Drop in a metastable vapor: Laplace equation,surface tension,and minimum drop size

A van der Waals mean field theory is applied to a Lennard–Jones fluid for studying drop formation in a supersaturated vapor. A spherical surface separates the fluid particles in two homogeneous regions. The model provides densities, radii, minimum radii and excess pressure. By comparing the excess pressure with that given by the Laplace equation, surface tension is worked out. Its dependence on drop size, densities, and temperature, and its asymptotic values to planar interface are found. The model reveals the existence of an absolute minimum drop and drops with densities close to the supersaturated vapor.     

Surface tension of equilibrium spherical drops in the vapor phase

The properties of metastable and equilibrium drops that occur in the vapor phase and differ in the size and position of the dividing surface are compared. Using an equimolecular dividing surface, it was found that the total free energy and the total mass of the substance in drops of the same size over a broad temperature range differ by not more than 0.3%. A similar comparison for a dividing surface chosen from the equality condition of the moments of forces gives rather similar results. Using the surface where the maximum surface tension is attained as the dividing surface is impossible at low temperatures, because under these conditions, the notion of surface tension has no physical meaning. At high temperatures, the difference between the total mass of the substance for metastable and equilibrium drops does not exceed 0.6%. The calculations were carried out over a broad temperature range on the basis of the lattice-gas model in the quasichemical approximation.     

Metastable extension of the liquid-vapor phase equilibrium curve and surface tension

The method of molecular dynamics has been used to calculate the parameters of liquid-vapor phase equilibrium and the surface tension in a two-phase system of 4096 Lennard-Jones particles. Calculations have been made in a range from the triple point to near-critical temperature and also at temperatures below the triple point corresponding to the metastable equilibrium of a supercooled liquid and supersaturated vapor. To determine the surface tension, along with a mechanical approach a thermodynamic one has been used as well. The latter was based on calculation of the excess internal energy of an interfacial layer. It has been shown that in accuracy the thermodynamic approach is as good as the more sophisticated mechanical one. Low-temperature asymptotics of the phase-equilibrium curve and also of liquid and vapor spinodals have been considered in the Lennard-Jones and the van der Waals models. The behavior of the surface tension and the excess internal energy of an interfacial layer at T-->0 is discussed.     

Linear tension of two-dimensional drops on planar adsorbent faces

The size dependence of the linear tension of round two-dimensional equilibrium drops in the vapor phase on a homogeneous surface of an adsorbent is studied at the pressure of saturated two-dimensional vapor. The calculations are based on the lattice gas model in a quasi-chemical approximation with allowance for the correlation effects of the nearest interacting molecules. Methods for calculating linear tension using the equimolecular reference line are considered. Temperature dependences of the linear tension are studied for metastable and equilibrium drops. It is found that the differences between the thermodynamic properties of two types of drops are slight over a wide range of variation in drop radii.     

Thermal properties of the metastable supersaturated vapor of the Lennard-Jones fluid

p, rho, T data of the supersaturated vapor of the Lennard-Jones fluid are obtained by molecular dynamics simulations. The metastable state points are identified before a phase separation takes place. An estimation of the location of the spinodal is given. The results are compared to two theoretically based equations of state and one empirical equation of state which was parametrized also taking into account metastable state points. The pressure obtained by simulation is found to be lower than that from both theoretically based equations of state, which do not account for the inhomogeneous density distribution of the supersaturated vapor.     

Surface tension of a Lennard-Jones liquid under supersaturation

A formally exact Kirkwood-Buff virial formula for the surface tension of a supersaturated interface is derived. A modified Gibbs ensemble method is given that allows the creation of interacting supersaturated phases of equal chemical potential, and which enables the Kirkwood-Buff formula to be applied. The methods are tested by Monte Carlo simulation of a supersaturated Lennard-Jones fluid with a planar liquid-vapor interface. The Kirkwood-Buff results for the supersaturated surface tension are found to be in reasonable agreement with new results obtained here using the recently developed, formally exact, ghost interface method, [M. P. Moody and P. Attard, J. Chem. Phys., 2002, 117, 6705]. The surface tension is obtained as a function of supersaturation at four temperatures, and it is found to decrease with increasing supersaturation, and to vanish at the vapor spinodal. The relevance of the present results to the nucleation of droplets in a supersaturated vapor is discussed.     

