2011_reinheimer et al._fourier transform rheology as a universal non-linear mechanical...

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Fourier Transform Rheology as a universal non-linear mechanical characterization

of droplet size and interfacial tension of dilute monodisperse emulsions

Kathrin Reinheimer a, Massimiliano Grosso b, Manfred Wilhelm a,

a Karlsruher Institut fr Technologie (KIT), Technische Chemie und Polymerchemie, Engesserstr. 18, D-76131 Karlsruhe, Germanyb Universit degli Studi di Cagliari, Dipartimento di Ingegneria Chimica e Materiali, Piazza dArmi, I-09123 Cagliari, Italy

a r t i c l e i n f o

Article history:Received 24 January 2011

Accepted 2 May 2011

Available online 9 May 2011

Keywords:

Fourier Transform Rheology

Emulsion

Ellipsoidal model

Droplet size

Interfacial tension

a b s t r a c t

A new protocol to gain interfacial tension and droplet size of dilutemonodisperse emulsions from Fourier

Transform Rheology (FTR), is proposed. Specifically, a universal dimensionless quantity Ewas found at

small strain amplitudes to correlate with the droplet size of the emulsion where Eis inversely related

to the square of the capillary number Ca and directly proportional to the relative intensities of the fifth

and third harmonics, I5/I3. The limiting value E0 at small strain deformations can be used as a universal

parameter to calculate different emulsion properties. Different morphological constitutive models for

emulsions were used to establish the universality of the parameter E0. Preliminary analysis on experi-

mental data confirms the validity of this approach for the characterization of emulsion properties, includ-

ing the estimation of interfacial tension and droplet radius.

2011 Elsevier Inc. All rights reserved.

1. Introduction

Emulsions (and blends in general) are a class of materials where

two or more fluids or viscous constituents are blended together to

create a new composite material. Blends can be broadly divided

into three categories: miscible, partially miscible and immiscible

blends where immiscible blends are by far the most common

group and are ubiquitous in the polymer, food processing, pharma-

cology and cosmetics industries e.g. [15] to name prominent

examples. For dilute emulsions, the typical microstructure of an

immiscible blend at rest consists of spherical droplets immersed

in a continuous matrix. As the size and size distribution of these

globular domains strongly affect both the processing and the

mechanical properties of the final products, significant efforts were

made in the last decades to quantitatively measure and control the

morphology of blends using non-invasive measurement tech-

niques such as direct optical microscopy, light scattering and rhe-ological methods [69]. For rheological experiments the interest

has been primarily focused on establishing protocols to determine

the morphological properties of emulsions from rheological mea-

surements (for a review on this topic see e.g.[5,1013] and the ref-

erences therein). It is important to note that these experiments

typically used either dynamic Small Amplitude Oscillatory Shear

(SAOS) flows or steady shear flows and analyzed the mechanical

linear or steady state response, e.g. G0(x),G00(x) org _c.

Most rheological models of dilute emulsions are based on ellip-

soidal deformation models where the morphology of the included

phase is assumed to be globular and a single droplet is modeled as

an ellipsoid under deformation. When dealing with neutrally-

buoyant Newtonian droplets dispersed in a Newtonian fluid at

low Reynolds numbers, the relevant physical parameters under

periodic shear flow conditions include the external fluid viscosity

gm, the droplet viscosity gd, the interfacial tension C, the radiusof the undistorted spherical droplet Rand the maximum shear rate

of the macroscopic flow _cmax c0x1, where x1/2p is the excitationfrequency andc0is the strain amplitude. Two dimensionless num-bers are defined using these quantities, the capillary number

Ca gm _c R=C and the viscosity ratio k gd=gm. The capillarynumber describes the relative importance of viscous and interfacial

forces which directly affect the degree of droplet deformation un-

der shear as the interfacial tension resists the applied shear stress

to preserve the original droplet shape.The fact that there is a clear connection between the morphol-

ogy of the emulsions (i.e. the shape of the droplets and their orien-

tation in flow) and their rheological behavior has been previously

established [1622]. For example, solving the equations of motion

for the single droplet problem under shear flow leads to a very

accurate prediction. There are also numerical simulations based

on the boundary element technique [14,15]which also give very

accurate answers, but are unfortunately very time consuming

and require significant computational resources. Several analytical

models for droplet deformation have been proposed in the litera-

ture as an adequate and simple way to describe droplet dynamics

under generic flow conditions [1620]. One particularly effective

0021-9797/$ - see front matter 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2011.05.002

Corresponding author. Fax: +49 721 608 43153.

E-mail address: [email protected](M. Wilhelm).

Journal of Colloid and Interface Science 360 (2011) 818825

Contents lists available at ScienceDirect

Journal of Colloid and Interface Science

www.elsevier .com/locate / jc is
http://dx.doi.org/10.1016/j.jcis.2011.05.002mailto:[email protected]://dx.doi.org/10.1016/j.jcis.2011.05.002http://www.sciencedirect.com/science/journal/00219797http://www.elsevier.com/locate/jcishttp://www.elsevier.com/locate/jcishttp://www.sciencedirect.com/science/journal/00219797http://dx.doi.org/10.1016/j.jcis.2011.05.002mailto:[email protected]://dx.doi.org/10.1016/j.jcis.2011.05.002


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phenomenological model introduced by Maffettone and Minale

[16], hereafter referred to as the MM model, was proven to give

quantitative agreement with experimental results up to Ca values

slightly larger than unity [21]. The MM model is also a valuable

tool to infer emulsion properties such as the average droplet radius

of the emulsion under these conditions[22].

For the situation where Small Amplitude Oscillatory Shear

(SAOS) is applied, Palierne proposed a mathematical solution for

the deformation and orientation of liquid droplets in flow that is

suitable for the linear range of periodic oscillations [7,2325]. In

these studies, the dependence of the viscoelastic properties of

emulsions on parameters such as matrix viscosity, interfacial ten-

sion and the radius of the systemand its oscillationfrequency were

investigated in detail. The Palierne model led to a robust procedure

that could relate the linear viscoelastic properties of polymer

blends to their chemicalphysical properties. For example, the

model could estimate the volume-average droplet radius of a blend

[26]or, alternatively, its interfacial tension [2729]. The Palierne

model has been subsequently extended to estimate a finite volume

fraction[7,8], to calculate a droplet size distribution instead of an

averaged mean value[30]and to include the effect of surfactants

on the interfacial tension and the droplet deformation [31]. The

impact of the surfactant concentration at the interface on the

resulting droplet deformation is of great interest, for example, in

food science applications[32]. The Palierne model can also be ex-

tended to predict the dynamic viscoelastic behavior of multiple

emulsions[33].

