Molecular dynamics study of vacancy-like defects in a model glass : static behaviour

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Molecular dynamics study of vacancy-like defects in a model glass : static behaviour

J. Delaye, Y. Limoge

To cite this version:

J. Delaye, Y. Limoge. Molecular dynamics study of vacancy-like defects in a model glass : static behaviour. Journal de Physique I, EDP Sciences, 1993, 3 (10), pp.2063-2077. �10.1051/jp1:1993232�.

�jpa-00246852�

Classification

Physics

Abstracts

61.70B 61.40D 66.30

Molecular dynamics study of vacancy-like defects in

_a

model

glass

_:

static behaviour

~

J. M.

Delaye

and Y.

Limoge

Section de Recherches de

M£tallurgie Physique,

Centre d'Etudes de

Saclay,

91191 Gif-sur-Yvette Cedex, France

(Received 24 March 1993, accepted in

final

form 21 June 1993)

Rdsumd. Dans _cet article, _{nous nous}

penchons

de fagon

syst£matique

_sur la

possibilit6

de d£finir des d£fauts de type lacunaire dans _{un verre} de Lennard-Jones, h diff£rents niveaux de relaxation _et de

pression,

_{par une} mdthode de simulation

num£rique

_en

dynamique

mot£culaire h volume _ou h

pression

constants. Le d£faut _est cr££ _en

supprimant

un atome et en suivant la

r£ponse

du

systkme.

Nous observons trois comportements

typiques

_: la

persistance

_sur

place

du

« trou _» laiss£ par l'atome

supprimd,

_sa migration _par des sauts

atomiques

clairement identifi£s et enfin _sa

dissipation

sur

place.

Une

analyse

d£taillde de _ces trois comportements montre

qu'ii

est

possible

dans _{un verre} de Lennard-Jones de d£finir des d6fauts de type ^lacunaire.

Abstract. The

possibility

of

defining vacancy-like

defects in _a Lennard-Jone&

glass

is searched for in _a

systematic

_manner.

Considering

different relaxation levels of the _same system, as well _as

different extemal pressures, we use a Molecular

Dynamics

simulation method, to

study

_at constant volume _or extemal pressure, the relaxation of _{a «} piece _» of glass, after the sudden removal of _an

atom. Three

typical

kinds of behaviour _can be observed _: the

persistence

of the empty ^volume left

by

the

missing

atom, its

migration by clearly

identifiable atomic

jumps

and the

dissipation

_{« on} the spot». ^A careful

analysis

of the

probabilities

of these three kinds of behaviour shows that _a

meaningful

definition of

vacancy-like

defects can be

given

in _a Lennard-Jones

glass.

Introduction.

The existence of

defects,

_{e. g.}

vacancies,

in _a

glass

either _{as an}

entity

_{or as a}

possible

_partner in

a diffusion mechanism is still _a

highly

debated

question.

It is the aim of this work _to

provide

arguments

and,

_we

hope,

elements that _{are as} safe _as

possible

for _a clear conclusion on both aspects.

The main

problem

ⁱⁿ

defining

defects in _a

glass

is the lack of _a proper

referential,

in _contrast to the

crystalline

_case, where the lattice fulfils this purpose

fairly

well. However, it has been

recognized

for _a

long

time that _a

long-range

referential is _{not a}

prerequisite.

On the contrary, as

far _as the defect is _a local

entity,

the existence of local order is sufficient for _a

meaningful

definition of _a defect. In

amorphous

silicon for

example,

the _same defects _as in

crystalline

silicon _can be found

[I].

Do metallic

glasses

exhibit _a

high enough

local order _to sustain

defects, particularly vacancies,

which _are the most localized and the

simplest

defects ?

Experimental

_or

quasi-experimental information,

_comes either from the determination of the diffusion

properties,

from creep measurements and various effects like irradiation

experiments (for

a review _see

[2]),

or from numerical simulations

generally using

molecular

dynamics

(M.D.),

but also static relaxation.

The first attempt to answer the defect

question

^{is due} to Bennett et al.

[3].

These

authors, using M-D-,

introduced vacancies and vacancy clusters in _a Lennard-Jones

glass by removing

one atom, or a few

neighbouring

atoms. These vacancies _as well _as the small clusters _were

shown to

disappear

at a

quite

low temperature, ^{I-e- low with} respect to the

migration

temperature ^{of defects in the}

crystalline

Lennard-Jones

glass.

Some defect

jumps

were

probably

detected. However the lack of _a

systematic study

of the correlations between the

properties

of the introduction site and the behaviour of the defects

prevented

_one from

obtaining

_a

complete picture

as well _as the elements of _a

possible

diffusion model.

