Modeling of phase transitions in three-phase polymorphic systems: Part I. Basic equations and example simulation
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Development of phase composition in one-component, three-phase systems containing a liquid phase (melt) and two polymorphic solids has been discussed. Two types of polymorphic systems have been analyzed: enantiotropic systems composed of three thermodynamically stable phases and monotropic systems with two stable and one metastable phase. Detailed relations between transition rates, molecular characteristics, and external conditions have been derived. Simulation of isothermal crystallization of a model system has been performed and discussed. Gi ðTÞ 5 Hi T Si
I. INTRODUCTION
Existence of several phases with different physical properties plays important role in materials science. Various phases can appear and exist off thermodynamic equilibrium for a long time. It is the kinetic and thermodynamic factors that allow for temporary appearance of different phases at the same external conditions. Different crystallographic structures in metals, inorganic oxides, low-molecular organic compounds, and polymers may affect mechanical, electrical, and biological properties of materials. Understanding mechanisms governing creation of different phases is of great importance, practical as well as fundamental. The model of nucleation-controlled transitions in multiphase systems has been published by one of the present authors.1 The model concerns development of phase composition in various conditions and the ways in which phase structure can be controlled. The original model1 included an incorrect assumption, though. Kinetics of simultaneous and successive transitions implied processes with finite time steps rather than continuous transformations. The correct treatment will be presented below. Three-phase systems composed of an amorphous (liquid) phase and two polymorphic solid phases will be analyzed. This article includes basic equations and simulation of isothermal crystallization kinetics in various temperatures. In the forthcoming articles, effects of material characteristics and external conditions on phase composition will be discussed and model predictions will be compared with experimental data. II. THERMODYNAMICS
Isobaric free enthalpies (Gibbs’ free energies), Gi, for three phases are approximated by linear functions of temperature, T a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2011.195 J. Mater. Res., Vol. 26, No. 13, Jul 14, 2011
http://journals.cambridge.org
Downloaded: 14 Mar 2015
i 5 0; 1; 2 ;
ð1Þ
where enthalpy Hi and entropy Si are constants. The characteristic intersection temperatures, Tij, are defined by equal free enthalpies DGij ðT Þ [ Gj ðT Þ Gi ðTÞ 5 0
5 T 5 Tij 5
DHij DSij
: ð2Þ
Any thermodynamically admissible transition requires reduction of free enthalpy DGij ðTÞ 5 Gj ðTÞ Gi ðTÞ , 0 5 T DGij ðTÞ 5 DHij 1 ; Tij
“i” ! “j” ; ð3Þ
where DHji is enthalpy and Tij is equilibrium temperature of the transition. In a three-phase system, condition (3) can be satisfied at any temperature by three different transitions. For crystallization, T , Tij
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