Characterization on the Viscoelastic Property of PDMS in the Frequency Domain
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Characterization on the Viscoelastic Property of PDMS in the Frequency Domain Ping Du1, I-Kuan Lin1, 2, Hongbing Lu3, Xi lin1 and Xin Zhang1 1 Department of Mechanical Engineering, Boston University, Boston MA 02215, U.S.A. 2 Global Science & Technology, Greenbelt, MD 20770, U.S.A. 3 Department of Mechanical Engineering, University of Texas at Dallas, Richardson, TX 75080, U.S.A. ABSTRACT A key issue in using Polydimethylsiloxane (PDMS) based micropillars as cellular force transducers is obtaining an accurate characterization of mechanical properties. The Young’s modulus of PDMS has been extended from a constant in the ideal elastic case to a timedependent function in the viscoelastic case. However, the frequency domain information is of more practical interest in interpreting the complex cell contraction behavior. In this paper, we reevaluated the Young’s relaxation modulus in the time domain by using more robust fitting algorithms than previous reports, and investigated the storage and loss moduli in the frequency domain using the Fourier transform technique. With the use of the frequency domain modulus and the deflection of micropillars in the Fourier series, the force calculation can be much simplified by converting a convolution in the time domain to a multiplication in the frequency domain. INTRODUCTION In recent years, PDMS based micropillars have been extensively studied as biotransducers for measuring the cellular forces [1-3]. The accuracy of these sensitive devices depends on appropriate modeling to convert the micropillar deformations into the corresponding reaction forces. The classical Euler beam theory was proved to be inadequate especially for the low aspect ratio micropillars, where the shear-induced additional deformation could not be neglected [4]. In addition, PDMS exhibits inherently time-dependent material properties, i.e. viscoelasticity, which is also important for the accurate force conversion. These issues have been addressed by using finite element analysis [5, 6], and a more in-depth analytical formula, namely the extended viscoelastic Timoshenko beam formula [7]. In our previous work [7], the viscoelastic property, i.e. the Young’s relaxation modulus was obtained by nanoindentation stress relaxation tests in the time domain. The frequency domain information, such as storage and loss moduli, if of more practical interest as it reflects the ability to contemporaneously store and dissipate applied mechanical energy. In addition, the extended viscoelastic Timoshenko beam formula was only solved for the condition of constantrate displacement history. However, the actual contraction behavior of cells is much more complicated. Zhao et al. reported that the cardiac myocyte contracted in a smooth periodic manner, and the amplitude and frequency changed substantially under chemical stimuli [2]. It is well-known that all waveforms in nature can actually be decomposed into the Fourier series, i.e. a linear sum of simple sinusoidal functions with different amplitudes and frequencies. Therefore it is
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