Moreover,

this work _was later

generally

understood

by

_most workers _as

being proof

^of the

instability

of

vacancies in _a

non-crystalline

solid. On the contrary, some years

later, Doyama [4]

showed

clearly

that vacancies could be created in _a Lennard-Jones

glass, using

the _same M-D- method.

He also noticed that this

persistent

character _was

clearly

associated with the symmetry ^{of the} introduction site.

Later _on, Brandt

[5]

tackled the

problem using

_a static relaxation method. He showed that

persistent

vacancies _can be defined in _a model

glass

with _a local relaxation level around the vacancy

depending

_on the

specific

interaction used

(Morse,

iron-like Johnson

potential).

More

recently,

Laakonen _et al.

[6a]

resumed Bennett's

investigation by introducing

_{a vacancy} in _a Lennard-Jones

glass.

In all _{cases, at a}

low,

but

finite, temperature

^{of 5}

K,

the vacancy

disappears

in _{a more or} less continuous fashion. Once _more, the lack of _a statistical

study

and of any attempt to link the evolution of the vacancy ^to the characteristics of the introduction site

precluded

a more

enlightening

discussion. The _same comment

applies

to later calculations

by

the _same authors

[6b].

Moreover, in all these simulations the authors do _not pay attention _to the fact that defects in _excess of

thermodynamical equilibrium

will

disappear

if the temperature ^is

sufficient for

migration

and the solid contains defect sinks.

Experimentally,

the

question

of the existence of the vacancy is _a bit clearer. Radiation

damage

and irradiation

experiments [7],

of the _same kind _as those done in the sixties and seventies _on

crystals,

have been

performed

_on various metallic systems.

Although naturally indirect, they

_can be

interpreted using

the _same tools _as in the

crystalline

_case,

pointing

out in all likelihood the existence of the _same defects. The _use of the

positron

annihilation method has

been in this respect very useful

[8].

After _some debate

conceming

the presence or the absence of _a life time

corresponding

to the vacancies in

« as

quenched

_» or irradiated

materials,

it has been

recognized

that the strong

trapping

of

positrons

in the _numerous empty _spaces of a disordered solid

precluded

the direct observation of any

possible

concentration of vacancies _at thermal

equilibrium,

_as well _as irradiation created _ones, except at saturation.

Keeping

this

problem

in

mind,

it becomes clear that

positron

annihilation

experiments

_are

compatible

with the existence of vacancies with _an

equilibrium

concentration of the _same order of

magnitude

_as

typical

concentrations in

crystals.

Tuming

now to the vacancy considered _{as a}

possible

diffusion mechanism, the situation is less clear.

Still

using

M-D-

simulations,

_a lot of work has been

performed

with the aim of

studying directly

the diffusion coefficient in _a

strongly supercooled liquid [9].

As _concems the diffusion

mechanism in the

glass,

this attempt appears to be

quite hopeless,

since it

requires

_an

extrapolation

_over at least ten orders of

magnitude

for the diffusion coefficient D. The clear conclusion of this

approach

^is that, down _to the numerical

glass transition,

the

diffusivity

of the

supercooled liquid

_{more or} less follows _an

approximate Williams-Landel-Ferry

law

[10].

Below this

transition, diffusivity

vanishes _as

long

_{as a true}

equilibrated glassy phase

is

produced [11],

and

nothing

_can be said about diffusion in the

truly glassy phase.

However _some very involved studies have been

performed recently [12]

ⁱⁿ order _to detect the

scaling regime,

where D

m

IT T~)~

^{Y and T~}

T~,

^{which is}

predicted by

the so-called

«mode

coupling» approximation

of the

hydrodynamics

of

supercooled liquids.

These

experiments

_are

performed

at or

slightly

above the numerical

glass transition,

nevertheless

they

_seem to

provide

evidence for _{a new} kind of atomic movements,

by jumps, progressively emerging

above the collective behaviour of the

liquid,

which could be relevant _to the present

questions.

We shall discuss these results _more

precisely

later.

Experimental

data _are

contradictory

_as well.

Limoge [13]

determined _an activation volume in _a FeNiPB

alloy by measuring

the chemical diffusion under pressure and found _a value of _one

atomic volume. Later _on,

Faupel

et al.

[14]

also determined the pressure effect upon the diffusion of Co tracers in _a FeCONbB

glass

and found _an almost _zero value

(in

fact

slightly negative

?). More

recently

Hbfler et al.

[15] performed

the _same kind of measurements in various NiZr

alloys,

and obtained activation volumes between _one and _two atomic

volumes,

with _a clear

dependence

_on the

composition.

These three

experiments

_were

explained using

concepts of,

respectively,

_vacancy mechanism, collective mechanism

(still undefined)

and diffusion

by

Frenkel

pairs, remembering

that in

crystalline

Zr, late transition metals _are assumed _to diffuse

interstitially.

