Experimental Nucleation Rates Research Articles

Homogeneous Organic Crystal Nucleation Rates in Solution from the Perspective of Chemical Reaction Kinetics

It is demonstrated for 11 different combinations of organic solutes and solvents that the supersaturation dependence of homogeneous organic crystal nucleation rates from solution can be predicted from the solubility, bar a single empirical rate constant, when it is assumed that nucleation takes place in reversible aggregates of solvated solutes formed in supersaturated solutions. Reversible solute aggregation represents natural solute density fluctuations that take place in any solute/solvent system. For thermodynamically ideal solutions, the steady state size distribution, and thus the population of reversible aggregates in supersaturated solution, can be predicted quantitatively from the overall solute concentration by a simple mathematical expression. Supersaturation creates an excess of reversible aggregates with sizes exceeding that of the largest aggregate in saturated solution. It is shown that the number of these excess aggregates is proportional to experimental homogeneous nucleation rates, suggesting a rate equation for homogeneous nucleation that has only one empirical parameter, namely, a rate constant specific to the solute/solvent combination. This rate constant can be determined from standard nucleation rate data. The system-specificity of homogeneous nucleation rates thus appears to be encoded solely in a rate constant for the transformation of the large excess aggregates into crystal nuclei. The driving force for triggering nucleation events in these aggregates is likely the extremely high local supersaturation, which provides the conditions for spatiotemporally aligned bond-breaking (e.g., de-solvation) and bond-forming (e.g., solute–solute bonding) events that create stable crystal nuclei. The possible influence of heterogeneous nucleation by solid impurities is considered.

Critical Nucleus Size and Activation Energy of Ag Nucleation by Electrochemical Observation of Isolated Nucleation Events.

The induction times for electrodeposition of individual Ag nanoparticles on Pt nanodisk electrodes in acetonitrile were used to determine the critical nucleus size and activation energy barrier associated with the formation of Ag nuclei. Induction times for the nucleation and growth of a single Ag nanoparticle were determined following the application of a potential step to reduce Ag+ at overpotentials, η, ranging from -130 to -70 mV. Sufficiently small Pt electrodes (5.1 × 10-10-2.6 × 10-11 cm2) were used to ensure that the detection of a single Ag nucleation event occurred during the experimental observation time (150 ms-1000 s). Multiple measurements of Ag nucleation induction times were recorded to determine nucleation rates as a function of η using cumulative probability theory. Both classical nucleation theory (CNT) and the atomistic theory of electrochemical nucleation were employed to analyze experimental nucleation rates, without a requisite knowledge of the nucleus geometry or surface free energy. Using the CNT, the number of atoms comprising the critical size nucleus, Nc, was estimated to be 1-9 atoms for η ranging from -130 to -70 mV, in good agreement with 1-5 atoms obtained using atomistic theory. The experimental nucleation rates were also used to determine the activation energy barriers for nucleation from the CNT, which varied from 3.31 ± 0.05 to 13 ± 1 kT over the same range of η.

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.

A Review of Experimental Methods for Nucleation Rate Determination in Large-Volume Batch and Microfluidic Crystallization

Determination of the experimental nucleation rate for crystallization in solution has been acknowledged as an important topic for a long time, as it improves the design and control of industrial cr...

Crystal Nucleation Kinetics in Supercooled Germanium: MD Simulations versus Experimental Data.

