Crystal Size Distribution Function Research Articles

Two-step nucleation and crystal growth in a metastable solution

This study is concerned with a theory of two-step nucleation and growth of crystals in a metastable liquid. This mechanism is that crystalline nuclei formation occurs in dense liquid clusters suspended in the solution. These clusters contain higher solution concentration and viscosity, leading to a lower surface free energy barrier and faster phase transition route. The theory is based on growth laws of crystals during the two-step bulk phase transformation. At the initial stage, the crystals evolve in a diffusion-limited environment with almost unchanged supersaturation. At the second stage, they become larger, move beyond these clusters, and evolve in accordance with a hyperbolic tangent law. A generalized particle growth law joining the first and second stages is obtained by stitching the diffusion limited and hyperbolic tangent laws. On this basis, an integrodifferential model of the evolution of a polydisperse ensemble of crystals was formulated and solved. The crystal-size distribution function increases and the solution supersaturation remains practically unchanged until the particle size corresponds to a transition in the particle growth rate from a diffusion-limited branch to a hyperbolic tangent branch. This is followed by an increase in the crystal growth rate, a decrease in the distribution function and solution supersaturation. Then the distribution function increases up to the maximum size of crystals grown in the solution. A sufficiently long time interval of almost constant supersaturation and the N-shaped behavior of the distribution function are the consequences of a two-step nucleation and growth mechanisms.

Desupersaturation dynamics in solutions with applications to bovine and porcine insulin crystallization

Evolution of crystal ensembles in supersaturated solutions is studied at the initial and intermediate stages of bulk crystallization. An integro-differential model includes fluctuations in crystal growth rates, initial crystal-size distribution and arbitrary nucleation and growth kinetics of crystals. Two methods based on variables separation and saddle-point technique for constructing a complete analytical solution to this model are considered. Exact parametric solutions based on these methods are derived. Desupersaturation dynamics is in good agreement with the experimental data for bovine and porcine insulin. The method based on variables separation has a strong physical limitation on exponentially decaying initial distribution and leads to the distribution function increasing with time. The method based on saddle-point technique leads to a dome-shaped crystal-size distribution function decreasing with time and has no strong physical limitations. The latter circumstance makes this method more reasonable for describing the kinetics of bulk crystallization in solutions and melts.

Nucleation and Growth of an Ensemble of Crystals during the Intermediate Stage of a Phase Transition in Metastable Liquids

In this paper, an analytical method of solving the integro-differential system of kinetic and balance equations describing the evolution of an ensemble of crystals during the intermediate phase of the bulk crystallization process is described. The theory is developed for kinetic equations of the first- and second order corresponding to the absence and presence of fluctuations in particle growth rates. The crystal-size distribution function as well as the dynamics of metastability reduction in a supercooled melt (supersaturated solution) are analytically found using the saddle-point and the Laplace transform methods. The theory enables us to obtain the crystal-size distribution function that establishes in a supercooled (supersaturated) liquid at the beginning of the final stage of a phase transformation process when Ostwald ripening, coagulation and fragmentation of crystals are able to occur.

Dynamics of particulate assemblages in metastable liquids: a test of theory with nucleation and growth kinetics.

This manuscript is devoted to the nonlinear dynamics of particulate assemblages in metastable liquids, caused by various dynamical laws of crystal growth and nucleation kinetics. First of all, we compare the quasi-steady-state and unsteady-state growth rates of spherical crystals in supercooled and supersaturated liquids. It is demonstrated that the unsteady-state rates transform to the steady-state ones in a limiting case of fine particles. We show that the real crystals evolve slowly in a more actual case of unsteady-state growth laws. Various growth rates of particles are tested against experimental data in metastable liquids. It is demonstrated that the unsteady-state rates describe the nonlinear behaviour of experimental curves with increasing the growth time or supersaturation. Taking this into account, the crystal-size distribution function and metastability degree are analytically found and compared with experimental data on crystallization in inorganic and organic solutions. It is significant that the distribution function is shifted to smaller sizes of particles if we are dealing with the unsteady-state growth rates. In addition, a complete analytical solution constructed in a parametric form is simplified in the case of small fluctuations in particle growth rates. In this case, a desupercooling/desupersaturation law is derived in an explicit form. Special attention is devoted to the biomedical applications for insulin and protein crystallization. This article is part of the theme issue 'Patterns in soft and biological matters'.

