⇦Chinese Physics Letters, 2019, Vol. 36, No. 7, Article code 070701 Analysis and Verification of the Relationship between the Maximum Mass Sensitivity of Quartz Crystal Microbalance and Electrode Parameters * Wei Pan (潘威), Xian-He Huang (黄显核)**, Qiao Chen (陈桥), Zhi-Chao Fan (范智超), Yuan Xu (徐源) Affiliations School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731 Received 16 March 2019, online 20 June 2019 *Supported by the National Natural Science Foundation of China under Grant No 61871098.
**Corresponding author. Email: xianhehuang@uestc.edu.cn Citation Text: Pan W, Huang X H, Chen Q, Fan Z C and Xu Y et al 2019 Chin. Phys. Lett. 36 070701    Abstract We analyze the effect of electrode diameter and thickness on the mass sensitivity. Through the theoretical approximate calculation, we find that the mass sensitivity does not change monotonically with electrode diameter and there is a maximum point. The optimum electrode diameter corresponding to the maximum mass sensitivity varies with the electrode thickness. For a particular electrode diameter, a quartz crystal microbalance (QCM) with thick electrode has a higher mass sensitivity. A proper plating experiment using 35 QCMs with different electrode diameters and thicknesses verifies this finding. The present study further reveals how electrode size affects mass sensitivity and is helpful for QCM design. DOI:10.1088/0256-307X/36/7/070701 PACS:07.07.Mp, 07.75.+h, 43.25.Zx, 43.35.Ns © 2019 Chinese Physics Society Article Text Quartz crystal microbalance (QCM) is a resonant transducer derived from quartz crystal resonator. The small mass adsorbed on the electrode surface will lead to a QCM frequency decrease. In 1959, Sauerbrey found that the small mass change has a linear relationship with the frequency decrease and proposed the calculation formula, i.e., the Sauerbrey equation.[1] However, the mass must be rigidly adsorbed on the electrode surface, otherwise the formula cannot be applied. Since the advent of QCM, many researchers have studied the QCM principle and applications, and have achieved excellent results. Over the last three decades, due to its high sensitivity, low cost, convenient operation and so on, QCM has been widely applied for gas sensing,[2–7] electrochemical analysis,[8–13] biochemical investigation,[14–18] and research of environmental science.[19–23] Based on its high sensitivity to mass change, QCM technology can provide an accurate mass sensing capability at the nano-gram level. Therefore, mass sensitivity is very important for QCM and studying of QCM mass sensitivity is very significant for QCM design and improvement. As we all know, the electrode has a very large effect on QCM mass sensitivity, and it is wondered how electrode size affects the mass sensitivity. According to the energy trapping effect,[24–26] the electrode diameter has a notable impact on QCM mass sensitivity. In our previous work,[27] we investigated how electrode diameter affects mass sensitivity and found that QCM mass sensitivity does not change monotonically with electrode diameter, and there is an optimum diameter that corresponds to the maximum mass sensitivity. In this Letter, we further study how electrode thickness affects QCM mass sensitivity and the maximum mass sensitivity in diameter. The mass sensitivity is the key factor that determines the sensing performance of QCM. The QCM working principle and mass sensitivity come from the particle displacement of the quartz crystal and electrode. The local mass sensitivity of the QCM depends on the local intensity of the inertial field developed on the crystal surface during crystal vibration, which is proportional to the vibrating amplitude in that point.[28] The particle displacement amplitude function $A(r)$ of QCM is the solution to the following Bessel equation[29] $$\begin{align} r^{2}\frac{\partial^{2}A}{\partial r^{2}}+r\frac{\partial A}{\partial r}+\frac{k_{i}^{2}r^{2}}{N}A=0,~~ \tag {1} \end{align} $$ where $N$ depends on the material constants of the quartz crystal,[30] and $k_{i}^{2}=(\omega^{2}-\omega_{i}^{2})/c^{2}$, where $i$=E, U (E and U represent the fully electrode region and non-electrode region, respectively), $c=\sqrt {c_{66}/\rho_{\rm q}}$ is the acoustic wave velocity in the crystal with $c_{66}$ being the elastic stiffness constant and $\rho_{\rm q}$ the density of the quartz, and $\omega_{i}$ is the cut-off frequency of fully electrode region ($\omega_{\rm E}$) and non-electrode region ($\omega_{\rm U}$), respectively. The particle displacement amplitude is invariant with the angular direction $\theta$,[29] thus we do not need to consider the impact of the angle. According to the acoustic waveguide theory, waves with a frequency higher than the cut-off frequency can propagate freely, waves with a frequency lower than the cut-off frequency will decay exponentially, and this is called energy trapping.[24–26] On the basis of the energy trapping theory, we can solve the Bessel equation of the vibrating particles. Through the particle displacement function, we can obtain the mass sensitivity function of QCM[31,32] $$\begin{align} S_{\rm f}(r)=\frac{|A(r)|^{2}}{2\pi \int_0^\infty {r|A(r)|^{2}dr}}\cdot C_{\rm f},~~ \tag {2} \end{align} $$ where $S_{\rm f}(r)$ is the mass sensitivity function in units of Hz/kg, $C_{\rm f}$ is Sauerbrey's sensitivity constant,[1,30] and $r$ is the distance from the center of electrode. Both the particle displacement amplitude distribution and mass sensitivity curves follow a Gaussian function. The highest vibration amplitude and maximum sensitivity are in the electrode center of the QCM. Therefore, the mass sensitivity in the QCM center $S_{\rm f}(0)$ (namely, absolute mass sensitivity) could to some extent reflect the sensitivity level of the QCM. Take 10 MHz AT-cut QCM with gold electrode for example. Through theoretical calculation, the mass sensitivity function of QCMs was found to be different greatly depending on the size of electrode. The absolute mass sensitivity results are shown in Fig. 1, and 10 MHz QCMs with different electrode diameters and thicknesses have different absolute mass sensitivities. Both thickness and diameter of electrode significantly impact the absolute mass sensitivity of QCM. As shown in Fig. 1, for a particular electrode thickness, the absolute mass sensitivity of QCM does not monotonically change with electrode diameter, and there exists a maximum point in the diameter-sensitivity curve.[27] More importantly, regardless of electrode diameter, the absolute mass sensitivity of the thick electrode is higher than that of the thin electrode. The diameter corresponding to the maximum mass sensitivity is different for different electrode thicknesses.
