Mean-field models/theories

This model is based on a mean-field theory, according to which the mean-field potential tends to align chromophores along the director and also to 

13 (a) A 3D AFN image of a surface relief pattern inscribed with a linearly polarized, single Gaussian beam, (b) The theoretical simulation that reproduces the main features of the actual image. 

Furthermore, one should also emphasize that this mean-field-bascd model does not take into account any possible thermal effects. This is an important limitation t ause SRGs on LC polymers are inscribed using hi -laser powers, considerably higher than for the spin-coated, amorphous polymers, and therefore thermal effects might be expected to contribute. For the laser powers used in LC polymers, for example, it has been already shown that thermal effects are not negligible for the amorphous polymers. 

15 (a) Schematic diagram of Fukuda and Sumaru s model The velocity of the mass transport at the top layer (vj follows the directions predicted by the field-gradient model. The velocity of mass uansport Iot the layss undmteath, y,. decreases as they lie deeper In the sample bulk, and is zero at the substrate. (b)Thc coordinate system for the model. 

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Free Volume Model Field Gradient Force Model Mean-Field Theory Model Model ofViscous Mass Flo>v Diffusion Model... 

Heiman et al. (1975a), Heiman and Lee (1975) and Heiman et al. (1976) have reported the magnetic properties of R(Gd to Lu)-Fe alloys in order to explain the dependence of the data on the concentration and on the R species. They have chiefly studied the variations of and of the internal magnetic field (from Mossbauer data). They propose a mean-field theory model and show that three features are required to obtain agreement between calculations and data ... 

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. 

Figure A2.5.27. The effective coexistence curve exponent P jj = d In v/d In i for a simple mixture N= 1) as a fimction of the temperature parameter i = t / (1 - t) calculated from crossover theory and compared with the corresponding curve from mean-field theory (i.e. from figure A2.5.15). Reproduced from [30], Povodyrev A A, Anisimov M A and Sengers J V 1999 Crossover Flory model for phase separation in polymer solutions Physica A 264 358, figure 3, by pennission of Elsevier Science.

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... 

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. 

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. 

Chapter 7 discusses a variety of topics all of which are related to the class of probabilistic CA (PCA) i.e. CA that involve some elements of probability in their state and/or time-evolution. The chapter begins with a physicist s overview of critical phenomena. Later sections include discussions of the equivalence between PCA and spin models, the critical behavior of PCA, mean-field theory, CA simulation of conventional spin models and a stochastic version of Conway s Life rule. 

The LST is a finitely parameterized model of the action of a given CA rule, >, on probability measures on the space of configurations on an arbitrary lattice. In a very simple manner - which may be thought of as a generalization of the simple mean field theory (MFT) introduced in section 3.1.3. - the LST provides a sequence of approximations of the statistical features of evolving CA patterns. 

Prausnitz and coworkers [91,92] developed a model which accounts for nonideal entropic effects by deriving a partition function based on a lattice model with three categories of interaction sites hydrogen bond donors, hydrogen bond acceptors, and dispersion force contact sites. A different approach was taken by Marchetti et al. [93,94] and others [95-98], who developed a mean field theory... 

Figure 3a shows the mean-field predictions for the polymer phase diagram for a range of values for Ep/Ec and B/Ec. The corresponding simulation results are shown in Fig. 3b. As can be seen from the figure, the mean-field theory captures the essential features of the polymer phase diagram and provides even fair quantitative agreement with the numerical results. A qualitative flaw of the mean-field model is that it fails to reproduce the crossing of the melting curves at 0 = 0.73. It is likely that this discrepancy is due to the neglect of the concentration dependence of XeS Improved estimates for Xeff at high densities can be obtained from series expansions based on the lattice-cluster theory [68,69].

In order to understand the thermodynamic issues associated with the nanocomposite formation, Vaia et al. have applied the mean-field statistical lattice model and found that conclusions based on the mean field theory agreed nicely with the experimental results [12,13]. The entropy loss associated with confinement of a polymer melt is not prohibited to nanocomposite formation because an entropy gain associated with the layer separation balances the entropy loss of polymer intercalation, resulting in a net entropy change near to zero. Thus, from the theoretical model, the outcome of nanocomposite formation via polymer melt intercalation depends on energetic factors, which may be determined from the surface energies of the polymer and OMLF. 

The most essential step in a mean-field theory is the reduction of the many-body problem to a scheme that treats just a small number of molecules in an external field. The external field is chosen such that it mimics the effect of the other molecules in the system as accurately as possible. In this review we will discuss the Bragg Williams approach. Here the problem is reduced to behaviour of a single chain (molecule) in an external field. Higher order models (e.g. Quasi-chemical or Bethe approximations) are possible but we do not know applications of this for bilayer membranes. 

Relativistic mean field (RMF) models have been applied successfully to describe properties of rinite nuclei. In general ground state energies, spin-orbit splittings, etc. can be described well in terms of a few parameters ref. [18]. Recently it has lead to the suggestion that the bulk SE is strongly correlated with the neutron skin [19, 20] (see below). In essence the method is based upon the use of energy-density functional (EDF) theory. 

If the polymer concentration increases so that the number of high order bead-bead interactions is significant, c>>c =p, (when c is expressed as the polymer volume fraction. Op), the fluctuations in the polymer density becomes small, the system can be treated by mean-field theory, and the ideal model is applicable at all distance ranges, independent of the solvent quaUty and concentration. These systems are denoted as concentrated solutions. A similar description appHes to a theta solvent, but in this case, the chains within the blobs remain pseudoideal so that =N (c/c ) and Rg=N, i.e., the global chain size is always in-... 

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