Determination of the shear impedance of viscoelastic liquids using cylindrical piezoceramic resonators

In this paper, a new method for determining the rheological parameters of viscoelastic liquids is presented. To this end, we used the perturbation method applied to shear vibrations of cylindrical piezoceramic resonators. The resonator was viscoelastically loaded on the outer cylindrical surface. Due to this loading, the resonant frequency and quality factor of the resonator changed. According to the perturbation method, the change in the complex resonant frequency /spl Delta/~/spl omega/ = /spl Delta/w/sup re/ + j/spl Delta//spl omega//sup im/ is directly proportional to the specific acoustic impedance for cylindrical waves Zc of a viscoelastic liquid surrounding the resonator, i.e., /spl Delta/~w /spl sim/ jZ/sub c/, where j = (-1)/sup 1/2/. Hence, the measurement of the real and imaginary parts of the complex resonant frequency determines the real part, R/sub c/, and imaginary part, X/sub c/, of the complex acoustic impedance for cylindrical waves Z/sub c/ of an investigated liquid. Further-more, the specific impedance Z/sub L/ for plane waves was related to the specific impedance Z/sub c/ for cylindrical waves. Using theoretical formulas established and the results of the experiments performed, the shear storage modulus /spl mu/ and the viscosity /spl eta/ for various liquids (e.g., epoxy resins) were determined. Moreover, the authors derived for cylindrical resonators a formula that relates the shift in resonant frequency to the viscosity of the liquid. This formula is analogous to the Kanazawa-Gordon formula that was derived for planar resonators and Newtonian liquids.     

Compound piezoelectric cylindrical resonators as sensors of the rheological parameters of viscoelastic media

The electro-elastic behavior of a viscoelastically loaded layered cylindrical resonator (sensor) comprising two coupled hollow cylinders is presented. The inner cylinder is a piezoelectric ceramic tube. The outer cylinder is a non-piezoelectric (passive) metallic cylinder. An analytical formula for the electrical admittance of a compound layered cylindrical resonator loaded with a viscoelastic liquid is established. Admittance (conductance) diagrams were obtained using a continuum electromechanical model. The established analytical formulas enable the determination of the influence of the liquid viscosity, material, and geometrical parameters of a compound cylindrical resonator on the response characteristics of the compound sensor. In the paper, the sensor implications resulting from the performed analysis are described. Moreover, the algorithm of the method developed by the authors to evaluate the rheological parameters of a viscoelastic liquid is presented. Good agreement between the theoretical results and experimental data is shown. The analysis presented in this paper can be utilized for the design and construction of cylindrical piezoelectric viscosity sensors, annular accelerometers, filters, transducers, and multilayer resonators.     

Development of circuit for measuring both Q variation and resonant frequency shift of quartz crystal microbalance

Although frequency shift is often used for QCM (Quartz Crystal Microbalance) measurement, Q of a resonator also varies when viscous loading occurs. However, it is difficult to measure Q variation in real time in comparison with resonant frequency shift. Furthermore, oscillation frequency shift deviates from real resonant one in case of large viscous loading. Here, the authors have developed the circuit based upon motional admittance method for measuring both Q variation and real resonant frequency shift. It was applied to a quartz resonator gas sensor, and its Q variation was measured by this circuit. Moreover, it was compared with a conventional oscillation circuit and the frequency shift of the former was remarkably different from that of the latter in case of large Q variation. When a quartz resonator coated with PEG20M (PolyEthylene Glycol 20M) membrane was exposed to water vapor, its Q decreased from 38700 to 3650 and a frequency shift of the oscillation circuit was only about one sixth of this circuit.     

Analysis of piezoelectric bulk-acoustic-wave resonators as detectors in viscous conductive liquids

An analytical solution for the resonance condition of a piezoelectric quartz resonator with one surface in contact with a viscous conductive liquid is presented. The characteristic equation that describes the resonance condition and accounts for all interactions including acoustoelectric interactions with ions and dipoles in the solution is obtained in terms of the crystal and liquid parameters. A simple expression for the change in the resonance frequency is obtained. For viscous nonconductive solutions, the frequency change is reduced to a relationship in terms of the liquid density and viscosity. For dilute conductive liquid, the change in frequency is derived in terms of the solution conductivity and dielectric constant. The boundary conditions for the problem are defined with and without the electrical effects of electrodes. Experiments were conducted with various viscous and conductive chemical liquids using a fabricated miniature liquid flow cell containing an AT-cut quartz crystal resonator. The results, which show good agreement with the theory, on the use of quartz crystal resonators as conductivity and/or viscosity sensors are reported.     

