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1 Dynamic Shear Rheology of Thixotropic Suspensions: Comparison of Structure-Based Models with Large Amplitude Oscillatory Shear Experiments Matthew J. Armstrong, Antony N. Beris, Norman J. Wagner a) Center for Molecular and Engineering Thermodynamics, Department of Chemical and Biomolecular Engineering, University of Delaware, Newark, DE a Corresponding Author. Tel address: wagnernj@udel.edu Synopsis Submitted to Journal of Rheology Rheological measurements on a model thixotropic suspension developed by Dullaert and Mewis are performed, including large amplitude oscillatory shear (LAOS) flow, shear flow reversal, and a novel unidirectional LAOS flow, to provide an extended rheological data set for testing constitutive models. A new structure-based model is developed by improving the elastic stress component in the Delaware thixotropic model. Model parameters are determined from steady, small amplitude oscillatory and step shear rate tests, and the model predictions are tested against LAOS experiments. Similar comparisons are made for three other literature models: two of which use a scalar internal structural parameter and include the modified Jeffrey s model proposed by De Souza and Thompson, while the third model developed by Bautista et al. is based on fluidity additivity. A common weakness in all models is identified to be the use of scalar order parameters that cannot account for the reversal of flow directionality inherent in LAOS flow. This is further illustrated by comparison of the models in flow reversals and by contrasting with predictions obtained in a novel unidirectional LAOS experiment where flow reversal does not take place. Keywords: Thixotropy, Viscoelasticity, Flow Reversal, LAOS, Colloidal Suspension, Structure a Author to whom correspondence should be addressed, wagnernj@udel.edu I. INTRODUCTION Thixotropy is defined as the continuous decrease of viscosity with time when flow is applied to a sample that has been previously at rest, and the subsequent recovery of viscosity when the flow is discontinued [Mewis and Wagner (29)]. Ideal thixotropy is defined as a time-dependent viscous response to the history of the strain rate with fading memory of that history [Larson (215)]. It is generally accepted that thixotropic suspensions have an underlying microstructure that reversibly breaks down under shear flow and rebuilds upon its cessation [Mewis and Wagner (212); Dullaert (25); Mewis (1979); Morrison (21)]. Typical 1

2 thixotropic suspensions are comprised of colloidal particles, aggregates, or flocs, which associate at rest, but are broken apart upon an applied shear flow. Because thixotropic suspensions are dominated by viscous hydrodynamic interactions, the stress in an ideal thixotropic suspension relaxes immediately upon flow cessation. However, for real systems thixotropy is oftentimes convoluted with viscoelasticity and one can differentiate between ideal and nonideal thixotropic materials, the latter exhibiting an additional viscoelastic time scale for stress relaxation [Mewis and Wagner (29)]. Thixotropic suspensions are ubiquitous to everyday life and in industry. Many food products, paints, cleaning, and industrial waste products are thixotropic in nature. [Dullaert and Mewis (25a); Mewis (1979); Koopman (213); Lichtenstein (24)]. Blood is also a thixotropic suspension [Bureau et al. (1979); Bureau et al. (198); Sousa et al. (213)] and several thixotropic models have been proposed to model blood flow, most recently by Apostolidis et al. (215). Lastly there are large-scale superfund site cleanup efforts underway at two national labs based on the remediation of highly thixotropic suspensions, including transuranic waste [Koopman (213)]. Generally speaking, thixotropic models can be categorized into three broad classes with increasing order of microstructure resolution (and hence, complexity): a phenomenological continuum approach, a phenomenological structural kinetics model approach, and a particle-level microstructural approach. The structural kinetics approach balances the simplicity of the continuum approach with a simplified phenomenological model for structure against the complexity of a more fundamental model based on particle micromechanics. As such, structurekinetics models attempt to capture the key features of the coupling of material properties to structure while avoiding computational intractability when applied to time-dependent and complex flows. Necessarily this involves introducing parameters for various structure building and breaking phenomena as well as functional forms relating viscous and elastic material properties to the structure parameter. Such phenomenological parameters can, in principle be derived by coarse graining the more fundamental particle micromechanics approach [Grmela et al. (214); Cranford et al. (21)]. Structural kinetics modeling for thixotropy start with Goodeve s original treatise on the relationship between thixotropy and viscosity first published in 1939 [Goodeve (1939)]. There, a simple structural kinetics model is first proposed that links the change in a scalar structure parameter to shear forces acting on a microstructure as discussed above. Goodeve proposed the first thixotropic, structural evolution equation using number of links or bonds that form the structure. Since then many have built on this framework and much of this is surveyed in reviews [Mewis (1979); Barnes (1997); Mewis and Wagner (29)]. In most approaches, the structure is described by a scalar parameter with values typically ranging between and 1 [Moore (1959); Fredrickson (197); Dullaert (25); Mewis and Wagner (212)], with 1 corresponding to a fully structured material and to the completely broken down structure. The range of approaches in structural kinetic modeling is very broad and alternative formulations 2

