A complete analytical solution of the Fokker–Planck and balance equations for nucleation and growth of crystals

Makoveeva, Eugenya V.; Alexandrov, Dmitri V.

doi:10.1098/rsta.2017.0327

Cited by 69 publications

(47 citation statements)

References 48 publications

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“…where J = w −1 (1 − χτ 0 C 0 wt) −1 exp(pφ(w)) (p and φ(w) are given in §2 in [15] for supersaturated solutions and supercooled melts for the Meirs and Weber-Volmer-Frenkel-Zeldovich kinetics),…”

Section: Analytical Solutionmentioning

confidence: 99%

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Effects of nonlinear growth rates of spherical crystals and their withdrawal rate from a crystallizer on the particle-size distribution function

Makoveeva

Alexandrov

2019

Phil. Trans. R. Soc. A.

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Cite this article: Makoveeva EV, Alexandrov DV. 2019 Effects of nonlinear growth rates of spherical crystals and their withdrawal rate from a crystallizer on the particle-size distribution function. Phil. Trans. R. Soc. A 377: 20180210. http://dx.One contribution of 17 to a theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures' .In this paper, we show that the nonlinear growth rate of particles in a supersaturated solution or supercooled melt, as well as the rate of removal of crystals from the metastable liquid of a crystallizer, significantly change the size-distribution function of crystals. Taking these rates into account, we present a complete analytical solution of the integrodifferential model describing the transient nucleation of solid particles and their evolution in a metastable liquid. The distribution function and metastability degree (supersaturation or supercooling) are found by means of the separation of variables and saddlepoint methods. The nonlinear growth rates of crystals in supersaturated solutions and supercooled melts (single-component and binary) are summarized and compared with experimental data.

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Section: Analytical Solutionmentioning

confidence: 99%

“…and substituting (3.8) into (3.5), we get where ν k (t) and F 0k are determined by expressions (3.9) and (3.10) in [15], and…”

Section: )mentioning

confidence: 99%

Effects of nonlinear growth rates of spherical crystals and their withdrawal rate from a crystallizer on the particle-size distribution function

Makoveeva

Alexandrov

2019

Phil. Trans. R. Soc. A.

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“…In this form, the system is nonlinear and an approximate solution was given. Since the time of Buyevich and Mansurov's paper, variants of this model have been used to study a variety of crystal growth systems . The governing equations are

\frac{\partial f (t, r)}{\partial t} + \frac{\partial G (t) f (t, r)}{\partial r} = 0,

\frac{d s (t)}{d t} = - K \int_{0}^{\infty} r^{2} G false(t false) f false(t, r false) d r,

where s is the normalized supersaturation, r the crystal radius or some other characteristic length,

G = d r / d t

the linear crystal growth rate, f the size distribution function, and K a constant.…”

Section: Introductionmentioning

confidence: 99%

“…Since the time of Buyevich and Mansurov's paper, variants of this model have been used to study a variety of crystal growth systems. 2,[17][18][19][20][21][22][23] The governing equations are…”

Section: Introductionmentioning

confidence: 99%

The kinetics of homogeneous and two‐step nucleation during protein crystal growth from solution

Barlow¹,

Gregus

2019

Int J of Chemical Kinetics

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Many experimental reports for the kinetics of crystal nucleation and growth, from an isothermal solution, point to a sigmoidal‐like behavior for the process. Here we consider three different nucleation models from the literature and show that all lead to sigmoidal or sigmoidal‐like behavior for the kinetics of nucleation. A two‐step nucleation process is known to occur within certain supersaturated protein solutions, and it is demonstrated in this report how the sigmoidal law yields kinetic information for the two‐step and homogeneous nucleation modes. We propose here that two‐step solute‐rich associates form in the solution around seed nuclei that are already present at or near the point in time when the solution is prepared. Using this hypothesis, we are able to model the time‐dependent volume of the two‐step phase per unit volume of solution and show that this compares well with reported experimental data. A kinetic model is given for the proposed process, which leads to a sigmoidal rate law. Additionally, a relation between the initial and final nuclei densities and the induction time is derived. As a result of this study, the conclusion is that two‐step activity increases with increasing initial supersaturation or increasing salt concentration.

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“…In the paper written by Makoveeva & Alexandrov [47], the structure and typical regions appearing in the mushy layer are discussed in detail. Furthermore, the authors analyse the process of nucleation and growth of newly born crystals in the most undercooled part of a mushy layer.…”

Section: (C) the Macroscopic Modelling Of Dendrites And Grainsmentioning

confidence: 99%

From atomistic interfaces to dendritic patterns

Galenko

Alexandrov

2018

Phil. Trans. R. Soc. A.

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One contribution of 16 to a theme issue 'From atomistic interfaces to dendritic patterns' . Transport processes around phase interfaces, together with thermodynamic properties and kinetic phenomena, control the formation of dendritic patterns. Using the thermodynamic and kinetic data of phase interfaces obtained on the atomic scale, one can analyse the formation of a single dendrite and the growth of a dendritic ensemble. This is the result of recent progress in theoretical methods and computational algorithms calculated using powerful computer clusters. Great benefits can be attained from the development of micro-, meso-and macrolevels of analysis when investigating the dynamics of interfaces, interpreting experimental data and designing the macrostructure of samples. The review and research articles in this theme issue cover the spectrum of scales (from nano-to macro-length scales) in order to exhibit recently developing trends in the theoretical analysis and computational modelling of dendrite pattern formation. Atomistic modelling, the flow effect on interface dynamics, the transition from diffusion-limited to thermally controlled growth existing at a considerable driving force, two-phase (mushy) layer formation, the growth of eutectic dendrites, the formation of a secondary dendritic network due to coalescence, computational methods, including boundary integral and phase-field methods, and experimental tests for theoretical models-all these themes are highlighted in the present issue.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

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A complete analytical solution of the Fokker–Planck and balance equations for nucleation and growth of crystals

Cited by 69 publications

References 48 publications

Effects of nonlinear growth rates of spherical crystals and their withdrawal rate from a crystallizer on the particle-size distribution function

Effects of nonlinear growth rates of spherical crystals and their withdrawal rate from a crystallizer on the particle-size distribution function

The kinetics of homogeneous and two‐step nucleation during protein crystal growth from solution

From atomistic interfaces to dendritic patterns

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