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Simulation of the growth of a binary composite by a controlled thermal annealing
R. Pandey, R. Sinkovits, E. Oran, J. Boris
To cite this version:
R. Pandey, R. Sinkovits, E. Oran, J. Boris. Simulation of the growth of a binary composite by a controlled thermal annealing. Journal de Physique I, EDP Sciences, 1994, 4 (10), pp.1427-1438.
�10.1051/jp1:1994197�. �jpa-00247002�
Classification Physics Abstracts
05.50 64.75 81.20E
Simulation of the growth of a binary composite by a controlled thermal annealing
R-B- Pandey (~), R-S- Sinkovits (~), E-S- Oran (~) and J.P. Boris (~)
(~) The Program in Scientific Computing, Department of Physics and Astronomy, University of Southern Mississippi, Hattiesburg, MS 39406-5046, U-S-A-
(~) Laboratory for Computational Physics and Fluid Dynamics, Naval Research Laboratory, Washington, DC 20375, U-S-A-
(Received 3 June 1994, received in final form 13 June 1994, accepted 21 June 1994)
Abstract. The Metropolis algorithm is used to study the evolution of the density profile
of partiale species in a model binary composite material on a simple cubic lattice. A speci-
fied fraction of the lattice sites are occupied by partiales of type A and the remaining sites are occupied by partiales of type B. The Hamiltonian for the system includes both nearest-
neighbor partiale-particle interactions and the interaction of the partiale with
a gravitational
field. Partiale-partiale interaction strength, gravity, and the temperature govem the hopping of each partiale in our annealing process. Variation of the planar density of the system by partiale type is studied as a function of annealing time, temperature, and the volume fraction of the two components. We observe a variety of density distributions such as a hnear density gradient
with the thickness, staircase hke variation in mass distribution, and their combinations over the
length scales which depend on these pararneters. The simulations show that a graded material with a desired density distribution cari be designed by appropriately controlling the annealing period, temperature, and the volume fraction.
1 Introduction.
In recent years, a variety of expenmental methods [1-10] bave been used to design composite
materials with prespecified physical properties. Methods including vapor deposition, thermal treatments (annealing and quenching), sintering, and squeeze casting bave been applied to materials such as aluminum alloy metal-matrix composites iii, aluminum /short liber aluminum
powder composite [2], iron and iron-cobalt alloy ultrafine powders [3], yttrium-doped tetragonal
zirconia powder compacts [4], silicon carbide [Si etc. In addition, there have been efforts to model trie growth processes by computer simulations [11-13]. For example, trie growth of materials such as Sim Gem/Si superlattice and pnn+ doped homoepitaxial Si by molecular beam epitaxy bas been studied by experiments [13] and by computer simulations [12]. In such models
lot R'Al DE Pl11slQUL1 T J ' la o(TOBLR ltJ>J
il Ii, partiales are deposited on trie substrate by a variety of methods such as diffusion limited
aggregation, ballistic deposition, and random deposition both with and without relaxation or diffusion of trie partiales after they land on trie substrate. Trie temporal growth of trie surface and scaling of its roughness bave been trie primary interest in these studies [11-13]. Very little attention is focussed on the effect of temperature in such modeling particularly in deposition
models [12].
The kinetics of aggregation and segregation in a mixture of particles with various size distri- bution in continuum media bas been extensively studied by both Monte Carlo and molecular dynamics simulations. Several attempts bave been made recently to understand trie effect of
shaking on trie distribution of partiales in powder mixtures. Monte Carlo simulations of a mix- ture of spherical partides of different sizes show that shaking leads to a phase separation with trie larger partiales on trie top of trie sample [14]. Studies of trie segregation in a deposition
model of sliding sphencal partiales of various sizes ils] show that trie degree of segregation
increases rapidly as trie fraction of large partiales is increased. Effects of vibration on trie structure of the granular state is discussed by Herrmann [16] while the sintering of nor~cohesive
nonsphencal partiales is studied by Kohring Ii?i; these studies also support the segregation
process.
Here we consider the effects of temperature on particle aggregation, segregation and mixing
in a heat-treatment process such as annealing followed by abrupt quenching. For a granular system, we regard temperature as an effective granular vibrational temperature due to agitation
of trie system. Further, we incorporate the effects of gravity in this model fabrication process.
For simplicity we consider a binary mixture of two type of partiales, one in trie matrix of trie other as in substitutional alloys.
2 Mortel.
We consider a three-dimensional lattice of size L x L x L, where L is trie lir~ear size of trie system in units of trie lattice constant. Init1ally a fixed number of partiales of type À, Np,
are randomly distributed at trie lattice sites; two partides are not allowed to occupy the same lattice site. Thus, a fraction p of the lattice sites are randomly occupied by these partides.
