Molecular dynamics study of vacancy like defects in a model glass : dynamical behaviour and diffusion

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Molecular dynamics study of vacancy like defects in a model glass : dynamical behaviour and diffusion

J. Delaye, Y. Limoge

To cite this version:

J. Delaye, Y. Limoge. Molecular dynamics study of vacancy like defects in a model glass : dy- namical behaviour and diffusion. Journal de Physique I, EDP Sciences, 1993, 3 (10), pp.2079-2097.

�10.1051/jp1:1993233�. �jpa-00246853�

J. Phys. I France 3 (1993) 2079-2097 _OCTOBER 1993, PAGE 2079

Classification

Physics

Abstracts

61.708 61.40D 66.30

Molecular dynamics study of vacancy like defects in

_a

model

glass

_:

dynamical behaviour and diffusion

J. M.

Delaye

and Y.

Limoge

Section de Recherche de

Mdtallurgie Physique,

Centre d'dtudes de

Saclay,

91191 Gif-sur-Yvette Cedex, France

(Received 24 Marc-h 1993, accepted in final

form

2J June 1993)

Rdsumd. Nous _avons montr£ dans l'article associd ill,

qu'il

dtait possible de ddfinir dans _un

verre de Lennard-Jones des ddfauts prdsentant toutes les

propridtds

des lacunes dans les cristaux.

Ce premier article dtait focalisd _sur les

propridtds

statiques. Le prdsent article s'attache h ddcrire leurs propridtds

dynamiques

h tempdrature non nulle, ainsi

qu'au

calcul de la contribution qu'ils apportent ^{h la} diffusion. Nous dtudions donc leurs grandeurs thermodynamiques caractdristiques enthalpies, entropies et volumes de formation,

qui apparaissent

_sous forme de

propridtds

distribudes. Ces donndes _sont ensuite

intdgrdes

dans _un moddle de marche _au hasard pour ddterminer la contribution h la diffusion.

Abstract. We have shown in the

companion

_paper II, referred to as D-L- I hereafter, that it _was

possible

to introduce in _a glass of Lennard-Jones particles, which is _to be understood _{as a}

plausible

model for metallic glasses,

point

defects that have all the characteristics of vacancies in _a crystal.

These defects _can be introduced in the glass by

removing

_an atom, they can

jump

_{over one or}

several interatomic distances and

finally

the defects _can be annihilated by complex collective displacements once they have arrived _on

specific

sites. In the first part ^{of this} work, _we studied the static properties of the vacancy like defects _: in the present one we shall describe their

dynamics

at a higher temperature, as well _as delineate the contribution of these defects _to the transport m the Lennard-Jones

glass,

by calculating their

thermodynamical

properties, formation enthalpy, fornlation volume and formation entropy. ^{All the~e} characteristics _are used in _a random walk model.

Introduction.

In the first part of this article, hereafter denoted _as D,L. I

Ii,

we have shown that _a vacancy- type ^defect can be defined in _a Lennard-Jones

glass,

the latter

being

understood _{as a}

plausible

model for the _structure and the

dynamics

of

amorphous

^metallic

alloys.

Until _now the concept ^of defect _{was not} considered

interesting

for

modelling

the

properties

of

amorphous

metallic

alloys [2].

The main _reason for this is twofold _on the _one hand there is

a

conceptual difficulty

in

defining

defects in _a system

lacking

_{a proper} referential

and,

on the other

hand,

_an insufficient

insight

_as to the very effects of the disorder _upon the various

JOLRNAL DE PH~SIQLE -T ~ N' lo OCTOBER lQ9~ 76

2080 JOURNAL DE

PHYSIQUE

I N° lo

properties

of

possible

isolated defects which has lead to erroneous conclusions

conceming

the statistical mechanical

properties

of _an

equilibrium population

of defects

[2, 3]. Therefore,

_we decided to avoid the difficult

question

of _{an a}

priori

definition of the vacancy and _we followed the evolution of _a Lennard-Jones

glass

after _«

picking

out » an atom, at various

temperatures,

In D-L-

I,

_we have

shown, using

an extensive statistical

study

at 0

K,

that the vacant site left

by

the _«

picked

out » atom evolves

following

_one of three

possible

_ways. A

persistent

_vacancy

can be found for _an

arbitrary long

^duration after _an initial relaxation of the

neighbourhood.

A second

path

of evolution is the

jump

of _one of the

neighbours

of the vacancy into it. In this

second _case the vacancy has

jumped

_{over one} interatomic distance in the

glass

and the

topology

of the initial site is reconstituted. The last

possibility

is the annihilation of the vacancy

« on the spot»

involving

small

cooperative displacements

of _a

large

number of _atoms.

Approximately

_{one quarter} of the sites in the Lennard-Jones

glass belong

to this last category and three quarters to the other two.

In the present

article,

we shall describe the second part ^of our work, devoted to the

dynamic

behaviour _at 5 K and lo K, in order _to show the influence of the thermal

agitation

_on the three kinds of different static behaviour observed _{at zero} temperature ^{in the} first paper

ii.

