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

2018 ◽  
Vol 376 (2113) ◽  
pp. 20170327 ◽  
Author(s):  
Eugenya V. Makoveeva ◽  
Dmitri V. Alexandrov
Keyword(s):  
Distribution Function ◽  
Analytical Description ◽  
Supercooled Liquid ◽  
Intermediate Stage ◽  
Nucleation And Growth ◽  
Time Dependent ◽  
Supersaturated Solution ◽  
Nucleation Kinetics ◽  
Type Integral ◽  
Laplace Type

This article is concerned with a new analytical description of nucleation and growth of crystals in a metastable mushy layer (supercooled liquid or supersaturated solution) at the intermediate stage of phase transition. The model under consideration consisting of the non-stationary integro-differential system of governing equations for the distribution function and metastability level is analytically solved by means of the saddle-point technique for the Laplace-type integral in the case of arbitrary nucleation kinetics and time-dependent heat or mass sources in the balance equation. We demonstrate that the time-dependent distribution function approaches the stationary profile in course of time. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

scholarly journals Phase transformations in metastable liquids combined with polymerization

2019 ◽  
Vol 377 (2143) ◽  
pp. 20180215 ◽  
Author(s):  
Alexander A. Ivanov ◽  
Irina V. Alexandrova ◽  
Dmitri V. Alexandrov
Keyword(s):  
Distribution Function ◽  
Saddle Point ◽  
Analytical Solutions ◽  
Polymer Melt ◽  
Size Distribution Function ◽  
Dimensionless Temperature ◽  
Polymerization Degree ◽  
Nucleation Kinetics ◽  
Type Integral ◽  
Laplace Type

This paper is concerned with the theory of nucleation and growth of crystals in a metastable polymer melt with allowance for the polymerization of a monomer. A mathematical model consisting of the heat balance equation, equations governing the particle-radius distribution function and the polymerization degree is formulated. The exact steady-state analytical solutions are found. In the case of unsteady-state crystallization with polymerization, the particle-size distribution function is determined analytically for different space–time regions by means of the Laplace transform. Two functional integro-differential equations governing the dimensionless temperature and polymerization degree are deduced. These equations are solved by means of the saddle-point technique for the evaluation of a Laplace-type integral. The time-dependent distribution function, temperature and polymerization degree at different polymerization rates and nucleation kinetics are derived with allowance for the main contribution to the Laplace-type integral. In addition, the general analytical solution by means of the saddle-point technique and an example showing how to construct the analytical solutions in particular cases are given in the appendices. The analytical method developed in the present paper can be used to describe the similar phase transition phenomena in the presence of chemical reactions. This article is part of the theme issue ‘Heterogeneous materials: metastable and non-ergodic internal structures’.

scholarly journals Analytical solutions of mushy layer equations describing directional solidification in the presence of nucleation

2018 ◽  
Vol 376 (2113) ◽  
pp. 20170217 ◽  
Author(s):  
Dmitri V. Alexandrov ◽  
Alexander A. Ivanov ◽  
Irina V. Alexandrova
Keyword(s):  
Phase Transition ◽  
Supercooled Liquid ◽  
Crystal Nucleation ◽  
Supersaturated Solution ◽  
Pure Liquid ◽  
Particle Nucleation ◽  
Nucleation Kinetics ◽  
Differential Model ◽  
Phase Transition Interface ◽  
Phase Transition Boundary

The processes of particle nucleation and their evolution in a moving metastable layer of phase transition (supercooled liquid or supersaturated solution) are studied analytically. The transient integro-differential model for the density distribution function and metastability level is solved for the kinetic and diffusionally controlled regimes of crystal growth. The Weber–Volmer–Frenkel–Zel’dovich and Meirs mechanisms for nucleation kinetics are used. We demonstrate that the phase transition boundary lying between the mushy and pure liquid layers evolves with time according to the following power dynamic law: , where Z 1 ( t )= βt 7/2 and Z 1 ( t )= βt 2 in cases of kinetic and diffusionally controlled scenarios. The growth rate parameters α , β and ε are determined analytically. We show that the phase transition interface in the presence of crystal nucleation and evolution propagates slower than in the absence of their nucleation. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