However _none of these inferences was based _{on a} clear theoretical

analysis

of the necessary

properties

of these mechanisms in _a

non-crystalline solid,

but

only

_{on a}

comparison

with well- known

crystalline

_{cases, or} with

liquids

in

Faupel's experiment.

It is _now clear that in all _cases

a

thorough

and

theoretically grounded analysis

^of all

possibilities

is needed _to go further and solve these

apparently contradictory

results.

Therefore, as a first step ⁱⁿ a

systematic exploration

of all

possibilities,

_we will characterize what could be _a vacancy in the

simplest amorphous quasi-metallic glass, namely

a

glass

of

particles interacting through

_a Lennard-Jones

potential.

We shall define _a vacancy as a

«

picked

out » atom in _a well relaxed

glass, avoiding

the _a

priori

definition of _a vacancy, and

study

how it will evolve _at various temperatures.

Establishing

the

persistence

in many cases of the hole left

by

this

missing

_atom, we shall be able _to calculate its various

thermodynamical properties

and hence its

possible

contribution to diffusion.

This

study

is

organized

in _two

companion

_papers, the first of which is devoted _to the methods used and static

results,

and will be noted _as D.L. I in the

following.

The second article

[20]

^will be devoted to the results obtained at a non-zero temperature, to the

study

of the various

thermodynamical properties

of the

defects,

and

lastly

to an outline of the contribution of these defects _to atomic transport. It will be noted as D-L- 2 in the

following.

Method.

We shall use systems

containing

either

864,

two systems, or 10976, _one, atoms in _a cubic box with

periodic

conditions. All the simulations _are

performed using

molecular

dynamics,

the

potential

used is the 6-12 Lennard-Jones _:

m 12

m 6

~P(r)

₌ 4 _e

~

r r

The values of

e and _{« are} chosen in order to represent ^the argon f-c-c-

crystal:

e =

l19 K _x

k~

^where

k~

^{is the Boltzmann} constant and _«

=

3.405

h.

Some simulations will be

performed

in the NVE ensemble

(number

of atoms, volume and

energy

kept constant) using

the Verlet

leap-frog algorithm

_at fourth order. Other simulations

will be

performed

in the NPH ensemble

(number

of atoms, pressure and

enthalpy kept constant) using

the

equations

of motion derived from the Andersen Hamiltonian

[16].

Two

extemal parameters ^have to be

chosen,

a fictive _{« mass »} for which we use a value of

1.54 _x

lo? g/cm~

in the

present

case, and _an extemal pressure

depending

_on the

glass

_{we are}

working

with. A

predictor-corrector

method of fourth order is used to solve the

equations

of

motion

[23].

Since the velocities _are involved in the

equations

of motion

describing

the NPH

ensemble, they

_are calculated at time _t from the

previous positions by

the vector formula at order 4 _:

~ ~

llX(t-At)-18X(t-2At)+9X(t-3At)-2X(t-4At)

~ ~~

6 _x At

These

predicted

velocities allow _us to calculate the _new

positions

_at time t. A _recurrent process is then

applied

to refine the values of the

positions

at time t. More

precise

velocities _are

calculated at time t from the _new

positions by

the

equation

:

and

using

the _new velocities, the

positions

_are better

adjusted

and _{so on.} In _{our case, two} iterations _are

applied.

In each of the

experiments

described

below,

_we will

specify

^{which simulation} method

(in

NVE _or NPH

ensemble)

is used. In _a few _{cases we} have also checked that _we recovered the

same results

using

a fifth order Gear

algorithm.

PREPARATION _OF THE GLASS. The first step ^{of the}

study

consists in

preparing

_two small

glassy

systems, A and B, and _one

large, C,

_at different pressures and _at different temperatures.

The

liquids

_are

prepared by relaxing during

3 000 time

steps

^{of 0.5} x 10~ ^'~ _s the proper f-c-c- argon

crystal

at 400 K in the NVE

ensemble, rescaling

the temperature every lo time steps.

The

liquids

differ _at least

by

the random distribution of the atomic velocities

initially

chosen.

Then _we cooled the

liquids

at _a rate of

10'3

_K/s down _to 80 K.

Up

to this

point,

the

glass

obtained is

independent

of the

applied cooling

rate,

according

to the results of Yonezawa

[17].

The

glassy

_system A is obtained

by cooling

further _one of the _two small

liquid

_systems at a rate of

10'~

_{K/s from 80 K down to 20 K}

(using

the constant volume

algorithm)

and

by applying

_a further relaxation in the NPH ensemble

(the

extemal pressure

being equal

to 2.5

kbar)

of 6 000 time steps ^{of10~ '~} s.

