Hexagonal-cubic phase transitions in lipid containing systems : epitaxial relationships and cylinder growth

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Hexagonal-cubic phase transitions in lipid containing systems : epitaxial relationships and cylinder growth

Paolo Mariani, Lia Amaral, Letizia Saturni, Hervé Delacroix

To cite this version:

Paolo Mariani, Lia Amaral, Letizia Saturni, Hervé Delacroix. Hexagonal-cubic phase transitions in

lipid containing systems : epitaxial relationships and cylinder growth. Journal de Physique II, EDP

Sciences, 1994, 4 (8), pp.1393-1416. �10.1051/jp2:1994206�. �jpa-00248049�

Classification Physic-s Abstracts

61.30 64.70 87.15

Hexagonal-cubic phase transitions in lipid containing _systems

_:

epitaxial relationships and cylinder growth

Paolo Mariani

(I. *),

Lia

Q.

Amaral

(2),

Letizia Satumi

(I)

and Hervd Delacroix

(3)

('

Istituto di Scienze Fisiche, Facolth di Medicina _e

Chirurgia,

Universith di Ancona, ^via Ranieri.

60131 Ancona,

Italy.

(2) Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 20516, CEP 01498-970, Sao Paulo, SP, Brazil.

I')

Centre de Gdndtique Moldculaire (**), Laboratoire Propre du CNRS, 91198 Gif-sun-Yvette France

(Received 22 December 1993, revised 4 March 1994, accepted 26 April 1994)

Abstract. In order to approach the nature of the phase transitions, three lipid-water systems (namely PaLPC, OLPC and DTAC), which exhibit _a sequence of phases of type I, including

hexagonal

(which exists in the central region ^of the three phase

diagrams)

_as well _as cubic bicontinuous Q~"° (Ia3d) and/or cubic micellar

Q~~'(Pm3n)

phases, have been considered. On the basis of

a recently proposed alternative description ^{of the} hexagonal

phase

in _terms of finite micelles, the structural parameters as a function of the lipid concentration _are analysed in terms of micellar growth ^{inside the} hexagonal domain. Moreover, considering the

epitaxial relationships

occurring at the hexagonal to cubic phase transition, the growth of the hexagonal phase from the 3D networks of rods of the Q~~° cubic

phase

and from the

globular

micelles packed in the Q223 cubic symmetry ^{has been followed. As} a result, _a different behaviour has been observed in the three lipid systems. These differences _are discussed in relation _to the phase _sequence: in particular, from _a

simple

geometrical approach, based _on the

analysis

of the

projection

of the electron

density

distribution _on the cubic

phase

in the direction of the

cylinder growth,

a

mechanism for the process of transformation of the _structure elements is proposed.

1. Introduction.

Lipids

present an extended

polymorphism.

In

particular,

_among ^{the wide}

^variety

^of structures that can be observed in

lipid-containing

systems, non-lamellar

hexagonal

and cubic

organiza-

tions show an

increasing

theoretical and

biological

interest

(reviewed,

for

example,

ⁱⁿ

ii

and

[2]). Concerning

the cubic

phases,

several papers tackled

recently

a

systematic analysis

of their

(*) Author _to whom

correspondence

should be addressed.

(**) Assoc16 h l'Universitd P. _et M. Curie.

structures

[2-11]

_{as a}

result,

the _structure of the six cubic

phases

_so far identified has been determined. In

particular,

in this work _we will consider three different

lipid-water

systems

which show two cubic

phases

^of type ^{I in} our notation

Q~~° (space

_group

Ia3d)

and Q~~~

(space

group

Pm3n)

which

belong

_{to two} different classes. The _Q231~

phase

is

bicontinuous,

in the _sense that both the _water and the

hydrocarbon

media _are continuous

throughout

the

structure

[1, 4, 12],

while the Q~~~ ^cubic

phase

is micellar

[2,

5, 8,

9].

The other non-lamellar

organization

which will be considered is the

hexagonal

H

phase.

The

structure of this

phase

has been assumed since the initial works _to consist of

« infinite _»

cylindrical micelles,

with _a two-dimensional

hexagonal positional

order in the

plane perpendicular

to the

cylinder

_axes

[13, 14].

However, recent results relative _to the sodium

dodecyl

sulfate

(SLS)

water system ^leads to an alternative

description [15]

: after _a direct

isotropic (I)

to

hexagonal phase

transition, the

hexagonal

cell parameter a has been observed _to vary with the

amphiphile

volume concentration c~ as a oI c-j

'°,

_this

_evidencing

_{the presence of}

finite hard rods in the

hexagonal

^domain. Moreover, in the _same system, a detailed

analysis

of micellar

growth

in the

vicinity

of the

isotropic

to

liquid crystalline phase

^{transition evidenced}

that the micelles grow little in the I

phase [16].

^{Other results relative} to the columnar

hexagonal phase

observed in _a

deoxyguanosine-water

_system, which presents an intermediate cholesteric Ch

phase (the phase

sequence

being I-Ch-H),

show _a behaviour _{a oI} c-j '~~,

typical

of infinite _or

long

flexible

cylinders

in the H

phase [17].

All these results _are in agreement ^with recent theoretical

previsions

for these transitions,

taking

into _account the self-association

properties

of

lyotropic

_systems

[18-21]. Recently,

evidence for micellar

growth

within the H domain after

a direct

isotropic

to

hexagonal phase

^{transition has been obtained in another}

lipid-water

system

(oleoyl-lyso-phosphatidyl choline, OLPC) [22],

that presents a

I-H-Q~~°-L phase

_sequence.