Towards the theory of phase transformations in metastable liquids. Analytical solutions and stability analysis

The process of ongoing nucleation and evolution of particles in a metastable liquid caused by external mass sources and crystal withdrawal mechanism is considered. The steady-state and unsteady-state analytical solutions are found analytically. The crystal-size distribution function, as well as the standard Cauchy problem for the liquid supersaturation, are derived. The linear dynamic stability theory of steady-state solutions is developed. The neutral stability surface as a function of system parameters is determined.

On the theory of bulk crystallization in the moving phase transition layer

Abstract The paper deals with the process of directional steady-state crystallization in the presence of a metastable phase transition region, where nucleation and crystal growth occur. A new analytical solution describing such a nonlinear heat and mass transfer phase transformation process is constructed on the basis of the saddle-point method for a Laplace-type integral. The crystal-size distribution function, supercooling and solid fraction in a moving metastable two-phase region are found analytically.

Effects of external heat/mass sources and withdrawal rates of crystals from a metastable liquid on the evolution of particulate assemblages

A complete analytical solution of an integro-differential model describing the transient nucleation of solid particles and their subsequent growth at the intermediate stage of phase transitions in metastable systems is constructed. A functional Fokker–Plank type equation for the density distribution function is solved by the separation of variables method for the Weber–Volmer–Frenkel–Zeldovich nucleation kinetics. A non-linear integral equation with memory kernel connecting the density distribution function and the system supercooling/supersaturation is analytically solved on the basis of the saddle point method for the Laplace integral. The analytical solution obtained shows that the transient phase transition process attains its steady-state solution at large times. An exact analytical solution for the steady-state problem is found too. It is demonstrated that the crystal-size distribution function increases with increasing the intensity of external sources. In addition, the number of larger (smaller) particles decreases (increases) with increasing the withdrawal rate of crystals from a metastable liquid.

Quantitative Design of Seed Load Based on model identification of nucleation and crystal growth stages without transmittance data

How to quantitatively design the loading time and amount of seed is a critical problem in the seeded crystallization process. In this study, correlations for different moment of crystal size distribution (CSD) function were established by subsection induction. By using the established different moment of CSD function, a new algorithm for kinetic model that can estimate the nucleation and growth kinetic parameters was proposed. Compared with kinetic model based on the conventional moments method, the new algorithm has a great significance since the overall heat transfer coefficient U is no longer necessary in the entire calculation. Using experimental data from literature, kinetic parameters of crystallization of KNO3 aqueous solution and xylene solution were estimated by the new algorithm, which are in good accordance with those fitting with moments method. This agreement shows the feasibility of the new algorithm. Finally, since it is difficult to measure transmittance data r of some systems, a criterion K that can identify nucleation and crystal growth stages without transmittance data was established. By means of the K curve of para-xylene (PX) and vitamin C, which is based on kinetic simulation results from the new algorithm and different moment of CSD function established by subsection induction, the single step seed load and two steps seed load were designed, respectively.

Kinetics of Precipitation of Non-ideal Binary Solid-solutions in an Aqueous Medium

We have developed a formalism which accounts for the kinetics of precipitation of binary non-ideal solid-solutions, and includes nucleation, growth and/or re-dissolution processes. It allows prediction of the time evolution of the aqueous ion activities together with all characteristics of the solid phase: crystal size distribution function and composition profiles of particles. We show that different scenarios of precipitation are possible depending on the thermodynamic properties of the solid solution end-members, the degree of non-ideality of the solid solution and the initial conditions of precipitation. We illustrate the various scenarios by simulations performed under a few simplifying assumptions for some carbonate solid solutions of geochemical interest. To our knowledge, this is the first time in the field of geochemistry that these out-of-equilibrium precipitation processes of non-ideal solid solutions are fully described.

