Steady-state Nucleation Rate Research Articles

Impact of MgO-ZnO substitution on nucleation of aluminosilicate glasses

The nucleation mechanisms is of technically interest and important in glass-ceramics production. Here we investigate the nucleation behaviors of NiO-doped MgO-Al2O3-SiO2 and ZnO-Al2O3-SiO2 glasses. It is found that crystallization tends to takes place on the surface and the interior of glasses for the former and the latter, respectively. The origin of different nucleation behavior is studied by comparing aluminum coordination, configurational entropy and rheological properties of the glasses. The substitution of MgO by ZnO leads to increase of tetrahedral aluminum concentration, and decrease of configurational entropy above glass transition temperature, in particular, the latter is closely related to volume nucleation. In addition, the qualitative comparison of thermodynamic and kinetic barriers of steady-state nucleation rate of two glasses indicates that ZnO-bearing aluminosilicate glass possesses higher diffusivity and lower critical nucleus size, which favoring volume nucleation.

Diffusion proxies reveal the dynamic process in supercooled and glassy lithium diborate

Understanding the effective diffusion coefficient (D) is crucial for describing atomic transport during crystal nucleation in supercooled liquids and glasses. However, pinpointing the key structural units driving crystallization in intricate, multicomponent glass-formers remains a challenge. This study presents a novel analysis of lithium diborate (Li₂O·2 B₂O₃ - LB₂) crystallization in supercooled and glassy states by using available viscosity and crystallization data for samples of the same batch. An original key feature is that the nucleation rates were measured using a single-stage rather than the traditional double-stage heat treatment. We compared three proxies for D: Dη obtained from viscosity, DU derived from crystal growth rates, and Dτ estimated from nucleation time-lags. Our analysis revealed that at deep supercoolings, below the glass transition temperature, DU and Dτ yield similar steady-state nucleation rate estimates, which significantly exceed those predicted by using Dη (the most frequently used diffusion proxy). This result suggests that the atomic jumps governing crystal nucleation may be comparable to those in crystal growth, but distinct from those associated with cooperative rearrangements controlling viscous flow. Additionally, the estimated kinetic spinodal temperature (Tks) is above the Kauzmann temperature Tk, implying that crystal nucleation precedes relaxation to the supercooled liquid state, circumventing the entropy paradox and corroborating recent MD simulations for other substances. These findings are valuable for refining theoretical models of crystal nucleation and highlight the complexities of the glassy state.

Nucleation kinetics model for primary crystallization in Al-Y-Fe metallic glass.

The high density of aluminum nanocrystals (>1021m-3) that develop during the primary crystallization in Al-based metallic glasses indicates a high nucleation rate (∼1018m-3s-1). Several studies have been advanced to account for the primary crystallization behavior, but none have been developed to completely describe the reaction kinetics. Recently, structural analysis by fluctuation electron microscopy has demonstrated the presence of the Al-like medium range order (MRO) regions as a spatial heterogeneity in as-spun Al88Y7Fe5 metallic glass that is representative for the class of Al-based amorphous alloys that develop Al nanocrystals during primary crystallization. From the structural characterization, an MRO seeded nucleation configuration is established, whereby the Al nanocrystals are catalyzed by the MRO core to decrease the nucleation barrier. The MRO seeded nucleation model and the kinetic data from the delay time (τ) measurement provide a full accounting of the evolution of the Al nanocrystal density (Nv) during the primary crystallization under isothermal annealing treatments. Moreover, the calculated values of the steady state nucleation rates (Jss) predicted by the nucleation model agree with the experimental results. Moreover, the model satisfies constraints on the structural, thermodynamic, and kinetic parameters, such as the critical nucleus size, the interface energy, and the volume-free energy driving force that are essential for a fully self-consistent nucleation kinetics analysis. The nucleation kinetics model can be applied more broadly to materials that are characterized by the presence of spatial heterogeneities.