In this paper, we

analyse

the

growth

of the micelles in the

hexagonal phase focusing

OLPC

[22]

and other two

lipid-water

_systems,

namely palmitoyl-lyso-phosphatidyl

choline, PaLPC and

dodecyl-trimethyl-ammonium

chloride,

DTAC,

that present a

I-Q~~~-H-L

and

a1-Q223- H-Q23°-L phase

_sequence,

respectively.

In

particular, X-ray

diffraction

analysis

demonstrate

the existence of

epitaxial relationships occurring

at the

hexagonal

to cubic

phase

transitions _:

making

reference _to the results of

Ran~on

and Charvolin

[23]

and of Clerc _et a/.

[24],

^which show the direction of

growth

of the

hexagonal cylinders

from the 3-dimensional networks of rods of the

Q~~°

bicontinuous cubic

phase,

_{we propose} that

hexagonal phase

forms

along

the

same direction also from the

globular

micelles

packed

ⁱⁿ the Q~~~ ^cubic symmetry. ^{On the} basis of _a

geometrical approach,

these

relationships give

_us the

possibility

to follow the transformation of the structure elements and _to suggest a mechanism for the

cylinder growth.

2.

Experimental.

2, I NOTATIONS _AND ABBREVIATIONS

DTAC

dodecyl-trimethyl

ammonium chloride OLPC

oleoyl-lyso-phosphatidyl

choline

PaLPC

palmitoyl-lyso-phosphatidyl

choline _;

SDS sodium

dodecyl sulphate

2D,

3D

2-dimensional,

3-dimensional

L, H, Q" ID lamellar, 2D

hexagonal

and 3D cubic

phase

of _space group of number

n,

according

to the International Tables

[25]

isotropic liquid phase

_;

c

weight

concentration

(lipid/(lipid+water))

c~ volume concentration of the

lipid,

calculated _as

reported by

Luzzati

[13]

c~, ~~~

volume concentration of the

lipid paraffine moiety,

calculated _as

reported by

Luzzati

[13]

a unit cell parameter

r, s vectors

specifying position

in real and in

reciprocal

_{space r} and _{s are} their

moduli _s

=

(2

sin tl)/A

(where

2 tl is the

scattering angle

and is the

X-ray wavelength)

s~ vector

specifying

the

positions

of the

reciprocal

lattice

points,

h

=

(h,

k,

f)

F

(h)

structure factor _at

reciprocal

lattice

point

h

p coefficient of the Gaussian

apodization eventually applied

_to the

(F (h)) [3]

_:

'~fl(h)(

=

I(F(h)( eXp(-

2

/~~S~)j/[°ih'F(h)(~eXp(-

₂

_/~~S~)j~~~

p)(0)

curvature of the autocorrelation function _at the

point

_r

=

0

[3,

4]

p

(r)

3D electron

density distribution,

I-e- Fourier transform of

(F (h))

_;

IA)

_average value of the function A(r) over the volume V of the unit cell

IA )

= ( IV )

.

A (r

dv~

Ap

(r) normalized,

dimensionless

expression

of the function _p

(r),

also called the 3D map :

Ap (r)

=

lp (r) jp (r)j ill jp 2(r)j jp (r)j 2)j~/2

Ap ?D section of the electron

density

distribution Ap

(r) computed,

at one

specified

distance from the

crystallographic origin (0,

0,

0)

and

normally

to the

crystallo- graphic

direction

[nmf], using

structure factors

(F (h))

Ap _j,,~,tj

projection

of the electron

density

distribution Ap

(r) along

the

crystallographic

direction

[nmf], _{computed using}

_structure factors

(F(h))

_;

plane (nmf) _plane

_oriented

perpendicularly

to the

crystallographic

direction

[nmf]

d~~~

length

of the 3-fold

[I

I

I] (body-diagonal)

axis of the cube _:

djjj

=

a,,5.

2.2 THE _PHASES. This work focuses _on three different

phases

of type I, ^all characterized

by

disordered conformation of the

hydrocarbon

chains and all

displaying

one-parameter structu- res,

belonging

to the 2D

hexagonal

and to the 3D cubic systems. The

hexagonal phase

consists of _a

hexagonal

_array of

cylinders,

coated with the

polar headgroups

of the

lipid

molecules

[13, 14].

As the structure is of type I, ^{the rods} are filled

by

the

hydrocarbon

chains of

lipids

and

are embedded in the _water matrix.

The

two cubic

phases

that will be considered _{are one}

bicontinuous (the

Q~~°

cubic

phase)

and the other micellar

(the Q"~).

The

Q~~°

can be described in terms of two 3D networks of

joined

rods,

mutually

intertwined and unconnected

[2, 4, 12]

the rods

(of

_{type I)} have

equal length

and _are

coplanarly joined

3

by

3. This

phase

is

related to the Schoen's

Gyroid infinitely period

minimal surface without intersections

[2, 4].

The structure of the

Q"~

consists of

disjointed

micelles

(of

_{type I)} embedded in _a continuous

water matrix. The micelles

belong

to two different classes _: those of _one class _are

quasi

spherical

in

shape

while those of the other class are

disk-shaped [2, 8, 9].

The micelles of each class are centred _{at one} of the

special positions

of the space group

Pm3n, namely

a and _c

[25].

A schematic

representation

of the structure of the

hexagonal

and of the

Q~~°

^and Q~~~ ^cubic

phases

is

reported

ⁱⁿ

figure

I.

In the

following,

we will discuss about

shape

and dimensions of the _structure elements _: it must be observed that

they

are bound to be defined

by

the structural conformation of the

molecules,

by

^the lattice interactions and dimensions and

by

the chemical

composition

of the

phase. However,

few

hypotheses

have to be made in

particular,

the

hydrocarbon

chains _are assumed to cluster into

regions

from which water is excluded and the interface between water and

paraffin

is assumed to be covered

by

the

hydrophilic

_groups of the

lipid

molecules

[4,

13, 22,

29].

