# Photothermal Self-Excitation of a Phase-Controlled Microcantilever for Viscosity or Viscoelasticity Sensing

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

#### 2.1.1. Optomechanical Block

#### 2.1.2. Demodulating and Control Electronics

^{6}/2

^{28}= 0.0894 Hz.

#### 2.1.3. Electrical Scheme of the PLL

#### 2.2. General Modeling Equations of the PLL

#### 2.3. General PLL Model Applied to Purely Viscous Newtonian Fluids

_{1}= 1.0553, a

_{2}= 3.7997, b

_{1}= 3.8018, and b

_{2}= 2.7364 are constants to describe the hydrodynamic function [30], and $L$ and $W$ are the length and width of the microcantilever. These expressions are shown in Part I.1 of the Supplementary Materials.

_{0}= 164.98 kHz) and a fixed delay in the loop ($\tau =9.0$ µs). In this simulation, the viscosity of the Newtonian liquid was swept between $\eta =5\times {10}^{-4}$ Pa s and $\eta =3\times {10}^{-3}$ Pa s, in steps of $\mathsf{\Delta}\eta =5\times {10}^{-4}$ Pa s. The density of the solution was assumed to be constant and independent of the viscosity, $\rho =998$ kg/m

^{3}at 20 °C [23].

#### 2.4. General PLL Model Applied to Viscoelastic Non-Newtonian Fluids

## 3. Results

_{0}= 65.04 kHz and Q = 250.

#### 3.1. Newtonian Water/Glycerol Solutions

^{3}[23].

^{12}). This value is then applied to the other modelled curves in Figure 4a.

#### 3.2. Non-Newtonian Fluid

## 4. Discussion

#### 4.1. Sensitivity of the PLL Platform to Small Changes in Rheological Parameters

#### 4.1.1. Sensitivity to Viscosity Variations in Newtonian Fluids

#### 4.1.2. Sensitivity to Variations of Elastic and Viscous Modulus in Non-Newtonian Fluids

#### 4.2. Inversion Problem to Extract G′ and G″ from the Measured Frequency and Amplitude of the Oscillations in the PLL

#### 4.2.1. Analytical Method

#### 4.2.2. Issues to Implement the Inversion Method

## 5. Conclusions

## Supplementary Materials

**G**′ and

**G**″ from the measured frequency and amplitude of the oscillations in the PLL, II.1. Determining

**m**and

_{A}**c**from the oscillation frequency and amplitude of the PLL, II.2. Determining

_{A}**G**′ and

**G**″ from the calculated

**m**and

_{A}**c**, References, Figure S1: Schematic of the electrical signals through the developed PLL platform.

_{A}## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Frequencies of oscillation (

**left**) and quadrature component of deflection I (

**right**), as function of the imposed phase $\varphi $ in the PLL and the viscosity of a Newtonian fluid. Two branches are observed for $n=2$ (

**left branch**) and $n=3$ (

**right branch**). Viscosity values range between $\eta =5\times {10}^{-4}$ Pa s and $\eta =3\times {10}^{-3}$ Pa s, in steps of $\mathsf{\Delta}\eta =5\times {10}^{-4}$ Pa s.

**Figure 3.**Frequencies of oscillation and quadrature component of deflection I, as a function of the imposed phase in the PLL and the (

**a**) elastic modulus and (

**b**) viscous modulus of a viscoelastic fluid. The dashed burgundy line represents the case of purely viscous water. Only the branch with n = 2 is shown.

**Figure 4.**(

**a**) Frequency of oscillation and quadrature component of deflection I, as a function of the imposed phase in the PLL, for different concentrations of glycerol solutions (solid and dashed lines represent experimental and modelled results, respectively); (

**b**) frequency of oscillation, as a function of the imposed phase in the PLL, for water and a viscoelastic solution of 4000 ppm of PAM.

**Figure 5.**(

**a**) Sensitivity to variations in viscosity in a Newtonian fluid (solid and dashed lines represent experimental and modelled results, respectively); (

**b**) modelled sensitivity to variations in the elastic and viscous modulus in a non-Newtonian viscoelastic fluid.

**Figure 6.**Inversion problem: the frequency and normalised amplitude should be experimentally measured, as indicated in the left panel (these curves were analytically built, instead). From these curves, the added mass and damping terms are calculated, as plotted in the middle panel. Finally, the viscous and elastic modulus of the fluid can be determined, as shown in the right panel.

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**MDPI and ACS Style**

Mouro, J.; Paoletti, P.; Sartore, M.; Vassalli, M.; Tiribilli, B. Photothermal Self-Excitation of a Phase-Controlled Microcantilever for Viscosity or Viscoelasticity Sensing. *Sensors* **2022**, *22*, 8421.
https://doi.org/10.3390/s22218421

**AMA Style**

Mouro J, Paoletti P, Sartore M, Vassalli M, Tiribilli B. Photothermal Self-Excitation of a Phase-Controlled Microcantilever for Viscosity or Viscoelasticity Sensing. *Sensors*. 2022; 22(21):8421.
https://doi.org/10.3390/s22218421

**Chicago/Turabian Style**

Mouro, João, Paolo Paoletti, Marco Sartore, Massimo Vassalli, and Bruno Tiribilli. 2022. "Photothermal Self-Excitation of a Phase-Controlled Microcantilever for Viscosity or Viscoelasticity Sensing" *Sensors* 22, no. 21: 8421.
https://doi.org/10.3390/s22218421