From system

A,

_{a new system} A' is

prepared by cooling rapidly

from 25 K down to 0 K in 50 time steps,

correcting

the atomic velocities at each step. ^Thus system A' has the _same

atomic

« network _{» as} system ^A.

Experiments

_are

performed

with system ^A' at 0

K,

5 K, and lo K.

Experiments

of relaxation after the removal of _{one atom are} done _at 0 K

by rescaling

the atomic velocities

regularly,

at 5 K without any

velocity

correction and the temperature ^rises

naturally

to 5 K due to the transformation of the _excess

potential

_energy

(stored during

the very fast

cooling applied

between 25 K and

OK)

into kinetic energy, and _at lo K

by rescaling regularly

the temperature to maintain it _at lo K.

The second

glassy

system B is obtained

by cooling

down the second small

liquid

system at a rate of

lo'~ _K/s

_{from 80 K to obtain}

a

glass

at OK and

by applying

_a further relaxation of 500 time

steps

of 10~ ^'~ _s in the NPH ensemble

(the

external pressure

being equal

to 0.8

kbar).

From system

B,

a system ^{B' is}

prepared

at the temperature of 10 K

by heating

_system B in loo time steps,

increasing

the temperature ^{of 2 K each 20 time} steps.

Experiments

at OK _are made with

system

^{B and}

experiments

_at lo K _are made with system ^{B' without} any correction ir~

temperature

^{in both} cases. The rise of temperature

during

a relaxation after the removal of

one atom is very

small,

due _to the better relaxation of the

glassy

systems B _or B' _as

compared

to systems ^{A and A'. We will} see that this difference in the relaxation levels of the _two systems has nevertheless no

deep

^influence on the

dynamical

results. A few

experiments

have been done _{on a}

larger _system, C, containing

^{10 976} atoms,

prepared

in the _same conditions _as

systems ^{B and B'. Since the results obtained} on this

larger

system are

strictly equivalent

to those obtained _on B and

B', they

will _not be discussed

specifically

neither in D-L- I _nor in D-L- 2.

LOCAL CHARACTERIzATION OF THE ATOMIC SITES. In the rest of the paper, as well _as in

D-L-

2,

different criteria will be used _to characterize the local

topology

around the _atoms.

The local

hydrostatic

_pressure

p(I )

and the shear _stress

t(I)

at site _are defined _{as :}

where the

«,'s

_are the

diagonal

elements of the atomic stress tensor and 3 is the _sum of the three subdeterminants of the atomic _{stress tensor}

according

to the definition of Bom and

Huang [18].

A criterion of

sphericity

_s

ii )

is defined

according

to :

s(I )

=

l~~ (

(r,~

~)~j

x 000

"

j

where the

r,~'s

are the distances between _atom I and its

neighbours j

and _r~ is the average distance between _atom I and its 12 _nearest

neighbours.

The number of

neighbours

used for the

definition of the

sphericity, 12,

is

arbitrary (in

_an

amorphous

system, each atom can have from II to 14 first

neighbours). Nevertheless,

the 12 nearest

neighbours

used to calculate the

sphericity

of _{an atom are}

representative

of the local

topology. By

this

method,

_we overestimate the value of

s(I)

for the _atoms

presenting

II first

neighbours only,

if _we compare it _{to a}

criterion of

sphericity

based _on the II first atomic

neighbours.

The values of _s

(I

for the _atoms

presenting

13 _or 14

neighbours

_are not very different from the values calculated with _a criterion

using

the 13 _or 14 first

neighbours.

Finally,

_a criterion of

icosahedricity h(I )

of _a site I is obtained

by adjusting

in the best way the orientation of _a

perfect

icosahedron _on the site of the atom I, in order _to minimize the

expression

12

~

h

=

£

I _cos ~P,~

j

where the

~P,~'s correspond

_to the

angles

between the atomic bond I

j

and the _nearest

direction of the

perfect

icosahedron centred _on the site I. It _must be noticed that both the

sphericity

and the

icosahedricity

of _a site I _are defined

using

this site and the _same 12 _nearest

neighbouring

atomic

positions.

METHOD OF CALCULATION OF THE THERMODYNAMIC PROPERTIES. In D-L- 2, dedicated to

the

dynamical behaviour,

the vacancy

(stable

_{on some}

sites)

will be characterized

by

its formation

enthalpy

and entropy, ^its

migration

_energy and its formation volume. The formation

enthalpy

is calculated

by relaxing

the

glass

with and without the vacancy, in the NPH ensemble and

by correcting regularly

the temperture at 0 K. After _a first decrease due _to the elimination

of the _excess

potential

_energy, the

enthalpy

^stabilizes at an

equilibrium

^{value. The} difference of the two values

gives

the formation

enthalpy according

to :

~ ~ 864

~°~ _~~ ~

863 ~~Y~

where

H~~

is the total

enthalpy

of the system ^with a vacancy

(containing

863

atoms)

and

H~~~ is the total

enthalpy

of the system ^without vacancy

(containing

864

atoms).

