Conventional Nucleation Theory Research Articles

Exploration of non-trivial relations for the non-steady state nucleation rate: usefulness of the elliptic theta functions for its experimental estimations

It is known that the non-steady state nucleation rate can be expressed as the product of the steady state nucleation rate and an elliptic theta function. However, there has been no research where rich mathematical properties of the elliptic theta functions were utilized to analyze the non-steady state nucleation processes. In this paper, we achieved two objectives by using the properties. The first one is to derive non-trivial relations for the non-steady state nucleation rate by using the rich mathematical properties of the elliptic theta functions, which could not be discovered in the conventional classical nucleation theory because of its complexities. The second one is to find solutions to two problems by solving a difference equation for the non-steady state nucleation rate: (i) it requires a large amount of effort and cost to estimate the time evolution of the non-steady state nucleation rate under some special conditions, and (ii) it is impossible to measure the non-steady state nucleation rate under some special cases. It is shown that our result help reduce the estimation cost of the non-steady state nucleation rate and makes mechanical estimation of the non-steady state nucleation rate possible from the past to the future.Article highlightsNon-trivial relations for the non-steady state nucleation rate are found from the mathematical properties of the elliptic theta functions.It is shown that our result help reduce the estimation cost of the non-steady state nucleation rate.It is shown that our result makes mechanical estimation of the non-steady state nucleation rate possible from the past to the future.

Seeding method for ice nucleation under shear

Hydrodynamic flow can have complex and far-reaching consequences on the rate of homogeneous nucleation. We present a general formalism for calculating the nucleation rates of simply sheared systems. We have derived an extension to the conventional Classical Nucleation Theory, explicitly embodying the shear rate. Seeded molecular dynamics simulations form the backbone of our approach. The framework can be used for moderate supercooling, at which temperatures brute-force methods are practically infeasible. The competing energetic and kinetic effects of shear arise naturally from the equations. We show how the theory can be used to identify shear regimes of ice nucleation behavior for the mW water model, unifying disparate trends reported in the literature. At each temperature, we define a crossover shear rate in the limit of 1000 s-1-10 000 s-1, beyond which the nucleation rate increases steadily up to a maximum, at the optimal shear rate. For 235 K, 240 K, 255 K, and 260 K, the optimal shear rates are in the range of ≈106 s-1-107 s-1. For very high shear rates beyond 108 s-1, nucleation is strongly inhibited. Our results indicate that the optimal shear rates have a non-monotonic dependence on temperature.

Propagation topography of redox phase transformations in heterogeneous layered oxide cathode materials

Redox phase transformations are relevant to a number of metrics pertaining to the electrochemical performance of batteries. These phase transformations deviate from and are more complicated than the conventional theory of phase nucleation and propagation, owing to simultaneous changes of cationic and anionic valence states as well as the polycrystalline nature of battery materials. Herein, we propose an integrative approach of mapping valence states and constructing chemical topographies to investigate the redox phase transformation in polycrystalline layered oxide cathode materials under thermal abuse conditions. We discover that, in addition to the three-dimensional heterogeneous phase transformation, there is a mesoscale evolution of local valence curvatures in valence state topographies. The relative probability of negative and positive local valence curvatures alternates during the layered-to-spinel/rocksalt phase transformation. The implementation of our method can potentially provide a universal approach to study phase transformation behaviors in battery materials and beyond.

Predictive modeling of Time-Temperature-Transformation diagram of metallic glasses based on atomistically-informed classical nucleation theory

Theoretical prediction of glass forming ability (GFA) of metallic alloys is a key process in exploring metallic alloy compositions with excellent GFA and thus with the ability to form a large-sized bulk metallic glass. Molecular dynamics (MD) simulation is a promising tool to achieve a theoretical prediction. However, direct MD prediction continues to be challenging due to the time-scale limitation of MD. With respect to practical bulk metallic glass alloys, the time necessary for quenching at a typical cooling rate is five or more orders of magnitude higher than that at the MD time-scale. To overcome the time-scale issue, this study proposes a combined method of classical nucleation theory and MD simulations. The method actually allows to depict the time-temperature-transformation (TTT) diagram of the bulk metallic glass alloys. The TTT directly provides a prediction of the critical cooling rate and GFA. Although the method assumes conventional classical nucleation theory, all the material parameters appearing in the theory were determined by MD simulations using realistic interatomic potentials. The method is used to compute the TTT diagrams and critical cooling rates of two Cu-Zr alloy compositions (Cu50Zr50 and Cu20Zr80). The results indicate that the proposed method reasonably predicts the critical cooling rate based on the computed TTT.

