Monte Carlo simulations of critical cluster sizes and nucleation rates of water

Assessment of Anharmonicities in Clusters: Developing and Validating a Minimum-Information Partition Function

Precise thermodynamic calculations are essential for understanding the dynamics of cluster systems and new particle formation. However, the widely employed harmonic statistical mechanical approach often falls short in terms of accuracy. In this study, we present an improved statistical model that incorporates vibrational anharmonicity via a novel partition function that requires only one additional system-specific input parameter. In addition to considering vibrational aspects, we also account for anharmonicity related to the configurational space. The role of anharmonicities is thoroughly examined in the case of general clusters, where the complete sets of conformers, mechanically stable spatial arrangements, are known up to clusters composed of 14 monomers. By performing consistent Monte Carlo simulations on these systems, we benchmark the statistical model's efficacy in reproducing key thermodynamic properties (formation free energy and potential energy) in the classical limit. The model exhibits exceptional alignment with simulations, accurately reproducing free energies within a precision of 2kBT and reliably capturing cluster melting temperatures. Furthermore, we demonstrate the significance and applicability of the model by reproducing thermodynamic barriers in homogeneous gas-phase nucleation of larger clusters. The transferability of our developed approach extends to more complex molecular systems and bears relevance for atmospheric multicomponent clusters, in particular.

Nucleation Rate Theory for Coordination Number: Elucidating Water-Mediated Formation of a Zigzag Na2SO4 Morphology

Predicting and controlling nanostructure formation during nucleation can pave the way to synthesizing novel energy materials via crystallization. However, such control over nucleation and crystallization remains challenging due to an inadequate understanding of critical factors that govern evolving atomistic structures and dynamics. Herein, we utilize coordination number as a reaction coordinate and rate theory to investigate how sodium sulfate, commonly known as a phase-change energy material, nucleates in a supersaturated aqueous solution. In conjunction with ab initio and force field-based molecular dynamics simulation, the rate theoretical analysis reveals that sodium sulfate from an initially dissolved metastable state transits to a heterogeneous mixture of prenucleated clusters and finally to a large cylindrical zigzag morphology. Measurements of Raman spectra and their ab initio modeling confirm that this nucleated morphology contains a few waters for every sulfate. Rate processes such as solvent exchange and desolvation exhibit high sensitivity to the evolving prenucleation/nucleation structures, providing a means to distinguish between critical nucleation precursors. Desolvation and forming the first-shell interionic coordination structure via monomer-by-monomer addition around sulfates are found to explain the formation of large nuclei. Thus, a detailed understanding of the step-by-step structure formation across scales has been achieved. This can be leveraged to predict nucleation-related structures and dynamics and potentially control the synthesis of novel phase-change materials for energy applications.

Nonisothermal nucleation in the gas phase is driven by cool subcritical clusters

Nucleation of clusters from the gas phase is a widely encountered phenomenon, yet rather little is understood about the underlying out-of-equilibrium dynamics of this process. The classical view of nucleation assumes isothermal conditions where the nucleating clusters are in thermal equilibrium with their surroundings. However, in all first-order phase transitions, latent heat is released, potentially heating the clusters and suppressing the nucleation. The question of how the released energy affects cluster temperatures during nucleation as well as the growth rate remains controversial. To investigate the nonisothermal dynamics and energetics of homogeneous nucleation, we have performed molecular dynamics simulations of a supersaturated vapor in the presence of thermalizing carrier gas. The results obtained from these simulations are compared against kinetic modeling of isothermal nucleation and classical nonisothermal theory. For the studied systems, we find that nucleation rates are suppressed by two orders of magnitude at most, despite substantial release of latent heat. Our analyses further reveal that while the temperatures of the entire cluster size populations are elevated, the temperatures of the specific clusters driving the nucleation flux evolve from cold to hot when growing from subcritical to supercritical sizes and resolve the apparent contradictions regarding cluster temperatures. Our findings provide unprecedented insight into realistic nucleation events and allow us to directly assess earlier theoretical considerations of nonisothermal nucleation.

