Molecular-dynamics simulation of directional growth of binary mixtures
P. Z. Coura
Departamento de Fı ´sica, Instituto de Cie ˆncias Exatas, Universidade Federal de Juiz de Fora, Juiz de Fora, MG,
CEP 36036-330, Brazil
O. N. Mesquita and B. V. Costa
Departamento de Fı ´sica, ICEx, UFMG, Belo Horizonte, MG, CP 702, 30123-970, Brazil
Received 17 July 1998
We use molecular dynamics to simulate the directional growth of binary mixtures. Our results compare very
well with analytical and experimental results. This opens up the possibility to probe growth situations which
are difficult to reach experimentally, being an important tool for further experimental and theoretical develop-
ments in the area of crystal growth. S0163-18299902405-4
I. INTRODUCTION
The main aim of this work is to show the feasibility of
using molecular-dynamic computer simulations to study di-
rectional growth of binary mixtures at atomic level. In com-
puter simulations, we can easily vary parameters of this pro-
cess and investigate regions of parameter space which are
difficult to access experimentally. It is then possible to make
predictions that might be useful for basic science or techno-
logical purposes.
The rapid expansion of the use of high quality crystalline
materials in optical and electronic devices during the past
decades has strongly stimulated research, both theoretical
and experimental, on dynamics of crystallization. A better
understanding about solidification of metals and eutectic fi-
bers are of unquestionable technological interest. Computer
simulations have played an important role in the develop-
ment and understanding of models of crystal growth.
1,2
During growth, the crystal-fluid interface is not at thermo-
dynamic equilibrium. The moving interface is a dynamical
system, which can display a variety of dynamical instabilities
and pattern formation. It has become a very important model
system for studying complex spatiotemporal dynamics.
3
A crystal can grow from the adjacent fluid melt, vapor, or
solution by different mechanisms, depending on the struc-
ture of the interface rough or smooth, material purity,
growth rates, temperature gradients, and related factors. For
a crystal to grow: i atoms or molecules must be trans-
ported from the fluid phase towards the interface where the
phase transformation is taking place; ii transported atoms
or molecules must have a nonzero probability of sticking to
the crystal surface; iii the latent heat generated during crys-
tal growth as well as the excess solute components segre-
gated must be carried away from the interface.
These requirements can be met in a controlled way in
experiments of directional growth, where a sample in an ap-
propriate furnace is submitted to a temperature gradient and
pulled with a fixed speed towards the colder region of the
furnace. For practical crystal growth, the sample can be cast
into a quartz tube with chosen diameter and length. This is a
three-dimension Bridgman growth arrangement. However,
detailed studies about dynamics of crystal growth have been
conducted in very thin transparent samples sandwiched be-
tween glass slides, such that the crystal-fluid interface can
be visualized and recorded with the use of videomicroscopy
techniques.
4,5
Results of such experiments have been com-
pared with results of two-dimensional models of crystal
growth. Our computer simulations are also carried out in two
dimensions.
As far as we know this is the first attempt to simulate
directional growth of a binary mixture utilizing molecular-
dynamics MD simulation techniques. Some earlier results
of MD were reported by Nijemeijer and Landau on laser
heated pedestal growth of fibers.
6
Previous simulations con-
sisted of numerical solutions of differential equations for
transport of heat and mass, and Monte Carlo techniques to
simulate attachment kinetics.
1,2
With the use of molecular-dynamics techniques we simu-
late the solidification of a two-component system consisting
of solvent atoms a and solute atoms b interacting via a
modified Lennard-Jones LJ potential. Particles interact via
three different potentials:
a , a
,
b , b
, and
a , b
=
b , a
which we will describe in detail in Sec. III. By tuning the
parameters of the LJ potential, we can choose the structure of
the interface rough or smooth and the segregation coeffi-
cient.
In this paper, we simulate a binary system with a rough
crystal-fluid interface like in metals and with a segregation
coefficient of the order of 0.1. The results of these simula-
tions are then compared with well-known models of segre-
gation during directional growth, where diffusion is the only
transport mechanism present. This work is organized as fol-
lows. In Sec. II we develop the theoretical background on
crystal growth of binary mixtures. In Sec. III we discuss the
simulation method we have used. In Sec. IV we show our
results and discussion, and in Sec. V we present our conclu-
sions.
II. DIRECTIONAL GROWTH OF BINARY MIXTURES
Most models of directional growth are two-dimensional
models.
7
Therefore, for comparison with these models, a
great deal of experimental observations have been done in
thin samples of transparent materials, where presumably the
PHYSICAL REVIEW B 1 FEBRUARY 1999-I VOLUME 59, NUMBER 5
PRB 59 0163-1829/99/595/34086/$15.00 3408 ©1999 The American Physical Society