Size-dependent Growth Kinetics Research Articles

Size-dependent growth kinetics model for potassium chloride from seeded chloride solution

Abstract It was widely known that crystallization kinetics are the basis for crystallization behavior and crystallizer scale-up design. Cooling crystallization of potassium chloride was an essential unit operation in recycling industry. Some researchers have proposed the strategy of adding seed crystals for the intermittent cooling crystallization process to control the particle size and distribution of the target product. This paper studied the complex function relation between particle size and growth rate of KCI in the crystallization process in a continuous mixed-suspension mixed-product-removal (MSMPR) crystallizer at a steady state. Using the crystallization kinetics data, the mathematical models of coupling crystallization were established based on the population balance equations and mass balance equations. Since population density distributions of products behave multiform under different conditions, based on diffusion theory. The growth rate was obtained by a least square method for the multivariate linear regression, and the reliability of the kinetics model was validated experimentally. Fitting results indicated that some classical models, including several size-independent growth models and size-dependent growth models, such as Bransom, C-R, MJ2, ASL, and MJ3, could not model the size-dependent growth accurately. Based on this situation, an exponential growth model was proposed and confirmed to describe the size-dependent growth behavior. It was found that the model parameters have definite meanings and were strongly related to particle size. Compared with the classical models, this model showed good pertinency and adaptability to experimental results when used to describe the population density distribution and the size-dependent growth rate of KCI. This research could provide a theoretical guide for optimizing the crystallization process and designing industrial crystallizers.

Continuous Nucleation and Size Dependent Growth Kinetics of Indium Phosphide Nanocrystals

Aminophosphines derived from N,N′-disubstituted ethylenediamines (R–N(H)CH2CH2N(H)–R; R = ortho-tolyl, phenyl, benzyl, iso-propyl, and n-octyl) were used to adjust the kinetics of InP nanocrystal f...

Size-dependent growth kinetics of struvite crystals in wastewater with calcium ions

AbstractKinetic parameters describing continuous reaction crystallization of struvite from aqueous solutions containing also calcium ions (from 100 to 2000 mg Ca2+/kg) were estimated. Test results were compared with kinetic data of struvite manufactured from real cattle liquid manure. Kinetic model for ideal MSMPR (Mixed Suspension Mixed Product Removal) crystallizer was used assuming dependence of crystal growth rate G on its size L (size-dependent growth, SDG MSMPR model). Based on nonlinear regression and statistical analysis, one from within five considered G(L) models was selected – Rojkowski exponential (RE) model – rendering the experimental population density distributions the best. It was concluded, that calcium ions influenced all components of struvite manufacturing process disadvantageously. A rise of Ca2+ concentration in a feed from 100 to 2000 mg/kg increased nucleation rate ca. 160-time, whereas growth rate of nuclei up to macroscopic size G0 decreased more than 10-time. Linear (larger) crystal growth rate G¥ was nearly two-times smaller: 1.71·10–8 m/s (100 mg Ca2+/kg) – 9.10·10–9 m/s (2000 mg Ca2+/kg). Resulting in a product with deteriorated quality. Mean size of the crystals decreased nearly two-times (to 18.4 μm), non-homogeneity within product population enlarged and calcium fraction in the product increased. The product, beside struvite MgNH4PO4·6H2O, also contained hydrated amorphous calcium phosphate(V) Ca3(PO4)2·nH2O (ACP). It was observed, that 5-times smaller concentration of phosphate(V) ions in a feed and magnesium ions excess in relation to phosphate(V) and ammonium ions (1.2 : 1 : 1) influenced all kinetic parameters of continuous struvite reaction crystallization advantageously.

A computational investigation of attrition-enhanced chiral symmetry breaking in conglomerate crystals

Attrition-enhanced chiral symmetry breaking in crystals, also known as Viedma ripening, is a remarkable phenomenon from a variety of perspectives. By providing a direct route to solid-phase homochirality in a controllable manner, it is of inherent interest to those who study chiral symmetry-breaking/amplification mechanisms. When applied to intrinsically chiral molecules, Viedma ripening may have implications for the origin of biological homochirality, as well as applications in chiral drug resolution. Despite an abundance of research, the mechanistic details underlying this phenomenon have not been unambiguously elucidated. We employ a Monte Carlo algorithm to study this driven system, in order to gain further insights into the mechanisms capable of reproducing key experimental signatures. We provide a comprehensive numerical investigation of how the model parameters (attrition rate, liquid-phase racemization kinetics, and the relative rates of growth and dissolution kinetics) impact the system's overall behavior. It is shown that size-dependent crystal solubility alone is insufficient to reproduce most of the experimental signatures of Viedma ripening, and that some form of a solid-phase chiral feedback mechanism must be invoked in order to reproduce experimentally observed behavior. In this work, such feedback mechanisms can take the form of agglomeration, or of artificial modification of the size dependent growth kinetics.

Size selected growth of nanodots: analytical prediction for the selected size

In size selected growth of metal and semiconductor nanodots the selected mean size is frequently larger than the size corresponding to the minimum of free energy of dots. This “overshooting” effect is studied here using a reaction kinetic model with size dependent growth kinetics in adatom processes and thermodynamical free energy difference of nanodots. We calculate analytical prediction for the selected mean size of the dots using stationary state approximation. It is shown, that the overshooting depends on the skewness of the Gaussian size distribution.

