JOURNAL OF MATERIALS SCIENCE 37 (2002) 2645 – 2650
A CA/MC model for the simulation of grain
structures in solidification processes
TONGMING WANG, JUNZE JIN, XIANSHU ZHENG
Research Center of Foundry Engineering, Dalian University of Technology,
Dalian 116024, People’s Republic of China
E-mail: tmwang@mpd.ams.eng.osaka-u.ac.jp
A CA/MC model is established to simulate the grain structures of solidification processes. It
first describes the envelop of grain by using Cellular Automaton technique, then the
interior of grain is corrected by Monte Carlo method. In this paper, solute field is calculated
in microscopic scope, and the solute redistribution in solid/liquid interface is calculated by
using Actual Solute Partition Coefficient model. The CA/MC model is applied to simulate
the microstructure evolution of Al-4.5% (mass) Cu alloy in water-cooled Cu mould. It shows
that the simulation result is agreement with that obtained experimentally.
C
2002
Kluwer Academic Publishers
1. Introduction
In recent years, many models used for structure sim-
ulation have been developed. For example, Wang and
Beckermann developed the Multiscale and Multiphase
model, which is based on the volume averaging tech-
nique [1–3]. Brown, Spittle, Zhu and Smith established
Monte Carlo (MC) method based on the law of the
lowest energy [4–6]. Rappaz and Gandin developed the
Cellular Automaton (CA) technique to simulate struc-
ture formation [7], in which the physical mechanism of
growth of dendritic grains is involved. Subsequently, a
coupled Finite Element-Cellular Automaton model for
the prediction of grain structure is developed by Rappaz
et al. [8].
The aim of the present contribution is to present
CA/MC model, which combines CA technique and MC
method in an effective way. By CA/MC model, the evo-
lution of amount, size and appearance of grains can be
simulated better than those only by CA technique or
MC method.
2. Heat flow calculation
Thermal field is the necessary data for the simulation
of grain structures. In present study, thermal field at
any moment is calculated by Alternant Explicit Dif-
ference (AED) method, which absorbs the advantage
of Explicit Difference method and Implicit Difference
method. Here, the latent heat released is calculated ac-
cording to the solidification fraction f
S
, which can be
fed back from the results of structure simulation. The
heat latent released in unit volume,
˙
q,isgivenby
˙
q = ρ · h ·
∂ f
S
∂t
(1)
where ρ is density, h is heat latent.
3. Solute calculation
In this paper, solute diffusion is calculated by the Di-
rect Finite Difference method. In addition, a new model
based on structure simulation is proposed to calculate
the solute redistribution in the solid/liquid interface.
The rejected solute from the cell of phase change are
calculated according to the actual solute partition coef-
ficient k
a
, and the cell which phase change occurred can
be got by the structure simulation. It is because that so-
lute calculation and structure simulation share the same
meshes that are used for the microscopic simulation.
For a cell of phase change, the solute conservation
equation is
ρV
c
ω
L
+
n
i=1
ω
Li
= ρV
c
(ω
S
+ ¯ω
L
· n) (2)
where ω
L
is the solute mass fraction before phase
changing, and ω
S
is the solute mass fraction after phase
changing,
n
i=1
ω
Li
is the sum of solute mass fraction
of all the liquid cells neighbor to the current cell, and
n is the number of all the neighbor liquid cells, ¯ω
L
is
the averaging solute mass fraction of all the neighbor
liquid cells after phase changing, V
c
is the volume of
the current cell.
According to the definition of the actual solute par-
tition coefficient, there is
k
a
=
ω
S
¯ω
L
(3)
Here, ¯ω
L
is taken as the solute mass fraction of liquid
in the solid/liquid interface.
In addition, it should be pointed out that K
a
is not
equal to K
0
which is the solute equilibrium partition
coefficient.
In the near equilibrium situation, k
a
can be calculated
by the Button model [9]
k
a
=
k
0
k
0
+ (1 − k
0
)exp
−
R
D
L
δ
(4)
0022–2461
C
2002 Kluwer Academic Publishers
2645