Mean-field Free Energy Research Articles

Mean-field phase diagram for cubic-based praseodymium compounds

Calculating the exact mean-field free energy from a Heisenberg-like Hamiltonian we determine the magnetic phase diagram for cubic-based praseodymium compounds with a ${\ensuremath{\Gamma}}_{3}$ doublet ground state and a ${\ensuremath{\Gamma}}_{4}$ triplet first-excited state. We find lines of first- and second-order phase transitions, separated by a tricritical point. We also calculate the temperature dependence of the magnetic-order parameter.

Vacancy-Mediated Phase Separation of Binary Alloy

The most important element process of the phase separation of binary alloys is the vacancy diffusion mechanism. A set of phenomenological dynamical equations for the vacancy-mediated phase separation is formulated from a mean-field free energy and the numerical simulation of the model is performed. From numerical results, vacancies form clusters along phase boundaries between two degenerate stable domains. The width of the vacancy-cluster across the interface grows with the logarithmic function of time. If effects of vacancy-clusters are not dominant, the dynamics is not different from that by the exchange dynamics. If the effects are dominant, the characteristic length proportional to the average domain size grows with the logarithmic function of time. The analytical approach shows that the logarithmic growth of the characteristic length is related to the growth law of the width of the vacancy-cluster.

Structure of droplet microemulsions in the semi-dilute regime

We derive a functional of the microemulsion droplets size distribution function by adding the entropic free energy of a system of polydisperse hard spheres to the Helfrich interfacial free energy of the amphiphilic film. The latter is minimized by a monodisperse system of spherical (independent) aggregates, at the equilibrium radius, while the former is minimized by a random distribution of droplet sizes. As a result the equilibrium droplet size distribution yields a mean radius that is smaller than the equilibrium radius of isolated droplets. A link with molecular parameters is established by using our microscopic model for ternary amphiphilic mixtures and a (mean-field) free energy functional of the density profiles [J. Chem. Phys. 109, 1152 (1998)] to calculate the parameters of the Helfrich interfacial free energy for a single droplet. The theoretical results are compared with small-angle neutron scattering data reported by Gradzielski et al. [Phys. Rev. E 53, 3900 (1996)].

Formation of Interfaces in Incompatible Polymer Blends: A Dynamical Mean Field Study

The interdiffusion dynamics of a polymer interface towards its equilibrium profile is studied using a dynamical mean field method, in which the nonequilibrium chemical potentials are evaluated numerically using the full polymer mean field free energy. Our results demonstrated that the interfacial width grows as t 1/4 at early times and saturates to the equilibrium thickness at long times, in agreement with previous, more approximate analysis based on Cahn-Hilliard-type theories. Furthermore, we show that the dynamical exponent can only be obtained unequivocally via a scaling analysis. The interdiffusion of a polymer interface in the single phase regime is also studied.

A simple off-lattice model for microemulsions

We have developed a simple off-lattice density functional theory and applied it to ternary water–oil–amphiphile mixtures. Our approach is based on mean field free energy functionals calculated from hard-sphere perturbation theory, to which is added a contribution arising from molecular association between the water-like and the amphiphilic species. This latter term is treated using the Wertheim theory for associating liquid mixtures, and it gives rise to a re-entrant binary phase diagram in a very natural way. The resulting ternary phase diagrams resemble experimental data both qualitatively and quantitatively. We also calculate the structure and free energy of interfaces between phases in this system, and show that the presence of amphiphile dramatically lowers the surface tension between the water-rich and oil-rich phases. This simple model contains many of the features of real microemulsion systems, and can be extended to study lamellar and other complex phases.

Mean-field theory of Boltzmann machine learning

I present a mean-field theory for Boltzmann machine learning, derived by employing Thouless-Anderson-Palmer free energy formalism to a full extent. Using the Plefka expansion an extended theory that takes higher-order correction to mean-field free energy formalism into consideration is presented, from which the mean-field approximation of general orders, along with the linear response correction, are derived by truncating the Plefka expansion up to desired orders. A theoretical foundation for an effective trick of using ``diagonal weights,'' introduced by Kappen and Rodr\'{\i}guez, is also given. Because of the finite system size and a lack of scaling assumptions on interaction coefficients, the truncated free energy formalism cannot provide an exact description in the case of Boltzmann machines. Accuracies of mean-field approximations of several orders are compared by computer simulations.

Relationship between long timescales and the static free-energy in the Hopfield model

The Glauber dynamics of the Hopfield model at low storage level is considered. We analytically derive the spectrum of relaxation times for large system sizes. The longest timescales are gathered in families, each family being in one-to-one correspondence with a stationary (not necessarily stable) point of the static mean-field free energy. Inside a family, the timescales are given by the reciprocals (of the absolute values) of the eigenvalues of the free-energy Hessian matrix.

