Measuring kinetic coefficients by molecular dynamics simulation of zone melting

8 Pages·0.76 MB·English

by  F. Celestini

Measuring kinetic coefficients by molecular dynamics simulation of zone melting
∗
Franck Celestini and Jean-Marc Debierre
Laboratoire Mat´eriaux et Micro´electronique de Provence, Universit´e d’Aix-Marseille III and CNRS, Facult´e des Sciences et
Techniques de Saint-J´erˆome, Case 151, 13397 Marseille Cedex 20, FRANCE
2
Moleculardynamicssimulationsareperformedtomeasurethekineticcoefficientatthesolid-liquid
0
interfaceinpuregold. Resultsareobtainedforthe(111),(100)and(110)orientations. BothAu(100)
0
and Au(110) are in reasonable agreement with the law proposed for collision-limited growth. For
2
Au(111), stackingfault domainsform, as firstreported byBurke,Broughton and Gilmer[J. Chem.
n
Phys. 89,1030(1988)]. Theconsequenceonthekineticsofthisinterfaceisdramatic: themeasured
a
kinetic coefficient is three times smaller than that predicted by collision-limited growth. Finally,
J
crystallization and melting are found to be always asymmetrical but here again the effect is much
1
more pronounced for the(111) orientation.
2

PACS numbers: 81.30.Fb, 68.45.-v, 02.70.Ns
]
i
c
s
-
rl I. INTRODUCTION (EAM) [5], the glue model (GM) [4] and the effective
t medium theory (EMT) [6]. In the near future, the in-
m
Solidification of pure elements is of technological in- crease of computer power should open the possibility to
.
t terest because the way a given material solidifies usu- addressthecaseofmorecomplicatedmaterialslikesemi-
a
m ally affects its structure and, as a consequence, its final conductors, molecular crystals and organic compounds,
elastic and other macroscopic properties. From a fun- for which potentials do not simply reduce to pair inter-
-
d damental point of view, interest in free and directed so- actions. Very recently, the functions γℓmn and µℓmn(Ti)
n lidication comes from the underlying nonlinear physics, have been determined and used in phase field simula-
o morphological instabilities being at the origin of generic tions of dendritic growth for pure nickel [7]. The good
c
[ microstructures such as dendrites or cells. quantitative agreement found between experiments and
Important theoretical and numerical contributions simulationsispromisingandshouldstimulateinthenear
1
have been made to solve this difficult physical problem futuretheconstructionofothermaterial-dedicatedphase
v
1 [1]. Recently,a quantitative phasefield modelwasintro- field models.
7 duced[2]. Asubsequentrefinement,consistinginsolving New methods for the determination of the func-
3 thediffusionequationwiththehelpofBrownianwalkers, tions γℓmn and kℓmn(Vi) have been recently proposed
1 permitted to bridge the wide gap between the capillary [8,9]. In the present paper, we rather concentrate on
0 and diffusion lengths, allowing direct comparison with µℓmn(Ti). The kinetic response of a solid-liquid inter-
2
experiments [3]. As a consequence, there is currently face has been simulated quantitatively in the 80’s by
0
/ an increasing need for accurate values of the interfac re- Broughton, Gilmer and Jackson (BGJ) for a Lennard-
t
a sponse functions that are used as input parameters for Jones (LJ) potential and a (100) orientation [10]. These
m realistic phase field simulations. authors showed that growth is not diffusion-limited but
- Inthe caseofapureelement, thesurfacetensionγℓmn rather that the interface velocity is related to the mean
d must be known as a function of the interface orientation kinetic energy of the atoms. For this collision-limited
n
(ℓmn). In addition, the kinetic coefficient µ (T ) giv- growth regime, the growth rate should be directely pro-
o ℓmn i
c ing the relation between the interface velocity and the portional to the distance between two successive layers
v: interface temperature Ti, should also be known for the dℓmn. Indeed, since the liquid atoms do not diffuse to
