Nucleation, rate

The presence of foreign species in a solution may increase or decrease the nucleation rate of the crystallizing substance. Both effects can be understood in terms of classical nucleation theory, where the nucleation rate is given by 

Jo is a pre-exponential factor, the shape factor Cs accounts for the geometry of the nucleus, is the interfadal energy of the nucleus with respect to the solvent, Vmoi is 

Additives may also affect the equilibrium solubrhty of a solute and thereby modify the supersaturation, which in turn results in a change in nucleation rate [8]. 

The induction time t nd is the inverse of the nucleation rate. Classical nucleation theory requires that a plot of In (t d) against 1/(T In (S) ) is a straight line, the slope 

Inhibition of nucleation has been investigated for a range of different crystallization phenomena, most notably for calcium carbonate crystallization with the aim of preventing fouling [8,13,14], crystallization of calcium oxalate with a view to understanding possible mechanisms for preventing urolithiasis (kidney stone formation) [15,16], crystallization of salts from well waters (prevention of well blockage in oil and gas exploitation [17], and sea water desalination [18]. 

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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]. 

As a follow-up to Problem 2, the observed nucleation rate for mercury vapor at 400 K is 1000-fold less than predicted by Eq. IX-9. The effect may be attributed to a lowered surface tension of the critical nuclei involved. Calculate this surface tension. 

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]. 

Equations (4.24) and (4.29) are equivalent, except that the former assumes instantaneous nucleation at N sites per unit area while the latter assumes a nucleation rate of N per unit area per unit time. It is the presence of this latter rate which requires the power of t to be increased from 2 to 3 in this case. 

Pecrystallisation of a copolymer having 15 wt % VC has been found to be nucleated by material that survives the melting process plus new nuclei (74). The maximum crystallisation rate occurred at 373 K the maximum nucleation rate at 283 K. Attempts to melt all the polymer led to degradation that interfered with recrystallisation. 

The most important variables affecting nucleation rate are shown by equations 10 and 11 to be iaterfacial eaergy, temperature, and supersaturation. 

The metastable limit can provide an empirical approach to modeling primary nucleation. This limit, which was first observed in 1951 (6), must be determined through experimentation, and nucleation rate is correlated with the following equation... 

Correlations of nucleation rates with crystallizer variables have been developed for a variety of systems. Although the correlations are empirical, a mechanistic hypothesis regarding nucleation can be helpful in selecting operating variables for inclusion in the model. Two examples are (/) the effect of slurry circulation rate on nucleation has been used to develop a correlation for nucleation rate based on the tip speed of the impeller (16) and (2) the scaleup of nucleation kinetics for sodium chloride crystalliza tion provided an analysis of the role of mixing and mixer characteristics in contact nucleation (17). Pubhshed kinetic correlations have been reviewed through about 1979 (18). In a later section on population balances, simple power-law expressions are used to correlate nucleation rate data and describe the effect of nucleation on crystal size distribution. 

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. 

A pair of kinetic parameters, one for nucleation rate and another for growth rate, describe the crystal size distribution for a given set of crystallizer operating conditions. Variation ia one of the kinetic parameters without changing the other is not possible. Accordingly, the relationship between these parameters determines the abiUty to alter the characteristic properties (such as dominant size) of the distribution obtained from an MSMPR crystallizer (7). 

Example 3 Population, Density, Growth and Nucleation Rate. 18-43... 

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... 

While Eq. (18-26) has been popular among those attempting correlations between nucleation rate and supersaturation, recently it has become commoner to use a derived relationship between nucleation rate and growth rate by assuming that... 

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 ... 

Calculate the population density, growth, and nucleation rates for a crystal sample of urea for which there is the following information. These data are from Bennett and Van Biiren [Chem. Eng. Frvg. Symp. Ser., 65(95), 44 (1969)]. Slurry density = 450 g/L Crystal density = 1.335 g/cm ... 

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. 

Crystallizers with Fines Removal In Example 3, the product was from a forced-circulation crystallizer of the MSMPR type. In many cases, the product produced by such machines is too small for commercial use therefore, a separation baffle is added within the crystallizer to permit the removal of unwanted fine crystalline material from the magma, thereby controlling the population density in the machine so as to produce a coarser ciystal product. When this is done, the product sample plots on a graph of In n versus L as shown in hne P, Fig. 18-62. The line of steepest ope, line F, represents the particle-size distribution of the fine material, and samples which show this distribution can be taken from the liquid leaving the fines-separation baffle. The product crystals have a slope of lower value, and typically there should be little or no material present smaller than Lj, the size which the baffle is designed to separate. The effective nucleation rate for the product material is the intersection of the extension of line P to zero size. 

In a study where the uniform nucleation rate was varied over several values at a fixed strain rate of = 10 /s, the fragment number results shown in Fig. 8.21 were obtained. At lower nucleation rates nearly all fracture initiation sites grow to completion and provide the reduced number of fragments shown. 

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... 

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