A nonlinear system of time-dependent integro-differential equations that describes the processes of phase transforma- tions in metastable melts and solutions is investigated. Using the saddle-point method for calculating the Laplace-type integral, a complete analytical solution was constructed, which determines the dynamic dependencies of the crystal size distribution function and the supercooling (supersaturation) of the system. The maximum size of the growing crystals, the average number of crystals and their average size as functions of time are found. It is shown that to correctly describe the evolution of a metastable system, it is necessary to take into account both the fundamental contribution of the saddle-point method and the following four contributions of the asymptotic expansion. The theory under consideration is in good agreement with experimental data.

Topics
Nonlinear systems, Supercooling, Integro-differential equation, Mathematical modeling, Asymptotic analysis, Supersaturation, Public policy and governance
1.
Y. A.
 
Buyevich
 and
V. V.
 
Mansurov
,
J. Cryst. Growth.
 
104
,
861
–
867
(
1990
).
https://doi.org/10.1016/0022-0248(90)90112-X
Google Scholar
Crossref
Search ADS
 
2.
Y. A.
 
Buyevich
,
V. V.
 
Mansurov
, and
I. A.
 
Natalukha
,
Chem. Eng. Sci.
 
46
,
2573
–
2578
(
1991
).
https://doi.org/10.1016/0009-2509(91)80050-9
Google Scholar
Crossref
Search ADS
 
3.
D. V.
 
Alexandrov
,
Phys. Lett. A
 
378
,
1501
–
1504
(
2014
).
https://doi.org/10.1016/j.physleta.2014.03.051
Google Scholar
Crossref
Search ADS
 
4.
D. V.
 
Alexandrov
,
J. Phys. A: Math. Theor.
 
47
, p.
125102
(
2014
).
https://doi.org/10.1088/1751-8113/47/12/125102
Google Scholar
Crossref
Search ADS
 
5.
E. V.
 
Makoveeva
 and
D. V.
 
Alexandrov
,
Phil. Trans. R. Soc. A
 
377
, p.
20180210
(
2019
).
https://doi.org/10.1098/rsta.2018.0210
Google Scholar
Crossref
Search ADS
 
6.
D. V.
 
Alexandrov
 and
I. V.
 
Alexandrova
,
Phil. Trans. R. Soc. A
 
377
, p.
20180209
(
2019
).
https://doi.org/10.1098/rsta.2018.0209
Google Scholar
Crossref
Search ADS
 
7.
D. V.
 
Alexandrov
,
I. G.
 
Nizovtseva
, and
I. V.
 
Alexandrova
,
Int. J. Heat Mass Transfer
 
128
,
46
–
53
(
2019
).
https://doi.org/10.1016/j.ijheatmasstransfer.2018.08.119
Google Scholar
Crossref
Search ADS
 
8.
D. V.
 
Alexandrov
,
J. Phys. A: Math. Theor.
 
51
, p.
075102
(
2018
).
https://doi.org/10.1088/1751-8121/aaa5b7
Google Scholar
Crossref
Search ADS
 
9.
E. M.
 
Lifshitz
 and
L. P.
 
Pitaevskii
,
Physical Kinetics
(
Pergamon
,
Oxford
,
1981
).
Google Scholar
 
10.
I. M.
 
Lifshitz
 and
V. V.
 
Slyozov
,
J. Phys. Chem. Solids
 
19
,
35
–
50
(
1961
).
https://doi.org/10.1016/0022-3697(61)90054-3
Google Scholar
Crossref
Search ADS
 
11.
J. R.
 
Hunt
,
J. Fluid Mech.
 
122
,
169
–
185
(
1982
).
https://doi.org/10.1017/S0022112082002158
Google Scholar
Crossref
Search ADS
 
12.
Y.
 
Enomoto
 and
A.
 
Okada
,
J. Phys.: Condens. Matter
 
2
,
4531
–
4535
(
1990
).
Google Scholar
 
13.
A. V.
 
Alyab’eva
,
Y. A.
 
Buyevich
, and
V. V.
 
