# Microcantilever: Dynamical Response for Mass Sensing and Fluid Characterization

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Cantilever Mechanics and Dynamical Response

#### 2.1. Euler–Bernoulli Beam

#### 2.2. Harmonic Oscillations with a Single Degree of Freedom

#### 2.2.1. Simple Harmonic Oscillator

#### 2.2.2. Forced Damped Harmonic Oscillator

#### 2.2.3. General One-Degree-of-Freedom Equation of Motion for Microcantilevers

#### 2.3. Operation in Dissipative Fluids

_{1}= 1.0553, a

_{2}= 3.7997, b

_{1}= 3.8018, and b

_{2}= 2.7364. These expressions are valid for Reynolds numbers ranging between 1 and 1000 [20], which are the typical values for most of the microcantilever-based sensing applications, and can be applied for the first resonance modes. Equations (28)–(31) can be used in conjunction to obtain the dependence between the resonance frequency and quality factor of the resonant mode n and the rheological properties of the fluid, as will be shown in Section 5.

## 3. Excitation Schemes and Noise

#### 3.1. Excitation Strategies

#### 3.1.1. External or Open-Loop Excitation Mechanisms

#### 3.1.2. Feedback or Closed-Loop Excitation Mechanisms

#### 3.2. Detection Mechanisms

#### 3.3. Noise

#### 3.3.1. Time Domain—Allan deviation

#### 3.3.2. Frequency Domain—Spectral Densities

^{2}/Hz. ${S}_{y}\left(f\right)$ and ${S}_{\Phi}\left(f\right)$ are one-sided spectral densities, and apply over a Fourier (or sideband) frequency range f from 0 to $\infty $ [60,61]. The relation between these two quantities is given by [61]:

#### 3.3.3. Conversion between Frequency and Time Domain—Power Law Spectral Densities

#### 3.3.4. Physical Origins of Noise

#### 3.3.5. Minimum Detectable Frequency Shift, ${\delta \mathrm{f}}_{\mathrm{min}}$

## 4. Mass Sensing

#### 4.1. Dynamic vs Static Sensing Modes

#### 4.2. Mass Sensitivity

^{-1}) and can be evaluated from Equation (49) as:

^{th}mode of the intrinsically damped resonator, ${m}_{A}$ and ${m}_{0}$ are the added mass by the fluid and the mass of the resonator, both per unit length, with ${m}_{c}=L{m}_{0}=\frac{{m}_{\mathrm{eff},n}{\left({\beta}_{n}L\right)}^{4}}{3}={\beta}_{n}^{\prime}{m}_{\mathrm{eff},n}$, as shown in Equation (16).

^{th}mode, can then be obtained from Equation (51), by differentiating it with respect to these several parameters, $\delta {f}_{R}=\frac{\delta {f}_{R}}{\delta {k}_{\mathrm{eff}}}\delta {k}_{\mathrm{eff}}+\frac{\delta {f}_{R}}{\delta Q}\delta Q+\frac{\delta {f}_{R}}{\delta L{m}_{A}}\delta L{m}_{A}+\frac{\delta {f}_{R}}{\delta {m}_{\mathrm{eff}}}\delta {m}_{\mathrm{eff}}$. After some cumbersome calculations (see [93] for details) and admitting that the variations in the quality factor (defined in Equation (28b)) and stiffness are negligible ($\delta {k}_{\mathrm{eff}}~0$ and $\delta Q~0$), one obtains:

#### 4.3. Limits of Detection (LoD)

## 5. Viscosity Sensing

#### 5.1. Viscoelastic Materials

#### 5.2. Measuring Rheological Properties of Fluids Using Microcantilevers

#### 5.2.1. Newtonian Fluids

#### 5.2.2. Viscoelastic Fluids

## 6. Outlook and Further Challenges

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## List of Symbols (in Order of Appearance in the Text)

L | cantilever length |

w | cantilever width |

h | cantilever thickness |

t | time |

x | space coordinate (distance from the cantilever support) |

$q\left(x,t\right)$ | time-varying distributed load acting on the beam at a distance x from the support, per unit length |

