Comparison of selected mathematical functions for the analysis of growth behavior of items and physical interpretation of Avrami-Weibull function

Abstract Empirical data of sigmoidal-shaped y(t) growth behavior of different types of items, such as papers and citations earned by individual and all successively published papers of selected top-cited authors, germination of tomato seeds and three different bacteria, are analyzed and compared by Avrami-Weibull, Verhulst (logistic) and Gompertz functions. It was found that: (1) Avrami-Weibull function describes different types of the data better than Gompertz and Verhulst funtions, and (2), in comparison with Verhulst and Gompertz functions, Avrami-Weibull function, expressed in the form: y(t)/ymax = 1-exp[(t/Θ)q] (where ymax is the maximum value of y(t) when t→∞, and Θ and q are constants), is equally very versatile in explaining the generation rate dy(t)/dt of items in terms of its parameters Θ and q. Using the basic concepts involved in the derivation of Avrami-Weibull function for overall crystallization from melt and supersaturated solution, the growth behavior of cumulative number y(t) of items produced at time t by individual (simple) sources and collectives or groups of simple sources (i.e. complex or composite sources) is presented. Comparison of the process of receiving of citations by papers with the processes of occurrence of chemical reactions and crystallization of solid phases from melts and supersaturated solutions shows that this process is similar to that of overall crystallization of solid phases from melts and solutions. Analysis of growth of citations using Avrami-Weibull function to individual papers published by different authors shows that 1 < q < 4 for most cases. This suggests that the process of citations to individual articles is mainly determined by progressive nucleation mode involving both diffusion and integration of published knowledge. &nbsp

On the use of process analytical technologies and population balance equations for the estimation of crystallization kinetics. A case study.

International audienceThe batch cooling solution crystallization of ammonium oxalate was performed in water at various constant cooling rates. Measurements of the solute concentration were obtained using in situ attenuated total reflectance fourier transform infrared (ATR-FTIR) spectroscopy, and final estimates of the crystal size distribution (CSD) were computed; thanks to in situ image acquisition and off-line image analysis. The crystallization process was then simulated using population balance equations (PBEs). Estimates of the nucleation and the growth parameters were computed through model/experiments fitting. According to the cooling rate, the PBE model allowed distinguishing between two distinct crystallization regimes, separated by an "intermediate regime." The respective contributions and shortcomings of solute concentration measurements and granulometric data to the identification of nucleation and growth kinetic parameters are analyzed and discussed. It is shown in particular that no real separate estimation of nucleation and growth parameters can be obtained in the absence of CSD data

Holistic approach to dissolution kinetics : linking direction-specific microscopic fluxes, local mass transport effects and global macroscopic rates from gypsum etch pit analysis

Dissolution processes at single crystal surfaces often involve the initial formation and expansion of localized, characteristic (faceted) etch-pits at defects, in an otherwise comparatively unreactive surface. Using natural gypsum single crystal as an example, a simple but powerful morphological analysis of these characteristic etch pit features is proposed that allows important questions concerning dissolution kinetics to be addressed. Significantly, quantitative mass transport associated with reactive microscale interfaces in quiescent solution (well known in the field of electrochemistry at ultramicroelectrodes) allows the relative importance of diffusion compared to surface kinetics to be assessed. Furthermore, because such mass transport rates are high, much faster surface kinetics can be determined than with existing dissolution methods. For the case of gypsum, surface processes are found to dominate the kinetics at early stages of the dissolution process (small etch pits) on the cleaved (010) surface. However, the contribution from mass transport becomes more important with time due to the increased area of the reactive zones and associated decrease in mass transport rate. Significantly, spatial heterogeneities in both surface kinetics and mass transport effects are identified, and the morphology of the characteristic etch features reveal direction-dependent dissolution kinetics that can be quantified. Effective dissolution velocities normal to the main basal (010) face are determined, along with velocities for the movement of [001] and [100] oriented steps. Inert electrolyte enhances dissolution velocities in all directions (salting in), but a striking new observation is that the effect is direction-dependent. Studies of common ion effects reveal that Ca2+ has a much greater impact in reducing dissolution rates compared to SO42−. With this approach, the new microscopic observations can be further analysed to obtain macroscopic dissolution rates, which are found to be wholly consistent with previous bulk measurements. The studies are thus important in bridging the gap between microscopic phenomena and macroscopic measurements

Using an on-line image analysis technique to characterize sucrose crystal morphology during a crystallization run

The morphological forms and habits of crystals and agglomeration are important properties on crystallization processes. Online techniques for realtime measurement of these properties are mandatory for a better comprehension of crystal growth phenomenon. The present paper presents and describes a new online method to determine the complexity level of a crystal or a population of crystals during a crystallization process. An image analysis technique is combined with discriminant factorial analysis leading to results that allow the computation of the complexity of crystals through the parameter agglomeration degree of crystals. With this methodology, it has been possible to distinguish online and automatically among three different classes of crystals according to their complexity. It further describes the application of such methodology on the study of CaCl2, D-fructose, and D-glucose influence on the crystallization of sucrose, namely, on crystal size, morphology, and complexity. The effect of supersaturation, growth rate, and impurity concentration on the type, amount, and complexity level of the agglomerates was determined at different temperatures. The combination of image analysis and kinetic results allowed to understand better the crystallization phenomena in the presence and absence of impurities. The image analysis results suggest the possible application of this tool for process control, optimizing, by this way, laboratory and industrial crystallizers.This work was supported by Fundacao para a Ciencia e Tecnologia under program contract numbers SFRH/BD/11315/2002 and SFRH/BPD/45637/2008

Extended defects in natural diamonds: Atomic Force Microscopy investigation

Surfaces of natural diamonds etched in high-pressure experiments in H2O, CO2 and H2O-NaCl fluids were investigated using Atomic Force Microscopy. Partial dissolution of the crystals produced several types of surface features including the well-known trigons and hillocks and revealed several new types of defects. Besides well-known trigons and dissolution hillocks several new types of defects are observed. The most remarkable ones are assigned to anelastic twins of several types. The observation of abundant microtwins, ordering of hillocks and presence of defects presumably related to knots of branched dislocations suggests importance of post-growth deformation events on formation of diamond microstructure. This work confirms previous reports of ordering of extended defects in some deformed diamonds. In addition, the current work shows that natural diamonds deform not only by dislocation mechanism and slip, but also but mechanical twinning. The dominant mechanism should depend on pressure-temperature-stress conditions during diamond transport from the formation domain to the Earth surface.Comment: Submitted to special issue (1st European Mineralogical congress, Frankfurt, Germany, September 2012) of European Journal of Mineralogy. 21 page, 9 figure

Biogenic calcite particles from microalgae-Coccoliths as a potential raw material

Synthetic calcite (CaCO3) particles are found in a broad range of applications. The geometry of particles produced from limestone or precipitation are versatile but limited to basic shapes. The microalga Emiliania huxleyi produces micro-structured calcite platelets, called coccoliths. This article presents the results of an application-orientated study, which includes characteristic values also used in the calcite industry for particle evaluation. It is demonstrated that coccoliths are significantly different from all industrial particles produced so far. Coccoliths are porous particles, mainly consisted of calcium carbonate, with further elements such as Mg, Si, Sr, and Fe often embedded in their structure. Their structure is extremely sophisticated, while the overall particle morphology and particle size distribution are homogeneous. This study gives a first inside into the potential of these exceptional objects and may set further impulses for their utilization in specific calcite particle applications