Symmetry, structural phase transitions and phase diagram of Langmuir monolayers

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Symmetry, structural phase transitions and phase diagram of Langmuir monolayers

V. Kaganer, V. Indenbom

To cite this version:

V. Kaganer, V. Indenbom. Symmetry, structural phase transitions and phase diagram of Langmuir monolayers. Journal de Physique II, EDP Sciences, 1993, 3 (6), pp.813-827. �10.1051/jp2:1993169�.

�jpa-00247873�

Classification Physics Abstracts

68.90 61.50E 61.65

Symmetry, structural phase transitions and phase diagram of

Langmuir monolayers

V. M.

Kaganer

^{and V. L. Indenbom}

Institute of

Crystallographyi

Russian Academy of Sciences, Leninskii pr.59, 117333 Moscow, Russia

(Received

16 December 1992, accepted 3 March

1993)

Abstract. The Landau theory of phase transitions is developed for transitions between condensed phases of

Langmuir

monolayers. The phase diagram is explained by the coupling

between the order parameters describing collective tilt of the molecules and ordering of their backbone

(zigzag)

planes. If the latter ordering is considered _as occurring between hexatic

phasesi then the predictions do not agree with X-ray data _on the structure of phases of either

fatty acid

or long-chain alcohol monolayers

on the water surface. However this transition _can be

explained _as translational ordering with doubling of the unit cell realized either by herringbone

ordering

_or by alternating orientations of the heads of the molecules in adjacent _rows.

1. Introduction.

Monolayers

of

amphiphilic

molecules _on the water surface

(Langmuir monolayers)

_were first studied

by

_means of surface _{pressure area} isotherm measurements at the turn of the century

[1-6].

The isotherms _possess features which _were

interpreted

_as

phase

transitions. In subse- quent

worki Stenhagen

_{[71 8] and} later

Lundquist

_{[9, 10]}

reported phase diagrams containing

_as

many as seven distinct

phases

for

monolayers

of various

amphiphilic compounds.

Recent

X-ray

diffraction

experiments [11-23]

^have shown that all these transitions do

really correspond

to

changes

of the

monolayer

structure.

The substances whose

monolayer

structure has been studied in _most detail _are the

fatty

acids. The

phase diagrams

for molecules with different chain

lengths

do not differ

qualitatively

and _can be

brought

into

correspondence by appropriately shifting

the temperature axis

[24].

Figure

I represents the

phase diagram

of behenic acid

(docosanoici C20) [18].

^{The lines of the} transitions have been found from the surface _{pressure area} isotherm measurements while

the structure of the

phases

has been determined

by X-ray

diffraction

experiments

at the

points

marked

by

_crosses. The orders of the transitions

were not determined in [18]. ^{Those shown in}

figure

^I are the _ones

reported by Lundquist

_{[9, 10]} for the

analogous phases

of the

ethyl

and

acetate esters and _are consistent with the

X-ray

data.

~~ ~_,__~

~"~

p__Q

I

l~~ I~IS

, ,

- i i Ii i,

fi

i /1 ^{' '}

@ o

^'

o

_~s

~

~r"'~

'

'~

~

~----~

x x

ij~--~ji

fi

- x

,

~

6 ^~2

oJ ^"

2 I

m m

~~

~c~~$~~~~

^~~~~

i ,

oJ I ~~j ~

O 2

~n x ,

I

x

~$~"~~fl

~ LQ

10 x 20 30

O x x

Temperature [°C]

Fig-I-

The phase diagram of behenic acid (C20) _on the water surface [18]. The structure of the

phases has been deduced from the X-ray measurements at the points marked by _crosses. Solid lines denote the first order transitions and dashed lines indicate the second order _ones.

The structural features

important

for further considerations

are the

following.

Four of the

phases, LS,

_S~ L2 and

L[

, are hexatic

phases

with

quasi-long-range

bond orientational order and

short-range

translational order. In the

phases

LS and

S,

the molecules _are untilted _on average. The unit cell of the LS

phase

_possesses

hexagonal

symmetry

indicating

that the molecules rotate

freely

about their

long

_axes here _{we use} the _term "unit cell" for hexatic

phases

in the _{same sense as} in the above-cited

X-ray

structure

analyses

of the

phases,

to describe the local environment of each molecule. At the first order LS S transition the unit cell defornm to orthorhombic and its _area decreases. This has been

explained by

the

ordering

of the backbone

(zigzag) planes

of the

molecules~ although

there _{was no} direct evidence in the

X-ray

diffraction data for

herringbone ordering

of the backbone

planes.

This

ordering

has been marked in

figure

I

only by analogy

with the structure of 3D

crystals

of

aliphatic

chain derivatives [25].

Compared

to the

hexagonal

unit cell of the LS

phase,

in which the molecules have _no collective tilt the

phases

L2 and

L[ display

molecular tilt towards the _nearest

neighbor

(NN)

and the next-nearest

neighbor (NNN), respectively.

The

shape

oi the twc-dimensional unit cell is also of interest ior the present

investigation.

An

X-ray

diffraction

study

of the LS L2 transition in

monolayers

oi arachidic acid

(C22)

[19] ^{has revealed that the unit cell stretches in the direction oi tilt and shrinks in the} _perpen-

dicular direction.

