Rate of homogeneous nucleation

The essential problem in nucleation studies is to estimate the rate of formation of nuclei having the critical size. When r = rc, the maximum free-energy barrier that must be overcome to form a liquid drop can be found by combining Equations (355) and (356) 

By combining with the maximum free energy barrier,(AGT)max, given in Equation (359), we obtain rate, /, as 

For ideal gases, the gas phase collision frequency, f which represents the number of molecules striking a unit area of surface per unit time can be expressed as 

Equation (361) is a sample equation for the rate of formation of droplets in their vapor, and similar equations can be derived for the rate of crystal formation by freezing their liquids, or precipitate formation from supersaturated solutions, although diffusion controlled cluster formation kinetics in liquids, lattice strain and anisotropic growth in crystals must be considered whenever necessary. 

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Nucleation can occur either homogeneously or heterogeneously. Homogeneous nucleation occurs when random molecular motion in the molten state results in the alignment of a sufficient number of chain segments to form a stable ordered phase, known as a nucleus. The minimum number of unit cells required to form a stable nucleus decreases as the temperature falls. Thus, the rate of nucleation increases as the temperature of the polymer decreases. The rate of homogeneous nucleation also increases as molecular orientation in the molten polymer increases. This is because the entropy difference between the molten and crystalline states diminishes as molecular alignment in the molten state increases. 

Homogeneous Nucleation (a) Using Eq. 8.6-2, calculate the rate of homogeneous nucleation of styrene as a function of temperature at atmospheric pressure and a temperature range from 145°C to 325°C. In calculating the pressure in the bubble, assume that it equals the vapor pressure (extrapolate it from lower temperature values). Use the Eotvos equation a = 2.1 (p/M)1 (Tc — T — 6), where the surface tension is in erg/cm3, temperature is in °C, and density in g/cm3, to evaluate the surface tension as a function of temperature. The critical temperature... 

The rate of homogeneous nucleation has been derived in a number of classical papers [6, 7]. The final result may be expressed in the form... 

The exponent values in expressions (5.2)—(5.5) depend on the particular parameters in the exponent index, but they are usually much larger than the unit at r < 10-50 nm (see Table 5.1). In a homogeneous mother sys tern (free of seeds for the solid phase condensation), the process rate depends on the rate of homogeneous nucleation of the new phase from nonequilibrium (oversaturated) systems. The high partial pressure of the equilibrium vapor or solute over small particles allows the first condensed particles nuclei of the new phase) to form at a considerable oversaturation of the vapor in the initially homogeneous system. 

The factors that regulate nucleation are best appreciated by considering the equation for the rate of homogeneous nucleation from solutions ... 

In summary then, the rate of homogeneous nucleation can be regulated via two different mechanisms (1) kinetic barrier, Ag this includes the dynamic slowing of molecules in their movement toward the interface and chemical association as a depletion mechanism of monomers (2) structural barrier, AG. These combine to provide the... 

The rate of homogeneous nucleation of a sphere takes the following form, derived from an Arrhenius-type expression ... 

The crystallization of dilferent polymorphs and crystal forms is best understood in terms of nucleation, which is often the rate-determining step (16). In the absence of foreign particles for inducing heterogeneous nucleation, spontaneous homogeneous nucleation can be assumed to occur as a first step in the crystallization process. The rate of homogeneous nucleation J can be expressed as... 

FIGURE 14.3 Rate of homogeneous nucleation /hom for the formation of ice in pure water at ambient pressure as a function of the undercooling AT. The broken line is a rough estimate. The dotted lines indicate the homogeneous nucleation point. ... 

Girshick, S. L., and Chiu, C.-P. (1990) Kinetic nucleation theory A new expression for the rate of homogeneous nucleation from an ideal supersaturated vapor, J. Chem. Phys. 93, 1273-1277. 

This thermodynamic approach illustrates in terms of a limiting supersaturation the energy requirements involved in nucleation. Since the majority of precipitation reactions in biological systems are likely to occur under kinetic conditions, the rate of nucleation will be of upmost importance. The rate of homogeneous nucleation Jn, can be considered as the rate at which nuclei surmount the maximum in the free energy curve of Fig. 3.1 and can be expressed as,... 

Fig. 3.2. The relationship between the rate of homogeneous nucleation, Jn, and the degree of supersaturation. There is a rapid rise in Jn at a critical value S ...

The rate of the primary homogeneous nucleation 5o,hom can be derived by multiplying an impact coefficient 5 with the total surface of all clusters present in a given volume V. The impact coefficient is the number of molecules which are hitting the surface based on a unit of time and surface. The total surface of all critical clusters is given by the number of clusters in a volume V and the surface of a cluster. The rate of homogenous nucleation is... 

Formation of such droplets must then be an activated process whose rate is proportional to exp [—AF /(A 7 ]. We can estimate this rate using equation (4.2.6) for the interfacial energy y, and the result is that the rate of homogeneous nucleation we should expect for polymer systems is vanishingly small. In practice nucleation is usually aided by the presence of other interfaces, for example impurity particles such as dust or the container walls may well be able to nucleate critical droplets with much lower activation energies (heterogeneous nucleation) or indeed with no activation energy at all. We will return to this subject in section 5.3 when we discuss the effects of surfaces on phase separation. 

The embryos that trigger vapor formation in a superheated liquid are microscopic bubbles small regions where the density is smaller than in the bulk. To calculate the rate of homogeneous nucleation in a superheated liquid according to the classical theory, one must therefore consider the energetics of bubble formation. The contents of vapor embryos can be treated as an ideal gas except near the critical point. Let P be the pressure inside the critical nucleus. Then, P being the bulk pressure in the superheated... 

It is convenient to provide an abbreviated account of the theory of the nucleation of drops from the vapor. The rate of homogeneous nucleation in a one-compound system is usually expressed as J, the number of drops nucleated per cubic centimeter per second, and has the form (12,13). 

Of course, the width of the metastable zone is not a precisely defined quantity. The concept of having no nucleation is reasonably clear (to become completely clear, a time for which the observer is willing to wait to see a nucleation event becomes important as noted by Fisher [9]). In contrast, what constitutes an observable or worse yet, a rapid rate of nucleation is open to discussion. Conventionally, an observable rate of homogeneous nucleation (taking place in the bulk of the mother phase, well away from the influence of any surface) is taken to be 1 nucleation event in 1 cubic centimeter per second... 

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