Nucleation at changing density of monomers

The kinetics of nucleation at constant supersaturation are analyzed when the process occurs at changing density of those monomers (atoms or molecules) that start the nucleation chain of consecutive bimolecular reactions. The Zeldovich kinetic equation is solved for arbitrary time variation of the monomer density and exact general expressions are obtained for the time dependence of the nonstationary nucleation rate J and the density N of nuclei formed in the system up to time t. Exponential decrease and increase of the monomer density are considered as examples, and the corresponding J(t) and N(t) dependences are analytically found.     

Nuleation and Growth of Thin Ag Films Vacuum-deposited onto W

An attempt to analyse the nucleation and growth phenomenon in a typical “Stranski-Krastanov” (SK) growth system - Ag/W (PAUNOV , MICHAILOV 1979; MICHAILOV ), in terms of island growth theory was made. Ag-layers were analysed by the use of scanning electron-and optical microscopes. A strong dependence of the nucleation kinetics on supersaturation and grain orientation was found to exist. On the basis of known rate equations, the number of atoms in the critical cluster i and the value of nucleation potential barrier E were estimated. The comparison between the experimentally obtained size-, first nearest neighbour- and radial distributions and theory shows that: 1) nucleation and growth processes proceed through a surface diffusion of the atoms; 2) no coalescence between the crystallites and no crystal mobility exist.     

On the nucleation theory of electrochemical alloy formation II. Concentration dependence of the stationary nucleation rate

The concentration dependence of the stationary rate of binary electrochemical nucleation is theoretically studied. Experimental data are obtained for the nucleation rate in mixed electrolytes containing silver and mercury ions. It is shown that the atomistic approach to electrochemical nucleation can be applied to multi-component systems.     

Computer analysis of regularities of nucleation and crystal growth in a solid

An interplay between the homogeneous nucleation and growth processes during recrystallization and phase first-kind transformation was investigated. Computer simulation taking account of unsteadility of nucleation process and possibility of elimination of potential nucleation sites at a constant growth rate allowed to obtain the time dependence of nucleation rate of the new phase, Ṅ, in the total bulk of a sample. The various kinds of the dependence Ṅ including an increase, a constancy and a decrease are obtained.     

Simulated effects of transient nucleation on the crystallization of glass samples

The kinetics of crystallization have been re-examined by taking into account the transient nucleation time and numerical methods were used to obtain the dependence of fraction crystallized on temperature, heating rate and nucleation conditions (temperature and time). These solutions were used to analyze the applicability of approximate models which were based on the assumption of steady state nucleation. The transient nucleation time, which precedes nucleation in as quenched samples, was taken into account to re-examine the deviations expected for relatively short nucleation time, and/or for cases when the actual nucleation temperature is displaced from the corresponding nucleation peak. Additional solutions were computed to obtain the dependence of crystallization peak temperature on the nucleation temperature. It was found that the crystallization peak temperature reaches its lowest value after nucleation at a temperature which can be significantly higher than the peak temperature of steady state nucleation.     

The influence of time dependence of temperature on the kinetics of nucleation

In this work the influence of time dependence of temperature on kinetic processes during nucleation on a simple model is studied. The nucleation rate and the average effective radius of nuclei R is counted. It is proved that in the case of oscillating temperature the nucleation rate also oscillates and the rate of growth of radius R is directly proportional to frequency of temperature change.     

On slow protein crystal nucleation: cluster‐cluster aggregation on diffusional encounters

With a view to experimental results on protein crystal nucleation the effects of cluster coalescence are probed semi‐quantitatively. The steric association restriction, which stems from the patchy surface of the protein molecules, explains both experimentally measured low crystal nucleation rate and coalescence limitations for crystalline clusters of protein molecules. The conclusion is that due to its action, and the impact of rotational diffusion, the coalescence of critical (and/or supercritical) clusters should be rejected as a conceivable alternative for explaining the slow nucleation of protein crystals. Besides, the analysis of cluster‐cluster aggregation on diffusional encounters may be of more general interest; it may be helpful by considering the coalescence of structured bio‐nano‐particles. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

First measurements of time-dependent nucleation as a function of composition in Na2O · 2CaO · 3SiO2 glasses

First measurements of the composition dependence of the time-dependent nucleation rate are presented. Nucleation rates of the stoichiometric crystalline phase, Na₂O · 2CaO · 3SiO₂, from quenched glasses made with different SiO₂ concentrations were determined as a function of temperature and glass composition. A strong compositional dependence of the nucleation rates and a weak dependence for the induction times were observed. Using measured values of the liquidus temperatures and growth velocities as a function of glass composition, these data are shown to be consistent with predictions from the classical theory of nucleation, assuming a composition-dependent interfacial energy.     