In general, most of the models mentioned above are valid only

in the linear regime, meaning that a Small Amplitude Oscillatory

Shear is applied to the sample under investigation and G0(x),G00(x) org _c are analyzed. However, in this work, the focus is onextending the characterization of emulsions into the non-linear re-

gime using Large Amplitude Oscillatory Shear (LAOS) flow instead.

In the literature, this type of analysis is often referred to as Fourier

Transform Rheology (FTR)[34]because the stress response is usu-

ally analyzed in the Fourier domain. FTR is useful because it con-

verts the measured stress data in the time domain into afrequency dependent spectrum that can detect even very weak

nonlinearities in the stress response. In addition, both the intensi-

ties and phases of the higher harmonics in the power spectrum

provide valuable information about the sample being evaluated.

FT-Rheology methodology has been applied to many types of com-

plex fluids and can, for example, distinguish between linear and

branched polymers [3537] and infer the droplet morphologies

in polymer blends[38,39]. The importance of FT-Rheology for the

characterization of dispersions of colloidal particles is shown in a

recent study [40] where the non-linear standard steady shear

experiments cannot distinguish between two different aqueous

polymeric dispersions, but differences between these samples are

clearly seen in the nonlinear oscillatory experiments.

The purpose of this work is to establish for emulsions a newnon-linear mechanical coefficient, E, where E is proportional to

the ratio of the 5th harmonic intensity to the 3rd harmonic inten-

sity scaled by c20 respectively Ca2. This proposed Ecoefficient quan-

tifies the nonlinearities arising from the interfacial stress

contribution, which is, after all, mainly responsible for the non-

Newtonian behavior of the emulsions under investigation.

In this paper, the mathematical model used (Section 2.1) and

the main principles of FT-Rheology (Section 2.2) are presented in

the Methods section. Afterwards, Sections 2.32.6summarize the

definition and development ofEas a function of the capillary num-

berCa and the viscosity ratio between the two emulsion compo-

nents k gd=gm: The experimental setup used to validate thismodel is then introduced (Section 2.7). Results are given in Section

3starting with a description of how the highest achieved values ofEcan be used to calculate the characteristic properties of an emul-

sion or blend. In the conclusion, the effectiveness of this new emul-

sion quantity,E, as a novel approach to characterize emulsions and

its experimental validation are summarized.

2. Theory and experiment

2.1. The model

Since FT-Rheology will be used in this study for determining

emulsion characteristics such as the droplet size (or, alternatively,

its interfacial tension) the LAOS experiments in FTR analysis must

be compared with the relevant theoretical predictions.

To make this comparison, a theoretical model based on a dilute

system consisting of buoyancy free droplets of a single Newtonian

liquid dispersed in a different, but also Newtonian, matrix is con-

sidered. The deformation forces acting on the isolated droplets un-

der oscillatory shear are balanced by the restoring effects caused

by the interfacial tension. In this scenario, break-up of the droplets

is excluded due to the high viscosity ratio and coalescence can also

be neglected because of the assumption of diluteness. As a conse-

quence of this assumption, the droplet volume distribution re-

mains unchanged under shear flow and is not time dependent.The measured shear stress is therefore a linear superposition of

the matrix contribution and the contributions originating from

each droplet[38,39], as predicted by Batchelor[41]:

rt pI|{z}isotropic term

gmrv rvT|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

external contribution

gmV

ZA

nu undA|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}viscous part

C

V

ZA

nn 1

3I

dA|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

elastic part|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflffl}interfacial contribution

1

In Eq. (1), C is the interfacial tension,pis the pressure,rvis theundisturbed velocity gradient tensor and rvT its transpose, gm is

the viscosity of the continuous phase, Vis the total volume of thesystem, n is the unit vector orthogonal to the interface between

the two phases,u is the velocity at the interface,dA is the area of

an interfacial element and the integrals are evaluated over the

whole interfacial area of the system, A . The unit vectors n andu

are orthogonal to the interface and computed with the MM model.

The MMmodel uses a second rank, symmetric and positive definite

tensor Sto describe the time dependent shape behavior of an ellip-

soid under non-linear oscillatory shearing, Eq. (2)[16,42]:

dS

dt f1k S

3III

h if2k;CatD S S D X S S X

f1k 40k1

2k319k16

f2k 5

2k3 3Ca

2

26Ca2

2

With the use of the emulsion times= gmR/C, whereR is the ra-dius, Eq.(2)is made dimensionless. In Eq. (2),I is the second rank

unit tensor, DandX are the deformation rate and the vorticity ten-

sors, respectively, and IIis the second scalar invariant of tensorS,

see also [16]. At rest, the droplet is spherical with S I. Please note

that droplet break-up will not occur under oscillatory shear flow as

long as the ratio of the viscosities is greater than or equal to 2.5, i.e.

that k gdgmP 2:5[43].

The MM model was proven to quantitatively describe the non-

linear response of emulsions up to Ca values slightly higher than

unity[21]. Therefore, the highest capillary numberCa experienced

by the largest radius Rmax of the droplet population using LAOSconditions is given by Camax

x1c0gm RmaxC

where _cmax c0x1. If the

K. Reinheimer et al. / Journal of Colloid and Interface Science 360 (2011) 818825 819


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strain amplitude of the oscillating experiment varies between

c0 = 0.0110 and the excitation frequency is chosen such thatx1/2p= 0.011 Hz a coverage of five decades of experimentalexcitation for the droplet behavior under LAOS can be compared

where the predictions from the MM model are still valid. A maxi-

mum capillary number Camax of about 1.3 is achieved when the

maximum droplet radius is estimated to be Rmax= 20lm (using

gm= 31 Pa s and C = 3 mN/m for these experiments, see below). If

n andu are known, Eq.(1)can be used to predict the shear stress

even under non-linear excitation. The prediction of the non-linear

response for LAOS excitation is based on the interfacial contribu-

tion as described in the Batchelor theory. In this analysis, the

viscous and elastic shear stress are referred to as the interfacial

shear stressrI. Further details can be found in the literature detail-ing the use of the MM model when combined with the Batchelor

theory[38,39]. From here on, the mathematical model formed by

combining the MMmodel, Eq.(2), with the Batchelor stress formu-

lation, Eq. (1), will be referred to as the MM-B model. As a first

approximation, the viscous part of the interfacial contribution in

Eq.(1)will be neglected.

