Low-dimensional intrinsic material functions for nonlinear viscoelasticity

Transcription

1 Author-created version, to be published in Rheologica Acta The final publication is available at (DOI: 1.17/s ) Low-dimensional intrinsic material functions for nonlinear viscoelasticity Randy H. Ewoldt, N. Ashwin Bharadwaj Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 6181, USA Abstract Rheological material functions are used to form our conceptual understanding of a material response. For a nonlinear rheological response, the possible deformation protocols and material measures span a high-dimensional space. Here we use asymptotic expansions to outline lowdimensional measures for describing leading-order nonlinear responses in large-amplitude oscillatory shear (LAOS). These intrinsic nonlinear material functions are only a function of oscillatory frequency, and not amplitude. Such measures have been suggested in the past, but here we clarify what measures exist and give physically meaningful interpretations. Both shear strain-control (LAOStrain) and shear stress-control (LAOStress) protocols are considered, and nomenclature is introduced to encode the physical interpretations. We measure experimentally the four intrinsic shear nonlinearities of LAOStrain for a polymeric hydrogel (PVA-Borax) and observe typical integer-power function asymptotics. The magnitudes and signs of the intrinsic nonlinear fingerprints are used to conceptually model the mechanical response and to infer molecular- and micro-scale features of the material.

2 Ewoldt & Bharadwaj - Intrinsic LAOS I. Introduction To understand mechanical properties of complex fluids or solids, we use descriptive material functions including linear viscoelastic moduli G ( ω ) and G ( ω ) +, transient shear viscosity η ( t, ɺ γ ), and many other measures (Dealy 1995). Material functions are the starting point to describe intensive material responses to various loading conditions, but they are not predictive models in themselves. Nonlinear rheological characterization can generate an overwhelming amount of information and material functions, with various measures that each map out a two-dimensional parameter space, e.g. stress relaxation modulus G( t, γ ) or the variety of material functions associated with large-amplitude oscillatory shear (LAOS); e.g. see (Giacomin et al. 11, Hyun et al. 11, Rogers 1, Ewoldt 13). One ever-present challenge is to identify and interpret the most appropriate material function(s) for the material and application of interest, choosing these from a high-dimensional space. Here we describe low-dimensional descriptions (material functions) that can be used to rheologically characterize any material of interest. Asymptotic expansions allow for low-dimensional descriptions and we will use the term intrinsic to refer to these asymptotically-defined nonlinear material functions. Here an intrinsic nonlinearity is interpreted as an inherent property of a system independent of the input magnitude causing the nonlinearity, e.g. independent of the shear amplitude for amplitude-intrinsic LAOS. This is consistent with, but not identical to, existing uses of the term intrinsic. Consider intrinsic shear viscosity [ η ], a material function that represents asymptotic changes to viscosity due to an additive at mass concentration c (mass per volume) (Ferry, 1976; Dealy 1995). The value of intrinsic viscosity is concentration-independent, or we might say concentration-intrinsic. A particular functional dependence could be assumed for both concentration and shear-rate dependence, (, c) = ( ) 1 + [ ]( ) c + O ( c ) η ɺ γ η ɺ γ η ɺ γ (1) where c is the mass concentration of additive, η ( γɺ ) is the baseline viscosity, and Oc ( ) represents terms of order c and higher. Standard definitions of intrinsic viscosity [ η ] neglect shear-rate dependence of the additive, therefore [ η]( γ ) = [ η] ɺ would be a constant. In general, an additive may cause rate-dependence of the viscosity, which prompts us to show this dependence

3 Ewoldt & Bharadwaj - Intrinsic LAOS 3 explicitly. We will see analogies between this and the four amplitude-intrinsic LAOS measures that exhibit frequency dependence. A low-dimensional description is possible within the asymptotic region of c ccrit (the precise limits of which depend on the desired accuracy). In such a region, instead of reporting the two-dimensional function η( ɺ γ, c), one only needs to report two functions, η ( γɺ ) and [ η] ( ) γɺ. Related asymptotic measures include concentration-intrinsic dynamic viscoelasticity (Kirkwood and Plock 1956; Paul 1969; Johnson et al. 197; Osaki 1973; Osaki et al. 197; Hair and Amis 1989), which on occasion has been expanded also in terms of LAOS shear strain amplitude for amplitude intrinsic measures (Paul 1969) Intrinsic properties can be a strong function of molecular or microstructural features, e.g. both the magnitude and sign of [ η] ( ) γɺ. Einstein was the first to relate its measurement to molecular dimensions (Einstein 196; 1911). Since then, the intrinsic viscosity [ η ] has been used to infer other micro-, nano-, and molecular-scale features, for example the aspect ratio of carbon nanotubes suspended in solution (Davis et al. 4). The sign of [ η ] is also sensitive to the underlying molecular features. It can be negative, for example, with some polymer melts with nanoparticle additives (Tuteja et al. 5). In addition to inferring structure, the intrinsic shear viscosity has a clear physical interpretation, readily apparent in Eq.(1), where it represents a leading-order change in the dissipative resistance to shear flow. Here we describe shear amplitude-intrinsic viscoelastic material functions that capture leadingorder nonlinearities in response to deformation; the motive is to obtain a clear physical interpretation and to associate intrinsic responses to structure. We consider simple shear deformation, and specifically oscillatory shear. The power of the oscillatory protocol is to conceptually decompose the response into dissipative effects (viscous or plastic) from the storage effects (elastic), e.g. linear viscoelastic moduli G ( ω), G ( ω) J ( ω), J ( ω). In contrast, stress relaxation modulus (, ), or linear viscoelastic compliances G t γ conflates viscous and elastic effects into a single measure, as does the creep compliance J( t, σ ). While these conflated material functions are valid, useful, and well defined, in this work we focus on the conceptually decomposed viscoelastic material functions from oscillatory deformation. Oscillatory deformation imposes a known frequency and amplitude, and therefore covers the two-dimensional viscoelastic response map of timescale and amplitude known as the Pipkin space

