Classical Homogeneous Nucleation Theory Research Articles

Nucleation Parameters of Pure and Impurity Added MgSO4·7H2O Single Crystals

Induction periods have been measured for various supersaturated aqueous solutions of magnesium sulphate heptahydrate (MgSO4·7H2O) added separately with KCl, KNO3, urea, and thiourea by the direct vision method. Various critical nucleation parameters have been calculated based on classical theory for homogeneous crystal nucleation. The critical nucleation parameters increase with the increase in concentration of impurity in the MgSO4·7H2O solutions.

Modified classical homogeneous nucleation theory and a new minimum in free energy change: 1. A new minimum and Kelvin equation

The main concern of classical homogeneous nucleation theory has been a thermodynamic description of initial stage of nucleation from embryo to nucleus with a little larger size over the critical one, thus, the change of parent phase in the system has been assumed to be negligible because of the largeness in volume and mass comparing that of nuclei. As a result, the nucleation curve (free energy change versus nucleus size) passes through well-known single maximum point corresponding to the critical size of the nucleus. In the present study, thermodynamics of the classical homogeneous nucleation was re-visited and developed a modified equation for multi-component solution and gas system with multi-component nuclei by taking into account the change of the free energy of parent phase. Using this equation, the calculation of nucleation curve beyond the size of critical nucleus became possible. A calculation of A–B binary solution system revealed a new minimum point in the nucleation curve, in addition to the maximum point. This minimum point indicates the theoretical possibility to stabilize a large amount of nano-nuclei in equilibrium with the supersaturated parent phase. In addition, Kelvin equation was proved at the extremum on the nucleation curve. Many scientists have misunderstood that Kelvin equation corresponds to the maximum state because they have unnoticed the presence of the minimum and its stability. At the minimum state, the nuclei should be more stable than those at the maximum state. Thus, Kelvin equation should correspond to the minimum state rather than the maximum state.

Experimental study of boiling-up kinetics and superheat limit for n-hexane on solid powder-like structures

The experimental setup is described and results are presented for measuring average boiling-up lag time for superheated n-hexane mixed with solid structures (activated coal, cellulose, silica gel) as function of temperature under atmospheric pressure. The “aging” of the cell with the filler was carried out before measurements: this was about 600–1000 boiling events. We developed a new method for analysing of “aging” procedure: comparison of average flux (frequency) of boiling-up events (processing of experimental data) and the frequency of nucleation obtained from exponential model. By the end of aging of the cell with silica gel the average empirical flux reduces by factor of four relative to the “exponential” value. But for activated coal and cellulose the difference in these fluxes is about 20 %. In all experiments, the event flux was nonstationary. For n-hexane in tested systems, the margin of superheating was Tn/Tcr ≅ 0.873–0.875, although it was Tn/Tcr ≅ ≅ 0.883 for n-pentane in systems filled by nickel powder (sintered porous nickel with grains of 1.5 or 5.0 micron size) and in the presence of a smooth copper plate. The average time of boiling-up lag in n-hexane at low normalized temperatures was also smaller than for n-pentane. For all systems, the lag time is almost the same for the temperature range Tn/Tcr ≅ 0.860–0.874 (plateau). Thus, a smaller amount of superheated liquid or its division into smaller liquid elements does not result in longer liquid lifetime for superheat liquid and the maximal superheat temperature, as one could expect from the classical theory of homogeneous nucleation.

Comparison between the classical theory predictions and molecular simulation results for heterogeneous nucleation of argon

We have performed Monte Carlo simulations of homogeneous and heterogeneous nucleations of Lennard-Jones argon clusters. The simulation results were interpreted using the major concept posing a difference between the homogeneous and heterogeneous classical nucleation theories-the contact parameter. Our results show that the multiplication concept of the classical heterogeneous nucleation theory describes the cluster-substrate interaction surprisingly well even for small molecular clusters. However, in the case of argon nucleating on a rigid monolayer of fcc(111) substrate at T=60 K, the argon-substrate atom interaction being approximately one-third as strong as the argon-argon interaction, the use of the classical theory concept results in an underestimation of the heterogeneous nucleation rate by two to three orders of magnitude even for large clusters. The main contribution to this discrepancy is induced by the failure of the classical theory of homogeneous nucleation to predict the energy involved in bringing one molecule from the vapor to the cluster for clusters containing less than approximately 15 molecules.

