Nucleation Theorem Research Articles

Nucleation of surface nanobubbles in electrochemistry: Analysis with nucleation theorem

The formation of single bubbles at nanoelectrodes during electrochemical reactions allows to accurately identify the critical nucleus for bubble formation. As demonstrated before, combining nanoelectrode experiments and an analysis approach based on classical nucleation theory (CNT) delivers useful insight into bubble nucleation. In this work we propose an alternative approach to analyze the critical nuclei by applying the nucleation theorem (NT), which is able to overcome the inherent shortcomings of CNT. The size of the critical nucleus can be calculated more accurately by fitting experimental data in a simple form of the NT. Simulating the local gas concentration using a finite element approach, and considering the effect of gas oversaturation on the interfacial tension and the real gas compressibility, we obtain a more realistic estimation of the critical nuclei morphology. With the NT-based analysis presented, we re-analyze the nucleation data reported before. The properties of the critical nuclei obtained here are roughly consistent with those obtained from the CNT-based approach. In addition, we confirm that the critical nucleus for bubble formation in high gas oversaturation is featured with a contact angle much larger than Young’s contact angle.

Critical cluster composition from homogeneous nucleation data: application to water in carbon dioxide\u2013nitrogen carrier gases

Knowledge on critical cluster composition is important for improving the nucleation theory. Thus, homogeneous water nucleation experiments previously carried out in nitrogen and 0%, 5%, 15% and 25% of carbon dioxide ( Campagna et al. 2020a, 2021) are analyzed. The tests were conducted at 240 K and 0.1 MPa, 1 MPa and 2 MPa. The observed nucleation rates are strongly dependent on supersaturation, pressure, temperature and mixture composition. These experimentally found dependencies can be used to derive the composition of critical clusters by means of the nucleation theorem. In this way, a macroscopic quantity, nucleation rate, reveals properties of critical clusters consisting of a few tens of molecules. Two novel methods are presented for the detailed application of the nucleation theorem. The first method extends to mixtures of ,,,,,,,N>2,,,,,, components the approach used in literature for two components. The second method not only applies to N>2 mixtures in a more straightforward manner, but it can also be used for unary as well as for binary and multi-component nucleation cases. To the best of our knowledge, for the first time the critical cluster composition is computed for high pressure nucleation data of a vapor (here water) in mixtures of two carrier gases (here carbon dioxide–nitrogen). After a proper parameterization of the nucleation rate data, both methods consistently lead to the same critical nuclei compositions within the experimental uncertainty. Increasing pressure and carbon dioxide molar fraction at fixed supersaturation leads to a decrease in the water content of the critical cluster, while the adsorbed number of nitrogen and carbon dioxide molecules increases. As a consequence, the surface tension decreases. This outcome explains the observed increase in the nucleation rate with increasing pressure and carbon dioxide molar fraction at constant supersaturation.Graphic abstract

A detailed numerical study on the evolution of droplet size distribution of dibutyl phthalate in a laminar flow diffusion chamber

A numerical model was used to study the homogeneous nucleation process of dibutyl phthalate (DBP) vapor in a laminar flow diffusion chamber (LFDC); the spatial and temporal evolution of DBP droplet size distribution was governed by the population balance equation (PBE). In the PBE, the nucleation rate was calculated by the self-consistent correction nucleation theory (SCCNT), droplet coagulation, vapor and droplet deposition losses were considered. The simulation results showed that the nucleation rate predicted by the SCCNT improved the underestimation of that predicted by the classical nucleation theory. Due to vapor deposition before nucleation and droplet deposition after nucleation on the wall, the DBP mass loss was severe, accounting for about 86.3% of the total inlet vapor mass, and the droplet size distribution shifted towards larger diameters. The simulation results agreed well with the experimental data in terms of the droplet size distribution and average number concentration at the outlet of the LFDC because of the detailed droplet dynamic, transport and deposition mechanisms treated in this model. Based on this model, the number of molecules in the critical cluster was calculated using the first nucleation theorem and found to be larger about 50% than that calculated using the Gibbs-Thompson equation.

