# A Review on Theory and Modelling of Nanomechanical Sensors for Biological Applications

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## Abstract

**:**

## 1. Introduction

## 2. Nanomechanical Sensors in Static Mode: Effect of Surface Stress

## 3. Nanomechanical Sensors in Dynamic Mode

#### 3.1. Effect of a Complete Layer

#### 3.1.1. Effect of Surface Stress

#### 3.1.2. Mass and Stiffness Effects

#### 3.2. Individual Particles

#### 3.2.1. Multimode Measuring

#### 3.2.2. Inertial Imaging

## 4. Hydrodynamic Loading

#### 4.1. Nanomechanical Resonators Immersed in Fluid

#### 4.2. Suspended Microchannel Resonators

## 5. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$x,y,z$ | Space coordinates |

$w\left(x,y\right)$ | Out-of-plane displacement of the plate |

$L$ | Length of the plate |

$b$ | Width of the plate |

$h$ | Thickness of the plate |

$M$ | Bending moment |

${E}_{c}$ | Young’s modulus of the resonator |

$\nu $ | Poisson’s ratio |

$\Delta \sigma $ | Differential surface stress |

$\kappa $ | Stoney’s curvature |

${m}_{effn}$ | Effective mass of the resonator associated with the nth mode |

${k}_{n}$ | Spring constant of the resonator associated with the nth mode |

$f$ | Frequency |

${\sigma}^{T}$ | Net surface stress |

$\eta $ | Ratio between the thickness of the analyte and the thickness of the plate |

${\rho}_{a}$ | Density of the analyte |

${\rho}_{c}$ | Density of the cantilever |

${E}_{a}$ | Young’s modulus of the analyte |

${m}_{a}$ | Mass of the analyte |

${m}_{c}$ | Mass of the cantilever |

${\psi}_{n}$ | Mode shape of the nth mode |

$X$ | Longitudinal coordinate normalized to the length of the cantilever |

${X}_{0}$ | Normalized adsorption position |

${V}_{a}$ | Volume of the analyte |

${V}_{c}$ | Volume of the cantilever |

${\lambda}_{n}$ | Eigenvalue associated with the nth mode |

${\gamma}_{flex}$ | Stiffness coefficient of the analyte for the flexural modes of the cantilever |

$\theta $ | Angle of orientation of the analyte with respect to the main axis of the cantilever |

${\gamma}_{mqrs}$ | Stiffness tensor of the analyte |

${\epsilon}_{mq}$ | Strain component $mq$ of the plate |

${\beta}_{n}$ | Coefficient related to the eigenvalue of the plate |

${\Delta}_{m}$ | Mass term |

${\Delta}_{s}$ | Stiffness term |

${\mathsf{\sigma}}_{Allan}$ | Allan deviation |

$\mathsf{\Sigma}$ | Covariance matrix |

${\mu}_{a}$ | Mass per unit length of the analyte |

${m}^{\left(k\right)}$ | Moment of order $k$ of the analyte mass distribution |

${\rho}_{f}$ | Density of the fluid |

$\omega $ | Angular frequency |

$\mathsf{\Gamma}$ | Hydrodynamic function for circular cross section |

$\widehat{w}$ | Out-of-plane displacement in the frequency domain |

$re$ | Reynolds number |

${\eta}_{f}$ | Viscosity of the fluid |

${\mathsf{\Gamma}}_{rec}$ | Hydrodynamic function for rectangular cross section |

