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Submitted on 1 Jan 1990
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On the molecular-dynamics study of a phase transition
in a quasicrystal model
K. Parlinski, E. Dénoyer, M. Lambert
To cite this version:
Short Communication
On the
molecular-dynamics study
of
aphase
transition in
aqua-sicrystal
model
K.
Parlinski(1,*),
EDénoyer(1)
and M.Lambert(2)
(1)
Laboratoire dePhysique
desSolides,
UniversitéParis-Sud,
91405Orsay,
France(2)
Laboratoire LéonBrillouin, CEA-CNRS, CEN,
Saclay,
91191 Gif surYvette,
France(Received
onMay
22,1990,
accepted
on June22,1990)
Abstract. 2014 A two-dimensional model of
large
(L)
and small(S)
atomsinteracting
via the Lennard-Jonespotential
has been studiedby
themolecular-dynamics technique.
Slowcooling
of the systemL0.435S0.565
from a disorderedconfiguration
leads to a defectedquasicrystalline phase.
Another runin which the L atoms were
additionally interacting by
the orientational threebody potential, produced
a
phase
withmicrocrystalline precipitates.
Thisphase
still preserves thepentagonal
symmetry of the diffractionpattern.
Thecrystalline precipitates
have a unit cell of thin Penrose rhombuses.Classification
Physics
Abstracts64.70K - 61.50E - 02.70
The
key
question
instudying
thequasicrystal
materials is to understandwhy
natureprefers
quasiperiodic
order toperiodic
crystalline long-range
order. In certainquasicrystalline
materi-als theenergy
considerations lead to a neardegeneracy
ofcrystalline
andquasicrystalline
phases.
However,
thequasicrystalline phase
offers states with muchhigher degeneracy
than thecrystalline
one. The
degeneracy
in thequasicrystal might
be sufficient toproduce
a finiteconfigurational
entropy
S.Consequently,
materials which at lowtempérature
remaincrystalline,
athigh
tem-perature
might undergo
aphase
transition to anentropy
stabilizedquasicrystal phases.
The freeenergy of the
quasicrystal phase
F = E - TS is then minimizedby
again
of theentropy.
Grains of thepresumed
icosahedralphase
of AlCuFe seem to begood
candidates forstudy-ing
the mentionedcrystal-quasicrystal phase
transition. The electronmicroscope
[1]
andX-ray
[2]
measurementsperformed
on smallsingle crystals
at differenttemperatures
revealed aphase
transition. The low
temperature
crystalline phase
possesses
rhombohedral unit cell(1, 3]
which isable,
due topeculiar
crystallographic
orientationalrelationships,
to from a microstructure(see
also[4] )
of overall icosahedralsymmetry.
Theoretically,
thequasicrystal
properties
have beenextensively
studied on a two-dimensionalmodel of a
binary alloy
[5-7].
The modelsystem
consists oflarge (L)
and small(S)
atoms1792
acting
via theisotropic
Lennard-Jonespair potential
where a
and ,Q
denote L or S atoms,Rap
is theseparation
between atoms aand ,Q.
Special
choiceof the
potential
parameters
°1.5 =1, uLL =
TLsCOs(7r/10)/cos(7r/5)
and o-ss =2c-Lssin(7r/10)
fulfils the condition of almost dense
packing
of two Penrose rhombuses.Figure
1 shows these fat and thin rhombusestogether
with an alternativeassignment
of two Penrose rhombuses in thebinary tiling.
The Penrosetiling, requires
that the ratio of concentration of fat to thin rhombuses isnF/nT
= T, where r =(1
+V5)
/2.
Hence,
the concentration of L and S atoms in the Penrosetiling
pattern
reads x =(2T
+1)/(4T
+3)
= 0.4472 and 1 - x =0.5528,
respectively.
One can,however, expect
otherphases
of thebinary alloy
L,,S,-.,
occuring
at différent concentrations.Thus,
thecrystal
with a unit cell ofonly
thin oronly
fat rhombusesrequires
the concentrationLO.33SO.66
orLo..50SO.50,
respectively.
Thesecrystals
are able to form twins of ten- or five-foldsymmetry
of the diffractionpattern.
It isimportant
to mention that othercrystalline phases,
with unit cellconsisting
of a few fat and thinrhombuses,
andcorresponding
toapproximant
structuresor other
periodic
structures with concentration 0.33 x0.50,
may alsoappear.
Fig.
1. -(a)
Decoration of the Penrose rhombuses(dashed
line)
and rhombuses in abinary tiling
(solid
lines). (b)
Basic move. Thisexchange
of atomspermits
to reach allpossible configurations
of the randomtiling
model.One can show
[b, 7J
that if in theLxSl-x
model the interactions are truncated tonearest-neighbours,
then theenergy
of theground
statedepends only
on the concentration of fat and thinrespectively.
