© 2015 The Korean Society of Rheology and Springer 151
Korea-Australia Rheology Journal, 27(2), 151-161 (May 2015)DOI: 10.1007/s13367-015-0015-y
www.springer.com/13367
pISSN 1226-119X eISSN 2093-7660
Effect of temporary network structure on linear and nonlinear viscoelasticity of
polymer solutions
Kwang Soo Cho1,*, Jae Woo Kim
1, Jung-Eun Bae
1, Ji Ho Youk
2, Hyun Jeong Jeon
2 and Ki-Won Song
3
1Department of Polymer Science and Engineering, School of Applied Chemical Engineering, Kyungpook National University, Daegu 702-701, Republic of Korea
2Department of Advanced Fiber Engineering, Division of Nano-Systems, Inha University, Incheon 402-751, Republic of Korea
3Department of Organic Material Science and Engineering, Pusan National University, Pusan 609-735, Republic of Korea
(Received January 16, 2015; final revision received April 27, 2015; accepted May 4, 2015)
We investigated the effects of temporary network structures on linear and nonlinear viscoelasticity of poly-mer solutions by use of oscillatory shear (LAOS) flow. We tested two different types of polymer solutions:entanglement systems and ion complex systems. It was found that the entanglement network is difficult toshow shear-thickening while network of ion complex gives rise to shear-thickening. The objectives of thispaper are the test of strain-frequency superposition for various polymer solutions and to suggest a newmethod classifying complex fluids consisting temporary networks using LAOS data.
Keywords: large amplitude oscillatory shear, strain-frequency superposition, temporary network structure,
shear-thinning fluids, shear-thickening fluids
1. Introduction
Rheological properties of polymeric fluids can be
described by Brownian motion of chain segments which
are constrained by the interconnection between them and
by surrounding chains. The constraint by adjacent chains
changes its nature dramatically as molecular weight of
polymer chain increases. When the molecular weight of
polymer in molten state exceeds the critical molecular
weight, the Brownian motion of a chain is confined in the
space whose dimension is less than three. The confined
space is called the conceptual tube and the motion of the
chain in the tube is called reptation. This entanglement
effect is also observed in polymer solution whenever con-
centration is much higher than the entanglement concen-
tration, ce. The entanglement effect on viscoelasticity of
polymer melts and solutions is nowadays well understood
by molecular theories (Rubinstein and Colby, 2003; Doi,
1995; Watanabe, 1999).
If the interaction between polymer segments and solvent
molecules is much stronger than thermal energy kBT, then
in addition to the entanglement effect, another effect is
expected to play a significant role in rheology of polymer
solution. If the interaction between segments in the same
chain is stronger than the thermal energy, then the polymer
chain cannot take all the conformations which may appear
without the specific interaction. If the interaction between
segments and additive molecules with the lower molecular
weight is strong enough, then these molecules play a role
as junctions and even short polymer chains may form tem-
porary networks which should be different from the net-
works formed by entanglement. Even though these network
are not originated from chemical bonding such as vulca-
nization, it is interesting to investigate how different struc-
tures of these temporary networks affect viscoelasticity of
polymeric fluids. There were remarkable achievements in
rheological modeling of telechelic systems (Marrucci et
al., 1993; Vaccaro and Marrucci, 2000). It was reported
that such fluids show shear-thickening (Pellens et al., 2004).
In general, dynamic moduli are measured in investiga-
tion of molecular motion of a polymer because dynamic
experiment in linear regime is a reliable and convenient
tool in several aspects. However, molecular weight distri-
bution may hide the relations between structural factors
and linear viscoelasticity. For this reason, one may con-
sider applying nonlinear viscoelasticity as an alternative to
the linear viscoelasticity. Nonlinear stress relaxation is a
good example because it is considerably easy to connect
nonlinear experiments with molecular theories such as the
Doi-Edwards model and its modified versions through the
damping function (Einaga et al., 1971; Osaki et al., 1982;
Larson, 1985; Archer et al., 2002; Inoue et al., 2002; Lee
et al., 2009; Kapnistos et al., 2009). However, the super-
position in nonlinear relaxation modulus is limited to the
long time period region and relaxation test usually suffers
from inherent errors in the short time region because a
perfect step function cannot be implemented as a strain in
any experiment. Furthermore, the signal of stress becomes
noisy in long time region because of the limit of torque
sensor. In order to observe structural relaxation, strain-rate
frequency superposition (SRFS) is suggested by Wyss et
al. (2007). SRFS being analogous to concept of time-tem-*Corresponding author; E-mail: [email protected]
Kwang Soo Cho, Jae Woo Kim, Jung-Eun Bae, Ji Ho Youk, Hyun Jeong Jeon and Ki-Won Song
152 Korea-Australia Rheology J., 27(2), 2015
perature superposition (TTS) extended observable fre-
quency range through “constant rate frequency sweep”
test. It also facilitates to obtain detailed information on
strain rate dependence of the relaxation process.
Oscillatory shear flow is a convenient type of viscom-
etric flow because reliable data can be obtained by easy
preparation of the experiments. Dealy group did pioneer-
ing work in the rheology of nonlinear oscillatory shear
flow. They developed sliding plate rheometer which gives
reliable data of large amplitude oscillatory shear (LAOS)
for various polymeric fluids (Tee and Dealy, 1975; Reimers
and Dealy, 1996). When strain amplitude is out of the lin-
ear regime, it has been difficult to interpret the viscoelastic
measurement until Fourier Transform (FT) rheology (Wil-
helm et al., 1998, 1999) and stress decomposition (SD)
(Cho et al., 2005) were developed. Although Wilhelm
group developed FT-rheology further by digital tech-
niques, there were earlier works done by Giacomin and
Oakley (1992) who obtained Fourier series graphically
from the Lissajous plot. For convenience, we call the
oscillatory shear in linear regime small amplitude oscilla-
tory shear (SAOS).
