Modification of the classical nucleation theory based on molecular simulation data for surface tension, critical nucleus size, and nucleation rate

Measuring the Surface Tension of Atmospheric Particles and Relevant Mixtures to Better Understand Key Atmospheric Processes

Aerosol and aqueous particles are ubiquitous in Earth's atmosphere and play key roles in geochemical processes such as natural chemical cycles, cloud and fog formation, air pollution, visibility, climate forcing, etc. The surface tension of atmospheric particles can affect their size distribution, condensational growth, evaporation, and exchange of chemicals with the atmosphere, which, in turn, are important in the above-mentioned geochemical processes. However, because measuring this quantity is challenging, its role in atmospheric processes was dismissed for decades. Over the last 15 years, this field of research has seen some tremendous developments and is rapidly evolving. This review presents the state-of-the-art of this subject focusing on the experimental approaches. It also presents a unique inventory of experimental adsorption isotherms for over 130 mixtures of organic compounds in water of relevance for model development and validation. Potential future areas of research seeking to better determine the surface tension of atmospheric particles, better constrain laboratory investigations, or better understand the role of surface tension in various atmospheric processes, are discussed. We hope that this review appeals not only to atmospheric scientists but also to researchers from other fields, who could help identify new approaches and solutions to the current challenges.

Particle size effect on surface/interfacial tension and Tolman length of nanomaterials: A simple experimental method combining with theoretical

Surface tension and interfacial tension are crucial to the study of nanomaterials. Herein, we report a solubility method using magnesium oxide nanoparticles of different radii (1.8-105.0 nm, MgO NPs) dissolved in pure water as a targeted model; the surface tension and interfacial tension (and their temperature coefficients) were determined by measuring electrical conductivity and combined with the principle of the electrochemical equilibrium method, and the problem of particle size dependence is discussed. Encouragingly, this method can also be used to determine the ionic (atomic or molecular) radius and Tolman length of nanomaterials. This research results disclose that surface/interfacial tension and their temperature coefficients have a significant relationship with particle size. Surface/interfacial tension decreases rapidly with a radius <10 nm (while the temperature coefficients are opposite), while for a radius >10 nm, the effect is minimal. Especially, it is proven that the value of Tolman length is positive, the effect of particle size on Tolman length is consistent with the surface/interfacial tension, and the Tolman length of the bulk does not change much in the temperature range. This work initiates a new era for reliable determination of surface/interfacial tension, their temperature coefficients, ionic radius, and Tolman length of nanomaterials and provides an important theoretical basis for the development and application of various nanomaterials.

A Deep Potential model for liquid-vapor equilibrium and cavitation rates of water

Computational studies of liquid water and its phase transition into vapor have traditionally been performed using classical water models. Here, we utilize the Deep Potential methodology-a machine learning approach-to study this ubiquitous phase transition, starting from the phase diagram in the liquid-vapor coexistence regime. The machine learning model is trained on ab initio energies and forces based on the SCAN density functional, which has been previously shown to reproduce solid phases and other properties of water. Here, we compute the surface tension, saturation pressure, and enthalpy of vaporization for a range of temperatures spanning from 300 to 600 K and evaluate the Deep Potential model performance against experimental results and the semiempirical TIP4P/2005 classical model. Moreover, by employing the seeding technique, we evaluate the free energy barrier and nucleation rate at negative pressures for the isotherm of 296.4 K. We find that the nucleation rates obtained from the Deep Potential model deviate from those computed for the TIP4P/2005 water model due to an underestimation in the surface tension from the Deep Potential model. From analysis of the seeding simulations, we also evaluate the Tolman length for the Deep Potential water model, which is (0.091 ± 0.008) nm at 296.4 K. Finally, we identify that water molecules display a preferential orientation in the liquid-vapor interface, in which H atoms tend to point toward the vapor phase to maximize the enthalpic gain of interfacial molecules. We find that this behavior is more pronounced for planar interfaces than for the curved interfaces in bubbles. This work represents the first application of Deep Potential models to the study of liquid-vapor coexistence and water cavitation.

