# Investigating a Detection Method for Viruses and Pathogens Using a Dual-Microcantilever Sensor

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## Abstract

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## 1. Introduction

## 2. Sensor Description and Problem Statement Formulation

## 3. Load and Dynamic Model of Cantilever Beam with Harmonic Base Excitation

## 4. Basic Concepts of the Considered Piezoresistor Detection Method

^{−15}g).

## 5. Experimental Study of the Dual-Microcantilever Sensor

## 6. An Investigation of the Sensitivity of the Method through the Possibilities of Determining the Frequency of the Cusp Point

^{−11}kg [23], and beam natural frequencies according to Table 1, a limiting sensitivity lower than 1.5 × 10

^{−17}kg or 15 fkg is obtained. Here, the mass of the microcantilever is 14.108 × 10

^{−10}kg. From Equation (54), it is concluded that to increase the limiting sensitivity, it is necessary to improve the measurement accuracy and increase the natural frequencies of the microcantilevers. At natural frequencies of the microcantilevers twice as high, for the considered case, the limit resolution is increased by one order of magnitude.

## 7. Experimental Determination of the Capabilities of the Method, Changing One of the Natural Frequencies of the Microcantilevers by Heating

## 8. Determine the Sensitivity of the Detection Method by Examining the Offset of the Cusp Point

^{−16}kg. This corresponds to 6.21344 × 10

^{5}fg. The mass of a SARS-CoV-2 virus is on the order of 1 fg = 1 × 10

^{−18}kg. One person was found to carry 10

^{10}to 10

^{11}viruses with a total mass of 1–100 µg during the peak of infection [39,40].

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic overview and topology of dual-cantilever microsensor: (

**a**) electromechanical schematic; (

**b**) Wheatstone bridge circuit; (

**c**) photo of sensor; (

**d**) close-up view of sensor topology.

**Figure 2.**Microcantilever beam with harmonically driven base: (

**a**) microcantilever diagram; (

**b**) lumped dynamic model of microcantilever beam.

**Figure 3.**Calculated amplitude–frequency responses of microcantilever 1 ${V}_{A1}$ [V] and microcantilever 2 ${V}_{A2}$ [V] as function of frequency $f$ [Hz].

**Figure 4.**The amplitude–frequency responses obtained by the voltage differences from the half-bridges of the two microcantilevers: (

**a**) the difference of the amplitudes of the two microcantilevers; (

**b**) the absolute value of the difference of the amplitudes of the two microcantilevers.

**Figure 5.**Experimental system for testing piezoresistive sensors with dual-microcantilever beams: (

**a**) general view; (

**b**) closer look at sensor and its actuation. 1. Sensor. 2. NI PXI system. 3. Ammeter, 4. Digilent wave generator. 5. Potentiometers for adjusting current in microcantilever heaters. 6. Batteries to power heaters. 7. Monitor. 8. Microchip. 9. Piezoelectric actuator. 10. Sensor housing.

**Figure 6.**Theoretical and experimental plots of amplitude–frequency response: (

**a**) theoretical and experimental amplitude–frequency response of microcantilever 1; (

**b**) theoretical and experimental amplitude–frequency response of microcantilever 2.

**Figure 7.**A graphical representation of the amplitude–frequency response results. The experimental results are plotted with a solid line, and theoretical results are represented by a circle symbol: (

**a**) the theoretical and experimental amplitude–frequency response of the output voltage of the Wheatstone bridge; (

**b**) the theoretical and experimental amplitude–frequency response of the absolute value of the output voltage of the Wheatstone bridge.

**Figure 8.**A schematic of the experiment to change the temperature of microcantilever 1 by Joule heating.

**Figure 9.**Experimental data processing for ${V}_{abs}$ from an Excel file obtained at current $i$ = 1053 µA. The frequency of the forced vibrations at the base of the two microcantilevers was varied in the range [65,520, 66,150] Hz.

