Proper order parameter

At high temperatures, ferroelectric materials transform to the paraelectric state (where dipoles are randomly oriented), ferromagnetic materials to the paramagnetic state, and ferroelastic materials to the twin-free normal state. The transitions are characterized through order parameters (Rao Rao, 1978). These order parameters are characteristic properties parametrized in such a way that the resulting quantity is unity for the ferroic state at a temperature sufficiently below the transition temperature, and is zero in the nonferroic phase beyond the transition temperature. Polarization, magnetization and strain are the proper order parameters for the ferroelectric. 

In the NET-GP analysis, the glassy polymer-penetrant phases are considered homogeneous, isotropic, and amorphous, and their state is characterized by the classical thermodynamic variables (i.e. composition, temperature, and pressure) with the addition of a single-order parameter, accounting for the departure from equilibrium. The specific volume of the polymer network, or, equivalently, the polymer density Pp, is chosen as the proper order parameter. In other words, the hindered mobility of the glassy polymer chains freezes the material into a non-equilibrium state that can be labeled by the... 

The last term gives the electrostatic contribution to the free energy. The coefficient e is afways positive and the pofarization is never the reason for the phase transition in chiral polar liquid crystals. The polarization P is an improper order parameter and appears only when the proper order parameter - the tilt - is non-zero. 

From symmetry consideration we should choose a proper order parameter for the lower symmetry phase (on account of molecular distribution functions). 

A thermodynamic model was recently proposed to calculate the solubility of small molecules in assy polymers. This model is based on the assumption that the densiQr of the polymer matrix can be considered as a proper order parameter for the nonequilibrium state of the system (7). In this chapter, the fundamental principles of the model are reviewed and the relation of the model to the rheological properties of the polymeric matrix is developed. In particular, a unique relation between the equilibrium and non-equilibrium properties of the polymer-penetrant mixture can be obtained on the basis of a simple model for the stress-strain relationship. 

By tradition, the order parameter in any order-disorder problem is always taken such that it is unity in the perfectly ordered phase and vanishes for the completely disordered phase. Examination of the average values described above shows that the proper order parameter for the nematic liquid crystal is... 

In order to develop the dyes for these fields, characteristics of known dyes have been re-examined, and some anthraquinone dyes have been found usable. One example of use is in thermal-transfer recording where the sublimation properties of disperse dyes are appHed. Anthraquinone compounds have also been found to be usehil dichroic dyes for guest-host Hquid crystal displays when the substituents are properly selected to have high order parameters. These dichroic dyes can be used for polarizer films of LCD systems as well. Anthraquinone derivatives that absorb in the near-infrared region have also been discovered, which may be appHcable in semiconductor laser recording. 

Figure 6.54 Diagram illustrating several types of order parameters involved in proper and improper ferroics. (After Newnham Cross, 1981.)...

A hexagonal representation of proper and improper primary ferroics as proposed by Newnham Cross (1981) is given in Fig. 6.54. The order parameter for proper ferroics appears on the diagonals of the hexagon, while the sides of the hexagon represent improper ferroics. They indicate the cross-coupled origin of ferroic phenomena. An improper primary ferroic in this classification is distinguished from a true secondary... 

This implies that, in general, temporal ordering parameters cannot be identified directly with physical time—they merely share one essential characteristic. This situation is identical to that encountered in the Lagrangian formulation of general relativity there, the situation is resolved by defining the concept of particle proper time. In the present case, this is not an option because the notion of particle proper time involves the prior definition of a system of observer s clocks—so that some notion of clock time is factored into the prior assumptions on which general relativity is built. 

Two sets of typical data experimentally measured are illustrated in Figs. 5.3(a) and (b), in which the curves represent the results calculated by Eq. (5.2) with properly regressed parameters Q. It should be noted that u.d in the figures is the airflow velocity at the exits of the nozzles which is quite different from the impinging velocity, u , in both the nature and the order of magnitude. 

In the highly anisotropic membrane environment, one can expect several different correlation times that correspond to the anisotropic membrane environment and/or the nonspheri-cal molecular probe as well as variations that exist along the membrane normal within the bilayer. Furthermore, the molecular motion is often limited by constraints imposed by the ordered surroundings of the probe. Properly designed spectroscopic experiments can, in many cases, extract both mobility and order parameters and can give a comprehensive picture of membrane fluidity. 

Often, the initial stages of Rietveld refinement are both important and difficult because the initial values of both the structural and profile parameters may be far fi-om the correct values. Hence, non-linear least squares may be less stable when compared to the same at the end of the refinement, i.e. when nearly all parameters are close to their accurate values. As mentioned above, variables should be refined in a proper order, usually starting from only a few most critical parameters and then adding other relevant variables, while continuously monitoring how previously refined parameters continue to change. Those that correlate or begin to diverge should be excluded from the refinement and, perhaps, constrained. 

Both the order parameter and the spontaneous strain for a phase transition have symmetry properties. The relationship between them is therefore also constrained by symmetry. Only in the case of proper ferroelastic transitions is the symmetry-breaking strain itself the order parameter. For most transitions, the order parameter relates to some... 

Proper identification of the order parameter of a particular system often needs detailed physical insight, and sometimes is complicated because different degrees of freedom are coupled. For example, there are many reports in the literature that an order-disorder transition of adsorbates on loose-packed substrates causes an adsorbate-induced reconstruction of the substrate surface. In such a situation, the order parameter of the adsorbate order-disorder transition is the primary order parameter whereas the lattice distortion of the substrate surface is a secondary order parameter . However, for pure surface reconstruction transitions (i.e. structural phase transitions of the surface of crystals where no adsorbates are involved) all considered degrees of freedom are atomic displacements relative to positions of higher symmetry. The proper distinction between primary and secondary order parameters is then much more subtle. 

In order to make contact with the Landau expansion, however, we consider now the special case q = 3 and expand F in terms of the two order parameter components 0] = ni — 1/3 and 02 = 2 — 1/3 (note that all rij — l/q in the disordered phase). One recognizes that the model for q = 3 has a two-component order parameter and there is no symmetry between 0,-and —0<. So cubic terms in the expansion of F arc expected and do occur, whereas for a properly defined order parameter, there cannot be any linear term in the expansion ... 

An exact knowledge of the microcanonical entropy, or the density of the states, of a protein model in both the native and nonnative states is crucial for a precise characterization of the folding process of the model, such as whether the folding is first-order or gradual, whether the model can fold uniquely to the native structure, whether there is a discontinuity in the order parameter of the conformation in the folding transition, and so on. Once the microscopic entropy function is accurately determined, the statistical mechanics of the protein folding problem is solved. The accurate determination of the entropy function of protein models by the ESMC method requires a proper treatment of the computational problems discussed in Section IV. 

The assignment of the order of the herringbone transition was discussed in the literature over many years. The LEED results [92, 93] (see Fig. 26) were consistent with an interpretation in terms of a first-order transition with pronounced rounding elfects. However, these data could not mle out a continuous transition so that this study was not accurate enough to decide the order of the transition. In addition, there is the problem to relate the LEED superlattice spot intensities to the proper long-range order parameter of the system [93, 108] (see the presentation in Section III.D.l). 

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