# New MEMS Tweezers for the Viscoelastic Characterization of Soft Materials at the Microscale

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## Abstract

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## 1. Introduction

## 2. The Adopted Microsystem

## 3. The Modeling Approach

- hat $^$ refers to a constant parameter, such as, for example, those referring to the initial configuration, as shown in Figure 5a;
- tilde
^{˜}refers to an actual parameter at the generic configuration, as represented in Figure 5b; - superscript$}^{0$ refers to the desired or target parameter;
- angles of bars are measured counterclockwise, starting from the positive abscissa;
- $\tilde{u}=\widehat{u}+u$ is the length of vector $\overrightarrow{BC}$ which is split in the initial length $\widehat{u}$ and the deformation $u$;
- $\vartheta}_{2$, $\vartheta}_{3$ and $\vartheta}_{4$ refer to the variations of the angular positions of the link vectors $\overrightarrow{AB}$, $\overrightarrow{BC}$ and $\overrightarrow{DC}$, with respect to their initial position; in this way their actual absolute angular positions will be $\tilde{{\vartheta}_{2}}={\widehat{\vartheta}}_{2}+{\vartheta}_{2}$, $\tilde{\vartheta}}_{3}={\widehat{\vartheta}}_{3}+{\vartheta}_{3$, $\tilde{\vartheta}}_{4}={\widehat{\vartheta}}_{4}+{\vartheta}_{4$, respectively;
- $\widehat{\vartheta}}_{2}=\pi -{\widehat{\vartheta}}_{4$ (as Figure 5a shows that they are supplementary angles)
- $\widehat{u}=d-2lcos{\widehat{\vartheta}}_{2}$, from geometry represented in Figure 5a;
- $l$ is the common length of the two vectors $\overrightarrow{AB}$ and $\overrightarrow{DC}$;
- $d$ is the length of the frame link $AD$;
- $k$ is the stiffness coefficient of the tissue sample;
- $k}_{2$ and $k}_{4$ are the two jaws torsional stiffness, which are related to the CSFH curved beam material and geometry;
- $r}_{b$, $b$, $h$ and $\beta$ are the radius, width, thickness and beam subtended angle of the CSFH flexure curved beam;
- $c$, $c}_{2$ and $c}_{4$ represent the viscous damping coefficients of the sample and of the two jaws;
- $I}_{2$ and $I}_{4$ represents the two jaws moments of inertia around $A$ and $D$, with $I}_{2}={I}_{4$;
- $v}_{2$ and $v}_{4$ are the tensions applied to the comb drives;
- $\chi$, $g$ and $w$ are the overlap angle, gap and width of the comb drive fingers;
- $z}_{0$ device-handle gap (silicon oxide layer thickness);
- $\mu$ air viscosity at 25 °C;
- $J}_{p\phantom{\rule{0.166667em}{0ex}}2,4$ polar moment of area exposed to air viscous damping, calculated around the rotation points;
- $\widehat{R}}_{a$ equivalent radius employed to model the air viscous damping.

## 4. The Adopted Electromechanical Model

## 5. The Identification of the Sample Stiffness and the Damping Coefficients

#### 5.1. Characterization of the Sample Stiffness

#### 5.2. Characterization of Sample Characteristic Damping

#### 5.3. The Adopted Operational Scheme

^{®}and Simulink

^{®}tools, can help to show the effectiveness of the proposed approach.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**A SEM image of (

**a**) the mechanical component of the microsystem and (

**b**) a detailed vew of the CSFH hinge.

**Figure 5.**The micro–system closed loop chain in the initial (

**a**) and in the generic (

**b**) configuration.

**Figure 8.**Dependency of the steady state angle $\theta}_{4,ss$ on stiffness $k$ (note that viscosity $c$ does not affect steady state balance).

**Figure 13.**Dependency of $\vartheta}_{4}/{\tau}_{2$ with respect to $c$ and $\omega$ for an assigned value of $k$.

**Figure 14.**Dependency of $\vartheta}_{4}/{\tau}_{2$ with respect to $c$ and $k$ for an assigned value of $\omega$.

**Figure 15.**Dependency of $\vartheta}_{4}/{\vartheta}_{2$ with respect to $c$ and $\omega$ for an assigned value of $k$.

**Figure 16.**Dependency of $\vartheta}_{4}/{\vartheta}_{2$ with respect to $c$ and $k$ for an assigned value of $\omega$.

**Figure 17.**Dependency of the phase delay $\Delta \varphi \left({\theta}_{4},{\theta}_{2}\right)$ with respect to $c$ and $\omega$ for an assigned value of $k$.

**Figure 18.**Dependency of the phase delay $\Delta \varphi \left({\theta}_{4},{\theta}_{2}\right)$ with respect to $c$ and $k$ for an assigned value of $\omega$.

Parameter | Value | Parameter | Value |
---|---|---|---|

$\widehat{\vartheta}}_{2$ | 1.44 rad | $\widehat{R}}_{a$ | $7.78\times {10}^{-4}$ m |

$\widehat{\vartheta}}_{4$ | 1.70 rad | $b$ | $5\times {10}^{-6}$ m |

$\beta$ | 4.20 rad | $h$ | $40\times {10}^{-6}$ m |

$\widehat{\vartheta}}_{3$ | $0$ rad | $J}_{p\phantom{\rule{0.166667em}{0ex}}2$, $J}_{p\phantom{\rule{0.166667em}{0ex}}4$ | $1.34\times {10}^{-13}$ m$}^{4$ |

$\widehat{u}$ | $150\times {10}^{-6}$ m | $m}_{2$, $m}_{4$ | $1.9\times {10}^{-8}$ kg |

$d$ | $5.47\times {10}^{-4}$ m | $I}_{2$, $I}_{4$ | $1.25\times {10}^{-14}$ kg$\xb7$m$}^{2$ |

$l$ | $1.496\times {10}^{-3}$ m | $\mu$ | $18.6\times {10}^{-6}$ kg/m$\xb7$s |

$r}_{b$ | $62.5\times {10}^{-6}$ m | $k}_{2$, $k}_{4$ | $0.30\times {10}^{-6}$ kg$\xb7$m$}^{2$/$s}^{2$$\xb7$rad |

$z}_{0$ | 2 $\times {10}^{-6}$ m | $c}_{2$, $c}_{4$ | $1.24\times {10}^{-12}$ kg$\xb7$m$}^{2$/s$\xb7$rad |

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## Share and Cite

**MDPI and ACS Style**

Di Giamberardino, P.; Bagolini, A.; Bellutti, P.; Rudas, I.J.; Verotti, M.; Botta, F.; Belfiore, N.P.
New MEMS Tweezers for the Viscoelastic Characterization of Soft Materials at the Microscale. *Micromachines* **2018**, *9*, 15.
https://doi.org/10.3390/mi9010015

**AMA Style**

Di Giamberardino P, Bagolini A, Bellutti P, Rudas IJ, Verotti M, Botta F, Belfiore NP.
New MEMS Tweezers for the Viscoelastic Characterization of Soft Materials at the Microscale. *Micromachines*. 2018; 9(1):15.
https://doi.org/10.3390/mi9010015

**Chicago/Turabian Style**

Di Giamberardino, Paolo, Alvise Bagolini, Pierluigi Bellutti, Imre J. Rudas, Matteo Verotti, Fabio Botta, and Nicola P. Belfiore.
2018. "New MEMS Tweezers for the Viscoelastic Characterization of Soft Materials at the Microscale" *Micromachines* 9, no. 1: 15.
https://doi.org/10.3390/mi9010015