Non-conserved order parameter

We start from the time-dependent Ginzburg-Landau equation for a non-conserved order parameter 0... 

The attachment-detachment model (see Fig. lb) can be described by using the probability function Eq. (23) but now assuming that the correlation length, L. Physically this means that infinite range conserved-order-parameter dynamics is the same as non-conserved order parameter dynamics. In that case the normalized... 

EVOLUTION EQUATIONS FOR CONSERVED AND NON-CONSERVED ORDER PARAMETERS... 

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 homogeneous mixture quenched from a stable phase into a thermodynamically unstable state within a phase diagram develops into an inhomogeneous system. Spinodal decomposition (SD) is induced by the instability of an order parameter, which is usually concentration [101]. In the early stages, the SD is interpreted within the framework of the Cahn theory for an isotropic SD [102,103]. In contrast, in the late stages, the SD is governed by diffusion or hydrodynamic processes and exhibits slow coarsening [104]. There are two types of order parameters that describe a system. One is a conserved order parameter such as concentration of binary mixtures. The other is a non-conserved order parameter such as the polarization of a ferroelectric... 

We consider mixtures of a flexible polymer and a nematogen described by one conserved order parameter (volume fraction of a nematogen) and one non-conserved order parameter (orientational order parameter of a nematogen). Since the orientational order parameter is a traceless symmetric tensor, its components can be expressed as [14] ... 

Here we shall consider two simple cases one in which the order parameter is a non-conserved scalar variable and another in which it is a conserved scalar variable. The latter is exemplified by the binary mixture phase separation, and is treated here at much greater length. The fonner occurs in a variety of examples, including some order-disorder transitions and antrferromagnets. The example of the para-ferro transition is one in which the magnetization is a conserved quantity in the absence of an external magnetic field, but becomes non-conserved in its presence. 

For small asymmetries, the superconducting state is homogeneous and the order parameter preserves the space symmetries. For most of the systems of interest the number conservation should be implemented by solving equations for the gap function and the densities of species self-consistently. In such a scheme the physical quantities are single valued functions of the asymmetry and temperature, contrary to the double valued results obtained in the non-conserving schemes. 

If all Cu atoms occupy (0,0,0) sites,/Cu = 1, consequently r - 1, and we have the completely ordered / brass. The symmetry is broken when /Cu = 1/2 or t = 0, and thus P and f become indistinguishable. Other (normalized) order parameters are in use lattice dimension, density, magnetization, polarization, or some function that describes the orientation of the molecular axes. Since the order parameter is a normalized extensive function (or a specific function, as for example, the mole fraction) and we are dealing with either an open or a closed system (/>., constant chemical potential or constant number of particles), r can be a non-conserved or a conserved quantity. 

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],... 

Let us briefly review the essential ingredients to this procedure (for more details of the method see [30] and for our model [42]). For a given system the hydrodynamic variables can be split up into two categories variables reflecting conserved quantities (e.g., the linear momentum density, the mass density, etc.) and variables due to spontaneously broken continuous symmetries (e.g., the nematic director or the layer displacements of the smectic layers). Additionally, in some cases non-hydrodynamic variables (e.g., the strength of the order parameter [48]) can show slow dynamics which can be described within this framework (see, e.g., [30,47]). 

Because the first models of the physics of superconductivity were based on the phenomenon of Bose-Einstein condensation (BEC) it is not surprising that the existence of the Josephson effect has also been postulated for cold-atom systems in the BEC state [11]. In both superconductors and BEC cold-atom systems, the Josephson effect arises from the approximation of non-conservation of particle number, which gives rise to a phase type of order parameter F(x) and concomitant wave phenomena, described at T = 0 by the Gross-Pitaevski equation " ... 

Basically, the time-dependent Ginzburg-Landau (TDGL) equation [12] relates the temporal change of a phase order parameter to a local chemical potential and a nonlocal interface gradient. With respect to a non-conserved phase field order parameter, the TDGL model A equation is customarily described as ... 

To elucidate the spatiotemporal emergence of crystalline structure and liquid-liquid phase separation in the crystalline-amorphous polymer blends, Rathi (80] employed the time dependent Ginzburg-Landau (TDGL Model-C) equations pertaining to the conserved concentration order parameter and the noncrystal order parameter. Model C is a combination of TDGL Model-A and Model-B, viz. ... 

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