Surface organization of dipole monolayers

89

JownaLofMoIecularLiquidn,51(1992) W-113 EltwvierScience PublieheraB.V.,Amsterdam

SURFACE OFZGANIZATION SILVANO

CINCOTTI.

MArlRO PARODI(*'.

Blophyslcal Via (Received

1 August

OF DIPOLE MONOLAYERS

&

Electronic

all'opera

Pla

and

ALEISSANDRO

Englnesrlng

lla.

16145

CHIAEtRERA

Department

Gancva,

ITALY

1991)

ABSTRACT

A

model

based

dlpoles tool

on

(closest

for

a

The

the

under

rather

rer llts

yielded

by

are

We

regarded

as

the

effects.

which

study

monolayer more

are

of 1s

been

two-dlmenslonal

is

withln

head

1s

are to

better

region.

It

interactions

among

To

whom

the

obtalnlng

an

Is

0167-732!2/en/3os.o0

01992

formed

a more

electric

mutual

that

the

lnteractlons

the

formation

the

of

surface to

should

121.

be

accuracy array

be

long-ranga

molecules

in

a

area

considerable

case.

characterization llpld

and

of

molecules

the In

a

phaso.

moment

associated

orientations

of

among

the

the

wlth llpld

hydrophobic

each heads llpld

domalns.

properly The

model

research

by

organization

take

account

blophyelcal

fluid

dipole

are

previous

lnto

predlctlng

lipid

accurate

the

molecules

correspondence

by

of

latter

for

necessary the

domalns

with

the

planar

nelghbours.

classical

In

the

and

for

understand

monolayuls.

('1

that

responsible

responsible

both

113 surrounded

thought

a uniform

tall8 Then

fllm

among

a

cellular

to

a bask

dlpoles

takes

and

adequate

propertles

for

pseudo-crystalline

usually

molecule

to

consistent that

In

such

short-range

propagatlon

electronics.

developed

are

among

as

dlpolas of

stablllties

model

self-organlzatlon topic

Is used

of

model

proposed

Llynamlcal through

Important

Langmulr-Blodgett It

a

They

static

the

lnteractlons

conflguratlons

conflguratlon that

achlcved

molecular

have

accurate

The

for

the

an

recent

efforts

basis

propertles

condltlons.

more

electrical

approximation)

array"

equlllbrlum

suggest

the

of

organlzatlon

general a

lnteractlons.

discussed.

The

surface

fundamental

ob'xlned

long-range

forn

nalghbours-"cellular

studying

monolayer.

simplified

use

of into

of

llpld account

a model

with

dlpolar

heads

the

electrical

good

predlctlve

addressed.

-ElaavierSciencePublSrheraB.V.

In

Allrightares~~-~ed

capabllltles research klnd

could

porslblllty orlentatlons

In

a a

monolayer

dlpolc

1s

each

dipoles. equations test

In

for

a

the

cellular

1s

shown

of

ldea

for

properly

can

nodes

interact

be

of

wlth

the to

slmplo

form

and

represents

also

Desplte

dipole by

surface

terms,

the

between

dipole

model

patterns

1s

that

limited

each

number

by

square

of

four

closest

a

very

the

more

rlgorous

1s

considerably

thls

equlllbrlum

direct

stability

of

square but

In

nelghbourlng

the

slmpllclty

for

assoclatlng

lattl-_e.

formulate

the

patterns

another

obtalned

a

Ieads

resulllng

some

molecular

correspondence

chosen.

approach

obtained

a

dlpole

general

this

this

those

at

the

as

array".

the

configurations.

reglon

equlllbrlum

141.

the

wlth

to

the

important

signals.

approach

wlth

assumed

resulting

array,

ldentlcal

this

llpld

in more

and.

study

a

two

lnformatlon

govern

a "cellular

dipoles

In

very

131.

basic

a

controlling

electric

with

of

to define

of

the

The

interact

the

dlpole

As

and

least

at

processing

that

for

constltutlng

of

of

rules

a model

lmplemontatlon

barycenters

case.

to

dipoles.

sl-nplest

the

the

presented.

assumed

nelghbour The

was

attempt

flelds

molecules

141.

towards

feaslblllty

electric

dlpolar

paper

the

the

determining

of

recent

polnt

capable

concerns

exogenous

of

starting

clrcult"

second

through

the

first'regards

"molecular The

lattice

In

The

goals.

of

scale.

be

the

chosen

lattice

complex

are

approach

[Sl. In

this

the

work,

rhombic

lattice

generallzotlon. of

the

to

In order

to

the

be

Increased in

c41-

study

to

this

the

structure

dipole the

elght

work.

are

Then

can hold

the

On

basic and

the

accuracy

of

values

obtalned

a very

large

thls

Improved.

considered be

in

regarded

for

as

First.