The validity of the classical nucleation theory (CNT), the most important tool to describe and predict nucleation kinetics in supercooled liquids, has been at stake for almost a century. Here, we carried out comprehensive molecular dynamics simulations of the nucleation kinetics of a fast quenched supercooled germanium using the Stillinger-Weber potential at six temperatures, covering a supercooling range of T/Tm = 0.70-0.86, where Tm is the equilibrium melting temperature. We used the seeding method to determine the number of particles in the critical crystal nuclei at each supercooling, which yielded n* = 150-1300 atoms. The transport coefficient at the liquid/nucleus interface and the melting point were also obtained from the simulations. Using the parameters resulting directly from the simulations, the CNT embraces the experimental nucleation rates, J(T), with the following fitted (average) values of the nucleus/liquid interfacial free energy: γ = 0.244 and 0.201 J/m2, for the experimental and calculated values of thermodynamic driving force, Δμ(T), respectively, which are close to the value obtained from n*(T). Without using any fit parameter, the calculated nucleation rates for the experimental and calculated values of Δμ(T) embrace the experimental J(T) curve. Therefore, this finding favors the validity of the CNT.

Theoretical Models of Decompression-Induced Plagioclase Nucleation and Growth in Hydrated Silica-Rich Melts

Microlites crystallization changes the flow properties of ascending magmas, with major implications on eruption dynamics. In order to predict microlite number density and size, we present a theoretical modelling of plagioclase nucleation and growth in rhyolitic melts, aimed at reproducing the decompression-induced isothermal crystallization experiments of Mollard et al. (2012). Thus, the modelling is valid for plagioclase crystallization in haplotonalitic/rhyolitic melts decompressed at 875 °C from 200 MPa to final pressures of 50, 75, or 100 MPa, which represents effective undercooling (Teff) of 110, 80 and 55 °C, respectively. The nucleation-rate calculation using the Classical Nucleation Theory (CNT; case of homogeneous nucleation) and values of crystal-melt interfacial tensions () either from literature or empirically calculated strongly disagree with the experimental nucleation rates. Inverting the CNT calculation by using the experimentally-determined nucleation rates suggests plagioclase-liquid from 0.041 to 0.059 J.m-2 with Teff increasing from 55 to 110 °C, which is about 2-3 times lower than  determined empirically and macroscopically. Further refining  by considering a non-smooth and non-spherical plagioclase nucleus leads to an elongated and rough morphology with a shape parameter () of 500-1350 and a rugosity parameter (ds) of 2.3. The experimental plagioclase growth rates have been modelled by atom diffusion in melt, following Fick’s second law adapted to multicomponent systems. The component with the smallest flux (i.e. limiting growth) is calcium, as determined from the analytical profiles of the experimental glasses. The crystal-growth model considers a crystal-melt interface advancing over time and a chemically-closed finite reservoir. The overall good agreement between the model and the experimental growth laws validates diffusion as the main process controlling isothermal decompression-induced plagioclase growth under moderate Teff <110 °C. The calculated CaO diffusion coefficients in silicic melt at 875 °C and water saturation pressures from 50 to 100 MPa ranges from 10-14 to 10-15 m2/s. The simulations may be relevant to predict nucleation and growth of plagioclase microlites in phenocryst-poor rhyolitic magmas that rapidly ascend from a storage zone and crystallize at shallower. This may be relevant to eruption dynamics such as dome-related blasts.

Vapor-phase nucleation of n-pentane, n-hexane, and n-heptane: Critical cluster properties.

The first and second nucleation theorems provide a way to determine the molecular content and excess internal energies of critical clusters, which rely solely on experimental nucleation rates measured at constant temperatures and supersaturations, respectively. Here, we report the size n* and excess internal energy Ex(n*) of n-pentane, n-hexane, and n-heptane critical clusters when particles form under the highly supersaturated conditions present in supersonic expansions. In summary, critical clusters contain from ∼2 to ∼11 molecules and exhibit the expected increase in the critical cluster size with increasing temperature and decreasing supersaturation. Surprisingly, the n* values for all three alkanes appear to lie along a single line when plotted as a function of supersaturation. Within the framework of the capillarity approximation, the excess internal energies determined for the n-heptane critical clusters formed under the low temperature (∼150 K) conditions in our supersonic nozzle are reasonably consistent with those determined under higher temperature (∼250 K) conditions in the thermal diffusion cloud chamber by Rudek et al. [J. Chem. Phys. 105, 4707 (1996)].

