Lattice based fields

Special simulation techniques have recently been devised to calculate free energies of these structure-forming fluids [43-45]. We have discussed several methods, which have been inspired by related approaches for calculating free energy of crystals in hard-condensed matter systems [43], rely on a field-theoretic representation via lattice-based fields [45] or exploit the possibility of simultaneously and accurately measuring the pressure and tdiemical potential due to the softness of the off-lattice potentials [44]. 

Influence of Mean-Field-Lattice-Based Theory on the... 

Choudhury et al. [36] in their work on hydrogenated nitrile butadiene rubber (HNBR)-nanoclay systems showed the thermodynamic aspects of nanocomposite formation using the mean-field-lattice-based description of polymer melt intercalation, which was first proposed by Vaia and Giannelis [37]. Briefly, the free... 

Returning to 3D lattice models, one may note that sine-Gordon field theory of the Coulomb gas should enable an RG (e — 4 — D) expansion [15], but this path has obviously not yet followed up. An attempt to establish the universality class of the RPM by a sine-Gordon-based field theory was made by Khodolenko and Beyerlein [105]. However, these authors did not present a scheme for calculating the critical exponents. Rather they argued that the grand partition function can be mapped onto that of the spherical model of Kac and Berlin [106, 297] which predicts a parabolic coexistence curve, i.e. fi — 1/2. This analysis was severely criticized by Fisher [298]. Actually, the spherical model has some unpleasant thermodynamic features, never observed in real fluids. In particular, it is associated with a divergence of the compressibility KTas the coexistence curve (rather than the spinodal line) is approached. By a determination of the exponent y, this possibility could also be ruled out experimentally [95, 97]. 

The treatment of polarization based on the assumption that water has a uniform dielectric constant involves a fundamental difficulty. Indeed, a uniform, continuous distribution of dipoles on a planar surface does not generate a field in a homogeneous medium and hence is not able to polarize the water. If the dipoles are distributed in the sites of a 2D planar square lattice, the field is... 

The formation and equilibrium structure of polymer layered silicate nanocomposites, in particular with organically modified layered silicates, has been shown to be a strong function of the nature of the polymer (polar or apolar), the charge carrying capacity of the layered silicate, as well as the chain length and structure of the cationic surfactant. However, both the polymer/silicate compatibility and hybrid equilibrium structure for these nanocomposites are observed to be independent of polymer molecular weight. The experimental results have been summarized by Vaia et al. and a lattice based mean field theory has been developed to explain these results [26]. 

It is, however, the self-consistent field method that has been most extensively applied to polymer adsorption and in particular the lattice-based discretisation of the Edwards modified diffusion equation associated with Scheutjens and Fleer (1979, 1980). In this model the solution up to the impenetrable adsorbing surface is modelled as a lattice of equal volume cells. All of the lattice layers... 

All methods of molecular ahgnment useful for building traditional lattice-based 3D-QSAR models can also be applied in the framewoik of the CMF approach. Meanwhile, thanks to the integrability of continuous functions describing molecular fields, the latter approach offers additional possibilities. 

CoMFA, introduced in 1988, assumes that a suitable sampling of the steric and electrostatic field around a set of aligned compounds might provide all the information necessary for understanding their biological properties, CoMFA came from earlier qualitative and quantitative lattice-based approaches to 3D-QSAR. 

It was found that the effective width of the interface between the PS and PMMA segments was 75 A, i.e., it was 50% broader than that found between the PS and PMMA homopolymers in the absence of the diblock copolymer (50 5 A [261]) and between the PS and PMMA lamellar microdomains of the pure PS-ft-PMMA in the bulk (50 4 A) [261,262]. The area occupied by the copolymer at the interface between the homopolymers is 30% larger than that of the copolymers in the bulk lamellar microstructure [39]. In that study, the amount of diblock copolymer at the interface was (approximately) equivalent ( 200 A) to half of the long period of the neat ordered copolymer. The same PS/PS-h-PMMA/PMMA system was subsequently investigated by a lattice-based self-consistent field model that was extended... 

A renormalization group calculation by Burch and Moore is based on analogies between lattice model field theories for magnetic systems and polymer solutions. By adjustment of a constant, their results agree with the perturbation expansion of F(z) in equation (82) to order z. With the same value of the constant, F(z) in the good solvent range, z > 0.75, is given as an implicit function of z by the relations... 

Because we do not have to follow a physical trajectory in an MC simulation, we can also use models that are further removed from the true physical or chemical reality. Such models include lattice models (see, e.g.. Refs. 28,61,62). With lattice models, the space of our system is (typically) evenly divided into cells, each of which are represented by one lattice site. Lattices can be very simple cubes or they can be specially adapted, highly connected grids. Here again, we need super-atoms, which, however, can occupy only lattice sites. In most lattice models every site is either singly occupied or empty, meaning that the interaction sites have an impenetrable hard core, which contrasts to Lattice-Boltzmann models used in studies of hydrodynamics in which every lattice site is occupied by a density, in which case, one deals with a density-based field theory. In lattice models, there exist only a fixed number of distances that can be realized. It makes no sense to distinguish between, say, a... 

A brief review is given on electronic properties of carbon nanotubes, in particular those in magnetic fields, mainly from a theoretical point of view. The topics include a giant Aharonov-Bohm effect on the band gap and optical absorption spectra, a magnetic-field induced lattice distortion and a magnetisation and susceptibility of ensembles, calculated based on a k p scheme. 

Electronic properties of CNTs, in particular, electronic states, optical spectra, lattice instabilities, and magnetic properties, have been discussed theoretically based on a k p scheme. The motion of electrons in CNTs is described by Weyl s equation for a massless neutrino, which turns into the Dirac equation for a massive electron in the presence of lattice distortions. This leads to interesting properties of CNTs in the presence of a magnetic field including various kinds of Aharonov-Bohm effects and field-induced lattice distortions. 

FIG. 13 Phase diagram of a vector lattice model for a balanced ternary amphiphilic system in the temperature vs surfactant concentration plane. W -I- O denotes a region of coexistence between oil- and water-rich phases, D a disordered phase, Lj an ordered phase which consists of alternating oil, amphiphile, water, and again amphi-phile sheets, and L/r an incommensurate lamellar phase (not present in mean field calculations). The data points are based on simulations at various system sizes on an fee lattice. (From Matsen and Sullivan [182]. Copyright 1994 APS.)... 

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