© 2017 The Korean Society of Rheology and Springer 269

Korea-Australia Rheology Journal, 29(4), 269-279 (November 2017)DOI: 10.1007/s13367-017-0027-x

www.springer.com/13367

pISSN 1226-119X eISSN 2093-7660

A sequence of physical processes quantified in LAOS by continuous local measures

Ching-Wei Lee and Simon A. Rogers*

Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, Urbana 61801, United States of America

(Received August 31, 2017; final revision received October 17, 2017; accepted October 20, 2017)

The response to large amplitude oscillatory shear of a soft colloidal glass formed by a suspension of multi-arm star polymers is investigated by means of well-defined continuous local measures. The local measuresprovide information regarding the transient elastic and viscous response of the material, as well as elasticextension via a shifting equilibrium position. It is shown that even when the amplitude of the strain is verylarge, cages reform and break twice per period and exhibit maximum elasticity around the point of zerostress. It is also shown that around the point of zero stress, the cages are extended by a nearly constantamount of approximately 5% at 1 rad/s and 7% at 10 rad/s, even when the total strain is as large as 420%.The results of this study provide a blueprint for a generic approach to elucidating the complex dynamicsexhibited by soft materials under flow.

Keywords: LAOS, nonlinear rheology, colloidal glass

1. Introduction

Understanding large amplitude oscillatory shear (LAOS)

has been the subject of much interest (Hyun et al., 2011).

Early attempts include Fourier transforming (FT) the data

in the time domain to obtain a spectrum of harmonic con-

tributions (Dodge and Krieger, 1971; Harris and Bogie,

1967). However, there are interpretation issues with the

FT approach relating to how we understand the frequency

spectrum rather than the time domain. Transforming the

response into the frequency domain is appealing, but a

generic method for gaining rheological meaning from the

frequency spectrum has yet to be proposed.

The stress decomposition (SD) of Cho et al. (2005) pro-

vides a geometric decomposition of the Lissajous figures

formed by parametrically plotting the oscillating stress

against the strain and the strain rate. Cho’s SD was used

by Ewoldt et al. (2008), who suggested describing the so-

called elastic and viscous stresses by a series of Cheby-

shev polynomials of the first kind. Ewoldt and co-workers

showed how the SD and Chebyshev formalisms are

related to the FT, and also defined a number of local mea-

sures to describe each of the elastic and viscous Lissajous

figures. To each figure they ascribed a secant and tangent

term at the maximum and minimum values of the input,

respectively. This approach poses a couple of interpreta-

tion problems. The first is that the two local measures are

widely separated, and the definitions cannot be clearly

extended to link them. The second problem is that there is

no clear way to simultaneously interpret secant and tan-

gent values even at the same point, without the addition of

more information. Interpreting secant and tangent metrics

when they are widely separated, without clear understand-

ing of how a response goes from one point to the next

remains an ongoing challenge.

Rogers and Lettinga (2012) showed that the SD makes

assumptions regarding the symmetries that are present in

LAOS responses that are too strict. As an example, Rog-

ers and Lettinga showed that when a response includes

yielding, the SD will average over both elastic and viscous

processes to form the so-called ‘elastic stress’ and the

‘viscous stress’. Given the interrelations between the FT,

SD, and Chebyshev approaches, the work of Rogers and

Lettinga (2012) showed that while they are mathemati-

cally robust, the physical interpretations that were attached

to them could not be generally applied. Another approach

was needed.

The original sequence of physical processes (SPP) work

(Rogers et al., 2011) described the response of a soft glass

comprised of star polymers suspended in squalene in a

phenomenological manner as a sequence in time, rather

than a linear superposition. In that work, the response to

LAOS was described as following a distinct sequence of

physical processes that began with purely elastic extension

of the cages. Once a critical amount of strain had been

acquired, the cages broke, and the glass flowed according

to the same processes followed in the steady state flow

curve. As maximum strain is approached, the rate decreases,

and eventually the cages reform when the rate goes to

zero. The process then repeats in the opposite direction.

To support this interpretation, it was shown that the deriv-

ative of stress with respect to strain in the large amplitude

regime resulted in values that matched the linear-regime

# This paper is based on an invited lecture presented by thecorresponding author at the 17th International Symposium on AppliedRheology (ISAR), held on May 25, 2017, Seoul.*Corresponding author; E-mail: sarogers@illinois.edu

Ching-Wei Lee and Simon A. Rogers

270 Korea-Australia Rheology J., 29(4), 2017

elastic modulus. This was interpreted as indicating that

when the cages reform, they do so in a configuration that

is very similar to that probed by the linear regime. This

particular term was called the ‘cage modulus’, and has

since been used in the study of other concentrated soft sys-

tems by van der Vaart et al. (2013), Poulos et al. (2013),

Kim et al. (2014), and recently by Radhakrishnan and

Fielding (2017).

While the phenomenology of the original SPP work lent

credence to the physical interpretation outlined above, the

metrics used were not defined in a universal manner. For

instance, the so-called ‘cage modulus’ is defined only at

the point of zero stress. There was no suggestion regard-

ing tracking the transience of the cages as they deform.

Likewise, even though specific points were interpreted as

reflecting static and dynamic yielding, there was no sug-

gestion of how to track the transition between elastic cage

extension and flow. Interpreting the phenomenological

metrics proposed by Rogers et al. therefore poses some of

the same problems as those faced by the use of the secant

and tangent metrics proposed by Ewoldt et al. (2008).

