Bulk rheometry at high frequencies: a review of experimental approaches
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Abstract
High-frequency rheology is a form of mechanical spectroscopy which provides access to fast dynamics in soft materials and hence can give valuable information about the local scale microstructure. It is particularly useful for systems where time-temperature superposition cannot be used, when there is a need to extend the frequency range beyond what is possible with conventional rotational devices. This review gives an overview of different approaches to high-frequency bulk rheometry, i.e. mechanical rheometers that can operate at acoustic (20 Hz–20 kHz) or ultrasound (> 20 kHz) frequencies. As with all rheometers, precise control and know-how of the kinematic conditions are of prime importance. The inherent effects of shear wave propagation that occur in oscillatory measurements will hence be addressed first, identifying the gap and surface loading limits. Different high-frequency techniques are then classified based on their mode of operation. They are reviewed critically, contrasting ease of operation with the dynamic frequency range obtained. A comparative overview of the different types of techniques in terms of their operating window aims to provide a practical guide for selecting the right approach for a given problem. The review ends with a more forward looking discussion of selected material classes for which the use of high-frequency rheometry has proven particularly valuable or holds promise for bringing physical insights.
Keywords
High-frequency rheology Rheometry Viscoelasticity Colloidal dispersions Polymers Supramolecular systemsIntroduction
Small amplitude oscillatory measurements probe the response of an equilibrium structure of a material and deliver its rheological fingerprint in the frequency domain. Changing the frequency also infers a variation in the length scale that is probed. Low-frequency moduli depend on slow processes which are typically reflecting collective, large-scale phenomena, whereas measurements at high frequencies will reveal the more local-scale dynamics. In fact, in order to obtain an integrated picture of many complex fluids or soft solids, oscillatory measurements over an extended frequency range are typically required. The exact location of the “high-frequency region” on an absolute frequency scale depends on the characteristic time scales of the material under investigation (Ferry 1980; Willenbacher and Oelschlaeger 2007). For thermorheologically simple materials with relaxation times governed by the same physical mechanism, time-temperature superposition can be applied to cover ranges outside of the instrumentation limits and rotational devices suffice (Ferry 1980). However, for many complex materials, this principle does not hold and the high-frequency region has to be reached physically by means of different measurement instruments. We will therefore need special “high-frequency rheometers” to extend the frequency domain of classical rotational devices.
Characterise local stress contributions (Schroyen et al. 2019) or determine interaction strengths in colloidal suspensions (Bergenholtz et al. 1998b; Fritz et al. 2002b);
Investigate local dynamics in emulsions (Liu et al. 1996; Romoscanu et al. 2003a);
Obtain detailed insights in the dynamics of glasses (Hecksher et al. 2017), hydrogels (Jamburidze et al. 2017), and wormlike micellar solutions (Willenbacher et al. 2007);
Identify fast relaxation processes in polymers and the effects of monomer friction (Kirschenmann 2003; Szántó et al. 2017) or bond characteristics in associative polymer systems (Goldansaz et al. 2016; Zhang et al. 2018).
Conventional rotational rheometers typically operate at maximal frequencies below f = 10 − 50 Hz, the limit being set by inertial effects of both instrument and sample. These inertial effects scale with ∝ f^{2} and despite software corrections they will eventually dominate. The exact limit depends on the properties of the measurement system and sample: the torque generated by instrument inertia is ∝ I ⋅ f^{2}, with I the inertial moment of the instrument, and is particularly problematic when measuring from the oscillating surface, while sample inertia generates a torque \(\propto \left (\frac {\rho f}{\eta } \right )^{2}\), with ρ and η the fluid density and viscosity (Ewoldt et al. 2015). A vast array of techniques has been developed since the middle of the previous century to access higher frequencies. In this review, we focus mainly on techniques that directly probe bulk material properties using well-defined kinematic conditions. Other useful methods include for instance surface fluctuation reflection spectroscopy until the intermediate kHz range (Tay et al. 2008; Pottier et al. 2013), Brillouin spectroscopy in the GHz range (Mather et al. 2012; Still et al. 2013; Hecksher et al. 2017), shear wave speed measurements (Joseph et al. 1986), or microrheological methods that span a large frequency region until the large kHz domain. Different microrheological techniques, e.g. using differential dynamic microscopy (Cerbino and Trappe 2008; Edera et al. 2017), diffusive wave spectroscopy (Pine et al. 1988; Scheffold et al. 2004), passive particle tracking (Mason et al. 1997), or active microrheology (Keller et al. 2001; Wintzenrieth et al. 2014), have been developed and successfully applied. However, they are indirect and therefore require the extrapolation of bulk properties, which can be an issue for instance for heterogeneous systems. Moreover, the local flow field generated in these systems is often a disturbance velocity field around a probe particle, which is non-viscometric and dependent on the probe material shape and its interaction with the sample (Cai et al. 2011). For an extensive review on microrheology which addresses these issues and gives an excellent overview of both fundamental and experimental issues, the reader is redirected to the recent monograph by Furst and Squires (2017).