i differentorientations. Forabinaryalloy,temperaturede- choose their adsorption sites but are almost instanta-
X pendenceofthesolutediffusioncoefficient,D(T),aswell neously incorporated into the solid, the larger dℓmn the
r asvelocityandorientationdependenceofthesegregation more effective and the faster the advance of the solid-
a
coefficient k (V ) are also necessary. liquid interface should be. The analytical expression for
ℓmn i
Both k and µ are hardly accessible in the experiments thegrowthvelocityofaroughsolid-liquidinterfacereads
and convection effects often lead to overestimated val- [11]:
uesofdiffusioncoefficients. Differentsimulationschemes
∆µ˜
have thus been proposed as an alternative. Such nu- V ∝dℓmnV˜h1−exp(cid:0)− kT (cid:1)i, (1)
merical experiments have been rendered possible by the
discovery of realistic interatomic potential models, such dℓmn being the interplane spacing, ∆µ˜ the chemical po-
as, in the case of metals, the embedded atom model tential difference between solid and liquid phases, T the
absolute temperature, k the Boltzmann constant, and V˜
1
the thermal velocity. This law is confirmed by molecu-
lar dynamics simulations for the (100) and (110) orien-
tations: the expected √2 ratio between the correspond-
ingkineticcoefficientsiswellrecoveredforseveralmetals
cristalizing in a face centered cubic (fcc) structure (Ni,
Ag and Au) [12,13]. Nevertheless, for these rough ma-
terials, growth of the (111) interface does not obey this
simple law: according to Eq. (1), the (111) orientation
shouldbe muchfaster,andwhatis foundis preciselythe
opposite. Burke, Broughton, and Gilmer [11] attribute
this slowing-down to the growth of competing fcc and
hcpdomainsinthesolidfyinglayer,followedbytheelim-
ination of the defect lines between the two phases.
Another question associated with solid-liquid inter-
facesisthatofsymmetrybetweensolidificationandmelt-
ing kinetics. Asymmetry has been already observed in
different systems. It is not really surprising for faceted
materialslikesiliciumwheresolidificationinvolvesnucle-
ation while melting does not. The question is more del-
icate when one considers rough materials with collision-
limited growth. Indeed, available results are controver-
sial: ifasymmetryhasbeenfoundforaNa(100)interface
[14], it has not been observed for a LJ(100) [15]. More
surprisingly, in the latter case an opposite asymmetry
(crystal growing faster than the melt) can be found, de-
pending on the way the solid germ is prepared.
Inthis paper,weaddresstheabovequestionsconcern-
ing the growth of a rough solid-liquid interface. We first
presentourimplementationofanon-equilibriummolecu-
FIG. 1. A typical simulation box with periodic bound-
lardynamicsschemeforazonemelting experiment. The
aryconditionsin allthreedirections ((111) solid-liquid inter-
secondsectionis devotedtothe study of(100)and(110)
faces). Atomsin dark grey are within thehot and cold slices
orientations. The special case of (111) growth is exam-
where temperature is fixed.
ined in section III and asymmetry between melting and
solidification in section IV. Finally, a summary of the
different results and a discussion are given in the last U is the glue term associating an extra potential energy
section. to atom i as a function of its coordination. This glue
potentialhasdemonstateditsefficiencyinpredictingthe
physical properties of gold as well as in describing sev-
II. SIMULATION PROCEDURE eral experimentally observed phenomena such as surface
melting and surface reconstructions [16].
For this study we use the Ercolessi glue potential for A distinctive feature of our method is to simulate a
Au [4]. In this formalismthe total potentialenergy for a zone melting experiment in which both a solidification
system of N atoms is given by: and a melting front are simultaneously advancing at a