cpl-36-7-070701-fig1.png
Fig. 1. The absolute mass sensitivities of 10 MHz QCMs with different diameters and thicknesses of electrode.
cpl-36-7-070701-fig2.png
Fig. 2. Schematic diagram of the experimental setup.
To verify the above theoretical results, a series of plating experiments were carried out using QCMs with different electrode thicknesses and diameters. All the experiments were performed in a class 10000 ultra-clean room of Wintron Electronic Co., Ltd (Zhengzhou City, China). The ambient temperature in the ultra-clean room is maintained at 23$^{\circ}\!$C. Thirty-five 'plano-plano' quartz wafers with a fundamental frequency of 10 MHz and a diameter of 8.7 mm were used in the experiment. Figure 2 is a schematic diagram of the experimental setup. To investigate the relationship between the mass sensitivity and the thickness and diameter of the electrode, these quartz wafers were divided into seven groups according to different diameters and thicknesses of the electrode and plated with thin gold films. In the first plating process, groups A, B, C, and D were plated gold electrodes with the same thickness $t_{1}=500$ Å and different diameters $d_{1}=5.1$ mm, 4.0 mm, 2.5 mm, and 1.0 mm, respectively, onto both sides; groups E, F, and G were the plated gold electrodes with the same thickness $t_{1}=1500$ Å and different diameters $d_{1}=2.5$ mm, 1.4 mm, and 1.0 mm, respectively, onto both sides. Through the first plating process, we obtained 35 QCMs with different electrode diameters and thicknesses, and their resonant frequencies were measured with an S&A250B-1 network analyzer (Saunders & Associates, LLC. Phoenix, Arizona 85050 USA) and recorded as $f_{1}$. In the second plating process, all the seven groups of QCMs were plated with thin gold films. All the plated films were onto the upper surface and had the same diameter $d_{2}=1.0$ mm and the same thickness $t_{2}=500$ Å. The resonant frequencies were measured and recorded as $f_{2}$. The gold film plated in the second plating process is used to simulate mass change on the electrode surface. Note that these quartz wafers and electrodes are circular, thus the angular direction $\theta$ has not been considered.[29] The equipment used in the plating process is S&AW-5600 base plating system (Saunders & Associates, LLC. Phoenix, Arizona 85050 USA). The coating thickness is set by the equipment program. The frequencies of all the 35 QCMs (seven groups) were measured and the results are listed in Table 1. Here ${\Delta f}_{\rm e}=f_{1}-f_{2}$ is the frequency shift caused by the thin gold film which was plated in the second plating process,[33] five values (obtained by five separate QCMs respectively) of ${\Delta f}_{\rm e}$ were recorded for each group, and $\bar{{\Delta f}_{\rm e}}$ and $\delta$ are the average value and the standard deviation of ${\Delta f} _{\rm e}$, respectively. The best standard deviation (29 Hz) in these experiments is obtained in group B, and the maximum standard deviation (258 Hz) is obtained in group G. These low standard deviations show the high stability of the experimental system and the environment. It should be noted that the standard deviations of groups G and D are larger because of their smaller electrode diameter, which results in larger resistance of quartz crystal resonator. Therefore, the test errors of groups D and G are larger than those of other groups.[34]
Table 1. Results of QCM plating experiments.
