# 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

- Morrison, B., III; Meaney, D.F.; McIntosh, T.K. Mechanical characterization of an in vitro device designed to quantitatively injure living brain tissue. Ann. Biomed. Eng.
**1998**, 26, 381–390. [Google Scholar] [CrossRef] [PubMed] - Edsberg, L.E.; Cutway, R.; Anain, S.; Natiella, J.R. Microstructural and mechanical characterization of human tissue at and adjacent to pressure ulcers. J. Rehabil. Res. Dev.
**2000**, 37, 463–471. [Google Scholar] [PubMed] - Wakatsuki, T.; Kolodney, M.S.; Zahalak, G.I.; Elson, E.L. Cell mechanics studied by a reconstituted model tissue. Biophys. J.
**2000**, 79, 2353–2368. [Google Scholar] [CrossRef] - Sacks, M.S.; Sun, W. Multiaxial mechanical behavior of biological materials. Annu. Rev. Biomed. Eng.
**2003**, 5, 251–284. [Google Scholar] [CrossRef] [PubMed] - Yang, W.; Fung, T.C.; Chian, K.S.; Chong, C.K. Viscoelasticity of esophageal tissue and application of a QLV model. J. Biomech. Eng.
**2006**, 128, 909–916. [Google Scholar] [CrossRef] [PubMed] - Allen, K.D.; Athanasiou, K.A. Viscoelastic characterization of the porcine temporomandibular joint disc under unconfined compression. J. Biomech.
**2006**, 39, 312–322. [Google Scholar] [CrossRef] [PubMed] - Levesque, P.; Gauvin, R.; Larouche, D.; Auger, F.A.; Germain, L. A computer-controlled apparatus for the characterization of mechanical and viscoelastic properties of tissue-engineered vascular constructs. Cardiovasc. Eng. Technol.
**2011**, 2, 24–34. [Google Scholar] [CrossRef] - Dagdeviren, C.; Shi, Y.; Joe, P.; Ghaffari, R.; Balooch, G.; Usgaonkar, K.; Gur, O.; Tran, P.L.; Crosby, J.R.; Meyer, M.; et al. Conformal piezoelectric systems for clinical and experimental characterization of soft tissue biomechanics. Nature Materials
**2015**, 14, 728–736. [Google Scholar] [CrossRef] [PubMed] - Botta, F.; Marx, N.; Gentili, S.; Schwingshackl, C.; Di Mare, L.; Cerri, G.; Dini, D. Optimal placement of piezoelectric plates for active vibration control of gas turbine blades: experimental results. In SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring; International Society for Optics and Photonics: San Diego, CA, USA, 2012; p. 83452H. [Google Scholar]
- Botta, F.; Dini, D.; Schwingshackl, C.; di Mare, L.; Cerri, G. Optimal placement of piezoelectric plates to control multimode vibrations of a beam. Adv. Acoust. Vib.
**2013**, 2013. [Google Scholar] [CrossRef] - Botta, F.; Marx, N.; Schwingshackl, C.; Cerri, G.; Dini, D. A wireless vibration control technique for gas turbine blades using piezoelectric plates and contactless energy transfer. In Proceedings of the ASME Turbo Expo, San Antonio, TX, USA, 3–7 June 2013. [Google Scholar]
- Kiss, M.Z.; Varghese, T.; Hall, T.J. Viscoelastic characterization of in vitro canine tissue. Phys. Med. Biol.
**2004**, 49, 4207–4218. [Google Scholar] [CrossRef] [PubMed] - Chen, K.; Yao, A.; Zheng, E.E.; Lin, J.; Zheng, Y. Shear wave dispersion ultrasound vibrometry based on a different mechanical model for soft tissue characterization. J. Ultrasound Med.
**2012**, 31, 2001–2011. [Google Scholar] [CrossRef] [PubMed] - Tavakoli, M.; Aziminejad, A.; Patel, R.; Moallem, M. Multi-sensory force/deformation cues for stiffness characterization in soft-tissue palpation. In Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society, New York, NY, USA, 31 August–3 September 2006; pp. 837–840. [Google Scholar]
- Boonvisut, P.; Çavuşoǧlu, M.C. Estimation of soft tissue mechanical parameters from robotic manipulation data. IEEE/ASME Trans. Mech.
