Time-resolved dynamics of the yielding transition in soft materials

Accepted Manuscript

Time-Resolved Dynamics of the Yielding Transition in Soft Materials Gavin J. Donley , John R. de Bruyn , Gareth H. McKinley , Simon A. Rogers PII: DOI: Reference:

S0377-0257(18)30215-5 https://doi.org/10.1016/j.jnnfm.2018.10.003 JNNFM 4052

To appear in:

Journal of Non-Newtonian Fluid Mechanics

Received date: Revised date: Accepted date:

24 June 2018 19 September 2018 13 October 2018

Please cite this article as: Gavin J. Donley , John R. de Bruyn , Gareth H. McKinley , Simon A. Rogers , Time-Resolved Dynamics of the Yielding Transition in Soft Materials, Journal of Non-Newtonian Fluid Mechanics (2018), doi: https://doi.org/10.1016/j.jnnfm.2018.10.003

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Highlights: The Frenet-Serret frame is the natural way to analyze material responses. The motion of an instantaneous phase angle is used to track the yielding transition. The yielding of soft materials is a gradual transition. Yielding can be time-resolved within a period of oscillation using this approach. The yield state of a Carbopol microgel is mapped in deformation and period time.

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Time-Resolved Dynamics of the Yielding Transition in Soft Materials Gavin J. Donleya, John R. de Bruynb, Gareth H. McKinleyc, and Simon A. Rogersa,* a

Department of Chemical and Biomolecular Engineering, University of Illinois at UrbanaChampaign, Urbana, IL, USA Department of Physics and Astronomy, University of Western Ontario, London, Ontario, Canada N6A 3K7

c

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b

Department of Mechanical Engineering, Hatsopoulos Microfluids Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

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Abstract

A rigorously-defined method that maps and quantifies the time-resolved dynamical yielding process of elastoviscoplastic materials is proposed and investigated. Building on the foundations of linear viscoelastic theory for oscillatory deformations, the method utilizes the motion of an instantaneous phase angle between the stress and strain of the rheological response within

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deformation space to quantify the yielding transition. Principal component analysis demonstrates that this phase angle velocity is based on the natural description for a material response in

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deformation space. The response of the Carreau model with constitutive parameters selected to correspond to a regularized viscoplastic fluid that undergoes an apparent yielding transition at a

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critical shear rate, and of a Carbopol microgel are investigated as canonical examples of theoretical and experimental model yield stress fluids. Calculation of the phase angle velocity

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clearly identifies the yield transition as being gradual. This approach provides a physicallymotivated understanding of the dynamical processes occurring during yielding of

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elastoviscoplastic materials. Keywords

Yield stress materials; Rheological measurements; Large amplitude oscillatory shear (LAOS); Sequence of physical processes (SPP); Time-resolved rheology; Yield threshold

* Corresponding author. Email address: [email protected]

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Declarations of interest: none

1. Introduction Soft materials that transition between solid-like and liquid-like behavior under sufficient applied

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deformation or stress are ubiquitous in daily life and numerous industrial and environmental processes. These materials, commonly referred to as yield-stress fluids, or, more formally, elastoviscoplastic fluids, are used in an enormous variety of applications, ranging from drilling fluids to body lotions, and from mucous to uncured concrete [1 – 4]. The transition from primarily solid-like to primarily liquid-like behavior under deformation is referred to as yielding

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and manifests as, for instance, the spreadability of mayonnaise and the squeezability of toothpaste. The phenomenon of yielding is also at the heart of the initiation of avalanches and mudslides. Furthermore, novel yielding materials show promise for applications such as stably suspending particles with a significant density mismatch within a fluid matrix [1, 5], and as building blocks or scaffolds in the precision manufacturing of soft materials [1, 6]. Each of these

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diverse applications gives the user the ability to control and dramatically vary the flow properties of the material by simply changing the magnitude of stress or strain applied to the material.

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Despite the ubiquity of yielding behavior in soft materials, it has remained difficult to describe in a quantitative manner. Many different definitions have been proposed in the literature [7 – 8].

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While these definitions vary with regard to specifics, they generally agree that yielding is a transition from solid-like, primarily elastic deformation to liquid-like, primarily viscous or

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plastic flow under continued loading history, and is associated with the reversible breakdown of microstructure within a given material. We adopt this general definition for the purposes of this

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work.

While the properties of yielding materials have been exploited for centuries, they were first investigated as their own class of materials by Bingham [9 – 10] in the early 1900‘s. Bingham chose to model materials exhibiting what he termed ―plastic‖ flow, the liquid-like flow of a deformable solid under sufficient stress, as being perfectly plastic (or non-deformable) solids below a critical value of stress, and as Newtonian liquids with a constant plastic viscosity above it [9, 11]. To determine the yield point, the stress applied to the material is extrapolated back to 3

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the zero-flow line. Bingham‘s now-eponymous model [11] was the first attempt to characterize yielding behavior, and suggested that yielding is an instantaneous phenomenon occurring at a specific value of stress. While modified versions of the Bingham model have been proposed to characterize increasingly complex behaviors [7], including the Herschel-Bulkley model [12] and the Casson model [13], the assumption of an instantaneous yield transition remained.

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Furthermore, these models did not consider the possibility of any elastic response below yield [14], instead maintaining that yielding materials are perfectly plastic below the yield point.

The idea of a yielding transition as laid out by Bingham remained relatively unchallenged for many years, with the first significant opposition coming from Barnes and Walters in their 1985 paper ―The Yield Stress Myth?‖ [15]. These authors claimed that yielding was not a transition at

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all, but rather an illusion caused by data of limited accuracy. Specifically, they claimed that prior measurements had insufficient accuracy at low shear rates, and that the yielding was therefore an experimental artefact. Their work ignited significant debate regarding the validity of the yield stress as a concept [1, 7, 10]. Notably, the quest to prove whether or not yielding behavior existed produced a proliferation in techniques for measuring the yielding transition [16]. This

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included a number of rheological techniques such as creep [4, 8, 16 – 18], stress relaxation [16], stress growth [4, 8, 16 – 17, 19 – 22], and amplitude sweeps [8, 17, 20, 23], as well as other

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pragmatic bench-top methods, including slump tests [4], inclined plane avalanche [4, 16, 24], and static equilibrium [16, 25] tests. A consensus was reached that the yield stress, and the

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yielding transition in general, is an engineering reality; it is a helpful metric for analysis and is valid on the time scale of the typical experiment [1, 7, 26]. Additionally, more recent work [27,

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28] has been able to definitively show the presence of a yielding transition in certain classes of material, with a given material transitioning from elastic or elastoplastic to elastoviscoplastic

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behavior with increasing imposed stress or deformation. While the range of tests now in existence for characterizing the yielding transition in soft materials is extensive, accurate determination of the yield point remains inconclusive. Numerous studies have shown that different methods give different values, and may reflect different parts of the yielding process [8, 17]. This is exacerbated by the fact that many of the tests have multiple proposed metrics for the determination of the yield point [17]. , For example, no fewer than five distinct metrics have been proposed for the location of yielding from an oscillatory 4

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strain amplitude sweep: 1) the end of the linear regime [8], or the point at which nonlinearities first begin to affect the dynamic moduli; 2) the crossover point between the small- and largeamplitude responses in

[17, 23]; 3) the crossover point between the small- and large-

amplitude responses in the stress amplitude [17]; 4) the point of maximum the crossover point between

and

[29 – 31]; and 5)

[8, 17]. This has led to the assertion by Rouyer et al. that

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―There is no unique and rigorously motivated criterion allowing a yield stress to be determined from oscillatory data‖ [23]. The variability between the metrics proposed for the yielding transition raises the question of which, if any, are correct, and what they physically represent. At least part of the discussion surrounding how to measure yielding stems from the assumption originating with Bingham that yielding is an instantaneous phenomenon occurring at a specific

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yield point. Such an assumption does not explain a number of experimental observations [14], including shear banding under uniform stress [32], shear localization [33], and multi-stage transitions [34]. Additionally, interpretation of such a model becomes difficult when measuring deformation under inhomogeneous shear, as yielded and unyielded regions can coexist [35]. Finally, while a sharp yielding transition has been seen in simulations of materials with a

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controlled dispersity [36], such a transition is unlikely even in a ―simple‖ amorphous yield stress material such as Carbopol due to its heterogeneous microstructure [37 – 38]. This is further

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corroborated by the fact that many experimental studies have shown that the yielding of Carbopol is a complex, multi-step process [18, 21 - 22, 39 – 40], indicating that the idea of a

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single yield point is a tenuous assumption at best. The instantaneous yielding assumption best lends itself to description by methods or metrics

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which are binary in nature, i.e. a material is either yielded or unyielded. Because of this, nearly all proposed metrics for yielding either utilize long-time, steady-state values or measures that

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have been averaged over some period of time [17]. While such measures would be adequate if yielding were indeed instantaneous, the measures themselves are unable to resolve whether this is the case, nor if yielding occurs over an extended interval of time. Furthermore, these techniques have no capacity to determine when yielding begins, or when it could be said to be complete, if it is a gradual process. Many tests also cannot distinguish yielding from the preyielding behavior of the system, a point which has been shown to be significant by multiple authors [20, 28]. Finally, while models such as the soft glassy rheology (SGR) model [41 – 43] 5

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and shear transformation zone (STZ) theory [44] allow the individual microstructural elements of a given material to yield in a distributed manner, almost all currently proposed metrics for experimental characterization of yielding assume that the stress or strain applied to a material on the macroscopic level is representative of the local, microscale stresses or strains throughout that material. Extensive transient velocimetry studies on the start-up of steady shear [21, 39] and

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creeping flows [18] have suggested that the microstructural processes leading to yielding are significantly more complex than the macroscopic rheological data can portray.

Many applications of yielding materials require an understanding of flow behavior on timescales far shorter than that required to reach the steady state [1], and many novel application areas for yield stress fluids would benefit from a precise, reproducible technique for measuring the yield

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point that reflects the dynamical processes occurring within a material [2, 45]. Such a method could enable the connection of desired macroscale flow properties to specific microscale dynamics, potentially allowing for the design of soft materials with specifically tailored yielding transitions. A technique that can quantitatively resolve what conditions trigger yielding, as well

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as when and how yielding occurs, is therefore desirable.

Any technique that successfully describes the dynamics of yielding must be able to provide a time-resolved picture of the properties of the material and ideally would do so through a

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mathematical framework that accurately represents the physical processes occurring in the material. In order to achieve the desired time resolution, the selected method must be able to

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process time-varying rheological data and independently resolve the instantaneous elastic and viscous components of the material response. This would allow the instantaneous contributions

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of solid- and liquid-like behavior to be quantified, and for the transition between the two asymptotic limits to be clearly denoted. Strain-controlled large-amplitude oscillatory shear

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(LAOS) stands out as an ideal experimental platform in this regard, as oscillatory measurements can be unambiguously decomposed into strain-dependent and rate-dependent contributions, providing the viscoelastic data needed. Further, the cyclic nature of the deformation and the ability to vary the amplitude allow for repeated observation of the yielding and unyielding transitions over a variety of timescales or test frequencies. It is worth noting that while extensive studies have been carried out on the dynamics of yielding under steady shear [20, 39] and on creeping flows [18], many of these studies do not reflect the application timescales of many 6

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yielding processes, nor do they allow for investigation of the reverse process (i.e. reestablishment of the unyielded or non-flowing state). Several recent studies have provided new insight into the complex dynamics of the yielding process, however, with [46] investigating the phenomena of delayed yielding under creeping flow, and [33] looking at the presence of multiphase yielding in slow flows of yield stress fluids.

