Kinetics of the intermediate stage of phase transition with elliptical crystals

Math Methods in App Sciences

2021

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In this paper, a complete analytical solution to the integro-differential model describing the nucleation and growth of ellipsoidal crystals in a supersaturated solution is obtained. The asymptotic solution of the model equations is constructed using the saddle-point method to evaluate the Laplace-type integral. Numerical simulations carried out for physical parameters of real solutionsshow that the first four terms of the asymptotic series give a convergent solution. The developed theory was compared with the experimental data on desupersaturation kinetics in proteins. It is shown that the theory and experiments are in good agreement.

“…It is important to note that the following approximations were used in deriving the growth rate ()

{(\frac{b_{0}}{Δ C_{0}})}^{2} ≫ 4 \ln^{2} (\frac{σ_{0} - 1}{σ_{0} + 1}) w, \frac{1}{β_{* *} (σ_{0}^{2} - 1)} ≫ (2 w + \frac{C_{p}}{Δ C_{0}}) \ln (\frac{σ_{0} - 1}{σ_{0} + 1}) .

These approximations follow from the estimates of the characteristic parameters for protein and insulin crystals 39,42 and our computations illustrated in Figure 2.…”

Section: Analytical Solutionsmentioning

confidence: 87%

“…The dimensionless rate G R as a function of crystal volume V 1 at fixed values of dimensionless supersaturation w . The system parameters are estimated as follows 39,42

C_{p} false/ normalΔ C_{0} = 2, β_{* *} = 0.01

[Colour figure can be viewed at wileyonlinelibrary.com]…”

Section: Growth Rates Of Ellipsoidal Particlesmentioning

confidence: 99%

A complete analytical solution to the integro‐differential model describing the nucleation and evolution of ellipsoidal particles

Nikishina

Math Methods in App Sciences

2021

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“…It should be especially noted that non-stationary effects in particle evaporation rates [25][26][27]39], the Gibbs-Thomson and attachment kinetic effects [35,40,41], the particle shape effects [42][43][44][45][46], as well as the influence of external sources and sinks [10,11,47,48] may also essentially change the evaporation dynamics by analogy with nucleation and crystal growth processes. For a more complete understanding of the question of the influence of these aspects on evaporation dynamics, further development of the theory is required, taking into account the simultaneous action of the above-mentioned effects.…”

Section: Discussionmentioning

confidence: 99%

Evaporation kinetics of a polydisperse ensemble of drops

Ivanov

Аlexandrova

Phil. Trans. R. Soc. A.

2021

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A mathematical model of the evaporation of a polydisperse ensemble of drops, with allowance for a nonlinear ‘diffusion’ term in the kinetic equation for the population density distribution function, is developed. The model describes the interaction of a gas phase with vaporizing drops: it has great potential for application in condensed matter physics, thermophysics and engineering devices (e.g. spray drying, cooling, power engineering). The kinetics of heat transfer between phases is theoretically studied. An analytical solution to the integro-differential equations of the process of droplet evaporation is found in a parametric form. Analytical solutions in the presence and absence of the ‘diffusion’ term are compared. It is shown that the fluctuations in particle evaporation rates (‘diffusion’ term in the Fokker–Planck equation) play a decisive role in the evolutionary behaviour of a polydisperse ensemble of vaporizing liquid drops. This article is part of the theme issue ‘Transport phenomena in complex systems (part 1)’.

“…The considered model of the boiling process of a superheated liquid can be generalized in future studies to take into account the “diffusion” of the distribution function in the space of bubble sizes, 40 the dependence of the growth rate of bubbles on their size, a more general law for the rate of bubble nucleation, the possible presence of an impurity dissolved in the liquid, and the other processes and phenomena. Such generalizations can be made by analogy with the theories of crystallization and dissolution in metastable liquids 41–50 …”

Section: Discussionmentioning

confidence: 99%

“…Such generalizations can be made by analogy with the theories of crystallization and dissolution in metastable liquids. [41][42][43][44][45][46][47][48][49][50]…”

Section: Discussionmentioning

confidence: 99%

Analytical solution of integro‐differential equations describing the process of intense boiling of a superheated liquid

Аlexandrova

Ivanov

Math Methods in App Sciences

2021

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In this article, an approximate analytical solution of an integro-differential system of equations is constructed, which describes the process of intense boiling of a superheated liquid. The kinetic and balance equations for the bubble-size distribution function and liquid temperature are solved analytically using the Laplace transform and saddle-point methods with allowance for an arbitrary dependence of the bubble growth rate on temperature. The rate of bubble appearance therewith is considered in accordance with the Dering-Volmer and Frenkel-Zeldovich-Kagan nucleation theories. It is shown that the initial distribution function decreases with increasing the dimensionless size of bubbles and shifts to their greater values with time.