Microcantilever: Dynamical Response for Mass Sensing and Fluid Characterization
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
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
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,
4. Mass Sensing
4.1. Dynamic vs Static Sensing Modes
4.2. Mass Sensitivity
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) |
time-varying distributed load acting on the beam at a distance x from the support, per unit length | |
time-varying deflection of the beam at a distance x from the support | |
shear forces acting on the element of the beam | |
bending moment acting on the element of the beam | |
density of the structural material | |
area of rectangular beam cross section | |
second moment of area of the rectangular cross section beam | |
E | Young’s modulus of the structural material |
temporal term solution of harmonic oscillation | |
spacial term solution of harmonic oscillation | |
constants of spacial term solution of harmonic oscillation | |
natural (undamped) resonance frequency of mode n | |
natural (undamped) radial resonance frequency of mode n | |
displacement of the one-degree-of-freedom microcantilever from the equilibrium position (z = 0) | |
velocity of the one-degree-of-freedom microcantilever | |
acceleration of the one-degree-of-freedom microcantilever | |
effective spring constant of the microcantilever | |
effective mass of the nth resonant mode of the microcantilever | |
total mass of the microcantilever | |
intrinsic viscous damping coefficient | |
Q | quality factor |
excitation frequency | |
excitation harmonic force at , with amplitude | |
amplitude of the motion at | |
phase between the applied external force and the motion at | |
resonance frequency of the nth mode of intrinsically damped resonators | |
mass of the cantilever per unit length | |
intrinsic viscous damping coefficient per unit length | |
time-varying distributed hydrodynamic load, acting on the beam at a distance x, per unit length | |
added mass by interactions with the surrounding fluid, per unit length | |
added damping coefficient by interactions with the surrounding fluid, per unit length | |
resonance frequency of the nth mode of extrinsically damped resonators with added mass and damping | |
quality factor of the nth mode | |
real part of the hydrodynamic load acting on a microcantilever with rectangular cross section | |
imaginary part of the hydrodynamic load acting on a microcantilever with rectangular cross section | |
density of the fluid | |
viscosity of the fluid | |
thickness of the layer in which the velocity of the fluid drops by a factor of 1/e | |
Reynolds number | |
a1, a2, b1, b2 | Maali’s constants for |
τ | integration time |
Allan deviation for time windows of duration τ | |
consecutive ith frequency measurements | |
nominal carrier frequency | |
spectral density of frequency fluctuations | |
spectral density of phase fluctuations | |
measured root mean squared (rms) value of normalized frequency | |
measured root mean squared (rms) value of normalized phase | |
width of the frequency band in Hz | |
power density in one single sideband due to phase modulation by noise, for a 1 Hz bandwidth (dBm/Hz) | |
total power of the carrier (dBm) | |
single-sideband phase noise, the ratio of to (dBc/Hz) | |
cut-off frequency of an infinitely sharp low-pass filter | |
, , , , | constants to fit power-laws to random walk frequency noise, flicker of frequency, white frequency noise, flicker of phase and white phase noise, respectively |
, , , , | numerical constants for conversion between frequency (spectral densities) and time (Allan deviation) domains |
minimum measurable frequency shift | |
LoD | limit of detection |
shift in the natural (undamped) resonance frequency | |
shift in the damped resonance frequency of microcantilevers with added mass and damping; | |
infinitesimal change of the effective stiffness of the cantilever induced by the adsorbate | |
infinitesimal change of the effective mass of the cantilever induced by the adsorbate | |
infinitesimal change of the added mass induced by the fluid | |
infinitesimal change in the viscosity of the fluid | |
infinitesimal change in the density of the fluid | |
S | sensitivity |
, | mass sensitivity in vacuum and in fluid |
viscosity sensitivity | |
, | applied shear stress and shear stress rate |
, | shear strain and shear strain rate of a viscous dashpot |
, | shear strain and shear strain rate of an elastic spring |
, | shear strain and shear strain rate of the spring-dashpot series |
elasticity constant of the fluid | |
characteristic relaxation time of the fluid | |
frequency of the applied shear stress and induced total strain response | |
phase between applied stress and total strain response | |
amplitude of the shear stress | |
amplitude of the total strain response | |
dynamic elastic modulus | |
, | elastic and viscous parts of the dynamic elastic modulus |
complex dynamic viscosity | |
, | viscous and elastic parts of the dynamic viscosity |
general ratio of amplitudes of the transfer function |
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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
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 StyleMouro, 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
APA StyleMouro, J., Pinto, R., Paoletti, P., & Tiribilli, B. (2021). Microcantilever: Dynamical Response for Mass Sensing and Fluid Characterization. Sensors, 21(1), 115. https://doi.org/10.3390/s21010115