Spinodal Decomposition - Fourier Components

Fourier Components

Cahn and Hilliard formulated a theory for the amplification (or attenuation) of an arbitrary composition fluctuation by considering, with Debye, the Fourier components of the composition rather than the composition itself. Thus, for a concentration fluctuation:

one obtains for the change in free energy on forming fluctuations:

The solution is then unstable (ΔF < 0) for all fluctuations of wave number β smaller than a critical wave number β_c given by:

or for all fluctuations of wavelength λ = 2π/β which are longer than a critical wavelength given by:

From these equations, it is seen that the incipient surface energy, reflected in the gradient energy term, prevents the solution from decomposing on too small a scale. This concept was first introduced by Hillert, and shows that as the spinodal is approached, the critical wavelength approaches infinity.