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.     

Main features of nucleation in model solutions of blood plasma

The regularities of nucleation in a model solution of human blood plasma under the conditions similar to physiological have been investigated. The induction order and constants are determined. It is shown that an increase in supersaturation leads to a transition from heterogeneous to homogeneous nucleation of crystallites. The critical nucleus size is estimated for a pure model system and for a system containing a number of additives. The impurities under study are found to form the following descending sequence with respect to their effect on nucleation: alanine > glucose > glycine > citric acid > milky acid > magnesium ions.

     

Isothermal nanocrystallization kinetics during polymorphic transformation

The experimental value of Avrami exponent during polymorphic nanocrystallization deviates from the theoretical prediction of the Kolmogorov-Johnson-Mehl-Avrami (KJMA) isothermal kinetics. The assumption of negligible nuclei volume, implicit in KJMA, is violated for all such transformations involving nano-sized grains. A discrete adaptation of the KJMA model is used to study the effect of the non-zero nuclei volume on nanocrystallization kinetics. Transformations initiated by homogeneous and heterogeneous nucleation are analyzed under constant nucleation and growth rate conditions. For homogeneous nucleation, a universal scaling relation with respect to finite nuclei size is found. In contrast to this, for the heterogeneous case a series of linear scaling relations are observed with respect to the nucleation conditions.     

Protein crystallization in low gravity by step gradient diffusion method.

Two-step crystallization experiments were conducted in low gravity employing a liquid-liquid diffusion method in an effort to eliminate problems associated with protein crystal growth under the supersaturating conditions required for nucleation. Experiments were performed in diffusion cells formed by the sliding of blocks on orbit. Step gradient diffusion experiments consisted of first exposing protein solutions in diffusion half-wells for brief periods to initiating buffer solutions of high precipitant concentrations to induce nucleation followed by expoure of the same protein solutions to solutions of lower precepitant concentration to promote growth of induced nuclei into crystals. To avoid convective disturbances that occur when solutions of discrepant densities are interfaced at normal gravity, crystallization of hen egg-white lysozyme and rabbit skeletal muscle aldolase by step gradient diffusion was investigated in low gravity on four NASA space shuttle flights. In general, the largest ctystals of both proteins formed at the highest initiating precipitant concentration used, which is consistent with nuclei formation upon brief exposure to high precipitant concentration, and that these nuclei are competent for sustained growth at lower precipitant concentration. The two-step approach dissociates nucleation events from crystal growth allowing parameters affecting nucleation kinetics such as time, precipitant concentration and temperature of nucleation to be varied separately from conditions used for post-nucleation growth.     

Improvement of Initial Conditions of the Crystal Growth by Dissolution and Refaceting

The influence of significant dissolution and refaceting of KDP and Rochelle salt crystals on their further growth is studied. It is shown that the crystal growth rate ‐ initial crystal size dependence is better defined for growth after refaceting. Initial conditions for growth of individual crystals after refaceting are better defined, than in the case of classical seed nucleation. By dissolution and refaceting, it is easy to provide the same initial conditions for crystal growth in each of several experiments, preformed under the same macroscopic external conditions.     

A study of primary nucleation of calcium oxalate monohydrate: II. Effect of urinary species

Kidney stones consist of various organic and inorganic compounds. Calcium oxalate monohydrate (COM) is the main inorganic constituent of kidney stones. However, the mechanisms for the formation of calcium oxalate kidney stones are not well understood. In this regard, there are several hypotheses including nucleation, crystal growth and/or aggregation of formed COM crystals. The effect of some urinary species such as oxalate, calcium, citrate, and protein on nucleation and crystallization characteristics of COM is determined by measuring the weight of formed crystals and their size distributions under different chemical conditions, which simulate the urinary environment. Statistical experimental designs are used to determine the interaction effects among various factors. The data clearly show that oxalate and calcium promote nucleation and crystallization of COM. This is attributed to formation of a thermodynamically stable calcium oxalate monohydrate resulting from supersaturation. Citrate, however, inhibits nucleation and further crystal growth. These results are explained on the basis of the high affinity of citrate to combine with calcium to form soluble calcium citrate complexes. Thus, citrate competes with oxalate ion for binding to calcium cations. These conditions decrease the amount of free calcium ions available to form calcium oxalate crystals. In case of protein (mucin), however, the results suggest that no significant effect could be measured of mucin on nucleation and crystal growth. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optical investigation of the self-organized growth of InAs/GaAs quantum boxes