2.2. FT-Rheology

LAOS data were analyzed according to the relevant FTR proto-

cols [34,44,45]. The imposed sinusoidal deformation is given by

c(t) = c0sin (x1t) where x1= 2pm1= 2p/T is the characteristicangular frequency with T the oscillation period and c0 the strainamplitude. The stress response r(t) retains the periodicity of theexciting strain e.g. r(t) = r(t+ T), but at large strain amplitudesthe stress can no longer be represented by a simple, single sinusoi-

dal waveform. The measured shear stress signal is therefore ex-

pressed in the Fourier domain as a superposition of higher

harmonics, since they fulfill ther(t) = r(t+ T) side constrain:

rt ()FT

~rx Z 1

1

rt expixtdt

Xj1

j1

odd j

cjdx jx1:

3

In Eq. (3), d(x jx1) is the Dirac delta located at x=jx1(j= 1, 2, 3, . . .),i is the imaginary unit and cj is the (complex) coef-

ficient of the jth harmonic. Since the tangential stress is an odd

function of the imposed strain, only odd terms of the Fourier series

can, in principle, be included in Eq. (3) [34]. The relationship be-

tween the applied strain deformation, the measured or simulated

shear stress and the related FT parameters is already explained

in detail in several references and leads to the result given in Eq.

(4) [44,46,47]:

rt X1

j1odd

cjx1 expijx1t

X1

j1odd

cjRx1 cosjx1t cjIx1 sinjx1t:4

wherei is the imaginary unit,cj(x1) = cjR+ icjIis the complex coeffi-cient of the jth harmonic in the Fourier domain and cjR(x1) andcjI(x1) are, respectively, the real and imaginary part ofcj(x1). Thecondition cjx1 cjx1 (with denoting the complex conju-gate) guarantees that the original time seriesr(t) is real valued.In addition, the absolute values Ij of the cj coefficients were previ-

ously found to give useful information and are calculated from:

Ij abscj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2jR c

2ji

q jcjj: 5

As Ij(j > 1) reaches appreciable values, i.e. for I3

I1: I3=1> 10

4,

this marks at the state of the art experimentally the point wherethe onset of nonlinearities in the stress response begins. Therefore,

FT-Rheology is demonstrated to be a powerful tool to detect and

quantify nonlinearities that are otherwise not observed in tradi-

tional linear rheological measurements [44,4749].

In addition, it should be noted that, at small amplitudes, the

scaling of the normalized intensity of thenth harmonic to the fun-

damental peak, InI1 In=1, is a function of strain as shown in the

following constitutive equations thus far investigated [48,51,52]:

In=1/ c

n1

0 as a consequence a new quantity can be defined:

nQ : In=1

cn10

limc0 !0

nQ : nQ0: 6

This allows for the definition of scalars based on the relative

intensities of the harmonics of the stress response, which can then

be used to infer inherent non-linear material properties of the sam-

ple. One example of this, that was used to evaluate polymer melts,

is the non-linear mechanical coefficient of the third harmonic3Q : Q [52]with:

3QI3I1

1

c20: Q: 7

According to Eqs.(6) and (7),the zero-strain non-linear coeffi-cientQ0is a constant value, also described as the intrinsic nonlin-

earity of a material at relatively small strain amplitudes

limc0!0

Q Q0

. This definition of Q0 allows for the introduction

and measurement ofQ0(x) spectra over a wide range of frequen-cies and material properties, allowing for quantitative comparisons

between experiments and simulations under non-linear oscillatory

shear conditions[52,53]. For emulsions, it was previously demon-

strated that the appearance of higher harmonics in the magnitude

spectrum is associated with the droplet size and interfacial tension

properties of the included phase in a straightforward manner[39].

To be more precise, the higher harmonics are easily evaluated by

taking into consideration only the non-linear elastic contribution,

which is valid because the other terms in Eq. (1) are either nil(the Newtonian external force and the isotropic pressure terms)

or, like the viscous part in the interfacial contribution Eq. (1), con-

sidered negligible. Therefore,

IkR 2pT

ZT

rIt expjkx1tdt

2pT

ZT

C

V

ZAR

nn 1

3I

dA

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflffl}

elastic contribution

expjkx1tdt: 8

Due to the non-linear nature of the interface deformation,

attention can be focused mainly on the behavior of the higher coef-

ficients of the magnitude spectrum, namely I3 and I5, therefore

being not influenced by the Newtonian response of the dispersed

phase or matrix viscosity:

I3 abs 1T

Rt0 Tto

rIt expi3x1tdt

; T 2px1 withk > 1; odd

I5 abs 1T

Rt0 Tto

rIt expi5x1tdt

; T 2px1 withk > 1; odd:

9

Forthe followinganalysis, it is importantto firstdefinea quantity

Efor emulsions that represents the non-linear ratio I5I3 : I5=3and to

note that Eoriginates purely from interfacial contributions. It is also

important to note that this non-linear coefficient does not involve

the fundamental harmonic I1 and that this is justified bythe hypoth-

esis [38]that, for dilute emulsions formed by two Newtonian com-

ponents, the fundamental harmonicI1in the magnitude spectrum

is mainly dominated by properties of the neat Newtonian matrix,i.e. the viscosity of the single phase. The information deduced from

820 K. Reinheimer et al. / Journal of Colloid and Interface Science 360 (2011) 818825


4/8

I1 and the phase information results in G0(x) and G00(x). Asa conse-

quencethePalierne model only analyses thefundamental harmonic

while FTR relates to the higher harmonic contribution. In contrast

the higher harmonics are affected solely by the interfacial stress,

whichin turn depends upon (i)the interfacialtension, (ii)the droplet

size and (iii) the viscosity ratio, among other factors. This scenario

has been previously confirmed by experiments [39], thus the I5/3ratio appears to be a useful tool for filtering out the linear part of

the experimental signal to focus only on the non-linear contribu-

tions caused by the deformed interface. As a consequence of Eq.

