Lattice field theory approach

R. D. Coalson, A. M. Walsh, A. Duncan, and N. Bental. Statistical mechanics of a Coulomb gas with finite size particles—a lattice field theory approach. J. Chemical Physics 102 4584-4594 (1995). 

Statistical-Mechanics of a Coulomb Gas with Finite-Size Particles—A Lattice Field-Theory Approach. 

Derivation of the Poisson-Boltzmann Equation and Associated Thermodynamical Relations A Lattice Field Theory Approach... 

DERIVATION OF THE POISSON-BOLTZMANN EQUATION AND ASSOCIATED THERMODYNAMICAL RELATIONS A LATTICE FIELD THEORY APPROACH... 

Tsonchev, S., Coalson, R.D., Duncan, A. Statistical mechanics of charged polymers in electrolyte solutions a lattice field theory approach. Phys. Rev. E 60, 4257-4267 (1999)... 

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. 

For further details regarding the development and interpretation of the order parameters, Refs. 20 and 26-28 consider on-lattice protein models, while Refs. 21, 23 and 29 consider field theory approaches. References 24, 25, and 30 provide overviews of and general arguments for the spin-glass approach to the protein folding problem. 

The other class of phenomenological approaches subsumes the random surface theories (Sec. B). These reduce the system to a set of internal surfaces, supposedly filled with amphiphiles, which can be described by an effective interface Hamiltonian. The internal surfaces represent either bilayers or monolayers—bilayers in binary amphiphile—water mixtures, and monolayers in ternary mixtures, where the monolayers are assumed to separate oil domains from water domains. Random surface theories have been formulated on lattices and in the continuum. In the latter case, they are an interesting application of the membrane theories which are studied in many areas of physics, from general statistical field theory to elementary particle physics [26]. Random surface theories for amphiphilic systems have been used to calculate shapes and distributions of vesicles, and phase transitions [27-31]. 

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... 

In practice, the scheme as explained above is not implemented. The consecutive generation of all possible chain conformations is a very expensive step. The reason for this is that there are of the order of ZN number of conformations, where Z is the lattice coordination number. A clever trick is to generate a subset of all possible conformations and to use this set in the SCF scheme. This approach is known in the literature as the single-chain mean-field theory, and has found many applications in surfactant and polymeric systems [96]. The important property of these calculations is that intramolecular excluded-volume correlations are rather accurately accounted for. The intermolecular excluded-volume correlations are of course treated on the mean-field level. The CPU time scales with the size of the set of conformations used. One of the obvious problems of this method is that one should make sure that the relevant conformations are included in the set. Typically, the set of conformations is very large, and, as a consequence, the method remains extremely CPU intensive. 

Simulations [73] have recently provided some insights into the formal 5c —> 0 limit predicted by mean field lattice model theories of glass formation. While Monte Carlo estimates of x for a Flory-Huggins (FH) lattice model of a semifiexible polymer melt extrapolate to infinity near the ideal glass transition temperature Tq, where 5c extrapolates to zero, the values of 5c computed from GD theory are too low by roughly a constant compared to the simulation estimates, and this constant shift is suggested to be sufficient to prevent 5c from strictly vanishing [73, 74]. Hence, we can reasonably infer that 5 approaches a small, but nonzero asymptotic low temperature limit and that 5c similarly becomes critically small near Tq. The possibility of a constant... 

There is a substantial body of theoretical work on micellization in block copolymers. The simplest approaches are the scaling theories, which account quite successfully for the scaling of block copolymer dimensions with length of the constituent blocks. Rather detailed mean field theories have also been developed, of which the most advanced at present is the self-consistent field theory, in its lattice and continuum guises. These theories are reviewed in depth in Chapter 3. A limited amount of work has been performed on the kinetics of micellization, although this is largely an unexplored field. Micelle formation at the liquid-air interface has been investigated experimentally, and a number of types of surface micelles have been identified. In addition, adsorption of block copolymers at liquid interfaces has attracted considerable attention. This work is also summarized in Chapter 3. 

There have been few attempts to generalize mean-field theories to the unrestricted case. Netz and Orland [227] applied their field-theoretical model to the UPM. Because such lattice theories yield quite different critical properties from those of continuum theories, comparison of their results with other data is difficult. Outhwaite and coworkers [204-206] considered a modification of their PB approach to treat the UPM. Their theory was applied to a few conditions of moderate charge and size asymmetry. 

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]. 

A wide variety of theories have been developed for polymer solutions over the later half of the last century. Among them, lattice model is still a convenient starting point. The most widely used and best known is the Flory-Huggins lattice theory (Flory, 1941 Huggins, 1941) based on a mean-field approach. However, it is known that a mean-field approximation cannot correctly describe the coexistence curves near the critical point (Fisher, 1967 Heller, 1967 Sengers and Sengers, 1978). The lattice cluster theory (LCT) developed by Freed and coworkers (Freed, 1985 Pesci and Freed, 1989 Madden et al., 1990 Dudowicz and Freed, 1990 Dudowicz et al., 1990 Dudowicz and Freed, 1992) in 1990s was a landmark. 

Here we steer a middle course, emphasizing analytical results extracted from the mean-field theories, in context with the lattice and scaling approaches, and explaining the relationship to macroscopic phenomena. Our treatment draws on Russel et al. (1989) in some places. 

In recent years there has been an explosion of interest in the electron properties of disordered lattices. The more common line of approach to this kind of problem is to study the mean resolvent of the random medium, and the memory function methods can be of remarkable help for this purpose. Otherwise one can investigate by the memory function methods (basically the recursion method) a number of judiciously selected configurations this line of approach is particularly promising because it allows one to overcome some of the limitations inherent in the mean field theories. In this section we de-... 

In the current paper, we discuss some of the new approaches and results that have been developed and obtained recently within the context of such molecular modeling research, and in particular with the mean field and Monte Carlo studies of a lattice model. The next section describes the Gaussian random field method (Woo et al, 2001), which provides a computationally efficient route to generate realistic representations of the disordered mesoporous glasses. Application of the mean field theory, and Monte Carlo simulations are described in Secs. 3 and 4, respectively. 

The finite temperature studies of Lennard-Jones lattice gas systems have been performed for the square [105,116], rectangular [106] and triangular [100,111,112] lattices using different approaches, including the simple mean-field theory, the renormalization group method, Monte Carlo simulation and Monte Carlo version of the coherent anomaly method. 

The van der Waals approximation discussed in Section LA applies the mean field approximation in the solid phase in the same way as in the fluid phase. Baus and co-workers [150,151] have recently presented an alternative formulation in which the localization of the molecules in the solid phase is taken into account. They have applied this to the understanding of trends in the phase diagrams of systems of hard spheres with attractive tails as the range of attractions is changed. For the mean field term in the solid phase they use the static lattice energy for the given interaction potential and crystal lattice. A similar approach was used earlier to incorporate quad-rupole-quadrupole interactions into a van der Waals theory calculation of the phase diagram of carbon dioxide [152]. 

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