The Nucleation Rate

The formation of nuclei of a new phase can be considered as a sequence of bimolecular reactions in which every cluster of one class transforms into the next higher or lower class by attachment or detachment of one atom  

At equilibrium, all rates of the attachment and detachment reactions are equal  

If an overpotential is applied, a net flux of clusters to higher classes is set. In the steady state the flux 

According to Becker and Doring, each one of these 5 equations except the first one are multiplied by the factor r[i=i rf(0/ a(0 where N varies from 1 to 5 - 1, increasing by unity for every subsequent equation. By adding these equations, all right-hand terms cancel except the first one cua(0)Zo, while the last one o)d S)Z S) can be neglected, so that  

This equation is quite general and does not depend on kind (two- or three-dimensional), state (liquid or crystalline), form and size of the clusters. If the clusters are crystalline, however, there are different ways of building them up to the size 5, so that for each one of these parallel going reaction sequences a different value of J may be expected. Every one of them contributes to the overall nucleation rate independently. 

Only the mechaiusm of simple condensation will be presented here. This will introduce a method that is also applicable to the potential mechanism of spot growth. 

looking at steps of the mechanism we see that in order to obtain the overall reaction and form moles of B, all the steps must be multiplied by the common multiplying coefficient  

The nucleation reaction [14.R1] has the same global equation as the transformation reaction. We define the rate of nucleation as the rate in relation to the newly-formed solid B. 

The rate coefficients of step i k and k are such that their ratio is the equilibrium constant of step i and we can write  

By applying relations [14.3] and [14.42], the superficial rate of condensation will 

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In principle, nucleation should occur for any supersaturation given enough time. The critical supersaturation ratio is often defined in terms of the condition needed to observe nucleation on a convenient time scale. As illustrated in Table IX-1, the nucleation rate changes so rapidly with degree of supersaturation that, fortunately, even a few powers of 10 error in the preexponential term make little difference. There has been some controversy surrounding the preexponential term and some detailed analyses are available [33-35]. 

The central quantity of interest in homogeneous nucleation is the nucleation rate J, which gives the number of droplets nucleated per unit volume per unit time for a given supersaturation. The free energy barrier is the dommant factor in detenuining J J depends on it exponentially. Thus, a small difference in the different model predictions for the barrier can lead to orders of magnitude differences in J. Similarly, experimental measurements of J are sensitive to the purity of the sample and to experimental conditions such as temperature. In modem field theories, J has a general fonu... 

Samples can be concentrated beyond tire glass transition. If tliis is done quickly enough to prevent crystallization, tliis ultimately leads to a random close-packed stmcture, witli a volume fraction (j) 0.64. Close-packed stmctures, such as fee, have a maximum packing density of (]) p = 0.74. The crystallization kinetics are strongly concentration dependent. The nucleation rate is fastest near tire melting concentration. On increasing concentration, tire nucleation process is arrested. This has been found to occur at tire glass transition [82]. 

Determination of Crystallization Kinetics. Under steady-state conditions, the total number production rate of crystals in a perfectly mixed crystallizer is identical to the nucleation rate, B. Accordingly,... 

Analysis of equation 48 shows that a single sample taken either from inside the crystallizer or from the product stream will allow evaluation of nucleation and growth rates at the system conditions. Figure 12 shows a plot of typical population density data obtained from a crystallizer meeting the stated assumptions. The slope of the plot of such data maybe used to obtain the growth rate, and the product of the intercept and growth rate gives the nucleation rate. 

Nucleation The mechanism of crystal nucleation from solution has been studied by many scientists, and recent work suggests that—in commercial crystallization equipment, at least—the nucleation rate is the sum of contributions by (1) homogeneous nucleation and (2) nucleation due to contaci between crystals and a) other crystals, h) the walls of the container, and (c) the pump impeller. If is the net number of new crystals formed in a unit volume of solution per unit of time. 

In order to treat crystallization systems both dynamically and continuously, a mathematical model has been developed which can correlate the nucleation rate to the level of supersaturation and/or the growth rate. Because the growth rate is more easily determined and because nucleation is sharply nonlinear in the regions normally encountered in industrial crystallization, it has been common to... 

A plot of In n versus L is a straight line whose intercept is In and whose slope is —l/Gt. (For plots on base-10 log paper, the appropriate slope correc tion must be made.) Thus, from a given product sample of known shiny density and retention time it is possible to obtain the nucleation rate and growth rate for the conditions tested if the sample satisfies the assumptions of the derivation and yields a straight hne. A number of derived relations which describe the nucleation rate, size distribution, and average properties are summarized in Table 18-5. 

