Early Stage Process of Pattern Formation Phenomena

Abstract

Pattern formation phenomena have been observed universally in various systems, such as metal alloys and polymer blends systems. 2) Entropy dominates at high temperatures, leading to a homogeneous, disordered phase, while energy becomes crucial with decreasing temperature, and this leads to the formation of the ordered phases. One of the most successful models is known as the Cahn-Hilliard (CH) equations. The familiar spinodal decomposition (SD) processes under a deep quench to an unstable state have been investigated by numerical simulations based on the above time-dependent Ginzburg-Landau ansatze. 4) Under a shallow quench, on the other hand, the fluctuations change the nature of phase transitions from the second to the first order. By using the Hartree approximation, Brazovskii has firstly shown about this effect under the long-range interaction caused by the nonzero wave number. 6) One of the important essences for this Brazovskii effect is implied here that the Landau free-energy fL(ψ) = −τLψ/2 + uLψ/4 is transformed as the effects of the fluctuations into fH(ψ) = −τHψ/2+uHψ/4+wHψ/36, where τL, uL, τH, uH and wH are phenomenological parameters. It leads to that the fluctuation-induced first order transition changes the early stages of ordering to the nucleation and growth (NCG) manner. In spite of that there exist many analytical predictions of the dynamics about fluctuationinduced pattern formation far from equilibrium [see the reference], this is not the case for numerical studies on the morphology transitions under a shallow quench. The purpose of this short note is to propose the simple numerical model, in which the Landau free-energy is extended to the Hartree free-energy so that the freeenergy can have another minimum between the local minima of the double well free-energy. In terms of the cell dynamical system (CDS) approach, Oono and Puri have proposed the hyperbolic tangent form of one-toone map, ∂fL/∂ψ = −α tanh(ψ) + ψ, by taking the Landau free-energy functional fL into account. 3) On the basis of the Hartree free-energy functional fH, in the present study, the CDS map is modified to ∂fH/∂ψ = −α tanh(βψ) + ψ. Notations α and β are adjustable parameters which are determined such that the modified map has the five intersection points with ∂f/∂ψ = 0; i.e.,