With the aid of FT and SD, LAOS is in general more
informative than SAOS. However, viscoelastic functions
of LAOS depend not only on frequency (angular fre-
quency) but also on strain amplitude. Hence, analysis of
LAOS is more difficult than that of SAOS. Hess and
Aksel (2011) combined SRFS and SD to analyze nonlin-
ear behavior of soft materials. It is helpful to understand
the nonlinear behavior for wider range of frequency or
time domain. Cho et al. (2010) developed a convenient
tool which simplifies such complexity, called strain-fre-
quency superposition (SFS). The SFS was obtained from
PEO (poly ethylene oxide) aqueous solutions which are
fully entangled polymer solutions. Their superposition is
valid regardless of polymer concentrations if the concen-
tration is higher than entanglement concentration ce. If
such superposition of nonlinear viscoelasticity is valid for
various polymeric fluids, one may develop an improved
probe to detect the difference in structures of complex flu-
ids. Hence in this study, we test the SFS for various com-
plex fluids that are considered to have different structures
of temporary networks.
In this study, we investigate the effects of structure on
linear and nonlinear viscoelasticity of complex fluids. The
three complex fluids are two entanglement polymer solu-
tions which show shear thinning and an unentangled poly-
mer solution with ion complex which shows shear
thickening. The two entangled polymer solutions have the
identical relaxation time spectrum except the scale factors
but have different interactions between chain segments
and solvent molecules. The shear-thickening solution is a
PVA (poly vinyl alcohol) aqueous solution without entan-
glement but with temporary network structures con-
structed by addition of sodium borate which makes ion
complex between boron ions and hydroxyl groups of poly-
mer segments.
2. Theoretical Background
Cho et al. (2005) developed a simple theory which
decomposes nonlinear shear stress into elastic and viscous
parts, which is called stress decomposition. It was
reported by two research groups that SD is equivalent to
FT in mathematics (Kim et al., 2006 and Ewoldt et al.,
2008). If a strain-controlled rheometer is considered, shear
stress of LAOS can be expressed by
(1)
where σ' is the elastic stress; is the viscous stress;
is the amplitude of shear strain; = sinωt and = cosωt.
It is easily understood that = and
where sinωt is shear strain and is
shear rate. Using the Chebyshev polynomial of the first
kind Tn(x), elastic and viscous stresses in Eq. (1) can be
expressed by (Ewoldt et al., 2008)
,
(2)
where and are, respectively, elastic and vis-
cous Fourier coefficients. Because , Eq.
(2) also implies
,
. (3)
Eq. (3) leads to
. (4)
To compare our notations with those of Reimers and
Dealy (1996) and Ewoldt et al. (2008), we have the
relation such that and
. Hence, the Fourier coefficients
and have the dimension of stress while the
generalized dynamic moduli and (Reimers
and Dealy, 1996) have the dimension of modulus, Che-
byshev coefficients e2n+1 and v2n+1 (Ewoldt et al., 2008)
have the dimension of modulus and viscosity, respectively.
Recently, Cho et al. (2010) found scaling rules of some
nonlinear viscoelastic functions of LAOS using following
dimensionless variables
σ t( ) = σ′ x, γo( ) + σ″ y, γo( )
σ″ γox y
x t( ) γo1– γ t( ) y t( ) = γo
1– ω 1– γ· t( )γ t( ) = γo γ· t( ) = dγ/dt
σ′ t( ) = n 0=
∞
∑ τ2n 1+′ γo, ω( )T2n 1+ x( )
σ″ t( ) = n 0=
∞
∑ τ2n 1+″ γo, ω( )T2n 1+ y( )
τ′2n 1+ τ″2n 1+
Tn cosθ( ) = cosnθ
σ′ t( ) = n 0=
∞
∑ 1–( )nτ2n 1+′ γo, ω( )sin 2n 1+( )ωt( )
σ″ t( ) = n 0=
∞
∑ τ2n 1+″ γo, ω( )sin 2n 1+( )ωt( )
I2n 1+ γo, ω( ) = τ2n 1+′( )2 τ2n 1+″( )2+
G2n 1+′ γo = 1–( )n τ2n 1+′ = 1–( )ne2n 1+ γoG2n 1+″ γo = τ2n 1+″ = ωγov2n 1+
τ2n 1+′ τ2n 1+″G2n 1+′ G2n 1+″
Effect of temporary network structure on linear and nonlinear viscoelasticity of polymer solutions
Korea-Australia Rheology J., 27(2), 2015 153
,
, (5)
where and are, respectively, the amplitudes of
elastic and viscous stresses and
, (6)
. (7)
We would like to emphasize that and
are
linear viscoelastic functions: storage and loss moduli.
Hence the normalizations described above are non-dimen-
sionalization of nonlinear viscoelastic functions by their
linear counterparts. Of course, it is understood that
and go to storage and loss moduli,
respectively as strain amplitude goes to zero.
Cho et al. (2010) showed that the scaling rules are valid
for PEO aqueous solutions with fully-developed entangle-
ments, regardless of both concentration and molecular
weight of PEO whenever the concentration is much higher
than ce. For the first harmonics, Cho et al. (2010) sug-
gested following empirical equations:
, (8)
and for higher harmonics:
(n = 1, 2, ...). (9)
Note that and are material parameters independent
of the polymer concentration. As for PEO aqueous solu-
tions, , and were reported.
Furthermore, it was found that the plot of as a
function of follows following empirical equation:
, (10)
where the material parameters k and of PEO aqueous
solution are, respectively, about 3 and about 5. It is note-
worthy that the nonlinear parameters , , k and
are intrinsic because they are not obtained by extrapola-
tion to zero-strain.
We will check whether the scaling rule is applicable to
the three complex fluids in this study.
