In fragmentation, particles of hydrophobic products may be produced by emulsification of the product in water, if it is a liquid, or grinding it down to a very small.
Phys.
J.
II
FYance
H4lAne Olivier
Aggregation
Hydrophobic
of
Chariol
RP, Physique
CEA
mixte
Laboratoire
de
Orsay, France Rhbne-Poulenc, 93308
1997,
FEBRUARY
(~), Anwar Hasmy (~), (3) and Bernard
Lannibois
Aguerre
Equipe
(~)
319-342
Limited
Surfactant in Water
(~
(1997)
7
Robert
Botet
Cabane
(~)
Rhbne-Poulenc, 93308 des Solides, Universit6
PAGE
319
Molecules
(~,*),
Aubervilliers, France Paris-Sud, Bhtiment 510,
91405
(~)
(Received
Aubervilliers,
July1996,
28
PACS.82.70.Kj
received
jn
France
final
form
1996, accepted 25
October
11
suspensions simulation modeling; random phenomena, processes,
Emulsions
1996)
October
and
Computational
PACS.82.20.Wt
PACS.05.40.+j
Fluctuation
and
Brownian
motion
hydrophobic molecules has been studied in presence of hydrophobic grow through aggregation of clusters molecules; the growth is terminated by adsorption of surfactant. accordThe particle qizes vary of hydrophobic molecules surfactant molecules. ing to the concentrations and of Two regimes have been found for the use of surfactant Inolecules: surfactant concentrations, an efficient at low adsorbed the surfaces of the growing particles; at reg1nle where all surfactant nlolecules are on high surfactant molecules left in concentrations, a wasteful reg1nle, where surfactant are excess Attempts to reduce the particle sizes by adding increasing anlounts of surfactant become water. inefficient surfactant These results point where nlost of the added renlains in water. at are sortie between aggregation s1nlulates the conlpetition explained by a kinetic aggregation model which well reThe results of exper1nlents of hydrophobic nlolecules and adsorption of surfactant. are where aggregation is allowed to proceed un1nlpeded for a time r, and produced by s1nlulations deternlined by conditions, particle sizes are adsorption of the surfactant In these then starts. the rate of aggregation and by the value of this t1nle delay. Abstract.
of
precipitation
The
Amorphous
surfactants.
added
in
of
water
particles
Introduction
1.
Many drugs currently reach
to
the
However, the is
much
too
amount
low
range from 10~8 the hydrophobic
development
under
sites,
active
these
drug
of
for
practical
g/g
to
molecules that
can
purposes:
g/g.
10~5
The
be
are
made
must
dissolved
typical classical
of
be
large hydrophobic
carried
across
way
to
achieve
this
in
the
In
order
environment.
aqueous
an
transported solubilities for large and
molecules.
medium
aqueous
hydrophobic transport
is
molecules to
dissolve
particles, I.e. supramolecular objects which may be dispersed in water. Typical carriers are polymeric particles, liposomes, micelles [I-4]. Still, carriers the of drug that be dissolved in most carriers is small, about 10$l of the amount can of where particles Therefore there is need design free delivery carrier system, to mass. a a (*)
©
Author
Les
for
iditions
molecules
correspondence
de
Phvsioue
into
carrier
(e-nlail:
1997
botettllps.u-psud.fr)
JOURNAL
320
drug could be covered with requirements are that the particle
pure
should
small
a
size
orier of100 m2 /g,
be of the
and
PHYSIQUE
DE
the
N°2
surfactant
of
amount
should
II
and
dispersed in specific
0.2 ~J, the
be
smaller
than
amount
of
surfactant
should
be
as
water.
The
surface
area
low
as
possihle.
dispersions may be obtained, either by fragmentation [5-8], or by growth processes fragmentation, particles of hydrophobic products may be produced by emulsification the product in water, if it is a liquid, or grinding it down to a very small size, if it is a solid. with growth processes, the hydrophobic product is dissolved in a solvent that is miscible
Very
fine
[9,10]. of In
In
and
water,
then
then
It is
solution
In all
is mixed
control
to
necessary
macroscopic
a
the
with
causing the
water,
precipitation
the
to
process
molecules
insoluble
precipitate.
to
particles
colloidal
obtain
instead
of
solid.
surfactants are needed, either to help the fragmentation (emulsification aggregation of the fragmented particles, or to control growth. The to prevent of large, sometimes comparable with the the surfactant amounts necessary are hydrophobic product. This reduces the advantage of such delivery systems over the specific toxicity of the formula. In this paper, we carriers, and may cause a non problem of the use of surfactants for one particular case: dispersions made through
these
processes,
grinding) or problem is that
of
of
amount
traditional address
the
precipitation
a
process.
hydrophobic product made of molecules that are nearly insoluble hydrophobic product is dissolved in a solvent that is miscible with The hydrophobic molecules water, and the solution is mixed with a large excess of water. are desolvated. instantly Because they are insoluble in water, they stick to each other whenever they meet, and dissociation do not there is no nucleation reactions In such conditions, occur. barrier, and aggregates of hydrophobic molecules grow immediately through a Diffusion Limited Cluster-cluster Aggregation (DLCA) mechanism [11-13]. If this aggregation is allowed to continue, the aggregates will become so large that they will consider
We
in
"model"
a
(10~8g/g).
water
This
reach
macroscopic
sizes
added
in
order
control
aggregates surfactant
their stick
nanometric
needed
aggregates
As will be
their
interior
adsorbed
obtained.
be
may a
po180ned aggregation final
the
control
to
surfaces.
(VH)
longer
description of
Th18
cause
no
fall
this
not
will
(precipitation). However, surfactants out of solution may be growth. Indeed, surfactant molecules adsorb on the growing may hydrophobic tails. Hydrophobic surfaces that are fully covered with saturated surfactant with to each other; hence, aggregates that are recombine with other aggregate8. In this way, a stable dispersion of
and
through may
molecules
by
to
shown
of the
sizes
below,
the
platicized by residual solvent and would require a ratio of surfactant
is
surfactant
given by:
volume
suggests
proces8
should aggregates aggregates are often
~ ~
be
that the
where
r
is
0.06.
thickness
the
desired
particle
However,
a
of the
diameters
100
are
general finding
is
that
nm,
the
which
of
surfactant covers
their
water.
