Phase separating/ordering systems conserved order parameter

We note that earlier research focused on the similarities of defect interaction and their motion in block copolymers and thermotropic nematics or smectics [181, 182], Thermotropic liquid crystals, however, are one-component homogeneous systems and are characterized by a non-conserved orientational order parameter. In contrast, in block copolymers the local concentration difference between two components is essentially conserved. In this respect, the microphase-separated structures in block copolymers are anticipated to have close similarities to lyotropic systems, which are composed of a polar medium (water) and a non-polar medium (surfactant structure). The phases of the lyotropic systems (such as lamella, cylinder, or micellar phases) are determined by the surfactant concentration. Similarly to lyotropic phases, the morphology in block copolymers is ascertained by the volume fraction of the components and their interaction. Therefore, in lyotropic systems and in block copolymers, the dynamics and annihilation of structural defects require a change in the local concentration difference between components as well as a change in the orientational order. Consequently, if single defect transformations could be monitored in real time and space, block copolymers could be considered as suitable model systems for studying transport mechanisms and phase transitions in 2D fluid materials such as membranes [183], lyotropic liquid crystals [184], and microemulsions [185],... 

A number of interesting effects occur in spatially periodically forced pattern forming systems with a nonconserved order parameter, which have been investigated during recent years [60-73, 120], Here we focus on nearly unexplored effects of spatially periodic forcing in system with a conserved order parameter, as they occur in phase separating systems which are forced by spatial temperature modulations and where thermodiffusion plays a crucial role. 

Now we shall examine the time evolution of scattering function in terras of the self-similarity between the phase separated structures (12. 14. 151 The scattered intensity at a given time t for the conserved order parameter system may be given by... 

Fig. 38. Schematic plots of the free energy barrier for (a) the mean field critical region, i.e. / 1, and (b) the non-mean field critical region, i.e. R,l( 1 -/ /7 ) 4— V2 i. When AF /Tc is of order unity, a gradual transition from nucleation to spinodal decomposition (in a phase-separating mixture) or spinodal ordering" (in a system undergoing an order-disorder transition with non-conserved order parameter distinct from tj>) occurs. From Binder (1984b).

A rather interesting method, motivated by successful applications to studies of liquid-gas phase separation of Lennard-Jones fluids and other fluids in the canonical ensemble, " starts from the observation that the temperature variation of the order parameter distribution can be analyzed in subsystems of linear dimension 1. Although in the total system (of linear dimension L) the order parameter is conserved, for different types of monomers (and chains). Of course, subsystems must be large enough, so that they still contain many chains. 

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