3. Experimental
3.1. MaterialsWe purchased PVAc (poly vinyl acetate) with weight
average molecular weight of 500 kg/mol measured by
GPC (gel permeation chromatography) from Aldrich Co.
We performed saponification of PVAc to obtain PVA by
addition of 40% KOH (potassium hydroxide) aqueous
solution (7 ml) to 5 g of PVAc in methanol (250 ml). As
the hydrolysis of the PVAc proceeded, PVA precipitated.
After being stirred at room temperature for 12 hours, the
precipitate was filtered and washed with an excess of
methanol at 50°C for 5 hours. The PVA was dried in a
vacuum oven at 50°C. We call this PVA ‘H-PVA’ because
its molecular weight is higher than that of other PVA(L-
PVA) which was purchased from Aldrich Co. The molec-
ular weight of L-PVA was in the range from 85 to 146 kg/
mol. Degrees of saponification of both PVA were about
99%.
We dissolved PVAc and H-PVA in DMSO (dimethyl
sulfoxide) in order that the concentrations of all polymer
solutions must be higher than the entanglement concen-
tration. Because H-PVA was obtained from PVAc by
saponification and the molecular weight of the PVA
monomeric unit is about 52% of that of PVAc monomeric
unit, we estimated the molecular weight of H-PVA as 260
kg/mol. We also assume that the molecular weight distri-
bution of H-PVA is nearly identical to that of PVAc. These
two polymer solutions are expected to form the temporary
networks of entanglements while they have different inter-
actions between the polymer segments and solvent mole-
cules. It is known that the two polymeric fluids are shear
thinning.
We dissolved L-PVA in pure water with concentration of
2 wt% and added Borax(sodium tetraborate) by various
contents in the L-PVA aqueous solution. We purchased
Borax from Aldrich Co. Because the concentration of L-
PVA aqueous solution (2 wt%) is less than the entangle-
ment concentration (c/ce ≈ 0.17), it is expected that 2 wt%
L-PVA aqueous solution would not show any entangle-
J2n γo, ω( )I2n 1+ γo, ω( )
G* ω( ) γo
---------------------------- = ϑ2n 1+ ζ( )≡
σE
m γo, ω( )G′ ω( )γo
------------------------ = Γ′ ζ( )σV
m γo, ω( )G″ ω( )γo
------------------------ = Γ″ ζ( )
σE
m σV
m
ζ γo, ω( ) γo cosδ ω( ) = γoG′ ω( )
G* ω( )
----------------≡
G* ω( ) ≡ G′ ω( )[ ]2 + G″ ω( )[ ]2
G′ ω( ) G″ ω( )
G1′ γo, ω( ) G1″ γo, ω( )
ϑ1 ζ( ) = expζζC
------–⎝ ⎠⎛ ⎞
ϑ2n 1+ ζ( ) = ϑ2n 1+
∞ ζ2n
1 ζ2+( )
n-------------------
ζC ϑ3
∞
ζC 5≈ ϑ3
∞0.061≈ ϑ5
∞0.013≈
logΓ′ ζ( )Γ″ ζ( )
logΓ′ k≈ Γ″ 1–( ) Γ″ ζ( ) = expζ
ζC″
--------–⎝ ⎠⎛ ⎞
ζC″
ζC″ ζC ϑ2n 1+
∞
Fig. 1. Schematic diagram of the formation of hydrogen bonds
between PVA chains and boron ions.
Kwang Soo Cho, Jae Woo Kim, Jung-Eun Bae, Ji Ho Youk, Hyun Jeong Jeon and Ki-Won Song
154 Korea-Australia Rheology J., 27(2), 2015
ment. However, addition of Borax to 2 wt% L-PVA aque-
ous solution makes a temporary network which is not
originated from entanglements but from ion bonds between
Borax and hydroxyl group of L-PVA. Fig. 1 is a schematic
structure of PVA-Borax system. The PVA polymer and
Borax form junctions based on ionic interaction between
boron ion and hydroxyl group of PVA. Intensive studies
on PVA-Borax system found that formation of ionic com-
plex is reversibly formed by (1) mono-diol or (2) di-diol
reactions Lin et al. (2005). Hence, the nature of temporary
network of L-PVA/Borax system is totally different from
those of H-PVA/DMSO and PVAc/DMSO. We summarize
the sample specification in Table 1.
3.2. Measurements of viscoelasticityWe used a rotational rheometer ARES (TA instruments).
We used a cone-and-plate fixture whose gap size between
cone and plate is 0.05 mm, diameter is 50 mm and angle
of the cone is 0.04 rad. Temperature was controlled at
25°C. For LAOS of the three polymer solutions, we used
an analog-digital converter for data acquisition. The shear
strain and shear stress were recorded as functions of time
with a sampling time of 0.02 second. As for PEO aqueous
solutions, we used the data of Cho et al. (2010) which
were measured in arbitrary function mode built in the ARES.
The frequencies for LAOS were 0.5, 1, 2, 5 and 10 rad/
s and strain amplitudes were 0.1, 0.25, 0.5, 1, 2.5, 5, 7.5
and 10. We fixed temperature at 25°C during measure-
ments of both linear and nonlinear viscoelasticity.
We measured zero-shear viscosity of PVA/DMSO and
PVAc/DMSO as a function of concentration. As for PVAc/
DMSO, zero-shear viscosity is proportional to 2.8 power
of concentration when concentration is less than 0.116 g/
cc while it is proportional to 5.7. Hence, we estimate the
entanglement concentration of PVAc/DMSO as 0.116. As
for PVA/DMSO, we could not obtain reliable zero-shear
viscosity when concentration is low and zero-shear vis-
cosity at higher concentration is proportional to 4.5 power
of concentration. Instead of determination of the transition
point, we calculated entanglement concentration by use of
entanglement molecular weight of PVA in literature (Me ≅
6944 g/mol, Fetters et al., 1994).
