A study of the relation between crystal growth kinetics and particle size distribution of crystals produced from solution

A new relation between crystal growth kinetics and particle size distribution based on statements in the literature is derived. It can be equally used for the determination of the characteristics of crystallization caused by cooling, evaporation or precipitation. Different kinetic constants (rate constants of nuclei formation k₁, k₂, k₃, and growth K₀, K₁, K₂) can be determined, and therefrom the particle size density curve. – The effect of kinetics characteristics on particle size distribution is summarized.     

Mathematical modeling of impurity distribution in the melt meniscus in the growth of profiled sapphire crystals

The distribution of impurities in the growth of profiled sapphire crystals is simulated. The distribution of impurities was calculated with the use of the diffusion equation with convective terms. The melt flow was found by solving the Navier-Stokes equation. The distributions of impurities over the melt meniscus are obtained at different crystallization rates. The maximum concentration supersaturation in the meniscus is studied as a function of its geometric parameters.     

Mathematical model of compensation for dispersion of light polarization states in electrooptical systems based on chiral liquid crystals

Spectral dispersion of the polarization states occurs after passage of linearly polarized light through a thin layer of a chiral nematic crystal along the helical axis. It is clearly pronounced for crystals with a high optical anisotropy and a helical pitch somewhat larger than the light wavelength. The results of numerical analysis of the spectral dependence of polarization states at the output of the liquid crystal layer and the method for compensating for the dispersion are presented for the first time. It is shown that polarization dispersion can be eliminated using phase plates of two types, one of which should have a high anomalous birefringence dispersion. The possibility of developing fast light modulators operating in a wide spectral range is discussed.     

Determination of size distribution and the activation energy of growth of the crystalline particle in TiNi thin films

Ti-49 at.% Ni thin films prepared by magnetron sputtering were annealed at 733-793 K for several time periods. Using small angle X-ray scattering, we have investigated the growth behavior of the crystalline particles in the TiNi films. We have obtained the size distribution of the samples annealed at 753 and 793 K for several time periods. The activation energy of growth of the crystalline particle was obtained.     

On the void distribution and size in shaped sapphire crystals

Ribbon and rod sapphire pulling has been performed in three different crystal growth equipments in order to study the effect of the installation, of the atmosphere, of the die shape, of the feed material and of the pulling rate on the distribution, number and diameter of the characteristic voids (micro‐bubbles) in the crystals. The location of the bubbles in the crystals depends on the die geometry; however, in most cases they are essentially located close to the crystal periphery and then can be efficiently removed by lapping. After statistical analysis of the results, it is demonstrated that the number of gas moles incorporated in the crystals, inside the voids, is totally independent of any growth parameter. It is also shown that the bubble diameter depends only on the pulling rate. Consequently, for a given pulling rate, the number of bubbles auto‐adjusts in order to satisfy the constant molar gas incorporation. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global analysis of heat transfer in CZ crystal growth of oxide taking into account three-dimensional unsteady melt convection: Effect of meniscus shape

A global model of heat transfer analysis for the Czochralski crystal growth of oxides, in which a three-dimensional unsteady melt flow was taken into account, was developed in our recent study. In the model, however, the focal point was the methodology of formulating the model by coupling a conventional global model of heat transfer, which is based on a pseudosteady axisymmetric assumption, with a model of a three-dimensional, unsteady melt flow via two interface models. Therefore, for simplicity, the shape of the melt free surface was assumed to be flat. In this study, the global model was further developed by considering the meniscus of the melt free surface. It was found that the meniscus of the melt free surface caused the melt flow to be more unstable and shifted the critical Reynolds number at which the melt/crystal interface inversion occurs toward a much lower value.     

Grain size distribution in the Kolmogorov model for nucleation and growth: Heterogeneous versus homogeneous nucleation

A theory of grain size distribution in nucleation and growth reactions described by the KOLMOGOROV model is developed. The distribution of agglomerates of growing and impinging nuclei is explicitely calculated for the two cases of heterogeneous and homogeneous nucleation. The results are compared and discussed with respect to the investigation of the nucleation mechanism in crystallization processes.     

Algorithm and program for precise determination of unit-cell parameters of single crystal taking into account the sample eccentricity

Crystallography Reports - A technique has been developed to refine the unit-cell parameters of single crystals with minimization of the influence of instrumental errors on the result. The...     

Mathematical model and numerical simulation of faceted crystal growth

A new mathematical macroscopic model is proposed to describe the nonstationary process of faceted crystal growth by the methods of directional crystallization with a slow change in external thermal conditions and low pulling rate of a cell through the growth system. The facet-growth rate is determined by the Stefan condition, integral over the face. Two boundary conditions are set for temperature: the continuity condition and the relation between the heat-flux jump and the supercooling at the facet points. The supercooling is determined by solving the entire heat problem. A facet is selected as a planar part of the phase boundary. The kinetic coefficient at the facet may depend on the supercooling. The energy conservation law is valid within the model developed. Examples of calculations of some model problems are presented.     

