Extended mean-field theory

The extended mean-field theory corresponds to the following approximate direct correlation function ... 

The present model may impose too strong an obstruction on a real system, and it will be of great interest to know whether the idea of a higher shell substitution effect can be extended and adjusted to the excluded volume problem. Certainly, nobody can characterize the cascade theory with substitution effects as a mean field theory. 

Section IV is devoted to excitons in a disordered lattice. In the first subsection, restricted to the 2D radiant exciton, we study how the coherent emission is hampered by such disorder as thermal fluctuation, static disorder, or surface annihilation by surface-molecule photodimerization. A sharp transition is shown to take place between coherent emission at low temperature (or weak extended disorder) and incoherent emission of small excitonic coherence domains at high temperature (strong extended disorder). Whereas a mean-field theory correctly deals with the long-range forces involved in emission, these approximations are reviewed and tested on a simple model case the nondipolar triplet naphthalene exciton. The very strong disorder then makes the inclusion of aggregates in the theory compulsory. From all this study, our conclusion is that an effective-medium theory needs an effective interaction as well as an effective potential, as shown by the comparison of our theoretical results with exact numerical calculations, with very satisfactory agreement at all concentrations. Lastly, the 3D case of a dipolar exciton with disorder is discussed qualitatively. 

To conclude, we can draw an analogy between our transition and Anderson s transition to localization the role of extended states is played here by our coherent radiant states. A major difference of our model is that we have long-range interactions (retarded interactions), which make a mean-field theory well suited for the study of coherent radiant states, while for short-range 2D Coulombic interactions mean-field theory has many drawbacks, as will be discussed in Section IV.B. Another point concerns the geometry of our model. The very same analysis applies to ID systems however, the radiative width (A/a)y0 of a ID lattice is too small to be observed in practical experiments. In a 3D lattice no emission can take place, since the photon is always reabsorbed. The 3D polariton picture has then to be used to calculate the dielectric permittivity of the disordered crystal see Section IV.B. 

The asymptotic approach has not yet been extended to interactions between stretched layers in theta and poor solvents, which control the incipient flocculation of polymerically-stabilized colloidal dispersions. Application of the mean-field theory to poor solvents produces an attractive minimum only for —nll2v/w112 > new112/l, when layers begin to interpenetrate rather than... 

Our purpose in these last two subsections has been to show how the simplest fundamental description of SEE for van der Waals solids can emerge from the hard-sphere model and mean field theory. Much of the remainder of the chapter deals with how we extend this kind of approach using simple molecular models to describe more complex solid-fiuid and solid-solid phase diagrams. In the next two sections, we discuss the numerical techniques that allow us to calculate SEE phase diagrams for molecular models via computer simulation and theoretical methods. In Section IV we then survey the results of these calculations for a range of molecular models. We offer some concluding remarks in Section V. 

At this point we emphasize that Eqs. (16H33) should not be understood as a formally rigorous derivation (such derivations by various techniques can be found in the literature [114, 116, 117]) but rather the present treatment (which follows Ref. 78) is a plausibility argument. In this spirit, one can also extend the theory to the interacting case, within the framework of mean field theory the inverse collective response function of the interacting system within RPA is always found from that of the noninteracting system by subtracting the Fourier transform of the interaction [118]. In our case we have a collective structure factor... 

The previously discussed theories were developed for monodisperse diblock copolymers, which are not TPEs. However, Leibler s mean-field theory has been extended to include polydispersity (Leibler and Benoit, 1981) and to include triblock, star, and graft copolymers (Olvera de la Cruz and Sanchez, 1986 Mayes and Olvera de la Cruz, 1989). In the former case, polydispersity corrections tend to lower x N corresponding to the ODT. As would be expected from the analogy between blends and diblocks, triblocks will phase separate at higher xN values than the corresponding diblocks. This theory predicts a monotonic increase in the critical value of x A as the symmetry of the triblock increases, to a maximum of about 18 for the symmetric triblock. Surprisingly, the minimum xN value that separates the order and disordered regions in triblocks does not necessarily correspond to the critical point. 

Optical second-harmonic generation experiments give a more detailed description of the anchoring at a microscopic scale [68,69 see also Chapter 5]. The molecule/surface interaction determines the orientational distribution in a thin surface layer extending up to 1 nm. The bulk uniaxial order develops on top of this layer via a transition layer of thickness which is well described by the usual mean-field theory, possibly including non-uniaxial components of the tensor order parameter. 

This mean-field theory to calculate the interfacial profile and interfacial free energy can be extended to compute also for a critical droplet the order parameter profile (for a binary mixture near the critical point,... 

Hence, away from the critical point (see Equation 15), the applicability range of Landau s approximation extends with increasing the long-range correlation distance. Increasing away from Tc is equivalent to an admission of a longer range of particle interactions in the system, which is the premise of applicability of the mean field theories (see section 1.5). 

We have briefly reviewed methods which extend the self-consistent mean-field theory in order to investigate the statics and dynamics of collective composition fluctuations in polymer blends. Within the standard model of the self-consistent field theory, the blend is described as an ensemble of Gaussian threads of extension Rg. There are two types of interactions zero-ranged repulsions between threads of different species with strength /AT and an incompressibility constraint for the local density. 

However, the number of liquid crystals that have been studied under pressure is very limited. In most cases neither the equation of state nor the pressure dependence of the order parameter is known. Only the mean-field theory of Maier and Saupe was extended to explain the dielectric properties of liquid crystalline phases. However, a recent approach by Photinos et al. analyzed the nematic reentrance and phase stability based on the variational cluster method. The lack of a full theoretical description as well as insufficient experimental data should stimulate further high-pressure investigations in this field. 

The Maier-Saupe mean field theory of nematics can be extended to smectic A liquid crystals following the development of McMillan [3.24]. The smectic A phase has a unique axis (the director) like the nematic phase, but it also possesses a one-dimensional translational periodicity. The centers of mass of the molecules tend to lie on planes normal to the director. The interplanar distance, d, is approximately a molecular length, twice the molecular length or in between these two length scales. There is no positional ordering of the centers of mass of the molecules within each plane. The single-molecule potential may be deduced from the Kobayashi s pair interaction potential [3.25]... 

The simple theory of the nematic-smectic A transition has been proposed by McMillan [59] (and independently by Kobayashi [60]) by extending the Maier-Saupe approach to include the possibility of translational ordering. The McMillan theory is a classical mean-field theory and therefore the free energy is given by the general Eq. (34). For the smectic A phase it can be rewritten as... 

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