Determination of individual rate constants of reaction and diffusion steps from over-all crystal growth coefficient

A universal method, being a solution of the problem formulated independently by NÝVLT and GARSIDE in 1971 is proposed. The method makes it possible — having at one's disposal the experimental growth rate constant — to determine individual rate constants of the main steps of the process of crystal growth i.e., the diffusion step and the surface integration step and thus indicate the step controlling the whole process of crystal growth. Experimental data on crystal growth of MgSO₄ and ZnSO₄ were evaluated.     

Crystallization kinetics of the borax decahydrate

The growth and dissolution rates of borax decahydrate have been measured as a function of supersaturation for various particle sizes at different temperature ranges of 13 and 50 °C in a laboratory-scale fluidized bed crystallizer. The values of mass transfer coefficient, K, reaction rate constant, k_r and reaction rate order, r were determined. The relative importances of diffusion and integration resistance were described by new terms named integration and diffusion concentration fraction. It was found that the overall growth rate of borax decahydrate is mainly controlled by integration (reaction) steps. It was also estimated that the dissolution region of borax decahydrate, apart from other materials, is controlled by diffusion and surface reaction. Increasing the temperature and particle size cause an increase in the values of kinetic parameters (K_g, k_r and K). The activation energies of overall, reaction and mass transfer steps were determined as 18.07, 18.79 and 8.26 kJmol⁻¹, respectively.     

Hydration of soda-lime glass

The hydration of soda-lime glass is studied using resonant nuclear reactions to measure the hydrogen and sodium profiles of hydrated glasses. The rate of growth of the surface layer of hydrated glass is initially proportional to the square root of time as is characteristic of diffusion controlled processes. After longer exposure a steady-state hydration profile is observed, which indicates that in addition to the diffusion controlled reaction there is a slow etching of the glass surface. The measured hydration profiles are discussed in relationship to the Doremus model of interdiffusing ions, which is found to be in good agreement with the data. This model is also discussed in relationship to measured hydration profiles of vacuum heated samples of hydrated glass.     

Micro/macro modeling of CVD synthesis

We propose a CVD reaction scheme in which a source material undergoes a gas-phase reaction to produce an intermediate, and then the intermediate diffuses to the solid surface and changes into a solid film through a surface reaction. A series of simple Monte Carlo (SMC) codes has been developed to simulate the observed film shape on micro-trenches and holes. By using these codes, surface reaction rate constants were determined so as to reproduce the experimentally observed film shape. By means of a macro-scale reactive transport analysis of a hot wall tubular reactor, gas-phase reaction rate constants for single component systems were determined to simulate the experimental growth rate distributions. The composition and growth rate of Yttria stabilized Zirconia (YSZ) film, a solid solution of Yttria and Zirconia, were qualitatively explained by a sum of single component's growth rates. As an application of these reaction models, we simulated a rotating-disk CVD reactor under low pressure. The simulations based on a quasi three-dimensional model revealed that the susceptor rotation suppresses the buoyancy convection and forms steeper gradients in temperature and concentration near the susceptor uniformly over wide area. At higher temperatures, the growth rate increased with rotation speed, but at lower temperatures the growth rate decreased with increasing rotation speed because the reduced retention time in the high-temperature region suppressed the gas-phase reaction.     

Improved reproducibility in zinc oxide single crystal growth using chemical vapor transport

The crystal growth process of zinc oxide (ZnO) by chemical vapor transport (CVT) using carbon as the transport agent is developed. The comparison between the chemical reaction rate and the diffusion velocity is our primary point of view. In ordinary CVT systems, the transport rate is diffusion-limited because the chemical reactions of both source and growth sides reach an equilibrium extremely faster than the diffusion velocity. Nevertheless, in our system, the transport rate is kinetics-limited because the estimated chemical reaction rates are slower than the diffusion velocity. Configurations of ampoules have been devised to decrease the diffusion velocity and to change from the kinetics-limited transport to the diffusion-limited one. Then the reproducibility of ZnO single crystal growth was considerably improved.     