141. a

a hexagonal

of

THE RHOMBICLATTICE: the

Jack"

case

the

all

for

deflnlng This

equlllbrlum with

Is

This

lattice.

a

1s

particular

the

1s

assessed the

by

accurate

the

the

cellular

a

case

which

related

of

the

possible

second

equations

those

values

all

the

conflguratlons

wlth

lnteractlng

BASIC

array).

equlllbrlum for

patterns

dipoles

compared

approach thls

of

basis.

and dlpole

the

number

consider

number

discussed

In

equillbrlum

("Union

obtalned

analyzed

US

lattice

consldoratlons

angles,

are

Let

approach

square

latter

Analogous

rhombic

crlterlon

the

array the

1s

reality.

of

used

as

former.

closer

cellulnr

replaces

Lo and

array

and the

the

stablllty

square

their

the

lnteractlon

achieved

by

Each

mesh

lattice

stablllties

angles.

values

must

generallzatlon

rhcmblc

comparing

values

Finally. energy

considering

dipoles.

DEFINITIONS

rhombic

mesh-grid

shown

In

Flgla.

1s

deflned

91

Y

j fd

~

x

j

n~j~ IJ .od~

J

f

(a)

(b)

I=lgurc 1. (a) the rhombic grid: (b) a dipole o f the grid. by

the angle

Owln~

to

trivial

In

included The is

~ betwoon

the

is assumed

deflnod

either

Fig.

in

Ib).For

and by

the

equlvalencles,

to e x t e n d by

the

~ a×es

length

possible

d of

its

sldes.

varlatlons

In

7

are

(0°,90°].

a

~.W

pair

infinitely of

(loJ~

numbers

system)

(~,9}

cosz]

in the ~ and W dlrectlons.

integer

reference

the orthogonal

l+j

and

geomotrlcal

range

grid

coordinates

the ~

or

coordinate

by

the

(associated

position

system,

Each

El j c a n

node

with

vector

[lj

be written

its (see

as

E,j = d

Each

(1) tJ s i n

; J

(l.J)

is

node

plane

and

h~ve

the positive

m

-

where

terms

modulus

of

the ~

p.

a

dlpole

Denotln g

axis

by

( s e e Flg.

~lj" ~iJ Ib),

All the

we

dipoles

angle

lie

between

o~

the

mt j

and

can write

(2)

enable

m_lj a n d

us

]gkl.

4~c 1~,,-~,13 <~.~>

dielectric

dipole

same

direction

of

Ls i n

two dipoles,

"

barycenter

[oo?1~lj

p

-~J These

the

the

stands

for

pormlttlvlty

mi j w i t h

a set

~lj

to

write

the

electric

interaction

enerEy

~!

tJ

between

as

%j.~,> - 3 < -Lj-~,.~,j>I~,j_~, <-~',-~''~'>1 ~ the

scalar

product

of

the

medium.

of

other

dipoles

between

Thus,

can

the

~

and

~.

Interactlon

be expressed

as

(3)

avd

c

energy

Is

the of

a

92

(4)

When PI,

contains

represents

w

all

the

the

whole

(lnflnlte)

electric

dipoles

of

lnteractlon

the

grid

except

of

81rJ In

properly

chosen

energy

the ).g lJ* the plane.

term When

1J 1s restricted

3

NJ those

in

a good

the

Union

contain

Jack

approxlmatlon

In both

the

U related all

to

the

to

and

all

terms

structure

for

exact

the

a flnlte

such

the

number

of

dlacussed

in

thls

paper).

WIJ

elements can

[like

be

used

as

energy.

approxlmate

dlpoles

In

cases.

the

plane

the

resulting

lnteractlon

energy

be

expressed

as

sum

can

half

the

of

W IJ

1 2

wIn

the

under wlth

!-

(51

LwylJ

rhombic

the

grid.

influence

the

dlpoles

solutlon

each

of of

to the

set

dlpole

the

the

@,)

electric set

can

rotate

force

3,).

orlglnated each

Then,

freely

round

by

the

equlllbrlum

Its

barycenter.

lnteractlon

of

conflguratlon

to

(6)

respect (5).

IJ a

of equations

G-/O with

m

la

the

to

the

orlentatlon

equlllbrlum

angles

equations

can

d

.Taklng IJ be written

lnto in

account

the

expressions

equivalent

(3)

form

dm

17) where

(8)

la a

term

3 lJ yields each 3 ,J

of 1s

that a

depends

corresponding

equatlons presented.

this

cellular

both

the

behavlour.

on

basic

(7).

In

For

any

array

allows

equlllbrlum

the

angles

reduction the

next

value one

ak,

of

in

the

section, of

to

the

set

4

Reducing IJof unknowns

number a falrly

slmple

the

mesh

angle

make

some

accurate

configurations

of

the

x

ln

the

range

prtdlctlons

dipoles

and

of

contained

cellular the

size

their

array

In for

(O",SOo

I.

concerning physical

9a

Ti-tE RHohmc

by

CELLULAR

dipole

of

I,,

the

dlpolsa

kl. The

numbers

(k-1). (1-J)

dlpolee

closest

Strictly

other

and

the

related

of 3

an

for

m -1-I.

and

m

for

E

and

B

on

of

both

for

the

the eight

(9)

l+l.J-1

l+l.J+l

@lJ

d

.

the

,

j+l

/

(8)

the

i-4

set

with

i

Figwe

i+l

ofsome dipoles in the rhombic of ‘~lr/; m : position of the nearest

2. 7%~ barycenters

htke. x_- position neighbows to gj_

Its

q,_ try

the

an

approxlmate

by

defining

crudest

the

evaluation YIJ

choice More

(Flg.21.

