Research Papers

Deformation of Microchannels Embedded in an Elastic Medium

[+] Author and Article Information
Vivek Ramachandran

Integrated Soft Materials Laboratory,
Carnegie Mellon University,
Pittsburgh, PA 15213
e-mail: vivek.ramachandran@epfl.ch

Carmel Majidi

Integrated Soft Materials Laboratory,
Carnegie Mellon University,
Pittsburgh, PA 15213
e-mail: cmajidi@andrew.cmu.edu

1Present address: Laboratory of Intelligent Systems, École Polytechnique Fédérale de Lausanne, Lausanne 1015, Vaud, Switzerland.

2Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 29, 2018; final manuscript received May 30, 2018; published online June 27, 2018. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 85(10), 101004 (Jun 27, 2018) (7 pages) Paper No: JAM-18-1244; doi: 10.1115/1.4040477 History: Received April 29, 2018; Revised May 30, 2018

The deformation of microfluidic channels in a soft elastic medium has a central role in the operation of lab-on-a-chip devices, fluidic soft robots, liquid metal (LM) electronics, and other emerging soft-matter technologies. Understanding the influence of mechanical load on changes in channel cross section is essential for designing systems that either avoid channel collapse or exploit such collapse to control fluid flow and connectivity. In this paper, we examine the deformation of microchannel cross sections under far-field compressive stress and derive a “gauge factor” that relates externally applied pressure with change in cross-sectional area. We treat the surrounding elastomer as a Hookean solid and use two-dimensional plane strain elasticity, which has previously been shown to predict microchannel deformations that are in good agreement with experimental measurements. Numerical solutions to the governing Lamé (Navier) equations are found to match both the analytic solutions obtained from a complex stress function and closed-form algebraic approximations based on linear superposition. The application of this theory to soft microfluidics is demonstrated for several representative channel geometries.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Stone, H. A. , Stroock, A. D. , and Ajdari, A. , 2004, “ Engineering Flows in Small Devices: Microfluidics Toward a Lab-on-a-Chip,” Annu. Rev. Fluid Mech., 36(1), pp. 381–411. [CrossRef]
Squires, T. M. , and Quake, S. R. , 2005, “ Microfluidics: Fluid Physics at the Nanoliter Scale,” Rev. Mod. Phys., 77(3), p. 977. [CrossRef]
Whitesides, G. M. , 2006, “ The Origins and the Future of Microfluidics,” Nature, 442(7101), pp. 368–373. [CrossRef] [PubMed]
Sackmann, E. K. , Fulton, A. L. , and Beebe, D. J. , 2014, “ The Present and Future Role of Microfluidics in Biomedical Research,” Nature, 507(7491), pp. 181–189. [CrossRef] [PubMed]
Xia, Y. , and Whitesides, G. M. , 1998, “ Soft Lithography,” Annu. Rev. Mater. Sci., 28(1), pp. 153–184. [CrossRef]
Quake, S. R. , and Scherer, A. , 2000, “ From Micro-to Nanofabrication With Soft Materials,” Science, 290(5496), pp. 1536–1540. [CrossRef] [PubMed]
Qin, D. , Xia, Y. , and Whitesides, G. M. , 2010, “ Soft Lithography for Micro-and Nanoscale Patterning,” Nat. Protoc., 5(3), pp. 491–502. [CrossRef] [PubMed]
Dickey, M. D. , Chiechi, R. C. , Larsen, R. J. , Weiss, E. A. , Weitz, D. A. , and Whitesides, G. M. , 2008, “ Eutectic Gallium-Indium (Egain): A Liquid Metal Alloy for the Formation of Stable Structures in Microchannels at Room Temperature,” Adv. Funct. Mater., 18(7), pp. 1097–1104. [CrossRef]
Cheng, S. , and Wu, Z. , 2012, “ Microfluidic Electronics,” Lab Chip, 12(16), pp. 2782–2791. [CrossRef] [PubMed]
Joshipura, I. D. , Ayers, H. R. , Majidi, C. , and Dickey, M. D. , 2015, “ Methods to Pattern Liquid Metals,” J. Mater. Chem. C, 3(16), pp. 3834–3841. [CrossRef]
Huh, D. , Kim, H. J. , Fraser, J. P. , Shea, D. E. , Khan, M. , Bahinski, A. , Hamilton, G. A. , and Ingber, D. E. , 2013, “ Microfabrication of Human Organs-on-Chips,” Nat. Protoc., 8(11), pp. 2135–2157. [CrossRef] [PubMed]
Bhatia, S. N. , and Ingber, D. E. , 2014, “ Microfluidic Organs-on-Chips,” Nat. Biotechnol., 32, pp. 760–772.
Wakimoto, S. , Ogura, K. , Suzumori, K. , and Nishioka, Y. , 2009, “ Miniature Soft Hand With Curling Rubber Pneumatic Actuators,” IEEE International Conference on Robotics and Automation (ICRA'09), Kobe, Japan, May 12–17, pp. 556–561.
Wehner, M. , Truby, R. L. , Fitzgerald, D. J. , Mosadegh, B. , Whitesides, G. M. , Lewis, J. A. , and Wood, R. J. , 2016, “ An Integrated Design and Fabrication Strategy for Entirely Soft, Autonomous Robots,” Nature, 536(7617), pp. 451–455. [CrossRef] [PubMed]
Kim, H.-J. , Son, C. , and Ziaie, B. , 2008, “ A Multiaxial Stretchable Interconnect Using Liquid-Alloy-Filled Elastomeric Microchannels,” Appl. Phys. Lett., 92(1), p. 011904. [CrossRef]
Unger, M. A. , Chou, H.-P. , Thorsen, T. , Scherer, A. , and Quake, S. R. , 2000, “ Monolithic Microfabricated Valves and Pumps by Multilayer Soft Lithography,” Science, 288(5463), pp. 113–116. [CrossRef] [PubMed]
Whitney, R. , 1949, “ The Measurement of Changes in Human Limb-Volume by Means of a Mercury-Inrubber Strain Gauge,” J. Physiol., 109(1–2), p. Proc–5.
Majidi, C. , Park, Y.-L. , Kramer, R. , Bérard, P. , and Wood, R. J. , 2010, “ Hyperelastic Pressure Sensing With a Liquid-Embedded Elastomer,” J. Micromech. Microeng., 20(12), p. 125029. [CrossRef]
Park, Y.-L. , Tepayotl-Ramirez, D. , Wood, R. J. , and Majidi, C. , 2012, “ Influence of Cross-Sectional Geometry on the Sensitivity and Hysteresis of Liquid-Phase Electronic Pressure Sensors,” Appl. Phys. Lett., 101(19), p. 191904. [CrossRef]
Tepáyotl-Ramírez, D. , Lu, T. , Park, Y.-L. , and Majidi, C. , 2013, “ Collapse of Triangular Channels in a Soft Elastomer,” Appl. Phys. Lett., 102(4), p. 044102. [CrossRef]
Huh, D. , Mills, K. , Zhu, X. , Burns, M. A. , Thouless, M. , and Takayama, S. , 2007, “ Tuneable Elastomeric Nanochannels for Nanofluidic Manipulation,” Nat. Mater., 6(6), pp. 424–428. [CrossRef] [PubMed]
Sparreboom, W. , Van Den Berg, A. , and Eijkel, J. , 2009, “ Principles and Applications of Nanofluidic Transport,” Nat. Nanotechnol., 4(11), pp. 713–720. [CrossRef] [PubMed]
Ozutemiz, K. B. , Wissman, J. , Ozdoganlar, O. B. , and Majidi, C. , 2018, “ Egain–Metal Interfacing for Liquid Metal Circuitry and Microelectronics Integration,” Adv. Mater. Interfaces, 5(10), p. 1701596. [CrossRef]
Sadd, M. H. , 2009, Elasticity: Theory, Applications, and Numerics, 2nd ed., Academic Press, Waltham, MA.
Muskhelishvili, N. I. , 1977, Some Basic Problems of the Mathematical Theory of Elasticity, 1st ed., Springer, Berlin. [CrossRef]
Smirnov, V. I. , 1964, A Course of Higher Mathematics, Vol. 5, Elsevier, Oxford, England.
Savin, G. N. , 1961, Stress Concentration Around Holes, Vol. 1, Pergamon Press, Oxford, England.
Kachanov, M. L. , Shafiro, B. , and Tsukrov, I. , 2003, Handbook of Elasticity Solutions, Springer Science & Business Media, Berlin, Germany. [CrossRef]
Anderson, T. L. , and Anderson, T. , 2005, Fracture Mechanics: Fundamentals and Applications, CRC Press, Bacon Raton, FL.
Adams, G. G. , 2015, “ Critical Value of the Generalized Stress Intensity Factor for a Crack Perpendicular to an Interface,” Proc. R. Soc. A, 471, p. 20150571. [CrossRef]


