Order parameter density

The systems undergoing phase transitions (like spinodal decomposition) often exhibit scaling phenomena [ 1—4] that is, a morphological pattern of the domains at earlier times looks statistically similar to a pattern at later times apart from the global change of scale implied by the growth of L(f)—the domain size. Quantitatively it means, for example, that the correlation function of the order parameter (density, concentration, magnetization, etc.)... 

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

The phase behavior of binary blends becomes much more complex if one relaxes the assumption of incompressibility. Then both liquid-liquid demixing and liquid-vapor phase separation are possible and the system is described by two scalar order parameters - density and composition (p, 4>) or the two densities of the species (pcos j Phd)- The interplay between the two types of phase... 

Pft(r) is the sixfold bond orientational order parameter density. 

Since the identification of universality classes for surface layer transitions needs the I-andau expansion as a basic step, we first formulate Landau s theory (Toledano and Toledano, 1987) for the simplest case, a scalar order parameter density

phase transition and slowly varying in space. It can be obtained by averaging a microscopic variable over a suitable coarsc-graining cell Ld (in d-dimensional space). For example, for the c(2x2) structure in fig. 10 the microscopic variable is the difference in density between the two sublattices I (a and c in fig. 10) or II (b and d in fig. 10), ,- = pj1 — pj. The index i now labels the elementary cells (which contain one site from each sublattice I, II). Then... 

For a system with a scalar order parameter (density, concentration difference), n = 1. 

In the nearest proximity around the binodal, there appear critical phenomena with their characteristically high level of correlated fluctuations of the order parameter (density for a substance or component concentration for a mixture ). By virtue of the universality principle, the properties of such fluctuations arc similar for both a one-component liquid-vapour system, a solution of low-molecular compounds, and a polymer solution. The critical phenomena in these systems aj e discussed iii this book in detail. The question as to the absence of any pretransition phenomena necir the liquidus is discussed as well. 

The universal singular behavior of the thermodynamic properties near the critical point is associated with the presence of long-range fluctuations of the order parameter (density in a one-component fluid). The size of the fluctuations is called the correlation length which diverges at the critical isochore in accordance with the asymptotic power law... 

In principle, atomistic studies with good quality force fields should be sufficient to represent liquid crystal phases or polymer melts to a high level of accuracy and most material properties (order parameters, densities, viscosities elastic constants etc.) should be available from such simulations. In practise, this is rarely (if ever) the case. For example, using molecular dynamics, the computational cost of atomistic simulations is such that it is rarely possible to simulate for longer than a few tens of nanoseconds for (say) 10000 atoms. Even these modest times often require several months of CPU time on todays fastest processors. 

We therefore expect to have to look for phase transitions in two coupled order parameters, density and orientation. To vary the density in the simulation we will employ a grand-canonical simulation technique using the configurational bias scheme for chain insertion and deletion... 

A liquid metal near its triple point is in equUibrium with a low-density, nomnetallic vapor phase (Fig. 2.2a). There is therefore a metal-nonmetal (MNM) transition that coincides exactly with the liquid-vapor transition—both the order parameter (density) and the... 

For a one-component fluid, the vapour-liquid transition is characterized by density fluctuations here the order parameter, mass density p, is also conserved. The equilibrium structure factor S(k) of a one component fluid is... 

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 ... 

The Maier-Saupe tlieory was developed to account for ordering in tlie smectic A phase by McMillan [71]. He allowed for tlie coupling of orientational order to tlie translational order, by introducing a translational order parameter which depends on an ensemble average of tlie first haniionic of tlie density modulation noniial to tlie layers as well as / i. This model can account for botli first- and second-order nematic-smectic A phase transitions, as observed experimentally. 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

The transition from smectic A to smectic B phase is characterized by tire development of a sixfold modulation of density witliin tire smectic layers ( hexatic ordering), which can be seen from x-ray diffraction experiments where a sixfold symmetry of diffuse scattering appears. This sixfold symmetry reflects tire bond orientational order. An appropriate order parameter to describe tlie SmA-SmB phase transition is tlien [18,19 and 20]... 

Positional Distribution Function and Order Parameter. In addition to orientational order, some Hquid crystals possess positional order in that a snapshot at any time reveals that there are parallel planes which possess a higher density of molecular centers than the spaces between these planes. If the normal to these planes is defined as the -axis, then a positional distribution function, can be defined, where is proportional to the... 

Moreover, the order parameter, which in the case of the gas-liquid transition is defined as the difference between the densities of both coexisting phases, A6 = 62 — 61, approaches zero when the temperature goes to (from below, since above T. the above order parameter is always equal to zero) as... 

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... 

Here the functions g(0) and /(0) are defined in a suitable way to produce the desired phase behavior (see Chapter 14). The amphiphile concentration does not appear expHcitly in this model, but it influences the form of g(0)— in particular, its sign. Other models work with two order parameters, one for the difference between oil and water density and one for the amphiphile density. In addition, a vector order-parameter field sometimes accounts for the orientional degrees of freedom of the amphiphiles [1]. 

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]. 

Random interface models for ternary systems share the feature with the Widom model and the one-order-parameter Ginzburg-Landau theory (19) that the density of amphiphiles is not allowed to fluctuate independently, but is entirely determined by the distribution of oil and water. However, in contrast to the Ginzburg-Landau approach, they concentrate on the amphiphilic sheets. Self-assembly of amphiphiles into monolayers of given optimal density is premised, and the free energy of the system is reduced to effective free energies of its internal interfaces. In the same spirit, random interface models for binary systems postulate self-assembly into bilayers and intro-... 

V. M. Nabutovskii, N. A. Nemov, Yu. G. Peisakhovich. Charge-density and order-parameter waves in hquid and sohd electrolytes in the vicinity of the critical point. Phys Lett 79 98-100, 1980. 

Recently efficient techniques were developed to simulate and analyze polymer mixtures with Nb/Na = k, k > I being an integer. Going beyond meanfield theory, an essential point of asymmetric systems is the coupling between fluctuations of the volume fraction (j) and the energy density u. This coupling may obscure the analysis of critical behavior in terms of the power laws, Eq. (7). However, it turns out that one can construct suitable linear combinations of ( ) and u that play the role of the order parameter i and energy density in the symmetrical mixture, ... 

Figure 3-4. Tlic order parameter A, Or) for the solilon solution (thick line) and the electron density IV/o(Jl) 2 for the intragap stale accompanying the solilon (dotted line) are shown to the left. To the right the spectrum of single-electron slates for the solilon lattice configuration is depicted.

While for strong interchain interactions large deviations of the order parameter from its average value are unlikely, for weak interactions the minimal-energy lattice configuration of disordered chains contains a finite density of kinks and anti-... 

Our first objective is going to be the determination of the finite order probability density function of Y(t) in terms of the known finite order probability densities for the increments of N(t). In preparation for this, we first note that, since N(s) — N(t) is Poisson distributed with parameter.n(s — t) for s > t, it follows that... 

We shall conclude this section by investigating the very interesting behavior of the probability density functions of Y(t) for large values of the parameter n. First of all, we note that both the mean and the covariance of Y(t) increase linearly with n. Roughly speaking, this means that the center of any particular finite-order probability density function of Y(t) moves further and further away from the origin as n increases and that the area under the density function is less and less concentrated at the center. For this reason, it is more convenient to study the normalized function Y ... 

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