Generalized model of resonant polymer-coated microcantilevers in viscous liquid media

Expressions describing the resonant frequency and quality factor of a dynamically driven, polymer-coated microcantilever in a viscous liquid medium have been obtained. These generalized formulas are used to describe the effects the operational medium and the viscoelastic coating have on the device sensitivity when used in liquid-phase chemical sensing applications. Shifts in the resonant frequency are normally assumed proportional to the mass of sorbed analyte in the sensing layer. However, the expression for the frequency shift derived in this work indicates that the frequency shift is also dependent on changes in the sensing layer's loss and storage moduli, changes in the moment of inertia, and changes in the medium of operation's viscosity and density. Not accounting for these factors will lead to incorrect analyte concentration predictions. The derived expressions are shown to reduce to well-known formulas found in the literature for the case of an uncoated cantilever in a viscous liquid medium and the case of a coated cantilever in air or in a vacuum. The theoretical results presented are then compared to available chemical sensor data in aqueous and viscous solutions.     

Responses of a Quartz Crystal Resonator Against Viscosity of Liquid up to 700 MPa

Conductance spectra of a quartz crystal resonator against sweep frequency were obtained in methanol and 1-pentanol up to 700 MPa using a piston-cylinder type cell. A resonance peak around a nominal frequency was observed on the conductance spectra. The frequency shift of the resonance peak increased linearly with pressure. The frequency shift against pressure was represented as the sum of the parts attributed to the viscosity of the liquid and that on the pressure effect of the resonator. The measurements for the resonator immersed in methanol and 1-propanol showed that the part attributed to the compression of the resonator on the frequency shift was similar for the two liquids. An empirical relation was developed on the electrical response of the resonance against the viscosity of the surrounding liquid up to 700 MPa based on these results. It may become possible to estimate the viscosity-density product of various liquids up to this pressure from the frequency shift.     

Resonator/Oscillator response to liquid loading

The resonant frequency of a thickness-shear mode resonator operated in contact with a fluid was measured with a network analyzer and with an oscillator circuit. The network analyzer measures changes in the device's intrinsic resonant frequency, which varies linearly with (ρη)(1/2), where ρ and η are liquid density and viscosity, respectively. The resonator/oscillator combination, however, responds differently to liquid loading than the resonator alone. By applying the operating constraints of the oscillator to an equivalent-circuit model for the liquid-loaded resonator, the response of the resonator/oscillator pair can be determined. By properly tuning the resonator/oscillator pair, the dynamic range of the response can be extended and made more linear, closely tracking the response of the resonator alone. This allows the system to measure higher viscosity and higher density liquids with greater accuracy.     

Effects of a liquid layer on thickness-shear vibrations of rectangular AT-cut quartz plates

Thickness-shear vibrations of rectangular AT-cut quartz with one face in contact with a layer of Newtonian (linearly viscous and compressible) fluid are studied. The two-dimensional (2D) governing equations for vibrations of piezoelectric crystal plates given previously are used in the present study. The solutions for 1D shear wave and compressional wave in a liquid layer are obtained, and the stresses at the bottom of the liquid layer are used as approximations to the stresses exerted on the crystal surface in the plate equations. Closed form solutions are obtained for both free and piezoelectrically forced thickness-shear vibrations of a finite, rectangular AT-cut quartz plate in contact with a liquid layer of finite thickness. From the present solutions, a simple and explicit formula is deduced for the resonance frequency of the fundamental thickness-shear mode, which includes the effects of both shear and compressional waves in the liquid layer and the effect of the thickness-to-length ratio of the crystal plate. The formula reduces to the widely used frequency equation obtained by many previous investigators for infinite plates. The resonance frequency of a rectangular AT-cut quartz, computed as a function of the thickness of the adjacent liquid layer, agrees closely with the experimental data measured by Schneider and Martin (Anal. Chem., vol. 67, pp. 3324-3335, 1995)     

Viscoelastic properties of low-viscosity liquids studied with thickness-shear mode resonators

The network analysis method was applied to AT cut quartz blanks (f(0) = 10 MHz), which were loaded with liquids of low and medium viscosity (water, methanol, ethanol, 1-propanol, 1-butanol, glycerol solutions). The shift of the resonance frequency Δf could be separated into a term due to rigidly coupled mass Δf(rig) and a term due to viscous damping Δf - Δf(rig). From the difference Δf - Δf(rig) and the broadening of the resonance curve, the complex shear modulus G = G' + iωη(L) was calculated. The viscosity coefficients η(L) are in good agreement with literature data. As G' > 0, it can be concluded that the examined fluids also reveal elasticity at shear frequencies in the MHz range. For the low-viscosity liquids, elastic contributions resulting from collective interactions of molecules are measurable but small and neglectable in most applications. The medium viscous liquid glycerol (98%) begins to exhibit considerable elasticity, resulting from the relaxation of separate molecules.     