In this work, we revisit the original data of Rogers et al.

(2011), with an additional data set that has not previously

been published. We apply the recently-proposed fully

quantitative SPP method (Rogers, 2017) to shed light on

the underlying physics that causes the particular macro-

scale responses, and also to demonstrate how the SPP

approach can be generally applied to studies of other soft

matter systems.

Within the SPP framework, the stress response to LAOS

is viewed as being a trajectory in deformation space defined

by the strain, strain rate, and stress. The SPP framework

uses the language of differential geometry of trajectories

in space laid out by Frenet (1852) and Serret (1851). The

orientation of the trajectory is used to define local elastic

and viscous measures at every point in the oscillation,

which allows for the tracking of transient processes in a

quantitative manner. Using the well-defined local mea-

sures of the SPP approach, we show here how the original

interpretation of Rogers et al. (2011) can be refined, and

how interpretations at all parts of the response can be

clearly and smoothly connected. This work can therefore

be used as a guide to using the SPP technique for research-

ers interested in LAOS flows in general.

Before discussing the results obtained from the SPP

approach, the salient features of the framework will be

presented. A full derivation and discussion can be found

in Rogers (2017).

Within the fully quantitative SPP approach, a material’s

response to oscillatory shearing is a trajectory in defor-

mation space given by

. (1)

The trajectory can be described by a set of three ortho-

normal vectors called the tangent, T, principal normal, N,

and binormal, B, vectors. The tangent vector points in the

direction of instantaneous motion, and the principal nor-

mal points in the direction of the derivative of the tangent:

(2)

(3)

The tangent and principal normal vectors span the oscu-

lating plane, which can be thought of as the plane in

which the trajectory sits on a local scale. The binormal

vector is given by the vector cross product of the tangent

and principal normal vectors and therefore details the ori-

entation of the osculating plane:

. (4)

The SPP framework defines two transient moduli,

and , which can be thought of as differential param-

eters: they represent the orientation of the trajectory in

deformation space. As discussed in Rogers (2017), a com-

plete description of any trajectory requires information

regarding the position of the osculating plane as well as

the plane’s orientation. We thus seek an equation of the

form:

, (5)

where the transient moduli, and , represent the ori-

entation of the osculating plane, and the displacement

stress, , represents its position in deformation space.

To determine what form the displacement stress must

have, we start with the point-normal form of the equation

of a plane, noting that the binormal vector is normal to the

osculating plane by definition:

. (6)

We choose to determine the position of the osculating

plane along the stress axis, and so we set the x and y com-

ponents of the plane to zero and solve:

. (7)

Substituting the form of the displacement stress in Eq.

(7) back into Eq. (5) leads to a description of the trajectory

that may be rearranged to the following:

. (8)

On the basis of Eq. (8), we may therefore define the

transient moduli as:

, (9)

A = Aγ Aγ /ω Aσ

[ ] = γ0sinωt γ0cosωt σ t( )[ ]

= γ t( ) γ t( )/ω σ t( )[ ]

T = A·

A------

N = T·

T-------

B = T N×

Gt′Gt″

σ = Gt′γ + Gt″γ· /ω + σ

d

Gt′ Gt″

σd

Bx x Ax–[ ] + By y Ay–[ ] + Bz z Az–[ ] = 0

σd =

Bγ

Bσ

------γ + Bγ·/ω

Bσ

---------γ· /ω + σ

Gt′Bγ

Bσ

------+ γ + Gt″Bγ·/ω

Bσ

---------+ γ·/ω = 0

Gt′ t( ) = −Bγ t( )Bσ t( )------------

A sequence of physical processes quantified in LAOS by continuous local measures

Korea-Australia Rheology J., 29(4), 2017 271

. (10)

In addition to defining transient moduli, the SPP frame-

work also provides explicit definitions of their derivatives,

which can be used to tell us not only whether the response

is softening, stiffening, thickening, or thinning, but also

when and by how much. The derivatives of the transient

moduli have slightly more complex forms than the tran-

sient moduli themselves, and require the principal normal

vector, the binormal vector, and the torsion, ,

which geometrically tells us how fast the osculating plane

rotates around the axis given by the tangent vector:

, (11)

. (12)

In addition to defining time-dependent moduli and

derivatives of the moduli, the SPP framework is unique

among oscillatory shear analysis techniques in that it

allows for unrecoverable strain via the inclusion of a mov-

ing strain equilibrium position, and a yield stress that is

not represented by the moduli. While the orientation of

the osculating plane gives information regarding the local

moduli, it is its displacement, σ d, that contains informa-

tion about the strain equilibrium position, γeq, and the yield

stress, σy. The displacement stress, defined by Eq. (7), is

physically interpreted as being equal to

. (13)

When the response is predominantly elastic, meaning

, the interpretation of the displacement

stress can be simplified to

, (14)

allowing for a straightforward determination of the equi-

librium position, and therefore also the recoverable and

unrecoverable components of the strain.

Application of the SPP framework results in transient

moduli that can be plotted against one another in a time-

dependent Cole-Cole plot. A legend for understanding the

transient moduli is presented in Fig. 1. The position of the

current state is interpreted via Fig. 1a, which provides

information on relative magnitudes of the elastic and vis-

cous contributions at a particular instant. The direction of

motion in the Cole-Cole plot is determined by the deriv-

atives of the moduli, as defined by Eqs. (11) and (12).