;

)o'~ l

( _j..j-. .@-..(...g.... qf;

^{o . o}

'I _{_)...I[-} .[~)j[[[I)[. -.§.

.-+-.

.""

O'°

."'"

'~

Q~~~ ^cubic phase

Fig,

I. Schematic

representation

of the non-lamellar _structures discussed in the text.

Hexagonal

phase : the structure is

represented

in _terrns of

parallel

tubes packed in _a 2D

hexagonal

_array (space _group p6 [25]). In the studied systems, ^the structure is type ^{I the}

cylinders

are filled by the

paraffin

chains and embedded in _a polar matrix. Cubic

Q~'° Phase

the _content of the box (defined by light lines) is shown in

perspective

(left frame) and in

projection along

the axis [100]

(right

frame). The black heavy lines mark the

projection

of the unit cell. The _structure is

represented

in _terms of _two 3D networks of rods, mutually intertwined and unconnected. The rods _are

joined

coplanarly 3~6y ³ and occupy positions _g, of symmetry 2 _; the rod junctions occupy positions b of symmetry ³² [25]. In the studied systems, the structure is

type ^{I the rods} are filled by the

hydrocarbon

chains and _are embedded in _a polar matrix. Cubic Q223 phase : perspective view (left frame) and

projection

along ^{the axis} [100] (right frame) of the structure.

The limits of the unit cell _are marked by ^thin lines. The _structure consists of 2 types ^of disjointed

globules

represented

by

circles. The unit cell contains 6 globules of symmetry 12m (open circles) centred _at

position

_c and 2

globules

of symmetry ^m3 (filled circles) centred _at position _a [25]. The structure is type ^{I the} globules, filled by the hydrocarbon chains, _are embedded in _a

polar

matrix.

2.3 ELECTRON _{DENSITY MAPS.} The structure of all the considered

phases

will be

presented

in the form of electron

density

_maps,

computed

from the

amplitudes

of the observed

reflections. As _a consequence of the data

normalization,

_we

really

_compute

by

Fourier

transform _a function Ap

(r

) which is _a normalized dimensionless

expression

of the fluctuations Of the electron

density

distribution _p (r) (for _more details _on the normalization

procedure,

_see

Refs

[3, 4]).

A number of electron

density

sections Ap and electron

density projections

Ap _j,,~,t will be shown in this paper :

they

_are

represented using equally spaced density

levels _;

negative

levels

(which correspond

to the low-electron

density regions

and then to the

paraffinic moiety)

_are

reported

_as dashed lines the level which

corresponds

to zero is drawn

by

_a full line. Moreover, _as the structures

investigated

_are of type ^I

(I.e.

the structural elements _are made of

paraffines),

for the sake of

clarity

the

positive

levels of electron

density

sections and

projections

are omitted in the

drawings

of

figures

9 and I. Note that we will present

only

electron

density

_maps of

hexagonal,

cubic Q~~~ ^{and cubic}

Q~~° Phases

relative _to the DTAC system ⁱⁿ

fact,

the information that could be extracted from the

corresponding

_maps from

OLPC and PaLPC _are

exactly

the _same (see also

[8]).

2.4 THE DATA. The

phase diagrams

of the DTAC _water system ^{and of OLPC and}

PaLPC _water systems were determined

respectively by

^Balmbra et a/.

[26]

and

by

Eriksson et a/.

[27]

and _are

reported

in

figure

2. These

phase diagrams

have in _common the presence of _a

hexagonal phase

in the central

region.

However, in the _case of DTAC and

PaLPC,

_a micellar Q~~~ ^cubic

phase

extends between the normal

hexagonal

and the

isotropic

fluid

phases,

while in the _case of DTAC and

OLPC,

_a

Q~~°

cubic

phase

_appears in the dried side of the

phase

diagram,

between the normal

hexagonal

and the lamellar

phases.

DTAC

p

^L

1

I H

I

#

0 50 100

Weight

Concentration (ib)

OLPC p

bd

~

i I H L

#

E

0 50 100

Weight

Concenwation (ib)

~°°

PaLPC

LJ fi e

I H L

50

u

o 50 loo

Weight

Concentration (ib)

Fig.

2. Phase

diagram

of the three systems ^{considered in this work. The} grey portions correspond to

crystalline _or to biphasic regions. The phases are labelled _as in the text. DTAC redrawn from Balmbra et a/. [26]. OLPC and PaLPC redrawn from Eriksson _et a/. [27].

The

X-ray

diffraction

analysis

of the structure of cubic Q~~~ ^and

hexagonal phases

observed in DTAC, PaLPC and OLPC _were

already reported by

Mariani _et al.

[4], by Vargas

et al.

[8]

and

by

Mariani and Amaral

[22].

In table I, _some relevant structural information _are resumed.

The structural

analysis

of the cubic

Q~~° Phase

^{observed in OLPC and DTAC will be}

presented

in the

following paragraphs. X-ray

diffraction

experiments

_were

performed by using

an Ital-Structures 1.5 kW

X-ray

_generator

equipped

with _a

Guinier-type focusing

_camera with

a bent quartz monochromator.

Samples containing

different

quantity

of water were

prepared by

Table I. Structuia/ data

of

the

her.agona/

and cubic

phases.