3 abound [Coussot et al. (1993); Roussel (24); De Souza (29), De Souza and Thompson (213)]. The structure-based models for thixotropic suspensions considered here incorporate a scalar parameter, λ, to represent structure and this structure parameter has an associated kinetic equation describing its evolution with time and shearing conditions. Early work by Goodeve associated λ with degree of bonding between particles or aggregates in the suspension [Goodeve (1939)], but in general this parameter does not have a quantifiable microstructure interpretation. In some cases, such as the modified Jeffrey s model, [De Souza and Thompson (213)] the structural parameter is allowed to range more broadly and can take on any positive value. Other formulations are possible, such as the BMP, or Bautista-Monero-Puig (1999) model [Bautista et al. (1999)], which uses the fluidity as a surrogate for the structure based on earlier work of Fredrickson (197). Experimentally, microstructure can be measured directly under flow through the use of structure-sensitive techniques such as x-ray scattering or small angle neutron scattering under flow, [Kim et al. (214); Gurnon et al. (214); Eberle et. al. (212)], small angle light scattering under flow [Pignon et al. (1997); de Bruyn et al. (27)] and confocal microscopy under flow [Xu et al. (214); Lin et al. (214b)]. The structure parameter satisfies a kinetic, evolution equation that combines physical processes acting to build or break the suspension microstructure. At the extreme of high rate of shear flow, the microstructure in a thixotropic suspension is assumed to be fully broken down; however, shear thickening in some stable suspensions may actually generate structure at high shear rates and this is not considered further here [Gurnon and Wagner (215)]. Upon flow cessation the microstructure rebuilds through processes such as Brownian motion and the process may be diffusion or reaction limited [Smoluchowski (1917)]. During flow shear-induced aggregation and Brownian aggregation compete with shear-breakage [von Smoluchowski (1916); von Smoluchowski (1917)]. Various models have been proposed for each of these processes in terms of the structure parameter, and their combination leads to an ordinary differential equation for the time evolution of the structure parameter under shear flow [Mewis and Wagner (29); Dullaert and Mewis (25c); Mewis and Wagner (212)]. Generally, structure-kinetics models have been fit to and validated against steady viscosity measurements combined with some time dependent flows, such as step up and step down in shear rate. More recently, structural kinetics models have been applied to other timedependent shear flows such as large amplitude oscillatory shear flow [Mujumbdar et al. (22); Dullaert and Mewis (26); Dullaert (25); de Souza and Thompson (213) ]. Large amplitude oscillatory shear (LAOS) applies a sinusoidal strain with specified amplitude and frequency, which can be independently varied to explore material response over a range of deformation amplitudes and frequencies. Early examples of LAOS include wok by Philoppoff to rheologically characterize viscoelastic materials [Philippoff (1966); Giacomin et al. (1993)]. Modern studies of LAOS combine rheological measurements with simultaneous measurements of microstructure [Rogers et al. (212); Helgeson et al. (21); Gurnon et al. (214); Lin et al. 3

4 (214a)]. In the absence of such direct evidence for the microstructural behavior under LAOS a number of techniques have been proposed to interpret the material s microstructure response during LAOS from the rheology alone [Giacomin and Dealy (1993); Ewoldt et al. (28); Cho (25); Rogers and Lettinga (212); Rogers (212)]. Such interpretations benefit from having robust and accurate structure-based constitutive models that are applicable to LAOS, which could further enable fitting LAOS measurements to predict material performance under other flow conditions [Hyun et al. (211); Gurnon and Wagner (212); Blackwell and Ewoldt. (214)]. Further developing and validating this capability for thixotropic suspensions is an important part of the work presented here. Structural evolution equations have been developed that focus on the difference between the present state of structure and an equilibrium state [Mujumbdar et al. (22); Dullaert (25); de Souza and Thompson (213); Mewis and Wagner (212)]. Recently, Dimitriou et al. (213) made several contributions to modeling thixotropic, elasto-viscoplastic materials by decomposing the strain and shear rate into elastic and plastic components, building on work by De Souza and Thompson (29). This technique assigns components of strain and shear rate to specific rheological behavior i.e., elastic (reversible, recoverable) behavior, and plastic (irreversible, dissipative) behavior. With regards to structure, the elastic component is associated with structure stretching and recovery, and the plastic component associated with structure destruction and rebuilding [De Souza and Thompson (29); Dimitriou et al. (213); Apostolidis et al. (215)]. We incorporate aspects of these works in the following in developing a new structure-based model called the Modified Delaware Thixotropic Model. Recently, similar structural evolution equations have been coupled to tensorial constitutive equations. In particular Jeyaseelan and Giacomin (28), and Jacob et al. (214), involved modified versions of the Maxwell and Giesekus models, respectively. In these tensorial approaches the structure parameter λ remains a scalar parameter with a range of values between zero and one. Other recent work modeling of elasto-visco-plastic, or EVP materials, by Stickel et al. (213) uses a modified version of the EVP model of Cheddadi et al. (212) and Saramito (29). The EVP model could not capture the thixotropic time scales inherent in our material, due to the fact that it only has a viscoelastic time scale, and does not admit a separate thixotropic time scale. The Blackwell and Ewoldt (214) model, based on a Jeffrey s framework, does not have a yield stress term, and therefore cannot fit the steady state data. However, these approaches can often be modified to include additional effects, such as the BMPM approach used here (results shown in Section IVb and in Supplemental Material) [Blackwell and Ewoldt (214)]. In the following we investigate the ability of structure-parameter based, thixotropic models to predict LAOS experiments conducted on a model thixotropic material. In the next section, Sec. II, we introduce the structure-based model for thixotropic suspensions, termed the Modified Delaware Thixotropic Model (MDTM), and compare it against some popular models from the literature. The experimental, model, thixotropic suspension used and the rheological 4