Nonlinear compensation effect of dynamic viscosity of multicomponent hydrocarbon systems

A nonlinear compensation effect determining the dependence of the energy of a flow of a Newtonian liquid on the preexponent of its viscosity was detected. Empirical dependences relating the softening temperature of a Newtonian high-molecular multicomponent hydrocarbon system with the activation energy of its viscous flow have been obtained. The results obtained are supported by the statistical data and allow one to predict the dynamic viscosity of oil and coal fractions and materials based on them.     

Electroelastic effect of thickness mode langasite resonators

Langasite is a very promising material for resonators due to its good temperature behavior and high piezoelectric coupling, low acoustic loss, and high Q factor. The biasing effect for langasite resonators is crucial for resonator design. In this article, the resonant frequency shift of a thickness-mode langasite resonator is analyzed with respect to a direct current (DC) electric field applied in the thickness direction. The vibration modes of a thin langasite plate fully coated with an electrode are analyzed. The analysis is based on the theory for small fields superposed on a bias in electroelastic bodies and the first-order perturbation integral theory. The electroelastic effect of the resonator is analyzed by both analytical and finite-element methods. The complete set of nonlinear elastic, piezoelectric, dielectric permeability, and electrostrictive constants of langasite is used in the theoretical and numerical analysis. The sensitivity of electroelastic effect to nonlinear material constants is analyzed.     

Nonlinear response of ferromagnetic film resonator under conditions of nonlinear damping of magnetization oscillations

Nonlinear response of a tangentially magnetized ferromagnetic film resonator has been experimentally studied under the conditions where the nonlinear effects in the resonator are related to four-wave parametric processes. It is shown that an increase in the microwave power supplied to the resonator leads to a nonlinear shift of the eigenfrequencies and broadening of the resonant absorption peaks. The latter phenomenon is related to the nonlinear damping of magnetization in the resonator. It is established that the nonlinear shift of the frequency is thresholdless, while the nonlinear damping has a threshold, the appearance of which also leads to limitation of the nonlinear shift of the resonator frequency.     

Optically driven resonance of nanoscale flexural oscillators in liquid

We demonstrate the operation of radio frequency nanoscale flexural resonators in air and liquid. Doubly clamped string, as well as singly clamped cantilever resonators, with nanoscale cross-sectional dimensions and resonant frequencies as high as 145 MHz are driven in air as well as liquid with an amplitude modulated laser. We show that this laser drive technique can impart sufficient energy to a nanoscale resonator to overcome the strong viscous damping present in these media, resulting in a mechanical resonance that can be measured by optical interference techniques. Resonance in air, isopropyl alcohol, acetone, water, and phosphate-buffered saline is demonstrated for devices having cross-sectional dimensions close to 100 nm. For operation in air, quality factors as high as 400 at 145 MHz are demonstrated. In liquid, quality factors ranging from 3 to 10 and frequencies ranging from 20 to 100 MHz are observed. These devices, and an all-optical actuation and detection system, may provide insight into the physics of the interaction of nanoscale mechanical structures with their environments, greatly extending the viscosity range over which such small flexural resonant devices can be operated.     

A 0-phase circuit for QCM-based measurements in highly viscous liquid environments

Currently, the series resonant frequency f/sub s/ and the motional resistance Rm of liquid loaded quartz crystal microbalance (QCM) sensors are extracted either directly, through network analyzer (NWA) impedance measurements, or from QCM-stabilized oscillator circuits. Both methods have serious drawbacks that may affect measurement accuracy, especially if the sensor is operated under highly viscous load conditions and Rm exceeds 1 k/spl Omega/. This paper presents a simple passive low-loss impedance transformation LC network which greatly reduces additional electrical loading of the QCM by the measurement system or sensor electronics and maintains a symmetric resonance and a steep 0-phase crossing at f/sub s/, even if Rm increases by several orders of magnitude as a result of liquid loading. A simple S21 transmission measurement allows direct f/sub s/ reading at the 0-phase frequency, while Rm is obtained from the circuit loss at f/sub s/. Circuit operation was verified at 9 MHz by QCM measurements in a liquid with known density and viscosity. The agreement between predicted and experimental data, which was obtained by a temperature-controlled measurement, was within 1%, even in very high viscosity ranges in which Rm exceeds 10 k/spl Omega/.     