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A number of data analysis frameworks have been proposed for LAOS data, including Fourier transform rheology [47], the stress decomposition approach [48], the waveform decomposition approach [49], the Chebyshev polynomial approach [50 – 51], and the sequence of physical processes (SPP) framework [52 – 55]. While each of these approaches can be used to interpret time-resolved oscillatory data, the assumptions underlying the various decompositions can lead

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to internal contradictions for elastoviscoplastic materials [51, 53, 56], resulting in measures with reduced physical meaning, especially when applied to yielding. In contrast, the SPP approach uses the derivatives of a material response to examine the changes which occur in a material at every instant in time within the cycle [53] and has been shown to connect time-resolved

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rheological data to the dynamic processes which occur within a material [57]. In the present work, we introduce a novel method to determine the yielding transition under strain-controlled LAOS. The goal of the investigation is to enable time-resolved characterization

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of the transition to address a number of currently unresolved questions: Is yielding an instantaneous, or gradual transition? If it is gradual, what is the characteristic timescale for

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yielding in a given elastoviscoplastic material?

2. Theory

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2.1 Oscillatory shear rheology When an oscillatory shear deformation, ( ) frequency

(

), with strain amplitude

and angular

, is applied to an elastic material the shear stress response is in phase with the

deformation. Conversely, if the material is a viscous Newtonian fluid with viscosity

, the

response is out of phase with the strain.

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When periodically deforming a material that is viscoelastic, the stress response can be represented as a combination of these two responses. For small enough deformations, the combination is linear and typically denoted by ( ) and

(

)

(

)]

(1)

are the frequency-dependent storage and loss moduli of the material,

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where

[

respectively. These dynamic moduli are interpreted as representing the solid-like (in-phase) and liquid-like (out of phase) components of the material response. Alternatively, eqn. 1 can be written in the form ( ) |

|

|

(

)

| is the magnitude of the complex modulus of the material, and

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where |

|

phase angle between the imposed strain and the measured stress. Small values of represent more solid-like responses, while values of

approaching

(2) is the

therefore

are indicative of more

liquid-like responses. The simple relationships expressed in eqns. 1 and 2 describe linear material responses only, and therefore do not hold once the deformation becomes large enough

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to leave the linear regime. As such, other methods must be used to accurately describe the

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nonlinear rheological properties of the material.

2.2 The Sequence of Physical Processes approach

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The sequence of physical processes (SPP) technique, introduced by Rogers [53], has been shown to provide physically valid interpretations of nonlinear rheological data. The technique uses the and

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mathematics of trajectories from differential geometry to define instantaneous moduli (

) throughout the time-varying response of the material to deformation. This is achieved by viewing the material response within a three-dimensional deformation space defined by the shear

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strain, the shear rate, and the shear stress. A complete derivation of the SPP parameters can be found elsewhere [53]; here we briefly review the central tenets of the approach. Within the SPP framework, the rheological response to oscillatory strain is viewed as a closed three-dimensional space curve in strain, rate, and stress space: ⃑

〈

̇⁄

〉.

(3)

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At each point along the curve, a set of three orthonormal vectors define the Frenet-Serret frame: the tangent, normal, and binormal vectors. These vectors define the instantaneous direction of motion, the instantaneous change in direction, and the vector cross product of these two, respectively [53]. A full derivation of the components of the Frenet-Serret frame can be found in appendix A.

define the instantaneous moduli:

(4) (5)

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̇⁄

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The binormal vector has projections along the strain, rate, and stress axes, which can be used to

The instantaneous moduli remain constant for any linear viscoelastic response which corresponds to a planar trajectory along which the binormal vector is constant. For rheologically nonlinear, non-planar responses, the binormal vector will change its orientation during the course of each imposed oscillatory cycle, and the instantaneous moduli will change accordingly. This

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connection allows changes in the orientation of the Frenet-Serret frame to be linked back to an

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underlying sequence of physical processes in the material. An example of a non-planar material response is displayed in Fig. 1(a), along with the resulting

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Frenet-Serret frame at three different points along that trajectory.

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Fig. 1. 3D representation of nonlinear response in deformation space as analyzed by: (a) FrenetSerret Frame and (b) Principal component analysis. The light blue arrow denotes the rotation of the frame as the properties of the material evolve with continued loading history.

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2.3 Elastic Strain Prior studies have shown that the oscillatory stress and strain measured during a material response to LAOS deformation can be decomposed into elastic and viscous contributions [48, 53, 57]. To definitively measure the elastic strain at any point in the cycle, a series of recovery tests would be needed at each point [57]. Using the SPP framework, however, it has been shown , the elastic

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that when the instantaneous response is predominantly elastic, i.e.,

recoverable strain at that instant can be estimated by treating the response as Hookean: ( )

( )

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A full derivation of this result can be found in [53].

(6)

2.4 The Frenet-Serret frame’s relation to the principal components of a rheological trajectory The most generic way to define an orthogonal frame for a given dataset is to subject the dataset to a principal component analysis (PCA) [58 – 59]. This analysis allows for the definition of a frame that is agnostic to the input dimensions of the data, as it is based solely on the directions in

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which the measured variations in the data are the most significant. The PCA analysis consists of an orthogonal linear transformation in which the first target dimension represents the direction of

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highest variance, the second dimension is in the direction of highest variance orthogonal to the first, and so on. PCA can thus be used to determine the natural dimensions for analyzing a given

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data set, which are subsequently referred to as its principal components. The same non-planar material response discussed in section 2.2 was subjected to a principal

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component analysis to determine the most natural coordinate system for the description of variances in the response. This was carried out using the PCA function in MATLAB on portions rad/s), centered on the target

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of the dataset 16 points long (i.e. 1/32 of the cycle, ~0.2 s for point. The resulting principal components are shown in Fig. 1(b).

Comparison of the Frenet-Serret frame with the principal components reveals that the tangent vector is the first principal component (inner product the second principal component (inner product

), the normal vector correlates with ), and the binormal correlates with the

third principal component at each point (inner product

). This is consistent with their

respective definitions, as the tangent vector represents the direction in which the material 11

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response moves with time, making it the most significant variation, and the normal vector represents changes in the tangent, making it the second most significant component. The binormal vector is the last remaining orthogonal direction in the Frenet-Serret frame, making it the least significant component within 3D space. The two frames are identical in definition and numerically within experimental error (

). The Frenet-Serret frame, upon which the SPP

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approach is based, is therefore the natural way to view and analyze responses within the deformation space of a LAOS experiment. 2.5 Identifying yielding in soft materials

Yielding is an inherently nonlinear rheological response that leads to strongly non-planar responses within the deformation space defined by strain, strain-rate, and stress. We seek a

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simple metric that will efficiently describe the geometrical/rheological transition that takes place during the yielding process.

An alternative to expressing a nonlinear response in terms of the instantaneous SPP moduli is to use the instantaneous magnitude and phase angle of the complex modulus [53], in an extension

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of eqn. 2. These quantities are defined in terms of the SPP moduli as |

√

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|

(7) )

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(

(8)

An illustration of this conversion can be seen in fig. 2(e). We interpret these parameters as time-

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resolved versions of their linear viscoelastic (cycle-averaged) counterparts: | instantaneous magnitude of the viscoelastic response, while

| is the

represents the ‗quality‘ or

instantaneous phase of the viscoelastic response. As in the linear viscoelastic regime, small represent solid-like responses, while large values of

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values of

indicate more viscous-like

instantaneous responses. Since, as noted previously, we broadly define yielding as the transition from a solid-like response to a liquid-like response, it can therefore be identified on a macroscopic scale as the point within the cycle when the instantaneous material response changes from primarily elastic,

⁄ , to primarily viscous,

crossover in the instantaneous moduli

and

⁄ . This is equivalent to a

, and can thus be seen as a time-resolved

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extension of a crossover in the average moduli in an amplitude sweep, which is widely regarded as the highest possible threshold for yielding in such a test [17]. Fernandes et al. [8] define yielding as the transition from a completely structured state to a completely unstructured state. We believe this definition overemphasizes the magnitude of the structural break-down during yielding, and that yield-stress materials retain a significant degree of internal structure

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immediately after yielding. However, the ideas presented by Fernandes et al. prompt us to define complete yielding as a process in which the phase angle goes to the viscous limit of

⁄ .

Similarly, we refer to any yielding process in which the phase angle is in the range ⁄

⁄ as incomplete yielding. We can make further distinctions by referring to a yielding transition that takes place instantly as ‗Bingham yielding‘, and one that takes place over a finite duration of

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time (or strain) as ‗non-Bingham yielding‘. Figure 2 illustrates these contrasts by comparing the response to deformation of a model Bingham material featuring elasticity below the yield stress with the experimentally measured response of a Carbopol microgel. Both show complete yielding, as

⁄ , but the Bingham fluid jumps instantaneously to that angle, while the

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Carbopol transitions gradually.

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Fig. 2. A comparison of the nonlinear responses between a hypothetical ideal elastic Bingham fluid (a – c) and an experimentally-measured Carbopol 980 microgel (d – f). Shown are: elastic Lissajous projections (a, d), Cole-Cole plots (b, e), and plots of instantaneous phase angle vs.

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strain (c, f). Green dots denote points at ~30 percent and ~80 percent of the period, to provide comparison across plots. The two equivalent ways of describing the instantaneous moduli are

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shown in (e): the instantaneous values of the time-resolved viscoelastic SPP moduli (eqns. 4 – 5) in red, and the complex magnitude and instantaneous phase angle (eqns. 7 – 8) in blue. The yield ⁄ is also shown in (b – c, e – f) as a gray dotted line.

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threshold of

While the instantaneous phase angle provides a quantitative threshold for identifying yielding,

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the gradual transition seen in fig. 2(d – f) suggests the utility of an additional measure to characterize the rate at which changes in the state of the material occur. It is therefore productive to focus on the rate of change in

to discern when, and how fast, yielding occurs. We propose

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to describe and study yielding via the phase angle and its time-derivative ( ̇ ), which we term the phase angle velocity. As

changes from a small value, corresponding to an elastic-dominated

state, to a large value, which corresponds to a viscous-dominated state, there will necessarily be an interval during which ̇ is positive. Conversely, when a flowing yield stress material resolidifies, unyields, or restructures, the instantaneous phase angle will transition from a large value to a small value, and ̇ will be negative. We therefore associate positive peaks in ̇ with

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the process of yielding, and negative peaks with reformation, unyielding, or restructuring. We refer to yielding as taking place when the phase angle increases beyond

⁄ , and we use the

phase angle velocity as a metric for how quickly yielding takes place. ‗Bingham yielding‘ will therefore be identifiable via instantaneous delta-function-like peaks in ̇ , while ‗non-Bingham yielding‘ will be easily identifiable by peaks in ̇ that are finite in their magnitude and duration.