Trie remaining sites are occupied by partides of type B. Trie initial configuration is ther~ a random mixture of two species and can represent a substitutional alloy ÀpBi-p, where p is trie concentration, 1.e., trie volume fraction of partides A. We assume that both types of partides
bave trie same size (with trie diameter equal trie lattice constant), but particles A are heavier than B. Then trie gravitational force on B is comparatively negligible if B is much lighter than A. Partides A are mobile in the background matrix of B partides.
We introduce a nearest neighbor repulsive interaction between A- A partides and attractive interaction between A B partides. This is described by the interaction Harniltonian,
H = J~j p~pj h~j p~i(A), il)
where a unit density p is assigned to each partide, such that p
= for partide A and p
= -1
for empty site ii-e- for partide B). Trie quantity J is trie interaction strer~gth which is positive,
and h is trie strength of an externat field, in this case gravity. Trie interaction energy betweer~
trie neighboring sites and j is defined as E~j = Jp~pj. Trie second sum in equation il) is taken
over trie particles of type A with1(A) as their z-coordinate. Trie z-axis is trie vertical direction, 1-e-, the direction of trie gravitational field. Such an interaction Hamiltonian of trie Ising type, but without the gravitational term, has been studied extensively by computer simulations [18].
The Ising model has been used for a long time to study the structural properties of bir~ary alloys, as described in standard texts in statistical mechanics and solid state physics. For a bir~ary composite with particle size much larger than the atomic scale where the microscopic
interaction (such as Coulomb, Lennard-Jones, etc.) is not well established, a pher~omenological
interaction such as shown in equatior~ il) is one possible startir~g point.
Without trie second term, equation il) is a simple Ising model with trie Kawasaki dyr~arnics
used here to conserve trie concentrations of partiales A ar~d B. Trie equilibrium thermodynamic
properties of such simple Ising model bave been studied extensively, usir~g both trie Glauber and Kawasaki dynamics particularly neon trie magnetic critica1point [18]. Usir~g trie sanie simple Ising model with the ferromagr~etic interaction ii-e- with r~egative J), a damage spreading phenomenon is studied by several researchers in recent years [19] in presence of a temperature gradient. Dynamic critical phenomer~a and the ir~terfce energy [20] between the domains of
oppositely aligned spins and the surface tension [21] have also beer~ ir~vestigated in detail in recent years. Most of these investigations are, however, concentrated on trie critical properties ii-e- trie critical exponent and trie universality of trie phase transitions) [19-21]; trie order parameter does not seem to be conserved in these studies. We would like to point out that our model differs from these studies: il we consider an antiferromagnetic interaction to enhance
mixing between A and B, (2) a spatial gradient field is consider to segregate heavy partiales,
and (3) we emphasize on the kinetics of restructuring trie density distributions due to interplay betweer~ trie competing effects of these two interactions and trie temperature. Trie study of trie thermodynamic phase transitions of this model may be ir~teresting. However, we limit
our studies here to trie structural properties when the system is far from equilibrium (due to
gradier~t field and quenching) in an attempt to develop models to study trie density distribution
in composites.
3. Results of simulations.
The computations described below were done on a 50 x 50 x 50 simple cubic lattice; calculations
performed on larger systems showed no finite-size effects (see below). Trie reduced temperature is measured in units of J/kB. Trie field is applied along trie z-direction with trie top of trie
simulation box defined as z
= 1; trie field strength h is set to unity. Figure shows a typical snapshot of partides quenched alter heating trie system for various penods of time at a fixed
temperature T
= 0.50 and concentration p = o.50. Trie init1al configuration is a random mixture of A and B in which partiales A are interdispersed almost uniformly into trie matrix of
partiales B. In 500 MCS time steps, trie density of A at trie bottom begins to increase. In 5000
MCS, trie mixture is quenched into a composite with a compact region of A at trie bottom,
a small B-ricin region on trie top and a large regime of A B mixed phase in between. On increasing trie annealing time (10,000 MCS), these three phases become comparable in their
size in trie composite. After a long anneahng time (100,000 MCS), trie thickness of trie A B mixed phase decreases considerably with only A in lower planes, B in trie upper planes, and
a rougir interface in between. This shows that trie annealing time plays a very important rote in distributing trie constituents in a binary composite material resulting in different density
distributions of A, B and A B mixed phase.
Since trie gravitational field drives A in trie z-direction, trie planar density in trie yz-planes along trie x-direction varies as a function of time. Trie planar der~sities in trie xz- and xy-planes along trie y- and z-directions remain almost unchanged from their init1al values as a fur~ction of
time. In trie remainder of trie paper we will consider only trie temporal and spat1al evolutions of trie density of A in yz-planes. Figure 2 shows trie evolution of a typical density profile, 1-e-,
Fig, l. Snapshots of the particle distributions ai annealing times t = 20, 160, 640, and 1,280 MCS.
