Vapor-to-droplet transition in a Lennard-Jones fluid: simulation study of nucleation barriers using the ghost field method

Machine-Learned Free Energy Surfaces for Capillary Condensation and Evaporation in Mesopores

Using molecular simulations, we study the processes of capillary condensation and capillary evaporation in model mesopores. To determine the phase transition pathway, as well as the corresponding free energy profile, we carry out enhanced sampling molecular simulations using entropy as a reaction coordinate to map the onset of order during the condensation process and of disorder during the evaporation process. The structural analysis shows the role played by intermediate states, characterized by the onset of capillary liquid bridges and bubbles. We also analyze the dependence of the free energy barrier on the pore width. Furthermore, we propose a method to build a machine learning model for the prediction of the free energy surfaces underlying capillary phase transition processes in mesopores.

Nucleation work, surface tension, and Gibbs-Tolman length for nucleus of any size

In the framework of the Gibbs approach to nucleation thermodynamics, expressions are derived for the nucleation work, nucleus size, surface tension, and Gibbs-Tolman length in homogeneous single-component nucleation at a fixed temperature. These expressions are in terms of the experimentally controlled overpressure of the nucleating phase and are valid for the entire overpressure range, i.e., for nucleus of any size. Analysis of available data for bubble and droplet nucleation in Lennard-Jones fluid shows that the theory describes well the data by means of a single free parameter, the Gibbs-Tolman length of the planar liquid/vapor interface. It is found that this length is about one-tenth of the Lennard-Jones molecular-diameter parameter and that it is positive for the bubble nucleus and negative for the droplet nucleus. In a sufficiently narrow temperature range, the nucleation work, nucleus radius, scaled surface tension, and Gibbs-Tolman length are apparently universal functions of scaled overpressure.

Evaluation of finite-size effects in cavitation and droplet formation

Nucleation of bubbles and droplets is of fundamental interest in science and technology and has been widely investigated through experiments, theory, and simulations. Giving the rare event nature of these phenomena, nucleation simulations are computationally costly and require the use of a limited number of particles. Moreover, they are often performed in the canonical ensemble, i.e., by fixing the total volume and number of particles, to avoid the additional complexities of implementing a barostat. However, cavitation and droplet formation take place differently depending on the ensemble. Here, we analyze the importance of finite-size effects in cavitation and droplet formation. We present simple formulas which predict the finite-size corrections to the critical size, the nucleation barrier, and the nucleation rates in the canonical ensemble very accurately. These results can be used to select an appropriate system-size for simulations and to get a more precise evaluation of nucleation in complex substances, by using a small number of molecules and correcting for finite-size effects.

Communication: superstabilization of fluids in nanocontainers

One of the main challenges of thermodynamics is to predict and measure accurately the properties of metastable fluids. Investigation of these fluids is hindered by their spontaneous transformation by nucleation into a more stable phase. We show how small closed containers can be used to completely prevent nucleation, achieving infinitely long-lived metastable states. Using a general thermodynamic framework, we derive simple formulas to predict accurately the conditions (container sizes) at which this superstabilization takes place and it becomes impossible to form a new stable phase. This phenomenon opens the door to control nucleation of deeply metastable fluids at experimentally feasible conditions, having important implications in a wide variety of fields.

Influence of system size on the properties of a fluid adsorbed in a nanopore: Physical manifestations and methodological consequences

Hysteresis and discontinuities in the isotherms of a fluid adsorbed in a nanopore in general hamper the determination of equilibrium thermodynamic properties, even in computer simulations. A way around this has been to consider both a reservoir of small size and a pore of small extent in order to restrict the fluctuations of density and approach a classical van der Waals loop. We assess this suggestion by thoroughly studying through Monte Carlo simulations and density functional theory the influence of system size on the equilibrium configurations of the adsorbed fluid and on the resulting isotherms. We stress the importance of pore-symmetry-breaking states that even for modest pore sizes lead to discontinuous isotherms and we discuss the physical relevance of these states and the methodological consequences for computing thermodynamic quantities.