DTAC

c 0.50 0.50* 0.58 0.60 0.65 0.72 0.76

0.895* 0.895

C-v, pa, 0.38 0.38 0.44 0.45 0.49 0.54 0.56 0.66 0.66

Phase Q22~ H H H H H H H

Q~~°

a

(A)

85.4 39.7 39.3 38.9 38.8 38.2 38,1 37.2 79.6

V,~~

(10~ h')

13.8

V~

(10~ A')

25. I

V~

(10~ h~

28.6

PaLPC

c 0.50 0.50* 0.51 0.52 0.54 0.55 0.57 0.60 0.65

c~_~~~ 0.27 0.27 0.28

0.285 0.295

0.30 0.31 0.33 0.36

Phase Q~~3 ^{H H H H H H H H}

a(h)

136.7 62.7 62.6 62.4 62.0 61.9 61.8 61.2 59.5

V~

(10~ l~)

55.3

V~

(10~ l~

87.5

OLPC

c 0.50 0.52 0.56 0.58 0.60 0.62 0.65 0.67 0.70

c~,~~,

0.285

0.30 0.32 0.33 0.34 0.36 0.37 0.39

0.405

Phase H H H H H H H H H

a

(1)

_{59.4 58.7 58,1 57.6 57.0 56.5 56.0 55.4 55.0}

v,~~

(

io3 l~

_i

OLPC,

contd.

c 0.71 0.77* 0.77

c~ _~~~ 0.41 0.45 0.45

Phase H H

Q~~°

a

(1)

54.8 53.4 113,I

V,~~

(10"1')

26.9

Data for DTAC and PaLPC relative _to the H and Q22~ phases _are taken from Mariani et al. [4] and from

Vargas

et al. [8]. Data for the

hexagonal phase

of OLPC _are taken from Mariani and Amaral [22]. The columns with _an asterisk _{on I} report ^{the value of} the parameters relative to virtual samples of phase H,

extrapolated

at the concentration of the cubic phase. The volume of _one hydrocarbon rods _in the Q2'~' cubic

phase

has been determined from the unit cell dimension and the paraffinic volume concentration by

using

V,,,~ = (a~_c~ ~~,)/24. V~ ^and V~ are respectively the volumes of any micelle centred in the

position

_a

and _c in the Q~~' cub)c phase (see text) _as calculated by Luzzati _et al. [2]. The absolute

error on the

lattice parameter ^is ± I I. Other notations and

symbols

as reported in the _text.

mixing

controlled _amounts of the

ingredients.

The relative

uncertainity

of the concentrations

was estimated to be 5 fl. The

samples

were mounted in

vacuum-tight

cells with thin mica

windows the cells _were

continuously

rotated

during

the exposure in order to reduce

spottiness

(which arises from the

tendency

of cubic

phases

to grow into

macroscopic

monodomains

[4]).

The films _were microdensitometered and the intensities measured and corrected _as described

by

Tardieu

[28].

2.5 CRYSTALLOGRAPHIC ANALYSIS OF THE OLPC-WATER _SYSTEMS. In order to determine

the structural data relative to the H

[22]

and

Q~'° _{Phases, X-ray}

diffraction

experiments

_were

performed

at room temperature as a function of _water concentration. In each

experiment,

a

number of

sharp low-angle

reflections _are observed and their

spacing

and intensities measured.

As

expected,

^{in the central}

region

of the

phase diagram

(from about

c. =

0.3 to _c

=

0.7) the observed

peaks

^show

spacing

ratios in the order _:

,5

:

,j

:

,I ,~,

_{which indicate the 2D}

hexagonal

symmetry

[13,

22,

25].

In the

region

between about _c

=

0.7 and _c

=

0.8,

the

spacing

ratios of the observed reflections _are in the

order,~

:

,~

:

,~ ,/16 ,$: ,)...

clearly indicating

the Ia3d cubic symmetry

[4, 12, 25].

Once the symmetry ^is obtained, the dimensions of the unit cell have been determined these data _are

reported

ⁱⁿ table I. It _can be noticed that

by using

the above described

experimental

device, the weak additional diffuse

scattering

around and away from the

Bragg peaks, expected considering

_many

previous

observations

(see

for

example

Ref.

[23]),

cannot be

clearly

observed.

2.6 CRYSTALLOGRAPHIC _{ANALYSIS OF THE} Q~~~ CUBIC PHASE OBSERVED IN THE DTAC-

wATER SYSTEMS. The

hexagonal

and the cubic Q223

phases

observed in the DTAC system

Table II.

Cij,sta/%graphic.

data

oJ

the DTAC .<_vstem.

Hexagonal phase

^Cubic Q~~~

Phase

c

0.89~*

_c

0.89~*

a

(h)

_37.2

a

(h)

79.6

F(

lo) 404

F(21

1) + 164

F(21)

29

F(220)

+ 158

F(20) + 40

F(3211

+ 14

F(3

19

F(400)

₊ 51

F(30)

20

F(420)

+ 17

F(322)

¹⁸

F(422)

19

F(431)

17

2

pi (h2

_{0 2}

pi (h2

~

0

pj (0 (10~~

A~

2)

13.7

p)(0)

^{(10~ 2}

^A~~

^15.5

( _{(Ap )~)}

2.634

( (Ap )~)

^2.907

~ ^~

~~~

i19.5

~

(j~[(~)jo-2 j-2)

~~.~

-~i~°~ ~'°~ ~~~'

~~

The _structure factor, _are normalized _to ~~

(F (h)(

^{~ 10~.}

The signs ^of the reflections _are those of the maps in

figures

3, ⁸ and 9.

(

(Ap

j~)

^{is the} moment of order 4 of the

histogram

of the maps Ap (r calculated from _raw data. Other notations and symbols as reported ⁱⁿ the text.

JOURNAL DE PHYSIQUE ,, T 4 N'~ AUGUST 1994 52

~

~

~

_~

O

~,

Q

Q Q

^O

^,(~~e,,

^O

',""""

"kl'l'(f,

~j Q'~i'Q"~t"

~

~

Q ',',,j"'',',,""'

',',,"",

Q

,

Q

^D _,,=,, ^{'*qfi$/$' O} ,=,,

%~~

'~'t((,'

'~"'~

'j*']jll'

~"""' ~"""' Q

"fi"'$Kli' ~j

_Q

"h"'$fir'

'~"'$ff' O '~"J$f$' O

Q Q ^{1/2'$$$' '~J~'$$$$'}