We will show that these kinds of

possible

behaviour for _a vacancy still

persist,

but the

respective probabilities

evolve with temperature.

Finally

we shall present ^{the outline of} a

possible

model of diffusion

involving

_vacancy defects,

Results.

METHOD,

Systems

A' and B' _are used for

experiments performed

at 5 K and lo

K,

for the

first,

and _at lo K for the second. The method which is used _to

keep

^the temperature constant has been detailed in D,L, I and will _not be reiterated here. A few observations of the behaviour of the defects have also been done _on the

large

system

C,

which did not show differences with the smaller _two.

The behaviour is followed

during

7 000 time steps ^{of lo~ '~} s for each vacancy studied, The set of introduction sites used in this part ^{is the} same as that used for the simulations

performed

at o K

ii,

The

analysis

^of the simulation _runs is the _{same as} that used for the simulations done _at o K.

The simulations

performed

_on system ^A' are

analyzed by using

the time evolution of the

average Voronoi volume

V~

of the _atoms

surrounding

the vacancy site and

visualizing

^their

displacements.

The simulations

performed

_{on system} B' _are

analyzed by

this

technique

_as well

as with the

help

of the _two criteria RECONS and

DISSIP,

which _were introduced in D.L. I.

RESULTS AT 5 K AND lo K. At these temperatures too, the results _can

clearly

be

separated

into three main classes _: sites where the vacancy still remains

present

on its introduction site

after the 7 000 simulation time steps, ^{and the}

dissipation

sites where the collective relaxation of the

neighbourhood

absorbs the

« vacant volume _» of the vacancy, and also reconstitution sites where _{one or more} atomic

jumps

_occur

allowing

for _a reconstitution of the _same local

topology

_as there _was before the introduction of the vacancy.

As

before,

we checked whether the evolution of the defect after its introduction could be linked with the characteristics of the site before introduction _we searched for correlations with the atomic stress level and the

topology

of the site

[il.

As

previously

observed _at OK _no correlations have been found at 5 K neither at 10K with the shear _{stresses nor} with the icosahedral

character,

but rather with the

hydrostatic

_pressure ^{value and} the

sphericity.

The

plots

^of the simulation results in the pressure

sphericity

_map, obtained with system A' at 5 K and 10 K _are shown

respectively

in

figures

I and 2, while those obtained with system B' at 10 K is shown in

figure

3. In these three

figures,

the filled dots represent ^the events of

N° lo VACANCY-LIKE DEFECTS IN A GLASS DYNAMICS 2081

lo

i7

fl

m 8

w , ^{o o}

j

6 f8

~

c~

^f8 f~

g

_~ ⁱ⁸

I

~

'p B~~msboDoflt~B

U .«W» iB . iB

G

^~

f

_{f8~ ~}

°

~£~e.~

~ .

X 2

o 2 4 6 8 lo 12

Spherlclty

^* 1000

Fig.

I. Statistics of the _events observed at, 5 K _on system A', in _a pressure x

sphericity

_map. The results of the simulations _are divided into four classes (O) for the

stability

events, (.) for the

reconstitution

by

atomic

jumps

events, (fl3) for the

dissipation

events, (+) for the uncertain events.

~

fl

t4

it

( "+

Cb [

₊

~

°

°+

# ,°.

^lB

^~lB i8~iB

11 i8~B +

"

. . fB

~ o O i8.

n 2 iB _.

fl

4l

f

_~

U

~

~

~

0 So loo 150 200 250

Spherlclty

Fig.

2. Statistics of the events observed at lo K _on system ^{A'. Same}

symbols

_as in

figure1.

=

© °

~ ~

W

; o

#

^~

f (

# ~

_.

1 , j

^fb

~~

^~

j ~

G '

~

⁺

G

2

. .

n. .

~

^~

20 40 So an loo 120 140

Spherlclty

F;g_ 3. Statistics of the events observed _at lo K on system B'. Same symbols _as in figure1.

2082 JOURNAL DE

PHYSIQUE

I N° lo

reconstitution

by jumps,

the open dots represent ^the

stability,

and the squares the

dissipation

ones. It is clear from the three

plots,

_as

already

observed at OK, that the

stability

or the

reconstitution

by jumps

_are favoured

by

_a

high

local

sphericity

and _a low local pressure. The

dissipation

_{ones are}

dispersed

_among the sites of _mean and

large

values of

hydrostatic

_pressure

and

sphericity. Again,

_{a very}

large hydrostatic

_pressure is not

incompatible

with _a

persistent

vacancy : but the small number of such sites prevents us from

being

_more affirmative. It must be noted that the number of stable sites is smaller than _at OK- This

point

will be further discussed with the

presentation

of the

probabilities

of each kind of behaviour at 5 K and lo K.

For the simulations

performed

_on system

B',

we also used the criteria RECONS and

DISSIP.