Translocation dynamics of freely jointed Lennard-Jones chains into adsorbing pores

Polymer translocation into adsorbing nanopores is studied by using the Fokker-Planck equation of chain diffusion along the energy landscape calculated with Monte Carlo simulations using the incremental gauge cell method. The free energy profile of a translocating chain was found by combining two independent sub-chains, one free but tethered to a hard wall, and the other tethered inside an adsorbing pore. Translocation dynamics were revealed by application of the Fokker-Planck equation for normal diffusion. Adsorption of polymer chains into nanopores involves a competition of attractive adsorption and repulsive steric hindrance contributions to the free energy. Translocation times fell into two regimes depending on the strength of the adsorbing pore. In addition, we found a non-monotonic dependence of translocation times with increasing adsorption strength, with sharp peak associated with local free energy minima along the translocation coordinate.

Monte Carlo simulation of cavitation in pores with nonwetting defects

We investigate the onset of cavitation in a metastable fluid confined to nanoscale pores with nonwetting defects present. Using grand canonical and gauge cell mesocanonical Monte Carlo simulations, we study the degree of metastability (relative vapor pressure), at which the critical bubble forms in a spherical pore with a circular nonwetting defect. It is shown that an increase of the defect size leads to a transition from homogeneous to heterogeneous nucleation of critical bubbles formed at the defect site. In this case, the desorption process may be initiated at larger relative vapor pressures than those predicted by the theories of homogeneous cavitation.

Nucleation of fluids confined between parallel walls: a lattice Monte Carlo study

Nucleation phenomena in lattice gas models of simple and chain molecules confined in slitlike pores are studied using Monte Carlo methods. Finite-size scaling is used to investigate the nature of phase transitions accompanying the formation of layers at the pore walls. It is demonstrated that nucleation leads to the symmetry breaking and the formation of nuclei at one wall only.

Nucleation versus spinodal decomposition in confined binary solutions

Basic features of spinodal decomposition, on one side, and nucleation, on the other side, and the transition between both mechanisms are analyzed within the framework of a generalized thermodynamic cluster model based on the generalized Gibbs approach. Hereby the clusters, representing the density or composition variations in the system, may change with time both in size and in their intensive state parameters (density and composition, for example). In the first part of the analysis, we consider phase separation processes in dependence on the initial state of the system for the case when changes of the state parameters of the ambient system due to the evolution of the clusters can be neglected as this is the case for cluster formation in an infinite system. As a next step, the effect of changes of the state parameters on cluster evolution is analyzed. Such depletion effects are of importance both for the analysis of phase formation in confined systems and for the understanding of the evolution of ensembles of clusters in large (in the limit infinite) systems. The results of the thermodynamic analysis are employed in both cases to exhibit the effect of thermodynamic constraints on the dynamics of phase separation processes.

Nucleation and cavitation of spherical, cylindrical, and slablike droplets and bubbles in small systems

Computer simulations are employed to obtain subcritical isotherms of small finite sized systems inside the coexistence region. For all temperatures considered, ranging from the triple point up to the critical point, the isotherms gradually developed a sequence of sharp discontinuities as the system size increased from approximately 8 to approximately 21 molecular diameters. For the smallest system sizes, and more so close to the critical point, the isotherms appeared smooth, resembling the continuous van der Waals loop obtained from extrapolation of an analytic equation of state outside the coexistence region. As the system size was increased, isotherms in the chemical potential-density plane developed first two, then four, and finally six discontinuities. Visual inspection of selected snapshots revealed that the observed discontinuities are related to structural transitions between droplets (on the vapor side) and bubbles (on the liquid side) of spherical, cylindrical, and tetragonal shapes. A capillary drop model was developed to qualitatively rationalize these observations. Analytic results were obtained and found to be in full agreement with the computer simulation results. The analysis shows that the shape of the subcritical isotherms is dictated by a single characteristic volume (or length scale), which depends on the surface tension, compressibility, and coexistence densities. For small reduced system volumes, the model predicts that a homogeneous fluid is stable across the whole coexistence region, thus explaining the continuous van der Waals isotherms observed in the simulations. When the liquid and vapor free energies are described by means of an accurate mean-field equation of state and surface tensions from simulation are employed, the capillary model is found to describe the simulated isotherms accurately, especially for large systems (i.e., larger than about 15 molecular diameters) at low temperature (lower than about 0.85 times the critical temperature). This implies that the Laplace pressure differences can be predicted for drops as small as five molecular diameters, and as few as about 500 molecules. The theoretical study also shows that the extrema or apparent spinodal points of the finite size loops are more closely related to (finite system size) bubble and dew points than to classical spinodals. Our results are of relevance to phase transitions in nanopores and show that first order corrections to nucleation energies in finite closed systems are power laws of the inverse volume.

