lissajous curve allows us to examine the overall
nonlinear stress response by comparing the curve shape. Fig. 2
illustrates some nonlinear stress responses with the Lissajous
curve. Ewoldt et al. [22] proposed a series of parameters describing
the deviation of Lissajous curve from ellipse in order to characterize
nonlinear viscoelasticity. The minimum-strain modulus GM and
large-strain modulus GL were defined as:
GMh
ds
dg
�
�
�
g
¼0
; GLh
s
g
�
�
�
g
¼�
g0
(1)
Similarly, the minimum- and large-strain rate viscosity hM and hL
were defined as:
hMh
ds
d _g
�
�
�
_g
¼0
;
hLh
s
_g
�
�
�
_g
¼�
_g0
(2)
where the oscillatory strain rate _gðtÞ ¼ g0ucosut ¼ _gcosut, where
_g0 was the amplitude of strain rate. The geometrical representation
of these parameters is illustrated in Fig. 5. GM and GL represent the
elastic (in-phase) response to the LAOS strain, while hM and hL the
viscous (out-of-phase) response. In the case of linear viscoelasticity,
the Lissajous curve is an ellipse, hence GM ¼ GL and hM ¼ hL. These
parameters were evaluated by the MITLaos program [33] using the
Fourier-transform relations derived by Ewoldt et al. [22], in which
the strain and stress waveforms were input.
Fig. 6 displays GM, GL, hM, and hL for the Laponite suspensions
with 0.63 wt% PEG of different Mw as a function of strain amplitude
g0. For all the samples, GM and GL curves are similar to those in
Fig. 1A, exhibiting a plateau in the linear regime (g0 � 30%) and
a decreasewith g0 in the nonlinear regime, except for the sample L-
4 k-0.63 which is a liquid. These features come from their nature of
the elastic component of viscoelasticity and the plateau value of GM
Fig. 5. Schematic representation for the definition of (A) minimum- and large-strain
modulus GM and GL, as well as (B) minimum- and large-strain rate viscosity hM and hL.
W. Sun et al. / Polymer 52 (2011) 1402e14091406Fig. 6. Minimum-strain modulus GM (A), large-strain modulus GL (B), minimum-strain rate v
at u ¼ 5.0 rad/s.iscosity hM (C), and large-strain rate viscosity hL (D) as a function of strain amplitude g0

and GL decreases as L > L-35 k-0.63 > L-10 k-0.63 > L-4 k-0.63, the
same sequence as G0 in Fig.1A. On the other hand, the appearance of
nonlinearity can be clearly recognized from the deflection of each
curve. However, GM and GL become indistinguishable for different
samples in the nonlinear regime (g0 > 30%).
There is a peak at each curve of hM and hL against g0 for the four
samples, similar to that in Fig. 1B due to the structural rearrange-
ment. But the difference in different samples is more obvious in
Fig. 6C and D. The peak position and peak value vary with the
samples. The peak of hM and hL appears at lower g0 with higher
value following the sample sequence as L-4 k-0.63, L-10 k-0.63, L-
35 k-0.63, and L, which is similar to that of G00 in Fig. 1B. But the
appearance of nonlinearity becomes easier to be identified from
these hM and hL against g0 curves. The hM and hL from the Lissajous
curve appear to be more sensitive to the slight structure change for
the Laponite suspension in the nonlinear regime. The g0 depen-
dence curve of GM, GL, hM, and hL for the Laponite suspensions
containing PEG-35 k at several concentrations is similar to the
corresponding one in Fig. 6 and not shown here for succinctness.
3.4. Parameters for overall nonlinear viscoelasticity
From the schematic illustration in Fig. 5, onemay expect that the
detail shape of the Lissajous curve, especially distortion from the
ellipse (linear response), can be deduced quantitatively by relating
the parameters GM and GL for elastic response and hM and hL for
viscous response. Because the difference between GM and GL and
between hM and hL characterizes the deviation of Lissajous curve
from the ellipse and hence the viscoelastic nonlinearity. Thus, we
suggest two parameters NE and NV to express the overall nonline-
arity in viscoelasticity as:
NE ¼
GL � GM
G01
; NV ¼
hL � hM
G001=u
(3)
where (see Appendix)
GL � GM ¼
P
N
n¼3;odd
G0n$
h
n � ð�1Þn�12
i
hL � hM ¼
P
N
n¼3;odd
G00n
u $
h
nð�1Þ
nþ1
2
�ð�1Þn
i
(4)
Therefore, NE and NV is the sum of all the higher harmonic storage
moduli and dynamic viscosities (n � 3) relative to the first ones
(n ¼ 1), respectively, characterizing the overall nonlinearity in
viscoelasticity. In the linear viscoelastic regime, NE ¼ NV ¼ 0. For
nonlinear viscoelastic response,NE andNV can be positive, negative,
or zero, depending on the shape of the Lissajous curve. The value of
NE and NV can be higher than unity because there is a multiplier in
the square bracket whichmay be large than unit. In Fig. 7,NE andNV
calculated from equation (3) were plotted against g0. For all
samples, NE, the elastic part of nonlinear response, increases from
0 at low g0 with increasing g0. In contrast, NV decreases with
increasing g0 even to negative values. We can observe the different
nonlinear responses for our samples at high g0 more clearly than
from other figures. The sample L shows the strongest nonlinearity.
