Critical Cluster Size Research Articles

The laminar flow tube reactor as a quantitative tool for nucleation studies: Experimental results and theoretical analysis of homogeneous nucleation of dibutylphthalate

A laminar flow tube reactor was designed and constructed to provide an accurate, quantitative measurement of a nucleation rate as a function of supersaturation and temperature. Measurements of nucleation of a supersaturated vapor of dibutylphthalate have been made for the temperature range from −30.3 to +19.1 °C. A thorough analysis of the possible sources of experimental uncertainties (such as defining the correct value of the initial vapor concentration, temperature boundary conditions on the reactor walls, accuracy of the calculations of the thermodynamic parameters of the nucleation zone, and particle concentration measurement) is given. Both isothermal and the isobaric nucleation rates were measured. The experimental data obtained were compared with the measurements of other experimental groups and with theoretical predictions made on the basis of the self-consistency correction nucleation theory. Theoretical analysis, based on the first and the second nucleation theorems, is also presented. The critical cluster size and the excess of internal energy of the critical cluster are obtained.

Structures and Stabilities of CaO and MgO Clusters and Cluster Ions: An Alternative Interpretation of the Experimental Mass Spectra

The structures and relative stabilities of doubly charged nonstoichiometric (CaO)nCa2+ (n = 1−29) cluster ions and of neutral stoichiometric (MgO)n and (CaO)n (n = 3, 6, 9, 12, 15, 18) clusters are studied through ab initio perturbed ion plus polarization calculations. The large coordination-dependent polarizabilities of oxide anions favor the formation of surface sites, making the critical cluster size where anions with bulk coordination first appear larger than that found in the related case of alkali metal halides. Thus, we show that there are substantial structural differences between alkali metal halide and alkaline earth oxide cluster ions, contrary to what is suggested by the similarities in the experimental mass spectra. An alternative interpretation of the magic numbers for the case of oxides is proposed, which involves an explicit consideration of isomer structures different from the ground states. A comparison with the previously studied (MgO)nMg2+ cluster ions shows that the emergence of bulklike structural properties with size is slower for calcium oxide. Nevertheless, the structures of the doubly charged clusters are rather similar for the two materials. By contrast, the study of the neutrals reveals interesting structural differences between MgO and CaO, similar to those found in the case of alkali metal halides.

Confirmation of classical nucleation theory by Monte Carlo simulations in the 3-dimensional Ising model at low temperature

We examine and visualize cluster formation, nucleation, and growth by Monte Carlo simulations in the three-dimensional Ising model at T/Tc=1/4, i.e., far from the critical temperature. Homogeneous nucleation rates as a function of supersaturation are obtained. From the slope of nucleation rate curves, model-free values for the critical cluster size are calculated, which agree with predictions of the Gibbs–Thomson relation for cubic nuclei. In fact, examining the shape of the critical clusters more closely, we find them to be cubical rather than spherical. Accordingly, the work of nucleus formation is accurately given by the classical nucleation theory, at least for the 3D-Ising model at the relatively low temperatures examined here. Furthermore, from the simulations subcritical equilibrium cluster number distributions are determined, which also agree with predictions for the exponential part of the classical nucleation theory.

Diffusion and cluster formation in supersaturated solutions of ammonium sulfate at 298 K

Diffusion coefficients of supersaturated aqueous solutions of ammonium sulfate have been measured at 298°C using Gouy interferometry and were observed to rapidly decrease with increasing concentration in the supersaturated region. Water activities in the supersaturated region were measured by electrodynamic levitation of single solution droplets. The number average cluster size, critical cluster size, degree of association and spinodal concentration were calculated employing these data. These results show that the true metastable zone width is much wider than previously thought with a calculated valve of the spinodal concentration at 69.63 molal.

Critical cluster size of metallic Cr and Mo nanoclusters

It is an intrinsic question whether or not the small metallic nanoclusters might have a structure different from the bulk structure. We observed a face-centered-cubic (fcc) structure for small Cr and Mo nanoclusters rather than a bulk body-centered-cubic (bcc) structure. We determined the critical cluster size ${n}_{\mathrm{crit}}$ to be $490\ifmmode\pm\else\textpm\fi{}100$ for metallic Cr nanoclusters and the range of the ${n}_{\mathrm{crit}}$ to be 1460--3900 for metallic Mo nanoclusters at which a structural transition from the fcc structure into the bulk bcc structure occurred.