Efficient Monte Carlo Sampling for Molecular Systems Using Continuous Normalizing Flow

Monte Carlo molecular simulation is a powerful computational method for simulating molecular behavior. It generates samples of the possible states of molecular systems. To generate a sample efficiently, it is advantageous to avoid suggesting extremely high-energy states that would never become possible states. In this study, we propose a new sampling method for Monte Carlo molecular simulation, that is, a continuous normalizing molecular flow (CNMF) method, which can create various probabilistic distributions of molecular states from some initial distribution. The CNMF method generates samples by solving a first-order differential equation with two-body intermolecular interaction terms. We also develop specific probabilistic distributions using CNMF called inverse square flow, which yields distributions with zero probability density when molecule pairs are in close proximity, whereas probability densities are compressed uniformly from the initial distribution in all other cases. Using inverse square flow, we demonstrate that Monte Carlo molecular simulation is more efficient than the standard simulation. Although the increased computational costs of the CNMF method are non-negligible, this method is feasible for parallel computation and has the potential for expansion.

Vapor-phase nucleation of n-pentane, n-hexane, and n-heptane: Critical cluster properties

The first and second nucleation theorems provide a way to determine the molecular content and excess internal energies of critical clusters, which rely solely on experimental nucleation rates measured at constant temperatures and supersaturations, respectively. Here, we report the size n^* and excess internal energy E_x(n^*) of n-pentane, n-hexane, and n-heptane critical clusters when particles form under the highly supersaturated conditions present in supersonic expansions. In summary, critical clusters contain from ∼2 to ∼11 molecules and exhibit the expected increase in the critical cluster size with increasing temperature and decreasing supersaturation. Surprisingly, the n^* values for all three alkanes appear to lie along a single line when plotted as a function of supersaturation. Within the framework of the capillarity approximation, the excess internal energies determined for the n-heptane critical clusters formed under the low temperature (∼150 K) conditions in our supersonic nozzle are reasonably consistent with those determined under higher temperature (∼250 K) conditions in the thermal diffusion cloud chamber by Rudek et al. [J. Chem. Phys. 105, 4707 (1996)].

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)].

An improved model of homogeneous nucleation for high supersaturation conditions: aluminum vapor

A novel model of stationary nucleation, treating the thermodynamic functions of small clusters, has been built. The model is validated against the experimental data on the nucleation rate of water vapor obtained in a broad range of supersaturation values (S = 10-120), and, at high supersaturation values, it reproduces the experimental data much better than the traditional classical nucleation model. A comprehensive analysis of the nucleation of aluminum vapor with the usage of developed stationary and non-stationary nucleation models has been performed. It has been shown that, at some value of supersaturation, there exists a double potential nucleation barrier. It has been revealed that the existence of this barrier notably delayed the establishment of a stationary distribution of subcritical clusters. It has also been demonstrated that the non-stationary model of the present work and the model of liquid-droplet approximation predict different values of nucleation delay time, τ_s. In doing so, the liquid-droplet model can underestimate notably (by more than an order of magnitude) the value of τ_s.

On the onset of surface condensation: formation and transition mechanisms of condensation mode

Molecular dynamics simulations have been carried out to investigate the onset of surface condensation. On surfaces with different wettability, we snapshot different condensation modes (no-condensation, dropwise condensation and filmwise condensation) and quantitatively analyze their characteristics by temporal profiles of surface clusters. Two different types of formation of nanoscale droplets are identified, i.e. the formations with and without film-like condensate. We exhibit the effect of surface tensions on the formations of nanoscale droplets and film. We reveal the formation mechanisms of different condensation modes at nanoscale based on our simulation results and classical nucleation theory, which supplements the 'classical hypotheses' of the onset of dropwise condensation. We also reveal the transition mechanism between different condensation modes based on the competition between surface tensions and reveal that dropwise condensation represents the transition states from no-condensation to filmwise condensation.

Low Temperature Kinetics of the First Steps of Water Cluster Formation

We present a combined experimental and theoretical low temperature kinetic study of water cluster formation. Water cluster growth takes place in low temperature (23-69 K) supersonic flows. The observed kinetics of formation of water clusters are reproduced with a kinetic model based on theoretical predictions for the first steps of clusterization. The temperature- and pressure-dependent association and dissociation rate coefficients are predicted with an ab initio transition state theory based master equation approach over a wide range of temperatures (20-100 K) and pressures (10^{-6}-10  bar).

