Dynamics of Interacting Brownian Particles in a Two-Dimensional Periodic Potential

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Dynamics of Interacting Brownian Particles in a Two-Dimensional Periodic Potential

M. Mazroui, A. Asaklil, Y. Boughaleb

To cite this version:

M. Mazroui, A. Asaklil, Y. Boughaleb. Dynamics of Interacting Brownian Particles in a Two- Dimensional Periodic Potential. Journal de Physique I, EDP Sciences, 1997, 7 (5), pp.675-690.

�10.1051/jp1:1997184�. �jpa-00247353�

Dynamics of Interacting Brownian Particles in

_a

Two-Dimensional Periodic Potential

M.

Mazroui, A. Asaklil and

Y.

Boughaleb _(*)

Laboratoire de

Physique

de la Matibre

Condens#e,

Facult# des Sciences Ben

M'sik,

PB

7955, Casablanca,

Morocco

(Received

7

August

1996, received in final form lo December 1§§6,

accepted

16

Jauuary 1997)

PACS.05.20.-y

Statistical mechanics

PACS.66.30.Dn

Theory

of diffusion and ionic conduction in solids

Abstract. The static structure factor and diffusion mechanism of N

interacting

Brownian particles immersed in _a two-dimensional

periodic potential

are discussed, in relation with supe- rionic

conductors, premelting

^surfaces and adsorbed

monolayers.

For the static structure factor

we report ^Brownian

dynamics

results

assuming

a repulsive ^Yukawa

pair potential.

For _some values of the

concentration,

the system can be

decomposed

in two

equivalent

_or in two different

subsystems along

the two directions of space. To get information about the diffusion

mechanism,

we have

computed

the full width at half maximum

(f.w.h.m.) A(q),

of the

quasi-elastic

line of the

dynamical

structure factor

S(q,

_{WI up to}

_large

values of _q

covering

several Brillouin _zones.

The entire

analysis

of

A(q)

with different

physical

parameters ^{shows that the} most

probable

dif- fusion _process in

good superionic

conductors

(concentration

of mobile ions _{c =}

2/3)

consists of _a

competition

between _a back correlated

hopping

in _one direction and forward correlated

hopping

in addition to

liquid-like

motions in the other

spatial

direction. The

analytical

treatment of this

investigation

is done

by using

^different approximations such _as the

homogeneous approximation

and the time

dependent

mean field

approximation

within the lattice _gas model.

1. Introduction

The diffusion of

particles

in _a two-dimensional

periodic

medium is _a

subject

of

increasing importance

for _a

variety

of

physical applications, including liquid

surfaces

[lj,

the

melting

process

[2j, superionic

conductors

[3-6j

or

submonolayer

films adsorbed _{on a}

crystalline

sub-

strate

[7j.

^{While the}

single particle problem

^is rather well understood

[8,9],

there is

only

limited information _on the

transport properties

in the

many-particle

_case. The latter

problem

is of

particular importance

in connection with the

superionic

conductors

[10,11].

These _ma- terials consist of two

species

of

particles:

one

species

is fixed around certain

equilibrium

sites and forms _a

regular

framework

through

^which the other

species

diffuses in

liquid-like

fashion.

These ionic solids _are of

great technological

interest _as

battery electrolytes

and _are stable _up to

quite hiih temperatures.

In _a

previous

_paper

[12],

we have shown that _an

important

^diffusion process in two~dimen- sional

superionic

conductors

(the

ionic motion is confined _to two-dimensional

layers),

_occurs

at the incommensurate concentration c =

2/3.

For this

specific

concentration the

system (*)

Author for correspondence

(e-mail: fsbcui©mail.cbi.net.ma)

@

Les

(ditions

_de

_Physique

₁₉₉₇

can be

decomposed

in two different

subsystems along

the two directions of space. The aim of the work

reported

here is to confirm

quantitatively

this behaviour. For this _{purpose we} focus _on the static structure factor

S(q),

which

gives

information about the distribution of

mobile ions _over the

periodic potential.

The

computation

of this

important quantity

is done within _a Brownian

model,

in which the

interacting

^mobile ions _are

subjected

to a

periodic potential.

Our

findings

for _c

=

2/3

show that the behaviour of

S(q)

is different

along

the two directions of _space,

reflecting

that the

arrangement

^{of mobile ions is} commensurate in xx-direction and incommensurate in

yy-direction. Furthermore, by varying

the

strength

of the

pair interaction,

we have established that surface diffusion _can

undergo

a continuous reversible

commensurate-incommensuratq

transition. On the other

hand,

in order to obtain information

on the

microscopic

diffusion

mechanism,

_we focus _on the

dynamical

structure factor

S(q,w)

which is

proportional

to the

qqasi-elastic scattering intensity,

both in _neutron and in atom

scattering experiments.

This

quantity

reflects both essential

dynamical

features: the collective excitations of the

particles

show up in its

high frequency _spectrum

while its low _energy

part (quasi-elastic line) gives

_us information about the diffusion _process. The full width at half maximum

(f.w.h.m.) /h(q)

of the

quasi-elastic peak

of

S(q, w),

at small _q, is

proportional

to the diffusion constant. Its behaviour at

larger

_q

depends

_on the diffusion mechanism. Much work has been devoted to the calculation of the diffusion coefficient

(or

_more

generally,

^of

the diffusion

tensor)

in the presence of _a

periodic potential. Although,

much information is contained in the

dependence

of the

frequency

width

/h(q)

for

large

_{q =}

(q(;

^the

problem

has been much less

thoroughly investigated.

