Characterizing Vibrating Cantilevers for Liquid Viscosity and Density Sensing

Abstract

Miniaturized liquid sensors are essential devices in online process or condition monitoring. In case of viscosity and density sensing, microacoustic sensors such as quartz crystal resonators or SAW devices have proved particularly useful. However, these devices basically measure a thin-film viscosity, which is often not comparable to the macroscopic parameters probed by conventional viscometers. Miniaturized cantilever-based devices are interesting alternatives for such applications, but here the interaction between the liquid and the oscillating beam is more involved. In our contribution, we describe a measurement setup, which allows the investigation of this interaction for different beam cross-sections. We present an analytical model based on an approximation of the immersed cantilever as an oscillating sphere comprising the effective mass and the intrinsic damping of the cantilever and additional mass and damping due to the liquid loading. The model parameters are obtained from measurements with well-known sample liquids by a curve fitting procedure. Finally, we present the measurement of viscosity and density of an unknown sample liquid, demonstrating the feasibility of the model.

1. Introduction

For many applications like online process or condition monitoring, the liquid parameters viscosity and mass density are of high relevance. The use of conventional laboratory equipment is often not applicable due to its cost, space requirements, and other preconditions, for example, vibration-free mounting. Furthermore, sample taking for such devices often involves manual labor, tending to be time consuming and error-prone.

Microacoustic sensors like quartz thickness shear mode (TSM) resonators [1, 2] and surface acoustic wave (SAW) devices, for example, [3] have proved particularly useful alternatives to traditional viscometers [4]. However, these devices measure viscosity at comparatively high-shear rates and small vibration amplitudes. For non-Newtonian liquids, the results are, therefore, not directly comparable to those obtained from conventional viscometers. For complex liquids such as emulsions, it has also been shown that microacoustic devices may not be sufficient to detect rheological effects which are present only on the macroscopic scale [5].

Micromachined vibrating structures usually feature lower resonance frequencies and higher vibration amplitudes, making them more suitable for non-Newtonian and complex liquids [6]. Microcantilevers commonly used in atomic force microscopy [7–9] have been successfully used as liquid property sensors. They allow for simultaneous measurement of the liquid's viscosity and mass density, requiring sample volumes of less than 1 nL [10]. However, a highly sensitive optical readout is required to determine the beam's vibration amplitudes. When immersed in liquid, the cantilevers face strong deterioration of the quality factor due to high-dissipative effects [10]. Consequently, the vibration amplitudes drops even more, limiting the sensor's measurement range to low-viscous liquids. In other works, micromachined cantilevers and doubly clamped beams driven by Lorentz forces [6, 11–13] or by the piezoelectric effect [14, 15] have been utilized as liquid property sensors, and the feasibility of these sensors has been demonstrated for viscosities in the range up to several Pa·s.

In this contribution, we characterize resonating cantilevers for the measurement of mass density and viscosity of liquids. They measure viscosity in a rheological domain which is more comparable to that probed by conventional laboratory instruments. The cantilevers feature piezoelectric excitation as well as piezoelectric readout. The vibrating part is about 55 mm long, but since only the cantilever tip is immersed in the liquid, the induced damping of the cantilever vibration is kept low. The sensors, therefore, exhibit high-quality factors, ranging from 20 to 60 even for highly viscous liquids. Consequently, the detection of the resonances could be accomplished by a simple readout electronics, and the measurement range is greatly extended. On the other hand, the sensitivity of the sensor is decreased. Furthermore, the sensor principle allows attaching different tips of well-defined geometries to the cantilevers.

When the cantilever tip is immersed in a liquid, the resonance frequency and the damping of the cantilever are influenced by the viscosity and density of the liquid. However, the cantilevers do not show a simple relationship between the result of such a measurement and the liquid parameters. Recently, several models have been devised to give a proper description of the interaction of a vibrating cantilever and the surrounding liquid, for example, [16–18] but most of these models assume fully immersed cantilevers, which are long and thin, that is, the width 𝑤 is much smaller than the length 𝐿. For the designs considered in this work, only the cantilever tip is immersed in the liquid. For modeling the sensor-fluid interaction, the cantilever length 𝐿 must therefore be replaced by the dipping depth 𝑑, which is in the same range as 𝑤. A solution for cantilevers featuring 𝑤≈𝐿, and accordingly 𝑤≈𝑑 is given in [19]. In [14] and other works, the influence of the liquid loading on the cantilever's frequency characteristics has been successfully modeled by approximating the forces acting on the cantilever tip by those acting on a sphere oscillating in a fluid. The results indicate that the cantilever tip is subject to an additional mass loading and an additional damping caused by the surrounding liquid.