5 tests conducted are presented, along with the model parameter fitting in Sec. III. In Sec. IV we present the LAOS experiments along with a comparison model predictions. We compare and contrast the models Sec. V and present our conclusions in Sec. VI. II. MODELS FOR THIXOTROPY Three thixotropic models incorporating a scalar structure parameter are used in this work, and in order of increasing complexity are the Simple Scalar Thixotropic Model (SSTM) presented in Chapter 7, Colloidal Suspension Rheology, [Mewis and Wagner (212)], a modification of the Delaware Thixotropic Model (MDTM) as originally derived by Mujumbdar, et al. (22) presented in detail in the following, and a modified Jeffrey s model, known as the Unified - Approach Model (UAM) from de Souza and Thompson, (213). Each of these models has a scalar parameter, λ, an evolution equation for structure in the form of an ordinary differential equation, as well as a constitutive equation for the stress in terms of elastic and viscous contributions. The fourth model examined in this investigation (BMPM) is from Bautista et al. (1999), and is a tensorial model based the evolution of inverse viscosity (fluidity) modes entering the description of a modified, multimodal Maxwell model. Fitting and predictions obtained for these three literature models are discussed in Sec. V, and are also to be found more extensively in the accompanying supplemental material. In the SSTM and MDTM models the total shear stress in shear flows is decomposed into an elastic stress, σ e and viscous stress σ v, both functions of the current level of microstructure, λ : σ tot =σ e(λ)+σ v(λ). (1) This respects the fundamentally different micromechanical sources of the stresses arising from hydrodynamic and interparticle interactions, respectively [Russel et al. (213)]. Alternatively, the UAM and BMPM models use a modified Jeffrey s constitutive equation for the deviatoric stress tensorσ : σ + η( λ ) G( λ ) σ = η( λ ) γ, (2) where the elastic modulus G, and the current viscosity, η are both functions of λ. A detailed description of the Modified Delaware Thixotropic Model is presented, while the details of the three other models are found in the literature, and in the Supplemental Material. 5

6 A. MDTM - Modified Delaware Thixotropic Model The Modified Delaware Thixotropic Model evolved from the original Delaware Model. [Mujumbdar et al. (22)], which was in turn a modification of the original Herschel-Bulkley model. The original Delaware Thixotropic Model consisted of the Herschel-Bulkley stress expression with material properties dependent upon a scalar parameter to describe the current level of structure. In the Modified Delaware Thixotropic Model (MDTM) we added an additional term to the structure evolution equation to account for shear aggregation processes, which introduces an additional time scale and an additional power law exponent: dλ! dt = k a -λ ˆt Brown r1 γ p +(1- λ ) 1+ ˆt r2 γ p d, (3) where k is a kinetic constant with units of inverse time, representing Brownian, structured Brown restoration effects and t r1, t r2 are two thixotropic constants, connected to the breaking and aggregation kinetic constants k and Break k : Aggr 1/a ˆ kbreak t r1 =, (4) k Brown 1/ d kaggr t ˆ r2 =. (5) k Brown Furthermore, the MDTM model incorporates aspects from the work of Dimitriou et al., who developed a framework to separate strain and shear rate into elastic and plastic components [Dimitriou et al. (213)] to better capture the elasto-viscoplastic behavior of soft solid materials. In particular, according to the theory of kinematic hardening of plasticity [Dimitriou et al. (213)], in such a way that strain and shear rate each have elastic and viscous components, and together add up to the total strain and shear rate respectively: γ =γ e +γ p γ = γ e + γ p. (6) In turn, the elastic strain is postulated to evolve as [Apostolidis et al. (215)]: γ e = γ p - γ e γ γ p, (7) max This adds an ancillary, ordinary differential equations for the MDTM modulus, 6

7 which includes an additional time constant dg f =k G (G f λ-g ) dt 1 k for this process. G, (8) Our model differs from that of Dimitriou et al. (213) in the following ways: 1. The MDTM uses an ordinary differential equation to evolve the elastic modulus in time, with k G representing the tie constant for rebuilding of the elastic modulus; 2. The MDTM incorporates a unique ordinary differential equation for elastic strain, γ e, that is a function of the current γ p, in such a way that the sum of plastic and elastic strain is always equal to total strain; 3. The MDTM incorporates an ordinary differential equation for the structural evolution that involves a term for shear aggregation; and 4. γ is set as a limit to ensure that the material cannot strain max more than the material limit. In the model, strain is the linear superposition of an elastic strain and a plastic strain, with shear rate correspondingly the linear superposition of elastic shear rate and plastic shear rate, as shown in Eq. (6). The calculation of the plastic component of shear rate is: γ p =! γ 2 γ e γ max γ 2+ γ e γ max!!!!!!!!!!!!! γ!!!!!!!!!!!!! γ <, (9) and is a function of γ, γ e and γ, where max γ is assumed to physically represent the max maximum elastic strain within the material, is postulated to be provided as a function of the structure parameter as, γco γ max(λ)= min,γ m λ 7. (1) Note that this expression requires the following connectivity at the zero deformation limit between the yield stress, σ, and the zero deformation strain, y γ appearing in Eq. (1) above, CO σ y γ CO =. (11) G Note that this identification of plastic and viscous strain and the associated evolution equations eliminate violations of the second law of thermodynamics associated with structural recovery upon flow cessation [Larson (215)]. With this understanding it is now possible to cast the

8 constitutive equation in terms of elastic and plastic processes, where the elastic components are aligned with stretching, reversible processes, and the plastic components are aligned with irreversible, dissipative processes. A version of this thixotropic model has recently been applied to successfully model and predict blood flow [Apostolidis et al. (215)]. The stress decomposition within MDTM has an additional viscous stress term, namely the structural viscosity: σ = G f γ e +λk ST γ p n 2 + K γ p n 1. (12) There are a total of 13 parameters in the MDTM model, which are tabulated, along with best fit values with uncertainties in Table I. The determination of these model parameters from experimental data is discussed further in Section IIIB. B. Other Models The other models are briefly discussed in the Supplemental material, with their respective equations and primary source provided. They are: the Simple Scalar Thixotropic Model (SSTM) Eq. (S.1-S.3) [Mewis and Wagner (212)], the Unified Approach Model (UAM) Eq. (S.4-S.14) [de Souza and Thompson (213)] and the Bautista-Manero-Puig Model (BMPM) Eq. (S.15-S.16) [Bautista et al. (1999)]. III. MATERIALS AND METHODS A. Thixotropic Suspension The model thixotropic suspension reported by Dullaert and Mewis (25a) was reformulated for this work, following as closely as possible the prescribed methods. The suspension consists of 2.9 vol% fumed silica (Evonik Aerosil R972) aggregates with primary particle radius of 16nm in a blend of 69 wt% paraffin oil, (Sigma Aldrich, density g/cc) and 31wt% polyisobutylene (TPC Group, molecular mass! Mw= 75 (g / mol). The particles as supplied are imaged with transmission electron microscopy and shown in Fig. 1, where it is apparent that the nominally 16nm primary particles are aggregated into large (micron) sized agglomerates. Note that the polyisobutylene molecular mass differs slightly from that used by Dullaert and Mewis (25a, 26) due to availability. The particles were dispersed in the blended solution using a Silverson Mixer, model L4Rt using the tubular mixing unit with squarehole, high-shear screen at the 8 rpm setting for 3 minutes, followed by degassing at 4 o C in a vacuum chamber for 1 hour. Samples were stored on the Wheaton benchtop roll-mixer. Fig. 2a compares the steady shear viscosity with that reported by Dullaert and Mewis (25a, 26). 8