^Our

aim,

in this second

section,

is to

employ

the lattice _gas models in combination with continuous

theory,

based _on the Brownian

model,

in order to calculate the

dynamical

structure factor

S(q,w)

and its f.w.h.m. up ^to

large

_q

values

extending

to several Brillouin _zones. The

analytical

treatment of this

investigation

is made

by using

different

approximations

such _as the

homogeneous approximation

and the time

dependent

_mean field

approximation.

2.

Langevin Equation

and Generalities

In this

section,

we shall

present

certain continuous models 1N.hich,

contrary

to

hopping models,

do not assume that the mobile ions _are located at the bottom of the

potential well,

do not

assume instantaneous

hops

and do not introduce _an artificial distinction between local motion

and diffusive behaviour.

They

thus allow _a unified

description

of both essential

dynamical

features of the motion of the mobile sublattice. In many

superionic conductors,

the

density

of the mobile

particles

presents a weak

spatial

modulation

along

the conduction

path [13].

^These

observations

suggest

a model where the ionic motion is described _{as a} continuous process. In this model

only

the mobile sublattice is treated

explicitly.

The ions of the

rigid

framework _are

assumed _to be fixed at their

equilibrium positions

and

generate

a

periodic potential

in which the mobiles ions _are assumed _to

perform

_a Brownian motion. The total

potential

consists of _a

periodic part Vi

^and ion~ion interactions i12.

Thus,

Tot

"

~ _{h (ri)}

₊

~j _%(ri

_rj

_(I)

1=1 1>j

where _ri is the

displacement

of the ith ion. The

particles

_are

independent

when

~§

" 0 and

one then deals with _a

problem

of Brownian diffusion in _a

periodic potential,

for which _some recent

general

results _are available

[14].

^{Since the mobile ions} are

charged,

_we

_expect

that the Yukawa term in

Tot

is of substantial

importance

l§(r)

₌

Q2 -exp(-r/rD). (2a)

r

The effective

charge Q

is

expected

to be less than the formal ionic

charge

due to

screening

and

charge

transfer effects. _rD is the

Debye screening length,

which is taken to be

equal

to

L/2,

where L

represents

^the

length

of the

system.

In _some of the most

extensively

studied

experimental

systems, ^the

periodic potential

is two-dimensional. This is the _case for _some

layered charge density

_wave

compounds

and _rare-gas

monolayers

adsorbed _on

graphite.

The

periodic potential

_can be

represented by

_a Fourier _series:

Vi (r)

=

£ AGcos(G r) (2b)

G

where G _are

reciprocal

lattice vectors of basic lattice. We shall return to this

potential

later.

The model consists then in

solving

2 N

coupled Langevin equations:

where I = _1, 2, 3, ... N and

m

_and ri epresent

respectively

the mass and

mobile ion

I.

~i is thefriction and _Ri It) is its responding white

The

time

dependent function

second as

jR~jtjR~ t'ii

= 2m~k~T>vdjt - ii)

where

fl

= _I/kBT, T

is

the emperature

and

kB

the

^{oltzmann constant.}

We

the

of

the

mobile

particles

with the

rigid

^amework _by

the

dimensionless ^uantity

T =

2ir~ilwo,

where wo represents the _ibration

of a

_single particle at

the

of the

potential well.

This

Brownian model

in

a potential has been

used

extensively

for

solid

electrolytes [15-18] and

can

be

applied

to

a number

of

vious examples

include

motions in a physisorbed

or adlayer, impurity

_diffusion

in

metallic

or

ionic lattices

and

intercalate

dynamics in

halcogenides or

graphite; less obvious xamples include Josephson _tunneling.

One

^of the essential

aspects

of the

present

study

concerns

the

evelopment of _a uantitative

description of the static properties. ecause of

its

3. Static

Properties

and Structure

3.I. STATIC STRUCTURE FACTOR

S(q).

In the

following,

_we

present

the

explicit

static results for the _square

egg-carton potential

obtained

by considering only

the three lowest order

terms of the Fourier

expansion

of the

periodic potential given by equation (2b)

Vi(z, y)

= A ₊

B(cos

_{qoz + cos}

qoy)

+ C

cos(qoz)cos(qoy) (4)

where _qo

" 2ir

la

denotes the

reciprocal

lattice vector with _a

being

the lattice constant which

is the _same for the _z and y directions. For numerical calculation _we will

mainly

choose the

parameters

A

= 0.04

eV,

B

= 0.02

eV,

C

= 0.01

eV,

in order to obtain the value 0.I eV for the barrier energy. The constant C

represents

the

coupling

term. It is

responsible

for the energy transfer between the _z and _y

degrees

of freedom. This

simple

form of

potential

^has been

largely

used _{as a} model

periodic potential

in theoretical studies of _man» different

problems,

such _as the non-linear conservative

dynamics

of _a classical

particle jig],

the collective diffusion of ions in

superionic

conductors

[12]

^{and the} noise~actived diffusion of _a classical

particle [20j.

The structure

representation

within _{q space}

(where

_q is the _wave

vector)

is

given by

^the

structure factor

S(q)

which is the Fourier transform of the

pair

correlation function and _can

be obtained

directly

and

exactly

^from molecular and

Langevin dynamics

sjq)

=

( ^{~exp(iqjr~ j)))} ₍₅₎

1,J o

where

(- -)

denotes the

averaging

over the

equilibrium distribution,

N is the number of the mobile

ions,

and _r~ the

position

vector of the ith ion. In this

section,

_we shall

study

_a different

situation where the structure of the

system

^is commensurate or

incommensurate,

in 2D space.