High-frequency (HF) rheometrical techniques span a range between 1 and 10^{8} Hz, starting from methods that directly extend the accessible frequencies of conventional rotational rheometers to setups operating in the MHz range, where fluid inertia and the limits of the continuum world become important. This review is structured as follows: wave propagation effects are discussed first as it plays a key role in instrument design considerations. Different techniques are then introduced and discussed based on their mode of operation, the complexity of the device, and the analysis method used. The review concludes with a few practical applications that illustrate the value of high-frequency rheometry for different complex material classes, which is hopefully motivating to select and use an adequate HF technique.
Shear wave propagation
Drag flow rheometers operating in oscillatory mode mostly probe material properties by means of a shear deformation generated by one oscillating surface. The sample’s response can be determined either from the force or torque on that moving surface or from a second, stationary surface. Oscillatory displacements in a bulk fluid lead to waves which propagate through the sample. For most classical rheometrical devices, a transverse wave is generated and assumed to be reflected immediately back. As a consequence, the magnitude of the velocity varies linearly over the gap, resulting in a constant shear rate. This is referred to as the gap loading limit in a landmark paper by Schrag (1977). However, this linear shear profile is not the general solution.
Gap or surface loading limitations
The wavelength and penetration depth are key design factors and will, for a certain material and frequency range, define the required gap for operation within either of the two limits. The penetration depth is of particular importance at very high frequencies since it defines the maximal distance over which the sample can be probed. Therefore, as a rule of thumb, it should be at least a factor 10 larger than the characteristic length scale of the microstructure.
Techniques
Comparison of dynamic ranges (min–max) in frequency, reported temperature ranges, complex viscosity/modulus, strain amplitude, and required sample volume for different techniques
f [Hz] | T [^{∘}C] | |η^{∗}| [Pa⋅s] | |G^{∗}| [Pa] | γ[%] | Vol [ml] | |
---|---|---|---|---|---|---|
(A.1) Piezo linear shear (monolithic; PSR-1) | 0.1 − 5 × 10^{3} | 25 − 150 | − | > 10^{2} | ≤ 0.01 | ≤ 0.1 |
(A.2) Piezo linear shear (stack; PSR-1) | 0.1 − 2 × 10^{3} | 10 − 50 | > 0.05 | 5 − 10^{9} | < 0.1 | ≤ 0.1 |
(A.3) Piezo rotation (PSR-1) | 0.5 − 2 × 10^{3} | − 100 − 300 | − | > 10^{2} | ≤ 0.1 | \(\sim 0.1\) |
(A.4) Piezo squeeze (PSR-2) | 0.5 − 7 × 10^{3} | 10 − 150 | 10^{− 3} − 10^{3} | 0.1 − 10^{6} | ≤ 0.1 | \(\sim 0.1\) |
(A.5) Electromagn. torsional | 10^{− 6} − 10^{4} | 2 − 60 (?) | − | 10^{4} − 10^{9} | < 0.01 | − |
(A.6) Electromagn. linear shear | 10 − 10^{4} | − | > 100 | − | < 20 | − |
(A.7) Thermal linear shear | 0.5 − 5 × 10^{2} | − | 1 − 100 | 50 − 10^{4} | − | ≤ 5 × 10^{− 6} |
(B.1) Piezo torsional | 10^{3} − 2.5 × 10^{5} | < 150 | 10^{− 3} − 1 | 10 − 10^{6} | ≤ 1 | ≥ 10 |
(B.2) Electromagn. torsional (f_{r}, Δf) | 10^{2} − 2 × 10^{4} | 20 − 25 (?) | 5 × 10^{− 4} − 0.06 | − | ≥ 1 | ≥ 15 |
(B.3) Electromagn. torsional (\(f_{r}, \omega _{\max \limits }\)) | 2 − 5 × 10^{2} | 20 − 25 (?) | 10^{− 3} − 2 | − | − | \(\sim 20\) |
(B.4) Torsional guided waves | 2 × 10^{4} − 3 × 10^{5} | 5 − 85 | 10^{− 3} − 10 | − | − | \(\sim 0.1\) |
(B.5) Planar shear | 6 × 10^{2} | − | 5 × 10^{− 3} − 0.5 | − | − | \(\sim 0.1\) |
(B.6) Wire vibrometer | \(\sim 10^{3}\) | 5 − 60 | 3 × 10^{− 3} − 0.1 | 5 − 10^{3} | − | \(\sim 60\) |
(C.1) Thickness shear | 10^{6} − 10^{8} | 10 − 60 | > 10^{− 4} | 10^{4} − 10^{9} (?) | ≤ 0.1 | ≤ 0.1 |
(C.2) Thickness shear (high temperature) | 5 × 10^{4} − 10^{8} | − 150 − 300 | > 10^{− 2} | 10^{6} − 10^{9} (?) | ≤ 0.1 | ≤ 0.1 |
