The process of nucleation of vapor bubbles from a superheated liquid and of liquid droplets from a supersaturated vapor is investigated using the Modified-Core van der Waals model Density Functional Theory (Lutsko, JCP 128, 184711 (2008)). A novel approach is developed whereby nucleation is viewed as a transition from a metastable state to a stable state via the minimum free energy path which is identified using the nudged elastic-band method for exploring free energy surfaces. This allows for the unbiased calculation of the properties of sub- and super-critical clusters, as well as of the critical cluster. For Lennard-Jones fluids, the results compare well to simulation and support the view that even at high supersaturation nucleation proceeds smoothly rather than via spinodal-like instabilities as has been suggested recently.
The nucleation of vapor bubbles within a superheated fluid is studied using density functional theory. The nudged elastic band technique is used to find the minimum energy pathway from the metastable uniform liquid to the stable uniform gas thus emphasizing the analogy between the the nucleation problem and that of chemical reactions. The result is both an accurate determination of the critical nucleus and an unbiased description of the density profile at various points along the path between the free energy extrema. This calculation is compared to two other methods: the use of parametrized pr…
In contrast to the abundance of work on the anomalous behavior of water, the relationship between the water's thermodynamic anomalies and kinetics of phase transition from metastable water is relatively unexplored. In this work, we have employed classical density functional theory to provide a unified and coherent picture of nucleation (both vapor and ice) from metastable water, especially at negative pressure conditions. Our results suggest a peculiar non-monotonic temperature dependence of liquid-vapor surface tension at temperatures where liquid-vapor coexistence is metastable with respect …
Results for the kinetics of vapor-liquid transitions, following temperature quenches with different densities, are presented from the molecular dynamics simulations of a Lennard-Jones system. For critical density, bicontinuous liquid and vapor domains are observed which grow with time obeying the prediction of hydrodynamic mechanism. On the other hand, for quenches with density significantly below the critical one, phase separation progresses via nucleation and growth of liquid droplets. In the latter case, Brownian diffusion and collision mechanism for the droplet growth is confirmed. We also…
The role of metastable liquid phases in vapor-crystal nucleation is studied using Density Functional Theory(DFT). The model gives a semi-quantitatively accurate description of both the vapor-liquid-solid phase diagram for both simple fluids (Lennard-Jones interactions) and of the low-density/high-density/crystal phase diagram for model globular proteins (ten Wolde-Frenkel interaction). The density profile is characterized by two local order parameters, the average density and the crystallinity. The bulk free energy model is supplemented by squared-derivative terms in these order parameters to …
An analytical description of nucleation stage in a supersaturated vapor with instantly created supersaturation is given with taking into account the vapor concentration inhomogeneities arising as a result of depletion due to non-stationary diffusion onto growing droplets. This description suggests that the intensity of the nucleation of new droplets is suppressed in spherical diffusion regions of a certain size surrounding previously nucleated droplets, and remains at the initial level in the remaining volume of the vapor-gas medium. The value of volume excluded from nucleation depends on the …
We theoretically investigate the nucleation of liquid droplets from vapor in the presence of a charged spherical particle. Due to field gradients, sufficiently close to the critical point of the vapor--gas system, the charge destabilizes the vapor phase and initiates a phase transition. The fluid's free energy is described by the van der Waals expression augmented by electrostatic energy and a square-gradient term. We calculate the equilibrium density profile at arbitrary temperatures, particle charges, and vapor densities. In contrast to classical nucleation theory, here, both liquid and vapo…
We introduce a general formalism to analyze nucleation phenomena in inhomogeneous media which considers the influence of the metastable phase, which is treated as a heat bath in which clusters are embedded, in the dynamics of the nucleation process. A kinetic equation for the evolution of the probability density of the clusters, taking into account possible inhomogeneities in the bath, is derived using mesoscopic nonequilibrium thermodynamics. From it, we obtain the nucleation rate and analyze the role played by the metastable phase in the process. We discuss in particular condensation and pol…
We investigate the influence of a shear flow on the process of nucleation. Mesoscopic nonequilibrium thermodynamics is used to derive the Fokker-Planck equation governing the evolution of the cluster size distribution in a metastable phase subjected to a stationary flow. The presence of the flow manifests itself in the expression for the effective diffusion coefficient of a cluster and introduces modifications in the nucleation rate. The implications of these results in condensation and polymer crystallization are discussed.