on

J+l

I-l.J-1

interactions

to

d

number

are

21 rf1.J

number

structure

distance

Inflnlte

and g+rdepends Q @J the distances Instance,

For

and

lnflnlte

occur ml, closest dipoles

Instance.

an

between

I,-%, I angle 1.

pi r.J*r

should

of

lattice

by

the

expr_

for

leads

surrounded

equlllbrlum

strongest

Thla

la

for

(7)

elements

rJThe

the

any

Then In

equatlono involve

1~

and

all

dipoles. appearlng

lattice

to ~9,~ (Fig-21

wlth

sum

rhombic

dlotance

speaking,

interacts

ARRAY

shorter

for

as

being

of

the

made

up

physical

properties

of

dlpolea

these

of only.

For

could be the four-element set at rJ choices include the two dipoles e

the

9

refined

the

I-l.J+l*

diagonal.

and,

subsequently.

could

be

those

on

the

longer

ml+l.J-l

Further

diagonal. shhpe"

centered

assume

all

For 4Cl

For

the

a given of

set

h

of

the

YIJ_

For

(whose

elght

set

sldes

dlpolos

a generic

each

between

corresponding

grld the

9 IJ letting

l.c..

1)' integer values

a rhombic

instance.

rhombic

L.

extensions at

are %l

L-order

-L and 9

generated number +L.

preservlng of

and

the

the

pair

excluding

the

"rhombic

(k-11. pair

1

(2L T 112 - 1 contains the rJ II 2Ld in length) centered in m

(1-J) (0.0).

elements

-1)'

in

Flg.2

rhombic

define set.

sum

the

lowest-order

(4) becomes

(L=l)

94

wlJ

= f,.,,

f,._,,

-L Any

choice

representation

choice yleld

al

of

(7)

for

to

stablllty

The

simplest

1J structure;

the contain

of

a

the

flrat

this

flnlte

step

number

of

[l-J1

* 0

of

step

touards

allows

each

unknowns

(10)

geometrical

properties

of

each

equilibrium

conflguration.

the

a

"cellular

of

only.

~I m

elements

the

of

choice

for

9O"I

whole

range

COO.

glvan

by

(Fig-Z).

In

+

the However.

that

1s

a

large

equllibrlum

array"

equlllbrlum proper

enough

to

configurations

dipoles;

b)

IJ

IS

3

3

evaluatlon

3

for

should include a number IJ accurate results concerning:

of

jk-11

-L

flnlte-element

equations

with

%

L-l

=

J’

%*I,

this

case.

dlfflcult though

%.

to

that IJ of values

that

taklng

(3).

to

IJ calculations

direct,

the

mesh

thase

angle

requirements

x

1s

the

Union

over

the

Jack

set

is

not

I!! l+l.Jfl

and

(91

and

W

of

meets

1s:

19I-l.Jkl'

Jfl'

obtaln

reasonably

3

put

(10) the

as

basic

result

In

expressS.ons.

compact

form.

It Some

tedious.

yield:

1 W

IIJ

4nc

111 I-1.J + mlrl,J)TBl

d3

+

l.J-1 + %, Jfl)TB3

where square

the

superscript

matrices

-2 BI =

deflned

0

[ 0

1

-12x4

T

stands as

1 +12x=

+

for

[%l,J-1

+ ml+l,J*I)TB2

[%l,J-1

+

%f,J+l

"transpose",

and

/x

"')

BI..

.BI

1

1 - 3x2

-3x

-3x

-2

i-F-2

Bz = 8 x3

- 2

(2~"

are

symmetrical

follows

-6x

/-C-z

J1-;;"

(2x"

B3 = -6x

+

-

1)

121~~ - 12X=

+

1

+ 3x2

1)

I

1 (12)

95

-2

1

8

B1 p and

(1 - x2)-

x depends

on

+

3x

3x

the

mash

2

3x

/-C-z

angle

/i-Y?

1 - 3x2 7 via

the

I

relation

-7

x

=

CO8

(131

-

2 Expression the

case

that

(11)

represents

a

four-element,

of

case

are

the

very

The

equlllbrlum

and

B1

same

a generallzatlon

B3

square

of

cellular

calculated

for

x

on

the

the

exprasslon

array.

m

l/-

Tha

given

matrlcea

(1-e..

7

=

in

141

for

obtained 900).

and

In play

role. equations

(7)

take

*PI

l+l.J-1

form

'J=o I-1. J+l

(141

da IJ

NOW,

since

d3 < I,J.-$-

>

dz! liZJ - --$

=

1J and

matrices

written

In

+B

X

Both

the

a)

can

at

+B

+

0

2

value

and

multlpllcatlve

equlllbrium

equations

can

+

%+I,J+l]

+

B3

[%.J-1

+

%,J+l]

+

(151

IJ

I

be

form

l-l.J-1

number. the

summarized

equlllbrlum.

the

symmetrical.

sultable

"KP!

