Modeling of phase transitions in three-phase polymorphic systems: Part II. Effects of material characteristics on transi
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Nonequilibrium phase composition in multiphase systems affects physical properties of many materials. Development of phase composition is controlled by external conditions and material characteristics. Based on the model presented in Part I [A. Ziabicki and B. Misztal-Faraj, J. Mater. Res. 26(13), 1585 (2011)], rates of phase transitions in a three-phase model monotropic system composed of an amorphous (liquid) phase and two solid polymorphs have been analyzed. Effects of material characteristics including activation energy of molecular mobility, heat and entropy of the transitions, interface tensions, and concentration of predetermined nuclei have been discussed.
I. INTRODUCTION
The kinetic model described in Ref. 1 offers the possibility to predict development of phase structure in three-phase systems. The kinetic transition functions Eij describing progress of the transition of phase i into phase j result in nucleation and crystal growth and are controlled by a number of molecular characteristics specific for the material and individual transitions. According to the classical theory of phase transitions,2–5 two basic classes of kinetic processes can be distinguished: “growth” (with a linear rate R_ ij ) of a constant number, N0ij , of “predetermined nuclei” present in the system at the start of the process and “sporadic nucleation” (with rate N_ ij ) combined with simultaneous, n-dimensional growth Eijpre
2 t 33 Z ðt; TÞ 5 N0ij 4 R_ ij ðsÞds5
Steady-state nucleation rate for a transition i / j reads1: " # 1=2 ij E 32 r3ij Tij2 12 rij kT D st e kT exp N_ ij ðTÞ 5 pffiffiffiffiffiffiffiffiffiffiffi 2=3 ; 2 h kT Dh2 Tij T 6pkT v 0
ij
ð3Þ and the linear growth rate reads: 1=2 ij Dhij ðTij TÞ kT ED e kT h pkTTij 3 2 1=3 2 7 6 4rij Tij v0 exp4 5 ; kT Dhij Tij T
5=6 R_ ij ðTÞ 5 2v0
ð4Þ
0
Zt Eijspc
ðt; TÞ 5
2 t 33 Z N_ ij ðsÞ4 R_ ij ðzÞdz5 ds
0
:
ð1Þ
s
In steady isothermal conditions, nucleation rate N_ ij and growth rates R_ ij are constants and Eq. (1) reduces to the socalled Avrami equations: Eijpre
isothermal
ðt; TÞ/N0ij
R_ 3ij ðTÞ t3
1 isothermal Eijspc ðt; TÞ/ N_ ij ðTÞR_ 3ij ðTÞ t4 4
:
ð2Þ
a)
where Dhij is heat of the transition; Tij is the equilibrium transition temperature; v0 is the volume of a single kinetic element; rij is the interface tension; and EDij is the activation energy of molecular motions, h and k, respectively, Planck and Boltzmann constants. The frequency function based on sporadic nucleation 1=4 spc 4 vspc ðTÞ 5 E =t ij ij 3=8 11=24 1=8 ij r Dgij v0 kT ED 1=8 ij kT e 5 96 h ðpkTÞ1=2 " # " # 1=3 8r3ij 3r2ij v0 exp exp kTDg2ij kT Dgij
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2011.196
and that controlled by predetermined nucleation
1596
Ó Materials Research Society 2011
J. Mater. Res., Vol. 26, No. 13, Jul 14, 2011
;
ð5Þ
B. Misztal-Faraj et al.: Modeling of phase transitions in three-phase polymorphic systems: Part II. Effects of material characteristics on transition rates
pre 3 1=3 vp
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