A test of classical nucleation theory: crystal nucleation of lithium disilicate glass

For several inorganic glasses predictions have been made of the homogeneous crystal nucleation rate using classical nucleation theory. However, in none of these cases were comparisons made with experiment, due primarily to the inability of being able to guarantee homogeneous conditions. Evidence has been provided that crystalline formation in Li₂O · 2 SiO₂ glass may initiate by a homogeneous mechanism. Thus, we have computed the nucleation rate curve of lithium disilicate crystals in this glass. It is found that not only do all forms of the classical theory predict nucleation rates many orders of magnitude smaller than those observed, but also that the temperature dependence of the theoretical rate is quite different from that observed.     

A kinetic treatment of glass formation V: Surface and bulk heterogeneous nucleation

An analytical method is described for calculating the detailed distribution of crystallite sizes in a supercooled liquid, and the changes in this distribution as a function of temperature (time) while the liquid is cooled from above the melting point.This method, termed the analysis of crystallization statistics, is applied to the calculation of continuous cooling curves for anorthite and o-terphenyl as representative of inorganic and organic systems. In addition to homogeneous nucleation, bulk as well as surface heterogeneous nucleation are considered. The effects of distributions of heterogeneities with contact angles between 40 and 100° as well as overall concentrations of heterogeneities between 10³ and 10⁹ cm^?3 are considered. Heterogeneities with contact angles higher than about 100° are shown not to have an effect on the critical cooling rate for typical concentrations of heterogeneities.For liquids containing distributions of heterogeneities, the nucleation behavior is dominated by small concentrations of heterogeneities having small contact angles. Theoretical log (I_vη) versus (T_r³ΔT_r²)^?1 curves have been constructed for homogeneous nucleation + heterogeneous nucleation with a single type of heterogeneity and for homogeneous nucleation + heterogeneous nucleation with the heterogeneities distributed with regard to contact angle. In the former case, the curve is composed of two linear portions; and in the latter case, the curve shows pronounced curvature. The curvature reflects a continous change in the frequency of heterogeneous nucleation.Surface heterogeneous nucleation was assumed to originate at discrete surface heterogeneities and was shown to give rise to continuous cooling curves similar to those calculated for bulk heterogeneous nucleation.     

Effect of various precipitants on the nucleation rate of tetragonal hen egg-white lysozyme crystals in an AC external electric field

It was observed that the effect of an external electric field on the nucleation rate of tetragonal hen egg-white lysozyme crystals varied depending on the precipitant used (NaCl, NiCl₂ or YbCl₃) and that the electric double layer (EDL) played an important role in generating an external electric field of the necessary strength to control the nucleation rate. This phenomenon depended on the ionic strength of the precipitant used; that is, a precipitant of greater ionic strength resulted in a thinner EDL and increased the effect of the external electric field as the driving force for nucleation. The dependence of the nucleation rate on the precipitant was attributed to the magnitudes of the external electric fields generated in EDLs of varying thickness which were formed in the presence of different precipitants.     