Determining Brown's Characteristic Curves Using Molecular Simulation

Brown's characteristic curves define lines on the thermodynamic surface where special thermodynamic conditions hold. These curves are an important tool for the development of thermodynamic models of fluids. Yet, practically no experimental data for Brown's characteristic curves is available. In this work, a rigorous and generalized method for determining Brown's characteristic curves based on molecular simulation was developed. As multiple thermodynamic equivalent definitions apply for the characteristic curves, different simulation routes were compared. Based on this systematic approach, the most favorable route for determining each characteristic curve was identified. The computational procedure developed in this work combines molecular simulation, molecular-based equation of state, and the evaluation of the second virial coefficient. The new method was tested on a simple model system (the classical Lennard-Jones fluid) and different types of real substances (toluene, methane, ethane, propane, and ethanol). It is thereby shown that the method is robust and yields accurate results. Moreover, a computer code implementation of the method is presented.

Determination of Interfacial Tension of Nanomaterials and the Effect of Particle Size on Interfacial Tension

The unique physical and chemical properties and performances of nanomaterials are closely related to the interfacial tension. However, there is no method to accurately measure the interfacial tension of nanomaterials. In addition, the effect of particle size on the interfacial tension of nanoparticles is unclear, and there exist conflicting conclusions about the value and sign of Tolman length. In this paper, a novel method of determining the interfacial tension (solid-liquid and solid-gas interfaces), temperature coefficient of interfacial tension, and Tolman lengths of nanomaterials by adsorption thermodynamics and kinetics was presented. The interfacial tension and its temperature coefficient of the solid-liquid interface of nano cadmium sulfide before adsorption were obtained, and further, the Tolman length was also obtained. The experimental results show that the particle size of nanoparticles has significant effects on the interfacial tension and its temperature coefficient. When the radius is larger than 10 nm, the interfacial tension and its temperature coefficient are almost constant with the decrease of the radius. When the radius is less than 10 nm, the interfacial tension decreases sharply and the temperature coefficient increases sharply with the decrease of the radius, and the temperature coefficient of the interfacial tension is negative. The Tolman length of the solid-liquid interface of nanoparticles is proved to be positive, and the particle size also has a significant effect on the Tolman length. The Tolman length decreases with the decrease of particle size. However, the effects of particle size on the Tolman length become significant only when the particle radius approach or reach the order of magnitudes of molecular (or atomic) radius. The effects of particle size on interfacial tension and Tolman length of nano cadmium sulfide obtained in this paper can provide significant references for the research and applications of interface thermodynamics of other nanomaterials.

Fcc vs. hcp competition in colloidal hard-sphere nucleation: on their relative stability, interfacial free energy and nucleation rate

Hard-sphere crystallization has been widely investigated over the last six decades by means of colloidal suspensions and numerical methods. However, some aspects of its nucleation behaviour are still under debate. Here, we provide a detailed computational characterisation of the polymorphic nucleation competition between the face-centered cubic (fcc) and the hexagonal-close packed (hcp) hard-sphere crystal phases. By means of several state-of-the-art simulation techniques, we evaluate the melting pressure, chemical potential difference, interfacial free energy and nucleation rate of these two polymorphs, as well as of a random stacking mixture of both crystals. Our results highlight that, despite the fact that both polymorphs have very similar stability, the interfacial free energy of the hcp phase could be marginally higher than that of the fcc solid, which in consequence, mildly decreases its propensity to nucleate from the liquid compared to the fcc phase. Moreover, we analyse the abundance of each polymorph in grown crystals from different types of inserted nuclei: fcc, hcp and stacking disordered fcc/hcp seeds, as well as from those spontaneously emerged from brute force simulations. We find that post-critical crystals fundamentally grow maintaining the polymorphic structure of the critical nucleus, at least until moderately large sizes, since the only crystallographic orientation that allows stacking close-packed disorder is the fcc (111) plane, or equivalently the hcp (0001) one. Taken together, our results contribute with one more piece to the intricate puzzle of colloidal hard-sphere crystallization.