Figure

4

displays

in _a RECONS DISSIP map the observed events. The values of

RECONS _are

larger

than those obtained at 0 K due _to the thermal vibrations and it becomes

more difficult to discriminate the three kinds of behaviour with the _mere

help

^of their

magnitude. Conversely

the criterion DISSIP remains relevant and allows _us to make the difference between the

stability

events,

corresponding

to small values, the

dissipation

events,

corresponding

to intermediate

values,

and the reconstitution events,

corresponding

to

large

values.

Again, analyzing

the results

by using

these criteria _or others, such _as Voronoi cells and atomic

displacements, gives

the _same

picture.

1#~

~

~~_~ ~

~~~~~ii~@~

w -

#

X I#'

io.7 _{to" lo" lo"} io.~

Dissip

Fig.

4. Statistics of the _events in system ^{B in} a RECONS*DISSIP map at lo K _on system ^{B'. Same} symbols _as in figure1.

PROBABILITIES OF EvoLu'rIoN. Table I presents ^{in the last two} rows the _true

probabilities

of the three kinds of behaviour _at 5 K and lo K

(applying again

a correction for the

oversampling

of _some parts ^{of the}

sphericity histogram,

_as

explained

ⁱⁿ

D.L.I).

An

interesting point

to be noted from the evolution of the

probabilities

with

temperature

^is

that, taking

into account the uncertainties inherent to the

probability calculations,

the fraction of

dissipation

events is

roughly

constant

amounting

_once more to 25 fb of the total.

The fraction of

stability

sites decreases

strongly

with temperature, ^{while the} percentage ^of sites of reconstitution increases for the

complementary

amount. This _means that the stable sites and the sites restored

by

atomic

jumps

_can be classified into the _same category, ^called « well ordered _» sites, while

dissipation

sites _are such that vacancies

dissipate

on them whatever the temperature ^and can be called _«

badly

ordered _{» ones.} In this model, the

sphericity

criterion appears ^{to be} a

quite pertinent geometrical

^{criterion in order} to

distinguish

_« well ordered

»

N° lo VACANCY-LIKE DEFECTS IN A GLASS DYNAMICS 2083

Table I. Probabilities

ofthe possible

events

for

the

decay ofa

_vacancy in _a

glass.

No values

are available

for

_system B at 5 K.

Stability

Event Reconstitution Event

Dissipation

Event

B or B' A _or A' B _or B' A _or A' B _or B' A _or A'

System

29fb 56fb 34.5fb 17fb 25fb 21fb OK

lsfb 45fb 29fb 5K

9fb 17fb slfb 50fb 28fb 27fb 10K

sites

(I.e.

sites with

a

good

^local

sphericity)

and _«

badly

ordered _» sites

(I.e.

sites with _a less

developed

local

sphericity).

^An

example

of such _a site

displaying stability

at 0 K and

giving

rise to a reconstitution event at lo

K,

is

given

in

figure 5,

which shows the evolution _{as a} function of time of the average Voronoi volume of the _atoms

surrounding

the vacancy site. At OK the vacancy remains stable

during

the 7 000 time steps ^{of the} simulation.

Conversely

at

lo K the average Voronoi volume decreases

quickly

after

approximately

5 000 time steps.

The final value is

equal

to the initial _one, before the introduction of the vacancy,

evidencing

an

atomic

jump.

The visualization of the atomic

displacements

^confirms the _occurrence of _a

jump

: in this _{case a} collective

jump involving

two atoms.

Nevertheless,

the distinction between _« well ordered _» and _«

badly

ordered _» sites is not clear cut, as can be

expected

^for a

non-crystalline

system, and we cannot exclude the existence of _some intermediate _cases.

Indeed, we observe _a few

dissipation

_{events at} 5 K and lo K on sites which _were

displaying stability

at 0 K. Such _an

example

has been _more

completely

described in _a

previous

article

[4].

Such unclear _{events are} nevertheless

quite

_rare, of the order of _a few

percent

^{of the total} number of the sites studied.

40

39.S T-OK

E ^~~'~

~

38

j~£,

37.S

: T.IOK

37

loon IOOO 3000 SOOO 7000

Time

Ii 0"'s)

Fig. 5. Average Voronoi volume, V~. of the atoms surrounding _{a vacancy} site. The square box shows

the value of V~ ^{before the} introduction of the vacancy. The graphs show the evolution of

V~ versus time _at T OK ( )_: the vacancy remains stable, and _at lo K (---) the vacancy

jumps

at t ₌ 5 000 and the site is

topologically

restored after the

jump.

2084 JOURNAL DE PHYSIQUE I N° 10

In order _to characterize the

geometrical organization

of the

spherical

order, a more detailed

study

has been done

conceming

the

spherical

sites _: the 15 fb of the sites

having

the smallest value of the

sphericity criterion,

and the 15 fb of the sites

having

the

largest

value _are shown in

figure

6. At first

sight

this

figure

indicates that the most

spherical

sites form _{more or} less 3d

clusters,

while less

spherical

_{ones seem} to be

grouped

as a

layer