%-l.J+l

real

are

more

1+1.J

is a

be

Ba

much

+I!!

ltl. J-l

where

which

Bl.... the

I--1,J

=

LJ

sign as

the factor

of

K

have

very

important

physical

meanings

[41.

follows

interaction 1s

a constant

energy for

W

1s proportlonal IJ the lattice:

to

X.

The

7, = b)

I

x

equitibrium

any

1

2

4 a E da

configuration

is

stable

when

only

K

is

a

negative

quantity.

The

first

of

of (15)

eq.

Hesslan

the into

above

two

expr.

statements

The

(111.

second

by

XCO. the

checking

The

forms

cellular

array.

the

as

verified

can

be

direct

proved

by

substitution

considering

the

provided

that

proof

hence,

they

schematIcally

to

the

shown

H

and

very not the

introduction

the

the

are wil1

(11)

have

related

mn

same “central”

in

positively

similar

be

given

to

of

those

I1

all

among

the

through

matricas

In

I41

for

eight

orthogonal

(15) the

can dipoles

of

m,,

the

matrices

take

on

of

the

cellular Pk,

Fig.3a.

(3)

F&m 3. (a) the general flow graph corresponding mm-ices is the same for ilte dipoles.

eight

reported

equation

relations

then Q

definite.

here.

equilibrium

modulus. dipole

Is

(a)

The

by

6

kl’

equilibrium,

the

after

dipoles

are

at

expression

simpler

array

of

lattice,

energy

All

that,

details

square

The

be

matrix

6

and

can

are

not

independent

IO eqm.

of

one

(16);

(6)

another.

the se1 of orthogonal

Due

to

the

97 arbltrarlneeh be

vleued

Jf

Al each of the boundary dlpoles 1J' the center of another cellular array.

as

situations.

the

thls

set

af

means

four

of

P

As

the

matrices

1s

he

can

be

for

least

In

array

simplest

numbered

for

from

any

1

to

dipole

the

celiclar

8

.as

of

shown

In

Flg.3b

step.

two

of

the

two

IPI

and

In

matrlces

P3)

terms

of

the

remalnlng

ones

as

(as

(171

Pn

can

(say.

be

P2

seen

and In

PI)

can

Flg.3bl.

be

put

In

terms

conssderlng

the

products

P

1

1

lmoaedlate

Pd are

p2

-

pl

p3

P

= P3 Pi'

(18)

(b)

to

4

verify

equivalent.

that

provided

both

exprosslons

for

P2

and

both

expressions

that

(191

p3 p1 - p1 p3 As

a

result.

orthogonal or

can

the

1.._.4

P4 = Pi' P3

Iti 1s

identical

put

n =

n

pa= P3

(a1

can

= p-' n+4

other

'chain"

At

cellular

lines);

these

a further

01

matrlcec

matrices

(continuous

the

that:

orthogonal

These

array.

of

whenever

matrices

of

P the

P3

and

1

set

can

fulfll be

constraint

obtalned

by

(19).

eqns.

the

(17)

and

remaining eqns_

(18al

(18b).

This

property

square

cellular.

equatlons

where

1s

(15)

matrix

and

Its

The

lnvarlance

Q

generalization

array. take

on

1s given

structure

equlllbrlum

a

la of

9

equation

of

The

first

the

eigenform

that

important

found

In

consequence

C41 Is

for

the

that

the

four-element equlllbrlum

by

lnvarlant results (20)

to in

must

l.J-

the

second

also

hold

lmportant for

any

consequence. of

the

eight

that

is,

the

"boundary"

dipoles.

lihlo condltlon

can

be

expressed

1 =

and

asslgns (22)

Eqns. Let

the

us

both

complete

now

interaction the

set

consider

A

d&ales.

same

complete

orthogonal

a1

COB

-sin

~3 1

P; sin

B

[ and by

studying

taklng

lnto

directly.

other

also

for

FIRST These

of

P

P 2’

eqns.

be

by

assumlng

proper

COB

d

sln

8

[

(19)

to

difficulty

(22).

lying

of

'In

the

proper

fulfilled.

This

In

however,

Ps...,

4’

cellular

tP

tha

n

I

PT)

sln

d

-cos

6

I 1

array.

organlzatlons

I 1

of

the

posslbillty

or

Improper

for (P,

1

=

(231

cases

four

a pair

the

task

be

the

computatlonal

are

equal

following

orthogonal while

can

to

the

117)

load_

those

for

and

~111

which,

(18)

The

obtained

dlscusslon

matrices

eqns.

accompllahed

by

glve

quite results for

be the

proper

the

focused way.

eq.

matrices

P.

I3

ORDER PERIODICITY CONFIGURATIONS conflguratlons

means

1. Under

It

1

case

ldentlcally

P,R IJ -n

%-

the

1

for

periodical

begin

P_, tD

dlpoles.

