Search

The richness of spatial structures of the diblock copolymer melts is demonstrated by the minimization of a free-energy functional of nonlocal type. The total chain length and the interface thickness are key factors in determining the morphology of the system, in addition to the constant mean curvature surface ansatz. We examine the transition between lamellar and double gyroid phases, in which the rhombohedral phase is also obtained in the weak segregation regime.

Application of the Saeki method [J. Phys. Soc. Jpn. 69 (2000) 679] to copolymer–homopolymer mixtures using the free energy proposed by Ohta and Ito [Phys. Rev. E 52 (1995) 5250] reveals the complicated three-dimensional morphologies, such as onion, helices and regular polyhedral structures.

The main purpose of this paper is to make it clear that the physical value obtained along the trajectory generated by the new numerical scheme [J. Phys. Soc. Jpn. 69 (2000) 679.] satisfies the canonical distribution. On the basis of our proof, the scattering form factors are evaluated from the time evolution of the spatial pattern that fluctuates around the equilibrium lamellar pattern. Beyond critical values of temperature, the transitions from a lamellar pattern to hexagonal and isotropic patterns occur in the vicinity of the morphological transition point.

The purpose of this letter is to propose an efficient numerical scheme to obtain the equilibrium pattern in a block copolymer system. Using this scheme, one can obtain well-defined periodic patterns from numerical simulations with a small amount of computational effort. This method is extended to the control of the fluctuations around the equilibrium pattern by keeping the kinetic energy of the system at a constant value.

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.