Elements of the topology of the region of the existence of the metastable states of a two-component vapor

Entire regions of the metastable states for binary and single-component systems were constructed. It was established that the regions include temperature intervals from absolute zero to critical temperatures, and from vapor—condensate equilibrium to the conditions of spinodal decomposition of supersaturated vapor in the system. Algorithms for determining the regions of metastable states are presented. The data obtained on their basis are noted to coincide qualitatively with known results. Correlations found earlier are employed to calculate the critical pressures and temperatures for the binary systems in the study.     

A theory of a curved vapor-liquid separation boundary: The lattice gas model

Molecular theory of curved vapor-liquid interphase boundaries was considered in terms of the lattice gas model. The theory uses the quasi-thermodynamic concept of curved layers of a separation boundary with a large radius. The transition from a rectangular lattice to such layers is performed by the introduction of a variable number of the nearest neighbors. The problems (1) of the transition from distributed molecular models to layer models reflecting macroscopic symmetry of the interphase boundary and (2) of a minimum linear size of the surface region to which thermodynamic approaches are applicable were considered. Equations for the quasi-equilibrium distribution of molecules at the vapor-liquid boundary in a metastable system were constructed in the quasi-chemical approximation taking into account direct correlations between the nearest interacting molecules. A metastable state is maintained by a pressure jump described by the macro-scopic Laplace equation on a separation surface inside the interphase region. Equations for local mean pressure values and normal and tangential pressure tensor components inside the interphase region were constructed. These equations were used to obtain microscopic difference mechanical equilibrium equations for curved boundaries of spherical and cylindrical drops in the metastable state. The relation between the micro-scopic difference mechanical equilibrium equations and similar differential equations and the macroscopic Laplace equation, which described pressure jump in a metastable system, was considered. Various definitions of surface tension are discussed.     

Experimental study of homogeneous nucleation from the bismuth supersaturated vapor: evaluation of the surface tension of critical nucleus

The homogeneous nucleation of bismuth supersaturated vapor is studied in a laminar flow quartz tube nucleation chamber. The concentration, size, and morphology of outcoming aerosol particles are analyzed by a transmission electron microscope (TEM) and an automatic diffusion battery (ADB). The wall deposit morphology is studied by scanning electron microscopy. The rate of wall deposition is measured by the light absorption technique and direct weighting of the wall deposits. The confines of the nucleation region are determined in the "supersaturation cut-off" measurements inserting a metal grid into the nucleation zone and monitoring the outlet aerosol concentration response. Using the above experimental techniques, the nucleation rate, supersaturation, and nucleation temperature are measured. The surface tension of the critical nucleus and the radius of the surface of tension are determined from the measured nucleation parameters. To this aim an analytical formula for the nucleation rate is used, derived from author's previous papers based on the Gibbs formula for the work of formation of critical nucleus and the translation-rotation correction. A more accurate approach is also applied to determine the surface tension of critical drop from the experimentally measured bismuth mass flow, temperature profiles, ADB, and TEM data solving an inverse problem by numerical simulation. The simulation of the vapor to particles conversion is carried out in the framework of the explicit finite difference scheme accounting the nucleation, vapor to particles and vapor to wall deposition, and particle to wall deposition, coagulation. The nucleation rate is determined from simulations to be in the range of 10(9)-10(11) cm(-3) s(-1) for the supersaturation of Bi(2) dimers being 10(17)-10(7) and the nucleation temperature 330-570 K, respectively. The surface tension σ(S) of the bismuth critical nucleus is found to be in the range of 455-487 mN/m for the radius of the surface of tension from 0.36 to 0.48 nm. The function σ(S) changes weakly with the radius of critical nucleus. The value of σ(S) is from 14% to 24% higher than the surface tension of a flat surface.     