Kinetics of diffusive decomposition in the case of several mass transfer mechanisms

An analytical description of the final stage of diffusive decomposition leaning upon the Slezov theory is developed for several mass transfer mechanisms. The process of formation and relaxation of the crystal size distribution function from the initial ripening stage to its final state corresponding to the universal distribution is studied. The boundary points of a transition region responsible for the tails of the distribution functions on the right of the relevant stopping points are found analytically. The explicit time-dependent analytical expressions for the distribution function and particle growth rates are derived with allowance for the plausible mechanisms of mass transfer.

Mass dissolution of polydisperse set of particles in a flow system

A mathematical model of continuous dissolution of salts based on a dynamic equation for crystal size distribution function was considered. A stationary solution was obtained by the moments method. For a nonstationary state, an evolution equation for the undersaturation of solution was derived. An approximate analytical method of solving this equation, which is in good agreement with the numerical solution, was proposed. The notion of the coefficient of continuous dissolution efficiency was introduced and analytical expressions for this value were obtained. The stability of the stationary solution was analyzed.

Dissolution of a Polydisperse System of Particles Shaped Like a Parallelepiped in a Nonflowing System

The process of mass dissolution has been investigated on the basis of the crystal-size distribution function in the case where the crystal faces traveled in accordance with the power law and the dissolution proceeded in a periodic regime. It has been proposed to reduce the dissolution problem to a system of equations that could be solved analytically in certain cases that are of interest in practice. The notion of dissolution efficiency has been introduced.

Petrogenesis of olivine-phyric shergottites Sayh al Uhaymir 005 and Elephant Moraine A79001 lithology A

Martian meteorites Sayh al Uhaymir (SaU) 005 and lithology A of EETA79001 (EET-A) belong to a newly emerging group of olivine-phyric shergottites. Previous models for the origin of such shergottites have focused on mixing between basaltic shergottite-like magmas and lherzolitic shergottite-like material. Results of this work, however, suggest that SaU 005 and EET-A formed from olivine-saturated magmas that may have been parental to basaltic shergottites. SaU 005 and EET-A have porphyritic textures of large (up to ∼3 mm) olivine crystals (∼25% in SaU 005; ∼13% in EET-A) in finer-grained groundmasses consisting principally of pigeonite (∼50% in SaU 005; ∼60% in EET-A), plagioclase (maskelynite) and < 7% augite. Low-Ti chromite occurs as inclusions in the more magnesian olivine, and with chromian ulvöspinel rims in the more ferroan olivine and the groundmass. Crystallization histories for both rocks were determined from petrographic features (textures, crystal shapes and size distributions, phase associations, and modal abundances), mineral compositions, and melt compositions reconstructed from magmatic inclusions in olivine and chromite. The following observations indicate that the chromite and most magnesian olivine (Fo 74–70 in SaU 005; Fo 81–77 in EET-A) and pyroxenes (low-Ca pyroxene [Wo 4–6] of mg 77–74 and augite of mg 78 in SaU 005; orthopyroxene [Wo 3–5] of mg 84–80 in EET-A) in these rocks are xenocrystic. (1) Olivine crystal size distribution (CSD) functions show excesses of the largest crystals (whose cores comprise the most magnesian compositions), indicating addition of phenocrysts or xenocrysts. (2) The most magnesian low-Ca pyroxenes show near-vertical trends of mg vs. Al 2O 3 and Cr 2O 3, which suggest reaction with a magma. (3) In SaU 005, there is a gap in augite composition between mg 78 and 73. (4) Chromite cores of composite spinel grains are riddled with cracks, indicating that they experienced some physical stress before being overgrown with ulvöspinel. (5) Magmatic inclusions are absent in the most magnesian olivine, but abundant in the more ferroan, indicating slower growth rates for the former. (6) The predicted early crystallization sequence of the melt trapped in chromite (the earliest phase) in each rock produces its most magnesian olivine-pyroxene assemblage. However, in neither case is the total crystallization sequence of this melt consistent with the overall crystallization history of the rock or its bulk modal mineralogy. Further, the following observations indicate that in both SaU 005 and EET-A the fraction of solid xenocrystic or xenolithic material is small (in contrast to previous models for EET-A), and most of the material in the rock formed by continuous crystallization of a single magma (possibly mixed). (1) CSD functions and correlations of crystal size with composition show that most of the olivine (Fo 69–62 in SaU 005; Fo 76–53 in EET-A) formed by continuous nucleation and growth. (2) Groundmass pigeonites are in equilibrium with this olivine, and show continuous compositional trends that are typical for basalts. (3) The CSD function for groundmass pigeonite in EET-A indicates continuous nucleation and growth (Lentz and McSween, 2000). (4) The melt trapped in olivine of Fo 76 to 67 in EET-A has a predicted crystallization sequence similar to that inferred for most of the rock and produces an assemblage similar to its modal mineralogy. (5) Melt trapped in late olivine (Fo ∼ 64) in SaU 005 has a composition consistent with the inferred late crystallization history of the rock. The conclusion that only a small fraction of either SaU 005 or EET-A is xenocrystic or xenolithic implies that both rocks lost fractionated liquids in the late stages of crystallization. This is supported by: (1) high pigeonite/plagioclase ratios; (2) low augite contents; and (3) olivine CSD functions, which show a drop in nucleation rate at high degrees of crystallization, consistent with loss of liquid. For EET-A, this fractionated liquid may be represented by EET-B.

