On the theory of phase transformations with a nonuniform nucleation rate

An expression for the phase volume fraction in a system with a nonuniform nucleation rate is derived by using the geometrical-probabilistic approach. Examples of such systems considered here are (1) a plane layer (with nucleation in the midplane) and random planes in space, (2) an infinitely long cylinder (with nucleation on the axis) and random lines in space, and (3) a sphere (with nucleation at the center) and nucleation at random points. In each case, an expression for the phase volume fraction is derived for the time-dependent rates of nucleation and growth. The equivalence of homogeneous nucleation and nucleation at points is established.