Keywords

2.1 Introduction

Rheology is the field of science that studies the deformation and flow of complex materials [1]. A complex fluid is a sort of complex material that does not obey Newton’s law of viscosity: it exhibits a nonlinear relationship between the imposed stress and the measured strain rate or vice versa [2]. In order to understand and model that relationship between stress and strain, we need to poke the material and observe how the material functions depend on the loading time scale, loading amplitude, external electromagnetic fields, temperature, etc. [3]. Rheometry is a branch of rheology dedicated to measuring the different material functions [4], which are determined under standard flows, such as shear and extensional flows that can provide complementary information to get a complete rheological characterization [5].

Shear thickening fluids (STFs) are complex fluids, typically consisting of dense suspensions of solid particles dispersed in an inert carrier fluid, that exhibit an increase in viscosity under the application of a shear rate/stress over a critical value [6]. In other words, the stress required to shear an STF increases faster than linearly with the shear rate [7], as shown in Fig. 2.1. In his seminal work of 1989 [10], Barnes stated that shear thickening behavior is normally conditioned by different parameters (Table 2.1). Subtle changes in the local particle arrangements are responsible for the transitions between shear thinning, Newtonian, and shear thickening regimes [9]. In recent years, it is common to distinguish between continuous shear thickening (CST) and discontinuous shear thickening (DST); the former is exhibited at intermediate packing fractions and the viscosity increase is relatively mild due to the formation of hydroclusters [11]; increasing the packing fraction, the viscosity increase becomes steeper up to a point where the viscosity exhibits a discontinuous increase beyond a critical shear rate, as result of the contact between particles [7].

Fig. 2.1
A set of 2 graphs. Graph a denotes the plots for apparent viscosity versus shear rate. It exhibits shear thinning in zones 1 and 3 along with shear thickening in zone 2. Graph b of shear stress versus shear rate highlights the shear thickening and thinning regions along with the lines for yield stress, random viscosity, D S T, and C S T across the Newtonian slope.

(a) Sketch of a viscosity curve exhibited by STF transiting from Newtonian to shear thinning and shear thickening (CST) regimes [8]. (b) Sketch of different flow curves exhibiting CST and DST [9]. Reprinted by permission from Elsevier

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Table 2.1 Parameters for conditioning the shear thickening behavior [10]
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Although shear thickening behavior may cause technical problems in flow processes, such as blockage of spraying nozzles or insufficient mold filling [12], thanks to their viscous dissipative nature and the possibility of tuning their viscosity curves, these fluids have allowed the development of important engineering solutions to different applications, namely in energy dissipation systems, such as vibration or impact absorption, which have accelerated in the last years [6, 13]. Even though STF has undergone dynamic loads in most of the applications, it is traditionally characterized by means of viscosity curves, which represent the equilibrium shear viscosity values reached at different steady shear rates. It is hard to believe that the STF will undergo steady simple shear flow in its final application, most probably the flow will be transient and complex, a combination of shear and extensional flows; this mismatch, both in terms of timescale and flow type, between the rheological information provided and the final deformation under real conditions, may result in a wrong performance of the fluid [14].

This chapter aims at providing a guide about how to perform a full rheological characterization of STF useful for both fitting and developing constitutive models and tweaking the formulations toward obtaining an optimal rheological performance of the fluid.

2.2 Shear Rheometry of STF

2.2.1 Steady Shear Flow

In shear rheometry, the fluid sample undergoes simple shear flow, which can be imposed either by rotational rheometers or by capillary (both at macro and microscales) ones [2]. Steady shear measurements allow obtaining either the viscosity curve, which provides information about the dependence of the steady shear viscosity on the shear rate, or the flow curve, which relates the steady shear stress with the shear viscosity.

There are two kinds of rotational rheometers: strain-controlled and stress-controlled rheometers, the latter commonly equipped with control systems allowing for operating in pseudo-strain-controlled mode. As highlighted by Mewis and Wagner [15], it is very important to bear in mind that “controlling stress allows one to move systematically through the transition, whereas in a strain-controlled device the discontinuous shear thickening region is generally not accessible”. Rotational rheometers enable the imposition of steady simple shear flow covering a wide range of shear rates/stresses by working with the right geometry, i.e., concentric cylinders, cone-plate or plate-plate, and the right dimensions. Whereas cone plate geometry ensures a homogeneous shear rate throughout the whole volume sample, for STF, plate-plate geometries are very convenient, particularly if wall slip would be an issue, as they can operate with grooved surfaces and different gaps to avoid it. Concentric cylinders can also be used with STF, as long as a small gap is being used to minimize the change of shear rate inside the gap and to avoid particle migration; however, loading a very viscous STF in a concentric cylinder geometry is more difficult than in other rotational geometries [16, 17]. Some other precautions must be considered when working with STF, not because of its rheological behavior, but because STF mostly consists of a dense suspension of particles [15]: regarding the gap size, either operating with cone-plate or plate-plate geometry, it is of paramount importance ensuring that the ratio of gap size to particle/aggregate size is large enough, between 10 and 50; when the formulation consists of particles not neutrally buoyant with the carrier fluid and large enough to avoid the Brownian motion so that sedimentation dominates, a vertical concentration gradient will be developed and the top moving plate will “feel” an apparent reduction in the viscosity of the sample; as in many suspensions, some shear thickening samples may require a pre-shear protocol to ensure reproducibility in the measurements [18]; partial evaporation of the carrier fluid may be also an issue when working with volatile solvents, which can be avoided either by fractioning the range of shear rates to be covered in a single experiment or by reducing the shearing time at each shear rate.

The experimentalist is responsible for being aware of the instrument’s resolution, instrument inertia, sample inertia, boundary effects, and volumetric effects to avoid bad data [3] when operating the rotational rheometer. Working with STF, the maximum torque of the rheometer may be reached with ease if working with a cone-plate or plate-plate geometry. To avoid misinterpretations of the steady shear flow measurements, the experimental window must be properly identified, because the data set obtained below the minimum torque line can be misinterpreted as shear thinning behavior and data beyond the onset of secondary flows can be misunderstood as shear thickening behavior.