4 Ewoldt & Bharadwaj - Intrinsic LAOS 4 (Pipkin 197). When the response is observably nonlinear (i.e. not scaling linearly with the input amplitude), the protocol has been termed large-amplitude oscillatory shear (LAOS). The linear counterpart is known as small-amplitude oscillatory shear (SAOS). The limit of linear viscoelasticity is arbitrary, depending on the sensitivity to nonlinear deviations. Experimental measurements may be described as linear when higher order effects are un-measurable. But theoretically speaking, purely linear viscoelasticity is defined only in the limit of zero amplitude (just as intrinsic viscosity is defined in the limit of zero concentration). The asymptotic deviation from SAOS linearity, in terms of a power-function expansion of shear stress, was considered theoretically by (Onogi et al. 197, their Eq.(9)). Some asymptotic nonlinearities were measured by (Davis and Macosko 1978) for solid PMMA. Asymptotic LAOS measures were used by (Vrentas et al. 1991) for checking the validity of certain constitutive models. Recently, some have started referring to the region of measurable leading-order deviations from linearity as medium-amplitude oscillatory shear (MAOS) (Hyun et al. 6, Hyun et al. 7, Hyun and Wilhelm 9, Wagner et al. 11). In the spirit of concentration-intrinsic viscosity denoted as [ η ], in this work will sometimes use the shorthand [LAOS] here for amplitude-intrinsic LAOS, and the associated material functions will also use brackets [ ]. For LAOS, the input amplitude may be either controlled-strain amplitude γ or controlled-stress amplitude σ. To distinguish between shear strain-control and shear stresscontrol tests, we will use the short-hand LAOStrain and LAOStress, respectively referring to large-amplitude oscillatory shear strain/stress (Dimitriou et al. 1, Ewoldt 13). We will use the acronym LAOS when the controlled input need not be specified. A typical LAOStrain response is shown in Figure 1, showing a simulated strain-amplitude sweep of a single-mode Giesekus model(appendix, Eq.(51)). The first-harmonic moduli are functions of the imposed strain amplitude. To illustrate conceptually the limit of linear viscoelasticity (SAOS) we use the criteria that the first-harmonic moduli change by.1%. (This arbitrary choice is for illustrative purposes only.) The response then enters the intrinsic nonlinear [LAOS] regime, where the leading-order nonlinearities suitably describe the response. The end of this intrinsic regime is loosely defined, and depends only on the accuracy required by the asymptotic expansion. For illustrative purposes, we limit the intrinsic LAOS regime to the region where the first-harmonic moduli have changed by 1% or less. Analogous nonlinear criteria for higher harmonic contributions can also be defined, e.g. when the third-harmonic amplitude is.1% or 1% of the

5 Ewoldt & Bharadwaj - Intrinsic LAOS 5 corresponding linear viscoelastic measure. The third-harmonic criteria shift the range of SAOS, [LAOS], and LAOS to slightly larger strain amplitudes. The small changes of the intrinsic regime can be difficult to see in Figure 1, and indeed difficult to measure, but there are many benefits to defining intrinsic LAOS material functions. First, intrinsic measures are a low-dimensional fingerprint of nonlinear material behavior, which can be related to microstructural differences (as with intrinsic viscosity [ η ]). Second, any physical interpretation of individual higher-harmonics is best applicable in the leading-order regime when the third-harmonics dominate the nonlinearity (Ewoldt et al. 8). Although leading-order changes may miss trends that occur at much larger strain amplitude, focusing on intrinsic nonlinearities allows for robust physical interpretations via Chebyshev coefficients and the deformation domain. Third, focusing on intrinsic nonlinearities will typically avoid the experimental artifacts associated with very large deformations. Experimental artifacts such as nonhomogeneous flow (Ravindranath et al. 11), wall slip, edge fracture, instrument inertia, and sample inertia can cast doubt on very large amplitude tests and limit how large the amplitude can be. Figure 1. Four different shear nonlinearities are possible in LAOStrain: changes to the first-harmonic moduli and emergence of third harmonics. This is demonstrated with the single-mode Giesekus model (Eq.(51)) at Deborah number De = λω = 1 with nonlinear parameter α =.3 (numerical simulation). Firstharmonic moduli are normalized by the plateau modulus G. Third-harmonic Chebyshev coefficients (Eq.(17)-(18)) are scaled by the corresponding linear viscoelastic material function at the same frequency, De = 1. The regions of linear SAOS, intrinsic nonlinear [LAOS], and fully nonlinear LAOS characterization are shown. The definitions of.1% and 1% changes are arbitrary and for illustrative purposes only.

6 Ewoldt & Bharadwaj - Intrinsic LAOS 6 I.A. Overview of intrinsic LAOS material functions What is new here is to give meaningful interpretation to the four leading-order nonlinearities of LAOS, emphasizing the importance of their signs. In contrast to some recent studies, we show that four separate intrinsic nonlinearities can be defined (and measured, Section V), giving more information than just a single measure (Hyun and Wilhelm 9). We introduce new variable nomenclature and sign conventions to encode the physical interpretation, for both LAOStrain and LAOStress. G For LAOStrain we will have the two familiar linear viscoelastic material functions ( ω), G ( ω), and four intrinsic nonlinear material functions [ e ]( ω), [ e ]( ω), [ v ]( ω), [ v ]( ω ) (Details in Section III; we use the letter e for elastic nonlinearities and v for viscous nonlinearities). All six of these measures have a functional dependence on frequency only; a powerfunction expansion allows for this low-dimensional representation. With strain input represented as ( t ) sin γ = γ ωt, we will see that the time-domain expansion of shear stress takes the form { } {[ e1 ] sin t [ 1] cos t [ e3] sin 3 t [ 3] cos3 t} 5 ( γ ) ( t;, ) = G ( ) sin t + G ( ) cos t 3 + ( ) + ( ) ( ) + ( ) σ γ ω γ ω ω ω ω γ ω ω ωv ω ω ω ω ωv ω ω () + O To be clear that the material functions are a function of frequency ω, we use e.g. [ e3 ]( ω ). The negative sign on the term with [ ]( ) and [ ] 3 [ v ] e3 ω, and the multiplicative factors of frequency ω on the 1 v terms, result from a coordinate change between the time-domain ( t;, ) deformation domain σ ( γ, γ; γ, ω) σ γ ω and the ɺ. Physical interpretations of these leading-order nonlinearities are revealed in the deformation domain, which represents stress as an instantaneous function of the imposed strain and strain-rate. The coefficients can be transformed into the time-domain representation, Eq.(), for ease of signal processing. Some analytical solutions are available for intrinsic material functions in LAOStrain. A recent summary of available LAOStrain analytical solutions for different constitutive models is given in Table 1 of (Giacomin et al. 11), and another has been recently published (Gurnon and Wagner 1). As an example we show the rheological fingerprint of the corotational Maxwell model in

