# Compressional-Wave Effects in the Operation of a Quartz Crystal Microbalance in Liquids:Dependence on Overtone Order

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}is the frequency of the fundamental mode and Z

_{q,eff}is the shear-wave impedance of the resonator (or some effective parameter close to that, in case flexural motion is taken into account). Again, Equation (2) only holds for semi-infinite media (no reflections, no waves returning to the crystal). Furthermore, Equation (2) builds on the small-load approximation. As discussed in Section 6.1.3 in Reference [13], the small-load approximation does not necessarily apply to flexural motion.

- The amplitudes of the compressional waves are weaker than one might think. Even at the fundamental mode (where they are strongest), they amount to less than 5% of the transverse amplitude in air.
- On the overtones (meaning on the higher eigenmodes, which contain more than one nodal plane) the flexural contributions are weaker than on the fundamental mode.
- The presence of a liquid may decrease the flexural contributions.

^{2}with n the overtone order. Section 3 describes the LDV instruments and Section 4 reports the results from LDV. Section 5 describes a separate experiment undertaken with a QCM immersed in a liquid, which quantitatively assesses the influence of compressional waves. Section 6 concludes.

## 2. Flexural Contributions are Most Pronounced on the Fundamental Mode

_{T}(z) is given as

_{T}is a prefactor to this function. The index T stands for transverse motion. The origin of the z-scale is in the center of the plate. h is half the plate’s thickness. n is an odd integer. This form of the equation applies to the odd modes of vibration (n = 1,3,5,…). Only the odd modes can be excited piezo-electrically. The sign ensures that the displacement always occurs into the direction of positive x at the upper surface. There is no third dimension (the model describes a beam, not a plate.)

_{T}(x,z) on x for simplicity (proportional to U

_{T}(1 − x/L)). Furthermore, we only consider one half of the bar (to the right in Figure 1B). We write

_{T}now is to be understood as the maximum transverse displacement amplitude. L is half the length of the bar. The factor U

_{T}(1 − x/L) produces a lateral gradient in the amplitude of shear and, in consequence, an extensional strain ε

_{xx}, and a corresponding stress σ

_{xx}. Given that the stress has opposite sign at the bottom and the top, there is a bending moment (Figure 1C), given as

_{c}. The extensional strain along x is

_{c}) covers curvature. The elastic energy contained in the extensional strain, W

_{el}

_{,ext}, is

_{el}

_{,ext}, with respect to ${R}_{c}^{-1}$ must vanish:

^{2}/R

_{c}) scales as the curvature and therefore scales as 1/n

^{2}(Figure 1D). The 1/n

^{2}-scaling is the consequence of the bending moment being proportional to 1/n

^{2}. It would also have been found had the calculation been carried out dynamically (searching the modes of vibration).

_{N}~L

^{2}/R

_{c}and insert R

_{c}from Equation (10)). Experiment, on the contrary, shows that the amplitude of normal motion amounts to a few percent of the transverse motion. The shortcomings of this model are the following:

- The quasi-static calculation misses inertial forces, which will reduce the amplitude of normal motion.
- The model treats a bar rather than a cylindrical plate.
- The characteristic lateral scale L decreases with overtone order because energy trapping becomes more effective [35].

## 3. Laser-Doppler Vibrometry

_{1}(see Appendix B). R

_{1}was determined with a separate setup. Because the wiring in this other setup was not strictly the same as the wiring in the LDV setup, these amplitudes are reported with single digits in Table 1. One needs to keep in mind that these estimated transverse amplitudes pertain to “equivalent parallel plates” (no energy trapping). Still, the analysis is perfectly valid for comparison between overtones and between air or water.

_{N}. The parameter U

_{N}served for comparison between overtones and between different experimental configurations.

## 4. Results from Laser-Doppler Vibrometry

#### 4.1. LDV Under Normal Incidence

_{N}.

_{T}(x,y) move inwards.

- The U
_{N}/U_{T}-ratio is smaller at 15 MHz (n = 3) than at 5 MHz (n = 1). However, the 1/n^{2}-scaling predicted by the model from Section 2 is not quantitatively confirmed. The U_{N}/U_{T}-ratio decreases between n = 1 and n = 3, but it does not decrease by a factor of 9. Comparing n = 3 and n = 5 (15 MHz and 25 MHz), the U_{N}/U_{T}-ratio does not even decrease. - The U
_{N}/U_{T}-ratio is larger for the larger crystal (with a diameter of 25.4 mm). While this is not expected, in principle (in-plane gradients decrease when the diameter of the plate increases), one needs to keep in mind that the thickness and the shape of the back electrode also plays a role in energy trapping. - The U
_{N}/U_{T}-ratio decreases when the resonator is immersed in water. This is to be expected based on the argument sketched in Figure 5. Bending reduces the extensional strain inside the crystal, but it also causes a pressure in the adjacent liquid. Given that water is nearly incompressible (compared to air), the pressure is substantial and reduces the bending. This argument invalidates the small-load approximation (Section 6.1.3 in Reference [13]). The small-load approximation implicitly claims the modes shape to be unaffected by the load. A side remark: The U_{N}/U_{T}-ratio might actually change sign when immersing the resonator into the liquid. The pressure exerted by the liquid might outweigh the consequences of the extensional stress inside the plate. The sign is not inverted here. This result contradicts reference [40] (the reasons being unclear).

