On the gibbs-thomson effect in concentrated binary systems
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Communications
1
dm2a 5
On the Gibbs–Thomson Effect in Concentrated Binary Systems
1 2 x2a 1 2 x2b b b V dP 2 V ma dPa m x2b 2 x2a x2b 2 x2a
2
1
2
[6]
MA QIAN and L.C. LIM
Note that when considering a solid binary or multicomponent precipitate/matrix system, it is usually assumed that the pressure of the matrix is fixed at the atmospheric value.*[5]
The Gibbs–Thomson expression for the solubility of a precipitate (b) in a concentrated binary solution matrix (a) has been well documented and is given by[1–4]
*Kulkarni and Dehoff[6] have recently shown the difference between Pa(`) and Pa(r) in a unary two-phase system. There is, however, no solution to the difference between Pa(`) and Pa(r) in a binary or multicomponent system.
1
x2a(r) 5 x2a(`) 1 1 x2a(r)
1 2 x2a(`) 2sabV mb 2 x2a(`) RTr e2a
x2b(`)
2
[1]
dm2a 5
x2a(`)
where and are the equilibrium atomic fractions of component 2 (the solute) in a at a curved interface of a spherical precipitate (b ) of radius r and a planar interface, respectively; x2b(`) the equilibrium composition of b corresponding to x2a(`); sab the specific interfacial free energy; V mb the molar volume of b; and e2a, the Darken factor, is given by e2a 5 1 1
1
ln g 2a ln x2a(r)
2
a
a
[3]
x1b dm1b 1 x2b dm2b 5 2Smb dT 1 V mb dPb
[4]
where mi denotes the chemical potential of i in a or b, and V ma is the molar volume of a . Note that at equilibrium, 2sab r
[5]
b
where P (r) and P (r) are the pressures in a and b, respectively, with b occurring as a spherical precipitate of radius r. Equation [5] reduces to Pa(`) 5 Pb(`) when r → ` or for a planar interface. Solving Eqs. [3] and [4] simultaneously for dm2a at constant temperature T by invoking MA QIAN, formerly Teaching Fellow, Department of Mechanical and Production Engineering, National University of Singapore, is Senior Research Fellow, Cooperative Research Center for Cast Metals Manufacturing, Department of Mining, Minerals and Materials Engineering, University of Queensland, St Lucia, QLD 4072, Australia. L.C. LIM, Associate Professor, is with the Department of Mechanical and Production Engineering, National University of Singapore, Singapore 119260. Manuscript submitted December 9, 1999. METALLURGICAL AND MATERIALS TRANSACTIONS A
2
[7]
dm2a 5 RT d ln (g 2ax2a)
1
5 RT 1 1
x2 (r)5x2 (`)
x1a dm1a 1 x2a dm2a 5 2Sma dT 1 V ma dPa
Pb(r) 2 Pa(r) 5
1
1 2 x2a V mb dPb x2b 2 x2a
Then, using mi 5 m0i 1 RT ln ai and ai 5 gi xi , where ai is the activity of i in the phase under consideration, we have, for the left-hand side of Eq. [7],
[2]
in which gi is the activity coefficient of i in the phase under consideration. Equation [1] was first proposed by Purdy[1] for the limiting case when (x2(r) 2 x2(`))/x2(`) ¿ 1, that is, when both (x2a(r) 2 x2a(`))/x2a(`) ¿ 1 for the composition of the matrix (a) and (x2b(r) 2 x2b(`))/x2b(`) ¿ 1 for that of the precipitate (b) are met. In this article, a generalized description of the Gibbs– Thomson effect in a concentrated binary system will be provided. As will be shown
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