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Handbook of Nonlocal Continuum Mechanics for Materials and Structures pp 241–287Cite as

Helical Buckling Behaviors of the Nanowire/Substrate System

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Abstract

When a nanowire is deposited on a compliant soft substrate or embedded in matrix, it may buckle into a helical coil form when the system is compressed. Using theoretical and finite element method (FEM) analyses, the detailed three-dimensional coil buckling mechanism for a silicon nanowire (SiNW) on a poly-dimethylsiloxne (PDMS) substrate is discussed. A continuum mechanics approach based on the minimization of the strain energy in the SiNW and elastomeric substrate is developed, and the helical buckling spacing and amplitude are deduced, taking into account the influences of the elastic properties and dimensions of SiNWs. These features are verified by systematic FEM simulations and parallel experiments. When the debonding of SiNW from the surface of the substrate is considered, the buckling profile of the nanowire can be divided into three regimes, i.e., the in-plane buckling, the disordered buckling in the out-of-plane direction, and the helical buckling, depending on the debonding density. For a nanowire embedded in matrix, the buckled profile is almost perfectly circular in the axial direction; with increasing compression, the buckling spacing decreases almost linearly, while the amplitude scales with the 1/2 power of the compressive strain; the transition strain from 2D mode to 3D helical mode decreases with the Young’s modulus of the wire and approaches to ~1.25% when the modulus is high enough, which is much smaller than nanowires on the surface of substrates. The study may shed useful insights on the design and optimization of high-performance stretchable electronics and 3D complex nanostructures.

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References

  • M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards Applied Mathematics Series 55. Tenth Printing. Engineering: 1076, Washington, DC, 1972)

    Google Scholar 

  • B. Audoly, A. Boudaoud, Buckling of a stiff film bound to a compliant substrate – part I. J. Mech. Phys. Solids 56(7), 2401–2421 (2008)

    Article  MathSciNet  Google Scholar 

  • C.P. Brangwynne, F.C. MacKintosh, S. Kumar, N.A. Geisse, J. Talbot, L. Mahadevan, K.K. Parker, D.E. Ingber, D.A. Weitz, Microtubules can bear enhanced compressive loads in living cells because of lateral reinforcement. J. Cell Biol. 173(5), 733–741 (2006)

    Article  Google Scholar 

  • B. Charlot, W. Sun, K. Yamashita, H. Fujita, H. Toshiyoshi, In-plane bistable nanowire for memory devices, in Symposium on Design, Test, Integration and Packaging of MEMS/MOEMS, 2008. MEMS/MOEMS 2008 (IEEE, 2008). http://ieeexplore.ieee.org/document/4752995/

  • X. Chen, J. Yin, Buckling patterns of thin films on curved compliant substrates with applications to morphogenesis and three-dimensional micro-fabrication. Soft Matter 6(22), 5667–5680 (2010).

    Article  Google Scholar 

  • Z. Chen, B. Cotterell, W. Wang, The fracture of brittle thin films on compliant substrates in flexible displays. Eng. Fract. Mech. 69(5), 597–603 (2002)

    Article  Google Scholar 

  • Y. Chen, Y. Liu, Y. Yan, Y. Zhu, X. Chen, Helical coil buckling mechanism for a stiff nanowire on an elastomeric substrate. J. Mech. Phys. Solids 95, 25–43 (2016)

    Article  MathSciNet  Google Scholar 

  • G. Crawford, Flexible Flat Panel Displays (Wiley, Chichester, 2005)

    Book  Google Scholar 

  • Y. Duan, Y. Huang, Z. Yin, Competing buckling of micro/nanowires on compliant substrates. J. Phys. D. Appl. Phys. 48(4), 045302 (2015)

    Article  Google Scholar 

  • J.W. Durham 3rd, Y. Zhu, Fabrication of functional nanowire devices on unconventional substrates using strain-release assembly. ACS Appl. Mater. Interfaces 5(2), 256–261 (2013)

    Article  Google Scholar 

  • K. Efimenko, W.E. Wallace, J. Genzer, Surface modification of Sylgard-184 poly(dimethyl siloxane) networks by ultraviolet and ultraviolet/ozone treatment. J. Colloid Interface Sci. 254(2), 306–315 (2002)

    Article  Google Scholar 

  • A. Goriely, R. Vandiver, M. Destrade, Nonlinear Euler buckling. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 464(2099), 3003–3019 (2008)

    Article  MathSciNet  Google Scholar 

  • H. Jiang, J. Zhang, Mechanics of microtubule buckling supported by cytoplasm. J. Appl. Mech. 75(6), 061019 (2008)