(6), a quadratic dependence on the strain deformation is expected

for I5=3/ c20as c0tends to zero.

2.3. Derivation of a universal behavior, preliminary definitions

The physical parameters that mainly affect the rheological

behavior of a dilute Newtonian dispersed phase in a Newtonian

matrix are: the viscosities of the matrix and the dispersed phase,

interfacial tension, the mean droplet diameter and the droplet size

distribution. For the sake of simplicity, only monodisperse blends

will be considered in the present study and the first goal of this

work is to generate a universal dependency which can relate the

dimensionless quantities identified in the model, i.e. the capillary

number Ca c0x1gm R

C and the ratio of the dispersed phase and ma-

trix viscositiesk gd=gm. A universal functionE(Ca,k) would havethe advantage that once such function is established via simula-

tions, it can then be used for every experimental emulsion. There-

fore, the amount of data needed to characterize the droplet size or

interfacial tension of an emulsion is reduced to a minimum.

It is possible to use FT-rheological analysis of the shear stress to

evaluate I3 and I5 in the resulting spectra as different physical

parameters are varied in experiment and simulation.Fig. 1repre-

sents the values I3/1 andI5/1 as a function ofc0 computed by theMM-B model. The strain deformation c0 is the logarithmic x-axisand normalization to the fundamental peak I1 is done to gain

experimentally an intense quantity. The influence ofI1and, there-

fore, the contribution of the Newtonian pure phases is lost by sub-

sequently establishing the ratio I5/3. The physical parameters of the

model used in the simulation are x1/2p = 0.1 Hz, gm= 60 Pa s,R= 10lm, C = 3 mN/m and k 3. As previously noted for low

strain amplitudes, the expected dependencies I3=1/ c20 andI5=1 / c40 are seen, confirming the asymptotic behavior predictedin other constitutive models [47,50]. The insets in Fig. 1 show

the derivation of the higher harmonics at the strain amplitude of

c0 = 3.135 used to generate this data. The unexpected occurrenceof a weak peak at 2x1 in the experiment is also clearly seen in

the inset. Although even harmonics are not predicted by the simple

theory, they are still frequently encountered in LAOS experiments.

Several explanations for the presence of even overtones in the

spectra have been proposed in the literature including wall-slip

phenomena[45,54] and time-dependent memory effects in the

system [34]. Recently [39], it was also proposed that these over-

tones originate from the instrument itself and might be unrelated

to the measured sample. They do not seem to give in the here pre-

sented case any information on the material itself and were there-

fore not studied further, especially as the intensity of the overtones

is so weak, i.e. the intensity ofI2is about a factor of 50 smaller than

that forI3.

The scalar quantity I5=3=c20 is thus a constant asc0tends to zero.However, of great interest is the fact that this limiting value con-

stant is a function of the emulsion properties gm,R, C and the exci-tation frequencyx1/2pover a wide range of parameter values.

2.4. Derivation of E as a function of capillary number

In starting this derivation, the viscosity ratio k is held constant

for the reasons mentioned previously. Fig. 2a displays a selection of

simulated curves showing the intrinsic non-linear ratioI5=3

=c20

for

different emulsion characteristics and for k 3. By varying the

parameters c0, x1, gm, R andC over a wide range and numerouscombinations of values, the following dependencies were found:

I5=3

c20/x21g

2mR

2

C2

Ca2

c20: 10

Taking into account the definition of the capillary number

Ca c0x1gm RC

, the value forI5/3 is clearly seen to be proportional to

the square of Ca. A superposition of all the curves shown in

Fig. 1. Normalized I3/1 and I5/1 as a function of strain amplitude for the MM-B

model. The physical parameters used in the simulation are x1=2p 0:1 Hz;gm 60 Pa s;R 10lm;C 3 mN=m and k 3. The insets show the time and

frequency data to gain the higher harmonics at x1=2p 0:1 Hz and a strainamplitude ofc0 = 3.135 for real experimental data.

Fig. 2. (a) Simulated ratio I5=3=c20as a function of strainamplitude. Each variationinan emulsion property results in a new curve with different plateau values for small

c0. The simulation conditions are, if not listed otherwise: x1=2p 0:1 Hz;gm 1 P a s;R 10lm;C 10mN=m. (b) The coefficient Ek I5=3=Ca

2as a func-

tionofCa. A universal curve Ekis describedin directanalogyto a Carreau like model,

Ek I5

=3

Ca2 E0

k1b Cac (where E

0k3 3:93; b 2:8 and c 1:88) for a viscosity ratio

k 3.



5/8

Fig. 2a is achieved by plotting the quantity Ek I5=3

Ca2as a function of

Ca, which leads to the creation of a single curve, as shown in Fig. 2b

where the simulation conditions are: x1/2p= 0.1 Hz, gm= 1Pas, R= 10lm and C = 10 mN/m. Similar to a Carreau model[4],

a three parameter model quantifies the dependence of Ek on Ca

for the shear rate dependent viscosity:

Ek E0k

1 bCac: 11

It can be easily demonstrated that the limiting behavior ofEkis

similar to that ofQ0 [52]in that:

limCa!0

Ek E0k :

As an example, for k 3, the parameters of Eq. (11) are esti-

mated by the least square method where it was found that

E0k3 3:93 (plateau value), b= 2.8 (representing the onset of

decreasing values at Ca= 1/b= 0.36) and c= 1.88 (describing the

slope of the curve at high Ca values).

First, it is important to emphasize that Ek is dimensionless as it

is the ratio of two dimensionless quantities. Furthermore, the sub-

script k is an important reminder that Ek depends on the viscosity

ratio and, therefore, that different values ofEk (andE

0

k) will be ob-served as k varies.