Direct Observation of Nanoscale Size Effects in Ge Semiconductor Nanowire Growth

Progress in the synthesis of semiconductor nanowires (NWs) has prompted intensive inquiry into understanding the science of their growth mechanisms and ultimately the technological applications they promise. We present new results for the size-dependent growth kinetics of Ge NWs and correlate the results with a direct experimental measurement of the Gibbs-Thomson effect, a measured increase in the Ge solute concentration in liquid Au-Ge droplets with decreasing diameter. This nanoscale-dependent effect emerges in vapor-liquid-solid Ge NW growth and leads to a decrease in the NW growth rate for smaller diameter NWs under a wide range of growth conditions with a cutoff in growth at sufficiently small sizes. These effects are described quantitatively by an analytical model based on the Gibbs-Thomson effect. A comprehensive treatment is provided and shown to be consistent with experiment for the effect of NW growth time, temperature, pressure, and doping on the supersaturation of Ge in Au, which determines the growth rate and critical cutoff diameter for NW growth. These results support the universal applicability of the Gibbs-Thomson effect to sub-100 nm diameter semiconductor NW growth.

Combined Quadrature Method of Moments and Method of Characteristics Approach for Efficient Solution of Population Balance Models for Dynamic Modeling and Crystal Size Distribution Control of Crystallization Processes

The paper presents a novel methodology for the estimation of the shape of the crystal size distribution (CSD) during a crystallization process. The approach, based on a combination of the quadrature method of moments (QMOM) and the method of characteristics (MOCH), provides a computationally efficient solution of the population balance equation (PBE) and hence a fast prediction of the dynamic evolution of the CSD for an entire batch. Furthermore, under the assumption that for supersaturation-controlled crystallization the main phenomenon is growth, an analytical CSD estimator is derived for generic size-dependent growth kinetics. These approaches are evaluated for the crystallization of potassium alum in water. The model parameters are identified on the basis of industrial experimental data, obtained using an efficient implementation of supersaturation control. The proposed methods are able to predict and reconstruct the dynamic evolution of the CSD during the batch. The QMOM−MOCH solution approach is eva...

Size-dependent growth kinetics of vitamin C crystals in water solutions of L(+)-ascorbic acid with the addition of methanol and ethanol

Size-dependent growth kinetics of vitamin C crystals in water solutions of L(+)-ascorbic acid with the addition of methanol and ethanol Growth kinetics of vitamin C crystals during the batch mass crystallization process in L(+)-ascorbic acid - methanol - ethanol - water system was determined. The linear growth rate values were estimated on the basis of the product crystal size distributions. The kinetic model of the continuous process in a MSMPR crystallizer was adopted for the batch mode description according to Nyvlt's conception, taking the sizedependent growth (SDG) rate effects into consideration. The kinetic parameter values were determined with a Rojkowski hyperbolic SDG model. A good compatibility between the experimental product crystal population density distributions and the SDG model predictions was observed. The interpretation of the kinetic data was presented and discussed.

Model reduction for prediction of silver halide precipitation

A fully mechanistic model was developed to simulate the turbulent mixing and precipitation of silver bromide in a semi-batch reactor. Instantaneous primary nucleation, multiple species thermodynamic equilibria, size-dependent growth kinetics, crystal population balances and accurate modeling of turbulent flow in a stirred tank are fully coupled in a multi-scale, Eulerian–Lagrangian approach. Several significant advances are achieved in the course of the model development. First, a novel symmetric geometric length scale is developed to allow resolution of the scales of nucleation, growth, and dissolution in a single model. This significantly improves the model's ability to predict the crystal size distribution, and eliminates a severe numerical instability in the population balance equations. Second, the multi-species equilibria are accurately represented by a linear equation over most of the range of interest. Third, nucleation occurs primarily in the feed plume, while growth occurs primarily in the bulk, although the two processes remain coupled through the mixing model and the population balances. Seeded and unseeded precipitation experiments were conducted to allow model validation. A comparison of the model predictions with the experimental crystal size distributions shows excellent agreement. The model is also able to isolate a weakness in the growth model in the Gibbs–Thompson controlled range, between the lower limit of the experimentally available data and the size of the nuclei. The validated model provides several insights into the precipitation process, and is suitable for further examining mixing effects on the precipitation process.

Effects of Bismarck Brown dye on the crystallization kinetics of potassium alum in a CMSMPR crystallizer.

This study examined the effects of an additive, Bismarck Brown dye, on the crystallization kinetics of potassium alum crystals in a well-mixed continuous crystallizer. In the data analysis, size-dependent growth kinetics was assumed. The addition of Bismarck Brown dye depressed crystal growth and nucleation rates. The supersaturation of solution at steady state in the crystallizer increased as the Bismarck Brown concentration increased under the constraint of constant retention time and magma density. In this case the growth rate was enhanced but the nucleation rate was reduced. The predicted kinetic behaviors influenced by the addition of Bismarck Brown dye were in good agreement with the experimental results.

The influence of gradual classification on continuous crystallization

Classification occurs when the mean residence time of a crystal of size L, τ( L), differs from that of the liquid in the crystallizer. The classification function p = τ/τ( L) characterises this difference and, when coupled with the population balance, predicts the crystal size distribution produced in a classifying crystallizer. This is exemplified here for size-dependent growth kinetics as well as when growth dispersion occurs. A contoured-base tank, characterised by a constant cross-section for magma circulation and unusually mild agitation, was operated continuously. In 13 runs potash alum was crystallized under MSMPR conditions, while in 17 runs lower stirrer speeds induced classification of the type τ( L) ≥ No discontinuities in growth and nucleation kinetics were observed. The mean crystal size increased substantially, while the coefficient of variation r only slightly in the classified regime. Classification had to be taken into consideration when predicting the crystal size distribution. The classifica function of potash alum crystals could not be predicted by measuring p for glass beads having the same settling rates. The limiting step in complet mixing the suspensions was lifting particles out of the boundary layer on the tank base.