fixed velocity V. This velocity is that of the virtual fur-
1 N N nace which imposes two symmetric thermal gradients.
U = 2 X Φ(rij)+XU(ni) (2) The particle coordinatesare defined in a reference frame
i,j=1 i=1 moving at velocity V in the z direction, so that after
equilibrationthe positions of the two interfaces are fixed
Thefirsttermisaclassicalpairinteraction. Inthesecond
in the simulation box. Heat transport from the furnace
term, n is the coordination of atom i,
i issimulatedbyimposingonetemperaturebelowandone
N above the melting point inside two distant slices, 20˚A
n = ρ(r ), (3) each in thickness (Fig. 1). Within each slice, tempera-
i X ij
j=1 tureiskeptconstantbyusingaclassicalvelocityrescaling
procedure[17]. Periodicboundaryconditionsareapplied
whereρ(rij)is afunctionofthe interatomicdistancerij, inthethreedirections. Moredetailsaboutthenumerical
with a cut-off radius of 3.9˚A here. The energy function methodcanbefoundinarecentstudyofsolutetrapping
2
−3.4
1400
K) s) V=0
m
(
o V=15 m/s
T t
a
1000 /
V
−3.5
e
(

Y
G
R
) −3.4 E
s N
m
E
o −3.6
t
a
/ −3.5
V
e 1000 1200 1400
(

E
T (K)
−3.6
−80 −40 0 40 80
o FIG. 3. Caloric curves for V = 0 (circles) and V = 15
Z (A) ms−1 (diamonds). For non zero velocity, the kinetic effects
split the curve in two parts. The dotted and full straight
FIG.2. Temperature and energy profiles along the z axis lines represent respectively the functions Ei(T), EL(T) and
perpendicular to theinterfaces. ES(T).
inaLJbinaryalloy,whereasimilarsimulationtechnique run lasts 106 MD steps (3.5 103 ps), so that the atoms
×
was used [9]. in the system solidify and melt severaltimes. According
First, the fcc solid and the liquid are equilibrated to a recent study by Tepper et al. [15], we know that
separatly at zero pressure and at a temperature close the melting kinetics can be affected by the way the solid
of the melting point. Our smallest system has a size is equilibrated. Thusmulti-cycling is necessaryto mimic
S0 20 20˚A2 in cross-section, that is about 64 atoms themeltingofarealsolid,usuallyresultingfromprevious
≃ ×
per layer. After equilibration, the solid and liquid are solidification(s).
broughtintocontactandplungedinthetemperaturegra- To conclude this section, the method used to estimate
dient imposed by the two temperature-controled slices. the interface temperature T from the caloric curves is
i
The total system is about 220˚A in height. After a sec- described. We assume the energy of atoms lying at the
ond equilibration period (during which the velocity of interace,E ,tobeaweightedaverageoftheperfectsolid
i
the furnace is zero), the two interfaces reach a station- and liquid energies at the same temperature T.
arypositionandweroughlyhave50%ofsolidandliquid
(see Fig. 1). Fig. 2 shows the temperature and energy Ei(T)=αES(Ti)+(1 α)EL(Ti). (4)
−
profiles along the z axis.
Combining the two profiles to eliminate the z coordi- Linearrelations,ES(T)=aST+bS,andEL(T)=aLT+
nate, oneobtains a caloriccurve,i.e., a plotofenergyas bL, are fitted to the data points obtained onthe low and
a function of temperature. In Fig. 3, the caloric curves high temperature side, respectively (Fig.3). The curve
obtainedfortwodifferentvaluesofthepullingvelocityV Ei(T) is thus a line with a slope
are displayed. For V =0, the data points corresponding
p=αa +(1 α)a . (5)
to the solidification and the melting fronts merge onto S − L
the same curve: no kinetic effects are at play and the
The value of coefficient α is then extracted from the
interfacetemperatureisthe equilibriummeltingtemper-
caloriccurveat zerovelocity,for which T must be equal
ature T0 1330K. When a velocity is imposed, a dy- i
≃ toT0 (Fig. 3). Finally,theinterfacetemperatureisgiven
namical hysteresis appears on the caloric curve. Kinetic
by the intersection of the line E (T) with the caloric
effects split the curve in two distinct parts: the interface i
curve. An alternative method consists in building an or-
temperature of the solidification front decreases while it
derparameterthatdistinguishesbetweensolidandliquid
increases on the melting front. We can deduce both in-
atoms[8,18]: aplotofthisorderparameterasafunction
terfaceundercoolingandinterfacesuperheatingfromthis
of temperature also gives an interface temperature. We
plot. An interest of this method is that, as said before,
have checked that the two methods give equivalent re-
the interface is fixed in the reference frame of the simu-
sults.
