Homogeneous nucleation theory

F. F. Abraham, Homogeneous Nucleation Theory, Academic, New York, 1974. 

Nucleation in solids is very similar to nucleation in liquids. Because solids usually contain high-energy defects (like dislocations, grain boundaries and surfaces) new phases usually nucleate heterogeneously homogeneous nucleation, which occurs in defect-free regions, is rare. Figure 7.5 summarises the various ways in which nucleation can take place in a typical polycrystalline solid and Problems 7.2 and 7.3 illustrate how nucleation theory can be applied to a solid-state situation. 

Gibbs considered the change of free energy during homogeneous nucleation, which leads to the classical nucleation theory and to the Gibbs-Tliompson relationship (Mullin, 2001). 

This paper presents the physical mechanism and the structure of a comprehensive dynamic Emulsion Polymerization Model (EPM). EPM combines the theory of coagulative nucleation of homogeneously nucleated precursors with detailed species material and energy balances to calculate the time evolution of the concentration, size, and colloidal characteristics of latex particles, the monomer conversions, the copolymer composition, and molecular weight in an emulsion system. The capabilities of EPM are demonstrated by comparisons of its predictions with experimental data from the literature covering styrene and styrene/methyl methacrylate polymerizations. EPM can successfully simulate continuous and batch reactors over a wide range of initiator and added surfactant concentrations. 

Here our interest is in the application of homogeneous nucleation theory to produce the comprehensive plots of meta-stable crystallization. Fig. 1 illustrates the meta-stable efflorescence paths(solid lines) of (NH4)2S04 and (NH4)3H(S04)2 particles as a function of RH with the decreasing rate of ARH = 0.005 min with the deliquescence paths(O). Fig. 2 shows the expectation time of the aqueous particle composed of (NH4)2S04 and H2SO4... 

According to homogeneous nucleation theory, the critical Gibbs energy to form a nucleus is given by... 

From nucleation theory (see Section IX), one can estimate the expected rate of formation of critical-sized vapor embryos in a liquid as a function of temperature. This rate is a very strong function of temperature emd changes from a vanishingly low value a few degrees below the homogeneous nucleation temperature to a very large value at this temperature. 

When a phase transition occurs from a pure single state and in the absence of wettable surfaces the embryogenesis of the new phase is referred to as homogeneous nucleation. What is commonly referred to as classical nucleation theory is based on the following physical picture. Density fluctuations in the pre-transitional state result in local domains with characteristics of the new phases. If these fluctuations produce an embryo which exceeds a critical size then this embryo will not be dissipated but will grow to macroscopic size in an open system. The concept is applied to very diverse phenomena ... 

Having described the equilibrium structure and thermodynamics of the vapor condensate we then re-examine homogeneous nucleation theory. This combination of thermodynamics and rate kinetics, in which the free energy of formation is treated as an activation energy in a monomer addition reaction, contains the assumption that equilibrium thermodynamic functions can be applied to a continuum of non-equilibrium states. For the purpose of elucidating the effects of the removal of the usual approximations, we retain this assumption and calculate a radially dependent free energy of formation. Ve find, that by removing the conventional assumptions, the presumed thermodynamic barrier to nucleation is absent. 

Having developed a self-consistent treatment of the structure and interfacial energy of the Lennard-Jones droplet in a finite volume which avoids the usual assumptions by which homogeneous nucleation theory is implementedf we return to original question and examine how this new information alters the physical picture of nucleation. It is common to apply the analog of Equation 6 for an isothermal-isobaric ensemble,... 

A central assertion of homogeneous nucleation theory is that interfacial free energy costs induce a spherical symmetry in the phase embryo. However, these simulation studies indicate that inter molecular interactions may not permit the development of spherical symmetry when these interactions are strong and highly asymmetric. 

Bubble Nucleation in a Liquid Phase The above classical nucleation theory can be easily extended to melt nucleation in another melt. It can also be extended to melt nucleation in a crystal but with one exception. Crystal grains are usually small with surfaces or grain boundaries. Melt nucleation in crystals most likely starts on the surface or grain boundaries, which is similar to heterogeneous nucleation discussed below. Homogeneous nucleation of bubbles in a melt can be treated similarly using the above procedures. Because of special property of gases, the equations are different from those for the nucleation of a condensed phase, and are hence summarized below for convenience. 

The atmospheric situation is complicated by varying conditions of temperature, relative humidity, and concentrations of other gases such as NH3 which can enhance nucleation rates over those expected for a well-mixed air mass at a fixed temperature and RH (e.g., see Nilsson and Kulmala, 1998). However, there is a general consensus that the observed rates of nucleation of H2S04 often, indeed usually, exceed those expected from classical binary homogeneous nucleation theory. (Note that this is not always the case. For example, Pirjola et al. (1998) reported that the measured formation of nuclei in the Arctic boundary layer... 

