Nucleation critical size

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. 

When the nucleus is formed on a solid substrate by heterogeneous nucleation the above equations must be modified because of the nucleus-substrate interactions. These are reflected in the balance of the interfacial energies between the substrate and the environment, usually a vacuum, and the nucleus-vacuum and the nucleus-substrate interface energies. The effect of these terms is usually to reduce the critical size of the nucleus, to an extent dependent on... 

In aggregate nucleation the oligomers reversibly associate with each other until the aggregate reaches a critical size above which it is thermodynamically stable and continues to grow. 

The first is diffusion capture. This theory was originally proposed by Fitch and Tsai (13) for the aqueous polymerization of methyl methacrylate. According to this theory, any oligomer which diffuses to an existing particle before it has attained the critical size for nucleation is irreversibly captured. The rate of nucleation is equal to the rate of initiation minus the rate of capture. The rate of capture is proportional to both the surface area and the number of particles. 

Knowledge concerning the mechanism of hydrates formation is important in designing inhibitor systems for hydrates. The process of formation is believed to occur in two steps. The first step is a nucleation step and the second step is a growth reaction of the nucleus. Experimental results of nucleation are difficult to reproduce. Therefore, it is assumed that stochastic models would be useful in the mechanism of formation. Hydrate nucleation is an intrinsically stochastic process that involves the formation and growth of gas-water clusters to critical-sized, stable hydrate nuclei. The hydrate growth process involves the growth of stable hydrate nuclei as solid hydrates [129]. 

Frei, E.H., Shtrikman, S. and Treves, D. (1957) Critical size and nucleation field of ideal ferromagnetic particles. Physical Review, 106 (3), 446-454. 

Fig. 2 Illustration of the induction and the steady (stationary) periods during the nucleation process. Small clusters exist in the supercooled melt at t = 0. During the induction period (t < r,), isolated nuclei of size N, smaller than the critical nuclei (named nanonuclei or embryo), are formed. The nuclei grow larger and larger with increase of time and some of them attain a much larger size than the critical size, N ...

There is a point at which these aggregates reach a critical size of minimum stability r and the free energy of formation AG is a maximum. Further addition of material to the critical nucleus decreases the free energy and produces a stable growing nucleus. The nucleation rate is the product of the concentration of critical nuclei N given by... 

Initiation of growth may also proceed by formation of metastable structures when nucleation is inhibited. Multiply twinned structures have been observed for a number of metals. Their presence indicates an icosohedral or decahedral precursor cluster which has decomposed to a multiply twinned crystal at a critical size [117, 118], Another example of metastable intermediate structures was reported by Dietterle et al. 

To summarize, in the present scenario pure hadronic stars having a central pressure larger than the static transition pressure for the formation of the Q -phase are metastable to the decay (conversion) to a more compact stellar configuration in which deconfined quark matter is present (i. e., HyS or SS). These metastable HS have a mean-life time which is related to the nucleation time to form the first critical-size drop of deconfined matter in their interior (the actual mean-life time of the HS will depend on the mass accretion or on the spin-down rate which modifies the nucleation time via an explicit time dependence of the stellar central pressure). We define as critical mass Mcr of the metastable HS, the value of the gravitational mass for which the nucleation time is equal to one year Mcr = Miis t = lyr). Pure hadronic stars with Mh > Mcr are very unlikely to be observed. Mcr plays the role of an effective maximum mass for the hadronic branch of compact stars. While the Oppenheimer-Volkov maximum mass Mhs,max (Oppenheimer Volkov 1939) is determined by the overall stiffness of the EOS for hadronic matter, the value of Mcr will depend in addition on the bulk properties of the EOS for quark matter and on the properties at the interface between the confined and deconfined phases of matter (e.g., the surface tension a). 