$W\left(x,t\right)$ | time-varying deflection of the beam at a distance x from the support |

${F}_{z}$ | shear forces acting on the element of the beam |

${M}_{y}$ | bending moment acting on the element of the beam |

$\rho $ | density of the structural material |

$A$ | area of rectangular beam cross section |

${I}_{z}$ | second moment of area of the rectangular cross section beam |

E | Young’s modulus of the structural material |

$\psi \left(t\right)$ | temporal term solution of harmonic oscillation |

$\mathsf{\Phi}\left(x\right)$ | spacial term solution of harmonic oscillation |

${c}_{1,2,3,4}$ | constants of spacial term solution of harmonic oscillation |

${f}_{0,n}$ | natural (undamped) resonance frequency of mode n |

${\omega}_{0,n}$ | natural (undamped) radial resonance frequency of mode n |

$z$ | displacement of the one-degree-of-freedom microcantilever from the equilibrium position (z = 0) |

$\dot{z}$ | velocity of the one-degree-of-freedom microcantilever |

$\ddot{z}$ | acceleration of the one-degree-of-freedom microcantilever |

${k}_{\mathrm{eff}}$ | effective spring constant of the microcantilever |

${m}_{\mathrm{eff}}$ | effective mass of the nth resonant mode of the microcantilever |

${m}_{c}$ | total mass of the microcantilever |

$c$ | intrinsic viscous damping coefficient |

Q | quality factor |

$\omega $ | excitation frequency |

${F}_{0}{e}^{i\omega t}$ | excitation harmonic force at $\omega $, with amplitude ${F}_{0}$ |

${A}_{0}$ | amplitude of the motion at $\omega $ |

$\varphi $ | phase between the applied external force and the motion at $\omega $ |

${\omega}_{res}$ | resonance frequency of the nth mode of intrinsically damped resonators |

${m}_{0}$ | mass of the cantilever per unit length |

${c}_{0}$ | intrinsic viscous damping coefficient per unit length |

${F}_{hydro}\left(x,t\right)$ | time-varying distributed hydrodynamic load, acting on the beam at a distance x, per unit length |

${m}_{A}$ | added mass by interactions with the surrounding fluid, per unit length |

${c}_{V}$ | added damping coefficient by interactions with the surrounding fluid, per unit length |

${\omega}_{R,n}$ | resonance frequency of the nth mode of extrinsically damped resonators with added mass and damping |

${Q}_{n}$ | quality factor of the nth mode |

${\mathsf{\Gamma}}_{rect}{}^{\text{'}}\left(\omega \right)$ | real part of the hydrodynamic load acting on a microcantilever with rectangular cross section |

${\mathsf{\Gamma}}_{rect}{}^{\u2033}\left(\omega \right)$ | imaginary part of the hydrodynamic load acting on a microcantilever with rectangular cross section |

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

$\eta $ | viscosity of the fluid |

$\delta $ | thickness of the layer in which the velocity of the fluid drops by a factor of 1/e |

$Re$ | Reynolds number |

a_{1}, a_{2}, b_{1}, b_{2} | Maali’s constants for ${\mathsf{\Gamma}}_{rect}\left(\omega \right)$ |

τ | integration time |

${\sigma}_{y}\left(\tau \right)$ | Allan deviation for time windows of duration τ |

${f}_{i}$ | consecutive ith frequency measurements |

${f}_{c}$ | nominal carrier frequency |

${S}_{y}\left(f\right)$ | spectral density of frequency fluctuations |

${S}_{\Phi}\left(f\right)$ | spectral density of phase fluctuations |

${y}_{rms}\left(f\right)$ | measured root mean squared (rms) value of normalized frequency |

${\mathsf{\Phi}}_{rms}\left(f\right)$ | measured root mean squared (rms) value of normalized phase |

$BW$ | width of the frequency band in Hz |

${P}_{noise\left(1Hz\right)}\left(f\right)$ | power density in one single sideband due to phase modulation by noise, for a 1 Hz bandwidth (dBm/Hz) |

