A review of thixotropy and its rheological modeling

The concept of thixotropy was introduced a century ago by Peterfi [1], but its study was soon thereafter overshadowed by an explosion of interest in linear and nonlinear viscoelasticity, especially of polymeric fluids. While the stresses in both viscoelastic and thixotropic fluids depend on their past deformation history, viscoelastic fluids are characterized by elasticity, whereas thixotropic fluids are characterized by the slow time dependence of their viscosity or yield stress. Long polymers deform elastically without breaking and produce viscoelasticity. Particle clusters connected by inextensible, fragile, bonds can break reversibly under stress, temporarily reducing viscosity, but not storing much elastic energy, leading to thixotropy. While constitutive theories for polymeric fluids have long been grounded in their microstructural physics of polymer deformation and relaxation, until recently, most constitutive equations for thixotropic materials have been phenomenological. In recent years, however, thixotropy has received increasing attention, since its importance and prevalence (see below), and its relative underdevelopment, have become increasingly recognized. It is thus time to review what has been accomplished, especially in recent years, and to lay out an agenda for future research in thixotropic phenomena.

Multiple excellent reviews of thixotropy are already available; namely, those of Bauer and Collins [2], Mewis [3], Barnes [4], Mujumdar et al. [5], Mewis and Wagner [6], and de Souza Mendes and Thompson [7]. Barnes’s review [4] focuses on the historical development of the topic from its earliest years, the definitions of thixotropy that have been given, its distinction from viscoelasticity, the typical behavior associated with thixotropy, a tabulation of publications describing various thixotropic materials, and a listing of the then-common models of thixotropy. The review of Mewis [3] defines thixotropy and its characteristics, lists materials commonly showing thixotropy, and then reviews constitutive equations based on the inelastic Reiner-Rivlin fluid, including integral generalizations, as well as viscoelastic constitutive equations to which thixotropy has been added. Mewis and Wagner [6] update this with a careful distinction between thixotropy and viscoelasticity and show typical rheological tests that display viscoelastic characteristics, along with a summary of models. Mujumdar et al. [5] nicely tabulate rheological phenomena associated with thixotropy, as well as functional forms used in constitutive equations for thixotropy. The review of de Souza Mendes and Thompson [7] focuses especially on the combination of thixotropy with elasticity and plasticity, which is a subject of growing interest. They also categorize thixotropic equations, for example, distinguishing ones that start with a viscoplastic stress equation, and add elasticity and thixotropy, from those that start with a viscoelastic equation, to which plasticity and thixotropy are introduced. De Souza Mendes and Thompson argue that the latter are superior and favor making the evolution of thixotropic structure dependent on stress rather than on strain rate. We discuss these options in more detail later.

This new review summarizes the most important aspects covered in the previous reviews, but then focuses attention on new developments and on recommended future directions. Readers are encouraged to read previous reviews to obtain a more complete picture of this complex topic.

The scope of this review is determined by the definition of what is, and is not, “thixotropy.” As noted by Mewis and Wagner [6], the name is derived from the Greek words θíξις (thixis: stirring, shaking) and τρéπω (trepo: turning or changing) and originally referred to a mechanically induced sol-gel transition. Since then, “thixotropy” has come to be generalized beyond a flow-induced solid-to-liquid transition to flow-induced changes in viscosity more generally. A number of more precise definitions have been given, including that by the IUPAC [8], which defines thixotropy 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 flow is discontinued.” (This definition could easily be extended to the reverse phenomenon of “anti-thixotropy,” in which flow causes a reversible increase in viscosity.) As noted by Mewis [3], a focus merely on a flow-induced decrease in viscosity does not clearly distinguish “thixotropy” from nonlinear viscoelasticity, which, like thixotropy, includes shear thinning. To distinguish “thixotropy” from viscoelasticity, it is of value to define “pure” or “ideal” thixotropy [9], even though few, or even no, fluids are purely thixotropic. Recent terminology including the designation “Thixotropic Elasto-Visco-Plastic” or “TEVP” fluids [10–13] is nonredundant only if the adjective “thixotropic” is something other than some combination of elasticity, viscosity, and plasticity. Below, we make a few comments to clarify our preferred definition of thixotropy and thereby also define the scope of this review. We also discuss simple models that can show thixotropy, viscoelasticity, and more general forms of “aging” that might be relevant for glasses and soft “glassy” materials.

Microstructured materials can show various responses to various deformation history that are called “viscoelasticity,” “plasticity,” “thixotropy,” “aging,” “jamming,” “kinematic hardening,” or “elastoplastic,” “thixotropic elasto-viscoplastic,” etc. Individuals or communities focusing on different classes of materials (such as polymers, glasses, colloids, and gels) have to some degree evolved different choices of terminology that can lead to confusion. While attempting to enforce precise definitions across all disciplines would be futile, some degree of clarity is required to avoid a complete muddle. As described in the next paragraph, the rheological taxonomy used here is based on ideal, distinctive, stress-strain history relationships.

We take rheological terms to describe “ideal” responses, which are only approximated by real materials under restricted flow conditions. This is in keeping with the well-known observation that even a “solid” or a “simple liquid” such as a glass or water shows, respectively, fluidlike or solidlike responses under suitable conditions. Rheological terms are therefore most precisely deployed in constitutive equations that reflect ideal behavior. It is a matter of pragmatic judgment whether a real material’s behavior is “close enough” to the ideal to warrant a particular terminology. Whether or not a “yield stress” truly exists in real materials, it clearly does exist within various useful constitutive equations. Likewise, no material truly meets our definition of “ideal thixotropy,” but many constitutive equations clearly do.

To maintain clear distinctions, we avoid using different rheological terms to describe the same phenomenon, unless they are explicitly recognized as synonyms. Thus, we restrict the term “thixotropy” to nearly inelastic behavior so that it is not confused with “nonlinear viscoelasticity” or “viscoelastic aging.”