Homogeneous nucleation of a vapor bubble from a bulk metastable phase is studied here for the first time using the multibody dissipative particle dynamics model. The van der Waals equation of state is used to define a conservative force that describes the model fluid used in this study. Additional computational tools were developed to help visualize the vapor bubble formed in the system. Continuum thermodynamic ideas are used to help identify the bubble interface and separate particles that constitute the vapor bubble and bulk liquid regions respectively. Using this idea, the density variation…
We use molecular simulation to calculate the nucleation free energy barrier for the freezing of a 456 atom gold cluster over a range of temperatures. The results show that the embryo of the solid cluster grows at the vapor-surface interface for all temperatures studied and that the usual classical nucleation model, with the embryo growing in the core of the cluster, is unable to predict the shape of the free energy barrier. We use a simple partial wetting model that treats the crystal as a lens shaped nucleus at the liquid-vapor interface and find that the line tension plays an important role …
The heterogeneous condensation of a Lennard-Jones vapor onto a nanoscale seed particle is studied using molecular dynamics simulations. Measuring the nucleation rate and the height of the free energy barrier using the mean first passage time method shows that the presence of a weakly interacting seed has little effect on the work of forming very small cluster embryos but accelerates the rate by lowering the barrier for larger clusters. We suggest that this results from a competition between the energetic and entropic features of cluster formation in the bulk and at the heterogeneity. As the in…
A simple method to get all main characteristics of nucleation process is proposed. The advantage of this method is an applicability to situations with non-linear behavior in time of effective external source of vapor. It is important because already existed objects of a new phase comsume vapor in non-linear manner and form an effective non-linear behavior of external conditions. To describe kinetics in this situatrion one can use a method proposed here. As the result some exact expressions for all main characteristics of nucleation kinetics are obtained.
Using the entropy $S$ as a reaction coordinate, we determine the free energy barrier associated with the formation of a liquid droplet from a supersaturated vapor for atomic and molecular fluids. For this purpose, we develop the $μVT-S$ simulation method that combines the advantages of the grand-canonical ensemble, that allows for a direct evaluation of the entropy, and of the umbrella sampling method, that is well suited to the study of an activated process like nucleation. Applying this approach to an atomic system such as $Ar$ allows us to test the method. The results show that the $μVT-S$ …
The Squared-Gradient approximation to the Modified-Core Van der Waals density functional theory model is developed. A simple, explicit expression for the SGA coefficient involving only the bulk equation of state and the interaction potential is given. The model is solved for planar interfaces and spherical clusters and is shown to be quantitatively accurate in comparisons to computer simulations. An approximate technique for solving the SGA based on piecewise-linear density profiles is introduced and is shown to give reasonable zeroth-order approximations to the numerical solution of the model…
The profile of vapor around the droplet growing in the diffusion regime with known moving boundary have been calculated and expressed in terms of special functions.
A new model to calculate the rate of nucleation is formulated. This model is based on the classical nucleation theory but considers also vapor depletion around the formed embryo. As the result the free energy has to be recalculated which brings a new expression for the nucleation rate.
We reconsider model II of [J. Chem. Phys. 1968, 49, 1778--1783], a two-dimensional lattice-gas system featuring a crystalline phase and two distinct fluid phases (liquid and vapor). In this system, a particle prevents other particles from occupying sites up to third neighbors on the square lattice, while attracting (with decreasing strength) particles sitting at fourth- or fifth-neighbor sites. To make the model more realistic, we assume a finite repulsion at third-neighbor distance, with the result that a second crystalline phase appears at higher pressures. However, the similarity with real-…
Hypothesis: As a fluid approaches three phase coexistence, adsorption may take place by the successive formation of two intervening wetting films. The equilibrium thickness of these wetting layers is the result of a delicate balance of intermolecular forces, as dictated by an underlying interface potential. The van der Waals forces for the two variable adsorption layers may be formulated exactly from Dzyaloshinskii-Lifshitz-Pitaevskii theory, and analytical approximations may be derived that extent well beyond the validity of conventional Hamaker theory. Calculations: We consider the adsorptio…
A simple model is proposed for the direct correlation function (DCF) for simple fluids consisting of a hard-core contribution, a simple parametrized core correction, and a mean-field tail. The model requires as input only the free energy of the homogeneous fluid, obtained, e.g., from thermodynamic perturbation theory. Comparison to the DCF obtained from simulation of a Lennard-Jones fluid shows this to be a surprisingly good approximation for a wide range of densities. The model is used to construct a density functional theory for inhomogeneous fluids which is applied to the problem of calcula…
We investigate vapor bubble nucleation in metastable TIP4P/2005 water at negative pressure via the Mean First Passage Time (MFPT) method using the volume of the largest bubble as a local order parameter. We identify the bubbles in the system by means of a Voronoi-based analysis of the Molecular Dynamics trajectories. By comparing the features of the tessellation of liquid water at ambient conditions to those of the same system with an empty cavity we are able to discriminate vapor (or interfacial) molecules from the bulk ones. This information is used to follow the time evolution of the larges…
Classical nucleation theory (CNT), linking rare nucleation events to the free energy landscape of a growing nucleus, is central to understanding phase-change kinetics in passive fluids. Nucleation in non-equilibrium systems is much harder to describe because there is no free energy, but instead a dynamics-dependent quasi-potential that typically must be found numerically. Here we extend CNT to a class of active phase separating systems governed by a minimal field-theoretic model (Active Model B+). In the small noise and supersaturation limits that CNT assumes, we compute analytically the quasi…
Recent theories of nucleation that go beyond Classical Nucleation Theory predict that diffusion-limited nucleation of both liquid droplets and of crystals from a low-density vapor (or weak solution) begins with long-wavelength density fluctuations. This means that in the early stages of nucleation, 'clusters' can have low density but large spatial extent, which is at odds with the classical picture of arbitrarily small clusters of the condensed phase. We present the results of kinetic Monte Carlo simulations using Forward Flux Sampling to show that these predictions are confirmed: namely that …
Vapor condensation on nanoparticle with radius smaller than the Kelvin radius is considered as fluctuation or as the heterogeneous nucleation. The expression for steady-state heterogeneous nucleation rate is obtained. Nucleation on negatively charged nanoparticles is discussed. The report was made at 17th International Nucleation and Atmospheric Conferences, Galway, Ireland, 2007.
