Critical concentration for nucleation

The nucleation rate depends strongly on the concentration of a non-excess reactant, so that a critical concentration for nucleation (nucleation threshold) is passed by the consumption of this reactant during the nucleation process. 

Figure 1. Nucleation and growth of actin filaments. Nucleation is shown here as a thermodynamically unfavored process, which in the presence of sufficient actin-ATP will undergo initial elongation to form small filament structures that subsequently elongate with rate constants that do not depend on filament length. Elongation proceeds until the monomeric actin (or G-actin) concentration equals the critical concentration for actin assembly.

The determination of the binding energy of DMPC molecule in the foam bilayer was carried out using the experimental results for the temperature dependence of the critical concentration for formation of foam bilayer (Fig. 3.95) and the theory of Kashchiev-Exerowa (see Section 3.4.4.2). The concentrations Cc and Ce (Eq. (3.129)) are specific constants of each system which determine the ability of a foam bilayer to exist in a metastable state within the concentration range Cc< C < Ce. When C >Ce the foam bilayer is thermodynamically stable (there is no driving force for the whole nucleation process in the foam bilayer). It follows from the theory that the critical concentration of amphiphile molecules in the solution equals the equilibrium one (Cc = Ce) in the case of a missing metastable region when only thermodynamically stable foam bilayers are formed. As mentioned above, the DMPC foam... 

This highly nucleating tubulin preparation had (a) a lowered rate of tubulin dissociation from the microtubule polymer, (b) a 19-fold decrease in the critical concentration for assembly (the concentration of tubulin required for nucleation to occur), and (c) the ability to copolymerize with and seed untreated tubulin assembly at concentrations below that typical for spontaneous nucleation in vitro (39). 

Laminin itself will aggregate in vitro into large polymers in a temperature-, time- and concentration-dependent manner. Aggregation exhibits both concentration and thermal reversibility and there is a critical concentration for polymerization of about 60 nM, reflecting cooperative nucleation-propaga-tion type assembly (Yurchenco et al., 1985). Diva-... 

Surface energy of amorphous silica (46 mJ. is considerably lower than that of quartz (335-385 mJ. m ). This difference in surface energy means that the grain size of amorphous silica is smaller than quartz which precipitated from same supersaturated solution. Critical concentration for the nucleation of amorphous silica is small. Thus, nucleation of amorphous silica occurs instead of quartz (Steefel and Cappellen 1990). Grain size of amorphous silica is very small. But it changes to quartz due to its crystal growth rate similar to quartz. 

Since then, the polymerization of various protein molecules was found to have similar features. There is a critical concentration for polymerization and the polymerization consists of nucleation and growth. Bacterial flagella are helically curved tubular polymers of flagellin molecules. Flagella are formed from purified flagellin molecules in solution, although spontaneous nucleation of flagella can hardly occur [10]. The polymerization of tubulin molecules to microtubules is... 

As an even more explicit example of this effect Figure 6 shows that EPM is able to reproduce fairly well the experimentally observed dependence of the particle number on surfactant concentration for a different monomer, namely methyl methacrylate (MMA). The polymerization was carried at 80°C at a fixed concentration of ammonium persulfate initiator (0.00635 mol dm 3). Because methyl methacrylate is much more water soluble than styrene, the drop off in particle number is not as steep around the critical micelle concentration (22.) In this instance the experimental data do show a leveling off of the particle number at high and low surfactant concentrations as expected from the theory of particle formation by coagulative nucleation of precursor particles formed by homogeneous nucleation, which has been incorporated into EPM. 

Several refinements of our experiments could test these theories further. By measuring etch pit densities as well as pit dimensions on sequentially-etched crystals, nucleation rate data and pit growth data could be collected, yielding information about the rate-limiting steps and mechanisms of dissolution. In addition, since the critical concentration is extremely dependent on surface energy of the crystal-water interface (Equation 4), careful measurement of Ccrit yields a precise measurement of Y. Our data indicates an interfacial energy of 280 + 90 mjm- for Arkansas quartz at 300°C, which compares well with Parks value of 360 mJm for 25°C (10). Similar experiments on other minerals could provide essential surface energy data. 

The threshold concentration of monomer that must be exceeded for any observable polymer formation in a self-assembling system. In the context of Oosawa s condensation-equilibrium model for protein polymerization, the cooperativity of nucleation and the intrinsic thermodynamic instability of nuclei contribute to the sudden onset of polymer formation as the monomer concentration reaches and exceeds the critical concentration. Condensation-equilibrium processes that exhibit critical concentration behavior in vitro include F-actin formation from G-actin, microtubule self-assembly from tubulin, and fibril formation from amyloid P protein. Critical concentration behavior will also occur in indefinite isodesmic polymerization reactions that involve a stable template. One example is the elongation of microtubules from centrosomes, basal bodies, or axonemes. 

The development of the freeze concentration process for fruit juices has been hampered by the fact that solute concentrate is entrained by the ice crystals. This incomplete separation of the entrained concentrate from the ice results in a considerable increase of the cost of the process. In this investigation sucrose solutions were concentrated by the formation of an ice layer on the externally cooled walls of the crystallizer. The formation of the layer was initiated by secondary nuclei induced by rotating ice seeds, at subcoolings smaller than the critical subcooling needed for spontaneous nucleation. A minimum in the amount of sucrose entrapped in the ice layer was observed at a subcooling smaller than the critical subcooling for spontaneous nucleation. The effect of soluble pectins on the minimum was also studied. 

Note that this predicts critical concentrations that are more than an order of magnitude below the experimentally observed concentrations of Wyslouzil et al. (1991) shown in Fig. 9.30 (see Problem 3). For a recent treatment of the binary homogeneous nucleation of H2S04 and H20, see Kulmala et al. (1998). 

Use Eq. (GG) to predict the critical concentration of gaseous H2S04 in ppb for nucleation at a relative humidity of 40% and a temperature of 25°C. How does this compare to the theoretical predictions and experimental observations shown in Fig. 9.30 ... 

The growth of calcite crystals to form speleothems is a delicately balanced process depending on the degree of supersaturation of the water and its total concentration of dissolved carbonates. Waters dripping onto speleothems require supersaturations on the order of Sk = +0.5 in order to overcome nucleation barriers (where Sic is the saturation index defined in the textbooks cited above). However, the critical supersaturation for 2-dimensional nucleation and the continued growth of a single ciystal is only slightly... 

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