(C.3) Reflectometry | 5 × 10^{6} − 5 × 10^{7} | 15 − 85 | − | 10^{5} − 10^{9} | − | ≤ 0.5 (?) |
(D) Cantilevers | 10^{3} − 10^{5} | 5 − 40 | 10^{− 3} − 2 | 10 − 10^{5} | − | ≤ 0.05 |
Subresonant drag flow rheometers
Subresonant rheometers working in the acoustic frequency range can be used to determine complex material properties at frequencies that directly extend the accessible frequency range of commercial rotational devices, starting from mid-range values of the latter (Fig. 5). The majority of such techniques makes use of piezotechnology, where a coupling is obtained between an electric field and mechanical response. Their measurement principle is similar to those of conventional rheometers, i.e. the material is confined in between 2 parallel surfaces with one being used to drag the material along. Gap loading operation can typically be assumed at least if the flow profile is homogeneous, which can be an issue for compressional instruments. Excitation of the plate is carried out by small piezoelectric elements instead of a heavy motor, which reduces the inertia of the measurement system drastically and allows for small strain amplitudes. The upper limit in frequency range is given by mechanical resonances of the system, which constitute an important design criterion. Below, different methods are discussed and compared based on their method of deformation and resulting flow field.
Piezo shear rheometers (PSR-1)
A different type of design was developed by Kirschenmann and Pechhold (2002) and is shown in Fig. 6c. The setup consists of a piezo-wheel with 6 spokes, used to translate the linear motion of the piezo elements into a rotation. Driving elements are attached to 3 spokes and create a rotational shear deformation, while elements on the other spokes are used for detection. The coupling of the excitation and detection reduces the sensitivity and complicates the analysis: fluid properties need to be determined by subtracting the response of the empty cell from the measured complex torque (Table 1 (A.3)). Furthermore, the complete measurement system had significant inertia, reducing the accuracy near or above only 1 kHz. Nonetheless, important advantages are the rotational symmetry and the possibility for decoupling the measurement cell from the piezoelements, hence facilitating the temperature control and allowing for a temperature range exceeding that of the piezoelectric material.
The resolution of these piezo shear rheometers (PSR-1) is largely determined by the accurate detection and analysis of the response signal. The most accurate results were obtained with linear shear rheometers using piezoelectric stacks and applying an extensive calibration step, with measurable complex viscosities as small as 0.05 Pa⋅s and a phase accuracy within 2^{∘} (Schroyen et al. 2017). A discussion on different sources of error can be found in Athanasiou et al. (2019). For coupled designs such as the piezo rotational rheometer (Kirschenmann and Pechhold 2002), the sensitivity decreases by a factor of 10. The upper limit is set by the mechanical compliance—or deformability—of the piezoelements or stacks and is typically high. A second important criterion is the maximum frequency limit, which depends on the successful suppression of inertial effects and is favoured by simple, robust designs. The accessible frequency range is typically limited below 5 kHz.
Piezo compressional rheometers (PSR-2)
Interest in integrated lab-on-a-chip devices has promoted the development of miniaturised setups as well. Sánchez et al. (2008) designed and empirically modelled a mini squeeze flow setup with decoupled excitation and detection elements for measurements on highly viscous samples with volumes < 10μ l. However, only the complex viscosity was reported and displayed considerable scatter. Alternatively, Cheneler et al. (2011) developed a microsqueeze flow rheometer for nanoliter volumes.
