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Research Papers

Theory on Bending in Cantilever Beams With Adsorbed Islands

[+] Author and Article Information
Chuangchuang Duan

LNM,
Institute of Mechanics,
Chinese Academy of Sciences,
Beijing 100190, China;
School of Engineering Sciences,
University of Chinese Academy of Sciences,
Beijing 100049, China
e-mail: duanchuangchuang@imech.ac.cn

Yujie Wei

LNM,
Institute of Mechanics,
Chinese Academy of Sciences,
Beijing 100190, China;
School of Engineering Sciences,
University of Chinese Academy of Sciences,
Beijing 100049, China
e-mail: yujie_wei@lnm.imech.ac.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 14, 2017; final manuscript received May 20, 2017; published online May 31, 2017. Assoc. Editor: Kyung-Suk Kim.

J. Appl. Mech 84(7), 071006 (May 31, 2017) (7 pages) Paper No: JAM-17-1203; doi: 10.1115/1.4036819 History: Received April 14, 2017; Revised May 20, 2017

Traction between adsorbed islands and the substrate is commonly seen in both living and material systems: deposited material gathers into islands at the early stage of polycrystalline film deposition and generates stress due to lattice mismatch, cells exert cellular traction to extracellular matrix to probe their surrounding microenvironment in vivo, and so on. The traction between these islands and the substrate can result in perceivable macroscopic deformation in the substrate and may be measurable if the substrate is a cantilever beam. However, currently broadly used Stoney equation is incapable of handling such boundary condition. In this paper, we give the closed-form expression on the resulted curvature in substrate beams by distributed tractions. Such a relationship could be employed to monitor the stress evolution during thin film deposition, to quantify the stress level of cell traction as cells adhere to cantilever beams, and other related mechanical systems like charging–discharging induced stress in island-patterned electrode films. Moreover, we found that follower traction induced by an array of islands could lead to negative curvature. It shields light on the early stage compressive stress during polycrystalline film deposition.