**2013**, 18, 1602–1611. [Google Scholar] [CrossRef] [PubMed] - Ebenstein, D.M.; Pruitt, L. Nanoindentation of biological materials. Nano Today
**2006**, 1, 26–33. [Google Scholar] [CrossRef] - Cox, M.A.J.; Driessen, N.J.B.; Boerboom, R.A.; Bouten, C.V.C.; Baaijens, F.P.T. Mechanical characterization of anisotropic planar biological soft tissues using finite indentation: Experimental feasibility. J. Biomech.
**2008**, 41, 422–429. [Google Scholar] [CrossRef] [PubMed] - Saxena, T.; Gilbert, J.; Stelzner, D.; Hasenwinkel, J. Mechanical characterization of the injured spinal cord after lateral spinal hemisection injury in the rat. J. Neurotrauma
**2012**, 29, 1747–1757. [Google Scholar] [CrossRef] [PubMed] - González-Cruz, R.D.; Fonseca, V.C.; Darling, E.M. Cellular mechanical properties reflect the differentiation potential of adipose-derived mesenchymal stem cells. Proc. Natl. Acad. Sci. USA
**2012**, 109, E1523–E1529. [Google Scholar] [CrossRef] [PubMed] - Ficarella, E.; Lamberti, L.; Papi, M.; De Spirito, M.; Pappalettere, C. Viscohyperelastic calibration in mechanical characterization of soft matter. In Mechanics of Biological Systems and Materialsz; Springer International Publishing: Cham, Switzerland, 2017; Volume 6, pp. 33–37. [Google Scholar]
- Mazza, E.; Nava, A.; Hahnloser, D.; Jochum, W.; Bajka, M. The mechanical response of human liver and its relation to histology: An in vivo study. Med. Image Anal.
**2007**, 11, 663–672. [Google Scholar] [CrossRef] [PubMed] - Zhao, R.; Sider, K.L.; Simmons, C.A. Measurement of layer-specific mechanical properties in multilayered biomaterials by micropipette aspiration. Acta Biomater.
**2011**, 7, 1220–1227. [Google Scholar] [CrossRef] [PubMed] - Choi, D.K. Mechanical characterization of biological tissues: Experimental methods based on mathematical modeling. Biomed. Eng. Lett.
**2016**, 6, 181–195. [Google Scholar] [CrossRef] - Nava, A.; Mazza, E.; Kleinermann, F.; Avis, N.J.; McClure, J.; Bajka, M. Evaluation of the mechanical properties of human liver and kidney through aspiration experiments. Technol. Health Care
**2004**, 12, 269–280. [Google Scholar] [PubMed] - Nava, A.; Mazza, E.; Furrer, M.; Villiger, P.; Reinhart, W.H. In vivo mechanical characterization of human liver. Med. Image Anal.
**2008**, 12, 203–216. [Google Scholar] [CrossRef] [PubMed] - Boudou, T.; Ohayon, J.; Arntz, Y.; Finet, G.; Picart, C.; Tracqui, P. An extended modeling of the micropipette aspiration experiment for the characterization of the Young’s modulus and Poisson’s ratio of adherent thin biological samples: Numerical and experimental studies. J. Biomech.
**2006**, 39, 1677–1685. [Google Scholar] [CrossRef] [PubMed] - Bosisio, M.R.; Talmant, M.; Skalli, W.; Laugier, P.; Mitton, D. Apparent Young’s modulus of human radius using inverse finite-element method. J. Biomech.
**2007**, 40, 2022–2028. [Google Scholar] [CrossRef] [PubMed] - Valero, C.; Navarro, B.; Navajas, D.; García-Aznar, J.M. Finite element simulation for the mechanical characterization of soft biological materials by atomic force microscopy. J. Mech. Behav. Biomed. Mater.
**2016**, 62, 222–235. [Google Scholar] [CrossRef] [PubMed] - Argento, G.; Simonet, M.; Oomens, C.W.J.; Baaijens, F.P.T. Multi-scale mechanical characterization of scaffolds for heart valve tissue engineering. J. Biomech.