By studying the optical properties of highly strained InAs/GaAs multilayers as a function of the deposited quantity of InAs, a high resolution probing of the change from two-dimensional to three-dimensional morphology of the InAs layers has been performed. We show that the critical thickness for the onset of three-dimensional growth is very well defined for given growth conditions. The nucleation of the islands is, however, asynchronous, due to the spatial fluctuations of the deposited quantity of InAs. A fast initial growth of the InAs islands leads within a few seconds to a quasi-equilibrium morphology for InAs. Asynchronous nucleation and fast initial growth combined are shown to be the major origin of the size fluctuations of InAs quantum boxes in GaAs obtained by self-organized growth under standard molecular beam epitaxy (MBE) growth conditions.     

Avrami exponent under transient and heterogeneous nucleation transformation conditions

The Kolmogorov-Johnson-Mehl-Avrami model for isothermal transformation kinetics is universal under specific assumptions. However, the experimental Avrami exponent deviates from the universal value. In this context, we study the effect of transient heterogeneous nucleation on the Avrami exponent for bulk materials and also for transformations leading to nanostructured materials. All transformations are assumed to be polymorphic. A discrete version of the KJMA model is modified for this purpose. Scaling relations for transformations under different conditions are reported.     

EDTA和KCl双掺杂下KDP溶液的稳定性研究

研究了EDTA与KCl不同掺杂浓度和不同过饱和比下KDP溶液的成核过程,测定了不同条件下KDP过饱和溶液的诱导期;根据经典成核理论计算了成核热、动力学参数,并分析了溶液稳定性随掺杂浓度的变化情况。利用化学腐蚀法对KDP晶体(100)面进行了腐蚀,得到了清晰的位错蚀坑,并使用光学显微镜观察了(100)面位错蚀坑的分布特点。结果表明,当过饱和度为4%、掺杂浓度为0.01 mol%EDTA和1 mol%KCl时,不仅KDP过饱和溶液的稳定性比较高,而且位错蚀坑的分布比较均匀、密度小,适合高质量的KDP晶体生长。     

Nucleation kinetics at strong supersaturation

Nucleation in an arbitrary d-dimensional system is considered. Two nucleation regimes are predicted within an original kinetic approach. The first regime (for a weak supersaturation) is characterized by the formation of a critical nucleus (critical nucleus regime). The second regime (for a strong supersaturation) corresponds to an uncontrollable growth of a cluster of any size (runaway nucleation regime). It is established that the nucleation regime is controlled by the metastability parameter M of the system under consideration. The analytical properties of the free energy of the nucleation system are investigated. It is demonstrated that a change in the parameter M leads to a considerable transformation of the nucleation barrier topography. The effect of the nucleation barrier height on the kinetics of nonstationary nucleation is described.     

Oriented growth of mullite from a glass melt using electrochemical nucleation

A dc-potential of 1.5 V between a platinum wire and a platinum crucible containing a magnesia–titania alumosilicate melt is sufficient to cause nucleation and subsequent crystal growth of mullite at the cathode at a temperature high enough to prevent nucleation under normal conditions. This crystal growth is fast in the crystallographic (0 0 1)-direction and leads to a glass–ceramic with a highly oriented microstructure. The mechanism of this process is discussed as well as the crystallization behavior under normal conditions.     