Therefore
volume
(Vs)
a
to
particles, bemonolayer coverage hydrophobic volume
~~
layer
surfactant
amount
spherical, amorphous
~~~
R
the
the amount
and
of the particles. Typically, Vs/l~H should be of the order of needed to control the growth of surfactant inefficient of a monolayer coverage. This use
and
thus
amount
R
the
the
radius
ratio
vastly in excess of surfactants have possible molecules available origins: either the surfactant two not may are the particle surfaces during the but they do they available, aggregation near are process, or not protect the surfaces efficiently. formation of nanoparticles by aggregation of hydrophobic In the following, we report the surfactants; of different competition molecules in conditions have been set for the presence between aggregation and poisoning by surfactants, and the resulting particle sizes have been of
aggregates
in such
conditions
is
SURFACTANT
N°2
measured.
In
molecules
and
poisoned
order to interpret these data, we have (or poisons) are moved surfactants
aggregation
probabilities
has
reproduced
been
AGGREGATION
LIMITED
with
designed set
a
kinetic
a
random
at
of
321
model a
on
that
rules
hydrophobic
where
lattice.
The
determine
process
of
sticking
the
all species. This simulation shows of surfactants: two regimes for the use where the regime, surfactant molecules efficiently, used but the aggregation an are proceeds to rather large sizes, and a "wasteful" regime, where a vast excess of surfactant is used aggregation at an earlier stage. The comparison with experimental results shows to stop the that the preparation of nanoparticles is usually conducted in the "wasteful" regime where of the fraction surfactant in the phase. Dispersions of nanometric remains a large aqueous particles (sizes below 100 nm) cannot be obtained in the "efficient" regime, unless the surfactant adsorbs efficiently on the growing aggregates than the hydrophobic molecules themselves. more
of
"efficient"
Materials
2.
hydrophobic product used in most precipitation experiments was cholesteryl acetate. This product was chosen because it has a very low solubility in water (2 x 10~8 g/g), a 8ubstantial solubility in polar solvents that are m18cible with water (e.g. 0.034 g/g in acetone at 23 °C), becau8e it of drugs currently in development. and resembles number It is a solid imp a obtained be of lls °C) which in number crystal [14-16], amorphou8 structures can a or in an microcrystalline 8tate. or We also used a liquid hydrophobic product, hexadecane imp 18 °C), in order to compare emulsification. the sizes of particles prepared through precipitation The solubility and through The
=
=
of
hexadecane For
deuterated
mass
x
10~l~ g lg.
experiments,
polystyrene to gift from Claude
generous AI = 15000
a
is I
water
scattering
neutron
used was
in
g/mol.
make
Picot
at
to
cores
of the
the
Institut
hereafter
noted
It is
intended the
determine
location
the
of the
hydrophobic nanoparticles. Charles Sadron; it had an
surfactant, we This polymer average
molar
PSD.
ethoxylated fatty alcohol with 12 carbons in the named C12E5, and a diblock copolymer groups, made of a styrene block (molar mass 1000) and an oxyethylene block (molar mass 1000), (type VPSE 1010). Goldschmidt hereafter PS-POE. obtained from named This polymer was hydrophobic surface, they from micellar solution When these surfactants adsorbed a on are a form monolayers where the area per molecule is 0.48 nm~ for C12E5 and 1.28 nm~ surfactant for PS-POE; the thickness of these layers are 0.5 nm and 2.5 nm, respectively. The hydrophobic molecules initially dissolved in polar solvents which were acetone (for were tetrahydrofuran and C12E5) and hexadecane cholesteryl with PSPOE), ethanol (for acetate surfactants
The
alkyl
linear
(THF) (for The
chain
PSD
variables
used
and 5
this study oxyethylene
in
were
an
hereafter
PS-POE).
and
which
define
the
composition
of the
mixtures
are
cH,
initial
concentration
of
solvent. of surfactant in the hydrophobic molecules in the solvent, and cs, initial concentration mainly g/g. of is found that the the precipitation Both expressed It in outcome was are determined by the value of the ratio cs/cH. Typical values for the composition were: cH, 10~3 molecules in of hydrophobic concentration to 3 x 10~~ g/g; ratio cs/CH, 5 x 10~~ to 10; final of solvent in water, 10~~ g/g. concentration water, 10~5 to 3 x 10~~ g/g; final
3.
Methods
3. I.
PRECIPITATION.
First, the
follows. Then
the
solution
precipitation of hydrophobic product and the surfactant injected into a large volume of The
hydrophobic was
molecules were
water.
in
water
was
dissolved
in the
At
moment
the
performed as polar solvent. of
injection,
JOURNAL
322
PHYSIQUE
DE
II
N°2
hydrophobic molecules was on the order of 105. This very high aggregation of the hydrophobic molecules, as described in determined by the relative introduction. the The aggregation stopped at a stage that was of hydrophobic molecules, cH, and of surfactant, cs. The final dispersion had concentrations droplets covered with a very small hexadecane colloidal stability in all ca8es, excepted for local supersaturation caused supersaturation
of
the
of
amount
surfactant.
QuAsi ELASTIC
3.2. were
calculated
[17,18].
It
immediate
was
SCATTERING.
LIGHT
from
their
found
that
the
The
sizes
were
sizes of the
through Quasi
measured
motions
in the
particles in the final dispersion Light Scattering (QELS)
Elastic
50 to 500
range
nm,
which
this
particles
were
for
method
is
adequate. 3. 3.
ANGLE
SMALL
NEUTRON
[17,18].
SCATTERING.
The
8tructures
of the
examined
by differences in scattering length between nuclei in the particles and nuclei in the phase. Good contrast between particles and continuous phase was achieved using D2U as the continuous phase. This contrast used to continuous was determine whether the particles were Alternatively, the hydrophobic molecules porous or dense. deuterated, while the surfactant remained hydrogenated. This and contrast water was were surfactant determine and the of the within the particles. Twcused to the location structure obtained Dll at ILL. These dimensional the instrument scattering patterns patterns were on of intensity, I, vs. magnitude, subsequently radially averaged to yield scattering curves were and to the Q, of the scattering vector. Q is related to the wavelength, I, of incident neutrons scattering angle, 9, by: through
SANS
Neutrons
scattered
are
Q The
general
(a)
features
phase
continuous
Q
of
are
as
scattering
curves
I
Sin
=
for
Ill
independent
12)
particles
dispersed
in
a
homogeneous
follows [19]:
intensity, Io, is proportional to particle mass and to the concentration in density of length, Ap, between D2U and the protonated scattering square molecules in the particles. Hence, the average of the particles may be determined; content (b) at low values, the rays scattered by nuclei at opposite ends of the particle interfere destructively. In this regime, the radius of the scattering the curvature curve measures average of gyration, Rg, of the particles. This determined by fitting the measured quantity may be intensity curve to an expansion formula. For globular particles the best expansion is the Guinier at
of the
-+
0
the
difference
formula:
~2j~2
IIQ)"Ioexp
~~
j3)
(c) at higher Q values, the intensity decay reflects interferences shorter within each distances at particle, I-e- their internal For dense globular particles (no internal structure), the structure. intensity follows Porod's law, from whidh the surface area A of the particles may be determined:
1(Q) particles only according
G3
AQ~~
hollow shells, the destructive interference is not as strong, Q~~i for rod-like particles it is Q~~ For bushy or porous dimension di the decay follows the power law Q~df Thus, from the slope fractal decay in a log-log plot, the dimensionality of the particles may be determined. For flat
or
is
to
(4) and
the
decay
particles with of the intensity
SURFACTANT
N°2
AGGREGATION
LIMITED
323
Q~ q.
d '
1,
~ll
,
'~
Q
l/Lm
~~
it
$~f~
~i
""
~
Fig.