We take rheological nomenclature to be descriptive of responses to various homogeneous flow histories, regardless of the microstructure. Thus, “thixotropy” can describe microstructurally different materials, such as gels, colloids, and waxy oils. The restriction to “homogeneous” deformations is required because a material that becomes inhomogeneous macroscopically, for example, by banding, slipping at the wall, inertial effects, sedimentation, or transition to turbulent flow, can produce time-dependent stresses on rheometer surfaces that, without this restriction, might be called “thixotropic.” When such inhomogeneities are unavoidable, the rheology of the material under a hypothetically homogeneous deformation must be inferred, sometimes by hypothesizing an appropriate constitutive equation that, along with boundary conditions, predicts the inhomogeneous response.

While the above falls far short of resolving all issues with terminology and may not be universally accepted, at least it should help the readers understand the definitions and distinctions used here.

Viscoelasticity includes continuous, reversible, time-dependent increases and decreases in viscosity upon start-up and cessation of flow. To distinguish viscoelasticity from thixotropy, it might be appropriate to define thixotropy as “a reversible, inelastic, time dependence of the viscosity or yield stress during and after flow.” The emphasis on inelastic, or viscous, behavior separates thixotropy from viscoelasticity. Thixotropy, then, is not associated with recoverable elastic strain. While most fluids regarded as “thixotropic” possess some elasticity, a fluid can nevertheless be regarded as “thixotropic” if the elasticity is small or occurs over a much shorter time scale than the thixotropic response. The most typical thixotropic response is a slow decline of shear viscosity over hundreds of strain units, with recovery of viscosity when the shear is slowed or halted. This thixotropic response is often accompanied by a much faster viscoelastic response on start-up or cessation of flow that can be recognized as viscoelastic by either its fast, but finite, rate of stress decay on cessation of flow or by the presence of recoil, or strain recovery, on removal of stress.

Figure 1, which is from Mewis and Wagner [6], illustrates nicely, for a sudden decrease in shear rate depicted in (a), the distinction between (b) a viscoelastic response and (c) a thixotropic response. Figure 1(d) shows a combination of viscoelastic and thixotropic responses. Figure 2 [14] shows this combination in a real material: Note that after a step-down in shear rate, the stress decreases rapidly but continuously—a typical viscoelastic response to the preceding shearing. This is followed by a much slower increase of stress, which is evidently caused by the rebuilding of suspension structure once the flow rate has become slow enough to make this possible. To make clear the distinction between thixotropy and viscoelasticity, Larson [9] has defined an “ideal” thixotropic fluid as one that possess a time-dependent viscous stress only, with no elasticity or strain recovery; a similar definition of “purely dissipative thixotropic materials” was proposed earlier by Goddard [15]. In this respect, “thixotropy” is similar to essentially all other rheological properties in that rarely, if ever, does it appear in ideal form in real materials.

While it might be reasonable to argue that any gradual change in viscosity with time under steady flow must be associated with a deformed microstructure that retains some viscoelasticity, “thixotropy” is an appropriate term when this elasticity is “small.” As mentioned in Sec. I, this most commonly occurs when relatively rigid aggregates of particles, wax crystals, or other particles break down gradually under prolonged strain and build up again at rest. It might be of value to define more precisely how “small” the elastic response must be to allow the term “thixotropy” to be applied. We therefore tentatively propose that behavior be considered “thixotropic” if, after stress is removed, elastic recovery of less than 0.1 strain units is obtained after application of more than 10 units of strain at a rate high enough to decrease the viscosity by a factor of two or more from its value at rest or during a previous shear at a lower rate. A related requirement is that after this strain has been applied and flow is suddenly halted rather than stress removed, 90% of the stress relaxes in a time less than 10% as long as was required to impose the deformation. If both of these requirements are satisfied, then behavior commonly recognized as “nonlinear viscoelastic” would have little danger of being confounded with thixotropy. Of course, even if these conditions are not completely met by some real materials, there may be pragmatic reasons to model the material as “thixotropic” nonetheless, using an appropriate constitutive equation.

Thixotropy should also be distinguished from viscoelastic aging, such as the one that occurs in glasses. Such aging is characterized by a slow-down in relaxation during rest, which can be reversed under flow as the glass is “rejuvenated.” This response should not be regarded as “thixotropic,” for two reasons. First, viscoelastic aging occurs in an underlying viscoelastic spectrum, wherein the relaxation times themselves change with time, either increasing under rest conditions or decreasing under flow. If thixotropy is to be understood as a viscous phenomenon, then relaxation processes and their time constants are not pertinent to it. As discussed in Sec. II A, one could use the criteria of significant elastic recoil after stress removal and/or a rate of stress relaxation after cessation of flow that is comparable to or slower than the rate of change of stress after start-up of flow to determine that a material’s response is viscoelastic rather than thixotropic. If it is viscoelastic, then any aging of the material would be “viscoelastic aging” rather than thixotropic. Second, in deeply quenched glasses, viscoelastic aging occurs over an indefinite period, or, more precisely, over time scales that extend beyond practical measurement times. After deformation ceases, an aging viscoelastic fluid will relax stress very gradually and may even hold a residual stress at arbitrarily long times. A thixotropic fluid can require a long time to reach a steady-state stress after start-up of flow or change in the flow rate, but loses that stress quickly upon flow cessation. Figure 3 illustrates idealized differences among (a) viscoelastic, (b) viscoelastic aging, (c) viscous, and (d) thixotropic behavior in start-up and cessation of shearing.

Note that the response shown in Fig. 3(b) as illustrative of “viscoelastic aging” might also be seen in a nonlinear viscoelastic material or an elastoviscoplastic one. Additional flow histories would be needed to determine the best rheological description of such a material. In addition, it is quite possible that multiple constitutive equations, some allowing “viscoelastic aging,” and some not, might equally well describe the material, especially if the “aging” eventually ceases.

Note that we have been careful to use the term “viscoelastic aging” to describe gradual changes in relaxation time in response to flow or its cessation. Common usage suggests that the bare term “aging” could describe any gradual change: in viscosity, modulus, or relaxation time. When it is the viscosity that ages, with relaxation times being much shorter than the time over which the viscosity changes, this can be termed “thixotropy.”

