Collective buckling of an elastic beam array on an elastic substrate for applications in soft lithography. (English) Zbl 1398.74121
Summary: We analyze the collective buckling of an array of vertical elastic beams with their lower ends built into an elastic substrate. The beams interact between themselves through the deformation of the elastic substrate. The present analysis is more sophisticated than previous ones on rigid beams on an elastic substrate in that the beams are regarded as elastic rather than rigid. From the linear theory for elastic beam buckling and the linear theory of elasticity, an eigenvalue problem is formulated and solved. Calculations show that the deformability of the beams lowers the critical height of the beams, but it does not affect the buckling pattern much. Our work also suggests that the collective buckling is dominated by the interaction of neighboring beams through the deformation of the substrate rather than whether the beams are rigid or elastic. The results are useful for the better understanding, design and application of the nanostructures produced by soft lithography.
References:
| [1] | Xia Y.N., Whitesides G.M.: Soft lithography. Ann. Rev. Mater. Sci. 28, 153–184 (1998) · doi:10.1146/annurev.matsci.28.1.153 |
| [2] | Chou S.Y., Krauss P.R., Renstrom P.J.: Imprint of Sub-25 Nm vias and trenches in polymers. Appl. Phys. Lett. 67, 3114–3116 (1995) · doi:10.1063/1.114851 |
| [3] | Chou S.Y., Krauss P.R., Renstrom P.J.: Nanoimprint lithography. J. Vac. Sci. Technol. B 14, 4129–4133 (1996) · doi:10.1116/1.588605 |
| [4] | Delamarche E., Schmid H., Michel B., Biebuyck H.: Stability of molded polydimethylsiloxane microstructures. Adv. Mater. 9, 741–746 (1997) · doi:10.1002/adma.19970090914 |
| [5] | Schmid H., Michel B.: Siloxane polymers for high-resolution, high-accuracy soft lithography. Macromolecules 33, 3042–3049 (2000) · doi:10.1021/ma982034l |
| [6] | Evans B.A., Shields A.R., Carroll R.L., Washburn S., Falvo M.R., Superfine R.: Magnetically actuated nanorod arrays as biomimetic cilia. Nano Lett. 7, 1428–1434 (2007) · doi:10.1021/nl070190c |
| [7] | Greenhill A.G.: Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and the greatest height to which a tree of given proportions can grow. Proc. Cambridge Phil. Soc. 4, 65–73 (1881) · JFM 13.0741.03 |
| [8] | Timoshenko S.P.: Theory of Elastic Stability. McGraw-Hill, New York (1936) |
| [9] | Alfutov N.A.: Stability of Elastic Structures. Springer, Germany (2000) · Zbl 0970.74001 |
| [10] | Hui C.Y., Jagota A., Lin Y.Y., Kramer E.J.: Constraints on microcontact printing imposed by stamp deformation. Langmuir 18, 1394–1407 (2002) · doi:10.1021/la0113567 |
| [11] | Lin H., Yang J., Tan L., Xu J., Li Z.: Collective buckling of periodic soft nanostructures on surfaces and promotion for nanolithography. J. Phys. Chem. C. 111, 13348–13356 (2007) · doi:10.1021/jp0737000 |
| [12] | Chen Z.G., Yang J.S., Tan L.: Collective buckling of a two-dimensional array of nanoscale columns. J. Phys. Chem. B 112, 14766–14771 (2008) · doi:10.1021/jp8046399 |
| [13] | Li, Z., Feng, K., Yang, J.S., Tan, L., Lin, H.: Collective buckling of nonuniform nanobeams interacting through an elastic substrate. Acta Mech. (to appear) · Zbl 1268.74024 |
| [14] | Sharp K.G., Blackman G.S., Glassmaker N.J., Jagota A., Hui C.Y.: Effect of stamp deformation on the quality of microcontact printing: theory and experiment. Langmuir 20, 6430–6438 (2004) · doi:10.1021/la036332+ |
| [15] | Chuang W.C., Ho C.T., Wang W.C.: Fabrication of a high-resolution periodical structure using a replication process. Opt. Express 13, 6685–6692 (2005) · doi:10.1364/OPEX.13.006685 |
| [16] | Hsia K.J., Huang Y., Menard E., Park J.U., Zhou W., Rogers J., Fulton J.M.: Collapse of stamps for soft lithography due to interfacial adhesion. Appl. Phys. Lett. 86, 154106 (2005) · doi:10.1063/1.1900303 |
| [17] | Gere J.M.: Mech. Mater. Broks/Cole, Pacific Grove, CA (2001) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