On the Vitality of the Classical Theory of Crystal Nucleation; Crystal Nucleation in Pure Own Melt; Atmospheric Ice and Snow; Ice in Frozen Foods

The main reason for the longevity of the Classical Nucleation Theory (CNT) is its firm thermodynamic basis; reviewing the discussion about the molecular-scale mechanism of crystal nucleation from solutions, and especially the mechanism of protein crystal nucleation, we note that the diverse nucleation pathways across the metastable phase cannot contradict the thermodynamic conclusions of the CNT. In this review paper, revisiting the basic postulates of CNT, we argue that not only the energy barrier for crystal nucleation but the entire dependence of Gibbs’ thermodynamic potential on the crystal size is worth interpreting. In doing so, two supplementations to CNT have been elaborated. The first one concerns the theoretical method employing Equilibration between the Bond energy (i.e., the intra-crystalline cohesive energy which maintains the integrity of a crystalline cluster), and the surface Destructive Energy (tending to tear-up the crystal) - abbreviated EBDE. Second, we show that the dependence of the Gibbs’ thermodynamic potential on the crystal size determines not only the birth, but also the initial growth (or dissolution during Ostwald ripening) of the just born nuclei of the new phase; this is predicted in the negative branch of the said dependence. Initially, EBDE was used for explaining crystal nucleation from solutions, but most recently, this method was redefined for considering crystal nucleation in melts. The purposively redefined EBDE was applied for considering ice nucleation, which is an important case of spontaneous melt crystallization in nature - the quantitative consideration of the ice crystal nucleation is needed for better understanding of atmospheric processes, such as snowfall, white frost, sleet, hail, and ice fog. By focusing on the action of ice nucleating particles (INPs), which engender heterogeneous nucleation of ice, the snowfall is elucidated in a new way - ice nucleation in the atmosphere is considered as a two-step process, the first one being vapor condensation in liquid droplets, and the second one - water freezing. Also, ice nucleation in frozen foods is re-considered applying EBDE. (It is known that freezing ensures a high-quality product and long shelf life of a wide range of food products, such as fish, meat, vegetables, tropical fruits, coffee, flavor essence, etc.) And because numbers and sizes of ice crystals are decisive for the degree of deterioration of food quality due to freezing, the mean sizes of the ice crystals (which depend on their number) are considered in a quantitative manner. Also, another consideration concerns ice crystal nucleation and growth occurring by freeze concentration of liquid foods. Although aimed at reviewing fundamental aspects of crystal nucleation, it is to be hoped that some results of the considerations in this paper may also be beneficial for practical applications; suggestions in this respect are mentioned throughout the paper. For instance, the direct comparison between ice crystal nucleation in pure water and in frozen foods suggests how the dynamic food freezing step may be optimized, etc. The review ends with a short paragraph presenting the advantages and disadvantages of CNT.     

Re-examination of effects of nucleation temperature and time on glass crystallization

The effects of nucleation temperature and time on the kinetics of non-isothermal glass crystallization have been re-examined to demonstrate the limitations of some approximate solutions used to extract kinetic parameters from differential thermal analysis (DTA) experiments. Those features were analyzed by numerical solutions of equations describing the dependence of fraction crystallized on the rates of nucleation and growth, and the corresponding transient time, reported for lithium disilicate. It was shown that the temperature of maximum nucleation rate varies on changing the nucleation time. Some guidelines were established to assist the selection of suitable conditions to perform crystallization studies by DTA, and to extract the values of activation energy and dimensionality of growth from the dependences of crystallization peak temperature on heating rate, and nucleation time. The main limitations of these methods were identified and discussed.     

Improvement of the theory of heterogeneous crystallization of metals and choice of the nanomodifier particle sizes

A numerical analysis of the joint effect that the radii of the nucleus and the insoluble impurity particle have on the nucleation work and the nucleation rate of crystallization centers has been performed. The extreme dependence of these parameters on the impurity particle size has been established for the first time and scientifically justified practical recommendations on the choice of nanomodifier particle sizes have been elaborated.     

Modelling of Growth Rate for MSMPR Crystallizer Data

One of the most important applications of the population balance approach to MSMPR crystallizer modelling is the recovery of crystal nucleation and growth rates data from steady-state crystal size distributions (RANDOLPH, LARSON). A large number of studies have confirmed that both size-dependent (growth rate is a function of crystal size) and growth rate dispersion (crystals of the same size do not have the same growth rate), causes nonlinearities which limit the usefulness of the RANDOLPH and LARSON approach. A discussion of modelling of crystal growth kinetics for simulated and real MSMPR crystallizer data is presented. In the former case, both linear and non-linear log population density distributions are used. The modelling of growth kinetics is done twice – once assuming that growth rate dispersion is a source of curvature in the log population density vs size data plot and, again when this curvature is caused by size-dependent growth. Calculations clearly indicate that even for crystallizing systems which follow the McCabe's δL law, both growth rate dispersion and size-dependent growth models lead to proper estimation of growth kinetics. When log population density vs size data plots exhibit curvature, however, use of the size-dependent growth rate approach gives more reliable growth kinetics across a broad crystal size range than those obtained from modelling of growth kinetics by growth rate dispersion.     