Pi =

cnses.Therefore.

case

1s

This

only

for

Cl91

B

account

three

the

the

the

obtalned

on

cos

1

should

and

the

relations

spatially

dlscusclon PI

to all

general

basic

matrices

(22)

1....8

energy

of

the

as

follows

when

all

the

1 =

1...,8

of

the

dlpoles

are

parallel.

that

Is

E241

IJ that

the

AZ=

occur

at

least

hypothesis

det

one that

Pr = 1

P1=Pi

clgcnvalucs

and

h,+

PS=Pi.

x 2

that

the

unit

elgenvalue

can

= 2 CO6

only

X1,

all

the

A2

of

P, are

PI

must

proper.

be

equal

Now,

to

as

i5,

be

of

multlpliclty

2.

and

that

JS,--O; hence.

PI-I.

Matrix

In

obtain

the

order

(eq-

20).

relations

to It

is

9 becomes

eigenvalues

worth

noting

X1.

that.

X2 as

and a

the

general

corresponding property.

elgenvectors the

following

hold:

(%+5) - ”

‘““’ _*“,,I

[

”

1

--

cl

4 x3

Rz = u

U

126)

1

0

I 8 x3

I,>) --x

Es4 =

u

0 3

2

u

(27)

Thus that the

(BI + B3]. the

same

eigenvalues

B2. result are

and

BI have

holds

for

%

and

@,. and

s

as

eigenvalues.From

thrlt the

corresponding

(251,

It

follows

expressfons

for

= 2

X,

2

- 6xa

-

3

14

4 *a

8 x3 = 2

x2

The

dlroctlons

1-e.. For

of

6,,=y/2 a

and

q

and

generic

"P%

%

- 4 + 1

6x2

3

-

and

G

are

7,

the

6r,+2+90°.

value

of

are

"I'J-E$

thoso

of

Constraint

shown

(28)

1

the

diagonals

(22)

of

1s fulfilled

equlllbrium

about

them

symmetry.

explained

negatlve

7

that. the

K1

on

fcr

In

and

X

this

results

for

Al

glven

percent

over

sum

The

related

K

zero,

of

expresslons

view.

number 1

and

In the

the

Xa

The the

of

(X8)

correctly

on

(28)

to any

the

value,

glve

equivalent

reason rate

of

1~

(28)

equal dlpole

convergence

K,(7)

with of

and

L-80

(which

In

instability

(X2-+

are

that,

elements increases_ IJ have been calculated with

range

predict

stability

3

Appendix.

whole

curves,

take

and

stable

conslderatlons

proportional

could

direc-

expected.

dlstrlbutions. the

121 becomes

eypresalons

point

whan

simple

case,

configuration

slow

of

6

qualltatlve perlodlclty

to

(b) shorter diagonal

angle

I.e..

2'

(y);

baals

7=90°.

"v W conflguratlon (Flg.4al and -1 (Flg.4b). However, expresslonn

4m)

direction

the

orientation

Moreover,

of as

approaches the

that

dktgoml

Juatlfied

observe

141.

values

for

easlly

and

configurations. When

be

In

corresponding

in Flg.4.

fmtterns. (a) longer

We

matrix.

ldentlty as

can

cell.

(W

Q,u~ 4. First order dip& tion (g).

of

rhombic

ldentlcally.

conflguratlons

(a?

Both

the

a01

acceptable the

of

case

of

the

computer,

ensures

using

errors

the

only

1Y sum IJ Therefore.accurate a

for

(X1--+ "v W

from

flrst-order (10)

1s very

numerical the

smaller

double than

7 values). Xa(71,

are

plotted

In Fig.5

a

(continuous

lines).

1

101 300

100

0

--zoo

I

-300

I-

-4Oc

i

-

-

-5OC ) -

-60C

I-

-7oc

I-

I

10

20

30

40

60

50

1

I

70

80

90

Y

Figure

5. PIor

patxerns Dashed

of Llte

of Figs.

“energy”

mrvcs

Kl(‘y)

4a and 4b respecrively.

lines: results given

and Kz(y)

Contintmtcs

limes:

corresponding

cantputcr rest&s (see

by expr. (28)_

(b)

00 Figure tice).

to rht dipok

6. Firsr order

dipole

palrerns for

y = 6U

(hexagonal

lar-

rcrr).

102 The

dashed

lines

shown

for

this

angle.

comparison

lattice.

the

The

two

specific

are

Pm I

the defining

which

into

means

16 P;'

the

general

1 being

the

condltlons

relations

(22)

Using

eqns. by

and

(-1)n

are

the general

r9=2

them

wlll

can

also

cellular

be

hexagonal

though

very

obtalned

by

a

P1

la

array-

l

unlt

elgenvaluc of

matrix.

least

P1

and,

identically 1181.

and P3_ Then,

one

elgenvalue

can

only

be

equal bf

to

1.

When

multlpllclty

2_

1s

for

any

mlJ

calculated

by

means

of

eq.

(201

fulfilled. the

whole

set

of

orthogonal

matrlces

can

be

taklng

I N - 0.

(301

1

I

structure

(-1)"

now

a

particular.

B1 +

of

matrlx

(-l)"+"

B2 +

\t 1s

(-1)"

B3

+

(-l)"*"

B4

{

and

of

for

-- 4

l

at

M.

p3 = (-1)H

thls

nodes

~60~:

are:

have

the

and

the

at

are

(17)

structure

(17) P1

eqns.

must

Identity

obtained

p1 -

P:

so

proper. Then.

that

of

hexagonal

l=l

account

wlth

Both

Kzlntersect

at

they

(28):

CONFIGURATIONS

= pl*q F IJ

-IJ

taklng

or.