$\mathsf{\Omega}$ | Correction factor for the hydrodynamic function |

${m}_{ad}$ | Added mass of the surrounding fluid |

${\gamma}_{f}$ | Damping factor |

$t$ | Time |

${k}_{effn}$ | Effective spring constant associated with the nth mode |

${F}_{th}$ | Non-correlated Langevin thermal force |

${m}_{b}$ | Buoyant mass |

${m}_{0}$ | Mass of the resonator filled with fluid |

${\chi}_{f}$ | Compressibility of the fluid |

${\chi}_{a}$ | Compressibility of the analyte |

## References

- Chien, M.-H.; Brameshuber, M.; Rossboth, B.K.; Schütz, G.J.; Schmid, S. Single-molecule optical absorption imaging by nanomechanical photothermal sensing. Proc. Natl. Acad. Sci. USA
**2018**, 115, 11150. [Google Scholar] [CrossRef] [Green Version] - Garcia, R. Nanomechanical mapping of soft materials with the atomic force microscope: Methods, theory and applications. Chem. Soc. Rev.
**2020**, 49, 5850–5884. [Google Scholar] [CrossRef] [PubMed] - Barson, M.S.J.; Peddibhotla, P.; Ovartchaiyapong, P.; Ganesan, K.; Taylor, R.L.; Gebert, M.; Mielens, Z.; Koslowski, B.; Simpson, D.A.; McGuinness, L.P.; et al. Nanomechanical Sensing Using Spins in Diamond. Nano Lett.
**2017**, 17, 1496–1503. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rugar, D.; Budakian, R.; Mamin, H.J.; Chui, B.W. Single spin detection by magnetic resonance force microscopy. Nature
**2004**, 430, 329–332. [Google Scholar] [CrossRef] - Chan, J.; Alegre, T.P.M.; Safavi-Naeini, A.H.; Hill, J.T.; Krause, A.; Gröblacher, S.; Aspelmeyer, M.; Painter, O. Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature
**2011**, 478, 89–92. [Google Scholar] [CrossRef] [PubMed] [Green Version] - O’Connell, A.D.; Hofheinz, M.; Ansmann, M.; Bialczak, R.C.; Lenander, M.; Lucero, E.; Neeley, M.; Sank, D.; Wang, H.; Weides, M. Quantum ground state and single-phonon control of a mechanical resonator. Nature
**2010**, 464, 697–703. [Google Scholar] [CrossRef] - Arlett, J.L.; Myers, E.B.; Roukes, M.L. Comparative advantages of mechanical biosensors. Nat. Nanotechnol.
**2011**, 6, 203–215. [Google Scholar] [CrossRef] [Green Version] - Kosaka, P.M.; Pini, V.; Ruz, J.J.; Da Silva, R.A.; González, M.U.; Ramos, D.; Calleja, M.; Tamayo, J. Detection of cancer biomarkers in serum using a hybrid mechanical and optoplasmonic nanosensor. Nat. Nanotechnol.
**2014**, 9, 1047. [Google Scholar] [CrossRef] - Hanay, M.S.; Kelber, S.; Naik, A.K.; Chi, D.; Hentz, S.; Bullard, E.C.; Colinet, E.; Duraffourg, L.; Roukes, M.L. Single-protein nanomechanical mass spectrometry in real time. Nat. Nanotechnol.
**2012**, 7, 602. [Google Scholar] [CrossRef] [PubMed] - Malvar, O.; Ruz, J.J.; Kosaka, P.M.; Domínguez, C.M.; Gil-Santos, E.; Calleja, M.; Tamayo, J. Mass and stiffness spectrometry of nanoparticles and whole intact bacteria by multimode nanomechanical resonators. Nat. Commun.
**2016**, 7, 1–8. [Google Scholar] [CrossRef] [PubMed] - Domínguez, C.M.; Ramos, D.; Mendieta-Moreno, J.I.; Fierro, J.L.G.; Mendieta, J.; Tamayo, J.; Calleja, M. Effect of water-DNA interactions on elastic properties of DNA self-assembled monolayers. Sci. Rep.
**2017**, 7, 536. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Del Rey, M.; Silva, R.A.D.; Meneses, D.; Petri, D.F.S.; Tamayo, J.; Calleja, M.; Kosaka, P.M. Monitoring swelling and deswelling of thin polymer films by microcantilever sensors. Sens. Actuators B Chem.
**2014**, 204, 602–610. [Google Scholar] [CrossRef] [Green Version] - Domínguez, C.M.; Kosaka, P.M.; Mokry, G.; Pini, V.; Malvar, O.; del Rey, M.; Ramos, D.; San Paulo, Á.; Tamayo, J.; Calleja, M. Hydration Induced Stress on DNA Monolayers Grafted on Microcantilevers. Langmuir
**2014**, 30, 10962–10969. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mertens, J.; Rogero, C.; Calleja, M.; Ramos, D.; Martín-Gago, J.A.; Briones, C.; Tamayo, J. Label-free detection of DNA hybridization based on hydration-induced tension in nucleic acid films. Nat. Nanotechnol.
**2008**, 3, 301–307. [Google Scholar] [CrossRef] [Green Version] - Yubero, M.L.; Kosaka, P.M.; San Paulo, Á.; Malumbres, M.; Calleja, M.; Tamayo, J. Effects of energy metabolism on the mechanical properties of breast cancer cells. Commun. Biol.
**2020**, 3, 590. [Google Scholar] [CrossRef] - Ligler, F.S.; Taitt, C.R. Optical Biosensors: Today and Tomorrow; Elsevier: Amsterdam, The Nethrelands, 2011. [Google Scholar]
- Waggoner, P.S.; Craighead, H.G. Micro-and nanomechanical sensors for environmental, chemical, and biological detection. Lab A Chip
