Classical Nucleation Theory Research Articles

Evaluation of transition rates from nonequilibrium instantons

Equilibrium rate theories play a crucial role in understanding rare, reactive events. However, they are inapplicable to a range of irreversible processes in systems driven far from thermodynamic equilibrium like active and biological matter. Here we develop an efficient numerical method to compute the rate constant of rare nonequilibrium events in the weak-noise limit based on an instanton approximation to the stochastic path integral and illustrate its wide range of application. We demonstrate excellent agreement of the instanton rates with numerically exact results for a particle under a nonconservative force. We also study phase transitions in an active field theory. We elucidate how activity alters the stability of the two phases and their rates of interconversion in a manner that can be well described by modifying classical nucleation theory. Published by the American Physical Society 2024

What will be the Euclidean dimension of an Ising ferromagnetic cubic shell?

The equilibrium and nonequilibrium properties of an Ising ferromagnetic cubic shell have been extensively studied by Monte Carlo simulation using Metropolis single spin flip algorithm. Although, geometrically the Euclidean dimension of the cubical shell is three, interestingly, the Ising ferromagnetic cubic shell undergoes ferromagnetic phase transition at a temperature which is very close to that for two-dimensional Ising ferromagnet. Surprisingly, the Ising ferromagnetic cubic shell shows a strange (neither exponential nor stretched exponential) kind of relaxation behaviour, instead of exponential relaxation as usually observed in the two dimensional Ising ferromagnet. The metastable lifetime of a ferromagnetic Ising cubical shell is studied as a function of the applied magnetic field. Here also, the cubic shell behaves more likely a two-dimensional object as found from statistical analysis and comparison with Becker-Döring prediction of classical nucleation theory.

Modeling phase separation in solids beyond the classical nucleation theory: Application to FeCr.

Despite a large amount of work being devoted to study the phase separation in solids, the underlying physical mechanism responsible for such diffusive first-order phase transitions remains difficult to model outside the spinodal regime, i.e., in the nucleation and growth regime. This work presents an alternative of the classical nucleation theory for modeling phase separation in this regime, even for systems far from the solubility limit, i.e., for high degree of meta-stability where the classical nucleation theory does not hold. This method then allows a direct comparison between simulations and experiments always performed in solids with a high degree of meta-stability.

On the theory of the nucleation of gas bubbles at grain boundaries and incoherent inclusions

On the base of the critical analysis of two-dimensional models of the nucleation of gas filled bubbles at grain boundaries of helium-implanted specimens under the action of tensile stresses, a new model is developed within the framework of the Reiss theory of homogeneous nucleation in binary systems. This approach considers that gas bubbles are formed as a result of agglomeration in a binary system of vacancies and gas atoms at grain boundaries, avoiding significant simplifications of previous models based on the classical nucleation theory for single-component (unary) systems. The new model is extended to consider the nucleation of Xe bubbles at grain boundaries in UO2 under irradiation conditions and can be used for numerical analysis of experimental observations after the foreseen implementation in a fuel performance code. A similar approach can be applied to the nucleation and growth of gas bubbles on incoherent inclusions, such as those observed in irradiated ODS steels.

Development of an advanced hydride reorientation model for Zircaloy cladding and its experimental validation

Hydride reorientation, which occurs under hoop stress during cooling, stands out as a primary mechanism for material degradation in spent fuel management. The radial hydride fraction (RHF) is strongly involved in the mechanical integrity of cladding, highlighting the necessity for a robust modeling framework for quantitative analysis. However, the predictability of previous thermodynamic models for hydride reorientation in reactor-grade Cold Worked Stress Relieved (CWSR) Zircaloy has been hindered due to the intricate nature of hydride reorientation and the difficulties in characterizing microstructures. Recent successful EBSD characterization of reactor-grade CWSR Zircaloy have revealed valuable insights into microstructural characteristics of hydrides, enabling advancements in the modeling framework of hydride reorientation. This study aims to develop a thermodynamic model specifically focused on predicting the RHF. The developed thermodynamic model, based on classical nucleation theory, integrates aforementioned microstructural findings, combined with the Hydride-Nucleation-Growth-Dissolution (HNGD) model to capture transient precipitation behavior during cooling. Extensive experimental validations demonstrate enhanced predictability of the model. Additionally, the study examines the sensitivities of hydride reorientation to hydrogen concentration, applied stress, and cooling rate. It also provides predictions on reorientation behavior for engineering implications such as extension of wet storage, matrix hardening, recrystallization, and thermal cycling, supported by plausible explanations rooted in the underlying physical mechanisms elucidated through the model.

