Continuous order

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

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If the continuous order parameter is written in terms of an order parameter density m(x), = f rn(x)dx and the Gibbs function... 

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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. 

Fine-scale, spatially periodic microstructures are characteristic of spinodal decomposition. In elastically anisotropic crystalline solutions, spinodal microstructures are aligned along elastically soft directions to minimize elastic energy. Microstructures resulting from continuous ordering contain interfaces called antiphase boundaries which coarsen slowly in comparison to the rate of the ordering transformation. 

The evolution of a system described by an equation related to eqn 2.26 was studied using cell dynamics simulations by Oono and co-workers (Bahiana and Oono 1990 Puri and Oono 1988). In the CDS method the continuous order parameter is discretized on a lattice and at time t is denoted where n labels... 

Fig. 6.36 Phase diagram calculated using SCFT for a blend of a symmetric diblock with a homopolymer with fl = 1 (see Fig. 6.32 for a blend with a diblock with / = 0.45) as a function of the copolymer volume fraction

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In principle, the crystallization of a protein, nucleic acid, or virus (as exemplified in Figure 2.2) is little different than the crystallization of conventional small molecules. Crystallization requires the gradual creation of a supersaturated solution of the macromolecule followed by spontaneous formation of crystal growth centers or nuclei. Once growth has commenced, emphasis shifts to maintenance of virtually invariant conditions so as to sustain continued ordered addition of single molecules, or perhaps ordered aggregates, to surfaces of the developing crystal. 

The most common continuous ordering parameters in physics and chemistry are position and time. For example, the water height in a lake on a windy day is a random function h(x,y, t of the positions x and y in the two-dimensional lake plane and of the time. For any particular choice, say xi,yi, /i of position and time A(xi,yi, Zl) is a random variable in the usual sense that its repeated measurements (over an ensemble of lakes or in different days with the same wind characteristics) will yield different results, predictable only in a probabilistic sense. 

As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not immediately obvious in view of the fact that the physical equations of motion are deterministic and this issue was discussed in Section 1.5.1. Random functions, ordered sequences of random variable, were discussed in Section 1.5.3. The focus of this chapter is a particular class of random functions, stochastic processes, for which the ordering parameter is time. Time is a continuous ordering parameter, however in many practical situations observations of the random function z(Z) are made at discrete time 0 < Zi < t2, , < tn < T. In this case the sequence z(iz) is a discrete sample of the stochastic process z(i). 

Tables 1 and 2, the variation of t (y) with y is reported in Fig. 16. It is significant that a similar change is observable for both systems for the whole range of composition. The rather continuous order-disorder transition appears close to y = 0.25. For y > 0.25, the linear variation of t (y) with y confirms the structural model and thus the previous results, i.e. the presence of alternating octahedral and tetrahedral layers in the structure.

Classification of continuous order-disorder transitions of commensurate superstructures in adsorbed monolayers at surfaces (from Schick, 1981). 

INFRARED PROBLEMS AND PHASE TRANSITIONS OF CONTINUOUS ORDER IN LOW DIMENSIONAL SYSTEMS... 

Infrared singular behaviour is characterized by power law singularities in thermodynamic quantities with singular (or critical) exponents which usually vary continuously with system parameters. Depending on the physical interpretation of these parameters in the different situations, this implies 1) and 2) singular ground-state properties, a special particle spectrum and instabilities for correlation functions 3) the phase transition of continuous order for 2-d magnetic systems and other systems with continuous... 

Phase transition of continuous order —with the following characteristic properties. The free energy is... 

Fig. 1. (a) A continuous ordered region of a crystal open circles represent holes or vacancies in the structure (b) small ordered domains surrounded by a continuum of noncrystalline thin shells. The nature of this noncrystalline portion is the topic of this paper. The thickness of the layers between the ordered domains is extremely thin. 

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