7 Ewoldt & Bharadwaj - Intrinsic LAOS 7 Figure. The equations were derived by (Giacomin et al. 11) and converted to the framework here (Appendix A) in order to define measures which depend only on De (and not a combination of De and relaxation time λ ). LAOStress intrinsic nonlinearities have not yet been defined or measured. We will have the analogous linear viscoelastic material functions J ( ω), J ( ω) functions [ c ]( ), [ c ]( ), [ f ]( ), [ f ]( ) , and four intrinsic nonlinear material ω ω ω ω (we use the letter c for compliance nonlinearities and f for fluidity nonlinearities (Dimitriou 1, Ewoldt 13). With input shear stress denoted as σ( t) = σ cosωt (Ewoldt 13), the time-domain strain expansion will take the form { } ( ) = + ( ) + ( ) 3 f ( ω) ( ) γ t; σ, ω γ ( ω, γ ) σ J ω cosωt J ω sinωt 5 ( σ ) ( ω) [ 1] [ f3] + σ [ c1 ] ω cosωt + sin ωt + [ c3] ( ω) cos 3ωt + sin 3ωt 3ω 3ω + O. (3) We see positive signs in front of all terms here for LAOStress, and this is due to a careful choice of representing the input as σ( t) = σ cosωt rather than σ( t) = σ sinωt. The seemingly arbitrary choice of the input trigonometric reference (cosine versus sine) has important consequences for definitions of material functions in both LAOStrain and LAOStress, primarily concerning sign issues of the material functions as discussed in detail by (Ewoldt 13). Of course, the relative phase of the input ought not to make a difference to the material functions if properly defined. The issue is resolved by clearly identifying the trigonometric reference. Furthermore, the issue exists only in the time-domain (Fourier coefficients), whereas the Chebyshev coefficients are immune to the ambiguity since they reference the deformation domain in which time is an unseen internal variable. The deformation domain also reveals the interpretations of the measures in Eq.(3), as we discuss in Section III.

8 Ewoldt & Bharadwaj - Intrinsic LAOS 8 Figure : Intrinsic rheological fingerprint of the corotational Maxwell model (single-mode) showing the common frequency-dependent linear viscoelastic moduli and the four intrinsic nonlinearities of LAOS (derived by (Giacomin et al. 11) and converted here to the intrinsic Chebyshev framework). All measures are only a function of the Deborah number, De = λω. The linear viscoelastic moduli are positive, whereas the nonlinearities can take either positive or negative sign as indicated by (+) and (-) labels, respectively. I.B. Pipkin space mapping The region of the intrinsic nonlinear regime is best shown with a Pipkin space. The Pipkin space is a regime map for viscoelastic material responses as a function of timescale and loading amplitude (Pipkin 197). For LAOStrain, the two deformation inputs define the Pipkin space: the strain amplitude γ and the Deborah number De = λω, as shown in Figure 3. A line delineating the linear from the nonlinear regime has been drawn by Pipkin and others (Pipkin 197, Dealy and Wissbrun 199; Macosko 1994, Giacomin et al. 11, Ewoldt et al. 1). Such a line can be interpreted as the maximum linear viscoelastic strain amplitude as a function of De. The sketched shape of this line varies depending on the author. Here we demonstrate how intrinsic LAOS nonlinearities can make quantitative predictions for the shape of this boundary. A second line can also be drawn, which shows the limit of the intrinsic nonlinear regime. In Figure 3 the boundaries for linear, intrinsic nonlinear, and nonlinear regimes are identified based on the corotational Maxwell model asymptotic response to LAOStrain (Figure ). We use the analytical results of (Giacomin et al. 11) (converted to our framework in (Appendix A), Eq.(47)- (5)). The linear viscoelastic boundary is based on the first occurrence of any.1% deviation from linear viscoelasticity. The upper limit of intrinsic nonlinearity is based on 1% deviation (similar to

9 Ewoldt & Bharadwaj - Intrinsic LAOS 9 Figure 1 for the Giesekus model). These percent deviations are chosen for illustration and the choices affect only the relative positions of the boundary and not the functional dependence. The boundaries are drawn by interpreting each intrinsic LAOS material function as a critical amplitude that will generate nonlinearity. Consider normalized nonlinearities less than an arbitrary small value ε, elastic nonlinearity viscous nonlinearity < ε, < ε. G ( ω) η ( ω) (4) With the four intrinsic shear stress nonlinearities, Eq.(), we have four possible criteria, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ e ] ω γ [ v ] ω γ ε, G ω η ω 1 1 ε [ e3 ] ω γ [ v3] ω γ ε, ε. G ω η ω (5) All of the material functions in Eq.(5) depend only on frequency (or Deborah number De = λω), * and therefore each equation can be written to define a line for the critical strain amplitude γ (De) at which the specified nonlinearity appears with relative magnitude ε. For example, in terms of the first-harmonic elastic nonlinearity, the critical strain amplitude expression is written γ * ( ) 1 ( De) ( ) 1 G De = ε. (6) [ e ] De A smaller intrinsic nonlinearity corresponds to a larger strain amplitude to generate a nonlinear rheological response. Different critical strain-amplitudes exist for each of the material functions ( ω) e ( ω) v ( ω) ( ω) [ e ], [ ], [ ], [ v ]. Although an arbitrary choice of ε is required, the boundary shapes depend only on the functional form of the frequency-dependent material functions. In Figure 3 we choose ε =.1 for the SAOS limit and ε =.1 for the [LAOS] limit of intrinsic leading-order nonlinearities. Figure 3 does not indicate which type of nonlinearity from Eq.(5) occurs first. We give this detail in Figure 4 for both the corotational Maxwell model and the Giesekus model. It is clear in Figure 3 that the critical strain amplitude is a function of De. We observe that at high De the critical strain is approximately constant, but at low De the critical strain scales as

