Modeling Relaxation Processes for Fluids in Porous Materials Using Dynamic Mean Field Theory: An Application to Partial Wetting

We review a recently developed dynamic mean field theory for fluids confined in porous materials and apply it to a case where the solid-fluid interactions lead to partial wetting on a planar surface. The theory describes the evolution of the density distribution for a fluid in a pore that has contact with the bulk during a quench in the bulk chemical potential. In this way the dynamics of adsorption and desorption can be studied. By focusing on partial wetting situation we can investigate influence of a weaker surface field on the mechanisms of capillary condensation and desorption. We have studied the dynamics of pore filling in a quench of the chemical potential between two states either side of the pore filling step, tracking the density distributions during the process. The pore filling process features an asymmetric density distribution where a liquid droplet appears on one of the walls. The droplet spreads and grows in size and this is followed by the appearance of a liquid bridge between the pore walls (for longer pores two liquid bridges are seen). The density distributions obtained in the dynamics resemble those obtained from static mean field theory in the canonical ensemble for an infinite pore without contact with the bulk.     

Particle number fluctuations in an ideal Bose gas

Abstract

We analyse occupation number fluctuations of an ideal Bose gas in a trap which is isolated from the environment with respect to particle exchange (canonical ensemble). We show that in contrast to the predictions of the grandcanonical ensemble, the counting statistics of particles in the trap ground state changes from monotonously decreasing above the condensation temperature to single-peaked below that temperature. For the exactly solvable case of a harmonic oscillator trapping potential in one spatial dimension we extract a Landau–Ginzburg functional which–despite the non-interacting nature of the system–displays the characteristic behaviour of a weakly interacting Bose gas. We also compare our findings with the usual treatment which is based on the grand-canonical ensemble. We show that for an ideal Bose gas neither the grand-canonical and canonical ensemble thermodynamically equivalent, nor the grand-canonical ensemble can be viewed as a small system in diffusive contact with a particle reservoir.     

Lattice Boltzmann simulations of contact line motion in a liquid-gas system

We use a lattice Boltzmann algorithm for liquid-gas coexistence to investigate the steady-state interface profile of a droplet held between two shearing walls. The algorithm solves the hydrodynamic equations of motion for the system. Partial wetting at the walls is implemented to agree with Cahn theory. This allows us to investigate the processes which lead to the motion of the three-phase contact line. We confirm that the profiles are a function of the capillary number and a finite-size analysis shows the emergence of a dynamic contact angle, which can be defined in a region where the interfacial curvature tends to zero.     

Solid3He magnetism in the classical approximation

The classical analog of the Hamiltonian proposed for³He spins by Stipdonk and Hetherington is studied. First- and second-order phase transitions similar to those found experimentally are obtained. This paper introduces the use of a Gaussian ensemble rather than a canonical ensemble in the Monte Carlo method. This ensemble is equivalent to the canonical ensemble in the limit of large systems except in the energy range inside of a first-order transition. This new method has general applicability, but is especially useful for studying first-order transitions, since it can eliminate the problem of hysteresis associated with them.     

Thermodynamic Anomalies of Small Quantum Systems Within a New Approach to Statistical Physics

Based on a new statistical theory, we investigate the thermodynamic anomalies of small quantum systems, such as the negative specific heat (NSH) and negative entropy (NE) within the generalized canonical ensemble. We consider the system–bath heat exchange and “uncompensated heat” in the thermodynamical level which is independent on the details of the system–bath coupling. For ideal finite systems, we calculate two thermodynamic quantities, i.e., the experimental specific heat and the entropy. The results show that the NSH and NE exist in quantum thermodynamics, particularly at low temperatures for small systems. They agree with the results of the reduced partition function theory and reveal that the finite boundary effects of the uncompensated heat and heat exchange of small quantum systems dominate the nonequilibrium irreversible processes.     

