Cooling crystallization from supersaturated solution is the crystallization method most frequently employed in the pharmaceutical industry. Nucleation represents the ﬁ rst step of the entire crystallization process, in which molecules arrange themselves in patterns characteristic of a crystalline solid, forming sites wherein additional particles attach and grow into crystals. It is well established that nucleation is a stochastic phenomenon, so that predicting deterministically where nucleation events will take place within a given volume is impossible. 1 For this reason, we may adopt two strategies to determine nucleation kinetics. We can work with a single, large volume, which owing to its size behaves deterministically, or with a large number of small, noninteracting volumes, which owing to their size behave stochastically. In the ﬁ rst case, at least in theory, one experiment su ﬃ ces for deriving nucleation rates, 2 while in the second case, to obtain the kinetics, one needs to consider the results of a large set of statistically independent, small-volume experiments. 3 In spite of the advantages that the deterministic approach o ﬀ ers in terms of (simple) experimental setup, this method makes it hard to operate isothermally under uniform ﬂuid dynamic conditions (owing to the large dimensions of the setup), rendering the system di ﬃ cult to operate, control, and analyze.
An effort to investigate the nucleation kinetics and optical constants of KCl doped (10 mol%) KAP crystal is done. Since nucleation is affected by width of the metastable zone, it is essential to measure it for designing products by crystallization processes. It is possible to obtain optimum crystallization processes by tuning the metastable zone width and actual operation point of the crystallizer within this zone . The findings are ex- pected to provide valuable information for designing optoelectronics devices intended for NLO applications.
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Data associated with the solubility determination is summarized in Fig. 1. Fig. 1(a) shows the dissolution temperatures of L-isoleucine in water determined at different rates, which shows that the heating rates applied to the solution does not give significant effect to the dissolution temperature. This result justifies the choice of heating rate of 0.7 °C/ min in determining other dissolution temperatures for concentrations of 21, 23, 25, 30, 35 g/L in Fig. 1(b). Fig. 1(b) shows the plot of the crystallization temperature, gravimetric solubility and dissolution so- lubility obtained from this study, together with the illustration of data retrieval for MSZW and supersaturation ratio, S = C/C* (detail equa- tions can be found in Supplementary Material, section S5). The results show that the gravimetric solubility is substantially higher than dis- solution solubility. Both data were extensively used by many previous researchers in describing nucleation kinetics and supersaturations of crystallization process [7,11–13,16,17] and in this paper, the impact of using these data, without validity assessment were discussed. The use of these data for MSZW calculation at infinitesimal slow cooling rate (≈0 °C/min) and at 0.7 °C/min were shown in Fig. 1(d). The gravi- metric solubility was determined without the effect of heating rates, but for the calculation of this data, the MSZW for gravimetric solubility was determined as the difference between the gravimetric solubility and the crystallization temperature obtained at the respective rates. The result in Fig. 1(d) shows that the use of gravimetric solution for MSZW cal- culation was underestimated as the MSZW calculated from dissolution solubility (the actual solubility) is always higher. The effect is also apparent for the calculation of supersaturation ratio, in which its value will be consistently underestimated, if calculated using gravimetric solubility. Fig. 1(c) presented in this paper, which was reproduced from Anuar et al.  depicts the change of MSZW with cooling/heating rates and concentration. The MSZW is highly dependent on the concentration and rates, and the MSZW used in Fig. 1(d) was obtained from Fig. 1(c). The MSZW at cooling/heating rate, b ≈ 0 °C/min is the intercepts of these lines and the y-axis. Fig. 1(e) shows the van’t Hoff plots of this solubility data in comparison to that expected from ideal solubility behaviour.