1nlage of
TEM
1.
dispersion, aggregation of
the
a
then
it
nlost
particles,
cholesteryl
dried
was
in
and
air the
dispersion
acetate on
collodion
a
in
water.
before
nlenlbrane
recrystallization
of
Uranyl acetate was observation; this
added caused
to
the
few.
a
ELECTRON MicRoscoPY. The particle shapes and sizes also observed through were Cryo-Transmission Electron Microscopy (CRYO-TEM) [20]. Thin liquid films were prepared perforated carbon films supported on 3 mm electron microscope grids. A specimen was on prepared by applying a small (about 3 ~Jl) drop onto the grid, blotting most of it to the desired thickness (under 200 nm) and plunging it into liquid ethane at its melting point. This ultrasolid-like (vitreous) fast cooling caused vitrification of the liquid phase, I.e., specimens became These without change of phase that leads to structural cryo-specimens rearrangement. were cooling-holder (Gatan 626) of the TEM stored and transferred under liquid nitrogen, to the (JEOL 2000FX), where they were equilibrated at -170 C, and examined with an acceleration voltage of 100 kV. 3.4.
°
3.5. how
ComPosiTioN surfactants
All
VARIABLES.
these
aggregation. The iv; second, as a normalized surface area, by the surface area that would be created the particle surfaces. These quantities are the
values
of the
compared 4.
with
composition the
Experimental
laws
variables
OBSERVATION
cholesteryl acetate dispersions The image (Fig. I) shows a
range,
a
broad
and
the
two
if all
or
in
the
cs/cHi
the
kinetic
as
surface
growth
area
in
order
that
an
molecules
used
were
final
dispersions,
laws
for
these
study
to
results
particle average of the dispersion
area
surfactant
the
measured
cs,
surface
of
first,
ways:
total
the
I.e.
performed
were
loss
made
divided to
cover
different
at
variations
from volume
then
are
models.
Results
4.I.
and
cH,
predicted by the
step, TEM.
measurements
aggregation process results expressed in are
the
limit
oF
THE
distribution
THROUGH
DisPERsioNs
of
were
dried
collection sizes.
It
ELECTRON
collodion
on
a
of
globular particles,
can
also
be
seen
MicRoscoPY.
membrane
that
with
aid
sizes
extensive
In
examined in
the
a
first
through 100
aggregation
nm
and
JOURNAL
324
"l'l I
PHYSIQUE
DE
N°2
II
fit,
pot J
'~G
~
[~
Fig.
10~~ g/g and cs/cH
"
fi1nl is
This
1nlage
CRYO-TEM
2.
(cH
particles 1nlportant
held in
are
a
vitrified
features
the
are
a
cholesteryl
dispersion
acetate
fi1nl of
The
vitrified
in
appears which is also
PS-POE
with
water
water
as
unifornl
a
as
surfactant
a
background. cholesteryl acetate grey
The perforation in a carbon fi1nl, seen. film, pushed by the meniscus against the edge of the fi1nl. water good dispersion of the particles in water, their globular shapes and
The the
crystallinity.
of any
absence
across
the
of
0.1).
=
0.2 /Lm
recrystallization
of All subsequent examination shows particles shown Figure 2 in image were confined in the meniscus of the vitrified the edge of one of the perforations in the water film, near and globular amorphous non-aggregated, with carbon film. The particles structure an appear of A CRYO-TEM. confirmed observation through This shapes. in measurement was every performed on an image where the particles were uniformly the particle sizes (diameters) was dispersed in the vitrified water film; the resulting size distribution is shown in Figure 3. some
dispersions
the
molecules
were
dissolved
were
100
were
particles
times
It
shows
the
values
of the
The the
was
acetone,
acetate, A smaller.
found
average initial
radii
initial
vary
that
and
acetate
and
dispersion
the
in
times.
for
in
between
The
In
surfactant
first
a
was
set of
the
precipitated
was
experiments, the hydrophobic copolymer PS-POE. Both
diblock
in
The
water.
10~~ g/g;
PS-POE, 5 x 10~~ to 10~~ g/g; colloidal obtained dispersion was in every immediately through QELS, measured were calculated
volumes cH
of
15 and 30 nm, cu
=
the
remained
sizes
concentration
concentration
drying.
of
result
a
SizEs. the
solution
the
these
particle
as
CRYO-TEM.
PARTICLE
oF
cholesteryl
cholesteryl
were:
through
STUDY
SYSTEMATIC
4.2.
occurred
have
made
same
from
cholesteryl and
10~~ g/g.
the
acetate
volumes
They
are
than
more
over
these
radii, in
at
initial
the
concentrations
final
case.
and
then
"
of the
sizes
again
at
later
Figure
month.
one
cs/cH
concentrations
The
I, for
4
different
acetone.
between
14
x
always larger
10~ and than
the
l10
x
10~ nm3,
volumes
that
SURFACTANT
N°2
AGGREGATION
LIMITED
325
o.06
~ 0.04 w
c
0.02
~~~ 0
20
lo
30 s
Fig.
3.