Groups $t_{1}$ (Å) $d_{1}$ (mm) ${\Delta f}_{\rm e}=f_{1}-f_{2}$ (Hz) $\delta$ (Hz) $\bar{{\Delta f}_{\rm e}}$ (Hz)
A 500 5.1 3143 3137 3140 3172 3217 30 3162
B 500 4.0 4010 3929 3959 3940 3942 29 3956
C 500 2.5 5098 4946 5122 5079 5085 62 5066
D 500 1.0 3915 3681 3830 3802 3741 79 3794
E 1500 2.5 6864 6700 6725 6701 6778 62 6754
F 1500 1.4 8304 8404 8478 8376 8458 62 8404
G 1500 1.0 6581 6294 6872 6793 6230 258 6554
In Table 1, for each group, the attached masses caused by the second plating process are equal, while the frequency shifts are quite different. Higher frequency shift caused by the same mass change means higher mass sensitivity. From groups A, B, C, D or E, F, G, we can find that the mass sensitivity of QCM does not monotonically change with electrode diameter ($d_{1}$) and exists an optimum electrode diameter at which we can obtain maximum mass sensitivity for a particular electrode thickness $t_{1}$. For electrode thickness 500 Å, the maximum mass sensitivity is obtained in group C. For electrode thickness 1500 Å, the maximum mass sensitivity is obtained in group F. Moreover, the optimum diameter corresponding to the maximum mass sensitivity is different for different electrode thicknesses. Regardless of the electrode diameter, the mass sensitivity of QCM with large electrode thickness is higher than that of QCM with small electrode thickness. In summary, through the theoretical analysis and experiment, we further verify the existence of the maximum mass sensitivity, and find that electrode thickness does affect QCM mass sensitivity and the maximum mass sensitivity significantly. QCM with thick electrode has a higher mass sensitivity than QCM with thin electrode. In diameter, QCM has a maximum mass sensitivity and the corresponding diameter values are different for different electrode thicknesses. This research reveals how the electrode diameter and thickness affect QCM mass sensitivity and will be very beneficial for QCM design, improvement and practical applications. References Verwendung von Schwingquarzen zur W�gung d�nner Schichten und zur Mikrow�gungInvestigation of the stability of QCM humidity sensor using graphene oxide as sensing filmsHigh-Performance, Flexible Hydrogen Sensors That Use Carbon Nanotubes Decorated with Palladium NanoparticlesToxicity of the organophosphate chemical warfare agents GA, GB, and VX: Implications for public protectionHigh-stability quartz crystal microbalance ammonia sensor utilizing graphene oxide isolation layerNanofibrous polyethyleneimine membranes as sensitive coatings for quartz crystal microbalance-based formaldehyde sensorsGeneral approach for electrochemical detection of persistent pharmaceutical micropollutants: Application to acetaminophenGravimetric and dynamic deconvolution of global EQCM response of carbon nanotube based electrodes by Ac-electrogravimetryIn vitro chloramphenicol detection in a Haemophilus influenza model using an aptamer-polymer based electrochemical biosensorUltrasensitive Immunoassay Based on Electrochemical Measurement of Enzymatically Produced PolyanilineQCM-D studies of polypyrrole influence on structure stabilization of β phase of Ni(OH)2 nanoparticles during electrochemical cyclingIn Situ Porous Structure Characterization of Electrodes for Energy Storage and Conversion by EQCM-D: a ReviewFacile fabrication of branched-chain carbohydrate chips for studying carbohydrate-protein interactions by QCM biosensorHighly Specific and Ultrasensitive Graphene-Enhanced Electrochemical Detection of Low-Abundance Tumor Cells Using Silica Nanoparticles Coated with Antibody-Conjugated Quantum DotsLabel-Free Quartz Crystal Microbalance with Dissipation Monitoring of Resveratrol Effect on Mechanical Changes and Folate Receptor Expression Levels of Living MCF-7 Cells: A Model for Screening of DrugsDetection of Aerosolized Biological Agents Using the Piezoelectric ImmunosensorEffect of trapped water on the frictional behavior of graphene oxide layers sliding in water environmentA Noncontact Dibutyl Phthalate Sensor Based on a Wireless-Electrodeless QCM-D Modified with Nano-Structured Nickel HydroxideDevelopment of a high-sensitivity plasticizer sensor based on a quartz crystal microbalance modified with a nanostructured nickel hydroxide filmQuartz crystal microbalance immunosensors for environmental monitoringRole of bovine serum albumin and humic acid in the interaction between SiO2 nanoparticles and model cell membranesEnergy trapping of thickness-shear vibration modes of elastic plates with functionally graded materialsEffects of electrodes with continuously varying thickness on energy trapping in thickness-shear mode quartz resonatorsTrapped‐Energy Modes in Quartz Filter CrystalsThe Exploration and Confirmation of the Maximum Mass Sensitivity of Quartz Crystal MicrobalanceIs quartz crystal microbalance really a mass sensor?Analysis of the Radial Dependence of Mass Sensitivity for Modified-Electrode Quartz Crystal ResonatorsThe modified design of ring electrode quartz crystal resonator for uniform mass sensitivity distributionScanning electrochemical mass sensitivity mapping of the quartz crystal microbalance in liquid mediaRadial mass sensitivity of the quartz crystal microbalance in liquid mediaUse of quartz vibration for weighing thin films on a microbalanceThe Resistance–Amplitude–Frequency Effect of In–Liquid Quartz Crystal Microbalance
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