Equilibrium Shapes of Solid Particles on Elastically Mismatched Substrates 1 Jeffrey W. Bullard 2 and Mohan Menon 3 Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 E-mail: jeff@puffin.mse.uiuc.edu Received April 26, 1999; accepted August 20, 1999 Small particles or islands bonded to a substrate can be pro- foundly influenced by both interfacial and elastic driving forces that tend to have opposing influences on the apparent wetting behavior. The superposition of these two driving forces can there- fore lead to a rich set of particle properties, most notably their equilibrium shapes. Here we present a variational analysis leading directly to an Euler–Lagrange equation that can be solved to yield the equilibrium shapes of partially wetting particles as a function of their size, interface energy densities, and elastic interaction with a rigid substrate. The solutions are used to gain insight into the variables that most significantly influence the equilibrium mor- phology, and to derive the approximate driving force for surface area reduction by coarsening among a dispersion of unequally sized particles. The relatively simple analytical model can also form a foundation upon which more realistic numerical simula- tions may be built and compared. © 1999 Academic Press Key Words: theory and modeling; particle shapes; elastic mis- match; wetting; thermodynamics. 1. INTRODUCTION The morphological evolution and equilibria of heterophase solids are sensitively dependent on interfacial and elastic driv- ing forces that originate at interphase boundaries. These two driving forces tend to oppose each other; in fact, it is the nature of their competing influences that governs the equilibrium and stability of systems ranging from stressed surfaces (1) to par- ticles that are coherently bonded to, or thermally mismatched with, an elastic matrix (2, 3) or substrate (4 –7). In this paper we consider equilibria within the latter type of system, namely, those composed of small solid particles bonded to a foreign substrate. The equilibrium particle shapes are of considerable theoretical and technological importance because the proper- ties of such particles— growth forms, electronic transport, quantum confinement, optoelectronic characteristics, catalytic activity, etc.— can depend critically on their dimensions and morphologies (8 –10). In many applications involving particles on substrates, the processing conditions can be far from equilibrium, and both the morphology and properties may be dictated by kinetic con- straints. But regardless of how near equilibrium a particle may be in practice, determination of its equilibrium shape is still important for at least two reasons. First, the free energy of the particle at equilibrium determines the driving force for, and trajectory of, any shape relaxation that may occur. Second, for a dispersion of unequally sized particles, the total free energy can be reduced by coarsening, during which the total surface area decreases by growth of larger particles at the expense of smaller ones. During coarsening, local equilibration of the particle shapes may well be rapid relative to the rates of interparticle diffusion. In such a scenario, the equilibrium shape of the particle will determine its chemical potential as a function of its volume, which in turn represents the driving force for mass transport between particles of different sizes. The theory of small particles in equilibrium with foreign substrates was limited, until recently, to a model derived by Winterbottom (11) which, although accounting for partial wet- ting conditions, excludes the possibility of elastic mismatch between substrate and particle. More recently, a few theoretical models of epitaxial thin film islands have included a contribu- tion of strain energy to predict island stability (6, 12–15). These models, while extremely insightful, are somewhat lim- ited because either they assume simplified distributions of elastic energy within the particle and substrate or they restrict the possible morphologies to a narrow family of polyhedra. Only one numerical model has been reported that fully ac- counts for both the details of the elastic field and the particle shape (16). That model simulates diffusion-limited mass trans- port to allow relaxation to a limiting shape in two dimensions, assuming a zero wetting angle and equal elastic moduli of both substrate and island. The assumptions invoked in that model make it strictly applicable only to nearly equally compliant, isotropic islands and substrates having a continuous wetting layer, such as those formed by a Stranski–Krastonov growth mode of germanium on silicon (5–7). 1 Supported by the National Science Foundation under Grant DMR- 9702610. 2 To whom correspondence should be addressed. 3 Currently with the Department of Inorganic Chemistry, Norwegian Insti- tute of Science and Technology, 7491 Trondheim, Norway. Journal of Colloid and Interface Science 219, 320 –326 (1999) Article ID jcis.1999.6495, available online at http://www.idealibrary.com on 320 0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.