Fig. 1b shows how the values of the derivatives lead to

an understanding of whether the response is instanta-

neously stiffening ( ), softening ( ), thicken-

ing ( ), or thinning ( ). By calculating the

values of the derivatives, we obtain accurate well-defined

measures of how quickly the stiffening, softening, thick-

ening, or thinning is occurring. Combining the position in

the Cole-Cole plot with the extra information provided by

the derivative leads to an enhanced understanding of the

transient rheology observed under LAOS.

2. Materials and Methods

Multi-arm star polymers, consisting of a weight average

functionality of 122 polybutadiene (mainly 1, 4-addition)

arms, where each arm has a molar mass of 72 kg/mol are

studied. The details of the synthesis of these star polymers

can be found elsewhere (Roovers, 1993). The star polymer

is suspended in squalene, a nearly athermal solvent for

polybutadiene at more than twice their overlap concentra-

tion, c*. The concentration of 2.5 c* has been shown to

lead to a soft colloidal glass (Christopoulou et al., 2009;

Helgeson et al., 2007; Rogers et al., 2008).

The rheological data were collected using a TA Instru-

ments ARES rheometer with a force rebalance transducer

100FRTN1 with a cone-and-plate geometry of diameter

25 mm and a cone angle of 0.04 rad. All experiments were

carried out at 20 ± 0.01°C. The temperature was con-

trolled via a Peltier plate. Other details of data collection

and handling can be found in Rogers et al. (2011).

3. Results and Discussion

3.1. Traditional characterizationA typical rheological characterization of any material

begins with determination of the steady-state flow curve

and the linear-regime frequency sweep. These two tests

form the axes of the space defined by Pipkin (1972), in

which all oscillatory responses can be said to reside. The

steady-state flow curve represents the response of the

material across a range of shear rates in the limit of zero

frequency. This test forms the ordinate axis of Pipkin’s

Gt″ t( ) = −Bγ·/ω

t( )

Bσ t( )----------------

τ = A·

– N B·

⋅

G·t′ = τ A

· Nγ

Bσ

-------Bγ Nσ

Bσ

2-------------–⎝ ⎠

⎛ ⎞

G·t″ = τ A

· Nγ·/ω

Bσ

----------Bγ·/ω

Nσ

Bσ

2-----------------–

⎝ ⎠⎜ ⎟⎛ ⎞

σd

t( ) = σy t( ) − Gt′ t( )γeq t( )

Gt′ t( ) >> Gt″ t( )

σd

t( ) = Gt′ t( )γeq t( )

G·t′ > 0 G

·t′ < 0

G·t″ > 0 G

·t″ < 0

Fig. 1. Positions and trajectories through -space

in time-dependent Cole-Cole plots can be interpreted via these

legends. The position at an instant is understood in terms of (a),

which assigns physical labels to the specific positions, while

motion through this space is interpreted by (b), which plots the

derivatives of the transient moduli against each other.

Gt′ t( ) Gt″ t( )[ ]

Ching-Wei Lee and Simon A. Rogers

272 Korea-Australia Rheology J., 29(4), 2017

space. The linear-regime frequency sweep represents the

response of the material across a range of frequencies in

the limit of zero shear rate, and makes the abscissa of Pip-

kin’s space. The study of LAOS is usually said to probe

the interior of Pipkin’s space, making a determination of

the responses along the axes an important first step.

The response of the soft glass to these two tests is shown

in Fig. 2. Erwin et al. (2010) described the steady-state

flow curve shown in Fig. 2a by the Carreau-Yasuda vis-

cosity model (Bird et al., 1987) with multiple modes cor-

responding to distinct physical processes. Contributions

were determined that were assigned to β-relaxation, and

transitions between shearing-thinning, shear-banding, and

soft/solid states. In determining the fundamental processes

that lead to the response observed in the linear-regime fre-

quency sweep, Erwin et al. fit the data of Fig. 2b to the

Havriliak and Negami equation (Havriliak and Negami,

1967) with 2 modes representing alpha and beta relaxation

processes.

Strictly speaking, Pipkin defined a space bound by the

Deborah number (equal to the product of the relaxation

time and the angular frequency) and an un-specified ‘shear

amplitude’, which is typically understood to be a Weis-

senberg number (defined as the product of the relaxation

time and the shear rate amplitude). While this approach

has been adopted in many studies, obtaining relaxation

times to form Deborah and Weissenberg numbers is dif-

ficult for the current system without relying on fitting,

such as that performed by Erwin et al. (2010). Regardless

of the lack of a clear relaxation time, some clear state-

ments can still be made about the magnitudes of the Deb-

orah and Weissenberg numbers explored in this study. The

frequencies explored under LAOS in this study are 1 rad/

s and 10 rad/s. From the data of Fig. 2b, this quite clearly

corresponds to large Deborah numbers, because G’ is

nearly an order of magnitude larger than G” at both fre-

quencies. Further, we apply strain amplitudes up to 420%,

which generates shear rates of 4.2 s−1 and 42 s−1, respec-

tively. The data of Fig. 1a indicate that these values are

Fig. 2. The two limiting axes of Pipkin’s space for the star poly-

mer soft colloidal glass: (a) The steady-shear flow curve rep-

resents the response of the material to a range of shear rates at

vanishing frequency. Shown in the inset is the steady shear dif-

ferential viscosity as a function of the shear rate. (b) The linear-

regime frequency sweep is that of a soft glass, and represents the

response across a range of frequencies at vanishing shear rates.

