Conserved order parameter

For a conserved order parameter, the interface dynamics and late-stage domain growth involve the evapomtion-diffusion-condensation mechanism whereby large droplets (small curvature) grow at tlie expense of small droplets (large curvature). This is also the basis for the Lifshitz-Slyozov analysis which is discussed in section A3.3.4. 

The function / incorporates the screening effect of the surfactant, and is the surfactant density. The exponent x can be derived from the observation that the total interface area at late times should be proportional to p. In two dimensions, this implies R t) oc 1/ps and hence x = /n. The scaling form (20) was found to describe consistently data from Langevin simulations of systems with conserved order parameter (with n = 1/3) [217], systems which evolve according to hydrodynamic equations (with n = 1/2) [218], and also data from molecular dynamics of a microscopic off-lattice model (with n= 1/2) [155]. The data collapse has not been quite as good in Langevin simulations which include thermal noise [218]. 

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... 

Spinodal decomposition and certain order-disorder transformations are the two categories of continuous phase transformations. Both arise from an order parameter instability in the case of spinodal decomposition, it is a conserved order parameter for continuous ordering, it is a nonconserved order parameter. 

Because composition is a locally-conserved order parameter, it cannot change in one location without affecting its neighborhood—fluxes are required to change a composition field. For example, in a binary alloy, the concentration field cb is re-... 

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

The Cahn-Hilliard equation applies to conserved order-parameter kinetics. For the binary A-B alloy treated in Section 18.1, the quantity in Eq. 18.22 is the change in homogeneous and gradient energy due to a change of the local concentration cB and is related to flux by... 

Numerical models of conserved order-parameter evolution and of nonconserved order-parameter evolution produce simulations that capture many aspects of observed microstructural evolution. These equations, as derived from variational principles, constitute the phase-field method [9]. The phase-field method depends on models for the homogeneous free-energy density for one or more order parameters, kinetic assumptions for each order-parameter field (i.e., conserved order parameters leading to a Cahn-Hilliard kinetic equation), model parameters for the gradient-energy coefficients, subsidiary equations for any other fields such as heat flow, and trustworthy numerical implementation. 

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. 

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