lationbox,sothatstatisticsareeasytorecord. Atipical
3
30
1.0
(100)
s) (100) 0.8
/
m
Y ( 20 (110) N 0.6
T µ
I
C
O 0.4
L 10
E (111)
V 0.2 V=15 m/s
0.0
0
0 50 100 1 10 100
T − T (K) S/So
o i
FIG.4. Velocity of thesolid-liquid interfaceas afunction FIG. 5. Normalized kinetic coefficient as a function of
ofundercoolingfor(100)and(110)orientations. Thestraight system size S for the (100) and (111) orientations.
lines are best fits to a linear kinetic law.
and we obtain roughly the same behavior for the (110)
III. GROWTH OF (100) AND (110) INTERFACES interface. This size effect has never been reported in the
past for (100) and (110)orientations: the fact that Hoyt
In this section we compute the kinetic coeficient for andco-workersdonofindsizeeffectsforthesetwoorien-
the Au(100) and Au(110) interfaces, using the method tations [8] is certainly due to the fact that their smaller
described above. We concentrate here on pulling veloc- system is larger than ours. We can now propose extrap-
ities ranging between V = 5 ms−1 and V = 30 ms−1, olated values for the kinetic coefficients:
for which kinetics remain linear. We also perform a
−1 −1
few simulations at higher velocities, where kinetics de- µ100 =18.8 1.0cms K (9)
±
viates from linearity, but comments on nonlinear effects
are postponed to the concluding section. In Fig. 4, we µ110 =12.6 1.0cms−1K−1 (10)
±
plot the interface velocity as a function of the measured
undercooling T0−Ti. Linear fits to the law Tgohoedcoargrreesepmoenndtinwgitrhatitoheµ1v0a0/luµe11√0 2=p1r.e4d9ic±ted0.1b5yisEiqn.
V =µℓmn(T0 Ti) (6) (1). Hence, the assumption of collision-limited growth
−
for (100) and (110) orientations is confirmed to be the
give the following estimates for the two kinetic coeffi- relevant one. At this point, we can compare our results
cients: with those of Hoyt et al. for gold [13]. If they also find
a √2 ratio between their two orientations, their µ val-
µ⋆ =23.1 1.0cms−1K−1 (7)
100 ues are larger than ours by a factor 1.8. Linearizing the
±
expression given by BGJ, we find
and
µ⋆ =15.5 1.0cms−1K−1. (8) V ∼T0−1Ti−1/2(T0−Ti) (11)
110
±
for the interface velocity. The potential used by Hoyt
However, finite-size effects are expected to bias these et al. gives a melting point T0 of 1090K [19] much
estimates because the system cross-section area, S0 = smallerthanthevalue1330KobtainedwithEroclessipo-
20 20˚A2, is rather small. tential. Introducing this temperature shift in Eq. (11)
×
Additionnalrunsarethusperformedinordertoquan- roughly accounts for the discrepancy between the values
titativelyevaluatefinite-sizeeffects. Thepullingvelocity ofµ. SinceErcolessipotentialgivesameltingpointmuch
is fixed to V =15ms−1, the system heightto H 222˚A, closerto the experimentalone,it shouldbe alsothe case
≃
and the cross-section area S is progressively increased. for our estimates of the kinetic coefficients.
WedefinethenormalizedkineticcoefficientµN(S)asthe In order to understand the origin of the size effects on
ratio of the kinetic coefficient obtained at size S to that the value of the kinetic coefficient, we take now a closer
obtained at size S =S0 [Eqs (7-8)]. As shown in Fig. 5, look at the in-plane structure of gold layers in the vicin-
the size effects are important and the kinetic coefficients ity of the solid-liquid interface. We compute a density
appear to converge only for S 100 100˚A2. For the profile along the z axis from which we are able to sepa-
≃ ×
(100) direction, there is a decrease of about 20 percent rateatomsbelongingtodifferentlayers. Deepinthesolid
4
the in-plane square structure of the (100) orientation is
effectively recovered without any significant amount of
defaults and vacancies. For the two solid layers just be-
low the interface the situation is more complex. To dis-
tinguish between different symmetries, we first perform
aVoronoiconstructionforallthe atomsinthe layer. We
then collect the set of first neighbors for each atom.