A similar method is used _to calculate the formation volume

by comparing

the

equilibrium

volume of the

glass

^with and without the vacancy. The formation volume is then

given by

:

~f0nn

" "

(~wv Vsys)

where

V~~

^{is the volume of the}

system

^with a vacancy and

V~~~ is the volume of the

complete

system. ^{The formation volume will be}

given

ⁱⁿ terms of the _mean atomic volume 1l.

The vibrational part ^{of the formation} entropy ^of a vacancy in _a

glass

^is calculated from the

eigen frequencies

of the

glass

with and without vacancy. If u~~~ and _u~~~~ are the

eigenfrequen-

cies with and without vacancy,

respectively,

then the formation entropy can be written

>

j

^2~

~i

+ ~n ~~~

T/hvwv))

^~

l

^''

(i

₊ Ln ~~B

~/~»"Y~~~

,-j "

~64

s~

₌ ⁸⁶⁴ x kB

°m ~~~

where the _{sums are}

performed

over the modes: 3 _x

IN -2)

in the first _sum and

3 _x

(N

I in the second _one

(N

is the total number of _atoms and the formula takes into

account the existence of three translational

modes).

The above

expressions

do _not take into

account the absence of the low

frequency

part of the

density

of _state induced

by I)

the finite size

of the calculation box and

it)

the _use of

periodic boundary

conditions. Nevertheless this

correction,

which is very

simple

to

introduce, corresponds

to _a small term. Moreover since _we

calculate

only

entropy differences, the formation entropy of defects is no more concem.

The

eigenfrequencies

_are calculated _at 0 K and the

validity

of the above formula is based _on

an assumed small

dependence

of the

eigenfrequencies

_on the temperature. ^The presence of _one mode

having

_{a pure}

imaginary frequency

in system ^{A' is}

probably

due to a less

complete

relaxation of this system ^{because of} its

higher quenching

rate. In the calculation of the formation

entropies,

this

mode, occurring

in the

glass

with and without vacancy, can be

safely ignored.

A set of

migration energies

has been determined

using

the sites where the vacancy remains stable,

by forcing

the

jump

of _one of its first

neighbour

_atoms into the vacancy site

(a

method

previously

used

by

Bennett

[19] consisting

in the addition of _an extra

potential

to the L-J-

potential forcing

the

jump along

a

particular path).

The

path

of the

jump

is chosen either _{as a}

straight

line _{or as a}

parabola,

and

during

the

jump,

_{we measure} the evolution of the

potential

energy of the atomic interactions within _a

sphere

centered _on the vacancy site. The _curve of the

potential

_energy as a function of time exhibits _a hill

corresponding

to the _overcome of the

migration

barrier. For each

jump,

we search for the _curvature of the

parabola

which minimizes the

potential barrier, allowing

_us to find the lowest saddle

point.

Examination of the atomic

displacements during

the forced

jump

shows that

only

the _atom which is forced _to

jump

has _a noticeable

displacement,

the others

remaining

_near their initial

position.

Therefore

forcing

a

jump

does not break the structure of the

glass

and the

height

of the barrier is

undoubtedly

_a

migration

barrier.

Characterization of the

glass.

The two

systems

^{A' and B} at 0 K _are used for simulations of vacancy

creations,

_as well _as the system ^{C in} a few _cases. To check the

stability

of these systems on a

long

time, we have

performed

a relaxation, in the NPH

ensemble,

of _a duration of 40000 time steps ^of

10~~s

_{and measured the}

mean

squared displacement

of the _{atoms i,ersus} time. The

temperature

^{is left free}

during

both relaxations. The evolution of the _mean

squared displacement

in both systems ^{is shown in}

figure

I. We _note that the first system ^{is stable}

during

7 000 time steps ^and

crystallizes

afterwards. The total duration of its life time

(13

000 time

steps)

^is in reasonnable agreement ^with

previous

calculations made under similar conditions

[21].

We chose therefore _to

perform

vacancy simulations of this duration in system A'. The second

system

^{is stable}

during

the whole of the 40 000 time steps, ^{and shows} no

sign

of

crystallization.