4. Results and Discussion
4.1. Linear viscoelasticityAs shown in the section 2, SFS requires the information
of linear viscoelasticity. Hence it is necessary to measure
linear viscoelasticity. It is worthwhile to compare the scal-
ing in linear viscoelasticity with that in LAOS.
From the data of terminal region, we can determine the
mean relaxation time λm and the mean modulus Gm by use
of following equations:
, . (11)
Note that the Gm defined in Eq. (11) is the inverse of the
steady-state compliance Je. It is noteworthy that λm and Gm
can be easily determined whenever well-developed termi-
nal behavior are observed.
Fig. 2 shows the plot of reduced moduli ( and
) as functions of reduced frequency λmω for (a)
PVAc/DMSO, (b) H-PVA/DMSO and (c) PEO aqueous
solutions. It is clear that the reduced plots of linear vis-
coelasticity for the three systems are independent of con-
centrations.
It is common to use the plateau modulus Ge as a
scaling factor of dynamic modulus and disentanglement
time λd from the reptation theory as a scaling factor of fre-
quency. Recently, scaling theory of polymer physics
shows that the plateau modulus of polymer solution fol-
lows , where is the plateau modulus
of the melt and φ is the volume fraction of polymer
(Rubinstein and Colby, 2003). However, the scaling by Gm
and λm gives better quality of superposition than that by
Ge and λd for PEO aqueous solutions.
Because H-PVA was obtained from the saponification of
PVAc, it is a reasonable assumption that the relaxation
λm = G′ ω( )
ωG″ ω( )-------------------
ω 0→lim Gm =
G″ ω( )[ ]2
G′ ω( )----------------------
ω 0→lim
G′ ω( )/Gm
G″ ω( )/Gm
Ge φ( ) Ge 1( )φ2.3≈ Ge 1( )
Table 1. Specifications of sample solutions.
Samples Concentrationa,b
c/ce Mw (kg/mol) ce (g/cc)
PVAc/DMSO
cp = 0.14 1.2
500 0.116ecp = 0.20 1.7
cp = 0.22 1.9
cp = 0.25 2.2
H-PVA/DMSO
cp = 0.103 1.5
260d 0.071fcp = 0.132 1.9
cp = 0.148 2.1
cp = 0.165 2.3
2 wt% L-PVA
aqueous
solution/Borax
cB = 0.25
0.17c 85~146
cB = 0.37
cB = 0.50
cB = 0.75
cB = 1.0
a) Note that cp stands for the concentration of polymer in g per cc.b)
cB stands for the concentration of Borax in wt%.c) The value 0.17 was obtained by calculation. Molecular weight
of L-PVA is 116 kg/mol in the calculation of c/ce.d)
Although the molecular weight of PVA from the saponification
of PVAc was not measured, it can be considered as about 52%
of PVAc. Degree of saponification was about 99% and the ratio
of molecular weight of the monomer of PVA to that of PVAc is
about 0.52.e) The entanglement concentration was calculated from the plot of
zero-shear viscosity as a function of concentration in g/cc.f) The entanglement concentration was calculated from the literature
value of entanglement molecular weight of PVA. See Appendix.
Effect of temporary network structure on linear and nonlinear viscoelasticity of polymer solutions
Korea-Australia Rheology J., 27(2), 2015 155
time spectrum of H-PVA/DMSO is very similar to that of
PVAc/DMSO. Fig. 3 shows the superposition of data in
Fig. 2. Fig. 3 reveals that the two relaxation time spectra
of H-PVA/DMSO and PVAc/DMSO are related by
(12)
where and are, respectively the relax-
ation time spectra of PVAc/DMSO and H-PVA/DMSO,
and H0 and λ0 are shift factors. It also reveals that the lin-
ear viscoelasticity of two polymer solutions with similar
molecular weight distributions are independent of detailed
structure of the repeating units. Hence, small difference in
interactions between the segment and solvent has little
effect on linear viscoelasticity except scaling factors. Of
course the scaling factors are the mean relaxation time and
the mean modulus (or the plateau modulus). The scaling
factors must be a function of concentration and usually
has the form of and . From the theory of
polymer physics, the values of exponents depend on inter-
action between polymer segments and solvent molecules
(Rubinstein and Colby, 2003).
It is interesting that linear viscoelastic data of PEO aque-
ous solution are superposed on the viscoelastic curves of
PVAc/DMSO and H-PVA/DMSO. It is difficult to say that
the molecular weight distributions of PEO of different
average molecular weights are similar to those of the
PVAc and H-PVA. It is noteworthy that the frequency
ranges of the viscoelastic data are lower than the cross-
over frequency at which loss tangent is unity. Hence, the
difference in relaxation time distribution does not clearly
appear in this frequency range although relaxation time
distribution depends on molecular weight distribution.
However, the scaled linear viscoelasticity of L-PVA/Borax
solutions is absolutely different from those of shear-thin-
ning fluids because of the difference in the origins of tem-
porary network.
The 2 wt% L-PVA aqueous solutions with Borax were
reported as shear-thickening fluids by Hyun et al. (2002)
while PVAc/DMOS and H-PVA/DMSO are shear-thin-
ning fluids. Because the polymer concentration of 2 wt%
HPVAc λ( ) = 1
H0
------HPVA
λλ0
-------⎝ ⎠⎛ ⎞
HPVAc λ( ) HPVA λ( )
λ φα∝ G φβ∝
Fig. 2. Normalized linear viscoelasticity of shear-thinning fluids.
(a) PVAc/DMSO; (b) H-PVA/DMSO; (c) PEO aqueous solutions.
For PEO aqueous solutions, molecular weights of samples are
denoted (400 K means 400 kg/mole and 1 M means 1000 kg/mol).
