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Quartz Tuning Fork (QTF)-based Scanning Probe Microscopy (SPM) is an important field of research. A suitable model for the QTF is important to obtain quantitative measurements with these devices. Analytical models have the limitation of being based on the double cantilever configuration. In this paper, we present an electromechanical finite element model of the QTF electrically excited with two free prongs. The model goes beyond the state-of-the-art of numerical simulations currently found in the literature for this QTF configuration. We present the first numerical analysis of both the electrical and mechanical behavior of QTF devices. Experimental measurements obtained with 10 units of the same model of QTF validate the finite element model with a good agreement.

In the past several decades, Atomic Force Microscopy (AFM) has been employed to measure the nano-scale properties of both soft [

As an alternative to classical AFM nanosensors, the use of Quartz Tuning Forks (QTFs) has been proposed. QTFs are piezoelectric devices that are commonly known for their application in customer electronics. In 1989, QTFs started to be used in microscopy [

Finite Element Analysis (FEA) has been widely used in sensor analysis and design [

We present herein a 3D finite-element model of the QTF, which models the coupling between the mechanical and the electrical behavior of the device by implementing the electric part, composed of a: voltage source, electrodes, and compensation circuit. In the first section, the mechanical model is presented. In the second section, the electromechanical model highlights the importance of accurately defining the geometry of the two electrodes. Finally, experimental measurements, which validate our finite element model, are shown.

The tuning fork is a bimorph cantilever based on the piezoelectric properties of the quartz. The sensor consists of two prongs attached to a base, which normally is clamped to a holder. Each of the prongs is coated with a thin layer of a conductor material, which permits the resonator to be driven either electrically or mechanically. In the former case, a potential difference between the electrodes is applied, whereas in the second one a dither is attached to one of the prongs of the QTF. Given the type of driver employed, either the electrical current through the device—for electrical excitation—or the generated piezoelectric voltage—for mechanical excitation—are related to the fork's vibration amplitude.

In microscopy applications, the QTF is driven to its resonance frequency by either mechanical or electrical excitation. There are two main vibrational modes for the same mechanical deformation: in-phase and anti-phase modes. For electrical excitation, the generated current can just be measured in the anti-phase mode because the current is only generated in the system when both prongs are vibrating in opposite directions.

In order to properly characterize the QTF dynamics, analytical models have been proposed in the literature [

It is worthwhile to mention that the calculation of the spring constant has led to great controversies and discrepancies in the research community due to the lack of a generalized model. Although the cantilever-based model for studying the dynamics of QTF is the most accepted model, a two-coupled oscillator model has been utilized in [

In the case of electrical excitation, the amplitude of oscillation can be calculated by using the following [_{rms}_{rms}_{0}

In this work, a finite element model (FEM) of the QTF is proposed. In order to make an accurate 3D model, dimensions of the QTF are set in accordance to measurements carried out in an optical microscope (B-353MET model, Euro-Microscopes). In contrast to commercial AFM sensors where cantilevers present a rotation with respect to the X coordinate, the tuning fork model rotates with respect to the Z coordinate; thus, the width (

As it was previously stated, the QTF is based on piezoelectric phenomena. Therefore, linear piezoelectricity equations of elasticity have to be defined and coupled to the electrostatic charge by means of piezoelectric constants [

{_{p}

{_{i}

{_{K}

{_{q}

[_{pk}

The different physical properties of quartz have been widely studied [

The piezoelectric constant matrix, which permits the structural and electrical behavior of the material to be coupled, is defined as follows:

The piezoelectric behavior of the material is accomplished by using the element type SOLID226 in ANSYS.

The model is defined in uMKS units according to ANSYS nomenclature. The model is composed of 12,384 hexahedral elements with a size of 60 μm, which is translated into 59,340 nodes. In addition, the bottom of the base is constrained to emulate the clamped structure of the real device.

For the implementation of the electrical part and to couple it with the mechanical behavior of the QTF, the loaded and the grounded electrodes needed to be defined. In addition, the electrodes also define the way in which the deformation occurs when an electric field is applied, and hence the type of acoustic wave generated [

As shown in

Therefore, two groups of nodes are created. Each electrode is defined by 2,317 elements that are translated into 6,390 nodes. An independent voltage source links one node from each of the two groups together implemented by using the element CIRCU94. One group is defined as the loaded electrode such that a symmetric sinusoidal voltage wave is supplied to it, while the other electrode is grounded.

The electrical part of the QTF is modeled by an equivalent circuit (Butterworth-Van Dyke model) based on a serial RLC circuit with a parallel capacitor [

One of the main problems is that the current flowing through the QTF becomes dominant away from the resonance frequency due to parasitic capacitance; this phenomenon is responsible for asymmetries and shifts in the frequency response. Thus, it is required to compensate for the parasitic current. In the proposed model, an inductor element with CIRCU94 is implemented within ANSYS for this very reason, whereas a means to compensate in the experimental setup will be explained later. Therefore, the voltage source and the inductor determine the final excitation circuit.