Water nucleation at extreme supersaturation.

We report water cluster formation in the uniform postnozzle flow of a Laval nozzle at low temperatures of 87.0 and 47.5 K and high supersaturations of lnS ∼ 41 and 104, respectively. Cluster size distributions were measured after soft single-photon ionization at 13.8 eV with mass spectrometry. Critical cluster sizes were determined from cluster size distributions recorded as a function of increasing supersaturation, resulting in critical sizes of 6-15 and 1, respectively. Comparison with previous data for propane and toluene reveals a systematic trend in the nucleation behavior, i.e., a change from a steplike increase to a gradual increase of the maximum cluster size with increasing supersaturation. Experimental nucleation rates of 5 · 1015 cm-3 s-1 and 2 · 1015 cm-3 s-1 for lnS ∼ 41 and 104, respectively, were retrieved from cluster size distributions recorded as a function of nucleation time. These lie 2-3 orders of magnitude below the gas kinetic collision limit assuming unit sticking probability, but they agree very well with a recent prediction by a master equation model based on ab initio transition state theory. The experimental observations are consistent with barrierless growth at 47.5 K, but they hint at a more complex nucleation behavior for the measurement at 87.0 K.

Nucleation rates of droplets in supersaturated steam and water vapour–carrier gas mixtures between 200 and 450 K

We compare experimental nucleation rates for water vapour in various carrier gases, estimated nucleation rates for steam, and nucleation rates obtained from molecular simulations. The data for steam are deduced from empirical adjustments of the classical nucleation theory developed by various authors to reproduce pressure and optical data for condensing steam flows in converging-diverging nozzles and turbine stages. By combining the data for nucleation in carrier gases and the data for steam nucleation, an unprecedented temperature range of 250 K is available to study the temperature dependence of nucleation rate. Original results of molecular dynamic simulations for TIP4P/2005 force field in the NVE (system constrained by number of particles, volume and energy) conditions are provided. Correction of classical nucleation theory for non-isothermal nucleation conditions is applied to experimental and simulated data. The nucleation rate data for steam follow a similar temperature trend as the nucleation rate data for water vapour in carrier gases at lower temperatures. The ratio of observed nucleation rates to classical nucleation theory predictions decreases more steeply with temperature than the empirical correlation by Wölk et al. ( J Chem Phys 2002; 117: 4954–4960). On the contrary to experimental data, the ratios of nucleation rates computed from molecular simulations to classical nucleation theory predictions do not show a significant temperature trend.

Vapor phase nucleation of the short-chain n-alkanes (n-pentane, n-hexane and n-heptane): Experiments and Monte Carlo simulations.

We measured the nucleation rates of n-pentane through n-heptane in a supersonic nozzle at temperatures ranging from ca. 109 K to 168 K. For n-pentane and n-hexane, these are the first nucleation rate measurements that have been made, and the trends in the current data agree well with those in the earlier work of Ghosh et al. [J. Chem. Phys. 132, 024307 (2010)] for longer chain alkanes. Complementary Monte Carlo simulations, using the transferable potentials for phase equilibria-united atom potentials, suggest that despite the high degree of supercooling, the critical clusters remain liquid like under experimental conditions for n-pentane through n-heptane, but adopt more ordered structures for n-octane and n-nonane. For all three alkanes, the experimental and simulated nucleation rates are offset by ∼3 orders of magnitude when plotted as a function of ln S/(Tc/T - 1)1.5. Explicitly accounting for the surface tension difference between the real and model substances, or alternatively using the Hale [Phys. Rev. A 33, 4156 (1986); Metall. Mater. Trans. A 23, 1863 (1992)] scaling parameter, Ω, consistent with the model potential, increases the offset to ∼6 orders of magnitude.

Crystal nucleation in sedimenting colloidal suspensions.