Conventional glasses as well as pastes and gels, and other materials falling under the rubric of “soft glassy” materials, need not display a well-defined rest state [16] and have usually been considered to be aging, rather than thixotropic materials. However, we will see that materials commonly regarded as thixotropic can also lack a well-defined rest state. A recent review of experimental and theoretical studies on aging can be found in Joshi and Petekidis [17]. Joshi [18] also recently developed a relatively simple free-energy-based soft glassy model with a relaxation time that can age either faster than, or slower than, the progress of time itself. In the former case, a time-dependent yield stress can develop a situation that Joshi refers to as “thixotropic.” For reasons we now discuss, we would not call this model “thixotropic” except in the limit of a relaxation time that remains short. Phenomenologically, both thixotropy and aging are characterized by one or more structure parameters that change under flow and relax at least to some extent upon flow cessation. The key difference between traditional “thixotropic” materials and “viscoelastic aging” ones such as glasses is that in the former, the relaxation time, if present at all, is short and the effects of the structure parameter on viscosity and yield stress are dominant, while in the latter, the primary influence on stress is through the relaxation time, which, if it grows rapidly enough, can produce a yield stress. In the viscoelastic aging model of Radhakrishnan et al. [19], when the Maxwellian viscoelastic relaxation time (⁠ $G 0 / η 0$⁠) becomes much smaller than the inverse of the structure aging time, and the shear rate remains much less than the inverse relaxation time, the model begins to resemble that of an ideal thixotropic fluid [see Eqs. (3) and (4) in [19] and Eqs. (17), (18), and (21) in Sec. V B 2]. Thus, the same model can encompass behavior ranging from nearly ideally thixotropic to viscoelastic aging. Phenomenologically, however, the key distinction we would like to maintain is that the relaxation of stress after flow cessation is slow and continues to get slower for materials that undergo viscoelastic aging, but is rapid or nearly instantaneous for a thixotropic fluid.

There are, in addition, macroscopically amorphous solids, including hard metallic glasses, and some soft pastes, whose response to stress is to yield locally either in shear bands or in local plastic zones, allowing macroscopic flow, but then freeze-in these local structures on unloading [20–23]. Such materials can behave differently in each subsequent deformation cycle, no matter how much rest time is allowed in between. The lack of change during rest disqualifies these materials from being “aging” materials, and yet the lack of a reproducible rest state makes it inappropriate to describe their time dependence as “thixotropic.” Models for such materials have been developed, including the “kinetic elastoplastic” (KEP) model of Bocquet et al. [22,23] and the “shear transformation zone” (STZ) model of Langer and co-workers [20,21].

Thixotropy is often accompanied by plasticity, i.e., a yield stress that must be exceeded to produce flow. The yield stress, how it is defined and measured either statically or dynamically, and its relationship to microstructure and to thixotropy, are discussed in a recent thorough review by Bonn et al. [24]. Recent work by Grenard et al. [25] describes time scales associated with yield stress in carbon black suspensions. Here, we simply note that when both plasticity and thixotropy are present, the yield stress is frequently sensitive to prior flow history, typically diminishing because of prior flow, but recovering a higher value after sufficient rest time. Thus, both the viscosity η and the yield stress σ_y of a thixotropic fluid depend on time under flow, where the viscosity augments the stress from any yield stress present. We can refer to these two manifestations of thixotropy as, respectively, thixoviscous and thixoplastic behavior. The early model of Moore [26] is a typical thixoviscous model, while the model of Nguyen and Boger [27] and the recent model of Dimitriou and McKinley [28] are examples of thixoplastic models. Most thixotropy models are both thixoviscous and thixoplastic. Real thixotropic materials typically show a yield stress, while anti-thixotropic materials are often flowable fluids at low stress that become more viscous or even solid-like during and after flow. We shall see some examples of the latter shortly.

Thixotropy is often characterized by performing transient rheological experiments in which the shear rate or stress undergoes a “step” or ideally instantaneous change. The initial fast transient response to this is usually dominated by inertial effects from the measuring instrument and testing fluid which should not be attributed to the constitutive response of the material tested. For example, in stress-start-up experiments with a Newtonian fluid, the shear rate accelerates gradually due to inertia and therefore the apparent viscosity decreases—which can be mistaken for thixotropy if care is not taken. Although some modern rheometers can make compensation for inertial effects [29], it is important to be aware of this issue.

Thixotropy has been identified in a very wide range of structured fluids, lists of which, along with references to the relevant publications, can be found in [3,6,30]. As Mewis and Wagner [6] pointed out, the term “thixotropy” is often used in a general sense to describe time-dependent shear-thinning behavior and the buildup of viscosity, yield stress, or modulus after cessation of shear flow, and some recent works still follow this convention. Thus, some of the materials described as “thixotropic” might, in fact, be better thought of as nonlinearly elastic or viscoelastic aging materials. With this caveat in mind, below is a list of fluids identified by various authors as thixotropic, drawn from the reviews described earlier:

• Colloidal suspensions: paints, coatings, inks, clay slurries, pharmaceuticals, cosmetics, agricultural chemicals, magnetic coatings, drilling muds, metal suspensions.

• Slurries: coal/lignite, mining slurries.

• Nanoparticle suspensions: asphaltene-containing crude oil.

• Emulsions: personal care products.

• Foams: mousses.

• Crystalline systems: waxy crude oils, waxes, butter/margarine, chocolate.

• Polymers: associating solutions/melts, starch/gums, sauces, unsaturated polyesters, nanocomposites.

• Fibrous suspensions: tomato ketchup, fruit pulps.

• Biological systems: fermentation broths, sewage sludges, bloods.

Comprehensive rheological data sets are very important in the studies of thixotropy and its modeling. In Table II, we list several representative works that provide well-defined data sets that can be used to examine constitutive models. To be included in this table, the data sets need to contain at least a steady-state flow curve, multiple transient tests such as shear-rate-jump tests, and at least one of the following features:

1. flow protocols covering a wide range of shear histories;

2. data in easily accessible forms;

3. test fluids formulated in multiple compositions;

4. different temperatures; and

5. rheology with simultaneous local velocimetry.