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By contrast, changes in the magnitude of the instantaneous complex modulus would indicate softening or stiffening, or thickening or thinning, but would not necessarily indicate a change from elastic to viscous behavior. Such changes may be associated with yielding, but do not characterize the yielding process as clearly and unambiguously as does the phase angle velocity.

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2.6 The Phase Angle Velocity

Combining eqns. 4, 5, and 8 gives the tangent of the instantaneous phase angle in terms of the components of the binormal vector as:

̇⁄

̇⁄

.

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( )

Differentiation of eqn. 9 and some algebra gives:

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̇

| ⃑⃑|

|

̇⁄

.

|

(10)

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The full details of this derivation are presented in appendix B. We remark that eqn. 10 is quite general and has no assumptions built into it, making it broadly applicable to any trajectory in

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to:

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deformation space. If we restrict the applied deformation to being sinusoidal, eqn. 10 simplifies

̇

̇ (⃛ ̈

̇) ̇

(11)

The phase angle velocity, as expressed in eqn. 11, does not depend on the magnitude of the deformation

, and only has units of [time]-1. The magnitude of the stress will not affect the

value of ̇ , as both the numerator and denominator are of order

. This lack of dependence on

the strain- and stress amplitudes is encouraging from a conceptual standpoint, because any effective metric for describing yielding should be independent of the overall extent of the 15

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deformation and should only depend on the instantaneous material response. Equation 11 depends only on time derivatives of the stress, scaled by the frequency so that the overall expression has units of

[time]-1. If the stress response is relatively independent of frequency,

as it would be in a perfectly viscoplastic material [51], the value of ̇ tends toward zero in the limit of zero frequency, as all changes in the material response occur infinitely slowly. Under

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similar conditions, ̇ tends toward infinity at high frequencies, as the time for any changes is reduced to zero. Additionally, if the rheological response to an imposed oscillatory deformation is linear, that is, the stress is sinusoidal with a constant phase shift (as described by eqns. 1 – 2), then it can be shown from eqn. 11 that ̇ will be identically zero, as the response exhibits a strain-invariant viscoelastic behavior.

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To generalize the expressions within this work, we use a non-dimensionalized time time ̃

and replace the components of eqn. 3 with versions that vary with the dimensionless time: 〈̃ ̃

̃̇ ⁄

̃ 〉. In this derivation, the variables are non-dimensionalized by time only, i.e.

( ̃ ). Since the amplitudes of the strain and stress do not affect ̇ , rescaling the time in eqn.

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11 gives

̃̇ ( ⃛̃ ̃̇ )

̃̈

̃̇

.

(12)

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̃̇

We use this dimensionless form of the phase angle velocity preferentially throughout this work.

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For a perfectly plastic material, eqn. 12 demonstrates that the value of ̃̇ will be independent of frequency. As such, it can be used at all frequencies without loss of generality. It should be noted however that if the stress response of the material is dependent on the frequency, as is the case

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for many real materials, the frequency may still indirectly affect the final value of ̃̇ .

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Equations 11 and 12 are the central theoretical results of this work. We propose that yielding under large amplitude sinusoidal straining can be accurately mapped and interpreted by the behavior of the phase angle velocity. We will show that a detailed understanding of yielding can be obtained by applying these compact expressions to experimental data and the computed response of a well-studied model.

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2.7 Comparison to the Mutation Number An existing measure for quantifying changes in material properties over time is the mutation number [60], which is defined as the ratio of the characteristic experimental timescale,

. For an arbitrary material property g

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timescale over which the material properties change,

, to the

the mutation number is defined as (

).

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Mours and Winter [60] focused on the storage modulus as the property of choice, which leads to

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a mutation number given by (

).

In order to define such a mutation number, the timescale ( over which the mutating property (

(14)

) cannot be less than the timescale

in the case of eqn. 14) is defined. In order to capture timeas the mutation property is problematic, as the

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resolved yielding behavior, the use of

transition from elastic to viscous response happens within a single cycle of the imposed

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deformation, while the conventional storage modulus requires at least one full period of oscillation to be well-defined. The SPP framework resolves this problem by defining

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instantaneous parameters with a temporal resolution that matches the resolution of the timeresolved experimental data.

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Unlike the dynamic moduli which vary over many orders of magnitude depending on the material in question, the instantaneous phase angle only varies over a limited range of values and

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is always of order unity or smaller. Because the 1/g term in the definition of the mutation number biases the resulting dimensionless metric toward smaller values of the phase angle, we modify the definition of the phase angle mutation number to provide a fair measure that reflects the changes of a parameter that can be zero. If the instantaneous SPP phase angle is used as the key property of interest for such a modified mutation number, we find that (

)

(15)

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For a time-resolved investigation of yielding within a single period of oscillation,

can be

defined in terms of the time between data points and the number of points required to compute the terms in eqn. 12: ,

points per period are collected and (a minimum of) four sampled points are required to

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where

(16)

compute the third derivative of the stress. Substituting this into eqn. 15, we get (

),

(17)

which shows that the phase angle velocity and the modified mutation number are trivially

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related. The phase angle velocity is thus the natural mutation number associated with yielding, as it provides quantitative information regarding the rate at which the material response to the imposed straining deformation changes from being predominantly elastic to predominantly viscous.

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2.8 Application to the Carreau Model

To illustrate how effectively the phase angle velocity describes yielding behavior, we calculate it

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for the response of a simple smooth constitutive model that approximates the discontinuous Bingham model. A number of generalized Newtonian fluid models have been proposed to do this, including the Carreau model [61], the Cross model [62], and Papanastasiou‘s regularization

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of the Bingham model [63]. Of these, the Carreau model is of particular interest because it does not involve piecewise operations. Its higher-order derivatives are therefore continuous, which

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allows for a simple calculation in terms of eqns. 11 and 12. ( ̃ )⁄

For a dimensionless sinusoidal input strain ̃( )

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expressed as ̃( )

̇ ( ̃ )⁄ ̇

̃ , the dimensionless rate is

̃ . With these simplifications, the dimensionless expression

for the stress given by the Carreau model is: ̃(

)

̃( ) (

(

̃( )) )

(18)

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where ̃ ( )

( ̃ )⁄

⁄ ̇ is the Carreau number, a dimensionless number

̇ and

representing the amplitude of the imposed strain rate relative to a characteristic shear rate beyond which the model predicts shear thinning. For the purposes of approximating Bingham-type yielding, we take the limit

, which

corresponds well to yield stress fluid behavior [51]. Evaluating the required derivatives of the , we find using eqn. 12 that the phase angle velocity for the Carreau

model can be expressed as ̃̇ ( ̃

̃(

)

̃

̃ ̃

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stress and substituting

̃ ̃

̃

̃

)

̃

(19)

We show how the instantaneous phase angle velocity at each point over a single oscillatory

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deformation cycle is affected by varying the Carreau number in fig. 3.

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Fig. 3: (a) Contours of ̃̇ calculated for the Carreau model (with n = 0) at different Carreau

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Numbers. Colored lines correspond to specific values of Cu shown in the three-dimensional deformation spaces below. Note that the dimensionless time ( ̃ ) does not change the shape of the

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trajectories, only the instantaneous value of the phase angle velocity ( ̃̇ ). (b) Response of the Carreau model (n = 0) to sinusoidal deformation at model (n = 0) to sinusoidal deformation at

. (c) Response of Carreau .

It can be seen in fig. 3 that the phase angle velocity undergoes two identical cycles throughout the period 0 ≤ ̃ ≤ 2 , regardless of the amplitude of deformation applied to the material. In these cycles, the peak in ̃̇ occurs at ̃

⁄ and ̃

⁄ , indicating that the highest phase 20

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angle velocities occur in the material at the extrema of the imposed strain and at zero imposed shear rate. For small imposed shear rate amplitudes (low

), the peak in ̃̇ is broad and low,

indicating that the yielding is gradual, but at large amplitudes, the peak becomes increasingly sharper and narrower, indicating a more instantaneous yield transition typical of perfectly plastic models [11, 23, 53]. Solving for the maximum value at

in eqn. 19, we find that the

̃̇

(

)

̇̇

(

)

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peak value of phase angle velocity as a function of Carreau number is:

(20)

Using similar means, we can determine that the phase angle velocity at the points of zero (

)

, for all values of the Carreau number.

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macroscopic strain is zero, ̃̇

3. Experiments 3.1 Materials

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The experiments in this study were performed using Carbopol 980, a model soft material known to exhibit yielding behavior [22, 38, 64]. Carbopol microgels consist of crosslinked polyacrylic

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acid polymer particles which swell when dissolved in water beginning at a pH of 4 – 7, depending on the specific variety of Carbopol [38, 64]. The particles are on the order of 10 µm in diameter once swollen, but the mean particle size and its distribution varies greatly based on

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preparation [65] and can be very polydisperse. The swollen microgel particles form a softjammed microstructure [38] which prevents significant deformation until a stress high enough to

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push the jammed particles past each other is applied, after which they can flow. Of the different Carbopol varieties available, Carbopol 980 was selected in particular for this work as prior

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research has suggested that it is a simple yield stress fluid [66]. That is, it exhibits behavior that is not significantly affected by shear history. The microgel samples for this study were prepared by dispersing 1 wt% of Carbopol 980 powder into ultrapure water (Milli-Q). Full dispersion was achieved by mixing the suspension for 25 minutes at 2000 rpm in a planetary orbital mixer (Thinky ARE-310). To swell the polymer particles, ~0.75 mL of a 10 % solution of NaOH was added to the mixture, which was then

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mixed for another 5 minutes at 2000 rpm. The use of the planetary orbital mixer mitigated the problem of air bubbles in the sample. 3.2 Rheology All rheological experiments were performed on an MCR 702 Twindrive rheometer from Anton

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Paar in strain-controlled mode. Experiments were conducted using a set of 50 mm diameter parallel plates. In order to eliminate the effects of wall slip, sandpaper with a particle size of ~60 µm was secured to the plates with double-sided tape. 3.2.1 Steady shear Measurements

The flow curve was collected through a series of peak hold tests at rates of 0.001 to 100

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with each rate being applied for 300 to 1000 s, based on the rate applied. All tests reached a steady stress plateau. 3.2.2 Oscillatory Measurements

A frequency sweep in the linear regime was performed at a strain amplitude of

over

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a frequency range of 10 to 0.01 rad/s, sweeping logarithmically from high to low frequencies. LAOS data were collected during amplitude sweeps with an amplitude range of

– 10 strain

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units. These sweeps were performed at three angular frequencies (0.316 rad/s, 1 rad/s, and 3.16 rad/s) to assess the impact of frequency on the phase angle velocity. Oscillations were allowed to

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proceed at each strain amplitude until a stable periodic signal (or alternance state) was achieved. Time-resolved data were also collected in a similar manner for a large-amplitude frequency

rad/s.