**2007**, 7, 1238–1255. [Google Scholar] [CrossRef] - Binnig, G.; Quate, C.F.; Gerber, C. Atomic Force Microscope. Phys. Rev. Lett.
**1986**, 56, 930–933. [Google Scholar] [CrossRef] [Green Version] - Binnig, G.; Rohrer, H. Scanning tunneling microscopy. Surf. Sci.
**1983**, 126, 236–244. [Google Scholar] [CrossRef] - Binnig, G.; Rohrer, H.; Salvan, F.; Gerber, C.; Baro, A. Revisiting the 7 × 7 reconstruction of Si(111). Surf. Sci.
**1985**, 157, L373–L378. [Google Scholar] [CrossRef] - Eom, K.; Park, H.S.; Yoon, D.S.; Kwon, T. Nanomechanical resonators and their applications in biological/chemical detection: Nanomechanics principles. Phys. Rep.
**2011**, 503, 115–163. [Google Scholar] [CrossRef] [Green Version] - Leng, H.; Lin, Y. A MEMS/NEMS sensor for human skin temperature measurement. Smart Struct. Syst.
**2011**, 8, 53–67. [Google Scholar] [CrossRef] - Liu, P.S.; Tse, H.-F. Implantable sensors for heart failure monitoring. J. Arrhythmia
**2013**, 29, 314–319. [Google Scholar] [CrossRef] [Green Version] - Boisen, A.; Dohn, S.; Keller, S.S.; Schmid, S.; Tenje, M. Cantilever-like micromechanical sensors. Rep. Prog. Phys.
**2011**, 74, 036101. [Google Scholar] [CrossRef] - Tamayo, J.; Kosaka, P.M.; Ruz, J.J.; San Paulo, Á.; Calleja, M. Biosensors based on nanomechanical systems. Chem. Soc. Rev.
**2013**, 42, 1287–1311. [Google Scholar] [CrossRef] [Green Version] - Barnes, J.R.; Stephenson, R.J.; Welland, M.E.; Gerber, C.; Gimzewski, J.K. Photothermal spectroscopy with femtojoule sensitivity using a micromechanical device. Nature
**1994**, 372, 79–81. [Google Scholar] [CrossRef] - Thundat, T.; Chen, G.Y.; Warmack, R.J.; Allison, D.P.; Wachter, E.A. Vapor Detection Using Resonating Microcantilevers. Anal. Chem.
**1995**, 67, 519–521. [Google Scholar] [CrossRef] - Thundat, T.; Wachter, E.A.; Sharp, S.L.; Warmack, R.J. Detection of mercury vapor using resonating microcantilevers. Appl. Phys. Lett.
**1995**, 66, 1695–1697. [Google Scholar] [CrossRef] - Fritz, J.; Baller, M.K.; Lang, H.P.; Rothuizen, H.; Vettiger, P.; Meyer, E.J.; Güntherodt, H.; Gerber, C.; Gimzewski, J.K. Translating Biomolecular Recognition into Nanomechanics. Science
**2000**, 288, 316. [Google Scholar] [CrossRef] [Green Version] - Ramos, D.; Tamayo, J.; Mertens, J.; Calleja, M.; Zaballos, A. Origin of the response of nanomechanical resonators to bacteria adsorption. J. Appl. Phys.
**2006**, 100, 106105. [Google Scholar] [CrossRef] [Green Version] - Tamayo, J.; Ramos, D.; Mertens, J.; Calleja, M. Effect of the adsorbate stiffness on the resonance response of microcantilever sensors. Appl. Phys. Lett.
**2006**, 89, 224104. [Google Scholar] [CrossRef] - Dominguez-Medina, S.; Fostner, S.; Defoort, M.; Sansa, M.; Stark, A.-K.; Halim, M.A.; Vernhes, E.; Gely, M.; Jourdan, G.; Alava, T.; et al. Neutral mass spectrometry of virus capsids above 100 megadaltons with nanomechanical resonators. Science
**2018**, 362, 918. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gil-Santos, E.; Ramos, D.; Martínez, J.; Fernández-Regúlez, M.; García, R.; San Paulo, Á.; Calleja, M.; Tamayo, J. Nanomechanical mass sensing and stiffness spectrometry based on two-dimensional vibrations of resonant nanowires. Nat. Nanotechnol.
**2010**, 5, 641. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Malvar, O.; Gil-Santos, E.; Ruz, J.; Ramos, D.; Pini, V.; Fernández-Regúlez, M.; Calleja, M.; Tamayo, J.; Paulo, A. Tapered silicon nanowires for enhanced nanomechanical sensing. Appl. Phys. Lett.