~~~j$#~' _Q

)]j$~#'

,,"' ,,','

<' Q ^<' Q

~ _~

~

^~

~

© ~

~

F)g. 3. DTAC system. Upper ^frame _: electron density _map calculated from the _raw data of the H

sample

extrapolated to the concentration of the cubic phase (see Tab. II). Lower frame sections Ap of the electron density _map of the Q~~°

Phase

calculated from the _raw data and the _« best _»

sign

combination _as determined by using the pattern recognition approach (see Tab. III. in the left side,

section normal to the ~-fold _axes and _to the

plane

(211)

containing

the point a/8,a/8,

a/8; in the

right

side, section normal _to the crystallographic axis [100]

containing

the point a/8.0,0. The bar represents 50 h.

were the

object

of recent

crystallographic

studies

[4, 8].

In

particular,

the

signs

of the reflections of

phase

H _were determined

by

a

systematic X-ray

diffraction

analysis

as a function

of the _{water content}

[4],

while the _structure of the cubic Q~~~

Phase

_was solved

by using

a

pattern

recognition approach [8]. By

contrast, the presence of _a cubic

Q~~° Phase

in the dried side of the

phase diagram

_{was up} to now

only

documented

[4, 26]. Therefore,

_we report here the

crystallographic analysis

of this

phase.

As

expected,

in the concentration range between about _c

= 0.82 and _c

=

0.90,

the

spacing

ratios of the observed reflections _are in the order

,I:,I:,/14:,@:,$fi...

indicating

the Ia3d cubic symmetry

[25] (see

Tabs. I and

II).

As

before,

due _to the used

instrumentation,

_no additional diffuse

scattering

was

clearly

detected.

To solve the

crystallographic phase problem,

and then _to reconstruct the structure of the

phase,

_we use the pattern

recognition approach

described

by

Luzzati _et al.

[3]. According

to

this

approach [3, 4, 8],

the

sign

of the reflections could be assessed

using

the axiom that the

histograms

of the electron

density

_maps of two different

phases, extrapolated

to the _same concentration and

properly

normalized in scale and

shape,

_{are very} similar _to each other. The

procedure

^involves the

comparison

^of two

histograms,

_one relevant _to the

phase

under

investigation

and the other to a reference

sample

of known structure. In the present case, a

virtual

hexagonal sample,

with the _same chemical

composition

like the cubic

sample,

has been used _as reference.

In order _to determine the _structure of this reference

sample,

we resorted _{to a}

swelling experiment [4, 8],

In the

hexagonal phase,

it is

expected

that, at all concentrations, the observed reflections

sample

the

circularly averaged

structure factor of the _cross section of the rods.

By plotting

the

intensity

of the reflections _{as a} function of _s, the

signs

of each lobe of the

circularly averaged

_structure factor have been in fact determined

[see Fig,

in Ref.

8].

The electron

density

_map of the reference H

sample

is then obtained

by linearly extrapolating

to the concentration of the cubic

phase

the

concentration-dependent

_structure factors of each reflection observed in the

hexagonal phase.

The so-obtained structure factors _are

reported

in table II the

corresponding

electron

density

_map is

reported

ⁱⁿ

figure

3.

According

_to the

procedure,

before the

comparison,

the intensities observed in the cubic

phase

and those of the reference

hexagonal sample

were

apodised,

such that the

pj(0)

^{takes the} same value in the _two

phases [3,

4, 7,

8] (see

Tab.

II).

All the

sign

combinations

compatible

with the diffraction data relative _to

Q~~° Phase

_were then

generated,

and each of the

corresponding histogram~

_was

compared

with the

histogram

of the map of the reference H

phase. According

to

Vargas

et al.

[8],

the

similarity

between the

histograms

was

assessed

by searching

for the minimal value of the parameter

Sl~

s, I-e- their distance in the six- dimensional space of the _moments,

((Ap )")

]'~~, for 3 _{m n m} 8. In the

figure

4, the

points

SIM g are

reported

as a function of

((Ap)~),

The best

sign

combination is the _one which

corresponds

to the minimum of

Sl~

g (see Tab.

II).

Some sections of the

corresponding

_map are

reported

in

figure

3. It is worth

noting

that the selected electron

density

_map is in excellent agreement ^with the well-known _structure of the

Q23° phase [see

also 3, 4, lo and I

?],

It is also

interesting

to note that the final result is invariant with respect to the choice of the

p)(0)

value at which the

comparison

is made

[P. Mariani, unpublished results].

iooo.o

~aa o oaa° ~°oaaoo° °

° ° o o

° a° on o

~ ~

~~ ~o o

loo,o

°° °°°

°& tjii~

~~~~IS

j

a

~ o

~~~8~

_°@~o

8

o °o ~~

~

~

° o

~

llf

1000

/~

DTAC 1.0

1.5 2 2.5 3

<(Ap)~>

Fig. 4. DTAC ~y~tem : plot of Mp

x ;ei iu~

(

_{l~p )~)} in searching for the sign combination of the

Q2'l'

phase who~e histogram be~t fits the hi,tograni oi the reference hexagonal phase. The apodized and

extrapolated data of table II _were u~ed. M~

x

and

(

(~p )~) are defined _in the _text. The _arrow point~ at the

minimal value of M~

x.

3.

Hexagonal phase.