${P}_{signal}$ | total power of the carrier (dBm) |

$\mathcal{L}\left(f\right)$ | single-sideband phase noise, the ratio of ${P}_{noise\left(1Hz\right)}\left(f\right)$ to ${P}_{signal}$(dBc/Hz) |

${f}_{h}$ | cut-off frequency of an infinitely sharp low-pass filter |

${h}_{-2}$, ${h}_{-1}$, ${h}_{0}$, ${h}_{1}$, ${h}_{2}$ | constants to fit power-laws to random walk frequency noise, flicker of frequency, white frequency noise, flicker of phase and white phase noise, respectively |

$A$, $B$, $C$, $D$, $E$ | numerical constants for conversion between frequency (spectral densities) and time (Allan deviation) domains |

$\delta {f}_{min}$ | minimum measurable frequency shift |

LoD | limit of detection |

$\delta {f}_{0}$ | shift in the natural (undamped) resonance frequency |

$\delta {f}_{R}$ | shift in the damped resonance frequency of microcantilevers with added mass and damping; |

$\delta {k}_{\mathrm{eff}}$ | infinitesimal change of the effective stiffness of the cantilever induced by the adsorbate |

$\delta {m}_{\mathrm{eff}}$ | infinitesimal change of the effective mass of the cantilever induced by the adsorbate |

$\delta {m}_{A}$ | infinitesimal change of the added mass induced by the fluid |

$\delta \eta $ | infinitesimal change in the viscosity of the fluid |

$\delta {\rho}_{f}$ | infinitesimal change in the density of the fluid |

S | sensitivity |

${S}_{mass,vac}$, ${S}_{mass,fluid}$ | mass sensitivity in vacuum and in fluid |

${S}_{viscosity,fluid}$ | viscosity sensitivity |

${\tau}_{A}$, ${\dot{\tau}}_{A}$ | applied shear stress and shear stress rate |

${\delta}_{D}$, ${\dot{\delta}}_{D}$ | shear strain and shear strain rate of a viscous dashpot |

${\delta}_{S}$, ${\dot{\delta}}_{S}$ | shear strain and shear strain rate of an elastic spring |

${\delta}_{tot}$, ${\dot{\delta}}_{tot}$ | shear strain and shear strain rate of the spring-dashpot series |

${G}_{0}$ | elasticity constant of the fluid |

$\lambda $ | characteristic relaxation time of the fluid |

$\omega $ | frequency of the applied shear stress and induced total strain response |

$\phi $ | phase between applied stress and total strain response |

${\tau}_{0}$ | amplitude of the shear stress |

${\delta}_{0}$ | amplitude of the total strain response |

${G}^{*}$ | dynamic elastic modulus |

${G}^{\prime}$, ${G}^{\u2033}$ | elastic and viscous parts of the dynamic elastic modulus |

${\eta}^{*}$ | complex dynamic viscosity |

${\eta}^{\prime}$, ${\eta}^{\u2033}$ | viscous and elastic parts of the dynamic viscosity |

$\left|\frac{H\left(\omega \right)}{{H}_{0}}\right|$ | general ratio of amplitudes of the transfer function |

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**Figure 1.**(

**a**) Schematic of a cantilever beam of length L, width w, and thickness h. The cantilever is subjected to a distributed time-changing load per unit length, q(x, t); (

**b**) longitudinal cross section of an infinitesimal element dx of the same cantilever (red part highlighted in (

**a**)), where the shear forces and bending moments act.

**Figure 2.**(

**a**) Plot of Equation (10) (orange line), whose first four solutions ${\beta}_{n}L$ (indicated in the text) are the crossings with zero. The blue line shows the asymptotic approximation $\mathrm{cos}\left({\beta}_{n}L\right)=0$, which can be solved analytically, and that agrees with the numerical solution for $n\ge 2$; (

**b**) mode shapes of the first four flexural modes of a cantilever.

**Figure 3.**(

**a**) Amplitude and (

**b**) phase responses as function of the normalized excitation frequency of a forced and damped harmonic oscillator. Four different levels of damping (Q) are considered, typically encountered in microcantilevers vibrating in air or liquid mediums.