Fig. 3. Superposition of the data of Fig. 2 shows that the three
shear-thinning fluids have the same relaxation time distributions
except scaling factors of relaxation time and relaxation intensity.
Kwang Soo Cho, Jae Woo Kim, Jung-Eun Bae, Ji Ho Youk, Hyun Jeong Jeon and Ki-Won Song
156 Korea-Australia Rheology J., 27(2), 2015
is much lower than the entanglement concentration (see
Table 1), it is difficult for the PVA-Borax solutions to have
chain entanglement. However, the ion complex due to
boron ion forms a temporary network which has different
nature from those due to the chain entanglements. Fig. 4
shows linear viscoelastic plots in terms of dimensionless
moduli and dimensionless frequency. Because the PVA-
Borax solutions do not show a fully-developed terminal
behavior in the frequency range permitted by the rheom-
eter, we could not determine Gm and consistently.
Although Fig. 4 shows terminal behavior, data do not
allow that Eq. (11) gives consistent values of Gm and ,
which provide good superposition when they are used for
the scaling. However, the loss moduli of these solutions
show the local maximum. Instead of Gm and λm, we used
the peak frequency and the peak modulus as scaling fac-
tors. The peak frequency ωp is the frequency at the local
maximum and the peak modulus Gp is the value of the loss
modulus at ωp. We can define the peak relaxation time λp
as the inverse of the peak frequency. As shown in Fig. 4,
the scaling by Gp and λp gives a fairly good superposition
for L-PVA/Borax solutions regardless of Borax concen-
tration cB.
In summary, scaling in linear viscoelasticity requires
two scaling factors: one for the magnitude of dynamic
moduli and the other for the characteristic time. Frequency
can be called controllable variable while dynamic modu-
lus measurable variables because we measure dynamic
modulus by controlling frequency. This scaling is a kind
of non-dimensionalization. Both measurable and control-
lable variables are non-dimensionalized by characteristic
modulus and time, respectively. As shown in section 2,
SFS is also a kind of non-dimensionalization. Note that in
SFS, nonlinear viscoelastic functions are normalized by
their counterparts in linear viscoelasticity. Since there are
two controllable variables in LAOS: strain amplitude and
frequency, simple non-dimensionalization of controllable
variables is not expected to result in good superposition.
This is a main difference between the scaling in linear vis-
coelasticity and SFS.
4.2. Nonlinear viscoelasticityIn this section, we apply the scaling rules for LAOS
developed by Cho et al. (2010) to the three complex fluids
such as PVAc/DMSO, H-PVA/DMSO, and L-PVA/Borax
aqueous solutions. We also compare the results from the
three polymer solutions with PEO aqueous solutions.
4.2.1. Dimensionless amplitudes of stress
In this section, we discuss the scaling of the two dimen-
sionless amplitudes of stresses and defined in Eq.
(5). Both and decrease as cosδ increases in
the case of shear-thinning fluids while they increase as ζ
increases in the case of shear-thickening fluids. As for
shear-thinning fluids, is observed as Cho
et al. (2010) have found it for PEO aqueous solutions
which are also shear-thinning fluids. As shown in Fig. 5,
the superposition of and as functions of ζ is fairly
valid for both PVAc/DMSO and H-PVA/DMSO. It is
noteworthy that the superposition shown in Fig. 5 is inde-
pendent of both frequency and strain amplitude because
the symbols of the same concentration include various fre-
quencies and strain amplitudes. The values of for
PVAc/DMSO, H-PVA/DMSO and PEO aqueous solutions
are, respectively, 5.3, 4.2 and 5.0.
However, in the case of PVA-Borax solutions, shear-
thickening fluids, both and are increasing functions
of ζ. As shown in Fig. 6, both elastic and viscous stress
amplitudes do not follow the superposition. Data points
are scattered at large ζ while the dimensionless amplitudes
maintain unity when ζ is sufficiently small. Both and
start to increase steeply at a certain value of ζ. The
empirical equation is not valid for the
dimensionless amplitude of viscous stress for shear-thick-
ening fluids. All the systems show strong nonlinearity
when ζ exceeds a certain level, say . Shear-thinning flu-
ids show decreasing and while shear-thickening
fluids show increasing and . Hence, the meaning of
is the onset of nonlinearity in both cases of shear-thin-
ning and shear-thickening fluids. Different from SAOS
data, the difference in the structures of temporary net-
works is outstandingly apparent in LAOS.
As shown in Fig. 5, it is difficult to find an empirical
equation that describes the functional relation of .
However, it is easier to find an empirical relation between
and as shown in Fig. 7. Shear-thinning fluids
show following relation:
. (13)
λm
λm
Γ′ Γ″Γ′ Γ″ ζ = γo
Γ″ exp ζ– / ζC″( )≈
Γ′ Γ″
ζC″
Γ′ Γ″
Γ′Γ″
Γ″ exp ζ– / ζC″( )≈
ζC″Γ′ Γ″Γ′ Γ″
ζC″
Γ′ ζ( )
Γ′ ζ( ) Γ″ ζ( )
logΓ′ k Γ″ 1–( )≈
Fig. 4. Normalized linear viscoelasticity of shear-thickening flu-
ids (PVA/Borax systems). The legend implies Borax concentra-
tions in weight fraction.
Effect of temporary network structure on linear and nonlinear viscoelasticity of polymer solutions
Korea-Australia Rheology J., 27(2), 2015 157
Combining Eq. (13) with , we have
. (14)
Hence, we can determine another nonlinear parameter k
from the plot of as a function of . As for PEO aque-
ous solution, Cho et al. (2010) determined the value of k
to be about 3. In this study, we have for PVAc/
DMOS solutions and for H-PVA/DMSO solu-
tions. It is necessary to investigate the origin of the 0.3 dif-
ference in k value: an experimental error or an effect due
to the interaction between solvent and polymer segment.
For this analysis, we need more data for various systems
of polymer solutions.
