Based on the model ( Figure 1c), we can derive the resonance curve function, the oscillation amplitude A e versus the driving frequency f ( = w / 2 π ), for the measured electrical signal I e: Therefore, the total signal I e includes both the motion-induced signal I m and the capacitive signal I c. While the current through the LRC circuit I m represents the vibrational motion of the TF or qPlus, the C 0 produces the stray capacitance current I c. In the equivalent circuit, the LRC circuit is connected in parallel with a capacitance C 0, and thus the total signal I e is given by the sum of I m and I c. Therefore, one should consider the nonlinearity when using a quartz sensor to quantify the tip–sample interaction potentials and forces with milli-electron volt and pico-Newton resolutions. Note that the equivalent circuit describes the linear motion of a TF or qPlus, although a recent study showed the amplitude dependence of the resonance frequency in such quartz resonators. The electrical responses of the ED-TF ( Figure 1a) and ED-qPlus ( Figure 1b) are described well by the equivalent circuit model shown in Figure 1c. Both the actuation and the detection of the qPlus can also be made electrically ( c) The equivalent circuit model for quartz resonators (e.g., tuning fork, qPlus). ( a) The quartz tuning fork with two prongs can be electrically actuated and its dynamic response can also be measured electrically, where the two prongs move in opposite directions, in an antisymmetric mode of vibration ( b) The qPlus sensor is made by fixing one prong firmly to a supporting wall so that the other prong is allowed to vibrate. Two working configurations of electrically driven quartz tuning forks and the traditional electrical circuit model for quartz resonators. Our results could be useful to optimize sensors’ dynamic characteristics for quantitative interaction measurements with qPlus or TF. Furthermore, we introduce two intrinsic constants that are independent of the probe type, TF or qPlus.
Our model predicts the changes in peak amplitude, resonance frequency, quality factor, and normalized capacitance from the TF to its qPlus configuration, and we confirm the changes experimentally. In this paper, we present an electromechanical model that comprehensively describes the motions of the ED-TF and ED-qPlus. Obviously, the one fixed prong affects the overall electromechanical characteristics in the qPlus configuration, but it is not quantitatively understood how the characteristics alter by transforming TF to qPlus. However, the dynamic characteristics of the ED-qPlus such as peak amplitude, resonance frequency, and quality factor are very distinct from those of the original form (i.e., bare TF), although they are both electrically driven and only one of the two prongs is fixed in the qPlus. The traditional equivalent circuit model for piezoelectric resonators can be used to describe the motion of both the ED-qPlus and electrically driven TFs (ED-TFs). While the mechanically driven qPlus allows analytical description of the probe dynamics, one could use the electrically driven qPlus (ED-qPlus) sensor to exploit the capability of self-actuation and self-detection. For TFs, the qPlus configuration is well-approximated as the harmonic oscillator when it is driven mechanically so that the qPlus sensor facilitates quantitative force measurement, and thus it is widely employed for dynamic force spectroscopy. Various mechanistic models of micro-cantilevers have been suggested based on a simple harmonic oscillator, a multi-mode oscillator, or normal and torsional deflections of a three-dimensional beam. For probes such as micro-cantilevers and quartz tuning forks (TFs), there have been long-investigated linear and nonlinear dynamics and associated models of the probes. Understanding the dynamics of a probe’s motion is important in order to use the probe as a quantitative force sensor in atomic force microscopy and spectroscopy.
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