Mansurov
,
J. Phys. II France
 
4
,
951
–
957
(
1994
).
https://doi.org/10.1051/jp2:1994176
Google Scholar
Crossref
Search ADS
 
14.
D. V.
 
Alexandrov
,
J. Phys. A: Math. Theor.
 
48
, p.
035103
(
2015
).
https://doi.org/10.1088/1751-8113/48/3/035103
Google Scholar
Crossref
Search ADS
 
15.
D. V.
 
Alexandrov
,
J. Phys. A: Math. Theor.
 
48
, p.
245101
(
2015
).
https://doi.org/10.1088/1751-8113/48/24/245101
Google Scholar
Crossref
Search ADS
 
16.
D. V.
 
Alexandrov
,
Phil. Mag. Lett.
 
96
,
355
–
360
(
2016
).
https://doi.org/10.1080/09500839.2016.1225996
Google Scholar
Crossref
Search ADS
 
17.
E. B.
 
Treivus
,
Kinetics of Growth and Dissolution of Crystals
(
Leningrad State Univ.
,
Leningrad
,
1979
).
Google Scholar
 
18.
R. F.
 
Strickland-Constable
,
Kinetics and Mechanisms of Crystallization
(
Academic Press
,
London
,
1968
).
Google Scholar
 
19.
D. V.
 
Alexandrov
 and
A. P.
 
Malygin
,
J. Phys. A: Math. Theor.
 
46
, p.
455101
(
2013
).
https://doi.org/10.1088/1751-8113/46/45/455101
Google Scholar
Crossref
Search ADS
 
20.
D. V.
 
Alexandrov
,
Phil. Mag. Lett.
 
94
,
786
–
793
(
2014
).
https://doi.org/10.1080/09500839.2014.977975
Google Scholar
Crossref
Search ADS
 
21.
M. V.
 
Fedoruk
,
Saddle-Point Method
(
Nauka
,
Moscow
,
1977
).
Google Scholar
 
22.
J. W.
 
Mullin
 and
C. J.
 
Leci
,
AIChE J. Symp. Ser.
 
68
,
8
–
20
(
1972
).
Google Scholar
 
23.
J.
 
Carside
,
C.
 
Gaska
, and
J. W.
 
Mullin
,
J. Cryst. Growth
 
13/14
,
510
–
516
(
1970
).
https://doi.org/10.1016/0022-0248(72)90290-4
Google Scholar
Crossref
Search ADS
 
24.
D. V.
 
Alexandrov
,
P. K.
 
Galenko
, and
L. V.
 
Toropova
,
Phil. Trans. R. Soc. A
 
376
, p.
20170215
(
2018
).
https://doi.org/10.1098/rsta.2017.0215
Google Scholar
Crossref
Search ADS
 
25.
P. K.
 
Galenko
,
D. V.
 
Alexandrov
, and
E. A.
 
Titova
,
Phil. Trans. R. Soc. A
 
376
, p.
20170218
(
2018
).
https://doi.org/10.1098/rsta.2017.0218
Google Scholar
Crossref
Search ADS
 
26.
D. V.
 
Alexandrov
,
A. A.
 
Ivanov
, and
I. V.
 
Alexandrova
,
Phil. Trans. R. Soc. A
 
376
, p.
20170217
(
2018
).
https://doi.org/10.1098/rsta.2017.0217
Google Scholar
Crossref
Search ADS
 
27.
E. V.
 
Makoveeva
 and
D. V.
 
Alexandrov
,
Phil. Trans. R. Soc. A
 
376
, p.
20170327
(
2018
).
https://doi.org/10.1098/rsta.2017.0327
Google Scholar
Crossref
Search ADS
 
28.
D. V.
 
Alexandrov
 and
I. G.
 
Nizovtseva
,
Phil. Trans. R. Soc. A
 
377
, p.
20180214
(
2019
).
https://doi.org/10.1098/rsta.2018.0214
Google Scholar
Crossref
Search ADS
 
This content is only available via PDF.
© 2019 Author(s).
2019
Author(s)