Numerical simulation of bulk condensation in gas-vapor mixtures flowing through a nozzle

The behavior of heavy water and n-nonane vapors mixed with an incondensable gas has been numerically simulated under the conditions of nonequilibrium bulk condensation. The efficiency of the employed mathematical model has been verified using available experimental data. The applicability of the conventional nucleation theory and modification thereof to the description of bulk condensation in a gas–vapor mixture flowing through a nozzle has been analyzed.

Nucleation and thermal stability of an icosahedral nanophase during the early crystallization stage in Zr-Co-Cu-Al metallic glasses

The early stage of crystallization in amorphous phases is a key step in the overall microstructural evolution. In a Zr-based amorphous alloy the formation of metastable nanophase was observed at the earliest crystallization stage, with an ultra-high nucleation density of 6.5 × 1024 m−3 and a size of ∼1–2 nm. The nanophase has an icosahedral structure and exhibits an extremely high thermal stability up to 150 K above the crystallization onset. The chemical and structural heterogeneity in the early stage of the glass crystallization were observed using high-resolution high-angle annular dark-field (HAADF) scanning transmission electron microscopy (STEM), which can be applied to study materials with atomic resolution. A nucleation kinetics analysis of the nanophase formation accounts for the high product number density and indicates a low interface energy value between the metastable clusters and the amorphous matrix, σa,i of 14 mJ/m2. The thermal stability and resistance to size change by growth or coarsening are attributed to a stalled growth due to solute depletion and the low σa,i value. With the high nucleation rate and the low barrier, it is remarkable that the kinetics follow conventional nucleation theory.

Overview: Understanding nucleation phenomena from simulations of lattice gas models.

Monte Carlo simulations of homogeneous and heterogeneous nucleation in Ising/lattice gas models are reviewed with an emphasis on the general insight gained on the mechanisms by which metastable states decay. Attention is paid to the proper distinction of particles that belong to a cluster (droplet), that may trigger a nucleation event, from particles in its environment, a problem crucial near the critical point. Well below the critical point, the lattice structure causes an anisotropy of the interface tension, and hence nonspherical droplet shapes result, making the treatment nontrivial even within the conventional classical theory of homogeneous nucleation. For temperatures below the roughening transition temperature facetted crystals rather than spherical droplets result. The possibility to find nucleation barriers from a thermodynamic analysis avoiding a cluster identification on the particle level is discussed, as well as the question of curvature corrections to the interfacial tension. For the interpretation of heterogeneous nucleation at planar walls, knowledge of contact angles and line tensions is desirable, and methods to extract these quantities from simulations will be mentioned. Finally, also the problem of nucleation near the stability limit of metastable states and the significance of the spinodal curve will be discussed, in the light of simulations of Ising models with medium range interactions.

Optimization algorithm for rate equations with an application to epitaxial graphene

We describe an algorithm that searches the parameter space of rate theories to optimize the associated rate coefficients based on a fit to experimental (or any other) data. Beginning with an initial set of parameters, which may be estimated, partially calculated, or indeed random, the algorithm follows a path, calculating the error at each point, until a minimum error is reached. We illustrate our method by correcting a previously proposed rate theory for the nucleation and growth of graphene on Ru(0 0 0 1) and Ir(1 1 1) to account for the temperature dependence of the graphene island density. This quantity shows an exponential decrease as the temperature is raised, in contrast to the power law decrease predicted by conventional nucleation theory, which indicates that a qualitatively different mechanism is operative for graphene island formation. Other applications of our method are also discussed.