Efflorescence Relative Humidity for Ammonium Sulfate Particles

The classical homogeneous nucleation theory was employed to calculate the efflorescence relative humidity (ERH) of airborne ammonium sulfate particles with a wide size range (8 nm to 17 microm) at room temperature. The theoretical predictions are in good agreement with the experimentally measured values. When the ammonium sulfate particle is decreased in size, the ERH first decreases, reaches a minimum around 30% for particle diameter equal to about 30 nm, and then increases. It is for the first time that the Kelvin effect is theoretically verified to substantially affect the ERH of ammonium sulfate particles smaller than 30 nm, while the aerosol size is the dominant factor affecting the efflorescent behavior of ammonium sulfate particles larger than 50 nm.

A Structural−Kinetic Approach to Model Face-Specific Solution/Crystal Surface Energy Associated with the Crystallization of Acetyl Salicylic Acid from Supersaturated Aqueous/Ethanol Solution

Classical homogeneous nucleation theory has been integrated with molecular modeling techniques to model the effects of cluster shape-anisotropy associated with nucleation from solution. In this approach, the geometric shape of a crystal nucleus is modeled assuming it equates to the predicted growth morphology of the resultant macroscopic crystal with the later simulated via an attachment energy model using empirical intermolecular force calculations adopting the atom−atom approximation. A new coupled model integrating nucleation theory, morphological simulation and solvent binding calculations, has been developed and combined with experimental nucleation data for the case example of acetyl salicylic acid (aspirin) crystallizing from an aqueous/ethanol solution. Nucleation parameters, such as critical nucleus size and specific surface energy, have been calculated and compared for the cases of both isotropic (spherical) and anisotropic (polyhedral) nucleus models. A comparison between nucleation data modeled on the basis of both shaped polyhedral and spherical clusters reveals a larger critical nucleus size for the anisotropic (Dequivalent = 16.54 Å at 40 °C) compared to the isotropic (D = 13.07 Å at 40 °C) shape model. Theoretical molecular modeling calculations of the specific surface energy anisotropy in solution show the specific surface energy for the habit planes of aspirin to vary from 3.65 mJ/m2 for the dominant {100} facet to 113 mJ/m2 for the minor {1̄11} facet. A comparison between the dominant crystal habit planes, notably, the hydrophilic {100} and hydrophobic {002}, reveals them to be more differentiated in terms of their specific surface energy in solution (3.65 and 10.90 mJ/m2) than in a vacuum (829 and 904 mJ/m2), respectively, in good agreement with their known surface chemistry. The total, specific surface energy (41.9 mJ/m2) in solution calculated via molecular modeling for the polyhedral-shaped clusters was found to be somewhat larger, but still in pleasing general agreement with that calculated from experimental nucleation data as determined using induction time measurements assuming a spherical nucleus (3.99 mJ/m2). The potential for the further development of this overall modeling approach is reviewed.

Kinetics of slow collapse process: Thermodynamic description of rate constants

Insoluble monolayer formed at the air/water interface compressed at a surface pressure above the equilibrium spreading pressure is unstable. In presented paper the classical theory for homogeneous nucleation is modified adding the activation energy term. It allows quantitative thermodynamic interpretation of the slow collapse phase transformation. The collapse of stearic acid Langmuir films has been carried out by systematic measurements of the area loss–time isobaric dependencies at various temperatures and isothermal dependencies at various pressures. Activation energies (activation enthalpies) and activation entropies have been evaluated for the nucleation ( E α = 1.32 eV) and the growth ( E β = 1.55 eV) processes. The experimental data for various pressures are discussed on the basis of the Gibbs energy analysis.

Effect of ammonia on new particle formation: A kinetic H2SO4‐H2O‐NH3 nucleation model constrained by laboratory measurements