Distribution
10~~ g/g vitrified
of
cs/cH
and
would
The
water.
automatically
by
a
obtained
be
particle
surfaces:
surfactant efficient
would use
molecules
of this
of
"
of particle sizes (diameters) ns nleasured CRYO-TEM on a 1). The 1nlage contained 1000 particles, unifornlly distributed particles were selected individually and their diameters s were
conlputer
as
a
cs/CH much
then
=
fi1nl
nleasured
the
cs/cH
surfactant, increased.
system;
it is:
The
=
they
rise
law for
this
and
performed experiments were surfactant, both dissolved in "
the
surfactant had been used to limit aggregation by covering the I, the particle volume corresponding to an efficient use of the be 4000 nm3. At high dilution, the particle approach the limit of volumes if all
at
is
0.4), the larger than
1nlage (cH across
progranl.
away rise
from it when
measured
the
volumes
volumes
with
were
C~
high
At
larger for
an
to
the
concentration
be the
same
at
all
cH of hydrophobic compositions cs/cH
(5)
CH.
hexadecane
ethanol.
calculated
found
was
v~v Other
40
(nm)
than
as
hydrophobic
a
surfactant those
efficient
of use
molecule,
concentration
C12E5
and
(cs/cH
"
I
and
cholesteryl acetate particles, and of surfactant; again, they follow
(cs/cH 0.04), However, at low surfactant concentration 10~~ g/g) of efficient successive observed: regimes large dilutions regime two (cH < at were a of the (denoted by arrows in Fig. 4), and at higher concentrations, again the surfactant use (rising straight line in the log-log representation of regime of wasteful use of the surfactant Fig. 4). the
described
law
in
equation (5).
"
the particle volumes generally larger than those predicted for an efficient are surfactant, and they rise with the of hydrophobic molecules in the concentration initial solution, according to equation (5). We have also initial measured systematically the particle volumes obtained with the same dilution cH but different values of the ratio cs/cH (Fig. 5). The data for and C12E5 hexadecane surfactant show a fast decrease of the that the volumes at cs/cH < 0.05, indicating average additional is used efficiently, followed by a slow decrease at cs/cH > 0.05, indicating that the surfactant is mostly wasted. In
use
summary,
of
the
JOURNAL
326
PHYSIQUE
DE
N°2
II
io~
tl
~
io~
'
D
n
57
E 10
> I
o
io~
io~
cH Fig. ispersions
surfactant; overall
dilution
squares:
S-POE,
an
slope
order it is
to
these
put
useful
to
cs/cH
notions
calculate
cH,
by
1.
Filled
of
squares:
in
arrow and
log-log scales,
of
efficient
the
amount
or
been
as
wasteful
of
predicted
a
by
fit
linear
nlodel.
kinetic
has
that
volunles
the
a
are
surfactant
of the
surface
kept
cs/cH
at
of
alues
lines
filled
use
particle
C12E5
the
indicates
the
been cholesteryl
changed. Hollow
and
hexadecane
The
has
nlolecules
has
surfactants,
the
product and
hydrophobic
to
surfactant
0.04.
"
efficient use is 1 data
the
of
of
"
at
12E5
for
The
In
cs/cH
at
and
calculated
ontaining
measured
olution,
given
a
cs/cH
the
of
and exadecane
solutions
ratio
the
acetate
basis,
of
precipitation
ade
quantitative a protected from
on
been
aggregation by
calculated the adsorption of surfactant. The surface of the particles may be area efficiently, surface radii, assuming a globular shape. If the surfactant used this was surface surfactant Thus would equal the of monolayer the molecules. all containing area area a The values of quantities is an indicator of the effectiveness of the surfactant. the ratio of both presented in Figure 6. this ratio are from
their
factant
0.05, choose
droplets
hexadecane
For
molecule
followed a
by
surface
The
wasteful
a
area
by C12E5,
covered
[21,22].
calculated
regime.
of1.28
nm~
For
ratios
show
cholesteryl surfactant
per
choose
we
a
a
surface
regime of
acetate
molecule
particles [23].
of 0.47
area
efficient
The
use
covered calculated
nm~
per
sur-
cs/cH
to
up
by PS-POE, ratios
=
we
show
a
surface is kept than what is covered by the surfactant. In regime where, surprisingly, more fact, cholesteryl acetate particles can be obtained in the absence of any added surfactant. This cholesteryl acetate the particles to repel each in is because contains ionic impurities that cause medium where dissociation ionic possible. Therefore cholesteryl dispersions is the acetate any the always regime of surfactant. in are excess In
summary,
sufficient
amounts
it is of
not
possible
surfactant:
controlling the aggregation. of the wasted surfactant; this
to in
This
in
is
one
particles of arbitrarily small fraction of the cases, only a snlall result be improved unless one cannot of the goals of the next section. obtain
such
volumes surfactant
understands
by adding is
efficient
the
fate
SURFACTANT
N°2
AGGREGATION
LIMITED
327
~~9 tl
n
a "
~
io
~
©1f
n
~S ~
~ ~ "
l~~
C
~
>
o
~
io~
o
O
io~ io
~
io
io
~
C~/C~ Fig.
of surfactant concentration particle volumes iv. Each set of data the average cs on dispersions made from the precipitation of solutions containing a given hydrophobic surfactant; the overall dilution of the solution, measured by cH, has been kept a constant, but the ratio cs/cH of surfactant hydrophobic nlolecules has been changed. Hollow dianlonds: to Cholesteryl acetate and PS-POE, at cH C12E5 0.01 g/g. Filled hexadecane and at cH squares: 0.04 g/g. Hollow hexadecane and C12E5 at cH 0.1 g/g. The lines are a guide to the eye. squares: 5.
Effect
corresponds product and
to
=
=
=
STRUCTURES
4.3.
poisoned The
.
final
of the
sition
PARTICLES
OF
aggregation
can
aggregation
voids .
less
preserved
structures
uous
collapse The
amount
than
the
of
reordering
of
Quantitative contrasts
surfactants A first without is are
The
between shown
initial
Where
is
subunits, answers
observe
to
the
in
become
to
either
surface
should the
have
or
these the
the
to
number
a
Diffusion is
there
of
structures
The and
to
process
the
of
compo-
reasons.
Aggregation of Clusters [11-13]. Are such tenreordering process by which the Limited
structures
porous
or
or
SURFACTANT.
THE
a
dense? from
protected by
the
surfactant: remain
the particle radii is usually much adsorption of surfactant (see the been trapped inside the particles by reasons? phase for kinetic
calculated
that is been
wasted did it
fractal
particles,
final
structures
that
with
aggregates
hydrophobic
amount
previous Sect.). the
the
and
interesting for
particles through
colloidal
of
(DLCA) usually produces
OF
according
characterized
This is
aggregates.
LOCATION
AND
also be
in
the
has
it
aqueous
obtained through SANS, using the appropriate questions were scattering from hydrophobic molecules and surfactants, or from
alone.
particles made of cholesteryl acetate, with or phase made of D20. In this case the contrast nanoparticles. These scattering and all the molecules that form the water curves followed by a steep decay. downward in Figure 7: they show an initial curvature measured is related to the particle sizes according to equation (3) the sizes curvature
set
of
PS-POE
experiments as
a
was
surfactant,
performed in
an
with
aqueous
JOURNAL
328
PHYSIQUE
DE
N°2
II
o
~ p
g 10° fi
~w ~
Q
_i
.
