Classical Nucleation Model Research Articles

A general model to evaluate the energy barrier to nucleation in vapour system by considering real gas approaches of the equation of state

Researchers have advanced theories and models of spontaneous homogeneous nucleation which result in the so-called “classical nucleation theory” (CNT). However, the corresponding algebraic equations to calculate the classical droplet radius and nucleation rate (classical nucleation model) widely used in engineering calculations e.g. to predict condensation in nozzles and turbomachines consider assumptions regarding the ideal gas law. Due to the discussion over real gases in recent years, many works apply the CNT probably with modifications for gases like CO2 whose working state is in many cases much closer to the critical point rather than e.g. water. However, a clear review of the modelling process might not be shown which discusses the choice of assumptions and the approaches of the equations of state (EOS).As the first step of understanding the impact of the “distance to the critical point” on nucleation processes, this work aims to model the energy barrier to nucleation ΔG∗, through two derivations. The first one assumes a constant liquid density and considers cubic, virial and free-energy-based approaches of EOS respectively. The second counts a free energy-based approach and a varying liquid density. Eventually, an evident deviation between the presented and several existing models is detected. For CO2, the isothermal compression term should be considered and a constant liquid density can be counted within a deviation of 7 %. Both cubic and virial models do not correspond to the free-energy-based one in the vicinity of the critical point.

Location-Dependent Phase Transformation Kinetics During Laser Wire Deposition Additive Manufacturing of Ti–6Al–4V

This work models the microstructural evolution as a function of position in laser-wire deposited Ti–6Al–4V parts using a classical multi-component JMAK nucleation and growth model with additive isothermal time-steps. Model predictions are compared to experimental observations. The model can be used to interpret nucleation and growth kinetics of various characteristic features of the microstructure that are inaccessible by experiments. It explains the presence of “layer bands” at specific locations unrelated to the weld bead structure. The last re-heating (of multiple thermal cycles) of a solidified layer, and where it peaks, plays a key role in increasing the nucleation density at specific locations, resulting in full α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-colony microstructures constituting the “layer bands,” while the rest of the build is predominantly basketweave. Additionally, changes in the basketweave α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-lath thickness as a function of the distance from the substrate are studied and compared with an Arrhenius-type function, with results highlighting a more complex relationship than an Arrhenius-type function would suggest.

Pathway of Room-Temperature Formation of CdSeS Magic-Size Clusters from Mixtures of CdSe and CdS Samples.

The synthetic application of prenucleation-stage samples of colloidal semiconductor quantum dots (QDs) is in its infancy. It is shown that when two prenucleation-stage samples of binary CdSe and CdS are mixed, ternary CdSeS magic-size clusters (MSCs) grow at room temperature in dispersion. As the amount of the CdS sample increases, the optical absorption of the CdSeS MSCs blueshifts from ≈380 to ≈360 nm. It is proposed that the cluster in the CdSe sample reacts with the CdS monomer from the CdS sample. The monomer substitution reaction of CdSe by CdS can proceed continuously; thus, CdSeS MSCs with tunable compositions are obtained. The present study provides compelling evidence that clusters formed in the prenucleation stage of QDs. The clusters are precursor compounds (PCs) of MSCs, transforming at room temperature with the thermoneutrality principle of isodesmic reactions. The nucleation and growth of QDs follows a multi-step non-classical instead of one-step classical nucleation model.

Deciphering the effect of nucleation/growth kinetics on the morphology of Li plating/stripping on vertically aligned carbon nanofibers - A low-tortuosity 3-D nanostructured carbon host