The microstructures and their re-arrangements under flow that lead to “thixotropy” are nicely described by Jamali et al. [12], in the context of a broad class of fluids that they designate as “thixotropic elastoviscoplastic” or “TEVP” fluids. This microstructural description not only helps identify the causes of thixotropy, but also helps distinguish thixotropy from other, accompanying, rheological phenomena. Jamali et al. [12] envision a TEVP fluid as one in which a microstructural network exists at rest, that deforms and then yields upon start-up of flow. The stresses produced prior to yielding justify the prefix “elasto” in the TEVP designation; the yielding corresponds to the “plastic” adjective; and the viscous resistance during flow justifies the designation “viscous” [65]. For some fluids, there is no need to add the further designation “thixotropic.” However, if the network breakdown under flow does not lead quickly to a final steady-state structure, but instead there are gradual changes in structure due to further breakdown or recombination of network fragments, and/or the development of spatial heterogeneities in structure, such that the macroscopic viscosity can change gradually over prolonged time periods, this warrants the additional appellation “thixotropic.” If the change in structure leads to a prolonged stress relaxation on cessation of flow and significant recoverable strain, then the term “viscoelastic aging” might be more appropriate than “thixotropic.”

Most definitions of thixotropy require that the shear-induced changes in viscosity be reversible, implying that, after a sufficient time after flow ceases, the fluid recovers a well-defined “rest state.” More precisely, the requirement is that after a long enough resting period, subsequent rheological measurements are the same as those obtained after any earlier resting period. (Since relaxation processes frequently show exponentially long tails, this requirement must be interpreted in the asymptotic sense that the difference between repeated measurements decays exponentially with prolonged rest time.) It is questionable whether real thixotropic fluids commonly satisfy this requirement. For waxy crude oils, for example, it has been suggested that changes in the fluid structure introduced by a high shear rate may be irreversible: The fluid will “remember” indefinitely the highest shear rate it has experienced [47]. For waxy crude oils, this “permanent” memory can be erased by raising the temperature to the melting point of the wax crystals, and a repeatable initial structure can then be attained by quenching under a well-defined thermal and shearing protocol [47]. Other thixotropic fluids, such as suspensions of colloidal particles, usually cannot be rejuvenated this way, and rheological results for such fluids might have significant variations in behavior from batch to batch or from one loading in the rheometer to another. In fact, one might suspect that the yield stress itself might resist the changes needed to bring the fluid back to a well-defined rest state. However, it is observed that upon rest, yield stress can build up with time suggesting that some re-arrangement of the structure is possible even when the material is below the stress needed for macroscopic flow [66]. The reproducibility of thixotropic fluids to repeated measurements or repeated preparation of the fluid from its constituents has not much been reported on, although rheological “lore” suggests that problems are common. A careful study of the reproducibility of common thixotropic fluids to repeat measurements, and repeat formulations, in the same, and different laboratories, would be a worthwhile project. In practice, lack of a rest state is often dealt with by subjecting the material to a prolonged (often hours-long) preshear to “erase” the prior loading and flow histories, prior to any rheological measurements.

As discussed elsewhere [9], it is not necessary for real materials to meet the definition of an ideally thixotropic material for the concept of thixotropy to be of value, as long as the thixotropic aspects of the fluid’s rheology can be identified. Thus, a gradual, shear-induced, change in a fluid’s viscosity that is at least partially reversed during a rest period could meaningfully be referred to as a “thixotropic” response. The lack of a complete recovery to a well-defined rest state could be attributed to other phenomena that accompany thixotropy, such as “eternal aging.” Such would be in keeping with conventional treatment of other rheological phenomena, including the concepts of “fluid” and “solid,” which are idealizations that do not exhaustively describe the rheology of any real material.

Anti-thixotropy refers to a gradual increase in the viscosity of a fluid under shear, rather than the more typical decrease. An example is shown in Fig. 4 of the viscosity as a function of time following start-up of shearing at a constant stress of 2.74 Pa for a partially hydrolyzed polyacrylamide solution in a water/glycerine mixture, studied by Buitenhuis and Springer [67]. Note that, after the initial rise in viscosity at short times up to 5 s, which is due presumably to viscoelasticity, a much later second rise occurs over the period of 20–50 s, before a modest decrease. Since strain can be calculated from the stress and viscosity, we infer that the second rise in viscosity occurs over a period of 60–150 strain units, too many to be attributed to ordinary viscoelasticity. If both the early-time viscosity, after the initial fast increase, and the long-time viscosity, after the second rise and overshoot, are plotted together for various concentrations of polyacrylamide, one obtains the results shown in Fig. 5. Note the critical shear stress above which the second, higher, viscosity first appears. This increase of stress, reflecting a flow-induced build-up rather than breakdown of structure, justifies the term “anti-thixotropic” for this fluid. The molecular origins of the flow-induced build-up of the network structure of these polymers remain mysterious. Note that this form of “thixotropy” is not accompanied by a yield stress; the viscosity does not appear to diverge at low shear stress in Fig. 5. (Or if it does diverge, it is with a very weak power law.) The gradualness of the build-up of viscosity, in units of strain, distinguishes anti-thixotropy from “jamming,” which is a more sudden shear thickening, which is sometimes discontinuous when plotted against strain rate or stress [68–71].

Coussot et al. [72] observed in bentonite clay suspensions the viscosity bifurcation phenomenon shown in Fig. 6. For shear stresses of 20 Pa or higher, the shear rate rises with time, a typical thixotropic response. For lower shear stresses, 12 Pa or less, the shear rate eventually collapses toward zero, either monotonically or after a period of rising shear rate. Thus, there is a minimum shear stress required for continuous flow; i.e., a yield stress. However, in contrast with simple plasticity, this yield stress depends on the rest time following a previous flow; the same applied stress can lead to either continuous flow or eventual cessation of flow, depending on the rest time [72]. Moreover, there are stresses for which the material seems to “change its mind,” initially showing an accelerating shear rate, but then collapsing to a zero shear rate. All of this is evidence of structures within the material that gradually heal at rest or are under a slow flow and can break when the flow is fast enough. This nonmonotonic behavior could possibly be explained by structures at multiple levels, some of which continue to heal even as others are breaking down during shearing at intermediate rates. As thixotropic yield stress materials are prone to banding, further studies are needed to examine the impact of shear bands on the bulk rheological response.