Numerical simulations of freezing behaviors of water droplets impacting cold hydrophobic surfaces

Freezing, as a prominent phase transition phenomenon, occurs frequently in nature and industry. This study employs a combination of the Volume of Fluid(VOF) model, solidification model, and a dynamic contact angle model to simulate the dynamics and solidification process of droplets impacting supercooled surfaces. Two distinct freezing morphologies, central-concave and central-convex, are observed. These different morphologies are attributed to the competition between the time scales associated with droplet impact dynamics and solidification (retraction time tr and solidification time tf,p). Thus, we propose a scaling law to predict the occurrence of different morphologies: when tr < tf,p, the final freezing morphology exhibits a central-convex shape, while when tr > tf,p, it exhibits a central-concave shape. In order to explore effects of nucleation on solidification, the classical nucleation theory is adopted to determine the nucleation time, and thus the freezing behaviors can be well captured. Furthermore, we examine the effect of volume expansion during phase transition on the freezing behaviors, and find that the increase in freezing time due to volumetric expansion can be well predicted by the scaling law tf ∼ Δ(H2)/ΔT.

Crystallization Kinetics of Tacrolimus Monohydrate in an Ethanol–Water System

Nucleation and growth during the crystallization process are crucial steps that determine the crystal structure, size, morphology, and purity. A thorough understanding of these mechanisms is essential for producing crystalline products with consistent properties. This study investigates the solubility of tacrolimus (FK506) in an ethanol–water system (1:1, v/v) and examines its crystallization kinetics using batch crystallization experiments. Initially, the solubility of FK506 was measured, and classical nucleation theory was employed to analyze the induction period to determine interfacial free energy (γ) and other nucleation parameters, including the critical nucleus radius (r*), critical free energy (∆G*), and the molecular count of the critical nucleus (i*). Crystallization kinetics under seeded conditions were also measured, and the parameters of the kinetic model were analyzed to understand the effects of process states such as temperature on the crystallization process. The results suggested that increasing temperature and supersaturation promotes nucleation. The surface entropy factor (f) indicates that the tacrolimus crystal growth mechanism is a two-dimensional nucleation growth. The growth process follows the particle size-independent growth law proposed by McCabe. The estimated kinetic parameters reveal the effects of supersaturation, temperature, and suspension density on the nucleation and growth rates.

Modeling and Experimental Validation of Cell Morphology in Microcellular-Foamed Polycaprolactone.

This study investigates the modeling and experimental validation of cell morphology in microcellular-foamed polycaprolactone (PCL) using supercritical carbon dioxide (scCO2) as the blowing agent. The microcellular foaming process (MCP) was conducted using a solid-state batch foaming process, where PCL was saturated with scCO2 at 6 to 9 MPa and 313 K, followed by depressurization at a rate of -0.3 and -1 MPa/s. This study utilized the Sanchez-Lacombe equation of state and the Peng-Robinson-Stryjek-Vera equation of state to model the solubility and density of the PCL-CO2 mixture. Classical nucleation theory was modified and combined with numerical analysis to predict cell density, incorporating factors such as gas absorption kinetics, the role of scCO2 in promoting nucleation, and the impact of depressurization rate and saturation pressure on cell growth. The validity of the model was confirmed by comparing the theoretical predictions with experimental and reference data, with the cell density determined through field-emission scanning electron microscopy analysis of foamed PCL samples. This study proposes a method for predicting cell density that can be applied to various polymers, with the potential for wide-ranging applications in biomaterials and industrial settings. This research also introduces a Python-based numerical analysis tool that allows for easy calculation of solubility and cell density based on the material properties of polymers and penetrant gases, offering a practical solution for optimizing MCP conditions in different contexts.

Metastable water at several compression rates and its freezing kinetics into ice VII

Water can be dynamically over-compressed well into the stability field of ice VII. Whether water then transforms into ice VII, vitreous ice or a metastable novel crystalline phase remained uncertain. We report here the freezing of over-compressed water to ice VII by time-resolved X-ray diffraction. Quasi-isothermal dynamic compression paths are achieved using a dynamic-piezo-Diamond-Anvil-Cell, with programmable pressure rise time from 0.1 ms to 100 ms. By combining the present data set with those obtained on various ns-dynamical platforms, a complete evolution of the solidification pressure of metastable water versus the compression rate is rationalized within the classical nucleation theory framework. Also, when crystallization into ice VII occurs in between 1.6 GPa and 2.0 GPa, that is in the stability field of ice VI, a structural evolution over few ms is then observed into a mixture of ice VI and ice VII that seems to resolve apparent contradictions between previous results.

