Gibbs classical nucleation

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). 

In classical nucleation theory the Gibbs energy of a nucleus is considered as the sum of contributions from the bulk and the surface. Let us consider nucleation of a spherical crystal from its liquid below its melting temperature at 1 bar. The difference in Gibbs energy between a nucleus with radius r and its liquid is... 

The Gibbs free energy of formation for small values of r using classical nucleation theory (- - -)> classical nucleation theory with Tolman s representation of the... 

FIGURE 6JS Classical nucleation theory dependence of nuclei size on Gibbs free energy at a function of saturation ratio, S. AS > 0 4> nonspontaneous formation, AG = 0 transient equilibrium, AG < 0 spontaneous formation of a solids phase. Redrawn with permission from Dirksen and Ring [4a]. Reprinted from [4a], oop3nn t 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK. 

Sbhnel and Mullin, Garside and recently Barlow and Haymet have discussed the molecular Interpretation of induction times from the standpoint of classical nucleation theory. Crystal nuclei with a critical size must be formed before the new solid phase is visible. According to the model there exists a free energy barrier, AG to the formation of the crystal nuclei. AG is proportional to (InS), where S is the supersaturation ratio. The Gibb s free energy, AG of the supersaturated solution is equal to -RTlnS (R=gas constant T=temperature). The induction time is a function of AG and thus AG according to the following equation... 

According to the classical nucleation theory, the Gibbs energy for the formation of a critical 2D nucleus of a 2D Meads phase on an atomically flat terrace (T), AGcrit,T, is given by [3.249] (cf. eq. (4.25a)) ... 

The rate for homogeneous nucleation J) of spherical assemblies can be expressed by the classical nucleation equation (Gibbs, 1948 Volmer, 1939 Becker and Doring, 1935 Turnbull and Fisher, 1949) ... 

It should be noted, however, that most conditions of deposition from the vapor phase have been shown to be such that classical nucleation theory is not well-suited to describe the nucleation kinetics of diamond, since the critical nucleus size is on the order of a few atoms.P The small size of the critical nucleus makes it quite inappropriate to use the classical thermodynamic variables to describe the nucleation processes. Under such conditions, the Gibbs free-energy of the formation of a critical nucleus carmot be expressed... 

The above kinetic equations, developed based on the thermodynamic approach of Gibbs (1928), Volmer and Weber (1926), and Becker and Doring (1935), belong to the so-called classical nucleation theories. They have been criticized for the use of surface energy (interfacial tension), cr, which is probably of little physical significance when applied to small molecular assemblies of the size of critical nucleus. 

Homogeneous nucleation - concept of the critical cluster A metal surface that is homogeneously flat, like mercury or quasiperfect silver single crystals [2], offers no specific adsorption sites. In analogy to 3D phenomena, the nucleation of a new 2D phase may be treated on the basis of the classical nucleation theory (CNT) as formulated by Gibbs [102] and further developed by Volmer [103,104], Parkas [105], Stransld and Kaishev [106-109], Becker and Doring[110], Zeldovich [111], and... 

In agreement with the classical nucleation theory (CNT), calculations show that the Gibbs free energy barrier of a critical nucleus developing on a solid surface is... 

The classical nucleation theory, based on Gibbs thermodynamics statements, uses the macroscopic properties characteristic of bulk phases, such as free energies and surface tensions, for the description of small clusters. Contradictory results arose in early studies of electrochemical nucleation [9], where the size of a critical mercury nucleus on a platinum substrate amounted to only a few atoms, with properties that could substantially differ... 

Other nucleation thermodynamic parameters such as the interfacial energy between Ndl23 or Y123 and melt, metastable zone width, Gibbs free energy, critical nucleation radii etc. were estimated by Paul et al. (1999, 2000) using classical nucleation theory. 

A different approach to the problem of nucleation in the presence of flow was proposed in Ref. (Blaak, 2004) by extending the classical nucleation theory in an intuitive and interesting form. Starting from Eq. (49), a macroscopic nonequilibrium equation of state was proposed for the Gibbs free energy difference by expanding in powers of the shear rate about the equilibrium values of both, the chemical potential difference and the surface energy... 

The following treatment of nucleation in classical nucleation theory (CNT) provides a glimpse of the methods and reasoning of classical theories. A supersaturated solution is assumed to contain spherical nuclei of radius R and with ric particles, and the Gibbs free energy change on formation of these nuclei includes a term in the difference between the free energies of the solid and the liquid, Ap, and a term that represents... 

Coppola et al. [142] modified the classical nucleation theory of Hoffman and Lauritzen by adding the flow-induced increase in the Gibbs free energy of the amorphous phase to the Gibbs free energy difference between the quiescent amorphous phase and the crystalline phase. This can be expressed as follows. 

The fact that a liquid can be supercooled is best understood qualitatively in the framework of classical nucleation theory (CNT) (see e.g. Ref. [3]). According to CNT the free energy of a spherical nucleus that forms in a supersaturated solution contains two terms. The first term accounts for the fact that the solid phase is more stable than the liquid. This term is negative and proportional to the volume of the nucleus. The second term is a surface term. It describes the free energy needed to create a solid/liquid interface. This term is positive and proportional to the surface area of the nucleus. The (Gibbs) free energy of a spherical nucleus of radius R has the following form ... 

Before we present the simulation results, we briefly discuss the effect of a wall on crystal nucleation in the context of Classical Nucleation Theory (CNT). Turnbull [99] extended CNT to the case of heterogeneous nucleation of a crystal that forms on a plane substrate. The difference with the homogeneous case is that there are now two interfaces present. The Gibbs free energy of a crystal containing n particles is given by ... 

The classical nucleation theory, as pioneered by Volmer and Weber [25, 26] and by Becker and Doring [27], deals with the nucleation process based on Gibbs free energy. For a single-solute system, at the initial stage, solute atoms (A) collide and form small clusters in the solution atom-by-atom ... 

Jhe general expression for the nucleation work, equation (1.32), says that AG(n) has a minimal value for clusters formed with a minimal excess energy ( ). In terms of the classical nucleation theory developed in the pioneering works of Gibbs [1.1], Volmer [1.11], Volmer and Weber [1.16], Kossel[1.17], Stranski [1.12, 1.13, 1.18, 1.19], Farkas [1.20], Stranski and... 

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