Nucleation classical theory

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  

On average, the number of the clusters with radius r (N,) following equations [22-24] can be obtained by the 

In this relationship, the excess free energy for the formation of clusters AG first increases then decreases with the radius of the particles, r. The critical radius, r, is associated with a maximum excess free energy, AG. Accordingly, there exists a critical number of atoms, , in cluster A at the radius of r. When r r, the system can lower its free energy by dissolution of clusters. Thus, these clusters are not thermodynamically stable and dissolve quickly, whereas some new clusters form due to spontaneous collisions. These unstable particles (A A ) are known as clusters or embryos, and their numbers follow the Boltzmann distribution and decrease exponentially with increases of AG as described in Equation 10.4. When the radius of a cluster is larger than the critical value (r r ), it becomes stable and is referred to as a nucleus. Thus, the expressions of critical radius r and maximum excess free energy AG can be obtained mathematically when dAG,/dr is equal to zero [22, 23, 28]  

Similarly, the nitmber of dusters that reach the critical size, and nucleation rate, dN i, /dt can be given by the following two equations  

The AG, Npir and /dt values can be used to describe how easy or difficult it is for a solute to nucleate. Nuclei form easily when a system has small values of and since the clusters need to overcome only a small energy barrier and incorporate few atoms to become stable. In contrast, if r- is large and is very positive, the formation of stable nuclei is difficult. Only a small portion of clusters can then grow into stable nuclei, and a slow nucleation rate is expected. 

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The resistance to nucleation is associated with the surface energy of forming small clusters. Once beyond a critical size, the growth proceeds with the considerable driving force due to the supersaturation or subcooling. It is the definition of this critical nucleus size that has consumed much theoretical and experimental research. We present a brief description of the classic nucleation theory along with some examples of crystal nucleation and growth studies. 

In the classic nucleation theory, the free energy of forming a cluster of radius r containing n atoms or molecules is the sum of two terms ... 

The classic nucleation theory is an excellent qualitative foundation for the understanding of nucleation. It is not, however, appropriate to treat small clusters as bulk materials and to ignore the sometimes significant and diffuse interface region. This was pointed out some years ago by Cahn and Hilliard [16] and is reflected in their model for interfacial tension (see Section III-2B). 

Classic nucleation theory must be modified for nucleation near a critical point. Observed supercooling and superheating far exceeds that predicted by conventional theory and McGraw and Reiss [36] pointed out that if a usually neglected excluded volume term is retained the free energy of the critical nucleus increases considerably. As noted by Derjaguin [37], a similar problem occurs in the theory of cavitation. In binary systems the composition of the nuclei will differ from that of the bulk... 

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). 

The MD simulations provided the necessary thermodynamic information to obtain the equilibrium configurations of the films. Often the deposition process will produce films which are not in the equilibrium configuration, and then the problem is to determine the stablity of these films against changes in morphology. Here simulations can also be helpful, since data on the surface energies and chemical potentials of strained films can be used to calculate the probability of cluster nucleation, using classical nucleation theory. 

Transfomation from a meta-stable phase, such as supersaturated solution, to a thermodynamically more favorable phase requires first the crystal nucleation of a germ of the new phase. According to the classical nucleation theory, the volume nucleation rate J (cm" sec ), describing the number of nuclei(i.e., a critical germ) formed per volume per time, is given by ... 

In this work, we developed the safeguard active-set method by modifying the active-set method for thermodynamic equilibrium in order to include the classical nucleation theory. At tn, assume that the partition ( (r ), M(t ), N(t ) and the crystallization time tciyst(t ) forM(t ) are known. For a new feed vector and RH at Vu compute W(tn+i), M(t i), N(t + )) and tciyst(t +i) as follows ... 

Crystallization can be divided into three processes the primary nucleation process, the growth process, and the overgrowth process. The growth process is mainly controlled by the secondary nucleation mechanism. The steady (stationary) primary and secondary nucleation mechanisms of atomic or low molecular weight systems have been well studied since the 1930s by applying the classical nucleation theory (CNT) presented by Becker and Doring, Zeldovich, Frenkel and Turnbull and Fisher and so on [1-4]. 