Dimensionality crossover of the4He superfluid transition in a slab geometry

The superfluid density of liquid⁴He is computed from vortex renormalization theories for the case of a slab geometry with a slab height L. With increasing temperature there is a crossover from three dimensions to two as the size of the largest vortex rings approaches L and they intersect with the walls, forming vortex pairs. The superfluid density becomes anisotropic, with the component parallel to the slab undergoing the universal Kosterlitz-Thouless jump, while the perpendicular component remains finite. The results are in agreement with both finite-size scaling and Monte-Carlo simulations.     

Engineering Dirac electrons emergent on the surface of a topological insulator

Abstract

The concept of the topological insulator (TI) has introduced a new point of view to condensed-matter physics, relating a priori unrelated subfields such as quantum (spin, anomalous) Hall effects, spin–orbit coupled materials, some classes of nodal superconductors, superfluid ³He, etc. From a technological point of view, TIs are expected to serve as platforms for realizing dissipationless transport in a non-superconducting context. The TI exhibits a gapless surface state with a characteristic conic dispersion (a surface Dirac cone). Here, we review peculiar finite-size effects applicable to such surface states in TI nanostructures. We highlight the specific electronic properties of TI nanowires and nanoparticles, and in this context we contrast the cases of weak and strong TIs. We study the robustness of the surface and the bulk of TIs against disorder, addressing the physics of Dirac and Weyl semimetals as a new research perspective in the field.     

Mixed convection under conditions of weak external flow in a vertical channel with a finite-size heat source

Mixed convection is investigated in a vertical plane-parallel channel from a finite-size heat source located on one of the walls. Numerical calculations are performed for the Prandtl number value of 0.7, in the ranges of values of Grashof number from zero to 10⁵ and Reynolds number from zero to 10. Two-dimensional unsteady-state Navier-Stokes equations in the Boussinesq approximation are used as model. The model is formulated in the “speed eddy-stream function-temperature” variables and solved by the finite element method.     

Determination of the phase diagram from interatomic potentials: The iron–chromium case

Prior to applying any interatomic potential, it is important to know the stability of the different phases it describes. In the literature many methods to determine the phase diagram from an interatomic potential are described. Although for pure elements the procedure to obtain the thermodynamic functions is well established, for alloys it is not. In this work a method is developed to determine the phase diagram, i.e., solubility limits and spinodal gap, for the case of miscibility gaps. The method combines Monte Carlo simulations in the isobaric semi-grand canonical ensemble, full thermodynamic integration and Redlich–Kister expansions to parameterize the Gibbs free energy. Besides numerical inaccuracies, this method does not rely on any physical approximations to determine the phase diagram of a given interatomic potential. The method is applied to two different Fe–Cr potentials that are widely used in the literature. The resulting phase diagrams are discussed by comparing them to the experimental one and ones obtained in other works from the same potentials.     

Defect Formation in Superconducting Rings: External Fields and Finite-Size Effects

Consistent with the predictions of Kibble and Zurek, scaling behaviour has been seen in the production of fluxoids during temperature quenches of superconducting rings. However, deviations from the canonical behaviour arise because of finite-size effects and stray external fields. Technical developments, including laser heating and the use of long Josephson tunnel junctions, have improved the quality of data that can be obtained. With new experiments in mind we perform large-scale 3D simulations of quenches of small, thin rings of various geometries with fully dynamical electromagnetic fields, at nonzero externally applied magnetic flux. We find that the outcomes are, in practise, indistinguishable from those of much simpler Gaussian analytical approximations in which the rings are treated as one-dimensional systems and the magnetic field fluctuation-free.     