Our considerations _are

particularly

relevant for intercalation

compounds

and

submonolayer

adsorbed

films,

where the concentration of mobile ions _can be varied to _a certain extent. As

quasi-periodic structures,

incommensurate structures _can be defined in terms of _{two or more}

mutually incompatible periodicities.

These

periodicities

_are sometimes inherent in the nature of the

system

and _can be

readily

identified. In the _case of _rare gas

monolayers

adsorbed _{on a}

crystalline surface,

the adsorbed atoms

experience

a

periodic potential

from the

substrate,

the

periodicity

of which

provides

_one of the

length

scales

la)

of the

system.

The other

length

scale

(b) corresponds

to the

equilibrium

distance between adsorbate atoms. This is defined _as the interatomic distance in the absence of _a

periodic potential,

I._e., in the limit of _a

perfectly

smooth substrate. In the

opposite

limit of _a

strong

^substrate

potential

the adsorbed atoms _are drawn toward the bottom of the

potential

wells and form _a commensurate structure. Incommensurate

structures _are obtained for intermediate

potential strengths

and

general

values of the ratio

a16.

In addition to

epitaxial

and intercalated

compounds,

_a number of nonstoichiometric inter-

gro,vth systems

_are known to show

compositional

incommensurabilities.

Examples _[21]

_are

Hg~_~AsF6

and

TTF7-~I5.

In all cases,

competing periodicities

_are associated with differ- ent

subsystems.

We examine here the structure of the mobile ionic array in twc-dimensional

framework ionic conductors such _as

p-alumina-

We

study

the static _structure factor

S(q)

_as

it is affected

by temperature

and

by

the various

potentials

to which the ions _are

subjected.

We

employ

stochastic

Langevin dynamics

which consist of

obtaining

correlation functions for

S(q), by solving numerically

the

coupled equation

set

(3).

Let _us first

give

an

extremely

brief

description

of

Langevin

simulations

(method

of

computation),

which has been described exten-

sively

in connection with several

problems [22j.

The model

system

we refer to is

composed

of N Brownian

particles interacting through

the Yukawa

potential

of

equation (2a),

and _are sub-

jected

to the

periodic potential given by equation (4) (inhomogeneous system). Initially,

the N

particles

_are

placed

_{on a square} lattice array enclosed in _a square box of

length

L ₌

aN~/~

The number N of

particles depends

_on the

stoichiometry,

The

particles

_are then allowed _to

move

according

to

equation (3)

until

equilibrium

is reached. The convergence of the average

configuration potential

_energy of the

system

to a constant value is usedlas the

equilibrium

criterion. The time step ^is assumed to be short

enough

and _was taken to be 3 _x

10~~~

_s,

since,

for

large

time steps the

stability

of

integrating

routine breaks down and

large

_error is incurred.

The friction coefficient _~i is taken to be constant and

isotropic.

Its value is fixed

by

the di- mensionless parameter ^T =

2ir~ilwo

which is

chosen,

in most cases, ^to be in the intermediate friction

regime (0.5

< T _<

2).

The simulation is then carried out

by solving numerically

the set

of

2N

coupled Langevin equations

of motion. In order to minimise the

edge

^effects due to the finite size of the simulated

system,

we use conventional

periodic boundary

conditions.

Numerical solutions consist in stochastic

trajectories,

from which the desired observables _are calculated. The reader who desires further detailed

description

of these

techniques

_as

applied

to framework solid

electrolytes

_can refer _to

[23j.

2

0

0,0 0.5 1.0 1-S 2.0 2.5 3.0

q/q°

Fig.

1. Static structure factor evaluated

separately along

the two directions of _space for

commen-

surate concentration _c

= 0.5, T

= 400 K. The effective

charge

of the Yukawa

pair

interaction is equal to 1 in atomic units

(Q~

=

1).

The barrier _energy of the

periodic potential

is equal to 0.1 eV (Vo = 0.1

eV)

for all

figures.

We have the _same behaviour of static structure factor

along

the two directions:

S(q~, 0)

₌

S(0,

_qy and

reflecting

that the system ^is commensurate

along

the two directions.

go "

2irla.

The static

properties

have been evaluated to obtain information about the correlated _mo- tion of the ions in the _presence of interactions. We first consider commensurate stoichiometries

(c~~

=

integer),

where the backward correlation dominates in the _sense that when _a

parti-

cle at the

equilibrium position

tries _to

jump

to the next

site,

the

periodic potential

and the

pair

interaction collaborate to restore the

particle

to its initial

position.

The stochastic tra-

jectories

of the two-dimensional model for framework ionic conductors show that for _a

global

commensurate concentration _c

= 0.5

(and

_more

generally

for _cases where

c~~

is _an

integer),

the

system

can be

decomposed

in two directions where the concentration

along

each direction remains commensurate with _c

= 0.5. This

qualitative picture

is confirmed

quantitatively by

the

Langevin dynamic computation

of

S(q).

In

fact,

the _same behaviour of

S(q)

is obtained

along

the two directions of _space

(Fig. 1).

From the

position

of the

peaks,

it is _seen that

nearest

neighbour

distances close to the lattice _{constant a are}

favoured,

which characterizes

commensurate densities.

