Click to return to Lecture 2.
The references for this lecture are here.
However, much of materials science is concerned with kinetics, where the rate of change of metastable structures (or their inability to change) is dominant. Here, we describe the elements of the kinetics of crystal growth, emphasising the role of the surface and surface defects. Models of adsorbed atoms on surfaces, in equilibrium with either the gas and/or the solid are then developed via problems, continuing into lecture 2.
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. For example, we know that surfaces can be reconstructed, as described in many textbooks, and in my ASU lecture notes as Section 1.4. We also know that vibrations at surfaces can be very different, with vibration amplitudes which are both larger than the bulk, and anisotropic, depending on the specific crystal face involved. These effects depend in detail on the interatomic forces and atomic masses of the solids concerned. The example of rare gas solids interacting via a Lennard Jones potential is illustrated in diagrams 24-27 (Allen and DeWette, 1969, Lagally, 1975).
The schematic model of a crystal surface is a simple cubic crystal interacting via nearest neighbor pair bonds. This is a Kossel crystal, as in the Terrace-Ledge-Kink model described in Section 1.2. At finite T, this model can be visualized by Monte Carlo (or equivalent) simulations, as indicated in diagram 12. At low T, the terraces are almost smooth, with few adatoms or vacancies (see diagram 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.
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.
2. A ledge, or step on the surface captures arriving atoms within a zone of width x 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). 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.
Consider the (001) face of a fcc 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.
b) Express the local equilibrium between the bulk crystal and the surface adatoms, to obtain their equilibrium concentrations at the same temperature, ignoring arrival from, or sublimation to the vapor. Hence decide whether the case described in a) correponds to under- or over-saturation, and calculate the thermodynamic driving force in units of kT.
b) Show that the steady state profile of adatoms between the steps depends on the ratio cosh (x/X)/ cosh(d/2X), where X is the BCF length. Show that the fraction of atoms which get incorporated into the steps, the condensation coefficient, is given by (2X/d)tanh(d/2X). Evaluate the limits (2X/d) >> 1 and << 1, and give reasons why these limits are sensible.