[67], and over frequencies ranging from 0.1 to 3.16

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sweep at a fixed rate amplitude of 10

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3.3 Data Processing

All time-resolved data were processed by our freely-available MATLAB-based SPP software [68]. The SPP analysis was performed on data reconstructed via Fourier-domain filtering, using all

odd

harmonics

identifiable

above

the

noise

floor.

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4. Results and Discussion 4.1 Characterization Results To confirm that the Carbopol behaves as a yield stress fluid, and to characterize the linear viscoelastic response and the steady-shear flow behavior, a small-amplitude frequency sweep,

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steady shear flow curve, and amplitude sweep were performed. For the frequency and amplitude sweeps, the reported values of the dynamic moduli have been averaged over the course of a cycle of deformation. The linear regime frequency sweep shown in fig. 4(a) shows that the material is significantly more elastic than viscous when small deformations are applied, as evidenced by the storage modulus ( ) being at least an order of magnitude larger than the loss modulus (

) over

the entire range of frequencies studied. Additionally, the frequency sweep demonstrates that the

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viscoelastic properties of the Carbopol do not depend significantly on frequency over the experimentally accessible window. The flow curve, shown in fig. 4(b), also shows typical yield stress fluid behavior, identified by a non-zero value of stress at vanishingly low shear rates. The steady flow data are fit well by a Herschel-Bulkley model, which gives a dynamic yield stress of

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87.5 Pa.

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Fig. 4. (a) Linear viscoelastic frequency sweep and (b) steady shear flow curve for Carbopol 980 showing typical yield stress behaviors. The flow curve is well described by a Herschel-Bulkley (

AC fit. (

)

(

) ̇

(

)

) (solid line in (b))

The amplitude sweep shown in fig. 5(a) acts as further confirmation of the elastoviscoplastic nature of the material [18]. A simple interpretation of the data of fig. 5, in terms of a comparison of the extreme responses, shows that the material changes from being predominantly solid-like (

) at small amplitudes to increasingly liquid-like (

) at large amplitudes. Hyun et

al. [69] designate this type of LAOS behavior as type III: exhibiting a weak strain overshoot. 24

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Additionally, the response of the stress amplitude, shown in fig. 5(b), to the increase in strain amplitude shows a change in slope at a strain amplitude around 0.2 strain units, which corresponds to a stress amplitude of around 90 Pa, which is broadly consistent with the yield

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stress obtained from the Herschel-Bulkley fit to the steady flow curve.

Fig. 5. Strain amplitude sweep at

1 rad/s for Carbopol 980, shown in terms of: (a) dynamic

moduli, and (b) stress amplitude. Metrics commonly used to determine the yield point in the literature are shown as the numbered crosses. The range of potential locations for yielding is bounded by the blue dotted lines shown in (a).

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While the amplitude sweep provides convincing evidence that yielding occurs as larger strain amplitudes are applied, attempts to determine a single point for yielding from this plot alone are inconclusive [17]. The blue lines in fig. 5(a) demarcate the range of possible amplitudes for the yield strain, ranging from the end of the linear regime to onset of the highly non-linear regime. Also shown in fig. 5 are five metrics commonly reported in the literature as identifying the yield

include: the point at which -

in

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point. The resulting values of the yield stress and yield strain are given in Table 1. These metrics varies from its linear regime value by > 5 % (referred to here as

); the intersection point between the asymptotic small- and large-amplitude responses

(

); the point of maximum

(

); the point at which

(

); and

the intersection point between the small- and large-amplitude responses in the stress amplitude ). These yield stress values range from 2 – 109 Pa, and the corresponding yield

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(

strain values range from 0.005 – 1.3 strain units. This range of values for the yield stress or yield strain is comparable to what was reported by [17] for similar systems. Based on the cycleaveraged data alone, it appears not to be possible to determine definitively a unique condition for

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yielding, only the range of stresses and strains within which it could possibly occur. Yield Strain (-)

2.5 ± 0.5

0.0056 ± 0.0016

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Yield Stress (Pa)

0.337 ± 0.006

91 ± 7

0.57 ± 0.15

109 ± 7

1.3 ± 0.5

82.8 ± 2.5

0.192 ± 0.004

87.5 ± 0.6

-

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( ̇

89 ± 7

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Table 1: Alternate measures for the yield point of the Carbopol gel extracted from the strain amplitude sweep and steady flow curve 4.2 Time-resolved results The time-resolved data from the amplitude sweep at 1 rad/s corresponding to amplitudes of are displayed in figs. 6(a – c). In particular, we show in figs. 6(a) and 6(b) the stress-

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strain (elastic) and stress-rate (viscous) projections of the full three-dimensional trajectories

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displayed in fig. 5(c).

Fig. 6. Oscillatory stress responses from strain amplitude sweep at

1 rad/s. The nonlinear

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time periodic responses are shown in: a) Elastic Lissajous plot, b) Viscous Lissajous plot, and c) 3D Lissajous figure in deformation space. d) shows the Cole-Cole plot of the instantaneous

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moduli calculated via the SPP analysis. The responses for selected strain amplitudes have been colored to show progressive changes in the shape and size of the trajectory with increasing strain amplitude.

By examining the shape of the trajectories in fig. 6, it becomes clear why viewing

and

in

isolation is insufficient to describe the yielding process. The dynamic moduli displayed on a standard amplitude sweep are cycle-averaged values: the storage modulus,

, represents the

average energy stored over the course of an oscillation, while the loss modulus,

, represents 27

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the average energy dissipated over that same oscillation. For these values to accurately reflect the material‘s instantaneous properties over the entire oscillation, the response of that material in the deformation space of fig 6(c) must be planar [53], so that the elastic and viscous Lissajous projections are complementary ellipses. Inspection of the measured LAOS data in fig 6(c) reveals that this is not the case, particularly at medium and large amplitudes. Given the

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significant distortions in the Lissajous figures, time-resolved measures are required to accurately represent the instantaneous rheological state of the material.

At smaller amplitudes, the elastic Lissajous projections in fig 6(a) are nearly single-valued, indicating a material response dominated by elastic effects [53]. As the amplitude increases, the elastic projections begin to differ during loading and unloading, showing that unrecoverable

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energy dissipation is occurring [53]. Two particular features to note on the larger amplitude curves are a nearly constant maximum slope near the strain extrema, from which the cage ⁄

modulus [54] can be defined (

|

), and a nearly flat top and bottom,

which indicates nearly perfectly plastic flow [11, 23, 51].

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The viscous Lissajous projections also demonstrate significant changes in behavior as a function of strain amplitude. At the largest amplitudes, a similar path is taken when the instantaneous

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shear rate is decreasing from its maximum value, indicating a consistency of flow conditions. Coupled with the confirmation of elasticity above, we see that a sequence of physical processes occurs within the material which causes it to transition from primarily elastic to primarily

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viscous and back twice within each cycle of oscillation when the imposed amplitude of deformation is sufficiently large. The material is therefore clearly yielding within a given cycle.

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The question becomes, when does the yielding take place, and how quickly does it complete? Using the SPP analysis [53, 57],

and

, the instantaneous elastic and viscous moduli, are

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determined from the 3D trajectories of the material response. Presented in fig. 6(d) are the timedependent Cole-Cole plots displaying these instantaneous moduli for the time-resolved data in Fig. 6(c). A full discussion of how to read these plots can be found elsewhere [57]. Here, we add to the current interpretations of the Cole-Cole plots by noting that the plots can also simply be read in terms of |

| and

as suggested by eqns. 7 and 8. At any point along the Cole-Cole

plot, the distance away from the origin corresponds to the magnitude of the instantaneous elastic modulus |

|, while the angle subtended between the positive

axis and the line joining the 28

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point and the origin corresponds to the instantaneous phase angle

. This is seen more clearly in

fig. 7(d). The phase angle velocity therefore corresponds to the rate at which the point on the Cole-Cole plot rotates about the origin, with positive values interpreted as corresponding to the yielding transition, and negative values corresponding to transitions such as restructuring, reformation, and unyielding. We refrain from presenting the SPP results in a time-dependent van

and

against |

|, in favor of retaining the

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Gurp-Palmen plot, which would consist of plotting

representation, which connects more closely to the most commonly used methods to

determine yielding as indicated by fig. 5.

At the smallest amplitudes, the nonlinearities in the oscillatory responses are negligible. At strain amplitudes of and

to vary throughout the oscillation on a scale that can be easily visualized,

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values of

and larger, the nonlinearities become significant enough to cause the

with the magnitude of the variation being a non-monotonic function of the imposed strain amplitude. Once an imposed strain amplitude of large enough to briefly cause larger than

to exceed the

is reached, the variations have become

⁄ yielding threshold, or for

to become

, indicating an instantaneous state that we identify as having incompletely yielded.

At strain amplitudes

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This is the smallest amplitude at which the material can be said to yield on a macroscopic scale. and larger, the maximum value of

approaches a constant value

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of approximately 250 Pa, which corresponds to the cage modulus described above.

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4.3 Phase Angle Velocity Analysis Results Utilizing this time-dependent nonlinear data, the phase angle and corresponding phase angle velocity were calculated for the reconstructed response of Carbopol to imposed LAOStrain

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deformation. The experimental values of ̃̇ were computed using the dimensionless formulation given in eqn. 12. We show in fig. 7 the data for the representative case of an oscillation at a

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strain amplitude of

and an angular frequency of 1 rad/s. From the plot of the time-

dependent phase angle in fig. 7(b), it is seen that this deformation far exceeds the yield threshold laid out in section 2.5. There is some oscillatory noise in the phase angle and the phase angle velocity that comes from the dependence on higher order derivatives of the shear stress as shown in eqn. 12, as well as the finite number of harmonics used in the reconstruction of the oscillatory stress signal. There are two significant peaks in the ̃̇ data (labelled 1 and 2 in figs. 7 – 8)

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located close to the points where the strain rate reverses direction and the stress is changing rapidly with the strain. The phase angle prior to the peaks is momentarily close to zero, indicative of a nearly perfectly elastic response. Immediately following the peaks, the phase angle goes through an interval of near constancy at a value just over

⁄ , indicating a

predominantly liquid-like response. The peaks in ̃̇ are therefore the transitions between the

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elastic and viscous responses that we identify as yielding. These peaks start while both the total stress and total strain are still negative for positive shear rates, indicating that the Carbopol is elastoviscoplastic. The fact that the transition is not marked by a sharp delta peak in ̃̇ indicates that the Carbopol undergoes non-Bingham yielding, i.e. a gradual yielding transition which occurs over some finite interval of time identified by the width of the peak in ̃̇ . Such

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observations are consistent with a local yield condition that triggers plastic rearrangements within the material, as demonstrated in prior work [70, 71] for theoretical responses of colloidal gels and glasses. To probe this further, we investigate below the elastic (recoverable) strain, a

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measure that can be obtained with the SPP framework [53].

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Fig. 7. Map of the phase angle velocity for Carbopol 980 at an angular frequency of 1 rad/s and a strain amplitude of 1.78. Peaks in ̃̇ are numbered on all plots. (a) shows the elastic Lissajous

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plot with the regions of positive ̃̇ color-mapped. The red regions show areas with a positive ̃̇ , with lighter regions having a higher numerical value. Black regions on the color map indicate a

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negative ̃̇ . (b) shows the plot of instantaneous phase angle vs. strain with the same mapping. (c) shows the plot of phase angle velocity vs. strain. (d) shows the time-dependent Cole-Cole plot

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for this response.