The conventional theory of homogeneous and heterogeneous nucleation in a supersaturated vapor is tested by Monte Carlo simulations of the lattice gas (Ising) model with nearest-neighbor attractive interactions on the simple cubic lattice. The theory considers the nucleation process as a slow (quasi-static) cluster (droplet) growth over a free energy barrier $ΔF^*$, constructed in terms of a balance of surface and bulk term of a "critical droplet" of radius $R^*$, implying that the rates of droplet growth and shrinking essentially balance each other for droplet radius $R=R^*$. For heterogeneous…
We analyze the properties of naturally formed nano-bubbles in Lennard-Jones molecular dynamics simulations of liquid-to-vapor nucleation in the boiling and the cavitation regimes. The large computational volumes provide a realistic environment at unchanging average temperature and liquid pressure, which allows us to accurately measure properties of bubbles from their inception as stable, critically sized bubbles, to their continued growth into the constant speed regime. Bubble gas densities are up to 50$%$ lower than the equilibrium vapor densities at the liquid temperature, yet quite close to…
Heterogeneous nucleation is studied by Monte Carlo simulations and phenomenological theory, using the two-dimensional lattice gas model with suitable boundary fields. A chemical inhomogeneity of length b at one boundary favors the liquid phase, while elsewhere the vapor is favored. Switching on the bulk field favoring the liquid, nucleation and growth of the liquid phase starting from the region of the chemical inhomogeneity is analyzed. Three regimes occur: for small fields, the critical droplet radius is so large that a critical droplet having the contact angle required by Young's equation i…
Using molecular simulation, we study the nucleation of liquid droplets from binary mixtures and determine the free energy of nucleation along entropic pathways. To this aim, we develop the $μ_1 μ_2 VT-S$ method, based on the grand-canonical ensemble modeling the binary mixture, and use the entropy of the system $S$ as the reaction coordinate to drive the formation of the liquid droplet. This approach builds on the advantages of the grand-canonical ensemble, which allows for the direct calculation of the entropy of the system and lets the composition of the system free to vary throughout the nu…
We investigate a new mechanism for the cosmological QCD phase transition: inhomogeneous nucleation. The primordial temperature fluctuations, measured to be $δT/T \sim 10^{-5}$, are larger than the tiny temperature interval, in which bubbles would form in the standard picture of homogeneous nucleation. Thus the bubbles nucleate at cold spots. We find the typical distance between bubble centers to be a few meters. This exceeds the estimates from homogeneous nucleation by two orders of magnitude. The resulting baryon inhomogeneities may affect primordial nucleosynthesis.
Unlike fluids at thermal equilibrium, biomolecular mixtures in living systems can sustain nonequilibrium steady states, in which active processes modify the conformational states of the constituent molecules. Despite qualitative similarities between liquid--liquid phase separation in these systems, the extent to which the phase-separation kinetics differ remains unclear. Here we show that inhomogeneous chemical reactions can alter the nucleation kinetics of liquid--liquid phase separation in a manner that is consistent with classical nucleation theory, but can only be rationalized by introduci…
Structural defects in ion crystals can be formed during a linear quench of the transverse trapping frequency across the mechanical instability from a linear chain to the zigzag structure. The density of defects after the sweep can be conveniently described by the Kibble-Zurek mechanism. In particular, the number of kinks in the zigzag ordering can be derived from a time-dependent Ginzburg-Landau equation for the order parameter, here the zigzag transverse size, under the assumption that the ions are continuously laser cooled. In a linear Paul trap the transition becomes inhomogeneous, being th…
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 nuclea…
The theoretical description of the heterogeneous nucleation kinetics is presented. This description takes into account the perturbation of the vapor phase initiated by the growing droplets. The form of the density profile around the growing droplet is analyzed which leads to some special approximations. Then the process of nucleation in the whole system is described. As the result all main characteristics of the process are determined analytically.