Application of the Nucleation Theorem to Crystallization of Liquids: Some General Theoretical Results

Different aspects in applying the nucleation theorem to the description of crystallization of liquids are analyzed. It is shown that, by employing the classical Gibbs’ approach in the thermodynamic description of heterogeneous systems, a general form of the nucleation theorem can be formulated that is valid not only for one-component but generally for multi-component systems. In this analysis, one basic assumption of classical nucleation theory is utilized. In addition, commonly employed in application to crystallization, it is supposed that the bulk properties of the critical clusters are widely identical to the properties of the newly evolving crystal phase. It is shown that the formulation of the nucleation theorem as proposed by Kashchiev [J. Chem. Phys. 76, 5098-5102 (1982)], also relying widely on the standard classical approach in the description of crystal nucleation, holds for multi-component systems as well. The general form of the nucleation theorem derived by us is taken then as the starting point for the derivation of particular forms of this theorem for the cases that the deviation from equilibrium is caused by variations of either composition of the liquid phase, temperature, or pressure. In this procedure, expressions recently developed by us for the curvature dependence of the surface tension, respectively, its dependence on pressure and/or temperature are employed. The basic assumption of classical nucleation theory mentioned above is, however, in general, not true. The bulk and surface properties of the critical crystal clusters may differ considerably from the properties of the evolving macroscopic phases. Such effects can be incorporated into the theoretical description by the application of the generalized Gibbs approach for the specification of the dependence of the properties of critical crystal clusters on the degree of metastability of the liquid phase. Applying this method, it is demonstrated that a similar formulation of the nucleation theorem, as derived based on classical nucleation theory, holds true also in cases when a dependence of the state parameters of the critical clusters on the degree of deviation from equilibrium is appropriately accounted for.

Critical Size of Secondary Nuclei Determined via Nucleation Theorem Reveals Selective Nucleation in Three-Component Co-Crystals

The critical size of the secondary nuclei plays an important role in determining the crystal growth rate. In the past, the Nucleation Theorem has been applied to determine the number of molecules in the critical nuclei of a single-component crystal via variation of the crystal growth rate with dilution by the non-crystallizable component. In this work, we extend the method to the three-component co-crystal poly (ethylene oxide)/urea/thiourea inclusion compound. The theoretical crystal growth kinetics were deduced and the dependence of the radial growth rate of the inclusion compound spherulites on the mass fraction of urea in urea/thiourea was measured. The results reveal that the secondary nuclei of the poly (ethylene oxide)/urea/thiourea inclusion compound consist mainly of ethylene oxide repeating units and urea molecules. We propose that only urea molecules and ethylene oxide repeating units are selected to form the secondary nuclei while co-crystallization of the three components happens at the lateral spreading stage. As a result, the composition of the critical secondary nuclei is different from that of the bulk inclusion compound crystals. The work is expected to deepen our understanding of the nucleation of multi-component co-crystals.

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)].

Study of the Crystal Growth Mechanism and Critical Secondary Nucleus Size of Poly(ethylene oxide)/Urea Inclusion Compound

This study focuses on the crystal growth mechanism and the critical nucleus size of the inclusion compound formed between poly(ethylene oxide) and urea (PEO-U-IC) β spherulites crystallized from the melt. The surface nucleation mechanism during the crystal growth process is proposed based on the exponential dependence of the spherulite radial growth rate G on the negative reciprocal of product of the isothermal crystallization temperature multiplied by the degree of supercooling (−1/(TcΔT)). The radial growth rates G of PEO-U-IC β spherulites crystallized from the mixed melt with the gradual addition of the diluting agent N,N′-dimethyl urea (2MeU) were measured under a polarized optical microscope. On the basis of the first nucleation theorem proposed by Kashchiev, the critical secondary nucleus size n* during the surface nucleation is determined from the slope of linear fitting line of ln G versus ln x, where x is the mass fraction of urea in U/2MeU. The critical secondary nucleus consists of 3–9 entities with each entity of the inclusion compound crystal containing 1 urea and 1.5 ethylene oxide repeating units when the crystallization temperature ranges from 65 to 85 °C. The relationship that n* is proportional to 1/(ΔT)2 confirms that the critical secondary nuclei at the solid–melt interface are two-dimensional.

Crystallization of hard spheres revisited. II. Thermodynamic modeling, nucleation work, and the surface of tension

Combining three numerical methods (forward flux sampling, seeding of droplets, and finite-size droplets), we probe the crystallization of hard spheres over the full range from close to coexistence to the spinodal regime. We show that all three methods allow us to sample different regimes and agree perfectly in the ranges where they overlap. By combining the nucleation work calculated from forward flux sampling of small droplets and the nucleation theorem, we show how to compute the nucleation work spanning three orders of magnitude. Using a variation of the nucleation theorem, we show how to extract the pressure difference between the solid droplet and ambient liquid. Moreover, combining the nucleation work with the pressure difference allows us to calculate the interfacial tension of small droplets. Our results demonstrate that employing bulk quantities yields inaccurate results for the nucleation rate.