~
1
O
o Z
~
tl
io°
cs/C~ Fig.
Surface
6.
nlolecules at
cH
0.01 g lg.
=
C12E5
and
yield
a
values
at
cH
indicate
that
through
g/g.
0.1
"
area
must
A
second
set
of
surfactant, hydrogenated high ratios of
in
low
at
ratios
vanished
has
a
in
curves
was
made
there
are
size
on
the
order
of I
flat
objects (Q~~ These
tation
is
are
molecules as
follows.
particles In
is wasted; higher aggregation process.
the
with
agreement
those
particles: according to Therefore, aggregation give dense
to
of PSD, the
case
of
good
made
this
PS-POE,
hexadecane squares: would the surfactant
structures.
using is
contrast
PS-POE between
as
the
and
"
"
"
other
IQ
objects
and the
observed
only. The
that
A fit of the
law)
Power
features
surfactant
l1nlit
in
surfactant
and
surfactant
sortie
the
to
shoulder
that
are
Hollow use
if all
acetate
obtained the deuterated phase. The scattering at curves aqueous hydrophobic molecules (cs /cH 0.5 or Ii show a Q~~ decay, and Q of samples made 0.3 nm~~ (Fig. 8). The scattering near curves 0.05 to 0.25) show the but the shoulder initial law decay, same power
surfactant
surfactant
shows
with
D20.
of
features transcription of these into real decay comes from flat such structures
A initial
Q
of
range related to
that
surfactant
of
this
is
performed
phase
aqueous
(cs/cH (Fig. 8).
adsorption
the
kept
be
Cholesteryl
0.04 g lg. Efficient
indicate
values
would
that
area
dianlonds:
and C12E5 at cH = guide to the eye.
Lower
besides
experiments an
characteristic
a
by
a
surface
the
Hollow
shape of the law indicates that the particles are dense. reordering process that collapses the voids
Power
followed
are
steep decay
The
be
a
lines
unity.
scattering
QELS. equation (4), the Q~~ from
events
to
forces
other
fit of the
a
obtained
The
equal
ratio
surfaces.
hexadecane
squares:
by
divided
the
on
Filled
surface
particles, particle
of the
area
adsorbed
were
In
hollow
also
shells
is
follows.
as
hollow
to
The
The
shells.
contribute
the
Q~~
shoulder
scattering;
of the are
where location
surfactant
the of the
coherent
these
of the
Q 0.3 nm~l objects have a scattering from =
of 7.6
nm.
scattering is caused by the molecules, the interprethe nanoparticles. that cover
surfactant
monolayers
law
power at
is obtained by adding the curve scattering from small spheres with a diameter
scattering
dispersions
in terms
space as
N°2
SURFACTANT
AGGREGATION
LIMITED
329
~ io~
~
*i
°
O
§p
(
O
O
~3
~
~ d~ 10~ j ~
slope=.4
~~i
io° Q
Fig. 7. nanoparticles
Angle
Snlall
D20.
in
cholesteryl io-~ g/g).
dianlonds:
lcu
=
The
change
have
an
thickness calculated
cs/cH
"
Q~~ (Fig. 7) to Q~~ (Fig. 8) particles. The fit to the scattering
to
be
8.2
thickness,
uniform
nm
These
of 24 than
for rather
at Q (Fig. 8);
show a
and
that
as
=
indicates
of at
the
a
that
this
in
standard
a
thickness
dispersion
the
results
but
nm
the
acetate
PS-POE
with
dispersions cholesteryl containing acetate particles, without surfactant. Hollow surfactant, conlposition ratio cs/cH a
of
curves
cholesteryl
pure
particles
acetate
thickness average is much larger 0.10.
diamonds:
from
of the
interior
Scattering
Neutron
Filled
(nm~)
there
range
deviation
cs/cH
layer formed of thick
"
0.25
and
surfactant
and
is
thin
thickness
PS-POE 3.I does
no
obtained
of the
monolayer of
adsorbed
is
nm
not
surfactant with
of 5
the
in
shells
that
nm.
This
molecules, which is for the dispersion at form a monolayer of
patches.
nm~~ is
observed in the dispersions that contain a large interpretation is that this scattering originates from surfactant micelles that Indeed, the average molar mass of these micelles, meain water. are sured through gel permeation chromatography, is 300000 g/mol. From this value, the aggregation number is calculated to be 150 PS-POE molecules, and the radius of the PS core is calculated This is in good agreement with the size of the small spheres used in to be 3.9 nm. (the water swollen POE shell would the fit of the scattering contribute only at smaller curve Q values). The
excess
In
small
excess
of
scattering
surfactant
sunlmary,
micelles
the
that
surfactant are
"
an
is
dispersed
0.3
obvious
located in
water.
in
irregular
monolayers covering
the
particles
and
in
JOURNAL
330
PHYSIQUE
DE
N°2
II
io~
io~
~~3 W
~ C
~ ~
10
W ~
fi d~ ~~o
I
i ~o
Q(nm
Fig. 8. surfactant,
Scattering
~)
PS-POE dispersions containing PSD nanoparticles covered with as a Taking final concentrations equal to 5 x 10~~ g/g for both PS-POE ii; THF, 0.1 g/g. The full lines correspond to the calculated and PSD (cs/cH for scattering curves (outer dispersion containing large shells diameters beyond, standard deviation thickness 24 a on nnl, thickness, 5 nm) and snlall micelles (dianleter 7.6 nm). Filled circles: Concentrations equal to 5 x 10~~ g/g and 10~~ g/g for PS-POE and PSD, respectively (cs/cH 0.05) THF, 0.1 g/g. The dashed line corresponds to the calculated diameters for a dispersion containing large shells (outer scattering curve nnl). beyond 175, thickness 24 nnl, standard deviation thickness, 5 on in
D20.
of
curves
Hollow
circles:
=
=
Numerical
5.
Simulations
of
the
Poisoning
Growth
MODEL. coalescence have been inNumerical models for droplets growing by accounted for in mechanism previously [24], however, the poisoning have not been growth poisoning these models. introduce lattice model to simulates In the following, we a a mechanism exhibed by our experimental system. as THE
5.I.
troduced
System. We have considered lattice limited in a cubic box of edge length sites considered. For the particle species H (hydrophobic molecule) and S (surfactant) are starting configuration, time t 0, a number NH of particles H and Ns of particles S, each one of a cubic site size of unit length, are randomly and uniformly in the box, avoiding distributed Concentration, or overlaps. Each particle is then a unit cell containing six faces of unit area. related to Ns and NH by: volume fraction, of species are 5.I.I.