When performing steady shear flow measurements with STF in a rotational rheometer equipped with crosshatched, serrated, or grooved plates, it is important to perform several preliminary tests: the experimentalist should test different gap sizes in order to ensure that the measurements are gap independent [19], as the roughness of the plates may introduce gap dependency; surplus of material around the plates should be carefully trimmed, otherwise the presence of a few milliliters of suspension left on the bottom plate in contact with the paste between the two plates may strongly affect the critical shear rate and induce gap dependency [20].

Any complex fluid would be fully characterized under steady shear flow with three material functions [21], i.e., the variation with shear rate of the shear viscosity and the first and second normal stress differences, N1 = σ11 − σ22 and N2 = σ22 − σ33 respectively [22], where by convention 1, 2, and 3 are the velocity, velocity gradient, and vorticity directions, respectively, in a viscometric shear flow. For isotropic materials, N1 has always been found to be positive (unless it is zero) and N2 negative and smaller than N1 in terms of magnitude [23]. If the material is an ideal solid (purely elastic), then the value of N2 would be zero. Typically, a negative value of N1 means that one is dealing with a shear thinning behavior. When dealing with STF, however, it has been found that the value of N1 can also be negative and with an absolute value equal to the applied stress [24]. Andrade et al. [25] reported in their work that the transition from N1 ≈ 0 to a negative value of the first normal stress difference in their colloidal suspensions with DST behavior appeared at rates lower than the onset of the shear thickening regime and was also related to a lower volume fraction, indicating contributions from lubrication forces, instead of the transition N1 ≈ 0 to positive (beyond critical shear rate) that suggested frictional interactions from contacts between particles. Pan et al. [26] also observed that increasing particle diameters in shear thickening granular suspensions led to the growing importance of hydrodynamic interactions, resulting in negative normal stresses. N2, however, generally remained very much neglected [27]; Laun [24] back in 1990s reported that N2 ≈  − N1/2 in opposition to most of the reported measurements and numerical simulations [28]. Thus, as the measurement of both normal stress differences may be limited by capillary stresses as discussed by Brown and Jaeger [7, 29] and shown by Garland et al. [30], the information regarding the normal stresses can alternatively obtained by means of extensional experiments.

Capillary rheometry has been typically used for determining the viscosity curves of polymer melts, providing reliable information at high shear rates, beyond the limits of rotational rheometers, and in similar conditions to the industrial process. The latter reasoning could be also applied to STF, as in many industrial processes, such as injection of pastes or ceramics inside molds, dense suspensions are extruded through a capillary and the shear thickening or jamming transition may happen in the process. The functioning principle of a capillary rheometer is very simple: the fluid sample flows at a controlled flow rate (Q) through a tube having a well-defined radius (r), and the pressure drop (∆p) between two points, within the fully developed region and separated by a certain distance (L), is measured. A non-slip condition is assumed to be held at the wall. As the shear rate at the wall (\( {\dot{\gamma}}_w \)) is directly proportional to the flow rate and the shear stress at the wall (τw) is also proportional to the pressure drop, the shear viscosity is just given by dividing one by the other \( \left(\eta ={\tau}_w/{\dot{\gamma}}_w\right) \). The basic principle of a capillary rheometer is illustrated in Fig. 2.2.

Fig. 2.2
A set of 2 schematic diagrams. Diagram a denotes the position of the piston, barrel, and die from top to bottom. Diagram b denotes V subscript z, sigma subscript r z, and sigma subscript r across the vertical section.

(a) Schematic diagram of a high-pressure capillary rheometer [31]. (b) Velocity and shear stress profiles [32]. Reprinted by permission from Elsevier

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This works well for Newtonian fluids; however, when dealing with shear thinning fluids, the Weissenberg-Rabinowitsch-Mooney (WRM) correction is required [23] because the shear rate at the wall is not linearly related to the flow rate. In principle, this correction also remains valid when working at the microscale with planar slit microfluidic devices [33]. However, the recent experimental results reported by Bossis and co-workers [34] discourage the use of capillary rheometry with STF, at least for those exhibiting DST. They realized that upon jamming, there is no longer a suspension flowing through a capillary, but instead a less concentrated suspension flows through a porous medium developed by the jammed particles; moreover, they also reported that the jamming does not occur inside the capillary but rather before its entrance, and finally, the fluctuations of pressure observed at constant shear rate are probably produced by the intermittent collapse of the jammed structure at the entrance of the die. However, they managed to get useful results [16] and, based on them, successfully modified the model proposed by Wyart and Cates [35]. Another interesting result from Bossis et al. [36] is that the jamming transition, which in the rotational rheometer occurs in a fraction of the inverse shear rate, exhibited in the extrusion a much slower dynamics than in the case of stress imposed on a rotational geometry. This latter observation supports the idea that characterizing STF under simple steady shear experiments in a rotational rheometer is by far insufficient to provide the required information to understand and predict the behavior of these fluids in practical applications.

Besides the information provided above, it is also important to note that capillary rheometry requires a larger amount of fluid sample than rotational rheometry, slip can often be a problem difficult to determine, and capillary rheometers are much less versatile than rotational ones since neither transient, oscillatory, nor normal stress measurements are possible [15]. Because of all the argued reasons, capillary rheometry does not seem to be the most convenient approach for steady shear flow measurements with STF.

