Short communication Stoney equation limits for samples deformed as a cylindrical surface J.M. Pureza a, *, F. Neri b , M.M. Lacerda b a Department of Mechanical Engineering, Santa Catarina State University, UDESC, Joinville, Brazil b Department of Physics, Santa Catarina State University, UDESC, Joinville, Brazil In a previous work, Pureza et al. [1] proposed an approach for the evaluation of the stress in thin films deposited on substrates much thicker than itself. This problem is a relevant issue for several areas of science [2–9], being the theme of a recent review [10] in commemoration of 100 years of the first description of the problem, proposed by Stoney [11] who considered the sample as an one-dimensional plate. The Stoney equation indicates a linear relationship between the film stress (s) and its bending, being straightforwardly modified for two-dimensional systems with small deformation by including the substrate Poisson ratio (n s ) [12], as seen in Eq. (1). s ¼ E s  t 2 s 6 ð1  n s Þ t f K (1) where E s is the substrate Young modulus, t s and t f are the substrate and film thickness, respectively, and K is the film curvature. It has become the standard expression for the analysis of this problem with reasonable agreement with experimental results. However, Stoney equation does not take into account relevant aspects like the non-uniformity of the stresses as well as the tri-dimensionality and the boundary conditions of the samples. Such aspects gave rise to semi-empirical modifications, with little experimental agree- ment [12–14] and more elaborated models (see e.g. [15] and references therein [1,16–18]), as well as finite-element simula- tions that present local dependence to stress distribution [3,19]. Finot et al. [3] using a finite-element analysis identified three distinct regimes for the evolution of curvature, which are dependent on the quantity A ¼ st f l 2 s t 3 s , where l s is the character- istic size of the sample: (I) for lower values of A, the deformation has a spherical shape and Stoney equation is satisfied considering 10% of error as acceptable; (II) as the parameter A increases, deformation maintains the spherical shape but Stoney equation looses validity and (III) for even larger values of A, Stoney equation is no longer valid and the sample undergoes two abrupt changes, initially to an ellipsoidal shape and finally to a cylindrical shape. In the previous work [1], by means of the minimization of the deformation energy, we derived an expression for the thin film stress when the deformation maintains the spherical shape (regimes I and II in Finot et al. [3]). We considered samples with thickness ratio t f / t s  1, a assumption that allows the simplification of the deforma- tion energy of the sample and consider that the substrate thickness change is a lower order term. So, the stress of the film is written in terms of the curvature K of the sample, as seen in Eq. (2), s u ¼ s ’ ¼ E s 6 1 1  n s  2n 2 s t 2 s t f K (2) which is quite similar to Stoney’s. Fig. 1 shows agreement between this equation and Stoney’s at regime I but, at regime II, results stay in between the finite element (obtained by Finot et al. [3]) and the other formulations [12,14–16]. However, the analysis of the values obtained by Finot [3] using finite-element simulation, see Fig. 1, indicate a important change on the dependence between the curvature and the stress at regime III. This situation motivated us to apply the same approach to a thin film deposited on a thick substrate that is deformed as a cylindrical surface. In this situation, Applied Surface Science 256 (2010) 4408–4410 ARTICLE INFO Article history: Received 28 July 2009 Accepted 2 February 2010 Available online 6 February 2010 PACS: 62.20.Dc, 81.40.Jj, 68.60.Bs Keywords: Stress Strain Stoney equation Deformation energy ABSTRACT This paper proposes a three-dimensional system for modelling stress in thin films deposited on thick substrates deformed as a small cylindrical surface by means of the minimization of the deformation energy. The results show the validity limits of the well-established Stoney equation and indicate the necessity of a correction term for substrates with Poisson ratio (n s ) in the range of 0.25 < n s  0.4. ß 2010 Elsevier B.V. All rights reserved. * Corresponding author. Tel.: +55 47 4009 7821; fax: +55 47 4009 7940. E-mail address: pureza@joinville.udesc.br (J.M. Pureza). Contents lists available at ScienceDirect Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc 0169-4332/$ – see front matter ß 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2010.02.002