During

the LS S transition the unit cell deiorms in the _{same way} [18]~

i-e-, the distance to two oi the

neighbors

increases whereas the distance to the

remaining

^iour

decreases. As the suriace _pressure decreases below the S

L[

transition~ the deiormation

monotonically

decreases. The

low-temperature high-pressure phase

CS is _a 2D

crystal

oi _un- tilted molecules with

quasi-long-range

translational order. The

low-temperature low-pressure phase L(

observed

by

Lin et al. [17] ^and not

reported by

Kenn et al. [18] ^{is also} a 2D

crystal

but with its molecules tilted to the nearest

neighbor.

The

high-temperature phase

_{Li [8],} also

absent in

figure

I, is _a two-dimensional

liquid.

A recent

X-ray

diffraction

study

oi heneicosanol

monolayers

_on the water surface [23] ^{has revealed the} same sequence of

high-pressure phases

as in

fatty

acids.

However,

in contrast to the

acids,

it _was iound

that,

at lower _pressure, the molecules tilt

only

towards their next-nearest

neighbor.

No

swivelling

transition

analogous

to the L2

L[

transition _was detected.

The aim of the present work is _to

analyze

the

phase

transitions in

Langmuir monolayers

with the aid of the Landau

theory [26].

The

phase

transitions

involving

molecular tilt and the

ordering

of the backbone

planes

_are

investigated,

_as is the

coupling

between the two. The

transitions between

LS, S,

L2 and

L[ phases

_are thus described.

The first step ^{in the}

application

of the Landau

theory

is _to determine the symmetry _group ^of the _more

symmetrical phase.

The

point

symmetry group of LS

phase

^is

obviously

the _group C6v

generated by

reflections in two

orthogonal planes

and sixfold symmetry about the axis _common to them. To describe the translational order in this

phase,

the choice exists between the _group of

discrete translations of _a 2D

hexagonal

lattice and the group of continuous translations of _a 2D

liquid.

The

phases

under consideration _possess

short-range

translational order with correlation

lengths ranging

from 10 to 60 intermolecular distances [18]. ^These are

sufficiently large

to

justify describing

^{the order} as

crystalline,

at least _as far _as the local

packing

of the molecules is concerned. We consider the

possibility

oi both continuous and discrete translational order oi the

phases.

The main results of the present work _are the

following.

The

phase diagram

is

explained by

the

coupling

between

just

two order parameters. ^{One of them describes the second order}

tilting phase

transitions

(LS

L2 and S

L[

whereas the

other, responsible

for the first order transition LS

S,

involves the

ordering

oi the backbone

planes.

A

quite general

consequence of the

coupling

between the two is the shift of the line of S

L[

transition _to

higher

^suriace

pressures with respect to the line of LS L2

transition,

both lines

being approximately parallel.

There _are two ways for the backbone

planes

to

order,

which _are allowed

by

the Landau

theory,

consistent with the

experimentally

observed unit cell deformations and do not

give

rise to any incommensurate

density

_waves.

Firstly,

the backbone

(zigzag) planes

^{of the} hexatic

phase

_may

develop

nematic-like

ordering which,

within the framework of the Landau

theory,

is

equivalent

to

equitranslational ordering

in

crystalline phases.

The number of molecules per unit cell is

unchanged

and

equal

to

unity.

This

ordering

is described

by representations

of the

point

symmetry _group. However the

predictions

obtained for this

ordering

do _{not agree} with the structural features of the

phases (directions

of tilt and unit cell

deformation)

observed for

fatty

acid and

long-chain

alcohol

monolayers

_on the water surface. The second _way~ which

gives good agreement

with the structural

data,

is translational

ordering by doubling

of the unit cell.

This _occurs most

probably

_as

herringbone

alternation of the

zigzag planes.

2.

Equitranslational ordering.

In this section _we consider

simultaneously

the transitions between hexatic

phases

and the transitions between

crystalline phases

which do not

change

the number of the molecules in the unit cell. These transitions involve

changes

of the orientational order in the

monolayer,

the translational order

being

inessential. Let _us describe first the

tilting phase

transition.

Introducing

^the unit vector _n

along

the _mean direction of the

long

axes of the

molecules,

_one

can consider its components _{n~, ny} in the

plane

of the

monolayer

as two-dimensional order parameter

describing

the collective tilt of the molecules. It is convenient to convert to

polar

coordinates _q,

fl:

n~ = qcos

fl,

_ny

= q sin

fl (1)

Here

fl

is the azimuthal

angle

oi the collective tilt and q = sin_9~ where 9 is the tilt

angle.

One has q = 0 in the

phase

oi untilted molecules and

vi

0 ior collective tilt.

The Landau

theory

is based _on the

expansion

oi the iree _energy

4l(p, T) (here

_p is the suriace pressure and T is the

temperature)

_over the _powers oi the order parameters, ^each term

being

invariant _over the symmetry group oi the

more

symmetrical phase.

Neither rotation

through

the

angle jr/3 according

to

fl

-

fl

+

jr/3,

_nor reflections in the symmetry

planes changing

the

sign

oi either _{n~ or} ny,

changes

the iree energy oi the

monolayer.