The homogeneous nucleation theory may suggest that the dependence of particle number on the concentrations of emulsifier and initiator changes from monomer to monomer. In fact, z in Eq. (4) is 0.6 for a water-insoluble monomer such as styrene (as predicted by Smith-Ewert) but decreases with increasing solubility of monomer in water or increasing transfer of radicals out of particles (12) ... 

Homogeneous nucleation may be described by assuming that critical-size nuclei will be formed from ideal vapor (water or air) at a rate, I, given by classical nucleation theory [4]. The equation is... 

The Cu-Co system is a particularly simple precipitation system in which a Corich /3 phase precipitates in a Cu-rich terminal a phase. The f.c.c. lattices of both phases are well matched in three dimensions, so that the precipitate interfaces are coherent with respect to either lattice as a reference structure and the interfacial energy is sufficiently isotropic so that they are almost spherical, as in Fig. 19.2. Both the interfacial energy and strain energy are therefore relatively low and the nucleation of the f3 phase is therefore relatively easy and occurs homogeneously. This system has been used to test the applicability of the classical nucleation theory (Section 19.1.1) [11, 12]. In this work, the experimental conditions under which... 

In the past half-decade, extensive studies have focused on aerosol nucleation in aircraft exhaust plumes [79]. This interest has brought attention to the formation of volatile aerosols that might eventually evolve into cloud condensation nuclei [80], Measurements of ultrafine particles reveal remarkably high abundances in jet wakes at very early times (within 1 second of emission) (e.g., [81]). As in the background atmosphere, the classical homogeneous nucleation theory has been applied to explain the number and size distribution of these volatile microscopic particles [82,83], However, while achieving some initial success, the theory has not been able to explain more recent, detailed observations. 

Homogeneous nucleation (HON) is rarely encountered in the real world. However, despite its shortcomings, the classical nucleation theory (first originating from the work of Volmer and Weber, 1926) still forms the basis of most modern treatments of nucleation. Therefore, only a brief discussion of the fundamental concepts of homogeneous nucleation is included here for completeness. 

The kinetics of nucleation of one-component gas hydrates in aqueous solution have been analyzed by Kashchiev and Firoozabadi (2002b). Expressions were derived for the stationary rate of hydrate nucleation,./, for heterogeneous nucleation at the solution-gas interface or on solid substrates, and also for the special case of homogeneous nucleation. Kashchiev and Firoozabadi s work on the kinetics of hydrate nucleation provides a detailed examination of the mechanisms and kinetic expressions for hydrate nucleation, which are based on classical nucleation theory. Kashchiev and Firoozabadi s (2002b) work is only briefly summarized here, and for more details the reader is referred to the original references. 

The dependence of final particle number on the initiator or emulsifier concentrations according to the micellar and homogeneous nucleation theories is given by the following equation ... 

Such large undercoolings are normally not observed in metals. Undercoolings are usually so small that they are not noticed. The reason for the difference between the theory and practice is that the theory assumes nucleation occurs homogeneously (i.e., randomly throughout the liquid), whereas nuclei usually form on preexisting solid surfaces. 

The first attempt to formulate a homogeneous nucleation theory to predict the absolute number concentration of particles, N, was made by Fitch and Tsai in 1970 (21). This was supported by a large number of experiments on the polymerization of MMA. 

According to the aggregative and coagulative nucleation mechanisms which have been derived originally from the homogeneous nucleation theory of Fitch and Tsai [128], the most important point in the reaction is the instant at which colloidally stabilized particles form. After this point, coagulation between similar-sized particles no longer occurs, and the number of particles present in the reaction is constant. As shown in Fig. 6, the dispersion copolymerization with macromonomers is considered to proceed as follows. (1) Before polymerization, the monomer, macromonomer, and initiator dissolve completely into the... 

Formation of metal clusters by gas aggregation, in which metal atoms are evaporated or sputtered into a cooled inert gas flow at relatively high pressure, has been well established in last decade. By repeated collisions with the carrier gas, the supersaturated metal vapor nucleates and forms clusters. The mechanism of cluster formation can be explained with homogeneous and heterogeneous nucleation theories. The gas aggregation methods have been applied extensively to produce small clusters of metals such as zinc, copper, silver etc. [23-26]. In some cases this method was used in combination with a mass filter such as a quadruple or a time-of-flight spectrometer [27, 28], The metal vapor for cluster source can be produced by either thermal evaporation [23-28] or sputter discharge [22, 29]. 

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