In our work, we present a vibrating cantilever sensor and a setup for the measurement of viscosity and density. The interaction of the sensor and the liquid in which the tip is immersed is modeled by an oscillating sphere. Possible simplifications of the model are discussed, and a general model is devised. This model allows for simple calibration of the sensor in a set of liquids with well-known properties. Therefore, the knowledge of mechanical and electrical properties of the cantilever is not required, the model parameters are instead obtained from the calibration procedure.

2. Sensor Fabrication

The cantilever sensors used in this work are based on commercially available lead zirconate titanate (PZT) bimorph bending actuators (Argillon GmbH, Redwitz, Germany). They feature a length of 49.95 mm, a width of 7.2 mm, and a total thickness of 0.8 mm [20]. The cantilevers consist of two piezoelectric PZT layers on both sides of a carbon fiber substrate (Figure 1(a)). The PZT layers are polarized in thickness direction. Electrodes (1, 2, 3) allow for excitation of the actuator. Applying a voltage between the top electrode (electrode 1) and the center electrode (2 in Figure 1(a)), which is used as ground electrode (Figure 1(b)) causes a contraction or elongation in the upper PZT layer but not in the substrate, and, therefore, deforms the cantilever. In our setup (Figure 2(a)) a sinusoidal voltage is applied, which leads to bending vibrations of the beam. A maximum voltage of 200 V can be applied to the cantilever. The actual beam deflection is determined by measuring the voltage at the sensing electrode.

(a)

(b)

Figure 1 

(a) PZT bending actuator with attached tip. The bending actuator consists of a carbon fiber substrate and two piezoelectric layers. The electrodes (1, 2, 3) allow for excitation and readout of the sensor. (b) Applying a voltage to the upper layer leads to contraction or elongation of the upper layer as indicated by the arrows.

Figure 2 

Measurement setup. The PZT bending actuator is rigidly clamped at one end. To the free end of the cantilever, a tip of well-defined geometry (width 𝑤) is attached and immersed in the sample liquid (dipping depth 𝑑). The interaction of the cantilever tip and the liquid changes the frequency characteristics, which is measured by the lock-in amplifier (LIA).

The bending actuator is clamped at one end, whereas different tips of well-defined cross-sections are attached to the free end. These tips are immersed in the sample liquid. The tip geometries and materials are given in Table 1. The clamping fixture is mounted on a rigid frame allowing for vertical (𝑥-direction) positioning of the sensor and preventing vibrations of the entire setup. A lock-in amplifier (LIA) measures the sensor voltage 𝑉_s, resulting in the cantilever's frequency response. As voltage source 𝑉_d, the internal oscillator of the lock-in amplifier is used.

Table 1 

Tip geometries (rectangular cross-section) and tip materials.

The cantilever exhibits several resonant vibration modes. As an example, Figure 3(a) shows the deflection (𝑧-direction) for a cantilever with tip A (Table 1) vibrating in air. The corresponding sensing electrode voltage is shown in the diagram below (Figure 3(b)). The resonance frequencies are 100 Hz and 851 Hz for the first and second modes, respectively. Further resonances were found at 2456 Hz (3rd mode) and 4870 Hz (4th mode). The mode shapes for a uniform cantilevered beam without a load at the tip are given in Figure 4 [21].

(a)

(b)

Figure 3 

Frequency characteristics of a PZT cantilever with tip A (Table 1), 𝑉d=400 mVrms driving voltage, vibrating in air. (a) The tip deflection (peak-peak) was measured using a Polytec OFV-5000/OFV-505 laser vibrometer. (b) The diagram below shows the corresponding sensing electrode rms voltage.

(a)

(b)

(c)

(d)

Figure 4 

Mode shapes of a uniform cantilevered beam. The mode shapes have been obtained from solutions of the Euler-Bernoulli beam equation and are also roughly valid for the piezoelectric bimorph.