Coussot et al. [72] proposed that such materials have nonmonotonic flow curves, as illustrated in Fig. 7. At a constant stress above a critical value (which itself depends on shear histories), for different initial conditions, the system either evolves toward the stable portion on the flow curve and flows continuously or evolves toward the unstable portion, which leads to flow stoppage. If structures at multiple levels are present, evolutions toward opposite directions might be present, which is likely to result in the observed nonmonotonic response of the shear rate.

Viscosity bifurcations are also seen in other thixotropic fluids, such as waxy crude oils, where a viscosity bifurcation was observed by Tarcha et al. [73]. These authors found that a critical strain better characterizes the condition for structural breakdown than does a critical stress, at least for their material.

A commonly observed phenomenon associated with thixotropy and anti-thixotropy is hysteresis in curves of shear stress versus shear rate, upon ramping up the shear rate and then ramping it down, at a prescribed rate. The first report of such a hysteresis loop in viscosity appears to be that of Weltman [74]. An example of a fumed silica suspension is shown in Fig. 8 [75]. This figure shows that the stress is higher on increasing the shear rate than on the shear-rate-reduction part of the cycle. This is consistent with ordinary thixotropy, in which shear breaks down structures, leading to a reduced viscosity that does not have time to recover before the second half of the cycle occurs.

Another example (Fig. 9), from the work of Krystyjan et al. [76], illustrates both thixotropy and apparent “anti-thixotropy” in waxy potato starch pastes. In these pastes, the stress is initially lower when the shear rate begins to decrease, but eventually is higher than that in the first half of the cycle. Thus, in this case, the shear has built up the viscosity rather than reducing it. This change in behavior occurs at the cross-over point of the shear-rate-rising and shear-rate-falling curves in Fig. 9. A similar behavior has been observed by Radhakrishnan et al. [19] and captured in a model that includes significant viscoelasticity. Thus, this behavior may be the result of viscoelasticity, rather than “anti-thixotropy.” The behavior reflects the complexity of starch, which contains associating molecules that are organized into granules that can swell, and interact with each other. The preparation of the starch, or the degree to which it has been pasted, affects the thixotropic characteristics and presumably also contributions of viscoelasticity. Note the lack of any obvious yield stress in these results, whether the fluid is thixotropic or anti-thixotropic, indicating that thixotropy need not be accompanied by a yield stress. A review of hysteresis loops in thixotropic fluids can be found in Mujumdar et al. [5].

Thixotropic materials often show shear banding [41], in which an initially homogeneous fluid separates into layers of different shear rates under an imposed shear flow. Shear banding greatly alters the local shear histories and impacts the interpretation of the bulk rheological properties. Shear banding also potentially has a major influence on industrial applications that involve processing flows of thixotropic fluids. Fielding and co-workers [77–80] have made intensive theoretical studies of the onset and evolution of shear bands in time-dependent flows including shear start-up, step stress, rapid strain ramp, and large amplitude oscillatory shear; see [81,82] for two good reviews. Puisto et al. [83] have shown banding and stress hysteresis in down-and-up shear-rate ramps in both viscoelastic and purely viscous (i.e., thixotropic) fluids with time-dependent viscosity. A review of experimental work on banding in polymers and soft glass materials can be found in Divoux et al. [84]. Transient shear banding has been found to be a general consequence of stress overshoots on start-up or step-up of shearing [77].

A critical value of shear rate below which shear banding occurs has been reported in several materials. Paredes et al. [41] studied the banding behavior of a thixotropic emulsion and a normal emulsion, the latter of which was a simple yield stress material without thixotropy. The authors found that when wall-slip was suppressed by roughening the geometry surface, the normal emulsion did not exhibit shear banding while the thixotropic emulsion formed a banded flow when sheared below a critical shear rate. Martin and Hu [85] found such a critical shear rate in Laponite suspensions and reported that aged Laponite suspensions show transient shear banding—bands that grow shortly after shear start-up and then disappear as the fluid evolves toward a uniform shear rate. Dimitriou and McKinley [28] confirmed the existence of a critical shear rate for banded flow in waxy crude oils, below which shear bands and wall-slip velocity fluctuate irregularly at a constant overall shear rate. The recent theoretical study of Jain et al. [86] showed that inertia plays an important role in both the short-term banding dynamics in shear start-up experiments and the shear bands at steady states. In shear start-up experiments, inertia seeds heterogeneity which further grows due to inhomogeneous aging and rejuvenation. The critical shear rate determined from the constitutive flow curve alone was found to be an insufficient criterion for steady-state banding.

The study of Divoux et al. [87] provides insights into the relationship between time scales of rheology and of shear banding. The authors measured the flow curve and local velocity profile of several soft materials including both simple yield stress fluids and thixotropic fluids. In their flow protocol, the shear rate was first swept down from $10 3$ to $10 − 3 s − 1$ in 90 logarithmically spaced steps of duration $δt$ and then swept up over the same range. Divoux et al. [87] found that both the hysteresis area in the flow curves (⁠ $A σ$⁠) and the overall degree of banding (⁠ $A v$⁠) depend strongly on $δt$⁠. For thixotropic materials such as laponite suspensions, $A σ$ and $A v$ exhibit a maximum at a critical value of $δt$⁠, as shown in Figs. 10(b) and 10(c). The authors also observed this phenomenon in mayonnaise and carbon black suspensions. This robust maximum indicates that both the bulk rheology and the banding dynamics depend strongly on an intrinsic material restructuring time. (For simple yield stress fluids such as Carbopol microgel, this time scale is very small; therefore, $A σ$ and $A v$ decrease monotonically as $δt$ increases.)

Recently, Radhakrishnan et al. [19] studied this phenomenon theoretically based on the fluidity models and the soft glassy rheology model [82], which qualitatively capture the dependence of $A σ$ and $A v$ on $δt$⁠. According to the models, this phenomenon is triggered by the nonmonotonic underlying stationary constitutive curve and the strongly time-dependent thixotropic response.

In another recent contribution, Garcia-Sandoval et al. [88] modeled band formation in worm-like micellar solutions using a viscoelastic (upper-convected Maxwell) constitutive equation whose relaxation time and viscosity are controlled by a fluidity parameter analogous to an inverse structure parameter $1/λ$⁠. The evolution of the fluidity is governed by relaxation and breakdown terms, similar to those in standard thixotropy models, as well as a diffusion term that penalizes gradients in fluidity. The resulting theory predicts formation and migration of bands after start-up of shear.