inbetween _« well ordered _» volumes. This

qualitative picture

is confirmed

by counting

the atomic

neighbourood

between both types ^of atoms, a « bond _»

being

counted whenever the distance between _{two atoms} is less than the distance

corresponding

to the first minimum of the

pair

distribution function. Such

« bonds _{» are more numerous} between the _more

spherical

_atoms than between the

badly

spherical

_ones. However these results _are

only

_very

preliminary,

and _{a more} detailed

study

is needed. To draw _a firmer conclusion it would be better to _use a criterion based

directly

_on the vacancy

behaviour,

and not the criterion of

sphericity

which has

only

a statistical

meaning.

*

Fig.

opology black and

the

15 9b atoms the _least spherical ^{one (grey toms)are} shown.

The

more

spherical

atoms seem to be _more or _less

grouped in 3d clusters while the _less spherical are _slightly

dispersed in between.

COLLECTIVE JUMPS. A

peculiar

feature of the

dynamic

behaviour, after the introduction of the vacancy, ^must be

emphasized

_:

namely

the

frequent

_occurrence of

multiple

atomic

jumps, involving

2 to 5 atoms,

moving simultaneously

_or

nearly

so. Even if _{a more} detailed

study

of this

phenomenon

_were to be necessary, the sequences of

jumps

_can

already

be characterized in the

following

manner the first _atom

jumps

_{over one} interatomic distance and

occupies

the vacancy

site,

the second _one

replaces

the first _one in _a

slightly

more

imprecise

_manner, and the vacancy is

dissipated

at the end of the

jump path.

This observation is _a

rough

indication of the

size of the _« ordered»

region along

the

jump,

since _a

multiple jump

of several _atoms

reconstituting

the initial

topology corresponds

to the existence of _a

significantly large

« ordered _»

region.

As _we have

already

shown

[4],

if the vacancy is

directly

introduced _at the end of the

jump

_{sequence, we} shall observe

systematically

_a

dissipation

event, and _{never a}

stability

_{or a}

jump.

^This behaviour

points clearly

to a

separation

of the sites of the Lennard- Jones

glass

into two

categories

which have _a permanent ^{character. This} permanency has to be

judged

with respect to the life time of _a vacancy.

Therefore,

the 25 fb of the sites where the vacancy

disappears

_can be named vacancy sinks and _sources.

N° lo VACANCY-LIKE DEFECTS IN A GLASS _: DYNAMICS 2085

In _a

preliminary study

_we have searched for the

eigenmodes

of system ^B without _vacancy

(864 atoms)

and

compared

them to those of the _same system

containing

_a stable vacancy

(863 atoms)

at

OK,

after _a proper relaxation of the

neighbours

of the vacancy site before

calculating

the

eigenmodes.

We have chosen _a site where the vacancy is stable _at 0 K and where _a collective atomic

jump

_occurs at 5 K. The introduction of the vacancy induces the creation of localized

eigen

modes of low

frequency

_on the _atoms

jumping

at 5

K,

with _a

large amplitude

_on the first _one. So the

frequent

_occurrence of collective

jumps

is

probably

linked with the _onset of _a few low

frequency

localized modes induced

by

the presence of the _{« vacant}

volume _» of the vacancy.

«SPONTANEOUS _{» VACANCY CREATION.}

-Coming

back to our initial

topic,

that

is,

the

possible

existence of vacancies in _a

glass

and their

dynamical behaviour,

_{we can} consider the

regions

where the vacancies

dissipate

_as the _sources and sinks for the

vacancies,

with the vacancy

travelling during

its life time _across the _« well-ordered _»

region.

A collective motion

of _atoms in _a

dissipation region

creates a « vacant volume _{» we can} call _a vacancy. This

vacancy has the

possibility

of

reaching by jumps

a

region

of _« well ordered _»

sites,

where it

jumps

several times until

reaching

_{a new}

dissipation

_zone. Then it is absorbed

by

a second collective

reorganization.

In this model, the _« well-ordered _{» zones} remain _{more or} less _« well- ordered _» after the passage of _a vacancy.

Naturally

such _{an event cannot} be observed _{to occur}

spontaneously

in _our system, ^due to the small box size and the finite duration of the simulation, except ^{if its formation}

enthalpy

_were

exceedingly

small which is

apparently

not the _case

(see later).

Nevertheless _we have simulated the creation and the annihilation of _a vacancy

using

the

constant volume

algorithm.

We first searched for _a

site,

labelled S in the

following,

where _a

jump

occurred when _a vacancy was introduced _on

it,

before

being dissipated

in _a

neighbouring dissipation

_zone, after _some

jumps.

Then _we

performed

_{a reverse} time

simulation, starting

from the final stage ^{of the} vacancy

dissipation, inverting

the _two final consecutive atomic

positions

and

