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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:

(2)

Short Communication

On the

molecular-dynamics study

of

a

phase

transition in

a

qua-sicrystal

model

K.

Parlinski(1,*),

E

Dénoyer(1)

and M.

Lambert(2)

(1)

Laboratoire de

Physique

des

Solides,

Université

Paris-Sud,

91405

Orsay,

France

(2)

Laboratoire Léon

Brillouin, CEA-CNRS, CEN,

Saclay,

91191 Gif sur

Yvette,

France

(Received

on

May

22,1990,

accepted

on June

22,1990)

Abstract. 2014 A two-dimensional model of

large

(L)

and small

(S)

atoms

interacting

via the Lennard-Jones

potential

has been studied

by

the

molecular-dynamics technique.

Slow

cooling

of the system

L0.435S0.565

from a disordered

configuration

leads to a defected

quasicrystalline phase.

Another run

in which the L atoms were

additionally interacting by

the orientational three

body potential, produced

a

phase

with

microcrystalline precipitates.

This

phase

still preserves the

pentagonal

symmetry of the diffraction

pattern.

The

crystalline precipitates

have a unit cell of thin Penrose rhombuses.

Classification

Physics

Abstracts

64.70K - 61.50E - 02.70

The

key

question

in

studying

the

quasicrystal

materials is to understand

why

nature

prefers

quasiperiodic

order to

periodic

crystalline long-range

order. In certain

quasicrystalline

materi-als the

energy

considerations lead to a near

degeneracy

of

crystalline

and

quasicrystalline

phases.

However,

the

quasicrystalline phase

offers states with much

higher degeneracy

than the

crystalline

one. The

degeneracy

in the

quasicrystal might

be sufficient to

produce

a finite

configurational

entropy

S.

Consequently,

materials which at low

température

remain

crystalline,

at

high

tem-perature

might undergo

a

phase

transition to an

entropy

stabilized

quasicrystal phases.

The free

energy of the

quasicrystal phase

F = E - TS is then minimized

by

a

gain

of the

entropy.

Grains of the

presumed

icosahedral

phase

of AlCuFe seem to be

good

candidates for

study-ing

the mentioned

crystal-quasicrystal phase

transition. The electron

microscope

[1]

and

X-ray

[2]

measurements

performed

on small

single crystals

at different

temperatures

revealed a

phase

transition. The low

temperature

crystalline phase

possesses

rhombohedral unit cell

(1, 3]

which is

able,

due to

peculiar

crystallographic

orientational

relationships,

to from a microstructure

(see

also

[4] )

of overall icosahedral

symmetry.

Theoretically,

the

quasicrystal

properties

have been

extensively

studied on a two-dimensional

model of a

binary alloy

[5-7].

The model

system

consists of

large (L)

and small

(S)

atoms

(3)

1792

acting

via the

isotropic

Lennard-Jones

pair potential

where a

and ,Q

denote L or S atoms,

Rap

is the

separation

between atoms a

and ,Q.

Special

choice

of 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 rhombuses

together

with an alternative

assignment

of two Penrose rhombuses in the

binary tiling.

The Penrose

tiling, requires

that the ratio of concentration of fat to thin rhombuses is

nF/nT

= T, where r =

(1

+

V5)

/2.

Hence,

the concentration of L and S atoms in the Penrose

tiling

pattern

reads x =

(2T

+

1)/(4T

+

3)

= 0.4472 and 1 - x =

0.5528,

respectively.

One can,

however, expect

other

phases

of the

binary alloy

L,,S,-.,

occuring

at différent concentrations.

Thus,

the

crystal

with a unit cell of

only

thin or

only

fat rhombuses

requires

the concentration

LO.33SO.66

or

Lo..50SO.50,

respectively.

These

crystals

are able to form twins of ten- or five-fold

symmetry

of the diffraction

pattern.

It is

important

to mention that other

crystalline phases,

with unit cell

consisting

of a few fat and thin

rhombuses,

and

corresponding

to

approximant

structures

or other

periodic

structures with concentration 0.33 x

0.50,

may also

appear.

Fig.