As shown in Fig. 7c, the shear-thickening fluids show
different behaviors in the plot of as a function of
such that increases exponentially as increases for
. Note that the point denotes the lin-
ear behavior. Compared with Fig. 6, the quality of super-
position of Fig. 7c is improved.
4.2.2. Dimensionless Fourier Intensities
Fig. 8 shows the dimensionless Fourier intensity of the
first harmonic, . As Cho et al. (2010) showed for
PEO solutions, the shear-thinning fluids follow
. (15)
The critical ζ for is denoted as in order to distin-
guish it from the critical ζ for , . Just as PEO aque-
ous solutions, the shear-thinning fluids in this study, show
that . Although for shear-thinning fluids is
a decreasing function of ζ, L-PVA/Borax solutions show
that increases as ζ. The Fourier intensity of shear-
thickening fluid also follows Eq. (15) approximately, but
the sign of the critical value is opposite to that of shear-
thinning fluids. This difference in the sign of can be
considered as an additional indicator that the L-PVA/
Borax system is shear-thickening. Combining the results
of , and , LAOS resolves the different complex
fluids more clearly than linear viscoelasticity.
Fig. 9 shows the dimensionless Fourier intensities of the
third harmonic as functions of ζ. The superposition of
looks like valid while the quality is lower than that
of the first harmonics. The lower quality of superposition
Γ″ exp ζ– / ζC″( )≈
Γ′ ζ( ) exp≈ k eζ– /ζ
C″
− 1( )( )
Γ′ Γ″
k 3≈k 3.3≈
Γ′ Γ″Γ′ Γ″
Γ″ 2< Γ′, Γ″( ) = 1, 1( )
ϑ1 ζ( )
ϑ1 ζ( ) = expζζC
-----–⎝ ⎠⎛ ⎞
ϑ1 ζC
Γ″ ζC″
ζC″ ζC≈ ϑ1 ζ( )
ϑ1 ζ( )
ζC
Γ′ Γ″ ϑ1
ϑ3 ζ( )
Fig. 5. Reduced stress amplitudes of shear-thinning fluids as
functions of ζ: (a) PVAc/DMSO; (b) PVA/DMSO.
Fig. 6. Reduced stress amplitudes of shear-thickening fluids
(PVA/Borax) as functions of ζ: (a) elastic amplitude ; (b) vis-
cous amplitude .
Γ′Γ″
Kwang Soo Cho, Jae Woo Kim, Jung-Eun Bae, Ji Ho Youk, Hyun Jeong Jeon and Ki-Won Song
158 Korea-Australia Rheology J., 27(2), 2015
in compared with is originated from that the
magnitude of I3 is much smaller than that of I1. The lines
in Fig. 9 are calculated from Eq. (9). It is interesting that
for both shear-thinning and shear-thickening fluids, the
functional forms of are nearly identical as shown in
Eq. (9) which was obtained from PEO aqueous solutions.
One may expect that higher harmonics do not distinguish
the structural differences.
ϑ3 ζ( ) ϑ1 ζ( ) ϑ3 ζ( )
Fig. 7. Reduced amplitude of elastic stress as a function of
reduced amplitude of viscous stress: (a) PVAc/DMSO; (b) H-
PVA/DMSO; (c) L-PVA/Borax systems.
Fig. 8. Fourier intensity of the first harmonic as a function of ζ:
(a) PVAc/DMSO; (b) H-PVA/DMSO; (c) L-PVA/Borax sys-
tems.
Effect of temporary network structure on linear and nonlinear viscoelasticity of polymer solutions
Korea-Australia Rheology J., 27(2), 2015 159
The saturation values of , are 0.050 for PVAc/
DMSO, 0.066 for H-PVA/DMSO, 0.061 for PEO aqueous
solutions and 0.17 for L-PVA/Borax systems. Shear-thin-
ning fluids show similar values while the shear-thickening
fluid shows higher value. This implies that contributions
from higher harmonics of shear-thickening fluids are
stronger than those of shear-thinning fluids.
4.2.3. Classification of complex fluids
We have investigated linear and nonlinear viscoelastic
behaviors of shear-thinning and shear-thickening fluids
according to the classification of shear-thinning and shear-
thickening proposed by Hyun et al. (2002). Their classi-
fication of complex fluids was also confirmed in this
study. Our approach includes the effect of both frequency
and strain amplitude in a unified framework by use of
strain-frequency superposition proposed by Cho et al.
(2010). The scaling permits the plot of as a function of
to classify shear-thinning and shear-thickening fluid
such that: the shear-thinning fluids form a straight line in
the 3rd quadrant of the plane of and while the
shear-thickening fluids form a curve in the 1st quadrant of
the plane regardless of concentration, frequency and strain
amplitude. The complex fluids showing an overshoot may
be expected to form a curve in the 2nd or the 4th quadrant
depending on which stress component shows the over-
shoot although we did not test such class of complex flu-
ids in this study.
Although we developed an improved method of classi-
fication, we have to explain why PVAc/DMSO, H-PVA/
DMSO and PEO aqueous solutions show shear-thinning
behavior and why L-PVA/Borax solutions show shear-
thickening behavior.
It is clear that the temporary network of the shear-thin-
ning fluids in this study shows the entanglements while
that of the shear thickening fluid shows strong ion bonds.
For entanglement systems, large strain amplitude induces
both chain extension and slips between the adjacent chains.
Slips between chains can be negligible in the linear vis-
coelastic region because it is difficult to expect that a
small strain amplitude breaks the topology of the tempo-
rary network. Because stress is developed mainly by the
extension of a chain or a partial chain and the effect of
strain is divided into chain stretching and slips, the stress
in the nonlinear regime does not increase in a manner pro-
portional to the strain.