Here,
foranyx,
nF =(3x - 1)/2
and nT =1- 2x are the numbers of fat and thinrhombuses,
respectively,
dividedby
the total number of atoms in thesystem,
and4nF
+3nT
= 1. Theparameters
cLs, cLL and 6ss denote thebinding
energy
at the minimum of thepotential,
equation (1).
Theground
stateenergies
of the threephases
areequal
if full = css and stable if’EU > ELL. We shall use 6Ls =
1,
andELL = css = 0.5. In the nearest
neighbour
approximation
the
ground
stateenergies
areequal
EF
=ET
=EQ
= -2.50per
atom.Since the Penrose
tiling
is constructedby
a deterministicalgorithm,
itsconfigurational
entropy
is zero.
However,
the stucture of thequasicrystal
is notnecessarily
the Penrosetiling
but it canhave a
configuration
which is best describedby
thespace
filling by
randomtiling by
randomtilings
using
the same two Penrose rhombuses. The randomtiling
model of thebinary alloy
describesthermodynamically
stablequasicrystal
withsharp
diffractionpeaks. Using
the transfer matrix method theconfigurational
entropy
has been found[8, 9].
Theentropy
density
is maximized at the atomic concentrationLo 4472So 5528
corresponding
to the Penrosetiling.
This concentrationproves
to be also critical for theintensity
behaviour of the diffractionpeaks
in the randomtiling
model of twotypes
of Robinsontriangles
[10] .
The
molecular-dynamics
simulations of thatsystem
[5,6]
have shown that a slowcooling
froma
liquid phase
lead to theimperfect quasicrystal configuration
with diffractionspots
forming
apentagonal
symmetry
in the diffractionpattern.
Fastquench
from theliquid
stateproduces
atwo-dimensional
glass
with diffuserings
in the diffractionpattern.
Monte Carlo studies[7,11,12]
of the samesystem
has confirmed that thequasicrystal
appears
to be anequilibrium
state of thesystem,
althrough
the structurecorresponding
totiling
of theplane
withrhombuses,
is a randomtiling.
The basicprocess
assumed in these studies was anexchange
of one L and two S atomssimultaneously,
as illustrated infigure
Ib. Thiselementary
move isresponsible
for anytiling
rearrangement
which may occur in this modelquasicrystal.
7b enhance the
stability
of acrystal phase having
a unit cell of a thin rhombuse wepropose
tosupplement
the Lennard-Jonespotential
by
a threebody
potential
which favours the rowconfig-uration of three L atoms
(0 =180° ) being
the nearestneighbours.
Thepotential
iswhere a,
(3, ’1
are three L atoms and(}(RaP’ Ray)
is theangle
between vectorsRap
andRay.
We havechosen 17
= 0.6.1tuncating
the interaction(3)
at the nearestneighbours,
theground
state
energy
of thecrystal
with fat or thin rhombuses as a unit cell increasesby ÔEF
= 0.006per
particle
of decreasesby
AET
= -0.188per
particle,
respectively.
Theground
stateenergy
of arandom
tiling quasicrystal
will nowdepend
on aparticular tiling.
The simulations have been
performed
using
standardmolecular-dynamics technique.
Themicrocanonical ensemble and free
boundary
conditions,
to avoidintroducing
artificialphason
strains,
have been used. To avoid loss ofparticles,
which couldevaporate
at thesystem
edges,
the atoms were confined in a·circular box of sizesubstantially
greater
than that of thecrystallite.
The masses of L and S atoms were chosenML
=2Ms
=1.5Mo.
Fixing
the 0-, S =1,
the massMo
= 1 and theenergy fIS = 1
(or
temperature
To
= 1 andkB
=1),
the unit of time wasTo
= (MoUÍ5/fLS) 1/2.
The main runs wereperformed
for asystem
of 354particles.
First,
we have checked that thecrystal phases
Lo 3380 ee
andLo, 50 So, 50
are stable. When the threebody potential (3)
YLLL
= 0 was switchedoff,
theground
stateenergies
of the relaxedsingle
1794
per
particle
andEF
= -2.77per
particle,
respectively.
Including
the threebody potential,
thecorresponding
ground
stateenergies
were shiftedby
ÔET
= -0.153per
particle
andÔEF
= 0.031per
particle,
respectively.
Note small decrease of theground
state energy of thincrystal
incomparison
to that in the nearestneighbours
approximation (
AET
= -0.188).
The difference iscaused
by
the nextneighbours
interaction.The initial
configuration
of the simulations with concentrationLo 43580 565
close to theper-fect
quasicrystal
concentration,
washighly
disordered state attemperature 0.40To,
just
above theliquid-gas
phase
transition. Afterequilibration,
thesystem
wasslowly
cooled down with thetem-perature
ratedT/dt
= -0.0005.During cooling
process
the atomsgathered
to asingle
cluster,
inwhich diffusion of atoms was still observed. Below T =
0.10T,
any diffusion motion ceased out. The finalconfigurations
of atoms in two suchcooling
runs are shown infigure
2a,b.