An exact measurement of nucleation incubation times in isothermal crystallizations of liquid metal Al via configuration heredity

Nucleation induction time τ∗ is a crucial parameter in the classical nucleation theory, which is intimately associated with the steady-state nucleation rate I0 in the homogenous nucleation. To examine the impact of temperatures T on τ∗(T) in the isothermal crystallizations of liquid metal Al, a series of molecular dynamics simulations are performed at different undercoolings ΔTm. With the help of cluster type index method based on Honeycutt–Anderson bond-type index, a cluster analysis method based on the continuous heredity of crystal clusters is developed to further distinguish nuclei from various embryos. By means of the transient heredity of crystal clusters, a nucleation incubation time τc different from nucleation induction time, i.e., a sum of incubation time of embryo τe and its effective growth time τgeff prior to arriving the critical size, is introduced and exactly measured. For the critical nuclei emerged in the transient nucleation stage, the τc are found to be long compared with the steady-state nucleation stage and mainly depend on an effective growth of incubated embryos, while those in steady-state nucleation stage are mostly decided by a successful incubation of initial embryos. Also, relative to the τ∗(T) derived from I0(T) at different ΔTm, these measured τc(T) are very small, but their average times τ¯c(T) in the whole region are mirror symmetric to I0(T).

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.

On the description of the microdomains within carbon fiber precursory mesophase pitch: a mesoscopic continuum approach

The present paper proposes a mesoscopic continuum approach in order to describe the behavior of microdomains within carbon fiber precursory mesophase pitch. The microdomains are assumed to have an orientation, which is determined by the average orientation of the particles that form it. On the mesoscopic space, balance equations for the microdomains are presented. Evolution equations for the density and for the orientation of the crystalline microdomains are proposed. In order to determine the temporal variation of the microdomain density, it was deduced a quite simple relation between mass production, critical density of microdomains and a mesoscopic operator acting on the orientation distribution function. As presented in the present work, the mass production can be determined by the crystallization kinetics theory via the steady-state nucleation rate. Specific forms for the mesoscopic operator are proposed in this work, although they may be extended to other models that describe oriented microstructures. There are not yet enough experimental data to test the mesoscopic model deduced here, but in turn, it is presented as a new tool for experimental studies, since it can estimate the time rate of microdomain property changes. Possible extensions of this model could be applied to describe mechanical and rheological properties of carbon fibers.

Anomalous crystallization kinetics of ultrafast ScSbTe phase-change memory materials induced by nitrogen doping

Phase-change random access memory has been served as a bridge across the memory wall within the conventional von Neumann computing architecture, owing to its excellent non-volatility and rapid switching. Nitrogen doping in Ge2Sb2Te5-like phase-change materials (PCMs) was a routine method for enhancing non-volatility of amorphous phases, although sacrificing crystallization speed markedly. Here we introduced N element into Sc0.3Sb2Te3 PCM that possesses sub-nanosecond crystallization speed, trying to cope with stability/reliability issues raised by backend thermal-budget processing. However, we observed an abnormal deterioration in thermal stability of N-doped amorphous Sc0.3Sb2Te3 films. As compared to the undoped counterparts, they exhibit a weakened fragile-to-strong crossover in viscosity that gives rise to a higher atomic mobility at elevated temperatures while a less rigid vitreous network at low temperatures. Therefore, N doping leads to an accelerated crystal growth rate and a higher steady-state nucleation rate in heterogeneous manner, causing a faster crystallization, whereas the kinetic suffocation contributed by original Sc element as approaching room temperature must have been partially counteracted by N dopant. This acceleration in crystallization kinetics delivers N-doped Sc0.3Sb2Te3 as a promising candidate in working memory.

Influence on crystal nucleation of an order-disorder transition among the subcritical clusters.

Studies of nucleation generally focus on the properties of the critical cluster, but the presence of defects within the crystal lattice means that the population of nuclei necessarily evolve through a distribution of precritical clusters with varying degrees of structural disorder on their way to forming a growing stable crystal. To investigate the role precritical clusters play in nucleation, we develop a simple thermodynamic model for crystal nucleation in terms of cluster size and the degree of cluster order that allows us to alter the work of forming the precritical clusters without affecting the properties of the critical cluster. The steady state and transient nucleation behavior of the system are then studied numerically, for different microscopic ordering kinetics. We find that the model exhibits a generic order-disorder transition in the precritical clusters. Independent of the types of ordering kinetics, increasing the accessibility of disordered precritical clusters decreases both the steady state nucleation rate and the nucleation lag time. Furthermore, the interplay between the free-energy surface and the microscopic ordering kinetics leads to three distinct nucleation pathways.

Nucleation parameters of SPC/E and TIP4P/2005 water vapor measured in NPT molecular dynamics simulations.

Nucleation rates for droplet formation in water vapor are measured in molecular dynamics (MD) simulations of SPC/E and TIP4P/2005 water by monitoring individual nucleation events. The nucleation process is simulated in the NPT ensemble to evaluate the steady-state nucleation rate in accordance with the assumptions of classical nucleation theory (CNT). Nucleation rates measured between 300 and 425 K for the SPC/E model, and between 325 and 475 K for the TIP4P/2005 model, agree with the CNT predictions roughly within the standard deviation of the MD measurements of the nucleation rates.