Renormalization group flow of the stiffness matrix - free-energy relation

In order to satisfy exact sum-rule requirements for correlation function structure at complete wetting, a two-field Hamiltonian , modelling the coupling of order-parameter fluctuations near the wall and unbinding interface, has been introduced. The model is characterized by a stiffness matrix, , whose bare (unrenormalized) elements are related to the mean-field free energy. We extend previous renormalization group studies to include the position dependence of the matrix elements and derive an elegant operator relationship which shows that the flow of the cross-coupling term parallels that of the free energy. This establishes the validity of a stiffness matrix - free-energy relation in the presence of fluctuation effects at the marginal dimension d = 3 for systems with short-ranged forces. We further show that an analogous relation exists for systems with long-ranged molecular interactions.

Bilayer coupling in cuprates based on t- J model

To study coupling between the 2-D layers of antiferromagnetically correlated electrons for the case of underdoped cuprates. We employ the t- J Hamiltonian by allowing phase fluctuations. Based on the present approach we obtain an mean field effective action and free energy for boson degrees of freedom. Finally we obtained the effective action for gauge fluctuations.

Sine-Gordon mean field theory of a Coulomb gas

Sine-Gordon field theory is used to investigate the phase diagram of a neutral Coulomb gas. A variational mean field free energy is constructed and the corresponding phase diagrams in two (2d) and three dimensions (3d) are obtained. When analyzed in terms of chemical potential, the Sine-Gordon theory predicts the phase diagram topologically identical with the Monte Carlo simulations and a recently developed Debye-H\"uckel-Bjerrum (DHBj) theory. In 2d we find that the infinite order Kosterlitz-Thouless line terminates in a tricritical point, after which the metal-insulator transition becomes first order. However, when the transformation from chemical potential to the density is made the whole of the insulating phase is mapped onto zero density.

Generalization of the mean-field Ising model within Tsallis thermostatistics

In this study, the mean-field Ising model, using the Bogolyubov inequality which has been obtained in the framework of the generalized statistical thermodynamics (GST), suitable for non-extensive systems, has been investigated. Generalized expressions for the mean-field magnetization and free energy have been established. These new results have been verified by the fact that they transform to the well-known Boltzmann-Gibbs results in the q → 1 limiting case. For the index q which characterizes the fractal structure of the magnetic system, an interval has been established where the generalized mean-field free energy has a minimum and mean-field magnetization has a corresponding finite value. The interval of q is consistent with paramagnetic free spin systems.

Undulating Charged Fluid Membranes and their Bending Constants

We consider harmonically undulating ionic surfactant membranes, calculating a general expression for the bending modulus valid for all salt and surfactant concentrations, and surface charge densities. This is achieved through a perturbative expansion of the mean-field electrostatic potential and free energy about a planar reference state, consisting of two parallel planar membranes with intervening salt solution. For a given choice of undulation mode, the result for the bending modulus is seen to be generally the same as that obtained from considerations of membranes deformed into a cylindrical geometry. Thus, we show that the bending modulus is independent of global aggregate geometry, at a general system composition. Specializing to the limit of excess added salt (equivalent to a single undulating membrane in contact with bulk electrolyte), we are able to extend the free energy calculation to fourth order in undulation amplitude, and derive the bending constants of the curvature expansion to this order. These have been suggested previously to be of importance in explaining the stability of bicontinuous crystalline and disordered phases, and the formation of passages in lamellar phases. We also discuss the breakdown of the curvature description at shorter wavelengths.

Curvature elasticity and thermodynamic stability of electrically charged membranes

The elastic behavior and stability of electrically charged amphiphilic membranes is investigated. In the present study, we address the question whether the electrostatic contribution to the curvature elastic moduli of a uniformly charged symmetric membrane leads to a curvature instability. To this end we consider a membrane in which the overall number of molecules is conserved during any deformation. In order to estimate both the molecular equilibrium area and the position of the neutral surface of each monolayer during bending, we include in the expression of the bilayer free energy beside an electrostatic, also a nonelectrostatic contribution. The former is described within the Gouy–Chapman theory of the diffuse double layer. The latter is a sum of a chain, an interfacial, and a nonelectrostatic head group contribution. The chain part is described using a detailed mean-field conformational free energy which is based on a molecular chain model. For the interfacial and nonelectrostatic head group contribution we use simple but general phenomenological expressions. It is shown that for medium and high membrane surface charge densities the electrostatic contribution to the bending moduli is not negligible. For highly charged membranes, the model predicts an instability with respect to a spherical deformation. This is discussed referring to the experimentally observed process of spontaneous vesiculation upon jump in pH of certain ionizable amphiphilic molecules.

Protein Adsorption on Lipid Monolayers at their Coexistence Region

We investigate theoretically the behavior of proteins as well as other large macromolecules which are incorporated into amphiphilic monolayers at the air-water interface. We assume the monolayer to be in the coexistence region of the ``main'' transition, where domains of the liquid condensed phase coexist with the liquid expanded background. Using a simple mean-field free energy accounting for the interactions between proteins and amphiphilic molecules, we obtain the spatial protein distribution with the following characteristics. When the proteins preferentially interact with either the liquid condensed or liquid expanded domains, they will be dissolved in the respective phase. When the proteins are energetically rather indifferent to the density of the amphiphiles, they will be localized at the line boundary between the (two-dimensional) liquid expanded and condensed phases. In between these two limiting cases, a delocalization transition of the proteins takes place. This transition is accessible by changing the temperature or the amount of incorporated protein. These findings are in agreement with recent fluorescence microscopy experiments. Our results also apply to lipid multicomponent membranes showing coexistence of distinct fluid phases.

