# 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 |

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**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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**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