1. -

(a)

Decoration of the Penrose rhombuses

(dashed

line)

and rhombuses in a

binary tiling

(solid

lines). (b)

Basic move. This

exchange

of atoms

permits

to reach all

possible configurations

of the random

tiling

model.

One can show

[b, 7J

that if in the

LxSl-x

model the interactions are truncated to

nearest-neighbours,

then the

energy

of the

ground

state

depends only

on the concentration of fat and thin

(4)

respectively.

Here,

for

anyx,

nF =

(3x - 1)/2

and nT =1- 2x are the numbers of fat and thin

rhombuses,

respectively,

divided

by

the total number of atoms in the

system,

and

4nF

+

3nT

= 1. The

parameters

cLs, cLL and 6ss denote the

binding

energy

at the minimum of the

potential,

equation (1).

The

ground

state

energies

of the three

phases

are

equal

if full = css and stable if

’EU > ELL. We shall use 6Ls =

1,

and

ELL = css = 0.5. In the nearest

neighbour

approximation

the

ground

state

energies

are

equal

EF

=

ET

=

EQ

= -2.50

per

atom.

Since the Penrose

tiling

is constructed

by

a deterministic

algorithm,

its

configurational

entropy

is zero.

However,

the stucture of the

quasicrystal

is not

necessarily

the Penrose

tiling

but it can

have a

configuration

which is best described

by

the

space

filling by

random

tiling by

random

tilings

using

the same two Penrose rhombuses. The random

tiling

model of the

binary alloy

describes

thermodynamically

stable

quasicrystal

with

sharp

diffraction

peaks. Using

the transfer matrix method the

configurational

entropy

has been found

[8, 9].

The

entropy

density

is maximized at the atomic concentration

Lo 4472So 5528

corresponding

to the Penrose

tiling.

This concentration

proves

to be also critical for the

intensity

behaviour of the diffraction

peaks

in the random

tiling

model of two

types

of Robinson

triangles

[10] .

The

molecular-dynamics

simulations of that

system

[5,6]

have shown that a slow

cooling

from

a

liquid phase

lead to the

imperfect quasicrystal configuration

with diffraction

spots

forming

a

pentagonal

symmetry

in the diffraction

pattern.

Fast

quench

from the

liquid

state

produces

a

two-dimensional

glass

with diffuse

rings

in the diffraction

pattern.

Monte Carlo studies

[7,11,12]

of the same

system

has confirmed that the

quasicrystal

appears

to be an

equilibrium

state of the

system,

althrough

the structure

corresponding

to

tiling

of the

plane

with

rhombuses,

is a random

tiling.

The basic

process

assumed in these studies was an

exchange

of one L and two S atoms

simultaneously,

as illustrated in

figure

Ib. This

elementary

move is

responsible

for any

tiling

rearrangement

which may occur in this model

quasicrystal.

7b enhance the

stability

of a

crystal phase having

a unit cell of a thin rhombuse we

propose

to

supplement

the Lennard-Jones

potential

by

a three

body

potential

which favours the row

config-uration of three L atoms

(0 =180° ) being

the nearest

neighbours.

The

potential

is

where a,

(3, ’1

are three L atoms and

(}(RaP’ Ray)

is the

angle

between vectors

Rap

and

Ray.

We have

chosen 17

= 0.6.

1tuncating

the interaction

(3)

at the nearest

neighbours,

the

ground

state

energy

of the

crystal

with fat or thin rhombuses as a unit cell increases

by ÔEF

= 0.006

per

particle

of decreases

by

AET

= -0.188

per

particle,

respectively.

The

ground

state

energy

of a

random

tiling quasicrystal

will now

depend

on a

particular tiling.

The simulations have been

performed

using

standard

molecular-dynamics technique.

The

microcanonical ensemble and free

boundary

conditions,

to avoid

introducing

artificial

phason

strains,

have been used. To avoid loss of

particles,

which could

evaporate

at the

system

edges,

the atoms were confined in a·circular box of size

substantially

greater

than that of the

crystallite.

The masses of L and S atoms were chosen

ML

=

2Ms

=

1.5Mo.