Incubation of Amyloidogenic Peptides in Reverse Micelles Allow Active Control of Oligomer Size and Study of Protein-Protein Interactions.

Studies of the structure and dynamics of oligomeric aggregates of amyloidogenic peptides pose challenges due to their transient nature. This concept article provides a brief overview of various nucleation mechanisms with reference to the classical nucleation theory and illustrates the advantages of incubating amyloidogenic peptides in reverse micelles (RMs). The use of RMs not only facilitates size regulation of oligomeric aggregates but also provides an avenue to explore protein-protein interactions among the oligomeric aggregates of various amyloidogenic peptides. Additionally, we envision the feasibility of preparing brain tissue-derived oligomeric aggregates using RMs, potentially advancing the development of monoclonal antibodies with enhanced potency against these pathological species in vivo.

Metastable zone width and nucleation kinetics of vanillyl alcohol crystallization in various solvents

In this study, linear cooling batch crystallization of vanillyl alcohol using a laser power system was carried out to experimentally measure the metastable zone widths of vanillyl alcohol in selected solvents (ethanol, 2-propanol, and methyl cyanide) at various saturated temperatures (40, 50, and 60 °C), cooling rates (0.5, 1, 1.5, and 2 °C/min), and agitation speeds (300, 400 rpm). Besides, the sonocrystallization of vanillyl alcohol in ethanol was conducted for measuring MSZW at a various ultrasonic amplitudes (0 %, 25 %, 50 %, and 100 %) at a fixed temperature of 40 °C, cooling rate of 1 °C/min, and agitation speed of 300 rpm. For all three solvents, the MSZW decreases with saturation temperature, while it increases with cooling rate, and this trend doesn’t change for different agitation speeds. Three models—the self-consistent Nývlt-like model, the classical 3D nucleation theory model, and the simplified linear integral model based on classical nucleation theory (CNT)—were employed to estimate the nucleation kinetic parameters for vanillyl alcohol in three solvents. The goodness fit of the models were checked by the coefficient of determination (R-squared). The R-squared values reflected a very good fit between the experimental and predicted values and implied that the models are reliable to estimate nucleation kinetic parameters. Additionally, the interfacial energy between vanillyl alcohol and the solvents was observed to decrease with increasing temperature. Overall, the results indicate that a low nucleation order and low interfacial energy suggest weak solute–solvent interactions, making nucleation easier in the following order: methyl cyanide > ethanol > 2-propanol.

A microscopic approach to crystallization: Challenging the classical/non-classical dichotomy.

We present a fundamental framework for the study of crystallization based on a combination of classical density functional theory and fluctuating hydrodynamics that is free of any assumptions regarding order parameters and that requires no input other than molecular interaction potentials. We use it to study the nucleation of both droplets and crystalline solids from a low-concentration solution of colloidal particles using two different interaction potentials. We find that the nucleation pathways of both droplets and crystals are remarkably similar at the early stages of nucleation until they diverge due to a rapid ordering along the solid pathways in line with the paradigm of "non-classical" crystallization. We compute the unstable modes at the critical clusters and find that despite the non-classical nature of solid nucleation, the size of the nucleating clusters remains the principle order parameter in all cases, supporting a "classical" description of the dynamics of crystallization. We show that nucleation rates can be extracted from our formalism in a systematic way. Our results suggest that in some cases, despite the non-classical nature of the nucleation pathways, classical nucleation theory can give reasonable results for solids but that there are circumstances where it may fail. This contributes a nuanced perspective to recent experimental and simulation work, suggesting that important aspects of crystal nucleation can be described within a classical framework.

A model for the precipitate transformation of Mg–Si-rich clusters into Mg5Si6 β″ in Al–Mg–Si aluminum alloys

Abstract A model is developed that describes the kinetics of precipitate transformations in the course of natural and artificial aging of Al alloys containing Mg and Si additions. In our approach, the disordered Mg–Si-rich clusters, which form during natural aging in the highly supersaturated Al matrix, can directly transform into the monoclinic Mg5Si6 (β″), without prior dissolution of the clusters and independent nucleation of β″ in the Al matrix. The transformation rate is evaluated with classical nucleation theory (CNT), assuming that the clusters represent an infinitely large matrix phase in which the β″ precipitates can nucleate. The adapted CNT model is described, and the basic features of the precipitate transformation are discussed in a parameter study. The model can also account for the observation that, during natural aging, the parent clusters occur in a variety of Mg to Si ratios, all of which have a characteristic probability of either transforming into the β″ phase or dissolving.