2.5. Influence of the viscosity ratio k

The viscosity ratio k includes the temperature dependence of

the viscosity and this causes, therefore, shifts in the plateau values

of E0k with variable k, see Fig. 3, for emulsions under non-linear

shear. The MM-B model is a suitable choice to describe emulsions

with viscosity ratios k > 2:5. At lower values ofk, droplet breakup

becomes significant and the MM-B model is less able to describe

the experiments. Using this simulation, it was also found that the

plateau valueE0k increases with increasing viscosity ratio.

The dependency of the plateau valueE0k on the viscosity ratio is

modeled in terms of a power law dependency:E0k E

0kp: 12

The parametersE0 andp in Eq.(12)were estimated by the pla-

teau values ofE0k evaluated from Eq.(11)at different kvalues using

a non-linear regression. With this approach, it was found that

E0 = 0.64 0.01 and p= 1.63 0.02. The excellent agreement be-

tweenE0k values and the power law is shown inFig. 3. In addition,

considering that the confidence intervals are very small and that

the adjusted R-square statistics (R2a ) [55] of the regression was esti-

mated to be R2a 0:99885, this further corroborates the appropri-

ateness of the power law assumption formulated in Eq. (12).

Thus, a new non-linear, dimensionless coefficient Ecan be final-

ly introduced and is given by the following relation:

EEkkp

I5=3

Ca2

1

kp: 13

It should be noted that this dimensionless coefficient is univer-

sally applicable as it now includes an explicit dependence on the k

parameter.With Eq.(13), a superposition of all factors into a single curve is

possible by plottingEas a function ofCa k0:82, seeFig. 4.

At lowCa k0:82 (or equivalentlyc0) values, it is possible to esti-mate the limiting value E0 for theE coefficient:

limCa!0

E limCa!0

Ekkp

E0kkp

E0kp

kp E0: 14

A crucial point is that this estimated E0 value, determined via

the MM-B model, should be universal for all dilute, monodisperse,

NewtonianNewtonian emulsions and can therefore be used to

determine the physical properties C or R. In fact, the following

relationship for the dimensional quantities implicitly appearing in

Eq.(13)can be written as:

I5=3

c20x21

E0kpg2mR

2

C2

And, taking into account the above results:

I5=3

c20x21

0:64k1:63g2mR

2

C2 15

It is important to note that the left side of Eq. (15)can be eval-

uated from LAOS experiments using FT-Rheology analysis. With

knowledge of the matrix viscosity gm and the viscosity ratio k,the plateau value E0 is a useful tool to calculate the droplet size

R when the interfacial tension C is known or vice versa. Thus,

the following course of action is proposed: (i) perform LAOS expe-riences at (relatively) small amplitude values; (ii) detect a strain

amplitude interval where I3 and I5 clearly evidence power law

behavior with respect toc0; (iii) evaluate the related 3Q0 and

5Q0coefficients as defined in Eq. (6); (iv) calculate the droplet size R

by exploiting the knowledge ofC or vice versa through Eq. (15).

In the Results section, an illustrative example is given where the

average droplet radius is estimated from a known value of the

interfacial tension. Alternatively, it is possible to estimate the

interfacial tension once the droplet distribution is determined

from, for example, optical measurements[56].

As will be later shown, one benefit of the proposed procedure is

that a few experiments are sufficient to estimate E0 (and, thus, R or,

Fig. 3. Computed E0k values as a function of the viscosity ratio k (black squares)

reported with the power law fit E0 E0k kp (solid line) with E0= 0.64 0.01 and

p= 1.63 0.02.Fig. 4. MastercurveE I5=3=Ca

2

k1:63

as a function ofCa k0:82

for different viscosityratiosk.



6/8

alternatively, C), which requires, in principle, less effort than the

traditional frequency sweep measurements.

Obviously, the methodology introduced here is limited to

emulsionsfulfillingtheassumptions of theadopted model,i.e. emul-

sions that are monodisperse, dilute and with two Newtonian

components. However, this technique could be extended to cover

polydisperse samples and non-Newtonian compounds if the proper

constitutive equations are taken into account.

Finally, it should be noted that the results obtained with the

MM-B model are quantitatively confirmed by a second ellipsoidal

model proposed by Yu et al. [57] as reported in detail in the Appen-

dix, see alsoFig. 6.

2.6. Experiments: materials and sample preparation

The model emulsions consisted of a matrix of poly(isobutylene)

(PIB) purchased from Ineos (Indopol H300) and a droplet phase of

poly(dimethylsiloxane) (PDMS) supplied by Rhone-Poulenc (Rho-

dorsil 47V200000). Relevant physical properties of the polymers

are listed inTable 1. Both PIB and PDMS exhibited pure Newtonian

behavior with a constant viscosity in the investigated shear rate

range at the temperatures used in this investigation (LAOS typicallyc0x1< 6 s1 and steady shear _c< 10 s1 were utilized). The viscos-

ity ratio is equal to 3.5, which is a large enough value to avoid sig-

nificant droplet break-up phenomena under pure shear flow.

Buoyancy effects are negligible in view of the almost identical poly-

mer densities and their high viscosities. All the experiments were

carried out with a volumetric fraction/ of the dispersed phase fixed

at 0.1, which creates the desired globular morphology. This value of

/is still small enough to neglect effects caused by droplet coales-

cence. Accordingly, the emulsion is assumed to be stable meaning

that its microstructure does not significantly vary during the exper-

iments (i.e. negligible droplet breakup and coalescence). Typically,

30 mL samples were prepared by mixing the two phases with a stir-

rer at 135 rpm in a 100 mL beaker until a homogeneous white

emulsion was obtained. Air entrapped during stirring was then re-

moved by allowing the blend to sit under vacuum overnight. The

rheological experiments were carried out at 30 C on samples that

were not subjected to any pretreatment or particular flow history.

The interfacial tension between PDMS and PIB was measured using

thependant drop method fromDataphysics (OCA 15EC) which gave

a value for the interfacial tensionC = 3.54 mN/mat 23 C. With the

temperature coefficient differential dC/dt approximated by

0.03 mN/m/K, which is the upper limit for immiscible homopoly-

mer blends [58], an interfacial tension of C = 3.33 mN/m was

estimated for 30 C. This result is in good agreement with other

estimations previously reported in the literature for this system

[59].