On the applicability of Young–Laplace equation for nanoscale liquid drops

Debates continue on the applicability of the Young–Laplace equation for droplets, vapor bubbles and gas bubbles in nanoscale. It is more meaningful to find the error range of the Young–Laplace equation in nanoscale instead of making the judgement of its applicability. To do this, for seven liquid argon drops (containing 800, 1000, 1200, 1400, 1600, 1800, or 2000 particles, respectively) at T = 78 K we determined the radius of surface of tension R_s and the corresponding surface tension γ_s by molecular dynamics simulation based on the expressions of R_s and γ_s in terms of the pressure distribution for droplets. Compared with the two-phase pressure difference directly obtained by MD simulation, the results show that the absolute values of relative error of two-phase pressure difference given by the Young–Laplace equation are between 0.0008 and 0.027, and the surface tension of the argon droplet increases with increasing radius of surface of tension, which supports that the Tolman length of Lennard-Jones droplets is positive and that Lennard-Jones vapor bubbles is negative. Besides, the logic error in the deduction of the expressions of the radius and the surface tension of surface of tension, and in terms of the pressure distribution for liquid drops in a certain literature is corrected.     

The free energy of the metastable supersaturated vapor via restricted ensemble simulations. II. Effects of constraints and comparison with molecular dynamics simulations

Extensive restricted canonical ensemble Monte Carlo simulations [D. S. Corti and P. Debenedetti, Chem. Eng. Sci. 49, 2717 (1994)] were performed. Pressure, excess chemical potential, and excess free energy with respect to ideal gas data were obtained at different densities of the supersaturated Lennard-Jones (LJ) vapor at reduced temperatures from 0.7 to 1.0. Among different constraints imposed on the system studied, the one with the local minimum of the excess free energy was taken to be the approximated equilibrium state of the metastable LJ vapor. Also, a comparison of our results with molecular dynamic simulations [A. Linhart et al., J. Chem. Phys. 122, 144506 (2005)] was made.     

Concentration profile of vapor—liquid interface for spherical drops

The modified lattice-gas model is used to construct the model and the concentration profiles of the liquid—vapor for spherical drops of different size. A monotonic character of the temperature dependence of the width of the transition layer between the liquid and the vapor is demonstrated. The dependence of the surface tension on the drop size was studied. The calculation was carried out in the quasichemical approximation, which takes into account the correlation effects of the neighboring interacting molecules. The results are in qualitative agreement with earlier density functional theory calculations and calculations by the van der Waals theory of capillarity. The numerical analysis of the molecular model of the spherical drop gives two types of solution, which correspond to equilibrium and metastasble states of the drop.     

Sensitivity of nucleation phenomena on range of interaction potential

Theoretical and computational investigations of nucleation have been plagued by the sensitivity of the phase diagram to the range of the interaction potential. As the surface tension depends strongly on the range of interaction potential and as the classical nucleation theory (CNT) predicts the free energy barrier to be directly proportional to the cube of the surface tension, one expects a strong sensitivity of nucleation barrier to the range of the potential; however, CNT leaves many aspects unexplored. We find for gas-liquid nucleation in Lennard-Jones system that on increasing the range of interaction the kinetic spinodal (KS) (where the mechanism of nucleation changes from activated to barrierless) shifts deeper into the metastable region. Therefore the system remains metastable for larger value of supersaturation and this allows one to explore the high metastable region without encountering the KS. On increasing the range of interaction, both the critical cluster size and pre-critical minima in the free energy surface of kth largest cluster, at respective kinetic spinodals, shift towards smaller cluster size. In order to separate surface tension contribution to the increase in the barrier from other non-trivial factors, we introduce a new scaling form for surface tension and use it to capture both the temperature and the interaction range dependence of surface tension. Surprisingly, we find only a weak non-trivial contribution from other factors to the free energy barrier of nucleation.     