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References

Zhang, Z. , and Lagally, M. G. , 1997, “ Atomistic Processes in the Early Stages of Thin-Film Growth,” Science, 276(5311), pp. 377–383. [CrossRef] [PubMed]
Thompson, C. V. , and Carel, R. , 1996, “ Stress and Grain Growth in Thin Films,” J. Mech. Phys. Solids, 44(5), pp. 657–673. [CrossRef]
Spaepen, F. , 2000, “ Interfaces and Stresses in Thin Films,” Acta Mater., 48(1), pp. 31–42. [CrossRef]
Cammarata, R. C. , 1994, “ Surface and Interface Stress Effects in Thin Films,” Prog. Surf. Sci., 46(1), pp. 1–38. [CrossRef]
Stoney, G. G. , 1909, “ The Tension of Metallic Films Deposited by Electrolysis,” Proc. R. Soc. London Ser. A, 82(553), pp. 172–175. [CrossRef]
Freund, L. B. , and Suresh, S. , 2004, Thin Film Materials: Stress, Defect Formation and Surface Evolution, Cambridge University Press, Cambridge, UK.
Chason, E. , and Floro, J. , 1996, “ Measurements of Stress Evolution During Thin Film Deposition,” MRS Proc., 428, p. 499. [CrossRef]
Floro, J. A. , Chason, E. , Cammarata, R. C. , and Srolovitz, D. J. , 2002, “ Physical Origins of Intrinsic Stresses in Volmer–Weber Thin Films,” MRS Bull., 27(1), pp. 19–25. [CrossRef]
Tello, J. S. , and Bower, A. F. , 2008, “ Numerical Simulations of Stress Generation and Evolution in Volmer–Weber Thin Films,” J. Mech. Phys. Solids, 56(8), pp. 2727–2747. [CrossRef]
Cammarata, R. , Trimble, T. , and Srolovitz, D. , 2000, “ Surface Stress Model for Intrinsic Stresses in Thin Films,” J. Mater. Res., 15(11), pp. 2468–2474. [CrossRef]
Rajamani, A. , Sheldon, B. W. , Chason, E. , and Bower, A. F. , 2002, “ Intrinsic Tensile Stress and Grain Boundary Formation During Volmer–Weber Film Growth,” Appl. Phys. Lett., 81(7), pp. 1204–1206. [CrossRef]
Tello, J. S. , Bower, A. F. , Chason, E. , and Sheldon, B. W. , 2007, “ Kinetic Model of Stress Evolution During Coalescence and Growth of Polycrystalline Thin Films,” Phys. Rev. Lett., 98(21), p. 216104. [CrossRef] [PubMed]
Sheldon, B. W. , Bhandari, A. , Bower, A. F. , Raghavan, S. , Weng, X. , and Redwing, J. M. , 2007, “ Steady-State Tensile Stresses During the Growth of Polycrystalline Films,” Acta Mater., 55(15), pp. 4973–4982. [CrossRef]
Chason, E. , Shin, J. , Chen, C.-H. , Engwall, A. , Miller, C. , Hearne, S. , and Freund, L. , 2014, “ Growth of Patterned Island Arrays to Identify Origins of Thin Film Stress,” J. Appl. Phys., 115(12), p. 123519. [CrossRef]
Chason, E. , Engwall, A. , Miller, C. , Chen, C.-H. , Bhandari, A. , Soni, S. , Hearne, S. , Freund, L. , and Sheldon, B. , 2015, “ Stress Evolution During Growth of 1-D Island Arrays: Kinetics and Length Scaling,” Scr. Mater., 97, pp. 33–36. [CrossRef]
Sethuraman, V. A. , Chon, M. J. , Shimshak, M. , Srinivasan, V. , and Guduru, P. R. , 2010, “ In Situ Measurements of Stress Evolution in Silicon Thin Films During Electrochemical Lithiation and Delithiation,” J. Power Sources, 195(15), pp. 5062–5066. [CrossRef]
Pharr, M. , Suo, Z. , and Vlassak, J. J. , 2013, “ Measurements of the Fracture Energy of Lithiated Silicon Electrodes of Li-Ion Batteries,” Nano Lett., 13(11), pp. 5570–5577. [CrossRef] [PubMed]
Bucci, G. , Nadimpalli, S. P. , Sethuraman, V. A. , Bower, A. F. , and Guduru, P. R. , 2014, “ Measurement and Modeling of the Mechanical and Electrochemical Response of Amorphous Si Thin Film Electrodes During Cyclic Lithiation,” J. Mech. Phys. Solids, 62, pp. 276–294. [CrossRef]
Sethuraman, V. A. , Chon, M. J. , Shimshak, M. , Van Winkle, N. , and Guduru, P. R. , 2010, “ In Situ Measurement of Biaxial Modulus of Si Anode for Li-Ion Batteries,” Electrochem. Commun., 12(11), pp. 1614–1617. [CrossRef]
Soni, S. K. , Sheldon, B. W. , Xiao, X. , Verbrugge, M. W. , Dongjoon, A. , Haftbaradaran, H. , and Huajian, G. , 2011, “ Stress Mitigation During the Lithiation of Patterned Amorphous Si Islands,” J. Electrochem. Soc., 159(1), pp. A38–A43. [CrossRef]
Xiao, X. , Liu, P. , Verbrugge, M. , Haftbaradaran, H. , and Gao, H. , 2011, “ Improved Cycling Stability of Silicon Thin Film Electrodes Through Patterning for High Energy Density Lithium Batteries,” J. Power Sources, 196(3), pp. 1409–1416. [CrossRef]
Haftbaradaran, H. , Xiao, X. , Verbrugge, M. W. , and Gao, H. , 2012, “ Method to Deduce the Critical Size for Interfacial Delamination of Patterned Electrode Structures and Application to Lithiation of Thin-Film Silicon Islands,” J. Power Sources, 206, pp. 357–366. [CrossRef]
He, Y. , Yu, X. , Li, G. , Wang, R. , Li, H. , Wang, Y. , Gao, H. , and Huang, X. , 2012, “ Shape Evolution of Patterned Amorphous and Polycrystalline Silicon Microarray Thin Film Electrodes Caused by Lithium Insertion and Extraction,” J. Power Sources, 216, pp. 131–138. [CrossRef]
Kumar, R. , Tokranov, A. , Sheldon, B. W. , Xiao, X. , Huang, Z. , Li, C. , and Mueller, T. , 2016, “ In Situ and Operando Investigations of Failure Mechanisms of the Solid Electrolyte Interphase on Silicon Electrodes,” ACS Energy Lett., 1(4), pp. 689–697. [CrossRef]
Discher, D. E. , Janmey, P. , and Wang, Y. L. , 2005, “ Tissue Cells Feel and Respond to the Stiffness of Their Substrate,” Science, 310(5751), pp. 1139–1143. [CrossRef] [PubMed]
Kubow, K. E. , Vukmirovic, R. , Zhe, L. , Klotzsch, E. , Smith, M. L. , Gourdon, D. , Luna, S. , and Vogel, V. , 2015, “ Mechanical Forces Regulate the Interactions of Fibronectin and Collagen I in Extracellular Matrix,” Nat. Commun., 6, p. 8026. [CrossRef] [PubMed]
Harris, A. K. , Stopak, D. , and Wild, P. , 1981, “ Fibroblast Traction as a Mechanism for Collagen Morphogenesis,” Nature, 290(5803), pp. 249–251. [CrossRef] [PubMed]
Rape, A. D. , Guo, W. H. , and Wang, Y. L. , 2011, “ The Regulation of Traction Force in Relation to Cell Shape and Focal Adhesions,” Biomaterials, 32(8), pp. 2043–2051. [CrossRef] [PubMed]
Harris, A. K. , Wild, P. , and Stopak, D. , 1980, “ Silicone Rubber Substrata: New Wrinkle in the Study of Cell Locomotion,” Science, 208(4440), pp. 177–179. [CrossRef] [PubMed]
Lee, J. , Leonard, M. , Oliver, T. , Ishihara, A. , and Jacobson, K. , 1994, “ Traction Forces Generated by Locomoting Keratocytes,” J. Cell Biol., 127(6), pp. 1957–1964. [CrossRef] [PubMed]
Dembo, M. , and Wang, Y. L. , 1999, “ Stresses at the Cell-to-Substrate Interface During Locomotion of Fibroblasts,” Biophys. J., 76(4), pp. 2307–2316. [CrossRef] [PubMed]
Tan, J. L. , Tien, J. , Pirone, D. M. , Gray, D. S. , Bhadriraju, K. , and Chen, C. S. , 2003, “ Cells Lying on a Bed of Microneedles: An Approach to Isolate Mechanical Force,” Proc. Natl. Acad. Sci. U.S.A., 100(4), pp. 1484–1489. [CrossRef] [PubMed]
He, S. J. , Su, Y. W. , Ji, B. H. , and Gao, H. J. , 2014, “ Some Basic Questions on Mechanosensing in Cell-Substrate Interaction,” J. Mech. Phys. Solids, 70, pp. 116–135. [CrossRef]
Mertz, A. F. , Banerjee, S. , Che, Y. , German, G. K. , Xu, Y. , Hyland, C. , Marchetti, M. C. , Horsley, V. , and Dufresne, E. R. , 2012, “ Scaling of Traction Forces With the Size of Cohesive Cell Colonies,” Phys. Rev. Lett., 108(19), p. 198101. [CrossRef] [PubMed]
Gardel, M. L. , Sabass, B. , Ji, L. , Danuser, G. , Schwarz, U. S. , and Waterman, C. M. , 2008, “ Traction Stress in Focal Adhesions Correlates Biphasically With Actin Retrograde Flow Speed,” J. Cell Biol., 183(6), pp. 999–1005. [CrossRef] [PubMed]
Timoshenko, S. P. , and Goodier, J. , 1970, Theory of Elasticity, McGraw Hill, New York.
Braun, M. , and Golubitsky, M. , 1983, Differential Equations and Their Applications, Springer, New York.
Thornton, J. A. , and Hoffman, D. , 1989, “ Stress-Related Effects in Thin Films,” Thin Solid Films, 171(1), pp. 5–31. [CrossRef]
Shull, A. L. , and Spaepen, F. , 1996, “ Measurements of Stress During Vapor Deposition of Copper and Silver Thin Films and Multilayers,” J. Appl. Phys., 80(11), pp. 6243–6256. [CrossRef]
Haftbaradaran, H. , Soni, S. K. , Sheldon, B. W. , Xiao, X. , and Gao, H. , 2012, “ Modified Stoney Equation for Patterned Thin Film Electrodes on Substrates in the Presence of Interfacial Sliding,” ASME J. Appl. Mech., 79(3), p. 031018. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