In addition to ﬂ uid shear, exposure of solutions to a solid surface such as the walls of the glass Couette cell is known to have an in ﬂ uence on nucleation. 1,12 It is widely claimed that true homogeneous nucleation is uncommon, so nucleation that occurs heterogeneously is most likely. 1 The overall free energy barrier for heterogeneous nucleation is lower than the corresponding free energy barrier for homogeneous nucleation, and the overall free energy barrier for heterogeneous nucleation decreases with increasing surface wettability. 13 The e ﬀ ect of interfacial area had not been explored in our previous research. In the present work, supersaturated glycine solutions were also exposed to a range of di ﬀ erent glass − liquid interfacial areas to investigate the eﬀect on nucleation kinetics. The results of repeated experiments allowed a statistical analysis to be performed, and the possibility of the measured induction times scaling with shear rate and surface area was investigated. The second element of this paper involves gaining a better understanding of the mechanism by which nucleation takes place. Classical nucleation theory has some well-known shortcomings, and experimental and theoretical work has led to the proposal of alternative theories, including a two-step nucleation process. 14−18 The two-step mechanism involves the production of intermediate, disordered, metastable liquid-like clusters of solute molecules, in which solute molecules then order themselves into crystalline nuclei. 14−18
Table presents the nucleation parameter of LHB for various super saturation using classical nucleation and modified theory at different temperatures. Interfacial energy has been calculated using the expression given by Sangwal . The interfacial energy of LHB solution crystal interface to decrease with increase of temperature. It is observed that the critical size and the critical energy barrier for the formation of nucleus is considerably reduced and constantly the nucleation rate is enhanced as super saturation is increased. After applying correction to classical theory, ∗ and ∆ ∗
Using classical nucleation theory and modified classical nucleation theory, the nucleation parameters have been calculated for the TSCCB crystal at different supersaturations. Table - 1 presents the values of and using the above theories. It is observed that the volume free energy change per unit volume decreases with supersaturation at a fixed temperature. Consequently, the
Department of Physics, Bharath Institute of Higher Education And Research, Chennai – 600073, India ABSTRACT: Theoritical investigations have been made to calculate the nucleatiion thermodynamical parameters like interfacial energy,critical energy barriers and nucleation rate of L-Arginine tetrafluroborate using the solubility data and applying the classical nucleation theory . in the present study both classical nucleation theory and modified classical nucleation theory have been employed to study the nucleation parameters of L-AFB crystal using various solvents water, acetone & ethanol. A comparative study has been made with respect to the solvents and the results are analyzed the successful growth of large size crystals of good quality L-AFBby gradual temperature lowering technique using suitable solvent, requires knowledghe of fundamental nucleation parameters that influence the growth in super saturation. In other words optimum crystallization processes can only be accomplished if proper super saturation level and suitable solvent chosen during the growth technique is decided by the theoretical estimation of nucleation parameters based on the classical and present theory. Results are dissussed to understand the growth kinetics of L-AFB crystals from low temperature solution growth .
The solubility of the crystals was measured by gravimetrical method (Selvarajan et al., 2009). Fig.1 shows the solubility curves for undoped and glycine doped LACC samples and it is observed that the solubility of samples in water increases with temperature, exhibiting a positive temperature coefficient of solubility. Nucleation kinetic studies have been carried out by measuring the induction period and it is a nucleation parameter and it is defined as the amount of time elapsed between the achievement of a supersaturated solution and the observation of nuclei. The critical nucleation parameters were determined using the following relations (Kanagadurai et al., 2010; Siva Dhas et al., 2009; Mohan Kumar et al., 1999). The expression of the induction time () can be written for critical nucleus in terms of interfacial tension as ln = -B+ (16 3 v 2 N 3 / (3R 3 T 3 (ln S) 2 ) where B is a constant, R is the universal gas constant, S is the super saturation ratio, v is the volume of unit cell, T is absolute temperature of the solution, is the interfacial tension and N is the Avogadro’s number. The slope (m) of the plot of 1 / (ln S) 2 against ln is given by m = (16 3 v 2 N 3 / (3R 3 T 3 ). The Gibbs free energy change for critical nucleus is G *
The classical homogeneous nucleation theory has been successfully verified for different crystal systems. Paul and Joshi  studied the effect of supersaturation on the induction period of KDP crystal. They calculated the surface energy of 546 erg cm-2 at the phase boundary which separates the growing crystals from the solution. Shanmugham et al.  studied the nucleation of KDP crystal. The nucleation rates were studied for supersatu- rated solution of potassium dihydrogen orthophosphate with and without the addition of soluble impurities in the temperature range from 20˚C to 40˚C. They also dis- cussed the effects of temperature, supersaturation and impurity content more extensively. The interfacial ten- sion, energy of formation and critical radius of nuclei were calculated on basis of classical nucleation theory. It was found that the selected soluble impurities such as K 2 CO 3 , K 2 PO 4 , K 2 C 2 O 4 , Na 2 B 4 O 7 and K 2 CrO 4 were re-