For a fcc solid with lattice parameter a, on the square
lattice of the (100) orientation an atom has four nearest
neighbors at a distance a/√2 and four second nearest
neighbors at a distance a. On the other hand, for a tri-
angular lattice (as the one of the (111) plane) the six
neighbors all lie at the same distance a/√2. In Fig. 6,
we showasnapshotofthe interfacesolidlayerwherethe
Delaunay triangulationis only drawnfor the atoms that FIG. 6. Snapshot showing the atoms in the (100) solid
have six first neighbors at comparable distances, in or- layer next to the interface. The Delaunay triangulation is
der to reveal the local triangular structure. It is clear only drawn in regions with triangular underlyingsymmetry.
that most of the atoms have reached their positions on
the square lattice but several islands with a triangular 30
symmetry remain. Note that the number of atoms in
the layerhasalreadyattainedthe value itwillhavedeep
in the solid with a perfect square structure. To compen-
20
sateforthehigherdensityofthetriangularstructure,the
)
s
correspondingislands are surrounded by a border region m/
where the density is very low. This coexistence of two V (
symmetries is not observed in our smallest system: one 10
can imagine that for a small area the square structure
is easily formed and hence triangular islands do not ap-
pear. This phenomenon is very close to the well known
0
reconstruction of the (100) solid-vapor interface where
0 50 100 150
thefirstlayeradoptsatriangularstructure[21]. Turning
T − T (K)
back to the solid-liquid interface, the system apparently b i
uses some of the solidification driving force to eliminate FIG. 7. Velocity of the (111) solid-liquid interface as a
oneofthe twophasesandfinallyreachanalmostperfect function of undercooling.
square symmetry. Hence, the interface velocity is lower
for larger systems.
orientation. The extrapolated value of the kinetic coeffi-
Such an in-plane ordering is not taken into account
cient,
in the collision-limited model but in spite of this we re-
coverthepredicted√2valuefortheratioµ100/µ110. This µ111 =7.0 1.0cms−1K−1, (12)
suggests a similar effect, roughly of the same order, for ±
the (110) orientation. We have not been able to visual- is now 60 percent below its value for S =S0. Relatively
ize ordering at (110) interfaces but one could imagine a to the two other orientations, we find
mechanismreminiscentofthemissingrowreconstruction
observed for (110) solid-vapor interfaces. µ111 0.37µ100 0.56µ110. (13)
≃ ≃
These ratios largely differ from the values predicted by
IV. THE SPECIAL CASE OF (111) INTERFACE Eq. (1), respectively 2/√3 1.15 and 2 2/3 1.63.
≃ p ≃
The (111) orientation, expected to grow faster because
We now turn to the case of the (111) orientation. In of a larger interlayer spacing, is surprisingly found to be
the same way as above for the (100) and (110) orienta- theslowestone. Thisdiscrepencytellsusthatthegrowth
tions we calculate the interface temperature for different mechanismforthe(111)orientationisnot,oratleastnot
velocities. As can be seen in Fig. 7, a linear kinetic only, a collision-limited one.