In FIg.6.

exprasslons

and

located

are

a six-element

by

X1

assoclatcd

presented

using

glvan

curves

baryccnters

- ORDEFZ PERIODICITY

In this case,

The

configurations

approach

SECOND

values

purposes.

dipole

case

important.

the

represent

be

specialized

for

the

different

M.N

pairs.

(311

103

(I)

(II)

The

case

sed

In

M=N=O

the

yields

previous

Uhen

M=N-1.

matrix

From

the

general

elgenvectors along

of

the

configurations

and

It

discus-

Q becomes

relations

9

are

v

(261

and

of

v

the

(271.

again

(I.e..

rhombic

the

follows dlpoles

are

The

structure).

that

tha

oriented

correapondlng

are

4 x3 K/2

equlllbrium

section;

dlagonals

the

elgenvalues

first-order

[ 6x2-2-L+

2

-X

&-]

3

1 x-cos -7

(321 ~-

2J

Xa

= 2

The

4 - 6xa

equlllbrlum

shown

-r--p8 x3 1

3I

conflguratlons

Ln Fig.7

for

corresponding

a generic

value

of

to

and In

X2 the

Fig.8

the

prevlous

can

be

the

continuous

the

following

section,

compared

Appendix, shows

aa,=,+%

are

(b)

Figure 7. Second order dipole patterns when M=N=I. ter diagonal direction (w_ In

and

r.

(a)

As

m _I,-~&

ulth

that

the

ones

the

with

(a) longer diagonal direction @I),-

results

the

given

computer

by

expressions

results

based

on

(321 sum

A2

(b) shor-

for

X1

given

L==SO. "approximate" (obtalned

propertles:

ulth

(dashed) a

curves

computer]

to

are

close

represent

enough

to

correctly

104 400

300

200 100

0

-300

-400

-500

-600

--700

u

10

20

30

40

50

60

70

60

90

Y Figure 8. PIor of flue “energy” ~w-ves Kl(y) arrd K2(9 corresponding m rhe dipore parrerns of Figs. 7a arld 76 respcctivciy. Conrinuous lines: computer results (see rem). Dashed

lines: resdts

given by clrpr. (32).

(a) Figure 9. Second order dipole parrems for y= W rice).

(b) (hexagonal

lac-

105 the longer

the

dlagonal

"v -

stable

for

for shown

(III)

Wren

for

M-1

(unstable)

(hexagonal

In Flg.9.

and

N-0,

for

are any

oriented

value

dlagonal

of

As

in

using

conflguratlons ulth

those

lattice).

the

the

- order

flrst

a six-element

matrix

g

have

the

deacrlbed

two

in

dipoles case,

hexagonal

same

both

of

cellular

1s

-

EC )-

5[ I-

ul

U)34 a-

21)-

11o-

10

20

40

30

50

t-u3

e-0

70

m

Y

Figure

10.

(expr433)) computer rhombic

of

Hot for

M=I

results cclhdar

lo

energy.

and

the @netion and N=O

(see array.

text).

e(y)

dcfming

UI

und

(f3l.W (111)).Continuous

Dashed

line:

results from

them array.

ml

0

Fig.7b1

conflguratlons

t

o-

the

141;

7

7a,

along

r;

dlrectlon.

- B2 + Ba - B4

1

00

(shorter

colnclde

by

dipoles

r<81°;

both

obtalned

the

Is unstable

(Fig-7al.

structures

HO0

where

conflguratlon

only

~0".

thelr

be

conflguratlon.

-;"

~2

line: the

can

are also

106 In

thls

case

sign.

As

along

the

(as

a

d

and

When

the

evaluated

The

by

L=80.

Then

al-n

7.

are

opposite

In

directed

longer

and

ei,=6

X2(7]

(33)

and

have

6,,++90°.

been

the

obtained

terms

6.

X1

and

Xz

function

by

are

This

I+' as given by expr.A3 IJ values of 6. XI and X2 can

(see

slmple

and

the

numerical

approaches.

array.

the

performlng

lnitlal

B3 are

no

cellular

numerically

results

In

the

11.

takes

accurate

an

to

different

In Fig.10

by

as

a)

XI(r)

of

approach

and

1

sin(6+90°)

approach,

means

curves

second

flrst

V

of

(19+90°)

corraspondlng

two

B1

by

-[

taklng the

elgenvectors

%

elgenvaluea

uslng

dashed

the

N41.

M-O.

co5

elgenvectors

computations,

case

Denoting

1

8

[ sin

a(7)

"dual"

consequence.

%”

two

the

diagonals.

cos

the

In

the

folloulng

value

of

be

Appendix).

obtained.

with

for

any

steps:

the

6.

take

by

expr.A3.

as

to

one

by

obtalned

the

flrat

approach;

calculate

b1

X

and

1

K

2

taking

m,J=P!+

and

~l,~=q

respectively;

CI

modlfy

the

value

Return

to

the

statlonary the

The

of

6

80

prevlol-;

for.

step first

the

say.

reduce until

the the

elght

in

varlatlons values

of

slgnlflcant

X1

X1

and

X2.

and

xa

are

dlglts.