A consistent formation free energy definition for multicomponent clusters in quantum thermochemistry

Quantum chemical calculations have proven to describe the thermodynamics of sub-nanometre clusters more accurately than the liquid drop model used in classical nucleation theory (CNT). However, the standard quantum chemical free energies of multicomponent clusters are fundamentally incompatible with the quantity appearing in the exponential of the CNT expression for the nucleation rate. The origin of this incompatibility is known to be in the statistical thermochemistry, but it is also connected to a much debated issue of self-consistency within CNT. Although these issues do not affect the main results of previous theoretical studies of nucleation, a thorough analysis and discussion about the nature of free energy in quantum chemistry of cluster formation has been missing. In this study, we present a consistent definition for the formation free energy using the law of mass action and a general equilibrium cluster distribution function, and apply this definition to gas-phase quantum thermochemical calculations. This internal consistency allows us to integrate the high-level thermochemical data to the basic framework of CNT and the liquid drop model contained therein. Moreover, based on our analysis, we derive a simple analytical expression for the steady-state nucleation rate compatible with the widely used numerical acdc model. As an illustrative example, the consistent formation free energies and nucleation rates are presented and analysed for atmospherically relevant H2SO4–NH3 clusters at various temperature and monomer vapour pressure conditions.

The best diffusivity proxy for crystal nucleation in stoichiometric oxide glasses

Due to the difficulty in measuring the mobility of the “structural units” of supercooled liquids and glasses during crystal nucleation, the effective diffusivity (D) has been frequently estimated via viscosity (Dη) or nucleation time-lags (Dτ). However, it has been recently reported that, below the glass transition temperature (Tg), the experimental steady-state nucleation rates (Ist) can reach much higher values than those predicted by the Classical Nucleation Theory (CNT) when assuming D ≈ Dη or D ≈ Dτ. Hence, another possibility would be inferring D from experimental crystal growth velocity data (D ≈ DU), which is a natural approach since nucleation and growth are likely controlled by the same mass transfer process through the crystal/liquid interface. Therefore, the approximations D ≈ Dη and D ≈ DU were tested in this work using lithium disilicate glass as a model system. To avoid the influence of compositional changes, we measured the nucleation rates, growth velocity, and viscosity using samples of the same glass batch. With this strategy, we found that the very long experimental nucleation times implemented below Tg shifted the experimental temperature of maximum nucleation rate to temperatures lower than those previously reported for this glass former. Most important is that the CNT gives a much better description of the experimental Ist(T) data in the whole analyzed temperature range if D is approximated by DU instead of Dη, indicating that crystal growth is indeed an adequate proxy for nucleation diffusivity.

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.

On the Fokker–Planck approximation in the kinetic equation of multicomponent classical nucleation theory

We examine the validity of the Fokker–Planck equation with linear force coefficients as an approximation to the kinetic equation of nucleation in homogeneous isothermal multicomponent condensation. Starting with a discrete equation of balance governing the temporal evolution of the distribution function of an ensemble of multicomponent droplets and reducing it (by means of Taylor series expansions) to the differential form in the vicinity of the saddle point of the free energy surface, we have identified the parameters whereof the smallness is necessary for the resulting kinetic equation to have the form of the Fokker–Planck equation with linear (in droplet variables) force coefficients. The “non-smallness” of these parameters results either in the appearance of the third or higher order partial derivatives of the distribution function in the kinetic equation or in its force coefficients becoming non-linear functions of droplet variables, or both; this would render the conventional kinetic equation of multicomponent nucleation and its predictions inaccurate. As a numerical illustration, we carried out calculations for isothermal condensation in five binary systems of various non-ideality at T=293.15 K: 1-butanol–1-hexanol, water–methanol, water–ethanol, water–1-propanol, water–1-butanol. Our results suggest that under typical experimental conditions the kinetic equation of binary nucleation of classical nucleation theory may require a two-fold modification and, hence, the conventional expression for the steady-state binary nucleation rate may not be adequate for the consistent comparison of theoretical predictions with experimental data.