In the runsof
figure
2a andfigure
2b the threebody potential
was swiched off(VLLL
=0)
and on(VILL =F 0),
respectively.
Theseparts
of theconfigurations
which could berecognized
as themicrocrystallites
of a
crystal
with a unit cell of thin rhombuseT,
have been decoratedby
thebinary tiling.
Thevol-umes of thèse
microcrystalline précipitâtes
consits of about 13% and50%,
respectively.
Note also that infigure
2b,
in order topreserve
the concentration of themicrocrystallites
which isLo 3380 66)
more L atoms have been
pushed
out to the surface. One could have also decorated the finalcon-figurations by
the Penroserhombuses,
but because ofhigh density
ofdefects,
the decoration ispossible
only
in a limited area,being
less than the area decoratedby
thebinary tilings.
The final
configurations
are full of defects anddislocations,
visible infigure
2c,d
as untilablerégions.
Thetilings
infigure
2c and 2d have been madeby assigning
thebinary
tiling
to the atomconfigurations
offigure
2à
and2b,
respectively. Inspection
offigure
2a,c
andfigure
2b,d
assures usthat a defect
appears
at theplace
where either L or S atom ismissing.
In themolecular-dynamics
technique
the atomicrearrangement
ismainly resulting
from a diffusion of asingle
vacancy. Thismechanism does not
preserve
the basic move,figure
Ib,
required
for the randomtiling
model,
and hence it is the source of defects. Thedensity
of defects in bothconfigurations
wasapproximately
the same.
The diffraction
pattern
of the disorderedsystem
athigh
temperature
of0.32To
consited ofdiffuse
rings
like thepattern
of theglassy
system.
The diffractionpatterns
of the two finalcon-figurations, figure
2a,b,
withVLU
= 0 andVLLL =F
0 are shown infigure
3a,b,
respectively.
Asa reference we show in
figure
3c the diffractionpattern
of aperfect quasicrystal
[6,11] .
Com-paring
the threefigures,
one couldrecognize
the overallpentagonal
symmetry
of the diffractionpatterns,
although
thespots
are not at allsharp,
theangles
of the five foldsymmetry
and thein-tensity
ofequivalent
spots
which should beequal,
are notexactly respected.
These are the naturalconsequences
of the very limited sizes of domains in our smallsystem.
’Ib
prove
the existence of thecrystal-quasicrystal
phase
transition one should heat the finalconfiguration containing
précipitâtes
of thincrystals
and should observe thequasicrystal
growth.
This is difficult to achieve
by
themolecular-dynamics
technique
because themicrocrystalline
pre-cipitates
should desolveby
the diffusion mechanism which is slow incomparison
to the availablesimulation time. Th make the diffusion faster one
usually
elevates thetemperature,
but then onecréâtes new defects which
destroy
both the orientational correlations in thesystem
and thepen-tagonal
symmetry
of the diffractionpattern.
Indeed,
in thepresence
of the threebody potential
we have heated the
configurations
ofparticles
shown infigure
2b,
up
to0.40To,
equilibrated
andquenched
rapidly
with thetemperature
ratedT/dt
= -0.010. In thequenched configuration
themicrocrystalline
précipitâtes
of the thincrystals
could berecognized
in the volume ofonly
6% ofthe
system.
But thedensity
of defects has increased almost twice incomparison
to thedensity
inthe first two runs, and this amount of defects
destroys
the orientational correlations so much that thepentagonal
symmetry
of the diffrationpattern
cannot be seen anylonger.
Fig.
2. - Finalconfigurations
ofbinary alloy
atlow-temperature
0.08To with(a)
Lennard-Jonespotential,
(b)
Lennard-Jonespotential
supplemented by
orientational threebody
potential acting
betweenlarge
atoms.(c)
and(d)
Binary tiling
corresponding
to(a)
and(b)
particle
configurations,
respectively.
Lennard-Jones
system
whichspontaneously
forms aquasicrystalline
state.Adding
a threebody
potential
one can favour thecrystalline
state. Thissystem
under slowcooling
arrives to theequi-librium state which contains
microcrystalline precipitates
andpreserves
the five-foldsymmetry
of1796
Fig.
3. - Diffractionpatterns of atom
configurations
at low temperature with(a)
Lennard-Jonespotential
and(b)
Lennard-Jonessupplemented
by
orientationalthree-body potential acting
betweenlarge
atoms. The(a)
and(b)
diffraction patternscorrespond
toconfigurations of (a)
and(b)
offigure
2,
respectively. (c)
The diffractionpattern
of theperfect quasicrystal.
Acknowledgements.
One of us
(K.P)
would like to thank the staff of Laboratoire dePhysique
desSolides,
University
ParisSud,
Orsay,
for theirhospitality
and assistanceduring
hisstay
there and CNRS for a financialsupport.
Fruitful
comments on thecomputer
program
by
EAugier,
J.R Gouriou and J.M. ’Iéulerare
gratefully acknowledged.
References