Fig. 3. (Color online) The raw LAOS data and the decomposed SPP time-dependent transient moduli. The elastic Lissajous figures

(a) and (d) are traced clockwise and the viscous Lissajous figures (b) and (e) are traced anti-clockwise. The transient Cole-Cole plots

(c) and (f) are traced anticlockwise. The arrows in (c) and (f) indicate the effect of strain amplitude on the responses, while the dashed

lines indicate the values of the linear viscoelastic moduli.

A sequence of physical processes quantified in LAOS by continuous local measures

Korea-Australia Rheology J., 29(4), 2017 273

found in the central region of the flow curve, where the

differential viscosity (shown in the inset) is a small frac-

tion of the zero-shear value, indicating large Weissenberg

numbers.

3.2. Characterization by the SPP approachShown in Fig. 3 are the Lissajous figures and Cole-Cole

plots of the LAOS data obtained at 1 rad/s (Figs. 3a-3c)

and 10 rad/s (Figs. 3d-3f). The elastic and viscous Lissa-

jous figures are formed by parametrically plotting the

stress against the strain and the rate, respectively. The

elastic Lissajous figures are traced clockwise, while the

viscous Lissajous figures are traced counter-clockwise, as

indicated by the arrows in Figs. 3a and 3b. The Cole-Cole

plots in all cases are deltoid-like curves that are traced in

a counter-clockwise manner. The Cole-Cole trajectories

corresponding to the larger strain amplitudes are indicated

by the arrows in Figs. 3c and 3f. Also shown in Figs. 3c

and 3f are the linear-regime values of the dynamic moduli.

Within the SPP framework, linear viscoelastic responses

have constant moduli, while the moduli change during

nonlinear responses. While not all of the Lissajous figures

are shown for the 1 rad/s experiments for clarity, we indi-

cate what an entire amplitude sweep looks like in the

Cole-Cole representation in Fig. 3c. The starred curve in

Fig. 3c corresponds to the inner-most Lissajous figure

shown in Figs. 3a and 3b.

The data included in the Cole-Cole plots shown in Figs.

3c and 3f are very information-dense. The remainder of

this work will show how specific information can be

extracted from such plots, and how that information leads

to a rich understanding of the dynamic behavior of soft

materials under LAOS. We will begin by ‘unpacking’ spe-

cific elastic and viscous responses from the data of Figs.

3c and 3f, before closely examining a single case to iden-

tify the sequence of processes the material goes through

during a LAOS cycle.

3.3. The cage modulus and elastic strainOne of the remarkable successes of Rogers et al. (2011)

was the definition of the ‘cage modulus’ as the derivative

of the stress with respect to the strain at the point of zero

stress. Rogers et al. (2011) showed how this term could be

calculated for all oscillatory responses, even those well

into the nonlinear regime, and how it provides information

regarding the linear elasticity of the cages. This metric has

since been used by van der Vaart et al. (2013), Poulos et

al. (2013), Kim et al. (2014), and recently by Radhakrish-

nan and Fielding (2017). Despite the correlation between

the cage modulus at all amplitudes and G’ in the linear

regime, there is a significant limitation that is addressed

by the fully quantitative SPP approach. The definition of

the cage modulus assumes the cages extend in a perfectly

linear elastic manner. That is, despite there being a non-

zero G” in the linear regime where intact cage dynamics

are probed, the cage modulus does not account for any

viscous contribution to cage extension.

The fully quantitative SPP approach provides simulta-

neous measures of elastic and viscous processes at all

times during an oscillation. Given the desire to understand

the elastic extension of the reformed cages, it is reasonable

to ask about the position and value of maximum elasticity

as determined by the SPP approach. This is equivalent to

locating the point of greatest in Figs. 3c and 3f. In

fact, when the point of greatest is located in Fig. 3c for

the largest amplitudes, it is immediately clear that there is

a nearly constant maximum elasticity. This value is indi-

cated by the grey arrow in Fig. 3c.

As well as indicating the value of the elastic modulus at

the point of maximum elasticity, the SPP approach also

provides information regarding the viscous modulus at

that point. We show this information in Figs. 4c and 4f. It

is clear from this data that cages do not extend in a purely

elastic manner, but rather extend viscoelastically. The vis-

cous modulus is much smaller than the elastic modulus,

and decreases further with increasing strain amplitude. As

evidenced by the correlation between the open stars and

open circles in Figs. 4c and 4f, the instantaneous viscous

modulus at the point of maximum elasticity is nearly iden-

tical to the traditionally-defined loss modulus that is cal-

culated using the entire waveform.

Further, as per the discussion in Rogers (2017), and pre-

sented here regarding Eq. (14), the SPP approach provides

information regarding the equilibrium position of the

strain when the elastic modulus is much larger than the

viscous modulus. It is clear from the data of Figs. 3c and

3f that the elastic modulus is much larger than the viscous

modulus at the point of maximum elasticity, and so the

simplification of Eq. (14) may be employed.

Collecting the maximum value of the elastic modulus,

the value of the viscous modulus at the same point, and

the equilibrium position determined by the displacement

stress and Eq. (14), we gain a detailed picture of the cage

extension process. Shown in Fig. 4 are the results of this

enhanced cage straining picture. As with previous figures,

Figs. 4a-4c shows data from 1 rad/s, while Figs. 4d-4f

shows data from 10 rad/s. We show in Figs. 4a, 4b, 4d,

and 4e the elastic and viscous Lissajous figures. Shown in

Figs. 4c and 4f are the results of the strain amplitude

sweeps with the values of the transient moduli at the point

of maximum elasticity.

