TABLE OF CONTENTS We have already in Chapter XII become acquainted with several types of instability-induced patterns: dendrites, concentration cells, eutectic lamellae. However, we have not provided any reason for expecting such patterns other than they are a manifestation of morphological instability.
Patterns exist in natural phenomena. Thermodynamics, via stability considerations, governs one group of such patterns. However, as noted in the previous chapter, other patterns are found which have no origin in thermodynamics since they are associated with kinetic processes. Still other patterns depend upon both thermodynamics and kinetics. We will explore the bases of these classes of patterns in this chapter. Our discussion of patterns will be brief and omit many classes of patterns due to the limitation of space. Nevertheless, we will provide an understanding of the origins of the class of spatial periodic patterns in Nature.
For systems of particles at thermodynamic equilibrium, periodicity occurs because the periodic array of the particles is the array having the minimum free energy. Crystals represent one class of matter for which the ground states of minimum free energy are periodic arrays. That periodic arrays of minimum free energy can exist when the interaction between units is of repulsive character only is not obvious. A simple proof can be given which shows that a periodic array is the minimum energy configuration for the case of a concave upwards repulsive potential between particles. This is carried out in Appendix 1 of this chapter. Figure 13.1 is...
