Nucleation on a sphere: the roles of curvature,confinement and ensemble

ABSTRACT

By combining Monte Carlo simulations and analytical models, we demonstrate and explain how the gas-to-liquid phase transition of colloidal systems confined to a spherical surface depends on the curvature and size of the surface, and on the choice of thermodynamic ensemble. We find that the geometry of the surface affects the shape of the free energy profile and the size of the critical nucleus by altering the perimeter–area ratio of isotropic clusters. Confinement to a smaller spherical surface results in both a lower nucleation barrier and a smaller critical nucleus size. Furthermore, the liquid domain does not grow indefinitely on a sphere. Saturation of the liquid density in the grand canonical ensemble and the depletion of the gas phase in the canonical ensemble lead to a minimum in the free energy profile, with a sharp increase in free energy for additional growth beyond this minimum.     

Mean Field Magnetic Phase Diagrams for the Two Dimensional <Emphasis Type="Italic">t</Emphasis> — <Emphasis Type="Italic">t</Emphasis>′ — <Emphasis Type="Italic">U</Emphasis> Hubbard Model

We study the ground state phase diagram of the two dimensional t — t′ — U Hubbard model concentrating on the competition between antiferro-, ferro-, and paramagnetism. It is known that unrestricted Hartree–Fock- and quantum Monte Carlo calculations for this model predict inhomogeneous states in large regions of the parameter space. Standard mean field theory, i.e., Hartree–Fock theory restricted to homogeneous states, fails to produce such inhomogeneous phases. We show that a generalization of the mean field method to the grand canonical ensemble circumvents this problem and predicts inhomogeneous states, represented by mixtures of homogeneous states, in large regions of the parameter space. We present phase diagrams which differ considerably from previous mean field results but are consistent with, and extend, results obtained with more sophisticated methods. PACS: 71.10.Fd, 05.70.Fh, 75.50.Ee     

Charge orderings in the atomic limit of the extended Hubbard model

The extended Hubbard model in the atomic limit (AL-EHM) on a square lattice with periodic boundary conditions is studied with use of the Monte Carlo (MC) method. Within the grand canonical ensemble the phase and order-order boundaries for charge orderings are obtained. The phase diagrams include three types of charge ordered phases and the nonordered phase. The system exhibits very rich structure and shows unusual multicritical behavior. In the limiting case of t_ij=0, the EHM is equivalent to the pseudospin model with single-ion anisotropy , exchange interaction W in an effective magnetic field . This classical spin model is analyzed using the MC method for the canonical ensemble. The phase diagram is compared with the known results for the Blume-Capel model.     

Phase transitions in small systems: Microcanonical vs. canonical ensembles

We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (non-analyticities) separating different microscopic phases. These microscopic phases may be identified as different microphysical dissociation states of the small system. The microscopic oscillations of microcanonical thermodynamic quantities (e.g., temperature, heat capacity, or pressure) should in principle be observable in suitably designed evaporation/dissociation experiments (which must realize the physical preconditions of the microcanonical ensemble). By contrast, singular phase transitions cannot occur, if a small system is embedded into an infinite heat bath (thermostat), corresponding to the canonical ensemble. For the simple model systems under consideration, it is nevertheless possible to identify a smooth canonical phase transition by studying the distribution of complex zeros of the canonical partition function.     

Canonical Analysis of Condensation in Factorised Steady States

We study the phenomenon of real space condensation in the steady state of a class of mass transport models where the steady state factorises. The grand canonical ensemble may be used to derive the criterion for the occurrence of a condensation transition but does not shed light on the nature of the condensate. Here, within the canonical ensemble, we analyse the condensation transition and the structure of the condensate, determining the precise shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as L ^1/2 where L is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. Our results are asymptotically exact and can also be interpreted within the framework of sums of random variables. We further analyse two additional cases: one where the condensation transition is somewhat different from the usual second order phase transition and one where there is no true condensation transition but instead a pseudocondensate appears at superextensive densities. PACS numbers: 05.40.-a, 02.50.Ey, 64.60.-i.     

Classification of Phase Transitions and Ensemble Inequivalence, in Systems with Long Range Interactions

Systems with long range interactions in general are not additive, which can lead to an inequivalence of the microcanonical and canonical ensembles. The microcanonical ensemble may show richer behavior than the canonical one, including negative specific heats and other non-common behaviors. We propose a classification of microcanonical phase transitions, of their link to canonical ones, and of the possible situations of ensemble inequivalence. We discuss previously observed phase transitions and inequivalence in self-gravitating, two-dimensional fluid dynamics and non-neutral plasmas. We note a number of generic situations that have not yet been observed in such systems.     