    Article  Google Scholar 

  • H. Jiang, D.Y. Khang, J. Song, Y. Sun, Y. Huang, J.A. Rogers, Finite deformation mechanics in buckled thin films on compliant supports. Proc. Natl. Acad. Sci. U. S. A. 104(40), 15607–15612 (2007)

    Article  Google Scholar 

  • S.J. Kalita, V. Somani, Al2TiO5–Al2O3–TiO2 nanocomposite: structure, mechanical property and bioactivity studies. Mater. Res. Bull. 45(12), 1803–1810 (2010)

    Article  Google Scholar 

  • D.H. Kim, J. Song, W.M. Choi, H.S. Kim, R.H. Kim, Z. Liu, Y.Y. Huang, K.C. Hwang, Y.W. Zhang, J.A. Rogers, Materials and noncoplanar mesh designs for integrated circuits with linear elastic responses to extreme mechanical deformations. Proc. Natl. Acad. Sci. U. S. A. 105(48), 18675–18680 (2008)

    Article  Google Scholar 

  • H.C. Ko, M.P. Stoykovich, J. Song, V. Malyarchuk, W.M. Choi, C.J. Yu, J.B. Geddes 3rd, J. Xiao, S. Wang, Y. Huang, J.A. Rogers, A hemispherical electronic eye camera based on compressible silicon optoelectronics. Nature 454(7205), 748–753 (2008)

    Article  Google Scholar 

  • S.P. Lacour, J. Jones, S. Wagner, T. Li, Z. Suo, Stretchable interconnects for elastic electronic surfaces. Proc. IEEE 93(8), 1459–1467 (2005)

    Article  Google Scholar 

  • T. Li, A mechanics model of microtubule buckling in living cells. J. Biomech. 41(8), 1722–1729 (2008)

    Article  Google Scholar 

  • H. Mei, C.M. Landis, R. Huang, Concomitant wrinkling and buckle-delamination of elastic thin films on compliant substrates. Mech. Mater. 43(11), 627–642 (2011)

    Article  Google Scholar 

  • S.G. O’Keeffe, D.E. Moulton, S.L. Waters, A. Goriely, Growth-induced axial buckling of a slender elastic filament embedded in an isotropic elastic matrix. Int. J. Non Linear Mech. 56, 94–104 (2013)

    Article  Google Scholar 

  • W.A. Oldfather, C.A. Ellis, D.M. Brown, Leonhard Euler’s elastic curves. Isis 20(1), 72–160 (1933)

    Article  Google Scholar 

  • X.H. Peng, A. Alizadeh, S.K. Kumar, S.K. Nayak, Ab initio study of size and strain effects on the electronic properties of Si nanowires. Int. J. Appl. Mech. 1(3), 483–499 (2009)

    Article  Google Scholar 

  • Q. Qin, Y. Zhu, Static friction between silicon nanowires and elastomeric substrates. ACS Nano 5(9), 7404–7410 (2011)

    Article  Google Scholar 

  • J.A. Rogers, T. Someya, Y. Huang, Materials and mechanics for stretchable electronics. Science 327(5973), 1603–1607 (2010)

    Article  Google Scholar 

  • S.Y. Ryu, J. Xiao, W.I. Park, K.S. Son, Y.Y. Huang, U. Paik, J.A. Rogers, Lateral buckling mechanics in silicon nanowires on elastomeric substrates. Nano Lett. 9(9), 3214–3219 (2009)

    Article  Google Scholar 

  • R.N. Sajjad, K. Alam, Electronic properties of a strained 100 silicon nanowire. J. Appl. Phys. 105(4), 044307 (2009)

    Article  Google Scholar 

  • V. Slesarenko, S. Rudykh, Microscopic and macroscopic instabilities in hyperelastic fiber composites. J. Mech. Phys. Solids 99, 471–482 (2016)

    Article  MathSciNet  Google Scholar 

  • T. Someya, T. Sekitani, S. Iba, Y. Kato, H. Kawaguchi, T. Sakurai, A large-area, flexible pressure sensor matrix with organic field-effect transistors for artificial skin applications. Proc. Natl. Acad. Sci. U. S. A. 101(27), 9966–9970 (2004)

    Article  Google Scholar 

  • J. Song, Y. Huang, J. Xiao, S. Wang, K.C. Hwang, H.C. Ko, D.H. Kim, M.P. Stoykovich, J.A. Rogers, Mechanics of noncoplanar mesh design for stretchable electronic circuits. J. Appl. Phys. 105(12), 123516 (2009)

    Article  Google Scholar 

  • C.M. Stafford, C. Harrison, K.L. Beers, A. Karim, E.J. Amis, M.R. VanLandingham, H.C. Kim, W. Volksen, R.D. Miller, E.E. Simonyi, A buckling-based metrology for measuring the elastic moduli of polymeric thin films. Nat. Mater. 3(8), 545–550 (2004)