10 Ewoldt & Bharadwaj - Intrinsic LAOS 1 γ 1. This scaling is the signature of a critical Weissenberg number, ~ De moderate De the criteria transitions between a critical Wi and a critical γ. Wi = γ De. At Co-plotted on Figure 3 are the regions in which some typical constitutive models are applicable, including the Newtonian, Generalized Newtonian Fluid (G.N.F.), and Second Order Fluid. Illustrative but arbitrary limits on De are used for these boundaries. Since the Newtonian and G.N.F. models are purely viscous, with no elasticity, they apply for very small De; here we use De 1 3. The Second Order Fluid is an expansion with respect to De. It does not involve shearthinning, and only captures the terminal viscoelastic regime at low De. We therefore choose De 1 for the Second Order Fluid. Since the Second Order Fluid does not predict shearthinning, we limit the critical Wi to be the same as the limit of the linear viscoelastic regime. Figure 4 indicates the four different critical strain amplitudes as a function of De for the corotational Maxwell model and Giesekus model. For the corotational Maxwell model, the lowest critical strain is always determined by the quantity nonlinearity. [ e ] γ 1 G, i.e. the relative first-harmonic elastic The Giesekus model critical strains are more complex. At low De the subdominant elastic nonlinearity is the first to have ε relative change, due to [ e1 ]( ω ). In other words, [ e1 ]( ω ) is the strongest relative nonlinearity, and therefore the critical strain required to observe this nonlinearity is smaller. At high De the sub-dominant viscous nonlinearity appears first indicating that [ ]( ω) η ( ω) is the strongest relative nonlinearity. The critical strain amplitude due to [ e3 ]( ω ) is v 1 noteworthy for this model, as it does not plateau at high De like the other intrinsic nonlinearities, but instead its critical strain amplitude increases with De. A large critical strain correlates with a small nonlinearity. In this case, the relative nonlinearity [ e3 ]( ω) G ( ω) decays as De increases. In Figure 3 the critical strain lines are smooth, based on the corotational Maxwell model. If these lines were instead based on the Giesekus model, a bump would appear near De.5. This is shown by the intersecting lower lines in Figure 4b where the critical strain changes from [ e1 ]( ω) G ( ω) to [ v 1]( ω) η ( ω) at this location.

11 Ewoldt & Bharadwaj - Intrinsic LAOS 11 The mapped region of intrinsic nonlinearity (Figure 3) frames the limit of applicability for the rest of the work here. We derive the theory of intrinsic material functions for LAOS, including LAOStrain and LAOStress. We start with expressions based in the time domain (involving Fourier series). Here the coefficients are well defined, and easily calculated with signal processing, but a physical interpretation is not available in the time domain. We therefore derive a representation in the deformation domain (involving Chebyshev polynomials of the first kind), and present an interpretation of the four intrinsic shear nonlinearities in both LAOStrain and LAOStress. Finally, we demonstrate that these four intrinsic nonlinearities can be experimentally measured, using a PVA-Borax polymeric hydrogel. We observe the integer power function expansion, and the measures are used to infer molecular and microstructural information about the material.

12 Ewoldt & Bharadwaj - Intrinsic LAOS 1 Figure 3: The general two-dimensional map of nonlinear rheology as a function of strain amplitude γ and De, known as the Pipkin space. The corotational Maxwell model is used for the boundaries of the linear viscoelastic and intrinsic nonlinear regi, using the criteria of the first observed.1% and 1% changes in a linear viscoelastic measure. The corotational Maxwell model quantitatively shows straininduced nonlinearity at high De ( γ = 15 ) and strain-rate-induced nonlinearity at low De (Weissenberg number Wi = λγɺ = 1 15 ). Some constitutive equations are shown within their region of applicability, including Newtonian, Generalized Newtonian Fluid (G.N.F.), and Second Order Fluid.

13 Ewoldt & Bharadwaj - Intrinsic LAOS 13 Figure 4. Critical nonlinear strain amplitude in LAOStrain as a function of De for single-mode models, (a) corotational Maxwell, and (b) Giesekus model ( α =.3 ). Shown are the critical strains associated with each of the four intrinsic shear nonlinearities of LAOStrain. The lowermost boundaries of the corotational Maxwell model (smallest critical strain amplitude) associated with.1% and 1% changes are used for the boundaries in Figure 3.

14 Ewoldt & Bharadwaj - Intrinsic LAOS 14 II. Theory for LAOStrain II.A. Time Domain LAOStrain In LAOStrain, it is common convention (Ewoldt 13) to represent the shear strain input as γ ( t) = γ sin ωt, (7) where γ is the strain amplitude and ω is the frequency. This consequently imposes an orthogonal strain rate ɺ γ ( t) = ɺ γ cosωt (8) where γɺ is the strain rate amplitude. The resulting shear stress response can be represented as a Fourier series (Dealy and Wissburn 199) { n n } σ ( t; ω, γ ) = γ G ( ω, γ ) sin nωt + G ( ω, γ ) cos nωt, (9) n: odd where the Fourier moduli Gn ( ω, γ ) and Gn ( ω, γ ) are functions of input frequency ω and strain amplitude γ. We have assumed a stress signal that has attained a time-periodic response, or alternance (Giacomin et al 11; Gurnon and Wagner 1), and is shear symmetric, hence the inclusion of only odd-harmonics n in Eq. (9) (Hyun et al. 11). An alternate representation of the resultant stress response is an infinite power series expansion in both strain amplitude and frequency (Onogi et al. 197; Pearson and Rochefort 198) { jn jn } (1) j ( t ) = G ( ) ( n t) + G ( ) ( n t) σ ; ω, γ γ ω sin ω ω cos ω. j: odd n: odd j Importantly, the coefficients G ( ω) and G ( ω) are only a function of input frequency ω, and jn jn all strain-dependence is assumed to originate from the integer power series expansion (the need for non-integer power expansions is addressed in detail in a separate work (Ewoldt & Bharadwaj 13)). In this notation the linear viscoelastic moduli are G ( ω) = G 11( ω) and G ( ω) = G 11( ω). Since the expansion is with respect to dimensionless strain amplitude, the coefficients G ( ω) and G ( ω) maintain the dimensions of stress (F/L ), or Pa in SI units. Comparing Eq. (9) and Eq.(1), the jn jn