performing

a

long

_run

starting

with these two inverted

positions.

We therefore observed first the creation of the vacancy

by

_{a sequence} of atomic

displacements

which is the

exact inverse of the sequence which

produced

^the elimination

during

the direct calculation. We

therefore observed the _« spontaneous » appearance of _a vacancy on the site S

initially

chosen.

After that, the _{same atom} which had been

jumping

into the vacancy in the direct time

simulation, again performed

the _same

jump

^{in the} reverse time simulation and the

decay

of the vacancy followed the _same

path

^{in both the} direct and _reverse simulations. This is shown in

figure

⁷

by plotting

the coordinates _{as a} function of time of the concemed atom

starting

with the final

positions (inverted)

at the end of the direct calculation. After the first

jump corresponding

to the appearance of the vacancy ^on site S, the exact reverse

jump

_occurs. This

was to be

expected

since the

migration

barrier of the

jumping

atom must be the lowest among those of the

neighbours

of site S _as this

jump

occurs

spontaneously.

Therefore _at the

quite

low

temperature ^{of the}

experiment,

the

probability

for this _{atom to}

jump

is much

larger

than that for the other _ones.

THERMODYNAMIC CHARACTERIZATION OF A VACANCY. In two

glasses,

A' and B, we have

calculated the various

thermodynamical properties

of a vacancy on a

quite large

number of sites. The distributions obtained _are

naturally

not exhaustive but

certainly

reflect

correctly

the

true ones. A _more

developed study,

^which tums out to be

quite

time

consuming,

is

already

under way. In order to delineate the

respective

roles of the level of relaxation and of extemal pressure, system ^{B has been also studied} at a zero pressure and _at 2.5 kbars

(same

_pressure as

in system

Al.

At these _two other pressures, the _same sites as in the

original

_{system at} 0.8 kbar have been used.

Therefore,

the three sets of data

correspond

to the _same vacancies _at different

2086 JOURNAL DE PHYSIQUE I N° lo

o.6

0.4

0. 2

f Y-coordinate

0

~i

~

0. 2 o

° 0. 4

o. 6

X-coordinate

0.8

0 500 100015002000 25003000

Time

Fig. 7. Illustration of _{a reverse} time simulation. The xyz coordinates of the

jumping

atom _are plotted

versus time. The 000 first time steps correspond to the vacancy creation and its migration _up to the site

under examination, only the last jump ^of the vacancy being visible for the atom itself (site S in the text).

After _a very rapid thermalization of the defect _on S, the _{exact reverse} jump occurs and the vacancy

disappears

by ^the same way it

appeared, given

the very

high probability

of that

precise jump.

By ^chance the

jump

path is almost

perfectly

along _y.

pressures, these vacancies

being persistent

_ones at 0.8 kbar. We gave in D.L. I the formulae used _to calculate the formation

enthalpy, AH~,

and

volume, AV~.

In the f.c.c. argon

crystal

at the _same pressure as in system ^A' we got a value of 0.134 eV,

including

a

large

contribution of the extemal pressure, P

AV~ amounting

to 0.05 eV. The

potential

_energy contribution amounts then _to 0.084 eV in

good

_agreement with the value calculated

by

Bennett

[26]

for

crystalline

argon under zero pressure

(0.088 eV).

We have summarized the results obtained _on both systems ^{A' and B in table II and}

figure

8.

The distribution of the formation

enthalpies

is

quite large ranging

from 0.o21 eV _to 0.12 eV with _an average of 0.07 eV in system A', from 0.047 eV _to 0.112 eV with _an average value of 0.08 eV in system B at 0.8 kbar and from 0.034 eV to 0.08 eV _at 0 kbar the _mean value

being

0.062 eV. The distribution of the formation

enthalpies

in this system ^is narrower than in system

A' and the average value calculated for B is

larger

than the

corresponding

value for

A'. This difference between the _two distributions is

mainly

due to the presence of _{a set} of stable vacancies

displaying

small formation

energies

and volumes in A'. Since in both systems ^the sites have been selected _at random, system ^{A' contains}

certainly

_a set of _«

stability

_»

sites,

which transform in system B into _«

jump

_» sites,

according

to what _was

already

found in

D.L.I. This difference is

likely

to be due _to the effect of pressure on the

mobility

which stabilizes the more

easily jumping

atoms, thanks to the effects of the

migration

volume.

However the role of the different levels of relaxation cannot be

discarded,

but _we think it _to be less

probable

since _a

highly

relaxed state is _more prone ^to

produce

_more stable vacancies than

the _reverse. On the contrary, the role of pressure in system ^{B is} very clear _{: on} the _one hand,

under pressure the formation volume decreases thanks _to the well-known vacancy compress-