Research on homogeneous nucleation and microstructure evolution of aluminium alloy melt

In this paper, based on the embedded atom method (EAM) potential, molecular dynamics simulations of the solidification process of Al-4 at.%Cu alloy is carried out. The Al-Cu alloy melt is placed at different temperatures for isothermal solidification, and each stage of the entire solidification process is tracked, including homogeneous nucleation, nucleus growth, grain coarsening and microstructure evolution. In the nucleation stage, the transition from high temperature to low temperature manifests a change from spontaneous nucleation mode to divergent nucleation mode. The critical nucleation temperature of the Al-Cu alloy is determined to be about 0.42 T _m (T _m is the melting point of Al-4 at.%Cu) by calculating the nucleation rate and the crystal nucleus density. In the nucleus growth stage, two ways of growing up are observed, that is, a large crystal nucleus will absorb a smaller heterogeneous crystal nucleus, and two very close crystal nuclei will merge. In the microstructure evolution of the isothermally solidified Al-Cu alloy, it is emerged that the interior of all nanocrystalline grains are long-period stacking structure composed of face centred cubic (FCC) and hexagonal close-packed (HCP). These details provide important information for the production of Al-Cu binary alloy nano-polycrystalline products.

Bond order redefinition needed to reduce inherent noise in molecular dynamics simulations

In this work, we present the bond order redefinition needed to reduce the inherent noise in order to enhance the accuracy of molecular dynamics simulations. We propose defining the bond order as a fraction of energy distribution. It happens due to the character of the material in nature, which tries to maintain its environment. To show the necessity, we developed a factory empirical interatomic potential (FEIP) for carbon that implements the redefinition with a short-range interaction approach. FEIP has been shown to enhance the accuracy of the calculation of lattice constants, cohesive energy, elastic properties, and phonons compared to experimental data, and can even be compared to other potentials with the long-range interaction approach. The enhancements due to FEIP can reduce the inherent noise, then provide a better prediction of the energy based on the behaviour of the atomic environment. FEIP can also transform simple two-body interactions into many-body interactions, which is useful for enhancing accuracy. Due to implementing the bond order redefinition, FEIP offers faster calculations than other complex interatomic potentials.

Parasitic crystallization of colloidal electrolytes: growing a metastable crystal from the nucleus of a stable phase

Colloidal particles have been extensively used to comprehend the main principles governing liquid-crystal nucleation. Multiple mechanisms and frameworks have been proposed, through either experiments or computational approaches, to rationalise the ubiquitous formation of colloidal crystals. In this work, we elucidate the nucleation scenario behind the crystallization of oppositely charged colloids. By performing molecular dynamics simulations of colloidal electrolytes in combination with the Seeding technique, we evaluate the fundamental factors, such as the nucleation rate, free energy barrier, surface tension and kinetic pre-factor, that determine the liquid-to-solid transition of several crystalline polymorphs. Our results show that at a high packing fraction, there is a cross-over between the nucleation of the CsCl structure and that of a substitutionally disordered fcc phase, despite the CuAu crystal being the most stable phase. We demonstrate that the crucial factor in determining which phase nucleates the fastest is the free energy cost of the cluster formation rather than their kinetic ability to grow from the liquid. While at a low packing fraction, the stable phase, CsCl, is the one that nucleates and subsequently grows, we show how at moderate and high packing fractions, a disordered fcc phase subsequently grows regardless of the nature of the nucleating phase, termed parasitic crystallization. Taken together, our results provide a panoramic perspective of the complex nucleation scenario of oppositely charged colloids at moderate temperature and rationalise the different thermodynamic and kinetic aspects behind it.

Characteristic Curves of the Lennard-Jones Fluid

Equations of state based on intermolecular potentials are often developed about the Lennard-Jones (LJ) potential. Many of such EOS have been proposed in the past. In this work, 20 LJ EOS were examined regarding their performance on Brown's characteristic curves and characteristic state points. Brown's characteristic curves are directly related to the virial coefficients at specific state points, which can be computed exactly from the intermolecular potential. Therefore, also the second and third virial coefficient of the LJ fluid were investigated. This approach allows a comparison of available LJ EOS at extreme conditions. Physically based, empirical, and semi-theoretical LJ EOS were examined. Most investigated LJ EOS exhibit some unphysical artifacts.