A cantilever beam with an array of islands bonded to its upper surface, where the lengths of the beam and one island are L and l, respectively. N denotes the number of islands and s denotes the length of the spacing between islands. τ refers to the maximum of the traction and EI refers to the flexural rigidity of the beam.

Grahic Jump Location
Fig. 2

(a) A cantilever beam subjected to tangential force f(x), where the length of the beam is l and the eccentricity of the tangential force is e. (b) An element cut from the beam with sides normal to the deflected axis of the beam.

Grahic Jump Location
Fig. 3

Schematic of the mth element consists of island and spacing: (a) free-body diagram of spacing between islands and (b) free-body diagram of the mth island

Grahic Jump Location
Fig. 4

The curvature of the beam with patterned islands, where X denotes the coordinate from the fixed end of the beam (see Fig. 1): (a) curvature along the beam with 111 islands, (b) curvature within a period (with length 2πl6/τ¯ρ)), and (c) curvature within one island (with length l)

Grahic Jump Location
Fig. 5

Nondimensional deflection curve of the beam: (a) the solid line stands for analytical solution and dots are FEM results, (b) ρ=0.5,N=4,20,100, ymax corresponds to the maximal deflection of the beam when ρ=0.5,N=4, (c) N=1000,ρ=0.05,0.1,0.15, ymax is the maximal deflection of the beam when ρ=0.15,N=1000, and (d) N decreases with increase in ρ, ymax refers to the maximal deflection of the beam when ρ=0.5,N=4

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