ibility,

and _on the other hand, the external pressure enhances the formation

enthalpy by

the trivial P

AV~

_term.

Figure

9 shows up a correlation between the formation

enthalpy

and the local

hydrostatic

pressure : the formaiion

enthalpy

increases when the local pressure ^decreases. Indeed _a low local pressure site is

likely

to be _a site where the interatomic distances _{are near} the

crystalline equilibrium

value and

therefore,

where atoms will have _a

large

^cohesive energy. It is thus _more

difficult to create a vacancy on this site.

N° lo VACANCY-LIKE DEFECTS IN A GLASS DYNAMICS 2087

Table II. Formation

entha/pies

and volumes _{as a}

function of

local _pressure and

sphericity

in system ^A'.

Formation Volume Local Pressure Formation

Enthalpy

(9b

atomic

volume) (kbar) (eV)

^~~~~~~~~~~

83 fb -0.26 0.12 1.19

76 fb 0.09 0.116 0.98

18 fb 1.8 0.036 3.18

34fb 2.17 0.063 5.22

20 fb 2.46 0.054 3.34

20 fb 2.47 0.058 0.97

58 fb 2.49 0.09 3.00

44 9b 2.57 0.108 6.01

26fb 2.57 0.042 3.56

38 fb 2.65 0.072 7.37

46 fb 2.99 0.086 1.47

16 fb 3.16 o.04 3.29

16fb 5.8 0.052 1.51

33 fb 5.89 0.058 5.43

40 fb 6.05 0.066 5.09

15 fb 6.24 0.032 6.62

16 fb 6.87 0.021 10.24

40fb 8.12 0.061 5.89

The vacancy formation volume _we found in _an argon

crystal

at 2.5 kbars _{amounts to} 92 fb of

an atomic volume. In the

glass (Tab. II)

the distribution extends from 15 fb _to 80 fb of _an atomic volume in system ^{A' with} an average of 35. 5 fb, and from 39 fb _to 88 fb of _one atomic

volume in system B, with an average of 66 fb. The

larger

the formation volume, the lower the local pressure. Low local pressure sites can therefore be

compared

to

crystalline

sites _:

large

formation volume and

enthalpy.

2088 JOURNAL DE PHYSIQUE I N° lo

8.14

~

%3 _~

e0.12 _iB

D

~ D

q~

^iB

fi ^o-1

~@ . *

~°

^~

_~

_fi

li ~~

_M. o

~ O ~

g

O °

~ °

E~~ * ^°

~§

_~ ^O

D

~

0 2 0 4 0 6 o 8 o 18 8

Formatlon Volume

(°An)

Fig.

8. Formation

enthalpies,

in eV, versus formation volumes, fb of _an atomic volume.

System

A': (D) _system B lo) at 0 kbar (.) at 0.8 kbar (fl3) at 2.5 kbars.

O.14

O-la q~

O

_

O-I ,e

> Q

~O.OS ~_

^~

* O

" .

Q .~ O

@ °.°S

~

o °

5 _.

. °

#

O.04

~ Go

O

O.02 O

2 O 2 4 S 8 16

Hydros18tlc _pressure (kbar)

Fig,

9. Same formation

enthalpies

_as in

figure

8 but _versus the local pressure, System A': (o) system ^B (.) _at 0.8 kbar.

We _can also check

that,

_as

explained

in the

description

of the

preparation

of the

glass [I]

_; system ^{B is} more relaxed than system ^A'.

Considering

the correlations, on the _one

hand,

between the formation volumes and

enthalpies (Fig. 8),

on the other hand, between the

formation

enthalpies

and the local _pressures

(Fig. 9),

_we observe that at a

given

value of volume _or pressure the correlations _{are more}

pronounced

for system

B,

where the

dispersion

is

clearly smaller,

than for system ^A'.

The calculation method of the

migration enthalpies

has been

explained

^{in D.L. I.} However the value _we calculate

depends

_on the duration of the forced

jump.

This

dependence

is shown in

figure

lo both for the

crystal,

where the

jump

is forced

along

_a

straight path,

and for the

glass,

where the

jump

^{is forced}

along

^either a

parabolic

_{or a}

straight trajectory.

We _{can see} that the choice of _a

parabolic path

for the

jump

in the

glass greatly

decreases the

migration enthalpies, especially

for the

already

low

migration

barriers. In table III _are summarized the

values of the

migration enthalpies

for ten atoms around a stable vacancy site,

using

either _a

straight

or a

parabolic jump path, together

^with the distance of the

jumping

atom from the vacancy site. The distribution extends from 0.0055 eV _to 0. ii 8

eV,

to be

compared

to a value

~f o 067 ev in f,c,c_ argon at OK

[26].

We can note the existence of very low

migration

N° lo VACANCY-LIKE DEFECTS IN A GLASS _: DYNAMICS 2089

o-i

>-~

° o.oa

~

~

#

a-as

I

O.04 U

f

KO

~ ^~'~~

o

o &oo iooo i&oo

Jump Duration

Fig.

10.

Migration

barriers in eV _{as a} function of the duration of the

jump

for _a

straight jump

path in the crystalline _argon (h), in the

glassy

system A' (.), and for _a parabolic

jump

path in A' ID).

Table III. Values

(in eV), of

the

migration

barriers

for

^{all the} atoms

surrounding

a stable

vacancy site. The

first

column

corresponds

to the

migration

barrier

for

_a

straight path,

the

second

for

_a

parabolic path,

and the third _to the

distance,

in

h, _of

_the

_jumping

_atom

_from

_the

vacancy site.

Migration

_energy

Migration

_energy Distance of the

jumping

(eV) (eV)

atom to the vacancy

Straight

line

path

Parabolic

path (h)

0.018 0.0055 3.494

0.057 0.0511 3.519

0.073 0.0676 3.543

0.028 0.0101 3.554

0.047 0.0456 3.563