L.
The
Two
=
cH
~) =
L
(6a)
SURFACTANT
N°2
cs
In
algorithm
our
introduce
we
(S specie) adsorption by cluster iH (formed by possible.
Algorithm.
The
5.1.2. iteration
include
to
r
one
more
ore
a time delay particles of the
in
H
which
surfactant
species)
becomes
iterative procedure, in which at a given consists in an (under the assumption that random is is picked at or a up coefficient proportional to n°, a is the kinetic is exponent equal to -) for a compact system of n particles), according to probability distributions, for H
diffusion
its
(6b)
algorithm
3-dimensional
The
iH
cluster
a
331
)~.
"
parameter
time
a
AGGREGATION
LIMITED
surfactant
species:
~j
£~~ nj(
~~"
for S species
and
~~~~
+
ns
~n ~Jn
~~~
~~~~
~ ~~
Then, a displacement of a unit step is tried by possibilities (+1, 0, 0), (0, +1, 0), (0, 0, +1), taking (PBC). If the moving object does not try to occupy a site of periodic boundary care occupied by other object, the displacement is performed. In the opposite case, no displacement surfactant is performed and: (I) if the collision is between a free and a surface site of specie H (belonging to a cluster iH), a bond is established them provided the time t is larger between than r, and the incremented by one. (ii) if the collision takes places mass n~~ of the cluster iH is between different clusters, and two (or more) sites (H or S species) belonging to two (or more) each one with at least one free surface site, we proceeds to the "coalescence" of the clusters by configuration with minimum surface. arranging the new mass (= n~>~ + n~ii~ +.. in a compact Then, we arrange on the new surface the total number of surfactants isl + isii +.. adsorbed previously by the coalescing clusters. If this number exceeds the surface, the exceeding new surfactants distributed randomly in the empty sites of the box. (iii) no change is again are where
at
adsorbed After another
between
is
in
According
to
free
a
surfactant
a
situations,
above
the
each
below 5.1.3.
t
=
are
goes
has
been
by choosing randomly
on
is stopped at a given surface by the joisoning effect, or if there is no (7) the time t is calculated adding:
described experiments regime, i.e. when cH +
have
been
calculated
Two-Dimensional
configurations
of
if its
time more
tend in than
which
all
cluster
one
~~ £~~
+
~~~ ns
iteration.
As the diluted
algorithm
the
simulation
The
~~
at
(even
surfactant
another
and
given cluster).
surfactant.
or
the six
among conditions
saturated
are
box.
the
collision
surfactants.
direction
a
by the surface of considering one of cluster
clusters
of free
number
random
if the
made
in
the
is
ns
choosing
at
randomly
coalescence,
and
we
been
the
All
addressed results
in
the
reported
=
We
our
white
or
observe
have
percolation threshold). cs < cp (cp the end of the growth process (t tend)-
with
in
calculations
numerical
our
is
Illustration.
obtained
0, H (depicted in grey, distributed
above,
the that
box. some
show in
algorithm when
a
For
0
of
three
cluster
is
c
0.5
~ ~ 0
20
60
40
V 2.0
(b) 1.5
~
I-o
c
o.5
o-O 0
50
40
30
20
lo
V
Fig.
14.
and
Cs
All
these
Size =
+
H
weight
results
curves
adsorbed H
distribution
on or
is
H
its +
attached
front
surface.
an
At
S, leading to
each
for CH
curves
4CH (hollow circles).
to
In
=
(b) Cs
average
2000
over
each
time
a
different
event,
=
0.001
4CH,
step,
and L r
=
=
50.
In
la)
(circles),
r
r =
=
0, Cs
2500
=
2CH (black circles)
(squares)
and
r
=
8000.
s1nlulations.
investigates
all
by
coalescence
or
~
~~~~
one
state
0
possible binary events: A probability adsorption.
the
namely:
Kin,jn
~
l~~
~
~ ~
(14) ~~
338
for
of two
coalescence available
is
DE
PHYSIQUE
radii R~~
and
Rj~, provided
II
N°2
least
at
adsorption
free
one
site
cluster, and:
each
on
of
clusters
JOURNAL
kj~
s
l)
+
Rm
"
'
(R~~
~~"
II
I)
+
(15)
EIH
(assumed here to be of radius I). The terms I/R~~ are due adsorption of a surfactant of the diffusion, clusters during their and the sections to R~~ + Rj~ are just the cross Brownian path. Then one event is chosen according to these probability weights, and the time increased by the is amount: for
Stokes
dt This
leads
~
equations:
kinetic
the
to
j16)
~ ~~
=
dnj,s~ it) ~~~~~
dt
~
~
I~J',J-J'"J',s~>11 j-j',s~_~i
j',s~i,s~_~i
~
~?l~~
~JS"J,sj
=
equations
These
difficult
too
seem
RESULTS.
6.2.
There
ment.
The
is
that
advantage
geometry,
no
"~'~? "~
~'~
model
there
is
of the
is
then
more
no
are
kinetic
this
validity for ordinary three-dimensional space is not the results. In Figure 15, we have plotted the on poisoning ratio cs/cH, for T 0, and NH 256. =
for
which well
are
recovered.
for the
-IA
surfactants
the
kj,slip,sj
-illp
lip
other
And
we
one.
Note
"
used
are
obtain
that
ji~~~
Ej+j ).