2.2.2 Transient Shear Flows

As mentioned before, STF will find practical applications in which its time-dependent behavior is of paramount importance, irrespective of whether it will be used in anti-impact protective equipment or in anti-vibration systems. Moreover, it is known that STF exhibits a much different rheological response when undergoing transient impact experiments than under the steady shear state, as described in the previous section; for instance, DST suspensions support stress orders of magnitude larger under impact conditions than inferred from steady state rheometer measurements [37,38,39]. Again, these transient results support the idea that any generalized Newtonian fluid model [8, 40,41,42,43,44,45,46,47,48,49] will provide information with limited utility from the practical point of view [50]. Thus, a more complete rheological characterization including both the steady state and transient behaviors would be required to provide theoreticians with useful data sets to develop a constitutive model able to predict time-dependent behaviors such as kinetic models [51].

Transient shear experiments cannot be executed with a capillary rheometer, but by means of rotational rheometers, and, ideally, they should always be performed with a cone-plate geometry, since the shear rate is constant throughout the volume sample. The use of parallel plates is discouraged because the shear strain and stress vary in the radial direction, which is even more inconvenient in transient experiments than in steady shear measurements because it causes a variable shear history throughout the volume sample that complicates the interpretation of the results [15].

In this section, transient experiments are referred to as unsteady shear flows under continuous rotation of the geometry in the rotational rheometer. Oscillatory experiments will be considered as a different group of unsteady shear flow. The variety of transient experiments available in the literature is quite wide [1]; hereafter, the different experiments, the information that they can provide, the precautions that should be considered, and the usefulness for characterizing STF will be considered.

Startup experiments, when applied to STF, should be preferably performed in control-stress mode, for the same reason argued in the previous section. Before the beginning of the experiment (t < 0), the fluid sample is at rest; at t = 0 constant shear stress is imposed on the fluid: τ(t ≥ 0) = τ0. This experiment aims at looking at the time evolution of the transient viscosity (η+(t, τ0)), first normal stress coefficient \( \left({\varPsi}_1^{+}\left(t,{\tau}_0\right)\right) \), and second normal stress coefficient \( \left({\varPsi}_2^{+}\left(t,{\tau}_0\right)\right) \), which will also depend on the magnitude of the applied shear stress. This sort of experiment allowed Rathee et al. [52] to study localized stress fluctuations in dense suspension, for instance.

Tassieri and his co-workers [53] have developed a Fourier transform-based method (i-Rheo) to determine the viscoelastic moduli from raw startup experimental data in control-stress mode. The efficacy of i-Rheo was validated with colloidal suspensions, both in the fluid and in the glass states [54]; nevertheless, they did not validate it for STF. In this sense, i-Rheo might be an excellent tool for determining the viscoelastic moduli of STF at high frequencies, beyond the limits of rotational rheometers, which are related either to the instrument or to the fluid inertia [3].

From the practical point of view, it may be important to understand how the STF relaxes upon releasing the load [50], how long it will take to be ready for the next load, or even whether it comes to the same initial structure prior to the released load (anti-thixotropy or rheopexy) [55,56,57,58]. Stress relaxation experiments consist of observing how the shear stress relaxes with time when the flow is abruptly stopped, which allows to determine a relaxation time in shear; this kind of experiment was used by Cho et al. [59] to discover that STF exhibiting DST retains its memory of the level of frictional contacts prior to the flow cessation even after the relaxation ends, despite the presence of the inter-particle repulsions that enable DST.

Concatenated stepwise experiments have been typically used for characterizing thixotropy, but they have been also proven to be useful for characterizing anti-thixotropy or rheopexy (Fig. 2.3). Rubio-Hernández et al. [51] used this experimental protocol to verify that fumed silica suspensions with CST behavior exhibited thixotropy within the shear thinning region and anti-thixotropy in the shear thickening region [51].

Fig. 2.3
A set of 4 graphical representations is arranged in 2 columns. The graphs in the first column denote thixotropy, or shear thinning and shear thickening. The graphs in the second column denote anti-thixotropy, or shear thinning and shear thickening, respectively. It indicates the lines for transitory apparent viscosity, constant shear stress, and steady apparent viscosity.

Possible evolution in the apparent viscosity with time in a three-step test applied to shear thinning and shear thickening materials [51]. Reprinted by permission from AIP Publishing

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Hysteresis loops have also been traditionally used to determine thixotropic behavior, although the existence of hysteresis loops is not a synonym of thixotropy; as aforementioned, the existence of thixotropy can only be determined with concatenated stepwise experiments. The hysteresis loop experiment consists of imposing an increasing ramp in shear rate with time followed by a decreasing ramp with the same slope, and register the time evolution of the transient viscosity for each ramp, i.e., \( {\eta}_1^{+}\left(t,\kern0.5em {\ddot{\gamma}}_1\right) \) for the increasing ramp with slope \( {\ddot{\gamma}}_1 \) and \( {\eta}_2^{+}\left(t,{\ddot{\gamma}}_2\right) \) for the decreasing ramp with slope \( {\ddot{\gamma}}_2=-{\ddot{\gamma}}_1 \). Results are typically represented as η+ vs \( \dot{\gamma} \), and if the curves given by \( {\eta}_1^{+} \) and \( {\eta}_2^{+} \) do not overlap, it is said that there is a hysteresis loop; nevertheless, they can also be found in terms of stress versus shear rate (Fig. 2.4).

Fig. 2.4
A set of 2 graphs of shear stress versus shear rate. Graph a indicates the lines for C S S 10 s points and C S R. All the lines have an increasing trend. Graph b denotes the plots of thixotropic and rheopectic fluids.