The sensitivity of PSR-2 is significantly higher compared with PSR-1 as a result of the compressional rather than shear motion (8) (Table 1 (A.4)). By playing with and optimising the gap height between 20 and 200 μ m, Crassous et al. (2005) succeeded in accurately measuring complex viscosities as low as 10^{− 3} Pa⋅s. The force amplification also reduces the upper limit for the measurable modulus, which is set by the measurement system’s compliance. Squeeze flow rheometers are easy to construct and frequencies until the low kHz region can readily be obtained, similar to linear shear flow devices. Furthermore, longitudinal piezoelements have a higher temperature resistance compared with shear stacks. The main disadvantage stems from the nature of the applied flow field: compressional flow inherently induces pressure gradients and can create non-homogeneous flow profiles and apparent, system-dependent properties (Hébraud et al. 2000). This can be worrisome in particular for more complex microstructures which are unknown a priori.
Other shear rheometers
Shear rheometers with different methods of excitation/detection have been developed as well, although their practical use has yet been limited. For instance, Chen and Lakes (1989) performed broadband measurements on viscoelastic solids by means of an electromagnetically driven torsional apparatus (Table 1 (A.5)). Effects of resonances were reduced by decoupling the detection of the displacement via interferometry and by numerically correcting for the resonance behaviour. Recently, Koganezawa et al. (2017) developed an electromagnetically driven planar shear setup for the characterisation of viscoelastic materials. The displacement is detected independently with a laser Doppler vibrometer. A first attempt was made at modelling the first resonance modes of the setup to extend the measurable frequency range ≥ 10 kHz, but needs refinement. Verbaan et al. (2015) suggested a more direct approach for extending the upper limit of subresonant techniques by introducing a high-viscous damper into the design. In addition, double-leaf springs were implemented to guide the sliding of a plate and prevent out-of-plane displacements. The damper effectively alters the behaviour around the resonance frequency, but adaptations to the setup are required to improve the sensitivity for fluids with lower viscosities (Table 1 (A.6)).
Recent years have also seen a growing interest for materials characterisation using only small sample volumes or in confined environments. Christopher et al. (2010) designed a MEMS-based nanopositioner stage for in-plane oscillations (Table 1 (A.7)). The sample is confined at gaps < 20μ m, which reduces the required sample volume but requires very careful alignment. Excitation of the oscillation is carried out by a thermal actuator, which is connected to the stage via parallel dual lever nanopositioners. The displacement is detected optically. While several MEMS-based viscosity probes have been developed recently, the setup developed by Christopher et al. (2010) seems to be the only subresonant shear probe able to measure both amplitude and phase (at least to some extent).
Bulk resonators and immersed rheometers
A second main category involves devices which operate at high acoustic or low ultrasonic frequencies (Fig. 5). Besides a few exceptions, setups in this range make use of resonating structures that undergo well-defined deformations. Their measurement principle is based on the effect of the viscoelastic fluid on a shift in the resonance frequency. The resonance can usually be detected precisely, but has a disadvantage that only discrete frequencies can be probed. Because of the reduced penetration depth and wavelength at kHz frequencies, operation of such devices will be mostly in the surface loading limit, so that the distance between the resonating structure and stationary wall or sample surface needs to be large (3). The penetration depth d_{s} is still sufficiently large so that bulk properties are being probed (4). Similar to the previously discussed subresonant rheometers, different designs have been considered (Table 1 (B)).
Torsional resonators
Torsional resonators consist of cylindrical elements that are immersed in the fluid and undergo a purely torsional deformation. The motion creates a propagating shear wave that is attenuated by the viscoelastic fluid. The deformation profile of the shear wave in the surface loading limit is well understood and rheological properties can readily be derived, but may be coupled with inertial effects. Inertia becomes even more important for resonators operating at high ultrasonic frequencies, such as the quartz crystal microbalance, which are described below (ultrasound rheometers). Resonance modes of torsional setups can either be driven piezoelectrically, magnetostrictively, or electromagnetically.
Electromagnetic or magnetostrictive torsional resonators oscillate in an electric or magnetic field. Specific examples include the multiple lumped resonator (Schrag and Johnson 1971) or the torsional pendulum (Blom and Mellema 1984). Contrary to piezo-driven resonators, detection of the oscillatory displacement is generally performed independently, i.e. either optically or electromagnetically via separate detection coils. Larger strain amplitudes are typically required to obtain sufficient signal-to-noise ratios (Table 1 (B.2)). Romoscanu et al. (2003c) succeeded in performing calibration-free measurements by measuring the phase shift digitally, via a single excitation/detection coil, and by accurately modeling the system’s response. In order to optimise the Q-factor as a function of the material properties, the resonating structure can be adapted, for instance by introducing coaxial segments (Nakken et al. 1995). More recently, Brack et al. (2018) made use of such coaxial resonators to implement a method for simultaneous measurements at multiple discrete frequencies. Alternatively, rather than using a fully immersed structure, Poulikakos et al. (2003) developed a torsional resonator loaded only at the free end for characterising soft solids (Valtorta et al. 2007). Torsional resonators were combined with rotational devices as well for simultaneous steady shear flow and high-frequency measurements (van den Ende et al. 1992; Dinser et al. 2008), which enable probing materials under non-equilibrium conditions.