The Influence of Lubrication and the Solid-Fluid Interaction on Thermodynamic Properties in a Nanoscopic Scratching Process

Liquid lubricants play an important role in contact processes; for example, they reduce friction and cool the contact zone. To gain better understanding of the influence of lubrication on the nanoscale, both dry and lubricated scratching processes in a model system are compared in the present work using molecular dynamics simulations. The entire range between total dewetting and total wetting is investigated by tuning the solid-fluid interaction energy. The investigated scratching process consists of three sequential movements: A cylindrical indenter penetrates an initially flat substrate, then scratches in the lateral direction, and is finally retracted out of the contact with the substrate. The indenter is fully submersed in the fluid in the lubricated cases. The substrate, the indenter, and the fluid are described by suitably parametrized Lennard-Jones model potentials. The presence of the lubricant is found to have a significant influence on the friction and on the energy balance of the process. The thermodynamic properties of the lubricant are evaluated in detail. A correlation of the simulation results for the profiles of the temperature, density, and pressure of the fluid in the vicinity of the chip is developed. The work done by the indenter is found to mainly dissipate and thereby heat up the substrate and eventually the fluid. Only a minor part of the work causes plastic deformation of the substrate.

Derivation of an exact, nonequilibrium framework for nucleation: Nucleation is a priori neither diffusive nor Markovian

We discuss the structure of the equation of motion that governs nucleation processes at first order phase transitions. From the underlying microscopic dynamics of a nucleating system, we derive by means of a nonequilibrium projection operator formalism the equation of motion for the size distribution of the nuclei. The equation is exact, i.e., the derivation does not contain approximations. To assess the impact of memory, we express the equation of motion in a form that allows for direct comparison to the Markovian limit. As a numerical test, we have simulated crystal nucleation from a supersaturated melt of particles interacting via a Lennard-Jones potential. The simulation data show effects of non-Markovian dynamics.

Studying vapor-liquid transition using a generalized ensemble

Homogeneous vapor-liquid nucleation is studied using the generalized Replica Exchange Method (gREM). The generalized ensemble allows the study of unstable states that cannot directly be studied in the canonical ensemble. Along with replica exchange, this allows for efficient sampling of the multiple states in a single simulation. Statistical Temperature Weighted Histogram Analysis Method is used for postprocessing to get a continuous free energy curve from bulk vapor to bulk liquid. gREM allows the study of planar, cylindrical, and spherical interfaces in a single simulation. The excess Gibbs free energy for the formation of a spherical liquid droplet in vapor for a Lennard-Jones system is calculated from the free energy curve and compared against the umbrella sampling results. The nucleation free energy barrier obtained from gREM is then used to calculate the nucleation rate without relying on any classification scheme for separating the vapor and liquid.

Three-dimensional phase field modeling of inhomogeneous gas-liquid systems using the PeTS equation of state

Recently, an equation of state (EoS) for the Lennard-Jones truncated and shifted (LJTS) fluid has become available. As it describes metastable and unstable states well, it is suited for predicting density profiles in vapor-liquid interfaces in combination with density gradient theory (DGT). DGT is usually applied to describe interfaces in Cartesian one-dimensional scenarios. In the present work, the perturbed LJ truncated and shifted (PeTS) EoS is implemented into a three-dimensional phase field (PF) model which can be used for studying inhomogeneous gas-liquid systems in a more general way. The results are compared with the results from molecular dynamics simulations for the LJTS fluid that are carried out in the present work and good agreement is observed. The PF model can therefore be used to overcome the scale limit of molecular simulations. A finite element approach is applied for the implementation of the PF model. This requires the first and second derivatives of the PeTS EoS which are calculated using hyper-dual numbers. Several tests and examples of applications of the new PeTS PF model are discussed.