(a) Up and down flow curves displaying hysteresis of shear thickening suspensions. Filled symbols for increasing stress and open symbols for decreasing stress sweeps [60]. Reprinted by permission from American Physical Society. (b) Schematic shear-stress rate behavior for time-dependent fluid behavior [61]. Reprinted by permission from Elsevier

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In STF, if the first and second ramps lie within the limit of the shear thinning region \( \left(\ \dot{\gamma}<{\dot{\gamma}}_c\right), \) then the presence of a hysteresis loop would mean that there is a difference between the kinematics of the destruction of the microstructure during the increasing ramp and the kinematics of construction of the microstructure during the decreasing ramp; if the first and second ramps go within the limits of the shear thickening region \( \left(\ {\dot{\gamma}}_c<\dot{\gamma}<{\dot{\gamma}}_{\textrm{max}}\right), \) then the kinematics of construction of the shear-induced microstructure during the increasing ramp is different from the kinematics of destruction of the microstructure dominated by repulsive forces during the decreasing ramp. If the experiment provides overlapping curves, then it would mean that both kinematics responsible for building and destroying microstructure are essentially equal. It is recommendable to test different slopes as in [62].

Flow reversal tests consist of imposing a shearing flow in one direction and instantaneously changing the direction of the flow while the magnitude of the shear rate remains the same [63]. This kind of test has been traditionally used to validate constitutive models for thixotropic fluids, which typically exhibit a two-step relaxation of shear stress caused by the presence of a viscoelastic relaxation and a kinematic hardening relaxation [58]. Nevertheless, there is no fundamental reason for not using the flow reversal tests in validating kinematic models for time-dependent behavior of STF; however, to the best of the author’s knowledge, the state of the art lacks this experimental data sets.

It can be found in the literature that unsteady shear flow tests are typically complemented with simultaneous confocal microscopy [64], boundary stress microscopy [65] X-ray [66] pressure measurements [67], or ultrasound imaging [68] in order to provide extra information regarding the presence or the evolution of microstructures, heterogeneities, vortices bands, etc., developed in the fluid sample along the experiment.

2.2.3 Oscillatory Shear Flows

Oscillatory shear flow experiments allow to decouple the viscoelastic response of a complex fluid into the elastic contribution and the viscous contribution; these viscous and elastic forces lead to an additional time scale and help to provide more insight into a controllable alternative shear history in comparison to steady shear experiments [69].

Typically, a sinusoidal shear stress or strain is imposed, and the resulting strain or stress is recorded; therefore, the material functions (G′ and G′′) are defined with the output signal. This way, when a sinusoidal strain signal is applied, the stress wave that is in phase with the applied strain wave divided by the amplitude of the strain wave is called the storage modulus (G′). On the other way, the amplitude of the stress wave that is out of phase with the strain wave divided by the strain amplitude is called loss modulus (G′′). If the sample is a Hookean solid, both signals are in phase (G′′ = 0); in the case of a Newtonian liquid, the signals are shifted by \( \frac{\pi }{2} \) (G′ = 0), and in the case of a viscoelastic liquid, the signals are shifted by a phase angle \( 0<\delta <\frac{\pi }{2}\ \left({G}^{\prime}\textrm{and}\ {G}^{{\prime\prime}}\ne 0\right) \) as illustrated in Fig. 2.5.

Fig. 2.5
A set of 3 graphs exhibits the trend of maximum stress and strain at different conditions. Graph a denotes elastic, where delta = 0 degree. Graph b denotes viscous, where delta = 90 degrees. Graph c denotes viscoelasticity, where the value of delta is greater than 0 and less than 90 degrees.

Stress and strain wave relationships for (a) a purely elastic (ideal solid), (b) a purely viscous (ideal liquid), and (c) a viscoelastic material [70]. Under the Creative Commons license

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The first step to analyze a sample by means of an oscillatory test is to perform an amplitude sweep. This test consists in applying a strain or stress amplitude sweep at a constant frequency and the typical results are represented in Fig. 2.6. These results allow one to determine the limit of the linear viscoelastic region (LVR), in which the viscoelastic moduli do not vary with the strain amplitude indicating that the structure is still preserved.

Fig. 2.6
A set of 4 line graphs of g prime slash g prime subscript 0, g double prime slash g double prime subscript 0, and gamma slash gamma subscript 0. The lines indicate the trend of g prime and g double prime. All the graphs, except graph b, denote a downward trend.

Types of LAOS behavior: (a) strain thinning, (b) strain hardening, (c) weak strain overshoot, (d) strong strain overshoot [71]. Reprinted by permission from Elsevier

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The frequency sweep test must then be developed by choosing a strain within the LVR. The frequency sweep test gives information on the time-dependent behavior of the sample in a non-destructive interval of strain/stress and consists in applying a frequency sweep at a constant strain from the LVR. However, the complex behavior of STF is not observed in this kind of experiment because of the small magnitude of the applied strain. Under certain conditions (high deformation, high volume fraction, high frequency, etc.), the signal response is no longer sinusoidal indicating the limit of the LVR and more harmonics appear that can give valuable information about the microstructure of the sample. Experimental tests performed in this region are called large oscillatory shear tests (LAOS) as opposed to their counterparts developed at low deformations, small amplitude oscillatory tests (SAOS). Several studies have been carried out during decades to understand the dynamic behavior of STF using oscillatory tests: e.g., Laun et al. [72], Boersma et al. [73], Raghavan and Khan [74], Yziquel et al. [75], Mewis and Biebaut [69], Lee and Wagner [76], Fischer et al. [77], Chang et al. [78], Khandavalli and Rothstein [79], Lee et al. [80], Rathee et al. [65].

Hyun et al. [71] wrote about the importance of LAOS experiments to characterize complex fluids such as STF. They basically classified the complex fluids into four types: type I, with G′ and G′′ decreasing (strain thinning); type II, with G′ and G′′ increasing (strain hardening); type III, with G′ decreasing and G′′ increasing followed by a decrease (weak strain overshoot); and type IV, with both G′ and G′′  increasing followed by a decrease (strong strain overshoot). Figure 2.6 is a schematic representation of these types of LAOS behaviors. Typically type IV is very representative of STF as is shown in Raghavan and Khan [74] and Galindo-Rosales et al. [18], for example.