Effect of anharmonicity of the crystal potential on ferroelectric-antiferroelectric phase transitions

The ferroelectric-antiferroelectric phase transitions in lead zirconate-titanate-based solid solutions have been considered with allowance made for anharmonicity of the crystal potential. In the phase diagram of lead zirconate-titanate, the boundary separating the regions of the ferroelectric and antiferroelectric states are shifted toward higher titanium concentrations. The calculated and experimental phase diagrams are presented for such cases.     

Boron nitride phase diagram. State of the art

Abstract

From recent experimental data on BN thermodynamic propeties, the equilibrium phase diagram for boron nitride has been plotted, which differs from the generally accepted Bundy-Wentorfs one. At atmospheric pressure cubic boron niride has been shown to be a thermodynamically stable modification up to temperatures of 1600 K, which drastically changes the established notions of BN polymorphism, based on assumed analogy of phase diagrams for carbon and boron nitride. These studies have shown that according to the proposed equilibrium phase diagram the threshold pressure of cBN crystallization can be reduced from 4 down to 2 GPa with the supercritical fluids present, which opens new fields for developing methods for cBN low-pressure synthesis.     

A comparison of the efficiency of Monte Carlo (MC) and molecular dynamics (MD) calculations at first order phase transitions

As a rule the canonical ensemble is used in the statistical analysis of phase transitions. There the system of interest is coupled to an infinite bath. It will be demonstrated that it is of considerable advantage to use a different ensemble where the size of the bath is comparable to the size of the system of interest. The advantages will be demonstrated for the example of a first order phase transition by making use of the density of states of the modifiedxy-model which has recently been determined in an MD-calculation for several lattice sizes. In contrast to the canonical ensemble the probability distribution does not suffer from the double hump structure and therefore allows an accurate determination of the properties at the transition. In the end we shall argue that the corresponding ensemble can also be realized in an MC-approach.     

Phase behaviour and interfacial properties of ternary system CO2 + n-butane + n-decane: coarse-grained theoretical modelling and molecular simulations

ABSTRACT

This work illustrates the application of a three-party approach based on theoretical modelling, molecular dynamic (MD) simulations and available experimental data for describing the phase equilibrium and interfacial properties for the ternary system: carbon dioxide + n-butane + n-decane and its corresponding binary sub-systems at 344.3 K. Specifically, a coarse-grained force field is employed for both theoretical predictions and MD. The interfacial region is described by the square gradient theory where the homogenous Helmholtz energy density contribution is provided by the Statistical Associated Fluid Theory equation of state for potentials of variable range for molecules conformed of segments interacting through the Mie potential (SAFT-VR Mie) and MD simulations in the canonical ensemble where the molecules are represented by a coarse-grained Mie force field. The novelty here is that both the theory and the simulations uniquely share the same underlying intermolecular potentials; hence, the experimental data are employed to verify both the theory and simulations. In this schema, the ternary mixture is full predictive as its parameters are only based on pure fluids parameters and binary interactions. It is observed that the phase equilibria and the interfacial properties are equally well represented by the used approach.     

一维强相互作用极化费米子的相图

文章首先简要评述了目前强相互作用的极化冷费米原子体系的研究现状.在三维,人们对该体系基态存在着不同认识.为对这个问题有进一步了解,文章探讨了一维强相互作用极化费米气体.在均匀情况下,这是一个可积系统,可以得到该体系的一个严格相图.作者发现了一种非均匀的超流相在相空间占主导地位.在有外加束缚势的实验情况下,通过局域密度泛函近似,作者发现了两种新颖的相分离相.     

Ground state phase diagrams for the mixed Ising 3/2 and 5/2 spin model

We calculate the ground state phase diagrams of a mixed Ising model on a square lattice where spins S (± 3/2, ± 1/2) in one sublattice are in alternating sites with spins Q (± 5/2, ± 3/2, ± 1/2), located on the other sublattice. The Hamiltonian of the model includes first neighbor interactions between the S and Q spins, next-nearest-neighbor interactions between the S spins, and between the Q spins, and crystal field. The topologies of the phase diagrams depend on the values of the parameters in the Hamiltonian. The diagrams show some key features: coexistence between regions, points where two, three, four, five and six states can coexist. Besides being very useful as a way to check the low temperature limit of the finite-temperature phase diagram, often obtained by mean-field theories, the richness of the ground state diagrams for certain combinations of parameters can be used as a guide to explore interesting regions of the finite-temperature phase diagram of the model.     