Very recent work by O’Bryan et al. [38] has shown that while “normal” Carbopol suspensions are simple yield-stress fluids, when stirred for a long time they can become thixotropic, and then display both banding and hysteresis in shear-rate ramps. A simple purely viscous fluidity model [similar to Eq. (18)] was able to predict the observed hysteresis.

Coussot et al. [72] provided a very simple thixotropic model, similar to Eqs. (1)–(4), with constant coefficients, that shows a viscosity bifurcation and banding. All that is required is that the sample be insufficiently aged so that a low enough shear stress is able to allow the sample to continue to age so that its yield stress eventually exceeds the applied stress, and flow then stops. If the stress is high enough to prevent aging, then flow continues indefinitely. However, the simple thixotropic model of Coussot et al. [72] was unable to show the nonmonotonic shear rate shown in Fig. 6. A nonmonotonic shear rate under a constant shear stress could presumably be obtained if the viscosity and yield stress dependent differently on $λ$ so that at some shear stresses the viscosity drops but the yield stress rises. This would allow the shear rate to increase initially due to the decreasing viscosity, but eventually the yield stress would exceed the applied stress and flow would stop. Perhaps, a physically more realistic way of obtaining a nonmonotonic viscosity bifurcation would be by the use of more than one thixotropy parameter, as we now discuss.

The above models rely on a single structural parameter $λ$ to encode all structural information needed for rheological prediction. As might be expected, this assumption is a gross over-simplification. Just as typical viscoelastic relaxation phenomena are only crudely captured by a single relaxation time $τ$ and a corresponding single modulus G, typical thixotropic behavior is only crudely captured by a single $λ$⁠. Accurate description of viscoelasticity usually requires multiple relaxation times $τ i$ and their corresponding weights or moduli $G i$⁠, reflecting relaxation occurring on multiple structural scales. A similar consideration applies to thixotropic phenomena. The existence of multiple viscoelastic relaxation times can be revealed through a nonmonotonic response to a multiple rate-jump test, and the same test can reveal multiple thixotropic structural parameters [61] (Fig. 12). Mewis and co-workers showed that thixotropic responses to single-step-rate jumps can be represented by a stretched exponential relaxation of the structural parameter $λ$⁠. They modeled this phenomenon by adding a time-dependent prefactor, $t − α$⁠, to the kinetic equation of $λ$⁠. Their model predicts the stretched exponential thixotropic evolution in step-shear-rate tests but is not applicable in arbitrary shear histories due to the inclusion of a time-dependent prefactor $t − α$ which violates the time-invariance symmetry. In addition, this model fails to predict the nonmonotonic response following multiple jumps of shear rates because it has only a single thixotropic structure parameter.

Figure 12 shows the rheological response of a fumed silica suspension after two jumps in shear rate separated by a time interval $Δt$ and the corresponding predictions of a multi-lambda model, the ML-IKH model (see Table IV for its full form, which includes elasticity and complex plasticity described later). For intermediate values of $Δt$⁠, experiments show nonmonotonic stress transients at time scales around 10 s. The ML-IKH model captures these nonmonotonic transients only when $β$ is less than unity.

de Souza Mendes and Thompson [7] described two basic methods of incorporating viscoelasticity into thixotropic viscoplastic models. In Type I models, one starts with the Bingham model, but with the yield stress written as the product of an elastic modulus and an elastic strain. One then introduces evolution equations for the elastic strain and viscosity. Some models [60] also include an evolution equation of the elastic modulus.

In Type II models, one starts instead with a viscoelastic stress equation, such as the Jeffreys model, to which plasticity and thixotropy are introduced by replacing the model parameters with some functional forms that depend on the structural state. Figure 13 shows the mechanical analog of a family of thixotropic viscoelastic models that are based on the Jeffreys model. de Souza Mendes and Thompson [7] show an analogous schematic for Type I models. Type II models are able to incorporate a yield stress by allowing the viscosity $η s (λ)$ to become unbounded at a critical level of structure $η s (λ)$⁠. Joshi and Petekidis [17] have explored models that build up a yield stress in this way.

As discussed in Sec. II B, the concept of “aging” is frequently invoked to describe slowly changing rheological properties, particularly in the field of glasses and “soft glassy” materials. In Sec. II B, we suggested that “aging” be considered as a general term that can refer to slow changes in modulus, yield stress, viscosity, or viscoelastic relaxation time. Under that definition, thixotropy would be a form of “aging” in which it is primarily the viscosity or yield stress that changes, with little, or ideally, no, elasticity present. Thus, aging of a viscoelastic material can be called “viscoelastic aging” to distinguish it from thixotropy. “Aging” can be subcategorized as “thixoviscous aging” (growth of viscosity), “thixoplastic aging” (growth of yield stress), and “viscoelastic aging” (growth of viscoelastic relaxation time). The first two of the above can be considered as forms of “thixotropy,” while the third one should not be, in our view. An additional distinction can be made between aging that finally ceases, and “eternal aging,” which does not, and for which there is no rest state. “Ideal” thixotropic materials do not age forever and have a rest state, although this situation may not be very common. If an otherwise thixotropic material lacks a rest state, it may be sensible to refer to it as an “eternally aging thixotropic” material. Under prolonged aging a material may become significantly viscoelastic, in which case we believe it would be better to call it a material with “viscoelastic aging” rather than “thixotropic.”

To illustrate this, we leave aside yield stress and discuss a simple viscoelastic Maxwell model with aging of a “Type II” form (see Sec. V B 1), motivated by a similar model in Radhakrishnan et al. [19], and show how it approaches several of the simple limits shown earlier in Fig. 3 depending on the structure and flow conditions.

An eternally aging material defined in Eqs. (17) and (26) will, if one waits long enough, eventually become viscoelastic, since the relaxation time defined by $τ(λ)=λ τ s$ eventually becomes large no matter how small $τ s$ is. Thus, a material following these equations, with eternal aging, may behave as a nearly ideally thixotropic material for a long time, but eventually ages into a viscoelastic one. As we have already emphasized, pragmatic considerations, rather than fundamental ones, generally determine whether a real material should be considered a “thixotropic” or a “viscoelastic aging” one.