Real-time study of nucleation, growth, and ripening during Fe/Fe(100) homoepitaxy using ion scattering

Real-time studies of submonolayer epitaxy via scattering of fast ions are applicable over a wide range of growth temperatures and deposition rates. Computer simulations of ion trajectories and nucleation theory yield quantitative information on atomistic growth processes. For homoepitaxy of Fe on Fe(100), we deduce island densities, monomer diffusion barrier, cluster binding energies, and post-deposition island ripening. Detailed information on transitions in critical cluster size is obtained.

In situ observation of nucleation and subsequent growth of clusters in silane radio frequency discharges

Growth processes of clusters in low-pressure and low-power silane radio frequency discharges are studied by using the newly developed double-pulse-discharge method which realizes in situ measurement of their size and density in a size range of 0.5–4 nm. The clusters begin to be composed of two size groups at about 10 ms after the discharge initiation: clusters in the small size group have an almost constant average size of about 0.5 nm through the discharge period, while those in the large one grow at about 4 nm/s in a monodisperse way. Time evolution of the measured average sizes and densities in the groups is transformed into that of size distributions assuming that the density of SinHx clusters for the small group decreases exponentially with the increase in the number of Si atoms, n, of them, and the size distribution for the large group is the lognormal one. The results show that a critical cluster size for nucleation is SinHx (n∼4).

Epitaxial growth of 3C-SiC on Si(001) using hexamethyldisilane and comparison with growth on Si(111)

Growth of 3C-SiC on Si substrates has been carried out by atmospheric pressure chemical vapor deposition using hexamethyldisilane as the source and a H 2/Ar gas mixture as the carrier gas. No separate carbonization step was used. Single crystalline 3C-SiC on Si(001) was grown using a two-step growth technique. For Si(001) substrates, the nucleation stage was optimum in the temperature range 1250–1300°C for a growth time of approximately 2 min. A high concentration of H 2 (12%) in the carrier gas favored epitaxial growth on these substrates. On the other hand, the optimum temperature range for the nucleation stage on Si(111) was 1150–1200°C and the optimum time for the nucleation stage was longer (∼5 min). Additionally, it was possible to achieve single crystallinity on Si(111) substrates with a lower H 2 content (8%) in the carrier gas. The second stage of growth (at higher temperature of ∼1380°C) was identical for both substrate orientations. The difference in the nucleation stage can be attributed to the lower surface free energy of the Si(111) substrate resulting in a smaller critical cluster size. Additionally, the presence of microtwins in the nucleated clusters enhances growth of the SiC(111) oriented nuclei on Si(001) substrates.

Migration of a Proton as a Function of Solvation within {ROH}n{H2O}H+ Cluster Ions: Experiment and Theory

Metastable and collision-induced decompositions of mass-selected {ROH}n{H2O}H+ cluster ions (where R ≡ CH3−, CH3CH2−, CH3CH2CH2−, and (CH3)2CH−) were observed to exhibit distinct size-dependent behavior. We observe that loss of a water molecule is dominant for n ≤ 8, whereas loss of multiple ROH molecules is the favored decomposition channel for n ≥ 9, resulting in the eventual formation of a stable {ROH}9{H2O}H+ cluster ion. We believe this is evidence for two distinct cluster geometries which explicitly depend on the number of ROH molecules present. That is, below a certain critical size the proton resides on the molecule with the highest proton affinity, the ROH. However, above that critical cluster size the proton will now preferentially reside on the water molecule, if there are sufficient alcohols to completely and symmetrically solvate the central H3O+. The structural implications of these results will be discussed in light of new theoretical calculations which have been performed on this system.

Fractal growth of LiF on Ag(111) studied by low-temperature STM

We studied the fractal growth of LiF on Ag(111) at 77 K using low-temperature scanning tunneling microscopy. The observed island density as a function of the flux and the island density distribution reveal consistency with nucleation theory for the critical cluster size corresponding to i=1. The migration barrier for LiF monomers was estimated to be of the order of several tens of meV. The ramified (100)-oriented LiF aggregates exhibit a fractal dimension of d=1.75±0.01. No shape transition to dendritic growth was observed for fluxes ranging from F=2.4×10 −2 to F=2.0×10 −5 ML s −1 which is attributed to the different symmetries of adsorbate and substrate.