Spherical seed mediated vapor condensation of Lennard-Jones fluid: A density functional theory approach

Vapor to liquid condensation in presence of spherical seed particle of any arbitrary radius ranging from zero to infinity has been investigated using density functional theory, by modeling the local Helmholtz free energy density functional as well as the density profile at the vapor-liquid interface. A general theory is, thus, obtained which provides the different modes of nucleation based on the size of the seed ranging from zero (corresponding to the homogeneous mode of nucleation) to infinity (corresponding to the heterogeneous nucleation on flat surface). The theory is applied to the Lennard-Jones fluid and the optimized shape (i.e., contact angle) and formation free energy of droplets of any arbitrary size have been obtained in this work. The change of the shape (optimized) with the variation of the size of the liquid droplet as well as with the size of the solid substrate has been studied, thus predicting the shape-size relationship in the course of vapor to liquid heterogeneous nucleation on a spherical solid substrate of any particular size. The spinodal decomposition of vapor has also been observed at higher strength of the solid-fluid interaction. The results have been compared with the results of the conventional classical nucleation theory.

Enhanced nucleation of Al islands on H-dosed Si(100)-2 × 1 surface: A combined density functional theory and kinetic Monte Carlo study

Using a combined density functional theory (DFT) and kinetic Monte Carlo (kMC) approach, we show that the presence of small dose of hydrogen atoms on Si(100)-2×1 surface can increase Al chain density by as much as four times compared to the clean Si(100)-2×1 value. Though the observed enhanced nucleation can be explained by trapping of Al adatoms by H and their subsequent stabilization into islands, we find that a similar increase in island density can be achieved even when H merely blocks (but does not trap) Al adatoms. For H-bound islands, our DFT analysis suggests that Al adatoms are pinned preferentially on one side of H. Additionally, we argue that despite the high ratio of diffusion to adsorption rates which should favor H-bound islands over homogeneously nucleated islands, the former is not as numerous conventional nucleation theory would predict.

Monte Carlo tests of nucleation concepts in the lattice gas model

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 (quasistatic) 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 nucleation at surfaces, the barrier is reduced by a factor depending on the contact angle. Using the definition of physical clusters based on the Fortuin-Kasteleyn mapping, the time dependence of the cluster size distribution is studied for quenching experiments in the kinetic Ising model and the cluster size ℓ(*) where the cluster growth rate changes sign is estimated. These studies of nucleation kinetics are compared to studies where the relation between cluster size and supersaturation is estimated from equilibrium simulations of phase coexistence between droplet and vapor in the canonical ensemble. The chemical potential is estimated from a lattice version of the Widom particle insertion method. For large droplets it is shown that the physical clusters have a volume consistent with the estimates from the lever rule. Geometrical clusters (defined such that each site belonging to the cluster is occupied and has at least one occupied neighbor site) yield valid results only for temperatures less than 60% of the critical temperature, where the cluster shape is nonspherical. We show how the chemical potential can be used to numerically estimate ΔF(*) also for nonspherical cluster shapes.

Numerical approaches to determine the interface tension of curved interfaces from free energy calculations

A recently proposed method to obtain the surface free energy σ(R) of spherical droplets and bubbles of fluids, using a thermodynamic analysis of two-phase coexistence in finite boxes at fixed total density, is reconsidered and extended. Building on a comprehensive review of the basic thermodynamic theory, it is shown that from this analysis one can extract both the equimolar radius R(e) as well as the radius R(s) of the surface of tension. Hence the free energy barrier that needs to be overcome in nucleation events where critical droplets and bubbles are formed can be reliably estimated for the range of radii that is of physical interest. It is found that the conventional theory of nucleation, where the interface tension of planar liquid-vapor interfaces is used to predict nucleation barriers, leads to a significant overestimation, and this failure is particularly large for bubbles. Furthermore, different routes to estimate the effective radius-dependent Tolman length δ(R(s)) from simulations in the canonical ensemble are discussed. Thus we obtain an instructive exemplification of the basic quantities and relations of the thermodynamic theory of metastable droplets/bubbles using simulations. However, the simulation results for δ(R(s)) employing a truncated Lennard-Jones system suffer to some extent from unexplained finite size effects, while no such finite size effects are found in corresponding density functional calculations. The numerical results are compatible with the expectation that δ(R(s) → ∞) is slightly negative and of the order of one tenth of a Lennard-Jones diameter, but much larger systems need to be simulated to allow more precise estimates of δ(R(s) → ∞).