Coherent description of transport across the water interface: From nanodroplets to climate models

Transport of mass and energy across the vapor-liquid interface of water is of central importance in a variety of contexts such as climate models, weather forecasts, and power plants. We provide a complete description of the transport properties of the vapor-liquid interface of water with the framework of nonequilibrium thermodynamics. Transport across the planar interface is then described by 3 interface transfer coefficients where 9 more coefficients extend the description to curved interfaces. We obtain all coefficients in the range 260-560 K by taking advantage of water evaporation experiments at low temperatures, nonequilibrium molecular dynamics with the TIP4P/2005 rigid-water-molecule model at high temperatures, and square gradient theory to represent the whole range. Square gradient theory is used to link the region where experiments are possible (low vapor pressures) to the region where nonequilibrium molecular dynamics can be done (high vapor pressures). This enables a description of transport across the planar water interface, interfaces of bubbles, and droplets, as well as interfaces of water structures with complex geometries. The results are likely to improve the description of evaporation and condensation of water at widely different scales; they open a route to improve the understanding of nanodroplets on a small scale and the precision of climate models on a large scale.

A variational approach to nucleation simulation

We study by computer simulation the nucleation of a supersaturated Lennard-Jones vapor into the liquid phase. The large free energy barriers to transition make the time scale of this process impossible to study by ordinary molecular dynamics simulations. Therefore we use a recently developed enhanced sampling method [Valsson and Parrinello, Phys. Rev. Lett.113, 090601 (2014)] based on the variational determination of a bias potential. We differ from previous applications of this method in that the bias is constructed on the basis of the physical model provided by the classical theory of nucleation. We examine the technical problems associated with this approach. Our results are very satisfactory and will pave the way for calculating the nucleation rates in many systems.

Understanding the sensitivity of nucleation free energies: The role of supersaturation and temperature

Simulations are used to investigate the vapor-to-liquid nucleation of water for several different force fields at various sets of physical conditions. The nucleation free energy barrier is found to be extremely sensitive to the force field at the same absolute conditions. However, when the results are compared at the same supersaturation and reduced temperature or the same metastability parameter and reduced temperature, then the differences in the nucleation free energies of the different models are dramatically reduced. This finding suggests that comparisons of experimental data and computational predictions are most meaningful at the same relative conditions and emphasizes the importance of knowing the phase diagram of a given computational model, but such information is usually not available for models where the interaction energy is determined directly from electronic structure calculations.

Effect of surface free energies on the heterogeneous nucleation of water droplet: a molecular dynamics simulation approach

Heterogeneous nucleation of water droplet on surfaces with different solid-liquid interaction intensities is investigated by molecular dynamics simulation. The interaction potentials between surface atoms and vapor molecules are adjusted to obtain various surface free energies, and the nucleation process and wetting state of nuclei on surfaces are investigated. The results indicate that near-constant contact angles are already established for nano-scale nuclei on various surfaces, with the contact angle decreasing with solid-liquid interaction intensities linearly. Meanwhile, noticeable fluctuation of vapor-liquid interfaces can be observed for the nuclei that deposited on surfaces, which is caused by the asymmetric forces from vapor molecules. The formation and growth rate of nuclei are increasing with the solid-liquid interaction intensities. For low energy surface, the attraction of surface atoms to water molecules is comparably weak, and the pre-existing clusters can depart from the surface and enter into the bulk vapor phase. The distribution of clusters within the bulk vapor phase becomes competitive as compared with that absorbed on surface. For moderate energy surfaces, heterogeneous nucleation predominates and the formation of clusters within bulk vapor phase is suppressed. The effect of high energy particles that embedded in low energy surface is also discussed under the same simulation system. The nucleation preferably initiates on the high energy particles, and the clusters that formed on the heterogeneous particles are trapped around their original positions instead of migrating around as that observed on smooth surfaces. This feature makes it possible for the heterogeneous particles to act as fixed nucleation sites, and simulation results also suggest that the number of nuclei increases monotonously with the number of high energy particles. The growth of nuclei on high energy particles can be divided into three sub-stages, beginning with the formation of a wet-spot, increase of contact angle with near-constant contact line, and finally growth with constant contact angle. The growth rate of nuclei also increases with the size of high energy particles.