law is also valid for the (111) orientation. Results of the Here again we look at the symmetries inside the lay-
finite-size analysis, presented in Fig. 5, show that the ers close to the interface. For a (111) layer, there are
sizeeffectsaremuchmorepronouncedthanforthe(100) threepossibleorderedphaseslyingonthreedifferentbut
equivalentsub-latticesthatwewillcalla,bandc. Asthe
5
20 With our simulation scheme, this study is straightfor-
ward,since both a melting anda solidificationfronts are
simulated at once: no additional calculations are thus
s) 0 required. Fig. 9 represents the velocities of both the
m/
V ( meltingandsolidificationfrontsasfunctionsofT0−Tifor
the(111)orientation(inourconventionsapositiveveloc-
−20 ity corresponds to solidifation). The data are obtained
in a system of size S = S0 and corrected according to
thefinite-size analysisreportedabove. Itisimportantto
−100 0 100 200
notethatnosizeeffectsareactuallyfoundforthemelting
T − T (K)
o i front: in contrast with the solidification front, the melt-
inginterfacetemperatureremainsthesamewhateverthe
FIG. 8. Snapshots of the three solid layers immediately
belowtheinteface(toplayerincontactwiththeliquidphase). system size. This can be understood if one remembers
Thegrey levelscorrespond tothethreedifferentsub-lattices: thatforsolidification,especiallyforthe(111)orientation,
a (white), b (light grey), c (dark grey). growth is not only collision-limited but also requires in-
plane ordering. This is no longer the case for melting,
which justifies the absence of size effects. The asymme-
stacking fault energy is weak for gold (it is actually zero
try shown in Fig. 9 is larger for the (111) orientation.
for the potential we use), once a perfect, say a, layer is
The same analysis is also made for the two other orien-
formed, the next layer to form is either b or c. In Fig. 8,
tations and we find the following degrees of asymmetry:
we show a snapshot of the three uppermost solid layers
and we distinguish between atoms belonging to a, b or µm111 =25 4cms−1K−1 3.6µs111 (14)
± ≃
c phases. For the lowest solid layer, phase a is selected
anditoccupiesthewholeplane. Forthelayerjustabove, µm100 =39 2cms−1K−1 2.1µs100 (15)
∗± ≃
thereiscoexistencebetweenbandcsub-lattices. Finally,
inthehighestsolidlayer,allthreephasescoexist. Were- µm110 =20 2cms−1K−1 1.6µs110 (16)
± ≃
cover here the effect first observed by Broughton et al.
wherethesuperscriptssandmreferrespectivlytosolid-
[11]foraLJpotential. Fora(111)orientationthesystem
ificationandmelting kinetics. An asymmetryis revealed
hesitatesbetweenthe differentphasesitcanequivalently
inthethreecasesbutitismorepronouncedforthe(111)
form. Here again, the system dissipates a part of the
orientation in the same way as size effects observed dur-
available driving force to select one of the phases. As a
ingsolidification. Weconcludeherethatthisasymmetry
consequence, the velocity of the interface is reduced as
is directly related to the ordering within the interface
compared to the value expected for a purely collision-
layers. The asymmetry is strong for (111) because of
limited growth. The size effect is easily understood be-
thepeculiargrowthmechanismdiscussedintheprevious
cause in a small system, coexistence is strongly reduced.
section.
It would be of interest to determine the amount of driv-
The melting front is found to be faster than the solid-
ing force spent in this in-plane organisation in order to
ification interface in agreement with the idea that disor-
estimate the correspondingdecrease inV111. To perform dering is an easier task than ordering. Our results con-
this, one could for instance use a 3-state Potts model
firm the majority of experimental and numerical studies
in three dimensions with ferromagnetic intra-plane and
[14,22–25]. We also confirm the conclusions of a debate
anti-ferromagnetic inter-plane interactions. To conclude
betweenRichards[26]andOxtobyandco-workers[27,28]
this section we have to point out that phase coexistence
on the importance of density change on the asymmetry
is related to the value of the stacking fault energy E .
s betweenmeltingandsolidificationkinetics. Inagreement
ForamaterialwithlargeE phasecoexistenceshouldbe
s with the conclusions of Oxtoby, the gold density change
less probable and the front velocity in better agreement
at melting is small ( 2%) and can not be responsible
with the prediction of Eq. (1). ≃
for such an important asymmetry. On another hand,
Tepper [15] does not find asymmetry for the growth of
a (100) LJ solid. Even if the materials differ, they both
V. ASYMMETRY BETWEEN MELTING AND
belong to the same class of rough materials and such a
SOLIDIFICATION
qualitative difference may be surprising. Nevertheless,
one should remember the strong tendency to surface re-
Asdiscussedintheintroduction,asymmetryisobvious
constructionin Au, as observedfor the (100)orientation
for faceted materials but is not as clear when consider-
where triangular-like regions are formed. This tendency
ing roughmaterials like metals. The question is to know
isfuthermoreenhancedbytheuseofErcolessigluepoten-
if, at equal absolute undercooling and superheating, the
tial but is weaker for a LJ potential, what could explain
solid-liquidandliquid-solidfrontshavethesamevelocity.
the different behaviors observed.