Addition of PEG with lower Mw or higher concentration weakens
the nonlinearity of the suspension at high g0. Although the physical
meaning of NE and NV is still unknown, like the case of parameters
GM, GL, hM, and hL, it is more effective to reveal and compare the
nonlinear viscoelastic response of the Laponite suspensions con-
taining PEG and NaCl with these parameters.
W. Sun et al. / Polymer 52 (2011) 1402e1409 1407Fig. 7. Overall nonlinear viscoelasticity parameters NE (A and C) and NV (B and D) plotted against g0 at u ¼ 5.0 rad/s.

[1] Ramsay JDF. J Colloid Interface Sci 1986;109:441e7.
er 54. Conclusions
Laponite suspension is a yielding fluidwith aggregated structure
of the “house-of-cards”. PEG chains in the Laponite suspension are
adsorbed on the platelets to change the interparticle interaction and
to hinder the formation of platelet networks. This is the first
experimental observation of the LAOS rheology on the nonlinear
viscoelastic behavior of the Laponite suspension containing PEG and
NaCl. Increase inPEGMwordecrease inPEGconcentration enhanced
the nonlinear viscoelasticity of the clay suspension, showing
macromolecule effect. The relative intensity I3/1 of the 3rd harmonic
reflected the delicate difference in the nonlinear viscoelasticity
induced by added PEGwith differentMw and concentrations, which
washardlydistinguishedbyconventional dynamic strain sweep.We
proposed two parameters NE and NV for describing the Lissajous
curvemodified from the geometric parametersGMandGL.NE andNV
can reveal the difference in the overall nonlinear viscoelasticity
more clearly andmore effectively. Consequently, the LAOS rheology
provided the information about the microstructure change in the
clay-polymer suspension and the appropriate interpretation was
desired to explore this unclear information.
Acknowledgments
The financial support from the NSF of China (50773024 and
21074040), the National High Technology Research and Develop-
ment Program of China (2009AA03Z102), and the Fundamental
Research Funds for the Central Universities, SCUT is gratefully
acknowledged.
Appendix. Derivation of NE and NV
The Lissajous curve is obtained by plotting s(t) against g(t). For
an incompressible fluid under uniform shear flow, the equation of
the Lissajous curve can be expressed by the following functions:
8
<
:
g
ðtÞ ¼ g0sinðt0Þ
s
ðtÞ ¼
P
N
n¼1;odd
snsinðnt0 þ dnÞ
where t0 ¼ ut, t0 > 0. From the definitions of GM and GL,
GMh
�ds
dg
�
�
�
�g
¼0
¼
ðds=dt0Þ
ðdg=dt0Þ
�
�
�
t0¼kp
¼
P
N
n¼1;odd
nsncosðnkpþ dnÞ
g0coskp
; ðk ¼ 0;1;2;.Þ
GLh
s
g
�
�
�
g
¼�
g0
¼
s
g
�
�
�
t0¼kpþp2
¼
P
N
n¼1;odd
snsin
�
n
�
kpþ p2
�
þ
dn
�
g0sin
�
kpþ p2
�
¼
P
N
n¼1;odd
snsin½nðkpþp2Þþdn�
g0coskp ; ðk ¼ 0;1;2;.Þ
rGL � GM ¼
P
N
n¼1;odd
sn
g0$an
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[30] Klein C, Venema P, Sagis L, van der Linden E. J Non-Newtonian Fluid Mech
2008;151:145e50.an ¼
ncosðnkpþ dnÞ � sin
�
n
�
kpþ p2
�
þ
dn
�
coskp ; ðk ¼ 0;1;2;.;n
¼ 1;3;5;.Þ
Note that
a1 ¼ 0
a3 ¼ 4cosd3
a5 ¼ 4cosd5
«
an ¼
h
n � ð�1Þn�12
i
cosdn; ðn ¼ 1;3;5;.Þ
rGL � GM ¼
P
N
n¼3;odd
G0n$
h
n � ð�1Þn�12
i
where
G0n ¼
sn
g0
cosdn
Similarly, we can obtain that
hL � hM ¼
X
N
n¼3;odd
G00n
u $
h
nð�1Þ
nþ1
2
�ð�1Þn
i
where
G00n ¼
sn
g0
sindn
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