Universal Critical Behavior of Curvature-Dependent Interfacial Tension

From the analysis of Monte Carlo simulations of a binary Lennard-Jones mixture in the coexistence region, we provide evidence that the curvature dependence of the interfacial tension can be described by a simple theoretical function σ(R)ξ(2)=C(1)/[1+C(2)(ξ/R)(2)], where ξ is the correlation length and R is the droplet radius. The universal constants C(1) and C(2) are estimated. In the model, a Tolman length is strictly absent, but, since its critical behavior is believed to be much weaker than ξ, we argue that it only provides a correction to scaling and does not affect the leading critical behavior, which should be described by the above function for any system in the Ising universality class. The large value of C(2)≃32 implies that conventional nucleation theory becomes inaccurate even for a significantly large droplet radius.

Fatigue Insensitivity of Nanoscale Freestanding Aluminum Films

We demonstrate a microelectromechanical-system-based setup for fatigue studies on 200-nm-thick freestanding aluminum specimens in situ inside the transmission electron microscope. The specimens did not show any sign of fatigue damage even at 1.2 ×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sup> cycles under nominal stresses about 80% of the static ultimate strength. We show direct evidence to propose that the conventional theory of fatigue crack nucleation through slip bands does not work for nanoscale freestanding thin films, which gives rise to the anomalous fatigue insensitivity.

Liquid-phase Synthesis of CaF&lt;sub&gt;2&lt;/sub&gt; Particles and Their Low Refractive Index Characterization

The purpose of the present study was to reveal the details of the preparation of CaF2 particles with controllable size (30–900 nm) and shape (spherical, hexagonal, and cubical forms) using a liquid-phase synthesis method, and to demonstrate that a change in the composition of the reactants and crystalline structure of the CaF2 product could improve material performance. The particles were synthesized from the reaction of CaCl2 and NH4F in an aqueous solution in the absence of any additional components (e.g., chemicals, surfactants). Monodispersed particles were achieved by the optimization of the reaction condition parameters: temperature, mixing rate, and reaction time. Control of the particle size was accomplished mainly by changing the concentration of the reactants, which is qualitatively explained by the conventional nucleation theory. Flexibility of the process in controlling particle morphology, from a spherical to a hexagonal and/or a cubical form, was predominantly achieved by varying the concentration of CaCl2. Since the identical XRD pattern was detected in particles with varying morphologies, the shape transformation was due to changes in particle growth. A theoretical background to support how the particles changed was also added and was compared with an analysis of the number of nuclei. In addition, sufficient adjustment of the reactant compositions made it possible to produce a material with an ultralow refractive index (nCaF2 was near to ntheoretical CaF2), which was confirmed by the measurement of the refractive index and the material crystallinity.

Nanometer to Submicrometer Magnesium Fluoride Particles with Controllable Morphology

Magnesium fluoride particles with controllable size (from several nanometers to submicrometers) and morphology (spherical and cubic forms) were successfully prepared via liquid-phase synthesis. The particles were synthesized from the reaction of MgCl(2) and NH(4)F in an aqueous solution at 75 degrees C for 1 h under a nitrogen atmosphere. Control of particle size was accomplished mainly by changing the concentration of the reactants, which could be qualitatively explained by conventional nucleation theory. Flexibility of the process in controlling particle morphology, from a spherical to a cubical form, was predominantly achieved by varying the concentration of MgCl(2). Since the same XRD pattern was detected in particles with varying morphologies, the shape transformation was due to changes in particle growth. With the ability to control particle size and morphology, the creation of other inorganic particles is possible and has potential for many field applications.