**2013**, 103. [Google Scholar] [CrossRef] [Green Version] - Yen, Y.-K.; Chiu, C.-Y. A CMOS MEMS-based Membrane-Bridge Nanomechanical Sensor for Small Molecule Detection. Sci. Rep.
**2020**, 10, 2931. [Google Scholar] [CrossRef] [PubMed] - Burg, T.P.; Manalis, S.R. Suspended microchannel resonators for biomolecular detection. Appl. Phys. Lett.
**2003**, 83, 2698–2700. [Google Scholar] [CrossRef] [Green Version] - Godin, M.; Bryan, A.K.; Burg, T.P.; Babcock, K.; Manalis, S.R. Measuring the mass, density, and size of particles and cells using a suspended microchannel resonator. Appl. Phys. Lett.
**2007**, 91, 123121. [Google Scholar] [CrossRef] [Green Version] - Malvar, O.; Ramos, D.; Martínez, C.; Kosaka, P.; Tamayo, J.; Calleja, M. Highly sensitive measurement of liquid density in air using suspended microcapillary resonators. Sensors
**2015**, 15, 7650–7657. [Google Scholar] [CrossRef] - Lee, I.; Park, K.; Lee, J. Note: Precision viscosity measurement using suspended microchannel resonators. Rev. Sci. Instrum.
**2012**, 83, 116106. [Google Scholar] [CrossRef] - Linden, J.; Thyssen, A.; Oesterschulze, E. Suspended plate microresonators with high quality factor for the operation in liquids. Appl. Phys. Lett.
**2014**, 104, 191906. [Google Scholar] [CrossRef] - Dohn, S.; Sandberg, R.; Svendsen, W.; Boisen, A. Enhanced functionality of cantilever based mass sensors using higher modes. Appl. Phys. Lett.
**2005**, 86, 233501. [Google Scholar] [CrossRef] [Green Version] - Xu, W.; Choi, S.; Chae, J. A contour-mode film bulk acoustic resonator of high quality factor in a liquid environment for biosensing applications. Appl. Phys. Lett.
**2010**, 96, 053703. [Google Scholar] [CrossRef] [Green Version] - Ramos, D.; Mertens, J.; Calleja, M.; Tamayo, J. Phototermal self-excitation of nanomechanical resonators in liquids. Appl. Phys. Lett.
**2008**, 92, 173108. [Google Scholar] [CrossRef] - Burg, T.P.; Godin, M.; Knudsen, S.M.; Shen, W.; Carlson, G.; Foster, J.S.; Babcock, K.; Manalis, S.R. Weighing of biomolecules, single cells and single nanoparticles in fluid. Nature
**2007**, 446, 1066–1069. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Favero, I.; Karrai, K. Optomechanics of deformable optical cavities. Nat. Photonics
**2009**, 3, 201–205. [Google Scholar] [CrossRef] - Fong, K.Y.; Poot, M.; Tang, H.X. Nano-Optomechanical Resonators in Microfluidics. Nano Lett.
**2015**, 15, 6116–6120. [Google Scholar] [CrossRef] [Green Version] - Gil-Santos, E.; Baker, C.; Nguyen, D.T.; Hease, W.; Gomez, C.; Lemaître, A.; Ducci, S.; Leo, G.; Favero, I. High-frequency nano-optomechanical disk resonators in liquids. Nat. Nanotechnol.
**2015**, 10, 810. [Google Scholar] [CrossRef] - Gil-Santos, E.; Ramos, D.; Pini, V.; Llorens, J.; Fernandez-Regulez, M.; Calleja, M.; Tamayo, J.; San Paulo, A. Optical back-action in silicon nanowire resonators: Bolometric versus radiation pressure effects. New J. Phys.
**2013**, 15. [Google Scholar] [CrossRef] [Green Version] - Gil-Santos, E.; Ruz, J.J.; Malvar, O.; Favero, I.; Lemaître, A.; Kosaka, P.M.; García-López, S.; Calleja, M.; Tamayo, J. Optomechanical detection of vibration modes of a single bacterium. Nat. Nanotechnol.
**2020**, 15, 469–474. [Google Scholar] [CrossRef] - Kosaka, P.M.; Calleja, M.; Tamayo, J. Optomechanical devices for deep plasma cancer proteomics. Semin. Cancer Biol.
**2018**, 52, 26–38. [Google Scholar] [CrossRef] - Zhang, H.; Xj, Z.; Wang, Y.; Huang, Q.; Xia, J. Femtogram scale high frequency nano-optomechanical resonators in water. Opt. Express
**2017**, 25, 821. [Google Scholar] [CrossRef] - Cleland, A.N.; Roukes, M.L. Noise processes in nanomechanical resonators. J. Appl. Phys.