Resonators that analyse the frequency shift f_{r} and bandwidth Δf typically have excellent resolution, making them suitable for low-viscous materials (> 10^{− 4} Pa⋅s). Although the design of the resonating structure can be optimised, the limit set by the Q-factor reduces the accuracy in case of excessive damping (Table 1 (B.1–B.2)). The measurable complex viscosity range is therefore limited < 1 Pa⋅s. The operation frequency depends on the natural frequency of the resonator and typically varies between 0.1 and 100 kHz; by combining different resonating structures multiple discrete frequencies can be sampled.
Wang et al. (2008) adapted an electromagnetically driven pendulum apparatus for measurements on more viscous materials. It was observed that analysing the maximum angular velocity \(\omega _{\max \limits }\) rather than the bandwidth Δf increases the accuracy for materials with |η^{∗}| > 0.2 Pa⋅s and can extend the accessible viscosity range > 1 Pa⋅s (Table 1 (B.3)). In a follow-up paper, Wang et al. (2010) derived an analysis of the entire resonance curve for operation in a continuous frequency window around f_{r} rather than at a single discrete value. Important drawbacks were the presence of parasitic in-plane displacements, a rather limited frequency range (\(f_{r} \sim 10^{2}\) Hz), and severe design limitations due to the gap loading assumption at such low frequencies (4). In spite of these drawbacks, the proposed method can be of interest for extending the measurement range of resonators.
Traveling torsional waves
In addition to resonator methods, McSkimin (1952) introduced a method for determining bulk viscoelastic properties in the kHz region by means of guided waves transmitted along an immersed cylindrical rod. The fluid impedance Z_{f} (12−13) could be derived semi-empirically from the measured attenuation and phase shift of the wave after reflection from the free end. Excitation and detection are carried out either using piezoelectric or magnetostrictive materials (McSkimin 1952; Nakajima and Wada 1970; Glover et al. 1968; Poddar et al. 1978). While most setups make use of immersed rods, hollow tube instruments with increased sensitivity and reduced sample volume were developed as well (Glover et al. 1968). These instruments however require a full analysis of the viscoelastic wave propagation since surface loading conditions cannot be assumed for the fluid inside the tube. Generally, reflected wave methods show an increased applicability towards more viscous fluids (\(\eta ^{*} \sim 10^{-3} - 10\) Pa⋅s) compared with torsional resonators with freely damped waves (Table 1 (B.4)). Operation outside of discrete resonance frequencies allows continuous operation and facilitates measurements at elevated temperatures. However, a major drawback was found to be the existence of non-linearities in the torsional motion, which causes systematic errors (Glover et al. 1968; Gaglione et al. 1993).
Planar resonators
In-plane oscillations of platelike structures create well-defined planar shear waves. Romoscanu et al. (2003b) developed an electromagnetically driven parallel plate resonator (Fig. 7b). Fluids are confined between the two plates with an adjustable gap thickness of order \(\sim 100 \mu \)m. At such gaps, wave propagation effects cannot be ignored (3), and wave propagation and fluid-structure interactions need to be accounted for. Viscoelastic properties are obtained from the frequency and phase shift of a damped reflected wave around discrete resonance frequencies, hence interpolating between different methods (Mason 1947; McSkimin 1952). The instrument performed well in terms of accessible viscosity range compared with other resonator methods (Table 1 (B.5)), although a correction factor was required for viscosities > 0.1 Pa⋅s. Since surface loading is not a requirement, the setup uses much smaller sample volumes. However, it could only be operated at a single frequency and below 1 kHz, which limits its advantages as a resonator technique.
Other deformations
Resonators with other types of deformation modes have been constructed as well. Determining viscoelastic properties from resonating structures with more complex deformation modes requires an often non-trivial determination of the fluid-structure interaction. One example is the vibrating wire rheometer, which makes use of an immersed wire in a magnetic field (Retsina et al. 1987; Hopkins and de Bruyn 2016). The exact resonance mode (\(\sim 1\) kHz, Table 1 (B.6)) can be altered via the amount of pretension on the wire and needs to be modelled for deriving material properties. Although the setup showed good accuracy as a viscometer, a critical evaluation of the measured phase is still missing. In a recent follow-up study, strain-sensitive optical sensors were implemented for enhancing the accuracy on the measured displacement (Malara et al. 2017).
