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.

This chapter is intended to review concepts that the reader has some familiarity with and introduce high level descriptions of linear marine systems analysis. An initial discussion on the similarity between mechanical vibration equations of motion and marine dynamical systems is made. Mechanical vibrations are defined as vibrations in the absence of fluids. Examples of static and dynamic coupling between the various modes of motion or degrees of freedom are presented. The differences between frequency domain and time domain representations are given by introducing the concept of response amplitude operators (RAO’s). Complex arithmetic and linear, second order differential equations are briefly reviewed. Two examples of mechanical vibrations that are relevant to marine dynamics are developed and solved. The first example has to do with base excitation, similar to what a high speed planing craft may experience in long waves. The second example addresses one method for vibration isolation/suppression, that may, or may not, be useful in shock/impact mitigation schemes.

Previous chapters presented linear models for responses of marine systems in regular, harmonic waves and various probabilistic properties of random processes, e.g. ocean waves. This chapter combines the two topics - a system’s deterministic response in the frequency domain and the statistics of that system’s random response when excited by a random, irregular sea. Several models for ocean wave spectra are presented and input/output relations for linear systems subject to stochastic excitation developed. The ocean wave environment is described by a single-sided wave spectrum based on various empirical formulae: P-M spectrum (single parameter, wind speed or significant wave height for the North Atlantic); ISSC spectrum (two parameter, significant crossing period and wave height); JONSWAP spectrum (six parameter, fetch limited, typical of the North Sea); and the Ochi six parameter spectrum (combined wind and swell). Short crested seas are defined and their effects discussed. The output spectrum of a linear system subject to stochastic input is derived and its Gaussian PDF given. By invoking a narrow banded assumption, PDF’s of the output follow the Rayleigh most probable extremes.