It iollows that the

expansion

oi the iree _{energy over} the _powers oi q invariant _over these rotations and reflections is

4lq =

Aq~

+

Bq~ Dq~

_cos

6fl (2)

When B _> 0,

equation (2) ^predicts

a second order

phase

transition at A

= 0: ior A > 0~

the minimum oi the free energy is achieved at q = 0; _as A

changes sign,

the minimum shiits

continuously

to q~ =

-A/28.

The last term in

equation

_(2)~ small in

comparison

with the other terms~ is the lowest order _term

depending

_on the

angle fl.

For D _>

0,

mininfization oi 4lq over fl

gives fl

=

jrm/3 (m

is

integer),

_I-e-, the tilt

occurs in the direction to the _nearest

neighbor.

Ii D _< 0, the minimum is achieved at

fl

₌

jr/6

+

jrm/3,

I.e.~ ior tilt towards the next-nearest

neighbor. Thus~ equation (2)

with B _> 0 and D _> 0 describes the second order

tilting phase

transition LS L2 in

iatty

acid

monolayers (Fig. I).

These

signs

oi the coefficients

are used in further

analysis.

The term

Din~

is oi the _same order _as the last _term in

equation (2)

^{and is oirfitted since it does not}

qualitatively

influence the

phase

transition.

The

ordering

oi the backbone

planes

oi the molecules _can be described

as the _appearance oi

strain ei~

(I,

j = 1,

2)

in the twc-dimensional lattice. The

hydrostatic

_{pressure e~~ + eyy} does not describe the

ordering transition,

and the

remaining

two components _{e~~ eyy} and 2e~y oi the strain tensor

gives

_a twc-dimensional order parameter ior this transition. For hexatic

phases~

_{one can} describe the

ordering

oi the backbone

planes

_as 2D nematic-like

ordering

with

the nematic order parameter t~ a

symmetrical

traceless 2D tensor.

Denoting

the director

by

N

(the

vector in the

plane

oi the

monolayer parallel

to the backbone

plane

orientation; N and

-N _are

equivalent)

_one has QI~ =

NiNj )bij

_(i~

j

_{= l~}

2).

ⁱⁿ the iramework oi the present paper, tensors t~ ^and can be considered

equivalent.

For the sake oi

definiteness,

_we iollow the

precedent

oi the

experimental

_papers and express our results in ternm of

crystalline

order.

Let _us introduce the

polar

coordinates (~cx defined

by

e~~ egg = ( _{cos cx} 2e~y = ( sin _cx

(3)

Then ( ₌ 0

corresponds

to the

hexagonal

symmetry oi the LS

phase

whereas

( #

0

gives phases

with distorted unit cells. On rotation oi the

monolayer through

the

angle jr/3

the

tensor components

(3)

^transiorms

according

to _{cx -} o +

2jr/3.

Each term in the

expansion

of the free energy ^must be invariant with respect to this transformation. In

increasing

_powers

off:

4lj

=

F(~ _G(~

cos 3cx ₊

H(~ ₍₄₎

The presence oi the third order term in equation

(4)

leads to _a first order

phase

transition oi the order parameter

(

_[26] the iree energy

(4)

has two local

minima~

( ₌ ^{0 and} (

#

_0~ ^and _as F

changes,

the transition _occurs when the minimum at

( #

0 becomes

deeper. Iii #

_0~ the iree energy

(4) depends

_on the

angle

_a. Its rr~inimum _occurs at o = 0 ior G _> 0 and at _cx

= jr ior

G _< 0

(equivalent angles differing

from the stated _ones

by 2jrm/3

_are not mentioned hereafter ior the sake oi

simplicity).

More

accurately, taking

into account the

a-dependent

term of next

highest

order G~(~ cos6cx in the iree energy

expansion~

there is _a first order transition irom

cx = 0 to cx = jr when G~ < 0 and G

changes

irom

positive

to

negative

values. Both phases

..

--

~~

$~~ ^"'

..

a b

"c

44~ _33~ II,

"

d

~~

e

,,

Fig.2.

The top view of the hexagonal unit cell

(a),

the ordering of the backbone planes of the molecules parallel to each other

(b,c),

the herringbone

ordering (d)

and opposite directions ofordering

of the transverse vectors in adjacent _rows (e,

f).

possess orthorhombic symmetry, and the unit cell either stretches _or

shrinks, respectively,

in the direction of the nearest

neighbor.

On the other

hand,

if G' _>

0,

^these

phases

_are linked

by

two second order

phase

transitions at G

= +

4G~(~.

In the intermediate

phase

_cx varies

continuously,

and the unit cell has monoclinic symmetry.

Let _{us assume} that the transition _over

( corresponds

to the LS S transition in

fatty

acid

monolayers

_on the _water surface. To agree with the direction of the deformation of the unit

cell,

which

corresponds

to _{cx =}

0,

_we must take G _> 0. At

sufficiently large F, equation (4)

has its minimum at

(

= 0. As F decreases to Fo _"

G~/4H,

the minimum at

to

"

G/2H

becomes

deeper, eventually causing

_a first order transition from

(

= 0 to

to

The

microscopic picture

of the

phases

is shown in

figure

2. In the

hexagonal phase ((

₌

₀₎

the molecules rotate about their

long

_axes

freely

with respect to the

neighbors (Fig. 2a).