The rate of nucleation is dependent on the degree of supersaturation as described in section 2.4.1, and because this will always be larger for Form 1 it may be incorrectly assumed that Form I will always precipitate first. The true situation is somewhat more complicated because the critical size, activation energy and nucleation rate also depend on the solid state that is being formed [6]. It is quite feasible and a regular occurrence, that a less stable polymorph will have a higher rate of nucleation than a more stable form, as illustrated in figure 6. 

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. 

The basic reason why superheated liquids can exist is that the nucleation step requires that a vapor embryo bubble of a minimum size must be achieved. Vapor embryos less than the critical size are unstable and tend... 

Increasing the temperature or lowering the pressure on a superheated liquid will increase the probability of nucleation. Also, the presence of solid surfaces enhances the probability because it is often easier to form a critical-sized embryo at a solid-liquid interface than in the bulk of the liquid. Nucleation in the bulk is referred to as homogeneous nucleation whereas if the critical-sized embryo forms at a solid-liquid (or liquid-liquid) interface, it is termed heterogeneous nucleation. Normal boiling processes wherein heat transfer occurs through the container wall to the liquid always occur by heterogeneous nucleation. 

This brief commentary on superheated liquids has indicated that they are readily formed if one prevents heterogeneous nucleation of vapor embryos. Also, there is a limit to the degree of superheat for any given liquid, pure or a mixture. This limit may be estimated either from thermodynamic stability theory or from an analysis of the dynamics of the formation of critical-sized vapor embryos. Both approaches yield very similar predictions although the physical interpretation of the results from both differ considerably. 

Availability change to form embryo Initial bubble diameter Frequency factor in nucleation Enthalpy of vaporization Rate of formation of critical-sized embryos per unit volume Jacob numter [Eq. (17)] Boltzmann s constant or thermal conductivity... 

At a microscopic scale, a single coalescence event proceeds through the nucleation of a thermally activated hole that reaches a critical size, above which it becomes unstable and grows [29]. We shall term E(r) the energy cost for reaching a hole of size r. A maximum of E occurs at a critical size r, E r ) = Ea being the activation energy of the hole nucleation process (Fig. 5.2). 

The regime governed by coalescence was examined in more detail. The process of film rupture is initiated by the spontaneous formation of a small hole. The nucleation frequency. A, of a hole that reaches a critical size, above which it becomes unstable and grows, determines the lifetime of the films with respect to coalescence. A mean field description [19] predicts that A varies with temperature T according to an Arrhenius law ... 

A subcritical aggregate having fewer subunit components than a nucleus. When this term is applied in the kinetics of precipitation, n refers to the number of subunits in a particle and n defines the number of subunits in a particle of critical size. This definition avoids confusion by distinguishing between subcritical (n < n subunits), critical (n = n subunits), and supercritical (n > n subunits) particle sizes. If a nucleus is defined as containing n n subunits, then an embryo contains n n subunits. Note that in this treatment, we are not using a phase-transition description to describe nucleation, and we are focusing on the smallest step in the process that leads to further aggregation. 

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 ... 

In the following derivation we will assume an almost complete wetting of the substrate by the material, in such a way that a continuous amorphous condensed film is formed at a thickness h smaller than the critical size of nucleation. In order to evaluate dG/dN of the process of incorporation of molecules from the amorphous condensed film to the spherulite, that is the ordered phase, we will hypothesize that the thickness of the amorphous film increases linearly with time, h(t) = Uhf, where the velocity is a constant, and that the spherulite has a cylindrical shape of radius R and height h, as illustrated in Fig. 5.10. 

Other computer simulations were made to test the classical theory. Recently, Ford and Vehkamaki, through a Monte-Carlo simulation, have identified fhe critical clusters (clusters of such a size that growth and decay probabilities become equal) [66]. The size and internal energy of the critical cluster, for different values of temperature and chemical potential, were used, together with nucleation theorems [66,67], to predict the behaviour of the nucleation rate as a function of these parameters. The plots for (i) the critical size as a function of chemical potential, (ii) the nucleation rate as a function of chemical potential and (iii) the nucleation rate as a function of temperature, suitably fit the predictions of classical theory [66]. 

---