**2012**, 45, 2893–2898. [Google Scholar] [CrossRef] [PubMed] - Rodriguez, M.L.; McGarry, P.J.; Sniadecki, N.J. Review on cell mechanics: Experimental and modeling approaches. Appl. Mech. Rev.
**2013**, 65, 060801. [Google Scholar] [CrossRef] - Ekpenyong, A.; Whyte, G.; Chalut, K.; Pagliara, S.; Lautenschläger, F.; Fiddler, C.; Paschke, S.; Keyser, U.; Chilvers, E.; Guck, J. Viscoelastic properties of differentiating blood cells are fate- and function-dependent. PLoS ONE
**2012**, 7. [Google Scholar] [CrossRef] [PubMed] - Gossett, D.R.; Tse, H.T.K.; Lee, S.A.; Ying, Y.; Lindgren, A.G.; Yang, O.O.; Rao, J.; Clark, A.T.; Di Carlo, D. Hydrodynamic stretching of single cells for large population mechanical phenotyping. Proc. Natl. Acad. Sci. USA
**2012**, 109, 7630–7635. [Google Scholar] [CrossRef] [PubMed] - Hodgson, A.; Verstreken, C.; Fisher, C.; Keyser, U.; Pagliara, S.; Chalut, K. A microfluidic device for characterizing nuclear deformations. Lab Chip
**2017**, 17, 805–813. [Google Scholar] [CrossRef] [PubMed] - Pajerowski, J.D.; Dahl, K.N.; Zhong, F.L.; Sammak, P.J.; Discher, D.E. Physical plasticity of the nucleus in stem cell differentiation. Proc. Natl. Acad. Sci. USA
**2007**, 104, 15619–15624. [Google Scholar] [CrossRef] [PubMed] - Pagliara, S.; Franze, K.; McClain, C.; Wylde, G.; Fisher, C.; Franklin, R.; Kabla, A.; Keyser, U.; Chalut, K. Auxetic nuclei in embryonic stem cells exiting pluripotency. Nat. Mater.
**2014**, 13, 638–644. [Google Scholar] [CrossRef] [PubMed] - Guck, J.; Schinkinger, S.; Lincoln, B.; Wottawah, F.; Ebert, S.; Romeyke, M.; Lenz, D.; Erickson, H.; Ananthakrishnan, R.; Mitchell, D.; et al. Optical deformability as an inherent cell marker for testing malignant transformation and metastatic competence. Biophys. J.
**2005**, 88, 3689–3698. [Google Scholar] [CrossRef] [PubMed] - Otto, O.; Rosendahl, P.; Mietke, A.; Golfier, S.; Herold, C.; Klaue, D.; Girardo, S.; Pagliara, S.; Ekpenyong, A.; Jacobi, A.; et al. Real-time deformability cytometry: On-the-fly cell mechanical phenotyping. Nat. Methods
**2015**, 12, 199–202. [Google Scholar] [CrossRef] [PubMed] - Guido, I.; Jaeger, M.; Duschl, C. Dielectrophoretic stretching of cells allows for characterization of their mechanical properties. Eur. Biophys. J.
**2011**, 40, 281–288. [Google Scholar] [CrossRef] [PubMed] - Kamble, H.; Vadivelu, R.; Barton, M.; Boriachek, K.; Munaz, A.; Park, S.; Shiddiky, M.; Nguyen, N.T. An electromagnetically actuated double-sided cell-stretching device for mechanobiology research. Micromachines
**2017**, 8, 256. [Google Scholar] [CrossRef] - Verotti, M.; Crescenzi, R.; Balucani, M.; Belfiore, N.P. MEMS-based conjugate surfaces flexure hinge. J. Mech. Des.
**2015**, 137, 012301. [Google Scholar] [CrossRef] - Belfiore, N.P.; Broggiato, G.B.; Verotti, M.; Balucani, M.; Crescenzi, R.; Bagolini, A.; Bellutti, P.; Boscardin, M. Simulation and construction of a MEMS CSFH based microgripper. Int. J. Mech. Control
**2015**, 16, 21–30. [Google Scholar] - Balucani, M.; Belfiore, N.P.; Crescenzi, R.; Verotti, M. The development of a MEMS/NEMS-based 3 D.O.F. compliant micro robot. Int. J.Mech. Control
**2011**, 12, 3–10. [Google Scholar] - Belfiore, N.P.; Pennestrì, E. An atlas of linkage-type robotic grippers. Mech. Mach. Theory
**1997**, 32, 811–833. [Google Scholar] [CrossRef] - Dochshanov, A.; Verotti, M.; Belfiore, N.P. A comprehensive survey on microgrippers design: operational strategy. J. Mech. Des.