For _an incommensurate concentration

(c

₌

2/3),

in which ions do _not sit at lattice sites but

are moved off

them,

the behaviour of the static structure factor

S(q)

in xx-direction

S(qz, 0)

differs

strongly

from the _one in

yy-direction S(0,qy) (Figs.

2a and

b).

^In addition to the

liquid

contribution

S(0, _q~) displays Bragg peaks coming

from the

periodic

modulation of the

density-, reflecting

the incommensurate situation in this

specific

direction. This state becomes very clear in the

strong coupling regime.

In

fact, by varying

the

strength

of the

pair

interaction

over a wide _{range, we} have observed _a continuous reversible commensurate-incommensurate

z

6

~

~4

$

~

2

~o.0 o.5 1,o 1,5 2,o 2.5 3,o

~~

q~q°

8

6

~

#4

d

~

i

o

o-o o-s i-o i-s i-o i-s 3.o

b) ^q'q°

Fig.

^{2. Same} as in

Figure

1 but here we consider incommensurate concentration _c

=

2/3

and

Q~

₌ 0.5. The situation is commensurate

along

the xx-direction

(a)

and incommensurate

along

the other

(b). Vo/kBT

= 3.

transition

(Fig, 3) S(0, _q~)

exhibits maxima at _wave vectors which _are

multiples

of _{qi m}

1.33qo (qo

= 2~

la,

where _a is the lattice

constant) indicating

the

strong

^order established

by

the

mutual

repulsion

of the ions. When the framework

potential

dominates V2, ^it may be favourable for the lattice to relax into a comnlensurate structure. The nlobile ions lose Coulombic _energy, but

gain potential

energy. But in the _case where the

pair potential

dominates the framework

potential,

^it creates an a1nlost

regular arrangement

^of the ions

by forcing

_sortie of thenl _to

move away from the minima of the

periodic potential: giving

rise to an inconlmensurate

,'I i i

i

j i

'i

C' ^{' '} '

cS ^,

",

~^

, , i

~

'

' '

~',

^~

2

,' ,'

_", ~

/

,

<' _/

',

'

~'(,

'

0

0.0 0.5 1.0 1-S 2.0 2.5 3.0

q/qo

Fig.

3.

S(0,

_qy

describing

_a situation of

competition

_{c =} 2

/3

and for different values of the

strength

of

pair

interaction.

(-)

small

Q~

values

(Q~

#

0.1); (.

intermediate

Q~

values

(Q~

=

0.5);

(.. ..) higher

Q~ values

(Q~

"

1),

the ratio

Vo/kBT

is

equal

to 3. Commensurate-incommensurate

transition is clear.

structure in this

specific

direction. Such _a situation _may result in _an effective

potential

with _a

complicated

structure _{as we} shall _see in the _next

paragraph.

In _a diffraction

experiment

_one would observe

Bragg spots

at

positions

_"

2~N/b (where

b is the

interpartide separation

and N

integer),

None of these

spots

^coincide with the

Bragg

_spots of the

periodic potential

at

positions

G

=

2~Mla,

where _a is the lattice constant.

In the other

direction,

the structure is commensurate

independently

of the

strength

of the

pair

interaction.

3.2. EFFECTIVE POTENTIAL

Vej.

The

equilibriuni density p(r)

of

diffusing

ions in supe- rionic conductors is _an

interesting quantity

in several respects.

Exper1nlents

on

a-AgI type conlpounds

_[6] ^show _a substantial delocalization of

diffusing

ions around the

preferred

inter- stitial

sites,

which reflects the structure of the donlinant

pathways. Moreover,

the variation of the

density along

the conduction

paths

is

closely

related _to

transport properties.

To discuss

this,

let _us introduce the effective

potential l~~

via the relation:

Pi~)

_" C°~St

~XPi~flifii~)). ₁₆₎

This effective

potential

includes the effects of both

potentials lfi

and _V2 in _a static way. It is

independent

of the friction and contains _no nlenlory effects which arise

essentially

front the fluctuations of the

pair

interactions.

Moreover,

the barrier

height

of

Vej

_can be very useful in

interpreting

the

transport properties

of the

systenl. By conlparing ii

with the bare

single particle potential l§ originating

from the host ions

only,

one can deduce

quantitative

infornlation _on the role of the

conlnlensurability effect,

I-e- _on the

degree

of

conlpetition

/

' ,'

', /

' _,,, «'

",_

,"

"' ;"'

".__,-' ",__-"'

I

44

46

~.8

0.0 0.2 0A 0.6 0.8 1.0

yla

Fig.

4. Effect of the temperature on the structure of the effective

potential resulting

from _a Yukawa

pair

interaction

(Q~

"

0.5).

The effective

potential

is

computed

in

yy-direction,

for _{c =}

2/3. (-

Vo/kBT

= 6,

(-)

_Vo

/kBT

₌ 3.

between

short-range

correlation _among the mobile ions and the ion-host interaction. The effective

potential concept simplifies greatly

the

many-body problem

so that _a broader _range of fast ion conductors may be studied and ionic

transport

more

fully

understood _on the basis of

this

simple

_concept.

Langevin dynamics

simulation

provides trajectories

to calculate

p(I)

for any

given pair

interaction and thus to obtain

ii directly.

The rest of this section is devoted to the discussion of _some results

concerning Vej

of the twc-dimensional Brownian

model,

for

a Yukawa

pair

interaction.

In

Figure

4 _we have

reported

the variations of Ve~ with different temperatujes for incom- mensurate concentration _c

=

2/3.

