Moleular Dynamis Simulation of Morphologial
Instabilities in Solid-Fluid Interfaes
B.V. Costa a
, P.Z. Coura b
and O.N.Mesquita
Departamento deFsiaICEX,UniversidadeFederaldeMinasGerais
CaixaPostal702, 30123-970,BeloHorizonte,MG,Brazil
a
bvsia.ufmg.br; b
pablosia.ufmg.br;
mesquitasia.ufmg.br
Reeivedon1February,2001. Revisedversionreeivedon23May,2001
We perform moleular dynamissimulationsof diretionalgrowth of a binary alloy. We x the
temperaturegradient,pullingspeed,impurityonentrationandonlyvarytheimpuritysegregation
oeÆient. ByhangingtherangeoftheLennardJonespotentialofimpurityatomsasomparedto
therangeofthepotentialofsolventatoms,theelastienergyostausesadereasesofsolubilityof
theimpurityatomsintothematrix(solvent)solidphaseandonsequentlyadereaseintheimpurity
segregation oeÆient. Within ertain range of segregation oeÆients, the growing interfae is
planar;belowit,theinterfaebeomesunstableandaellularstrutureemerges.
I Introdution
Rapid growth of a solid-uid interfae under
dire-tional solidiation onditions is a ommon method
of investigating solidiation morphologies as well as
segregationduringrystalization[1℄. Morphologial
in-stabilities inthesolid-uidinterfaehavebeenstudied
foralongtime[2℄. Itisknownthatduringgrowth,the
rystal-uid (melt, vapor or solution) interfae, is not
at thermodynami equilibrium. Themoving interfae
is a dynamial system, that an display a variety of
dynamialinstabilitiesandpatternformation[3,4℄.
Therapidexpansionoftheuseofhighquality
rys-tallinematerialsinoptialandeletronideviesduring
thelastdeadeshasstimulatedresearh,both
theoret-ial and experimental, on dynamis of rystallization.
A better understandingonsolidiation ofmetalsand
eutetibersareofunquestionabletehnologial
inter-est. Computer simulations haveplayed an important
role on the development and understanding of reent
modelsofrystalgrowth[5-9℄. Arystalangrowfrom
theadjaentuid(melt,vapororsolution)bydierent
mehanisms,dependingonthestrutureoftheinterfae
(rough orsmooth),materialpurity, growthrates,
tem-peraturegradients andrelated fators. These
require-ments an be met in aontrolled wayin experiments
of diretional growth, where asample in an
appropri-pulledwith axed speed towardstheolder region of
the furnae. For pratialrystal growth, the sample
an be ast into a quartz tube with hosen diameter
and length. This is a 3-dimension Bridgman growth
arrangement. However,detailedstudiesabout
dynam-isof rystalgrowthhavebeenonduted inverythin
transparentsamples(sandwihedbetweenglassslides),
suh that the rystal-uid interfae an be visualized
and reorded with the use of videomirosopy
teh-niques [10, 11℄. The results of suh experiments have
beenompararedwithresultsoftwo-dimensional
mod-elsofrystalgrowth. Ouromputersimulationsarealso
arriedoutintwodimensions. Withtheuseof
moleu-lardynamistehniqueswesimulatethesolidiationof
atwo-omponentsystemonsistingofsolvent(atomsa)
andsolute(atomsb)interatingviaLennard-Jones(LJ)
potentials:
a;a ,
b;b and
a;b =
b;a
whih hasbeen
shown to desribe verywell the diretional growth of
abinary mixture [7, 8, 9℄. the method is desribe in
detailsin setion III.Bytuning the parametersofthe
LJpotential,weanhangethestrutureof the
inter-faeand thesegregationoeÆientsuhthat aplanar
interfaeangounstable.
In this paper, wesimulate a binary systemwith a
roughrystal-uid interfaeandwithseveral valuesof
the LJ interation potentialas disussed below. This
work is organized as follow. In setion II we briey
used. Insetion IVweshowourresultsand insetion
Vwepresentouronlusions.
II Diretional Growth of Binary
Mixtures
Agreatdealofexperimentalobservationsindiretional
growth have been done in thin samples of
transpar-ent materials, where presumably the third dimension
isnotimportant,andtherystal-uidinterfaeanbe
followed in time, by using videomirosopy with
digi-talimage analysis. Thegrowthfurnaeonsistsoftwo
metalbloksatontrolledtemperatures:oneblokwith
temperatureabovethesamplemeltingtemperatureand
the other below. A thin sample of the material to be
studiedissandwihedbetweenglassslideswithspaers,
whosethiknessisingeneralofafewmira,tokeepthe
systemasloseaspossibleofa2dgeometry,and
guar-anteethatthetransportofmassismainlydiusivewith
noonvetionintheuidphase. Thesampleisthenput
ontopofthemetalbloks,withgoodthermalontat.
