Chemical nucleation involves cluster growth by chemical reactions. In the case where clusters grow via a simple sequence of reversible chemical reactions, a summation expression for the steady-state nucleation rate can be derived. However, in many cases the chemical pathway to cluster growth is more complicated, and requires solving a set of species population balance equations that depend on the specific chemical system. Two examples are considered: soot nucleation in hydrocarbon combustion and nucleation of silicon particles in thermal decomposition of silane. In both cases, chemical kinetic mechanisms have been developed that allow for numerical simulations of particle formation. Soot nucleation is believed to proceed through the formation of polycyclic aromatic hydrocarbons. Models have been developed for the formation of the first aromatic ring and for subsequent growth, either through reaction with small molecules or by coagulation. Silicon nucleation from silane involves a large set of silicon hydride species, which can be grouped into classes according to their structure and reactivity, facilitating estimates of their free energies and reaction rate constants.

In single-component homogeneous nucleation, the summation expression for the steady-state nucleation rate requires values of the forward rate constants and Gibbs free energies of cluster formation. If atomistic data are available for these quantities, then these could be used instead of CNT. In an atomistic approach, clusters are treated as distinct molecular species, rather than as a small piece of the bulk condensed phase. Examples are presented of atomistic data generated by means of computational chemistry for water clusters up to size 10, and for aluminum clusters up to size 60. In both cases, the free energy of cluster formation is found to be a multimodal function of cluster size, both quantitatively and qualitatively different than in CNT. Condensation rate constants can be affected by the need for a third body as a collision partner, and by attractive intermolecular forces in collisions between clusters and monomers. An approach is suggested for constructing a “master table” of free energies of cluster formation, based on a hybrid of atomistic data, experimental values inferred by means of the nucleation theorem, and extrapolations to larger cluster sizes based on CNT.

Particle nucleation in plasmas occurs under a wide range of conditions. In some cases, such as thermal plasma synthesis of metal nanoparticles, nucleation may follow the conventional scenarios of single-component homogeneous or ion-induced nucleation. In other cases, such as dust formation in nonthermal plasmas of the type used in semiconductor processing, the paths to nucleation are specific to the chemistry of the gases introduced into the processing chamber. In such cases, nucleation typically involves a mix of phenomena that combine chemical nucleation with plasma physics, with the chemistry being driven by electron impact, and the charging of small clusters by free electrons and ions playing an important role in cluster growth. The charging and transport of clusters and particles affect the electric field profile, causing the plasma and the aerosol phase to be strongly coupled. An example is considered of silicon particle nucleation in silane-containing plasmas, the most studied system because of its importance in semiconductor processing. Cluster growth in this system is dominated by reactions between anion clusters and neutral molecules.

In most experimental nucleation studies, it is assumed that steady-state nucleation exists, meaning that the time required to reach steady state is much shorter than the timescales for changes in temperature and saturation ratio. To examine this assumption, we solve the time-dependent equations for cluster number densities and nucleation currents, for clusters of each size. Assuming CNT values for condensation and evaporation rate constants, one finds that the time lag for achieving steady-state nucleation is typically on the order of microseconds. This is usually short enough to justify the steady-state assumption, a possible exception being systems such as rapid nozzle expansions. If instead atomistic values are used for rate constants, one finds that it may take much longer to achieve steady state. This is at least partially due to the multimodal nature of atomistic free energy profiles, with local minima corresponding to “magic numbers,” clusters that are more stable than clusters of adjacent sizes. The existence of magic numbers can significantly slow the approach to steady-state nucleation and may therefore invalidate the steady-state assumption in many cases.

The late nineteenth-century Wilson cloud chamber experiments found that the presence of ions caused water vapor to nucleate at lower saturation ratios than in air free of ions. The classical theory of ion-induced nucleation is based on the Thomson model of an ionic droplet, in which an ionic core is surrounded by liquid of the condensing substance. A potential difference exists between the ionic core and the droplet surface, which introduces an electric work term into the Gibbs free energy of cluster formation. This term leads to the existence of stable prenuclei that are smaller than the critical size clusters and more abundant, at steady state, than the bare ions. With assumptions otherwise the same as in CNT for neutral self-nucleation, an expression can be derived for the steady-state rate of ion-induced nucleation. Deficiencies of this theory, in addition to those of CNT for neutral self-nucleation, include that it neglects the effect of the ion on condensation rate constants. Moreover, the theory predicts that the sign of the ion makes no difference in the nucleation rate, in contradiction to the results of most experimental studies for various substances.

