NAN/PHY/MSE 546 (Venables) Sect 1.3

Notes for NAN/PHY/MSE 546 Sect 1.3 (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 August 2012, reformatted.

1.3 Thermodynamics versus Kinetics

Equilibrium phenomena are described by thermodynamics, and on a microscopic scale by Statistical Mechanics. However, much of Materials Science is concerned with Kinetics , where the rate of change of metastable structures (or their inability to change) is dominant. Many of you will know more examples of such effects than I do: but here I want to draw this distinction sharply in physicist's terms. An equilibrium effect is the vapor pressure of a crystal (pure element). A kinetic effect is crystal growth from the vapor. These are treated separately below, and in the book, section 1.3. There are more references in the book, but currently there are no web-updates for this section.

Thermodynamics of the Vapor Pressure

Refs: most Statistical Mechanics books, but not all, e.g. T.L. Hill, Introduction to Statistical Mechanics, p79-80; F. Mandl, Statistical Physics, p182-3; R. Baierlein, Thermal Physics, p 276-8.

The sublimation of a pure solid at equilibrium is given by the condition µv = µs. It is a standard result that the chemical potential of the vapor at low pressure p is

µv = -kT ln (kT/pλ3), where λ = h/(2πmkT)1/2 is the thermal de Broglie wavelength.

This can be rearranged to give the equilibrium vapor pressure pe in terms of the chemical potential of the solid, as

pe = (2πm/h2)3/2 (kT)5/2 exp(µs/kT).

Thus, to calculate the vapor pressure, we need a model of the chemical potential of the solid. A typical µs at low pressure is the 'quasi-harmonic' model, which assumes harmonic vibrations of the solid, at its (given) lattice parameter. This free energy per particle

F/N = µs = U0 + <3hν/2> + 3kT<ln(1-exp(-hν/kT))>,

where we use angle brackets < > to mean average values. The (positive) sublimation energy at zero T , L0 = -(U0 + <3hν/2>), where the first term is the (negative) energy per particle in the solid relative to vapor, and the second is the (positive) energy due to zero-point vibrations.

The vapor pressure is significant typically at high temperatures, where the Einstein model of the solid is surprisingly realistic (provided thermal expansion is taken into account in U0). Within this model (all 3N ν's are the same) and in the high T limit, we have

<ln(1-exp(-hν/kT))> = <ln (hν/kT)>, so that exp (µs/kT) = (hν/kT)3exp(-L0/kT).

This gives pe = (2πmν2)3/2 (kT)-1/2 exp (-L0 /kT), so that peT1/2 follows an Arrhenius law, and the pre-exponential depends on the lattice vibration frequency as ν3. The absence of Planck's constant h in the answer shows that this is a classical effect, where equipartition of energy applies.

This equation is closely followed in practice over many decades of vapor pressure. The T1/2 term is slowly varying, and many tabulations of vapor pressure simply express log10(pe) = A - B/T, and give the constants A and B. Calculations along these lines yield values for L0 and ν. Some curves are given in diagrams 10 (book figure 1.9) and 11 (book figure 1.10), and extracted values using the above equations are given in Table 1. Note that the sublimation energies are accurately known: the frequencies are good to maybe 20%, but depend on the use of (approximate) Einstein model.

Element Lattice Constant
(a0) nm
Sublimation Energy
(L0) eV or K
Einstein Frequency
ν (THz)
Metals: Ag
0.4086 (fcc) at RT
2.95 ± 0.01 eV
0.2866 (bcc)
4.28 ± 0.02
Semiconductors: Si
0.5430 (diam)
4.63 ± 0.04
3.83 ± 0.02
Van der Waals: Ar
0.5368 (fcc) at 50K
84.5 meV or 981 K
120 meV or 1394 K
167 meV or 1937 K

Click to return to section C1

The Kinetics of Crystal Growth

Original refs: J.D. Weeks and G.H. Gilmer, Adv. Chem. Phys. 40 (1979) 157-227; W. K. Burton, N. Cabrera and F.C. Frank, Proc. Roy. Soc. A243 (1951) 299-358.