Nucleation is an important phenomenon in crystal growth and it is the precursor of the overall crystallization process. Nucleation is consequently a study of the initial stages of the kinetics of crystallization. Nucleation may occur spontaneously or it may be induced artificially. There are homogeneous and heterogeneous nucleation and these are initial stage of the crystallization process. On the other hand, nuclei are often generated in the vicinity of crystals or dust particles present in the supersaturated system. This phenomenon is referred to as secondary nucleation. Primary nucleation is believed to be initiated in a series of bimolecular collisions that forms an aggregate of a small number of molecules of the dissolved material. Embryos below a critical cluster size are unstable and may disintegrate, whereas embryos that exceed this critical cluster size will become stable nuclei and will grow. One should keep in mind that the size of these embryos is still beyond the limit of detection, even by dynamic light scattering. When few atoms, ions or molecules join together in a supersaturated solution, a cluster or nucleus is formed and the overall excess free energy (∆G ) between the nucleus and solute in the supersaturated solution is given by
with 𝐴 𝐻𝐸𝑁 𝑉 nearly one or two orders of magnitude lower than the value of 𝐴 𝐻𝑂𝑁 𝑉 across all conditions. This may be due to the lower number of potential nucleation sites available in heterogeneous nucleation (foreign particles) compared with homogeneous nucleation (molecules/molecular clusters) . The pre-exponential factors, 𝐴𝑉 , increase with averaged shear rates (from the CFD simulation results) in homogeneous and heterogeneous nucleation, respectively (Figure 11). This is due to enhancing the attachment frequency and probably decreasing the energy barrier for diffusion or desolvation. The shear rate has a stronger influence on the pre-exponential factors in heterogeneous nucleation (𝐴 𝐻𝐸𝑁 𝑉 in
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A new model for the overall transformation kinetics of bainite has been developed. Based on the displacive mechanism for the bainite transformation, the model distinguishes between the nucleation kinetics of bainitic ferrite in prior austenite grain boundaries, and at tips and adjacent positions of previously formed subunits. Some geometrical aspects of the development of the transformation have been used in the modelling. The theoretical results show that the tendencies obtained with the model are in agreement with experience. The second part of this work deals with the experimental validation of this model. [doi:10.2320/matertrans.47.2465]
Another common feature of polyQ diseases is the neuro- nal accumulation of the mutant protein in nuclear or cyto- plasmic inclusion [2,7]. In addition to the length dependence of disease onset, the length of polyQ sequence also predicted the propensity toward aggrega- tion of polyQ-containing peptides [8,9]. The aggregation of polyQ peptides in vitro follows a simple nucleated growth polymerization pathway, implying crystallization or, in some cases, amyloid fibril formation [9,10]. Nucle- ated growth polymerization is a two-stage process consist- ing of the energetically unfavorable formation of a nucleus (i.e., nucleation), followed by efficient elongation of the nucleus via sequential additions of monomers . Its kinetics is exemplified by long lag time followed by rapid aggregate growth, with a strong dependence of aggregation lag time on monomer concentration . Nucleated growth polymerization has been proposed to govern disease progression kinetics in Alzheimer's and prion-related diseases . We and others have previously suggested a linkage between the biophysics of polyQ aggregation nucleation and HD onset [9,12]. The actual mechanism of the generation of nuclei based on polyQ sequences will be structurally complex, but a kinetic parameter of nucleation is expected to be an exponential function of repeat length . The polyQ length depend- ence of disease onset correlates strongly with the tendency of expanded polyQ proteins to aggregate in disease mod- els [14,15]. Accordingly, we have focused on this length dependence of age-of-onset and nucleation kinetics to derive a stochastic mathematical model describing geno- type-phenotype correlations in polyQ diseases.
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In this work, we aim to understand precritical ﬂuctuations and their connection to nucleation kinetics by comparing cluster ﬂuctuations on two model substrates that enhance the nucleation of ice to the same extent. To this end we perform an extensive set of molecular dynamics simulations. From these it emerges that the traditional hetCNT picture can break down, because a substrate can facilitate the formation of different polymorphs. As a consequence, when using hetCNT, one must choose a different bulk-reference to describe the nucleation process correctly. Although here we illustrate the potential role of precritical ﬂuctuations in the context of CNT, in principle they could be used with any theory that provides a free energy proﬁle for nucleation. We hope that the new insight obtained furthers the theoretical understanding of het. nucleation, polymorph selection, and the role of precritical ﬂuctuations.
of chloride plating baths might diminish the cost related to the synthesis of cobalt nanoclusters by electrodeposition. To the best of our knowledge, there is no information regarding the nucleation kinetics of the cobalt nanoclusters onto HOPG from chloride solutions. We consider that a good understanding of the kinetics of cobalt electrodeposition will provide a good control of size and morphology of cobalt nanoparticles. Thus, in the present paper, a study of the cobalt electrodeposition onto HOPG electrode from chloride solutions is carried out by electrochemical, Atomic Force Microscopy (AFM) and Magnetic Force Microscopy (MFM) characterization techniques, in order to gain a deeper insight into this process.