Nucleation kinetics of selenium (+4) precipitation from an acidic copper sulphate solution

The removal of selenium from copper sulphate solution prior to the electrowinning of copper is desirable in order to minimise contamination of the copper cathodes by selenium and other impurities. The selenium removal is effected by a precipitation process that takes place under high supersaturation conditions, which favour nucleation over any other particle formation processes. There is currently no fundamental information on the nucleation kinetics of this important process. In this study, the nucleation kinetics of selenium (+4) precipitation from an acidic copper sulphate solution was determined using the classical nucleation theory (CNT). Experiments were carried out by varying the levels of supersaturation from 8.66×10¹⁵ to 4.33×10¹⁷ at a temperature of 95 °C under atmospheric pressure. The nucleation rates for four different levels of supersaturation, the nucleation work and the nucleus size were determined. The kinetic constant A was found to be 3.92×10²⁷ m⁻³ s⁻¹, which shows that the nucleation process takes place through a homogeneous mechanism. The associated thermodynamic parameter (B) was determined to be 8.98×10⁰⁴.     

Determination of activation energies for nucleation and growth from isothermal differential scanning calorimetry data

Differential scanning calorimetry (DSC) is usually adopted to analyze solid-state phase transformation incorporating nucleation, growth and impingement. Then, for isothermal transformation, time-dependent Avrami exponent and overall effective activation energy can always be deduced using recipes, which are derived from an analytical phase transformation model. On this basis, a concise and reliable approach to determine time-independent activation energies for nucleation and growth is proposed. Numerical calculations have demonstrated that the new approach is sufficiently precise under different conditions of transformation (e.g. nucleation: mixed nucleation and Avrami nucleation; growth: interface-controlled growth and diffusion-controlled growth; impingement: randomly nuclei dispersed, anisotropic growth and non-random nuclei distributions). Application of the approach in crystallization of Zr₅₅Cu₃₀Al₁₀Ni₅, Zr₅₀Al₁₀Ni₄₀ and Cu₄₆Zr₄₅Al₇Y₂ bulk amorphous alloys as measured by isothermal DSC was performed.     

Crystallization of Silver — Germanium Alloys under Microgravity II. Primary Crystallization

Results of a study on primary crystallization in hypoeutectic and hypereutectic alloys under different convection conditions during solidification are presented. The conditions of suppressed buoyancy driven convection are assumed in the microgravity samples. The intensity of convection was shown to affect the resulting microstructures significantly. The number and the size of primary dendrites in both hypoeutectic samples was found to be comparable. In the microgravity processed hypoeutectic sample the primary dendrites exhibit a greater tendency to branch compared with the ground-based sample. In eutectic samples, a small amount of primary solid solution, due to the existence of an asymmetric coupled zone, was observed. In hypereutectic samples the number of primary germanium crystals is higher in the space sample. The influence of convection on the nucleation of primary crystals thus depends on the composition of such samples. In the case of the primary solid solution the effect of convection on the growth stage of crystallization appears. On the other hand, in the case of primary germanium the intensity of convection affects mainly the nucleation stage of crystallization. The influence of convection on the growth is only secondary. The reasons for such a difference are based on the transport requirements during the nucleation stage.     

Study of film crystallization kinetics initial stage by Monte Carlo simulation

The Monte Carlo Simulation has been used for studying the initial stage characteristics of film crystallization during the layer-by-layer film growth by two-dimensional nucleation. It is shown that the non-monotonous changes of the growing film perimeter and mass correspond to the reflection intensity oscillation from the growth surface. The observed changes of the overgrowing part of the surface are in agreement with the probability-statistical calculations. The vizualization of surface relief changes during the growth process under different conditions has been provided.     

Surface growth mechanisms and structural faulting in the growth of large single and spherulitic titanosilicate ETS-4 crystals

Morphological, surface and crystallographic analyses of titanosilicate ETS-4 products, with diverse habits ranging from spherulitic particles composed of submicron crystallites to large single crystals, are presented. Pole figures revealed that crystal surfaces with a-, b- and c- axes corresponded to 110, 010 and 001 directions, respectively. Thus, technologically important 8-membered ring pores and titania chains in ETS-4 run along the b-axis of single crystals and terminate at the smallest crystal face. Height of the spiral growth steps observed on 1 0 0 and 0 0 1 surfaces corresponded to the interplanar spacings associated with their crystallographic orientation, and is equivalent to the thickness of building units that form the ETS-4 framework. Data suggest that the more viscous synthesis mixtures, with a large driving force for growth, increased the two- and three-dimensional nucleation, while limiting the transport of nutrients to the growth surface. These conditions increase the tendency for stacking fault formation on 1 0 0 surfaces and small angle branching, which eventually results in spherulitic growth. The growth of high quality ETS-4 single crystals (from less viscous synthesis mixtures) occurred at lower surface nucleation rates. Data suggest that these high quality, large crystals grew due to one-dimensional nucleation at spiral hillocks, and indicate that under these conditions un-faulted growth is preferred.     