9 Error bars are calculated from four independent experiments using the t-distribution at the 95% confidence interval. Differences, which are most evident at high shear rates, arise primarily from differences in the solvent viscosity (1.84 ( Pa s ) for Dullaert and Mewis (25a, 26) and 1.17 ( Pa s ) here), which is due to the different molecular mass of the polyisobutylene. Also shown are the elastic stress components determined from stress jump measurements as described in the next section and the results are in semi-quantitative agreement with the prior work. FIG. 1: TEM micrograph of the fumed silica (Degussa R972) aggregates as supplied (image courtesy of Ryan Murphy, Univ. of Delaware). B. Rheological Measurement Protocol Rheological measurements were performed with the ARES G2, strain controlled rheometer (TA Instruments) using a 5mm cone and plate geometry with.44 rad cone angle Dynamic yield stress experiments were conducted on a DHR3 stress controlled rheometer (TA Instruments) with 4mm cone and plate geometry with a.4 rad cone angle. For the steady flow curves we performed a preshear of 3 (s -1 ) for 3 (s) before each new shear rate set point. Following the completion of each preshear the new shear rate was set and allowed to equilibrate for a varying amount of time depending on the shear rate itself. For the highest shear rate (3 (s -1 ) ) this time was determined to be 1 minutes. For the lowest shear rate (1-3 (s -1 )) this time was determined to be 2 hours. Maximum equilibration times were determined for each shear rate. All of the transient step-up and step-down series were preceded by a preshear of 3 (s -1 ) for 3 (s), followed by a period of time to arrive at a steady stress value for a given initial shear rate (6-9 (s) for all shear rates). The LAOS flows were preceded by a preshear of 3 (s -1 ) for 3 seconds in the same manner. The LAOS results presented here are for the alternance state, which is independent of the initial state. Values of the viscosity obtained during the preshear were used before and after each experiment to determine that no irreversible changes occurred to the sample during testing. The rheological data presented here was found to be reproducible from loading to loading and batch-to-batch of sample. 9

10 σ (Pa) (a) Total Solvent Dullaert + Mewis Total Elas. Stress Dullaert + Mewis Elas. Stress γ (s -1 ) FIG. 2: Steady state flow curve experimental results of model thixotropic system (2.9vol% fumed silica in paraffin oil and PIB), compared against the results of Dullaert and Mewis (25a, 26) and the solvent mixture. Also shown are the contributions from the elastic stress as obtained from stress jumps, with the error bars, compared against the results from Dullaert and Mewis (25a, 25b). The measurement of the elastic stress uses a strain jump experiment following the procedure outlined by Dullaert and Mewis and is shown in the Supplemental Material [Dullaert and Mewis (25b)], Fig. S.1b. This experiment rapidly reduces the shear rate from a steady, constant value of shear rate to a value near zero. From literature [O Brien and Mackay (2)] and theory [Mewis and Wagner (212)] the viscous forces go to zero with the shear rate, such that extrapolation of the residual stress back to the time of the rate reduction yields a measure of the elastic component of the total stress derived from the rheometer s torque measurement. More details of this procedure are provided by Dullaert and Mewis (25b). B. MDTM Parameter Determination Procedure The thirteen parameters of the MDTM are listed in Table I, are listed generally in the order they were determined, which can be characterized as a two step process. The first five parameters listed in Table I were fit to specific aspects of the experimental data as follows: the three parameters σ,η,n were determined using the steady state flow curve and its limiting y 1 values at zero and high shear rate, while G and m are evaluated by fitting to the SAOS, steady σ y flow curve and using the relationship γ CO = G. This value for the yield stress is corroborated with the G G crossover amplitude shown in Figure S.2 (Supplemental 1

11 Material). The critical strain γ was shown to compare well with the strain at the limit of CO linearity as determined from selected amplitude sweep experiments shown in the Supplemental Material, Figure S.2. The remaining eight parameters are then determined simultaneously with the parallel simulated annealing algorithm by fitting to 18 sets of experimental data by defining an objective function to be minimized F OBJ, which is a summation of the L 2 norm applied to each respective data set, shown in Eq.(13), divided by number of points in each dataset. In Eq. (14) F OBJ is the sum of the relative error of each data set divided by number of data sets, and M is the total number of data sets, while P [Armstrong et al. (215)], 1 ( σmodel σdata ) Rel. Error =, (13) P σ Data 2 F OBJ N 1 = (Rel. Error). (14) M k= 1 k Table I: Parameter Values Determined for MDTM. Par. Units Meaning Range Guess Initial Limiting Vals. Optimal µ: σ: (+/-) σ y Pa Yield stress (-) (-) 11 (-) (-) η Pa s n1 Infinite shear viscosity (-) (-) 1.17 (-) (-) n 1 (-) Power law η (-) (-) 1 (-) (-) G Pa Elastic Modulus (-) (-) 45 (-) (-) m (-) Power law par. of γ e evol. (-) (-) -1.5 (-) (-) K ST Pa s n2 Consistency par. [9-11] > n 2 (-) Power law par. K ST [.5-1] [.5-1] Power law par. shear a (-) break. [1-2] [1-2].36 [ d (-) Power law par. shear aggr. [ ].75].5 Char. time of Brownian k Brown s -1 mot. [1e ] >.41 tˆ r (k break /k Brown ) s Char. time of λ breakdown [1e -3-1] >.225 tˆ r (k aggr /k Brown ) s Char. time of λ buildup [1e -3-1] >.21 k G s -1 Char. time of G evol. [1e -3-1] >