The effect of transient nucleation behavior on thermal stability of Fe48Co32P14B6 metallic glass

The experimental isothermal kinetic curves for Fe48Co32P14B6 glass measured in the temperature range 683–714 K have been analyzed with using the analytical equation derived as a combination of the integral form of Kolmogorov’s equation of mass crystallization kinetics and the Kashchiev’s transient nucleation model. The transient nucleation rates show the trend to the steady-state values with increasing of annealing temperature. From a combination of the kinetic data and the values of crystal growth velocity, U, determined from the structural data, the steady-state nucleation rate values have been estimated. These data are analyzed within the classical nucleation theory and the work of the critical nucleus formation and the specific interfacial free energy values have been calculated. It has been established that transient behavior of nucleation rate in Fe48Co32P14B6 glass results in an increase of the onset crystallization time by 6.4 to about 3 times at 683 and 714 K, respectively. The temperature dependence of the transient time in the glass investigated is compared with that of the effective diffusivity as well with the similar data found for Fe40Ni40P14B6 and Fe40Co40P14B6 glasses and the possible reason of the observed changes in the degree of non-stationarity of nucleation rate has been discussed.

Relaxation Effect on Crystal Nucleation in a Glass Unveiled by Experimental, Numerical, and Analytical Approaches

Understanding the mechanisms and dynamics of crystal nucleation and growth in glass-forming materials is fundamental to avoid or control crystallization - the critical steps for developing and producing novel glasses and glass-ceramics. Below the temperature Tmax, corresponding to the maximum of experimentally measured nucleation rates, the reported nucleation rates are frequently smaller than the theoretical predictions by the Classical Nucleation Theory (CNT). This discrepancy dramatically increases with decreasing temperature, and has been denoted as the “breakdown” of the CNT. For understanding this alleged breakdown, it is important to stress that the experimental Tmax is located somewhat above or at the glass transition temperature, Tg. Here, using a 5BaO·8SiO2 (B5S8) glass, first, we showed a significant change in the characteristic relaxation time with temperature in the Tg region: it is four times shorter for the supercooled liquid than for the glass. Then a combination of detailed experiments and theoretical (analytical and numerical) analyses for the B5S8 glass confirmed our recent findings for Li2O·2SiO2 and 2Na2O·CaO·3SiO2 glasses. The “breakdown” is just an artefact, since the experimental nucleation times used are often not long enough to allow the complete structural relaxation of the glass and hence reach the ultimate steady-state nucleation rate. Finally, for the first time, we also show that due to the almost complete crystallization of the B5S8 glass, and likely of other glasses that exhibit sufficiently high nucleation and growth rates, the structural relaxation process cannot be finished and, correspondingly, the ultimate steady-state nucleation regime at T

Molecular dynamics simulations of spontaneous and seeded nucleation and theoretical calculations for zinc selenide

Understanding and controlling the liquid to crystal transformation is a central topic for numerous natural phenomena and technological applications. However, the microscopic mechanism of crystal nucleation is still elusive, which leads to strong controversies regarding the ability of the most used model, the Classical Nucleation Theory (CNT), to describe nucleation rates in supercooled liquids. In this work, we were able to deeply supercool Zinc Selenide (ZnSe), and determine spontaneous homogeneous steady-state nucleation rates, JMD, by MD simulations using the mean lifetime method. At moderate supercoolings, where the nucleation rates are much smaller, we used the seeding method to compute the nucleation rates by the CNT formalism, JCNT, without any fitting parameter, using the physical properties obtained by MD simulations: the melting temperature, Tm, density, melting enthalpy, diffusion coefficient, D+, and the critical nucleus size, N∗, combined with two expressions for the thermodynamic driving force, Δμ. The values of interfacial free energy, γ, calculated by the CNT expression using the MD simulation data, via both the seeding method and the mean lifetime method at moderate and deep supercoolings show a weak temperature dependence, which is in line with the Diffuse Interface Theory. The extrapolated values of γ, from the spontaneous nucleation regime to the seeding nucleation region cover the range of values of γ calculated via the seeding method and the CNT formalism. Finally, the JCNT extrapolated from moderate supercoolings to deep supercoolings are in good agreement with the JMD. These results confirm the validity of the CNT.

Effects of Glass Transition and Structural Relaxation on Crystal Nucleation: Theoretical Description and Model Analysis.