The above analysis indicates that even for simple models that include both viscoelasticity and aging, a wide range of responses can be obtained during start-up and cessation of shearing, making it difficult, in general, to distinguish viscoelasticity, viscoelastic aging, and thixotropy except in the limits of small or large parameter values. It was noted in earlier work [9] that nonlinear viscoelasticity can often encompass a relaxation time that grows with time following cessation of flow. For example, a simple dilute solution of nearly fully extended polymers relaxes with a relaxation time that increases with time as the polymers relax. It is questionable whether this should be called “aging,” and even more questionable to call it “thixotropy.” The above discussion did not even introduce plasticity or yield stress, which can further complicate the discussion. For example, a yield stress might be generated in a suspension containing little viscoelasticity, or in a fluid whose viscoelastic relaxation time ages nonlinearly and reaches infinity in finite time, thus leading to a solid-like response after a finite time. In the former case, the yield stresses, if dependent on prior shear, might be called “thixotropic,” while the latter might be considered as a form of “viscoelastic aging.”

Given the complexities and overlaps that easily occur in the use of rheological terms, we believe it is best to be guided by the ideal limits of the various phenomena. “Ideal” thixotropy is purely viscous or plastic, with no elasticity or recoverable strain, and eventually recovers a rest state after cessation of flow. “Ideal” viscoelasticity shows prolonged stress relaxation after cessation of flow and elastic recoil on stress removal, but the relaxation time itself (or the spectrum of relaxation times) remains constant. “Ideal” viscoelastic aging displays a relaxation rate that slows on time scales much longer than the relaxation time itself, in principle continuing to slow indefinitely. And, of course, “ideal” viscous fluids show instantaneous attainment of steady stress after start-up of shear, instantaneous loss of stress after cessation of shear, and no elastic recovery on stress removal. These four ideal responses are illustrated in Fig. 3. Real materials often at best only approximate these behaviors, and sometimes do so only for limited ranges of strain rates and times. Hence, it may be better to refer to viscoelastic or thixotropic “responses” rather than “materials,” as the same material may display quite different behavior depending on conditions of the experiment. To avoid confusion or redundancy in terminology, we believe it is important to hold firmly to the definition of an “ideal” thixotropic material as one that is completely inelastic, although time dependent. Even if real materials refuse to comply with this definition, at least we can be guided by a clear definition of the ideal case. And even if viscoelasticity inevitably accompanies thixotropy, there are pragmatic reasons one might want in some situations to model only the thixotropic aspects of the response and ignore the viscoelastic. For one, nonlinear viscoelasticity is notoriously difficult to model, and thixotropy comparatively easy, especially in three-dimensional flows.

Earlier, we defined “ideal thixotropy” as a flow-history dependence of viscosity and yield stress, but without any elasticity, and illustrated it with a simple constitutive equation involving a Bingham yield stress. The Bingham law of plasticity, even when we allow for a yield strain as described in “Type I” models in Sec. V B 1, is typically too simple to describe in detail the plastic response of real materials, especially in complex flow histories. So, to go beyond both “ideal thixotropy” and “Type I” and “Type II” models of TEVP materials, we will first describe briefly more advanced concepts from the theory of plasticity, including the Bauschinger effect and kinematic hardening, and then show how these can be incorporated into more realistic constitutive equations for thixotropic materials. Inevitably, the introduction of advanced concepts of plasticity will bring consideration of elastic effects, which will lead us to beyond “ideal thixotropy” to equations describing TEVP fluids. Note that kinematic hardening is an important subject of study in the plasticity field that mainly focuses on compressive and tensile deformation of metallic materials. Dimitriou et al. [28,90] showed that kinematic hardening is also an important rheological response in soft TEVP materials under shear deformation and the previous scalar kinematic hardening equations can be conveniently used in shear flows. The recent simulation studies by Fraggedakis et al. [91,92] demonstrate the necessity of sophisticated TEVP models in flows with complex kinematics.

The Bauschinger effect is the change in the yield stress as a result of strain history. Perhaps, the most common example is the decrease in the compressive yield stress of a material as a result of an increase in the tensile yield stress due to a prior tensile strain; see Fig. 14. The increase in yield stress in the original straining direction with a consequent decrease in the opposite direction is known as kinematic hardening [93]. (In isotropic hardening, the yield stress increases in all directions upon straining in one direction.) Thus, kinematic hardening is a shift of the yield surface in the direction of the most recent straining, so that when the strain is reversed, the yield stress in the opposite direction is temporarily reduced. Upon continued straining in the opposite direction, hardening eventually occurs in that direction as well. This phenomenon was discovered in polydomain solids, such as metals, for which strain-induced orientation of grains leads to an increase in yield stress in the direction of the straining, but a decrease in the opposite direction, which is revealed once the strain direction is reversed.

Figure 15 shows mechanical models that include the yield stress, back stress, and an elastic stress, in various combinations, but that omit the viscous stress $η γ ˙ p$ (Table III). The behavior of each of these models in periodic reversing shear flow is shown in Fig. 16. The behavior of Eq. (30) (with $η$ = 0), namely, $C[ 1 − exp ⁡ ( − q γ p ) ]/q+ σ y$⁠, is represented by the initial start-up segment of Fig. 16(c), beginning at zero strain and stress $σ y$ and continuing onto the plateau region $σ y +C/q$⁠.

Note that the back stress introduces a dependence of stress on “plastic strain” $γ p$⁠. We put “plastic strain” in quotes because this dependence on strain introduces elasticity into the equation. This can be seen by suddenly reversing the direction of shearing flow in the above model. If the strain prior to this was small enough to use the linearized version of Eq. (30) (for small $γ p )$⁠, then the stress will remain initially positive when $γ ˙ p$ changes sign (as long as $γ ˙ p$ is smaller in magnitude than $σ y$/ $η$⁠), as $γ p$ then gradually decreases in magnitude. The decreasing positive stress during reverse straining is a recoil phenomenon from which work might be extracted, unless it is blocked by the yield stress. Hence, the back strain is a form of elasticity, despite the label “plastic” attached to the strain $γ p$⁠. The nonlinearity within the general expression $1−exp⁡( − q γ p )$ is not elastic, as it arises from the term $−qA| γ ˙ p |$ in Eq. (29), which is irreversible owing to the dependence on the absolute value of the strain rate. Thus, kinematic hardening introduces elasticity into plasticity theory and should be considered as a theory of elastoplasticity, rather than of plasticity alone.