**2002**, 92, 2758–2769. [Google Scholar] [CrossRef] [Green Version] - Stoney, G.G. The tension of metallic films deposited by electrolysis. Proc. R. Soc. Lond. Ser. A Contain. Pap. A Math. Phys. Character
**1909**, 82, 172–175. [Google Scholar] - Gruber, K.; Horlacher, T.; Castelli, R.; Mader, A.; Seeberger, P.H.; Hermann, B.A. Cantilever Array Sensors Detect Specific Carbohydrate−Protein Interactions with Picomolar Sensitivity. ACS Nano
**2011**, 5, 3670–3678. [Google Scholar] [CrossRef] [PubMed] - Mader, A.; Gruber, K.; Castelli, R.; Hermann, B.A.; Seeberger, P.H.; Rädler, J.O.; Leisner, M. Discrimination of Escherichia coli Strains using Glycan Cantilever Array Sensors. Nano Lett.
**2012**, 12, 420–423. [Google Scholar] [CrossRef] - McKendry, R.; Zhang, J.; Arntz, Y.; Strunz, T.; Hegner, M.; Lang, H.P.; Baller, M.K.; Certa, U.; Meyer, E.; Güntherodt, H.-J.; et al. Multiple label-free biodetection and quantitative DNA-binding assays on a nanomechanical cantilever array. Proc. Natl. Acad. Sci. USA
**2002**, 99, 9783. [Google Scholar] [CrossRef] [Green Version] - Sader, J.E. Surface stress induced deflections of cantilever plates with applications to the atomic force microscope: Rectangular plates. J. Appl. Phys.
**2001**, 89, 2911–2921. [Google Scholar] [CrossRef] [Green Version] - Tamayo, J.; Ruz, J.J.; Pini, V.; Kosaka, P.; Calleja, M. Quantification of the surface stress in microcantilever biosensors: Revisiting Stoney’s equation. Nanotechnology
**2012**, 23, 475702. [Google Scholar] [CrossRef] [Green Version] - Thundat, P.I.O.; Warmack, R.J. MICROCANTILEVER SENSORS. Microscale Eng.
**1997**, 1, 185–199. [Google Scholar] [CrossRef] - Ilic, B.; Czaplewski, D.; Craighead, H.G.; Neuzil, P.; Campagnolo, C.; Batt, C. Mechanical resonant immunospecific biological detector. Appl. Phys. Lett.
**2000**, 77, 450–452. [Google Scholar] [CrossRef] - Ruz, J.J.; Tamayo, J.; Pini, V.; Kosaka, P.M.; Calleja, M. Physics of nanomechanical spectrometry of viruses. Sci. Rep.
**2014**, 4, 1–11. [Google Scholar] [CrossRef] [Green Version] - Ruz, J.J.; Malvar, O.; Gil-Santos, E.; Calleja, M.; Tamayo, J. Effect of particle adsorption on the eigenfrequencies of nano-mechanical resonators. J. Appl. Phys.
**2020**, 128, 104503. [Google Scholar] [CrossRef] - Ramos, D.; Mertens, J.; Calleja, M.; Tamayo, J. Study of the origin of bending induced by bimetallic effect on microcantilever. Sensors
**2007**, 7, 1757. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ramos, D.; Tamayo, J.; Mertens, J.; Calleja, M. Photothermal excitation of microcantilevers in liquids. J. Appl. Phys.
**2006**, 99, 124904. [Google Scholar] [CrossRef] [Green Version] - Toda, M.; Inomata, N.; Ono, T.; Voiculescu, I. Cantilever beam temperature sensors for biological applications. IEEJ Trans. Electr. Electron. Eng.
**2017**, 12, 153–160. [Google Scholar] [CrossRef] [Green Version] - Wu, L.; Cheng, T.; Zhang, Q.-C. A bi-material microcantilever temperature sensor based on optical readout. Measurement
**2012**, 45, 1801–1806. [Google Scholar] [CrossRef] - Kang, K.; Nilsen-Hamilton, M.; Shrotriya, P. Differential surface stress sensor for detection of chemical and biological species. Appl. Phys. Lett.
**2008**, 93, 143107. [Google Scholar] [CrossRef] [Green Version] - Sang, S.; Zhao, Y.; Zhang, W.; Li, P.; Hu, J.; Li, G. Surface stress-based biosensors. Biosens. Bioelectron.
**2014**, 51, 124–135. [Google Scholar] [CrossRef] - Godin, M.; Tabard-Cossa, V.; Miyahara, Y.; Monga, T.; Williams, P.J.; Beaulieu, L.Y.; Lennox, R.B.; Grutter, P. Cantilever-based sensing: The origin of surface stress and optimization strategies. Nanotechnology
**2010**, 21, 075501. [Google Scholar] [CrossRef] - Haiss, W. Surface stress of clean and adsorbate-covered solids. Rep. Prog. Phys.
**2001**, 64, 591–648. [Google Scholar] [CrossRef] - Ibach, H. Adsorbate-induced surface stress. J. Vac. Sci. Technol. A Vac. Surf. Film.
**1994**, 12, 2240–2245. [Google Scholar] [CrossRef] - Timoshenko, S.; Woinowsky-Krieger, S. Theory of Plates and Shells; McGraw-Hill: New York, NY, USA, 1959. [Google Scholar]
- Zeng, X.; Deng, J.; Luo, X. Deflection of a cantilever rectangular plate induced by surface stress with applications to surface stress measurement. J. Appl. Phys.