Homogeneous crystal nucleation from a metastable hard-sphere colloidal liquid has been extensively studied in simulations and experiments. A 12 order of magnitude difference between simulated and experimental nucleation rates is observed, the origin of which remains a puzzle. Here, we experimentally study crystal nucleation at the single particle level in suspensions of hard-sphere-like colloids under the influence of sedimentation. We find that sedimentation significantly enhances the nucleation rate, but contrary to what was previously thought, this is not due to simple density fluctuations, as the nucleation barriers become independent of the local density in a sedimentating fluid. Instead, we find an enhancement of the local dynamics, which suggests that local convective cells are responsible for shear-enhanced crystallization at low volume fractions.

Predictions of homogeneous nucleation rates for n-alkanes accounting for the diffuse phase interface and capillary waves.

Homogeneous droplet nucleation has been studied for almost a century but has not yet been fully understood. In this work, we used the density gradient theory (DGT) and considered the influence of capillary waves (CWs) on the predicted size-dependent surface tensions and nucleation rates for selected n-alkanes. The DGT model was completed by an equation of state (EoS) based on the perturbed-chain statistical associating fluid theory and compared to the classical nucleation theory and the Peng-Robinson EoS. It was found that the critical clusters are practically free of CWs because they are so small that even the smallest wavelengths of CWs do not fit into their finite dimensions. The CWs contribute to the entropy of the system and thus decrease the surface tension. A correction for the effect of CWs on the surface tension is presented. The effect of the different EoSs is relatively small because by a fortuitous coincidence their predictions are similar in the relevant range of critical cluster sizes. The difference of the DGT predictions to the classical nucleation theory computations is important but not decisive. Of the effects investigated, the most pronounced is the suppression of CWs which causes a sizable decrease of the predicted nucleation rates. The major difference between experimental nucleation rate data and theoretical predictions remains in the temperature dependence. For normal alkanes, this discrepancy is much stronger than observed, e.g., for water. Theoretical corrections developed here have a minor influence on the temperature dependency. We provide empirical equations correcting the predicted nucleation rates to values comparable with experiments.

Phase transformations in PuGa 1 at.% alloy: New valuable insight into the isothermal martensitic δ→α′ transformation

This paper deals with the results of an investigation of low-temperature martensitic transformation in a metastable PuGa 1 at.% alloy. The kinetics of this process were studied experimentally via in situ XRD characterizations performed during isothermal holds. This revealed a double-C Time–Temperature–Transformation diagram similar to that reported in the literature for enriched alloys. The originality of our results lies in the demonstration of the existence of different martensite morphologies corresponding to the upper and lower parts of the TTT diagram. These microstructural variations seem to be the result of different accommodation mechanisms.The experimental nucleation rate study has suggested an autocatalytic character of the isothermal martensitic nucleation process as well as a large sensitivity to elastic and plastic strain. Indeed, the incubation period and subsequent increase in nucleation rate could be related to the appearance of new embryos rather than the growth of existing plates, while the partial character of the transformation may be attributed to interactions between plates that are being formed in the two-phase δ + α′ alloy. These hypotheses have been confirmed by the classical heterogeneous nucleation approach by Pati and Cohen, which takes into account the autocatalytic character of the transformation. This approach has also indirectly revealed the existence of restraining forces that reflect a high sensitivity of the energy barrier for nucleation to the α′ phase amount.

Theoretical evidence for inconsistencies in experimental bubble nucleation rates of propane/CO2 and R22/CO2 mixtures

A binary formulation of the classical nucleation theory (CNT) is developed for homogeneous bubble nucleation in systems composed of a liquid solvent and a dissolved gas. The CNT predictions coincide with experimental nucleation data from the literature for diethylether/N2 and isobutane/CO2 mixtures, while several inconsistencies are identified for propane/CO2 and R22/CO2 experimental datasets.

