Assessment of New Relation for the Elastic Compliance of a Film-Substrate System
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Assessment of New Relation for the Elastic Compliance of a Film-Substrate System Andrei Rar1,2, H. Song3, and G.M. Pharr1,2 1 The University of Tennessee, Department of Materials Science & Engineering, Knoxville, TN 2 Oak Ridge National Laboratory, Metals and Ceramics Division, Oak Ridge, TN 3 Licenergy Corp., Houston, TX. ABSTRACT A new closed-form relation for the elastic compliance of a film-substrate system indented by a flat cylindrical punch is presented. The relation is an extension of the work of Gao at al. [1] modified to increase the applicable range of modulus mismatch between the film and substrate. The accuracy of the relation is assessed by comparison to finite element simulations, other approximate numerical solutions, and nanoindentation experiments performed on a fluorinated silicate glass (FSG) film on a silicon substrate. The benefits and limitations of the relation are discussed.
INTRODUCTION The decreasing size of electronic and MEMS devices requires new analytical methods to determine mechanical properties of nanostructured materials. A commonly encountered problem is the determination of the elastic (or Young) modulus of materials deposited as very thin films on substrates. One popular approach for obtaining substrate independent elastic modulus measurements is to is use well established nanoindentation methods for bulk materials [2], while constraining the indentation depths to less than 5 - 10% of the film thickness [3]. However, such an approach is often not feasible, especially for films with thicknesses less than 100 nm. Moreover, even for films with thickness of a few microns, direct measurements of mechanical properties may be difficult because of surface roughness, oxides films or other surface contaminants. One approach for obtaining substrate-independent film properties is based on the measurement of the composite response of the film/substrate system as a function of indentation depth, and extrapolating to small indentation depths where the film properties dominate. However, such an approach requires an appropriate analytical expression for the composite property as a function of depth through which the extrapolation can be facilitated. Numerous analytical solutions have been developed for the composite elastic modulus of film/substrate systems for flat punch indentation [1, 4, 5]. The analysis of Gao at al. [1] has proven particularly useful because it is a closed-form solution with no adjustable parameters. The analysis is based on first order perturbation method which relates the effective compliance Ceff= (1-ν)/µ of the film/substrate system to the elastic properties of the film and substrate through the simple relation Ceff =
1− νs − (ν f − ν s)I1 µ s + ( µ f − µs)I0
L10.10.1
(1)
Here, µ is the shear modulus, ν is Poisson's ratio, and the subscripts "f" and "s" represent the film and substrate, respectively. I0 and I1 are weighting functions given by I0 =
2 1 t 1 + (t/ a) (1− 2ν) ln arctan(t/ a) + π 2π(1− ν) a (t/ a) 2
2
−
t/ a 1 + (t/ a)
2
(2)
and I1
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