The time evolution of temperature ⁱⁿ system ^{B is shown in}

figure

2, After _a first small increase due _to the transformation of

potential

into kinetic energy, the temperature

stabilizes around 0.12K,

starting

from 0K. If system ^{A' is relaxed}

by maintaining

the

temperature at 0,12 K, _no

crystallization

is observed,

1

O

E

~

~

W tL

~$

~

°L

~ 0

0

ime (1 0"' ~s)

Fig.

I. -

^ean

during a 000 time steps _imulation.

0.20S ~

o.ids -

~

- ~

~ o.ids

@ 6

~

$

_o.17s

i~

- '*

t4 C

~j o.iss 5 d

©k

(

-

o.iss ~

~ 4

0.14S

~

O.13S 3

0 10000 2000a 30000 40000

Time

Ii 0"~~s)

Fig. ^2. Temperature versus time for the system ^B during the _same 40 000 time steps ^simulation as

used for

figure

1.

13

11

s

~ syst.m e

~t

* s

~ system a.

i

1

0 2 4 6 8 10

r (Angstroems)

Fig. 3. Calculation of the radial distribution function g jr) for both systems ^{A' and B.} The atomic

positions

are averaged _over 300 time steps ^and a shift is

applied

to separate the two

graphs.

The two radial distribution functions of systems ^{A' and B} are shown in

figure

3. The first

peak

and the

split

second

peak

are the usual

fingerprints

of _an

amorphous

structure.

Nevertheless,

_as

usually

observed too, the differences in

preparing

systems ^A' and B

(which explain

the differences in the

crystallization behaviour),

_are not reflected in _a clear manner in the

pair

distribution function, which is well-known not to be _a very sensitive test of the

glassy

structure.

It is

interesting

to calculate the distributions of the local criteria defined in the

previous

section. The two groups of distributions _are similar for the _two systems

(except

for _a shift in the

hydrostatic

_pressure due to the different extemal pressures

applied)

_; as a consequence

only

the distributions of system ^B are

displayed

in

figures

4-7.

These

criteria,

defined from the local atomic interactions and the

topology

around an atomic site, exhibit _some noticeable correlations. First the

hydrostatic

_pressure and the shear _{stress are}

related _{: a}

large hydrostatic

_pressure

implies

_a

large

shear stress. On the other

hand,

the low

hydrostatic

_pressures sites

generally display

_a

good sphericity.

This _can be

easily

understood

30 100

25

20

60

15

40 lo

~ 20

o o

0 2 3 4 5 6 7 8 9 lo 11 12 0.05 0.55 1.05 1.55 2.05 2.55 3.05 3.55 4.05 4.55

Spherlclty

^* 1000

(Reduced Units) lcosahedrlclty

Fig.

4.

Fig.

5.

Fig.

4.

Histogram

of the atomic

sphericities

in system ^B.

Fig.

5.

Histogram

of the atomic icosahedricities in system ^B.

60 40

50 ~5

30

25

30 20

20 ~~

io io

5

I.05

_0.65

1.25 1.85 2.45 3.05 3.65 4.25 4.85 5.45

~2.95.1.85-0.75

_{0.35 1.45 2.55 3.65 4.75}

5.85 6.95

Shear Stress

(kbar)

_{Local Pressure}

(kbar)

Fig.

6.

Fig.

7.

Fig.

6.

Histogram

of the atomic shear _stresses in system ^B.

Fig.

7.

-Histogram

of the atomic local pressures in system ^B.

by

the

following

_{argument any} characterized deviation from _a

spherical

environment will induce _a

high

level of shear stress as well _{as a}

high

_pressure component.

Static behaviour of vacancies at 0 K.

In this section _we shall report ^{the observed behaviour of the}

glass

after the removal of _{an atom.}

Three kinds of behaviour _are

clearly

observed from the simulations

performed

_: the first is the

persistence

of the vacancy on its introduction site

during

the

complete relaxation,

the second is

the _occurrence of _an atomic

jump (an

_atom

taking

the

place

of the vacancy in such _{a manner}

that the local

topology

is

restored)

and the third is the

dissipation

_on the site of the vacancy

by

_a

collective

collapse

of the

neighbouring

atoms.

QUALITATIVE

AND QUANTITATIVE METHOD OF ANALYSIS. A group of simulations has been first

performed

_{on system} A' maintained at o K. To

keep

the temperature constant, corrections of the atomic velocities _are

performed

^{each lo time}

steps.

^{The reaction of the} system ^{has been} in _a first step

analyzed using

two sources of information _:

I)

the atomic motions of the _atoms

surrounding

the introduction site of the vacancy and

it)

the _use of the Voronoi

polyhedra [22]

of the

neighbours

of the introduction site. The average of the volumes of the Voronoi

polyhedra,

hereafter called

V~

is a measure of the local _« vacant volume _» present in the

vicinity

of the introduction site and its evolution

during

the simulation is _a

representation

of the reaction of the system to the vacancy introduction.