The

ordering

of the backbone

planes

of the molecules deforms the unit

cell,

the

phases

_{o =} 0 and _cx

= 1r

arising during

the

ordering along

^different reflection symmetry

planes (Figs. 2b,c).

In terms of hexatic

phases, figures 2b,c

_can be treated _as nematic

ordering

with _two

possible

orientations of the director

with respect to the bond direction.

The line A

= 0 of the

tilting phase

transition and the line F

= Fo of the backbone

plane ordering

transition _{can cross} in the

phase diagram.

The interaction between these transitions is of

special

interest. The

coupling

terms in the free _energy

expansion

invariant _over the

transformation _cx

- o +

2jr/3, fl

_-

fl

+

jr/3

mentioned above _are

4ljq

₌

-J(q~

_cos(cx

_2fl)

₊

K(~q~ cos(20

₊

_2fl)

₊

L(q~

_{cos(cx +}

4fl) (5)

The

products

of invariants

f~q~

and

fq~

_cos(cx

2fl)

omitted in

(5)

are not essential. Mini- mization of the total free energy

* *o ₌

*n

+

*f

_{+ Win 16)}

leads _to the

phase diagram

shown in

figure

3. The coefficients A and F _are considered _as variables

depending

_on temperature and surface _pressure whereas other coefficients in

equations (2-5)

_are taken _as constants within the

region

of the

phase diagram

considered.

Fortunately,

the transitions LS -S and LS L2 at

figure

^I _occur at

approximately

constant temperature ^and surface _pressure,

respectively~

which

simplifies

the

comparison

of _our

phase diagram (Fig. ₃₎

with the observed _one. Variables A and F

can be referred to _as surface

pressure-like

and

temperature-like, respectively.

A

a

11

~

i

~

^{d e}

iii /v

F~ ^F

Fig.3.

Theoretical phase diagram. Phase I possess the hexagonal unit cell. In phase II the ordering of the backbone planes of the molecules _occurs accompanied with spontaneous deformation of the unit cell. Phase IV _possess the collective tilt of the molecules accompanied with induced deformation of the unit cell. In phase III molecules tilt in spontaneously deformed unit cell. Solid lines denote the first order phase transitions and dashed lines indicate the second order _ones.

The

phase

I is the most

symmetrical phase possessing

_a

hexagonal

lattice of untilted molecules

((

₌ 0,_{q =}

0).

^{At the line ab there is} a first order transition

leading

to backbone

plane

_or-

dering,

_as described above. In

phase

II the spontaneous strain

(

is

equal

to

(o

_near the line ab and increases

monotonically

_as F

decreasesj

_q

= 0. At the line de the second order

tilting phase

transition takes

place.

In

phase

IV the tilt _causes induced strain

(

_~- q~ ^which renor-

malizes the coefficients of

equation (2).

The lowest-order

angle-dependent

term of

equation (5) gives

_{cx =}

2fl

ior J _> 0 and _cx

=

2fl

_{+ jr} ior J _< 0. Then the strain

(

is determined

by

the

terms

F(~-

J

(q~

which

give

the minimum at

(

_{= (j} J

/2F)q~. _Inserting

the strain

(

and the

angle

cx in the iree energy

expansion (6)

reduces it to

equation (2)

with renormalized coefficients

~~ ~~~ ~_~~ _~~

~~ ~

4F ^'

~~ ~

8F3 ^~

~F2

^~ _2F

instead oi B and D. To describe the second order

tilting phase

transition in

iatty

acid _monc-

layers,

_we take B~ _> 0 and D~ _> 0. Then

fl

= 0 and ior J _> 0 _one has

a = 0, I.e.~ ^{the unit} cell stretches in the direction oi

tilt~

_as has been observed

experimentally jig].

Hence _we take J _> 0 in iurther

analysis.

In contrast to the spontaneous strain

(o

in

phase II, phase

IV shows

an induced strain

proportional

to q~.

Taking

A ₌

a(p pc) (here

_pc is the suriace pressure ^at the

phase

transition

line)

_one has _q

+~ p and

(

_{+~ (pc}

p).

On the line bc the

tilting phase

transition _occurs in the _presence oi spontaneous deiormation.

The

q-dependent

low-order terms

give

^the contribution

(A J()q~

₊

Bq~

to the iree energy, where ( Ge (o ^{is the minimum oi}

(4).

Hence the

tilting

transition takes

place

at A

=

J(.

I-e-, it shiits to

higher

_pressures ^with respect to the line de. The

angles

_cx and fl are determined

by

the

signs

oi G and J~ which _are

already

fixed

by

the directions oi the deiormation in

phases

II and IV: ior

iatty

acid

monolayers

these _are G _> 0 and J _> 0. This

means that the minimum

oi the iree _energy

(6)

ⁱⁿ

phase

III _occurs ior _cx

= 0 and

fl

₌ 0.

Uniortunately

the latter

angle disagrees

with the observed

tilt,

which in the L2

phase

oi the

iatty

acids

(Fig. I)

is towards the

next-nearest

neighbor (fl

₌

jr/2).

However

expression (6)

ior the iree _energy

implies

_a strong

coupling

between the tilt direction and the strain. It _may be concluded that the

ordering

described above is

unlikely

to be _a correct

description

ior the

iatty

acids.