15 Ewoldt & Bharadwaj - Intrinsic LAOS 15 Fourier moduli, which are easier to calculate from measured data, can be related to the power expansion coefficients as ( ω, γ ) = ( ω) + ( ω) γ + 4 ( γ ) ( ω, γ ) = ( ω) + ( ω) γ + 4 ( γ ) ( ω, γ ) = ( ω) γ + 4 ( γ ) ( ω γ ) = ( ω) γ + 4 ( γ ) G G G O 1 31 G G G O 1 31 G G O 3 33 G, G O In the limit of small amplitude, the first-harmonic moduli reduce to G1 ( ω, γ) G ( ω) and G ( ω, γ ) G ( ω). 1 The four O( γ ) terms in Eq.(11) produce leading-order nonlinearities at finite γ. The physical interpretation of these four intrinsic nonlinearities is not clear from the time-domain representation. We will change the coordinate frame from the time domain to the deformation domain (Figure 5) and use appropriate orthogonal polynomials to define physically-meaningful intrinsic nonlinearities, as detailed in the following Section II.B. The scaling with respect to strain-amplitude in Eq.(9)-(1) could instead be with respect to strain-rate amplitude ɺ γ = γ ω. The Fourier coefficients are then dynamic viscosity coefficients { n ( ) n ( ) } σ ( t) = ɺ γ η ω, γ sin nωt + η ω, γ cos nωt. (1) n: odd For this viscous perspective the power function expansion is (Fan and Bird 1984; Giacomin et al. 11) (11) j j ɺ { jn ( ) ( ) jn ( ) ( )} (13) j: odd n: odd σ ( t) = γ η ω sin nωt + η ω cos nωt. In the expansion with respect to strain-rate, the coefficients η ( ω) and η ( ω) have dimensions that depend on the power j. For j = 1 the dimensions are that of viscosity (F.T/L ), or Pa.s in SI units. But in general the dimensions are (F.T j /L ). Due to these peculiar dimensions, the expansion with respect to dimensionless strain amplitude γ may be preferred, as in Eq.(1). It is for this reason that the strain-amplitude expansion will be used in the deformation-domain Chebyshev jn jn

16 Ewoldt & Bharadwaj - Intrinsic LAOS 16 representation given in the following section, for shear strain-controlled LAOS. We will see that for shear stress-controlled LAOS the expansion is with respect to stress amplitude σ, and the peculiar dimensions are unavoidable for the intrinsic coefficients. Figure 5. LAOStrain response of the Giesekus model ( α =.3 ) at De = 1 and * γ =.685 chosen such that [ e ] γ / G =.1. Representation of the nonlinear response is possible in (a) the time domain as * 1 periodic waveforms and in (b) the deformation domain as parametric loops; dashed and dotted lines are e v elastic stress σ ( γ ) and viscous stress σ ( ɺ γ ), respectively. The deformation domain Lissajous curves are more insightful and provide meaningful physical interpretations to each kind of nonlinearity depending on their signs and magnitudes whereas this information is uninterpretable in the time domain representation. II.B. Deformation domain LAOStrain Meaningful interpretation of LAOS nonlinearities comes from a coordinate transformation to the deformation domain (Figure 5) with time as an internal variable. Starting with LAOStrain, we use normalized deformation-domain parameters

17 Ewoldt & Bharadwaj - Intrinsic LAOS 17 x γ ( t) γ = sinωt (14) y ɺ γ ( t) ɺ γ = cos ωt (15) to represent the total stress σ e v as a superposition of elastic stress σ ( γ ) and viscous σ ( γɺ ) stresses (Cho et al. 5) e ( x y ) = ( x ) + ( y ɺ) v σ, ; ω, γ σ ; ω, γ σ ; ω, γ. (16) This decomposition is based on the idea that the elastic and viscous stresses are functions of instantaneous strain γ and strain rate γɺ, respectively. This interpretation of elastic and viscous stress is well-established in the linear viscoelastic regime, and we therefore expect it to be welldefined for the asymptotic nonlinear regime as well. Challenges do exist with the decomposed interpretation at very large deformations, especially for yield stress fluids, in which case local measures of nonlinearities have proven useful (Rogers and Lettinga 1), including for LAOStress (Dimitriou et al. 1). The decomposed stresses in Eq.(16) can be represented in terms of orthogonal Chebyshev polynomials of the first kind (Ewoldt et al. 8) e ( x ) e ( ) T ( x ) σ ; ω, γ = γ ω, γ n (17) n n: odd v σ ( y; ω, ɺ γ ) = ɺ γ v ( ω, γ ) Tn ( y) (18) n n: odd where en ( ω, γ ) and v n( ω, γ ) are functions of the LAOS input parameters ω and γ. The Chebyshev basis functions T ( ) 1964) n x are defined by the recurrence relation (Abramowitz and Stegun T ( x) = 1 T ( x) = x 1 T ( x) = xt ( x) T ( x). n+ 1 n n 1 (19)

18 Ewoldt & Bharadwaj - Intrinsic LAOS 18 The third harmonic is the most important for intrinsic nonlinearities, T x = x x. The ( ) Chebyshev coefficients are directly related to the Fourier coefficients of Eq.(9) and Eq.(1) by the relations for n odd (Ewoldt et al. 8). e n ( 1) ωη ( 1) ( 1) ( n 1) n = Gn = n v n Gn = = ηn ω () Some interpretations of first- and third-harmonic coefficients were given previously (Ewoldt et al. 8). The first-harmonic material functions e1 ( ω, γ) = G1 ( ω, γ) and v ( ω, γ ) = η ( ω, γ ) 1 1 are measures of average elasticity and average dissipation, respectively. They are coefficients of a linear basis response, but this basis response is only equivalent to the linear viscoelastic response of G ( ω ) and η ( ω) in the limit of zero strain amplitude. The third-harmonic measures add the basis function T x = x x, and indicate local ( ) changes and distortion of the decomposed stresses. The coefficients e3 ( ω, γ ) and v ( ω, γ ) 3 determine the leading order convexity of the decomposed elastic and viscous stresses, respectively. Adding a positive function T ( ) 3 x results in curves with positive convexity for positive instantaneous strains. This convexity of the decomposed elastic curve has been interpreted as strain-stiffening for e3 ( ω, γ ) >, or strain-softening with e3 ( ω, γ ) <, whereas for the viscous curve the convexity indicates rate-thickening if v ( ω, γ ) > 3 or rate-thinning if v ( ω, γ ) < 3 (Ewoldt et al. 8). In this work we will show that third-harmonics can also be interpreted in the context of first-harmonic intrinsic nonlinearities, indicating if strain or rate-of-strain is predominantly causing the average changes. This will become clear after defining the intrinsic Chebyshev nonlinearities. To define low-dimensional intrinsic Chebyshev nonlinearities, we use an integer power function expansion of en ( ω, γ ) and v n( ω, γ) with respect to dimensionless strain amplitude. This is similar to Eq.(11) which expands the Fourier coefficients Gn ( ω, γ ) and Gn ( ω, γ ). The leading-order Chebyshev expansion, with the inter-relation to the Fourier coefficients, is