3. Theoretical Model

The vibration behavior of the piezoelectric bending actuator is described by the Euler-Bernoulli beam equation [21]:𝜕𝐸𝐼⁴𝜓(𝑥,𝑡)𝜕𝑥⁴𝜕+𝜇²𝜓(𝑥,𝑡)𝜕𝑡²=𝜕²𝑀_p(𝑥,𝑡)𝜕𝑥², (1) where 𝐸𝐼 and 𝜇 are the effective bending stiffness and the effective mass per unit length of the composite beam, 𝜓(𝑥,𝑡) is the beam deflection in 𝑧-direction (Figure 2), and 𝑀_p is the actuating moment due to the piezoelectric effect. Since 𝑀_p is considered to be a constant moment along the entire beam, 𝜕²𝑀_p/𝜕𝑥²=0. Consequently, the boundary conditions for the clamped-free beam are𝜓(0,𝑡)=0,𝜕𝜓(0,𝑡)𝜕𝜕𝑥=0,₂𝜓(𝐿,𝑡)𝜕𝑥²=1𝑀𝐸𝐼^p,𝜕³𝜓(𝐿,𝑡)𝜕𝑥³1=−𝐸𝐼𝐹, (2)where 𝐿 is the length of the beam and 𝐹 is the force acting on the tip due to the interaction with the surrounding liquid. The actuating moment 𝑀_p is given by [22]𝑀_p=_𝐴p𝑌_p𝑑₃₁𝐸_z𝑧d𝐴=𝑤_b𝑌_p𝑑₃₁𝑉_d𝑧_m, (3)where 𝐴_p, 𝑌_p, and 𝑑₃₁ are the cross-sectional area, Young's modulus, and the piezoelectric modulus of the actuating layer and 𝐸_z is the electric field in this layer in 𝑧-direction. 𝑤_b is the width of the bending actuator, 𝑧_m is the mean distance of the actuating layer from the beam center, and 𝑉_d is the excitation voltage (Figure 2(a)).

The measured voltage 𝑉_s is calculated by applying𝐷_𝑧=𝑑₃₁𝑇_𝑥𝑥+𝜀𝐸_𝑧,𝑇_𝑥𝑥=1𝐾₁₁(𝑆_𝑥𝑥−𝑑₃₁𝐸_𝑧) (4)to the sensing PZT layer, where 𝐷_𝑧, 𝑇_𝑥𝑥, and 𝑆_𝑥𝑥 are electric displacement, stress, and strain. 𝜀 and 𝐾₁₁ are the permittivity and the compliance of the PZT layer. Since 𝑉_s is measured by means of a voltage amplifier in our setup, the current 𝐼_s from the sensing electrode vanishes, and, therefore, the charge:𝑄=_𝐴e𝐷_𝑧d𝐴=0, (5)where 𝐴_e is the surface area of the sensing electrode. From (4), (5), and 𝑆_𝑥𝑥=−𝑧𝜕²𝜓(𝑥,𝑡)/𝜕𝑥²,𝑉_𝑠(𝑡)=𝐾_v𝜕𝜓(𝐿,𝑡)𝜕𝑥 (6)is obtained, where 𝐾_v is a constant depending on material and geometry parameters [22]. The voltage at the sensing electrode is given by the slope of the beam deflection 𝜓(𝑥,𝑡) at the free end, 𝑥=𝐿, of the cantilever. This fact is greatly confirmed by the measurements given by Figure 3. Despite the lower tip deflection amplitude of the second mode the resulting output voltage is higher than for the first mode of vibration.

The interaction with the liquid surrounding, the cantilever tip can be modeled by approximating the vibrating cantilever as an oscillating sphere immersed in a liquid [14], exhibiting an effective sphere radius 𝑅 and an effective sphere mass. The force 𝐹 acting on such a sphere is given by [23]𝑅𝐹=6𝜋𝜂𝑅(1+𝛿)d𝑢d𝑡+3𝜋𝑅²2𝜂𝜌𝜔(1−2𝑅)d9𝛿²𝑢d𝑡², (7)where 𝑢 is the sphere displacement, 𝑅 is the sphere radius, 𝜔 is the angular oscillation frequency, and 𝛿 is the depth of penetration of the acoustic wave, which is given by𝛿=2𝜂𝜔𝜌. (8)The tips are attached to the free ends of the cantilevers, therefore, the sphere displacement is 𝑢=𝜓(𝐿,𝑡).