With a linear frequency sweep measurement in the range ofx1/2p= 0.110 Hz, the characteristic relaxation timesrelof the emul-sion was determined to be srel= 0.404s [24]. According to thePalierne model[24], the volume average droplet radius of the sys-

tem can then be obtained from Eq.(16):

hRiV4Csrelgm

10k 1 2/5k 2

19k 162k 3 2/k 1 16

which results in hRiV= 9.4 lm.

2.7. LAOS experimental setup

The FT-Rheology experimental setup is based on a Rheometrics

Scientific Advanced Rheometer (ARES G2), which is a strain con-

trolled rotational rheometer using a separate motor and transducer

technology. The motor is an air bearing supported, brushless DCmotor, where position and rate are measured with an optical

encoder including a position feed-back loop. The benefits of this

motor are a non-invasive motor current, inertia and friction on

sinusoidal strain controlled rheological measurements [47]. The

new generation of rheometers from TA Instrument already in-

cludes a setup for FT-Rheology that is based on the published work

of Wilhelm [44,46]. For the ARES G2 software TRIOS, there are two

possible methods to analyze the LAOS response. The first one is a

real time correlation to determine the magnitude and phase of

the fundamental frequency as well as the contributionof odd high-

er harmonics as a function of experimental time. A second possibil-

ity is the post processing of the acquired raw strain and stress data

in the time domain detected via TRIOS software using a discrete

Fourier Transformation with a specially written Matlab routine.

As first introduced by Wilhelm[34],the higher harmonic intensi-

ties are typically presented as the ratio of the magnitude of the

nth harmonic to the magnitude of the fundamental peakInI1 Inx1

Ix1 In=1 withn= odd andn > 1. To reduce mechanical noise,

a rigid and mechanically stable environment was ensured by using

a heavy stone table with carpet sheets under the legs. Within the

applied frequency domain, the maximum signal to noise ratio for

the prominent first harmonic is typically S/N= I1/Nffi 105.

The rheometer consists of a Force Rebalance Transducer (FRT),

which is suitable for measuring torques between 50 nN m and

200 mN m as specified by the manufacturer. Theinstalled brushless

DCmotorwithjewelled air bearings is capableof applying rotational

angular velocities from 106 rad/s to 300 rad/s and deformation

amplitudes of 1 lradto an unlimitedmaximum. Theapplied oscilla-

tion frequency can be varied between 107 rad/s and 628 rad/s.Measurements were performed in a 50 mm cone plate fixture

from TA Instruments with an angle ofa = 0.02, which is the largestdiameter cone and plate fixture available from TA Instrument. This

was selected to maximize the contact surface between the sample

and test fixture and, consequently, maximize the torque signal. The

temperature control was set to 30 C with a force convected oven.

The excitation frequency was set to x1/2p= 0.1 Hz and the defor-mation amplitude varied in the range ofc0 2 [0.1; 10] in absolutevalues. A comparison of the post processing of detected time cycles

of the shear stress with the real time correlation of the higher har-

monics from the ARES G2 resulted in a perfect overlay of the curves

for In/1(c0). The two softwares calculated a value for the relativehigher harmonic contribution that varied only by a factor of

I5=3TRIOSI5=3Matlab

1:007. Therefore, as the online detection is the faster

procedure, this was preferred. Droplet breakup can be considered

to be negligible due to the low capillary numbers used (Camax 1).

A frequency sweep test performed at both the beginning and end

of the LAOS measurements monitored the stability of the emulsion

morphology. An overlap of G0(x) and G00(x) at these two stages(before and after) was considered proof of the stability of the emul-

sion microstructure[39].

3. Results

3.1. Calculation of droplet size

In the previous section, the concept of a universal emulsioncurve ECa k0:82 and its limiting value E0 as Ca tends to zero

Table 1

Main physical properties of the PDMS/PIB system.

Polymer Formula Molecular

weight

(g/mol)

Density

(g/cm3)

at 23 C

Viscosity

(Pa s)

at 30 C

Interfacial

tension (mN/m)

at 30 C

PDMS [Si(CH3)2O]n 200,000 0.896 110 3.33

PIB [CH2C(CH3)2]n 1300 0.973 31



7/8

was introduced as a general description for emulsions. An illustra-

tive experiment will now be presented to confirm the usefulness of

this emulsion curveEfor evaluating the droplet radius or, alterna-

tively, the interfacial tension.

For these experiments, therange of strain amplitudes that can be

used has a lower limit due to the fact that nonlinear behavior only

starts at a certain minimum strainamplitude and also has an upper

limit caused by the onset of measurement artifacts such as loss of

sample volume. Using the correct frequency ofx1/2p = 0.1 Hz, re-sulted in curves wherethepredicted strain amplitude dependencies

ofc20and c40 for, respectively,I3/1and I5/1were measured [51] as re-

ported in Fig. 5. Fig.5a shows in detail the measured I3/1 and I5/1 val-

ues as a function of strain amplitudec0(filled symbols). For a givenrange ofc0 values, both the power law relationshipsI3=1 / c20 andI5=1/ c40 are observed, whereas this dependence is lost at lowerstrain rates dueto experimental noise. ThetorqueL at theminimum

value for I3/1= 5.74 104 is L= 863 106 N m and reflects I1.

Therefore, I3 has a torque of I3/1L = 495 nNm. The torque ofI5 at

the minimum value for I5/1= 9.3 105 is calculated using

L= I1= 1850 106 N m resulting in a value of 172 nN m, which is

close to the measurement limit of the current rheometer specified

at 50 nN m by the manufacturer. The measured I3/1 and I5/1 are

capped three points below the minimum measured torque for I3and I5. In this light, it should be remarked that the power lawbehav-

ior observed at intermediate strain rates is experimentally repro-

ducible, while the irregular values observed for I3/1 and I5/1 at

smaller strain rates are dominated by noise. These experiments,

therefore, allow for a reasonable estimation of the coefficients 3Q0and 5Q0 introduced in Eq. (6). It should also be noted that the onsets

of power law behavior are seen for the two higher harmonics at

different c0values. Specifically, for I3it was found atc0< 1 whereasI5 exhibits power lawbehavior only at slightlylarger values of strain

deformation (c0< 3). Thus, the scalar 5/3Q0=

5Q0/3Q0 can be evalu-

ated later with ease. For sake of clarity, the extrapolated curves at

small strain deformations are also reported in Fig. 5. Conversely,

Fig. 5b compares the measured (filled symbols) and extrapolated

(open symbols) curves ofEtogether with the universal emulsioncurve obtained with the MM-B model simulation. To calculate the

droplet radius, Eq. (15) can be transformed leading to the following

explicit relationship for the droplet radius:

R

ffiffiffiffiffiffiffiffiffiffiffiffi5=3Q0

E0

s C

x1gm

1

kp=2 17

The terms on the right hand side are known from experiments

using various strain amplitudes: the measured plateau value of

the intrinsic non-linear ratio is 5/3Q0= 0.013, x1/2p= 0.1 Hz,k 3:5, gm= 31 Pa s and C = 3.33 mN/m, E

0 = 0.64 and p/2 = 0.82.

