Kinetic Pathways of Order-Disorder and Order-Order Transitions in Weakly Segregated Microstructured Systems

Abstract

The kinetics of hexagonal to disordered and hexagonal to body-centered-cubic phase transitions in weakly segregated, microstructured systems (e.g., diblock copolymers) is studied using a time-dependent Ginzburg-Landau (TDGL) approach. Both computer simulation of the TDGL equation and analysis of a simplified two-mode model reveal nontrivial pathways during the transition.PACS numbers: 81.10. Aj, 81.30.Hd, 83.70.Hq A wide variety of chemical and physical systems, such as Langmuir films, ferrofluids, and diblock copolymers, exhibit ordered periodic domain structures [1]. Irrespective of differences in the systems, the domain structures have surprisingly similar appearance: stripes and circular droplets in two dimensions, and lamellae, hexagonal (HEX) cylinders, and body-centered-cubic (bcc) spheres in three dimensions. Although the specific origins may differ from system to system, the formation of spatially periodic patterns can be attributed to the competing shortrange and long-range interactions. Near the order-disorder transition, these systems can be phenomenologically described by an order parameter free energy functional of the formwhere c͑ r͒ is the order parameter, e.g., the local magnetization in magnetic systems, or the local density contrast between the two types of monomers in diblock copolymers. t is related to the distance from the order-disorder transition temperature, and the coefficients b, c, u, and y are phenomenological parameters which can be computed from more microscopic models. The last term in Eq.(1) represents the long-range repulsion, which penalizes longwavelength inhomogeneities. The equilibrium properties of systems described by Eq. (1) have been the subject of extensive experimental and theoretical studies [1,2]. In this Letter, we address the phenomenology of the kinetics of the various order-order and order-disorder transitions in weakly segregated, microstructured systems, using a time-dependent Ginzburg-Landau approach. Specifically we study the kinetic pathways of HEX to disordered and HEX to bcc phases after a sudden temperature jump. This study is motivated by the general intrinsic interest in understanding kinetics of phase transitions involving spatially modulated phases, in particular, by recent experiments on diblock copolymers [3][4][5][6][7]. To be concrete, we shall use diblock copolymers as the context; however, we believe the phenomenology is quite general for the class of systems described by the free energy Eq. (1), and will not limit our choice of parameters specifically to those for diblock copolymers [8].For conserved order parameters, as is appropriate for diblock copolymers, we may write the time-dependentHere M is a mobility coefficient, which we assume to be a constant; h͑ r, t͒ is a random force, which for a system in equilibrium at temperature T , satisfies the fluctuationdissipation relationAs a minimal model, we ignore any hydrodynamic effects and possible nonlocality in the mobility coefficient [10,11]. We will also ignore the noise term in subsequen...

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