+ SJ
efficiently, the
An
alternative
quite similar geometry.
the
to
study
to
way
model
described
in
approach is that it is simpler to impleMoreover, since one can write down be tempted in simple Nevertheless, cases. described in the preceding section, and its ascertained. This validity has to be checked
Smoluchowski
simulations
equations, analytical investigation can approach is a mean-field theory of the model
the
l~J
~J
analytically.
handled
the
the
so
~
be
to
simulations:
preceding section, except
the
kj,sllj,sj
+
J"SJ
~'~J
Monte-Carlo
is
~
"p +
~'~~
them
Kj',jllj',s~> llj,sj
j',s~i
these
two
cluster
average
We
that
see
exponents
versw
the
regimes (small- cs/cH,
two
of surfactants) is excess cs/cH values regime, and slightly larger than those obtained
for
-2.7
(Eq. (9))
volume
the
large- cs/cH,
and
exponents
fast.
the
are
that
small-
simulations, possibly as a of the absence of geometry in the consequence Smoluchowski approach. The behaviour of all other computed data are quite similar and are not reproduced here. This shows clearly that the Smoluchowski approach can be used in this process, and that the two regimes meet at about the same point. For example, the ratio of the surface lost by poisoning surface lost by coalescence obtained from equation (13), decreases with cH (whqn to the sp /sc, aggregation becomes faster) and increases with cs /cH (when poisoning becomes efficient). more
in
the
This
last
trend
linear
law.
most
of the
of
Carlo
Monte
surface
surface
in Figure 16: the rise explained in the following surface is lost by coalescence,
illustrated
is
This
be
can
initial sites
sites
in
in
the
an
H
molecule
geometrical
(here, algorithm
we
for
of
sp/sc
with
cs/cH
At
very
low
way..
choose cluster
therefore a
=
sizes
sc
6.5 up
m
aNH
match
to to
3).
remarkably close to a concentrations,
is
surfactant where
the All
a
the
is
values surfactants
of
number
number are
used
of to
SURFACTANT
N°2
AGGREGATION
LIMITED
339
~~3
slope=-2.7
$
~>
lope=-lA
~~o
10
c~
Fig.
log-log plot
15.
Snloluchowski to
the
different
power
the
of
quantity
calculations.
nunlerical
donlains
law
iv We
that
-1
best
fit
the
versus
NH
considered
=
data.
our
poisoning 256
This
cs/cH
ratio
and
r
=
result
curve
The
0.
front
full an
obtained lines
from
the
correspond
average
over
sp/sc
versus
1000
simulations.
io'
slope=1
~o
~
~~~~Jf
~~~~
~
~
o ° o
~
,
io~ c
Plot of the Fig. 16. poisoning ratio cs/cH the slope1.
surface obtained
lost
from
by poisoning the
to
Smoluchowski
s/c
the
~
surface nunlerical
lost
by
coalescence
calculations.
The
full
line
the
indicates
JOURNAL
340
aggregation,
stop
therefore
sp
Ns.
t
~~ ~
sc
high
very
ing
more
to
while
cH,
concentration,
surfactant
than 2
In this
molecules.
that
of
forming
an
limit
H-S
II
N°2
Thus, ~~
At
PHYSIQUE
DE
~~
(~8)
~f
cH
hydrophobic molecules probability of forming is proportional to cs.
the the bond
again proportional to cs /cH. The complete that aggregation is the dominant process
aNH
in
/sc,
variation
of sp
most
practical
form
never
H-H
an
clusters
bond
Therefore
shown
in
situations
the
is
demonstrates
cs/cH
to
sp/sc
ratio
Figure 16,
(up
contain-
proportional
is
"
4).
Discussion
7.
We
this
start
compared for
the
discussion
with
the
of the
role
with
a
from
results surfactant
in
brief
of experimental summary Finally, simulations.
numerical
precipitation
results. we
try
to
These
results
put
together
are a
then model
processes.
solution The precipitation of hydrophobic insoluble yields particles that are completely amorare Stable obtained when particles phous, dense and globular. dispersions the are are aqueous function surfactants. surfactants In these kept apart by adsorbed by impurities that or as by two parameters: stable dispersions, the particle sizes are determined the poisoning ratio cs/cH, I-e- the ratio of surfactant dilution molecules ratio to hydrophobic molecules, and the hydrophobic molecules the original solution. the of solvent in ratio I.e. to cH, The of particle sizes according to these variations parameters show that there are 2 regimes, surfactants the differ in the way used. At very low of surfactants, which concentrations are indicating that surface of the particles matches the of a monolayer of the surfactant, area area surfactant used to keep particles apart and limit their aggregation. At higher all molecules are of surfactant, the surface of the particles is much less than the of a concentrations area area monolayer of the surfactant (Fig. 6). Neutron scattering experiments on dispersions made in this regime reveal that part of the surfactant is adsorbed the particles, and part of it forms micelles in water. on excess Thus, attempts to reduce the particle sizes by adding increasing of surfactant amounts become inefficient at some point where added surfactant remains in most of the water. 7.I.
SUMMARY
molecules
7.2.
in
EXPERIMENTAL
oF
conditions
COMPARISON
oF
where
RESULTS.
they
EXPERIMENTS
wiTH
NUMERICAL
SIMULATION.
The
kinetic
model
aggregation of hydrophobic molecules and The two-dimensional poisoning by surfactant. illustration presented in Figure 9 shows qualexperimental particles covered with adsorbed surfactant itative with the results: agreement with single molecules, also covered by surfactant, Quancoexist and with surfactant. excess (Figs. 10 and II) show a regime of efficient results from the 3-dimensional simulation titative of the surfactant followed by a regime where most surfactant Comparison with is wasted. use experiments (Figs. 5 and 6) shows that, in the simulation, the regime of efficient use extends (cs/cH of 0.04); as a result, the parsurfactant concentrations 4 instead to much higher ticle volumes have been reduced to extremely low values (taking I nm as the diameter of a iv t 10 instead of 107 at the crossover). This discrepancy originates from the time monomer, delay between aggregation and surfactant adsorption, as explained below. assumed that surfactant In the efficient regime of this simulation, it was adsorption starts at molecules. there is a time the In the experiments, time as the aggregation of hydrophobic same delay between the two caused by the finite rate at which water diffuses into the solvent that A similar delay was introduced surfactants. initially contains hydrophobic molecules and in demonstrates
the
effects
of the
competition
between
"
SURFACTANT
N°2
AGGREGATION
LIMITED
341
concentrations, this simulation yields the same results aggregation leaves surface enough bind all to as one area (Fig. 10). However, at high surfactant the surfactant concentration, the particle sizes remain those obtained with no time delay, because the initial above aggregation is not prevented by Comparison surfactant. the with the shows that the actual experiments time delay is excess that used in the still much larger than simulations. A simulation conditions would in these have to handle particles of extremely large sizes. Alternatively, the hydrophobic molecule used could be rescaled to be an aggregate simulation in the containing 10~ molecules. In the wasteful regime, the experiments show that the volume per particle increases average linearly with concentration (Fig. 4). The numerical simulation reproduces this trend and provides a simple explanation for it. Indeed, the particle volume rises linearly with concentration only in the limit of long delays (Fig. 12). Therefore the rise in the final volume reflects the unimpeded aggregation during the time delay before the beginning of surfactant adsorption. previous
the
The
At
simulation.