Acoustic Wave Flow Sensor Using Quartz Thickness Shear Mode Resonator

A quartz thickness shear mode (TSM) bulk acoustic wave resonator was used for in situ and real-time detection of liquid flow rate in this study. A special flow chamber made of 2 parallel acrylic plates was designed for flow measurement. The flow chamber has a rectangular flow channel, 2 flow reservoirs for stabilizing the fluid flow, a sensor mounting port for resonator holding, one inlet port, and one outlet port for pipe connection. A 5-MHz TSM quartz resonator was edge-bonded to the sensor mounting port with one side exposed to the flowing liquid and other side exposed to air. The electrical impedance spectra of the quartz resonator at different volumetric flow rate conditions were measured by an impedance analyzer for the extraction of the resonant frequency through a data-fitting method. The fundamental, 3rd, 5th, 7th, and 9th resonant frequency shifts were found to be around 920, 3572, 5947, 8228, and 10 300 Hz for flow rate variation from 0 to 3000 mL/min, which had a corresponding Reynolds number change from 0 to 822. The resonant frequency shifts of different modes are found to be quadratic with flow rate, which is attributed to the nonlinear effect of quartz resonator due to the effective normal pressure imposing on the resonator sensor by the flowing fluid. The results indicate that quartz TSM resonators can be used for flow sensors with characteristics of simplicity, fast response, and good repeatability.     

FE-investigation of shear localization in granular bodies under high shear rate

Shear localization in granular materials under high shear rate is analysed with the finite element method and a micro-polar hypoplastic constitutive model enhanced by viscous terms. We consider plane strain shearing of an infinitely long and narrow granular strip of initially dense sand between two very rough walls under conditions of free dilatancy. The constitutive model can reproduce the essential features of granular materials during shear localization. The calculations are performed under quasi-static and dynamic conditions with different shear rates. In dynamic regime, the viscosity terms are formulated based on a modified Newtonian fluid and according to the formula by Stadler and Buggisch (Proceedings of the conference on Reliable flow of particulate solids, EFCE Pub. Series, vol 49. Chr. Michelsen Institute, Bergen, 1985). Emphasis is given to the influence of inertial and viscous forces on the shear zone thickness and mobilized wall friction angle.     

EFFECTS OF LIQUID VISCOSITY ON RHEOLOGY OF CONCENTRATED SUSPENSIONS

This paper presents the experimental results on the effects of liquid viscosity on the rheology of concentrated suspensions of solid particles in Newtonian liquids. Specifically, the relative viscosity of a pseudoplastic suspension decreases as the viscosity of the suspending liquid increases, indicating excess energy dissipation in a less viscous liquid. In contrast, the relative viscosity of a Newtonian suspension is only slightly affected by the liquid viscosity. It is in excellent agreement with the value predicted from the rigid sphere model which neglects nonhydrodynamic interactions, and assumes zero particle-to-liquid relative velocity. The flow behavior indices of both concentrated suspensions are independent of the liquid viscosity.     

EFFECTS OF LIQUID VISCOSITY ON RHEOLOGY OF CONCENTRATED SUSPENSIONS

ABSTRACT

This paper presents the experimental results on the effects of liquid viscosity on the rheology of concentrated suspensions of solid particles in Newtonian liquids. Specifically, the relative viscosity of a pseudoplastic suspension decreases as the viscosity of the suspending liquid increases, indicating excess energy dissipation in a less viscous liquid. In contrast, the relative viscosity of a Newtonian suspension is only slightly affected by the liquid viscosity. It is in excellent agreement with the value predicted from the rigid sphere model which neglects nonhydrodynamic interactions, and assumes zero particle-to-liquid relative velocity. The flow behavior indices of both concentrated suspensions are independent of the liquid viscosity.     

Doubly rotated resonators for sensing the properties of liquids

When a doubly rotated resonator is operated in a liquid, the displacement of the surface is partly out of the plane of the plate of the resonator. The out-of-plane component of the displacement propagates a damped compressional wave into the liquid, and the in-plane component propagates a damped shear wave. In this paper, we report the measurements of the series resonant frequency and the motional arm resistance of doubly rotated quartz resonators (theta approximately 35 degrees and phi = 7 degrees) in liquids to compare with singly rotated AT-cut resonators (theta approximately 35 degrees and phi = 0 degrees). A modified Butterworth-Van Dyke (BVD) equivalent circuit model is suggested to analyze doubly rotated cut resonators under liquid loading.