The variation of

l~~

is

only

_seen in the direction where the

system

^is incomnlensurate. From these

plots

_we notice

that,

in the low

temperature reg1nle (dashed line),

the effective

potential

exhibits _a

complicated

_structure, characterized

by

the appearance of _new

equilibrium

sites

(most

ions _are

displaced

_away from the lattice

sites).

This

structure cannot be derived from lattice _gas models based _on the

assumption

that the ions _are

nlostly

located at the bottonl of the

potential

wells and

junlp

to the next vacant site instan-

taneously.

The

shape

of the effective

potential

_can

strongly

affect the

dynanlical properties

of

systenl,

as we shall _see in Section 4.2. In the

high

tenlperature

reg1nle

the variation of

l~~

is snloothed without _structure

(full line).

As the

tenlperature increases,

the thernlal fluctuations of the ions beconle

1nlportant allowing

the bare

periodic potential

to reinforce its donlination

over the interionic interaction: the _structure is

conlpletely s1nlplified.

Such structure for the effective

potential

_was observed

recently by

the _nleans of

X-ray

diffraction

along

the tunnel axis deduced

by

Weber and Schuldz

[24].

These

interesting

results indicate the

strong

role of correlation effects in

reducing

^effective

potential

barriers and in

nloving

the centre of the ion

displacenlent

off of the fornlal lattice site

(nlin1nla

of the

periodic potential).

4.

Dynamic Properties

The

dynamic properties

of

superionic

conductors _can be

conveniently

studied

by

neutron scat-

tering [25].

^From the theoretical

point

^of view this

implies

the calculation of the

quasi-elastic part

^{of the}

dynamic

structure

factor,

_{as a} function of wave-vector _q and of the

frequency

_uJ.

This

important quantity

which _can reflect the correlation effects between the

diffusing

ions in both _space and time is defined _as the Fourier transform of the

density-density

correlation

function:

~~~'"~ 2~N /

~~~~~~

~~~~'

_{~~~~ ~' ~~~ ~~~}

where

plq, t)

₌

~j _{exp(iqrj (t))}

j

S(q,

_uJ) determines the cross-section for neutron

scattering.

In

general,

for

superionic

^conduc- tors, it consists of _a

quasi-elastic

line which broadens _as the stochastic _or the diffusive motion increases and of

oscillatory

side

peaks

associated with

phonons

modes of the mobile sublat- tice. For _non

interacting particle systems,

^the

dynamical

structure factor and its f.w.h.m. _are

calculated _up to

large

values of the _momentum

transfer, covering

in several Brillouin _zones,

by solving

the two-variable Fokker-Planck

equation,

in relation with

premelting

surfaces and adsorbed

nlonolayers [26, 27].

^{But in} our case we wish to

investigate

the influence of the in- teractions of

diffusing

ions with thenlselves and with the lattice in two-d1nlensional _space. For

interacting particles,

the

analytical

and numerical

conlputation

^of

S(q,uJ)

within this nlodel renlains

conlplex

and difficult _even for

s1nlple

one-d1nlensional

systenls. So,

in order to de- tern1ine the

dynanlic properties,

we consider another

s1nlple

alternative

approach,

which _we

introduce in the

following

section.

4.I. LATTICE GAS MODELS. Correlation effects in the ionic niotion

beyond

the

validity

of nlean-field theories represent an

interesting subject

in the

physics

of ionic conduction. The lattice gas nlodel _assunles that the

diffusing

ions _are distributed _over certain lattice sites and

can

perforn1jumps

to other sites under the influence of _a heat-bath with the

junlp

t1nle far smaller than the t1nle between

jumps.

In this

approach,

the state of nlobile ions at _a

given

instant of t1nle is

specified by

a set of

occupation

nunlbers _n

=

(ni,

_{n2, n3, ni,}

).

This number takes

respectively

the values 0 and I

according

to the site if it is empty or

occupied.

The

properties

of the

systenl

are

determined

by

the lattice gas

Hamiltonian,

written in terms of these

occupation

numbers _as:

H(n)

=

~j _ij-i,njni,

~J

~j

_al

(8)

~ l,I'

where

h-if

denotes the

pair

interaction and

~J is the chemical

potential

which determines the average

occupation jai

" c. Lattice gas models _are

closely

connected with

Ising

models.

So,

let _us introduce

pseudo-spin

variables which

distinguish

two states at each site:

ai =

f°r _a

particle

at -I for _{a vacancy}

ni "

1/2(1+ ai)

is the total

occupation (for

_more details _see Ref.

[28] ).

The

probability P(a, t)

to find the

configuration

_{a =}

(al,

_{a2, a3, ai,}

)

_at t1nle t is

governed by

the _nlaster

equation:

aPjj't)

=

~jw(a>

_~

a)pja>, t) w(a

~

a>)pja, t)j

₌

Lp(a, t) jg)

front which the

equation

^of nlotion for _any function

f(a,t)

can be obtained. In

(9)

L is the evolution operator and a' denotes the

configuration

which results front _a

by interchanging

the

occupation

nunlber of two

nearest-neighbour

sites. For _an isolated

systenl

^{the transition} rates W

obey

the detailed balance condition

W(a

_~

a')exP(-flH(a))

=

W(a'

_~

a)exPl-flH(a')); fl

₌

lkBT)~~ (lo)

We should _note that different choices of rates

conlpatible

with the detailed balance condition

are

possible _[29], giving

a different

dynanlical

behaviour.

Therefore,

_sortie additional

physical inputs

are

required

when

applying

the nlodel to an actual nlaterial. In the _case of the Coulonlb

pair interaction,

Bunde and Dieterich

[28, 30]

^{assunled that the local barrier}

along

the bond

(I,

_{+ &)} ^has a contribution

proportional

to the local electric

field, leading

to the

following

expression

of the transition rate

Wi,i+&ia)

₌

W°ia)expj-fliHia) Hia'))j. ill)

The nlain limitation of the stochastic lattice gas model formulated in this way lies in the fact that the

underlying

lattice is assunled _to be

rigid

and the effects of

particle

inertia _are

neglected.

Still the

general

nlodel

equation (9)

^is

extrenlely

useful in

describing

correlated

low-frequency

atonlic nlotions in nlany different _areas,

including hydrogen-diffusion

ⁱⁿ

nletals,

two-d1nlensional diffusion of adsorbed atonls or kinetics of

clustering

near a first-order

phase

transition. Much of the progress in these _areas is due to the

increasing

role of Monte-Carlo s1nlulations

[31].

4.2. HALF-WIDTH oF THE

QuAsi-ELASTIC

PEAK. In this

section,

_we shall be

niainly

concerned with the diffusive nlotion of

particles

in

superionic

conductors. This is

conveniently

studied

through

the

quasi-elastic peak

in the

dynanlic

structure factor

S(q,uJ)

which is de-

fined,

as mentioned

above,

_as the Fourier transform of the

density-density

correlation function

(n(q, ^{t), n(q,} 0)).

Its width is found _to contain valuable information _on the mechanism for the diffusion _process. The coherent

dynamical

structure factor _can be written in the

following

form

using

the

Mori-Zwanzig projector

method

[32]

Slq,W)

₌

)Slq)Rei-iLJ

⁺

^Alq)

⁺

^{Mlq, -iW)i~~ l12)}