Molecular Dynamics Calculation of the Viscosities of Biaxial Nematic Liquid Crystals

We have evaluated the Green–Kubo relations for the viscosities of a biaxial nematic liquid crystal by performing equilibrium molecular dynamics simulations. The viscosity varies by more than two orders of magnitude depending on the orientation of the directors relative to the streamlines. The molecules consist of nine fused Gay–Berne oblates whose axes of revolution are parallel to each other and perpendicular to the line joining their centers of mass. This gives a biaxial body, the length-to-width-to-breadth ratio of which is equal to 5:1:0.4. The numerical evaluation of the Green–Kubo relations for the viscosities is facilitated by the application of a Gaussian director constraint algorithm that makes it possible to fix the directors in space. This does not only generate an inertial director-based frame but also a new equilibrium ensemble. In this ensemble the Green–Kubo relations for the viscosities are simple linear combinations of time correlation function integrals, whereas they are complicated rational functions in the conventional canonical ensemble.     

Finite-size effects in surface tension: Thermodynamics and the Gaussian interface model

It has been suggested by Kayser that finite-size effects associated with capillary waves might play a significant role in some surface tension measurements; for capillary rise between plates a distance D apart, an effect varying as 1/D and apparently observable in measurements, was proposed. In reconsidering this problem, one must analyze the thermodynamics of finite-size corrections to surface tension. In particular, one sees that capillary rise between plates does not measure the interfacial free energy density but, rather, a derivative of the interfacial free energy with respect to a system dimension. The quantity needed to draw definite conclusions, the finite-size residual free energy, can be calculated within the harmonic or Gaussian capillary wave model in d spatial dimensions with the aid of Poisson summation techniques and should yield the correct leading asymptotic behavior. For d=3 and experimentally relevant parameter values, the results are independent of the short-wavelength cutoff needed in the model and can be checked against the theory of conformai covariance at two-dimensional critical points. It is found that the finite-size effects in capillary-rise measurements of surface tension vary as 1/D ² (with a universal coefficient) but are too small to be seen in current experiments.Invited paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.     

Finite element simulation of combined buoyancy and thermocapillary driven convection in open cavities

Thermocapillary-induced and buoyancy-driven convective flows that commonly occur in crystal growth are numerically simulated using Galerkin finite element method. The physical domain comprises of a open cavity with aspect ratio one and differentially heated vertical walls. The top gas–melt interface is free to deform subject to 90° contact angle boundary conditions at the two vertical walls. The unsteady two-dimensional Navier–Stokes equations are discretized in time using Chorin-type splitting scheme and pressure is determined from the Poisson's equation. The free surface is taken to be resting on vertical spines and its evolution in time is determined from the kinematic free surface equation. The governing equations for heat and momentum are solved in the Arbitrary Lagrangian Eulerian frame of reference to handle the moving boundary. The influence of Grashof number, Marangoni number, Bond number, Ohnesorge number and Prandtl number on the flow field and heat transfer is investigated.     

Wedge disclinations in the shell of a core–shell nanowire within the surface/interface elasticity

The elastic behaviors of a two-axes dipole of wedge disclinations and an individual wedge disclination located inside the shell of a free standing core–shell nanowire is studied within the surface/interface elasticity theory. The corresponding boundary value problem is solved using complex potential functions, defined through modeling the disclination dipole by two finite walls of infinitesimal edge dislocations. The stress field, disclination strain energies and image forces acting on the disclinations, are calculated and studied in detail. It is shown that the stresses are rather inhomogeneous across the nanowire cross section, change their signs and reach local maxima and minima far from the disclination lines in the bulk or on the surface of the nanowire. For negative values of the surface/interface modulus and relatively small values of the ratio of the shell and core shear moduli, the surface/interface effect manifests itself through non-classical stress oscillations along the shell free surface in the case of a disclination dipole and core–shell interface in both the cases of a disclination dipole and an individual disclination. The non-classical solution for the strain energy deviates from the classical solution with different effects caused by the surface/interface moduli on the wedge disclination dipole and an individual disclination. When the core is softer than the shell, the dipole with radial orientation of its arm has an unstable equilibrium position in the shell. In general, if the surface/interface modulus is positive, the surface/interface effects are rather weak; however, if it is negative, the effect can be very strong, especially near the shell surface.     