3. ANALYSIS oF THE CYLINDER GROWTH. The structure of the

hexagonal phase

^{has been}

usually

assumed _to consist of _« infinite _»

cylindrical

micelles

packed

in _a

planar hexagonal

cell with parameter a.

Recently,

on the basis of results obtained in the SDS system

[15]

and of statistical-mechanical calculations for the direct

isotropic

to

hexagonal phase

transition

[18- 2l],

an alternative

description

has been

proposed

in _terms of finite micelles. In

particular,

the finite

cylinders pack

as a fluid in the third dimension

(normal

_to the 2D

hexagonal cell),

with

an average distance C between micellar centers.

Polydispersity

in

cylinder length

is _a basic property of micellar systems ^{and in} this

description

is _a necessary condition for the _occurrence of _a direct

isotropic-to-hexagonal phase

transition

II

5,

18-21].

Note that the usual absence of

diffraction in the direction

perpendicular

to the

hexagonal plane

shall therefore be ascribed _to

polydispersity

of

cylinder length

and not

necessarily

to the presence of

infinitely long cylinders.

Let _us consider

initially

that the radius and the average

length

of the

cylindrical

micelle be R and L,

respectively.

The

amphiphile-to-water

volume ratio has _to be the _same in the unit cell

as in the whole

sample,

therefore

(L/C

)

=

(,5/2

ar c~

a~

_R~ ~ l

where _c~ is the

amphiphile

volume fraction. However, since the

lipid polar region

contains

water, we shall consider R, L and the volume concentration of the

paraffin

core

(C'v,pa<)-

Assuming

a constant R value, and

estimating

it from the

length

of the extended chain

[15],

it is

possible

to obtain from the

experimental

values (a and c, ~~,) the ratio L/C, that

gives

essentially

the fraction of water in C direction. It is evident that for _« infinite

»

cylinders,

_we

have L/C

= I, so that

a oI cj

)j,.

(2)

In order to describe the system as

containing

finite

cylinders,

the

hypotesis

^of a uniform

decrease in

interparticle

distances in all three dimensions has been made

[18-20]

:

L/C = 2Rla.

(3)

From

equation

(I), we then get :

~ ~ ~

(2 ,/jlar

_)- '/3

~.- ^)/3

~. M'

However, _as discussed in _a

previous

_paper

II

5], the

approach

of _{a «}

cylinder

_{» may} lead _to inconsistencies in the _case of finite

objects.

It is better in this _{case to} consider _a

sphero-cylinder

with radius R,

cylinder length f,

total

length

L

=

(f

₊ 2 R and

anisometry

_v

=

(f/2

_R

_).

The condition for

amphiphile-to-water

volume ratio becomes then

(instead

of

Eq. (I))

_:

i~

~~'P~'~~

~3~/~v

~~~

Under the condition of uniform decrease in

interpartide

distances in all three dimensions

(Eq. (3)),

it results

[15]

a - 2 R

fi

~i«++~ii~

°

cv

pi, (6)

If

experimentally

it is observed _a

=

Acj j~,,

^the

anisometry

becomes

[22]

«

~'~. l~l, ^{~i~ll (7)}

where K=

(2,star)(A/2R)~.

_{It is evident that for}

_cylinders

_with

constant

radius,

K is _a constant. If.i

= 1/3, I-e- if the

cylinders

_are finite and the condition of uniform decrease in

interparticle

distances in all three dimensions is fulfilled, the average

anisometry

is

constant

K 2/3

" ~

(8)

l K

By

contrast, if,i is smaller than 1/3,

equation (7) gives

a law for the

cylinder growth

with

concentration

[22].

In

particular,

the

(Kc(' j~l'')

value must respect ^{the condition} :

' _~ (fi~l'~~

'~)

~ 2/3

(~)

PA'

as it

corresponds

to an average

anisometry

_v

ranging

^from o~ to 0 (I.e, from

cylinders

of infinite diameter to

spherical particles

of diameter

equal

to 2 R).

Therefore,

the

analysis

of the function

(Kc[~ j~/

^{'~ in the domain of existence of the H}

phase,

^and

particularly

at the transition

points

to

neighbouring phases,

may throw

light

on the process of

cylinder growth

and _on the

nature of the

phase

transition. Moreover, we stress that the condition

given by equation (31

cannot be valid when

cylinders

become _« infinite _», and in this limit the

equation (7)

is _no

longer

^valid,

The

analysis

of the OLPC system

[22] gives

_a first clear evidence of micellar

growth

within the

hexagonal

domain, with _a

phase

sequence

I-H-Q~~°

Since the SLS system ^evidences

about constant

anisometry

^{in the H}

region,

with,i

=

1/3 within

experimental

_error

[15],

and has a

phase

_sequence I-H-M

(hexagonal distorted),

_we focus in this paper on systems ^{that show}

cubic

phases

^{before and/or after the}

hexagonal phase.

3.2 RESULTS AND INTERPRETATION. The _{curves a versus} c~,

~~, are

reported

in

figure

5 table III

gives

the values obtained from the

fittings

to the

experimental points

_; results for

70

~mw~n__~ ~

~q ⁵⁰

m

-+- PaLPC

-- oLPc

~ DTAC

30

0.3 0.5

J

0.7

C v,par

Fig. ^5. Hexagonal parameter a as a function of the volume concentration of the

paraffinic

core c~ ~~,

for the three considered systems. Data for DTAC and PaLPC _are taken from references [4] and [8].

Data for OLPC _are from reference [22]. The values obtained from the fittings to the

experimental points

are reported in table III.

OLPC have been taken from Mariani and Amaral

[22].

It could be observed that the

equation (2)

does not fit the data in any case. As it has been shown

theoretically [18-21]

that the exponents ^{1/2 and 1/3} are the

fingerprint

of,

respectively,

infinite (or

flexible) objects

(with volume around

changing

in two

dimensions)

and hard finite

objects (with

volume around the

particles changing

in all the three

dimensions),

we can conclude that, in all the three

different systems, ^the structure elements in the

hexagonal phase

behave like finite

cylinders.

Figure

⁶ shows L/C values calculated in the

hexagonal phase by using equation (I)

on the

assumption

of _{a constant} R value, as estimated from Tanford relation

[30]

and molecular models from the

length

of the

lipid

extended chain

(see

Tab,

III).

Note that due to the

expected polydispersity

in the

cylinder length,

the calculated L/C ratios represent an average value.

Even if the results _are

critically dependent

_on the value

adopted

for the

radius,

_a clear trend of increase of L/C with

increasing

concentration _can be observed for all the three

lipids.

It is clear that DTAC and

PaLPC,

both

presenting

the cubic Q223

phase,

show similar behaviour.

By

contrast, in the _case of

OLPC,

where _an I-H

phase

transition exists, lower values of

L/C _are observed. Both PaLPC and DTAC reach the limit L/C

=

I (« infinite _» micelles, and therefore

Eqs. (3)

and

(7)

no

longer

valid). But PaLPC reaches this limit _at the H-L

phase

Table III.

Analysis of cylinder grow>tli

in the

he,ragona/ phase.

PaLPC OLPC DTAC

A 49.66 ₌ 0.62 44.70 ₌ 0, I1 35,52 ₌ o,14

,1 0.187 _± o-U 0.226 _± 0.002 0. I 18 _± 0.006

c-f. 96.0 fl 99.8 fl 99.0 fl

R

(1)

_{20.5 ±1.0 21.0}

± 1.5 15.4 _± 1-U

i~,

~~,(I

^{H )} 0,lsl _± 0.017

Kc(' jl''

_0.72

± 0.15

v

~

0.2

c~ ~_,,(Q?~~ Hi ^0.261 ± 0,012 0.402 _± 0.008

Kc-[

j/

^'~ l.09 _± 0,16 0.94 _± 0,13

v

~

4.5

c,

~~,(H

^{Q~~'~) 0.423} ± U-U15 0.606 _± U-U14

Kc(

^'

_j~/

^'~ l.00 _± 0.21 .22 _± 0.24

v o~

i~,

~,,,(H

^{L) 0.453} ± 0.008

Ki[~ p/

' l .39 _± 0. 20

V

c-f- is the correlation factor obtained

fitting

the experimental data by

a=Acjj~,

(see text).

R _is the radius oi the

paraffinic cylinder

_)n the

hexagonal

phase, assumed _{constant, as} estimated from the e~tended chain of the lipid~ [30]. In the _case of OLPC, where the double bond bends the chain, the length has been corrected

by using

molecular models. _c, ~,,,lI HI, _c,

~~,(Q~~'-

HI. _i~ ~~,(H Q~~~~) and

~

~,jH L _are

respectively

the

paraffin)c

volume concentrations where the indicated phase transitions

ocjur

[4~ 8, 26. ?7]. The anisotropy _~L ^{ha~ been calculated}

disregarding

the _{errors on} the

Ki(' jl''

values.

Other notations and

symbols

_as reported in the text.

.o

q

~

0.6

--- PaLPC

--- OLPC

-~ DTAC

0.2

0.2 0.4 0.6 0.8

c

Fig.

6. L/C values (see text) calculated in the hexagonal phase on the assumption ^of _a constant radius (see Tab. III) for the three considered systems as a function of the weight concentration _I. Note that for

infinitely

long cylinders the L/C ratio must be equal _to I. The reported lines _are only guides to the eyes.

transition, while DTAC passes

through

this condition still in the H

phase,

before

entering

the

Q~~°

cubic

phase.

Thus OLPC and

DTAC,

while both

presenting

the

H-Q~" Phase

transition, have different L/C values _at this transition.

Let us now

analyze

results

through equation (7).

The values of the function

(Kc(' _j~l'~)

and the

corresponding

anisometries

v, calculated _at the

phase

transitions _on the

assumption

of

constant radius, _are

reported

in table III. The data for OLPC

[22]

_are consistent with the

presence of

nearly spherical

micelles at the I-H transition (v ₌ ^0.2

),

^which grow

along

the H

phase. Infinitely long cylinders

exist _at the H to

Q~"

cubic

phase

transition, The observed

Kc[) j~l''

values indicate _a

growth

of micelles in the

hexagonal

domain also for the DTAC. At the

Q~~~-H phase

transition the finite micelles show _a

large anisometry (about 4.5),

while _at the

H-Q23° phase

transition the micelles appear

infinitely long.

In the case of

PaLPC,

the values of

Kc() j~l'~

^observed at the _two

phase

transitions _are

larger

than the

unity,

_so that their anisometries _cannot be determined _; it should be stressed however that the

large

_errors in

Kc)'pl'~

leave open the

possibility

of finite micelles at the

Q223-H

transition, which grow

along

the

hexagonal region

also for PaLPC.

Results from table III show that, while micelles have small anisometries after the I-H

phase

transition in

OLPC, they

are

quite large

after _a cubic

Q~~~-H phase

transition in both PaLPC and DTAC.

Moreover,

all the three

lipid

systems ^show

long

micelles, _{near to} the _« infinite _» limit, in the

higher

concentration side of the

hexagonal

domain, both at the

H-Q~" Phase

transition for OLPC and DTAC and at the H-L

phase

transition for PaLPC. What _{seems to}

characterize the

H-Q~~° Phase

transition is _{an « extra »} micellar

growth

in the H

phase just

before the transition for OLPC such

growth

is

exponential [22],

while for DTAC the value

L/C becomes

larger

than I. Such _{« extra »}

growth

seems not to occur for PaLPC, ^that has

larger

anisometries _at the

Q223-H phase

transition, but presents a H-L transition

just

before L/C reaches I. In the next sections, the transition

Q~~~-H

and

H-Q~~°

will be

investigated

in

more detail.

4.

HexagonaLcubic phase

transition _:

epitaxial relationships.

4,

HEXAGONAL-Q~~°

CUBIC PHASE TRANSITION. The

epitaxial relationships

between the

hexagonal phase

and the cubic

Q~" _Phase

_were first established

by Ran~on

and Charvolin

65

PaLPC

°~'~

~' 55

I

fl#

~lll ~~~~

45

0.2 0.4 0.6 0.8

c

60

oLpc

~~

~ ~°

d Q~~°

~ ^~~' ~m

40

0.2 0.4 0.6 0.8

c

40

di,,Q~^~' DTAC

~

3

i~

~

i

h~

~ ~Q

~l

I1