Grahic Jump Location
Fig. 1

(a) Examples of soft microfluidics for emerging applications: (i) Quake valve (reproduced with permission from Unger et al. [16] Copyright 2000 by AAAS), (ii) fluidic soft robot (reproduced with permission from Wehner et al. [14] Copyright 2016 by Springer Nature), (iii) LM electronics (reproduced with permission from Ozutemiz et al. [23] Copyright 2018 by Wiley—VCH Verlag GmbH & Co. KGaA), and (iv) tunable nanoparticle filtration (reproduced with permission from Huh et al. [21] Copyright 2007 by Springer Nature). (b) Soft microfluidic channel modeled as a prismatic opening embedded inside of an elastic sheet subject to surface pressure p. (c) Selected channel cross section, which is representative of geometries typically used in soft microfluidics.

Grahic Jump Location
Fig. 2

(a) Deformation of an elliptical channel with w = 0.2, h = 0.1, p = 0.05, and ν = 0.49: (gray) initial shape, (markers) numerical solution for deformed shape, (dashed) approximate solution for deformed shape. (b) Cross-sectional area as a function of normalized pressure: (circular markers) numerical solution, (dashed) approximate solution, and (solid) analytic solution. (c) Deformation of a rectangular channel with w = 0.2, h = 0.1, p = 0.05, and ν = 0.49: (gray) initial shape, (markers) numerical solution for deformed shape, (dashed) approximate solution for deformed shape. (d) Cross-sectional area as a function of normalized pressure: (circular markers) numerical solution, (dashed) approximate solution, and (solid) analytic solution.

Grahic Jump Location
Fig. 4

(a) The solution to the classical Michell problem of a circular opening in an elastic plate under far-field stress is used to estimate the deformation of an elliptical channel with principle dimensions a and b. (b) The deformation of a rectangular channel is estimated by superposing uniform deformation of a homogenous Hookean solid with the opening of a slit under far-field stress.

Grahic Jump Location
Fig. 3

Gauge factor G versus channel aspect ratio α. (a) Circular cross section with a = 0.1 and varying b: (circular markers) numerical solution, (solid) analytic solution, (dashed) approximate solution. (b) Rectangular cross section with w = 0.1 and varying h: (square markers) numerical solution, (solid) analytic solution, and (dashed) approximate solution. (c) Numerical solutions for triangular (triangle markers) and diamond-shape (diamond markers) also exhibit a similar monotonic dependency; the dashed line corresponds to a fit of G=1.2/α.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In