Ice nucleation in supercooled water under shear

Classical nucleation theory (CNT) provides a good framework for understanding the nucleation phenomenon in metastable conditions. However, there is still no satisfactory understanding of the shear effect. Here, we propose an ice nucleation model considering the rebounded deformation energy to analyze the ice nucleation rate. It is concluded that: (1) The effect is more pronounced by changing the contact angle to regulate the free energy when both the supercooled degree and the shear rate are small. (2) The variation trend of the nucleation ratio Jhet,sh/Jhet,no increases at first but it may decrease when the contact angle is small, or keeps increasing when the contact angle is large. (3) The probability of the shear nucleation is about several ten orders of magnitude to that of static nucleation. (4) The key factor in the acceleration of the supercooled liquid nucleation is the released deformation energy after encountering the obstruction.

Theoretical analysis of the initial nucleation characteristics of trace water vapor frosting on a cryogenic surface

In certain extreme conditions characterized by ultra-low water vapor content and ultra-low temperatures (e.g. cryogenic wind tunnel), trace water vapor frosting can also pose significant hazards. Considering the crucial role of initial nucleation on subsequent frosting processes, this study firstly investigated and compared the nucleation characteristics of frosting at different water vapor contents (0.1−1000 ppmv) from humid air to trace water vapor based on classical nucleation theory. The preferred nucleation phase diagrams and critical nucleation conditions were analyzed. Sensitivity of the nucleation mass transfer rate, as well as the frosting pathways were further identified. Results show that the nucleation characteristics of trace water vapor frosting (<100 ppmv) is significantly different from that of humid air frosting (>1000 ppmv). Trace water vapor frosting is more inclined towards desublimation nucleation due to its lower nucleation temperature and larger critical contact angle. The critical contact angles for 1000 ppmv, 100 ppmv and 0.1 ppmv are 49°, 75° and 120°, respectively. Furthermore, trace water vapor nucleation requires a greater subcooling degree, has a smaller critical nucleation radius, and is highly sensitive to surface contact angle and further-subcooling degree. At a surface contact angle of 120°, the nucleation subcooling degree of 0.1 ppmv is 2.9 K greater than that of 1000 ppmv. This study helps understanding the nucleation characteristics under different water vapor content conditions, which indicates that reducing subcooling degree and increasing contact angle are more effective anti-frosting methods than reducing water vapor content for trace water vapor frosting.

Bridging classical nucleation theory and molecular dynamics simulation for homogeneous ice nucleation.

Water freezing, initiated by ice nucleation, occurs widely in nature, ranging from cellular to global phenomena. Ice nucleation has been experimentally proven to require the formation of a critical ice nucleus, consistent with classical nucleation theory (CNT). However, the accuracy of CNT quantitative predictions of critical cluster sizes and nucleation rates has never been verified experimentally. In this study, we circumvent this difficulty by using molecular dynamics (MD) simulation. The physical properties of water/ice for CNT predictions, including density, chemical potential difference, and diffusion coefficient, are independently obtained using MD simulation, whereas the calculation of interfacial free energy is based on thermodynamic assumptions of CNT, including capillarity approximation among others. The CNT predictions are compared to the MD evaluations of brute-force simulations and forward flux sampling methods. We find that the CNT and MD predicted critical cluster sizes are consistent, and the CNT predicted nucleation rates are higher than the MD predicted values within three orders of magnitude. We also find that the ice crystallized from supercooled water is stacking-disordered ice with a stacking of cubic and hexagonal ices in four representative types of stacking. The prediction discrepancies in nucleation rate mainly arise from the stacking-disordered ice structure, the asphericity of ice cluster, the uncertainty of ice-water interfacial free energy, and the kinetic attachment rate. Our study establishes a relation between CNT and MD to predict homogeneous ice nucleation.

Unveiling the nucleation and growth of Zr oxide precipitates in internally oxidized Nb3Sn superconductors

We report on atomic-scale analyses of nucleation and growth of Zr oxide precipitates and the microstructural evolution of internally oxidized Nb3Sn wires for high-field superconducting magnet applications, utilizing atom probe tomography (APT), transmission electron microscopy (TEM), and first-principles calculations. APT analyses reveal that oxygen and zirconium are already segregated at grain boundaries (GBs) in the unreacted Nb-1Zr-4Ta (at%) alloy prior to forming Nb3Sn through reacting the Nb alloy with Sn and SnO2. After forming Nb3Sn, Zr oxide precipitates nucleate both at the Nb3Sn/Nb heterophase interfaces and in the Nb3Sn grains, driven by the small solubilities of Zr and O in Nb3Sn compared to their value in Nb. A high number density (Nv) of Zr oxide nanoprecipitates is observed in the Nb3Sn layers, ∼1023 m−3, with a mean diameter <10 nm for a heat treatment at 625 °C. Quantitative APT and TEM analyses of the Zr oxide precipitates in the reacted Nb3Sn layers elucidate details of the nucleation, growth, and coarsening processes of the Zr oxide precipitates in Nb3Sn. First-principles calculations and classical nucleation theory are employed to study the nucleation of Zr oxide precipitates in Nb3Sn and to estimate the maximum energy barrier and critical radius for nucleation. Our research unveils the kinetic pathways for nucleation and growth of Zr oxide precipitates and the microstructural evolution of Nb3Sn layers, which helps to understand and improve the superconducting properties of internally oxidized Nb3Sn wires for use in high-field superconducting magnets.