Observation of nucleation effect on crystallization in lithium aluminosilicate glass by viscosity measurement

The crystal nucleation effect in lithium aluminosilicate glasses was investigated by the viscosity measurement with aid of the fiber elongation method. The abrupt increase of viscosity due to the crystallization of glass was observed in viscosity-temperature curve but the minimum viscosity temperature (T_η) related with crystallization showed a strong dependence on the nucleation state such as nucleation temperature, nucleation time and heating rate. The results by viscosity agreed well with those of DTA. The nucleation effect on the microstructure of glass-ceramics was also discussed. Finally, the nucleation effect on the crystallization kinetics was approached quantitatively by calculating the crystal volume from viscosity value.     

Pressure and concentration dependence of nucleation kinetics for crystallization of subtilisin

Nucleation kinetics of subtilisin was studied as a function of supersaturation (10–70 mg/ml) and pressure (0.1–34 MPa). The nucleation was found to have a 1.6 order dependence on supersaturation. Extent of nucleation decreased by a factor of 60 as the pressure increased from 0.1 to 34 MPa. High pressure was also used as a probe to understand the nucleation behavior of subtilisin and an estimation of the pressure-dependence of nucleation rate provided a value of 330 cm³/mol for the activation volume for nucleation. The residues in the contact region were determined to be hydrophilic by protein structure analysis. Responsible mechanisms are hypothesized.     

A Calorimetric Measurement of Phase Transformation Kinetics in Solid Octaphenylcyclotetrasiloxane: Evidence for Nucleation as the Rate Determining Process

The kinetics of transformation from the metastable (at high temperature) solid 3-phase of octaphenylcyclotetrasiloxane to the stable 2-phase have been measured by a calorimetric technique. The results of the kinetic study and several related experiments are consistent with the view that nucleation is the rate determining process. Transformation time constants for large and small crystallites were found; the temperature dependence of each is in accord with the Turnbull-Fisher nucleation theory. From that theory γ, the interfacial energy between the 3- and 2-phases, has been found to be -3 ergs/cm², an order of magnitude smaller than that for typical liquid-solid interfaces. A study of the dependence of the transformation kinetics on crystallite size suggests that the phase change is surface nucleated.     

On the interpretation of metastable zone width in anti‐solvent crystallization

Considering nonlinear dependence of solute concentration on anti‐solvent–solvent composition a modified Nývlt‐like equation based on traditional power‐law relation between nucleation rate and developing supersaturation and a new equation based on the classical theory of three‐dimension nucleation are proposed to explain the dependence of anti‐solvent addition rate on metastable zone width defined as excessive anti‐solvent composition in anti‐solvent crystallization. The experimental data on the metastable zone width in anti‐solvent crystallization of benzoic acid are analyzed and discussed from the standpoint of these equations. It is found that the new approach based on the classical nucleation theory provides better insight into the processes involved in anti‐solvent crystallization. Analysis of the experimental results on anti‐solvent crystallization of benzoic acid [D. O'Grady, M. Barret, E. Casey, and B. Glennon, Trans. IChemE A 85 , 945, (2007)] revealed that: (1) the value of metastable zone width for a solvent–anti‐solvent system is determined by the solute–solvent and solute–anti‐solvent interactions, (2) the dependence of the metastable zone width on stirring is associated with the enthalpy of mixing, and (3) the new approach predicts a threshold anti‐solvent addition rate associated with the setting up of an equilibrium between solvent and anti‐solvent and a maximum anti‐solvent addition rate connected with the induction period t_ind for the onset of crystallization. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)