Reversible lithium metal anodes (LMAs) are the holy grail for future rechargeable lithium metal batteries. Three-dimensional (3-D) conductive hosts have been extensively explored as an effective approach to suppressing dendrite formation and enabling reversible Li plating/stripping. However, the microscopic morphologies of Li plating and their correlation with the cell performance are not clear. Herein we unravel these issues using the vertically aligned carbon nanofiber (VACNF) array as a model 3-D conductive carbon host which has a well-defined vertical low-tortuosity structure allowing observation of the intrinsic Li morphologies infiltrated into the 3-D host. The VACNF array indeed provides much higher stability and reversibility for Li plating/stripping due to its high surface area and lithiophilic properties. We found that Li plating on both VACNF array and planar Cu electrodes follows the classical nucleation and growth model. Though the low plating current density (≤0.10 mA/cm2) provides better cycling stability consistent with the Sand’s equation, it forms sparse irregular grains stacked with dendrite-like long Li fibers. In contrast, the moderate to high plating current densities (1.0 − 5.0 mA/cm2) produce more uniform Li morphologies consisting of smaller micro-columns or micro-spheres. By decoupling the plating and stripping current densities, we unravel that the more uniform micro-columnar Li infiltrated in the VACNF array obtained at the moderate plating current density (∼1.0 mA/cm2) indeed exhibits the highest cycling performance. This provides new insights into the relationship between macroscopic electrochemical tests and microscopic Li morphologies, aiding in optimizing the performance of LMAs based on 3-D conductive hosts.

Fully time-dependent cloud formation from a non-equilibrium gas-phase in exoplanetary atmospheres

Context. Recent observations suggest the presence of clouds in exoplanet atmospheres, but they have also shown that certain chemical species in the upper atmosphere might not be in chemical equilibrium. Present and future interpretation of data from, for example, CHEOPS, JWST, PLATO, and Ariel require a combined understanding of the gas-phase and the cloud chemistry. Aims. The goal of this work is to calculate the two main cloud formation processes, nucleation, and bulk growth consistently from a non-equilibrium gas phase. The aim is also to explore the interaction between a kinetic gas-phase and cloud microphysics. Methods. The cloud formation is modelled using the moment method and kinetic nucleation, which are coupled to a gas-phase kinetic rate network. Specifically, the formation of cloud condensation nuclei is derived from cluster rates that include the thermochemical data of (TiO2)N from N = 1 to 15. The surface growth of nine bulk Al, Fe, Mg, O, Si, S, and Ti binding materials considers the respective gas-phase species through condensation and surface reactions as derived from kinetic disequilibrium. The effect of the completeness of rate networks and the time evolution of the cloud particle formation is studied for an example exoplanet, HD 209458 b. Results. A consistent, fully time-dependent cloud formation model in chemical disequilibrium with respect to nucleation, bulk growth, and the gas-phase is presented and first test cases are studied. This model shows that cloud formation in exoplanet atmospheres is a fast process. This confirms previous findings that the formation of cloud particles is a local process. Tests on selected locations within the atmosphere of the gas-giant HD 209458 b show that the cloud particle number density and volume reach constant values within 1 s. The complex kinetic polymer nucleation of TiO2 confirms results from classical nucleation models. The surface reactions of SiO[s] and SiO2[s] can create a catalytic cycle that dissociates H2 to 2 H, resulting in a reduction of the CH4 number densities.

Crystallization thermodynamic properties and nucleation kinetics of Sr(OH)2·8H2O in an Sr(OH)2·8H2O-H2O system

There have been few reports in literature on the crystallization thermodynamic properties and nucleation kinetics of strontium hydroxide octahydrate despite their great importance to the reactor design and the regulation of crystal properties. In this work, the crystallization thermodynamic properties of Sr(OH)2·8H2O in pure water were investigated using a dynamic temperature-controlled process and an equilibrium process. The metastable zone width (MSZW) was examined under different crystallization process conditions including stirrer speed, solution concentration, cooling rate, and different strategies in adding seed crystals. In addition, models for the nucleation kinetics and MSZW were successfully constructed based on the classical nucleation model. Based on the data provided, it can be concluded that the dissolution of Sr(OH)2·8H2O in pure water is an entropy-driven endothermic process and its MSZW was gradually narrowed with an increase in stirring speed and concentration and with a decrease in cooling rate. Additionally, adding seeds was an effective strategy to narrow the metastable zone. Therefore, this endeavor provided a rational principle for subsequential investigations of Sr(OH)2·8H2O and other compounds.