In principle, solving the equations given above yields the frequency characteristics of the vibrating cantilever with a tip immersed in liquid for all modes of vibration. As the cantilever represents a composite structure involving layers featuring different material properties, effective parameters have to be used in the beam equation, for example, for Young's modulus. As also the material parameters of the layers are not, or only partly, available in the required accuracy and since the solution of the beam equation would require the solution of a higher order system in order to determine the coefficients by a suitable expansion (e.g., eigenmode expansion), we have approximated the resonance behavior of the sensor in the vicinity of the first mode resonance frequency as a second-order system, given by(𝑀_e+𝑀_i)d²𝑢d𝑡²+(𝑏_e+𝑏_i)d𝑢d𝑡+𝐾𝑢=𝐹₀𝑒^j𝜔𝑡, (9)where 𝑢 is the deflection of the cantilever tip in 𝑧-direction (Figure 2), 𝑀_e and 𝑏_e are the effective mass and the intrinsic damping of the cantilever and the tip, 𝐾 is the spring constant, and 𝐹₀ and 𝜔 are the driving force amplitude and angular frequency. 𝑀_i and 𝑏_i are the induced mass and damping due to the liquid loading, given by (7)𝑀_i=3𝜋𝑅²2𝜂𝜌𝜔(1−2𝑅𝑏9𝛿),_i𝑅=6𝜋𝜂𝑅(1+𝛿). (10)(11)

For sample liquids exhibiting high viscosities, and low-vibration frequencies, 𝑅 is negligible compared to 𝛿, leading to a simplification of (10) and (11) [14]. For the considered cantilever tips and dipping depths (Table 1), the effective sphere radius 𝑅 is in the range of a few millimeters. The expected penetration depths 𝛿 for the sample liquids used in this work were calculated for a vibration frequency of 100 Hz and are given in Table 2. The results show that 𝛿 is in the same range as 𝑅, and the prerequisites for said simplification are not fulfilled.

Table 2 

Reference values (dynamic viscosity 𝜂 and mass density 𝜌) of the sample liquids used in the measurements and the expected depth of penetration 𝛿 at an angular frequency of 𝜔=2𝜋⋅100Hz, see (8). The dynamic viscosity was measured using a Brookfield LVDV+II-CP cone/plate rheometer at an ambient temperature of 23^°C.

Consequently, the consideration of the characteristic penetration depth 𝛿 leads to a frequency dependence of the liquid mass loading 𝑀_i, (cf. (10)), and the liquid damping 𝑏_i (cf. (11)). In the following, both 𝑀_i and 𝑏_i are considered constant in the vicinity of the resonance frequency, which is justified by the high-quality factors and the accompanied narrow bandwidths of the resonances. Therefore, solving the differential equation (9) using the Laplace transform yields1𝑈(𝑠)=1+(2𝐷_𝑛/𝜔_𝑛)𝑠+(1/𝜔²_𝑛)𝑠²𝐹(𝑠), (12)where 𝑈(𝑠) and 𝐹(𝑠) are the Laplace transforms of the tip deflection 𝑢 and the driving force:1𝜔²_𝑛=𝑀_e+𝑀_i𝐾,2𝐷_𝑛𝜔_𝑛=𝑏_e+𝑏_i𝐾, (13)where 𝜔_𝑛 is the resonance frequency and 𝐷_𝑛 the damping factor of the respective vibration mode. From (7) and (13), one obtains𝜔²_𝑛=𝜔²_𝑛,air11+(2𝜋𝑅³/3𝑀_e)𝜌+(6𝜋𝑅²/√2𝑀_e)(1/√𝜔_𝑛)√,𝜂𝜌2𝐷_𝑛𝜔_𝑛=2𝐷_𝑛,air𝜔_𝑛,air(1+6𝜋𝑅𝑏_e𝜂+6𝜋𝑅²√2𝑏_e√𝜔_𝑛√𝜂𝜌), (14)where 𝜔_𝑛,air and 𝐷_𝑛,air are the resonance frequency and damping factor of the cantilever without liquid loading. The oscillating sphere model shows that the cantilever's resonance frequency is affected by a term related to the liquid density, and a second term containing the viscosity-density product, whereas the time constant 𝑇_𝑛=2𝐷_𝑛/𝜔_𝑛 depends on the viscosity and the viscosity-density product.