**2012**, 111, 083531. [Google Scholar] [CrossRef] - Timoshenko, S.P. Theory of Elasticity; McGraw-Hill Education (India) Pvt Limited: Noida, India, 2010. [Google Scholar]
- Ziegler, C. Cantilever-based biosensors. Anal. Bioanal. Chem.
**2004**, 379, 946–959. [Google Scholar] [CrossRef] [PubMed] - Karabalin, R.B.; Villanueva, L.G.; Matheny, M.H.; Sader, J.E.; Roukes, M.L. Stress-induced variations in the stiffness of micro-and nanocantilever beams. Phys. Rev. Lett.
**2012**, 108, 236101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lachut, M.J.; Sader, J.E. Effect of surface stress on the stiffness of thin elastic plates and beams. Phys. Rev. B
**2012**, 85, 085440. [Google Scholar] [CrossRef] [Green Version] - Chen, G.Y.; Thundat, T.; Wachter, E.A.; Warmack, R.J. Adsorption-induced surface stress and its effects on resonance frequency of microcantilevers. J. Appl. Phys.
**1995**, 77, 3618–3622. [Google Scholar] [CrossRef] - Gurtin, M.E.; Markenscoff, X.; Thurston, R.N. Effect of surface stress on the natural frequency of thin crystals. Appl. Phys. Lett.
**1976**, 29, 529–530. [Google Scholar] [CrossRef] - Lagowski, J.; Gatos, H.C.; Sproles, E.S. Surface stress and the normal mode of vibration of thin crystals:GaAs. Appl. Phys. Lett.
**1975**, 26, 493–495. [Google Scholar] [CrossRef] - Lu, P.; Lee, H.P.; Lu, C.; O’shea, S.J. Surface stress effects on the resonance properties of cantilever sensors. Phys. Rev. B
**2005**, 72, 085405. [Google Scholar] [CrossRef] - Pini, V.; Tamayo, J.; Gil-Santos, E.; Ramos, D.; Kosaka, P.; Tong, H.-D.; van Rijn, C.; Calleja, M. Shedding Light on Axial Stress Effect on Resonance Frequencies of Nanocantilevers. ACS Nano
**2011**, 5, 4269–4275. [Google Scholar] [CrossRef] [Green Version] - Lachut, M.J.; Sader, J.E. Effect of surface stress on the stiffness of cantilever plates. Phys. Rev. Lett.
**2007**, 99, 206102. [Google Scholar] [CrossRef] - Pini, V.; Ruz, J.; Kosaka, P.; Malvar, O.; Calleja, M.; Tamayo, J. How two-dimensional bending can extraordinarily stiffen thin sheets. Sci. Rep.
**2016**, 6. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ruz, J.J.; Pini, V.; Malvar, O.; Kosaka, P.M.; Calleja, M.; Tamayo, J. Effect of surface stress induced curvature on the eigenfrequencies of microcantilever plates. AIP Adv.
**2018**, 8, 105213. [Google Scholar] [CrossRef] - Ramos, D.; Tamayo, J.; Mertens, J.; Calleja, M.; Villanueva, L.G.; Zaballos, A. Detection of bacteria based on the thermomechanical noise of a nanomechanical resonator: Origin of the response and detection limits. Nanotechnology
**2007**, 19, 035503. [Google Scholar] [CrossRef] [PubMed] - Hanay, M.S.; Kelber, S.I.; O’Connell, C.D.; Mulvaney, P.; Sader, J.E.; Roukes, M.L. Inertial imaging with nanomechanical systems. Nat. Nanotechnol.
**2015**, 10, 339–344. [Google Scholar] [CrossRef] [PubMed] - Naik, A.K.; Hanay, M.S.; Hiebert, W.K.; Feng, X.L.; Roukes, M.L. Towards single-molecule nanomechanical mass spectrometry. Nat. Nanotechnol.
**2009**, 4, 445–450. [Google Scholar] [CrossRef] [PubMed] - Ramos, D.; Malvar, O.; Davis, Z.J.; Tamayo, J.; Calleja, M. Nanomechanical Plasmon Spectroscopy of Single Gold Nanoparticles. Nano Lett.
**2018**, 18, 7165–7170. [Google Scholar] [CrossRef] - Sage, E.; Brenac, A.; Alava, T.; Morel, R.; Dupré, C.; Hanay, M.S.; Roukes, M.L.; Duraffourg, L.; Masselon, C.; Hentz, S. Neutral particle mass spectrometry with nanomechanical systems. Nat. Commun.
**2015**, 6, 1–5. [Google Scholar] [CrossRef] [Green Version] - Ekinci, K.L.; Roukes, M.L. Nanoelectromechanical systems. Rev. Sci. Instrum.
**2005**, 76, 061101. [Google Scholar] [CrossRef] [Green Version] - Sansa, M.; Sage, E.; Bullard, E.C.; Gély, M.; Alava, T.; Colinet, E.; Naik, A.K.; Villanueva, L.G.; Duraffourg, L.; Roukes, M.L.; et al. Frequency fluctuations in silicon nanoresonators. Nat. Nanotechnol.
**2016**, 11, 552–558. [Google Scholar] [CrossRef] [Green Version] - Kiracofe, D.; Raman, A. On eigenmodes, stiffness, and sensitivity of atomic force microscope cantilevers in air versus liquids. J. Appl. Phys.