The data for the instantaneous phase angle and dimensionless phase angle velocity collected in the amplitude sweep at 1 rad/s is displayed in fig. 8. Comparable plots of the data collected for the amplitude sweeps at 0.316 and 3.16 rad/s, and the high amplitude frequency sweep can be found in appendix C. It can be seen that, unlike the purely viscoplastic response of the Carreau model (with n = 0) in fig. 3, the time position of the peak in ̃̇ begins at smaller amplitudes close 31

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to ̃

, and moves closer to the point of rate reversal ( ̃

) as the strain amplitude

increases. This clearly illustrates the elastoviscoplastic character of the Carbopol‘s response to

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large amplitude oscillatory deformations.

Fig. 8. Contour plots of (a) instantaneous phase angle and (b) phase angle velocity of Carbopol 980 for the amplitude sweep at an angular frequency of 1 rad/s. The primary peaks in ̃̇ are numbered, and the black dotted lines indicate the strain amplitude at which the material first yields (i.e.

⁄ ).

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From this point forward, we compare the yielding behavior across these different tests by introducing a number of different descriptions of the primary peak in ̃̇ (peaks 1 & 2 in figs. 7 – 8). Each of these metrics provide a complementary piece of information regarding the yielding transition and how it proceeds. These metrics fall into two broad categories relating to the positions and the dimensions of the peaks. The positions are quantified by the shear strain at the approximate elastic strain ( ⁄

) at the point of maximum

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peak, the shear stress at the peak, the normalized time within the cycle (t/T) at the peak, and the [53], which correlates to the start

of the peak. The dimensions of the peaks are quantified by the maximum phase angle velocity, which indicates how fast the yielding proceeds, and the full width at half maximum (FWHM) of the peak in strain, stress, and time. The FWHM data provide detailed information regarding the

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duration of the yielding transition in terms of each measure. The above listed metrics were determined from the analysis of the time-resolved data for the amplitude sweeps and largeamplitude frequency sweep described in section 3.2.2. The amplitude sweep results are shown in fig. 9 for peak positions and fig. 10 for peak dimensions. The corresponding results for the large-

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amplitude frequency sweep are shown in figs. 11 and 12, respectively.

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Fig. 9. Metrics for the position of the peak in ̃̇ for amplitude sweeps at angular frequencies of 3.16 (blue), 1 rad/s (black), and 0.316 rad/s (red) respectively. The black dotted lines

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indicate the strain amplitude at which the material yields (

⁄ ) for the first time. The

figures show (a) the strain at the peak, (b) the stress at the peak, (c) the percentage of the period at the peak, and (d) the estimated elastic strain, ⁄

, at the point of maximum

, which is the

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start of the peak. The grey dotted line in (d) shows the limit of a purely elastic response. The strain attained at the phase angle velocity peak, shown in fig. 9(a), is identical within experimental error between the three frequencies across most of the analyzed amplitude range. This is consistent with the frequency sweep shown in fig. 4(a), which indicated that the material response was relatively independent of frequency. The macroscopic strain at the peak is a slightly super-linear function of the amplitude, and is approximately equal to ~80 percent of the 34

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imposed amplitude. This indicates that the peak in ̃̇ , the point at which the rate of yielding is highest, manifests itself at a large fraction of the amplitude, which occurs very shortly after the strain rate reverses direction. The data of fig. 9(b) show that the shear stress at the peak is a nonmonotonic function of the imposed strain amplitude. The stress at the peak continues to rise until strain amplitudes

are applied. The maximum value of the stress at the peak occurred at

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the smallest amplitude which exhibited yielding for each frequency tested. At very large strain amplitudes, the stress at the peak is a decreasing function of the strain amplitude.

The data of fig. 9(c) demonstrate that all peaks occur between the point of shear rate reversal (t/T = 0.25) and the point of zero strain (t/T = 0.50), with the points shifting closer to the point of rate reversal as the amplitude increases. This indicates that yielding takes place at macroscopic

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strains that have opposite signs from the instantaneous shear rate. The strain position of the yielding peak also suggests that there is an approximately constant amount of strain accumulated from reversal that dictates the yielding transition [54]. This is corroborated by the slight increase in slope of the data in fig. 9(a). We show in fig. 9(d) the elastic (recoverable) strain estimated from the SPP approach (see eqn. 6), at the point of maximum

(which marks the start of the

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peak in ̃̇ by definition). The estimated elastic (recoverable) strain approaches a constant of around 0.3 strain units at large strain amplitudes, a value which is independent of frequency. This

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is a macroscopic measure which correlates to the instantaneous elastic deformation to applied shear in the material, and a constant recoverable strain at the point where yielding begins

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indicates that a yield strain criterion may be useful in describing time-resolved yielding, in contrast to the yield stress concept invoked in steady-shear tests. Such a criterion is similar to

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that defined in some recent studies [70, 71] using Brownian dynamics simulations and the SGR model to study the out-of-equilibrium dynamics of colloidal gels and glasses with yielding of

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mesoscopic elements represented by the escape of a microstructural unit from an energetic well. It is important to note that the elastic strain at the start of the yielding process is below the purely elastic limit at all amplitudes, providing more evidence that the material is elastoviscoplastic below the yield threshold. This is in contrast to elastic Bingham and elastic Hershel-Bulkley type responses, where any deformation below the yield stress is assumed to be perfectly elastic.

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Fig. 10. Metrics for the position of the peak in ̃̇ for amplitude sweeps at angular frequencies of

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3.16 (blue), 1 rad/s (black), and 0.316 rad/s (red). The black dotted lines indicate the strain amplitude at which the material yields for the first time (

⁄ ). The figures show: (a) the

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height of the peak (the maximum phase angle velocity); and the FWHM of the peak in (b) time, (c) strain, and (d) stress. The outlying point in the strain sweep at the lowest oscillatory

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frequency (0.316 rad/s) is due to a double peak in ̃̇ , shown in appendix C. The dimensions of the peaks in the amplitude sweeps give additional insight into the yielding transition. We show in fig. 10(a) the maximum phase angle velocity (the height of the peak) as a function of the strain amplitude. The peak phase angle velocity shows slightly super-linear scaling with the applied strain amplitude, indicating that the rate at which the phase angle changes is faster at larger strain amplitudes. We note that this simple relation is not necessarily expected, as the expressions for the phase angle velocity (eqns. 11 and 12) contain no 36

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information regarding the amplitude of the deformation. Additionally, the use of eqn. 12 produces a result that is independent of

, indicating that the relationship between the strain

amplitude and phase angle velocity is dependent only on the material itself. The material first crosses the yield threshold (

⁄ ) at approximately the same amplitude (

),

regardless of oscillatory frequency. This point also appears to occur at the amplitude where ̃̇ ̃̇

. We therefore

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exceeds a critical threshold located between 1 and 2, when ̃̇

propose that the dimensionless phase angle velocity may also be used as a proxy for yielding, and that values greater than approximately 1.8 can be used to indicate yielding.

We show in fig. 10(b) that the duration of the yielding transition, as indicated by the FWHM in

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dimensionless time, approaches a constant value of ~1 at small amplitudes across all frequencies, and is approximately equal to 16 percent of the period. If taken in dimensional time, the width of the peak would be inversely proportional to the applied frequency. In contrast, fig. 10(c) demonstrates that the FWHM in strain is essentially constant at around 0.1 strain units at large imposed strain amplitudes

, with the value being independent of frequency. Taken

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together, these results indicate that the processes which lead to yielding change from being time(or frequency-) limited at small amplitudes to being strain-limited at large amplitudes. It is only when the process becomes strain-limited that the material yields for the first time (i.e.

⁄ ).

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The width of the peak in ̃̇ given by the FWHM in stress, shown in fig. 10(d) is nearly identical

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to the FWHM in strain shown in fig. 10(c). In addition to performing traditional amplitude sweeps at fixed frequencies, we have also , to

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performed a sweep in frequency at a constant deformation rate amplitude of ̇

check the frequency dependence of the yielding metrics in the highly nonlinear regime. This experiment is reminiscent of the strain-rate frequency superposition of Wyss et al. [67] in which

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the strain rate amplitude is kept fixed and the frequency is swept, but we do not interpret our results in the same manner. We perform this experiment to investigate the effects of changing the frequency (and thus inversely affecting the strain amplitude) at a fixed value of the shear rate, and whether doing so will lead to more ideal (instantaneous) yielding behavior. The largeamplitude frequency sweep results show similar trends to those observed in the amplitude sweeps at fixed frequency. These metrics are shown in figs. 11 and 12.

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Fig. 11. Metrics for the position of the peak in ̃̇ for a high-amplitude frequency sweep at a constant shear rate amplitude of ̇

10 s-1. The figures show (a) the strain at the peak, (b) the

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stress at the peak, (c) the percentage of the period at the peak, and (d) the estimated elastic strain at the point of maximum

.

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The macroscopic strain at the peak is an inverse function of the angular frequency (fig. 11(a)). As the rate amplitude is being held constant, this means that the response is also a function of strain amplitude, with a similar scaling to that seen in fig. 9(a) where the peak occurs at strains very close to the maximum. The elastic strain at the start of the peak, as estimated by the SPP approach, is also independent of frequency at the large strain amplitudes tested (fig 11(d)), and is identical to the limit seen in the strain amplitude sweep (fig. 9(d)) with around 0.3 strain units of elastic recoverable strain being acquired before yielding begins. Additionally, the stress at the 38

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peak (fig. 11(b)) is relatively constant at the fixed rate amplitude, and the relative time through the period at the peak (fig. 11(c)) tends towards the plastic limit

0.25 in the limit of low

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frequency.