Then

stop

calculation.

resulting

curves

lndlcated

are

wlth

continuous

lines

in

Flgs.10

and

11. A

comparison

approxlmatlon despite

based

Its

on

of

and

12b.

respectively

the

flrst 1s

dipole

klnd

7=90°.

wlth

those

When

7dO.O

the

arld continuous

"cellular

distributions

of

always

when

dashed

llnes

allows

array"

proves

accurate

that

the

predlctlons

slmpllclty.

Examples

sscond

the

between

6=3S01.

distribution

is

p%

and

Owing

stable

JL~ to

for

are

the

any

glven

signs value

In

of of

Figs.12a

K1 7.

and

X2.

while

the

unstable'.

6=90°

and

described then

(7=4S9,

for

the In

6-=60°.

two

portlnent

dipole

dlstributlons

colnclde

141. In

this

case.

the

two

dipole

dlstrlbutlons

are

107 600

500

\;\

K-

400

300

xm

200

x-

loo

, -

._

a I-

-1oc

h -

-2oc

I-

-3ot

)-

K /

0

I

I

10

20

1

I

30

I

I

40

50

I

60

I

70

I

60

90

Y Figure 11. Plot of the “energy” curves Kl(y) and K2(* for cases (III) and (f V). Continuous lines: computer results (see text). Dashed lines: results from the rhombic cellular array.

Figure 12. Second order &pore distributions for M=Z and N=O, with y= 45. “‘energy” level Kl); (b) 8y = 8 + 9CY= 1ZsCyr, “energy” level K2);).

(a) 9~ = 8 = 35’ @I,

108 equlvalent

to

those

In

Flgs.9b

and

9a

respectively.

use

of

the

apart

from

trivial

rotatlons.

(IV1

The

remaining

MPO.

0

=

can

case

N=f

B1 -

2

easily

once

Ba - B3

be

studled

the

the

and

+6

10

yield

by

making

for

the

rhombic

equlllbrium

equllfbrlum

8

B4

L

number

assigned. the

-

values

set

values eO,

of

following

9

6 for

and IJ the

of

SIO

N-O.

property:

the

angle

angle

N=l

are

been

;T have

6 for

H-1.

related

N=O

by

=;r

01

the

and

(34)

same

value

of

the

interaction

energy

IY IJ'

This

statement

structure 6

IO

of

into

can

be

(see



sum

A4

verlfled

makes

this

by

observing

Appendix). sum

that,

substituting

equivalent

to

sum

thanks the

A3,

the

to

value

evaluated

bol=

rI *lo_

for

1

Then

the

values

trlvlally resulting have

the

same

as

of

6

derlved

from

plot

shown

1s

same

doflnlng those

meanlngs

the

related

In Flg.13. a'3 in

elgenvectors to

case

where

the

F'lg.10.

The

+

[III).

and

using

continuous

curves

can yz eqn.(341.

for

and X1

dashed

and

X2

be The

lines are

the

For

the

in Flg.11.

Examples

of

sake

comparison.

of

dipole

Here

yc4!S0). same

energles

first

kind

dlstrlbutlons the

6r1C1°. as of

and

those

are

value the ln

of

glven r

Is

conflguratlons Figs.lZa

conflguratlon

1s

and

always

In the In 12b.

Flgs.14a

and

same

in

as

Flgs.14a

14 b. Flg.12

and

14b

respectively.

stable,

while

(1. e-. have

Again. the

second

the the IS

unstable. The

cases

r=90°

completely the

previous

and

analogous case.

;r=60° yield (apart

from

6r0°.

and

trlvlal

the

related

rotations)

conflguratlons to

those

obtained

are in

109

15

0

10

20

30

40

50

60

70

00

90

Y Figure 13. Plot of (expr.(34)) for M=O

the jknction Cl(yl) defining ~1 and 9 and N=I (case (IV)). Coruinuous line: rescclts from the computer rcsulcs (see text). Darhed line: rhombic cellular array.

0a

(W

Figure 14. Second order dipofe distributions for M=O und N = I, with y= 45‘. (a) 9~ = 8 = 1W (?I, “energy” level XI); (b) 8y = 8 + W = lCtU&?, “energy” lcvcl K2);).

110

DISCUSSIONAND CONCLUSION The

configurations

patterns for

Union

the

the

Jack

first

two

rhombic

(I)

the

- in

for

one

the

are

(W-1,

and

N=Ol

direction

and

[4]

elements

effective

of

the

method

The

short-range predicting

square

[41

for

above-shown

161.

when

the

tool

The

temporal

terms

indicate

the

diagonal

[Fig.

4a)

;y ranges

when

obtained

two

diagonals

of

stabillty

the by

(341.

that

the

(M=l.

