Investigation of nucleation kinetics and crystal defects of HMX

The nucleation kinetics of HMX (cyclotetramethylene tetranitramine, C₄H₈N₈O₈) in γ‐butyrolactone was studied in cooling process by induction time method. The laser scattering method was used to measure the solubility data and metastable region of HMX in γ‐butyrolactone. The induction time was measured over a range of supersaturation at different temperatures. Then, the nucleation mechanism of HMX in γ‐butyrolactone was investigated by analysis the relationships between induction time and supersaturation. The results indicated homogeneous nucleation dominated at high supersaturation of S >1.35, while the heterogeneous nucleation dominated at low supersaturation of S < 1.35. The values of interfacial tension at different final temperatures were calculated to indicate the ability of HMX to be crystallized. The growth mechanism of HMX was investigated by the data fitting applying different growth mechanism models and identified as two‐dimensional nucleation‐mediated (2D) growth. Finally, the effects of supersaturation and temperature on the crystal defects were analyzed based on the nucleation kinetics. When the temperature is below 303.15K, homogeneous nucleation dominated the nucleation process at higher supersaturation. Fine HMX crystals with more defects were produced. On the contrary, heterogeneous nucleation mechanism dominated at lower supersaturation. large regular HMX crystals with fewer defects were formed when the temperature is above 318.15K.     

Kinetics of (R,S)- and (R)-mandelic acid in an unseeded cooling batch crystallizer

The objective of this work is to determine the nucleation and growth kinetics of (R,S)-mandelic acid ((R,S)-MA) and (R)-mandelic acid ((R)-MA) in aqueous solutions using an unseeded cooling crystallization process. To obtain the nucleation and growth kinetics, the solubility, metastable zone limits, and supersaturation were measured by in-situ attenuated total reflection Fourier transform infrared (ATR-FTIR) spectroscopy and focused beam reflectance measurement (FBRM). The nucleation rate and growth rate parameters were determined by a nonlinear optimization algorithm. The effects of initial concentration and cooling rate on supersaturation and the nucleation rate are also discussed.     

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 kinetics on the (100) and (001) faces of TGS crystals

The growth kinetics and mechanisms on the (001) and (100) faces of TGS crystals were investigated. A phase contrast microscope with a CCD camera was used to observe the growth of the crystal. We found the growth on the (001) and (100) faces at high supersaturation was mainly controlled by a BCF surface diffusion mechanism. The kinetic data for the (100) face were also fitted by the nucleation and layer growth model of two-dimension nucleation at high supersaturation. Some important growth parameters for TGS crystals, such as edge energy, activation energy, and so on, were estimated.     

Study of phosphoric acid crystallization using a focused beam reflectance measurement method

A way for restoring the crystal size distributions (CSD) from measured chord length distributions (CLD) was reported in this paper. The kinetics of phosphoric acid crystallization process was investigated in cooling mode using focused beam reflectance measurement (FBRM) and digital photo technique. In order to restore the CSD from measured CLD and verify the reliability of FBRM data, digital photo technique in real time and optical microscope were applied in large crystal size and small range, respectively. Results indicated a converting constant A existed between CLD and CSD when crystal growth follows size‐independent growth (Mcabe's ΔL law) law. It was verified by Malvern particles size analysis method. The converting constant A varied with crystal morphology. The crystal growth order increased with the stirring increasing speed during phosphoric acid crystallization process. The trend was especially notable at higher speed situations. It can illustrate that the state of phosphoric acid hemihydrate crystal growth was controlled by both diffusion and surface‐integration with the increasing stirring speed. (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nuleation and Growth of Thin Ag Films Vacuum-deposited onto W

An attempt to analyse the nucleation and growth phenomenon in a typical “Stranski-Krastanov” (SK) growth system - Ag/W (PAUNOV , MICHAILOV 1979; MICHAILOV ), in terms of island growth theory was made. Ag-layers were analysed by the use of scanning electron-and optical microscopes. A strong dependence of the nucleation kinetics on supersaturation and grain orientation was found to exist. On the basis of known rate equations, the number of atoms in the critical cluster i and the value of nucleation potential barrier E were estimated. The comparison between the experimentally obtained size-, first nearest neighbour- and radial distributions and theory shows that: 1) nucleation and growth processes proceed through a surface diffusion of the atoms; 2) no coalescence between the crystallites and no crystal mobility exist.     