Effect of entropy on the nucleation of cavitation bubbles in water under tension.

Water can exist in a metastable liquid state under tension for long times before the system relaxes into the vapor via cavitation, i.e., bubble nucleation. Microscopic information on the cavitation process can be extracted from experimental data by the use of the nucleation theorem, which relates measured cavitation rates to the size of the critical bubble. To apply the nucleation theorem to experiments performed along an isochoric path, for instance, in cavitation experiments in mineral inclusions, knowledge of the bubble entropy is required. Using computer simulations, we compute the entropy of bubbles in water as a function of their volume over a wide range of tensions from free energy calculations. We find that the bubble entropy is an important contribution to the free energy that significantly lowers the barrier to bubble nucleation, thereby facilitating cavitation. Furthermore, the bubble entropy per surface area depends on the curvature of the liquid-vapor interface, decreasing approximately linearly with its mean curvature over the studied range of bubble volumes. At room temperature, the entropy of a flat liquid-vapor interface at ambient pressure is very similar to that of critical bubbles over a wide range of tensions, which justifies the use of the former as an approximation when interpreting data from experiments. Based on our simulation results, we obtain an estimate for the volume of the critical bubble from experimentally measured cavitation rates.

Statistical Thermodynamics of an “Open” Hard Sphere System on the Equilibrium Fluid Isotherm: Study of Properties of the Freezing Transition Without Direct Involvement of the Equilibrium Solid Phase

Using several theoretical tools $$\ldots $$ (i) the nucleation theorem, (ii) an equivalent cavity, (iii) the reversible work of adding a cavity to an open hard sphere system, and (iv) the theory of “stability”... the authors estimated the density at which the hard sphere freezing transition occurs. No direct involvement of the equilibrium solid phase is involved. The reduced density $$\uppi a^3\rho _f/6$$ (where a is the hard sphere diameter and $$\rho _f $$ is the actual density at which freezing occurs) is found to be 0.4937 while the value obtained by computer simulation is 0.494. The agreement is good, but the new method still contains some approximation. However, the approximation is based on the idea that at a density just below $$\rho _f $$ the fluid adopts a distorted structure resembling the solid, but different enough so that long-range order vanishes. Initial loss of stability may not be involved in every fluid–solid transition, but it may be an early step in the hard sphere and related systems.

Lifetime of metastable states in a Ginzburg-Landau system: Numerical simulations at large driving forces.

We developed a "brute-force" simulation method and conducted numerical "experiments" on homogeneous nucleation in an isotropic system at large driving forces (not small supersaturations) using the stochastic Ginzburg-Landau approach. Interactions in the system are described by the asymmetric (no external field), athermal (temperature-independent driving force), tangential (simple phase diagram) Hamiltonian, which has two independent "drivers" of the phase transition: supersaturation and thermal noise. We obtained the probability distribution function of the lifetime of the metastable state and analyzed its mean value as a function of the supersaturation, noise strength, and volume. We also proved the nucleation theorem in the mean-field approximation. The results allowed us to find the thermodynamic properties of the barrier state and conclude that at large driving forces the fluctuating volumes are not independent.

Communication: kinetics of scavenging of small, nucleating clusters: first nucleation theorem and sum rules.

Despite recent advances in monitoring nucleation from a vapor at close-to-molecular resolution, the identity of the critical cluster, forming the bottleneck for the nucleation process, remains elusive. During past twenty years, the first nucleation theorem has been often used to extract the size of the critical cluster from nucleation rate measurements. However, derivations of the first nucleation theorem invoke certain questionable assumptions that may fail, e.g., in the case of atmospheric new particle formation, including absence of sub-critical cluster losses and heterogeneous nucleation on pre-existing nanoparticles. Here, we extend the kinetic derivation of the first nucleation theorem to give a general framework to include such processes, yielding sum rules connecting the size dependent particle formation and loss rates to the corresponding loss-free nucleation rate and the apparent critical size from a naïve application of the first nucleation theorem that neglects them.

Direct simulations of homogeneous bubble nucleation: Agreement with classical nucleation theory and no local hot spots.