Resonators that undergo more ill-defined deformation modes may generate non-viscometric flow fields. They have mainly been applied as viscometers and are therefore not included here; some examples are briefly discussed below (cantilevers and resonating viscometers).
Ultrasound rheometers
High-frequency ultrasound rheometers employ resonance modes in the MHz range (Fig. 5). At such high frequencies, the penetration depth d_{s} (4) reduces to the order of μ m or smaller, depending on the fluid properties and exact frequency range. Consequently, only the microstructure near the wall of the resonator is probed and care must be taken that bulk properties are still being measured. Required sample volumes for surface loading operation on the other hand are much smaller, typically < 0.1 ml.
Shear waves
Despite the experimental difficulties, thickness-shear mode rheometers are relevant to obtain rheological information of viscoelastic fluids and films in the MHz range. Similar to torsional resonators, they are sensitive towards low-viscous materials (\({|\eta ^{*}|}_{\min \limits } \sim 10^{-4}\) Pa⋅s) but require much less sample (Table 1 (C.1)). More advanced configurations were developed as well, such as a high temperature- and pressure-resistant setup (< 1 GPa, Table 1 (C.2)) (Theobald et al. 1994), or more recently a resonator with interdigitated electrodes for simultaneous measurements of mass, viscosity and conductivity (Muramatsu et al. 2016).
Reflectometry
Cantilevers and resonating viscometers
Immersed resonating structures can be used to determine bulk rheological properties at frequencies > 100 Hz (Table 1). However, they generally require a large sample volume to fulfil surface loading conditions. Recent years have seen an increased interest for fluid characterisation at small scales, e.g. to reduce sample volumes or perform in situ measurements. In combination with advances in fabrication methods, this has pushed the development of miniature fluid sensors.
Cantilevers
AFM microcantilevers have been used for various applications where high sensitivities are required, including force measurements or spectroscopy (Berger et al. 1997). Furthermore, experimental investigations have shown that cantilevers operated at their resonance frequency can be used for deriving fluid viscosities based on the oscillation displacement, which is used as a feedback parameter (Bergaud and Nicu 2000; Ahmed et al. 2001). Such derivations generally incorporate a theoretical analysis based on the work of Sader (1998) on the frequency response of an immersed beam cantilever, driven by a thermal driving force and undergoing flexural vibration modes. The size of the oscillating structure is typically much smaller than the wavelength of motion. As a result, hydrodynamic length scales were found to be very small (\(\sim 10 \mu \)m), reducing the required sample volume (≤ 1 nl) but limiting the structural length scale that can be studied. Smaller cantilever dimensions showed an enhanced sensitivity for the fluid viscosity (Boskovic et al. 2002).
Resonating viscometers
In addition to flexural cantilevers, other types of complex resonating structures have been developed for measuring fluid properties at elevated frequencies. For instance, Fig. 9b shows a MEMS metallic suspended plate resonator for viscosity sensing of complex low-viscous fluids (Reichel et al. 2010, 2011). Different membrane and immersed resonators, using either Lorentz force or piezo actuation, have been developed for viscosity and density measurements on low-viscous fluids (Reichel et al. 2009; Clara et al. 2016; Lu et al. 2017; Lucklum et al. 2011). Other examples include parallel plate and waveguide techniques for probing viscosities > 1 Pa⋅s (Abdallah et al. 2016; Kazys et al. 2013). Miniaturised probes in particular offer substantial potential for in situ measurements in confined environments (Ruiz-Díez et al. 2015), and novel fabrication methods have enabled the creation of elegant three-dimensional mesostructures (Ning et al. 2018). Nonetheless, a major drawback compared with more traditional resonance methods remains the often ill-defined flow field surrounding these devices. The fluid-structure interaction is hence non-trivial, especially for viscoelastic materials, and their applicability has been limited to detecting density and viscosity. A complete description of viscosity probes lies outside of the scope of this review.