Evaluation of nucleation rate by in‐situ focused beam reflectance measurement in an unseeded batch cooling crystallization

Nucleation kinetics in the cooling crystallization of hexahydro‐1,3,5‐trinitro‐1,3,5‐triazine (RDX) from γ‐butyrolactone was studied by converting total counts/s measured by in situ focused beam reflectance measurement (FBRM) into number of crystals. The classical nucleation rate model, which is derived exclusively from the thermodynamic state for the nucleation and molecular collision frequency, was found to be inadequate to describe the experimentally measured nucleation rates. However, the nucleation rates predicted by the modified classical nucleation rate model, inclusive of an additional temperature term, were found to be in good agreement with those measured in the present work. Furthermore, the metastable zone widths are also found to be more accurately predicted by the modified classical nucleation rate model than the classical approach, which assumes that the mass‐based nucleation rate is an exponential function of supersaturation and is equal to the supersaturation rate.     

Metastability and crystal growth kinetics on L‐arginine phosphate

Stability of saturated L‐Arginine Phosphate (LAP) solution studied as a function of supercooling rate and crystal growth kinetics investigated as a function of supersaturation are reported in this communication. Solution stability was studied by observing the metastable zone width at different cooling rates employing a polythermal method. Analysis of the experimental data yielded the kinetic constant of nucleation and the order of nucleation. Crystal growth rates studied on small seed crystals with regular morphology, under normal growth conditions and at different supersaturation levels were found to satisfy BCF surface diffusion model. Crystal growth rates were investigated normal to the (100), (010) and (00 ) faces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Advancements (and challenges) in the study of protein crystal nucleation and growth; thermodynamic and kinetic explanations and comparison with small-molecule crystallization

This paper reviews advancements and some novel ideas (not yet covered by reviews and monographs) concerning thermodynamics and kinetics of protein crystal nucleation and growth, as well as some outcomes resulting therefrom. By accounting the role of physical and biochemical factors, the paper aims to present a comprehensive (rather than complete) review of recent studies and efforts to elucidate the protein crystallization process. Thermodynamic rules that govern both protein and small-molecule crystallization are considered firstly. The thermodynamically substantiated EBDE method (meaning equilibration between the cohesive energy which maintains the integrity of a crystalline cluster and the destructive energies tending to tear-up it) determines the supersaturation dependent size of stable nuclei (i.e., nuclei that are doomed to grow). The size of the stable nucleus is worth-considering because it is exactly related to the size of the critical crystal nucleus, and permits calculation of the latter. Besides, merely stable nuclei grow to visible crystals, and are detected experimentally. EBDE is applied for considering protein crystal nucleation in pores and hydrophobicity assisted protein crystallization. The logistic functional kinetics of nucleation (expressed as nuclei number density vs. nucleation time) explains quantitatively important aspects of the crystallization process, such as supersaturation dependence of crystal nuclei number density at fixed nucleation time and crystal size distribution (CSD) resulting from batch crystallization. It is shown that the CSD is instigated by the crystal nucleation stage, which produces an ogee-curve shaped CSD vs. crystal birth moments. Experimental results confirm both the logistic functional nucleation kinetics and the calculated CSD. And even though Ostwald ripening modifies the latter (because the smallest crystals dissolve rendering material for the growth of larger crystals), CSD during this terminal crystallization stage retains some traces of the CSD shape inherited from the nucleation stage. Another objective of this paper is to point-out some biochemical aspects of the protein crystallization, such as bond selection mechanism (BSM) of protein crystal nucleation and growth and the effect of electric fields exerted on the process. Finally, an in-silico study on crystal polymorph selection is reviewed.     

On the kinetic of crystallization of zinc oxalate (II). Determination of the specific surface energy at the crystal solution interface by kinetic studies

An apparatus based on turbidimetry was constructed for studying the kinetics of crystallization of sparingly soluble salts. The kinetics of crystallization of zinc oxalate was investigated at various supersaturations and pHs of the medium. The specific surface energy at the crystal-solution interface was determined from the induction periods of the S-shaped curves by means of the classical theory of nucleation. The calculations were made using the slope of the logarithmic dependence of I or τ, respectively, on supersaturation (equations 1 and 2) and the critical supersaturation which was also determined on the basis of kinetic data. Various values of σ were found for the different supersaturations. Different values for this quantity were also obtained when supersaturation was presented by the concentration ratio (S = C/C₀) or by the ratio of the product of the two ions concentrations in the supersaturated solution to that in the saturated one (S = ab/L_p). These σ values were lower than those obtained using the critical supersaturations at the two different solution pHs. For the present it is impossible to give a definite explanation of the results obtained. The experiments for determining the specific surface energy at the crystal-solution interface will be continued.     