19 Ewoldt & Bharadwaj - Intrinsic LAOS 19 ( ) ( ) ( ) ( ) v ( ) v ( ) 4 1 ω γ = 1 ω γ = ω + 1 ω γ + γ e (, ) G (, ) G [ e ] O( ) 4 3 ω γ = 3 ω γ = 3 ω γ + γ e (, ) G (, ) [ e ] O( ) 4 1 ω γ = η1 ω γ = η ω + 1 ω γ + O γ v (, ) (, ) [ ] ( ) v 4 3 ω γ = η3 ω γ = 3 ω γ + O γ (, ) (, ) [ ] ( ) (1) where the elastic coefficients G ( ω ), [ e1 ]( ω ), [ e3 ]( ω ), and the viscous coefficients η ( ω), [ v1]( ω ), and [ v3]( ω ) are functions of only frequency ω. The four nonlinearities of Eq.(1) are summarized graphically in Figure 6. The nonlinearities are decomposed to show how each independently rotates or bends the Lissajous curves and decomposed stress curves. The first-harmonic nonlinearities each cause rotation of the underlying linear basis function (increasing slope for positive values of [ e1 ]( ω ) and [ v ]( ω )). The underlying 1 basis function is linear, but it changes for each cyclic loading at increasing strain amplitude this has been termed intercycle nonlinearity (Ewoldt et al. 8). Physically, the rotation is interpreted as changes in the average elasticity or viscosity, revealed when plotted in the domain of stress versus strain or stress versus strain rate. The third-harmonic intrinsic nonlinearities bend and twist the decomposed stress response, by adding the third-harmonic basis function T x = x x. This basis function represents local ( ) deviation from the linear basis function. It tells of the relative differences in local nonlinear effects, i.e. relative nonlinearities within a single cycle this has been called intracycle nonlinearity (Ewoldt et al. 8). The bending and twisting can be described as intracycle strain-stiffening/softening or intracycle rate-thickening/thinning, but an additional interpretation is possible. For any non-zero third-harmonic, there will be regions where the total response is higher and lower than the linear basis. Sinusoidal oscillations put strain and rate-of-strain out of phase, therefore instantaneous relative deviations can be associated with either input. By identifying locations where the relative intracycle deviation is consistent with the average intercycle change, one can identify the driving cause of the average change. Figure 7 outlines this additional interpretation. We now interpret the Giesekus model response shown in Figure 5b, De = 1. At this frequency, the chosen strain amplitude ( γ =.685) resides in the low-dimensional intrinsic LAOS regime (see Figure 4). The emergence of nonlinearity changes the shape of the elastic and viscous decomposed stress curves, which is explained by the signs and magnitudes of the intrinsic nonlinearities. The

20 Ewoldt & Bharadwaj - Intrinsic LAOS [ e 1] nonlinearity dominates in this example. The negative values of [ e 1] and [ e3 ] signify average elastic softening (clockwise rotation) and instantaneous elastic strain softening (mild distortion) in the decomposed elastic stress curve. Independently, negative [ v ] 1 signifies average viscous thinning (clockwise rotation). Positive [ v ] 3 captures relative local changes separate from the average; [ v ] > is intracycle rate-thickening, bending the curve such that the response is relatively thicker at 3 ɺ γ = ɺ γ and therefore relatively thinner at ɺ γ =. Since the average change is thinning, [ v ] < 1 prefer here to interpret [ v ] > as an indicator of the location of relatively more thinning, which is 3 near ɺ γ =, or equivalently near large strains γ = ± γ. Sinusoidal oscillations put strain and rate-ofstrain out of phase, which means this response with [ v ] > can be called rate-thickening, or 3 equivalently, strain-thinning. The latter interpretation gives context to the first-harmonic average thinning, as shown in Figure 7. The definitions of our amplitude-intrinsic measures are related to the concept of concentrationintrinsic viscosity, as in Eq.(1). An important distinction is that Eq.(1) does not use a scaling with respect to the linear viscoelastic material functions. Such a scaling is possible, and for example this would result in modifying Eq.(1) to take the form (for just the first-harmonic elastic modulus), we [ e ]( ω) 1 4 G 1 ( ω, γ ) = G ( ω) 1 + γ + O( γ ). G ( ω) () The ratio of amplitude-intrinsic [ e1 ]( ω) / G ( ω) is analogous to concentration-intrinsic viscosity [ η]( γɺ ) of Eq.(1). This is certainly a useful quantity as shown in Figure 4 and Eq.(5) for interpreting critical strain amplitudes. However, we avoid defining this ratio as the fundamental material function in order to simplify the definition and also minimize numerical errors associated with division involving two measured quantities with finite precision error.

21 Ewoldt & Bharadwaj - Intrinsic LAOS 1 Figure 6. Schematic showing possible nonlinear viscoelastic contributions to LAOStrain deformation. Lissajous curves are normalized by strain amplitude γ and the linear viscoelastic stress amplitude σ. Each intrinsic nonlinearity is shown separately, at 1% nonlinearity compared to the linear response. Positive values are shown for illustration. Negative values of [ e 1] and [ v 1] would instead rotate the curves clockwise, and negative values of [ e 3] and [ v 3] would cause the opposite convexity. Figure 7. Flowchart for interpreting intrinsic Chebyshev LAOStrain nonlinearities. The symbols + and indicate positive and negative values of the quantity, respectively.

