Equilibrium shape of a buoyant particle
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http://journals.cambridge.org
J. Mater. Res., Vol. 5, No. 7 Jul 1990
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specifically including the possibility that the particle can partially sink into the substrate. We will specifically not consider the kinetics of the process for the moment, which could readily lead to quite different results. It is important to define our notation for the different free energy terms: T— the free energy per unit area of the substrate, explicitly a function of the surface normal. y—the free energy per unit area of the particle, implicitly a function of the facet normal. £—the free energy of creating an interface by joining a unit area of free particle surface and free substrate surface. This is also implicitly a function of the facet normal for a given epitaxial orientation, and is implicitly negative. hj—the normal distance to t h e ; facet. hs—the normal distance to the substrate surface. Aj—the area of t h e ; facet. As—the cross-sectional area of the particle in the substrate surface plane. For the model we will employ two approximations: (1) We will assume that the substrate is large compared to the size of the particle. With this approximation, one can readily show that the change in surface energy due to redistribution of the volume displaced by the particle of substrate material is of the order of TVp/V113 where Vp is the volume displaced and V the total substrate volume; this is small and can be neglected. This assumption should be valid in most real systems. (2) We will assume that £ and V are proportional to y with proportionality constants independent of facet orientation. This is a relatively severe approximation, but without it no simple analytical solution appears to exist. Note that if we were to use a simple model, such as a broken bond model, this approximation is not too severe. We will later on discuss some of the possibilities if this assumption is relaxed. Using an approach similar to that of van Laue,2 we can write the problem to be solved as one of minimizing: F =
j - TAS - 2A
(1)
See, for instance, Fig. 2. We now employ an approach similar in spirit to that used for a modified Wulff construction;6'7 namely, we make a mathematical cut of the © 1990 Materials Research Society
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L. D. Marks and P. M. Ajayan: Equilibrium shape of a buoyant particle
1nm
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FIG. 1. High resolution electron microscopy images of a small gold particle sinking into MgO substrate material during 40 min of observation, taken from Ajayan and Marks.8
particle in the plane of the surface, partitioning the term involving F into two terms aT for below the substrate and (1 - a)T for above the substrate. We can then rewrite the problem as involving minimizing: F = Fa + Fb
(2)
Fa and Fb are independently minimized or only the sum of the two. We will first consider independent minimization, which is simple since both Eqs. (3) and (4) are Wulff problems for which the solutions are given by: Jj
where Fa is for the region above the surface:
=
aT (3)
and Fb is for the r
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