effect of temporary network structure on linear and ... · polymer melts and solutions is nowadays...

© 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 Kim1, Jung-Eun Bae1, Ji Ho Youk2, Hyun Jeong Jeon2 and Ki-Won Song3

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′ x̃, γo( ) + σ″ ỹ, γo( )

σ″ γox̃ ỹ

x̃ t( ) γo1– γ t( ) ỹ t( ) = γo

1– ω 1– γ· t( )γ t( ) = γo γ· t( ) = dγ/dt

σ′ t( ) = n 0=

∞

∑ τ2n 1+′ γo, ω( )T2n 1+ x̃( )

σ″ t( ) = n 0=

∞

∑ τ2n 1+″ γo, ω( )T2n 1+ ỹ( )

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+ γo

G2n 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( )≡

σEm γo, ω( )G′ ω( )γo

------------------------ = Γ′ ζ( )σVm γo, ω( )

G″ ω( )γo------------------------ = Γ″ ζ( )

σEm 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+∞ ζ

2n

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.



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 Gmcan 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 Gmand 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( )/GmG″ 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.



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.



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 λpas 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.



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 ζCΓ″ 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″



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.



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.



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.



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.

φeMe

M------

⎝ ⎠⎛ ⎞

μ

≈

c ce

effect of temporary network structure on linear and ... · polymer melts and solutions is nowadays...

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