Parasitic capacitance can be obtained by doing harmonic simulation over a broad frequency range by interpreting the contribution of this capacitance as the slope of a linear fit dependent on the frequency response. Using

In order to perform the harmonic simulation, one more input parameter is needed: the quality factor of the QTF. The quality factor is defined in numerical simulations through the damping ratio (

For our model, the inductance was calculated to be 26.5 H by using

Parasitic contribution in the finite element model is only due to the electrodes rather than contacts, cables,

Concerning the way in which the measurements are taken, the device is electrically driven, and the amplitude of oscillation is obtained by measuring the current through the QTF. An AC voltage source at the resonant frequency of the fork is applied, and the current is measured by a transimpedance amplifier (TIA) with a gain R_{G} = 10^{6} (V/A).

In the experimental setup, a capacitor-compensated circuit was implemented to drive the QTF [

As a consequence, only the current through the nanosensor is amplified by the TIA, and it is translated into voltage to measure the extent of oscillation with a lock-in amplifier [

The first objective in validating our model was to obtain the resonance frequency of the QTF, for both the in-phase and anti-phase vibrational modes.

For the in-phase mode, when both prongs are vibrating in the same direction, our model demonstrates a resonance frequency of 27,433 Hz. However, we have not verified this value experimentally due to the small readout signal that is not distinguishable from the intrinsic noise of the equipment. Nevertheless, in study [

Regarding the anti-phase mode, when both prongs are vibrating in the opposite direction, the resonance frequency is completely observed and well correlated with the nominal value of 32,768 Hz provided by the manufacturer [

In order to validate the dynamics response of the model proposed, ten QTFs of the same model and manufacturer (

The first step to validate the model was to compare experimental and simulation data for one tuning fork with several _{drive}_{drive}

In order to properly validate the proposed model, several QTFs have to be characterized. Comparison of the ten different QTFs was performed between measured and simulation data for a single _{drive}

The amplitude of oscillation can be experimentally obtained by interferometric techniques [

As shown in _{drive}

Finally the validation of the QTF as a sensor was done by small-mass loading one of the prongs of the device in the finite element model and measuring the shift in the resonant frequency produced by the added mass. A small cube of solid material with a known mass was coupled to the left prong of the QTF, in a similar way that the experiments conducted in [

A new model of quartz tuning fork with two free prongs and electrically excited is presented based on finite element analysis developed in ANSYS by incorporating the electrical part: excitation, compensation circuit and the electrodes. The model has been validated by measuring ten separate quartz tuning forks at different driving amplitudes—from 10 mV to 100 mV—which exhibit a strong agreement between 91% and 98%. The remaining error can be attributed to small geometric differences between the electrodes of the model and the electrodes from the actual quartz tuning fork. In addition, the parasitic capacitance cannot always be completely compensated in the experimental measurements.

Finally, the model proposed herein allows from the comparison between experimental and simulation data, which is complicated to achieve from other models in the literature. These analytical models are normally developed for mechanically excited quartz tuning forks implying that the dither driving energy must be determined. However, this is no easy task due to the appearance of mechanical losses and the coupling between the sensor and dither, which are difficult to quantify. Our developed model also overcomes the difficulty of measuring certain parameters, such as the amplitude of oscillation and the sensitivity. With the results obtained, the model could be used to calculate the effective spring constant of the device as deeply discussed in [

This work was supported in part by the Spanish Ministerio de Educación under project TEC2009-10114 and also by the regional Catalan authorities under project VALTEC09-2-0058.

The authors declare no conflict of interest.

(

(

(

Implemented circuit for parasitic capacitor compensation.

(_{drive}

Results of the amplitude of oscillation of the ten QTFs from the finite element model.

Results of the frequency shift produced by the added mass from the finite element model.

Resonance frequency for the in-phase and anti-phase vibrational modes.

Castellanos |
27,800 | 32,766 |

Experimental Results | - | 32,768 |

FEM Results | 27,433 | 32,768 |

Manufacturer [ |
- | 32,768 |

Quality factor of the ten QTF.

QTF1 | 104542 | 4.78e-06 |

QTF2 | 112800 | 4.43e-06 |

QTF3 | 113077 | 4.42e-06 |

QTF4 | 102144 | 4.89e-06 |

QTF5 | 110724 | 4.51e-06 |

QTF6 | 109778 | 4.55e-06 |

QTF7 | 101471 | 4.92e-06 |

QTF8 | 135054 | 3.70e-06 |

QTF9 | 107489 | 4.65e-06 |

QTF10 | 120019 | 4.16e-06 |