    Article  Google Scholar 

  • T. Su, J. Liu, D. Terwagne, P.M. Reis, K. Bertoldi, Buckling of an elastic rod embedded on an elastomeric matrix: planar vs. non-planar configurations. Soft Matter 10(33), 6294–6302 (2014)

    Article  Google Scholar 

  • S.P. Timoshenko, J.M. Gere, Theory of Elastic Stability (Courier Corporation, North Chelmsford, 2009)

    Google Scholar 

  • S.P. Timoshenko, J.N. Goodier, H.N. Abramson, Theory of elasticity (3rd ed.) J. Appl. Mech. 37(3), 888 (1970)

    Article  Google Scholar 

  • Y. Wang, J. Song, J. Xiao, Surface effects on in-plane buckling of nanowires on elastomeric substrates. J. Phys. D. Appl. Phys. 46(12), 125309 (2013)

    Article  Google Scholar 

  • E.A. Wilder, S. Guo, S. Lin-Gibson, M.J. Fasolka, C.M. Stafford, Measuring the modulus of soft polymer networks via a buckling-based metrology. Macromolecules 39(12), 4138–4143 (2006)

    Article  Google Scholar 

  • J. Xiao, H. Jiang, D.Y. Khang, J. Wu, Y. Huang, J.A. Rogers, Mechanics of buckled carbon nanotubes on elastomeric substrates. J. Appl. Phys. 104(3), 033543 (2008)

    Article  Google Scholar 

  • J. Xiao, S.Y. Ryu, Y. Huang, K.C. Hwang, U. Paik, J.A. Rogers, Mechanics of nanowire/nanotube in-surface buckling on elastomeric substrates. Nanotechnology 21(8), 85708 (2010)

    Article  Google Scholar 

  • F. Xu, W. Lu, Y. Zhu, Controlled 3D buckling of silicon nanowires for stretchable electronics. ACS Nano 5(1), 672–678 (2011)

    Article  Google Scholar 

  • S. Xu, Z. Yan, K.I. Jang, W. Huang, H.R. Fu, J. Kim, Z. Wei, M. Flavin, J. McCracken, R. Wang, A. Badea, Y. Liu, D.Q. Xiao, G.Y. Zhou, J. Lee, H.U. Chung, H.Y. Cheng, W. Ren, A. Banks, X.L. Li, U. Paik, R.G. Nuzzo, Y.G. Huang, Y.H. Zhang, J.A. Rogers, Assembly of micro/nanomaterials into complex, three-dimensional architectures by compressive buckling. Science 347(6218), 154–159 (2015)

    Article  Google Scholar 

  • J. Yin, X. Chen, Buckling of anisotropic films on cylindrical substrates: insights for self-assembly fabrication of 3D helical gears. J. Phys. D. Appl. Phys. 43(11), 115402 (2010)

    Article  Google Scholar 

  • E. Zeeman, Euler buckling, in Structural Stability, the Theory of Catastrophes, and Applications in the Sciences (Springer, 1976), pp. 373–395. http://www.springer.com/la/book/9783540077916

  • Y. Zhao, J. Li, Y.P. Cao, X.Q. Feng, Buckling of an elastic fiber with finite length in a soft matrix. Soft Matter 12(7), 2086–2094 (2016)

    Article  Google Scholar 

  • C. Zhou, S. Bette, U. Schnakenberg, Flexible and stretchable gold microstructures on extra soft poly(dimethylsiloxane) substrates. Adv. Mater. 27(42), 6664–6669 (2015)

    Article  Google Scholar 

  • Y. Zhu, F. Xu, Q. Qin, W.Y. Fung, W. Lu, Mechanical properties of vapor–liquid–solid synthesized silicon nanowires. Nano Lett. 9(11), 3934–3939 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. International Center for Applied Mechanics, State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an, China

    Youlong Chen

  2. International Center for Applied Mechanics, State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an, China

    Yilun Liu

  3. Department of Earth and Environmental Engineering, Columbia Nanomechanics Research Center, Columbia University, New York, NY, USA

    Xi Chen

Authors
  1. Youlong Chen
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  2. Yilun Liu
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  3. Xi Chen
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Correspondence to Youlong Chen .

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Editors and Affiliations

  1. Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA, USA

    George Z. Voyiadjis

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Chen, Y., Liu, Y., Chen, X. (2019). Helical Buckling Behaviors of the Nanowire/Substrate System. In: Voyiadjis, G. (eds) Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Springer, Cham. https://doi.org/10.1007/978-3-319-58729-5_47

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