Sample size 50 x 50 x 50 was used with p
= 0.50 and T
= 1.00.
trie planar density of A versits trie thickness of trie sarnple as a function of ar~r~ealing time at constant temperature T
= 1.00 with 50Sl concentration of each type of particle. Trie ir~it1al
density of A is approximately 0.50 in ail yz-planes. In a short annealing time (20 MCS), trie density of A remains almost trie same in trie bulk (planes 10 to 40), with a drarnatic change in
density at the few planes at the top and at trie bottom. Trie planar density begins to decrease ai trie top and increase at the bottom. After a longer annealing lime (160 MCS), there are three distinct density regimes: a higher-density regime which increases at trie bottom ix = 35-50), a lower-density (about 0.50) regime in trie bulk ix
= 20-35), and a regime where trie densities decrease at trie top in trie remaining yz-planes.
Trie qualitative features of trie growth and decay of trie density profile, shown in figures 2a and b remain trie sonne for a number of independent calculations. Therefore, we will use only
one ir~dependent calculation in trie following analysis. Simulations of system with lattices of
o-s
-=- (t m 20 mcs)
-à- (t - 40 mes)
-c- (t
m 80 mes)
~-- (t m 160 mes)
- (t 320 mes)
-A- (t 640 mes)
- (t m1280 mes)
10 20 30 40 50
X
a)
/ O,
, ,
j
,
f
f j £ 1"
q # /
/
~ / fi
Î ~ '
T '
à
© ,
0. ;
c i
É~ ,À j ,f ~+- jt zo Mcs)
/
,~fi ÎÎ ~ÎÎÎ~~
O.Z
,
b é ( 6. (t 160 MCS)
£ / O A.~ (t 320 MOS)
, O ~o- (t 640 MOS)
f O ~W~ (t 1280 MCS)
o-o
10 20 30 40 SO
X
b)
Fig. 2. Planar density of partiales A versus depth ~ at trie temperature T
= 1.00 and concentration
p = 0.50 with vanous anneahng lime (20 to 1,280 MCS). A 50 x 50 x 50 sample was used with the
number (NRUN) of the mdependent ruas NRUN
= 1 la) and 20 (b).
different sizes (Figs. 2 and 3) do not show a significant difference in trie qualitative behavior from that of the 50 x 50 x 50 sample.
The different density regimes, A~rich, B-rich and A-B mixed, exhibit a well defined interface at low temperatures (T < 1.00). The sharp change in density at the interfaces leads to the
staircase~like density profiles seen in the figures. Figure 4 shows a typical growth rate of the À-rich planes in the lower part of the sample at T
= 1.00. Trie growth of the front Xf shows the power-law behavior,
Xf = Clé (2)
where C and k are constants that depend on both trie temperature and trie species concen-
tration; trie power-law exponent k
~- 0.70 at p = 0.50, depends on trie temperature ar~d trie concentration. At higher temperatures Ii.e. T
= 5.00), even the identification of an interface is difficult when the annealing time is long and the density varies from plane to plane throughout
the sample.
When trie temperature is lowered to T
= 0.50, the development of trie density profile slows down considerably. There is no substant1al change in trie density profile even after 1000 MCS:
a few more planes become denser at trie bottom and less dense at trie top (see Fig. 5). Trie
growth and decay pattems of trie density profile of À in trie lower and trie upper planes behave
in a manner very similar to that observed at trie higher temperature T = 1.00. However, trie
time in which trie density distribution shows a similar change in profile is increased drastica1Iy.
When the temperature is increased to T = 5.00, trie shape of the density profile changes quickly, as shown in figure 6. From 640 to 1280 MCS, the density gradient is nearly linear with
a very low density at the top and high density at the bottom. This shows that an annealing
process at high enough temperature, followed by quenching, leads to a graded material that has a linear density gradient. These calculations indicate that a composite materai with a
desired density profile could thus be designed by monitoring the temperature, annealir~g time, and the concentration of its constituents in a controlled fashion.
Figure 7 shows the planar density of À in the yz-planes as a function of z from a number of calculations performed at various temperatures and at a very long annealing time (12,888 MCS). The density increases almost hnearly as a function of depth at higher temperature
(T = 8.00). Àt low temperatures, there is a steep gradient in the density of À: there is almost
no À in the upper Iayers, a slow increase m density towards the bottom (Iast 10-15 Iayers),
and a linear density gradient in between (planes 20-35) before the density saturates at its high
value at the bottom. The slope of the hnear density gradient decreases as the temperature increases.
When the concentration of À is lower, as shown in figure 8 for p
= 0.30, there is no significant change in trie growth rate of the density profiles with short annealing time except that trie mixed regime has a lower concentration of À. However, at trie long annealing times (1,280 MCS), there
is a linear density gradient in the lower half of trie sample (from z
= 30 to 50). We would hke to point out that trie variations of our planar density profiles seem to be different from that of trie cross-section profiles of trie order parameter of a recent study of a related model by Puri et al. [22].