A statistical understanding of nucleation

In order to study a stochastic phenomenon such as nucleation it is necessary to collect a large enough set of nucleation data to obtain nucleation statistics. This is done by performing nucleation experiments with the same solution under exactly the same conditions many times N (N∼150–300). Such an experiment, based on simultaneous levitation of N∼150–300 identical microdroplets (1–20 μm in diameter) of supersaturated solutions in a solvent atmosphere, is possible by employing the linear quadrupole electrodynamic levitator trap (LQELT). The LQELT is supplemented with a special optical system which is based on scattering of monochromatic polarized light. This will enable fast observation of nucleation and, thus, induction times in each of the levitated microdroplets. The N different induction times, counted from the moment t₀ at which supersaturation is established are recorded. This data provides nucleation statistics (induction time statistics). The numerical and analytical studies of nucleation statistics and parameters provide an insight into statistical properties of the underlying nucleation phenomenon.     

Growth and Kinetic Studies of Spherulitic Gadolinium Tartrate Crystals from Silica Gel

Spherulitic growth of gadolinium tartrate dihydrate crystals by controlled diffusion in silica gel is reported. The influence of growth parameters e.g., reactant concentration, gel pH and gel ageing, on the size, quality and nucleation density of the spherulites has been studied. It is observed that under varying experimental conditions involving change in reactant concentrations, gel pH and gel ageing, the material crystallizes in the form of spherulites. The results of growth kinetics are discussed. Scanning electron microscopy confirms spherulites to be aggregates of single crystallites, with each micro-sized crystal exhibiting a cuboid morphology. Surface structures exhibited by the micro-sized crystals indicate their growth by two-dimensional nucleation mechanism.     

Study on crystallization kinetics and the crystal internal defects of HMX

In this paper, the solubility data of HMX (1,3,5,7‐tetranitro‐1,3,5,7‐tetrazocane) in acetone from 323.15 K to 293.15 K were accurately measured by use of the laser‐monitoring observation technique. Intermittent dynamic method was utilized to study crystallization kinetics of HMX in acetone. The data of crystallization kinetics were obtained by moment analysis, and the parameters of the growth rate and nucleation rate equations were derived by using multiple linear least squares method. Subsequently, growth rate and nucleation rate at different conditions were calculated according to these equations. In addition, Optical Microscopy Images (qualitative) and Particle Apparent Density (quantitative) experiments were applied to study the crystal internal defects of HMX under different crystallization conditions. It can be found that the crystal apparent density of HMX is in the range of 1.8993 g·cm⁻³ to 1.9017 g·cm⁻³, very close to the theory density of HMX; the internal defects and the crystal size do not increase after 25 °C, from which we predict that the HMX crystal growth reaches the steady growth segment. These results suggest that the nucleation rate is a significant factor influencing the crystal internal defects, and larger nucleation kinetics can reduce crystal internal defects.     

Simulated effects of transient nucleation on the crystallization of glass samples

The kinetics of crystallization have been re-examined by taking into account the transient nucleation time and numerical methods were used to obtain the dependence of fraction crystallized on temperature, heating rate and nucleation conditions (temperature and time). These solutions were used to analyze the applicability of approximate models which were based on the assumption of steady state nucleation. The transient nucleation time, which precedes nucleation in as quenched samples, was taken into account to re-examine the deviations expected for relatively short nucleation time, and/or for cases when the actual nucleation temperature is displaced from the corresponding nucleation peak. Additional solutions were computed to obtain the dependence of crystallization peak temperature on the nucleation temperature. It was found that the crystallization peak temperature reaches its lowest value after nucleation at a temperature which can be significantly higher than the peak temperature of steady state nucleation.