**2010**, 107, 033506. [Google Scholar] [CrossRef] [Green Version] - Lee, J.W.; Tung, R.; Raman, A.; Sumali, H.; Sullivan, J.P. Squeeze-film damping of flexible microcantilevers at low ambient pressures: Theory and experiment. J. Micromech. Microeng.
**2009**, 19, 105029. [Google Scholar] [CrossRef] - Ricci, A.; Canavese, G.; Ferrante, I.; Marasso, S.L.; Ricciardi, C. A finite element model for the frequency spectrum estimation of a resonating microplate in a microfluidic chamber. Microfluid. Nanofluid.
**2013**, 15, 275–284. [Google Scholar] [CrossRef] - Ricciardi, C.; Canavese, G.; Castagna, R.; Ferrante, I.; Ricci, A.; Marasso, S.L.; Napione, L.; Bussolino, F. Integration of microfluidic and cantilever technology for biosensing application in liquid environment. Biosens. Bioelectron.
**2010**, 26, 1565–1570. [Google Scholar] [CrossRef] [PubMed] - Papi, M.; Arcovito, G.; De Spirito, M.; Vassalli, M.; Tiribilli, B. Fluid viscosity determination by means of uncalibrated atomic force microscopy cantilevers. Appl. Phys. Lett.
**2006**, 88, 194102. [Google Scholar] [CrossRef] [Green Version] - Paul, M.R.; Clark, M.T.; Cross, M.C. The stochastic dynamics of micron and nanoscale elastic cantilevers in fluid: Fluctuations from dissipation. Nanotechnology
**2006**, 17, 4502–4513. [Google Scholar] [CrossRef] - Sader, J.E. Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. J. Appl. Phys.
**1998**, 84, 64–76. [Google Scholar] [CrossRef] [Green Version] - Van Eysden, C.A.; Sader, J.E. Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope: Arbitrary mode order. J. Appl. Phys.
**2007**, 101, 044908. [Google Scholar] [CrossRef] - Maali, A.; Hurth, C.; Boisgard, R.; Jai, C.; Cohen-Bouhacina, T.; Aimé, J.-P. Hydrodynamics of oscillating atomic force microscopy cantilevers in viscous fluids. J. Appl. Phys.
**2005**, 97, 074907. [Google Scholar] [CrossRef] - Son, S.; Tzur, A.; Weng, Y.; Jorgensen, P.; Kim, J.; Kirschner, M.W.; Manalis, S.R. Direct observation of mammalian cell growth and size regulation. Nat. Methods
**2012**, 9, 910–912. [Google Scholar] [CrossRef] [Green Version] - Martín-Pérez, A.; Ramos, D.; Tamayo, J.; Calleja, M. Real-Time Particle Spectrometry in Liquid Environment Using Microfluidic-Nanomechanical Resonators. In Proceedings of the 2019 20th International Conference on Solid-State Sensors, Actuators and Microsystems & Eurosensors XXXIII (TRANSDUCERS & EUROSENSORS XXXIII), Berlin, Germany, 23–27 June 2019; pp. 2146–2149. [Google Scholar]
- Olcum, S.; Cermak, N.; Wasserman, S.C.; Christine, K.S.; Atsumi, H.; Payer, K.R.; Shen, W.; Lee, J.; Belcher, A.M.; Bhatia, S.N.; et al. Weighing nanoparticles in solution at the attogram scale. Proc. Natl. Acad. Sci. USA
**2014**, 111, 1310. [Google Scholar] [CrossRef] [Green Version] - Martín-Pérez, A.; Ramos, D.; Tamayo, J.; Calleja, M. Coherent Optical Transduction of Suspended Microcapillary Resonators for Multi-Parameter Sensing Applications. Sensors
**2019**, 19, 5069. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pini, V.; Ramos, D.; Domínguez, C.M.; Ruz, J.; Malvar, O.; Kosaka, P.; Davis, Z.; Tamayo, J.; Calleja, M. Optimization of the readout of microdrum optomechanical resonators. Microelectron. Eng.
**2017**, 183, 37–41. [Google Scholar] [CrossRef] - Martín-Pérez, A.; Ramos, D.; Gil-Santos, E.; García-López, S.; Yubero, M.L.; Kosaka, P.M.; San Paulo, Á.; Tamayo, J.; Calleja, M. Mechano-Optical Analysis of Single Cells with Transparent Microcapillary Resonators. ACS Sens.
**2019**, 4, 3325–3332. [Google Scholar] [CrossRef] [PubMed] - Han, K.; Suh, J.; Bahl, G. Optomechanical non-contact measurement of microparticle compressibility in liquids. Opt. Express
**2018**, 26, 31908–31916. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Schematic depiction of the two different operation modes, static and dynamic, in which a nanomechanical sensor can function, showing their basic principle of operation when used for biological applications. (