Kinetic analysis of homogeneous droplet nucleation using large-scale molecular dynamics simulations

Studies on homogeneous nucleation have been conducted for decades, but a large gap between experiment and theory persists when evaluating the nucleation rate because the classical nucleation theory (CNT) with all its modifications still cannot fully incorporate the kinetics of homogeneous nucleation. Recent large-scale molecular dynamics (MD) simulations on homogeneous nucleation estimated a nucleation rate around the same order of magnitude as that obtained in experiments. This immensely improved agreement between experiment and theory is exciting because MD can provide detailed information on molecular trajectories. Therefore, a better understanding of the kinetics of homogeneous nucleation can now be obtained. In this study, large-scale MD simulations on homogeneous nucleation were performed. Through kinetic analysis of the simulation results, the nucleation rate, free energy barrier, and critical cluster size were found. Although the nucleation rates directly obtained from the simulations differed from those calculated from the CNT by 8-13 orders of magnitude, when the parameters calculated from the molecular trajectories were substituted into the classical theory, the discrepancy between the nucleation rates decreased to within an order of magnitude. This proves that the fundamental formulation of the theoretical equation is physically sound. We also calculated the cluster formation free energy and confirmed that the free energy barrier decreases with increasing supersaturation ratio. The estimated barrier height was twice that determined by theory, whereas the critical cluster size showed very good agreement between simulation and theory.

Near surface nucleation and particle mediated growth of colloidal Au nanocrystals

During non-classical growth of nanostructures via assembly of primary nuclei, nucleation and assembly are assumed to be distinct processes: nanoparticles nucleate randomly and aggregate to form extended structures through Brownian motion in the presence of long-range attractive interactions. Here we investigate the relationship between these two processes by using in situ AFM, in situ, ex situ and cryo TEM and UV-Vis spectroscopy to observe growth of colloidal gold and simulations to develop a mechanistic model of the process. Our results reveal an inexorable link between nucleation and assembly with nuclei forming almost exclusively within a ∼1 nm interfacial region of existing particles. The new particles immediately close the gap either through a diffusive jump or via growth of a neck between the seed and new particle, generating aggregates exhibiting features commonly attributed to oriented attachment of independently nucleated particles. The results demonstrate that creation of initial particle interfaces leads to local environments that redirect growth towards non-classical processes.

Liquid state isomorphism, Rosenfeld-Tarazona temperature scaling, and Riemannian thermodynamic geometry

Aspects of isomorph theory, Rosenfeld-Tarazona temperature scaling, and thermodynamic geometry are comparatively discussed on the basis of the Lennard-Jones potential. The first two approaches approximate the high-density fluid state well when the repulsive interparticle interactions become dominant, which is typically the case close to the freezing line. However, previous studies of Rosenfeld-Tarazona scaling for the isochoric heat capacity and its relation to isomorph theory reveal deviations for the temperature dependence. It turns out that a definition of a state region in which repulsive interactions dominate is required for achieving consistent results. The Riemannian thermodynamic scalar curvature R allows for such a classification, indicating predominantly repulsive interactions by R>0. An analysis of the isomorphic character of the freezing line and the validity of Rosenfeld-Tarazona temperature scaling show that these approaches are consistent only in a small state region.

Deviation from equilibrium conditions in molecular dynamic simulations of homogeneous nucleation

We present a comparison between Monte Carlo (MC) results for homogeneous vapour-liquid nucleation of Lennard-Jones clusters and previously published values from molecular dynamics (MD) simulations. Both the MC and MD methods sample real cluster configuration distributions. In the MD simulations, the extent of the temperature fluctuation is usually controlled with an artificial thermostat rather than with more realistic carrier gas. In this study, not only a primarily velocity scaling thermostat is considered, but also Nosé-Hoover, Berendsen, and stochastic Langevin thermostat methods are covered. The nucleation rates based on a kinetic scheme and the canonical MC calculation serve as a point of reference since they by definition describe an equilibrated system. The studied temperature range is from T = 0.3 to 0.65 ϵ/k. The kinetic scheme reproduces well the isothermal nucleation rates obtained by Wedekind et al. [J. Chem. Phys. 127, 064501 (2007)] using MD simulations with carrier gas. The nucleation rates obtained by artificially thermostatted MD simulations are consistently lower than the reference nucleation rates based on MC calculations. The discrepancy increases up to several orders of magnitude when the density of the nucleating vapour decreases. At low temperatures, the difference to the MC-based reference nucleation rates in some cases exceeds the maximal nonisothermal effect predicted by classical theory of Feder et al. [Adv. Phys. 15, 111 (1966)].