22 Ewoldt & Bharadwaj - Intrinsic LAOS II.C. Interrelations with other measures: In this section we relate the four intrinsic material functions to other nonlinear viscoelastic measures, including local measures of nonlinearity introduced by (Ewoldt et al. 8). This gives context to the intrinsic measures and supports their physical interpretation. We also give interrelations with other notation to facilitate conversion to a common framework. Local measures of nonlinear viscoelasticity were introduced by (Ewoldt et al. 8), showing particular locations on Lissajous curves that can be used to define viscoelastic moduli and dynamic viscosities. At the minimum strain γ = (equivalent to maximum strain rate ɺ γ = ± ɺ γ), the local slope of the stress with respect to strain is a measure of elastic modulus, dσ ( n 1) G M = nen ( ω, γ )( 1 ). dγ γ = ɺ γ =± ɺ γ (3) A corresponding measure at the largest instantaneous strain γ = ± γ (equivalent to minimum strain rate ɺ γ = ) is also a measure of local elastic modulus σ G L = γ γ =± γ ɺ γ = e n ( ω γ ),. (4) In terms of the intrinsic elastic nonlinearities, the various measures of elastic modulus are ( ) ( ) { } ( ) { } 4 1 ω γ = ω + γ 1 ω + γ G, G ( ) [ e ]( ) O( ) 4 = G ω, γ G ( ω) γ [ e ]( ω) 3[ e ]( ω) O( γ ) M 4 = G ω, γ G ( ω) γ [ e ]( ω) [ e ]( ω) O( γ ). L (5) The average measure G 1( ω, γ ) is influenced only by [ e1 ]( ω ), but the local measures G (, ) M ω γ and G (, ) L ω γ are dependent on a combination of [ e1 ]( ω ) and [ e3 ]( ω ), with a different sign dependence on [ e3 ]( ω ) for each. For finite [ e3 ]( ω ), at leading order, the average measure G 1( ω, γ ) will always lie between G (, ) M ω γ and G (, ) L ω γ.

23 Ewoldt & Bharadwaj - Intrinsic LAOS 3 Analogous intrinsic relations can be derived for average viscosity η ( ω, γ ) viscosity η ( ω, γ ), and large-rate viscosity η ( ω, γ ) M measures [ v1]( ω ) and [ v3]( ω ) in the same form as Eq.(5). L, minimum-rate 1. These are functions of intrinsic viscous There are several different measures to characterize a nonlinear response to LAOS, but in the intrinsic regime only four measures are required. This is the power of the low-dimensional representation with intrinsic material functions. Deeper into the nonlinear regime the 4 O( γ ) and higher terms appear, and the local measures of GM ( ω, γ ), η (, ) M ω γ, etc. become of increasing importance. Only four intrinsic nonlinearities are required for the shear response, but various nomenclature exists to represent them, as discussed in Section II.A. Below we give inter-relations for converting other nomenclature to the Chebyshev nomenclature here, [ e1 ]( ω ), [ e3 ]( ω ), [ v ]( ω ) 1, [ v ]( ω ) 3, which used the strain-amplitude expansion to define each measure, Eq.(1). We prefer the Chebyshev coefficient representation not only because it allows for a physical interpretation (Ewoldt et al. 8), but it also is immune to the trigonometric reference of the input which can cause ambiguous signs of higher-harmonic Fourier coefficients (Ewoldt 13). Previous work has represented intrinsic nonlinearities in the time-domain, with either the elastic or viscous scaling (Eq.(1) or Eq.(13)). To convert these results from the Fourier representation to the Chebyshev representation, we use the inter-relation of Eq.() and compare coefficients of the power function expansions, from Eq.(1), Eq.(13), and Eq.(1). The relation is 3 1 ω = G31 ω = ω η31 ω 3 3 = G33 = 33 [ e ]( ) ( ) ( ) [ e ]( ω) ( ω) ω η ( ω) G31 ( ω) [ v1 ]( ω) = = ω η31 ( ω) ω G33 ( ω) [ v3]( ω) = = ω η33 ( ω) ω (6) The conversions result in sign changes (to change from the time-domain to the deformationdomain), and factors of ω which occur for two reasons. First, because the viscous Chebyshev coefficients v1( ω, γ ) and v3( ω, γ ) are defined by a shear-rate scaling, giving the coefficients

24 Ewoldt & Bharadwaj - Intrinsic LAOS 4 dimensions of viscosity. Second, the intrinsic coefficients [ v1]( ω ) and [ v3]( ω ) also have dimensions of viscosity since the power function expansion is still with respect to the dimensionless strain amplitude γ, Eq.(1). Although four independent intrinsic nonlinearities exist, some studies have chosen to combine the third harmonic terms into a lumped measure (Hyun and Wilhelm 9). In this case the relative intensity of the third harmonic is normalized by the first harmonic, I3/1 = I3 I1. Based on intrinsic regime scaling, a nonlinear coefficient is defined as small strain amplitudes, or intrinsic LAOS, this lumped intrinsic measure as I3/1 γ Q( ω, γ ) = /. In the asymptotic limit of Q ( ω) = lim Q( ω, γ ). Using our notations, we rewrite γ Q = ( ωv ) [ e ] + [ ] 3 3 G + G 1 1. (7) Such lumped parameters are intrinsic, and can be useful, but they omit first-harmonic nonlinearities, combine the elastic and viscous third-harmonic measures, and remove the sign information. We will see that all of these are important aspects of distinguishing the experimentally measured rheological fingerprints in Section V. Translating nomenclature is more important for LAOStrain than LAOStress. This is because asymptotic analytical solutions are available for LAOStrain in the literature, and some notation has been introduced which can be converted to the deformation-domain framework. For LAOStress, no analytical asymptotic solutions have been published to date, and we are unaware of any intrinsic regime notation. We define this in the next section. III. Theory for LAOStress Here we introduce the framework, notation, and interpretation for LAOStress. Choosing to represent the input stress as(ewoldt 13) the strain and strain rate response are represented as σ( t) = σ cosωt (8)