Fixing

the 0-, S =

1,

the mass

Mo

= 1 and the

energy fIS = 1

(or

temperature

To

= 1 and

kB

=

1),

the unit of time was

To

= (MoUÍ5/fLS) 1/2.

The main runs were

performed

for a

system

of 354

particles.

First,

we have checked that the

crystal phases

Lo 3380 ee

and

Lo, 50 So, 50

are stable. When the three

body potential (3)

YLLL

= 0 was switched

off,

the

ground

state

energies

of the relaxed

single

(5)

1794

per

particle

and

EF

= -2.77

per

particle,

respectively.

Including

the three

body potential,

the

corresponding

ground

state

energies

were shifted

by

ÔET

= -0.153

per

particle

and

ÔEF

= 0.031

per

particle,

respectively.

Note small decrease of the

ground

state energy of thin

crystal

in

comparison

to that in the nearest

neighbours

approximation (

AET

= -0.188).

The difference is

caused

by

the next

neighbours

interaction.

The initial

configuration

of the simulations with concentration

Lo 43580 565

close to the

per-fect

quasicrystal

concentration,

was

highly

disordered state at

temperature 0.40To,

just

above the

liquid-gas

phase

transition. After

equilibration,

the

system

was

slowly

cooled down with the

tem-perature

rate

dT/dt

= -0.0005.

During cooling

process

the atoms

gathered

to a

single

cluster,

in

which diffusion of atoms was still observed. Below T =

0.10T,

any diffusion motion ceased out. The final

configurations

of atoms in two such

cooling

runs are shown in

figure

2a,b.

In the runs

of

figure

2a and

figure

2b the three

body potential

was swiched off

(VLLL

=

0)

and on

(VILL =F 0),

respectively.

These

parts

of the

configurations

which could be

recognized

as the

microcrystallites

of a

crystal

with a unit cell of thin rhombuse

T,

have been decorated

by

the

binary tiling.

The

vol-umes of thèse

microcrystalline précipitâtes

consits of about 13% and

50%,

respectively.

Note also that in

figure

2b,

in order to

preserve

the concentration of the

microcrystallites

which is

Lo 3380 66)

more L atoms have been

pushed

out to the surface. One could have also decorated the final

con-figurations by

the Penrose

rhombuses,

but because of

high density

of

defects,

the decoration is

possible

only

in a limited area,

being

less than the area decorated

by

the

binary tilings.

The final

configurations

are full of defects and

dislocations,

visible in

figure

2c,d

as untilable

régions.

The

tilings

in

figure

2c and 2d have been made

by assigning

the

binary

tiling

to the atom

configurations

of

figure

and

2b,

respectively. Inspection

of

figure

2a,c

and

figure

2b,d

assures us

that a defect

appears

at the

place

where either L or S atom is

missing.

In the

molecular-dynamics

technique

the atomic

rearrangement

is

mainly resulting

from a diffusion of a

single

vacancy. This

mechanism does not

preserve

the basic move,

figure

Ib,

required

for the random

tiling

model,

and hence it is the source of defects. The

density

of defects in both

configurations

was

approximately

the same.

The diffraction

pattern

of the disordered

system

at

high

temperature

of

0.32To

consited of

diffuse

rings

like the

pattern

of the

glassy

system.

The diffraction

patterns

of the two final

con-figurations, figure

2a,b,

with

VLU

= 0 and

VLLL =F

0 are shown in

figure

3a,b,

respectively.

As

a reference we show in

figure

3c the diffraction

pattern

of a

perfect quasicrystal

[6,11] .

Com-paring

the three

figures,

one could

recognize

the overall

pentagonal

symmetry

of the diffraction

patterns,

although

the

spots

are not at all

sharp,

the

angles

of the five fold

symmetry

and the

in-tensity

of

equivalent

spots

which should be

equal,

are not

exactly respected.

These are the natural

consequences

of the very limited sizes of domains in our small

system.

’Ib

prove

the existence of the

crystal-quasicrystal

phase

transition one should heat the final

configuration containing

précipitâtes

of thin

crystals

and should observe the

quasicrystal

growth.