It is noteworthy that entanglement is not a bond but a
constraint for chain movement. In shearing, the constraint
becomes more compliant for flow direction because of the
chain orientation than the perpendicular direction to the
orientation. As for PVA-Borax system, strain is also divided
into chain stretching and slips. However, the nature of the
temporary network is different from that of entanglement
systems. The slip in PVA-Borax system can occur when
some ion bonds are broken. After a slip, rapid motion of
ions reforms a new bond quickly. Although the slip in
entanglement systems contributes little to stress, the slip in
the ion-bonding systems requires increase in stress to
break the bonds and reformation of new bonds also increases
ϑ3 ζ( ) ϑ3
∞
Γ′Γ″
logΓ′ Γ″
Fig. 9. Fourier intensity of the third harmonic as a function of
ζ: (a) PVAc/DMSO; (b) H-PVA/DMSO; (c) L-PVA/Borax sys-
tems.
Kwang Soo Cho, Jae Woo Kim, Jung-Eun Bae, Ji Ho Youk, Hyun Jeong Jeon and Ki-Won Song
160 Korea-Australia Rheology J., 27(2), 2015
the stress. This explains why nonlinear stress in PVA-
Borax systems is larger than the linear stress of the cor-
responding strain while that of shear-thinning systems is
smaller.
One may doubt that even for entangled polymer solu-
tion, LAOS response at high frequency results in shear-
thickening because high frequency at does not
allow enough time for a slip and gives a rise to chain
extension severely. The report by Hyun et al. (2002) for
xanthan gum is a good example for this argument. Xan-
than gum is a polysaccharide with a long chain branch.
Their measurement of LAOS for xanthan gum was done
at the frequency where storage modulus is larger than loss
modulus. However, the xanthan gum solution does not
show the shear-thickening behavior. Instead, the xanthan
gum solution shows overshooting of loss modulus and
thinning of storage modulus when strain sweep test is
done while L-PVA/Borax system shows increases of both
moduli as the strain amplitude increases. Hyun (provided
unpublished data to the authors) also showed that 1%
hyaluronic acid solution with 1 M NaCl (sodium chloride)
is shear-thinning in strain sweep test at the frequency
higher than the cross-over frequency. Hyaluronic acid
does not have a long chain branch.
As for the long chain branch, it is noteworthy to mention
the damping functions of comb polymers (Lee et al., 2009
and Kapnistos et al., 2009). The damping functions of
comb polymers are still decreasing functions of strain just
as those of linear polymers although structures of comb
polymers are reflected in the damping behavior. However,
the LAOS response of xanthan gum shows overshooting
of loss modulus even though it has long chain branches.
Hence, we conclude that LAOS provides higher resolution
power than the nonlinear relaxation test in classification of
complex fluids. We expect that LAOS test will make an
outstanding contribution for identification of chain struc-
ture of nonlinear polymers in near future.
5. Conclusions
We have confirmed the strain-frequency superposition
developed by Cho et al. (2010) in both shear-thinning flu-
ids such as PVAc/DMSO and H-PVA/DMSO and shear-
thickening fluid such as L-PVA/Borax solutions. Differ-
ence in the structure of temporary network results in sig-
nificant difference in both linear and nonlinear visco-
elasticity such that the entanglement network shows shear-
thinning while the bonding network shows shear-thicken-
ing.
We also propose a plot which can classify complex flu-
ids in a unified framework including the effects of fre-
quency and strain amplitude. We showed that two material
parameters, and k, are useful for a nonlinearity mea-
sure and classification of complex fluids.
Acknowledgments
This work (2013R1A1A2055232) was supported by
Mid-Career Researcher Program through NRF grant funded
by the MEST.
Appendix A: Entanglement Concentration
It is known that the specific viscosity of polymer solu-
tion is proportional to the power of concentration:
(A1)
where is the viscosity of the solvent and is the zero-
shear viscosity of polymer solution. The exponent n has
different values according to concentration range. If con-
centration is lower than the overlap concentration then it
is expected that n is unity. When concentration is between
the overlap concentration and the entanglement concen-
tration, the exponent n is about 2 for theta solution. When
concentration is higher than the entanglement concentra-
tion, the exponent is 14/3 for theta solution and 3.9 for
athermal solution (Rubinstein and Colby, 2003). When
molecular weight is high, the separation between the over-
lap concentration and the entanglement concentration is
G′ G″>
ζC
ηo ηs–
ηs
--------------- φn∝
ηs ηo
Fig. 10. Plot of zero-shear viscosity as a function of concentra-
tion. The concentration is normalized by entanglement concen-
tration. (a) PVAc/DMSO; (b) PVA/DMSO.
Effect of temporary network structure on linear and nonlinear viscoelasticity of polymer solutions
Korea-Australia Rheology J., 27(2), 2015 161
observed in the plot of specific viscosity as a function of
concentration. However, as for moderate molecular weight,
the change of exponent occurs continuously in semi-dilute
region because the separation of the two characteristic
concentrations is not large.
It is known that the entanglement concentration in vol-
ume fraction is given by
. (A2)
The exponent m is about 0.76 for athermal solution and 3/
4 for theta solution (Rubinstein and Colby 2003). Hence it
is expected that the exponent m is nearly independent of
solvent and polymer.
Fig. 10 shows zero-shear viscosity as a function of con-
centration. As for PVAc/DMSO, it is observed as stiff
change of slope. Hence, we can guess that entanglement
concentration is about 0.116 g/cc at which the exponent n
in Eq. (A1) changes from 2.8 to 5.7. However, we could
not observe such change of exponent as for PVA/DMSO.
The slope of 4.5 is found. As for PVA/DMSO, we used
entanglement molecular weight of PVA from Fetters et al.