Nucleation of NaCl from Aqueous Solution: Critical Sizes, Ion-Attachment Kinetics, and Rates

Nucleation and crystal growth are important in material synthesis, climate modeling, biomineralization, and pharmaceutical formulation. Despite tremendous efforts, the mechanisms and kinetics of nucleation remain elusive to both theory and experiment. Here we investigate sodium chloride (NaCl) nucleation from supersaturated brines using seeded atomistic simulations, polymorph-specific order parameters, and elements of classical nucleation theory. We find that NaCl nucleates via the common rock salt structure. Ion desolvation-not diffusion-is identified as the limiting resistance to attachment. Two different analyses give approximately consistent attachment kinetics: diffusion along the nucleus size coordinate and reaction-diffusion analysis of approach-to-coexistence simulation data from Aragones et al. ( J. Chem. Phys. 2012, 136, 244508 ). Our simulations were performed at realistic supersaturations to enable the first direct comparison to experimental nucleation rates for this system. The computed and measured rates converge to a common upper limit at extremely high supersaturation. However, our rate predictions are between 15 and 30 orders of magnitude too fast. We comment on possible origins of the large discrepancy.

Communication: Tolman length and rigidity constants of water and their role in nucleation.

A proper understanding of nucleation is crucial in several natural and industrial processes. However, accurate quantitative predictions of this phenomenon have not been possible. The most popular tool for calculating nucleation rates, classical nucleation theory (CNT), deviates by orders of magnitude from experiments for most substances. We investigate whether part of this discrepancy can be accounted for by the curvature-dependence of the surface tension. To that end, we evaluate the leading order corrections for water, the Tolman length and the rigidity constants, using square gradient theory coupled with the accurate cubic plus association equation of state. The Helfrich expansion is then used to incorporate them into the CNT-framework. For water condensation, the modified framework successfully corrects the erroneous temperature dependence of the nucleation rates given by the classical theory and reproduces experimental nucleation rates.

Molecular dynamics simulation on the wetting characteristic of micro-droplet on surfaces with different free energies

The wetting characteristic of micro-droplets on surfaces with different free energies is crucial to heterogeneous nucleation theory and the growth mechanism of micro-droplets during vapor condensation. In this paper, the spreading processes and wetting characteristics of nanoscale water droplets on various surfaces are explored by molecular dynamics simulation method. The surfaces are constructed from face centered cubic copper-like atoms with different Lennard-Jones potential parameters. The Lennard-Jones interaction energy well-depth of the surface atoms is adjusted to acquire different surface free energies, and the ratio of surface-water interaction energy well-depth to the water-water interaction energy well-depth is defined as the interaction intensity. In the present study, the relationship between interfacial free energies and solid-liquid interaction intensities is evaluated using molecular dynamics simulations. The wetting characteristics of TIP4P/2005 water droplets on surfaces with various free energies are simulated and analyzed as well, using molecular dynamics simulations under an NVT ensemble. Results indicate that the solid-liquid interfacial free energy increases as the solid-liquid interaction intensity increases, with different spreading processes and wetting characteristics achieved for the water droplets on these surfaces. For the surfaces with lower interaction intensities, water cannot spread on the solid surfaces and hydrophobic surfaces are obtained when the interaction intensity ratio between surface atoms and water molecules is lower than 1.6. As the interaction intensity increases, the surface translates from hydrophobic into hydrophilic, and finally into a complete wetting state as the interaction intensity reaches up to 3.5. Due to the limitation of nanoscale dimensions, the forces that exert on droplet surface are non-continuous and asymmetric. As a result, significant fluctuations of liquid-vapor interface and local solid-liquid contact line can be observed for the droplet in nanoscale. The transient contact angle of nano-droplets is also fluctuating within a certain range, which is different from that observed for macro-droplets. From the viewpoint of statistics, an apparent contact angle can be obtained for the droplet on each surface. The contact angle decreases with solid-liquid interaction intensities linearly, which is in accordance with the calculated results of classic Young's theory using the interfacial free energies obtained from molecular dynamics simulations. The fact that an apparent contact angle is already established for a droplet in nanoscale, supporting the capillary assumption that is widely adopted in classic nucleation theory. The fluctuation of liquid-vapor interface and contact angle also provides a qualitative explanation for the discrepancy between experimental nucleation rates and predictions in classic nucleation theory.