This is difficult to achieve

by

the

molecular-dynamics

technique

because the

microcrystalline

pre-cipitates

should desolve

by

the diffusion mechanism which is slow in

comparison

to the available

simulation time. Th make the diffusion faster one

usually

elevates the

temperature,

but then one

créâtes new defects which

destroy

both the orientational correlations in the

system

and the

pen-tagonal

symmetry

of the diffraction

pattern.

Indeed,

in the

presence

of the three

body potential

we have heated the

configurations

of

particles

shown in

figure

2b,

up

to

0.40To,

equilibrated

and

quenched

rapidly

with the

temperature

rate

dT/dt

= -0.010. In the

quenched configuration

the

microcrystalline

précipitâtes

of the thin

crystals

could be

recognized

in the volume of

only

6% of

the

system.

But the

density

of defects has increased almost twice in

comparison

to the

density

in

the first two runs, and this amount of defects

destroys

the orientational correlations so much that the

pentagonal

symmetry

of the diffration

pattern

cannot be seen any

longer.

(6)

Fig.

2. - Final

configurations

of

binary alloy

at

low-temperature

0.08To with

(a)

Lennard-Jones

potential,

(b)

Lennard-Jones

potential

supplemented by

orientational three

body

potential acting

between

large

atoms.

(c)

and

(d)

Binary tiling

corresponding

to

(a)

and

(b)

particle

configurations,

respectively.

Lennard-Jones

system

which

spontaneously

forms a

quasicrystalline

state.

Adding

a three

body

potential

one can favour the

crystalline

state. This

system

under slow

cooling

arrives to the

equi-librium state which contains

microcrystalline precipitates

and

preserves

the five-fold

symmetry

of

(7)

1796

Fig.

3. - Diffraction

patterns of atom

configurations

at low temperature with

(a)

Lennard-Jones

potential

and

(b)

Lennard-Jones

supplemented

by

orientational

three-body potential acting

between

large

atoms. The

(a)

and

(b)

diffraction patterns

correspond

to

configurations of (a)

and

(b)

of

figure

2,

respectively. (c)

The diffraction

pattern

of the

perfect quasicrystal.

Acknowledgements.

One of us

(K.P)

would like to thank the staff of Laboratoire de

Physique

des

Solides,

University

Paris

Sud,

Orsay,

for their

hospitality

and assistance

during

his

stay

there and CNRS for a financial

support.

Fruitful

comments on the

computer

program

by

E

Augier,

J.R Gouriou and J.M. ’Iéuler

are

gratefully acknowledged.

References

[1]

AUDIER M. and GUYOT

R,

Third International

Meeting

on

Quasicrystals:

Incommensurate Structures in Condensed

Matter,

27

May-

2 June

1989,

Vista

Hermosa, Mexico,

to be

published.

[2]

BANCEL

P.A.,

Phys.

Rev. Lett. 63

(1989)

2741.

[3]

DÉNOYER

F.,

HEGER

G.,

LAMBERT

M.,

AUDIER M. and GUYOT

P., J.

Phys.

France

5i

(1990)

651.

[4]

LAMBERT

M.,

DÉNOYER

F.,

C.R.Acad. Sci. Paris 309

(1989)

1463.

[5]

LANÇON

F. and BILLARD

L.,

Europhys.

Lett. 2

(1986)

625.

[6]

LANÇON

F. and BILLARD

L.,

J.

Phys.

France 49

(1988)

249.

[7]

WIDOM

M.,

STRANDBURG K.J. and SWENDSEN

R.H.,

Phys.

Rev. Lett. 58

(1987)

706.

[8]

HENLEY

C.L.,

J.

Phys.

A21

(1988)

1649.

[9]

WIDOM

M.,

DENG D.R and HENLEY

C.L.,

Phys.

Rev. Lett. 63

(1989)

310.

[10]

WOLNY

J.,

PYTLIK L. and LEBECH

B.,

J.

Phys.

21

(1988), J. Phys.

Condensed Matter 2

(1990)

785.

[11]

STRANDBURG

K.J.,

Phys.

Rev. B40

(1989)

6071.

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