Multiscale Modeling of Theta ' Precipitation in Al-Cu Binary Alloys

2976 V. Vaithyanathan et al. / Acta Materialia 52 (2004) 2973–2987
2.4. Phase-field model
In the phase-field methodology, a precipitate microstructure
is described by a set of continuum order parameter
fields characterizing the difference in composition
and structure between the precipitate and matrix [11]. In a
two-dimensional representation of h 0 precipitates in a
binary Al–Cu alloy, the precipitate microstructure is described
by one composition field and two orientation-related
long range order parameter fields.
The chemical free energy of the microstructure, including
the bulk and interfacial energy, is then expressed
as
Z
F bulk þ F int ¼ f ðc; g 1 ; g 2 ; g 3 Þ þ a 2 ðrcÞ2
V
þ 1 X
b
2 ij ðpÞr i g p r j g p
!dV ; ð4Þ
p
where a and b ij ðpÞ are gradient energy coefficients for
composition and order parameters, respectively. b ij ðpÞ is
retained as a second rank tensor to incorporate the
(tetragonal) anisotropy in interfacial energy. The
anisotropy in interfacial energy of the precipitate is dependent
on the orientational variant, g, of the precipitate
(see Fig. 3). Since the phase-field model used in this
study is a dislocation-free model, semi-coherent interfaces
are treated effectively as coherent interfaces with a
larger interfacial energy.
b ij ð1Þ ¼ b
11ð1Þ 0
;
0 b 22 ð1Þ
b ij ð2Þ ¼ b
11ð2Þ 0
0 b 22 ð2Þ
c
sc
(η 1 = 0,
η 2 =1 or -1)
¼ b
22ð1Þ 0
:
0 b 11 ð1Þ
c
(η 1 =1 or -1,
η 2 =0)
ð5Þ
Fig. 3. Schematic of the h 0 precipitates in 2D showing the two variants
along with their equilibrium order parameters. The interfaces of the
variants are marked as coherent (c) and semi-coherent (sc) to show the
symmetry and the interface orientation dependence on the variants.
sc
The bulk free energy describing the phase-separation
and order-disorder transformation involved during
precipitation of the h 0 phase is approximated using the
following polynomial in c and g:
f ðc; g 1 ; g 2 ; g 3 Þ
¼ A 1 ðc C 1 Þ 2 þ A 2 ðc C 2 Þðg 2 1 þ g2 2 þ g2 3 Þ
þ A 41 ðg 4 1 þ g4 2 þ g4 3 ÞþA 42ðg 2 1 g2 2 þ g2 2 g2 3 þ g2 1 g2 3 Þ
þ A 61 ðg 6 1 þ g6 2 þ g6 3 Þ
þ A 62 fg 4 1 ðg2 2 þ g2 3 Þþg4 2 ðg2 3 þ g2 1 Þþg4 3 ðg2 1 þ g2 2 Þg
þ A 63 ðg 2 1 g2 2 g2 3 Þ;
ð6Þ
where C 1 and C 2 are constants close to the equilibrium
compositions of solid solution matrix and h 0 precipitates,
respectively. The coefficients A 41 and A 61 are constrained
to have negative and positive values, respectively, to
describe a first order phase transition [20].
The elastic energy contribution to the total free energy
of the system is obtained analytically using the
method of Khachaturyan [21,22]. The stress-free misfit
strain, 0 ijðrÞ, between precipitates and matrix, is expressed
as a function of the structural order parameters,
0 ij ðrÞ ¼X 0 ij ðpÞg2 p ðrÞ;
ð7Þ
p
where 0 ijðpÞ represents the stress-free strain of the pth
variant. With the homogeneous modulus approximation
(matrix and precipitates have similar elastic constants),
the elastic energy in terms of order parameters can be
expressed as [23]
E el ¼ V 2 k X
ijkl ij kl V k ijkl ij 0 kl ðpÞg2 p ðrÞ
þ V 2 k X X
ijkl 0 ij ðpÞ0 kl ðqÞg2 p ðrÞg2 q ðrÞ
1 X
2
p
p
q
X
Z
q
p
d 3 g
ð2pÞ 3 B pqðnÞfg 2 p ðrÞg g fg2 q ðrÞg g ;
ð8Þ
where k ijkl is the elastic stiffness tensor. B pq ðnÞ ¼
n i r ij ðpÞX jk ðnÞr kl ðqÞn l and X jk ðnÞ is the inverse matrix of
X 1
jk ðnÞ ¼ n ik ijkl n l . n ¼ g=jgj is the unit vector in reciprocal
space, fg 2 q ðrÞg g
is the Fourier transform of the
square of order parameter [g 2 q ðrÞ] and fg2 p ðrÞg g
is the
complex conjugate of fg 2 p ðrÞg g . ij ¼ 0 ij þ a ij
is the sum of
homogeneous strain caused by transformation and external
constraints or applied stress, respectively. In the
absence of applied strain, the fourth term in the elastic
energy (Eq. (8)) is the dominant term controlling the
precipitate morphology.
The temporal evolution of the microstructure in the
phase-field model is obtained by numerically solving the
Cahn–Hilliard and Allen–Cahn equations [24]
ocðr; tÞ
ot
¼ Mr 2 dF
dcðr; tÞ ;
ð9Þ