In situ observation of the phase transformation kinetics of bismuth during shock release

A time-resolved x-ray diffraction technique is employed to monitor the structural transformation of laser-shocked bismuth. Results reveal a retarded transformation from the shock-induced Bi-V phase to a metastable Bi-IV phase during the shock release, instead of the thermodynamically stable Bi-III phase. The emergence of the metastable Bi-IV phase is understood by the competitive interplay between two transformation pathways towards the Bi-IV and Bi-III, respectively. The former is more rapid than the latter because the Bi-V to B-IV transformation is driven by interaction between the closest atoms while the Bi-V to B-III transformation requires interaction between the second-closest atoms. The nucleation time for the Bi-V to Bi-IV transformation is determined to be 5.1±0.9 ns according to a classical nucleation model. This observation demonstrates the importance of the formation of the transient metastable phases, which can change the phase transformation pathway in a dynamic process.

Insights Into Curie‐Temperature and Phase Formation of Ferroelectric Hf1−xZrxO2 with Oxygen Defects from a Leveled Energy Landscape

AbstractThe phase composition of HZO thin films is critical for the ferroelectric and electrical properties of the films and the devices they are integrated into. Optimization is a major challenge since the phase formation depends significantly on many influencing variables that are only partially understood so far. The Curie temperature is identified as an important parameter for understanding the behavior, since it depends sensitively on Zr content, the density of oxygen‐related defects, layer thickness, and external stress. A two‐step process, phase formation by pure kinetic transformation followed by nucleation, is proposed for phase formation. This is necessary because nucleation theory alone cannot explain the experimentally observed dependence on oxygen content. The classical nucleation model is modified at two crucial points. First, the polycrystalline structure is incorporated which allows the size effect to be implemented. Furthermore, the interface energies between the child and parent phase, which result from static ab initio calculations, are rescaled from dynamical effects. The resulting model is used to calculate the phase fractions during thermal processing. The results for the most important influencing variables are discussed and compared with experimental results. The causes of the undesired monoclinic phase are further analyzed.

Investigation on the age-hardening behavior of Al–Si–Cu–Mg cast alloys by mechanical testing, microstructural characterization, and a CNGT based precipitation model

Abstract In the present contribution, the age-hardening behavior of two commercial Al–Si–Cu–Mg alloys was studied during heat treatments at 200 °C and 240 °C. Size and volume fraction of precipitates were characterized by means of transmission electron microscopy, small-angle X-ray scattering, and small angle neutron scattering. Additionally, tensile tests were performed to evaluate the mechanical behavior. Finally, a multi-component multi-phase classical nucleation and growth theory precipitation model was developed to predict the evolution of Mg- and Cu-containing precipitates and their hardening effect. The simulations were in good agreement with experimental findings.

Deconvolution of main hydration kinetic peaks in properly sulfated Portland cements with boundary nucleation and growth models and relation to early-age concrete strength development

When the hydration kinetics of properly sulfated Portland cements is monitored by heat flow calorimetry, the initial dissolution peak and the dormant period is followed by the main hydration peak – formation of calcium-silicate-hydrates and portlandite, the so-called silicate peak. This main peak is superimposed by a peak associated to further tricalcium aluminate dissolution and a surge in precipitation of sulfoaluminates (the so-called sulfate depletion peak) right after sulfate added for setting control is depleted in the material system. In this paper strategies for the deconvolution of silicate peak and sulfate depletion peak are presented. Hereby, the boundary nucleation and growth model and the classical bulk nucleation and growth model serve as tools for deconvolution of calorimetric data. We discuss obtained kinetic parameters and their temperature dependency. The boundary nucleation and growth model for constant nucleation rate emerged as the most suitable model for the silicate reaction, the bulk nucleation and growth model with an order of three, indicating site saturation, as adequate for the sulfate depletion peak. The reaction enthalpy for tricalcium silicate hydration was obtained as 503 J/g, remarkably close to literature values. Calorimetric data related to the sulfate depletion peak is corroborated by X-ray diffraction analysis of formation history of crystalline hydration products. Furthermore, the history of the (silicate) degree of reaction as predicted by the boundary nucleation and growth model for different temperature histories is related to experimentally obtained early-age strength histories. • Investigation of hydration kinetics of commercial, well-sulfated Portland cement. • Main peak (C-S-H and portlandite formation) superimposed by sulfate depletion peak. • Strategies for deconvolution of peaks with Avrami–Cahn models as tools for analysis. • Degree of silicate reaction history is linked to early-age strength gain.