ls1 mardyn: The Massively Parallel Molecular Dynamics Code for Large Systems

The molecular dynamics simulation code ls1 mardyn is presented. It is a highly scalable code, optimized for massively parallel execution on supercomputing architectures and currently holds the world record for the largest molecular simulation with over four trillion particles. It enables the application of pair potentials to length and time scales that were previously out of scope for molecular dynamics simulation. With an efficient dynamic load balancing scheme, it delivers high scalability even for challenging heterogeneous configurations. Presently, multicenter rigid potential models based on Lennard-Jones sites, point charges, and higher-order polarities are supported. Due to its modular design, ls1 mardyn can be extended to new physical models, methods, and algorithms, allowing future users to tailor it to suit their respective needs. Possible applications include scenarios with complex geometries, such as fluids at interfaces, as well as nonequilibrium molecular dynamics simulation of heat and mass transfer.

Nucleation in mesoscopic systems under transient conditions: peptide-induced pore formation in vesicles

Attachment of lytic peptides to the lipid membrane of virions or bacteria is often accompanied by their aggregation and pore formation, resulting eventually in membrane rupture and pathogen neutralization. The membrane rupture may occur gradually via formation of many pores or abruptly after the formation of the first pore. In academic studies, this process is observed during interaction of peptides with lipid vesicles. We present an analytical model and the corresponding Monte Carlo simulations focused on the pore formation in such situations. Specifically, we calculate the time of the first nucleation-limited pore-formation event and show the distribution of this time in the regime when the fluctuations of the number of peptides attached to a vesicle are appreciable. The results obtained are used to clarify the mechanism of the pore formation and membrane destabilization observed recently during interaction of highly active α-helical peptide with sub-100-nm lipid vesicles that mimic enveloped viruses with nanoscale membrane curvature. The model proposed and the analysis presented are generic and may be applicable to other meso- and nanosystems.

A perspective on the interfacial properties of nanoscopic liquid drops

The structural and interfacial properties of nanoscopic liquid drops are assessed by means of mechanical, thermodynamical, and statistical mechanical approaches that are discussed in detail, including original developments at both the macroscopic level and the microscopic level of density functional theory (DFT). With a novel analysis we show that a purely macroscopic (static) mechanical treatment can lead to a qualitatively reasonable description of the surface tension and the Tolman length of a liquid drop; the latter parameter, which characterizes the curvature dependence of the tension, is found to be negative and has a magnitude of about a half of the molecular dimension. A mechanical slant cannot, however, be considered satisfactory for small finite-size systems where fluctuation effects are significant. From the opposite perspective, a curvature expansion of the macroscopic thermodynamic properties (density and chemical potential) is then used to demonstrate that a purely thermodynamic approach of this type cannot in itself correctly account for the curvature correction of the surface tension of liquid drops. We emphasize that any approach, e.g., classical nucleation theory, which is based on a purely macroscopic viewpoint, does not lead to a reliable representation when the radius of the drop becomes microscopic. The description of the enhanced inhomogeneity exhibited by small drops (particularly in the dense interior) necessitates a treatment at the molecular level to account for finite-size and surface effects correctly. The so-called mechanical route, which corresponds to a molecular-level extension of the macroscopic theory of elasticity and is particularly popular in molecular dynamics simulation, also appears to be unreliable due to the inherent ambiguity in the definition of the microscopic pressure tensor, an observation which has been known for decades but is frequently ignored. The union of the theory of capillarity (developed in the nineteenth century by Gibbs and then promoted by Tolman) with a microscopic DFT treatment allows for a direct and unambiguous description of the interfacial properties of drops of arbitrary size; DFT provides all of the bulk and surface characteristics of the system that are required to uniquely define its thermodynamic properties. In this vein, we propose a non-local mean-field DFT for Lennard-Jones (LJ) fluids to examine drops of varying size. A comparison of the predictions of our DFT with recent simulation data based on a second-order fluctuation analysis (Sampayo et al 2010 J. Chem. Phys. 132 141101) reveals the consistency of the two treatments. This observation highlights the significance of fluctuation effects in small drops, which give rise to additional entropic (thermal non-mechanical) contributions, in contrast to what one observes in the case of planar interfaces which are governed by the laws of mechanical equilibrium. A small negative Tolman length (which is found to be about a tenth of the molecular diameter) and a non-monotonic behaviour of the surface tension with the drop radius are predicted for the LJ fluid. Finally, the limits of the validity of the Tolman approach, the effect of the range of the intermolecular potential, and the behaviour of bubbles are briefly 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.