Analysis of the dissolution of a polydisperse system of particles having the shape of a parallelepiped

A mathematical model based on a crystal size distribution function is proposed for the continuous dissolution of particles having the shape of a parallelepiped. An evolution equation for the undersaturation of the solution is derived. Results of calculations using this equation are presented. A stability analysis of the steady‐state solution obtained is carried out.

Stability Analysis of the Growth of Particles of a Polydisperse System in a Solution under Nonisothermal Conditions

A mathematical model of continuous saline crystallization from solutions based on the crystal–size distribution function is considered. A steady solution is obtained and its stability is studied.

Description of mass growth of crystals from solutions with allowance for the disappearance of crystal faces during crystal growth

Mass crystallization was studied on the basis of a crystal-size distribution function for a kinetic law of growth of faces (when the growth rate of a crystal face does not depend on the size and shape of the face) and with allowance for vanishing of individual faces during crystal growth. It is proposed that the crystallization problem should be reduced to a system of equations that admit analytical solutions in some cases of practical interest.

Instability and unsteady processes of the bulk continuous crystallization—I. Linear stability analysis

The physical nature of the instability of stationary regimes of continuous crystallization from supersaturated solutions is explained. The system of equations with distributed parameters, comprising the mass and number balance laws expressed in terms of the crystal size distribution function, is reduced to a single functional integro-differential equation describing the evolution of the saturation for arbitrary mass flux of a supersaturated solution into a volume under consideration, nucleation and crystal growth kinetics and withdrawal rate of crystals out of the volume. The mechanism of instability of a stationary crystallization process is studied. Dependencies of the neutral stability surface and of the period of initial self-oscillations occurring as a result of the instability upon physico-chemical and operating parameters are obtained. A comparison of theory and experiments is presented.

Prediction of the zeolite crystal size distribution in batchwise hydrothermal synthesis

Several ways of characterizing the crystal size distribution during batchwise hydrothermal synthesis of molecular sieve zeolites are shown. In particular, we show how the crystal size distribution function can be predicted once the transient birth rate and growth rate functions have been determined, and how average size, standard deviation, and skewness can be used to characterize the size distribution. Simulation results are presented which show the effect of manipulating several key parameters in the synthesis system (e.g. initial reagent concentration and nucleation rate). It is shown how one might change the operating policy of the batch crystallizer to obtain larger particle sizes and/or narrower crystal size distribution.

The effects of crystal size distributions on the crystallinity analysis of bone mineral.

The mechanisms by which crystal size distributions affect the usual method of quantities X-ray diffraction analysis of bone mineral have been determined on synthetic crystals. It was observed that each component of a crystal-size distribution diffracts independently. This independence causes systematic nonlinear behavior in the plot of integrated intensity vs. broadening parameter curves. The nonlinearity resulted in an overestimation of the amount of nondiffracting material present in bone mineral. Because crystal size distributions may vary for different crystallographic directions, it is strongly suggested that the usual practice of adding the c-axis and a-axis integrated intensities to estimate the crystallinity of the sample be discontinued. Methods of understanding the crystal size distribution function in bone mineral are discussed and evaluated.