Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

Critical mean field

In addition, a reorientation of the Jahn-Teller domains occurs in the magnetic field, and at some critical mean field the sample reaches a single domain state (Cooke et al. 1972, Kazey and Sokolov 1986, Melcher et al. 1973). in TmVO is, apparently, equal to 70 kOe, because at this mean field the following condition is fulfilled phonons are scattered on the sample grain boundaries but not on the boundaries which separate the domains (fig. 100, dashed line). [Pg.182]

This is the well known equal areas mle derived by Maxwell [3], who enthusiastically publicized van der Waal s equation (see figure A2.3.3. The critical exponents for van der Waals equation are typical mean-field exponents a 0, p = 1/2, y = 1 and 8 = 3. This follows from the assumption, connnon to van der Waals equation and other mean-field theories, that the critical point is an analytic point about which the free energy and other themiodynamic properties can be expanded in a Taylor series. [Pg.445]

Table A2.3.4 simnnarizes the values of these critical exponents m two and tliree dimensions and the predictions of mean field theory. Table A2.3.4 simnnarizes the values of these critical exponents m two and tliree dimensions and the predictions of mean field theory.
We now turn to a mean-field description of these models, which in the language of the binary alloy is the Bragg-Williams approximation and is equivalent to the Ciirie-Weiss approxunation for the Ising model. Botli these approximations are closely related to the van der Waals description of a one-component fluid, and lead to the same classical critical exponents a = 0, (3 = 1/2, 8 = 3 and y = 1. [Pg.529]

Table A2.3.5 Critical temperatures predicted by mean-field theory (MFT) and the quasi-chemical (QC) approximation compared with the exact results. Table A2.3.5 Critical temperatures predicted by mean-field theory (MFT) and the quasi-chemical (QC) approximation compared with the exact results.
An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. [Pg.536]

This implies that the critical exponent y = 1, whether the critical temperature is approached from above or below, but the amplitudes are different by a factor of 2, as seen in our earlier discussion of mean-field theory. The critical exponents are the classical values a = 0, p = 1/2, 5 = 3 and y = 1. [Pg.538]

The assumption that the free energy is analytic at the critical point leads to classical exponents. Deviations from this require tiiat this assumption be abandoned. In mean-field theory. [Pg.538]

Jacob J, Kumar A, Anisimov M A, Povodyrev A A. and Sengers J V 1998 Crossover from Ising to mean-field critical behavior in an aqueous electrolyte solution Phys. Rev. E 58 2188... [Pg.553]

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. [Pg.636]

The coexistence curve is nearly flat at its top, with an exponent p = 1/8, instead of the mean-field value of 1/2. The critical isothemi is also nearly flat at the exponent 8 (detemiined later) is 15 rather than the 3 of the analytic theories. The susceptibility diverges with an exponent y = 7/4, a much stronger divergence than that predicted by the mean-field value of 1. [Pg.644]

A2.5.7.2 CROSSOVER FROM MEAN-FIELD TO THE CRITICAL REGION... [Pg.653]

However, for more complex fluids such as high-polymer solutions and concentrated ionic solutions, where the range of intemiolecular forces is much longer than that for simple fluids and Nq is much smaller, mean-field behaviour is observed much closer to the critical point. Thus the crossover is sharper, and it can also be nonmonotonic. [Pg.655]

Syimnetrical tricritical points are predicted for fluid mixtures of sulfur or living polymers m certain solvents. Scott (1965) in a mean-field treatment [38] of sulfiir solutions found that a second-order transition Ime (the critical... [Pg.659]

Little is known about higher order critical points. Tetracritical points, at least imsynnnetrical ones, require four components. However for tetracritical points, the crossover dimension mean-field, or at least analytic. [Pg.660]

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches. Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.
In the LS analysis, an assembly of drops is considered. Growth proceeds by evaporation from drops withi < R and condensation onto drops R > R. The supersaturation e changes in time, so that e (x) becomes a sort of mean field due to all the other droplets and also implies a time-dependent critical radius. R (x) = a/[/"(l)e(x)]. One of the starting equations in the LS analysis is equation (A3.3.87) withi (x). [Pg.750]

FIG. 9 Changes of the monolayer film critical temperature with the concentration of impurities obtained from the Monte Carlo simulations (open circles) and resulting from the mean field theory (solid line). (Reprinted from A. Patrykiejew. Monte Carlo studies of adsorption. II Localized monolayers on randomly heterogeneous surfaces. Thin Solid Films, 205 189-196, with permision from Elsevier Science.)... [Pg.274]

The mean field treatment of such a model has been presented by Forgacs et al. [172]. They have considered the particular problem of the effects of surface heterogeneity on the order of wetting transition. Using the replica trick and assuming a Gaussian distribution of 8 Vq with the variance A (A/kT < 1), they found that the prewetting transition critical point is a function of A and... [Pg.279]


See other pages where Critical mean field is mentioned: [Pg.126]    [Pg.228]    [Pg.241]    [Pg.126]    [Pg.228]    [Pg.241]    [Pg.424]    [Pg.437]    [Pg.461]    [Pg.510]    [Pg.531]    [Pg.534]    [Pg.651]    [Pg.653]    [Pg.658]    [Pg.746]    [Pg.753]    [Pg.755]    [Pg.2368]    [Pg.2371]    [Pg.2371]    [Pg.63]    [Pg.103]    [Pg.107]    [Pg.118]    [Pg.122]    [Pg.273]    [Pg.279]    [Pg.296]    [Pg.306]   
See also in sourсe #XX -- [ Pg.355 ]

See also in sourсe #XX -- [ Pg.143 ]




SEARCH



Critical exponent mean-field value

Critical field

Critical phenomena and the Ising-mean field crossover

Ionic fluid criticality mean-field theories

Mean field critical region

Mean-field

Mean-field-like criticality

© 2024 chempedia.info