A particularly simple case to consider is that of negligible yield stress $σ y$ and negligible viscous stress $η γ ˙ p$ either because of small-strain rate or small viscosity. In this case, the resulting hysteretic stress-strain relationship is shown in Fig. 16(b). This limit shows that the back stress contribution stores elastic energy, which can be partially extracted upon flow reversal. Figure 16 also shows the effect of inclusion of a yield stress $σ y$ to the shear and reversal predictions, both without a yield strain [Fig. 16(c)], and with it [Fig. 16(d)]. The corresponding mechanical models are shown in Figs. 15(c) and 15(d). Note that the inclusion of a high enough yield stress (without a yield strain) changes the slope of the stress-strain curve during flow reversal from finite [Fig. 16(b)] to infinite [Fig. 16(c)], implying that the recovery of elastic work in Fig. 16(b) is lost when the yield stress is high, as in Fig. 16(c).

The existence of kinematic hardening in thixotropic suspensions is demonstrated in reversing flows, the simplest of which is a reversal in the direction of steady shear after steady state has been attained in unidirectional shear. Figure 17 shows the stress evolution after such a reversal for a fumed silica suspension for a range of shear rates, where the straining prior to reversal is arbitrarily taken to be in the negative direction so that the final shear direction is positive. Figure 17 should be compared with the left-most leg of the schematic stress-strain curve in Fig. 16(d), where the negative straining direction is suddenly reversed to straining in a positive direction. Thus, the steep part of the curve with slope G and negative stress in Fig. 16(d) corresponds to the steep initial portion of the curve in Fig. 17, with negative stress. The portion with positive stress and a less steep slope in Fig. 16(d) corresponds to the portion of the curves in Fig. 17 with positive stress. In Fig. 17, the response at small strains after reversal is thus primarily elastic, with the negative stress from the initial shearing transitioning to positive stress as the small amount of elastic strain (<0.1, or <10%) is removed from the sample. [This small-strain elasticity is described in Eq. (31).] Shortly thereafter, a “knee” appears in the response, corresponding roughly to the point of yield of the sample in the positive direction, after which the stress initially grows more slowly with strain. This yield stress after reversal is lower in magnitude than the yield stress required to initiate plastic strain upon start-up after a long period of rest, consistent with a combination of a reduced isotropic yield stress and a shift of the yield surface toward the direction of the previous strain. That the yield surface has indeed shifted is indicated by the significant rise in the slope of the stress versus log strain at a strain of around unity, consistent with a reversal of the back strain from a negative value creating by the prior flow to a positive one created by the continued straining in the positive direction. [This rise in slope after the knee shows up when stress is plotted against log strain; when plotted against linear strain, the result is shown in the inset of Fig. 17 and resembles closely the left leg of the stress-strain cycle in Fig. 16(d).] Once the back strain has reversed sign and come to steady state, it ceases to increase, and the stress quickly levels off to a constant value (albeit higher than in the absence of kinematic hardening), consistent with the attainment of a steady-state back stress. From a microstructural point of view, back strain represents an orientation of elements of the material, causing resistance to yielding to increase. By the same token, this orientation stores elastic energy that might be partially recovered, depending on the magnitude of the viscosity and yield stress. We emphasize that the existence of kinematic hardening in the thixotropic fluid described in Fig. 17 is clearly revealed in the shape of the stress-strain curve (i.e., the “double plateau” on a semilog plot, the first due to yielding and the second to kinematic hardening) after reversal of the straining direction.

We note that addition of details such as back stress greatly complicates the modeling of thixotropy, especially in full tensor theories, and the improvement thereby gained is mainly in providing better predictions in more complex flow histories at relatively small accumulated strain following changes in the flow direction. In many cases of industrial importance, the improved predictions (which are still by no means quantitative) may not be worth the greatly increased complexity of the equations and the numerical expense of solving them. Still, with increasing computer power, inclusion of more accurate modeling of plasticity may become increasingly common, even for industrial applications. We also note that full tensor forms of Eqs. (27)–(31) are available in the literature [13,90], but not presented here, owing to their complexity. Table IV provides a summary of the scalar models used for the predictions in Fig. 17.

Thixotropy is caused by reversible structural breakage under flow and its buildup when flow decreases or stops. Structural kinetics models (SKMs) described above capture these physics phenomenologically through one or more structure parameters and corresponding evolution equations. While some models of this kind predict experimental data well, they provide no clear information about the microstructures; the structure parameter is only conceptually related to the microscopic structural state. Population balance models (PBMs), however, take account of the flow-induced changes of the size distribution of microstructures and the impact of these changes on the bulk rheology. PBMs have long been used to model aggregation and fragmentation kinetics in colloidal dispersions. Many PBMs belong to the category of ideal thixotropic models but they focus mainly on the dynamics of aggregate mass distribution instead of the rheological properties. Recent years have seen growing interest in constructing quantitative thixotropic constitutive models from PBMs [96,97]. Here, we briefly review the framework of PBMs and discuss their connections with SKMs.

In suspensions, colloidal particles aggregate and form fractal structures over a wide range of length scales. PBMs take a mean-field description of the structural state by considering only the number density distribution of aggregates of different mass. PBMs are constructed based on the principle of mass conservation—a floc of mass x can stick to one of mass y and form a larger floc of mass $x+y$⁠. (Following conventions, the primary particle or cluster has unit mass.) This process can be represented as $x,y→x+y$⁠. The rate of this coagulation process is assumed to be proportional to the number density of flocs of mass x and y, $n( x , t )$ and $n( y , t )$⁠. The rate coefficient is often described by a collision kernel $K( x , y )$ that is a positive and symmetric function of x and y. The collision kernel accounts for the effects of flow, the Brownian motion, and other interactions among particles. PBMs usually do not consider coagulation events that involve the simultaneous collision of three or more flocs.