Research Papers

A Model of a Trapped Particle Under a Plate Adhering to a Rigid Surface

[+] Author and Article Information
George G. Adams

e-mail: adams@coe.neu.edu
Department of Mechanical and Industrial Engineering,
Northeastern University,
Boston, MA, 02115

Manuscript received June 28, 2012; final manuscript received December 17, 2012; accepted manuscript posted February 11, 2013; published online July 12, 2013. Assoc. Editor: George Kardomateas.

J. Appl. Mech 80(5), 051016 (Jul 12, 2013) (6 pages) Paper No: JAM-12-1274; doi: 10.1115/1.4023625 History: Received June 28, 2012; Revised December 17, 2012; Accepted February 11, 2013

Micro and nanomechanics are growing fields in the semiconductor and related industries. Consequently obstacles, such as particles trapped between layers, are becoming more important and warrant further attention. In this paper a numerical solution to the von Kármán equations for moderately large deflection is used to model a plate deformed due to a trapped particle lying between it and a rigid substrate. Due to the small scales involved, the effect of adhesion is included. The recently developed moment-discontinuity method is used to relate the work of adhesion to the contact radius without the explicit need to calculate the total potential energy. Three different boundary conditions are considered—the full clamp, the partial clamp, and the compliant clamp. Curve-fit equations are found for the numerical solution to the nondimensional coupled nonlinear differential equations for moderately large deflection of an axisymmetric plate. These results are found to match the small deflection theory when the deflection is less than the plate thickness. When the maximum deflection is much greater than the plate thickness, these results represent the membrane theory for which an approximate analytic solution exists.

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Schmidt, M. A., 1998, “Wafer-to-Wafer Bonding for Microstructure Formation,” Proc. IEEE, 86(8), pp. 1575–1585. [CrossRef]
Gösele, U., and Tong, Q. Y., 1998, “Semiconductor Wafer Bonding,” Ann. Rev. Mater. Sci., 28(1), pp. 215–241. [CrossRef]
Pamp, A., and Adams, G. G., 2007, “Deformation of Bowed Silicon Chips Due to Adhesion and Applied Pressure,” J. Adhes. Sci. Technol., 21(11), pp. 1021–1043. [CrossRef]
International Roadmap Committee, 2011, “International Technology Roadmap for Semiconductors, 2011 Edition,” Semiconductor Industry Association, http://www.itrs.net/Links/2011ITRS/2011Chapters/2011ExecSum.pdf
Zong, Z., Chen, C. L., Dokmeci, M. R., and Wan, K., 2010, “Direct Measurement of Graphene Adhesion on Silicon Surface by Intercalation of Nanoparticles,” J. Appl. Phys., 107(2), p. 026104. [CrossRef]
Yamamoto, M., Pierre-Louis, O., Huang, J., Fuhrer, M. S., Einstein, T., and Cullen, W. G., 2012, “Princess and the Pea at the Nanoscale: Wrinkling and Unbinding of Graphene on Nanoparticles,” Phys. Rev. X, 2(4), p. 041018. [CrossRef]
Pamp, A., and Adams, G. G., “Effect of Adhesion on Wafer Separation Due to Trapped Particles,” ASME/STLE 2007 International Joint Tribology Conference, San Diego, CA, October 22–24, ASME Paper No. IJTC2007-44157, pp. 809–811.
Wan, K.-T., and Mai, Y. W., 1995, “Fracture Mechanics of a Shaft-Loaded Blister of Thin Flexible Membrane on Rigid Substrate,” Int. J. Fract., 74, pp. 181–197. [CrossRef]
Komaragiri, U., Begley, M. R., and Simmonds, J. G., 2005, “The Mechanical Response of Freestanding Circular Elastic Films Under Point and Pressure Loads,” ASME J. Appl. Mech., 72(2), pp. 203–212. [CrossRef]
Su, C. Y., Tsao, S., Huang, L. Y., and Hu, C. T., 2010, “Distinguish Various Types of Defects in Bonded Wafer Pairs With the Dynamic Blade Insertion Method,” J. Electrochem. Soc., 157, pp. H792–H795. [CrossRef]
Bollmann, D., Landesberger, C., Ramm, P., and Haberger, K., 1996, “Analysis of Wafer Bonding by Infrared Transmission,” Jpn. J. Appl. Phys., Part 1, 35, pp. 3807–3809. [CrossRef]
Horn, G., Mackin, T. J., and Lesniak, J., 2005, “Trapped Particle Detection in Bonded Semiconductors Using Gray-Field Photoelastic Imaging,” Exp. Mech., 45(5), pp. 457–466. [CrossRef]
Horn, G., Mackin, T., Lesniak, J., and Boyce, B., 2004, “A New Approach for Detecting Defects in Bonded MEMS Devices,” Exp. Tech., 28(5), pp. 19–22. [CrossRef]
Scarpa, F., Adhikari, S., Gil, A. J., and Remillat, C., 2010, “The Bending of Single Layer Graphene Sheets: The Lattice Versus Continuum Approach,” Nanotechnology, 21(12), p. 125702. [CrossRef]
Duan, W. H., and Wang, C. M., 2009, “Nonlinear Bending and Stretching of a Circular Graphene Sheet Under a Central Point Load,” Nanotechnology, 20(7), p. 075702. [CrossRef]
Majidi, C., and Adams, G. G., 2009, “A Simplified Formulation of Adhesion Problems With Elastic Plates,” Proc. R. Soc. London, Ser. A, 465(2107), pp. 2217–2230. [CrossRef]
Majidi, C., and Adams, G. G., 2010, “Adhesion and Delamination Boundary Conditions for Elastic Plates With Arbitrary Contact Shape,” Mech. Res. Commun., 37(2), pp. 214–218. [CrossRef]
Timoshenko, S., and Woinowsky-Krieger, S., 1959, Theory of Plates and Shells II, McGraw-Hill, New York.
Mansfield, E. H., 1964, The Bending and Stretching of Plates, Pergamon, New York.
Adams, G. G., 1993, “Elastic Wrinkling of a Tensioned Circular Plate Using von Kármán Plate Theory,” ASME J. Appl. Mech., 60(2), pp. 520–525. [CrossRef]


Grahic Jump Location
Fig. 1

Perspective view of deflected plate

Grahic Jump Location
Fig. 4

Compliant clamp boundary condition

Grahic Jump Location
Fig. 5

Force versus particle height

Grahic Jump Location
Fig. 6

Contact radius versus particle height

Grahic Jump Location
Fig. 2

Full clamp plate deflection

Grahic Jump Location
Fig. 3

Partial clamp boundary condition



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