The whole of the

experimental

data _on

fatty

acids is

explained

in the _next section

by

another free _energy

expansion

with another order parameter.

However,

since the first order transition LS S

displays

_a finite

change

of

strain,

_we cannot

entirely

exclude

equitranslational ordering

irom consideration. For

example,

the second term of

equation (5)

_may be

comparable

with the first _one at finite

f.

However _we do not consider this

possibility

further.

The

disagreement

found above _can

readily

be

expressed

in terms

appropriate

for hexatic

phases.

In

phase IV,

both the tilt and induced nematic order _occur

along

the bond

direction, implying

that J _> 0. In

phase II,

the spontaneous nematic order is also

aligned

with the bond

direction, leading

to G > 0. Hence the

phase adjacent

to II is

expected

to show tilt in the

same direction.

However,

the observed tilt takes

place

normal to the bonds.

Since

equitranslational ordering

_may take

place

for

monolayers

of another

amphiphiles,

we

describe

briefly

further features of the

phase diagram following

from the free _energy

(6).

^{If the}

coefficients

D,

G and J _are

positive,

_as has been considered

above,

the

phases

III and IV do not differ in their symmetry. ^{As the tilt}

angle

increases the difference between

spontaneous

^and induced deformation

disappears

and the line df of the first order transition

(Fig. 3)

^terminates

at the critical

point.

If

phases

III and IV _possess different tilt directions

owing

to _a

negative

coefficient

D, G,

_or

J,

the line df cannot terminate. As A decreases and _q increases in the

phase III,

_so that the

equality 9DGf~q~

=

J(Gf~ Dq~)

is

satisfied,

_a second order transition

occurs to _a

phase

with intermediate value of _cx

giving

_a ^monoclinic unit cell. The line of this

transition _crosses the line bdf.

In the Landau

theory

_[26],

phase

transitions _are classified

by

the irreducible

representations

of

symmetry

_groups. Let _us describe the transitions considered above from this

standpoint.

Equitranslational ordering

involves the

representations

of the

point

symmetry _group of the

more

symmetrical phase.

The

point

_group C6v

has,

besides the

identity representation,

three one-dimensional

representations

and _two two-dimensional _ones

([27], Chap. XII). Determining

the symmetry ^elements

surviving

at a transition

by

their characters

[28],

^{it is}

easily

shown that the

representation

A2 ^{describes the transition} to the _group

C6,

whereas the

representations

Bi ^and 82 ^lead to the _group C3v

(the

notation is that of

[27]).

The former transition _can be caused

by

the _appearance of molecular

chirality

and the latter _ones

by

the

ordering

of the molecules with three-fold _axes.

They

_are not relevant to known

Langmuir monolayers

and _are not considered in the present paper.

Proceeding

to twc-dimensional

representations,

it _can be

easily

shown in the _{same way} that the

representation

Ei ^leads to the _group

Cs

whereas the

representation

E2 describes the transition _to the _group

C2v. Expanding

the vector

representation

V and the

representation

[V~] ^{of the}

symmetrical

second-rank tensor _over the irreducible

representations,

_{one can} find the

physical interpretation

of the order parameters [28]. ^The

representation

Ei ^transforms

z- and y-components ^of a vector. Then the

in-plane

_{components n~, ny}

(I)

of the vector

n directed

along

the

long

_axes of the molecules

gives

_an

appropriate

order parameter. ^The

representation

E2 transforms the combinations _a~~ ayy and

2a~y

^{of the} components ^of a

symmetrical

second-rank tensor I. The components ^ofeither the strain tensor _or 2D-nematic order parameter tensor t~ _can be used _as the order parameter,

depending

_on the _{presence or} the absence of translational order. The number of the invariants of _any power over

f

and _q in the free energy

expansions (2),(4),

and

(5)

has been checked with the aid of the characters of the symmetry elements. It is

simpler

to derive the invariants

by

direct

application

of the

symmetry ^than

using

the normal

procedures

of _group

representations theory. However,

the latter

procedure

is the

only

_way to determine the invariant _terms in the free _energy

expansion during

translational

ordering

of the

monolayer

considered in the next section.

JOURNAL DE PHYS>0uE ii -T ~, N'6, JUNE 1991 ~2

3. Translational

ordering.

Let _us consider

ordering accompanied by

_an increase in the number oi molecules _per unit cell. These transitions _are described

by representations

oi the _space syrnrnetry _group with

non-zero wave vectors k [26]. ^Wave vectors at

general positions

in the Brillouin _zone

give

rise to

density

_waves incommensurate with the

spacing.

However there _{are a} small number oi k-vectors at

symmetrical positions

in the Brillouin _zone, chosen

using

the Liishitz condition, which lead _to

periodic

structures. The

analysis

oi the _{space group}

representations satisiying

the Lifshitz condition has been

performed recently by Loginov

et al. [29] ^for

phase

transitions between ordered

phases

oi smectic

liquid crystals. Compared

to the _group

C(~

relevant ior the present _paper, the _space group

D(~

^considered

by Loginov

et al. contains _an additional

generator,

namely,

reflection in the

plane

oi the

layer.

The results ior

in-plane ordering

_can

be

directly applied

to

monolayers.