The classical theory of multicomponent nucleation, including binary nucleation, ternary nucleation, etc., makes similar key assumptions as in CNT for single-component nucleation. Clusters are modeled as spherical liquid droplets consisting of an ideal multicomponent solution. The surface tension is assumed to equal that of a flat surface of liquid having the same composition, in equilibrium with its multicomponent vapor. The Gibbs free energy of cluster formation is a function of the cluster’s size and composition and of the gas-phase partial pressures of each component. The critical size and composition are found at a saddle point on the multidimensional free energy surface, where the free energy of cluster formation is a maximum in one direction and a minimum in all orthogonal directions. Several improvements have been proposed within the framework of classical theory. These include models of the cluster growth trajectory near the critical point; studies that account for composition-dependent surface tension; and models that consider the existence of surface-active layers that cause the chemical composition near the cluster surface to be different than in the core.

Clusters can form and grow from a supersaturated vapor by successive reactions in which molecules (or “monomers”) of the vapor collide with the cluster and stick. In general, these reactions are reversible. The net forward rate of each of these reactions is termed the “nucleation current” of clusters of the size formed by the reaction. If a steady-state cluster size distribution exists, then the nucleation currents for clusters of all sizes are identical and can be equated to the steady-state (or “stationary”) nucleation rate. In that case, one can derive a closed-form expression for the nucleation rate in terms of a summation over clusters of all sizes up to some arbitrarily large size. The key terms in this summation are the forward rate constants and the Gibbs free energies of cluster formation from the monomer vapor. Evaluating the summation requires size-dependent values of these terms. For saturation ratios that lie within the condensation–evaporation regime, the free energy of cluster formation has a maximum at the critical cluster size. The nucleation theorem relates this size to the dependence of the nucleation rate on saturation ratio.

Gas-phase nucleation of condensed-phase particles is important in many contexts, including interstellar dust formation, air pollution, global climate change, combustion and fires, semiconductor processing, and synthesis of nanoparticles for practical applications. Nucleation occurs via the growth of atomic or molecular clusters to “critical size” – the size where further growth is irreversible. These critical-size clusters are the nuclei for particle formation, and the growth of clusters to the size of nuclei is the concern of nucleation theory. Various scenarios occur, including single-component homogeneous nucleation from a supersaturated vapor, multicomponent nucleation, ion-induced nucleation, chemical nucleation, and nucleation in plasmas. Classical nucleation theory, which treats small clusters as having the same properties as the bulk condensed phase, is still widely used to estimate nucleation rates for many kinds of systems. However, it is anticipated that atomistic approaches based on computational chemistry will increasingly be used to facilitate more accurate predictions of gas-phase nucleation rates for substances and chemical systems of interest.

Classical nucleation theory (CNT) models clusters of all sizes as structureless, spherical liquid droplets, having the same surface tension as a flat surface of the bulk liquid in equilibrium with its vapor at the same temperature – the “capillarity approximation.” The cluster free energy is divided into volume and surface contributions, and the rate of monomer addition to a cluster per unit area is equated to the flux of molecules to a plane in an ideal gas. Under these assumptions, together with several mathematical approximations, the summation expression for the steady-state nucleation rate is converted to a closed-form analytical expression for the nucleation rate as a function of temperature, saturation ratio, and substance properties. Comparing the nucleation rate predicted by CNT to experimental results for many substances, one finds considerable disagreement in terms of the magnitude of the nucleation rate as well as the qualitative dependence of nucleation rate on both temperature and saturation ratio. Analyzing the possible sources of this discrepancy, by far the major source of error is the liquid droplet model for the Gibbs free energy of cluster formation.