Applications
Colloidal dispersions/gels
Early work on colloidal dispersions dates back half a century, e.g. Hellinckx and Mewis (1969) applied a guided wave method (McSkimin 1952) to characterise pigment dispersions at high processing shear rates. In general, the complex rheological behaviour of concentrated colloidal dispersions is set by the interplay between different interaction forces, either thermodynamic or hydrodynamic in origin, and the (shear rate dependent) microstructure (Mewis and Wagner 2012). These different contributions act on time and length scales which depend on the specific relaxation processes. Being able to vary the frequency above 10 Hz using bulk high-frequency rheometry is a powerful tool to directly probe local properties. Dispersions can be studied irrespective of the size, properties, or volume fraction of the dispersed phase. Figure 10a shows the applicability of different techniques to determine the high-frequency response of colloidal dispersions and gels. Subresonant piezorheometers offer the highest versatility since they allow to scan the frequency-dependent stress contributions in a continuous manner. When relaxation frequencies are > 1 kHz, or for studying low-viscous dispersions, resonator methods can be applied. Ultrasound rheometers are of lesser interest; the penetration depth will decrease to the order of the structural length scales of the material at higher ultrasonic frequencies (4).
The high-frequency response can be used to obtain quantitative information for various applications. Knowledge on the relaxation behaviour is important for the design, handling and understanding of complex materials. In case of a dilute Brownian suspension, the characteristic relaxation time can be predicted through the diffusional motion of a particle inside the medium. Relaxation frequencies typically decrease with volume fraction due to a reduced self-diffusion coefficient (Brady 1993; Bergenholtz et al. 1998a) and are influenced by colloidal interactions, particle shape, and structural effects, necessitating experimental characterisation (Bergenholtz et al. 1998b; Kirkwood and Auer 1951; Schroyen et al. 2017). At volume fractions above the glass transition, colloidal dispersions display multiple relaxation mechanisms: high-frequency measurements are highly useful to directly determine the fast ‘beta’ relaxation processes related to local in-cage rearrangements (Hecksher et al. 2017; Athanasiou et al. 2019). In emulsions, interfacial contributions resulting from surface tension give rise to additional relaxation times as well. High-frequency rheology has been employed to study the fast, local scale relaxation dynamics (Liu et al. 1996; Mason 1999; Romoscanu et al. 2003a, 2003c). In general, for soft systems, these techniques have a lot of potential for elucidating the interplay between the ‘softness’ of the dispersed phase and, for example, hydrodynamic effects.
Colloidal interactions alter the rheological response directly via the interaction potential and indirectly through the microstructure (Mewis and Wagner 2012). The latter can be applied to obtain direct, quantitative information on the microstructure of the dispersion. At frequencies that are very high compared with the characteristic relaxation frequency, the viscous response is dominated by bulk hydrodynamic stresses depending only on the effective volume fraction taken up by the fillers (Sierou and Brady 2001). The limiting hydrodynamic high-frequency response can therefore be used to quantify the local microstructure and dispersion state in attractive, partially dispersed suspensions (Bossis et al. 1991; Vermant et al. 2007; Schroyen et al. 2017). This is of large practical interest, e.g. for product evaluation or for scanning the efficiency of dispersion methods (Potanin 1993; Schroyen et al. 2017). Furthermore, the ability of separating microstructural contributions would be of large interest for studying non-equilibrium structures and thixotropy in attractive systems, but requires an integrated high-frequency and large-shear instrument.
Polymers and supramolecular materials
For polymeric and macromolecular solutions as well as melts, the high-frequency response can be used to investigate the microstructure on a local scale, such as the Rouse dynamics, monomeric friction effects, associative interactions, and transient forces. In case of dilute solutions, the fluid viscosity is generally low, rendering torsional resonator methods highly suitable for probing bulk properties (Fig. 10b). The Birnboim-Schrag multiple-lumped resonator (Schrag and Johnson 1971) was used extensively for determining the viscoelastic behaviour of various dilute polymer solutions, such as star or comb-polymers (Mitsuda et al. 1973a, 1973b), and was applied for instance for estimating small degrees of long-chain branching based on the measured relaxation times (Mitsuda et al. 1974). Furthermore, both torsional resonators and oscillatory squeeze flow instruments have proven useful to study the short-time dynamics and structure of wormlike micellar solutions (Buchanan et al. 2005; Willenbacher and Oelschlaeger 2007; Oelschlaeger et al. 2009) or polysaccharide solutions (Oelschlaeger et al. 2013). By combining different rheometrical techniques together with DWS microrheology, Willenbacher and Oelschlaeger (2007) provided an integrated picture of the viscoelastic response of a micellar solution, spanning a frequency range of 8 decades. The different techniques showed good agreement and proved that, through the high-frequency response, characteristic features of the wormlike micelles such as the persistence length can be derived accurately.