The Brillouin _zone ior the

hexagonal

unit cell is also _a

hexagon.

There _are

only

two _wave vectors which

satsiiy

the Liishitz condition: these

join

the center of the

hexagon

either to its _{corner or} to the center of its side. The first of these

gives

rise to transitions to

phases possessing hexagonal

symmetry. However the LS S transition in

Langmuir monolayers

characterized

by

_an orthorhombic deformation of the unit cell cannot be

explained

in this _way.

The second of these _wave vectors leads to

eight representations

Ti T8 of the _space group

D(~

^which

degenerate

in

pairs

when

proceeding

to the group

C(~.

The transition via the

representations Ti

₌ T4

preserving

^the

hexagonal

symmetry ^{of the unit cell is} not of interest

here for the _reasons mentioned above. The

representations

T2

"

T3,T5

_# T8 and T6 _" T7

give

rise to iree energy

expansions

which coincide _{up to} the _terms of the sixth order _over the three-dimensional order parameter ~Jj(I = 1,

2, 3)

[29]

4~~ "

~i'~

₊

_~li~~

₊

~2(i'~

+

i'~

+

i'()

₊

~li'~

₊

~2i'~(i'~

+

i'(

₊

i'()

₊

~3§~)i'(i'(i (~)

where _~J~

= ~J]

+~J] +~J].

When _~J~ is fixed the minimum oi

(7)

with respect to the components

~Ji

depends

_on the

sign

oi G2. For G2 > 0 _one has _~Ji

# §~2 # §~3. This _{case preserves} the

hexagonal

unit cell and _can be excluded ior the _{same reasons as} above. For G2 _< 0, the minimum _occurs at ~Ji # §~j §~2 # §~3 " 0. This _case is oi

primary

interest ior the

purposes oi the present paper. The symmetry groups oi the ordered

phases,

the iorbidden

reflections and the _structures oi the

phases

ior each oi the

representations

_are iound in

[29].

Each

phase

contains two molecules _per unit cell and consists oi

alternating

_rows oi

uniiormly

ordered molecules

(Figs. 2d-I).

The

possibilities

_are

herringbone

order

(Figs. 2d)

and

opposite

transverse vectors in

adjacent

_rows

(Figs. 2e,I).

All these types oi

ordering give

rise to the _same iree _energy

expansion (7)

and thus in the iramework oi the present

theory

cannot be

distinguished

irom the

thermodynamic

behavior oi the system. ^The transverse vectors for the molecules in _a

Langmuir monolayer

_are defined to be the

projection

of the heads of the molecules onto the

plane

of the

monolayer.

Their

ordering

induces molecular tilt~ also in

opposite

directions in

adjacent

_rows~ since it is allo~i,ed

by

the

phase

symmetry shown in

figures

2e,f. The tilt _can be small

enough

to be detected in the

X-ray

diffraction ~'rod scans"

(normal

to the

monolayer plane)

_as

out-of-plane

maxima in the

intensity

distributions. The

X-ray

diffraction studies _are not accurate

enough

to

distinguish

the

phases

shown in

figures

2d-f

by

their forbidden reflections. For the sake of

definiteness~

_we

refer to the

ordering

_as

herringbone ordering,

which is

commonly

observed in 3D

packings

of

aliphatic

chain derivatives [25] ^{and smectic}

liquid crystals

_[30] and _seems the most

plausible

choice for the S

phase

of

Langmuir monolayers.

However all results _are

equally

valid ior the two other types ^oi

ordering. Equation (7)

reduces for G2 _< 0 to

4~~ = F~7~

G~4

+

H~6 ₍₈₎

with G

=

-(Gi

+

G2)

and H

=

Hi

+ H2 + H3. To obtain _a first order transition LS S in

Langmuir monolayers,

_we take the coefficients

F, G,

H to be

positive.

Ii G

changes sign,

^the

transition is second order.

The

coupling

between the order parameter _~J; and strain _was also derived in

[29].

^It was

shown that the term

(~i'~ i'( i'()(~zz

_{Egg +}

~"(i'( _{i'()£zY (~)}

is invariant with respect to the symmetry _group

C(~.

With the aim of

determining

the

coupling

between the

herringbone

order and

tilt,

_we

replace

the strain terms in

equation (9) by

^the

terms

q) q]

and _qin2 which _are transformed under the action of the symmetry ^{elements in}

the _{same way as} the

corresponding

^strains.

Taking

_{~Ji " §~} and ~J2 " §~3 "

0,

_one has the

coupling

term

J~J~q~

cm

2fl.

The

choice,

_{e-g- ~J2}

" §~, §~1 ^" §~3 "

0, gives

the

equivalent

term

J~J~q~ cos(2fl

+

2jr/3). Adding

the

product

of the invariants _~J~ and

q~,

one has

finally

4l,

₌ l~J~q~ +

J~J~q~

_cos

2fl (10)

Thus,

_we have the

unique

free energy

expansion

for the translational

ordering

of the _monc-

layer giving

rise to the deformation of the unit cell. Let _us consider the

phase diagram following

from the

coupling

of this

ordering

with the collective tilt of the molecules. The free _energy

expansion

is

16 4lo _" 4lq +

lb~

+ lb~q

(11)

with the terms introduced

by equations (2), (8),

and

(10).