The fabrication process descnbed by this model is a nonequihbrium phenomenon, as can be
seen from trie energy profiles shown in figure 9. Figure 10 shows the rms displacement of each À
particle on average and that of their center of mass as a functior~ of time. These displacements show a power-law behavior in limited time regimes. Furthermore, trie power-law exponents depends on the temperature. As we have seen above, the density distribution depends on the annealing time and the temperature. Therefore, the collective as well as ir~dividual partide dynamics during the annealing process affect the mass distribution in the composite.
,Z
,' É
O.B ,
, / ,'
~i 1 ,1
;,' / A '
ç~ ,,p f £
~ ji À'
2~ " /
T 4
fi
© à
~
~ , ~
,~ ,~ i~~ ~Î. Î ÎÎÎÎÎÎÎ
A j ~jf ~* lt 80 MCS)
/ ,, -Ô- (t 160 MOS)
,~ ,À Î$. Î Î ÎÎÎÎÎÎÎ
,A A ~*. (t 1280 MOS)
£
5 10 1S 20 25 30
X
a)
' é
ù-B 4
O j
-
à
~i ) ] ~
-
2~
àT
© à à 0,4
5Î
O Î
~~ (t-20MCS)
£ ~ ~" (t 40 MOS)
0.2 f ~= ( jj~m~jj~
ÎÎ Î ÎÎÎÎÎÎÎ
+~ lt 1280 MOS)
20 40 50 80 100
X
b)
Fig. 3. Same as figure 2a with samples 30 x 30 x 30 (a) and 100 x 100 x 100 (b).
~~~~,«'""'"
i o
~,ù"""'
a ,,""'
é
~~~~,'
""""
4
000 Tme
Fig. 4. Position of a growth front plane versus time on a log~log scale. We have considered the front plane position ai the density of A-partiales about 1/3. The statistics are the same as m figure 2a.
Ô4 i-
d~
2f
Cà
~
# 0.4 éÎ
-C- (tm 20mcs)
-à- (t 320 mes)
o~ ~- lt 640 mes)
+ (tm1280mcs)
10 20 30 40 50
X
Fig. 5. -Planar density of partiales A versus ~ ai the temperature T
= 0.50, and concentration p = 0.50 with the same statistics as figure 2a.
o-s
~é4
2~
7 j à fi
°~ ~-
-o-
(t - mes)
-c- (t m160 mes)
0.2 - - 320
-A- (t - 640
- (t
10 20 40 50
X
Fig. 6. -Planar
density of
p
o-s
ç~
°~
2~
T
fi
©à
à
°-
-à-
RH (T =
o-Z -A- =
- (T 8.00)
10 20 30 50
X
Fig. 7. -
Planar density
(t
o-s
+H (t m 20 MOS)
-6- (t m 80 MOS) RH (tm160MCS)
+ (t 320 MCS)
-
0.6 -à- (t 640 MCS)
~i - (t 1280 MOS)
2~
1 fi
~ ) o-
10
X
ig.
-Q- lT O.Sù)
-à- (T 1.00)
R- (T 5,00)
s
~
i~1
uJ 6
4
2
200
(me
Fig.
à
E
%
ô W
~ l
OE
= (T 0.50)
- (T 0.50)
-à- (T 1.00)
-A- (T 1.00)
R- (T S.00)
- jT 5.00)
io ioo iooo
Tme
Fig. 10. rms displacement of each partiale (open symbols) and that of their center of mass (filled symbols) versus lime on a log~log scale at different temperatures at a fixed concentration p
= 0.50
using the same statistics as figure 2a.
4. Summary.
We have presented a computer simulation mortel for designing a composite mater1al of two
types of partides. Thermal annealing of a random mixture is modeled using the Metropolis algorithm in a Monte Carlo simulation. We have shown that a graded model composite can be
fabricated by controlled annealing, varying trie temperature and concentration of trie particles.
It is diflicult to compare such qualitative findings with trie quantitative experimental data due to lack of our knowledge about trie interaction among trie partides. However, this work points
out trends that con be venfied and tested.
Although it is often difficult to relate trie Monte Carlo time step to a realtime, a rough order of magnitude estimate can be made based on the partiale properties. For example, for
an aluminum partide of diameter one micron and mass 2.5 x 10~~~ kg at the temperature T = 1000 K with the strength of interaction unity, time to hop a partide is of trie order of 10~~ s. Therefore, one MCS is of trie order of10 s and a typical annealing time of1000 MCS is equivalent to about three hours.
Acknowledgements.
This work was done at the Naval Research Laboratory where RBP spent a summer in a NAVY-
ASEE summer faculty research program. We thank the referee for useful commer~ts.