Effect of electrical field on dendritic growth of boric acid

Growth rates of boric acid crystals have been measured both in pure aqueous solution and under the influence of electrical field as a function of supersaturation and electrical field intensity in a laboratory scale fluidized bed crystallizer at the temperature of 30°C. The effect of electrical field was estimated from the growth rate data to evaluate the relative magnitude of two resistances, diffusion and integration. In absence of electrical field, the obtained results indicate that the controlling mechanism is mainly integration. However, in presence of electrical field, growth rate of boric acid were controlled by diffusion and reaction steps. In the absence of electrical field, boric acid crystals grow dendritically at any level of supersaturation from pure boric acid solution. In case of electrical field, the dendritic structure was depressed.     

CFD modelling of single crystal growth of potassium dihydrogen phosphate (KDP) from binary water solution at 30 °C

Crystal growth rate depends on both diffusion and surface reaction. In industrial crystallizers, there exist conditions for diffusion-controlled growth and surface reaction-controlled growth. Using mathematical modelling and experimental information obtained from growth studies of single crystals, it is possible to separate these phenomena and study how they are affected by concentration, slip velocities of particles, temperature and finally estimate the parameters for crystal growth models.In this study, a power-law growth model using activity-based driving force is created. Computational fluid dynamics (CFD) was used to evaluate the thickness of a diffusion layer around the crystal. Parameters of the crystal growth model were estimated using a non-linear optimization package KINFIT. Experimental data on growth rate of the (1 0 1) face of a potassium dihydrogen phosphate (KDP) single crystal and simulated data on the thickness of a diffusion layer at the same crystal face were used in parameter estimation. The new surface reaction model was implemented into the CFD code. The model was used to study the effect of flow direction on growth rate of the whole crystal with various slip velocities and solute concentrations.The developed method itself is valid in general but the parameters of crystal growth model are dependent on the system. In this study, the model parameters were estimated and verified for KDP crystal growth from binary water solution.     

Determination of linear growth rates of crystals (I). Calculation of linear growth rates of individual crystal faces from overall rates

The presented paper (Part I and II) is devoted to the fundamental principles of the kinetics of crystal growth. The number of studies in this field has been increasing and some authors have introduced different terms and definitions. To avoid possible mistakes and confusions, the individual definitions of the rate of crystal growth are specified and their interrelations are discussed in detail. Basic methods for measuring the kinetics of crystal growth are described. The rate of growth of individual crystal faces or the average rate of crystal growth can be determined using the selected experimental method. The presented study demonstrates that the results obtained by either of the two measuring methods (linear rate of growth of crystal faces and average rate of crystal growth) can be interrelated. The established method of calculating linear growth rates of individual crystal faces is based on an analysis of the time dependence of the volume of a growing crystal. The relationship between the linear growth rates of individual crystal faces and the over-all crystal growth rate is presented and a method for assessing the linear growth rates of individual crystal faces from over-all growth rate data, which can be measured readily, is suggested.     

Kinetics of crystal growth of strontium tungstate from solutions in sodium tungstate melts by continuous cooling

The crystallisation kinetics of strontium tungstate from unstirred saturated solutions in sodium tungstate melts was studied by continuous cooling from initial crystallisation temperatures T₀ = 1000° to 800°C to room temperature at cooling rates R_T = 0.67° to 3.3°C min⁻¹. The main crystal growth was diffusion rate-controlled; the final crystal growth was rate-controlled by the development rate of excess solute concentration. The estimated diffusion rate constant (k_D) values increased with cooling rates and initial crystallisation temperatures. They are higher than the rate constants for diffusion-controlled growth of calcium tungstate from sodium tungstate melts, but very much smaller than those for strontium tungstate from lithium chloride melts.     

Inhibition of Calcium Oxalate Monohydrate Crystal Growth in High and Low Ionic Strength Solutions

The effect of additives on the kinetics of growth of calcium oxalate monohydrate crystals has been studied. Conductivity and potentiometry measurements have been compared. Growth rates were calculated from precipitate curves by a cubic spline method. An approach consisting on the calculation of rate constants and orders of reaction from logarithmic plots of growth rate versus supersaturation has been followed to study crystal growth kinetics. This method revealed that the presence of additives is causing not only a decrease on the rate constant but an increase on the order of reaction as well. The effect of additives (EDTA, citrate and phytate) was considerably weaker in high ionic strength media. Phytate produced a complete blockage of crystal growth in concentrations as low as 2 × 10^—6 mole/L in both methods.     