Using these values, the droplet size radius is thus estimated to be

R= 8.8 lm. This value is remarkably close to the dropletradius esti-mated via the Palierne model, hRiV= 9.4 lm, confirming the good

agreement between these two procedures. One reason that the

agreement is not even better is that the calculation for droplet

radius via an estimation ofE0 is based on simulations for monodis-

perse emulsions, however, the actual sample is likely to exhibit

some polydispersity.

It is also interesting to note that there is a reasonable agreement

between the model simulations (solid line) and experiments (black

points) even at higher strain rate deformations, as shown in Fig. 5b

forCa k0:82 >0:8, thus confirming the validity of the MM-B model

even at higher strain rates.

4. Conclusions

A new approach to characterize emulsions towards droplet size

and interfacial tension based on FT-rheological measurements is

presented. The non-linear behavior of monodisperse dilute emul-

sions, subjected to LAOS conditions is investigated for emulsions

composed of two Newtonian constituents. In this case, the nonlin-

earity of the response under LAOS is thought to originate exclu-

sively from the presence of the polymerpolymer interface. The

Maffettone and Minale constitutive model coupled with Batchelor

theory (MM-B) was selected for the analysis and investigated in

the Fourier domain at LAOS conditions.

Via numerical simulations using the MM-B model, a general

functionEfor emulsions could be established as a universal corre-

lation among (i) the ratio of the higher harmonics I5/3, (ii) the cap-illary numberCa and (iii) the viscosity ratio k. Its limiting value E0

Fig. 5. (a) MeasuredI3/1andI5/1values as a function of strain amplitude c0(filledsymbols). (b) Measured (filled symbols) and extrapolated (open symbols) curves of

I5=

3

Ca2k1:63 together with the universal emulsion curve of the simulation made using theMM-B model.

Fig. 6. Comparison of the universal emulsion curve Esimulated either with MM-B

model (line) and Yu and Bousmina model (symbol).



8/8

as the strain deformation tends to zero is universal and holds for

any dilute emulsion with two Newtonian constituents. This univer-

sal value is useful for inferring blend properties such as the droplet

radius or, alternatively, the interfacial tension between the two

emulsion components.

Preliminary experimental results on a model blend (PDMS 10%

in PIB 90%) confirm the validity of this approach in that the average

droplet radius estimated with E0 agrees well with the average

droplet radius estimated from traditional linear rheological meth-

odologies (i.e. Palierne method), excluding effects caused by

polydispersity.

In conclusion, the non-linear dimensionless quantity Eappears

to be a promising parameter to characterize blend properties such

as droplet sizeand interfacial tensionand this procedure can be also

extended in the future to other mathematical models to take into

account, for example, the presence of non-Newtonian components

and/or concentrated emulsions. With respect to this a general func-

tion Eis a comprehensive value in combination with FT-rheological

experiments to analyzeemulsions towards radius or interfacial ten-

sion in a few steps.

Acknowledgments

The authors would like to thank Ivanka Koleva from the group

of Professor Rehage at the Technical University Dortmund (Ger-

many) for measuring the interfacial tension. Furthermore, the

authors would like to thank Ineos and Rhone-Poulenc for the dona-

tion of PIB and PDMS, respectively. Finally, we wish to acknowl-

edge the input from the referees who helped to improve this

article and stimulated future work as well as Mrs. Kbel for her

improvements as a native English speaker.

Appendix A

A.1. Comparison with the Model of Yu and Bousmina

As mentioned previously, the concept introduced here uses

ubiquitous quantities, which are not specific to any given model.

Therefore, other approaches can also be compared with the results

here. For example, the model of Yu et al. uses the Maffettone and

Minale model to simulate the droplet shape behavior, but calcu-

lates the shear stress with the following equation [57]instead of

using the Batchelor theory:

rI2f2K

II IS S S

2

3III 18

where Sis the non-dimensional droplet shape tensor, I the unit ten-

sor, IandII the first and second scalar invariant ofS, respectively,

andK 6 C5R k12k3 /5k15k2 /

. The dependency ofEas a function ofCa

with these two different models shows quantitative agreement, as

shown inFig. 6. Finally, it should be noted that this result supports

the assumption made in the calculation of the interfacial shear

stress that only the elastic contribution must be considered, see

Eq. (1), and that any viscous contribution to the interfacial contribu-

tion can be neglected in the first approximation using the MM-B

model.

References

[1] R. Brummer, Rheology Essentials of Cosmetic and Food Emulsions, Springer,

2006.

[2] D.J. Mc Clemens, Food. Emulsions, Principles, Practices, and Techniques, CRC

Press, University of Massachusetts, Amherst, 2004.

[3] S.R. Derkach, Adv. Colloid Interface 151 (2009) 1.

[4] R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University

Press, New York, Oxford, 1999.

[5] C.L. Tucker, P. Moldenaers, Annu. Rev. Fluid Mech. 34 (2002) 177.[6] M. Fahrlnder, C. Friedrich, Rheol. Acta 38 (1999) 206.

[7] J.F. Palierne, Rheol. Acta 29 (1990) 204.

[8] J.F. Palierne, Rheol. Acta 30 (1991) 497.

[9] D. Rusu, E. Peuvrel-Disdier, J. Rheol. 43 (6) (1999) 1391.

[10] J.M. Rallison, Annu. Rev. Fluid Mech. 16 (1984) 45.

