Lecture notes by John A. Venables. Revised for Spring '03;

which leads to mininum total surface excess free energy. Equilibrium therefore corresponds to one crystal with {hkl} faces exposed such that

If γ is isotropic, then the form of the crystal is a sphere in the absence of gravity (liquid drop). With gravity, for larger drops, the 'sessile drop' method is one method of measuring surface tension/ energy (see Adamson), but beware impurities...

So, one needs to know γ(hkl) to deduce the shape of small crystals, or, if you know the shape, you might be able to say something about γ(hkl). Explore this in the next section.

Terrace atoms have extra energy/area, with respect to bulk, e_{t} (6 bonds
minus 5 bonds, every a^{2}). So

Ledge atoms have extra energy/length over terrace atoms (5 bonds minus 4 bonds, every a). This gives

Finally kink atoms have (4-3)φ/2 relative to the ledge
atoms = e_{k}/atom; more interestingly a kink atom has
3φ relative to bulk atoms. This is the same as L/atom,
so adding (or subtracting) an atom from a kink site is equivalent to condensing
(or subliming) an atom from the bulk.

This last result may seem surprising, but it arises because moving a kink around on the surface leaves the number of T, L and K atoms, and the energy of the surface, unchanged. The kink site is thus a 'repeatable step' in the formation of the crystal. Impress your friends by using the original German expression 'wiederhohlbarer Schritt'. This schematic simple cubic crystal is referred to as a Kossel crystal, and the model as the TLK model, shown in diagram 4 (book figure 1.4, and also see web update for section 1.2.3). The original references are W Kossel, Nach. Ges. Wiss. Gottingen (1927) p 135, and I.N. Stranski, Z. Phys. Chem. 136 (1928) 259; but unless both your German and your Library are particularly good, it may be better to approach this topic via more recent books and reviews.

Within this model, we can work out the surface energy as a function of (2D or 3D) orientation. For the 2D case (diagram 3, or book figure 1.3), we can show (so can you) that

But 1/n = tan |θ|. Therefore
e_{s} = e_{t} cos|θ| + e_{l}/a sin|θ|,
or, within the model

Draw this function and show that we have cusps in the (010) directions/ plane normals.

Plot polar diagram of γ(θ), gamma(theta). The Wulff theorem says that the minimum of ∫γdA, the surface integral of gamma.dA, results when you draw the perpendicular through γ(θ) and take the inner envelope: this is the equilibrium form. This is easy to see qualitatively, not so easy mathematically. The cusps are present in the equilibrium form (singular faces). Other faces may or may not be, depending in detail on γ(θ), see diagram 5 (book figure 1.5).

A full set of 3D bond-counting calculations has been done by J.K. MacKenzie, A.J.W. Moore and J.F. Nicholas, J. Phys. Chem. Solids 23 (1962) 185-196 and 197-205; these papers include general rules for nearest neighbor and next nearest neighbor interactions. There is also an Atlas of 'ball and stick' models by these authors. The actual forms of small crystals is a specialist topic, which we may do later. For now, see diagrams 6 and 7 (book figure 1.6), and note that close-packed faces tend to be present in the equilibrium form. For fcc, these are {111}, {100}, {110}..; for bcc {110}, {100}...

For really small particles the discussion has to take the discrete size of the faces into account. This extends up to particles with a million atoms or so, and favors {111} faces in fcc crystals still further (L.D. Marks, Surf. Sci. 150 (1985) 358-366).

The effect of T is interesting. Singular faces have low energy and low entropy; Vicinal (stepped) faces have higher energy and entropy. Thus for increasing T, we have lower free energy for non-singular faces, and the equilibrium form is more rounded. Realistic finite T calculations are relatively rare, and there is still quite a lot of uncertainty (e.g. C. Rottman and M. Wortis, Phys. Reports 103 (1984) 59-79).

Several experiments have been done on the anisotropy of surface energy, and on its variation with T. These experiments require low vapor pressure materials, and have typically been done on Pb, Sn and In, which melt at a lowish temperature; they require very long equilibration times. An example is shown for Pb in figures 8 (book figure 1.7) and 9 (book figure 1.8), taken from the work of J.C. Heyraud and J.J. Metois (Surf. Sci. 128 (1983) 334-342; see also A. Pavlovska et al, Surface Sci. 221 (1989) 233-243).

We notice that the anisotropy is quite small (much smaller than in the Kossel crystal calculation), and that it decreases as T approaches the melting point. This is due to two effects: 1) a calculation with the realistic fcc structure gives a smaller anisotropy than the Kossel crystal (see diagram 7, book figure 1.6); 2) realistic interatomic forces may give still smaller effects. There are more comments and references in my book, and also access to web-based displays of particular crystal surfaces, accessible from the web update to section 1.2.3 and from my web page on simulations.

A point to note for later is that the surface may be under tensile or compressive stress. These stresses give rise to differences between surface tension (or free energy) and surface stress, as discussed in Zangwill, Chap 1, pages 8-12, and also in Cammarata's review. An example exploring the effect of interatomic forces on this difference is D. Wolf, Surf. Sci. 226 (1990) 389-406. But before that we should consider surface reconstructions, which also affect such quantities.

- Consider the (012) face on a Kossel (simple cubic) crystal with 6 nearest neighbor bonds.
- a) Use the analysis of section 1.2 to consider the surface energy of this crystal in terms of the sublimation energy L and the lattice parameter a. Find the ratio of the surface energy of the (012) and the (001) face.
- b) Repeat this exercise for the (012) face of a f.c.c crystal with 12 nearest neighbor bonds. Compare your result with diagrams 7 and 9 (book figures 1.6 and 1.8), and comment on the relative values.
*Note: this problem can be done most readily by drawing the structure and counting bonds. However, there is a general approach you could consider, described by MacKenzie et al. (1962). If the stereograms shown in figure 1.6 are not familiar, consult a crystallography book such as Kelly and Groves (1970), or use the web-based book comments for section 1.2.3 or go directly to a 1998 Sussex student project on the Properties of stereograms.*