**2017**, 139, 070801. [Google Scholar] [CrossRef] - Verotti, M.; Dochshanov, A.; Belfiore, N.P. A comprehensive survey on microgrippers design: mechanical structure. J. Mech. Des.
**2017**, 139, 060801. [Google Scholar] [CrossRef] - Verotti, M.; Belfiore, N.P. Isotropic compliance in E(3): Feasibility and workspace mapping. J. Mech. Robot.
**2016**, 8, 061005. [Google Scholar] [CrossRef] - Verotti, M.; Masarati, P.; Morandini, M.; Belfiore, N.P. Isotropic compliance in the Special Euclidean Group SE(3). Mech. Mach. Theory
**2016**, 98, 263–281. [Google Scholar] [CrossRef] - Belfiore, N.P.; Simeone, P. Inverse kinetostatic analysis of compliant four-bar linkages. Mech. Mach. Theory
**2013**, 69, 350–372. [Google Scholar] [CrossRef] - Belfiore, N.P.; Verotti, M.; Di Giamberardino, P.; Rudas, I. Active joint stiffness regulation to achieve isotropic compliance in the euclidean space. J. Mech. Robot.
**2012**, 4. [Google Scholar] [CrossRef] - Verotti, M.; Dochshanov, A.; Belfiore, N.P. Compliance synthesis of CSFH MEMS-based microgrippers. J. Mech. Des.
**2017**, 139, 022301. [Google Scholar] [CrossRef] - Cecchi, R.; Verotti, M.; Capata, R.; Dochshanov, A.; Broggiato, G.; Crescenzi, R.; Balucani, M.; Natali, S.; Razzano, G.; Lucchese, F.; et al. Development of micro-grippers for tissue and cell manipulation with direct morphological comparison. Micromachines
**2015**, 6, 1710–1728. [Google Scholar] [CrossRef] - Bagolini, A.; Ronchin, S.; Bellutti, P.; Chiste, M.; Verotti, M.; Belfiore, N.P. Fabrication of novel MEMS microgrippers by deep reactive ion etching with metal hard mask. IEEE J. Microelectromechanical Syst.
**2017**, 26, 926–934. [Google Scholar] [CrossRef] - Lim, C.; Zhou, E.; Quek, S. Mechanical models for living cells—A review. J. Biomech.
**2006**, 39, 195–216. [Google Scholar] [CrossRef] [PubMed] - Verotti, M. Analysis of the center of rotation in primitive flexures: Uniform cantilever beams with constant curvature. Mech. Mach. Theory
**2016**, 97, 29–50. [Google Scholar] [CrossRef] - Verotti, M. Effect of initial curvature in uniform flexures on position accuracy. Mech. Mach. Theory
**2018**, 119, 106–118. [Google Scholar] [CrossRef] - Rao, S.S. Mechanical Vibrations; Addison-Wesley: Reading, MA, USA, 1993. [Google Scholar]
- Tu, C.C.; Fanchiang, K.; Liu, C.H. 1 × N rotary vertical micromirror for optical switching applications. Proc. SPIE
**2005**, 5719, 14–22. [Google Scholar] - Hou, M.T.K.; Huang, J.Y.; Jiang, S.S.; Yeh, J.A. In-plane rotary comb-drive actuator for a variable optical attenuator. J. Micro/Nanolithography MEMS MOEMS
**2008**, 7. [Google Scholar] [CrossRef]

**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 $\Delta \varphi \left(\right)open="("\; close=")">{\theta}_{4},{\theta}_{2}$ with respect to $c$ and $\omega$ for an assigned value of $k$.

**Figure 18.**Dependency of the phase delay $\Delta \varphi \left(\right)open="("\; close=")">{\theta}_{4},{\theta}_{2}$ 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