Nucleation time

The linear term is absent because the first derivative is zero at the maximum. The term inside the square brackets is negative around the free energy maximum. Defining 

If the free energy barrier is sharply peaked (as shown in Section 8.5), the saddle-point approximation can be used to approximate the integral inside the square brackets to get 

The steady-state flux follows from Equation 10.31 as 

Any particular form of the free energy barrier height, as described in Chapter 5, can be used in evaluating the nucleation rate. As an example, the nucleation time for a flexible polyelectrolyte chain of N segments to escape from a confining spherical cavity to the outside world is obtained by combining Equations 5.31, 10.33, and 10.34 as 

Using the equations for a general stochastic process with the drift and diffusion factors (A and B in Equation 6.84), derived in Section 6.6, explicit expressions 

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Two nucleation processes important to many people (including some surface scientists ) occur in the formation of gallstones in human bile and kidney stones in urine. Cholesterol crystallization in bile causes the formation of gallstones. Cryotransmission microscopy (Chapter VIII) studies of human bile reveal vesicles, micelles, and potential early crystallites indicating that the cholesterol crystallization in bile is not cooperative and the true nucleation time may be much shorter than that found by standard clinical analysis by light microscopy [75]. Kidney stones often form from crystals of calcium oxalates in urine. Inhibitors can prevent nucleation and influence the solid phase and intercrystallite interactions [76, 77]. Citrate, for example, is an important physiological inhibitor to the formation of calcium renal stones. Electrokinetic studies (see Section V-6) have shown the effect of various inhibitors on the surface potential and colloidal stability of micrometer-sized dispersions of calcium oxalate crystals formed in synthetic urine [78, 79]. 

We assume, for our analysis on small particles, that the overall crystallization time is limited by the nucleation time for a single germ. 

As a rule, short nucleation times are the prerequisite for monodisperse particle formation. A recent mechanistic study showed that when Pt(acac)2 is reduced by alkylalu-minium, virtually all the Pt cluster nuclei appear at the same time and have the same size [86]. The nucleation process quickly consumes enough of the metal atoms formed initially to decrease their concentration below the critical threshold. No new metal cluster nuclei are created in the subsequent diffusion-controlled growth stage. 

Nucleation Number N = 0.5A//mn0 l D(fpIT) Compare nucleation time to impact time Dykhuizen [390]... 

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... 

Figure 2. Nucleation time as a function of the maximum gravitational mass of the hadronic star. Solid lines correspond to a value of a = 30 MeV/fm2 whereas dashed ones are for a = 10 MeV/fm2. The nucleation time corresponding to one year is shown by the dotted horizontal line. The different values of the bag constant (in units of MeV/fm3) are plotted next to each curve. The hadronic phase is described with the GM1 model.

The nucleation time r, i. e.. the time needed to form a critical droplet of deconfined quark matter, can be calculated for different values of the stellar central pressure Pc which enters in the expression of the energy barrier in Eq. (5). The nucleation time can be plotted as a function of the gravitational mass... 

As we can see, from the results in Fig. 2, a metastable hadronic star can have a mean-life time many orders of magnitude larger than the age of the universe Tuniv = (13.7 0.2) x 109 yr = (4.32 0.06) x 1017 s (Spergel et al. 2003). As the star accretes a small amount of mass (of the order of a few per cent of the mass of the sun), the consequential increase of the central pressure lead to a huge reduction of the nucleation time and, as a result, to a dramatic reduction of the HS mean-life time. 

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). 

B11 < B < l>1. Now, in addition to pure HS, there is a new branch of compact stars, the hybrid stars but the nucleation time r(MHs,max) to form a droplet of Q -matter in the maximum mass hadronic star, is of the same order or much larger than the age of the Universe. Therefore, it is extremely unlikely to populate the hybrid star branch. Once again, the compact star we can observe are, in this case, pure HS. 

Samples of gallbladder bile obtained in this way were analysed for bile acids, phospholipids and cholesterol (from which the cholesterol saturation indices were derived). Biliary bile-acid composition was then measured by HPLC. The vesicles were separated from micelles by sucrose density gradient ultra-centrifugation and the cholesterol microcrystal nucleation time measured as described above. 

Graph 8.4 Group data for the cholesterol microcrystal nucleation time (NT) measures in days. The value at 10 days represents the limit above which nucleation time is normal. Data taken from reference 18. 

A kinetic study in a well-stirred semi-batch reactor was conducted to determine the rate of methane as shown in Figure 4 (Lee et al., 2005b). As seen, the system with TBME has the shortest nucleation time and fastest hydrate growth rate followed by NH and MCH. This trend... 

The occurrence of coagulative nucleation does not alter the -power dependence of N on R,. However, the coagulative nucleation mechanism indicates a more complex dependence of N on S. The exponent of S decreases monotonically from 1.2 to 0.4 with increasing S. The concentration of polymer particles is higher and the nucleation time is longer for systems with high surfactant concentrations. Polymer particle formation becomes less efficient at longer... 

The induction time is marked as 1 and includes the time taken for crystal nuclei to form which are not visible to macroscopic probes. The induction time is defined in practice as the time elapsed until the appearance of a detectable volume of hydrate phase or, equivalently, until the consumption of a detectable number of moles of hydrate former gas. The induction time is often also termed the hydrate nucleation or lag time (Section 3.1). (The induction or lag time is the time taken for hydrates to be detected macroscopically, after nucleation and onset of growth have occurred, whereas nucleation occurs on too small a size scale to be detected. Therefore, the term nucleation time will not be used in this context. Instead, the term induction time or induction period will be used. The induction time is most likely to be dominated by the nucleation period, but also includes growth up to the point at which hydrates are first detected.)... 

In agreement with the findings reported in another study (36). seeding proved efficient by considerably shortening the nucleation period of ZSM-20 "Figure 4". Moreover, the nucleation time is even more reduced when ZSM-20 seed crystallites (0.5 wt. % with respect to dry gel) were added after the complete evaporation of ethanol, thus confirming the inhibiting role of the latter in ZSM-20 crystallization. 

Correlation between nucleation time (days) and the average ZSM-20 particle size (pm), as a function of the initial A1 content. 

Figure 11 shows the kinetics of crystallization of samples 2,4,and 5. Obviously die nucleation time for sample 5 is shorter than that characterizing samples 2 and 4, prepared in presence of larger TEAOH concentrations after only 5 days heating at 100°C. It also appears that for a TEA+/Al203 ratio of 17.5, the source of silica does not affect the nucleation time of samples 2 and 4. However, sample 2 (Si(OEt)4) achieves more rapidly a 100% crystallinity than sample 4 (Aerosil), suggesting a more efficient utilization of the Si(OH)4 monomers stemming from a slow hydrolysis, to build up the final framework. 

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