**Figure 10.**Experimental data on the effect of heating microcantilever 1 on the frequency of the experimental cusp point, natural frequency, and their approximation: (

**a**) the cusp point frequency as a function of the heater current and its approximating linear relationship; (

**b**) tge dependence of the natural frequency of microcantilever 1 on its heating current i and the approximating line.

**Figure 11.**Added thought mass that causes the same variation in the natural frequency of microcantilever 1 as that caused by current heating.

Parameter | Symbol | Value | Unit |
---|---|---|---|

Length of microcantilever 1 | ${l}_{11}$ | 294 × 10^{−6} | m |

Length of microcantilever 2 | ${l}_{12}$ | 292 × 10^{−6} | m |

Width of microcantilever 1 | ${l}_{21}$ | 150 × 10^{−6} | m |

Width of microcantilever 2 | ${l}_{22}$ | 172 × 10^{−6} | m |

Height of microcantilever 1 | ${l}_{31}$ | 4 × 10^{−6} | m |

Height of microcantilever 2 | ${l}_{32}$ | 4 × 10^{−6} | m |

Basis resistance of a piezoresistor | ${R}_{0}$ | 1000 | Ω |

Density of the silicon | $\rho $ | 2329 * | kg/m^{3} |

Young’s modulus of the n-silicon in [110] direction | ${E}_{110}$ | 170 * | GPa |

Piezoresistivity coefficient for direction 11 n-Si | ${\pi}_{11}$ | −102 × 10^{−11} ** | Pa^{−1} |

Piezoresistivity coefficient for direction 12 n-Si | ${\pi}_{12}$ | 53 × 10^{−11} ** | Pa^{−1} |

Piezoresistivity coefficient for direction 44 n-Si | ${\pi}_{44}$ | −14 × 10^{−11} ** | Pa^{−1} |

Stiffness for n-type silicon plane 100 in axis [110] | ${C}_{11}$ | 165.65 × 10^{9} *** | Pa |

Stiffness for n-type silicon plane 100 in axis [010] | ${C}_{12}$ | 63.94 × 10^{9} *** | Pa |

Stiffness for n-type silicon plane 100 in axis [001] | ${C}_{44}$ | 79.51 × 10^{9} *** | Pa |

Natural circular frequency of microcantilever 1 | ${\varpi}_{1}$ | 10,402.535 | s^{−1} |

Natural circular frequency of microcantilever 2 | ${\varpi}_{2}$ | 10,568.028 | s^{−1} |

Natural frequency of microcantilever 1 | ${f}_{s1}$ | 65,361.057 | Hz |

Natural frequency of microcantilever 2 | ${f}_{s2}$ | 66,400.888 | Hz |

Supplying voltage | ${v}_{cc}$ | 8 | V |

Effective amplitude of the external vibrations | ${a}_{e}$ | 9.92 × 10^{−8} | m |

Damping factor of microcantilever 1 | ${\beta}_{1}$ | 1554.755 | s^{−1} |

Damping factor of microcantilever 2 | ${\beta}_{2}$ | 1675.886 | s^{−1} |

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**MDPI and ACS Style**

Banchelli, L.; Todorov, G.; Stavrov, V.; Ganev, B.; Todorov, T.
Investigating a Detection Method for Viruses and Pathogens Using a Dual-Microcantilever Sensor. *Micromachines* **2024**, *15*, 1117.
https://doi.org/10.3390/mi15091117

**AMA Style**

Banchelli L, Todorov G, Stavrov V, Ganev B, Todorov T.
Investigating a Detection Method for Viruses and Pathogens Using a Dual-Microcantilever Sensor. *Micromachines*. 2024; 15(9):1117.
https://doi.org/10.3390/mi15091117

**Chicago/Turabian Style**

Banchelli, Luca, Georgi Todorov, Vladimir Stavrov, Borislav Ganev, and Todor Todorov.
2024. "Investigating a Detection Method for Viruses and Pathogens Using a Dual-Microcantilever Sensor" *Micromachines* 15, no. 9: 1117.
https://doi.org/10.3390/mi15091117