Effect of supersaturation on the crystallization of phenylbutazone polymorphs

The article describes the effect of degree of supersaturation, σ, on the crystallization of specific polymorphs of phenylbutazone from its methanolic solution at 20 °C. At low initial supersaturation, σ ≤ 2.0, the fraction of the metastable α polymorph in the crystallized product exceeds that of the δ polymorph, while at σ ≥ 5.0, the fraction of the stable δ polymorph increases in the crystallized product. The results are explained by the effect of supersaturation on the relative rates of nucleation and crystal growth of the polymorphs. Furthermore, the mechanism of nucleation and crystal growth also change with supersaturation. Supersaturated methanolic solutions of phenylbutazone exhibit a critical temperature at which the nucleation rates of the polymorphs decrease drastically. This effect is partly explained by the decreased mobility of phenylbutazone molecules at lower temperatures. Nucleation is most rapid when the crystallization temperature is close to the transition temperature, T_t(α ⟷ δ), between the polymorphs, α and δ. The nucleation rate decreases as the temperature difference between T_t(α ⟷ δ) and the crystallization temperature increases. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nucleation Studies in Supersaturated Triglycine Sulphate Solutions

Rates of nucleation of supersaturated aqueous solutions of Triglycine sulphate are measured by the induction period over the temperature range 30–50 °C. The effects of temperature and supersaturation on the induction period are reported. The interfacial tension, energy of formation and radius of the critical nuclei of the crystal are calculated in accordance with classical nucleation theory. Nucleation rate increases with increase in supersaturation and temperature, while interfacial tension decreases with increase in temperature.     

Spatiotemporal protein crystal growth studies using microfluidic silicon devices

Fundamental investigations of protein crystallization using miniaturized microfluidic silicon devices were presented towards achieving spatiotemporal nucleation and subsequent post-nucleation growth. The developed microfluidic silicon device was typically composed of crystal growth cell, reservoir cell, and optionally of heater elements for supersaturation control. A specific fine pattern area in the growth cell which was fabricated on the silicon substrate with doped p- and n-type silicon layers, served as spatially selective nucleation site of dissolved protein molecules through electrostatic attractive force. In a model material, hen egg white lysozyme, a large number of crystals were grown on the defined nucleation site evenly spaced from each other, whereas parasitic crystal growth positioned around the selective nucleation site, was suppressed by the effects of electrostatic repulsive force between the doped silicon surface and charged protein molecules. A possible crystallization mechanism of describing the heterogeneous nucleation during the initial stage and during the growth of the crystal at the electrolyte–semiconductor silicon surface is proposed and discussed.     

The mechanism of the formation of enantiotropic polymorphs of carisbamate in solution crystallization

The nucleation kinetics, as a function of supersaturation level, was studied for carisbamate, a polymorphic crystalline compound, in methanol, ethanol, 2-propanol and water. The induction times in nucleation kinetics varied markedly with respect to relative supersaturation in the range 1.3–2.3. At the same relative supersaturation, the induction time for carisbamate in methanol is the shortest, increasing in order from ethanol, 2-propanol, and then water. The interfacial tensions γ between carisbamate and methanol, ethanol, 2-propanol, and water were estimated from their induction times based on nucleation theory and were found to be about 3.6, 4.1, 4.5, and 5.7 mJ/m², respectively. These values were of same order of magnitude as those obtained from solubility data. The equation that displays the influence of interfacial tension, supersaturation and temperature on crystallization kinetics was derived, and found to be consistent with experimental observations. The mechanism of enantiotropic polymorphism for carisbamate in the solvents is illustrated. Using interfacial tension values determined for single solvents, the polymorphic form resulting from crystallization in mixed and pure solvent systems could be predicted with good accuracy.     

Part 1: Kinetics and mechanism of the crystallization process

The kinetics of spontaneous crystallization of sodium chloride from aqueous‐ethanol solutions were studied. During the crystallization the electrical conductance and optical transmission of the supersaturated solutions were measured automatically. For monitoring of the total surface of growing potassium chloride crystals at the crystallization the turbidimetric method was used. The growth rate and activation energy were determined. The crystal growth rate was proportional to supersaturation. When the volume fraction of ethanol in solution increased from 14.85 to 29.72%, the activation energy of the growth process did not change and was about 50 kJ· mol^‐1. Aggregation of the crystals was found. The aggregation kinetics of the crystals may be described approximately by the famous Smoluchowski equation for coagulation of colloidal particles. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kinetics of Mixed Crystal K2(SO4, CrO4) Growth from Aqueous Solution

The investigation of crystal growth kinetics and phase equilibria in the system K₂SO₄–K₂CrO₄–H₂O was carried out using the method for the determination of the saturation temperature and the growth rate by microscopic observation of the seed crystal behaviour. The kinetics of mixed crystal growth can be characterized by the expression V = 450.24σ^1,6,2, where σ is the value of effective supersaturation of the solution calculated by taking into account the specific features of the phase equilibria diagram. Use of this criterion allows the control of the rate of delta crystal growth from solutions of variable composition.     