Isothermal Nucleation Rates of n-Propanol, n-Butanol, and n-Pentanol in Supersonic Nozzles: Critical Cluster Sizes and the Role of Coagulation.

We follow the nucleation of n-alcohols, n-propanol through n-pentanol, in two sets of supersonic nozzles having differing linear expansion rates. Combining the data from static pressure trace measurements with small-angle X-ray scattering we report the experimental nucleation rates and critical cluster sizes. For n-propanol, position resolved measurements clearly confirm that coagulation of the 2-10 nm size (radius) droplets occurs on the time scale of the experiment but that the effect of coagulation on the results is minimal. Under the conditions of the current experiments, our results suggest that alcohols have critical clusters that range from the dimer (n-pentanol) to the hexamer (n-propanol). We then compare the experimental results with classical nucleation theory (CNT), the Girshick-Chiu variant of CNT (GC), and the mean field kinetic nucleation theory (MKNT). Both CNT and MKNT underestimate the nucleation rates by up to 5 and 7 orders of magnitude, respectively, while GC theory predicts rates within 2 orders of magnitude. Correspondingly, the critical cluster size for all alcohols is overpredicted by factors of 2-9 with agreement improving with increasing chain length. An interesting byproduct of our experiments is that we find that the coagulation rate is enhanced by a factor of 3 over the value one would calculate for the free molecule regime.

Description of fluid dynamics and coupled transports in models of a laminar flow diffusion chamber

The aim of this study is to assess how much the results of nucleation experiments in a laminar flow diffusion chamber (LFDC) are influenced by the complexity of the model of the transport properties. The effects of the type of fluid dynamic model (the steady state compressible Navier-Stokes system for an ideal gas/parabolic profile approximation) and the contributions of the coupled terms describing the Dufour effects and thermodiffusion on the predicted magnitude of the nucleation maxima and its location were investigated. This study was performed on the model of the homogeneous nucleation of an n-butanol-He vapor mixture in a LFDC. The isothermal dependencies of the nucleation rate on supersaturation were determined at three nucleation temperatures: 265 K, 270 K, and 280 K. For this purpose, the experimental LFDC data measured by A. P. Hyvärinen et al. [J. Chem. Phys. 124, 224304 (2006)] were reevaluated using transport models at different levels of complexity. Our results indicate that the type of fluid dynamical model affects both the position of the nucleation maxima in the LFDC and the maximum value of the nucleation rate. On the other hand, the Dufour effects and thermodiffusion perceptibly influence only the value of the maximal nucleation rate. Its position changes only marginally. The dependence of the maximum experimental nucleation rate on the saturation ratio and nucleation temperature was acquired for each case. Based on this dependence, we presented a method for the comparison and evaluation of the uncertainties of simpler models' solutions for the results, where we assumed that the model with Navier-Stokes equations and both coupled effects taken into account was the basis. From this comparison, it follows that an inappropriate choice of mathematical models could lead to relative errors of the order of several hundred percent in the maximum experimental nucleation rate. In the conclusion of this study, we also provide some general recommendations concerning the proper choice and setting of the mathematical model of transport processes in the LFDC.

Parametrization of the homogeneous ice nucleation rate for the numerical simulation of multiphase flow

AbstractHomogeneous ice nucleation in supercooled water is studied with the classical nucleation theory (CNT). The nucleated ice particles are assumed to possess physical properties of metastable cubic ice. The comparison with literature data on homogeneous ice nucleation rates shows a good agreement between CNT predictions and experimental nucleation rates. The CNT predicted nucleation rate, critical cluster radius, and critical nucleation work are parametrized for use in numerical simulations of the freezing process of supercooled water droplets. (© 2012 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)