For the interpretation of our measurement results involving rectangular cross-sections, we devised a more generalized model. Based on (14) and by introducing four independent coefficients 𝑐₁, 𝑐₂, 𝑐₃, and 𝑐_4, we have 𝜔²_𝑛=𝜔²_𝑛,air11+𝑐₁𝜌+𝑐₂(1/√𝜔_𝑛)√,𝑇𝜂𝜌_𝑛=𝑇_𝑛,air(1+𝑐₃𝜂+𝑐₄√𝜔_𝑛√𝜂𝜌), (15)(16)where 𝑇_𝑛,air is the in-air time constant 2𝐷_𝑛,air/𝜔_𝑛,air. The values of the parameters 𝑐_𝑖 are determined by the tip size and geometry, the effective cantilever mass and damping, the dipping depth 𝑑 (Figure 2(b)), and the respective mode of vibration 𝑛.

Figure 5 elucidates the model given by (15) and (16). We consider four liquids L_A, L_B, L_C, and L_D of which L_A and L_B are of the same density 𝜌₁, whereas L_C and L_D are of the same viscosity 𝜂₃. The expected values of 𝜔_𝑛 and 𝑇_𝑛 are given by markers in the figure. The curves in the figure represent the results expected for liquids of the same viscosity and density, respectively.

(a)

(b)

Figure 5 

Time constant 𝑇𝑛 and resonance frequency 𝜔𝑛 of a resonant cantilever dipping in 4 different sample liquids LA, LB, LC, and LD as given by the generalized models (15) and (16).

For the determination of viscosity and density of an unknown liquid, (15) and (16) have to be solved for 𝜂 and 𝜌. The system of equations can be written as𝐴=𝐵𝜌+𝐶√√𝜂𝜌,𝐷=𝐸𝜂𝜌+𝐹𝜂, (17) where 𝐴=𝜔²_𝑛,air/𝜔²_𝑛−1, 𝐵=𝑐₁, 𝐶=𝑐₂/√𝜔_𝑛, 𝐷=𝑇_𝑛/𝑇_𝑛,air−1, 𝐸=𝑐₄√𝜔_𝑛, and 𝐹=𝑐₃. Solving the system of equations yields 𝜌_1,21=−2𝐴𝐶𝐸−2𝐴𝐵𝐹−𝐶²𝐷±𝐵(𝐵𝐹−𝐶𝐸)14(𝐴𝐶𝐸−2𝐴𝐵𝐹−𝐶²𝐷)²𝐵²(𝐵𝐹−𝐶𝐸)²−𝐴²𝐹,𝜂𝐵(𝐵𝐹−𝐶𝐸)_1,21=−2𝐶𝐷𝐸−2𝐵𝐷𝐹−𝐴𝐸²±𝐹(𝐵𝐹−𝐶𝐸)14(𝐶𝐷𝐸−2𝐵𝐷𝐹−𝐴𝐸²)²𝐹²(𝐵𝐹−𝐶𝐸)²−𝐵𝐷².𝐹(𝐵𝐹−𝐶𝐸) (18)For the cantilevers used in this work, it turns out that 𝐵𝐹−𝐶𝐸=𝑐₁𝑐₃−𝑐₂𝑐₄<0. Furthermore, all the coefficients of (17) are positive, that is, 𝐴,𝐵,𝐶,𝐷,𝐸,𝐹>0. Therefore, in (18), only that 𝜌_1,2 and 𝜂_1,2, respectively, are positive which exhibit a plus before the root sign, that is,1𝜌=−2(𝑐₂𝑐₄−2𝑐₁𝑐₃𝜔)(^2𝑛,air𝜔²_𝑛𝑐−1)−²²𝜔_𝑛(𝑇_𝑛𝑇_𝑛,air−1)𝑐₁(𝑐₁𝑐₃−𝑐₂𝑐₄)1+{4[(𝑐₂𝑐₄−2𝑐₁𝑐₃𝜔)(^2𝑛,air𝜔²_𝑛𝑐−1)−²²𝜔_𝑛(𝑇_𝑛𝑇_𝑛,air−1)]²𝑐²₁(𝑐₁𝑐₃−𝑐₂𝑐₄)²−𝑐₃(𝜔²_𝑛,air𝜔²_𝑛−1)²𝑐₁(𝑐₁𝑐₃−𝑐₂𝑐₄)}^1/2,1𝜂=−2(𝑐₂𝑐₄−2𝑐₁𝑐₃𝑇)(^𝑛𝑇_𝑛,air−1)−𝑐²₄𝜔_𝑛(𝜔²_𝑛,air𝜔²_𝑛−1)𝑐₃(𝑐₁𝑐₃−𝑐₂𝑐₄)1+{4[(𝑐₂𝑐₄−2𝑐₁𝑐₃𝑇)(^𝑛𝑇_𝑛,air−1)−𝑐²₄𝜔_𝑛(𝜔²_𝑛,air𝜔²_𝑛−1)]²𝑐²₃(𝑐₁𝑐₃−𝑐₂𝑐₄)²−𝑐₁(𝑇_𝑛𝑇_𝑛,air−1)²𝑐₃(𝑐₁𝑐₃−𝑐₂𝑐₄)}^1/2. (19)

