NAN/PHY/MSE 546 (Venables) Sect 1.1

Lecture notes for NAN/PHY/MSE 546 Sect 1.1 (Venables)

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

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Lecture notes by John A. Venables. Notes revised Spring '03. Latest version 22 Aug 2012, reformatted to avoid the need for the symbol font.

1. Introduction to Surface Physics

I'm not giving you a general introduction here: you can get that by reading the prefaces of the recommended books; we can come back to 'motivation' via questions you pose. The references given are for further exploration: I am not expecting you to charge off and look all of them up in order to satisfy me.

However, some words of explanation are probably needed. Section 1.1 introduces some of the thermodynamic ideas which are used to discuss small systems. Section 1.2 explores this in more detail for small crystals, both within the TLK model, and with examples taken from real crystals. Section 1.3 explores the differences between thermodynamics and kinetics, as exemplified by the vapor pressure and ideas of crystal growth, introducing the role of atomic vibrations.

Section 1.4 discusses the ideas behind reconstruction of crystal surfaces, and section 1.5 introduces some concepts related to surface electronics. These sections are needed for the sections that follow, though they could be studied after section 2, for example, without much loss. We may wish to come back to individual topics later; for example, although the thermodynamics of small crystals is studied here, we have not covered many experimental examples, nor more than the simplest models. The reason is that not everyone will want to study this topic in detail. This section 1 has been reworked into my book, chapter 1, which has contents and updates here. I have prepared some related appendices, which are in the book, and these may also be useful.

1.1 Elementary Thermodynamic Ideas

Refs: Blakely, Introduction to the Properties of Crystal Surfaces, Chaps 1,3; Hudson, Chap 5; Luth, Chap 3; Zangwill, Chap 1; C. Herring (1953) article, quoted by Blakely. Several review articles, such as C. Rottmann and M. Wortis, Physics Reports 103 (1984) 59-79.

Thermodynamic Potentials and the Dividing Surface

Ideas go back to J.W. Gibbs in the 1870's and 80's (collected works, 1928). I will follow Blakely. Plot concentration profile in the vicinity of a phase boundary between phases 1 and 2, and sketch idea of the 'dividing surface' (diagram 1, or book figure 1.1).

Extensive thermodynamic potentials (e.g. U, F, G...) can be written as a contribution from phases 1, 2 plus a surface term:
e.g. Fbulk = F(N1,N2) and we know that dFbulk = - SdT - PdV + μdN = 0 at constant T, V, N.

Now with the surface (area A), Ftot = F(N1,N2,A) and dFtot = dFbulk + fsdA... if N1, N2 remain constant. This fs is the extra Helmholtz free energy per unit area. This represents Gibbs' idea of the 'dividing surface'. Although the concentration varies in the neighborhood on the surface, we consider the system as uniform up to this interface: fs is then the surface excess free energy.

Surface Tension and Surface Energy

Define the surface tension, γ (gamma) as the reversible work done in creating unit area of new surface:
i.e. γ = lim(dA→0) (dW/dA) = (dFtot/dA)T,V,
e.g. as in diagram 2 (book figure 1.2): ΔF = F1 - F0 = 2γA; dFtot = γdA.
At const T, V: dFtot = -SdT - PdV + Σμi dNi + fsdA = fsdA + ΣμidNi. Therefore γdA = fsdA + ΣμidNi.

In a 1-component system, e.g. metal-vapor, we can choose the dividing surface such that dNi = 0, and then γ ≡ (is equivalent to) fs. Otherwise, the introduction of a surface can cause changes in Ni, i.e. we have N1 + N2 in the bulk, and dNi → surface, so that dNi, the change in the bulk number of atoms in phase i, is negative. We then write

dN = -ΓdA and γ = fs - ΣΓiμi,

where the second term is the free energy contribution of atoms going from the bulk to the surface. As Blakely puts it on page 5, γ is the surface density of (F-G): think about that - it is related to Statistical Mechanics in the Grand Canonical Ensemble...!

This simple example shows that care is needed. The most common phenomena of this type are:

If you need more details of multi-component thermodynamics, see Blakely, Chap 3, or A.W. Adamson 'Physical Chemistry of Surfaces', e.g. 4th or 5th edition, 1982/1990, Chap 3.5., or Hudson 'Surface Science: an introduction', 1992, chap 5. But for now, I don't; thus γ = fs for 1-component systems. So, we can go on to define surface excess internal energy, es; entropy ss. And hence, using the usual thermodynamic relationships:

es = fs +Tss = γ - T(dγ/dT)V; ss = -(dfs/dT)V.

The entropy ss is typically positive, with value a smallish number of Boltzmann's contant (k) per atom. One reason is that surface atoms are less strongly bound, and vibrate with lower frequency and larger amplitude that bulk atoms. Hence es > fs at T > 0.

Surface Stress and Surface Energy

You may note that I have not taken the trouble to distinguish surface energy and surface stress at this stage, because of the complexity of the ideas behind surface stress. Both quantities have the same units, but surface stress is a second rank tensor, whereas surface energy is a scalar quantity. Zangwill makes the distinction in his Chapter 1, pages 8-12, but it feels very early in the course to worry about such ideas. If you need to consider surface stress in your work, a good review is by R.C. Cammarata, Prog. Surf. Sci. 46 (1994) 1-38. I consider some of the issues further in my book, chapter 7, section 7.3.1, and the corresponding updates. But stress is now a hot topic, and there are many different phenomena to be considered; it would be suitable for a mini-project later in the semester.

Some questions to get us started

Continue to section 1.2

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