#### 4.2. LDV Under Oblique Incidence

## 5. Cuvette Resonances

^{2}-scaling (Section 2) than the U

_{N}/U

_{T}-ratios from Table 1. Similar to Table 1, the radii of the circles in the polar diagram increase when going from 15 MHz to 25 MHz. Note that there is no straightforward quantitative relation between the U

_{N}/U

_{T}-ratio and the magnitude of the coupled resonance. They are correlated, but the details involve integrations over the area of the plate. We do not elaborate on these.

_{N}/U

_{T}-ratio and the radii of the circles in Figure 7C decrease when going from the fundamental mode to the third overtone. That decrease does not continue when going to the 5th overtone at 25 MHz, though. This finding is in contrast with results from measurements in the dry, where effects of flexural contributions can also be found. We elaborate on these experiments in Appendix D.

_{shear}= (i ω ρ η)

^{1/2}[42] and the compressional wave-impedance are additive in their effect on frequency and bandwidth. The parameter β accounts for the small amplitude of normal motion. While β depends on U

_{N}/U

_{T}, in principle, it was left as a fit parameter, here. The parameter φ accounts for a relative phase between bending and shear. As experiments shows, this phase is nonzero. Z

_{CW,bulk}= (P ρ)

^{1/2}(with ρ the density and P the P-wave modulus) is the compressional-wave impedance of the semi-infinite medium. The denominator in the second line accounts for the multiple reflections. R

_{1}and r

_{2}are the complex reflectivities at the two surfaces of the cavity. (Only |r

_{1}r

_{2}| is a fit parameter, because the phase α is not linearly independent from the phase −2π i t

_{off}/T). T is the time between to maxima of the cuvette resonances, t

_{off}is some offset in time. The fit parameters are shown in Figure 9B.

**Note added in proof:**One might think that compressional waves can be avoided by abandoning energy trapping. To this end, we tested a crystal with a front electrode covering the entire area and with the back electrode removed. One may drive a resonator without electrodes by exciting the vibration across an air gap. In the absence of energy trapping, the O-rings holding the resonator at the rim damp the resonance. That certainly is a disadvantage, but it can be dealt with. Subjecting this resonator to the experiment described in Section 5 (decreasing water levels), we found cuvette resonances similar to what is shown in Figure 7. The problem was not solved. Evidently, even this resonator did not vibrate in a pure thickness-shear mode. The reasons for this are unclear.

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Treatment of Compressional-Wave Effects in the Frame of the Small-Load Approximation

_{L}, is a ratio of stress to velocity. In its general form, the load impedance is a 3rd-rank tensor defined by

_{eff}is an effective mass. M

_{eff}is equal to half the plate’s mass for the thickness-shear motion of a parallel plate, where the factor of 1/2 originates from the nodal planes inside the resonator.

**r**

_{S}is the location on the surface. v denotes a complex amplitude of a velocity (either tangentially, index x, or normal to the surface, index z). All velocities have been normalized to a reference velocity, v

_{ref}. The reference velocity is defined, such that the integral of v

^{2}/v

_{ref}

^{2}equals unity (meaning that v

^{2}/v

_{ref}

^{2}is a statistical weight).

_{L,xzx}is the shear-wave impedance. Z

_{L,zzz}is the compressional-wave impedance. β is some numerical factor. Its value is an outcome of the integrations from Equation (A2). For the geometry under consideration, the transverse impedance, Z

_{L,xzx}, is given [42] as

^{3}, and η = 10

^{−3}Pa s, one finds the absolute value of Z

_{L,xzx}to be 5.6 × 10

^{3}kg/(m

^{2}s). If the liquid were truly semi-infinite even with regard to compressional waves (it is not), one would insert the bulk-compressional-wave impedance for Z

_{L,zzz}and write

_{CW}is the speed of sound. With c

_{CW}≈ 1000 m/s and ρ = 1000 kg/m

^{3}, one finds Z

_{CW,bulk}≈ 10

^{6}kg/(m

^{2}s).