KINETICS OF OSTWALD RIPENING: CROSSOVER FROM WAGNER'S MODE TO THE LIFSHITZ–SLEZOV MODE

2007 ◽  
Vol 21 (15) ◽  
pp. 941-953 ◽  
Author(s):  
P. YU. GUBANOV ◽  
I. L. MAKSIMOV ◽  
V. P. MOROZOV
Keyword(s):  
Grain Size ◽  
Distribution Function ◽  
Ostwald Ripening ◽  
Early Stage ◽  
Intermediate Stage ◽  
Size Distribution Function ◽  
Supersaturated Solution ◽  
Kinetics Of ◽  
New Phase ◽  
Monomer Diffusion

The kinetics of the Ostwald ripening in a homogeneous supersaturated solution is studied both numerically and analytically. The time evolution of the grain-size distribution function in a new phase is theoretically described, taking into account the finite value of the maximal size of a grain. Two situations are considered: the kinetics of grain growth is controlled either by the grain-monomer reaction process (an early stage) or by the monomer diffusion process (a late stage). A transition to the final distribution is shown to take place through an intermediate-asymptotical mode of the Ostwald ripening kinetics, the crossover of the kinetic indices is demonstrated, and the duration of intermediate stage is evaluated.

Modelling Solid Nucleation and Growth In Supercooled Liquid

1999 ◽  
Vol 580 ◽  
Author(s):  
J.P. Leonard ◽  
James S. Im
Keyword(s):  
Random Number ◽  
Thermal Evolution ◽  
Supercooled Liquid ◽  
Nucleation And Growth ◽  
Nucleation Kinetics ◽  
Time Step ◽  
Si Films ◽  
Solid Nucleation ◽  
Random Nature ◽  
Induced Crystallization

AbstractIn this paper, we present a numerical model that incorporates algorithms to simulate nucleation and growth in supercooled liquid in a manner that properly accounts for the stochastic nature of nucleation. The basis of our model relies on a discretization of space and time to address thermal evolution, rapid growth of the undercooled interface, and nucleation in supercooled liquid. The present formulation of nucleation permits the spatially and temporally random nature of the phenomenon to be manifested in the transformation and resultant microstructure. This is accomplished by (1) calculating the probability of nucleation in each and every liquid node during each time step using the Poisson expression, and (2) triggering nucleation if and only when the random number assigned to a node for the time step is less than the calculated nucleation probability. No empirical or deterministic conditions for nucleation are imposed; nucleation occurs spontaneously and solely based on nucleation kinetics. We demonstrate the effectiveness of the overall model by analyzing conditions similar to those encountered in pulsed laser-induced crystallization of thin Si films, and discuss the generality of the proposed stochastic formulation of nucleation.

DETERMINATION OF THE DISTRIBUTION FUNCTION OF THE TIME BETWEEN FAILURES OF A WEAPON MODEL WITHOUT TAKING INTO ACCOUNT THE ENEMY’S FIRE IMPACT

2019 ◽  
pp. 39-45
Author(s):  
M. Sliusarenko ◽  
O. Semenenko ◽  
T. Akinina ◽  
O. Zaritsky ◽  
V. Ivanov
Keyword(s):  
Distribution Function ◽  
Analytical Description ◽  
Time To Failure ◽  
Missile Defense ◽  
Incorrect Choice ◽  
Military Equipment ◽  
Time Between Failures ◽  
The Military ◽  
Fire Impact