6
20 extractthe amountofdrivingforcespentforphasesepa-
rationin order to modify Eq. (1) and find an acceptable
expression for the interface velocity.
For melting we find the following order between the
s) 0 differentkineticcoefficients: µm100 >µm111 >µm110. Toour
m/ knoweldgethis hierarchydoes notobey anyexistinglaw.
V ( Thisresultwillhopefullystimulatefurtherinvestigations
to reach a clear understanding of the specifities of melt
growth as compared to crystal growth.
−20
The presentstudy is devotedto the linearrelationship
between velocity and undercooling. For all the orienta-
−100 0 100 200 tionsconsideredhere,nonlineareffectsappearatvelocity
V 30 ms−1 and undercooling ∆T 200K. It is not
T − T (K) ≃ ≃
o i possible to explain this deviation using either the diffu-
sion limited [29,30] or the collision-limited growth law.
FIG. 9. Velocity of the (111) solidification and melting Thissuggestsapossiblechangeintheinterfacestructure
interfacesasafunctionofundercooling. Notethattheresults forsuchlargedeviationsfromequilibrium. Densitydiffer-
for solidification incorporate finite-size corrections. encebetweentheliquidandsolidphasesshouldalsocon-
tributetotriggernonlinearbehavior[26]. Understanding
this cross-over would be of importance in the context of
Finally, comparing the melting kinetic coefficients in
very rapid solidification.
the different orientations, we find µm > µm > µm .
100 111 110
Finally, we would like to stress that the kinetic effects
We presently do not have a satisfactory explanation for
can contribute to the anisotropy of the segregationcoef-
this hierarchy in the melting kinetics.
ficientk(V)forabinaryalloy. Atsufficientlylargeveloc-
ity, one expects an important difference in the interface
temperaturesfor (111)and(100)orientations. As a con-
VI. DISCUSSION
sequence, the diffusivity of solvent atoms and hence the

segregationcoefficient, as predictedby the Aziz law [31],
Our molecular dynamics simulations of zone melting
should also differ. This effect may cause solute trapping
experiments allow us to measure simultaneously the so-
to appear at lower velocities for (111) than for (100) or
lidification and melting kinetics for a pure element.
(110) orientations. We are currently investigating such
For (100)and(110)orientations,growthis apparently
segregation effects induced by kinetic anisotropy in the
well described by a collision-limited process. Neverthe-
Al-Cu system.
less, we observe small 2D islands with triangular sym-
metry to form in the solid layer at the (100) solid-liquid
interface. As a consequence, size effects and asymmetry
ACKNOWLEDGMENTS
betweenmeltingandsolidificationarefound. Wecannot
decide whether this effect is solely due to the tendency
Itisapleasureforus tothank M.Asta, B.Billia,J.J.
of the glue potential to overestimate surface reconstruc-
Hoyt and A. Karma for fruitfull discussions.
tion, or if it is an intrinsic property of gold and/or other
Electronic address: [email protected]
metals. ∗
The case of the (111) orientation is rather special.
Phasecoexistenceofthreetriangularsub-lattices,asfirst
proposedby Broughtonet al. [11],is recovered. This pe-
culiar behavior has a strong influence on the kinetics of
the interface. Our finite-size analysis show that in order
to measure a realistic value of the kinetic coefficient one [1] K. Kassner, Pattern Formation in Diffusion-Limited
has tosimulate systems witha solid-liquidinterfacearea Crystal Growth (World Scientific,Singapore, 1996).
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×
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