Forms and applications of the nucleation theorem

The nucleation theorem is a general relation between the nucleation work, the nucleus size, and the supersaturation or other thermodynamic parameters of the old phase. The theorem appears in different forms, depending not only on the chosen set of independent variables describing the nucleation work but also on which of these variables is changed while the others are held fixed. This paper gives a rigorous, systematic, and comprehensive presentation of various forms of the nucleation theorem and shows how some of them can be applied to concrete cases of nucleation. Both theoretical and experimental applications of the theorem to nucleation in unary, binary, or ternary phases are considered.

A simulation method for the calculation of chemical potentials in small, inhomogeneous, and dense systems

We present a modification of the gauge cell Monte Carlo simulation method [A. V. Neimark and A. Vishnyakov, Phys. Rev. E 62, 4611 (2000)] designed for chemical potential calculations in small confined inhomogeneous systems. To measure the chemical potential, the system under study is set in chemical equilibrium with the gauge cell, which represents a finite volume reservoir of ideal particles. The system and the gauge cell are immersed into the thermal bath of a given temperature. The size of the gauge cell controls the level of density fluctuations in the system. The chemical potential is rigorously calculated from the equilibrium distribution of particles between the system cell and the gauge cell and does not depend on the gauge cell size. This scheme, which we call a mesoscopic canonical ensemble, bridges the gap between the canonical and the grand canonical ensembles, which are known to be inconsistent for small systems. The ideal gas gauge cell method is illustrated with Monte Carlo simulations of Lennard-Jones fluid confined to spherical pores of different sizes. Special attention is paid to the case of extreme confinement of several molecular diameters in cross section where the inconsistency between the canonical ensemble and the grand canonical ensemble is most pronounced. For sufficiently large systems, the chemical potential can be reliably determined from the mean density in the gauge cell as it was implied in the original gauge cell method. The method is applied to study the transition from supercritical adsorption to subcritical capillary condensation, which is observed in nanoporous materials as the pore size increases.

Monte Carlo simulation study of droplet nucleation

A new rigorous Monte Carlo simulation approach is employed to study nucleation barriers for droplets in Lennard-Jones fluid. Using the gauge cell method we generate the excess isotherm of critical clusters in the size range from two to six molecular diameters. The ghost field method is employed to compute the cluster free energy and the nucleation barrier with desired precision of (1-2)kT. Based on quantitative results obtained by Monte Carlo simulations, we access the limits of applicability of the capillarity approximation of the classical nucleation theory and the Tolman equation. We show that the capillarity approximation corrected for vapor nonideality and liquid compressibility provides a reasonable assessment for the size of critical clusters in Lennard-Jones fluid; however, its accuracy is not sufficient to predict the nucleation barriers for making practical estimates of the rate of nucleation. The established dependence of the droplet surface tension on the droplet size cannot be approximated by the Tolman equation for small droplets of radius less than four molecular diameters. We confirm the conclusion of ten Wolde and Frenkel [J. Chem. Phys. 109, 9901 (1998)] that integration of the normal component of the Irving-Kirkwood pressure tensor severely underestimates the nucleation barriers for small clusters.