Boersma et al. [73] studied the nonlinear viscoelastic behavior of dispersions of silica (SiO2) particles in a glycerol/water mixture and dispersions of glass particles in a glycerol/water mixture. They found that G′′ dominates over G′, indicating a liquid-like behavior because of the absence of flocculation. A nonlinear response was observed after a critical combination of frequency and deformation. When larger strain amplitudes and frequencies are applied to STF, the viscoelastic modulus increases abruptly (strain hardening). Raghavan and Khan [74] analyzed the rheological behavior under oscillatory tests of suspensions of fumed silica in polypropylene glycol. They reported strain hardening behavior and demonstrated that this behavior at high strain amplitudes is quite similar to shear thickening in steady flow experiments. This correlation can be represented using a modified Cox-Merz rule (proposed by Doraiswamy et al. [81]) in which the results of the complex viscosity as a function of dynamic shear rate (γ0 ∙ ω) can be superposed with the results of the viscosity in steady shear experiments. Also, three different fumed silica particles suspended in polypropylene glycol and in paraffin oil were used by Yziquel et al. [75] to analyze the linear and nonlinear rheological behavior. They observed solid-like behavior (G′ > G′′) for the three different silica particles at 8.2 wt% in paraffin oil. It was not possible to observe the linear regime as the loss modulus increased with the strain amplitude at low values of strain, possibly due to the breakdown of the structure. On the contrary, for the suspension in polypropylene glycol, the loss modulus was higher than the elastic modulus, indicating a liquid-like behavior. In the nonlinear regime, they observed excess dissipation energy (calculated from amplitudes sweeps) because of the breakdown of the 3D structure of the fumed silica suspensions. Mewis and Biebaut [69] observed also a strain hardening in oscillatory flow for monodispersed silica core particles with a grafter layer of poly(butyl methacrylate) (PBMA) in octanol. They found that the critical stress at which the strain hardening occurs is independent of the frequency and is also the same as the critical shear stress in steady flow measurement when the shear thickening appears. They also concluded that for their systems, the onset for the strain hardening is independent of the flow history, since in the oscillatory flow, the critical conditions are obtained periodically and lead to the same effect as reaching the critical conditions in continuous regime. Lee and Wagner [76] investigated the critical strain amplitude able to provoke shear thickening in oscillatory flow and the frequency dependence. They showed that neither the extent nor the modified Cox-Merz rules were able to correlate the dynamic and the steady state viscosities in the shear thickening regime or in the onset of the shear thickening. They proposed a superposition of the two viscosities at the point when the shear thickening regime can be observed by plotting the data versus the average stress magnitude. For low frequencies, the critical strain varies inversely with the frequency, but for high frequencies, an apparent plateau appears because of a large amount of slip. Fischer et al. [77] used in their studies concentrated suspensions of hydrophilic fumed silica in polypropylene glycol. They performed frequency sweep experiments for a wide range of strain amplitudes and found a power law correlation showing a decrease in the viscosity with frequency before and after the transition, being the effect more marked at frequencies after the transitions, leading to think that the shear thickening could disappear at sufficiently high frequencies. Galindo-Rosales et al. [18] used hydrophobic fumed silica in polypropylene glycol. Fig. 2.7 shows the results of the amplitude sweep tests: the loss modulus (G′′) is higher than the elastic modulus (G′), typical of a liquid-like behavior, but both moduli present relatively low values according to the non-flocculated nature of the suspensions. As it was reported previously, at a certain value of the stress amplitude, both moduli increase abruptly revealing the strain hardening behavior.

Fig. 2.7
A set of 2 line graphs represents the plots for g prime, g double prime versus shear stress amplitude. Each graph denotes 6 lines indicating different values of G prime and G double prime. Both graphs exhibit an increasing trend up to a point and a subsequent decrease.

Amplitude sweeps of Aerosil R816 suspensions in (a) PPG400, (b)PPG2000 [18]. Reprinted by permission from Springer

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A deep-inside analysis of the LAOS experiments was developed by Khandavalli and Rothstein [79]; they evaluated the rheological properties of three different suspensions—fumed silica in polyethylene oxide, fumed silica in polypropylene glycol, and cornstarch in water—and found a type III and type II LAOS behaviors for G′ and G′′ for the fumed silica in polyethylene oxide and in polypropylene glycol respectively (according to the classification of Hyun et al. [71]). For the cornstarch in water suspension, the results were G′ ≥ G′′, and as the strain amplitude increased, the moduli declined, but at large strain amplitudes, G′ and G′′ increased showing a strain-rate thickening behavior. The Lissajous-Bowditch curves were useful to analyze the viscoelastic nonlinearities which revealed strong differences between the samples. Rathee et al. [65] also analyzed the LAOS response of colloidal suspensions formulated with silica particles in a glycerol/water mixture in a regime of discontinuous shear thickening, obtaining results consistent with the ones of Khandavalli and Rothstein [79].

The analysis of the viscoelastic properties in the nonlinear regime by means of LAOS experiments is of great importance for the development of damping materials and their applications to energy absorption and vibration control. The analysis of the energy dissipation capacity of a damper is normally done through a hysteresis curve in which the enclosed area is the consumed energy of the structure in one cycle, similarly to the Lissajous-Bowditch curves obtained in LAOS experiments. Some work related to the development of optimized dampers aimed to develop dynamic models for dampers with STF (Zhao et al. [82], Zhang et al. [83], Lin et al. [84], Zhao et al. [85]). The hysteretic curves with different excitation frequency and amplitude developed by Zhao et al. [85] are shown in Fig. 2.8 and can be easily compared with the data from LAOS.

Fig. 2.8
A set of 3 graphs of z slash N versus x slash micron. It indicates the trend of 3 lines denoting the values of alpha and beta as, 1.00, 0.0. 0.25, negative 0.75. 0.15, negative 0.85.