A clear understanding of the nucleation mechanisms is important to understand the lifecycle of atmospheric particles and to provide reliable predictions of climate change associated with aerosols. On the basis of the classical ternary homogeneous nucleation (THN) theory developed in recent years, the presence of ppt level NH3 concentration significantly enhances nucleation rates. Here we compare the predicted NH3 enhancement with that derived from several reported experiments. We find that the differences between results predicted and observed are very large. Compared to the binary homogeneous nucleation of H2SO4‐H2O (no NH3), the classical THN theory predicts up to ∼30 orders of magnitude enhancement in the nucleation rates in the presence of ppt level NH3, while the laboratory measurements show only one to two orders of magnitude enhancement in the presence of several ppt to several ppm level of NH3. Furthermore, the classical THN theory predicts a decrease in nucleation rates as RH increases, but the measurements indicate an opposite trend. Since experimental results are probably more reliable than the theoretical model, we developed a kinetic THN model so that the nucleation enhancement effect of NH3 derived from these experimental results could be included. The kinetic model assumes that H2SO4 is the principal specie dominating the cluster growth and nucleation, while H2O and NH3 are secondary species influencing the cluster compositions and hence the evaporation coefficient of H2SO4 from the clusters. The kinetic model is a new approach to simulating THN, and it can take into account the thermodynamic data of molecular clusters derived from density functional theory study and prenucleation cluster measurements. The preliminary simulations using the kinetic THN model constrained by the experimental results indicate a negligible contribution of THN to new particle formation in the boundary atmosphere. Our analysis suggests that the kinetic THN model is unlikely to underpredict the real ternary nucleation rates by much as long as there is no significant error in the experimental results. The implication of this finding and areas for further investigations are discussed.

Paraffin Polydispersity Facilitates Mechanical Gelation

Incipient wax−oil gel deposits form in crude oil transport pipelines when long-chain n-paraffins precipitate at the cold interior surface of the pipe wall. The kinetics of paraffin gel formation was studied using model fluids consisting of monodisperse and polydisperse n-paraffin components dissolved in petroleum mineral oil. Classical homogeneous nucleation theory was applied to investigate the supersaturation conditions necessary for crystal formation. Differential scanning calorimetry was used to monitor paraffin crystallization rates and to provide solid-phase fraction measurements. Gelation occurs when growing n-paraffin crystals interlock and form a volume-spanning crystal network which entrains the remaining liquid oil among the crystals. Paraffin wax−oil gels exhibit a mechanical response to an imposed oscillatory stress, which is characterized by the elastic storage modulus G‘ being greater in magnitude than the viscous loss modulus, G‘ ‘. Low-temperature rheological gels can form from model fluids with n-paraffin contents as low as 0.5 wt %. Images of wax−oil gel morphologies were obtained using a cross-polarized microscope fitted with a z-drive and indicated crystal lengths of ∼10−20 μm. A microstructural gelation model based on percolation theory was introduced to provide predictions of gel formation conditions among randomly oriented paraffin crystals. The structural model provides correlations of crystal morphologies and solid fractions at the percolation threshold condition. Comparison of the initial wax contents required for gelation of monodisperse and polydisperse n-paraffin wax indicates that sharp crystal edges and ordered crystal faces hinder the paraffin crystal−crystal “anchoring” interactions which result in mechanical gelation.

Pyrogenic Iron(III)-Doped TiO2 Nanopowders Synthesized in RF Thermal Plasma: Phase Formation, Defect Structure, Band Gap, and Magnetic Properties

Iron(III)-doped TiO(2) nanopowders, with controlled iron to titanium atomic ratios (R(Fe/Ti)) ranging from nominal 0 to 20%, were synthesized using oxidative pyrolysis of liquid-feed metallorganic precursors in a radiation-frequency (RF) thermal plasma. The valence of iron doped in the TiO(2), phase formation, defect structures, band gaps, and magnetic properties of the resultant nanopowders were systematically investigated using Mössbauer spectroscopy, XRD, Raman spectroscopy, TEM/HRTEM, UV-vis spectroscopy, and measurements of magnetic properties. The iron doped in TiO(2) was trivalent (3+) in a high-spin state as determined by the isomer shift and quadrupole splitting from the Mössbauer spectra. No other phases except anatase and rutile TiO(2) were identified in the resultant nanopowders. Interestingly, thermodynamically metastable anatase predominated in the undoped TiO(2) nanopowders, which can be explained from a kinetic point of view based on classical homogeneous nucleation theory. With iron doping, the formation of rutile was strongly promoted because rutile is more tolerant than anatase to the defects such as oxygen vacancies resulting from the substitution of Fe(3+) for Ti(4+) in TiO(2). The concentration of oxygen vacancies reached a maximum at R(Fe/Ti) = 2% above which excessive oxygen vacancies tended to concentrate. As a result of this concentration, an extended defect like crystallographic shear (CS) structure was established. With iron doping, red shift of the absorption edges occurred in addition to the d-d electron transition of iron in the visible light region. The as-prepared iron-doped TiO(2) nanopowders were paramagnetic in nature at room temperature.