Dynamic Scaling of Surface Growth in the Influence of Small Clusters Mobility

The deposition, diffusion and coalescence processes appeared in the surface growth of compact or liquid-like clusters system are studied. By introducing a critical cluster size ic, such that clusters of size s ⩽ ic are mobile while clusters of the size s > ic are immobile, the effects of mobility of small clusters on cluster size distribution are simulated. As ic increases, the exponent γ which relates the maximum cluster density to the normalized deposition rate is increased. Numerical simulation results indicate that the scaled cluster-size distributions are independent of the deposition rate, but closely associated with the small clusters mobility. As ic increases, it is found that the distributions become narrow.

Critical cluster size and droplet nucleation rate from growth and decay simulations of Lennard-Jones clusters

We study a single cluster of Lennard-Jones atoms using a novel and physically transparent Monte Carlo simulation technique. We compute the canonical ensemble averages of the grand canonical growth and decay probabilities of the cluster, and identify the critical cluster, the size for which the growth and decay probabilities are equal. The size and internal energy of the critical cluster, for different values of the temperature and chemical potential, are used together with the nucleation theorems to predict the behavior of the nucleation rate as a function of these parameters. Our results agree with those found in the literature, and roughly correspond to the predictions of classical theory. In contrast to most other simulation studies, we are able to concentrate on the properties of the clusters which are most important to the process of nucleation, namely those around the critical size. This makes our simulations computationally more efficient.

Transition from superparamagnetism to ferromagnetic single-domain in a Heisenberg model for nano-cluster magnetic system

The transition behaviour of a magnetic system consisting of numerous nano-sized clusters with a zero external field was studied. We investigated the transition from superparamagnetism (SP) to ferromagnetic single-domain (SD) state due to the increase of cluster size by Monte Carlo simulation of Heisenberg and Ising models which are the two extreme cases in anisotropy of spin exchange. The counter-effect of uniaxial anisotropy energy against thermal activation plays an important role in determining the degree of polarization of magnetic moments along easy axis as well as the total anisotropic energy of the system. Based on the sigmoidal anisotropic energy dependence on cluster size, we studied the relationship of the dynamic growth exponent P and critical cluster size N C of the transition with reduced temperature t and uniaxial anisotropic constant A. We found that the transitional growth exponent P, exhibiting a maximum at certain temperature, is not a universal constant as some previous investigators suggested.

Femtosecond dynamics of final-state effects in the valence band photoemission of silver clusters on a graphite substrate

We studied the electronic structure of quasi size-selected silver clusters grown in nanopits of a graphite surface using ultraviolet photoelectron spectroscopy at T = 40 K. For clusters with more than 9 × 102 atoms the initial state of the d-electrons has already reached the bulk limit, but there is a dynamic final-state effect which we can explain with a simple model taking into account the finite lifetime of the photohole (corresponding to a charged cluster) and the cluster-substrate interaction on a femtosecond timescale. Below a critical cluster size of about 5 × 102 atoms we have observed additional initial-state effects.

Scaling properties of InAs/GaAs self-assembled quantum dots

We studied the scaling law for volume distribution of the self-assembled quantum dots grown by molecular-beam epitaxy of the Stranski-Krastanow mode. The scaling law was found to hold regardless of the annealing time and growth temperature. Also, we found the scaling law for the pair distribution of self-assembled quantum dots. Both the volume distribution and pair distribution agreed with the scaling functions for two-dimensional submonolayer homoepitaxy model with critical cluster size $i=1,$ which excludes adatom detachment from clusters.

THE INFLUENCE OF A CRITICAL SIZE ON THE SCALING BEHAVIOR OF CLUSTERS COALESCENCE

By introducing a critical cluster size N c , the irreversible and reversible cluster–cluster aggregation are studied with Monte Carlo simulation method. In a long time limit the average size of cluster <S>∞ reaches its stationary value which depends on the critical size N c and the breakup constant k. Our results indicate that in the presence of critical size the critical exponent y, which is defined as <S(k,∞)>~k-y, increases as the critical size increases and is lower than the value of (α+ξ+2)-1, where the exponents α and ξ associate with the detachment and attachment of clusters.