This first method allowed _{us to} separate ^{three different} types ^{of reaction} to a vacancy introduction in system

A',

which will be described later. The

knowledge

of these kinds of reaction induced _{us to} introduce _two other criteria _to

analyse

_more

precisely

the simulations

performed

_on the system ^B : the criteria _« RECONS _» and _« DISSIP

». The _use of these criteria allows for _{a more}

rapid analysis

of the nature of the response of the system to the

introduction of _a vacancy. It becomes therefore

possible

to

perform

an extensive

study

which

would have been otherwise _too much time

consuming.

The first _one is

designed

to evidence atomic

jumps

and therefore consists in _a

comparison

between the initial and final local

topologies

around the vacancy site. Just before the introduction of the vacancy, the 13 sites made up

by

the vacancy site and its 12 nearest

neighbours

_are book

kept.

^{At the end of the} relaxation, each of the 12 atoms nearest to these sites is associated with the _nearest site. RECONS is then defined _as the _mean

squared

distance

between the _atoms and their associated site

12

RECONS

=

1/12

£ d)

, =i

where _d~ is the distance between the _atom I and its associated site.

RECONS is then _{a measure} of the local reconstitution of the initial _« network _» after the reaction of the system to the introduction of the vacancy. In the _case of _a

single

atomic

jump

with _a

perfect

reconstitution of the initial site _one would observe _a RECONS value of _zero. The second criterion DISSIP is defined from the

displacements

of the 12 _{nearest atoms} of the vacancy site between the

beginning

and the end of the simulation _: it is the second _moment of the distribution of the atomic

displacements.

12

DISSIP

=

1/12

£ ((r, r,o)~ (&r~)~)

,

where r, is the

position

of the _{atom at} the end of the simulation and r,o is its

position

at t ₌

0, &r~

is the _mean

displacement

of the _nearest

neighbours

_;

d,

and r, are measured in

intemal

units,

here 3 _x

« in A' and B systems.

DISSIP is small when all the atoms move shift

roughly along

the _same distance

during

the relaxation and is

large

when _one

particular

atom moves more than the others

(in

the _case of _an atomic

jump

for

example).

BEHAVIOUR AFTER A VACANCY INTRODUCTION. A series of 96 simulations has been

performed

_on system ^A' and of 59 simulations _on system ^{B. As told}

before,

three different kinds of reaction of the

glass

to the vacancy introduction have been observed.

The first _one is the

persistence

of the vacancy on the introduction site. The relaxation of the

atoms around the vacancy site is not strong

enough

to

dissipate

the vacancy which remains

40.S 40.S

40 40

t 39.S #~ ^~~.S

~

~~

~

~~

@ @

38.5

)

38.S

g

^~

38 > 38

37.S 37.S

37 37

1000 1000 3000 5000 7000 1000 1000 3000 Soon 7000

Time 11

0'~ ~s)

_{Time 0"~}

~s)

Fig. 8. Fig. 9,

Fig. 8. Evolution of the _mean Voronoi volumes, V~, of the neighbours of _a vacancy versus time. This event corresponds to the

stability

of the vacancy on its introduction site. The average volume before the

introduction of the vacancy is shown by the black square.

Fig.

9. -Same

graph

as in

figure

8 for _a

dissipation

of the vacancy on the spot. ^{The volume}

V~

decreases

continuously

with time,

corresponding

_{to a}

progressive dissipation

of the vacant volume of the vacancy absorbed

by

the relaxation of the

surrounding

atoms.

present

la graphic

visualization of the final state of the system ^{confirms the} presence of the

vacancy).

The

stability

events are characterized

by

a

graph

of the average Voronoi volume

V~

of the form shown in

figure

8 _: after _a first increase due to the vacancy

introduction,

the average volume reflects the relaxation of the atomic

neighbours

but its value at the end of the simulation remains much

larger

than the initial value. It _means that _{a vacant}

volume,

_e.g. the vacancy, is still present on the introduction site.

Obviously,

_no atomic

jump

can be observed in

such _{a case.}

The second

possible

behaviour is the

dissipation

of the vacancy on the spot. ^This occurs

generally

when the relaxation of the atoms around the vacancy is

sufficiently large

to absorb

the _{« vacant} volume _» of the vacancy. The

dissipation

events

correspond

_{to a}

progressive

decrease of

V~ (Fig. 9)

after the increase due _to the vacancy introduction and the atomic motions do not reflect any

jump.

A decrease _{occurs more or} less

rapidly depending

_on the

characteristics of the site.