(1994) (Me ≅ 6944 g/mol). Use of Eq. (A2) gives the
entanglement concentration is about 0.071 g/cc. Our expe-
rience, significant nonlinear behavior in LAOS is not
observed whenever even at high strain amplitude of
10 for various polymer solutions such as PS in ethyl ben-
zene and PMMA in DMF. Hence, we think that nonlin-
earity in LAOS indicates that concentration is high enough
to form well developed entanglement. Most effective indi-
cation of nonlinearity is whether the plot of elastic stress
against strain is straight or not.
References
Archer, L.A., J. Sanchez-Reyes, and Juliani, 2002, Relaxation
dynamics of polymer liquids in nonlinear step shear, Macro-
molecules 35, 10216-10224.
Cho, K.S., K. Hyun, K.H. Ahn, and S.J. Lee, 2005, A geomet-
rical interpretation of large amplitude oscillatory shear
response, J. Rheol. 49, 747-758.
Cho, K.S., K.-W. Song, and G.-S. Chang, 2010, Scaling relations
in nonlinear viscoelastic behavior of aqueous PEO solutions
under large amplitude oscillatory shear flow, J. Rheol. 54, 27-63.
Doi, M., 1995, Introduction to Polymer Physics, Oxford Univer-
sity Press.
Einaga, Y., K. Osaki, M. Kurata, S. Kimura, and M. Tamura,
1971, Stress relaxation of polymer solutions under large strain,
Polym. J. 2, 550-552.
Ewoldt, R.H., A.E. Hosoi, and G.H. McKinley, 2008, New mea-
sured for characterizing nonlinear viscoelasticity in large
amplitude oscillatory shear, J. Rheol. 52, 1427-1458.
Fetters, L.J., D.J. Lohse, D. Richter, T.A. Witten, and A. Zirkel,
1994, Connection between polymer molecular weight, density,
chain dimension, and melt viscoelastic properties, Macromol-
ecules 27, 4639-4647.
Giacomin, A.J. and J.G. Oakley, 1992, Structural network models
for molten plastics evaluated in large amplitude oscillatory
shear, J. Rheol. 36, 1529-1546.
Hess, A. and N. Aksel, 2011, Yielding and structural relaxation
in soft materials: Evaluation of strain rate frequency superpo-
sition data by the stress decomposition method, Phys. Rev. E
84, 051502.
Hyun, K., S.H. Kim, K.H. Ahn, and S.J. Lee, 2002, Large ampli-
tude oscillatory shear as a way to classify the complex fluids,
J. Non-Newton. Fluid 107, 61-65.
Inoue, T., T. Uematsu, Y. Yamashita, and K. Osaki, 2002, Sig-
nificance of the longest Rouse relaxation time in the stress
relaxation process at large deformation of entangled polymer
solutions, Macromolecules 35, 4718-4724.
Kapnistos, M., K. M. Kirkwood, J. Ramirez, D. Vlassopoulos,
and L. G. Leal, 2009, Nonlinear rheology of model comb poly-
mers, J. Rheol. 53, 1133-1153.
Kim, H., K. Hyun, D.-J. Kim, and K.S. Cho, 2006, Comparison
of interpretation methods for large amplitude oscillatory shear
response, Korea-Aust. Rheol. J. 18, 91-98.
Larson, R.G., 1985, Nonlinear shear relaxation modulus for a lin-
ear low-density polyethylene, J. Rheol. 29, 823-831.
Lee, J.H., P. Driva, N. Hadjichristidis, P.J. Wright, S.P. Rucker,
D.J. Lohse, 2009, Damping behavior of entangled comb poly-
mers: experiment, Macromolecules 42, 1392-1399.
Lin, H.-L, Y.-F. Liu, T.L. Yu, W.-H. Liu, and S.-P. Rwei, 2005,
Light scattering and viscoelasticity study of poly(vinyl alco-
hol)-borax aqueous solutions and gels, Polymer 46, 5541-5549.
Marrucci, G., S. Bhargava, and S.L. Cooper, 1993, Models of
shear thickening behavior in physically cross-linked networks,
Macromolecules 26, 6483-6488.
Osaki, K., K. Nishizawa, and M. Kurata, 1982, Material time
constant characterizing the nonlinear viscoelasticity of entan-
gled polymeric systems, Macromolecules 15, 1068-1071.
Pellens, L., K.H. Ahn, S.J. Lee, and J. Mewis, 2004, Evaluation
of a transient network model for telechelic associative poly-
mers, J. Non-Newton. Fluid 121, 87-100.
Reimers, M.J. and J.M. Dealy, 1996, Sliding plate rheometer
studies of concentrated polystyrene solutions: Large amplitude
oscillatory shear of a very high molecular weight polymer in
diethyl phthalate, J. Rheol. 40, 167-186.
Rubinstein, M. and R. Colby, 2003, Polymer Physics, Oxford
University Press.
Tee, T.-T., and J.M. Dealy, 1975, Nonlinear viscoelasticity of
polymer melts, Trans. Soc. Rheol. 19, 595-615.
Vaccaro, A. and G. Marrucci, 2000, A model for the nonlinear
rheology of associating polymers, J. Non-Newton. Fluid 92,
261-273.
Watanabe, H., 1999, Viscoelasticity and dynamics of entangled
polymer, Prog. Polym. Sci. 24, 1253-1403.
Wilhelm, M., D. Maring, and H.-W. Spiess, 1998, Fourier-trans-
form rheology, Rheol. Acta 37, 399-405.
Wilhelm, M., P. Reinheimer, and M. Ortseifer, 1999, High sen-
sitive Fourier-transform rheology, Rheol. Acta 38, 349-356.
Wyss, H.M., K. Miyazaki, J. Mattsson, Z. Hu, D.R. Reichman,
and D.A. Weitz, 2007, Strain-rate frequency superposition: A
rheological probe of structural relaxation in soft materials,
Phys. Rev. Lett. 89, 2383030.
φe
Me
M------
⎝ ⎠⎛ ⎞
μ
≈
c ce<