Femtosecond photoionization of (H2O)n and (D2O)n clusters

Cluster ion distributions of water in a molecular beam are investigated by femtosecond ionization at 780 nm and reflectron time-of-flight mass spectrometry. The electric field strength generated by the ultrashort laser pulses is sufficient to efficiently ionize most of the molecules that are present in the molecular beam. In this work ion signals of large water clusters containing up to 60 monomers are reported. Upon ionization rapid proton transfer is observed, leading to the formation of protonated water cluster ions. Unprotonated clusters (H2O)n+(n>2) are not observed in the mass spectra. The configurational energy imparted to the protonated clusters induces unimolecular dissociation on the μs time scale. These metastable reactions are characterized by modeling the ion trajectories in the mass spectrometer. The numerical procedure in conjunction with the integrated parent and daughter intensities results in unimolecular dissociation rates as a function of cluster size. Additional information about proton transfer reactions is obtained by the investigation of deuterium substitutions. Even though these substitutions correspond to large relative changes in the mass of the atom as well as in the zero point energy, unprotonated (D2O)n+ clusters of significant abundance are not produced in supersonic expansions of deuterated water. An additional result of this work is the observation of doubly charged ions above a critical cluster size (n=37).

Morphology development and crystal growth in nanocrystalline aggregates under hydrothermal conditions: insights from titania

High-resolution transmission electron microscopy (HRTEM) and X-ray diffraction (XRD) peak broadening (Scherer) analysis were used to study coarsening and morphology development of nanocrystals. Experimental work used synthetic, equidimensional anatase (TiO 2) particles about 5 nm in diameter. Under hydrothermal conditions (100–250°C, 15–40 bars), rapid growth occurs along [001], driven in part by the relatively high surface energy of (001) and in part by a kinetic effect involving a cyclic generation of highly reactive adsorption sites. Rapid growth along [001] depresses the <101> growth rate until (001) surfaces shrink to a critical cluster size. A second major coarsening mechanism significantly reduces the surface energy, especially under acidic conditions. This mechanism involves topotactic attachment of primary particles at high energy surfaces (most commonly {112}, less commonly (001)) and can result in elongate single crystals. Oriented attachment may be an important coarsening mechanism under a wide range of conditions encountered in the laboratory and in natural environments.

Critical oxide cluster size on Si(111)

The initial stage of oxide growth and subsequent oxide decomposition on Si(111)-7×7 at temperatures between 350 and 720°C are studied with the optical second harmonic generation for O2 pressures (Pox) between 5×10−9 and 4×10−6Torr. The obtained pressure dependencies of the initial oxide growth rate (Rgr) and the subsequent oxide decomposition rate are associated with the cluster-forming nature of the oxidation process. For the model of oxide cluster nucleation and growth, a scaling relationship is derived among the critical oxide cluster size, i, and the experimentally measurable values of Rgr and Pox. The critical oxide cluster size, i, thus obtained from the kinetic data increases with temperature. This correlates with an increase of desorption channels and their rates in that the competition between growth and decomposition requires more stable oxide clusters, i.e. clusters with a larger critical size, for oxide to grow at higher temperatures. The increase of i with decreasing Pox is related with a decrease of Rgr: a decreased Rgr requires critical clusters with a longer lifetime, i.e. clusters with a larger size.

Formation free energy of clusters in vapor-liquid nucleation: A Monte Carlo simulation study

The formation free energy of clusters in a supersaturated vapor is obtained by a constrained Monte Carlo technique. A key feature of this approach is to set an upper limit to the size of cluster. This maximum cluster size serves essentially as an extra thermodynamic variable that constrains the system. As a result, clusters larger than the critical cluster of nucleation in the supersaturated vapor can no longer grow beyond the limiting size. Like changing the overall density of the system, changing the maximum cluster size also results in a different supersaturation and thereby a different formation free energy. However, at the same supersaturation and temperature it is found that the formation free energy has a unique value, independent of the upper limit of cluster size. The predicted size of critical cluster of nucleation is found to be consistent with the nucleation theorem as well as previous results using different simulation approaches.