Diffusion of intrinsic defects in dielectric and semiconductor crystals with impurities

This paper reports on the results of detailed theoretical investigations into the diffusion of intrinsic defects in impurity crystals doped with mixed-valence ions. The special case of diffusion stimulated by variations in the redox properties of the atmosphere at the crystal boundary during high-temperature annealing is analyzed. The major consideration is given to the following fundamental problems: (i) the dynamics of valence transitions and the structure of the chemical reaction zone, (ii) the possibility of determining the type of chemical reaction at the crystal-atmosphere interface and the type of diffusing defects, (iii) the effect of dilatation mechanical stresses arising in the reaction zone on the reaction-zone structure and on the dynamics of diffusion processes, and (iv) the determination of the diffusion parameters of intrinsic defects and the constants of their interaction with impurity centers.     

Kinetic studies on the influence of temperature and growth rate history on crystal growth

Crystallization experiments of sucrose were performed in a batch crystallizer to study the effect of temperature and growth rate history on the crystal growth kinetics. In one of the growth methods adopted, the isothermal volumetric growth rate (R_V) is determined as a function of supersaturation (S) at 35, 40 and 45 ºC. In the other, crystals are allowed to grow at constant supersaturation by automatically controlling the solution temperature as the solute concentration decreased. Using the latter method R_V is calculated as the solution is cooled. The obtained results are interpreted using empirical, engineering and fundamental perspectives of crystal growth. Firstly, the overall activation energy (E_A) is determined from the empirical growth constants obtained in the isothermal method. The concept of falsified kinetics, widely used in chemical reaction engineering, is then extended to the crystal growth of sucrose in order to estimate the true activation energy (E_T) from the diffusion‐affected constant, E_A. The differences found in the isothermal and constant supersaturation methods are explained from the viewpoint of the spiral nucleation mechanism, taking into account different crystal surface properties caused by the growth rate history in each method. Finally, the crystal growth curve obtained in the batch crystallizer at 40 ºC is compared with the one obtained in a fluidized bed crystallizer at the same temperature. Apparently divergent results are explained by the effects of crystal size, hydrodynamic conditions and growth rate history on the crystallization kinetics of sucrose. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Growth and etching of silicon in chemical vapour deposition systems; The influence of thermal diffusion and temperature gradient

In the growth of epitaxial silicon using SiCl₄ as source material it is known, that the growth rate at first increases with increasing p_SiCl4, but at higher p_SiCl4 decreases and finally becomes negative. The exact shape of the growth curve and especi ally the point of zero growth, where the etch rate counterbalances the growth rate, differs considerably amongst various investigators. This phenomenon was subject to theoretical considerations. Three models were evaluated: (i) a simple equilibrium model in which gas transport limited kinetics are used with one single diffusion coefficient for all species; (ii) an extended model using individual diffusion coefficients; (iii) a model as in (ii) but including the effect of thermal diffusion. It is shown that models (i) and (ii) do not give a general solution of the problem. The best results are obtained with model (iii). The growth rate appears to be very sensitive to the temperature gradient ▽T. As ▽T depends on the kind of apparatus used in the deposition process, it provides a reasonable explanation for the observed differences.     

Growth Kinetics of Potassium-Dihydrogen Phosphate Crystals in Solution

Growth rate data of (101) faces of KDP crystals as a function of the relative supersaturation of the solution are analyzed on the basis of BURTON , CABRERA and FRANK surface-diffusion model applied to the growth from solution. The characteristic constants in the BCF function are determined for 50°C. Approximate values of the free activation energies for dehydration, desorption, surface diffusion and entering the kinks from the surface are calculated. The experimental data previously obtained by MULLIN and AMATAVIVADHANA and by BELOUET are also discussed.     

