Order parameter models

An important example is the one-order-parameter model invented by Gompper and Schick [77], which describes a ternary mixture in temis of the density difference between water and oil ... 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

The first step in quantitative description of pure polyamorphic fluid is a selection of the model that can qualitatively describe a possible multiplicity of critical points in wide range of temperatures and pressures. A great many of explanations of multicriticality in monocomponent fluids (perturbation theory models semiempirical models lattice models, two-state models, field theoretic models, two-order-parameter models, and parametric crossover model has been disseminated after the pioneering work by Hemmer and Stell Here we test more extensively the modified van der Waals equation of state (MVDW) proposed in work and refine this model by introducing instead of the classical van der Waals repulsive term a very accurate hard sphere equation of state over the entire stable and metastable regions... 

While the agreement between the experimental data and the theoretical calculation is very satisfying, other types of volume and enthalpy recovery experiments show that the single-ordering parameter model is not sufficiently flexible to rationalize many of the important aspects of the glass transition phenomenon. As a result, more complex models based on multiple ordering parameters are now under development.8... 

In the present conribution, we develop a continuum-based model to describe experimentally observable interphases in thin adhesive films. The model is based on an extended contiuum theory, i.e. the mechanical behaviour in these interphases is captured by an additional field equation. The introduced scalar order parameter models the microscopical mechanical properties of the film phenomenologically. 

In some instances, the single-order parameter model (16) is not sufficient. This is the case when fluctuations of the amphiphile concentration play an important role, such as in the calculation of the scattering intensity in film contrast. A second scalar order parameter field, p(r), which describes the deviation of the local amphiphile concentration from the average p, has to be included in the model. For a balanced system, the free energy functional... 

Blount s theorem seems to rale out odd-parity states in UPt3 at first glance since there is strong evidence for node lines on the Fermi surface. As a consequence, the majority of early order parameter models for UPts adopted multicomponent even-parity states. However the anisotropy of thermal conductivity, reversal of upper critical field anisotropy and Knight shift results in UR3 are better accounted for by an odd-parity order parameter. For an extensive discussion of this problem we refer to sect. 4.1. Another more recent case is UNi2AI3 where evidence for an odd parity state exists. It seems that Blount s theorem is not respected in real HF superconductors. 

BCS ratio 2A 0)/kTc and specific heat ratio ACsjCn for some common order parameter models Einzel (2002) characterised by the gap form factor /( >, ip) as function of polar angles of k and normalised on the unit sphere... 

Here we explain how the thermodynamic and kinetic anomalies can be explained in the same framework of our two-order-parameter model. 

Here we consider a simple two state model of liquid, which corresponds to the weak-coupling limit of our two order-parameter model [25,37]. We first estimate how the average fraction of locally favored structures, S, increases with a decrease... 

So-called order parameter models are being currently used to understand a wide variety of systems that evolve outside of linearization regimes. These models can be motivated in several different ways, but we do not know yet of any systematic derivation that starts from first principles. Whereas linearized transport equations are well understood, their nonlinear extension... 

The order parameter was arguably first introduced by Landau at the equilibrium thermodynamic level to study phase behavior. Order parameter models can also be motivated through classical bifurcation theory, and several physical systems have been recently modeled in this way. They include Rayleigh-Benard convection, Faraday waves,or pattern formation in optical systems. Close to a bifurcation point, these models asymptotically describe the system under study, but they are now routinely used, in a phenomenological feshion, to describe highly nonlinear phenomena. 

Perhaps a sign of the growing importance of this approach is a recent review article in the journal Reviews of Modern Physics under the title The world of the complex Ginzburg-Landau equation (by I. Aranson and L. Kramer), just on solutions and properties of a particular type of order parameter model. 

Chakraborty S (2007) Order parameter modeling of fluid dynamics in narrow confinements subjected to hydrophobic intoac-tions. Phys Rev Lett 99 094504(1 )... 

In liquid crystals intermolecular interactions are short-range and Cp is dominantly related to short-range order fluctuations. Since fluctuations in local order occur above and below the transition, one expects, in contrast to the mean-field situation, anomalous Cp behavior above as well as below T. Unfortunately models taking short-range order parameter fluctuations into account are much more difficult to handle and analytic solutions are only available for some special cases. However, sophisticated renormalization group (RG) analyses have been carried out for several types of n vector order parameter models of which the three-dimensional cases are of particular relevance for liquid crystals [7, 10, 11]. 

R.J. Braun, J.W. Cahn, G.B. McFadden and A.A. Wheeler, Anisotropy of Interphaces in an Ordered Alloy A Miltiple-Order-Parameter Model, Phil. Trans. Roy. Soc. London A 355(17.30), 1787-1833 (1997). 

In the past there have been attempts to treat the kinetics of glass formers in terms of single ordering parameter models. (13) For experiments which measure volume, it is convenient to define a new parameter called delta as the normalized departure from equilibrium. 

Such single ordering parameter models have been proposed in the literature (2) and lead more or less directly to equations of the form... 

Nevertheless, this is precisely what is attempted when single ordering parameter models are applied to glassy systems. Here all of the response of the system is associated with a single fundamental recovery process, an approximation which, in our opinion, is too simplistic to be of very much use. 

Figure 6. Distribution of normalized instantaneous departures from eauilibrium, g. , as a function of recovery times Two distributions are shox-m, the single ordering parameter model as a dot and a simple equal intensity partitioning as the two crosses.

With this information, one can determine the behavior for the single ordering parameter model this is shown in Figure 7 as the dotted line. Again, as was noted above, the straight continuous line for both expansion and contraction is far from what is observed experimentally (Figure 5). 

Figure 7. eff associated with the distribution functions shown in Figure 6. The dotted line corresponds to the single ordering parameter model and the solid line renresents the equal intensity partitioning case.

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