The minimization of the free _energy

over two order parameters, ^{and the}

resulting phase diagram,

_are

quite

similar to those of

figure

3 derived in the

previous

section.

However,

the appearance of the order parameter _~J

describing

the

monolayer

_on the

microscopic level,

instead of the

macroscopic

_one

(strain), changes

the behavior of the strain and the tilt directions in the

phases

under consideration.

Let _us

again

consider A and F _as parameters

depending

_on the surface _pressure and the temperature, ^{whereas all other} coefficients

B,D,G,H,I,J

are taken _constants. The most

symmetrical phase

I

(Fig. 3)

possesses a

hexagonal

lattice of untilted molecules: _~J

=

0,

_{q =} 0.

The free energy

(8)

has three local

minima,

_{at ~J =} 0 and + _~J(~J

# 0).

The

phases differing

in the

sign

of _~J do not differ

physically,

_so that _we consider further

only

_{~J >} 0. As F decreases to the value Fo _"

G~/4H,

the first order transition I-II _occurs from _~J

= 0 to ~J = ~Jo with

~J] =

G/2H.

The

phase

II _{possesses ~J}

# 0,

which increases

monotonically

_as F

decreases,

from the value _~Jo at the line ab~ and q = 0.

At the transition

I-IV,

_~J remains _zero. » second-order

tilting phase

transition takes

place

at A ₌

0,

_as described in

previous

section.

Taking

D _>

0,

the free _energy minimum _occurs at

fl

₌ 0, in accordance with the

experimental

observations for

fatty

acids

(Fig. I).

^The

transition II-III is the

tilting phase

transition

occurring

at _~J

#

0. The tilt direction in

phase

III is determined

by

the lowest-order

p-dependent

term in the free energy

expansion,

which is

now

J~J~q~ cos2fl. Taking

J _>

0,

this term is minimum at

fl

₌

jr/2,

also in accordance with observations.

Expanding

the free _energy

(8)

for F close to Fo and _~J

slightly differing

from ~Jo,

one can represent the free energy

(11)

as

~b *o ₌

An~

+

Bn~

+

(

lid

iao)~ +

IF Fo)ia~ lJ I)ia~n~. l12)

Here the term

Dq6

_cos

6fl

is small in

comparison

with the terms oi

equation (12)

and is omitted.

The minimum of

(12)

over ~J is achieved at

i' " i'o I +

$llJ _{I)Q~ IF} ^o)1) ₍₁₃₎

As the tilt

angle

_q increases in the

phase III,

_~J also increases it J _> I and decreases otherwise.

Then

equation (12)

reduces to

4l 4lo "

IA Ao)n~

+ B~q~ +

~2((F Fo) (14)

with Ao ₌

(J I)~J]

and B~

= B

(J -1)~/2G.

Ii B~ _> 0, the

tilting

transition remains second order. It _occurs at A

=

Ao.

When J _>

I,

the line bc oi the transition shiits to

positive

A with respect to line

de,

_as shown in

figure

3 and _as observed

experimentally (Fig. I).

We restrict ourselves to this _case and do not consider iurther the

opposite inequality

J _< I when the line bc shiits to

negative

A.

The minimum oi the free energy

(14)

in

phase

III is

equal

to

, ,~ ₌

(A j/0)~

+

~iif _{Fo) (IS)}

At the line bd oi the transition

I-III,

the

right-hand

side of

equation (IS)

is

equal

to _zero:

F Fo _#

~~

~

~j~ (16)

4 ~§~o

~

This is the

equation

oi _a

parabola

with its vertex at the

point

b. The

position

oi the

point

d is deternfined

by equation (16)

^with A

= 0.

At the first order transition III-IV the

herringbone

order oi

phase

III (~J m ~Jo) ialls ^discon-

tinuously

to ~J = 0 in

phase

IV. At the line di oi the transition the iree _energy

(15)

oi

phase

III is

equal

to the minimum oi

equation (2), -A~/48

in

phase

IV.