Steady-State Cavitating Nozzle Flows With Nucleation

Quasi-one-dimensional steady-state cavitating nozzle flows with homogeneous bubble nucleation and nonlinear bubble dynamics are considered using a continuum bubbly liquid flow model. The onset of cavitation is modeled using an improved version of the classical theory of homogeneous nucleation, and the nonlinear dynamics of cavitating bubbles is described by the classical Rayleigh-Plesset equation. Using a polytropic law for the partial gas pressure within the bubble and accounting for the classical damping mechanisms, in a crude manner, by an effective viscosity, stable steady-state solutions with stationary shock waves as well as unstable flashing flow solutions were obtained, similar to the homogeneous bubbly flow solutions given by Wang and Brennen [J. Fluids Eng., 120, 166–170, 1998] and by Delale, Schnerr, and Sauer [J. Fluid Mech., 427, 167–204, 2001]. In particular, reductions in the maximum bubble radius and bubble collapse periods are observed for stable nucleating nozzle flows as compared to the nonnucleating stable solution of Wang and Brennen under similar conditions.

Simple correction to the classical theory of homogeneous nucleation

Formation of the new disperse phase via homogeneous nucleation plays a fundamental role wherever the first-order phase transitions occur. Inconsistent temperature dependence of the nucleation rates and poor agreement of theoretical critical supersaturations with experimental data for a number of substances are fundamental problems of the classical nucleation theory (CNT). Here we show that these problems can be solved with a simple empirical correction to CNT. Despite its simplicity, the corrected CNT (CCNT) accurately predicts temperature dependences and absolute values of the critical supersaturations for both organic and inorganic substances with widely varying properties at different ambient conditions and it works surprisingly well in a wide size range down to few molecules. The difference in predictions of CCNT and other versions of the classical nucleation theory commonly used in analyzing experimental data is discussed. It has been found that CCNT consistently gives better agreement with experimental data than other versions of classical nucleation theory.

Determination of induction period and crystal growth mechanism of dexamethasone sodium phosphate in methanol–acetone system

The induction period of dexamethasone sodium phosphate at different supersaturation was experimentally determined in a methanol–acetone system. The laser monitoring observation technique was used to determine the appearance of the first nucleus in solution. The effect of solution composition on induction period was discussed. Based on classical homogeneous nucleation theory, the solid–liquid interfacial tension and surface entropy factor were calculated from the induction period data. The experimentally determined values of interfacial tension are in agreement with the theoretical values predicted by the Mersmann equation. It was found that the nucleus of dexamethasone sodium phosphate grows continuously in pure methanol and turns from continuous growth to birth and spread growth with increasing acetone content in a methanol–acetone mixture.

Thermal melting of solid materials induced by ultrafast laser pulse irradiation as explosively homogeneous nucleation

By using the classical nucleation and growth dynamics to describe the melting process of superheated solid materials induced by ultrafast laser pulse irradiation, we found that the melting process of highly superheated solid materials is governed by explosively homogeneous nucleation. For the superheating degree of 1.6–1.9, the melting time for aluminum is several tens of picoseconds. In this case the lattice is heated within a few picoseconds, thus the total melting time of the solid materials is governed by the explosively homogeneous nucleation. However, for the very short laser pulse irradiation, such as femtosecond laser pulses with high fluences, the nonthermal mechanism plays a critical role in the melting process, and the classical homogeneous nucleation theory and growth dynamics is not useful. That is to say, the classical homogeneous nucleation theory and growth dynamics is useful for describing the rapid thermal melting process, however, is not suitable for explaining the nonthermal melting process.

Formation of binary ion clusters from polar vapours: effect of the dipole-charge interaction

Abstract. Formation of binary cluster ions from polar vapours is considered. The effect of vapour polarity on the size and composition of the critical clusters is investigated theoretically and a corrected version of classical Kelvin-Thomson theory of binary ion-induced nucleation is derived. The model predictions of the derived theory are compared to the results given by classical binary homogeneous nucleation theory and ion-induced nucleation theory. The calculations are performed in wide range of the ambient conditions for a system composed of sulfuric acid and water vapour. It is shown that dipole-charge interaction significantly decreases the size of the critical clusters, especially under the atmospheric conditions when the size of critical clusters is predicted to be small.

