often found that such mathematical descriptions provide at least a close approximation. For an array of spherical grains the mean radius R at a time t is (Humphreys and Hatherly 2004): where t0 is the nucleation time and G is the growth rate dR/dt. If N nuclei form in the time increment dt and the grains are assumed to be spherical then the volume fraction will be: This equation is valid in the early stages of recrystallization when f<<1 and the growing grains are not impinging on each other. Once the grains come into contact the rate of growth slows and is related to the fraction of untransformed material (1-f) by the Johnson-Mehl equation: While this equation provides a better description of the process it still assumes that the grains are spherical, the nucleation and growth rates are constant, the nuclei are randomly distributed and the nucleation time t0 is small. In practice few of these are actually valid and alternate models need to be used. It is generally acknowledged that any useful model must not only account for the initial condition of the material but also the constantly changing relationship between the growing grains, the deformed matrix and any second phases or other microstructural factors. The situation is further complicated in dynamic systems where deformation and recrystallization occur simultaneously. As a result, it has generally proven impossible to produce an accurate predictive model for industrial processes without resorting to extensive empirical testing. Since this may require the use of industrial equipment that has not actually been built there are clear difficulties with this approach. Factors influencing the rate The annealing temperature has a dramatic influence on the rate of recrystallization which is reflected in the above equations. However, for a given temperature there are several additional factors that will influence the rate. The rate of recrystallization is heavily influenced by the amount of deformation and, to a lesser extent, the manner in which it is applied. Heavily deformed materials will recrystallize more rapidly than those deformed to a lesser extent. Indeed, below a certain deformation recrystallization may never occur. Deformation at higher temperatures will allow concurrent recovery and so such materials will recrystallize more slowly than those deformed at room temperature e.g. contrast hot and cold rolling. In certain cases deformation may be unusually homogeneous or occur only on specific crystallographic planes. The absence of orientation gradients and other