Hysteric curves with different excitation frequency and amplitude: (a) f = 35 Hz, F = 2 N, (b) f = 50 Hz, F = 2 N, (c) f = 50 Hz, F = 5 N [85]. Reprinted by permission from Elsevier

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2.2.4 Superposition Rheology

Superposition in rheology means that during the experiment, both steady shear deformation and oscillatory motion are simultaneously applied to the sample. The directions of the velocity vectors of each motion can be either parallel, thus designating a parallel superposition, or at right angles, known as transverse or orthogonal superposition. The parallel shear stress superposition is represented in Fig. 2.9 and given by \( {\tau}_{yx}\left(\dot{\gamma},{\gamma}_0,t\right)=\eta \left(\dot{\gamma}\right)\bullet \dot{\gamma}+{G}_{\Big\Vert}^{\ast}\left(\omega, \dot{\gamma},{\gamma}_0\right)\bullet {\gamma}_0\bullet \sin \left(\omega \bullet t\right) \), where \( {G}_{\Big\Vert}^{\ast } \) is the parallel complex modulus, \( \dot{\gamma} \) is the steady shear rate, \( \eta \left(\dot{\gamma}\right) \) is the steady shear viscosity, and γ0 and ω are the amplitude and frequency of the superimposed oscillation, respectively [87, 88].

Fig. 2.9
A graph of applied shear stress versus time indicates the regions for pre-shear, rest, bottom shear stress, and tau subscript 0, which is an element of S A O S from left to right.

Parallel superposition (steady shear rate and SAOS) protocol [86]. Reprinted by permission from Elsevier

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Although data interpretation regarding superposition rheology is still under investigation since the viscoelastic superposition response on a microscopic level is rather complex, the technique is very powerful as it provides valuable information regarding the material’s viscoelasticity under nonlinear perturbation [89]. The effects of parallel superposition have been gradually investigated in shear thickening suspensions, such as in the work of Mewis and Biebaut [69] where suspensions of silica particles in poly(butyl methacrylate) (PBMA) were subjected to parallel superposition eliminating or reducing the contributions from the longest relaxation times during the experiments and evidencing shear thickening behavior before the steady state viscosities start to increase. Recently, Rubio-Hernández [86] tested a concentrated fumed silica suspension in polypropylene glycol (PPG) with parallel superposition rheology. They concluded that, for simple oscillatory measurements, the loss modulus dominated G′′ > G′, contrary to what happened with superimposed testing, where G′ > G′′, suggesting that the microstructure of the STF evolved from few and big hydroclusters on the onset of the shear thickening region to many and small ones when shearing increases.

2.3 Extensional Rheometry of STF

Even though most of the real flow conditions are complex—i.e., they consist of a combination of shear, rotation, and extensional flows, and it has become evident that just by shear rheometry, it is not possible to properly model the behavior of complex fluids—the truth is that extensional rheometry is underdeveloped when compared to shear rheometry. The reason is that imposing a surface-free uniaxial extensional flow is challenging, particularly for mobile liquids, such as suspensions or polymeric solutions. Only the Filament Stretching Extensional Rheometer (FiSER™) and the Capillary Breakup Extensional Rheometer (CaBER®) have given proof of merits for rheometric purposes. Both devices are based on the filament stretching approach, but whereas the FiSER™ imposes a constant extension rate, the CaBER® device imposes a step-strain deformation out of the equilibrium and the filament thinning process is performed under capillary forces and at an uncontrolled extension rate. In any case, they are both considered accurate methods for characterizing viscoelastic polymeric solutions, particularly when operated with simultaneous high-speed imaging [5]; nevertheless, in recent years, their use has been extended to other complex fluids such as emulsions [90, 91] or suspensions [25, 92,93,94,95,96,97]. The extensional properties of STF have also been characterized with success by means of the FiSER™ [25, 49, 98,99,100] and the CaBER® [25, 101, 102] devices in the recent years.

In both devices, the initial aspect ratio \( \left({\varLambda}_0=\frac{h_0}{d_0}\right) \) introduces an important shearcomponent in the flow at the early stages of the experiment. In order to minimize the shear effects, the FiSER™ device can impose two steps in the stretching protocol: in the first step, the liquid bridge is stretched at a small extension rate, well below the onset of extensional thickening and when the aspect ratio is large enough, ide-ally \( \frac{1}{{\varLambda_0}^4}\sim 0 \); and in the second step, the desired extension rate (\( \dot{\epsilon} \)) is imposed [25]. Further investigations about the influence of systematically introducing a controlled pre-deformation history can be found in [103]. The force sensors provide information about the time evolution of the pulling force exerted by the liquid on the plates, F(t), and then the tensile stress difference generated within the filament canbe calculated: \( {\tau}_{zz}-{\tau}_{rr}=\frac{4F(t)}{\pi {d}_{\textrm{min}}^2(t)}+\frac{1}{2}\frac{\rho g\pi {h}_0{d_0}^2}{4\pi {d}_{\textrm{min}}^2(t)}-\frac{2\sigma }{d_{\textrm{min}}(t)} \). Thus, the transient extensional viscosity is given by \( {\eta}_E^{+}(t)=\frac{\tau_{zz}-{\tau}_{rr}}{\dot{\epsilon}} \). Although CaBER® can provide interesting information about the extensional rheometry of STFs, it possesses three significant drawbacks: (i) it cannot perform the two-step stretching protocol and, therefore, the experimentalist must take that into account for the analysis of the results; (ii) it can only control the initial extension rate until reaching the final separation, upon when then the filament thinning process undergoes an uncontrolled extension rate, which has to be calculated from the time evolution of the minimum diameter as \( \dot{\epsilon}=\frac{-2}{d_{\textrm{min}}(t)}\frac{d\left({d}_{\textrm{min}}(t)\right)}{dt} \); (iii) CaBER® is not originally equipped with a force transducer, thus the extensional viscosity can only be estimated by means of surface tension and the time evolution of the filament radius \( \left({\eta}_E^{+}\approx \frac{-\sigma }{d\left({d}_{\textrm{min}}(t)\right)/ dt}\right) \).