In the article, based on the analysis of the requirements for the readiness of weapons and military equipment during combat use and the reliability of their operation in the course of combat operations, it was discovered that one of the reasons that causes a discrepancy between the declared failures and real ones may be the incorrect choice and justification of the time distribution function up to the refusal of military means. As a rule, during the development of these tools, the function of distribution of time to failure is chosen by analogy with similar patterns of weapons and military equipment. In the theory of reliability, special attention is given to choosing the function of time-breaking non-response (failures or failures). Therefore, the article deals with the questions of evaluating the effectiveness of functioning of complex systems and methods of modeling the processes of their functioning, taking into account the laws of the distribution of random variables. The discrepancy between the declared irregularity of the military apparatus and the fact that is actually observed in the troops can be explained by the incorrectly accepted hypothesis about the distribution of time to failure. Therefore, the article analyzes the order of the justification of such a function without taking into account the enemy's fire impact and the proposed variant of determining the function of distribution of the time of work until the refusal of the model of military equipment. The article also cites the reasons for the discrepancy between the claimed missile defense equipment and what is actually observed in the troops. The proposed mathematical model of faultlessness, which at stages of designing and design will allow to set requirements to the model of technology with the help of analytical description. The sequence of calculations of non-failure indexes based on the use of Weibull distribution is substantiated.

The solution of an infinite set of differential-difference equations occurring in polymerization and queueing problems

1963 ◽  
Vol 59 (1) ◽  
pp. 117-124 ◽  
Author(s):  
A. Wragg
Keyword(s):  
Difference Equations ◽  
Queueing Theory ◽  
Bessel Functions ◽  
Formal Solution ◽  
Monomer Concentration ◽  
Arrival Rate ◽  
Time Dependent ◽  
Type Integral ◽  
Dependent Parameter ◽  
Infinite Set

AbstractThe time-dependent solutions of an infinite set of differential-difference equations arising from queueing theory and models of ‘living’ polymer are expressed in terms of modified Bessel functions. Explicit solutions are available for constant values of a parameter describing the arrival rate or monomer concentration; for time-dependent parameter a formal solution is obtained in terms of a function which satisfies a Volterra type integral equation of the second kind. These results are used as the basis of a numerical method of solving the infinite set of differential equations when the time-dependent parameter itself satisfies a differential equation.

Formulation of a Mathematical Process Model for the Foaming of a Mesophase Carbon Precursor

1992 ◽  
Vol 270 ◽  
Author(s):  
S. S. Sandhu ◽  
J. W. Hager
Keyword(s):  
Process Model ◽  
Analytical Description ◽  
Nucleation And Growth ◽  
Bubble Surface ◽  
Bubble Nucleation ◽  
Mesophase Pitch ◽  
Experimental Production ◽  
Mathematical Equations ◽  
Experimental Effort ◽  
Mathematical Process

ABSTRACTMathematical equations have been formulated to guide an experimental effort to produce an open-celled mesophase pitch foam. The formulation provides an analytical description of homogeneous bubble nucleation and growth, diffusion of the blowing gas through the liquid to the bubble surface, and the average material thickness between bubbles. Implications of the formulation for the experimental production of mesophase pitch foam are discussed.

scholarly journals Properties of Projection Lines in the Space of the Orientation Distribution Function

1989 ◽  
Vol 10 (3) ◽  
pp. 243-264 ◽  
Author(s):  
A. Morawiec ◽  
J. Pospiech
Keyword(s):  
Distribution Function ◽  
Pole Figure ◽  
Orientation Distribution Function ◽  
Analytical Description ◽  
Orientation Distribution ◽  
Orientation Space ◽  
The Relationship ◽  
Symmetry Operations

The relationship between the orientation distribution function (ODF) and the pole figure is based on the geometry of projection lines in the orientation space.The paper presents an analytical description of the projection lines and their transformations by symmetry operations. Using simple algebraical rules some properties of the projection lines as well as some properties of the associated projection lines (coupled due to the centrosymmetry of the pole figure) have been derived.

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