Materials Science and Engineering A 441 (2006) 68–72
Shape factors in modeling of precipitation
E. Kozeschnik
a,b,∗
, J. Svoboda
c
, F.D. Fischer
b,d,e
a
Graz University of Technology, Institute for Materials Science, Welding and Forming,
Kopernikusgasse 24, A-8010 Graz, Austria
b
Materials Center Leoben Forschungsgesellschaft mbH, Franz-Josef Straße 13, A-8700 Leoben, Austria
c
Academy of Science of the Czech Republic, Institute of Physics of Materials, Zizkova 22, Cz-61662 Brno, Czech Republic
d
Montanuniversit¨ at Leoben, Institute of Mechanics, Franz-Josef Straße 18, A-8700 Leoben, Austria
e
Austrian Academy of Sciences, Institute for Materials Science, Jahnstraße 12, A-8700 Leoben, Austria
Received 20 January 2006; received in revised form 8 June 2006; accepted 25 August 2006
Abstract
Recently, a model for the growth and dissolution kinetics of precipitates in multi-component multi-phase environments has been developed. In
the original treatment, the evolution equations for the size and chemical composition of the precipitates have been derived for spherical precipitates.
In this paper, shape factors are introduced, which provide extensions of the existing formalism to needle-shaped and disc-shaped precipitate
geometries. The mathematical formalism can be seamlessly incorporated into the existing model. The growth kinetics of precipitates with different
shapes are discussed and compared to the spherical case.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Precipitates; Precipitation kinetics; Shape factor
1. Introduction
In two recent papers [1,2], a new approach to the modeling
of precipitation kinetics in multi-component systems has been
developed. With the simplification that the precipitates have a
spherical shape, the Gibbs free energy G of a given volume of
matter with m precipitates and n components has been expressed
as
G =
n
i=1
N
0i
μ
0i
+
m
k=1
4πρ
3
k
3
λ
k
+
n
i=1
c
ki
μ
ki
+
m
k=1
4πρ
2
k
γ
k
.
(1)
The subscripts ‘0’ denote quantities related to the matrix. The
index ‘k’ denotes quantities related to the precipitate; ρ
k
is the
precipitate radius, γ
k
is the specific interfacial energy, and λ
k
accounts for the contribution of mechanical energy due to the
misfit volume between the precipitate and the matrix. N
0i
is the
number of moles of component i in the matrix, μ
0i
and μ
ki
are
∗
Corresponding author at: Graz University of Technology, Institute for Mate-
rials Science, Welding and Forming, Kopernikusgasse 24, A-8010 Graz, Austria.
Tel.: +43 316 873 4304; fax: +43 316 873 7187.
E-mail address: ernst.kozeschnik@tugraz.at (E. Kozeschnik).
the average values of chemical potentials in the matrix and in
the precipitates and c
ki
are the average values of concentrations
in the precipitates.
In the derivation of the model, it has been assumed that the
radius and the mean chemical composition of each precipitate
evolve according to the thermodynamic extremal principle of
maximum entropy production [3]. In non-equilibrium systems
under a constant temperature and pressure, the available Gibbs
energy is dissipated by three mechanisms: (i) interface migration
(Q
1
), (ii) diffusion in the matrix (Q
2
) and (iii) diffusion in the
precipitate (Q
3
). These quantities are discussed in detail in Refs.
[1,2].
Some experimental observations indicate, that the shape of
precipitating phases are often far from the spherical one and
they can be very well approximated by needle-shaped to disc-
shaped geometries with a fixed aspect ratio known from the
observations. The main aim of the paper is to develop shape
factors, incorporate them into the fundamental expressions for
G, Q
1
, Q
2
and Q
3
and provide an approximate treatment of
precipitation kinetics for needle-shaped to disc-shaped precip-
itates. Finally, the extended evolution equations are presented,
and the influence of the precipitate shape on precipitation kinet-
ics is discussed in detail on the base of comparison with the
spherical case.
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.msea.2006.08.088