Temperature control in the FiSER™ and CaBER® is not as good as in the rotational rheometers, where the bottom plate geometry is typically equipped with a Peltier temperature controller. Thus, if it is intended to work at a different temperature, it is recommended to have the sample in an external bath at the required temperature and quickly load the sample and run the experiments, which are very fast and temperature evolution could be neglected.

2.4 Field-Responsive STF

In magnetorheology [104] and electrorheology [105], an external magnetic and electric field is imposed, respectively, to the fluid sample while the rheological experiment is undergone. For this purpose, commercial rotational rheometers can be equipped with a magnetorheological cell or an electrorheological cell and, in both cases, the external field imposed is steady and perpendicular to the direction of the flow field. The presence of either the electric field or the magnetic field induces a microstructure of particles aligned in the direction of the field, which typically results in an increase of the yield stress. Very recently, M. Terkel and J. de Vicente have developed a new magnetorheological cell able to impose unsteady and triaxial magnetic fields [106, 107]. Consequently, exotic magnetic mesostructures can be generated, resulting in an enrichment of the magnetorheological response. Nevertheless, this degree of freedom in terms of control of the direction and unsteadiness of the electric field has not been reached yet for the electrorheological cells of the rotational rheometers.

Regarding the extensional rheometry, researchers from the Transport Phenomena Research Center [108] have recently developed a series of magnetorheological [92, 94] and electrorheological [94] cells able to impose steady and uniaxial external magnetic fields and electric fields, respectively, to the fluid sample when characterized under extensional flow in the CaBER® device. In the case of extensional magnetorheometry, two add-ons were designed allowing to choose the direction of the magnetic field either parallel or perpendicular to the extensional flow, whereas in the case of extensional electrorheometry, only one add-on was created and the electric field can only be imposed aligned with the direction of the flow. Table 2.2 summarizes the current state of the art in terms of possible configurations to perform magneto- and electrorheological measurements.

Table 2.2 Possible field flow configurations currently available in - and electrorheometry
Full size table

Despite the magnetorheological effect being traditionally linked to an increase in viscosity and yield stress under shear flow operating in the perpendicular configuration, it is possible to formulate magnetorheological STF (MRSTF) by dispersing carbonyl iron particles in a carrier fluid exhibiting shear thickening behavior [109,110,111,112,113], allowing to control the shear thickening behavior with the presence of the external magnetic field. Very recently, Bossis et al. [114] reported an outstanding magnetorheological effect based on discontinuous shear thickening. Their MRSTF was formulated by dispersing carbonyl iron particles in water using a superplasticizer molecule, which allowed to reach a volume fraction as high as 62% with a small yield stress and a still low plastic viscosity. The DST behavior is the consequence of particle-particle friction [115]. In their rheological characterization, they used parallel plates with serrated surfaces to avoid slip, and the results showed that by increasing the intensity of the magnetic field, the jamming transition occurred at smaller shear rates. Because the torque range in the rotational rheometer is too low for this kind of samples and because the radial stress developed during the jamming expels the particles from the suspension, they designed a viscometer able to hold ten times larger torques and based on a double helix geometry, which behaves approximately as a cylinder and avoids sedimentation, thanks to the induced small flow along the axis; the outer cylinder had vertical stripes to prevent wall slip.

In the electrorheological effect, the presence of an external electric field also results in an increase in the viscosity and yield stress under shear flow operating in the perpendicular configuration. Nevertheless, the application of an external electric field is also able to affect the shear thickening behavior of certain suspensions. That is the case of the E-FiRST (Electric Field Responsive Shear Thickening), which was first reported by Shenoy et al. [116]; instead of enlarging the shear thickening response of the fluid, in the E-FiRST effect, the presence of an external field allows to suppress the onset of shear thickening and, subsequently, the resistance to flow at high shear rates. In that study, non-serrated plates were used in a strain-controlled rheometer. Tian and co-workers [117, 118] reported a reversible shear thickening of electrorheological fluids above a low critical shear rate and a high critical electric field strength; their fluid sample consisted of a dispersion of NaY zeolite particles coated with glycerin in silicone oil, which was tested in a two concentric cylinder geometry by means of shear rate ramp tests.

2.5 Non-conventional Rheometry of STF

Apart from the latest works on the application of STF for polishing optical components and vibration control [119, 120], most of the vast research related to STF is dedicated to protective applications [6], such as the development of anti-impacts [121,122,123,124,125], anti-blast [126,127,128], bullet proof [129, 130], or stab/spike-resistant [131,132,133,134,135] materials.

In these impact protection applications, STF undergoes either low or high velocity impacts, which impose a flow to the liquid that is intrinsically transitory and, therefore, obviously far from the experimental conditions imposed in the rotational rheometer leading to the steady state viscosity/flow curves. Moreover, even if the rheologist wanted to measure the “instantaneous” flow curve in the rheometer, that curve would only be reliable if the time between points would be over 15 ms, because of the instrument and fluid’s inertias [14]. In low-velocity impact tests, the characteristic timescale for the composite providing the maximum force in its response is smaller than 5 ms, which will be much shorter in the case of bullet proof, stab/spike-resistant, or anti-blast applications. So, there is no doubt that new experimental approaches are required to provide the determination of the time-resolved viscosity curve of a liquid sample with a time resolution of the order of milliseconds at most.