(Invited) The Impact of Interface Layer on Li Plating and Stripping Morphology

Li metal is the preferred anode material for high energy Li batteries. However, a stable plating and stripping of Li metal at the anode−solid electrolyte interface (SEI) remains a significant challenge, particularly at practically feasible current densities. The formation of the SEI layer can significantly change the Li-nucleation density and growth morphology. With multiscale modeling, these factors can be investigated in more detail. Upon, lithiation, while a classical nucleation and growth model can describe the nucleation density of Li clusters and the charging rates in ultrahigh vacuum, a trace amount of reactive gas (oxygen for example) can change Li growth morphology. Reactive molecular dynamics simulations revealed that the fine balance between Li growth rate and the reaction rate of the thin lithium-oxide shell on the surface of the metallic Li can lead to Li nanowire formation. During delithiation, the accumulation of Li vacancies can lead to void generation and interface delamination. A DFT-informed kinetic Monte Carlo (KMC) model was developed to investigate the impact of the SEI layer, stack pressure, and alloying effects. The contact area is then incorporated into a 1D Newman battery model to simulate the impact of the contact area on the battery performance.

Promoting effect of AlN foreign particles on crystallization of sodium sulfate decahydrate

Crystallization is a major separation technology in chemical process engineering and wastewater treatment. The promotion of nucleation is a primary concern because it affects both product purity and separation efficiency. In this study, the effect of aluminum nitride foreign particles on the crystallization of sodium sulfate decahydrate was investigated. Also the influence of aluminum nitride foreign particles on kinetics and thermodynamics of sodium sulfate decahydrate nucleation was determined. Induction time, metastable zone width and solubility of sodium sulfate decahydrate in the aluminum nitride foreign particles were determined by the application of a turbidity monitoring technique in stable and uniform temperature conditions by drip titration method unlike the traditional polythermal and isothermal methods commonly applied in the unstable temperature conditions. In the presence of aluminum nitride foreign particles, induction time and metastable zone width decreased 30% and 13% respectively and solubility did not show significant changes. The experimental results were analyzed by empirical secondary nucleation model, classical nucleation theory and empirical Kashchiev heterogeneous nucleation model. It was found that the secondary nucleation model can better illustrate the influence of aluminum nitride foreign particles on nucleation. In the presence of aluminum nitride foreign particles, the surface tension decreased which results in lower induction time.

Robust Simulations of Nanoscale Phase Change Memory: Dynamics and Retention.

A robust simulation framework was developed for nanoscale phase change memory (PCM) cells. Starting from the reaction rate theory, the dynamic nucleation was simulated to capture the evolution of the cluster population. To accommodate the non-uniform critical sizes of nuclei due to the non-isothermal conditions during PCM cell programming, an improved crystallization model was proposed that goes beyond the classical nucleation and growth model. With the above, the incubation period in which the cluster distributions reached their equilibrium was captured beyond the capability of simulations with a steady-state nucleation rate. The implications of the developed simulation method are discussed regarding PCM fast SET programming and retention. This work provides the possibility for further improvement of PCM and integration with CMOS technology.

Secondary magnesite formation from forsterite under CO2 sequestration conditions via coupled heterogeneous nucleation and crystal growth