_{L,zzz}is computed similarly to how optical fields are computed for planar optical cavities (also: “Fabry-Perot filters”, see [44]). The total impedance at the resonator surface is a geometric series of the form

_{1}and r

_{2}are the reflectivities at the two surfaces of the cavity, k

_{C}is the wavenumber, d

_{liq}is the thickness of the liquid cell, and 2 k

_{C}d

_{liq}is the phase accumulated per round trip (not taking reflections into account).

_{off}is some offset in time. In the second line, r

_{1}r

_{2}was as expressed as |r

_{1}r

_{2}|exp(i α) with the phase α. The phase cannot be determined from the fit, because it is not linearly independent from the offset in time, t

_{off}. Equation (A7) inserted into Equation (A2) leads to the fit function (Equation (11)) used to produce the solid lines in Figure 9A.

## Appendix B. Estimation of the Transverse Amplitude from the Electrical Power

_{avg}= 10

^{15/10}mW = 31.6 mW. The amplitude I

_{0}of the current follows from the relations

_{1}is the motional resistance as determined from impedance analysis. The amplitude of the velocity at the resonator surface, v

_{T}, is given as [16]

_{q}= 330 µm is the thickness of the plate. e

_{26}= 9.65 × 10

^{−2}C/m

^{2}is the piezoelectric stress coefficient. The parameter A

_{eff}is the effective area, calculated from Equation 7.4.7 in [13] as

_{26}= 3.1 × 10

^{−12}m/V is the piezoelectric strain coefficient. Q is the quality factor. From the amplitude of the velocity, v

_{T}, the amplitude of the displacement, U

_{T}, follows as

**Table A1.**Parameters (determined by impedance analysis undertaken with the network analyzer E5100 from Agilent) entering the estimation of the transverse amplitudes and resulting transverse amplitudes in Table 1.

14 mm Diameter (Dry) | 14 mm Diameter (Wet) | 25.4 mm Diameter (Dry) | |||||||
---|---|---|---|---|---|---|---|---|---|

f (MHz) | A_{eff} (mm²) | R_{1} (Ω) | U_{T} (nm) | A_{eff} (mm²) | R_{1} (Ω) | U_{T} (nm) | A_{eff} (mm²) | R_{1} (Ω) | U_{T} (nm) |

5 | 25 | 19 | 124 | 46 | 280 | 18 | 41 | 16 | 84 |

15 | 27 | 24 | 35 | 38 | 590 | 5 | 32 | 42 | 22 |

25 | 25 | 52 | 15 | 30 | 980 | 3 | 25 | 52 | 15 |

## Appendix C. Zernike Decomposition of the Operating Deflection Shapes

**Figure A1.**A Zernike decomposition of the operating deflection shape shown in Figure 3C. The dominating mode carries index 7.

## Appendix D. Assessment of the Magnitude of Flexural Admixtures from Experiments in Air

_{q}+ m

_{f}with M

_{q}a “modal mass” of the resonator and m

_{f}the mass of a film deposited on the resonator surface.

_{f}<< M

_{q}, one may write

^{−1/2}≈ 1 − ε/2). At first glance, Equation (A13) appears to be in conflict with the Sauerbrey equation [46], which states that

_{q}here is the mass of the resonator plate. The solution to this puzzle is that the modal mass, M

_{q}, is only half the mass of the resonator plate, m

_{q}. This is because the parallel plate contains nodal planes. Averaging the square of the velocity distribution over the volume of the plate (the key step in the derivation of the modal mass) produces a factor of 1/2 (meaning that M

_{q}= m

_{q}/2).

^{2}(rather than n), because the slope in such plots is related to the film’s viscoelastic compliance. Importantly, flexural admixtures affect the frequency shift not only at the fundamental mode, but also at overtones 3 and 5. The main text suggested that flexural admixtures were mostly a problem at the fundamental mode. In this example, flexural admixtures are a problem on overtones 1 to 5 (and even 7).

**Figure A2.**Overtone-normalized shifts of frequency and bandwidth obtained after depositing a polymer film with a thickness of about 500 nm onto the resonator plate. The area in green denotes values of Δf/n, which are smaller than expected from the Sauerbrey relation. The difference goes back to flexural admixtures. Data from [47].

## Appendix E. The LDV Instrument Used to Measure Transverse Components of the Displacement

**Figure A3.**Sketch of the setup used to determine both the normal and the transverse component of the displacement from LDV.