Curvature-dependent surface tension of a growing droplet.

A ghost interface simulation technique is developed and applied to supersaturated Lennard-Jones liquid-vapor interfaces. It is shown that the surface tension decreases approximately linearly with the supersaturation ratio and that it vanishes at the spinodal. The effect leads to a curvature-dependent surface tension since, it is argued, the local supersaturation of the vapor above a droplet is greater than in the bulk due to slow diffusion in the vapor phase. An analytic approximation is given for the local supersaturation ratio, and an analytic expression for this contribution to Tolman's length is derived. The theory gives a smaller critical radius and reduces the free energy barrier to nucleation compared to classical homogeneous nucleation theory, which have important implications for the kinetics of droplet and bubble formation.

Effect of die temperature on the morphology of microcellular foams

AbstractA study on the extrusion of microcellular polystyrene foams at different foaming temperatures was carried out using CO2 as the foaming agent. The contraction flow in the extrusion die was simulated with FLUENT computational fluid dynamics code at two temperatures (150°C and 175°C) to predict pressure and temperature profiles in the die. The location of nucleation onset was determined based on the pressure profile and equilibrium solubility. The relative importance of pressure and temperature in determining the nucleation rate was compared using calculations based on classical homogeneous nucleation theory. Experimentally, the effects of die temperature (i.e., the foaming temperature) on the pressure profile in the die, cell size, cell density, and cell morphology were investigated at different screw rotation speeds (10 ∼ 30 rpm). Experimental results were compared with simulations to gain insight into the foaming process. Although the foaming temperature was found to be less significant than the pressure drop or the pressure drop rate in deciding the cell size and cell density, it affects the cell morphology dramatically. Open and closed cell structures can be generated by changing the foaming temperature. Microcellular foams of PS (with cell sizes smaller than 10 μm and cell densities greater than 10 cells/cm3) are created experimentally when the die temperature is 160°C, the pressure drop through the die is greater than 16 MPa, and the pressure drop rate is higher than 109 Pa/sec.

Topological defects and HCP nucleation in BCC helium

We propose a new model for the nature of the nucleation of HCP from BCC helium. The dynamic release of shear at the surface of the BCC–HCP phase boundary, sustains the simultaneous nucleation and growth of topological defects in the BCC phase. The topological defects are lines of dynamic shear in the BCC phase. The shear energy gain of this process balances the surface tension, as the growing HCP surface is quickly covered by many defect-loops. We show that this scenario gives better agreement with experiments, which differ with the classical theory of homogeneous nucleation by 6–10 orders of magnitude.

Diffusion–influenced nucleation: a case study of oxygen precipitation in silicon

The classical theory of homogeneous nucleation is an interface–limited theory and does not strictly apply for transformations where long–range diffusion is dominant. A new approach that takes account of the coupled stochastic fluxes of interface attachment and long–range diffusion is presented. Steady–state nucleation rates can be orders of magnitude smaller and the induction times for time–dependent nucleation correspondingly larger than expected from the classical theory. In solute precipitation, the regions of the parent phase near subcritical precipitate clusters are enriched (rather than depleted) in solute. Oxygen precipitation in silicon is chosen to illustrate these features. Measured precipitate densities following multi–step anneals show better agreement with predictions from the coupled–flux model than with those from the classical theory. The kinetics of formation for small oxide clusters and the dependence of their population density on the initial oxygen concentration agree well with data for thermal donor defects.

Homogeneous bubble nucleation in liquids: The classical theory revisited

The classical theory of homogeneous bubble nucleation is reconsidered by employing a phenomenological nucleation barrier in the capillarity approximation that utilizes the superheat threshold achieved in experiments. Consequently, an algorithm is constructed for the evaluation of the superheat temperatures in homogeneous boiling (tensile strengths in cavitation), the critical radii and steady-state nucleation rates. The nucleation theorem is written in this framework and is applied to the classical theory of homogeneous bubble nucleation for the phenomenological nucleation barrier employed. The superheat temperatures calculated show excellent agreement over a wide range of liquid pressures for most of the substances investigated. The steady-state nucleation rates are also altered by many orders of magnitude, in agreement with the results of previous investigators using different approaches.