Chaotic behavior in nonlinear Hamiltonian systems and equilibrium statistical mechanics

The relation between chaotic dynamics of nonlinear Hamiltonian systems and equilibrium statistical mechanics in its canonical ensemble formulation has been investigated for two different nonlinear Hamiltonian systems. We have compared time averages obtained by means of numerical simulations of molecular dynamics type with analytically computed ensemble averages. The numerical simulation of the dynamic counterpart of the canonical ensemble is obtained by considering the behavior of a small part of a given system, described by a microcanonical ensemble, in order to have fluctuations of the energy of the subsystem. The results for the Fermi-Pasta-Ulam model (i.e., a one-dimensional anharmonic solid) show a substantial agreement between time and ensemble averages independent of the degree of stochasticity of the dynamics. On the other hand, a very different behavior is observed for a chain of weakly coupled rotators, where linear exchange effects are absent. In the high-temperature limit (weak coupling) we have a strong disagreement between time and ensemble averages for the specific heat even if the dynamics is chaotic. This behavior is related to the presence of spatially localized chaos, which prevents the complete filling of the accessible phase space of the system. Localized chaos is detected by the distribution of all the characteristic Liapunov exponents.     

Phase transitions in four-dimensional AdS black holes with a nonlinear electrodynamics source

In this work we consider black hole solutions to Einstein's theory coupled to a nonlinear power-law electromagnetic field with a fixed exponent value. We study the extended phase space thermodynamics in canonical and grand canonical ensembles, where the varying cosmological constant plays the role of an effective thermodynamic pressure. We examine thermodynamical phase transitions in such black holes and find that both first- and second-order phase transitions can occur in the canonical ensemble while, for the grand canonical ensemble, Hawking–Page and second-order phase transitions are allowed.     

General diagram of the phase remtions of actinide metals under pressure

Abstract

The proposed diagram is based on the results of high-pressure X-ray diffraction work and visualizes the general trends in phase relations of actinide metals under pressure. The transition from dhcp to ccp, which occurs under the action of pressure in heavy actinides and light lanthanides, also occurs with decreasing atomic number at ambient pressure. The same structural sequence is found in the lanthanide and the actinide metals, but its first two members, hcp and Sm-type, are missing in the actinide metals, and the phase transitions are shifted to lower pressures and higher atomic numbers in the actinides.     

Phase diagram of Bose-Fermi mixtures in one-dimensional optical lattices

The ground state phase diagram of the one-dimensional Bose-Fermi Hubbard model is studied in the canonical ensemble using a quantum Monte Carlo method. We focus on the case where both species have half filling in order to maximize the pairing correlations between the bosons and the fermions. In case of equal hopping we distinguish among phase separation, a Luttinger liquid phase, and a phase characterized by strong singlet pairing between the species. True long-range density waves exist with unequal hopping amplitudes.     

Mesoscopic systems with fixed number of electrons

In this paper, we study the physics of mesoscopic systems with noninteracting electrons of fixed number. From a technical point of view, this means a discussion of the differences between the canonical and the grand canonical ensemble (fixed versus fluctuating number of particles). Such a discussion is not trivial since the grand canonical ensemble is the most convenient basis for the statistics of identical particles and one has to spend labour in order to retrieve the canonical ensemble. Specifically, we are considering ensembles of mesoscopic systems with disorder, either by atomic defects or by fluctuations in their geometric definitions and we discuss various forms of disorder averages.     

Stochastic-dynamical thermostats for constraints and stiff restraints

A broad array of canonical sampling methods are available for molecular simulation based on stochastic-dynamical perturbation of Newtonian dynamics, including Langevin dynamics, Stochastic Velo- city Rescaling, and methods that combine Nosé-Hoover dynamics with stochastic perturbation. In this article we discuss several stochastic-dynamical thermostats in the setting of simulating systems with holonomic constraints. The approaches described are easily implemented and facilitate the recovery of correct canonical averages with minimal disturbance of the underlying dynamics. For the purpose of illustrating our results, we examine the numerical application of these methods to a simple atomic chain, where a Fixman term is required to correct the thermodynamic ensemble.     

A new analytic solution for the Zwanzig-Lauritzen model of polymer chain folding

A two-dimensional model of polymer chain folding invented by Zwanzig and Lauritzen is here studied using a grand ensemble and transfer matrix method. Due to the character of the model, there are no extensive parameters in the grand ensemble and the dispersion in system size is large, raising doubts about the validity and usefulness of the ensemble. We find it possible to define a thermodynamic limit such that it leads to near equivalence between the canonical and grand ensembles in the limit of large systems. The transfer matrix in this case is a nonlocal operator on a space of L₂ functions, and the eigenvalue equation is a homogeneous Fredholm integral equation of the second kind which can be completely solved in terms of Bessel functions. The grand partition function can then be expressed as a sum of powers of the known eigenvalues. It is an easy matter to reproduce the second-order phase transition in the canonical ensemble found in the original work on the model. The investigation is extended to yield the probability densities describing the length of a segment and the correlations among segments. The concept of a local width of the folded chain is found to break down at higher temperatures, while critical correlations are characterized by infinite range, as expected. Apart from physical and methodological implications, the new solution provides striking illustrations of some basic ideas concerning phase transitions.