Effects of the Fluctuations on Thermodynamic Properties of the Mixed Valence Compounds11The project supported by National Natural Science Foundation of China.

Based on the Anderson lattice mode1 in the slave boson formalism, we study the effect of fluctuations around the saddle-point approximation on the thermodynamic properties for mixed valence lattice systems. The fluctuation component of slave boson field is explicitly introduced and its interactions with electrons are studied. By using the Green's function technique we obtain modified self-consistent equations of mean-field parameters and free energy. The fluctuation correction to the pseudophase transition temperature is obtained, and the validity of the mean-field approximation is examined. In the limit of low energy and small momentum transfer, a fluctuation correction to the specific heat proportional to at low temperature is extracted. The chemical potential shift is also studied. In order to describe the thermodynamic properties correctly, it is important to determine the chemical potential shift properly.

Boundary tension: From wetting transition to prewetting critical point

We develop a mean-field model free energy which we use in a van der Waals-like theory to study the prewetting transition in a system of two fluid phases when an incipient third phase may wet the interface between them. The line of prewetting transitions in the phase diagram is determined from the bulk wetting transition to the prewetting critical point. As the prewetting critical point is approached, the two coexisting surface phases become more and more alike, and they become identical at the prewetting critical point. The values of the boundary tension of the one-dimensional boundary formed by the edge-on meeting of two coexisting surface phases are calculated exactly (numerically) in a range between the wetting transition and the prewetting critical point. The data points obtained are extrapolated to a finite and positive boundary tension at the wetting transition and to a zero boundary tension at the prewetting critical point. These results are consistent with related earlier work. After scaling the dimensionless boundary tensions with appropriate force units, we determine that their values range from 0 at the prewetting critical point to 𝒪(10−12) N close to the wetting transition. These orders of magnitude compare well with recent experimental results.

Mean-field phase diagram of a coupled XY-Ising model for discotic liquid crystals.

A simple model Hamiltonian is derived to describe the hexagonal disordered ${\mathit{D}}_{\mathit{hd}}$ and hexagonal ordered ${\mathit{D}}_{\mathit{ho}}$ columnar phases of discotic liquid crystals with a restricted set of parameters. Characterizing a rigid helical column by an XY-like angular variable and by an Ising-like helicity variable reduces the three-dimensional problem to a coupled XY-Ising Hamiltonian on a triangular lattice. A Landau-type mean-field free energy expansion for two kinds of order parameter is described. The period-3 phase diagram exhibits many types of phases including those with total Ising (helicity) disorder with XY order. This feature is proposed as the structure of the hexa(hexylthio)triphenylene (HHTT) compounds in the ${\mathit{D}}_{\mathit{hd}}$ phase. The phase diagram also shows phases with two columns of opposite helicity to the third one and phases where 1/3 of the columns have helicity disorder. These helicity states are relevant to the ${\mathit{D}}_{\mathit{ho}}$ phase of (HHTT) compounds.

Line and boundary tensions at the wetting transition: Two fluid phases on a substrate

We develop and analyze a mean-field model free energy that describes two fluid phases on a substrate in order to calculate the (numerically) exact line and boundary tensions, on approach to the first-order wetting transition. A theory based on the van der Waals theory of gas–liquid interfaces is used. We implement a multigrid algorithm to determine the two-dimensional spatial variation of the density across the three-phase and boundary regions, and hence, the line and boundary tensions. As the wetting transition is approached, the tensions approach the same, finite, positive limit with diverging slopes. We compare our results with those of recent related work.

Harmonic corrections to the mean-field phase diagram for block copolymers

Allowance is made for harmonic corrections within mean-field theory for weakly segregated block copolymers. The theory of Marques and Cates [C. M. Marques and M. E. Cates, Europhys. Lett. 13, 267 (1990)] for the stability of phases near the ordering transition is applied to diblock copolymer melts. A wave-vector dependence of the coefficients in the Landau mean-field free energy is included. We find that the face centered cubic (fcc) and square phases predicted by Marques and Cates to be stable near the ordering transition depending on the structure factor for the system are not stable for diblock copolymers. In addition, hexagonal close packed (hcp) and ordered bicontinuous double diamond (OBDD) structures are found to be unstable with respect to the classical lamellar, cylinder, and body centered cubic phases. The hexagonal intermediate phases and complex bicontinuous structures recently observed near the order–disorder transition are not accounted for in this theory.

Polymerization-induced Phase Separation of a Liquid Crystal-Polymer Mixture

Abstract We study the phase separation, induced by polymerization, of a liquid crystal-polymer mixture by using the Flory-Huggins mean-field free energy and a simple kinetic model of polymerization. Equilibrium phase behavior and the kinetic processes are analyzed separately. These results are then combined under the assumption that the polymerization rate is sufficiently slow, or the diffusive processes sufficiently rapid, that phase separation occurs at a well defined degree of polymerization. This allows the calculation of the critical time at which phase separation begins.