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Fig. 12. Metrics for the dimensions of the peak in ̃̇ for a high-amplitude frequency sweep at a 10

. The figures show: a) the magnitude of the peak (i.e. the

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constant rate amplitude ̇

maximum phase angle velocity), and the FWHM of the peak in b) time, c) strain, and d) stress. The data of fig. 12 show the maximum phase angle velocity and the FWHM values of the peak in the large amplitude frequency sweep. Figure 12(a) shows that the maximum in phase angle velocity is inversely proportional to the frequency and shows comparable scaling with strain amplitude to that seen for the amplitude sweep in fig. 10(a). As this test is performed at a constant strain rate amplitude, the frequency dependence seen is nearly entirely due to the 39

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increase in strain amplitude. The duration of the yielding, shown in fig. 12(b), is a weakly negative, sub-linear function of the frequency, while the strain width and stress width (figs. 12(cd)) are relatively constant across the frequencies tested. The widths of the phase angle velocity peaks in both strain and stress plateau at low frequencies, suggesting that ideal instantaneous Bingham yielding is not an experimental reality, and that there is some minimum amount of

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strain that must be accumulated during the yielding process, reflecting the finite size of the material constituents. Additionally, the width of the peak in strain is quite close to that seen for the amplitude sweep (fig. 10(c)), strengthening the argument for a strain-driven yielding transition in Carbopol. 4.4 Summary of Time-Resolved Results

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Based on the phase angle velocity data obtained from the nonlinear time-periodic data, we are able to clarify the origins of the features of the cycle-averaged amplitude sweeps from changes in the instantaneous response. Within the linear viscoelastic regime, the dynamic moduli are nearly constant and single-valued over the course of a period, as expected. As the imposed deformation and

M

exceeds the limits of the linear regime, the instantaneous values of

begin to show

significant variations throughout the cycle. The emergence of this time-dependent behavior connects to the deviation from the limiting linear viscoelastic values of

and

seen in

ED

amplitude sweeps such as fig. 5. This connection between instantaneous and globally averaged measures is strengthened by the fact that a number of the time-varying measures, including the

PT

maximum elastic strain at the start of yielding (

), the transition point (

)

between the time-limited and strain-limited behavior in the width of the yielding peak, and the

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strain amplitude (

) at which the instantaneous moduli first cross the

lie relatively close to the peak in

(

). This suggests that the peak in

line, all is

AC

representative of the point at which the material yields on a macroscopic scale for the first time. From a physical viewpoint, the smooth shift in the behavior of the dynamic moduli (i.e. the simultaneous decrease in

and increase in

) indicates that the net balance of energy flow

shifts from storage to dissipation with increasing amplitude. This may reflect a microstructural interpretation where a small fraction of material elements within the system yield, without the material as a whole doing so [72]. The peak in

would then represent the point at which

enough of the elements yield within a cycle that the material as a whole can be said to exhibit 40

ACCEPTED MANUSCRIPT

yielding. The observed material trends subsequent to this peak (i.e. the decrease in both

and

) result from the system completing the strain-dependent yielding process in progressively shorter intervals of time as the amplitude increases, leaving increasing periods of primarily plastic flow during the remainder of the cycle. While the cycle-averaged

and

no longer

reflect the overall amounts of elastic and viscous deformations in the material, as the

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instantaneous moduli change throughout the period, their crossover can still be seen as a good estimate of the imposed strain amplitude at which the time spent in yielded and unyielded states of the material is of the same order within a cycle of deformation. However, this crossover of the dynamic moduli happens at strain amplitudes much larger than those required to enforce yielding.

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The time-varying phase angle velocity data suggest that yielding in soft materials can be interpreted as an ensemble effect of local rearrangements. In such an interpretation, nearly all the nonlinear behaviors of the material arise from some portion of the microstructural elements yielding locally, dissipating energy in the process. Deformations just outside of the linear regime show the initial effects of this dissipation, as they result in a small number of discrete

M

rearrangements of the constituent microstructural elements of the material. The macroscopic response of the material is still primarily dictated by the elastic deformation of the individual

ED

microstructural elements, until the amplitude increases to a point where the deformation of individual elements causes the elastic strain in the material to reach a critical threshold of 0.3

PT

strain units (see figs. 9(d) and 11(d)). Once this point is reached the material yields on a macroscopic scale, and shows a predominantly viscoplastic instantaneous response as indicated ,

and

. The yielding behavior transitions from being time-limited to strain-limited at

CE

by

this point, reflecting that the critical local strain to initiate yielding has been reached. Yielding is not instantaneous though, and proceeds while the material continues to acquire more strain,

AC

indicated in figs. 10(c) and 12(c) to be around 0.1 strain units. The phase angle velocity determined under LAOS conditions therefore gives an estimated measure of the distribution of local yield strains. Increasing the strain amplitude further speeds up the material fluidization. At sufficiently large strain amplitudes, the instantaneous rate of energy dissipation is strong enough and lasts for long enough over each period that the average viscous response dominates the average elastic response, producing the crossover observed in typical strain amplitude sweeps that report cycle-averaged viscoelastic quantities (e.g. fig. 5(a)). 41

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5. Conclusions We summarize the principal findings of this work in in terms of four main points: 1) Analysis of

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time-resolved rheological data is necessary to gain a full picture of yielding dynamics in a soft material; 2) Yielding in a soft elastoviscoplastic solid such as Carbopol gels is a gradual transition; 3) The instantaneous phase angle is an effective metric to discern the macroscopic yield state of a material; and 4) The phase angle velocity defined in eqns. 11 and 12 is an effective means of characterizing this gradual yielding transition and the relative width or

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sharpness of the peaks observed in ̃̇ may provide insight into the local distribution of yield strains within the material.

Time-varying data obtained in large amplitude oscillatory shear experiments thus allow for a deeper understanding of the yielding transition than available using existing methods. The gradual nature of the yielding process observed in LAOS not only provides insight into the

M

dynamics of how a material yields, but also explains why the cycle-averaged measures have the specific form they do. The condition of the instantaneous phase angle exceeding a critical value,

ED

⁄ , provides a clear and unambiguous threshold for instantaneous yielding on the macroscopic scale. Finally, the phase angle velocity has been shown to be an effective measure

PT

of the yielding transition, as it allows the progress of the yielding transition to be resolved with respect to the applied deformation, both before and after yielding is achieved macroscopically in

CE

the material. Due to its close interconnection with the dynamic moduli, ̃̇ also allows for decomposition of the signal into contributions corresponding to a physically-intuitive sequence

AC

of physical processes that characterize the yielding transition. Additionally, it provides a means by which changes in the yielding transition between various dynamic yielding test protocols can be quantitatively compared. All these features are unique to the phase angle velocity. The insights gained from this work enable progress toward the goal of forming structureproperty relationships that map the microstructural dynamics of a soft solid to the rheological response observed during yielding. The Carbopol microgel used here is a soft jammed material, and the dynamics by which the material particles escape their jammed state during yielding have 42

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been shown to be very complex [18, 21 – 22, 39 – 40]. Additionally, the Carbopol microstructure is not simple from an energetic standpoint [37 – 38], as the microgel elements have a dispersity of shapes and sizes, and are therefore also initially trapped in very different local energetic states. The results presented in figs. 9(d) and 10(c) support ideas that the yielding transition may be triggered by the local structural elements being strained beyond some critical threshold. The

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transition from time-limited to strain-limited behavior in the yielding behavior suggests that yielding on a macroscopic scale is composed of a series of local events.

These local dynamics are reminiscent of trap-based energetic models for amorphous materials such as the SGR model [41 – 43] for soft glassy materials or shear transition zone (STZ) theory [44] for metallic glasses. Such models utilize a landscape of energy wells to model the local

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glassy jamming of the material microstructure, using the local strain of a given structural element as the initiator for a ―hopping‖ transition from one energy well to another. The gradual yielding we have shown here can be viewed as a cumulative effect of microscale ―hops‖ of elements from one configuration to another. At small amplitudes of deformation, only a small number of the most weakly jammed elements are able to change their configuration. When the applied

M

deformation is increased to a point where a critical number of these local elements change their configurations, a cumulative change in the macroscopic viscoelastic properties of the material

ED

occurs. This manifests as a progressive change in the overall measured bulk rheology, which we can now identify via evaluation of the phase angle velocity.

PT

It is important to note that the analysis shown here, as with all other purely rheological methods, focuses entirely on insights gained from the macroscopic shear response. This means that, when

CE

used on its own, it does not resolve the local dynamics that occur during the yield transition, but rather indicates an ensemble average across the sample volume. The instantaneous phase angle

AC

and phase angle velocity do not require that the flow be homogeneous, as bulk rheological data corresponds to a sum over local configurational states within the material. To obtain fullyresolved local dynamics of the yield transition, we suggest that time-resolved phase angle velocity measurements be combined with a spatially-resolved technique such as rheo-XPCS [73], ultrasonic speckle velocimetry [18, 21 – 22, 39], or time-resolved rheo-NMR [74 – 75]. With the growing interest in LAOS methods, time-resolved nonlinear rheometry has become more accessible, and most commercial rheometers now provide user access to time-resolved 43

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strain, rate, and stress data. The phase angle velocity defined here can be readily computed in terms of the time-varying stress and strain measured by the rheometer. While we have focused our study on a soft jammed ‗ideal yield stress fluid‘, the concept of an instantaneous phase angle velocity is broadly applicable because its definition makes no assumptions regarding material form or response. It is therefore expected to be of significant interest in the study of a wide

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variety of elastoviscoplastic systems.

Acknowledgements

The authors thank Anton Paar for use of the MCR 702 rheometer through their VIP academic

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research program. The authors also thank Lubrizol for supplying the Carbopol used for this work. J. R. de B. acknowledges research support from the Natural Sciences and Engineering Research Council of Canada. S. A. R. acknowledges startup support from the Department of

M

Chemical and Biomolecular Engineering at the University of Illinois at Urbana-Champaign.

Appendix A: Derivative Forms of Frenet-Serret Frame in Deformation Space

ED

For the purposes of this appendix, we will denote derivatives as

[ ]

. Within deformation space,

the material response to applied deformation follows the trajectory of a space curve given by

PT

⃑( )

[ ]

〈 ( )

( )⁄

( )〉

(a1)

To determine the instantaneous properties of the material for a given applied deformation

CE

protocol, one must look at the instantaneous orientation of the trajectory. As shown above in section 2.4, the Frenet-Serret frame (FSF) is the natural way to view and analyze such a

AC

response. The FSF is defined by three orthonormal vectors: the tangent vector, normal vector, and binormal vector. The tangent vector is defined as the instantaneous derivative of the trajectory of the space curve; ⃑⃑

⃑

〈

[ ]

[ ]

⁄

[ ]〉

(a2)

The magnitude of the tangent vector is thus defined by

44

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√

| |

[ ]

[ ]⁄

(

[ ]

)

(a3)

The second component of the FSF, the normal vector, is defined as the derivative of the normalized tangent vector, ⃑⃑

⃑⃑ [ ]

[ ] [ ]

⁄ (

[ ]

(

[ ] [ ] [ ] [ ]⁄

[ ]

)

[ ] [ ] [ ]

[ ] [ ]

)

[ ] [ ] [ ]

(

[ ]

(

[ ]⁄

[ ] [ ]

)

[ ] [ ] [ ]

)

with a magnitude of

[ ] [ ]( [ ] [ ]

)

[ ] [ ]) [ ]

[ ]

[ ]

[ ]

[ ] [ ] [ ]

(

[ ]⁄

AN US

| | [ ]

[ ]

(

(a4)

√ [ ] ( [ ]

[ ]

[ ]

(

( [ ]

[ ]⁄

[ ]

[ ] [ ] [ ] [ ]

)

[ ] [ ] [ ]

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〈

(| |)

[ ]

[ ]

)

( [ ]

〉

)

[ ]

)

[ ]

)

(a5)

The final component of the FSF is the binormal vector. This is the vector perpendicular to the

M

instantaneous plane of the material response, and is defined as the cross-product of the tangent and normal vectors: ⃑⃑

⃑⃑

〈

[ ]

)

[ ] [ ]

( [ ]⁄ )

[ ] [ ]( [ ] [ ]

[ ]

[ ] [ ] [ ]

[ ] [ ]