Taking

B~ _m B~ one has

F Fo _" ~~

~

~~~'~ ~

~~A ⁽¹⁷⁾

Equation (17)

defines the

straight

line

df,

whose

slope

is

negative. However,

the _accuracy oi this

equation

is _poor, since the iree _energy in

phase

III will contain

higher-order

terms which

cannot be

neglected

at finite distances irom the II-III transition.

The tilt direction in

phase

III _near the line bc is determined

by

the term

J~J~q~

cos

2fl

whose minimum is

fl

₌

jr/2 (tilting

to the next-nearest

neighbor)

ior J _> 0. As _q increases, the term

-Dq6

_cos

6fl

should be taken into consideration. At

9Dq~

= J~J~ ^the second order

phase

transition _occurs

resulting

in continuous variation oi

fl.

The intermediate tilt direction

gives

rise to _a monoclinic unit cell. The line oi the transition _crosses the line bdi,

Let _{us now}

proceed

to

analyze

^the strains in the

phases

II,

III,

and IV. In contrast to the

previous section,

the strain is not included in the iree energy

expansion (II)

since it is not the order parameter.

However~

the order

parameter

_~J is not detected

directly

_{as a}

macroscopic quantity

and the strain is _an

important

manifestation oi the

ordering.

The iree

energy

expansion

_over

f

_can be restricted to the _terms

4~

₌

C(~

_{U~J~f cos}

cx

V(q~ cos(o 2fl). (18)

The first term here is the elastic energy of the

hexagonal lattice,

C _> 0. The second term iollows irom the invariant

(9)

transiormed to

polar

coordinates

(3).

The third term is the

coupling

term

(5). Strictly speaking~

the free _energy should be minirr£zed first _over

(,

_as the result will influence the coefficients oi the

expansion (II).

For this _{reason, we} consider the

coefficients oi

equation (11)

^{to have}

^already

^been renormalized

by (18).

In

phase

IV _one has

~J ⁼ 0 and fl ₌ 0. Minimization oi

(18)

_{over o}

gives

_{cx =} 0 ior

V > 0~ which

corresponds

to

stretching

oi the unit cell in the tilt direction in agreement ^with

experimental

observations

[19].

^{The induced strain}

f

=

(V/2C)q~. Taking

A ₌

a(p pc),

where _pc is the surface _pressure at the transition and _a is _a constant, one has _q

+~ pc p

and

(

~J

(pc p)

in

phase

IV. In

phase

_{II q =}

0,

the spontaneous strain is

(

₌

(U/2C)~J~

^with

cx = 0 for U _> 0. The unit cell stretches in the _same direction _as in

phase IV,

in

agreement

with observations

[18].

^The

signs

of U and V _are

already

determined

by

^the strains in

phases

II and

IV, respectively,

and allow the behavior of strain in

phase

III _to be

predicted. Taking

ior

phase

III the value

fl

=

jr/2 already found,

_one has from

equation (18)

_{cx =} 0 and

( 2C~'

^{~ 2 V}

i~

²

⁽¹⁹⁾

Variation of _~J,

equation (13),

_can be

neglected

in

comparison

with the second term of

(19)

and

~J can be taken

equal

to ~Jo. Hence the strain decreases _as the tilt

angle

increases. This result is in

good

agreement ^{with the}

experimental

observation [18] ^{that the two first order}

Bragg

peaks

_move

together monotonically

_as the tilt

angle

increases.

4. Phase transitions in l~eneicosanol

monolayers.

The iree _energy

expansions analyzed

in the _two

previous

sections _are

quite general

consequences of the symmetry ^{of the}

monolayer,

but the

particular

details

depend

in addition _on the

signs

oi the coefficients. These _were determined for the

fatty

acids

by examining

the structural features of their

phases.

A recent

X-ray

diffraction

study

of heneicosanol

monolayers

_on the surface of water [23] ^is as

complete

_as the studies of

fatty

acids and

permits

_a similar

analysis.

The

comparison

of the

long-chain

acids and their alcohols is of

special

interest since the interaction between the

long

^{chains of the molecules} remains the _same whereas the interaction between the

headgroups

at the water surface differs. We show in the present ^{section that}

equitranslational ordering

_can be ruled _out for heneicosanol

by

the _same

arguments

used for

fatty

acids. The

observed structural features

are described

by

translational

ordering,

and the

only qualitative

difference from the

fatty

acids is that the coefficient D in the

tilting

free _energy

(2)

^has

opposite sign.

The X-ray

study

_[23] revealed the _{same sequence} of

high-pressure phases

of untilted molecules

(LS

^S

CS).

However, ^when the _pressure is lowered the molecules tilt

only

towards the next- nearest

neighbor.

In the notation of the present _paper, at the second order

tilting phase

transition I-IV

(see Fig. 3),

^{the tilt} to the next-nearest

neighbor (fl

₌

jr/2)

is found to be

accompanied by stretching

of the unit

cell,

also in the NNN direction (cx =

jr). During

the first order transition I-II the unit cell stretches towards the nearest

neighbor

_{(cx =}

0),

in the _same

direction _as in

fatty

acids. At the second order transition II-III the tilt to the next-nearest

neighbor (fl

₌

jr/2)

_occurs in the unit cell

already

stretched in the NN direction _(cx

=

0).

Let _us show first that these structural features of the

phases

_are

incompatible

with the free energy

expansion (6)

for

equitranslational ordering.

To obtain _a minimum of the free _energy

(2),(5)

ⁱⁿ

phase

IV at _{cx = jr} and

fl

₌

jr/2,

_one should take J _> 0 and D <

0;

the strains in

phase

II

(o

=

0)

are described

by equation (4)

with G _> 0.

Then, considering

the transition II-III with _cx

= 0

already fixed,

_one would have from

equation (5)

that

fl

= 0, in contradiction with

experimental

observation.

The free energy

expansion (11)

for the translational

ordering

_agrees with the structural features. The tilt direction

fl =1r/2

in

phase

IV is described

by minimizing equation (2)

with D _< 0 whereas the _same tilt direction in

phase

III follows from

equation (10)

with J _> 0.

Proceeding

to strains

(18)

and

taking

U _> 0, V _> 0, _one finds _n

= 0 in

phases

II and III and

cx = 1r in

phase

IV. The square-root law _q

+~ p of increase of the tilt

angle

in

phase

III

accompanied

with linear decrease of deformation

f to

'~ P PC

following

from

equation (19)