~~~~~

~j -lnlq)/Lnlq)1

~ "

Ns(qj

=

lIVl ~j ~(j())~.

^~~~~

Here,

^b _are the

jump

vectors

joining

_a site to its

nearest-neighbour

sites and

M(q, -iuJ)

is the nlenlory function

describing

the collisional effects.

(W) _represents

the _average transition rate between two nlin1nla of the bare

potential l§.

This function

has,

of _course, the

periodicity

of the

reciprocal

lattice. The

analytical

calculation of

S(q)

in the

inhomogeneous _systenl

is very difficult. For the determination of static

properties,

in the one-dimensional _{case, one}

can sometimes

enlploy

a common

simpler procedure

which consists in

approximating

^the

pair

correlation function

g(x, x') by

the _go

(x x')

for the

homogeneous system

obtained

by putting

Vi

" 0. This

procedure

is referred to _as the

homogeneous approximation (H.A.) _[33]

and

is

widely

used in

calculating

the

density profiles

of

inhomogeneous

systems,

especially

in the

context of surface effects in

liquids _[34].

Within this

approximation S(q)

takes the

following

fornl

[35]

S(q)

= I +

~j

^~~'

_{[So(q k') ii}

_{+ N}

~j

^~~'

&kf,q

(14)

~, Po

~

~, Po

~

where

So (q)

represents ^the structure factor of the associated

honlogeneous

systenl ^and

consists, generally,

of

peaks

at

nlultiples

of _{qi "}

2~16.

k'

represents

^lattice

reciprocal

vectors

(k'

= nqo,

where _qo

" 2~

la

and _n

=

0, +1, +2, ).

^The Fourier coefficients _pkf of the

one-particle density

are

given

^elsewhere

[34].

^{In the}

high tenlperature reg1nle

_or ⁱⁿ the _case where the

periodic potential Vi

acts _{as a} snlall

perturbation

_on the

liquid systenl,

^{the first few} Fourier coefficients

are sufficient to describe the

conlplete

structure factor

S(q).

If the interaction between the ions is

strong enough,

_{we can}

expand

the interaction

potential

around the

equilibriunl positions

^of

the

homogeneous system

and _we retain

only

the

quadratic

terms

(harnlonic coupling).

This then leads to the Frenkel-Kontorova

potential [35],

^{with which} we have calculated the static

structure factor

equation (14)

^{and effective}

potential:

V2 =

°

~ _(xi+i

x~

b)~ (15)

2

characterized

by

_a

generalized

force _a. The

strength

of the

pair

interaction is nleasured

by

the

screening length ~~~

which is

given

in units of

b, by: ~~~

=

aflb~ /2~.

The framework

potential

has the

periodicity

_a. In

general

^this

periodicity

is different from

b,

the

periodicity

that the interaction between the ions would

produce

in the absence of the framework

potential.

The first authors to

study

the

equilibrium properties

of discommensurate _cases

la # b)

^with a free- end

boundary

condition _were Frank and _van der Merwe

[37]. Despite

its extreme

s1nlplicity,

the Frenkel-Kontorova

potential

exhibits nlost of the features to be discussed in the

following.

Let _{us return now to}

equation (13).

It shows the relation between the diffusion _process and

the structure features of the

systenl.

To

get qualitative picture

on the _nature of the

transport

nlechanisnl,

_{one can use} the nlean-field

approx1nlation (MFA).

This consists in

neglecting

all memory effects

(M(q, -iuJ)

=

0).

This

approximation

_seems to be

particularly appropriate

in the

high damping

limit _~ » coo- It _can be very useful to _our

problem,

in which we treat

only

correlated ion

dynamics

at

frequencies

much below the atomic vibrational

(long-time dynam- ics)

_ones.