0.078 0.0792 3.606

0.058 0.0355 3.607

0.074 0.0767 3.671

0.09 0.0835 3.728

0.12 0.l18 3.745

2090 JOURNAL DE PHYSIQUE I N° 10

barriers which _were

already predictable

^from the _occurrence of atomic

jumps

near zero

temperature.

Unfortunately obtaining

such _a distribution represents a

heavy

amount of

calculations,

which makes it difficult _to search for _a

probable

correlation between the characteristics of _a site and the distribution of the

migration

barriers around it. The

comparison

with the

crystalline

Lennard-Jones is however _not

completely meaningful

since the present

enthalpy

values have been obtained _{at a}

quite large

pressure. As _we have

already discussed,

there _{are some reasons} for

predicting

_non-zero

migration

volumes. Nevertheless this effect is not

likely

to cancel the

large

difference between the

migration

_energy in the

crystal

and the

highest enthalpy

obtained here. The Lennard-Jones

glass displays migration

barriers which _can be much

higher,

as well _as much lower, than in the

crystal.

Finally, using

the

eigenmodes

of _one system

(A'

_or

B)

with and without vacancy, we

calculated _a few formation

entropies, given

in table IV.

Figure

II represents ^{the values of the}

formation

entropies

obtained in system ^B versus the formation volume. The value of the

formation entropy ^{in the}

crystal

^of _argon is found _to be

equal

to 2.7 kbar. Burton found _a lower value of 2.24 kbar, but his calculation _was

performed

on small clusters of _atoms

[6].

In all

Table IV.

Vacanc.y formation entropies

calculated in system ^{A' and in} system

B

using

the

eigenfrequencies

with and without vacancy.

Formation

Entropy

System

A'

System

B

3.I k 3.I k

4.7k 4.9k

5.6k 6.3k

6.2 k 4.8 k

7. I k 4.5 k

7.I k 2.I k

8.0k 4.3k

3.6 k

5.9 k

5.5 k

3.9 k

3.8 k

4.6 k

N° lo VACANCY-LIKE DEFECTS IN A GLASS DYNAMICS 2091

7

-~ ~

«~ _s

.

* .

#

# ~ ~

c .

~

.

C 4

(

.

E

$ ~

2

0 20 40 SO 80 loo

Formation Volume (°AD)

Fig. II. Formation entropies ^{obtained in} system B _versus the corresponding formation volumes.

cases but _one, the formation entropy ^is

larger

in the

glass

than in the

crystal,

with a mean value

amounting

to 5.0 kbar in system A' and _to 4.4 kbar in B. A correlation _can be established between the formation entropy and the

strength

^of the relaxation of the

neighbouring

atoms of

the vacancy, as shown

by

the value of the formation volume in

figure11

_: the lower the

relaxation,

I.e. the

larger

the formation

volume,

the lower the formation entropy.

Discussion.

It is

quite surprising

that in

spite

of _a

significant

amount of work, the _{answers to} the

questions

raised _at the

beginning

of this article _are still controversial. Indeed _a correct

interpretation

of the simulation results _can

only

be obtained after _a careful

analysis

of the way

by

^{which the} defects evolve after their introduction, and _a

thorough

statistical

analysis

of the different

behaviour observed.

We had in fact

already suggested

this

difficulty

some years ago

[3, 7], pointing

out that,

given

_a reasonable formation energy, the vacancy concentration introduced in _a standard MD simulation box

by removing

a

single

atom is far in _excess of

thermodynamical equilibrium.

Therefore it should

disappear

as soon as

I)

defect _sources and sinks _are

present

^and

it)

_a sufficient thermal energy is

given allowing

the defect to

jump.

This is indeed what occurs in this very

simple glass.

Albeit smaller than in the

crystalline

Lennard-Jones, the _mean

formation

enthalpy

of a vacancy is still

sufficiently large

for the

supersaturation

due to a

single

vacancy ^to be

exceedingly large,

even in system ^{C. A}

quite large

fraction of the sites _{act as} vacancy sinks,

approximately

one quarter. ^{The situation is indeed similar} to the _case of the

crystal containing jogged

dislocations, ^{with the}

only

difference

that,

in the present _case, ^the wide distribution of

migration enthalpies

allows an efficient elimination _to take

place

_even at a

low temperature. ^{In this} respect, ^{it is} very

important

to underline the difference between the mechanical and the

thermodynamical stability

^{of defects.}

Our

present

^results are not

contradictory

to the

previous

ones. In Bennett's work

[5],

_a careful

reading

allows us to deduce that

dissipation

and reconstitution events could also have been detected. Given the

quite large

temperature ^{for the} simulations

performed,

the

rapid disappearance

of the monovacancy is in

good

agreement ^{with the} present

finding

of _a wide distribution of

migration enthalpies.

The _same comment

applies

to various calculations

performed

^otherwise

[8].

In all these simulations vacancies _are introduced in a Lennard Jones

glass using