We present results from direct, large-scale molecular dynamics simulations of homogeneous bubble (liquid-to-vapor) nucleation. The simulations contain half a billion Lennard-Jones atoms and cover up to 56 million time steps. The unprecedented size of the simulated volumes allows us to resolve the nucleation and growth of many bubbles per run in simple direct micro-canonical simulations while the ambient pressure and temperature remain almost perfectly constant. We find bubble nucleation rates which are lower than in most of the previous, smaller simulations. It is widely believed that classical nucleation theory (CNT) generally underestimates bubble nucleation rates by very large factors. However, our measured rates are within two orders of magnitude of CNT predictions; only at very low temperatures does CNT underestimate the nucleation rate significantly. Introducing a small, positive Tolman length leads to very good agreement at all temperatures, as found in our recent vapor-to-liquid nucleation simulations. The critical bubbles sizes derived with the nucleation theorem agree well with the CNT predictions at all temperatures. Local hot spots reported in the literature are not seen: Regions where a bubble nucleation event will occur are not above the average temperature, and no correlation of temperature fluctuations with subsequent bubble formation is seen.

Critical cluster size cannot in practice be determined by slope analysis in atmospherically relevant applications

The first nucleation theorem is the most widely used method to assess atmospheric new-particle formation mechanisms from particle formation rate measurements. The theorem states that the slope (∂logJ)/(∂logC) of the nucleation rate J versus the concentration C of a nucleating compound gives the number of molecules of that species in the critical cluster. In principle, the derivation of the theorem is solid, but it contains very restrictive assumptions, the validity of which is questionable in realistic situations. It applies only for systems where clusters grow by addition of single molecules, and there are no external losses. In addition, application of the theorem to experimental data requires that the nucleation rate can be determined from particle concentration observations. This work presents simulation results on particle formation rates in atmospherically relevant conditions. We show that the slope of the nucleation rate in realistic conditions differs from that in an ideal situation. The slope analysis can easily lead to erroneous conclusions on the critical cluster size, and should therefore not be used to interpret experimental data.

Free energy of cluster formation and a new scaling relation for the nucleation rate.

Recent very large molecular dynamics simulations of homogeneous nucleation with (1 - 8) × 10(9) Lennard-Jones atoms [J. Diemand, R. Angélil, K. K. Tanaka, and H. Tanaka, J. Chem. Phys. 139, 074309 (2013)] allow us to accurately determine the formation free energy of clusters over a wide range of cluster sizes. This is now possible because such large simulations allow for very precise measurements of the cluster size distribution in the steady state nucleation regime. The peaks of the free energy curves give critical cluster sizes, which agree well with independent estimates based on the nucleation theorem. Using these results, we derive an analytical formula and a new scaling relation for nucleation rates: ln J'/η is scaled by ln S/η, where the supersaturation ratio is S, η is the dimensionless surface energy, and J(') is a dimensionless nucleation rate. This relation can be derived using the free energy of cluster formation at equilibrium which corresponds to the surface energy required to form the vapor-liquid interface. At low temperatures (below the triple point), we find that the surface energy divided by that of the classical nucleation theory does not depend on temperature, which leads to the scaling relation and implies a constant, positive Tolman length equal to half of the mean inter-particle separation in the liquid phase.

A study of nucleation in supersaturated ibuprofen vapor in a flow diffusion chamber

Isothermal nucleation of supersaturated ibuprofen racemate vapor has been experimentally studied in a flow diffusion chamber at 293.3 and 301.2 K. Nucleation rates have been measured in the range of 104−104 cm−3 s−1 as functions of supersaturation. According to the first nucleation theorem, the numbers of molecules in critical nuclei have been found and used to determine the nucleation rate and supersaturation values as depending on the sizes of critical nuclei. The comparison of the experimental data with theoretical predictions has shown that the nucleation rates measured as functions of the numbers of molecules in critical nuclei are higher than the rates predicted by the classical theory by six to seven decimal orders of magnitude but, within one order of magnitude, coincide with the rates predicted by a theory previously proposed in a work by one of the authors, in which nucleation clusters were considered to be microscopic objects.

Influence of aerosol lifetime on the interpretation of nucleation experiments with respect to the first nucleation theorem

Abstract. The SAWNUC (Sulphuric Acid Water NUCleation) microphysical aerosol nucleation model is used to study the effect of reactor walls on the interpretation of nucleation experiments with respect to nucleation theory. This work shows that loss processes, such as wall losses, influence the interpretation of nucleation experiments, especially at low growth rates and short lifetimes of freshly nucleated particles. In these cases the power dependency of the formation rates, determined at a certain particle size, with respect to H2SO4 does not correspond to the approximate number of H2SO4 molecules in the critical cluster as expected by the first nucleation theorem. Observed ∂log(J)/∂log([H2SO4]) therefore can vary widely for identical nucleation conditions but different sink terms.