Oriented growth of large size calcium fluoride single crystals for optical lithography

We report (1 1 1), (1 1 0) and (1 0 0) growth of CaF₂ by the vertical Bridgman method. Crystals up to 250 mm diameter were grown and various growth parameters such as growth rate, temperature gradient and post-growth cooling rate were studied. It was found that the growth rate and the cooling rate are slower for the larger diameter crystals with a fixed temperature gradient. These growth parameters were optimized for growing the crystals along specific orientation after realizing that CaF₂ has a tendency to grow along an orientation close to 1 1 0. Degradation in optical transmittance was evaluated by irradiating the crystal to γ-rays up to a dose of 10⁵ rad. Optimized scavenger addition resulted in crystals with better radiation resistance and excellent VUV transmittance.     

The dispersion of the dynamic scattering of colourless nematic liquid crystals

Using a photogoniometer the dispersion of the dynamic scattering process for visible light in colourless nematic liquid crystals was studied depending on sample voltage, observation angle and primary beam polarization. The results may be interpreted by means of a scattering model based on the dispersion effects of the birefringence and of the elliptical polarization.     

A new example for the applicability of the relationship between the particle size and the kinetic of the crystal growth

Experiments have been made with PbCrO₄ precipitating system in a batch type tank. In the experiments the speeds of stirring and the residence times were changed. The particle size distribution of precipitates formed has been determined. The rate determining mechanism of the crystal growth were evaluated by two different methods (NIELSEN method and method elaborated in our institute). The two methods used have indicated the same informations about the rate determining processes. It also has been stated that the coagulation does not disturb the selection of the suitable growing mechanism.     

Numerical simulation of the LEC-growth of GaAs crystals with account of high-pressure gas convection

The influence of the inert gas pressure on the growth of 4 GaAs crystals by the liquid encapsulated Czochralski method (LEC) process is studied for a range of the Ar gas pressure up to 10 bar by using our finite-volume computer code STHAMAS. Up to the pressure of 0.6 bar we are considering laminar convection. For the pressure range from 5 to 10 bar we are using the buoyancy extended standard k-ε turbulence model with wall functions to simulate the gas flow. The numerical results show that the Argon gas pressure has a strong influence on the consumption of heater power in qualitative agreement with our experimental results. The convex curvature of the growth interface and the maximum thermal stress (von Mises criterion) are found to increase with increasing gas pressure both in the laminar and turbulent evaluations.     

Effect of precipitation procedure and detection technique on particle size distribution of CaCO3

The deposition of inorganic salts (“scales”) such as calcium carbonate is an important flow assurance problem during crude oil production. The knowledge of the features of the precipitated solids, mainly the particle size and morphology, is crucial to understand the nature of the solids and to avoid or reduce the effect of their deposition. For instance, the use of additives is one of the most usual procedures to mitigate this problem. Additives interact with scale-forming substances either by increasing the induction time, or by inhibiting crystal growth, changing the morphology of solids.     

Temperature field design,process analysis and control of SAPMAC method for the growth of large size sapphire crystals

In this paper, the relationship between quality of sapphire crystal and growing parameters of SAPMAC (Sapphire growth technique with micro‐pulling and shoulder‐expanding at cooled center) method was discussed. Optimized temperature distribution and technique control were proposed by theoretical analysis, numerical simulation computation and experimental validation to obtain large size sapphire crystals. For a‐axis crystallized direction, with 1.0‐5.0mm/h growth velocity and 10‐30K/h temperature decreasing speed, large sapphire single crystal (∅︁240mm×210mm, 27.5kg) having high optical quality was successfully grown. The absorption spectrum of standard samples was measured as well. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dendritic growth of snow crystals

The hypothesis that allows the interpretation of dendritic growth of a snow crystal in terms of diffusion-limited aggregation is criticized. The results of simulation of growth of quasi-two-dimensional crystals in two-and three-dimensional media based on the classical two-parametric model of diffusion-limited aggregation are used as an argument in this criticism. It is established that the model dimensionality considerably influences morphology of the grown crystal. The mechanism of dendritic growth of a snow crystal in which the main part is played by the surface processes at the ice/water interface is suggested.     

Iso-epitaxial growth of n-octacosane crystals

Iso-epitaxial(epilayer)growth of n-octacosane crystals is reported. The morphology of the epilayers is in the form of triangular growth islands with or without truncature. Epilayer growth precedes spiral growth mechanism which occurs at medium and low supersaturations, respectively. The occurrence of triangular and truncated rhombic platelets is found to be a manifestation of growth at medium supersaturations.     

Understanding the growth mechanism of CuI crystals during gel growth experiments

Cuprous iodide crystals have been grown with decomplexation method in silica gel. Various crystal morphologies, such as polycrystalline aggregate, skeletal, dendrite, hopper crystals and regular tetrahedron crystals, were observed in different growth regions with an optical microscope. Their surfaces were photographed using a binocular metallographic microscope and the results were explained with the crystal growth mechanism which was determined by supersaturation. These observations support the general hypothesis that the concentration of reactant affects the relative growth rate by controlling the nucleation and diffusion. The mutual influence of the crystals grown in different regions was also discussed. Additionally, the suitable condition for getting regular tetrahedron crystals or large hopper crystals was obtained by changing the concentration of CuI·HI complex in the later period of crystal growth. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)