Excess equimolar radius of liquid drops

The curvature dependence of the surface tension is related to the excess equimolar radius of liquid drops, i.e., the deviation of the equimolar radius from the radius defined by the macroscopic capillarity approximation. Based on the Tolman [J. Chem. Phys. 17, 333 (1949)] approach and its interpretation by Nijmeijer et al. [J. Chem. Phys. 96, 565 (1991)], the surface tension of spherical interfaces is analyzed in terms of the pressure difference due to curvature. In the present study, the excess equimolar radius, which can be obtained directly from the density profile, is used instead of the Tolman length. Liquid drops of the truncated and shifted Lennard-Jones fluid are investigated by molecular dynamics simulation in the canonical ensemble, with equimolar radii ranging from 4 to 33 times the Lennard-Jones size parameter σ. In these simulations, the magnitude of the excess equimolar radius is shown to be smaller than σ/2. This suggests that the surface tension of liquid drops at the nanometer length scale is much closer to that of the planar vapor-liquid interface than reported in studies based on the mechanical route.

A thermodynamically consistent determination of surface tension of small Lennard-Jones clusters from simulation and theory

We have determined the surface tension of small Lennard-Jones clusters using molecular dynamics and Monte Carlo simulation methods as well as density functional theory calculations. For the two simulation methods the surface tension is calculated via a rigorous thermodynamic route using simulation data as input. The capillary approximation of the classical nucleation theory, where the surface tension of a planar surface is used for cluster surface, is found to be quite reasonable even when the cluster size is as small as 100-150 atoms. For smaller cluster sizes the cluster surface tension is considerably lower than the planar value. We have also obtained an approximative value for the Tolman length by extrapolating to the planar limit the difference between the equimolar radius and the radius of the surface of tension. A negative Tolman length is suggested by all the methods used.

Cumulative distribution functions associated with bubble-nucleation processes in cavitation

Bubble-nucleation processes of a Lennard-Jones liquid are studied by molecular dynamics simulations. Waiting time, which is the lifetime of a superheated liquid, is determined for several system sizes, and the apparent finite-size effect of the nucleation rate is observed. From the cumulative distribution function of the nucleation events, the bubble-nucleation process is found to be not a simple Poisson process but a Poisson process with an additional relaxation time. The parameters of the exponential distribution associated with the process are determined by taking the relaxation time into account, and the apparent finite-size effect is removed. These results imply that the use of the arithmetic mean of the waiting time until a bubble grows to the critical size leads to an incorrect estimation of the nucleation rate.

Scaled vapor-to-liquid nucleation in a Lennard-Jones system

Scaling of the homogenous vapor-to-liquid nucleation rate, J, is observed in a model Lennard-Jones (LJ) system. The model uses Monte Carlo simulation-generated small cluster growth to decay rate constant ratios and the kinetic steady-state nucleation rate formalism to determine J at four temperatures below the LJ critical temperature, T{c}. When plotted vs the scaled supersaturation, lnS/[T{c}/T-1]{3/2}, the values of logJ are found to collapse onto a single line. A similar scaling has been observed for the experimental nucleation rate data of water and toluene.