Etude par Sonde Atomique Tomographique de la formation de nano ...

tel-00751814, version 1 - 14 Nov 2012
Chapter 2. Materials, experimental and simulation techniques
the precipitate volume related part of the Gibbs energy. The left term takes the interface
between precipitate and matrix into account.
In thermodynamic equilibrium, the Gibbs free energy is a minimum. Since real systems
during heat treatments are in a highly non-equilibrated state, driving forces exist for an
evolution of the precipitate microstructure such that G is minimized. Three dissipative
processes have been considered in MatCalc software:
with
Q1: Migration of interfaces with mobility, Mk,
Q2: Diffusion of all atoms inside the precipitates;
Q3: Diffusion of all atoms in the matrix
Q
3
� m
n
��
k�1
i�1
Q
2
Q
1
� m
�
k�1
� m
n
��
k�1
i�1
2
4� rk
(2.18)
M
k
5
4�RTrk
c
45c
D
ki
0i
ki
0i
2
ki
3
4�RTrk ( �k
( cki
� c0i
) � rkc
c D
ki
/ 3)
The Gibbs free energy dissipation rate G, is given by sum of these three terms
2
(2.19)
(2.20)
–G=Q1+Q2+Q3. With the total Gibbs free energy and the corresponding dissipation terms, the
thermodynamic extremum principle [44] is applied and a linear system of rate equations for
the change of radius and chemical composition of each individual precipitate is obtained.
Further details about the model and the numerical treatment of the evolution equations are
given in [41–43].
c) Evaluation of interfacial energies
The interfacial energy γ and effective driving force F are dominant quantities in the
simulation of nucleation and growth of precipitates. These quantities appear during evaluation
of critical nucleation energy ΔG* (see Table 2.11). In contrast to F, which can be estimated
from thermodynamic databases, γ is not accessible by direct experimental measurement and is
often considered as fitting parameter. In MatCalc, γ is determined from the “nearest-
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