In the application of classical nucleation theory (CNT) and all other theoretical models of crystallization of liquids and glasses it is always assumed that nucleation proceeds only after the supercooled liquid or the glass have completed structural relaxation processes towards the metastable equilibrium state. Only employing such an assumption, the thermodynamic driving force of crystallization and the surface tension can be determined in the way it is commonly performed. The present paper is devoted to the theoretical treatment of a different situation, when nucleation proceeds concomitantly with structural relaxation. To treat the nucleation kinetics theoretically for such cases, we need adequate expressions for the thermodynamic driving force and the surface tension accounting for the contributions caused by the deviation of the supercooled liquid from metastable equilibrium. In the present paper, such relations are derived. They are expressed via deviations of structural order parameters from their equilibrium values. Relaxation processes result in changes of the structural order parameters with time. As a consequence, the thermodynamic driving force and surface tension, and basic characteristics of crystal nucleation, such as the work of critical cluster formation and the steady-state nucleation rate, also become time-dependent. We show that this scenario may be realized in the vicinity and below the glass transition temperature, and it may occur only if diffusion (controlling nucleation) and viscosity (controlling the alpha-relaxation process) in the liquid decouple. Analytical estimates are illustrated and confirmed by numerical computations for a model system. The theory is successfully applied to the interpretation of experimental data. Several further consequences of this newly developed theoretical treatment are discussed in detail. In line with our previous investigations, we reconfirm that only when the characteristic times of structural relaxation are of similar order of magnitude or longer than the characteristic times of crystal nucleation, elastic stresses evolving in nucleation may significantly affect this process. Advancing the methods of theoretical analysis of elastic stress effects on nucleation, for the first time expressions are derived for the dependence of the surface tension of critical crystallites on elastic stresses. As the result, a comprehensive theoretical description of crystal nucleation accounting appropriately for the effects of deviations of the liquid from the metastable states and of relaxation on crystal nucleation of glass-forming liquids, including the effect of simultaneous stress evolution and stress relaxation on nucleation, is now available. As one of its applications, this theoretical treatment provides a new tool for the explanation of the low-temperature anomaly in nucleation in silicate and polymer glasses (the so-called “breakdown” of CNT at temperatures below the temperature of the maximum steady-state nucleation rate). We show that this anomaly results from much more complex features of crystal nucleation in glasses caused by deviations from metastable equilibrium (resulting in changes of the thermodynamic driving force, the surface tension, and the work of critical cluster formation, in the necessity to account of structural relaxation and stress effects) than assumed so far. If these effects are properly accounted for, then CNT appropriately describes both the initial, the intermediate, and the final states of crystal nucleation.

Comment on "Crystallization of Supercooled Liquids: Self-Consistency Correction of the Steady-State Nucleation Rate" by Abyzov et al., Entropy 2020, 22, 558.

It is shown that in the growth region (above the critical nucleation size) the transient distributions obtained numerically from the Becker-Döring equation (BDE) by Abyzov et al., Entropy 2020, 22, 558, are in accurate correspondence with the matched asymptotic (singular perturbation) solution by Shneidman, Sov. Phys. Tech. Phys. 1988, 33, 1338. The solution is unmodified by “self-consistency” corrections which affect only the steady state rate. Sensitivity of the results to selection of a specific form of the BDE (the “nucleation model”) also is briefly discussed.

Model for the Nucleation and Diffusion-Controlled Growth of Binary Alloy Nuclei under Potentiostatic Conditions

An approach is proposed to perform a mathematical description of the electrochemical formation and growth of binary alloy nuclei on the surface of an indifferent electrode under constant supersaturation (overpotential). This approach considers the total flux of two species of deposited ions in the electrolyte bulk to the surface of hemispherical new-phase nuclei. The concentration profile is averaged with allowance for the mole fraction ratio of the components. The cases of instantaneous and progressive nucleation with diffusion-controlled growth are studied. The overlap of the diffusion zones of the nuclei is analyzed in terms of the Scharifker–Hills model. Equations are derived to calculate the apparent diffusion coefficient of deposited ions, the number of nuclei per unit electrode surface, and the steady-state nucleation rate.

Transient nucleation in selective laser melting of Zr-based bulk metallic glass

The crystallization rate during selective laser melting (SLM) of bulk metallic glasses (BMG) is a critical factor in maintaining the material's amorphous structure. To increase the understanding of the interplay between the SLM process and the crystallization behavior of BMGs, a numerical model based on the classical nucleation theory has been developed that accounts for the rapid temperature changes associated with SLM. The model is applied to SLM of a Zr-based BMG and it is shown that the transient effects, accounted for by the model, reduce the nucleation rate by up to 15 orders of magnitude below the steady-state nucleation rate on cooling, resulting in less nuclei during the build process. The capability of the proposed modelling approach is demonstrated by comparing the resulting crystalline volume fraction to experimental findings. The agreement between model predictions and the experimental results clearly suggests that transient nucleation effects must be accounted for when considering the crystallization rate during SLM processing of BMGs.