12 The experimental data used during this nonlinear dynamic parameter fitting are the steady state flow curve (Fig. 3a), the elastic stresses, (Fig. 3a) and the sixteen sets of transient step-up and step-down data (see Figs. 4a-d as examples). In the global fitting, constraints were imposed on some parameters. The ranges used for the a, and d values are discussed in literature for a thixotropic material, where a has been recommended to range between one and two, [Mewis and Wagner (212); Dullaert and Mewis (25a); Dullaert and Mewis (26)] while it has been suggested by Dullaert and Mewis, that the value of d be set close to ½, therefore the range is feasible [Dullaert and Mewis (26)]. Additionally, due the fact that we know the material will behave as a shear thinning material we constrained the range of n2 to be between.5 and 1 [Mewis and Wagner (212)]. Importantly, the final fit values were not limited by the extremes of these ranges. Using randomly selected starting values, 15 independent fits were performed and statistics calculated, including best-fit parameter averages, and standard deviations. The parameter values determined for each of the 15 trials are reported in Table S.1. The parameter values giving the smallest value of F OBJ (Eq. 12) were used for the results presented in this work (Table I), which are presented along with the average and standard deviation of the 15 independent fits. The MDTM always converged to a single basin with respect to F OBJ as observed from the parameter values from the 15 trials. The parameter correlation matrix is shown in Table S.2 along with an analysis. Importantly, this metadata indicate that there is essentially one global minimum identified for the model fitting to these data. IV. EXPERIMENTAL RESULTS & MODEL COMPARISONS A. Experimental Results and MDTM Comparisons Fig. 3a compares the steady state flow curve and elastic stresses against with the best fit of the MDTM, over more than five orders of magnitude variation in shear rate. The data exhibit a significant yield stress at low shear rates and subsequent shear thinning to a nearly constant high shear viscosity. To help the interpretation of the data, the model predictions for the structure parameter λ are also shown. The structure parameter is close to 1 (corresponding to a fully structured material) until the stress significantly rises above the yield stress, whereupon it drops, to become fully unstructured ( λ =) at the highest shear rates probed. The elastic component of this steady shear stress is observed to decrease as the structure is broken down by the flow and the MDTM describes this effect, but at slightly lower shear rates than observed experimentally. Fig. 3b shows the small amplitude oscillatory shear measurements over four orders of magnitude of frequency. The amplitude sweep experiments used to define the linear regime are shown in the Supplemental Material (Fig. S.2&S.3). The experimental spectrum does not exhibit a terminal liquid regime at the lowest frequencies probed, but rather is characteristic of a material with an apparent yield stress. The small amplitude oscillatory shear frequency sweep is fit to a spectrum of relaxation times, and corresponding moduli to get an idea of the relaxation spectrum, as shown in Supplemental Material, corroborating that any terminal relaxation process to a 12

13 σ (Pa) G'; G" (Pa) thixotropic liquid must be on the order of at least 1 (s). Empirically, it is observed that the suspension will creep over the course of many days, demonstrating the fact that thixotropic relaxation processes are over longer periods of time than viscoelastic relaxation processes (a) σ total σ elastic σ MDTM σ MDTM-elastic γ (s -1 ) λλ 1 1 (b) G' Data G" Data G' MDTM G" MDTM ω (rad/s) FIG. 3: (a) Steady state flow curve of model thixotropic system with MDTM fit and λ ; (b) Small amplitude oscillatory shear (frequency sweep, γ =.1 ) results with MDTM best fit. Fits of the SAOS data by the MDTM model are also shown in Fig. 3b, where the zero frequency limit of the elastic modulus, is determined to be 225 (Pa) (dashed line in Fig. 3b). The critical yield strain γ CO is determined from the ratio of the yield stress and elastic modulus to be.24. This agrees well with the values identified in the strain sweep experiments (See Fig. S.2 in Supplemental Material), providing confidence in these parameter values. Note that the MDTM has a frequency independent elastic modulus due to the separation of strain into plastic and viscous terms that yields both an elastic and viscous contribution to the equilibrium modulus. The loss modulus also has an additional contribution from the viscous stresses, which becomes evident at higher frequencies. The model s simple representation of the elastic yield stress leads to a G that does not depend on frequency and hence, can only approximately represent the observed experimentally spectrum. Two basic protocols are followed for the step up and step down in shear rate experiments. The first is transitioning to a common final shear rate from different values of initial, steady shear rate, while the second is transitioning from a common initial steady shear rate to different final values of shear rate. The results of the step up and step down experiments are shown in Fig. 4 (a-d), where the initial stress values correspond to points on the steady flow curve (Fig. 3a) and only the transient stress responses from the time of the rate jump are shown in the figure so they can be presented on a logarithmic time scale for clarity. These results follow trends characteristic of a thixotropic material and closely follow the observations of Dullaert and Mewis (26). Comparison of these curves shows that there are different time scales for structure break down and structure build up, and that these processes depend on the rate of deformation during 13