4. Measurements

With the setup depicted in Figure 2(a), the frequency response of the cantilevers is examined. The lock-in amplifier (Stanford Research SR830) is used to drive the cantilever and to measure the sensor voltage 𝑉_s and the phase shift 𝜙(j𝜔) between the driving voltage, that is, the driving force and the sensor voltage, that is, the actual cantilever deflection. The measurements are carried out with a sinusoidal driving voltage of 200mV_rms and within a frequency range from 70 to 110 Hz. At the resonance frequency of the first mode, a maximum tip deflection of 19𝜇m (peak-peak) was measured in air by means of a Polytec OFV-5000/OFV-505 laser vibrometer.

A variety of oils are used as test liquids: AK150, AK350 (Wacker Chemie), and SIL300 are silicone oils, and Alcatel 120 (A120) oil. They exhibit liquid densities in the range from 881 to 1078kg/m³ and viscosities from 145 to 440mPa⋅s. These nominal liquid parameters are obtained from data sheets and measurements by means of a Brookfield LVDV+II-CP cone/plate rheometer (Table 2). The parameters of the liquids vary with temperature. As especially the viscosity is highly temperature dependent, temperature control of the liquid container has been established by means of a peltier heater/cooler system. All results presented here have been obtained at 23^°C.

The dipping depth (Figure 2(b)) was adjusted by lowering the sensor until the tip touches the liquid surface, and then adding the desired 𝑑=2mm, using a micrometer screw. For liquids exhibiting low-surface tensions, like those considered in this work, a concave meniscus is formed at the liquid-tip interface. We expect the resulting liquid surface shape to increase the effective dipping depth, and, therefore, to influence the sensor's behavior. Consequently, the sensor principle would be limited to liquids with similar surface tensions, which can be assumed for the considered oils.

Figure 6 summarizes the measurement results at mode 1 for tip A in air and immersed in the sample liquids. As expected from (15) and (16), the liquid loading decreases the resonance frequency 𝜔₁ and increases the damping factor 𝐷₁.

(a)

(b)

Figure 6 

Magnitude and phase response of a cantilever with tip A vibrating in air and immersed in different liquids (Table 2). The dipping depth 𝑑 is 2mm, the driving voltage 𝑉d=200mVrms.

From the cantilever's frequency response, we extract the first mode resonance frequency 𝜔₁=2𝜋𝑓₁ and the damping factor 𝐷₁ by fitting a second-order transfer function:1𝜙(j𝜔)=arg{1+j𝜔(2𝐷₁/𝜔₁)−(𝜔²/𝜔²₁)} (20)to the phase shift measurements (Figure 6) with respect to the parameters 𝜔₁ and 𝐷₁.

Figure 7 shows the results for the cantilevers with tips A, B, C, and D (Table 1) and the sample liquids (Table 2). The left column of Figure 7 shows the measured time constant 𝑇₁=2𝐷₁/𝜔₁ versus the dynamic viscosity 𝜂 of the sample liquids. The results for each cantilever appear to be nearly lying on a single trend curve, indicating that 𝑇₁ is dominantly influenced by the liquid's dynamic viscosity. According to model equation (16), the value of 𝑐₄ thus must be small compared to 𝑐₃.

(a) Tip A

(b) Tip B

(c) Tip C

(d) Tip D

Figure 7 

Resonance frequency 𝜔1 and time constant 𝑇1 obtained for the four different tips and a variety of sample liquids. The crosses (+) in the diagrams represent the calculated values obtained from the parameter fit and show good agreement between the experimental results and the model.

Figure 7, right column, depicts the resonance frequency 𝑓₁ versus the liquid density. The results are widely spread in the 𝜌-𝑓₁ plane. Obviously, the resonance frequency of the cantilever tip immersed in the liquid is strongly influenced by both the liquid's density and viscosity. In our model, this relationship is described by (15).