Free energies of molecular clusters determined by guided mechanical disassembly

The excess free energy of a molecular cluster is a key quantity in models of the nucleation of droplets from a metastable vapor phase; it is often viewed as the free energy arising from the presence of an interface between the two phases. We show how this quantity can be extracted from simulations of the mechanical disassembly of a cluster using guide particles in molecular dynamics. We disassemble clusters ranging in size from 5 to 27 argonlike Lennard-Jones atoms, thermalized at 60 K, and obtain excess free energies, by means of the Jarzynski equality, that are consistent with previous studies. We only simulate the cluster of interest, in contrast to approaches that require a series of comparisons to be made between clusters differing in size by one molecule. We discuss the advantages and disadvantages of the scheme and how it might be applied to more complex systems.

Kinetic Monte Carlo simulations of the assembly of filamentous biomacromolecules by dimer addition mechanism

In cells, several important biomacromolecules form oligomers through a dimer addition mechanism. Rate equations based on mean field approximations are usually employed to describe the assembly process. However, such equations often require multiple assumptions that mask some detailed changes of the biomolecular configurations during aggregations. Here, we present a Kinetic Monte Carlo simulation scheme to account for the diffusion and rotation of dimers on two-dimensional hexagonal lattices while naturally including the stochastic features. We investigate the effects of the interaction energy between dimers, the diffusion coefficient and the concentration of dimers on the aggregation by dimer addition mechanism. Our simulations identified unusual double-S shape evolutions of aggregation kinetics, which are probably associated with the formation of metastable clusters.

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.

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.

Quantum-mechanical solution to fundamental problems of classical theory of water vapor nucleation

The inconsistent temperature dependence of nucleation rates, disagreement of theoretical critical or onset supersaturations with experimental data, and insufficiently accurate predictions of nucleation rates are fundamental problems of the classical nucleation theory (CNT) of water vapors, which is a foundation of various multicomponent nucleation models widely used in the aerosol microphysics, physical chemistry, and chemical technology. In the present study, a correction to the CNT obtained from "first principles" has been derived and significant progress has been made in solving the fundamental problem of predicting nucleation rates of water vapors. The modified model with the quantum-mechanical correction incorporated is in very good agreement with experiments over the full range of temperatures (T=210-290K) , saturation ratios (S=2-100) , and nucleation rates (J= approximately 10{1}-10{17} cm-3).

Connection between the virial equation of state and physical clusters in a low density vapor

We carry out Monte Carlo simulations of physical Lennard-Jones and water clusters and show that the number of physical clusters in vapor is directly related to the virial equation of state. This relation holds at temperatures clearly below the critical temperatures, in other words, as long as the cluster-cluster interactions can be neglected--a typical assumption used in theories of nucleation. Above a certain threshold cluster size depending on temperature and interaction potential, the change in cluster work of formation can be calculated analytically with the recently proposed scaling law. The breakdown of the scaling law below the threshold sizes is accurately modeled with the low order virial coefficients. Our results indicate that high order virial coefficients can be analytically calculated from the lower order coefficients when the scaling law for cluster work of formation is valid. The scaling law also allows the calculation of the surface tension and equilibrium vapor density with computationally efficient simulations of physical clusters. Our calculated values are in good agreement with those obtained with other methods. We also present our results for the curvature dependent surface tension of water clusters.

Origin of the failure of classical nucleation theory: incorrect description of the smallest clusters

We carry out molecular Monte Carlo simulations of clusters in an imperfect vapor. We show that down to very small cluster sizes, classical nucleation theory built on the liquid drop model can be used very accurately to describe the work required to add a monomer to the cluster. However, the error made in modeling the smallest of clusters as liquid drops results in an erroneous absolute value for the cluster work of formation throughout the size range. We calculate factors needed to correct the cluster formation work given by the liquid drop model. The corrected work of formation results in nucleation rates in good agreement with recent nucleation experiments on argon and water.

Theoretical and experimental study on phase transitions and mass fluxes of supersaturated water vapor onto different insoluble flat surfaces

The heterogeneous nucleation and condensation of water vapor onto three different surfaces (newsprint paper, Teflon, cellulose film) was studied theoretically and experimentally. The theoretical framework included the use of the classical theory of heterogeneous nucleation, diffusion theory corrected with transition regime correction factors, and the theory of heat transfer. Experiments were carried out using an environmental scanning electron microscope (ESEM). The experimental results for newsprint paper were investigated more closely. Our results show that the measured onset supersaturations were smaller than the modeled ones when the experimentally determined contact angle was used. Furthermore, the measured condensational growth rates were smaller than the modeled ones, presumably resulting from the approximations that had to be made in the calculations.