Sliding plate rheometers [136,137,138] have proven to be an alternative to rotational rheometers for the characterization of shear thinning fluids under “large, rapid, transient shear deformations”, allowing the measurement of normal stresses [139] and reaching high frequencies when reducing the gap size down to the microscale [140,141,142]. This latter feature, characterizing STF at frequencies in the order of 10 kHz, is of paramount importance because, on the one hand, it enables a fine analysis of the interplay between local scale hydrodynamics and inter-particle forces [143], and that information can help in the development of new formulations of STF; on the other hand, it allows developing fluid models that can be used in practical dampers’ design for motion stages [144]. Sliding cylinder rheometers are similar to sliding plate rheometers, but they prevent edge effects and bearing friction issues. The principle of the sliding cylinder rheometer is also similar to the principle of the falling rod viscometer, which is considered a precise method for measuring the absolute viscosity of Newtonian fluids ranging from 10 to 107 Pa·s [145, 146]. In these three kinds of rheometers, when the relative gap between the plates or cylinders is very small, there is no need to know the constitutive equation of the fluid to calculate the shear strain and shear rate, as in the rotational rheometer. Although they have proven to be useful for highly viscous systems, such as polymer melts [147] and others [144, 145], to the best of the authors’ knowledge, it has not been reported in the literature their performance when dealing with STF.

Bikerman’s penetroviscometer [146] for determining the viscosity of Newtonian fluids under steady state conditions is able to provide viscosity measurements between 102 and 105 Pa·s with an error below 3.5%. As in the falling rod viscometer, the fluid sample is contained between two coaxial cylinders in which the inner one moves downward and the outer is fixed; however, unlike the falling rod rheometer, the area of contact does not remain constant in the penetroviscometer and the fluid is forced to flow upward through the annular gap between the two cylinders. This kind of flow is currently known as back extrusion flow or annular pumping flow, which is the basis of the back extrusion technique (BET) [148], which is mainly used in food engineering [149,150,151,152] to evaluate the rheological properties of the power-law or Herschel-Bulkley fluids in general, but it can also be used for determining the viscoelastic moduli of concentrated polymer solutions by imposing an axially oscillatory movement with a very small amplitude to the inner cylinder [153]. Based on the BET, Hoshino [154,155,156] has recently developed the Short Back Extrusion (SBE) method, which is able to provide good agreement with the steady viscosity curves measured with a rotational rheometer; however, SBE is not suited for high speed measurements. Fakhari and Galindo-Rosales [14] analyzed the possibility of measuring transient shear viscosity at large, rapid, and transient shear deformation by imposing a back extrusion flow; an analytical expression for the instantaneous viscosity relating the friction force at the inner cylinder wall to its velocity by means of a geometric factor was validated providing an accuracy of ~93% for Newtonian fluids and for the right set of parameters in the experiment within a timescale of the order of ∼2 ms; however, it has not been validated yet for STF, neither analytically, numerically, nor experimentally.

The split-Hopkinson pressure bar (SHPB) is typically used for the characterization of dynamic mechanical properties of materials and it has also been used for characterizing STF in transient squeeze flows at high compression rates, which are characteristics of an impact event [157,158,159,160]. It consists of a gas gun and three cylindrical bars, i.e., a striker bar, an incident bar, and a transmission bar; during the experiment, the incident wave, the reflection wave, and the transmission wave are recorded by means of strain gauges (Fig. 2.10). Lim et al. [162] discussed in detail the conditions under which classic SHPB data analysis is applicable for Newtonian fluid samples and they found a good agreement between the theory and the experiment obtained for thin specimens (∼1 mm) across a wide range of shear strain rates (over 105 s−1). Despite this technique providing useful information for the development of STF to be used under high-velocity impact applications, further work is still required for determining the material properties from the SHPB measurements, as it was done previously for Newtonian fluids [162]. For STF, this would require a robust constitutive model and/or independent measurements of the kinematics in the sample during deformation [157].

Fig. 2.10
A schematic diagram of the split-Hopkinson pressure bar indicates the position of the pressure gauge, gas gun, gas barrel, striker bar, incident bar, transmitted bar, strain gauges, oscilloscope, momentum trap, and P C. It also indicates the incident and transmitted signals going through the sample.

Schematics of the split-Hopkinson pressure bar (SHPB) [161]. Under the Creative Commons license

Full size image

Very recently, Madsen et al. [163] developed a non-intrusive technique able to provide instantaneous viscosity up to a timescale on the order of 20 μs. They developed a structured-light detection system that allows particle tracking over femtometer length scales and 16 ns timescales, which allows the determination of the instantaneous velocity of a trapped particle in an optical tweezer. In the ballistic regime, where the measurement is fast compared to the particle’s momentum relaxation time, the viscosity can be obtained more directly through its connection to kinetic energy dissipation. Neglecting hydrodynamic memory effects, collisions with molecules in the fluid exponentially damp the velocity of the particle over the momentum relaxation time. This momentum relaxation typically occurs at a much faster rate than the position relaxation, which allows for reducing the integration time required for precise viscosity measurements and, therefore, increases the speed of the measurement. The method was only validated for Newtonian fluids with viscosities ranging from 0.3 to 1.9 mPa∙s with temporal resolution between 20 μs and 1 ms; therefore, further research would be required to validate it for STF.

2.6 Future Perspectives

In the last decade, STF has proven to be very popular in protective applications [6]. Most of the published work aims the application of STF in fabric development for body protection technology. The range of STF application is quite extensive. There are several works in the literature regarding sports products (athletic rackets), medical products (surgical gloves or gowns), space technology, electronics and sensing, or even the petrochemical industry [164].

Rheology is a powerful tool that has been only exploited to its most in the characterization of STF when scientists tried to unveil the physics behind this fluid. The use of rheology has been typically limited to providing just their viscosity curves. It is essential to understand that STF will undergo transient and complex flows in real applications, and viscosity curves will fall short in predicting their response. Therefore, a complete rheological characterization can better predict how STF will perform in real flow conditions.

It becomes evident that to better predict the fluid-flow features of STF under real conditions allowing the development of new composites, it is of paramount importance to develop new and more realistic constitutive models for STF that goes beyond the current generalized Newtonian fluid models. These new constitutive models should be able to predict the viscoelastic nature based on the experimental information that a complete rheological characterization is able to provide. This is a challenge for the rheology community in the years to come.