Lecture notes by John A. Venables. Notes revised Spring '03.

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.

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.

Extensive thermodynamic potentials (e.g. U, F, G...) can be written as
a contribution from phases 1, 2 plus a surface term:

e.g. F_{bulk} = F(N_{1},N_{2}) and we know that
dF_{bulk} = - SdT - PdV + μdN = 0 at constant T, V, N.

Now with the surface (area A), F_{tot} = F(N_{1},N_{2},A)
and dF_{tot} = dF_{bulk} + f_{s}dA... if N_{1},
N_{2} remain constant. This f_{s} 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: f_{s} is then the
*surface excess free energy*.

i.e. γ = lim

e.g. as in diagram 2 (book figure 1.2): ΔF = F

At const T, V: dF

In a 1-component system, e.g. metal-vapor, we can choose the dividing surface such
that dN_{i} = 0, and then γ ≡ (is equivalent to)
f_{s}. Otherwise, the introduction of a surface can cause changes in N_{i},
i.e. we have N_{1} + N_{2} in the bulk, and dN_{i}
→ surface,
so that dN_{i}, the change in the bulk number of atoms in phase i, is negative.
We then write

dN = -ΓdA and γ = f_{s} - ΣΓ_{i}μ_{i},

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:

- 1) a soap film lowers the surface tension of water. Why? Because the soap molecules come out of solution and form (monomolecular) layers at the water surface (with their 'hydrophobic' ends pointing outwards);
- 2) a clean surface in Ultra-high vacuum has a higher free energy than an oxidised (or contaminated) surface. Why? Because if it didn't, there would be no 'driving force' for oxygen to adsorb, and the reaction wouldn't occur.

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 = f_{s}
for 1-component systems. So, we can go on to define surface excess internal energy,
e_{s}; entropy s_{s}. And hence, using the usual thermodynamic
relationships:

e_{s} = f_{s} +Ts_{s} = γ
- T(dγ/dT)_{V};
s_{s} = -(df_{s}/dT)_{V}.

The entropy s_{s} 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 e_{s} > f_{s} at T > 0.

- 1) Most of us don't have the definitions of the thermodynamic functions in foreground memory. Remind yourself of the definitions of entropy S, and the thermodynamic potentials U, F and G, and differentials dU, dF and dG. How are these functions used to discuss equilibrium phenomena?
- 2) We also don't usually carry around statistical mechanical ideas and distributions in our head. Describe briefly the characteristics of the Micro-Canonical, Canonical and Grand-Canonical distributions, and what fluctuations are allowed in arriving at these distributions. (If you haven't done a course on Statistical Mechanics, this material can be understood as the semester proceeds).
- 3) What is the pressure difference between the inside and outside of a small soap bubble of radius r? Put in numbers to see the effect of surface tension and size, taking care over the units of pressure. Soap bubbles have radii in the 1-10 cm range. But what if we are interested in nano-particles of metals in the size range 10-100 nm?
- 4) One reason for positive surface entropy is given above, due to slower atomic vibrations (weaker bonding). Can you think of any other reasons, which might argue in the same, or indeed the opposite, direction?
- 5) More advanced: Why is γ (gamma) the surface density of (F-G), as Blakely puts it?