## This content will become publicly available on March 4, 2020

## Phase transformations in metastable liquids combined with polymerization

## Abstract

Here, we report on 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 this paper can be used to describe the similar phase transition phenomena in the presence of chemical reactions.

- Authors:

- Ural Federal Univ. (Russian Federation)

- Publication Date:

- Research Org.:
- SLAC National Accelerator Lab., Menlo Park, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1547036

- Grant/Contract Number:
- AC02-76SF00515

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Philosophical Transactions of the Royal Society. A, Mathematical, Physical and Engineering Sciences

- Additional Journal Information:
- Journal Volume: 377; Journal Issue: 2143; Journal ID: ISSN 1364-503X

- Publisher:
- The Royal Society Publishing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; polymerization; crystallization; nucleation; metastable liquid

### Citation Formats

```
Ivanov, Alexander A., Alexandrova, Irina V., and Alexandrov, Dmitri V. Phase transformations in metastable liquids combined with polymerization. United States: N. p., 2019.
Web. doi:10.1098/rsta.2018.0215.
```

```
Ivanov, Alexander A., Alexandrova, Irina V., & Alexandrov, Dmitri V. Phase transformations in metastable liquids combined with polymerization. United States. doi:10.1098/rsta.2018.0215.
```

```
Ivanov, Alexander A., Alexandrova, Irina V., and Alexandrov, Dmitri V. Mon .
"Phase transformations in metastable liquids combined with polymerization". United States. doi:10.1098/rsta.2018.0215.
```

```
@article{osti_1547036,
```

title = {Phase transformations in metastable liquids combined with polymerization},

author = {Ivanov, Alexander A. and Alexandrova, Irina V. and Alexandrov, Dmitri V.},

abstractNote = {Here, we report on 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 this paper can be used to describe the similar phase transition phenomena in the presence of chemical reactions.},

doi = {10.1098/rsta.2018.0215},

journal = {Philosophical Transactions of the Royal Society. A, Mathematical, Physical and Engineering Sciences},

number = 2143,

volume = 377,

place = {United States},

year = {2019},

month = {3}

}