The thing to understand about the above calculation is that the vapor pressure does not depend on the structure of the surface, which acts simply as an intermediary: i.e., the surface is 'doing its own thing' in equilibrium with both the crystal and the vapor. The surface of a Kossel crystal can be visualized by Monte Carlo (or equivalent) simulations, as indicated in diagram 12 (book diagram 1.11). At low T, the terraces are almost smooth, with few adatoms or vacancies (see diagram 4 (book figure 1.4) for these terms). As T is raised, then the surface becomes rougher, and eventually has a finite interface width. We might look at these studies in more detail later: there are distinct roughening and melting transitions at surfaces, each of them specific to each {hkl} crystal face. The simplest MC calculations in the so called SOS (solid on solid) model show the first but not the second transition.

This picture of a fluctuating surface which doesn't influence the vapor pressure applies to the equilibrium case; what happens if we are not at equilibrium? The classic paper in this field is the second reference quoted, known as BCF, and much quoted in the Crystal Growth literature. We have to consider the presence of kinks and ledges, and also (extrinsic) defects, in particular screw dislocations. This paper, and the developments from it, are quite mathematical, so we will only look at a few simple cases here, in order to introduce some terms and establish some ways of looking at surface processes.

First, the idea of supersaturation S = (p/pe), and thermodynamic driving force, Δµ = kT ln S. Δµ is clearly zero in equilibrium, is positive during condensation, negative during sublimation / evaporation. The deposition rate or flux (R or F are used in the literature) is related, using kinetic theory, to p as R = p/(2πmkT)1/2.

Second, the idea that an atom can adsorb on the surface, becoming an adatom, with a (positive) adsorption energy Ea, relative to zero in the vapor. (Sometimes this is called a desorption energy, and the symbols for all these terms vary wildly). The rate at which the adatom desorbs is (very roughly) given by νexp-(Ea/kT), where we might want to specify the pre-exponential frequency as νa to distinguish it from other frequencies; it may vary relatively slowly (not exponentially) with T.

Third, the adatom can diffuse over the surface, with energy Ed and corresponding pre-exponential νd. We expect Ed < Ea, maybe much less. The adatom diffusion coefficient is then (jump distance a) approximately

D = (νda2/4) exp-(Ed/kT),
and the adatom lifetime before desorption,
τa = νa-1 exp(Ea/kT).
BCF then showed that
xs = (Dτa)1/2
is a characteristic length, which governs the fate of the adatom, and defines the role of ledges (steps) in evaporation or condensation. This is a suitable exercise to get familiar with the ideas of local equilibrium, and diffusion in one dimension, as set out in Assignment 1.

The main points that follow from the above considerations are:

1. Crystal growth (or sublimation) is difficult on a perfect terrace, and substantial supersaturation (undersaturation) is required. When growth does occur, it proceeds through nucleation and growth stages, with monolayer thick islands (pits) having to be nucleated before growth can proceed. Early MC studies of these effects are seen in diagram 13 (book figure 1.12).

2. A ledge, or step on the surface captures arriving atoms within a zone of width xs either side of the step, statistically speaking. If there are only individual steps running across the terrace, then these will eventually grow out, and the resulting terrace will grow much slower (as in point 1). In general, rough surfaces grow faster than smooth surfaces, so that the final 'growth form' consists entirely of slow growing faces.

3. The presence of a screw dislocation in the crystal provides a step (or multiple step), which spirals under the flux of adatoms (diagram 14, or book figure 1.13). This provides a mechanism for continuing growth at modest supersaturation. Detailed study shows that the growth velocity depends quadratically on the supersaturation for this mechanism, and exponentially for mechanism 1 (book figure 1.14).

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

Local equilibrium at the surface of a crystal at temperature T

Consider the (001) face of a f.c.c crystal with 12 nearest neighbor bonds, and (small concentrations of) adatoms and vacancies at this surface. The sublimation energy is 3eV and the frequency factor is 10 Thz. Use the appropriate formulations of section 1.3 to do the following:

Effects of vacancies and/or lattice vibrations on the sublimation pressure

Consider the model of the vapor pressure of a solid described in section 1.3, table 1.1 and diagram 10 (book figure 1.9). This model neglects the effects of vacancies, and the model of lattice vibrations is only a first approximation.

Note: this problem is useful for a discussion of points of principle and practicality, and can be expanded via detailed computation for a course project. The "Effects of Vacancies" were considered by Bala Ramadurai with an interactive Javascipt/Java program in 1999. This project was done again by Thomas Heaton using MatLab in 2009.

Continue to section 1.4

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