Crystal Nucleation Rate

Discussion of transient effects in the nucleation kinetics in polymers subjected to orienting stresses bases on a kinetic theory of nucleation proposed in [20-22] for the systems of oriented asymmetric elements. The theory converges to the classical approach in the limit of isotropic orientation. This fact explains apparent validity of the classical theory in the apphcations to unoriented polymers. 

In the systems of flexible chain polymers, asymmetric single elements are identified with the uniaxial statistical chain segments. Considering a cylindrical cluster of uniaxial elements oriented at 0 = (0, ), the condition of compliant orientation reduces fraction of the chain segments involved in the association process to w(6))A where 

Under time-dependent chain deformation and orientation of the chain segments, the angular distribution of the rate of production of clusters f3g at any instant of time t by the association-dissociation reaction reads 

In the case of A covering the entire orientation space we have w 0, t))A = 1 and (4.34) reduces to the form used in classical nucleation theory. For im-oriented systems we have w 0, t))A = AjA n independently of A, and (4.34) also assumes the classical form, but with the association rate constant reduced by the tolerance range, = yP (t)AjA n and k t) = k t). In terms of 

In the kinetic equation (4.35), transient free energy of the elastic chain deformation controls ratio of the rate constants (4.36) while effects of molecular orientation are accounted for by the concentration factor A w 0,i). The concentration factor reduces to unity for the case of isotropic systems, assumes values above unity in the range of enhanced orientation, and below unity in the range of reduced orientation. Equation (4.35) introduces effects of molecular deformation and orientation of chain segments. 

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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. 

Figure 7 shows the relation between the crystal nucleation rate and 0. decreases linearly with increase of 0 in... 

Figure 4-2 Calculated nucleation rate for Vc = 46 x 10 m /mol, E = 250 kj/mol, A5m-c = 50 J-K -moP, Al = 5 x 10 m, the equilibrium temperature of 1500K for (a) and (b), and the equilibrium pressure of 3 GPa for (c). (a) The dependence of crystal nucleation rate on the interface energy. Note that for a small change in interface energy from 0.300 to 0.295 J/m, the peak nucleation rate increases by more than one order of magnitude. If the interface energy changes from 0.3 to 0.2 J/m, the peak nucleation rate would increase by 17 orders of magnitude, (b) The nucleation rate of crystal and melt as a function of temperature, (c) The nucleation rate of crystal and melt as a function of pressure.

On a nucleation rate versus pressure diagram (Figure 4-2c), melt nucleation rate below the crystal-melt equilibrium pressure and crystal nucleation above the pressure are roughly S3mimetric. In Equation 4-9, only AG would vary with pressure or concentration. Hence, both melt nucleation rate and crystal nucleation rate increase monotonically with departure from equilibrium. There is no peak nucleation rate. 

Figure 1-14 Crystal nucleation rate versus growth rate 56...

Crystal nucleation rates, expressed as the number of nuclei formed per unit volume per unit time, increase with protein solubility. Higher solubility leads to increased molecular encounters in solution and reduced levels of supersaturation required for spontaneous nucleation. Nucleation rates typically show a high-power dependence on protein supersaturation, and so empirically increase rapidly above a critical value... 

Auer, S., and Frenkel, D., Prediction of absolute crystal-nucleation rate in hard-sphere colloids. Nature, 409, 1020, 2001. 

Auer, S., Frenkel, D. Quantitative prediction of crystal-nucleation rates for spherical colloids a computational approach. Annu. Rev. Phys. Chem. 55, 333-361 (2004)... 

The inhibiting effect of the additive on the growth of the affected enantiomorph has been proven also by direct comparison of the size of (R d and (S i single crystals grown in parallel from seeds under conditions close to equilibrium (Fig. 2). Although there are indications that the same kind of inhibition may also influence the crystal nucleation rate, such an effect is very difficult to isolate and quantify. 

See a critical analysis of the problem in Auer, S. Frenkel, D. Quantitative prediction of crystal-nucleation rates for spherical colloids a computational approach, Annu. Rev. Phys. Chem. 2004, 55, 333-361. See also Turner, G. W. Bartell, L. S. On the probability of nucleation at the surface of freezing drops, J. Phys. Chem. A2005,109, 6877-6879, and references therein. These papers rely on a statistical analysis of simulated nucleation trajectories of clusters of SeFe molecules to estimate nucleation rates. 

Abstract This article reviews the recent progress that has been made in the application of computer simulations to study crystal nucleation in colloidal systems. We discuss the concept and the numerical methods that allow for a quantitative prediction of crystal nucleation rates. The computed nucleation rates are predicted from first principles and can be directly compared to experiments. These techniques have been applied to study crystal nucleation in hard-sphere colloids, polydisperse hard-sphere colloids, weakly charged or slightly soft colloids and hard-sphere colloids that are confined between two plane hard walls. 

The crystal nucleation rate is given by the product of Pc and a kinetic factor F that describes the rate with which a critical nucleus grows. The CNT expression for the nucleation rate per unit volume is ... 

The above expression for the nucleation rate is the one most commonly used to analyze crystal nucleation rate experiments. The problem with the CNT approach is however that, in most cases, neither 1 nor y are accurately known. More often than not, both parameters are obtained by fitting the CNT expression to experimental nucleation data. 

Fig. 3. Measured crystal nucleation rates / as of function of volume fraction in a system of hard-sphere colloids. The data are taken from Ref. [5] (open circles) and Ref. [6] (filled cubes). The hues result from a two parameter fit of Eq. (4) to the experimental data. The inset shows the dimensionless nucleation rate densities plotted logarithmically versus /((f>lsp). The figure is taken from Ref [4]...

As experiments to determine absolute crystal nucleation rates are notoriously difficult, there is a clear need for a first principle prediction of a crystal nucleation rate. 

To estimate the crystal nucleation rate we computed the kinetic prefactor F as described before. The result for the crystal nucleation rates as a function of A is that the decrease in the nucleation barrier transforms into an increase of the crystal nucleation rate of about two orders of magnitudes. Our simulations can be compared directly with the experimental results of Harland and van Megen [5], who measured nucleation rates by time-resolved static light scattering for PMMA spheres of radius 201 nm To make this comparison, we show in Fig. 31 the crystal nucleation rate as a function of the rescaled volume fraction of the metastable fluid. Comparing... 

Auer S (2002) Quantitative prediction of crystal nucleation rates for spherical colloids A computational study, Ph.D. thesis, University of Amsterdam, Amsterdam, The Netherlands (available from http //www.amolf.nl) 173 Viisanen Y, Strey R, Reiss H (1993) J. Chem. Phys. 99 4680 173 Oxtoby DW, Kashchiev D (1994) J. Chem. Phys. 100 7665 173 Oxtoby D (2001) Nature 413 694 173... 

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