For

In

by

the

angle

r

in

r=90°

all

the

dipole

previous

the

81°.

equllibrium

plotted

in

only

dlrectlon

the

the

patterns,

occurs

r

33*

along

than

such

of

and

N=l)

the

s(a)

orthogonal

value 90°

lower

stability

discussed

any

between

patterns.

functions

cases.

for

r is

diagonals_

the

both

4bl

patterns

perlodlcal

while

Ida).

along

provided

of

In

played

a

given

a

by

energy

In

the

the

capability basis

square

evolution results

approach,

the

Flgs.10 in

(8+90°1

the

6

always

section.

further

characterization

of

the

model long-range

(closest

should

ln

to

be

is

beyond

be

the

made

viewed

up

of

the

as

a

Once can

this

easily

be

those

"energy" four

scope

of

simple

and

and

the

information

is

characteristics

patterns. energy

lie

with

obtalned

by

the

Appendlx. for for

predicting formulatlng this

lnitially

reported

31,

evaluation

interaction

coincide

differences

geometrical

dipole

lattice.

of

patterns

assuming

which the

equilibrium of

only

accurate

lnvestigatlng

values

The

calculated

an array

reliable

of

lattlco.

are

However.

simple

case

7b).

obtained

N=ll.

and

role

in

accurate

approach

equations

oriented

perlodlcal

those

Property

cellular

stabilitles

the

(M=O.

for

only.

proposed

known.

(Fig.

second-order

are

stated,

which

values.

periodical

patterns.

previously in

simplest

patterns

shorter

(Fig.

longer

12a

the

out

equllibrium

the

only

instability.

points

the

equilibrium

"energy'

second-order

6+90°

(Figs.

the

diagonal

two

13

the

are

periodical

of

no

dipoles

of

of

diagonal

6

implles

signs

longer

kind

directions

found

Jhe

remaining

directions

represent

rhombic-lattice

the

flgures.

the

shorter

As

the

direction

- along

For

with

first-order

the

Flgs.4.7.12.14

approximation.

mesh.

for

(II)

In

consistent

the

In

shown

this

approximate

dynamic

at

has

oriented

papur

nelghboura)

Interactions.

situations

possibility

randomly

in

equilibrium

been

dipoles

corroborate seems

least

in

to the

the

this

equations_

used in

makes

a

to

study

VISCOUS

evidence

be

adequate

static

limit.

In the

medium that

enough

a to

111 APPENDIX

Let

us first

consider

vector

can

the generic

be

wrltten

p+q 1

1J

-

COST

I<:

glven

by expr.(3).

The

posltlonal

be

speclallzed

as

1

p=k-1

wlth

--d

%I

term

_q sin

T

q-1-5

then

11,~ -

~~1~ - d2

(T~~-&,.E,~>

+ 2pq

P" + q2

COST 1

P cos flIJ + 4 cos

= - dP

1 The

general

expr.(lO)

for

the

L-order

(' -

e,J,)

rhombic

set

9

can

now

1J for

1)

the cases

First

For

of

interest..

- order

such

conflguratlons

configurations.

q,,qq ,, for any k.1.

Then

wlth

Al

2

= f

fP4

and

-_L p -_L q

1 f= w

3/a Pa + q2 (

21

Second

al

1 -3

- order

M-1,

+ 2pq

cosy )

[

p cos u IJ + q =os P" +q=

i

conflguratlons

N-l

In this

case.

deflnltlona

I301

yleld

PI=P,--1

and

+ 2pq

CT - *,,q cosy

112

q,

(- l)p+q m,,

=

then

w

=

1J

A2

with

A2

p

r

f

1y+q

(-

f*

_L p __L q

b)

M=l. In

N=O thls

case,

then

w

P1

In

and

I

=

%I

(- l)p ml,

A3

wlth

=

r

4azzd3 i

M=O.

P3 =

A3

= IJ

Cl

P21

= -I.

f

q

(-

l)p fw

(-

l)q

-I_ p -I_

N=l thla

then

case,

W

1J

P1 -

I. P3 = -

=

and

I

%1

A4

wlth

A4

=

(- '1"

=

F

-L

19,J

r

fm

p -_L 4

ACKNOWLEDGMENT

Work

supported

Eleitronica

by

ESPRIT

II

-

BRA

3200

OLDS.

CNR

-

NADESS

and

FIRST

40%

Piolecolare.

REFERENCES

1

2

H.

Mohwald,

H.

Hauser,

in Holecular

I.

Pascher.

Electronic

R.H.

Devices.

Pearson.

(1988),

S.

507.

Sundell.

in

Bfochimica

et

113 Biophysics

3

A.

s_

M.

A. the

M.

S.

(1981).

21.

Clncotti. the

A.

De

V Eurosensors

Parodl. Journal

Clncottl.

A.

Gloria. Cord..

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1991.

Liqufds.

Chlabrera.

in

H-

Dl

ZittX.

tin

press)-

"HodclIlng

Chlabrera.

of

E.

of

dipole

1991,

Journal

(In

of

Parodi.

S.

monolayers

as

press>.

Elccfrostatfca.

26

47.

Chlabrsra, 12th

of

arrays".

Parodi.


6

S_

Proc.

Cincotti.

cellular

5

650

Chlabrera.

Rldella.

4

Acta.

Ann.

S. Int-

Clncottl. Conf.

A. of

the

De

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IEEE

EBBS.

E. 12

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Zlttl.

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Proc,

of