The equilibrium shape of anisotropic interfacial particles

Abstract

The Wulff construction yields the equilibrium shape of a particle of fixed volume embedded in a single phase at constant temperature. When the particle attaches to an interface (boundary) between two distinct phases or grains of the same phase, the Wulff construction must be modified to account for the abutment of Wulff shapes at the boundary as well as the boundary energy that is replaced by the particle. If the boundary is non-deformable, a portion of the particle interface is constrained to replace the image (i.e. the exact position) of the boundary that is removed, and the Winterbottom construction yields the equilibrium particle shape. If the boundary can deform, the particle is not constrained to replace the image of removed boundary, and the particle shape is determined by a more general modification of the Wulff construction. In two dimensions, the particle attaches to an initially flat boundary and creates two disjoint segments that remain flat at equilibrium. In three dimensions, the boundary surrounding the particle is contiguous, and numerical calculations show that such a boundary is not necessarily flat but maintains a constant mean curvature of zero at equilibrium.

Acknowledgements

The authors would like to thank Ulrich Dahmen and Erik Johnson for enlightening discussions as well as Steve Langer and John Cahn for useful observations. Kenneth Brakke is also gratefully acknowledged for providing fitting answers to questions regarding the implementation of Surface Evolver. EJS is grateful for support from a National Defense Science and Engineering Graduate Fellowship and the Massachusetts Institute of Technology–France programme. Special thanks are due to Paul Wynblatt for valuable discussions and comments on early drafts of this manuscript. W. C. C. wishes to acknowledge National Science Foundation award DMR-0010062, Nanometer Scale Induced Structure Between Amorphous Layers and Crystalline Materials (NANOAM), in addition to the co-funding of the NANOAM project by EU contract G5RD-CT-2001-00586 and proposal GRD2-2000-30351. D. C. wishes to acknowledge support of a Technologies, Materials, and Thermal Hydraulics for Lead Alloys (TECLA) European contract.

Notes

§ Email: [email protected].

‡ Email: [email protected].

‡ Email: [email protected].

† Throughout this work, the particle is treated as a single crystal or amorphous (e.g. as a void). In other words, the particle contains no internal interfaces.

‡ A surface of zero mean curvature is also known as a minimal surface. A familiar example of a minimal surface occurs when a soap film spans a finite closed curve (such as a wire loop) and the pressure is the same on both sides of the film.

§ A surface of non-zero constant mean curvature minimizes the surface area under a volume constraint. When the pressure is greater on one side of a surface (as in a soap bubble), the surface will tend to adopt a surface of non-zero constant mean curvature.

† Both the Gibbs (chemical) free energy and are homogeneous of degree 1 (or HD1). If a function is homogeneous of degree n (or HDn), multiplying its argument by a number λ multiplies the entire function by that number raised to the nth power, that is λ ⁿ . A corollary to this theorem (Euler’s theorem) is that are HD0 functions.

† The Wulff centre is the point corresponding to the origin of the gamma and xi plots used to construct a particular Wulff shape. When has a centre of symmetry, it is useful, but not necessary, to choose that centre as the Wulff center. An example for which the selected Wulff centre does not coincide with the centre of volume of a Wulff shape is given in appendix A.

† A note on the equilibrium condition for anisotropic boundaries is given in § 6.

‡ Alternatively, the tangent plane at any point on the boundary will intersect the boundary elsewhere. Such points are anticlastic or hyperbolic.

† The mean curvature can also be interpreted as the rate of change in surface normal with an infinitesimal increment along the surface, averaged over all directions. Alternatively, it is the increase in volume for a local displacement along . can be interpreted as a measure of the span of surface normal directions for an infinitesimal patch of surface. This definition expressed as an extrinsic geometric invariant.

† The basic idea is that the exterior vertices provide a measure of the local deviation of the surface from its tangent plane at P, that is a measure of the surface curvature at P.

Additional information

Notes on contributors

Dominique Chatain §

§ Email: [email protected]. ‡ Email: [email protected].