Spinodal Decomposition - Fourier Transform

Fourier Transform

The motivation for the Fourier transform comes from the study of a Fourier series. In the study of a Fourier series, complicated periodic functions are written as the sum of simple waves mathematically represented by sines and cosines. Due to the properties of sine and cosine it is possible to recover the amount of each wave in the sum by an integral. In many cases it is desirable to use Euler's formula, which states that e2πiθ = cos 2πθ + i sin 2πθ, to write Fourier series in terms of the basic waves e2πiθ, with the distinct advantage of simplifying many unwieldy formulas.

The passage from sines and cosines to complex exponentials makes it necessary for the Fourier coefficients to be complex valued. The usual interpretation of this complex number is that it gives you both the amplitude (or size) of the wave present in the function and the phase (or the initial angle) of the wave. This passage also introduces the need for negative "frequencies". (E.G. If θ were measured in seconds then the waves e2πiθ and e−2πiθ would both complete one cycle per second—but they represent different frequencies in the Fourier transform. Hence, frequency no longer measures the number of cycles per unit time, but is closely related.)

If A(β) is the amplitude of a Fourier component of wavelength λ and wavenumber β = 2π/λ the spatial variation in composition can be expressed by the Fourier integral:

in which the coefficients are defined by the inverse relationship:

Substituting, we obtain on equating coefficients:

This is an ordinary differential equation that has the solution:

in which A(β) is the initial amplitude of the Fourier component of wave wavenumber β and R(β) defined by:

or, expressed in terms of the diffusion coefficient D:

In a similar manner, the new diffusion equation:

has a simple sine wave solution given by:

where R(β) is obtained by substituting this solution back into the diffusion equation as follows:

For solids, the elastic strains resulting from (in)coherency add terms to the amplification factor R(β) as follows:

where, for isotropic solids:

where E is Young's modulus of elasticity, υ is Poisson's ratio, and η is the linear strain per unit composition difference. For anisotropic solids, the elastic term depends on direction in a manner which can be predicted by elastic constants and how the lattice parameters vary with composition. For the cubic case, Y is a minimum for either (100) or (111) directions, depending only on the sign of the elastic anisotropy.

Thus, by describing any composition fluctuation in terms of its Fourier components, Cahn showed that a solution would be unstable with respect to sinusoidal fluctuations of a critical wavelength. By relating the elastic strain energy to the amplitudes of such fluctuations, he formalized the wavelength or frequency dependence of the growth of such fluctuations, and thus introduced the principle of selective amplification of Fourier components of certain wavelengths. The treatment yields the expected mean particle size or wavelength of the most rapidly growing fluctuation.

Thus, the amplitude of composition fluctuations should grow continuously until a metastable equilibrium is reached with a preferential amplification of components of particular wavelengths. The kinetic amplification factor R is negative when the solution is stable to the fluctuation, zero at the critical wavelength, and positive for longer wavelengths—exhibiting a maximum at exactly times the critical wavelength.

Consider a homogeneous solution within the spinodal. It will initially have a certain amount of fluctuation from the average composition which may be written as a Fourier integral. Each Fourier component of that fluctuation will grow or diminish according to its wavelength.

Because of the maximum in R as a function of wavelength, those components of the fluctuation with times the critical wavelength will grow fastest and will dominate. This "principle of selective amplification" depends on the initial presence of these wavelengths but does not critically depend on their exact amplitude relative to other wavelengths (if the time is large compared with (1/R). It does not depend on any additional assumptions, sinced different wavelengths can coexist and do not interfere with one another.

Limitations of this theory would appear to arise from this assumption and the absence of an expression formulated to account for irreversible processes during phase separation which may be associated with internal friction and entropy production. In practice, frictional damping is generally present and some of the energy is transformed into thermal energy. Thus, the amplitude and intensity of a 1-dimensional wave decreases with distance from the source, and for a three-dimensional wave the decrease will be greater.