Using the measurement results (Figure 7), the model parameters 𝑐₁, 𝑐₂, 𝑐₃, and 𝑐₄ can be determined. Figure 8 illustrates the model equations (15) and (16) and the parameter fit procedure. The values of 𝜔_1,air and 𝑇_1,air are obtained from a single measurement of the vibrating cantilever in air. Furthermore, each measurement in a sample liquid yields a pair of (𝜔₁,𝑇₁). The values of viscosity and density, 𝜂 and 𝜌, of each sample liquid are also known (Table 2). Finally, a fit algorithm applied to the model equations results in the model parameters 𝑐₁, 𝑐₂, 𝑐₃, and 𝑐₄ for a particular cantilever. The results of this parameter fit are depicted in Figure 9. The diagram shows the model parameters 𝑐₁, 𝑐₂, 𝑐₃, 𝑐₄ for the cantilevers with different tips. It is important to note that the parameters do not only depend on the width of the attached tip, but also on the mass and the intrinsic damping of the entire cantilever. Therefore, the parameter values do not necessarily decrease with the tip width, as shown by the location of the tip D marker in the 𝑐₃-𝑐₄ plane (Figure 9). The PZT bending actuator, with tip D has being attached, was not sealed by a protective coating, and thus features a lower intrinsic damping. However, the parameter fitting process described above yields parameter values considering such different behaviors of the cantilevers.

Figure 8 

Obtaining the model parameters 𝑐1, 𝑐2, 𝑐3, and 𝑐4 from measurements (Figure 7) in air and in reference liquids with known viscosities and densities (Table 2).

(a)

(b)

Figure 9 

Fitted parameter values 𝑐1, 𝑐2, 𝑐3, and 𝑐4 obtained by fitting the model equations to the measurement results.

For a validation of the model, the parameters determined above were used to calculate the values of 𝜔₁ and 𝑇₁ for each tip-liquid combination. These calculated values are indicated by crosses (+) in Figure 7 and are in good agreement with the experimental data.

At last, we use the model parameters obtained above to determine the viscosity and density of an olive oil sample. The measurement results, that is, the resonance frequencies 𝜔₁ and the time constants 𝑇₁ of the cantilevers immersed in the sample liquid are given in Table 3. From these results, the density and the viscosity of the liquid are calculated using (19), respectively.

Table 3 

Determination of viscosity and density of an olive oil sample by means of vibrating cantilevers. The density 𝜌 and the viscosity 𝜂 have been calculated from the measurement results 𝜔₁=2𝜋𝑓₁ and 𝑇₁ using (19). The density and viscosity obtained by means of weighting and a cone-plate rotational viscometer, respectively, are 𝜂_ref=66.4mPa⋅s and 𝜌_ref=913kgm⁻³.

5. Conclusion

The change of the dynamic behavior of a vibrating cantilever allows to investigate the physical properties of liquids. Various types of small tips of different geometries are attached to the cantilever and immersed into the sample solutions. The liquid surrounding the cantilever tip changes both the resonance frequency and the damping of the entire cantilever structure. To be able to conclude, from the measured frequency response to the liquid's parameters, an analytical model is needed. The developed model is based on the forces acting on an oscillating sphere in liquid, but generalized model parameters are used to consider the actual geometries of the applied cantilever tips. These parameters, furthermore, include the electrical and mechanical characteristics of the beam, which, therefore, must not be known. The model proved to be well suited for the characterization of various cantilevers and tip geometries by measuring in liquids with known density and viscosity. To extract the model parameters from the measured data, a curve fitting procedure was performed. The obtained parameters are specific for each cantilever tip and allow the subsequent simultaneous determination of density and viscosity of unknown liquids. Compared to other works, the sensor features high-quality factors of the considered resonance mode even for highly viscous liquids, greatly extending the measurement range.

Acknowledgments

The authors would like to thank the Institute for Measurement Technology of the Johannes Kepler University (JKU), Linz, Austria, for providing the laser vibrometer used, and Professor Hans Irschik and Dr. Manfred Nader of the Institute of Technical Mechanics (JKU Linz), and Roman Beigelbeck of the Research Unit for Integrated Sensor Systems, Austrian Academy of Sciences, for useful discussions on the beam theory. This work was supported by the Austrian Science Fund (FWF) Project L103-N07.