( [ ]⁄ )

[ ]

[ ] [ ] [ ]

[ ]

( [ ]⁄ )

[ ]

〉

(a6)

| | [ ] [ ]) [ ]

[ ]

( [ ]

( [ ]⁄ )

[ ]

)

[ ] [ ] [ ] [ ]

[ ]

( [ ]

[ ]

)

[ ]

(a7)

AC

CE

√ [ ] ( [ ]

[ ]

PT

and has a magnitude of

[ ] [ ]

ED

⃑⃑

Associated with the three vectors of the FSF, there are two scalar parameters which characterize the instantaneous rotation of the frame: the curvature, , which indicates the rotation of T around B, and the torsion , which indicates the rotation of B around T. These are defined as follows:

45

ACCEPTED MANUSCRIPT

⃑⃑

⃑⃑

(| |) [ ]

√

( [ ]

[ ]

)

[ ] [ ]( [ ] [ ]

[ ] [ ])

[ ]

( [ ]

(| |) ( [ ]

[ ]

( [ ]⁄ )

[ ]

[ ] [ ] [ ] [ ]

)

[ ]

( [ ]

[ ]

)

)

(a8) ⃑⃑

⃑⃑

[ ] [ ]

( [ ]

[ ]

)

( [ ] [ ]

[ ] [ ]

)

[ ] [ ]( [ ] [ ]

[ ]

(| |)

( [ ] [ ]

[ ] [ ])

[ ]

[ ] [ ]

( [ ]

[ ]

)

[ ]

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(| |)

( [ ] [ ]

[ ]

[ ] [ ] [ ] [ ]

)

)

[ ]

( [ ]

[ ]

)

(a9)

All parameters within the SPP framework can be defined as some combination of the

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components of these vectors and these scalar rotation parameters.

Appendix B: Derivation of the Phase Angle Velocity

In section 2.6, it was shown that the instantaneous phase angle can be written in terms of the

M

components of the binormal vector as

̇⁄ ̇⁄

ED

( )

.

(9)

PT

To differentiate eqn. 9, the following identity can be used: (

( )

( ))

( )

.

(b1)

CE

The derivative of the instantaneous phase angle is therefore ̇

AC

(

(

̇⁄

)

(

̇⁄

̇⁄

),

(b2)

which can be simplified to ̇

̇⁄

)

̇⁄

(

)

.

(b3)

The denominator of eqn. b3 is the magnitude of the binormal vector projection on the strain – strain-rate plane. We denote this as 46

ACCEPTED MANUSCRIPT

|

|

̇⁄

.

̇⁄

(b4)

We now invoke the definition of the derivative of the binormal vector within the Frenet-Serret frame, which involves the torsion, and the principal normal vector, ⃑⃑ : ⃑⃑,

(b5)

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( ⃑⃑ )

where, s is the arc length of the curve. Using the chain rule, we write ( )

( )

.

Additionally, for any curve in deformation space: ⃑

| ⃑⃑|,

AN US

| |

(b6)

(b7)

where ⃑ is the instantaneous position vector of the response in deformation space, and ⃑⃑ is the tangent vector from the Frenet-Serret frame. Substituting eqns. b4 – b7 into eqn. b3 we obtain the following expression for the phase angle velocity: | ⃑⃑|

|

M

̇

|

̇⁄

(

̇⁄

̇⁄

).

(b8)

ED

The last factor on the right-hand side of eqn. b8 is the component of the cross product of the

PT

normal and binormal vectors projected along the stress axis: ̇⁄

( ⃑⃑

̇⁄

⃑⃑ ) .

(b9)

CE

Based on the definition of the Frenet-Serret frame in deformation space, this cross-product must

AC

be parallel to the tangent vector. The specific scaling is proportional to the curvature, : ( ⃑⃑

⃑⃑ )

| ⃑⃑|

.

(b10)

Substituting eqn. b10 back into eqn. b8, we arrive at the simplified general expression for the phase angle velocity given in the text (eqn. 10), written in terms of the tangent and binormal vectors and both the torsion and the curvature of the trajectory in deformation space: ̇

| ⃑⃑| |

̇⁄

|

.

(10)

47

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Appendix C: Supporting Information

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The experimental data displayed here is shown in the same form as fig. 8 in the manuscript. The remaining amplitude sweep data is found in figs. c1 and c2, for angular frequencies of 0.316 rad/s and 3.16 rad/s respectively. The data for the high-amplitude frequency sweep at a constant strain rate amplitude of ̇

10 s-1 is shown in fig. c3. The noise in this data is higher than that

in figs. 8, c1, and c2 due to the reduced point density in time at lower frequencies. The peaks in

AC

CE

PT

ED

M

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this data are analyzed in the main body of the paper in figs. 9 – 12.

48

CE

PT

ED

M

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CR IP T

ACCEPTED MANUSCRIPT

Fig. c1. Contour plots of (a) instantaneous phase angle and (b) phase angle velocity of Carbopol

AC

980 for the amplitude sweep at an angular frequency of 0.316 rad/s. The primary peaks in ̃̇ are numbered, and the black dotted lines indicate the strain amplitude at which the material yields (i.e.

⁄ ).

49

CE

PT

ED

M

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ACCEPTED MANUSCRIPT

Fig. c2. Contour plots of (a) instantaneous phase angle and (b) phase angle velocity of Carbopol

AC

980 for the amplitude sweep at an angular frequency of 3.16 rad/s. The primary peaks in ̃̇ are numbered, and the black dotted lines indicate the strain amplitude at which the material yields (i.e.

⁄ ).

50

CE

PT

ED

M

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CR IP T

ACCEPTED MANUSCRIPT

Fig. c3. Contour plots of (a) instantaneous phase angle and (b) phase angle velocity of Carbopol

AC

980 for the high-amplitude frequency sweep at a constant shear rate amplitude ̇

10 s-1. The

primary peaks in ̃̇ are numbered, and the black dotted lines indicate the strain amplitude at which the material yields (i.e.

⁄ ).

In fig. 10(b – d), one of the peaks (

rad/s) showed FWHM values and

error ranges that did not correspond to the other points in the data set. This occurred due to the presence of a double peak in the computed ̃̇ , which did not reflect the Gaussian profile which 51

ACCEPTED MANUSCRIPT

the FHWM analysis requires. This profile, along with the profiles at the same amplitude from the

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other amplitude sweeps, are displayed in fig. c4.

M

Fig. c4. Comparison in the peak ̃̇ across different frequencies at a strain amplitude of

ED

0.316. The non-Gaussian profile of the peak at an angular frequency of 0.316 rad/s is responsible

References

PT

for the outlying point in fig. 10(b – d).

CE

[1] N.J. Balmforth. I.A. Frigaard. G. Ovarlez. ―Yielding to stress: Recent developments in viscoplastic fluid mechanics‖ Annu. Rev. Fluid Mech. 46 (2014), pp. 121-146

AC

[2] D. Bonn. M.M. Denn. L. Berthier. T. Divoux. S. Manneville. ―Yield stress materials in soft condensed matter‖ Rev. Modern Phys. 89 (2017), pp. 035005 [3] A. Malkin. V. Kulichikhin. S. Ilyin. ―A modern look on yield stress fluids‖ Rheol. Acta, 56 (2017), pp. 177-188 [4] P. Coussot. ―Yield stress fluid flows: A review of experimental data‖ J. Non-Newtonian Fluid Mech., 211 (2014), pp. 31-49 52

ACCEPTED MANUSCRIPT

[5] G. Ovarlez. F. Bertrand. P. Coussot. X. Chateau. ―Shear-induced sedimentation in yield stress fluids‖ J. Non-Newtonian Fluid Mech. 177 (2012), pp. 19-28 [6] X. Tang. X. Zhang. R. Tao. ―Flexible fixture device with magneto-rheological fluids‖ J. Intelligent Material Systems and Structures, 10 (1999), pp. 690-694

Newtonian Fluid Mech., 81,1-2 (1999), pp. 133-178

CR IP T

[7] H.A. Barnes. ―The yield stress—a review or ‗παντα ρει‘—everything flows?‖ J. Non-

[8] R.R. Fernandes. D.E.V. Andrade. A.T. Franco. C.O.R. Negrado. ―The yielding and the linear-to-nonlinear viscoelastic transition of an elastoviscoplastic material‖ J. Rheol., 61 (2017), pp. 893

AN US

[9] E.C. Bingham. ―An Investigation of the Laws of Plastic Flow‖ Bulletin of the Bureau of Standards., 13 (1916), pp. 309-353

[10] P. Coussot. ―Bingham‘s heritage‖ Rheol. Acta, 56,3 (2017), pp. 163-176 [11] A.S. Yoshimura. R.K. Prud‘homme. ―Response of an elastic Bingham fluid to oscillatory

M

shear‖ Rheol. Acta, 26,5 (1987), pp. 428-436

Z. 39 (1926), pp. 291-300

ED

[12] W.H. Herschel. R. Bulkley. ―Konsistenzmessungen von Gummi-Benzollösungen‖ Kolloid-

PT

[13] C.C. Mills (Ed.), Rheology of Disperse Systems, Pergamon Press, Oxford, 1959 [14] M.M. Denn. D. Bonn. ―Issues in the flow of yield-stress liquids‖ Rheol. Acta, 50,4 (2011),

CE

pp. 307-315

[15] H.A. Barnes. K. Walters. ―The yield stress myth?‖ Rheol. Acta, 24,4 (1985), pp. 323-326

AC

[16] Q. Ngyuen. D. Boger. ―Measuring the Flow Properties of Yield Stress Fluids‖ Annu. Rev. Fluid Mech., 24 (1992), pp. 47-88 [17] M. Dinkgreve. J. Paredes. M.M. Denn. D. Bonn. ―On different ways of measuring ―the‖ yield stress‖ J. Non-Newtonian Fluid Mech., 238 (2016), pp. 233-241 [18] T. Divoux. C. Barentin. S. Manneville. ―From stress-induced fluidization processes to Herschel-Bulkley behaviour in simple yield stress fluids‖ Soft Matter, 7 (2011), pp. 8409-8418 53

ACCEPTED MANUSCRIPT

[19] P. Coussot. L. Tocquer. C. Lanos. G. Ovarlez. ―Macroscopic vs. local rheology of yield stress fluids‖ J. Non-Newtonian Fluid Mech. 158 (2009), pp. 85-90 [20] M. Dinkgreve. M.M. Denn. D. Bonn.‖ ―Everything flows?‖: elastic effects on startup flows of yield-stress fluids‖ Rheol. Acta, 56,3 (2017), pp. 189-194 [21] T. Divoux. C. Barentin. S. Manneville. ―Stress overshoot in a simple yield stress fluid: An

CR IP T

extensive study combining rheology and velocimetry‖ Soft Matter, 7 (2011), pp. 9335-9349 [22] T. Divoux. D. Tamari. C. Barentin, S. Manneville. ―Transient Shear Banding in a Simple Yield Stress Fluid‖ Phys. Rev. Let., 104 (2010), pp. 208301

[23] F. Rouyer. S. Cohen-Addad. R. Hohler. ―Is the yield stress of aqueous foam a well-defined

116

AN US

quantity?‖ Colloids and Surfaces A: Physicochemical and Engineering Aspects. 26(2005) p.111-

[24] P. Coussot. Q.D. Nguyen. H.T. Huynh. D. Bonn ―Avalanche behavior in yield stress fluids‖ Phys. Rev. Lett. 88 (2002), pp. 175501

M

[25] J. Boujlel. P. Coussot. ―Measuring yield stress: a new, practical, and precise technique derived from detailed penetrometry analysis‖ Rheol. Acta, 51,10 (2012), pp. 867-882

ED

[26] G. Astarita. ―The engineering reality of the yield stress‖ J. Rheol., 34 (1990), pp. 275 [27] C.J. Dimitriou. R.H. Ewoldt. G.H. McKinley. ―Describing and prescribing the constitutive

PT

response of yield stress fluids using large amplitude oscillatory shear stress (LAOStress)‖ J.