various methods either

«

picking

out _» an atom, as in the present

work,

_or

2092 JOURNAL DE PHYSIQUE I N° lo

simulating

an irradiation

experiment by giving

an atom an energy sufficient to

displace

it. In the latter _case however the local energy

deposition

^{in the} cascade is well-known from radiation

damage

studies in

crystals

to enhance

strongly

^the

annealing

of the defects created. The

meaning

of the

rapid annealing

of defects in the

glassy systems

^{is therefore unclear}

[8c].

Brandt's _statement

[9],

to the contrary, ^{that vacancies} are

always persistent,

_even at finite

temperatures,

appears ^{to be}

partially

due _to the method he _uses for

relaxing

the defective system : an

algorithm

^{which does} not allow for

fully cooperative

^{atomic motions. In the} present

calculations it is this

cooperative

aspect ^which

gives

rise to the

dissipation

events. Moreover the temperature ^{is simulated}

by giving

^static random

displacements

to all atoms : this is

clearly

very far from the collective aspect ^{of the} localized modes _we

found, frequently resulting

in

multiple jumps.

The results of

Doyama [10]

_are

directly comparable

to ours, even if the lack of _a sizeable statistics prevents a detailed

comparison.

The

persistence

of the vacancies, created in this _case

by knocking

down _some atoms, as a simulation of _an actual irradiation of _a

glass,

was indeed noticed in the best ordered

sites, although

the

precise meaning

^of « ordered _» is not

explicitly given.

The fact that

glasses

are

locally

ordered is indeed _{a common}

belief,

and this order is at the root of most of the structural models of

glasses [I ii.

In the _case of the

simplest glasses,

such _as the

present

Lennard-Jones

glass,

the

early suggestion by

Frank

[12]

that the icosahedral order

should be

locally prevalent

in

liquids,

_was later

applied

to two component

glasses by

Sadoc

[13].

The icosahedral order has been searched for

by

various methods

[14, 15],

but it is

Probably

much less

developed

than earlier believed

[14, 16].

In the present ^{results the} icosahedral character of the local symmetry appears to be less

significant

with respect to the

dynamic behaviour,

than the

spherical

_one. More

precisely,

in _a

perfect icosahedron,

the twelve outer atoms are on a

sphere,

and

icosahedricity

_as well _as

SPhencitY

Would be

simultaneously perfect.

In _a disordered

icosahedron,

_as well _as in any dense

packing

of thirteen atoms, the distortion is

likely

to be _more

pronounced

either with respect to the

angular

coordinates of the _outer atoms, or to their distances with

respect

to the Central atom- In _our work

ordering

in

neighbour

distances is

clearly

_more

important

than

Ordering

ⁱⁿ

angular coordinates,

in order _to

explain

the behaviour of _a vacancy introduced _on that site. From this

picture

of the

dynamic properties

of the

amorphous Lennard-Jones,

_an

interesting description

_emerges for the static structure of the

glass.

We observe _zones which

display

_a

spherical

order in the above _sense, with the

largest

«diameter» up ^to 6 _or

7 interatomic

distances,

this value

coming coherently

both from the

length

of the

jump

sequences and from the atomic fraction of «vacancy sites» and «sink»

sites,

if the

dissipation

sites _are

supposed

to form _a

monolayer

between 3d ordered _zones. Nevertheless the

8e°rnetncal

characteristics of these ordered _{zones cannot} be _more

precisely

settled _at the

rf1°rflent. In

Particular

it is _not clear whether

they

_are

truly forming

3d clusters _or if

they

are

hl8hlY

ams°tr°PIG,

diSPlaYin8

_a

large

aspect ^{ratio and} an intermediate character between 3d and 2d. Here

also,

_{a more}

developed study

is

needed,

and is under way. We _can

naturally

_not

deduce fr°rfl the Present

study

whether the

packing

of the ordered _zones is random _or not, and whether the

connectivity

between them is random _{or not.}

However, it is

important

to

interpret

this

picture

not too

naively.

The present system ^is not akin _{to a}

micr°CrYstalline

_one which

displays

well defined

equivalent crystalline

sites. In a

disordered system ^all

properties

_are

govemed by

distributions of

values,

and there _cannot be _a clear cut distinction between vacancy and sink sites. On the contrary, ^{there is}

necessarily

_a

gradual

^{transition of the}

properties

between both groups of

sites,

which is obvious in

figures

II of D.L. I and 4 of the present paper. In this respect, ^the

important

result is that, in all but _a few

percent ^{of the}

sites,

the behaviour of the system ^{after the} introduction of _a defect _can be

clearly

classified either _{as a}

dissipation

event on a sink

site,

_{or as a}

persistence

_or reconstitution _event.