Flow-only and coupled flow-diffusion experiments at 95 °C and 100 bars pCO2 carried out in micro-capillary tubes packed with forsterite mineral grains were used to constrain a coupled classical heterogeneous nucleation and crystal growth reactive transport model describing the formation of secondary magnesite. The study made use of a novel experimental setup in which one capillary tube for flow is connected via a three-way tee to a perpendicular capillary tube sealed at the distal end in which only molecular diffusion is allowed to occur—an experimental analogue of a single fracture-rock matrix system. While the high flux of CO2 bearing fluids and their low pH did not result in the formation of secondary carbonates in the flow-dominated channels, as much as 2.7% magnesite formed in a diffusion-controlled capillary tube sample after 300 hours of reaction. The precipitation of secondary magnesite was not uniformly distributed, however, but showed a distinctive peak shaped pattern along the sample reacted as quantified by RAMAN spectroscopic analysis. About 50% of the total magnesite precipitation formed within a narrow interval of a few millimeters close to the middle of the 3 cm diffusion sample. To simulate the behavior of the diffusion–reaction column, and in particular the pronounced mm scale peak at approximately 1.8 cm, an interfacial free energy of approximately 70 mJ·m−2 combined with a relatively high crystal growth rate for secondary magnesite was required. In agreement with the observations based on RAMAN spectroscopy, the simulations suggest that more than 50% of Mg2+ dissolved from the primary forsterite precipitated as secondary magnesite. The magnesite precipitation at the central peak position in the diffusion sample showed such rapid growth that it created a local minimum in pore fluids Mg2+ concentration close to the peak, creating a “nucleation shadow” that focused continued growth there as nearby regions had no nucleation seeds available for crystal growth. Continued crystal growth on the initial magnesite band acted to further increase the reactive surface area at this point, thus enhancing the spatially focused crystal growth rate and creating a positive feedback leading to pattern formation. This highlights the potentially critical role of an initial nucleation event in controlling mineral precipitation patterns in subsurface porous media, patterns that may determine how the pore structure subsequently evolves physically and chemically over time due to reactive flow and transport.

X-ray studies bridge the molecular and macro length scales during the emergence of CoO assemblies

The key to fabricating complex, hierarchical materials is the control of chemical reactions at various length scales. To this end, the classical model of nucleation and growth fails to provide sufficient information. Here, we illustrate how modern X-ray spectroscopic and scattering in situ studies bridge the molecular- and macro- length scales for assemblies of polyhedrally shaped CoO nanocrystals. Utilizing high energy-resolution fluorescence-detected X-ray absorption spectroscopy, we directly access the molecular level of the nanomaterial synthesis. We reveal that initially Co(acac)3 rapidly reduces to square-planar Co(acac)2 and coordinates to two solvent molecules. Combining atomic pair distribution functions and small-angle X-ray scattering we observe that, unlike a classical nucleation and growth mechanism, nuclei as small as 2 nm assemble into superstructures of 20 nm. The individual nanoparticles and assemblies continue growing at a similar pace. The final spherical assemblies are smaller than 100 nm, while the nanoparticles reach a size of 6 nm and adopt various polyhedral, edgy shapes. Our work thus provides a comprehensive perspective on the emergence of nano-assemblies in solution.

Phase field model for multiphase alloys under arbitrary thermal history: An application to IN718 super-alloy

Currently, it is still a significant challenge to predict the microstructural evolution of multiphase alloys in complex manufacturing processes such as joining, welding and additive manufacturing due to their complex thermodynamics and thermal history. In this work, we present a phase-field framework, aiming to predict the evolution of precipitate microstructure under arbitrary thermal history. Our model incorporates nucleation, growth and transformation by taking into account interfacial, chemical and elastic energies. In order to quantitatively describe the nucleation of phases, we leverage on experimental time-temperature-transformation data to parameterize the classical nucleation model. For demonstration, we employ super-alloy IN718, where thermodynamic and kinetic data is obtained using CALPHAD method and major precipitate phases, i.e. γ′,γ″ and δ and their variants are considered. The predicted continuous-cooling-transformation diagram is consistent with data existing in literature, thus validating the approach. We further predict the precipitate evolution under rapid thermal cycling to mimic a typical additive manufacturing process.

Classical Nucleation and Lattice Model Unite

Classical nucleation theory predicts the limit of superheat of liquids quite well. To come up with an equation for the limit of superheat of polymer solutions, the lattice model for polymer solutions was used to give the surface tension of polymer solutions. A formula for bubble nucleation in polymer solutions was derived by Jennings with the precursor equation dlnA/dK=1/(6K) where J=AexpK gives the nucleation rate for liquids. The aim of this paper was to show that the precursor equation holds for monomer in the polystyrene-cyclohexane system. Thus, the precursor equation is true for all molecular weight polymer. This happens because the surface tension of polystyrene is significantly more than cyclohexane and the influence of the surface tension dominates.