The effect of molecule size (excluded volume) and the range of interaction on the surface tension, phase diagram and nucleation properties of a model globular protein is investigated using a combinations of Monte Carlo simulations and finite temperature classical Density Functional Theory calculations. We use a parametrized potential that can vary smoothly from the standard Lennard-Jones interaction characteristic of simple fluids, to the ten Wolde-Frenkel model for the effective interaction of globular proteins in solution. We find that the large excluded volume characteristic of large macrom…
The times required for reaching the power-law-in-time growth of the droplet radius after nucleation of a markedly supercritical binary droplet, are found analytically and estimated at diffusion-controlled or free-molecular regimes of isothermal binary vapor condensation. The process of setting steady concentration in the binary solution within the growing droplet at the same regimes of binary condensation has been analyzed. It has been shown that setting the steady droplet concentration has also a power-law character in time. The parameters of the power law are specified for each regime of bin…
Thanks to an expansion with respect to densities of energy, mass and entropy, we discuss the concept of thermocapillary fluid for inhomogeneous fluids. The non-convex state law valid for homogeneous fluids is modified by adding terms taking into account the gradients of these densities. This seems more realistic than Cahn and Hilliard's model which uses a density expansion in mass-density gradient only. Indeed, through liquid-vapor interfaces, realistic potentials in molecular theories show that entropy density and temperature do not vary with the mass density as it would do in bulk phases. In…
The search for local structures within a disordered medium has led to proposals of several methods for probing transient short-range symmetry in a homogeneous mono-atomic liquid. We offer a comparison of different characterizations of such local structure in an in-homogeneous liquid. We simulate the interfaces between a Lennard-Jones liquid and (i) a flat Lennard-Jones wall, (ii) a wall of face-centered cubic-packed Lennard-Jones particles, and (iii) a Lennard-Jones vapor. The interface density distributions in cases (i) and (ii) have oscillations extending several particle diameters into the …
A statistical field theory for an inhomogeneous liquid, a planar liquid/vapor interface, is devised from first principles. The grand canonical partition function is represented via a Hubbard-Stratonovitch tranformation leading, close to the critical point, to the usual $ϕ^4$ scalar field theory which is then rigorously considered at the one-loop level. When further simplified it yields the well-known capillary wave theory without any ad hoc phenomenological parameter. Internal coherence of the one-loop approximation is discussed and good overall qualitative agreement with recent numerical simu…
Recent very large molecular dynamics simulations of homogeneous nucleation with $(1-8) \cdot 10^9$ Lennard-Jones atoms [Diemand et al. J. Chem. Phys. {\bf 139}, 074309 (2013)] allow us to accurately determine the formation free energy of clusters over a wide range of cluster sizes. This is now possible because such large simulations allow for very precise measurements of the cluster size distribution in the steady state nucleation regime. The peaks of the free energy curves give critical cluster sizes, which agree well with independent estimates based on the nucleation theorem. Using these res…
We study the nucleation of the liquid phase from a supersaturated vapor in two dimensions (2D). Using different Monte Carlo simulation methods, we calculate the free energy barrier for nucleation, the line tension and also investigate the size and shape of the critical nucleus. The study is carried out at an intermediate level of supersaturation(away from the spinodal limit). In 2D, a large cut-off in the truncation of the Lennard-Jones (LJ) potential is required to obtain converged results, whereas low cut-off (say, $2.5σ$ is generally sufficient in three dimensional studies, where $σ$ is the…
Monte Carlo simulations of crystal nuclei coexisting with the fluid phase in thermal equilibrium in finite volumes are presented and analyzed, for fluid densities from dense melts to the vapor. Generalizing the lever-rule for two-phase coexistence in the canonical ensemble to finite volume, "measurements" of the nucleus volume together with the pressure and chemical potential of the surrounding fluid allows to extract the surface free energy of the nucleus. Neither the knowledge of the (in general non-spherical) nucleus shape nor of the angle-dependent interface tension is required for this ta…
We present results from large-scale molecular dynamics (MD) simulations of homogeneous vapor-to-liquid nucleation. The simulations contain between one and eight billion Lennard-Jones (LJ) atoms, covering up to 1.2 μs (56 million time-steps). They cover a wide range of supersaturation ratios, S=1.55 to 10^4, and temperatures from kT = 0.3 to 1.0 ε (where ε is the depth of the LJ potential, and k the Boltzmann constant). We have resolved nucleation rates as low as 10^{17} cm^{-3} s^{-1} (in the argon system), and critical cluster sizes as large as 100 atoms. Recent argon nucleation experiments p…