In situ AFM investigation of 2D nucleation on the (100) face of KDP

Surface morphology of the (100) face of potassium dihydrogen phosphate (KDP) crystals which were grown at different supersaturations at 25 °C was investigated by in situ atomic force microscopy (AFM). Various AFM images of 2D nucleation under different growth conditions were presented. It is found that the growth of KDP is controlled by polynuclear nucleation mechanism at the high supersaturation. With reduction of the supersaturation, the growth velocity of 2D nuclei becomes very slow and shows typical anisotropy. It is found that the process of coalescence of 2D nuclei does not lead to defect. The experiments show that the growth mechanism for KDP at 25 °C changes between step flow and 2D nucleation in the supersaturation range of 4.5‐5%. The triangular nuclei which are close to equilateral triangle are observed in the experiment at the supersaturation σ = 6% for the first time, showing typical anisotropic growth. Through observing the dissolution of 2D nuclei, the dissolving process can be regarded as the reverse process of growth. We also find that the microcrystals landing on the surface at σ = 9% would grow and coalesce with each other and there is no observable defect in the coalescence. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Stochastic Model for Nucleation Kinetics Determination in Droplet-Based Microfluidic Systems

The measured induction times in droplet-based microfluidic systems are stochastic and are not described by the deterministic population balances or moment equations commonly used to model the crystallization of amino acids, proteins, and active pharmaceutical ingredients. A stochastic model in the form of a Master equation is formulated for crystal nucleation in droplet-based microfluidic systems for any form of nucleation rate expression under conditions of time-varying supersaturation. An analytical solution is provided to describe the (1) time evolution of the probability of crystal nucleation, (2) the average number of crystals that will form at time t for a large number of droplets, (3) the induction time distribution, and (4) the mean, most likely, and median induction times. These expressions are used to develop methods for determining the nucleation kinetics. Nucleation kinetics are determined from induction times measured for paracetamol and lysozyme at high supersaturation in an evaporation-based high-throughput crystallization platform, which give low prediction errors when the nucleation kinetics were used to predict induction times for other experimental conditions. The proposed stochastic model is relevant to homogeneous and heterogeneous crystal nucleation in a wide range of droplet-based and microfluidic crystallization platforms.     

晶体生长机制和生长动力学的蒙特卡罗模拟研究

采用动力学蒙特卡罗方法,对在完整光滑界面上低过饱和度溶液中的晶体生长机制和动态过程进行计算机模拟,得到了晶体生长速率与溶液过饱和度之间的关系以及晶体生长的表面形态.对以二维成核为主要生长机制的动力学生长规律进行分析,发现了二维成核生长的生长死区以及单核生长转变为多核生长时的过饱和度临界值,讨论了热粗糙度、表面扩散、台阶平均高度以及表面尺寸对晶体平均生长速率的影响.     

Understanding supersaturation‐dependent crystal growth of L‐alanine in aqueous solution

The dependence of crystal growth rate of L‐alanine on solution supersaturation was investigated by combining experiments and molecular dynamics (MD) simulations. The experimental results show that lower supersaturated solution yields more elongated L‐alanine crystals along the c‐axis, i.e., the aspect ratio (c/b) of the crystal decreases with the increase of solution supersaturation, which is due to the higher supersaturation inducing a rise in the relative growth rate between the main side surface (the (120) surface) and the main end surface (the (011) surface). MD simulations on the two surfaces in contact with different supersaturated solutions revealed that the solute molecules tend to be more efficiently attached to the (011) surface than to the (120) surface at both supersaturations studied, as the interaction between the solute molecules and the L‐alanine molecules in the first layer of the (011) surface is stronger than that of the (120) surface. However, higher supersaturation leads to larger relative interaction energy between the (120) and (011) surfaces, suggesting an increase in the relative growth rate of the two surfaces (R₍₁₂₀₎/R₍₀₁₁₎) with supersaturation, which is in agreement with the experimental results.