25 Ewoldt & Bharadwaj - Intrinsic LAOS 5 { n ( ) n ( ) } γ ( t) = γ ( ω, γ ) + σ J ω, σ cos nωt + J ω, σ sin nωt (9) n: odd { n ( ) n ( ) } ɺ γ ( t) = σ φ ω, σ sin nωt + φ ω, σ cos nωt (3) n: odd where J, n J n are compliances and φ, n φ n are fluidities. The term γ ( ω, γ ) represents the zeroth harmonic accounting for the possibility of the output strain signal not being centered about zero. An alternate series expansion of the resulting strain in the powers of stress amplitude and frequency is j j j: odd n: odd { ( ) ( ) jn jn ( ) ( )} (31) γ ( t; ω, σ ) = γ ( ω, γ ) + σ J ω cos nωt + J ω sin nωt which can be expanded as { } { J ( ) cos t + J ( ) sin t + J ( ) cos3 t + J ( ) sin3 t} + O( ) ( ) ( ) ( ) ( ) γ t; ω, γ = γ ω, γ + σ J ω cosωt + J ω sinωt + σ ω ω ω ω ω ω ω ω σ (3) where the linear viscoelastic compliances are J ( ω) = J 11( ω) and J ( ω) = J 11( ω). Comparison of Eq.(9) and Eq.(3) gives a relation between the Fourier moduli and the power expansion coefficients as ( ω, γ ) = ( ω) + ( ω) σ + 4 ( σ ) ( ω, γ ) = ( ω) + ( ω) σ + 4 ( σ ) ( ω, γ ) = ( ω) σ + 4 ( σ ) ( ω, γ ) = ( ω) σ + 4 ( σ ). J J J O 1 31 J J J O 1 31 J J O 3 33 J J O 3 33 The first-harmonic measures again represent a cyclic average, but physical interpretations for the third-harmonic measures are not yet apparent. For the same reasons stated earlier, we switch to the deformation domain framework and introduce a Chebyshev representation (Ewoldt et al. 8) through a harmonic stress input σ ( t) made dimensionless as (33) σ ( t) z =. (34) σ The resulting strain response is decomposed into apparent elastic and viscous components,

26 Ewoldt & Bharadwaj - Intrinsic LAOS 6 e γ = γ ( σ) + γ v ( σ) (35) rewritten in a viscous perspective in terms of strain rate as e ɺ γ = ɺ γ ( σ) + ɺ γ v ( σ). (36) The elastic and viscous decompositions result from the idea that the elastic strain and viscous strainrate are both instantaneous functions of stress. The Chebyshev representation follows as e γ ( σ ) = σ c T ( z) ɺ v n: odd γ ( σ ) = σ f T ( z) n: odd n n n n (37) where c n stands for compliances and f n stands for fluidities. As is the case in LAOStrain control, a direct relationship can be identified between the Fourier and Chebyshev coefficients c f n n = J n = nω J. n (38) The leading-order Chebyshev expansion, with the inter-relation to the Fourier coefficients, is c ( ω, σ ) = J ( ω, σ ) = J ( ω) + [ c ]( ω) σ + O( σ ) c ( ω, σ ) = J ( ω, σ ) = [ c ]( ω) σ + O( σ ) f ( ω, σ ) = φ ( ω, σ ) = φ ( ω) + [ f ]( ω) σ + O( σ ) f ( ω, σ ) = φ ( ω, σ ) = [ f ]( ω) σ + O( σ ), (39) where the intrinsic compliances and fluidities are a function of frequency only. The signs are conveniently all positive in this inter-relation, which results from the choice of a cosine input, rather than a sine input, in Eq.(8) to define the Fourier coefficients (Ewoldt 13). The dimensions of the orthogonal Chebyshev coefficients c (, ) n ω γ are compliance (L /F) or 1/Pa in SI, but the intrinsic compliance coefficients [ c1 ]( ω ) and [ c3 ]( ω ) have dimensions (L 6 /F 3 ) due to the power function expansion with respect to stress amplitude σ. There is a similar difference between the dimensions of the orthogonal Chebyshev coefficients f (, ) n ω γ which have dimensions of fluidity, or inverse viscosity L /(FT), and the intrinsic fluidities [ f1]( ω ) and [ f3]( ω ) having dimensions of L 6 /(F 3 T). Such cumbersome dimensions are not uncommon for intrinsic

27 Ewoldt & Bharadwaj - Intrinsic LAOS 7 measures, e.g. when intrinsic viscosity [ η ] is defined from an expansion with respect to concentration, rather than volume fraction. The first-harmonic material functions [ c1 ]( ω ) and [ f1]( ω ) encode the average changes of compliance and fluidity as the stress amplitude is increased. As with LAOStrain, these LAOStress measures cause rotation of Lissajous curves (positive is counterclockwise, negative is clockwise). The intrinsic third-harmonics [ c3]( ω ) and [ f3]( ω ) tell of relative nonlinearities within a cycle, as a function of the input stress. Adding a positive third-harmonic basis function T ( ) 3 z will at high stress cause relatively higher compliance (or fluidity), and at low stress relatively lower compliance (or fluidity). The local changes (intracycle nonlinearities) should be interpreted in the context of the average changes, as indicated by [ c1 ]( ω ) and [ f1]( ω ). There is an important difference here between LAOStress and LAOStrain. In LAOStrain the elastic response is referenced to strain and viscous response to strain-rate, Eq.(17)-(18). In LAOStress, there is only one input to reference, the stress σ, as shown in Eq.(37). Moreover, stress is an absolute reference input. Zero stress and maximum stress are clearly defined, in contrast to the strain-input which can be reset by yielding events and confuse the interpretation of local responses as zero strain. In this way, LAOStress is perhaps easier to interpret than LAOStrain. For both LAOStrain and LAOStress, the low-dimensional description of the intrinsic regime is based on the assumption of a power-function expansion of the response. A subtle point is that an integer expansion has been assumed, and this is the case for all known prior work in this area. Such integer power expansions may not be adequate for some cases. We address this general need for non-integer power expansions in a separate work (Ewoldt & Bharadwaj 13), where we introduce non-integer power expansions and demonstrate their applicability with two common generalized Newtonian fluid models (Cross and Carreau-Yasuda). There we show that intrinsic nonlinearities are still associated with first-harmonic deviations and third harmonic emergence in the response, if the stress is shear symmetric. This shows that intrinsic nonlinearities can still be defined in general, but the amplitude-dependent scaling must be checked and specified. In the next section, we experimentally demonstrate that all four intrinsic nonlinearities can be measured, and frequency dependence shown. We characterize a physically-associating polymer gel for which standard integer power law scaling of material functions ~ γ is observed.