Although polymers often display a complex relaxation spectrum, as do macromolecular solutions, moduli are orders of magnitude higher and different measurement approaches are required (Fig. 10b). Subresonant piezo rheometers have been implemented for decades to study the properties of side-chain liquid-crystal polymers and/or composite materials (Gallani et al. 1994; Pozo et al. 2009; Roth et al. 2010). For instance, Zanna et al. (2002) investigated the influence of different molecular parameters on the rheological properties of cross-linked polysiloxane. Piezo rheometry was combined with time-temperature superposition to extend the accessible range even more, and the high-frequency region could be used to determine the onset of and scaling in the Rouse regime. Polymer gels display a power law regime at low frequencies governed by the overall network properties. Similar to colloidal gels, the high-frequency response can be used to investigate local scale properties and has been applied to assess the flexibility of the chains comprising the network based on scaling laws (Gittes et al. 1997). For homogeneous polymers with structural length scales that are sufficiently small (4), faster processes at or above the intermediate regime can be probed by performing measurements at even higher frequencies. Szántó et al. (2017) measured the entanglement relaxation time of polyethylene melts by means of quartz resonators operating in the MHz range, which was recently extended for copolymers of poly-1-butene and polyethylene (Liu et al. 2019).
In general, high-frequency rheometry is particularly useful for materials to which time-temperature superposition does not apply. It will therefore be very relevant for further work on the local dynamics of complex macromolecular systems. For instance, for semi-crystalline polymers, the rubbery plateau modulus can only be determined using high-frequency rheology (Struik 1987; Tajvidi et al. 2005; Szántó et al. 2017). In addition to obtaining this important information, the role of solvents as well as the challenging question of entanglement dilution can be addressed properly (Gold et al. 2019). In this regard, revisiting the linear viscoelasticity of block copolymers, where failure of time-temperature superposition was used by Rosedale and Bates (1990) to determine the disorder-to-order transition temperature, is of interest. Along the same lines, solutions of associating systems such as organogels or hydrogels are not amenable to time-temperature superposition, unless the solvent is athermal (Rubinstein and Colby 2003). This renders the utilisation of high-frequency rheometric methods necessary (Rodriguez Vilches et al. 2011; Collin et al. 2013). The obtained information on local dynamics may provide insights into the microscopic mechanisms governing the viscoelastic response of the investigated systems. Importantly, for associative systems, or even for concentrated solutions in complex solvents such as ionic liquids, it is unclear whether application of time-temperature superposition is appropriate even if a master curve can be obtained, since interpretation of the shift factors is complicated (Gold et al. 2017; Zhang et al. 2018). Independent high-frequency measurements can be very valuable in this regard.
Conclusion
The current review provided an overview of different high-frequency bulk rheometrical techniques that have been developed, spanning a frequency range from 0.1 to 10^{8} Hz. Table 1 presented a comparative summary of different device types. Although a multitude of techniques has been developed, not all are as practically useful. Wave propagation effects play an important role in high-frequency rheology and influence not only the experimental design parameters but also the structural length scales that can be probed. Subresonant piezorheometers can operate in a continuous manner until the low kHz range. Shear mode rheometers operate under a simple flow profile and are hence more accurate, but the resolution is limited and can be extended by using compressional instruments. Bulk resonators extend the frequency range further in a discrete manner. They are highly sensitive but excessive damping can be an issue. Guided wave methods on the other hand are less accurate but provide access to higher moduli. Ultimately, in the MHz range, both thickness shear mode resonators and reflectometers have been used. The penetration depth of the wave decreases < 10μ m and measured values do not necessarily represent bulk properties. When scanning for an applicable technique, different parameters have to be taken into account: the frequency range associated with the length/time scale at interest, the range in moduli, sample volume, and environmental parameters. In Fig. 10, practical instrument ranges were compared with the measurement ranges at interest for different materials. These ranges are a crucial design criterion.
Different techniques target specific material classes and frequencies, and identifying a suitable high-frequency method depends on the exact application that is addressed. For many complex materials such as colloidal dispersions, micellar solutions, or some polymer solutions and melts, access to low-to-intermediate kHz frequencies is highly relevant for probing interactions and specific features of the local scale microstructure. Examining the ultrasound MHz range can be useful for detecting very fast processes and relaxation times of homogeneous materials, i.e. polymer glasses, or to determine rheological properties of viscoelastic films. Furthermore, high-frequency rheology should be seen as complementary to conventional shear rheology. Combining high-frequency measurements with conventional rotational tests can offer an integrated view over the microstructure, ranging from collective properties at low frequencies to local scale properties at high frequencies.
Notes
Funding information
The authors acknowledge the European Union (EU) Horizon 2020-INFRAIA-2016-1, EUSMI grant agreement no 731019 and SIM Flanders (SBO-TRAP) for financial support.
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