14 the process. Furthermore, there are vestigial signatures of viscoelasticity at early times showing that the material is not purely thixotropic in its response. σ (Pa) σ (Pa) (a) =2.5 (s -1 ) =1. (s -1 ) =.5 (s -1 ) =.1 (s -1 ) t (s) (c) γ i =.5 (s -1 ) γ i =1. (s -1 ) γ i =2.5 (s -1 ) γ i =5. (s -1 ) t (s) σ (Pa) σ (Pa) 1 2 (b) = 5. (s -1 ) = 2.5 (s -1 ) = 1. (s -1 ) =.5 (s -1 ) t (s) (d) =.1 (s -1 ) =.5 (s -1 ) = 1. (s -1 ) = 2.5 (s -1 ) t (s) FIG. 4: (a) Data and MDTM fits of step down in shear rate from 5. (s -1 ) to 4 different values listed in plot; (b) Data and fits of step up in shear rate from.1 (s -1 ) to 4 different values listed in plot ; (c) Data and fits of step down to.25 (s -1 ) from 4 different values listed in plot (in this data the initial shear rates are the same while the final shear rates vary); (d) Data and fits of step up to 5. (s -1 ) from 4 different values listed in plot (in this data the final shear rates are the same while the initial shear rates vary). Structure predictions from the MDTM are shown in Fig. 5 (a-d) corresponding to Fig. 4(a-d). Using the steady state flow curves for reference (Fig. 3a), the MDTM predictions for structure evolution during these transient tests can be easily explained. Namely, structure rebuilds upon lowering the applied shear rate, leading to rebuilding of the structure viscosity and yield stress and vice versa. Furthermore, the structure evolution is monotonic during these processes, explaining the model s monotonic predictions of the stress. 14

15 (a) 1..8 (b).6 λ λ =.1 (s -1 ) =.5 (s -1 ) =1. (s -1 ) =2.5 (s -1 ) t (s) (c) γ i =.5 (s -1 ) γ i =1. (s -1 ) γ i =2.5 (s -1 ) γ i =5. (s -1 ) t (s) λ λ =.5 (s -1 ) = 1. (s -1 ) = 2.5 (s -1 ) = 5. (s -1 ) t (s) (d) γ i =.25 (s -1 ) γ i =.5 (s -1 ) γ i =1. (s -1 ) γ i =2.5 (s -1 ) t (s) FIG. 5: (a) MDTM Structure predictions of step down in shear rate from 5. (s -1 ) to 4 different values listed in plot; (b) Structure predictions of step up in shear rate from.1 (s -1 ) to 4 different values listed in plot; (c) Structure predictions of step down to.25 (s -1 ) from 4 different values listed in plot (in this data the initial shear rates are the same while the final shear rates vary).; (d) Structure predictions of step up to 5. (s -1 ) from 4 different values listed in plot respectively (in this data the final shear rates are the same while the initial shear rates vary); All predictions are calculated with the parameter values in Table 1. B. Comparison to literature models In addition to MDTM three additional thixotropic models have been used to fit the data from the literature: the SSTM [Mewis and Wagner (212)]; UAM [de Souza and Thompson (213)]; and BMPM [Bautista et al. (1999)]. The results were qualitatively similar to those obtained with MDTM (see Supplemental Material). In the following, we highlight some key comparisons of the four model fits to the experimental data sets presented in the previous section. Detailed comparisons of the best fits for each of the three literature models is presented in the same level of detail as for the MDTM model can be found in the Supplemental Materials. 15

16 σ (Pa) G'; G" (Pa) (a) σ total σ MDTM σ SSTM σ BMPM σ UAM γ (s -1 ) G' Data G" Data G' MDTM G" MDTM G' SSTM G" SSTM G' UAM G" UAM G' BMPM G" BMPM ω (rad/s) FIG. 6: (a) Steady state flow curve model fits and data for all four models; (b) Small amplitude oscillatory shear (SAOS) predictions and data for all four models. Fig. 6a compares the steady state flow curves and the elastic stresses with experiment. Not surprisingly, all of the models are have more than enough parameters to accurately model the steady state flow curve. Fig. 6b shows that the SSTM, and UAM fail consistently to approximate the actual SAOS spectra, whereas the multimode BMPM can very accurately fit the linear viscoelastic material response (by construction). The MDTM fits, while qualitatively capturing the principal trends, also fail to qualitatively fit the data. Previous work has identified important sources of elasticity in aggregated or flocculated suspensions to arise from both intraand inter-aggregate interactions [Shih et al. (199)]. The simplified expressions for the yield stress employed in single scalar parameter structural thixotropic models can only capture the overall behavior qualitatively at best (MDTM) and not the detailed frequency dependence of the moduli. However, as the focus of thixotropic modeling is the time dependent flow behavior under nonlinear conditions, the level of approximation afforded by the MDTM model is sufficient and superior to that of the SSTM and the UAM approaches. Examination of the step-down and step-up in shear rate results and model fits in Fig. 7a&b shows significant differences between the models (note that the result of the data sets are presented in detail in the supplemental and only two characteristic cures are provided here for brevity). The MDTM can fit the data well, while the SSTM is less satisfactory from a quantitative perspective and both the UAM and BMPM approaches yield qualitative inconsistencies. The advantages of the MDTM over the SSTM is also due to the addition of a third process to account for shear-induced aggregation, which is important for accurately modeling the differences in recovery processes involving changes in shear rate. In contrast, the UAM and BMPM include processes that always lead to both viscoelastic and thixotropic responses in rate jump experiments, and these are also clearly evident in some of the other comparisons shown in the supplemental material. This is especially true for the 5-mode BMPM, 16