The

dynamic properties depend strongly

_on the friction

parameter

_~

which,

in

general,

_may

depend

_on the

position (for s1nlplicity,

_{~ is} assunled to

honlogeneous

ⁱⁿ our

case).

Detailed _an-

alytical

studies have been

conlpleted

ⁱⁿ the

high-friction

l1nlit _on the basis of the Snloluchowski

equation [38j.

In this

reg1nle,

the

long-t1nle dynanlics

can be characterized _{as a} discretized _ran- don1 walk between

adjacent

lattice cells: _a

particle junlps

^{fron1one site I} to a

nearest-neighbour

site

j

and

thernlalizes, losing

all _nlenlory of _this

junlp.

While the above nlodel is not valid for

underdanlped

systenls as the

particles

then _conserve the nlenlory of their last

junlps,

I.e. the

junlps

_{are nlore} or less

strongly

correlated. It is known both for continuous

dynamics [39]

^and for lattice gas models

[40]

^{that the diffusion of} a

single particle subject

to _a

periodic potential

is faster than

that predicted by

classical rate

theory.

Due to the weak

dissipation

_process, the

nlean free

path

_can be

larger

than the lattice

spacing

a if the relation

(7r~Vo/uJokBT)

« 1

7

6

s

~

4

~/ 3

C/l

2

~0 0 0 5 1.0 1.5 2 0 2.5 3.0

q/qi

Fig.

5. Static structure factor in the

homogeneous approximation

for Frenkel-Kontorova model.

The

microscopic

parameters associated with this

figure

_{are: c}

=

2/3,

correlation

length

^~~~

= 3

(~~~

=

aflb~/27r~),

l§

=

6kBT,

r

= 10 and _qi

"

27r/b.

is satisfied. In this

limit,

the thermalization of the

particle requires

a time far

exceeding

_one

oscillation

period,

_so diffusive

jumps exceeding

_a in

length

_can be observed.

For

systems

^with incommensurate _or

high-order

commensurate

concentration,

some ions _are shifted _away from the

equilibrium

lattice site in the

ground

state,

leading

to a very structured effective

potential, especially

at low

temperature (see Fig. 4):

for such

situations,

_an extension of the

expression (13)

has to take into account different

jump

rates associated with different

jump lengths,

which _are in

general

less than _or

equal

to _a.

By using

the _same treatment

(Mori-Zwanzig projector method)

for this

situation,

we arrive then at the

general

form:

ji

~~q6j

Alq)

=

~j ~j~j~~ ^(W(b))- li6)

Now the vector b connects not

only

the minima of the framework

potential,

but also the

minima of the effective

potential (Fig. 4)

whose structure cannot be derived from lattice gas models _or

hopping

models.

So,

in order to

get

better detailed information about these

dynamic quantities,

_we consider _a combination of the _two models detailed before. Accurate

expressions

of _some

quantities

_are ^{available from}

analytic approximations

within the continuous models.

A few words about the

computational

effort

required

for the evaluation of

(W(b)).

This

important quantity

is calculated from the different barriers

corresponding

to different

jump lengths (the

effective

potential

_may exhibit a

complicated

structure at low temperatures and at inconlnlensurate

concentrations,

_see

Fig. 4).

The existence of barriers with

different height

influences the

dynamic

_process

strongly.

In the

high danlping limit,

these barriers _are inter-

preted

_as the activation energy.

In

Figure

5 _we consider

S(q)

for the incommensurate

density (c

₌

2/3)

derived from the

homogeneous approximation equation (14).

Its

pattern

^{could be}

composed

of

peaks

associated with the

liquid

structure nqi

(qi

"

27r16,

^{where b is the} _average ^{distance between mobile}

ions)

and others shifted from those of

So(q) by

different

reciprocal

lattice vectors

(q,im

_{= nqo}

+Tnqi).

Contrary

to

Figure 3,

_we wish _to nlake it very clear that

S(q),

obtained from the

honlogeneous approx1nlation,

is

conlputed

here

strictly along

_one

single chain,

without

taking

into account the effects of the others. The _appearance of the _new

peaks

and the shift in the

position

of

sortie of thenl in

Figure

3 with respect to

Figure

5 is

due, essentially,

to these

physical

effects

which _can be concei~,ed _{as a}

projection

of all other chains _{on one} chain. The two _{curves are}

obviously different; they

describe two different situations.

The half-width of the

quasi-elastic

line of

S(q,uJ)

contains valuable information about the interaction of the mobile ions with the

rigid

framework and with each other.

Also,

the

depen-

dence of

A(q)

_on the

wave-scattering

vector q can

provide

_a

good

indication of the

dynamical

correlation effects in the

system:

^{it therefore} characterizes the diffusion _process.

In

Figures

6a and b _we present results for

A(q)

obtained

by

several methods. The

parameters

used here _are realistic for ionic conductors

(T

₌ 200

K, ~i

₌ 0.I

eV;

_{Vo is} the activation energy in the absence of interaction between mobile

particles).