A temperature gradient appears along of the sample
ell. Thesampleis thenpulledtowardstheoldblok
in a verypreise and ontrolled way bya pulling
sys-temandstartstosolidify. Afteraninitialtransientthe
systemahievesasteady-statewheretheinterfae
posi-tionbeomesxedinthelaboratoryframe,thegrowth
speedis the same, but opposite, as the pulling speed
andthesoluteonentrationprolebeomessteadyin
thelaboratoryframe. Morphologialinstabilitiesofthe
planarinterfaeareinhibitedduringdiretionalgrowth
of pure materials. However, for binary systems,
de-pendingononentrationofsolute,temperature
gradi-ent and growth speed, morphologialinstabilities an
our: Theplanarinterfaebeomesellularand
even-tually dendriti. This isthe soalled Mullins-Sekerka
instability(MS).Inoursimulationsweareableto
ob-serve both regimes: planar and ellularinterfaes. In
thisworkwewillfousourattentionontheevolutionof
aninterfaewherewelearlyseesegregation,transport
of solute at the interfae and morphologial
instabili-ties.
III Simulation
Oursimulationisarriedoutusingmoleulardynamis
withallpartilesinteratingthroughthepotential.
i;j (r i;j )= i;j (r i;j ) i;j (r ) d i;j (r i;j ) dr i;j ri;j=r (r i;j r ) (1) where i;j
istheLJpotential
i;j (r i;j )= " r i;j 12 r i;j 6 # : (2)
Theindexesiandjstandsforpartilesinthepositions
~ r
i and ~r
j
respetively, and 0 i;j N, where N is
thetotalnumberofpartilesand~r
i;j =j~r
i ~r
i
j. A
ut-o, r
=2:5
aa
is introdued in thepotentialin order
to aelerate thesimulation. If thefore onapartile
is found by summing ontributions from all partiles
atingupon it,then thistrunation limitsthe
ompu-tational eortto anamount proportional to the total
numberofpartilesN. Ofoursethistrunation
intro-dues disontinuities both in the potential and fore.
Tosmooththis disontinuity we introduea onstant
term (r
). Another term (d=dr)
r=r (r r
) is
in-trodued to remove the fore disontinuity. Partiles
in our simulation move aording the Newton's law,
that generate a set of 2N oupled equations of
mo-tion whih are solved by inreasing forward in time
the physial state of the system in small time-steps
of size t = 0:02
aa (m
a =
aa
). The resulting
equa-tions are solved by using theBeeman's method of
in-tegration. In order to improve the method we use
a Verlet and a elular table. The Verlet table
on-sists of an address vetor whih ontains the number
and position of eah partile inside a irle of radius
r
v = 3
aa
. After some steps in time, the
neighbor-hood of eah partilehanges, sothat, wehave to
re-fresh the Verlet table. This refreshment proess an
take a long time. In order to make it shorter we
di-vide the system in ells of size
x z = (3:5 aa ) 2 ,
suh that in realulating theVerlet table we haveto
searh only in neighbor ells. Inittialy we distribute
N =n
x n
z
=162110partilesoverthetwo
dimen-sionalsurfaeL
x L
z
=1622 1=6
1102 1=6
. We
assumeperiodiboundaryonditionsinthexdiretion.
Inthez diretionwedivide thesystemin twodistint
regions,asolidandauidone.Inthesolidregion
parti-lesstandinitiallyintheirequilibriumpositionina
to-talof16230partiles. Ontheuidregionthedensity
isinitially
f =0:5
2
aa
,givingatotalof16280
par-tiles, randomlydistributed in atriangularlattieand
slightlydisloatedfromtheirequilibrium position. We
imposeatemperaturegradient,G,alongthezdiretion
using aveloity renormalizationapproah[7, 8, 9℄. In
onesideofthesystemdenedby170
aa
z151
aa ,
temperature is xed to T
0
= 0. In another region
0z30
aa
,temperatureisxedtoT
h =0:7 aa =k B ,
higher than the melting temperature T
f
= 0:4
aa =k
of the pure material. The melting temperature was
measured independently as desribed below. We let
the system evolute for 1:210 5
steps in time of size
t = 0:02
aa (m a = aa ) 1=2
, whih seems to be
enoughtoequilibratethesystem. Onetheequilibrium
is reahedwestartpullingthesystemin the+z
dire-tion, at apulling veloity v
p
= 410 3 ( aa =m a ) 1=2 .
The z = 0 layer works as a soure of partiles,
man-tainingaonstantpartileowtothematerial.