Binary nucleation rates for ethanol/water mixtures in supersonic Laval nozzles: Analyses by the first and second nucleation theorems

We performed pressure trace measurements and small angle x-ray scattering measurements to determine the vapor-liquid nucleation rates of EtOH/H2O mixtures including pure EtOH and pure H2O in two supersonic Laval nozzles with different expansion rates. The nucleation rates varied from 0.9 × 10(17) to 16 × 10(17) cm(-3) s(-1) over the temperature range of 210 K to 230 K, EtOH activity range of 0 to 11.6, and H2O activity range of 0 to 124. The first and second nucleation theorems were applied to the nucleation rates to estimate the sizes, compositions, and excess energies of the critical clusters. The critical clusters contained from 4 to 15 molecules for pure H2O and EtOH/H2O clusters, and from 16 to 23 molecules for pure EtOH clusters. Comparing the excess energies of the pure H2O critical clusters with the results of a quantum-chemistry calculation suggested that the pre-factor of the theoretical nucleation rate is almost constant regardless of the monomer concentration. One possible explanation for this result is that cooling of the critical clusters limits the nucleation rate under the highly supersaturated conditions. The results of the analyses also yielded the relation between the surface energy and the composition of the critical clusters, where the latter are predicted to consist only of surface molecules. Applying this relationship to the EtOH/H2O bulk liquid mixtures, we estimated the EtOH mole fraction in the surface layer and found it is higher than that derived from the surface tension based on the Gibbs adsorption equation when the EtOH mole fraction in the liquid is higher than about 0.2 mol/mol. This discrepancy was attributed to the existence of the EtOH depletion layer just below the surface layer of the liquid.

Large scale molecular dynamics simulations of homogeneous nucleation

We present results from large-scale molecular dynamics (MD) simulations of homogeneous vapor-to-liquid nucleation. The simulations contain between 1 × 10(9) and 8 × 10(9) Lennard-Jones (LJ) atoms, covering up to 1.2 μs (56 × 10(6) time-steps). They cover a wide range of supersaturation ratios, S ≃ 1.55-10(4), and temperatures from kT = 0.3 to 1.0ε (where ε is the depth of the LJ potential, and k is the Boltzmann constant). We have resolved nucleation rates as low as 10(17) cm(-3) s(-1) (in the argon system), and critical cluster sizes as large as 100 atoms. Recent argon nucleation experiments probe nucleation rates in an overlapping range, making the first direct comparison between laboratory experiments and molecular dynamics simulations possible: We find very good agreement within the uncertainties, which are mainly due to the extrapolations of argon and LJ saturation curves to very low temperatures. The self-consistent, modified classical nucleation model of Girshick and Chiu [J. Chem. Phys. 93, 1273 (1990)] underestimates the nucleation rates by up to 9 orders of magnitudes at low temperatures, and at kT = 1.0ε it overestimates them by up to 10(5). The predictions from a semi-phenomenological model by Laaksonen et al. [Phys. Rev. E 49, 5517 (1994)] are much closer to our MD results, but still differ by factors of up to 10(4) in some cases. At low temperatures, the classical theory predicts critical clusters sizes, which match the simulation results (using the first nucleation theorem) quite well, while the semi-phenomenological model slightly underestimates them. At kT = 1.0ε, the critical sizes from both models are clearly too small. In our simulations the growth rates per encounter, which are often taken to be unity in nucleation models, lie in a range from 0.05 to 0.24. We devise a new, empirical nucleation model based on free energy functions derived from subcritical cluster abundances, and find that it performs well in estimating nucleation rates.

Heterogeneous nucleation in the low-barrier regime

In simulations of the two-dimensional Ising model, we examine heterogeneous nucleation induced by a small impurity consisting of a line of l fixed spins. As l increases, we identify a limit of stability beyond which the metastable phase is not defined. We evaluate the free energy barrier for nucleation of the stable phase and show that, contrary to expectation, the barrier does not vanish on approach to the limit of stability. We also demonstrate that our values for the height of the barrier yield predictions for the nucleation time (from transition state theory) and the size of the critical cluster (from the nucleation theorem) that are in excellent agreement with direct measurements, even near the limit of stability.