The exact points of maximum elasticity are indicated in

the elastic Lissajous figures in Figs. 4a and 4d by open

symbols. In all cases, the point of maximum elasticity

occurs very near the point of zero stress, which is the pre-

cise point at which the cage modulus was originally

defined. The straight grey lines in the elastic Lissajous fig-

ures indicate the slope of the response expected if the cage

Gt′Gt′

Ching-Wei Lee and Simon A. Rogers

274 Korea-Australia Rheology J., 29(4), 2017

modulus was perfectly elastic with a modulus equal to that

of the linear regime. The red curved lines indicate the vis-

coelastic response of cage extension as determined by the

quantitative SPP approach by combining Eqs. (14) and

(5). It is clear from the amplitude sweeps shown in Figs.

4c and 4f that the cages extend with predominantly elastic

responses (shown by filled stars), but that there is also a

viscous modulus associated with their extension (shown

as open stars) that is very nearly equal to the value of the

average G”.

The clear correspondence between the maximum elas-

ticity determined by the SPP approach and the linear-

regime elasticity suggests that a similar physical process is

responsible. However, it is clear from Figs. 4a and 4d that

the point at which the cages exist and are being extended

according to something close to their linear viscoelastic

response is a long way from where the experiments began

at the point we, in the lab-frame, call zero strain. The sim-

plest explanation for this observation is that the memory

of the material is at least partially erased by flow. When

the material is flowing, it is acquiring strain in a viscous,

unrecoverable manner and therefore makes no special dis-

tinction regarding where the experiment began. There is

clearly some elastic strain associated with the straining of

the cages, but that takes place about an equilibrium point

that is far removed from zero strain. This observation led

Rogers (2017) to make a clear distinction between strains

in the material and lab frames. Using other language, we

may say that strains in the material frame are recoverable

strains, while lab-frame strains are a normalized amount

of deformation and include plastic strains as well as recov-

erable elastic strains. A measurement of strain in the lab

frame is not always equivalent to a measurement of strain

in the material frame and the strains that are responsible

for elastic rheology are material strains, not lab strains.

Any desire to accurately determine elastic contributions to

nonlinear viscoelasticity must therefore make this distinc-

tion. In this respect, the SPP approach is unique in its

allowance for unrecoverable strain via the movement of

the equilibrium position. Not only does the SPP approach

have the concepts of recoverable and unrecoverable strain

built into it, but there is also a mechanism for determining

the point from which strain is acquired recoverably.

As mentioned previously, when the transient elastic

modulus is much larger than the transient viscous modu-

lus, the SPP framework allows for the calculation of the

equilibrium position, and therefore the elastic recoverable

strain, which is determined as the difference between the

Fig. 4. (Color online) The raw LAOS data and the viscoelasticity of cages. The raw data are presented as elastic and viscous Lissajous

figures ((a) and (b) for 1 rad/s, (d) and (e) for 10 rad/s). The red lines correspond to a reconstruction of the stress using the maximum

value of the elastic modulus, the viscous modulus at the same instant, and the equilibrium strain determined from the displacement

stress. The locations at which maximum elasticity is reached are indicated with open circles. The grey lines in (a) and (d) indicate a

purely elastic response with the same elastic modulus as found in the linear regime. The filled and open red stars in (c) and (f) indicate

the maximum elasticity as a function of amplitude, along with the viscous modulus at the same point, respectively.

A sequence of physical processes quantified in LAOS by continuous local measures

Korea-Australia Rheology J., 29(4), 2017 275

total strain and the equilibrium position. Such conditions

hold at the open circles shown in Figs. 4a and 4d, and the

values of the elastic strain are determined and shown as a

function of the total strain amplitude in Fig. 5 for both the

1 rad/s and 10 rad/s cases. If all strain was acquired elas-

tically (recoverably), then the elastic strain would increase

with a slope of 1 in both cases. That is, purely elastic

materials have perfect memories. However, as can be seen

from Fig. 5, this is not the case with the colloidal glass

investigated here. In both frequency cases, the elastic strain

is nearly independent of amplitude in the LAOS regime,

with only weak power law dependences observed. When

oscillating at 1 rad/s, the elastic strain increases with the

amplitude raised to the power of 0.2. This already weak

scaling is reduced if one takes only the largest amplitudes

into account, when the exponent drops to 0.08. Over the

range of amplitudes tested at 10 rad/s, the elastic strain

increases even less with increasing amplitude, where the

power-law exponent is 0.04. In both cases, the average

value of the elastic strain (~5% at 1 rad/s and ~7% at 10

rad/s) corresponds to a strain amplitude that is smaller

than the peak in G” that is typical of type III materials

(Hyun et al., 2002), and indicates that only very small

amounts of strain are elastically acquired even at large

strain amplitudes and large values of the lab-frame strain.

3.4. Flow under LAOSThe major advantage of the new quantitative SPP anal-

ysis scheme over the previous phenomenological analysis

is that there need not be any input from the user to deter-

mine relative elastic and viscous contributions to the stress

at each point in the cycle. As shown in section 3.2, this

allows for the clear determination of viscoelastic exten-

sion of the cages, rather than purely elastic extension that

has been assumed previously. Further, there is a clear

mapping between each point in the cycle, rather than a

determination of viscoelastic responses at widely sepa-

rated points.