17 σ (Pa) σ (Pa) which is constructed on a viscoelastic framework. However, these additional viscoelastic shortterm transients that are predicted by the model are not present in the data. 25 =.5 (s -1 ) 2 15 MDTM SSTM UAM BMPM 1 2 (b) = 2.5 (s -1 ) MDTM SSTM UAM BMPM t (s) FIG. 7: (a) Single step-down in shear rate from 5. (s -1 ) to.5 (s -1 ) and data comparison with all four models; (b) Single step-up in shear rate from.1 (s -1 ) to 2.5 (s -1 ) and data comparison with all four models. In this section, we highlighted some key comparisons of the four model fits to the experimental data sets presented in the previous section. The values of the minimum relative error measures corresponding to these fits are compared in Table II. The steady flow curve is well represented by all models, but best fit by the MDTM. Not surprisingly, the BMPM model best fits the SAOS data, with the SSTM with its constant yield stress failing to represent the data adequately. The transient data is best captured by the MDTM approach, which therefore yields the most satisfactory overall fit. What is apparent from these fits to a range of experimental data is that accurate constitutive models for nearly ideal thixotropic materials under time dependent nonlinear shear flow require a dominant viscous response that is coupled to a time and rate dependent structure. Although viscoelastic models can capture some aspect of thixotropic systems, they fail to capture the dominant qualitative features inherent in thixotropic flows. However, models need to make accurate predictions for experiments not included in the fitting. Therefore, next we compare predictions of these models obtained with the same fit parameters against various additional time-dependent flows that are used for characterizing thixotropic materials. t (s) 17

18 Table II: Relative error measure comparisons for steady shear flow (SS), small amplitude oscillatory shear flow (SAOS), strain rate jumps (All Transient). Description MDTM SSTM UAM BMPM min(error, SS ) min(error, SAOS ) min(error, All Trans. ) VI. PREDICTING STRESS RESPONSES UNDER NON-LINEAR, TRANSIENT SHEAR FLOWS A. Large Amplitude Oscillatory Shear (LAOS) Figs. 8 (a-d) present new LAOS experimental results for a range of frequencies and strain amplitudes, shown in dimensionless form for ease of comparison. Inspection reveals characteristics of both thixotropic and viscoelastic behavior; for example, a yielding behavior can be clearly observed in Fig. 8a at ω =.1 rad / s. In addition, at the higher strain amplitudes there is clear evidence of stress overshoots leading to secondary LAOS loops in the viscous projections. This is a consequence of a lag time within the structured material such that the structure breakdown lags behind the instantaneous shear rate, so called thixotropic loops in classical thixotropic tests (e.g. see Chapter 7 of Mewis and Wagner (212)). This effect becomes less evident as the frequency is increased from.1 (rad/s) to 1 (rad/s), whereupon the sample exhibits a predominantly viscous stress response. This is expected as! γ is increasing, which leads to less structure, and consequently, more viscous behavior, as it is the structure directly contributes to the elastic nature of the material, while if the frequency is sufficiently high such that the structure cannot recover sufficiently during the cycle. Kim et al. (214) reported similar observations for colloidal gels, where direct measurements of the structure under flow were used to verify this interpretation. 18

19 (a) ω =.1 (rad / s) (b) ω =.1 (rad / s) (d) ω = 1 (rad / s) (c) ω = 1 (rad / s) FIG 8: (a) LAOS data at ω =.1 (rad/s) over a range of strain amplitudes ( LAOS data normalized by maximum values, then normalized stress values scaled by.5,.67,.83 and 1 for γ = 1,1,1,&1, respectively, for presentation purposes). (b) LAOS data at ω =.1 (rad/s) over a range of γ (LAOS data normalized by maximum values, then normalized stress values scaled by.33,.5,.67,.83 and 1 for γ =.1,1,1,1,&1, respectively, for presentation purposes). (c) LAOS data at ω =1 (rad/s) over range of γ (LAOS data normalized by maximum values, then normalized stress values scaled by.5,.67,.83 and 1for γ =.1,1,1,&1, respectively, for presentation purposes). (d) LAOS data at ω =1 (rad/s) over range of γ (LAOS data normalized by maximum values, then normalized stress values scaled by.67,.83 and 1 for γ =.1,1,&1, respectively, for presentation purposes). In the following model predictions are compared against the LAOS data. Model predictions for LAOS are performed by running the model until alternance is achieved, in a 19

20 manner similar to the experiments. Comparisons are made in both elastic and viscous projections, with respect to γ(t), and, respectively γ (t), of the Lissajous-Bowditch space. To illustrate a typical model behavior, we show the three dimensional Lissajous-Bowditch plot comparing the experimental data with predictions of the MDTM at ω= 1 (rad / s) ; γ = 1, and ω= 1 (rad / s) ; γ = 1 in Fig. 9 (a-e), and Fig. 1 (a-e) respectively. 3 2 (b) (c) σ (Pa) σ (Pa) (d) σ Data σ MDTM (e).8 3. σ Data σ MDTM λ λ λ MDTM -1 1 γ (-) γ (s -1 ) λ MDTM FIG. 9: (a) Three dimensional Lissajous-Bowditch curve for ω= 1 (rad/s); γ =1; (b) Two dimensional elastic projection; (c) Two dimensional viscous projection; (d) Two dimensional structural, elastic projection; (e) Two dimensional structural viscous projection. The arrows indicate the direction of time evolution during the cycle and the numbers correspond to specific states discussed in the text. 2

21 1 (b) 1. 1 (c) σ (Pa) σ (Pa) σ Data -1 σ Data σ MDTM σ MDTM 1. (d) (e) λ.8 λ.8 λ MDTM γ (-) γ (s -1 ) λ MDTM FIG. 1: (a) Three dimensional Lissajous-Bowditch curve! ω =.1!(rad / s);!γ = 1 ; (b) Two dimensional elastic projection; (c) Two dimensional viscous projection; (d) Two dimensional structural, elastic projection; (e) Two dimensional structural viscous projection. The arrows indicate the direction of time evolution during the cycle and the numbers correspond to specific states discussed in the text. The conditions shown in Fig. 9 give rise to states where both elastic stresses and viscous stresses are of significance at points during the cycle and the frequency is such that the suspension s structure can recover significantly during the cycle at alternance, and thus, it provides a good illustration of the rich LAOS behavior of thixotropic suspensions and a good test 21