The purpose of this section is to present direct evidence of nucleation during the induction period by means of synchrotron small angle X-ray scattering (SAXS). In the classical nucleation theory (CNT), the number density distribution function of nuclei of size N at time t, f(N, t), is expected to increase with an increase of t during the induction period and saturates to a steady f(N, t),fst(N) in the steady period. The change off(N, t) should correspond to that of the scattering intensity of SAXS. 

Classical nucleation theory (CNT) shows that I is a product of the probability of diffusion and that of formation of a critical nucleus [1,4],... 

Fig. 15 Plot of log I versus AT-2 of FCSCs for Mn = 30 x 103, 50 x 103, 71 x 103, 99 x 103, and 139 x 103. The solid lines represent the best fit of the plots, which corresponds to the classical nucleation theory. Iq is the intercept of the vertical axis at AT-2 = 0...

Within classical nucleation theory the free energy of the system as a function of the radius r of a spherical crystallite can be expressed as... 

Crystallization conditions can often be manipulated to favor the nucleation of alternate crystal forms. A metastable polymorph of metformin hydrochloride has been isolated using capillary crystallization techniques, and subsequently studied using thermal microscopy [24]. Calculations based on classical nucleation theory indicated that a metastable form could be obtained using high degrees of... 

The prediction made by the model calculations should be taken with some care for two reasons 1) H2O and H2SO4 are considered to be the condensing species, whereas other species may be active in experimental or domestic environments 2) the model uses classical nucleation theory, which is the only workable theory, but which is also to be criticized because it applies macroscopic entities to clusters that contain only a few molecules (3). 

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... 

Figure 3 shows a plot of the volume normalized nucleation time constant as a function of isothermal crystallization temperature for PEO droplets, taken from the work of Massa and Kalnoki-Veress [84]. As expected, droplets of different volumes have the same value of r V. The inset in Fig. 3 is a plot consistent with classical nucleation theory (see Eqs. 1, 4) only the last four data points correspond to the work of Massa and Kalnoki-Veress. The first... 

Fig. 3 Semi-logarithmic plot of the volume-normalised time constant, rV> as a function of temperature. The data shows a linear dependence when xV is plotted as a function of l/[Tc(Tm - Tc)2] as is expected from classical nucleation theory (see inset). (Reprinted with permission from [84]. Copyright 2004 by the American Physical Society)...

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 ... 

R. Although expressions for this parameter exist, they are derived by a hybrid of molecular mechanical and thermodynamic arguments which are not at present known to be consistent as droplet size decreases (8). An analysis of the size limitation of the validity of these arguments has, to our knowledge, never been attempted. Here we evaluate these expressions and others which are thought to be only asymptotically correct. Ve conclude, from the consistency of these apparently independent approaches, that the surface of tension, and, therefore, the surface tension, can be defined with sufficient certainty in the size regime of the critical embryo of classical nucleation theory. 

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

It is remarkable that the predictions of classical nucleation theory without any consideration of polymer connectivity are borne out in experiments. At higher supercooling, deviations are expected because of temperature dependence of the nucleation rate prefactor. 

Nucleation rate based on the classical nucleation theory The nucleation rate is the steady-state production of critical clusters, which equals the rate at which critical clusters are produced (actually the production rate of clusters with critical number of molecules plus 1). The growth rate of a cluster can be obtained from the transition state theory, in which the growth rate is proportional to the concentration of the activated complex that can attach to the cluster. This process requires activation energy. Using this approach, Becker and Coring (1935) obtained the following equation for the nucleation rate ... 

Failure of the Classical Nucleation Theory There are several suggested explanations for the failure of the classical nucleation theory to quantitatively predict the nucleation rate, including the following ... 

The classical nucleation theory itself is inadequate. Hence, effort has been made to develop nonclassical theories of nucleation, often by... 

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. 

Neilson G.F. and Weinberg M.C. (1979) A test of classical nucleation theory crystal nu-cleation of lithium disilicate glass. /. Non-Cryst. Solids 34, 137-147. 

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