Fermi–Bose Correspondence and Bose–Einstein Condensation in the Two-Dimensional Ideal Gas

The ideal uniform two-dimensional (2D) Fermi and Bose gases are considered both in the thermodynamic limit and the finite case. We derive May's Theorem, viz. the correspondence between the internal energies of the Fermi and Bose gases in the thermodynamic limit. This results in both gases having the same heat capacity. However, as we shall show, the thermodynamic limit is never truly reached in two dimensions and so it is essential to consider finite-size effects. We show in an elementary manner that for the finite 2D Bose gas, a pseudo-Bose–Einstein condensate forms at low temperatures, incompatible with May's Theorem. The two gases now have different heat capacities, dependent on the system size and tending to the same expression in the thermodynamic limit.     

Sequential Metropolis Algorithms for Fluid Simulations

In this work, the implementation of our recently proposed sequential Metropolis algorithm in the grand canonical ensemble, a case particularly relevant for continuum fluids, is considered. By performing Monte Carlo simulations for the two-dimensional lattice gas, it is shown that our algorithm converges faster than all known grand canonical algorithms that satisfy strict detailed balance. The main advantages of the new algorithm are its simplicity, generality, and the possibility of parallel implementation.     

Generalized hydrodynamics in the transient regime and irreversible thermodynamics

In this article the thermodynamically consistent formulation of generalized hydrodynamics is reviewed and applications to shock-wave structures, ultrasonic wave absorption and dispersion and microchannel flows of the generalized hydrodynamics so formulated are discussed. The kinematic terms of the constitutive equations in the generalized hydrodynamic equations for liquids, which have been calculated by means of non-equilibrium grand canonical ensemble, are also presented.     

Kosterlitz-Thouless Transition in Finite-Size BEC Systems

The Kosterlitz-Thouless (KT) transition in two-dimensional BEC systems is calculated taking into account the fact that in experiments these are finite-size systems. The outer boundaries of the condensate effectively act as hard walls, and this has a polarizing effect on the vortex pairs of the KT transition, causing the superfluid fraction to become strongly anisotropic. The decreased pair energy near the walls results in a strongly enhanced vortex density near the boundaries. Since the pair density can now be directly measured, we extend here our previous calculations to include the vortex density as a function of the distance from the boundaries. Possible experiments using sound propagation in the gases are proposed to probe the anisotropic properties of the superfluid density, including an unusual sharp dip in the superfluid density that is predicted to occur down the middle of a long superfluid strip.      

Molecular simulation of phase coexistence: Finite-size effects and determination of critical parameters for two- and three-dimensional Lennard-Jones fluids

Abstact The subject of this paper is the investigation of finite-size effects and the determination of critical parameters for a class of truncated Lennard-Jones potentials. Despite significant recent progress in our ability to model phase equilibria in multicomponent mixtures from direct molecular simulations, the accurate determination of critical parameters remains a difficult problem. Gibbs ensemble Monte Carlo simulations with systems of controlled linear system size are used to obtain the phase behavior in the near-critical region for two- and three dimensional Lennard-Jones fluids with reduced cutoff radii of 3, 3.5, and 5. For the two-dimensional systems, crossover of the effective exponent for the width of the coexistence curve from mean field ( = 1/2 in the immediate vicinity of the critical point to Ising-like (= 1/8) farther away is observed. Critical parameters determined by fitting the data that follow Ising-like behavior are in good agreement with literature values obtained with finite-size scaling methods. For the three-dimensional systems, no crossover to mean field-type behavior was apparent. Extrapolated results for the critical parameters are consistent with literature estimates for similar fluids. For both two- and three-dimensional fluids, system size effects on the coexistence curves away from the critical point are small, normally within simulation statistical uncertainties.Invited paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994, Boulder. Colorado, U.S.A.