In order to obtainan estimatefor T
f
wedid an
inde-pendent simulation with 1:510 3
partiles in a box
ofxeddimensionsanddensity710 3
aa
,belowthe
soliddensity. WeobtainedthatT
f 0:4 aa =k B . The
onsideredsystemonsistsoftwodierenttypesof
par-tiles. Oneisthesolvent(apartiles)andthesolute(b
partiles). Wedenethreetypesofinterations,
solute-solute (b - b), solvent-solvent(a- a), solute-solvent(b
- a). The initial solute onentration
0
is 5%. This
onentrationisdenedastheratiobetweenthe
num-ber of partiles in the solute and the total numberof
partiles inthesolvent. Toalulatethedensityalong
thegrowingrystalweusestripsofsizez=20
aa .
Forsimpliity,fromnowonwemeasureenergyinunits
of
aa
, distane in units of
aa
, mass in units of m
a
sothattemperatureandtimearemeasuredinunitsof
k B = aa and(m a 2 aa = aa ) 1=2 respetively.
Thesimulationisarriedoutfor
bb
=0:1and
ab =0:5.
Value of
ab
are 1:00;1:05;1:15;1:20. We always use
aa =
bb
. Typially, in oursimulationsof diretional
growth,thesolidphasedensity
s
was5:7timeslarger
than the uid phase density
f
. For a given pulling
speedthe timefor thesystemto ahievesteady-state,
andonsequentlythesimulationtime,dereasesasone
inreasestheratiobetweenthesolidanduiddensities.
Thereasonisthatthetypialtimeforreahing
steady-stateisoftheorderofD=v 2 f =D=v 2 p ( f = s ) 2 ,where
DisthediusionoeÆientandv
f
istheveloityofthe
uid phase. Sine D inreasesapproximatelylinearly
with dereasing density ratio, the omputation time
goes approximately linear with (
f =
s
). We heked
the veloity prole of the uid phase and determined
thatsolutetransportismainlydiusiveforour
simula-tions,withD=1:3
aa ( aa =m a ) 1=2 IV Results
We studied the evolution of ellular instabilities
dur-ingdiretionalsolidiationforxedtemperature
gradi-entG =510 4 ( aa =k B aa
) and xed pullingspeed
v
p
= 410 3 ( aa =m a ) 1=2
, but for dierent valuesof
preditions [2℄, the interfae is unstable if l
T < l D , where l T
is thethermal lengthand l
D
is thediusion
length. l T kBT f 2 LG 0 (1 k )
2 k and l D D s vpf 60 aa .
Sine L 0:59
aa
(latent heat) [7℄, one obtains that
l
T 27
(1 k ) 2
k
aa
. Thereforethe interfaeisstable for
k>0:25. Bydereasingk oneantriggertheellular
instability. The value of k estimated from Fig. 1 is
about0:3. The interfaeis stable as preditedabove.
Byinreasingthevalueof
ab
asomparedto
aa ,the
impurityto beabletoinorporateintotherystalhas
todeformitslattie. Thehighelastienergyostauses
a derease of the impurity solubility into the rystal
phase, resulting in a smaller k. From Figs. 2 and 3
we estimatek 0:22for
ab
= 1:05
aa
; k 0:04 for
ab
=1:15
aa
and k0:01for
ab
=1:20
aa
, all
val-uesare in the unstableregion. As k getssmaller, the
rystalbeomesleanerandimpuritiesonentrateinto
thegroovesoftheellularstruture.
800
600
400
200
0
z
Crystal
Fluid
Figure1. Finalongurationofthesimulatedrystal after
810 4
timestepsfor
ab
=1:0. Greyandblakirles
rep-resent solvent and solute atoms respetively. The solvent
(grey)rystaline strutureis hexagonal as expetedfor a
Lennard-Jonessolid. PullingdiretionisalongZ.
Morpho-logialinstability is notobserved. Thereis solute
Figure2. Fromtoptobotton theplots show thesurfae's
time evolution for
ab
= 1:05. The sequene starts at
t=1:610 4
followingintimestepsofsizet=4:010 4
.
Symbolsarethesameasingure1.Theestimatedk0:22
indiatesthattheinterfae isintheunstableregionas
ob-Figure3. Final ongurationofthesimulatedrystalafter
810 4
time steps. From top to botton ab = 1:05;1:15
and 1:20. Symbolsarethe sameas ingure1. As
ab
in-reases, segragation inreases (kdereases)failitating the
V Conlusions
Ouromputersimulationsofdiretionalgrowthofa
bi-nary mixture learly show: (1) The eet of impurity
segregationduringgrowth;(2)Segregationinreases(k
dereases) as
ab
beomes larger than
aa
; (3) For k
smallerthan0:25theellularinstabilityistriggeredfor
thevaluesof G,v
p and
0
used. Theimpurities
segre-gated at the interfae onentrateinto the groovesof
the ellular struture resulting in impurity inlusions
that auselargedefets intherystalmatrix.
Aknowledgements
This work was partially supported by CNPq,
FAPEMIG and FINEP. Numerial work was done in
theLINUX parallellusterat theLaboratoriode
Sim-ula~aoDepartamentodeFsia-UFMG.
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