In Chapter 2 we saw that a spherical front in a pure material in an undercooled liquid is either unstable or not, depending on the size of the sphere. There is no secondary control that allows one to mediate the growth. In Chapter 3, we saw that, in directional solidification of a binary liquid, the concentration gradient at the interface creates an instability; however, there is a secondary parameter, the temperature gradient, that opposes the instability and thus can be used to control the local growth beyond the linearized stability limit. Thus, much attention has been given to directional solidification both experimentally and theoretically, and it is in this chapter that nonlinear theory will be discussed.

There are two approaches to the nonlinear theory, depending upon whether the critical wave number ac of linear theory is of unit order, or is small (i.e., asymptotically zero). In the former case one can construct Landau, Ginzburg–Landau, or Newell–Whitehead–Segel equations to study (weakly nonlinear) bifurcation behavior. This gives information regarding the nature of the bifurcation (sub- or supercritical), the question of wave number selection, the preferred pattern of the morphology, and hence the resulting microstructure. If ac is small one must use a longwave theory that generates evolution equations governing the nonlinear development. Such longwave theories can be weakly or strongly nonlinear, depending on the particular situation. They, too, can then be analyzed to discover the nature of the bifurcation and the selection of preferred wave number and pattern.