[11] A.K. Chesters, Trans. I Chem. E 69 (1991) 259.

[12] H.A. Stone, L.G. Leal, J. Fluid Mech. 198 (1989) 399.

[13] J.M. Ottino, P. DeRoussel, S. Hansen, D.V. Khakhar, Adv. Chem. Eng. 25 (1999)

105.

[14] M.R. Kennedy, C. Pozrikidis, R. Skalak, Comput. Fluids 23 (1994) 251.

[15] A.Z. Zinchenko, A. Rother, R.H. Davis, Phys. Fluids 9 (1997) 1493.

[16] P.L. Maffettone, M. Minale, J. Non-Newtonian Fluid 78 (1998) 227.

[17] M. Bousmina, M.B. Aouina, R. Chaudhry, R. Gunette, R.E.S. Bretas, Rheol. Acta

40 (2000) 538.

[18] W. Yu, M. Bousmina, J. Rheol. 47 (2003) 1011.

[19] N.E. Jackson, C.L. Tucker, J. Rheol. 47 (2003) 659.

[20] A.S. Almusallam, R. Larson, M.J. Solomon, Polymers 48 (2004) 319.

[21] S. Guido, M. Grosso, P.L. Maffettone, Rheol. Acta 43 (2004) 575.

[22] T. Jansseune, J. Mewis, P. Moldenaers, M. Minale, P.L. Maffettone, J. Non-Newtonian Fluid 93 (2000) 153.

[23] D. Graebling, R. Muller, J .F. Palierne, Macromolecules 26 (1993) 320.

[24] D. Graebling, R. Muller, J.P. Palierne, J. Phys. IV 3 (1993) 1525.

[25] C. Lacroix, M. Bousmina, P.J. Carreau, B.D. Favis, A. Michel, Polymer 37 (1996)

2939.

[26] N.C. Das, H. Wang, J. Mewi, P. Moldenaers, J. Pol. Sci. Part B Polymer Physics 43

(2005) 3519.

[27] J. Huitric, P. Mederic, M. Moan, J. Jarrin, Polymer 39 (1998) 4849.

[28] P.C. Guschl, J.U. Otaigbe, J. Colloid Interface Sci. 266 (2003) 82.

[29] K.A. Vincze-Minya, A. Schausberger, J. Appl. Polym. Sci. 105 (2007) 2294.

[30] C. Friedrich, W. Gleisner, E. Korat, D. Maier, J. Weese, J. Rheol. 39 (1995) 1411.

[31] U. Jacobs, M. Fahrlnder, J. Winterhalter, C. Friedrich, J. Rheol. 43 (1980) 1495.

[32] E.J. Windhab, M. Dressler, K. Feigl, P. Fischer, D. Megias-Alguacil, Chem. Eng.

Sci. 60 (2005) 2101.

[33] R. Pal, J. Colloid Interface Sci. 313 (2007) 751.

[34] M. Wilhelm, D. Maring, H.W. Spiess, Rheol. Acta 37 (1998) 399.

[35] T. Neidhfer, S. Siuola, N. Hadjichristidis, M. Wilhelm, Macromol. Rapid

Commun. 25 (2004) 1921.

[36] G. Schlatter, G. Fleury, R. Muller, Macromolecules 38 (2005) 6492.

[37] K. Hyun, E.S. Baik, K.H. Ahn, S.J. Lee, J. Rheol. 51 (2007) 1319.

[38] M. Grosso, P.L. Maffettone, J. Non-Newtonian Fluid 143 (2007) 48.

[39] C. Carotenuto, M. Grosso, P.L. Maffettone, Macromolecules 41 (2008) 4492.

[40] S. Kallus, N. Willenbacher, S. Kirsch, D. Distler, T. Neidhfer, M. Wilhelm, H.W.

Spiess, Rheol. Acta 40 (2001) 552.

[41] G.K. Batchelor, J. Fluid Mech. 41 (1970) 545.

[42] S. Guido, M. Minale, P.L. Maffettone, J. Rheol. 44 (2000) 1385.

[43] H.P. Grace, Chem. Eng. Commun. 14 (1982) 225.

[44] M. Wilhelm, Macromol. Mater. Eng. 287 (2002) 83.

[45] K. Hyun, M. Wilhelm, C.O. Klein, K. Soo Cho, J. Gun Nam, K. Hyun Ahn, S. Jong

Lee, R.H. Ewoldt, G.H. Kinley, Prog. Polym. Sci., 2011, doi: 10.1016/

j.progpolymsci.2011.02.002.

[46] D. van Dusschoten, M. Wilhelm, Rheol. Acta 40 (2001) 395.

[47] A. Franck, Appl. Rheol. 18 (2008) 44.

[48] R.H. Ewoldt, A.E. Hosoi, G.H. McKinley, J. Rheol. 52 (2008) 1427.

[49] K. Cho, K.H. Ahn, S.J. Lee, J. Rheol. 49 (2005) 747.

[50] J.M. Brader, M. Siebenbrger, M. Ballauff, K. Reinheimer, M. Wilhelm, S.J. Frey,

F. Weysser, M. Fuchs, Phys. Rev. E 82 (2010) 061401.[51] J.G. Nam, K. Hyun, K.H. Ahn, S.J. Lee, J. Non-Newtonian Fluid 150 (2008) 1.

[52] K. Hyun, M. Wilhelm, Macromolecules 42 (2009) 411.

[53] M.H.Wagner,V.H. Roln-Garrido, K. Hyun, M. Wilhelm, J. Rheol. 55 (2011) 495.

[54] S.G. Hatzikiriakos, J.M. Dealy, J. Rheol. 35 (4) (1991) 497.

[55] N.R. Draper, H. Smith, Applied Regression Analysis, Wiley & Sons, New York,

1998.

[56] V. Ziegler, B.A. Wolf, Polymer 46 (2005) 11396.

[57] W. Yu, M. Bousmina, M. Grmela, C. Zhou, J. Rheol. 46 (2002) 1401.

[58] J.T. Koberstein, Interfacial Properties, Wiley, New York, 1990.

[59] S. Kitade, A. Ichikawa, N. Imura, Y. Takahashi, I. Noda, J. Rheol. 41 (1997) 1039.


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