Diffusion, nucleation and reaction in a three-component system: Fe on Si(111)-‘5×5’–Cu

The role of an ultrathin Cu layer on the nucleation and growth of Fe/Si nanostructures on Si(111) is investigated using scanning tunneling microscopy. While for deposition on the bare Si(111)-7×7 surface, the diffusion and nucleation of Fe atoms are significantly influenced by the reaction with the Si substrate, the Si(111)-‘5×5’–Cu surface causes a decoupling of the diffusion and nucleation on and reaction with the surface. For moderate temperatures (T⩽350 °C), this decoupling results in a diffusion behaviour, which can be adequately described by the conventional nucleation theory (CNT), using a single value for the effective diffusion barrier E*=0.49 eV and a critical nucleus size i=2. Furthermore, it is evidenced that only Fe atoms contribute to the critical nucleus and E* thus describes the diffusion and nucleation of Fe atoms. On the other hand, at temperatures above 400 °C, the island growth kinetics change considerably, i.e. the preferential step edge nucleation and three-dimensional growth are observed, and CNT is no longer applicable.

The Influence of an Adsorbate Layer on Adatom Diffusion and Island Nucleation: Fe on Si(111)-√3 x √3-Au

Using scanning tunneling microscopy, the influence of a thin Au layer on the diffusion of Fe adatoms and the subsequent island nucleation on a Si(111) surface is investigated. The adsorbate induces thestructure that increases the surface mobility of subsequently deposited Fe atoms, resulting in the formation well-defined nanoclusters. Surprisingly, the domain walls—inherent to the reconstruction—do not influence the surface diffusion, which demonstrates that the passivation is of much more importance for the self-assembly than the surface corrugation. Using the decoupling of the diffusion and nucleationonthe surface and the reactionwiththe surface and conventional nucleation theory, the activation energy for surface diffusionEd = 0.61 eV and the critical cluster sizei = 3 are determined, which reveal the microscopic details of the diffusion and nucleation processes.

The Development of “Macroscopic Composition Gradient Method” and its Application to the Precipitate-Nucleation Near the Edge of Miscibility Gap

A new characterization method, "Macroscopic Composition Gradient (MCG) Method" is proposed to investigate the phase transformations near the phase boundaries, such as the solubility limit, order/disorder line and so on. Since the macroscopic composition gradient in the alloy is prepared so as to step over the phase boundary, the morphological transition of critical phenomena at the phase boundary can be observed by means of analytical transmission electron microscopy. By utilizing this method, the critical minimum size of stable precipitate in the vicinity of edge of miscibility gap is experimentally determined for the Ni3Si in Ni-Si, Ni3Al in Ni-Al, Cu4Ti in Cu-Ti and Co in Cu-Co binary alloy systems. The results are as follows: The critical nucleus size shows a steep increase up to several tens of nm in a very narrow composition range less than 0.3at% from the phase boundary. The Gibbs-Thomson relation and the conventional nucleation theory statistically rationalize such the composition dependence of nucleus size change. However, the nucleus formation is kinetically never rationalized by the conventional nucleation theories. The phase decomposition of supersaturated solid solution progresses by a mechanism of spinodal phase decomposition, even in the composition range near the solubility limit, i.e. a so-called Nucleation- Growth region. Such the phase decomposition behavior is never rationalized by the Boltzmann- Gibbs free energy, which is based on the extensive entropy. The experimental facts obtained here are explained by Tsallis's non-extensive entropy. It should be noted that the present experiments can systematically be conducted in the composition range very near the solubility limit where they has hardly been examined in the past. The MCG method proposed here is considered to open a new way to investigate the microstructure evaluation, particularly for the critical phenomena near the phase boundary.

Anomalous Surface Diffusion of C60 and Anisotropic Growth of Nano Islands on Ni(111)

The migration behavior of C60 on Ni(111) has been inferred from its growth morphology at various substrate temperatures, as observed with scanning tunneling microscopy. The number density of islands increased and their average sizes decreased anomalously in the temperature range of approximately 573 K to approximately 973 K. This trend contradicts the prediction in conventional nucleation theory. At low and high temperatures, C60 commence nucleation on both sides of surface steps in a "bi-directional step flow" mode. However, anisotropy occurs within an intermediate temperature range, in which C60 nucleate predominantly at upper step edges. Surprisingly, in-situ growth measurements at this intermediate temperature range revealed that C60 actually start nucleating from lower step edges, with concomitant formation of Ni terraces underneath. These anomalous thermal dependence of diffusivity and the peculiar growth morphology of C60 on Ni(111) are attributed to C60-induced reconstruction of Ni(111) at higher temperature.