The results obtained

by

the

combination of two theories

(Eq. (16)

_are

compared

with the cosinelike behaviour of the

single jump theory [41]

^{and with results obtained}

by using

the _mean field

approximation

in which the f.w.h.n1. takes the

following

fornl:

~~~~ ~~2

Slql ~~~

The half-width associated with _a

system

^of non

interacting particles (Fig. 6,

curve

(. ))

is

presented.

For small values of _q

(q

_«

27rla), A(q)

increases _as

Dq~

where D

represents

the diffusion coefficient of the mobile sublattice.

In _our

inhonlogeneous

_systenls, the presence of correlation effects in the

dynanlical

behaviour of ions _can affect

strongly

the curve,

A(q).

In

fact,

for

interacting particles, A(q)

shows

strong

oscillations. It

presents

relative minima at

q's

which coincide with the maxima of the static structure factor

S(q), especially

at qi "

27r/b

and at the modulated

peak

_qn,m

= nqo ⁺ Tnqi

This behaviour is similar to de Gennes

narrowing.

The static _structure factor _occurs at _wave

vectors

corresponding

to the most

probable

interionic

separation.

These

spacings

_{are represen-}

tative of _a

highly

correlated ionic

arrangement,

^{which is}

long

lived because of the

cooperative

motions necessary for them to break _up. This

long

life-time is reflected _{as a}

narrowing

in the energy width of

S(q,uJ)

for those

q's.

We notice that the functions

S(q)

and

S(q,uJ)

_are

related via the _sum rule:

S(q)

=

fS(q,uJ)duJ.

On the other

hand, by comparing

the curve

(full lines) corresponding

to

equation (16)

with the _curve

(dash-dotted lines)

which is obtained

by employing

the _mean field

approximation (Eq. iii)),

_{we see}

clearly

that

equation (16) gives

more detailed information _on the diffusion mechanism. Its structure shows _some minima which

are not observed in the

other, especially

the minima at qo =

27rla.

As the

strength

of the

pair

interaction is

increased,

the

amplitude

of oscillations of

A(q) increases,

^{and the} nlin1nla associ- ated with

liquid peaks

becomes

clear,

as shown in

Figure

fib. This indicates that the diffusion process for this _case is _nlore

liquid-like (ions nloving continuously). By taking

into account the entire behavior of

A(q),

_we conclude that the nlost

probable

diffusion process in

good

two-d1nlensional

superionic

conductors

(c

₌

2/3)

consists of _a

conlpetition

between _a back correlated

hopping

in _one direction and _a

superposition

of both forward correlated

hopping

and

liquid-like

nlotions in the other direction of _space. These results _are in

good agreement

with those of

conlputer

simulations of one-dimensional

superionic

conductors

[42j.

oio

~°~

0.08

(3/2)qp,

/"'

~

°.06

~ ^/.

/

~

'i.'

13 _',1

~ qi

~l _0.04

~,,--, ^{,/" ",,}

~,--,~

0.02

~'~

j~°

",

,'

),,

^{,' ',}

~,' ~,/-"~.,

_~/ ., ,'

~'~

~0.0 'j

.5 .0 .5 2 .0 2 5 3 .0

a)

q

/q

0.20

q~/2j

(3/2)q~

0.15

,/"",

,, ^,,

1'

~;

,' _ii

/ ..

° 010 ~~

fl

_qi

~

,"~",

/"

",

0.05 ,' ,,

,' qo

/

'_~,

^/"

",

~,

',

~ ~

~0_.0

~i.5

_{.0 .5 2 .0 2 .5 3 0}

~~

q IQ

i

Fig.

6.

a)

The

q-dependence

of the half-width of the

quasi-elastic

line of

S(q,

WI for Frenkel- Kontorova system ^{in the} incommensurate situation

c =

2/3

from different methods:

(.. ..) homoge-

neous

approximation equation (17);

(.

single jump theory

and

(-) equation

where the effective

potential

is computed

by

numerical simulation. The parameters are the _{same as} in

Figure

5.

b)

Same

as in

(a)

but here _we consider stronger

pair

interaction: ~~~

= 5. The diffusion _process for this _case is _more

liquid-like.

5. Conclusion

In this

report,

we have discussed the static structure factor and diffusion mechanism of _con-

ducting

ions in two-dimensional

superionic

conductors

by

_mean of Brownian

theory.

The

essential aspect of this

investigation

is the

quantitative description

of these

properties,

^which

show

clearly

that the

system

with _c

=

2/3

_can be

decomposed

into two different situations

along

the two directions of _space: commensurate in _one direction and incommensurate in the other.

The character of the diffusion process is

quite clearly

revealed

through

the _q

dependence

of the width of the

quasi-elastic peak

of the

dynamical

structure factor

S(q, _uJ).

For _c

=

0.5,

the width _goes to zero at each

reciprocal

lattice

point (q

₌

2~nla). Thus,

the diffusion mechanism associated with this _case is characterized

by

the back correlated

hopping.

It is very different from the _one found for incommensurate concentration _{c =}

2/3

and

especially

when the interaction between mobile ions becomes

important.

The _q

dependence

of the width is

very sensitive to the existence of correlated

jumps

in addition to

liquid-like

motion.

So,

due _to the

decomposition

of _our

system

in two lD

subsystems

for _c

= 2

/3,

_we conclude that the most

probable

diffusion process of two-dimensional

superionic

^conductors consists of _a

competition

between _a back correlated

hopping

in _one direction and forward correlated

hopping

in addition to

liquid

motion in the other direction of _space.

References

[1] Evans

R.,

Adv.

Phys.

28

(1979)143.

[2] Ramakrishnan T-V- and Yussouf

M., Phys.

Rev. 819

(1979)

2775.

[3j ^Kleitz

M., Sapoval

B. and Ravaine

D.,

Solid State

Ionics-83,

Parts I and II

(Amsterdam:

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