Vortex flows of moist air with non-equilibrium and homogeneous condensation

A small-disturbance model is presented for the complex dynamics of vortex flows of moist air in a straight, circular pipe with non-equilibrium and homogeneous condensation. The model explores the nonlinear interactions among the vortex near-critical swirl ratio and the small amount of water vapour in the air. The condensation rate is calculated according to classical nucleation and droplet growth models. The asymptotic analysis gives the similarity parameters that govern the flow problem. These are the flow inlet swirl ratio , the inlet Mach number , the initial humidity , the number of water molecules in a characteristic fluid element , the inlet centreline super-saturation ratio and the ratio of characteristic condensation and flow time scales . Also, the flow field may be described by an ordinary first-order nonlinear differential equation for the flow evolution coupled with a set of four first-order ordinary differential equations along the pipe for the calculation of the condensate mass fraction. An iterative numerical scheme which combines the Runge–Kutta integration technique for the flow dynamics with Simpson’s integration rule for the calculation of the condensation variables is developed. Specifically, equilibrium states are determined, including the possibility of the appearance of multiple states under the same boundary conditions, and the stability characteristics of these states are described. The model is used to study the effects of humidity and of energy supply from nanoscale condensation processes on the large-scale dynamics of vortex flows as well as the effect of flow swirl on condensation processes in swirling flows.

Numerical studies of nitrogen spontaneous condensation flow in laval nozzles using varying droplet growth models

Spontaneous condensation of gas at transonic or supersonic flow is a great concerned phenomenon in many cryogenic engineering fields. An accurate model is of great significance to better understand the mechanism of such condensation and arising issues. In this paper, numerical studies for the effect of droplet growth model modification on the nitrogen condensation flow through Laval nozzles were performed in ANSYS CFX via user defined language. The used four profiles of Laval nozzle were designed with different expansion rate (3000, 5000, 7000 and 10000 s−1) in Ji’s experiment (Ji 1990). The throat size is 6.5 (height) × 3.2 (width) mm. Firstly, the classical nucleation model with non-isothermal correction was successful to accurately predict the location of condensation onset in experiment by the down calibration for nucleation rate. The supercooled degree is about 9 K and nucleation rate is at the order of 1020 m−3·s−1. Secondly, the simulations of condensation flow using different droplet growth models including Gyarmathy and Young models were accomplished. Based on the analysis of interphase heat and mass transfer, the influence mechanism caused by modification for droplet growth rate on two-phase flow fields was revealed. The results indicate that this modification has a considerable effect on the development of droplet size and relaxation time of gas back to equilibrium, but little influence on location prediction of condensation onset. Finally, the effect of inlet total pressure on two-phase flow parameters was also analyzed. At low pressure, a larger supercooled degree (∼11 K) is required to trigger the occurrence of spontaneous condensation. Correspondingly, the influence intensity resulting from the change of droplet growth model on the interphase mass transfer and thermal fields will obtain some enhancement, but still inappreciable compare to the expansion effect.

Nucleation observed at atomic scale

Crystals form in storm clouds, metals, pharmaceuticals, and even in diseased tissues. Despite their ubiquity, scientists still don’t fully understand what happens when a liquid solution first starts to form a solid crystal, a step called nucleation. Now researchers have gotten their first glimpse of the details of the process, imaging individual atoms during nucleation in metal nanoparticles (Nature 2019, DOI: 10.1038/s41586-019-1317-x). The classical model of nucleation, in which a small number of atoms or molecules form a distinct solid structure called a nucleus, the starting point for a crystal, is accurate for some systems. But in the past 2 decades scientists have found evidence for nonclassical models of nucleation. These break the classical rules in different ways—for instance, by proceeding through a two-step process starting when a disorganized but concentrated group of atoms or molecules comes together before forming a defined nucleus. No modes of nucleation have been observed