In this step, nucleation was not happen. As shown in figure 7, the highest fitting is for 50ºC and n=2 with R 2 < 0.8. Comparing the regression results in Table 3 also evident. Matching data in this model compared to the pre equilibrium core and autocatalytic model occurred with less regression. Due to the low compliance data for this model and that in this model the nucleation stage is not intended to aggregate, the existence of nucleation stage in the aggregation reaction enhances. As can be seen in figure the distance between the lines fitted to the data of the first stage is high but by going into the second phase of diagrams, especially at 50°C it is pulled to data. These results show an increase in the amount mass in the solution and going to the second stage of the reaction will increase the consistency of data in the model. Then the growth phase reaction occurs as expressed in the models.The curves obtained by fitting the data to the model showes that the change is ascending at the first and is omitted in nucleation (Fig. 7). The better result is happened in 50°C because of the existence of some aggregates in the nucleation step. Lower compliance at 25°C is due to the low reaction and changes in solution. The best fitting is happened in n=2. Such as result of pre equilibrium core model, that shows the initiation of aggregation by joining two reteplase monomer proteins together.
The electrocrystallization of Ni in 2HEAF was studied in a similar way to that used in aqueous electrolytes, assuming that the nucleation process can be either an instantaneous or progressive process, and the growth mode can involve one, two, or three-dimensional growth, with needles, disks cones or hemispheres shapes. In order to determinate the type of nucleation process involved in Ni deposition, it is assumed that the crystal growth is carried out according to a three-dimensional model with predominant hemispheres shapes. This assumption is supported with the obtained SEM images which also show the existence of hemispheres evenly distributed over the electrode surface. The analytical expression used for the instantaneous (5) and progressive(6) nucleation were defined by Scharifker and coworkers [43, 44], these equations define multiple nucleation phenomena followed by diffusion-controlled growth of three-dimensional islands.
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A schematic for the formation of new particles from sulphuric acid and water vapour is shown in Figure 2. For the nucleation events observed in boreal forests it has been found that the sulphuric acid concentration is usually too low (typically 10 6 –10 7 molecules cm −3 ) to contribute more than a few percent to the observed subsequent particle growth after the initial particle nucleation has occurred (overcoming of the nucleation barrier, see below) . Once formed and thermodynamically stable, other substances such as condensable oxidized organic compounds take part in the growth of the newly formed particles and contribute the largest mass fraction of the particles. Therefore, the formation of atmospheric new particles is often divided into a two-step process: First the nucleation itself where clusters overcome the nucleation barrier, and then the subsequent growth of these clusters. For the nucleation events that occur in boreal forest regions, it has been shown, for example, that terpene oxidation products from substances such as alpha-pinene, beta-pinene or limonene play an important role for the observed growth of particles after nucleation. The terpenes are emitted by the trees in fairly large amounts.
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In the literature there are several studies of isothermal annealing of GST interpreted using the JMAK theory. It is also worth to add studies to measure the activation energy EA using Kissinger analysis. Reported activation energies deviates generally around 2 eV with few exceptions: 2 eV, 1.8 eV, 2.26 eV, 2.23 eV, 2.3 eV, 2.24 eV, 2.15 eV, 3eV, 0.81 eV, 1 eV. Fig. 1a shows JMAK plots at 150ºC for ν=1022 s-1 and n=2.5 for different activation energy values. Although differences in the activation energy are small, variations in total crystallized material at a given time can be huge. As we see below this activation energy of about 2 eV is very close to activation energy of crystallization growth rate estimated from diffusion coefficient of GST. Analysis techniques used to determine EA are based on assumption of an Arrhenian temperature dependence for the crystallization kinetics. Since the kinetics is partly controlled by the nucleation rate, which is non-Arrhenius, the use of such analysis techniques is not justified and can only yield information on transformations controlled by growth rate. Hence the activation energy obtained is approximately equal to the activation energy of growth. Strictly speaking, also growth is non-Arrhenius so that the activation energy obtained would depend on the temperature range over which the experiments were carried out. It is, therefore, not justified to split the observed activation energy between nucleation and growth parameters using ad hoc assumptions, as suggested in Ref. .
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Figure 7 shows a plot of real and observed nucleation efficiencies plotted against cooling rate at the three levels of overall average nucleation undercooling. Generally, increasing the cooling rate increases nucleation efficiency. At the lower value of average nucleation undercooling (4 °C) the nucleation efficiencies were highest at all cooling rates. Given the power law form of the growth law (mentioned above), the growth rate is expected to have a stronger effect at the higher levels of undercooling. In the cases where high undercooling values were reached, real nucleation rates were lowered significantly. Fast growth effects (high 𝜁 𝑉 ) dominated the real nucleation rate equation, 𝑁̇ 𝑅𝐸𝐴𝐿 = 𝑁̇(1 − 𝜁 𝑉 ).
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