CE

Rheol. 57 (2013), pp. 27

[28] P.C.F. Moller. A. Fall. D. Bonn. ―Origin of apparent viscosity in yield stress fluids below

AC

yielding‖ EPL, 87 (2007), pp. 30084 [29] A. Le Grand. G. Petekidis. ―Effects of particle softness on the rheology and yielding of colloidal glasses‖ Rheol. Acta, 47 (2008), pp. 579-590 [30] F. Renou. J. Stellbrink. G. Petekidis. ―Yielding processes in a colloidal glass of soft star-like micelles under large amplitude oscillatory shear (LAOS)‖ J. Rheol. 54 (2010), pp. 1219 [31] A. Saint-Jalmes. D.J. Durian. ―Vanishing elasticity for wet foams: Equivalence with emulsions and role of polydispersity‖ J. Rheol. 43 (1999), pp. 1411 54

ACCEPTED MANUSCRIPT

[32] P.C.F. Moller. S. Rodts. M.A.J. Michels. D. Bonn. ―Shear banding and yield stress in soft glassy materials‖ Phys. Rev. E, 77(2008), pp. 041507 [33] F. Pignon. A. Magnin. J.M. Piau. ―Thixotropic colloidal suspensions and flow curves with minimum: identification of flow regimes and rheometric consequences‖ J. Rheol. 40 (1996), pp. 573-587

CR IP T

[34] P. Coussot. ―Slow flows of yield stress fluids: yielding liquids or flowing solids?‖ Rheol. Acta, 57,1 (2018), pp. 1-14

[35] J.J. Stickel. J.S. Knusten. M.W. Liberatore. ―Response of elastoviscoplastic materials to large amplitude oscillatory shear flow in the parallel-plate and cylindrical-Couette geometries‖ J.

AN US

Rheol. 57 (2013), pp. 1569

[36] P. Leishangthem. A.D.S. Parmar. S. Sastry. ―The yielding transition in amorphous solids under oscillatory shear deformation‖ Nature Communications, 8 (2017), pp. 14653 [37] G. Ovarlez. Q. Barral. P. Coussot. ―Three-dimensional jamming and flows of soft glassy

M

materials‖ Nature Materials, 9 (2010), pp.115-119

[38] J.M. Piau. ―Carbopol gels: Elastoviscoplastic and slippery glasses made of individual

ED

swollen sponges: Meso- and macroscopic properties, constitutive equations and scaling laws‖ J. Non-Newtonian Fluid Mech. 144,1 (2007), pp. 1-29

PT

[39] T. Divoux. D. Tamari. C. Barentin, S. Teitel. S. Manneville. ―Yielding dynamics of a Herschel–Bulkley fluid: a critical-like fluidization behavior‖ Soft Matter, 8 (2012), pp. 4151-

CE

4164

[40] Y. Liu. J.R. de Bruyn. ―Start-up flow of a yield-stress fluid in a vertical pipe‖ J. Non-

AC

Newtonian Fluid Mech. 257 (2018), pp. 50-58 [41] S.M. Fielding. P. Sollich. M.E. Cates. ―Aging and rheology in soft materials‖ J. Rheol. 44 (2000), pp. 323 [42] P. Sollich. ―Rheological constitutive equation for a model of soft glassy materials‖ Phys. Rev. E 58 (1998), pp. 738

55

ACCEPTED MANUSCRIPT

[43] P. Sollich. F. Lequeux. P. Hebraud. M.E. Cates. ―Rheology of Soft Glassy Materials‖ Phys. Rev. Let. 78 (1997), pp. 2020 [44] M.L. Falk. J.S. Langer. ―Dynamics of viscoplastic deformation in amorphous solids‖ Phys. Rev. E, 57 (1998), pp. 7192

problem‖ Soft Matter, 13 (2017), pp. 7578

CR IP T

[45] A.Z. Nelson. R.H. Ewoldt. ―Design of yield-stress fluids: a rheology-to-structure inverse

[46] B.J. Landrum. W.B. Russel. R.N. Zia. ―Delayed yield in colloidal gels: Creep, flow, and reentrant solid regimes‖ J. Rheol. 60 (2016), pp. 783

[47] M. Wilhelm. ―Fourier-Transform Rheology‖ Macromolecular Materials and Engineering.

AN US

287 (2002), pp. 83-105

[48] K.S. Cho. K. Hyun. K.H. Ahn. S.J. Lee ―A geometrical interpretation of large amplitude oscillatory shear response‖ J. Rheol. 49 (2005), pp. 747

[49] C.O. Klein. H.W. Spiess. A. Calin. C. Balan. M. Wilhelm. ―Separation of the Nonlinear

M

Oscillatory Response into a Superposition of Linear, Strain Hardening, Strain Softening, and Wall Slip Response‖ Macromolecules. 40 (2007), pp. 4250-4259

ED

[50] R.H. Ewoldt. A.E. Hosoi. G.H. McKinley. ―New measures for characterizing nonlinear viscoelasticity in large amplitude oscillatory shear‖ J. Rheol. 52 (2008), pp. 1427

PT

[51] R.H. Ewoldt. P. Winter. J. Maxey. G.H. McKinley. ―Large amplitude oscillatory shear of

CE

pseudoplastic and elastoviscoplastic materials‖ Rheol. Acta, 49,2 (2010), pp.191-212 [52] S.A. Rogers. ―A sequence of physical processes determined and quantified in LAOS: An

AC

instantaneous local 2D/3D approach‖ J. Rheol. 56 (2012) p. 1129 [53] S.A. Rogers. ―In search of physical meaning: defining transient parameters for nonlinear viscoelasticity‖ Rheol. Acta, 5,56 (2017), pp. 501-525 [54] S.A. Rogers. B.M. Erwin. D. Vlassopolous. M. Cloitre. ―A sequence of physical processes determined and quantified in LAOS: Application to a yield stress fluid‖ J. Rheol. 55 (2011) p. 435

56

ACCEPTED MANUSCRIPT

[55] S.A. Rogers. M.P. Lettinga. ―A sequence of physical processes determined and quantified in large-amplitude oscillatory shear (LAOS): Application to theoretical nonlinear models‖ J. Rheol. 56 (2012) p. 1 [56] M.R.B. Mermet-Guyennet. J.G. de Castro. M. Habibi. N. Martzel. M.M. Denn. D. Bonn. ―LAOS: The strain softening/strain hardening paradox‖ J. Rheol. 59 (2015), pp. 21-32

CR IP T

[57] C. Lee. S.A. Rogers. ―A sequence of physical processes quantified in LAOS by continuous local measures‖ Korea-Australia Rheology Journal, 29 (2017), pp.269-279

[58] H.Y. Chen. R. Liegeois. J.R. de Bryun. A. Soddu. ―Principal-component analysis of particle motion‖ Phys. Rev. E, 91 (2015), pp. 042308

AN US

[59] I. T. Jolliffe, ―Principal Component Analysis, 2nd ed.‖ (Springer,New York, 2002).

[60] M. Mours. H.H. Winter. ―Time-resolved rheometry‖ Rheol. Acta. 33 (1994), pp. 385-397 [61] R. Bird. R. Armstrong. O. Hassager. Dynamics of polymeric liquids: volume 1 fluid mechanics. John Wiley & Sons, Inc, New York (1987)

M

[62] M.M. Cross. Rheology of non-Newtonian fluids: a new flow equation for pseudoplastic

ED

systems. J Colloid Sci 20,5 (1965), pp. 417–437

[63] T.C. Papanastasiou. ―Flow of materials with yield‖ J. Rheol. 31 (1987), pp. 385

PT

[64] I.A. Gutowski. D. Lee. J.R. de Bruyn. B.J. Frisken. ―Scaling and mesostructure of Carbopol dispersions‖ Rheol. Acta, 51 (2012), pp. 441-450

CE

[65] M. Dinkgreve. M. Fazilati. M.M. Denn. D. Bonn. ―Carbopol: From a simple to a thixotropic yield stress fluid‖ J. Rheol. 62 (2018), pp. 773

AC

[66] G. Ovarlez. S. Cohen-Addad. K. Krishan. J. Goyon. P. Coussot. ―On the existence of a simple yield stress fluid behavior‖ J. Non-Newtonian Fluid Mech., 193 (2013), pp. 68-79 [67] H.M. Wyss. K. Miyazaki. J. Mattsson. Z. Hu. D.R. Reichman. D.A. Weitz. ―Strain-Rate Frequency Superposition: A Rheological Probe of Structural Relaxation in Soft Materials‖ Phys. Rev. Lett. 98 (2007), pp. 238303 [68] To request a free copy of SPPplus, please write to [email protected] 57

ACCEPTED MANUSCRIPT

[69] K. Hyun. S.H. Kim. K.H. Ahn. S.J. Lee. ―Large amplitude oscillatory shear as a way to classify the complex fluids‖ J. Non-Newtonian Fluid Mech. 107 (2002), pp.51-65 [70] J.D. Park. K.H. Ahn. N.J. Wagner. ―Structure-rheology relationship for a homogeneous colloidal gel under shear startup‖ J. Rheol. 61 (2017), pp. 117-137 [71] J.D. Park. S.A. Rogers. ―The transient behavior of soft glassy materials far from

CR IP T

equilibrium‖ J. Rheol. 62 (2018), pp. 869

[72] M.C. Rogers. K. Chen. L. Andrzejewski. S. Narayanan. S. Ramakrishnan. R.L. Leheny. J.L. Harden. ―Echoes in x-ray speckles track nanometer-scale plastic events in colloidal gels under shear‖ Phys. Rev. E, 90 (2014), pp. 062310

AN US

[73] R.L. Leheny. M.C. Rogers. K. Chen. S. Narayanan. J.L. Harden. ―Rheo-XPCS‖ Cur. Opin. Colloid Interface Sci. 20 (2015), pp. 261-271

[74] P.T. Callaghan. ―Rheo NMR and Shear Banding‖ Rheol. Acta, 47 (2008), pp. 243-255 [75] P. Coussot. H. Tabuteau. X. Chateau. L. Tocquer. G. Ovarlez. ―Aging and solid or liquid

AC

CE

PT

ED

M

behaviors in pastes‖ J. Rheol., 50 (2006), pp. 975

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