22 of model fidelity. At the high shear rates of states 1 & 3, the stress is largely viscous in nature. Reducing the rate of strain rate to zero during the flow reversal (states 2 & 4) shows small differences in stress response during the cycle depending on the position due to hysteresis in structure rebuilding and breakdown. Note that the data and the model predictions both cross during the cycle in the viscous projection (see Fig. 9c). The non-zero stress at zero shear rate is characteristic of the yield stress corresponding to the non-zero structure in the suspension. It is evident that the MDTM can qualitatively capture these effects and supports the typical microstructural interpretation through predictions of the structure evolution (Fig. 9d&e). Importantly, the thixotropic loop evident in the LAOS is also evident in the structure parameter, which is different than the steady-shear structure consistent with the instantaneous shear rate. Reducing the frequency and strain amplitude each by an order of magnitude puts the LAOS cycle at sufficiently low strain rates and strains such that the experiment is largely dominated by the elasticity leading to the yield stress. Note that this strain value is still above γ CO, so the suspension is in the weakly nonlinear regime. The experimental data shows predominantly elastic behavior with a stress overshoot and viscous flow evident as the maximum shear rates are reached (states 1 & 3). Very significant differences in stress at the states corresponding to flow reversals during the cycle (states 2 & 4) are a consequence of the yield stress. Fig. 1 (d,e) show that the sample remains nearly fully structured during the oscillation, such that the model s response is dominated by the modeling of the yield stress. Note that the model again qualitatively captures the stress overshoot, but now incorrectly predicts a loop in the viscous projection. This can be seen as a large overestimation of the yield stress as evident in the elastic projection (Fig. 1a), where the model lies well outside the experimental data. Similar observations were made in experimental studies of colloidal gels by Kim et al. [Kim et al. (214)], where LAOS can result in different structure at alternance as compared to that observed during steady shear flow. 22

23 (a) (b) γ 1 γ ω (rad/s) ω (rad/s) ω (rad/s) FIG. 11 Pipkin Diagram: (a) Elastic projections. (b) Viscous projections. Data (gray open circles) and Modified Delaware model predictions (blue lines). The data are presented in dimensionless form and the model predictions are made dimensionless by the same maximum values used for the data. The boxes (gold dashed frames) indicate the states shown in Fig. 9 & 1. A global summary of the LAOS experiments and MDTM predictions are presented in the Pipkin diagram in Fig 11 (a-b). The gold boxes identify the two frequency and strain amplitude combinations are shown in Fig. 9 and Fig. 1. Not surprisingly, in all case progressing diagonally to a higher frequency and higher strain amplitude (i.e., increasing the maximum shear rate) the MDTM predictions improve in accuracy as the stress response is dominated by the γ viscous behavior. However, as and ω are decreased the structure of the material becomes more and more relevant, and as the structure becomes more relevant the suspension s elastic response dominates the stress signal. For example, at frequency 1 rad/s and strain amplitude 1, the MDTM is able to accurately predict the Lissajous-Bowditch curve, even predicting the secondary loops, and overshoot caused by structure build up during the increasing portion of the shear rate on the viscous projection. However, when the frequency is reduced to.1 (rad/s) and the strain amplitude 1, the model is less accurate and misses qualitative features of the data. The maximum stress predicted is about twice the measured stress, and there are no secondary loops evident in the viscous projections while the model predicts minor secondary loops. From Fig. 1 23

24 (a-b) it can be observed that the MDTM consistently over-predicts the maximum stress at low frequency and strain amplitude, which we believe is due to the over-prediction of the structure, coupled with a simple isotropic model for the elastic stresses. Two important conclusions can be drawn from this analysis. First, the MDTM can quantitatively predict many features of the LAOS response of nearly ideal thixotropic suspensions, especially for states where the viscous response is relevant. The reduction in accuracy for low frequencies and strain amplitudes reveals the inadequacy of the phenomenological model to accurately capture the elastic stress contributions and the coupling to structure. These lead to the second conclusion, namely that careful consideration is required in order to use LAOS measurements for determining model parameters in constitutive equations. Clearly parameters that accurately fit low frequency and amplitude experiments will not accurately capture the behavior under stronger nonlinear flows. Fig. 12 (a-b) compares predictions of all four models for the two states considered in Fig. 9 & 1. All four models provide reasonable predictions for the state at higher frequency and amplitude, but differences are apparent. Most strikingly, the UAM predicts significant stress overshoots leading to a duck head pattern in the viscous projection that is not evident or so pronounced in the other models or the data. At the lower frequency and amplitude all of the models deviate significantly from the experiments, but capture the global features characteristic of probing the yield stress of the suspension. However, all models overpredict the strength of the elastic stress under these and related experimental conditions. 3 (a) 3 (b) σ (Pa) -1-2 σ Data σ SSTM σ MDTM σ (Pa) -1-2 σ Data σ SSTM σ MDTM σ UAM -3 σ UAM σ BMPM -3 σ BMPM -1 1 γ (-) -1 1 γ (s -1 ) FIG. 12: (a) Elastic projection, and (b) Viscous projection of all four models at ω= 1 (rad / s) ; γ = 1. Similar trends as observed for the MDTM in Fig.1 were also evident for the other three models explored in this work, with the other model results shown in detail in the supplemental materials with Fig. S.14. All of the models offer poor predictive capability for conditions of lower values of strain amplitude and frequency where the yield stress dominates the suspension s response, while all of the models offer more accurate predictions at high strain amplitudes and 24