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Register a new userOn the Origin of Logarithmic-Normal Distributions: An Analytical Derivation, and its Application to Nucleation and Growth Processes
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The logarithmic-normal (lognormal) distribution is one of the most frequently observed distributions in nature and describes a large number of physical, biological and even sociological phenomena. The origin of this distribution is therefore of broad interest but a general derivation from basic principles is still lacking. Using random nucleation and growth to describe crystallization processes we derive the time development of grain size distributions. Our derivation provides, for the first time, an analytical expression of the size distribution in the form of a lognormal type distribution. We apply our results to the grain size distribution of solid phase crystallized Si-films.
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We study the time dependence of the grain size distribution N(r,t) during crystallization of a d-dimensional solid. A partial differential equation including a source term for nuclei and a growth law for grains is solved analytically for any dimension d. We discuss solutions obtained for processes described by the Kolmogorov-Avrami-Mehl-Johnson model for random nucleation and growth (RNG). Nucleation and growth are set on the same footing, which leads to a time-dependent decay of both effective rates. We analyze in detail how model parameters, the dimensionality of the crystallization process, and time influence the shape of the distribution. The calculations show that the dynamics of the effective nucleation and effective growth rates play an essential role in determining the final form of the distribution obtained at full crystallization. We demonstrate that for one class of nucleation and growth rates the distribution evolves in time into the logarithmic-normal (lognormal) form discussed earlier by Bergmann and Bill [J. Cryst. Growth 310, 3135 (2008)]. We also obtain an analytical expression for the finite maximal grain size at all times. The theory allows for the description of a variety of RNG crystallization processes in thin films and bulk materials. Expressions useful for experimental data analysis are presented for the grain size distribution and the moments in terms of fundamental and measurable parameters of the model.
Structural aspects of crystal nucleation in undercooled liquids are explored using a nonlinear hydrodynamic theory of crystallization proposed recently [G. I. Toth et al., J. Phys.: Condens. Matter 26, 055001 (2014)], which is based on combining fluctuating hydrodynamics with the phase-field crystal theory. We show that in this hydrodynamic approach not only homogeneous and heterogeneous nucleation processes are accessible, but also growth front nucleation, which leads to the formation of new (differently oriented) grains at the solid-liquid front in highly undercooled systems. Formation of dislocations at the solid-liquid interface and interference of density waves ahead of the crystallization front are responsible for the appearance of the new orientations at the growth front that lead to spherulite-like nanostructures.
The incubation time preceding nucleation and growth of surface nanostructures is interesting from a fundamental viewpoint but also of practical relevance as it determines statistical properties of nanostructure ensembles such as size homogeneity. Using in situ reflection high-energy electron diffraction, we accurately deduce the incubation times for Ga-assisted GaAs nanowires grown on unpatterned Si(111) substrates by molecular beam epitaxy under different conditions. We develop a nucleation model that explains and fits very well the data. We find that, for a given temperature and Ga flux, the incubation time always increases with decreasing As flux and becomes infinite at a certain minimum flux, which is larger for higher temperature. For given As and Ga fluxes, the incubation time always increases with temperature and rapidly tends to infinity above 640 {deg}C under typical conditions. The strong temperature dependence of the incubation time is reflected in a similar variation of the nanowire number density with temperature. Our analysis provides understanding and guidance for choosing appropriate growth conditions that avoid unnecessary material consumption, long nucleation delays, and highly inhomogeneous ensembles of nanowires. On a more general ground, the existence of a minimum flux and maximum temperature for growing surface nanostructures should be a general phenomenon pertaining for a wide range of material-substrate combinations.
We report the results of x-ray scattering studies of AlN on c-plane sapphire during reactive radiofrequency magnetron sputtering. The sensitivity of in situ x-ray measurements allowed us to follow the structural evolution of strain and roughness from initial nucleation layers to fullyrelaxed AlN films. A growth rate transient was observed, consistent with the initial formation of non-coalesced islands with significant oxygen incorporation from the substrate. Following island coalescence, a steady state growth rate was seen with a continuous shift of the c and a lattice parameters towards the relaxed bulk values as growth progressed, with films reaching a fully relaxed state at thicknesses of about 30 nm.
The nucleation and growth of CdS nanoparticles within a polymer matrix was followed by in-situ synchrotron X-ray diffraction. The nanoparticles form by effect of the thermolysis of thiolate precursors at temperatures between 200 and 300 Celsius degrees. Above 240 Celsius degrees the precursor decomposition is complete and CdS nanoparticles grow in the polymer matrix forming a nanocomposite with interesting optical properties. The nanoparticle structural properties (size and crystal structure) depend on the annealing temperature.(abridged version)
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