The last

possible

behaviour is the _occurrence of the

jump

of _{one atom} from its site _to the vacancy site

producing

the reconstitution of the initial local order

prevailing

before the vacancy

introduction. This kind of behaviour will be called _a reconstitution event hereafter. It is characterized

by

the existence of _a clear atomic

jump

_over

nearly

_one interatomic distance

performed by

_one of the _atoms which _were

initially

nearest

neighbours

of the vacancy. At the end _{an atom} is located _{on or}

just

_near the vacancy site. The

plot

of

V~

_{as a} function of time

(Fig. lo) shows,

after _a first part

corresponding

to a relaxation

analogous

to that of

figure 8,

_a

sudden

drop corresponding

to the _occurrence of the

jump.

After the

jump

the Voronoi

polyhedra

^of the

neighbours

of the vacancy recover their initial characteristics.

This

separation

^of all the simulation results into three main classes

(a

few _{events are} intermediate between

stability

and

dissipation

_or

correspond

to a bad reconstitution and will be called

« other

»)

_can be

directly

visualized from the map of the events in _a RECONS _x DISSIP

plane,

for the _events observed _on system

B,

shown in

figure11.

41

40.S

40

#~

-t

- 39.S

@

39

~

~ 38.S

38

37.S

1000 o 1000 2000 3000 4000 sore sore 7000

Time

Ii 0"~~s)

Fig. lo. Evolution of the _mean Voronoi volumes, V~, of the neighbours ^of a vacancy versus time in the _case of _a reconstitution by

jump

events. Between 0 and t 2 800,'the

neighbouring

atoms of the vacancy relax but the vacancy still remains visible. Between _t 2 800 and t 4 000, a sudden decrease

of

V~

occurs,

corresponding

to an atomic

jump

and the vacancy moves away. The final value of

V~ is the _{same as at t}

=

0 without vacancy (the _square box),

confirming

the local reconstitution of the site.

~~-2

#~~j$&

++

'°~' _O

gb ~( ^~

( ~ ^Qf

^°

-~

o

..

« .«

'°~~

.

,~-s

ia~°

Dlsslp

Fig.

I I. Statistics of the _events in system B in _a RECONS _x DISSIP map of 0 K. The results of the simulations _are divided into four classes _: (o) for the

stability

events, (.) for the reconstitution

by

atomic jump events, (fl3) for the

dissipation

events, (+) for the uncertain _events. See _text for the explanation of the criteria.

Stability

_{events are} characterized

by

low values for both DISSIP and

RECONS,

which is the

sign

of small relaxations

harmoniously

shared between the atoms

surrounding

the vacancy site without any

jump,

the atoms

remaining

_near their initial sites.

Dissipation

events are characterized

by

_an intermediate value for DISSIP and _a

large

value for RECONS _: this is the result of _a strong ^{relaxation around the} vacancy

during

which the

atoms lose their initial sites. The initial atomic

positions

_are

wiped

out and _{a new} local

configuration

is built.

Reconstitution _events

correspond

to a

large

value for DISSIP and _a low value for RECONS due _to the existence of _an atom

jumping

from _one

neighbouring

site into the vacancy.

The

quantitative

criteria RECONS and DISSIP used for the

analysis

of the results obtained

on system ^B allow for _a

quantification

of the level of reconstitution and confirm the

qualitative analysis

based _on the determination of atomic motions and Voronoi

polyhedra.

ROLE _OF THE LOCAL STRUCTURE. We have also tried _to correlate the _nature of the relaxation

of the

glass

after the vacancy introduction with the local characteristics of the introduction site before. We have searched for

possible

correlations either with the atomic stresses, shear and

hydrostatic

pressure

level,

_or with the

topology,

icosahedral and

spherical

character. We have

not been able _to detect any correlation between the 3d icosahedral character of the

neighbourhood

of _a site and the

subsequent

evolution. The level of shear stresses also is

apparently

_not related to the vacancy behaviour. This last

point

was for _us rather

unexpected, given

^{the role} of the

sphericity,

and _{we want to}

develop

in the future _{a more} refined

study

of

this absence of correlation.

Figure

12 is _a

plot

in the

hydrostatic pressure-sphericity

_map of the _events observed at o K

on system ^{A'. 96} simulations have been

performed

and the set of introduction sites is

representative

of all the

possible

sites. However since _we have chosen the sites at random in ranges of pressure, some parts ^of the map are

empty

^at an intermediate

hydrostatic

_pressure

level.

Moreover,

the

frequency

for

choosing

^the extreme values of pressure and

sphericity

is not

representative

of their actual

probability

but

corresponds

to an

oversampling

of these

extreme classes. In the

region

of bad

sphericity

and low

hydrostatic

_{pressure, no}

points

are

shown

since,

_as said

above,

no atom has

simultaneously

_a low pressure and _a bad

sphericity.