NAN/PHY/MSE 546 (Venables) Sect 1.2

Lecture notes for NAN/PHY/MSE 546 Sect 1.2 (Venables)

Arizona Board of Regents for Arizona State University and John A. Venables

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Lecture notes by John A. Venables. Revised for Spring '03; Latest version 25 Aug 2012, reformatted to avoid the need for the symbol font.

1.2 Surface Energies and the Wulff Theorem

Refs: as above plus original ref: Zeitschrift f. Kristallographie 34 (1901) 449 (not everything was done last year, and the most thorough description is in Herring's 1953 article quoted by Blakely).

General Considerations

At equilibrium, a small crystal, vol V, with N atoms of 1 component: dF = 0 at const T,V leads to γdA = 0, or

∫γdA = minimum,

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

∫γ(hkl)dA(hkl), is minimum.

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.

The Terrace-Ledge-Kink Model

Take simple cubic lattice, with nearest neighbor (nn) bonds (diagram 3, or book figure 1.3), with surface inclined at angle θ to low index (010) plane. Bulk atoms have 6 bonds = 6φ. The sublimation energy L, per unit volume, is (6φ/2)(1/a3). The division by 2 is to avoid double counting: 1 bond involves 2 atoms. Units are say eV/nm3, or many (chemical) equivalents, such as kcal/mole. Useful conversion factors: 1eV ≡ 11604K ≡ 23.06 kcal/mole, see book Appendix C.

Terrace atoms have extra energy/area, with respect to bulk, et (6 bonds minus 5 bonds, every a2). So

et = (6-5)φ/2a2 = φ/2a2 = La/6 per unit area.

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

el = (5-4) φ/2a = La2/6 per unit length.

Finally kink atoms have (4-3)φ/2 relative to the ledge atoms = ek/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

es = (et + el/na) cosθ.

But 1/n = tan |θ|. Therefore es = et cos|θ| + el/a sin|θ|, or, within the model

es = (La/6)(cos|θ| + sin|θ|).

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

Wulff Construction and the forms of small crystals

Refs as above, plus historical handout from Herring's 1953 article, quoted by Blakely.

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.

Some problems to do, also on assignment #1 for the course


Continue to section 1.3

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