Introduction to Growth Kinetics Problems

Abstract

In these lectures I will give an introduction to the fascinating problems associated with the growth of order [1, 2] in unstable thermodynamic systems. To set the stage let me begin with the conceptually simplest situation. Consider the phase diagram shown in Fig.1 for a ferromagnetic system in the absence of an externally applied magnetic field. At high temperatures, above the Curie temperature T _c ,the average magnetization, the order parameter for this system, is zero and the system is in the paramagnetic phase. Below the Curie temperature, for the simplest case of an Ising ferrromagnet, one has a non-zero magnetization with two possible orientations: the net magnetization can point in say the + z-direction or the - z-direction. There are two degenerate equilibrium states of the system in the ferromagnetic phase. Now consider the experiment where we first prepare the system in an equilibrium high temperature state where the average magnetization is zero. We then very rapidly drop the temperature of the thermal bath in contact with the magnet to a temperature well below the Curie temperature. In this case the magnetic system is rendered thermodynamically unstable. It wants to equilibrate at the new low temperature, but it must choose one of the two degenerate states. Consider the schematic shown in Fig.2 where ψ is the local value of the order parameter and V is the “potential” governing this variable. Clearly the potential and the free energy are minimized by a uniform magnetization with value +ψ ₀ and -ψ ₀.