Some Problems Related to Scaling in Systems Undergoing Phase Separations: Fid-Bau Portal

Some Problems Related to Scaling in Systems Undergoing Phase Separations

Furukawa, Hiroshi

Abstract We consider the dynamics of phase separation where coarsening of droplets or domains proceeds continously [1–3]. We omit the nucleation and growth. In the late stage, however, no clear distinction exists between the nucleation and growth and the continous spinodal decomposition. The dynamics of phase separation of a material such as an alloy may he classified into three stages, i.e., the early, intermediate and late stages. When the average domain size becomes large enough, the interfacial thickness, s, of domains or droplets is set vanishingly small compared with the average domain size, R. This stage is called the late stage of the phase separation. In this stage the length scale of the system is only the average domain size, R. Then the dynamical scaling assumption is introduced. The process of phase separation is invariant under the rescalings of distance r and time t. Let F(r,t) be a function of r and t. Let t be assumed to be rescaled as b1/at, then r is rescaled as br and also F is rescaled as b−cF. Then F(br,b1/at)=b−cF(r,t). By setting b1/at=1, we find that 1.1 $$F(r,t) = {{t}^{{ac}}}F(r{{t}^{{ - a}}},1) = {{[R(t)]}^{c}}F(r/R),$$ where 1.2 $$R \propto {{t}^{a}}$$ The Fourier coefficient Fk(t) of F(r,t) is scaled as 1.3 $${{F}_{k}}(t) = {{R}^{{d + c}}}F(kR),$$ where d is the spatial dimension and k is the wave number.