The focus of this work so far has been on determination

of the viscoelastic extension of intact cages. The subject

now shifts to how the material flows once those cages

have been broken. We show in Fig. 6 the viscous modulus

divided by the frequency as a function of the rate for both

frequencies tested. Dividing the viscous modulus by the

frequency produces a transient viscosity that is compared

in Fig. 6 directly with the differential viscosity that was

previously shown in the inset of Fig. 2a.

A similar response is observed at both frequencies at

large amplitudes. The transient viscosity remains constant

as the shear rate initially increases from zero. The tran-

sient viscosity then increases and reaches a local maxi-

mum before decreasing rapidly, and ultimately following

the steady-state flow response. The transient viscosity

remains non-zero as the rate goes back down to zero.

Symmetry dictates that the value of the transient viscosity

at zero rate is the same whether the rate is increasing or

decreasing.

Even though the stress overshoot observed in the Lis-

sajous figures shown in Figs. 3 and 4 continues to increase

with increasing amplitude, the overshoot in the transient

viscosity actually has a global maximum value at each fre-

quency. Further, the position of the maximum in the tran-

sient viscosity is slightly earlier than the stress overshoot.

This indicates that the stress overshoot is a complex phe-

nomenon that is caused by a combination of elastic and

viscous effects, and cannot be accounted for by either

elastic or viscous processes in isolation.

3.5. A detailed sequence of physical processesThe previous sections have shown how the quantitative

SPP analysis scheme can be used to extract information

from the LAOS response of a soft colloidal glass regard-

Fig. 5. The elastic strain at the point of maximum elasticity. The

SPP approach is unique in its allowance for the amount of elastic

strain in the system to differ from the total strain. The elastic

strains are significantly smaller than the total strain amplitude,

and are only very weakly dependent on the strain amplitude.

Dashed lines indicate an elastic strain equal to the strain ampli-

tude, while solid lines are power law fits with exponents of 0.2

at 1 rad/s and 0.04 at 10 rad/s.

Fig. 6. (Color online) The viscous modulus divided by the fre-

quency plotted on top of the steady state flow curve differential

viscosity shown in Fig. 2a. The viscous modulus goes through a

maximum before decreasing toward, and then following the

steady-state flow curve. Lines are the LAOS experiments, while

hollow symbols represent the steady-shear flow curve response.

Ching-Wei Lee and Simon A. Rogers

276 Korea-Australia Rheology J., 29(4), 2017

ing the modulus of the cages pre-yielding, the extension of

the cages at the point of maximum elasticity, and the tran-

sient viscosity as the glass is forced to flow. In this final

section we carefully step through the entire sequence of

processes exhibited by the colloidal glass under a single

oscillatory condition. Even though we have chosen to

examine the case of a strain amplitude of 420% at an

angular frequency of 1 rad/s, the process we follow of

interpreting the SPP metrics is generally applicable. In this

section we make use of the elastic Lissajous figures, the

transient Cole-Cole plots, where the values of the transient

moduli are shown, as well as plots of the derivatives of the

transient moduli. By using all three display methods, we

can clearly identify and quantify the complex dynamics

associated with LAOS flow of the soft colloidal glass.

In order to be able to discuss a sequence of processes,

one must make an explicit choice of where to begin.

While we make the choice to start at a (lab-frame) strain

of zero, with positive rate, according to the traditional

description of strains being sinusoidal and rates being

cosinusoidal, we note that the SPP approach is immune to

problems associated with other approaches when other

choices of where one chooses to call ‘the start’ are made

(Rogers, 2017).

Beginning at a lab-frame strain of zero in the steady

state LAOS response shown in Fig. 7a, we note the mate-

rial response is in the fluidized state, flowing according to

the steady-state flow curve, as discussed in section 3.3.

This is evidenced by the transient elastic modulus, ,

having a value of approximately zero, but there being a

non-zero value of the transient viscous modulus, seen

in Fig. 7b. It is also apparent that there is very little change

in the moduli, given the fact that the derivatives have very

small values, as seen in Fig. 7c.

As the maximum lab-frame strain is approached, the rate

decreases to zero and the cages that dominate the linear

Gt′

Gt″

Fig. 7. The sequence of physical processes as supported by the quantitative SPP framework for a strain amplitude of 420% at 1 rad/

s. The grey lines indicate the whole response, while black sections indicate specific processes. The sequence of processes is discussed

in detail in the text.

A sequence of physical processes quantified in LAOS by continuous local measures

Korea-Australia Rheology J., 29(4), 2017 277

viscoelastic response reform. This part of the stress wave-

form is shown in Fig. 7d. The reformation of cages results

in a singificant and rapid increase in the elastic modulus

as seen in Fig. 7e, with only minor changes in the transient

viscous modulus. This is a nearly purely stiffening pro-

cess, as seen by the positive value of the time deriviate of

the elasticity in Fig. 7f. At the point of maximum elas-

ticity, where the transient elastic modulus is much greater

than the transient viscous modulus, the displacement

stress can be simplified as per Eq. (14), and the elastic

strain can be determined as discussed in section 3.2. The

fact that the elastic strain is significantly different from the

total strain, as seen in Fig. 5 is a clear indication that in the

flowing portion of the trajectory, strain is acquired vis-

cously, in an unrecoverable manner, breaking the symme-

try between lab-frame and material-frame strains.

Once cages are strained beyond a critical limit, they

break (yield) and any still intact elastic structure recoils.

The portion of the stress waveform during which this pro-

cess takes place is indicated by Fig. 7g, while the recoil is

seen in the rapid decrease, and then negative value of the

transient elastic modulus in Fig. 7h. During this same

breakage process, the transient viscosity rapidly increases.