another
linear
with
rise
surfactant
low
because
the
initial
concentration
indeed
is
expected
that
aggregation [I Ii. the results of experiments In conclusion, are allowed to proceed unimpeded for a time t,
from
equation for
Smoluchowski
the
kinetic
is
At
there is, concentration, molecules; thus aggregation
surfactant
low
well-reproduced by adsorption then point,
that
at
simulation
and
surfactant
enough
than
more
aggregation
if
of the
surface
starts.
adsorb
to
until it is limited because all remaining continues poisoned. Thus the surfactant molecules used efficiently, but the final partiare cle volumes large. At high surfactant adsorption of surfactant stops the concentrations, are immediately for Consequently, the aggregation after the time delay adsorption. aggregation numbers determined by the length of delay this time and by the rate of aggregation, I.e. are by the of hydrophobic concentration molecules. Adding more surfactant does not change these particle volumes; consequently the excess surfactant wasted. is
all
surfactant
8urfaces
7.3.
are
MODEL
and
FOR
PRECIPITATION
THE
hydrophobic
tion of
molecules
in
As in any
PRocEss.
water
can
be
divided
kinetic
stages:
three
into
precipitapropagation
the
process,
initiation,
termination.
with the solution containing the hydrophobic they hydrophobic molecules because aggregate, surfactants do bind the insoluble this mixed At however, the in solvent. this not to stage, are growing aggregates, because they are not sufficiently hydrophobic. According to Propagation proceeds through random collisions of the growing aggregates. with that is Smoluchowski volume linearly time, the equation, the average at a rate grows molecules. proportional to the of hydrophobic concentration Initiation
caused
is
and
molecules
Termination
ticles. In
The
The
practical
situations,
solution
final cause
will
be
high the
pressure most
the
may
a
set
the
dilute that
homogenizer could attractive
option.
to
it the
achieves
be
that
for
this
the
the
endeavour.
particles at will just be the
necessary between
to
be
time
delay
instantaneous to
surfactants
sets
therefore
and
will
adapted
the
dispersions
obtain
cover
aggregation, that
high
termination
surfactant
excess
Reducing
device
so
limitations
needed
of
so
is
content
severe
since rate
be
water
water
the
aggregation and usually desired to
amount
aggregation.
requires
surfactant a
reduce
dispersion further
it is
above
beyond the be efficient,
not
the
delay between
described
concentration
stage will
mixing of Immediately,
the
when
occurs
time
results
by
surfactant.
achieve
that
fine
as
are
Increasing the
aggregation
par-
particles. possible. surfactant
aggregation
Diluting sizes;
the
initial
however,
it,
reconcentrate
water
as
the
end of the
wasted.
the
on
size of the
particle
mixing of this
adsorb
final
which
may
adsorption and polar solvent.
instantaneous
and
mixing,
it
the
of If
would
JOURNAL
342
DE
PHYSIQUE
N°2
II
Acknowledgments This
work
the
BIOAVENIR
us
used
(A.H.)
the
beams
neutron
financed
program like to acknowledge
would
of ILL
by
in
Grenoble,
France.
RHONE-POULENC
support
from
It
with
CONICIT
was
the
performed
MRE
(Venezuela)
and
as
MICE.
and
of
part
One of
(France).
CNRS
References
[Ii [2] [3] [4]
[5]
F., E., Fessi H., Puisieux
G.,
Barratt
Vauthier
Al-Angary A. 1 (1992) 277.
and
C.
Halbert
P., Couvreur P., Devissaguet J-P-, Dubernet C., Fattal S., Polym. Biomater. (1994) 749. and G. W., Drug Targeting Delivery (Microencapsulation of drugs)
Couarraze
Benita
Pharm. Biopharm. 39 (1993) Allemann E., Gurny R. and Doelker E., Eur. J. Gregoriadis G., Liposomes as drug carriers (Wiley, 1988). Julienne M.C., Alonso M.J., Gomez Amoza J.L. and Benoit J.P., Drug Dev. Ind. 18
(1992)
173.
Pharmacy
1063.
Koosha F. and Muller R.H., Archiv. Pharm., suppl. 321 (1988) 680. A., Cabane B., Pharm. Res. 12 (1995) 39. Kaplun Talmon Y. and Sj6str6m B., [7] Controlled Veillard M., J. G., Chabot and Spenlehauer Vert M., Benoit J.P., F. [8] [6]
(1988)
Pharmace~t.
J.
[10]
StoInik 30
[I Ii
N.,
Alkhouri-Fallouh
[9]
S.,
(1994)
Jullien
Rel.
7
217.
R.
Davies
Roblot-Treupel
(1986)
28
M.C.,
L., Fessi H., Devissaguet
J-P-
and
Puisieux
F., Int.
125.
Illum
L., Davis S.S.,
Boustta
M.
and
Vert M., J.
Controlled
Ret.
57.
and
Botet
R., Aggregation and
Fractal
Aggregates (World Scientific, Singapore,
1987). Meakin P., Phase Critical Transitions and Phenomena, vol. 12 (Academic Press, New York, 1988) p. 335. Growth Phenomena (World Scientific, Singapore, 1989). [13] Vicsek T., Fractal [14] Wendorf J.H. and Price F.P., J. Phys. Chem. 75 (1971)18. [15] Wendorf J.H. and Price F.P., Mol. Cryst. Liq. Cryst. 25 (1974) 71. [16] Kunihisa K.S. and Gotoh M., Mol. Cryst. Liq. Cryst. 42 (1977) 97. [17] Cotton J.P., Neutrons, X-rays and Ligth Scattering, P. Linder and Th. Zemb, Eds. (ElsePublishers, 1991) p. 3. vier Science C. and G., Analyses chimiques et caractAiisations, Techniques de Weill [18] Strazielle l'IngAnieur PI, pp. 1065-1-1065-22. Surfactants Sciences Surfactants methods of investigations, solutions: [19] Cabane B., new Series, R. Zana, Ed., vol. 22 (Marcel Dekker Inc., New York, 1987) p. 57. Microz. Techniq~e10 [20] Bellare 3.R., Davis H. T., Scriven L. E. and Talmon Y., J. Electron
[12]
(1988) [21]
87.
Lichterfeld
F., Schmeling T. and Strey R., J. Phys.
Chem.
90
(1986)
[22] Olsson U., Wlrz U. and Strey R., J. Phys. Chem 97 (1993) 4535. O., private Sonneville communication. [23] Meakin P., Rep. Prog. Phys. 55 (1992) 157. [24] for a review see: [25] Family F. and Meakin P., Phys. Rev. A 40 (1989) 3836. [26] van Dongen P.G.J. and Ernst M.H., J. Stat. Phys. 50 (1988) 295.
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