The Role of Non-linearized Terms in the Diffusion Equation during a Decomposition Process

Abstract

For alloys whose free energy is given by [(c_e−0.5)²−(c−0.5)²]²×const., interdiffusion coefficient, \ ildeD, of solid solution including composition variation of q(x, t)=c(x, t)−c₀ can be written as \ ildeD(x, t)=D₀−D₁q(x, t)+D₂q²(x, t) where c₀(<0.5) and c_e are the average and equilibrium compositions respectively. The diffusion equation is, for such a case, written in Fourier space as
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\ oindentwhere F(h) is a Fourier spectrum representing a composition variation. G(h) and H(h) are functions obtained by convoluting F(h) once and twice, respectively. hβ represents wavenumber of Fourier waves and K is (gradient energy coefficient)×(mobility of atoms). The value of D₀ increases with decrease in c₀, and it is positive for alloys of c₀<c_s (spinodal composition). D₁ is zero at c₀=0.5 and increases with decrease in c₀, while D₂ is constant for all alloy compositions.
Through the diffusion process, the second term of eq. (1) and the third term produce the harmonic Fourier spectra with the components of the same sign and those of alternating sign, respectively. In composition variations of small amplitude behavior of F(h) is controlled by the first term. With increase in amplitude the controlling term for F(h) changes from the first to the second term, and then to the third term. The following results are obtained by numerical calculation of eq. (1). Composition variations in alloys of c₀>c_s amplify spinodally by virtue of the first term, while in alloys of c₀<c_s they change their configrations to those of nuclei by virtue of the second term after some incubation time. Composition variation peaks are squared and nuclei grow by virtue of the third term at a later stage of decomposition. The required condition for the formation of nuclei is that a part of composition variation exceeds c_s.