Mean field theories

The qualitative nature of the phase boundaries, however, is not simply related to the behavioral class of the deterministic rule corners. For example, while figure 7.7-c shows a phase transition to a class-3 rule, figures 7.7-a and 7.7-b show that the boundaries end at the class-3 rule. Similarly, while in figure 7.7-a the phase transition ends at a class-2 rule, there is only the absorbing stationary state close to the class-2 rule in figure 7.7-c. 

7 Three sections of the (pi,p2,P3)-phase plot (after [kinzel85b]) see text. 

Except for a few special cases (most notably, /inear rules (see below) and the P2 = I line for the isotropic peripheral PCA discussed above, which happens to be an endpoint of a disorder line of an exactly solved 2-dim Ising model on a triangular lattice and can be solved exactly [kinzel85b]), phase-diagrams for PCA 

In this section we briefly outline a general mean-field theory approach to arbitrary PCA and then apply the formalism to a particular class of one-parameter rules. We then compare the theoretical predictions to numerical simulations on lattices of dimension 1 d 4. 

The most drastic approximations leading to Eq. (1) are credited by the following arguments [39] a chain conformation of a polymer coil in dense melt sys- 

On the basis of standard criteria for equilibrium, stability limits, and criticality yielding coexistence curve (binodal), spinodal line, and critical point, the phase behavior maybe predicted using Eq. (1)  

The first two terms correspond to the combinatorial entropy terms of Eq. (1) and form the non-interacting part of the structure factor which is just a weighted average of the single-chain structure factors SA(q) and SB(q) of both blend components. SA(q) and SB(q) are characterized by the radius of gyration RgA= aA(NA/6)1/2 and RgB=aB(NB/6)1/2, where aA and aB are the statistical segment lengths of polymer A and polymer B, respectively. The last term of Eq. (4) yields the SANS determined interaction parameter %SANS  

The structure of the interface formed by coexisting phases is well described by the Cahn-Hilliard approach [53] (developed in a slightly different context by Landau and Lifshitz [54]) extended to incompressible binary polymer mixtures by several authors [4,49,55,56]. The central point of this approach is the free energy functional definition that describes two semi-infinite polymer phases ]), and f 2 separated by a planar interface (at depth z=0) and the composition ( )(z) across this interface. The relevant functional Fb for the free energy of mixing per site volume Q (taken as equal to the average segmental volume V of both blend components) and the area A of the interface is expressed by 

We may also note an analogy between mean field theory and classical mechanics, and treat the integrand of the Fb functional as the Lagrangian Then 

Equation (8.33) is similar to the LSW equation of Eq. (8.11), where the rate of change in the radius of the grain in Ostwald ripening is controlled by the interface reaction, so that the critical radius for the rate of change is given by [23]  

Therefore, all the theories indicate that the NGG in polycrystaUine solids can be treated as the Ostwald ripening governed by the interface reaction mechanism. However, in practice, the grain growth data cannot be always described by the parabolic law, so that a general grain growth equation is used  

2 Chemical Potential of a Polymer Chain in Solution When N 1, 

The deviation from the ideal solution is magnified by N. A small difference of x from 1 /2 shows up as a large nonideality when N is large. Thus the polymer solutions, especially those of high-molecular-weight polymers, can be easily nonideal. When x i/2, in particular, N(l - 2x) can be easily as large as to cause a dip in the plot of Apip. 

Equations 2.7, 2.14, and 2.15 serve as a starting point for further discussion of thermodynamics of the polymer solution in the lattice fluid model. Another approach to the thermodynamics based on these equations is given in Appendix 2.B. 

In the low concentration limit, Eq. 2.16 gives the osmotic pressnie Hyeai of the ideal solution  

Thus the ratio of II to Ilideai compared at the same concentration is 

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Although the exact equations of state are known only in special cases, there are several usefid approximations collectively described as mean-field theories. The most widely known is van der Waals equation [2]... 

The parameters a and b are characteristic of the substance, and represent corrections to the ideal gas law dne to the attractive (dispersion) interactions between the atoms and the volnme they occupy dne to their repulsive cores. We will discnss van der Waals equation in some detail as a typical example of a mean-field theory. 

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. 

Table A2.3.4 simnnarizes the values of these critical exponents m two and tliree dimensions and the predictions of mean field theory.

As a prelude to discussing mean-field theory, we review the solution for non-interacting magnets by setting J = 0 in the Ising Flamiltonian. The PF... 

Fluctuations in the magnetization are ignored by mean-field theory and there is no correlation between neighbouring sites, so that... 

The neglect of fluctuations in mean-field theory implies that... 

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. 

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. 

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. 

Plotting r versus 1/n gives kTJqJ as the intercept and (kTJqJ)( -y) as the slope from which and y can be determined. Figure A2.3.29 illustrates the method for lattices in one, two and tliree dimensions and compares it with mean-field theory which is independent of the dimensionality. 

Weeks J D, Katsov K and Vollmayr K 1998 Roles of repulsive and attractive forces in determining the structure of non uniform liquids generalized mean field theory Phys. Rev. Lett. 81 4400... 

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. 

Exponent values derived from experiments on fluids, binary alloys, and certain magnets differ substantially from all those derived from analytic (mean-field) theories. Flowever it is surprising that the experimental values appear to be the same from all these experiments, not only for different fluids and fluid mixtures, but indeed the same for the magnets and alloys as well (see section A2.5.5). 

Figure A2.5.26. Molar heat capacity C y of a van der Waals fluid as a fimction of temperature from mean-field theory (dotted line) from crossover theory (frill curve). Reproduced from [29] Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 532, figure 4, by pennission of Elsevier Science.

Levelt Sengers J M H 1999 Mean-field theories, their weaknesses and strength Fluid Phase Equilibria 158-160 3-17... 

The single-eonfiguration mean-field theories of eleetronie strueture negleet eorrelations among the eleetrons. That is, in expressing the interaetion of an eleetron at r... 

Poly(ethylene oxide). The synthesis and subsequent hydrolysis and condensation of alkoxysilane-terniinated macromonomers have been studied (39,40). Using Si-nmr and size-exclusion chromatography (sec) the evolution of the siUcate stmctures on the alkoxysilane-terniinated poly(ethylene oxide) (PEO) macromonomers of controlled functionahty was observed. Also, the effect of vitrification upon the network cross-link density of the developing inorganic—organic hybrid using percolation and mean-field theory was considered. 

Theories of Gelation. The classical or mean field theory of polymeri2ation (4) is useful for visuali2ing the conditions for gelation. This model yields a degree of reaction, of one-third at the time of gelation for chemical species having functionaUty equal to four. Two-thirds of the possible... 

P Koehl, M Delarue. Application of a self-consistent mean field theory to predict protein side-chains conformation and estimate their conformational entropy. J Mol Biol 239 249-275, 1994. 

Mean-field Theory of Adsorption in Mesoporous Media... 

Theoretically, several aspects of the Thommes-Findenegg experiment can be analyzed at the mean-field level [157]. A key quantity of a mean-field theory of confined fluids is the (Helmholtz) free energy, given by... 

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. 

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.)... 

I. Jensen, H. C. Fogedby. Kinetic phase transitions in a surface-reaction model with diffusion Computer simulations and mean-field theory. Phys Rev A 2 1969-1975, 1990. 

J. Mai, W. von Niessen. The CO -(- O2 reaction on metal surfaces. Simulation and mean-field theory The influence of diffusion. J Chem Phys 95 3685-3692, 1990. 

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