IOP PUBLISHING
NANOTECHNOLOGY
Nanotechnology 19 (2008) 435701 (5pp)
doi:10.1088/0957-4484/19/43/435701
Size and shape effects on creep and diffusion at the nanoscale G Guisbiers1 and L Buchaillot IEMN, UMR CNRS 8520, Scientific City, Avenue Henri Poincar´e, BP 60069, F-59652 Villeneuve d’Ascq, France E-mail:
[email protected]
Received 26 May 2008, in final form 7 August 2008 Published 22 September 2008 Online at stacks.iop.org/Nano/19/435701 Abstract In this paper, we have investigated the size and shape effects on creep and diffusion phenomena at the nanoscale. From a classical thermodynamic model, the higher diffusion of nanostructures is explained. As creep is particularly due to diffusion processes, it is therefore important to consider it at the nanoscale. Therefore, to be able to control creep in the nanoworld, temperature and stress thresholds, taking into account the size and shape of the nanostructure, are defined. (Some figures in this article are in colour only in the electronic version)
stress thresholds are then defined. In section 3, the results are discussed and compared with the literature. Section 4 deals with the conclusions and prospects.
1. Introduction The current scientific and technological interest in nanomaterials comes from the fact that matter at the nanoscale behaves differently from in the macroscopic world [1–5]. When the size of materials decreases down to the nanometer range, new properties arise due to size, shape and quantum effects. In nano-and microtechnology, the reliability of devices is of extreme importance and therefore it is necessary to know the thermomechanical behavior of the structures at these scales. Residual stress, yielding, creep, fatigue and fracture of materials exert a great influence on the performances of nanoand microstructures [6]. In particular, devices made out of low melting point materials are sensitive to flaw mechanisms like creep. For example, the mobile bridge of RF-MEMS (radiofrequency microelectromechanical systems), generally made in aluminum [7–9], which is a low melting point material, is then expected to creep, although it is well known that the diffusion activation energy is lower for nanostructured materials than the bulk ones, and that therefore the diffusion is enhanced at the nanoscale. Until now, to the best of our knowledge, there is no creep criteria developed at the nanoscale. Therefore, we propose to develop this issue by studying the creep behavior of nanostructured materials, inside a classical thermodynamic approach. This top-down approach discusses, in section 2, the size and shape effects on the creep mechanism through diffusion behavior at the nanoscale. In particular, the size effect on the diffusion activation energy is analyzed through the size effect on the melting temperature. Temperature and
2. Creep The creep of materials is a time-dependent inelastic deformation process at constant stress [4, 10]. The evolution of creep with time can be separated into three different stages: primary creep, secondary creep and ternary creep which ends up with fracture. The primary, secondary and ternary creep are due, respectively, to grain boundary diffusion (Coble creep), dislocation movements and lattice (bulk) diffusion (Nabarro– Herring creep). The creep rate of each stage varies differently with the grain size: in fact, ε˙ primary ∝ d −3 , ε˙ secondary ∝ d 0 , ε˙ ternary ∝ d −2 , where d is the grain diameter. Compared to bulk, at low sizes, the creep is enhanced. Therefore, to control creep at the nanoscale, it is necessary to investigate the influence of size and shape on diffusion. 2.1. Diffusion phenomena at the nanoscale Before analyzing diffusion at the nanoscale, let us remember what is diffusion at the macroscale. The displacement of atoms due to thermal energy, kT , is called diffusion. This process is governed by an Arrhenius equation [10]: Q∞ D∞ = D0,∞ e(− RT )
where D0,∞ is a pre-exponential factor, Q ∞ is the thermal activation energy without considering size and shape effects, R is the ideal gas constant and T is the temperature.
1 Author to whom any correspondence should be addressed.
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© 2008 IOP Publishing Ltd Printed in the UK
Nanotechnology 19 (2008) 435701
G Guisbiers and L Buchaillot
There are four types of diffusion: surface, grain boundary, dislocation and lattice (bulk) [10]. Each type of diffusion is characterized by a thermal activation energy. All of these thermal activation energies Q ∞ , are related to the melting temperature, Tm,∞ , via a coefficient C which is different for each type of diffusion, Q ∞ = C Tm,∞ . C is constant for a given class of materials and a given type of diffusion process (considering dislocation movements, C ∼ 18 R for metals) [11, 12]. At the nanoscale, the material properties can be different for two main reasons. First, nanomaterials have a high surface area over volume ratio. Second, quantum effects appear when one reaches the interatomic distance range. When looking at the nanoworld, there are two main approaches: top-down and bottom-up [1, 13, 14]. In the top-down approach that we are considering in this paper, one looks at the variation of the properties of systems that change when going from the macroscopic to the nanoscopic dimensions. This allows us to use the well-known physical laws and to extrapolate to the small dimensions. In the bottom-up approach, one starts from atoms and one adds more and more atoms in order to see how the properties are modified. Adopting a top-down approach and assuming no size effect on C , we can write the thermal activation energy at the nanoscale, Q , as Q = C Tm (2)
Figure 1. Diffusion activation energy versus the radius of the copper and gold spherical nanoparticle.
application of thermodynamics is ∼2 nm. This is the lower size limit that we will use in this work. Therefore any shape instability effects due to the thermal fluctuations above 3% are not addressed here, and other methods such as molecular dynamics simulations should be considered for such extremely small nanostructures (