**b**) Chronogram of the publication and citation rate in the nanomechanical sensing field, focused on biological applications, and splitting them into static (red) and dynamic (black) operation modes.

**Figure 2.**(

**a**) Schematic of a cantilever plate that has been bent by the action of differential surface stress between the upper and lower faces. (

**b**) Contour lines of the longitudinal and transversal curvatures normalized to Stoney’s curvature for a cantilever plate. It can be seen that, far from the clamped region, both curvatures follow Stoney’s equation, which is not the case for the region close to the clamping.

**Figure 3.**(

**a**) Theoretical prediction of the relative frequency shift due to net surface stress ${\sigma}^{T}$ for the fundamental mode of a silicon nitride cantilever of 100 nm in thickness and 20 µm in length for different values of the width. (

**b**–

**d**) The theoretical prediction of the relative frequency shift due to the differential surface stress $\Delta \sigma $ for the same cantilever and for the first, second, and third flexural modes, respectively, as well as for different values of the width. Insets show a representation of the mode shape.

**Figure 4.**(

**a**) Influence of the adsorbed layer on the flexural modes of a silicon beam as a function of the ratio between thicknesses $\eta $ and for different values of the Young’s modulus of the adsorbed material. (

**b**) Same graph, but at the range of values of $\eta $ where the linear approximation (21) can be applied.

**Figure 5.**(

**a**) Schematic depiction of a bacterium adsorbed longitudinally and transversally oriented on a resonator. (

**b**,

**c**) Relative frequency shift produced by the adsorption of a typical E. coli bacterium on a silicon nitride cantilever and a doubly clamped beam of 200 × 20 × 0.6 µm for the first three flexural modes. It can be seen that, because of the edge effects of the bacterium, the stiffness effect is considerably smaller when the bacterium is transversally oriented.

**Figure 6.**Schematic of the particle’s mass and stiffness determination using multimode measuring. In a first step, the frequencies of the resonator are tracked and the relative frequency shifts produced by the particle adsorption are recorded. In a second step, the joint probability density function ($JPDF$) is formed using the frequency noise for each of the modes being tracked. Finally, the $JPDF$ can be integrated in the normalized position ${X}_{0}$ and either in ${\Delta}_{s}$ or ${\Delta}_{m}$ to obtain the mass or the stiffness $PDF$, respectively.

**Figure 7.**(

**a**) Real (solid line) and imaginary (dashed line) parts of the hydrodynamic function as a function of the oscillation frequency for a rectangular cantilever of 200 μm in length and 20 μm in width immersed in water. (

**b**) Theoretical simulation of amplitude spectra of two equivalent nanomechanical resonators: one oscillating in vacuum (solid line) and one oscillating in liquid (dashed line). The effective mass and damping constant of the resonator in vacuum were consequently changed to obtain an equivalent liquid immersed resonator.

**Figure 8.**(

**a**) Schematic depiction of a hollow transparent suspended capillary with flowing particles inside. (

**b**) Theoretical prediction of the relative frequency shift for the first mode of the capillary for different masses as a function of the position.

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## Share and Cite

**MDPI and ACS Style**

Ruz, J.J.; Malvar, O.; Gil-Santos, E.; Ramos, D.; Calleja, M.; Tamayo, J.
A Review on Theory and Modelling of Nanomechanical Sensors for Biological Applications. *Processes* **2021**, *9*, 164.
https://doi.org/10.3390/pr9010164

**AMA Style**

Ruz JJ, Malvar O, Gil-Santos E, Ramos D, Calleja M, Tamayo J.
A Review on Theory and Modelling of Nanomechanical Sensors for Biological Applications. *Processes*. 2021; 9(1):164.
https://doi.org/10.3390/pr9010164

**Chicago/Turabian Style**

Ruz, Jose Jaime, Oscar Malvar, Eduardo Gil-Santos, Daniel Ramos, Montserrat Calleja, and Javier Tamayo.
2021. "A Review on Theory and Modelling of Nanomechanical Sensors for Biological Applications" *Processes* 9, no. 1: 164.
https://doi.org/10.3390/pr9010164