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**Figure 1.**In the idealized parallel plate model, the amplitude of the shear wave is the same, everywhere (

**A**). If the amplitude of shear varies across the surface (

**B**), a compressive stress and a tensile stress result at the two surfaces of the plate. These will bend the plate (

**C**). Bending will also occur on the overtones, but to a lesser extent (

**D**).

**Figure 2.**Sketch of the LDV microscope used to study the normal motion of the QCM surface. The unit OFV-353S is a laser-Doppler vibrometer supplied by Polytec (Waldbronn, Germany).

**Figure 3.**Operating deflection shape in normal direction (diameter of plate: 14 mm). (

**A**–

**C**): in air. (

**D**–

**F**): rear side in contact with water; Frequencies: (

**A**,

**D**): 5 MHz, (

**B**,

**E**): 15 MHz, (

**C**,

**F**): 25 MHz.

**Figure 5.**In liquids, the pressure exerted by the fluid reduces bending. There is extensional strain inside the crystal (causing bending) but the same extensional strain is present in the adjacent liquid as well. For volume conservation, extensional strain along x inside the liquid causes a corresponding strain along z, which exerts a normal stress onto the plate and thereby reduces bending.

**Figure 6.**Maps of the absolute values of the amplitudes of motion determined with an LDV setup based on scattering. (

**A**) Transverse motion, (

**B**) normal motion.

**Figure 7.**When mounting the resonator horizontally in an open cell (

**F**) and letting the liquid evaporate, one observes coupled resonances (

**A**–

**C**). The vertical enlargement (

**D**,

**E**) shows the coupled resonances on the overtones. Because of the reduced wavelength, these are more densely spaced than the coupled resonances on the fundamental mode. Furthermore, and more importantly, they occur at a much-reduced amplitude.

**Figure 9.**(

**A**): Fits of Equation (11) to a subset of the data from Figure 7 (5 MHz and 15 MHz). The cuvette resonances are reproduced. (

**B**) Corresponding fit parameters.

14 mm Diameter (Dry) | 14 mm Diameter (Wet) | 25.4 mm diameter (Dry) | |||||||
---|---|---|---|---|---|---|---|---|---|

f (MHz) | U_{N} (nm) | U_{T} (nm) | U_{N}/U_{T} (%) | U_{N} (nm) | U_{T} (nm) | U_{N}/U_{T} (%) | U_{N} (nm) | U_{T} (nm) | U_{N}/U_{T} (%) |

5 | 1.38 | 124 | 1.1 | 0.29 | 18 | 1.6 | 1.10 | 84 | 1.3 |

15 | 0.20 | 35 | 0.6 | 0.027 | 5 | 0.5 | 0.22 | 22 | 1.0 |

25 | 0.12 | 15 | 0.8 | 0.016 | 3 | 0.5 | 0.20 | 15 | 1.3 |

14 mm Diameter (Dry) | 14 mm Diameter (Wet) | 25.4 mm Diameter (Dry) | |
---|---|---|---|

f (MHz) | D (mm) | D (mm) | D (mm) |

5 | 5.3 | 5.8 | 5.7 |

15 | 4.5 | 4.4 | 4.0 |

25 | 4.1 | 4.1 | 3.4 |

**Table 3.**Diameters of circles seen in the polar diagram (Figure 7C).

f (MHz) | Diameter of Circle (Hz) |
---|---|

5 | 60 |

15 | 5 |

25 | 5 |

35 | 1 |

45 | 3 |

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## Share and Cite

**MDPI and ACS Style**

Kowarsch, R.; Suhak, Y.; Eduarte, L.C.; Mansour, M.; Meyer, F.; Peschel, A.; Fritze, H.; Rembe, C.; Johannsmann, D.
Compressional-Wave Effects in the Operation of a Quartz Crystal Microbalance in Liquids:Dependence on Overtone Order. *Sensors* **2020**, *20*, 2535.
https://doi.org/10.3390/s20092535

**AMA Style**

Kowarsch R, Suhak Y, Eduarte LC, Mansour M, Meyer F, Peschel A, Fritze H, Rembe C, Johannsmann D.
Compressional-Wave Effects in the Operation of a Quartz Crystal Microbalance in Liquids:Dependence on Overtone Order. *Sensors*. 2020; 20(9):2535.
https://doi.org/10.3390/s20092535

**Chicago/Turabian Style**

Kowarsch, Robert, Yuriy Suhak, Lucia Cortina Eduarte, Mohammad Mansour, Frederick Meyer, Astrid Peschel, Holger Fritze, Christian Rembe, and Diethelm Johannsmann.
2020. "Compressional-Wave Effects in the Operation of a Quartz Crystal Microbalance in Liquids:Dependence on Overtone Order" *Sensors* 20, no. 9: 2535.
https://doi.org/10.3390/s20092535