The rapid softening and subsequent recoil manifests as a

large negative value of the derivative of the transient elas-

tic modulus as seen in Fig. 7i. During the same process,

the rapid increase in the transient viscous modulus leads to

a large positive value of its derivative. At the peak, it can

be seen from Fig. 7i that the material response is softening

at a rate of more than 4500 Pa/s, and thickening at a rate

of around 1000 Pa/s.

The next processes that are undergone happen very

quickly and involve the overshoot feature observed in the

stress response as seen in Fig. 8a. Once the recoil has

reached a maximum, the transient elastic modulus begins

to increase again. This happens rapidly over an interval

Fig. 8. The sequence of physical processes as supported by the quantitative SPP framework for a strain amplitude of 420% at 1 rad/

s. These images directly follow those of Fig. 7. The grey lines indicate the whole response, while black sections indicate specific pro-

cesses. The sequence of processes is discussed in detail in the text.

Ching-Wei Lee and Simon A. Rogers

278 Korea-Australia Rheology J., 29(4), 2017

around a tenth of a second, with the value of the transient

elastic modulus changing by around 400 Pa, (Fig. 8b) at

approximately 4000 Pa/s (Fig. 8c). At the same time, the

structure becomes easier to flow, and the transient viscous

modulus decreases (Fig. 8b) by around 150 Pa, at a rate of

1500 Pa/s (Fig. 8c).

Immediately after the overshoot feature has occurred, as

seen in Fig. 8d), both the elastic and viscous transient

moduli are significantly different from their stable flow

values. The change that takes place next occurs over

another short interval, on the order of a tenth of a second,

when the transient elastic modulus drops by about 100 Pa

and the transient viscous modulus increases by about 60

Pa (Fig. 8e), leading to a softening rate of around 1000 Pa/

s and a thickening rate of around 600 Pa/s (Fig. 8f).

After the short intervals where the changes are very

large and rapid, the material settles into its natural flowing

state that also marked the beginning of the sequence (Figs.

8g-8i).

While it is not a particular focus of this work, we are

able to make some remarks regarding the yielding of the

material. According to the interpretations afforded by the

SPP approach, the time between the point of maximum

elasticity and stable flow is on the order of half a second.

Yielding has been the subject of much debate in the rhe-

ological literature, and is thought to occur gradually over

a finite time interval (Coussot, 2014). The SPP analysis

carried out here implies that the entire yielding process

occurs twice per period over an interval shorter than half

a second. Given that the SPP approach is designed to han-

dle transient rheological responses under flow, it is there-

fore the ideal tool with which to study yielding.

The sequence described above, with the help of the data

of Figs. 7 and 8, occurs twice per period, once in each

direction. This symmetry means that even though the Lis-

sajous figures are only traced out once per period, the

Cole-Cole plots, and the trajectories of the derivatives are

traced out twice per period.

4. Conclusions

Large amplitude oscillatory shear data from the response

of a soft colloidal glass of star polymers, initially pub-

lished by Rogers et. al. (2011), has been revisited with the

new quantitative sequence of physical processes (SPP)

analysis (Rogers, 2017). With the new technique, we are

able to provide a more in-depth quantitative picture of the

complex dynamics than previous phenomenological

approaches.

Mapping out the maximum elasticity per period over a

range of amplitudes at two different frequencies, and the

value of the transient viscous modulus at the same point,

we are able to determine the complete viscoelastic behav-

ior of cages of nearest neighbors. While the elasticity of

the cages is nearly independent of the amplitude, the

amount of viscous dissipation decreases with amplitude.

The point in the trajectory at which maximum elasticity is

determined is very nearly at the point of zero stress at all

amplitudes. This finding coincides with the previous defi-

nition of the so-called ‘cage modulus’, which was defined

as the derivative of the stress with respect to strain at the

point of zero stress. We note that while the cage modulus

required a particular choice to be made regarding where it

should be defined, the position of maximum elasticity in

the SPP approach is completely dictated by the material

response and requires no user input. This work therefore

provides confirmation of the utility of the cage modulus as

a useful metric for other concentrated systems, but also

suggests a more general alternative that does not require

the concepts of caging dynamics to be useful.

The point of zero stress occurs a long way from zero

strain in the lab frame in all cases. Further, the near equal-

ity of the linear viscoelastic moduli and the transient mod-

uli at the point of maximum elasticity suggests the similar

physical states dominated by caging dynamics are being

probed in each case. This further suggests that the equi-

librium strain from the perspective of the material is

located near the point of zero stress and that the total strain

in the material frame is significantly smaller than the total

strain in the lab frame. The SPP technique allows for the

calculation of the elastic (recoverable) strain at the point

where the transient elastic modulus is at its maximum

value. It has been shown that the elastic strain is nearly

independent of amplitude and is indeed much smaller than

the total strain that has been applied to the soft glass. The

small values of elastic strain, approximately 5% at 1 rad/

s and 7% at 10 rad/s, suggest that during the LAOS of the

soft glass, most of the strain is acquired via an unrecov-

erable process. We therefore conclude that LAOS is able

to erase a material’s memory by breaking the memory-

bearing structure and enforcing flow.

It has also been shown that when flow of the soft glass

is enforced by LAOS, the flow conditions followed are

transient versions of the steady-state flow curve.