Condensation of supersaturated water vapor on charged/neutral nanoparticles of glucose and monosodium glutamate

The effects of size, charge, dissolution, and dissociation on the condensation of supersaturated water vapor on monodisperse nanoparticles of glucose and monosodium glutamate (MSG) were investigated in a flow cloud chamber (FCC). The dependence of the critical supersaturation, S(cr), on particle size in the range of 30 to 90 nm and on temperature in the range of 10 to 50 degrees C were determined experimentally. The results show that the experimental S(cr) decreases with increasing particle size at a rate in reasonable agreement with the predictions of the Kohler and Volmer theories of nucleation for soluble particles, but decreases with increasing temperature at a rate higher than the prediction of the Volmer theory. The dissociation of MSG into ions lowers the experimental S(cr) to a value smaller than that for the more soluble glucose, agreeing with predictions. The experimental S(cr) is smaller than the predictions of both theories, and the discrepancy cannot be fully explained by the reductions in surface tension due to the dissolution of particles and curvature dependence. The condensation of supersaturated vapor on singly positively charged particles with diameters of 30, 60, and 90 nm was also examined, and no obvious charge effect on S(cr) was observed.     

Comparison of the properties of metastable and equilibrium drops in the vapor phase

Russian Chemical Bulletin - The properties of metastable and equilibrium drops that occur in the vapor phase and differ in the size and position of the dividing surface are compared. Using an...     

Analyzing the density fluctuation effect on molecular distributions in small drops

The characteristics of a vapor-liquid interface are calculated using the lattice gas model applied to the analysis of the states of metastable supersaturated vapor, depending on the size of a liquid drop and the system temperature. The interaction of molecules is considered in a quasi-chemical approximation describing the effects of direct correlations of nearby molecules. It is found that considering the density fluctuation increases the drop radius, corresponding to the condition of the generation of a new phase.     

Surface Tension: Mechanics,Thermodynamics, and Relaxation Times

A microscopic analysis is presented of the existing definitions of equilibrium surface tension, which can be divided into two types: mechanical and thermodynamic. Each type of definition can be studied from the presentation below according to thermodynamic hypotheses or molecular calculations. An analysis of the planar interface is given and its generalization for curved (spherical) interfaces is considered. The distinction between approaches describing the surface tension of metastable and equilibrium droplets is discussed. Based on nonequilibrium thermodynamics, it is shown that the introduction of metastable droplets is due to a violation of the relationship between the times of impulse and chemical potential relaxation in condensed phases. Problems of calculating the surface tension in nonequilibrium situations are created.     

Water in extremely narrow planar pores with crystalline walls. 2. Thermodynamics

The free energy, entropy, and work of water vapor adsorption in planar pores with widths of 0.62 and 1.25 nm located in a silver iodide crystal parallel to its basal face have been computed at the molecular level. In contrast to adsorption on a free surface, the adsorption in the pores proceeds in three stages, i.e., the formation of molecular films on the walls, coalescence of the films, and densification of the fluid in the pore volume. At the second stage, the equilibrium between the fluid in the pore and the vapor over the pore at temperatures corresponding to normal conditions is thermodynamically unstable and accompanied by the development of a free energy barrier and the existence of metastable states. As temperature is elevated, the instability is gradually evened out; however, its signs remain preserved even at the boiling temperature of water. Extremely narrow pores with widths smaller than 1 nm are always filled with water under conditions of even a rather dry natural atmosphere. The filling of pores several nanometers wide in strongly unsaturated water vapors overcomes the free-energy barrier; however, the fluid that has filled the pore remains stable with respect to evaporation in vapors with densities lower than the density of saturated vapor by several orders of magnitude. The existence of the free-energy barrier and metastable states in nanosized breaks in crystals creates conditions for hysteresis of adsorption-desorption cycles.