Free Energy Evaluation of Cavity Formation in Metastable Liquid Based on Stochastic Thermodynamics.

Nucleation is a fundamental and general process at the initial stage of first-order phase transition. Although various models based on the classical nucleation theory (CNT) have been proposed to explain the energetics and kinetics of nucleation, detailed understanding at nanoscale is still required. Here, in view of the homogeneous bubble nucleation, we focus on cavity formation, in which evaluation of the size dependence of free energy change is the key issue. We propose the application of a formula in stochastic thermodynamics, the Jarzynski equality, for data analysis of molecular dynamics (MD) simulation to evaluate the free energy of cavity formation. As a test case, we performed a series of MD simulations with a Lennard-Jones (LJ) fluid system. By applying an external spherical force field to equilibrated LJ liquid, we evaluated the free energy change during cavity growth as the Jarzynski's ensemble average of required works. A fairly smooth free energy curve was obtained as a function of bubble radius in metastable liquid of mildly negative pressure conditions.

In-situ synchrotron X-ray investigating crystallization of isotactic polypropylene with β-nucleating agent during injection molding

The coupling effects of mold temperature and packing pressure fields on structure evolution of isotactic polypropylene with β-nucleating agent (iPP/β) during injection molding were systematically investigated using high-throughput in-situ synchrotron X-ray diffraction/scattering technology for the first time. A small number of β-crystals are generated in pure iPP and iPP/β due to the shear flow introduced by the filling stage. The formation of β-crystals induced by β-nucleating agent (β-NA) in iPP/β commenced after about 1.8 s of melt filling. The two-stage growth of β-crystals in iPP/β exhibits differences in growth time and orientation. Both the decrease in the lattice spacing and the β-α transition in iPP/β occur at the packing stage and are suppressed at high mold temperatures (≥ 90 °C). The polymorphism competitive growth of iPP/β is revealed, and the effects of coupling external fields on it are well elucidated by combining finite element simulation and classical nucleation theory. Additionally, the scattering intensity and orientation of iPP/β lamellae initially increase to a maximum value and subsequently decrease until reaching the equilibrium with time. Our current work lays a solid foundation to precisely customize the crystal structure of iPP/β in practical processing and thus to produce high-performance iPP products.

A Multiscale Computational Modeling Framework to Quantify Microstructural Effects on Nucleation of Li Metal on Carbon Scaffold Architectures

Carbon materials with scaffold architecture have been shown to mitigate the non-uniform Li plating/stripping issues during cycling, which is critical for achieving significantly enhanced energy densities and improved safety of Li metal batteries. However, optimal design of the architecture and microstructure of the carbon materials requires fundamental understanding of the Li metal nucleation behavior at different length scales. In this poster, we present a multiscale computational modeling framework to quantify the microstructural effects of carbon materials on the tendency of Li metal nucleation during plating by integrating atomic-scale simulation, mesoscale microstructure-based homogenization, continuum-scale electrochemistry modeling, and classical nucleation theory. We modeled the architecture of the carbon scaffold and the electrochemistry of Li transport across the electrode and electrolyte by finite element simulation to determine boundary conditions for our mesoscale models. We then generated different porous microstructures of various carbon materials by stochastic and physics-based simulations informed from experiments and performed multiphysics-based simulations at mesoscale to identify the electro-chemo-mechanical hotspots which are sensitive to the microstructure morphology. We calculated the energy barriers for Li to form clusters on graphene surfaces as a function of local Li concentrations and electric fields by ab initio molecular dynamics simulations. The synergistic deployment of different models across scales allows us to theoretically estimate the Li metal nucleation tendency based on a modified classical nucleation theory for different microstructures under various electroplating conditions. Comparison of the theoretical predictions and experimental results will be briefly discussed. The insights obtained from this multiscale framework offer new understanding and strategies for the design of carbon scaffold architectures to mitigate the non-uniformity of Li plating and issues in Li metal anode deployment in solid-state batteries.This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and 22-ERD-043.