**Figure 4.**(

**a**) Experimental amplitude response of an acoustically excited microcantilever oscillating in water, showing a noisy peak not accurately fitted by the theoretical amplitude curve; (

**b**) dependence of the amplitude response of a microcantilever on the gain of the feedback loop in the Q-control method, for ${\omega}_{0}=1$ rad/s, original $Q=10$ (for G = 0), $\frac{{F}_{0}}{{m}_{\mathrm{eff}}}=1$ N/kg and $\varphi =\frac{\pi}{2}$ rad.

**Figure 5.**Power-laws of the spectral density of phase fluctuations, ${S}_{\Phi}\left(f\right)$, and corresponding noise mechanisms. A typical experimental curve of ${S}_{\Phi}\left(f\right)$ is plotted in light grey. The cut-off frequency (f

_{h}) is also indicated.

**Figure 6.**Illustration of a biosensing strategy with a microcantilever. The cantilever is functionalized with antibodies on the top surface (these are the biorecognition agents, which are chemically bound to the surface). Due to the specific binding of analytes (in this simple example, the red antigens are captured by the antibodies but the green ones are not), a resonance frequency shift or a static deflection will occur. On the bottom, the scale of the size and mass of bacteria, virus particles, proteins, and micro RNA strands is illustrated.

**Figure 7.**Mass sensing strategies. (

**a**) Dynamic mode: the mass adsorbed on the surface of the cantilever causes changes in both its effective mass and stiffness, resulting in a shift in the resonance frequency, $\delta {f}_{0}$, to lower or higher values, depending on which effect dominates (red and green spectra). (

**b**) Static deflection mode: the sensor deflects by a quantity $\delta z$ due to intermolecular interactions at the surface of the sensor. In the case of a DNA biosensing experiment, the surface stress is due to the electrostatic repulsion between the molecules at the surface when complementary DNA (red strands) hybridizes with the initially immobilized DNA probes (green strands).

**Figure 8.**(

**a**) Elastic (real) and viscous (imaginary) parts of the dynamic elastic modulus; (

**b**) viscous (real) and elastic (imaginary) parts of the complex viscosity for $\lambda =1$ s, ${G}_{0}=1$ Pa, and $\eta =1$ Pa s in a Maxwell fluid.

**Figure 9.**(

**a**) Added mass and (

**b**) viscous damping per unit length of a rectangular microcantilever (width w = 10 µm) as a function of the frequency of oscillation in the fluid (water, ρ

_{f}= 1000 kg/m

^{3}) and of the elastic and viscous parts of the dynamic modulus, calculated with Equations (70) and (71).

**Figure 10.**(

**a**) Elastic and (

**b**) viscous components of the dynamic modulus of a rectangular microcantilever (width w = 10 µm) as a function of the frequency of oscillation in the fluid (water, ρ

_{f}= 1000 kg/m

^{3}) and of the added mass and viscous coefficient, calculated with Equations (73) and (74).

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Mouro, J.; Pinto, R.; Paoletti, P.; Tiribilli, B. Microcantilever: Dynamical Response for Mass Sensing and Fluid Characterization. *Sensors* **2021**, *21*, 115.
https://doi.org/10.3390/s21010115

**AMA Style**

Mouro J, Pinto R, Paoletti P, Tiribilli B. Microcantilever: Dynamical Response for Mass Sensing and Fluid Characterization. *Sensors*. 2021; 21(1):115.
https://doi.org/10.3390/s21010115

**Chicago/Turabian Style**

Mouro, João, Rui Pinto, Paolo Paoletti, and Bruno Tiribilli. 2021. "Microcantilever: Dynamical Response for Mass Sensing and Fluid Characterization" *Sensors* 21, no. 1: 115.
https://doi.org/10.3390/s21010115