The level of quantitative detail provided by the SPP

approach in elucidating the complex dynamics that soft

materials exhibit during LAOS is expected to find great

utility in future investigations of transient yielding and

other time-dependent phenomena. Because of the signifi-

cant temporal resolution afforded by the SPP approach, it

is anticipated that dynamics that occur on timescales much

shorter than the period of oscillation will be clearly iden-

tified in a range of materials. Further, by separating the

lab-frame and material-frame strains, use of this approach

could provide a level of detail that would lead to further

theoretical and model developments in the study of soft

materials.

A sequence of physical processes quantified in LAOS by continuous local measures

Korea-Australia Rheology J., 29(4), 2017 279

Acknowledgements

SAR thanks the organizers of the ISAR meeting in

2017, at which some aspects of this work were presented,

for the invitation to present this work.

References

Bird, R.B., R.C. Armstrong, and O. Hassager, 1987, Dynamics of

Polymeric Liquids Vol. 1: Fluid mechanics, 2nd

ed., John Wiley

and Sons Inc., New York.

Cho, K.S., K.H. Ahn, and S.J. Lee, 2005, A geometrical inter-

pretation of large-amplitude oscillatory shear response, J.

Rheol. 49, 747-758.

Christopoulou, C., G. Petekidis, B. Erwin, M. Cloitre, and D.

Vlassopoulos, 2009, Ageing and yield behavior in model soft

colloidal glasses, Philos. Trans. R. Soc. Lond, Ser. A- Math.

Phys. Eng. Sci. 367, 5051-5071.

Coussot, P., 2014, Yield stress fluid flows: A review of experi-

mental data, J. Non-Newton. Fluid Mech. 211, 31-49.

Dodge, J.S. and I.M. Krieger, 1971, Oscillatory shear of nonlin-

ear fluids. I. Preliminary investigation, Trans. Soc. Rheol. 15,

589-601.

Erwin, B.M., D. Vlassopoulos, and M. Cloitre, 2010, Rheological

fingerprinting of an aging soft colloidal glass, J. Rheol. 54,

915-939.

Ewoldt, R.H., A.E. Hosoi, and G.H. McKinley, 2008, New mea-

sures for characterizing nonlinear viscoelasticity in large-

amplitude oscillatory shear, J. Rheol. 52, 1427-1458.

Frenet, F., 1852, Sur les courbes à double courbure, J. Math.

Pures Appl. 17, 437-447.

Harris, J. and K. Bogie, 1967, The experimental analysis of non-

linear waves in mechanical systems, Rheol. Acta. 6, 3-5.

Havriliak, S. and S. Negami, 1967, A complex plane represen-

tation of dielectric and mechanical relaxation processes in

some polymers, Polymer 8, 161-210.

Helgeson, M.E., N.J. Wagner, and D. Vlassopoulos, 2007, Vis-

coelasticity and shear melting of colloidal star polymer glasses,

J. Rheol. 51, 297-316.

Hyun, K., M. Wilhelm, C.O. Klein, K.S. Cho, J.G. Nam, K.H.

Ahn, S.J. Lee, R.H. Ewoldt, and G.H. McKinley, 2011, A

review of nonlinear oscillatory shear tests: Analysis and appli-

cation of large amplitude oscillatory shear (LAOS), Prog.

Polym. Sci. 36, 1697-1753.

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 Mech. 107, 51-65.

Kim, J., D. Merger, M. Wilhelm, and M.E. Helgeson, 2014,

Microstructure and nonlinear signatures of yielding in a het-

erogeneous colloidal gel under large amplitude oscillatory

shear, J. Rheol. 58, 1359-1390.

Pipkin, A.C., 1972, Lectures on Viscoelasticity Theory, Springer-

Verlag, New York.

Poulos, A.S., J. Stellbrink, and G. Petekidis, 2013, Flow of con-

centrated solutions of starlike micelles under large-amplitude

oscillatory shear, Rheol. Acta 52, 785-800.

Radhakrishnan, R., and S.M. Fielding, 2017, Shear banding in

large amplitude oscillatory shear (LAOStrain and LAOStress)

of soft glassy material, arXiv preprint, arXiv:1704.08332v1.

Rogers, S.A., 2017, In search of physical meaning: Defining tran-

sient parameters for nonlinear viscoelasticity, Rheol. Acta 56,

501-525.

Rogers, S.A., B.M. Erwin, D. Vlassopoulos, and M. Cloitre,

2011, A sequence of physical processes determined and quan-

tified in LAOS: Application to a yield stress fluid, J. Rheol. 55,

435-458.

Rogers, S.A., D. Vlassopoulos, and P.T. Callaghan, 2008, Aging,

yielding, and shear banding in soft colloidal glasses, Phys. Rev.

Lett. 100, 128304.

Rogers, S.A. and M.P. Lettinga, 2012, A sequence of physical

processes determined and quantified in large-amplitude oscil-

latory shear (LAOS): Application to theoretical nonlinear mod-

els, J. Rheol. 56, 1-25.

Roovers, J., L.L. Zhou, P.M. Toporowski, M. van der Zwan, H.

Iatrou, and N. Hadjichristidis, 1993, Regular star polymers

with 64 and 128 arms. Models for polymeric micelles, Mac-

romolecules 26, 4324-4331.

Serret, J.-A., 1851, Sur quelques formules relatives à la théorie

des courbes à double courbure, J. Math. Pures Appl. 16, 193-

207.

van der Vaart, K., Y. Rahmani, R. Zargar, Z. Hu, D. Bonn, and

P. Schall, 2013, Rheology of concentrated soft and hard-sphere

suspensions, J. Rheol. 57, 1195-1209.