Growth of equiaxed dendritic crystals settling in an undercooled melt,Part 1: Tip kinetics

Experiments are conducted to measure the dendrite tip growth velocities of equiaxed crystals of the transparent model alloy succinonitrile–acetone that are settling in an undercooled melt. The tip velocities are measured as a function of the crystal settling speed and the Eulerian angle between the dendrite arms and the flow direction relative to the crystal. The ratio of the settling speed (or flow velocity) to the tip growth velocity ranges from 62 to 572. The ratio of the measured tip velocity to that predicted from a standard diffusion theory for free dendritic growth ranges from almost zero for dendrite tips growing in the wake of the crystal, about unity for dendrite tips with an orientation close to normal to the flow direction, and up to two for dendrite tips growing into the flow. Despite the relatively strong flow relative to the crystal, the average tip growth velocity of the six primary dendrite arms of an equiaxed crystal is found to be in excellent agreement with the standard diffusion theory result. The individual tip velocities are correlated using a boundary layer model of free dendritic growth in the presence of melt flow that is modified to account for the flow angle dependence. Using the same dendrite tip selection parameter, σ^*, as established previously under purely diffusive conditions (0.02), good agreement is achieved between the measured and predicted tip velocities. The model is also found to predict well the variations in the tip velocity that occur during settling due to crystal rotation and settling speed changes.     

Crystallization Kinetics of Sodium Perborate

Some basic physicochemical properties of aqueous sodium perborate solutions, essential in designing an effective control, were measured. The kinetics of primary nucleation was determined by the measurement of the metastable zone width. The differential method was applied for the determination of the crystal growth rate equation. It was found that the solubility and the nucleation rate constants are in very good agreement with the literature data whereas the parameters of the growth rate equation show an influence of the experimental conditions.     

Dissolution kinetics of K-alum crystals as judged from the measurements of surface undersaturations

The normal dissolution rates, the slopes and the average velocities of dissolution steps of etch pits on the (111) face of a K-alum crystal, originating from dislocations with Burgers vector 110, were measured in relation to the surface undersaturations. From the mutual relations, it was shown that the dissolution was controlled by both diffusion and surface kinetic processes, although the contribution of the latter process was smaller than in the case of growth. It was also demonstrated that the normal dissolution rate was always larger than the growth rate. This is attributed to the fact that the slopes of growth hillocks are invariably smaller than those of etch pits.     

Application of diffusion couple technique for the determination of the Ti‐Bi‐Sn phase diagram

The system Ti‐Bi‐Sn has been investigated by solid/liquid diffusion couples at 500, 600, 700 and 800°C. A non‐negligible solubility of Sn into solid Bi is found. Diffusion layers of the recently revealed ternary compound Ti₃BiSn have been observed, thus its existence has been confirmed. Data about the homogeneity ranges of the binary end‐system compounds have been obtained. The assessed growth constants of the diffusion layers are comprised in the interval 10^‐12 –10^‐14 m².s^‐1. The phases participating in the hypothetical ternary eutectic reaction, probably are: (Bi), (Sn), Liquid and Ti₂Sn₃. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Electrolytic growth and dielectric properties of magnesium hydrogen phosphate single crystals

The growth of MgHPO₄, magnesium hydrogen phosphate, (MHP) single crystals in silica gels under the influence of an externally applied uniform electric field is described. A controlled reaction between magnesium chloride and orthophosphoric acid by a diffusion process in the gel medium is employed. The bright, transparent MHP crystals upto 4 × 2 × 1 mm³ in sizes are grown at room temperature. The dielectric measurements are carried out in the temperature range between 300 to 650 K. The effect of the applied frequency on the dielectric constants and the dielectric losses are investigated. It is observed that the mechanism of the dielectric behaviour is different in the lower and higher frequency and temperature ranges. The activation energy of the oscillators are calculated and found to be 1.105 eV. Attempt is made to draw some qualitative conclusions, taking in view the existing theories of various kinds of polarization, and implications are discussed.     

Crystallization of glaserite from aqueous solutions

Crystallization of glaserite (double salt of potassium and sodium sulphate) from aqueous solutions was studied by determining solubility, metastable zone width, growth and dissolution kinetics. Solubility measurements confirmed the presence of a large range of mother liquor compositions from which the precipitation of glaserite at about 73 – 76 wt% K₂SO₄ is obtained: inside this region the solubility lines resulted curvilinear. The metastability zone is quite large, ranging from 14 to 23 °C, and mainly increases by increasing the Na₂SO₄ content in the mother liquor. The growth rate of glaserite is second order, slightly size dependent and primarily controlled by diffusion as for K₂SO₄ but both growth and dissolution constants are quite lower than those for K₂SO₄.