Equation (18-31) contains no information about the ciystalhzer s influence on the nucleation rate. If the ciystaUizer is of a mixed-suspension, mixed-product-removal (MSMPR) type, satisfying the criteria for Eq. (18-31), and if the model of Clontz and McCabe is vahd, the contribution to the nucleation rate by the circulating pump can be calculated [Bennett, Fiedelman, and Randolph, Chem. E/ig, Prog., 69(7), 86(1973)] ... 

Nucleation due to crystal-to-ciystal contact is greater for equal striking energies than ciystal-to-metal contact. However, the viscous drag of the liquid on particle sizes normaUy encountered hmits the velocity of impact to extremely low values. The assumption that only the largest crystal sizes contribute significantly to the nucleation rate by ciystal-to-crystal contact permits a simple computation of the rate ... 

Here it can be seen that the nucleation rate is a decreasing function of growth rate (and supersaturation). The physical explanation is believed to be the mechanical influence of the crystallizer on the growing suspension and/or the effect of Bujacian behavior. 

The nucleation rate is, in fact, critically dependent on temperature, as Fig. 8.3 shows. To see why, let us look at the heterogeneous nucleation of b.c.c. crystals at grain boundaries. We have already looked at grain boundary nucleation in Problems 7.2 and 7.3. Problem 7.2 showed that the critical radius for grain boundary nucleation is given by... 

Unfortunately, both primary nucleation parameters cannot be predicted a priori as yet and in practice the nucleation rate must be measured and correlated empirically for each system. 

Secondary nucleation is an important particle formation process in industrial crystallizers. Secondary nucleation occurs because of the presence of existing crystals. In industrial crystallizers, existing crystals in suspension induce the formation of attrition-like smaller particles and effectively enhance the nucleation rate. This process has some similarity with attrition but differs in one important respect it occurs in the presence of a supersaturated solution. 

Hz and decreases at 33.4Hz. By comparing data of different feed point positions, it was found that the nucleation rates for the experiments with the feed point located near the impeller were higher than those for feed points outside the draft tube. 

The nucleation rate is plotted versus the supersaturation for different stirrer speeds in a log-log diagram (Figure 6.21). The kinetic order n in the correlating equation... 

The expression for the nucleation rate 5 in the compartment / is derived from the theory of primary nucleation and found to be (Mullin, 2001)... 

The size of crystals produced in the gas-liquid system varied from 10 to 100 pm by controlling the level of supersaturation, while the liquid-liquid system produced crystals of 5—30 pm. The wide variation of crystal size is due to the marked sensitivity of the nucleation rate on the level of supersaturation, while the impurity content is another variable that can affect the crystal formation. 

Note that this result is independent of dimension, apart from some constant prefactor, while the nucleation rate depends on dimension. The... 

Development of the theory of nucleation is the long-standing problem in the statistical physics. The kinetic approach to this problem was proposed by Zeldovich" . For the nucleation rate. 7. i.e. for the number of critical and supercritical embryos being formed in the unit volume per unit time at early stages of nucleation, he obtained the following expression... 

The present work aims to derive fully microscopic expressions for the nucleation rate J and to apply the results to realistic estimates of nucleation rates in alloys. We suppose that the state with a critical embryo obeys the local stationarity conditions (9) dFjdci — p, but is unstable, i.e. corresponds to the saddle point cf of the function ft c, = F c, — lN in the ci-space. At small 8a = c — cf we have... 

Application of Eqs. (21)-(27) to the calculations of the nucleation rates J for various alloy models revealed a number of interesting results, in particular, sharp dependence of J and embryo characteristics on the supersaturation, temperature, interaction radius, etc. These results will be described elsewhere. 

The uniformity of film thickness is dependent upon temperature and pressure. The nucleation rate rises with pressure, such that at pressures above atmospheric the high rate of nucleation can lead to comparatively uniform oxide films, while increase in temperature reduces the density of oxide nuclei, and results in non-uniformity. Subsequently, lateral growth of nuclei over the surface is faster than the rate of thickening until uniform coverage is attained, when the consolidated film grows as a continuous layer ... 

In particular, blends of PVDF with a series of different polymers (polymethylmethacrylate [100-102], polyethylmethacrylate [101], polyvinyl acetate [101]), for suitable compositions, if quenched from the melt and then annealed above the glass transition temperature, yield the piezoelectric [3 form, rather than the normally obtained a form. The change in the location of the glass transition temperature due to the blending, which would produce changes in the nucleation rates, has been suggested as responsible for this behavior. A second factor which was identified as controlling this behavior is the increase of local /rans-planar conformations in the mixed amorphous phase, due to specific interactions between the polymers [102]. 

At very low molecular weights process (iv) will become important [49]. Then at least two molecules are required to form a stable nucleus5, and they must both join the surface at approximately the same time so that a relatively large increase in concentration is needed to significantly enhance the nucleation rate, that is, y > 1. Although Sanchez and Di Marzio made this more quantitative there are too many other complicating effects to make general approximations. 

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