~~~~

25

0.2 0.4 0.6 0.8

c

Fig.

7.

Epitaxial relationships

variation of the (10)

spacing

of the

hexagonal

phase observed in the three considered systems as a function of the

weight

concentration _c. The _arrows point to the repeat distance between the (211j

crystallographic

planes observed in the corresponding cubic phases. The reported lines _are only guides to the eyes.

[23]

and

extensively analysed by

Clerc et al.

[?4].

The present results confirm these

relationships.

In fact, the two lattices appear commensurate : as

reported

in

figure

7 for both DTAC and OLPC systems, ^the repeat ^{distance between the}

lo) planes

in the

hexagonal phase

is continuou~ with the repeat ^{distance between the}

(?

ii)

planes

in the cubic

phase.

As

demonstrated in references

[23, 24],

the lo)

planes

of the

hexagonal phase

are in fact

parallel

to the

(211) planes

^of the

Q~"

cubic

phase

both _are the

planes

^of

highest density

in the _two

phases

[14, 23, _see also

10].

Therefore, the two

adjacent mesophases

are

strongly

related _: in

particular,

the

growth

of the

hexagonal phase

is

highly

favoured

along

the

[I

I] ^direction of the cubic structure, or, what is

equivalent,

the

Q~~°

cubic

phase develops along

the direction of the

cylinders

of the

hexagonal phase,

that transforms into a

[I I]

cubic direction

[23, 24].

In order _to obtain information about the mechanism of

phase

transformation, _we show in

figure

8 _some electron

density

_maps traced

perpendicularly

to the

ii

direction _as well _as the

fi

@

fi

Fig.

8. DTAC system perspective view of few sections Ap ^normal to the ill Ii direction and of the

projection

Ap _jjjj~ of the electron

density

_map of the

Q2'°

_phase calculated from the _raw data of table II.

The first section at the bottom contains the point 0,0,0, ^the last _one at the top ^the point

a/2,a/2,a/2 the distance between two successive sections is _a

(,5/24). _According

to the

projection,

the heavy lines indicate the _two different positions where rods form at the transition to the

hexagonal

phase

note that _one third of the

hexagonal cylinders

forms from the successive coplanar

junctions

(for

example,

along the heavy line passing

through

the _center of the sections), the last two thirds form from the threefold

helices (for example, along the heavy line traced in the

right

side). For sake of clarity, the structure elements of the cubic

phase

involved in the process of formation of _two

hexagonal

cylinders are

represented in grey.

JOURN~L DE PHYSIQUE ,i T 4 ~' x ~UGUST 1994 v

corresponding ^projection

^{of the electron}

^density

^{of the unit cell in this}

^direction, Apjj~jj.

In

particular,

the

projection Apjjjjj (see

also

Fig. 9)

shows _a

pseudo-hexagonal

symmetry ^and appears

perfectly superposable

to the electron

density

_map of the

neighbouring hexagonal phase. Considering

the content of the different

sections,

it is

possible

_to

analyse

from _a

geometrical point

of view how the cubic networks transform into

hexagonal

rods, In

particular,

3

hexagonal cylinders

derive from _one cubic unit cell. However, _as

pointed

_out

by

Clerc _et a/.

[24],

at the

transition,

_one third of

hexagonal cylinders

forms from the succession of

junctions

where three

coplanar

rods _are connected, in _tum

belonging

to each 3D networks.

By

contrast, the last two thirds of

cylinders

^{form from the infinite} threefold helices, each helix

being entirely

contained in

only

_one of the two 3D networks.

The first

transformation,

which has been

gracefully

described

by

Clerc _et a/.

[24]

to

imply

« tower _» fluctuations

[see

also Ref.

23],

is characterized

by

^the fusion of the _two

paraffinic

o o

o o

o o

o o

' ' ' ^'

',

' ,

_, ' ,

, ',-,

J

I

' I

J

,' , , , ,

,' . ,J ,

'

'

. I

,'

, '

, ^'

"

Fig. 9. DTAC system. Upper ^frame _: electron density _maps calculated from the _raw data of H sample~

respectively extrapolated

to. the concentration of the cubic

Q22'

_phase (left side) [8] and to the

concentration of the cubic

Q~~'

_Phase (right side). Lower frame projections

Apjjjjj

of the electron

density

_map of the cubic Q~~'

Phase

calculated from the _raw data reported _in reference [8] (left side) and of the electron

den~ity

_map of the cubic Q~~'

Phase

calculated from the _raw data indicated

in table II

(right side). In the maps~ the hexagonal and pseudo-hexagonal unit cells _are marked by heavy lines. The bar represents

501.