THE JOURNAL OF CHEMICAL PHYSICS 129, 134903 共2008兲

Lamellar to micelle transition of nonionic surfactant assemblies induced
by addition of colloidal particles
Yukiko Suganuma,1 Naohito Urakami,2 Rina Mawatari,1 Shigeyuki Komura,3Kaori Nakaya-Yaegashi,1 and Masayuki Imai1,a兲
1Department of Physics, Faculty of Science, Ochanomizu University, Otsuka,
Bunkyo, Tokyo 112-8610, Japan2
Department of Physics, Biology and Informatics, Yamaguchi University,
Yoshida, Yamaguchi 753-8512, Japan3
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University,
Minami-Osawa, Hachioji, Tokyo 192-0397, Japan

共Received 2 January 2008; accepted 14 August 2008; published online 7 October 2008兲
We have investigated the entropic interactions between lamellar membranes and spherical colloidal
particles using a small-angle neutron scattering 共SANS兲 technique. By adding colloidal particles
between lamellar sheets, the first lamellar peaks in SANS profiles become intense and the secondand higher order Bragg peaks begin to appear, indicating that the membrane fluctuations aresuppressed by the colloidal particles. We estimate the interlamellar interaction potential in thepresence of the colloidal particles from the layer compressibility obtained by the SANS profile
analysis and propose a phenomenological free energy model based on the restriction of membrane
fluctuations. By further addition of the colloidal particles, the lamellar membranes transform toprolate micelles. In order to release the strong frustration due to the restriction of membrane
fluctuations, the surfactant assemblies change the morphology from the two dimensional sheets tothe one dimensional prolate micelles. © 2008 American Institute of Physics.
关DOI: 10.1063/1.2978385兴
I. INTRODUCTION

Nowadays complex systems consisting of various kindsof soft materials, such as polymers, colloidal particles, liquid
crystals, and amphiphilic assemblies, have acquired great interests, because these complex materials are widely observedin biological systems and are extensively used in pharmaceutical, cosmetic, food, and other fine chemical products. By
doping guest particles into a host mesoscopic structure
共mesostructure兲 of soft matter, the dominant interactions thatstabilize the mesostructure are modified by additional entropic interactions originated from the geometrical restriction. The most familiar entropic interaction is the depletioninteraction,1–3 which is an attractive interaction between thehost colloidal particles. The depletion interaction was firstrecognized in the spherical colloid and polymer mixture systems and then observed in various soft matter complex systems, such as rod and sphere mixture systems.1,4–12 In thesesystems, by overlapping the depletion zone around the hostparticles, the guest particles obtain large free volume. Increase of the free volume raises the translational entropy ofguest particles but at the expense of lowering the entropy of
mixing. If the gain in free volume is sufficient, the host particles aggregate, i.e., the attractive interaction between thehost colloidal particles. This concept is easily extended tomore complicated systems, such as membrane/polymer13–17
and membrane/colloid18–20 systems. The polymer doped
lamellar membrane systems were investigated theoreticallya兲

Author to whom correspondence should be addressed. Electronic mail:imai@phys.ocha.ac.jp.

0021-9606/2008/129共13兲/134903/10/$23.00

by Daoud and de Gennes,13 and revisited by Brooks andCates.14 Their theoretical predictions were experimentallyconfirmed by Ligoure et al.15–17 By doping neutral polymer
chains between the lamellar sheets, the attractive interlamellar interaction is induced so as to obtain the large free volume for the guest polymer chains by overlapping the depletion zones of the adjacent membranes. This attractive
interlamellar interaction 共depletion interaction兲 destabilizes
the lamellar membranes and finally causes a phase separation
into a dilute lamellar phase where the polymer chains are
localized in a concentrated lamellar phase.
On the other hand, the repulsive interlamellar interaction
is induced by doping neutral colloidal particles without specific interactions with the membranes.18,19 The repulsive interaction originates from the restriction of the lamellar fluctuations by the presence of the colloidal particles. The
restriction causes strong frustration to the lamellar membranes and finally destroys the structure.19
On the basis of the above results, we systematically investigate the structural transition of lyotropic lamellar phaseinduced by the addition of colloidal particles. We use thenonionic surfactant lamellar system, where the interlamellarinteractions are governed by the Helfrich interaction21 andthe structural changes of lamellar membranes induced by theaddition of colloidal particles are followed by a small-angle
neutron scattering 共SANS兲 technique. The intermembrane interaction mediated by the presence of the colloidal particles
is estimated from the layer compression modulus B̄ obtainedby the SANS profile analysis. In order to explain the effectsof colloidal particles B̄, we propose a phenomenological free129, 134903-1© 2008 American Institute of Physics

134903-2

J. Chem. Phys. 129, 134903 共2008兲

Suganuma et al.

energy model. Further addition of colloidal particles brings amorphology transition of the surfactant assembly. In order todetermine the morphology of surfactant assemblies, we performed contrast matching experiments of SANS, where theinterference between the scattered neutrons from membranes
and colloidal particles can be neglected. We will discuss themorphology transition from entropic interactions induced bygeometrical restrictions.

to 0.01% and 0.02%, and varied ⌽s in the range of
3.1% ⬍ ⌽s ⬍ 8.3%. It should be noted that in both series themean interlamellar distance d̄ 共d̄ = d − ␦, d: lamellar repeat
distance and ␦: membrane thickness兲 is fairly larger than thesize of the colloidal particles. The samples in the lamellar
phase without colloidal particles were monophasic andslightly cloudy; however, in the presence of the colloidal
particles, the samples became transparent.

II. EXPERIMENTS

B. Instruments

A. Samples

SANS measurements were carried out using the
SANS-U instrument of Institute of Solid State Physics,
University of Tokyo at JRR3-M reactor of the Japan Atomic
Energy Research Institute in Tokai.23,24 In this experiment weused neutron beam having a wavelength 共␭兲 of 7.0 Å
with the resolution of ⌬␭ / ␭ = 15%. The diameter of theirradiated neutron beam was 5 mm and the scattered neutrons were detected by a two dimensional 共2D兲 position
sensitive detector with the sample to detector distances of
1, 4, and 12 m, which covers q 共magnitude of scatteringvector, q = 4␲ sin ␪ / ␭, 2␪: scattering angle兲 range from0.002 to 0.3 Å−1. The two dimensional isotropic scattering
patterns were corrected for heterogeneity of detector, background scattering, and the incoherent scattering intensityfrom the protonated substances. Then they were circularly
averaged to obtain one dimensional scattering profiles. The
calibration to the absolute intensity scale was performed using a standard sample of Lupolene.24

In this study we investigated ternary mixtures ofwater/C12E5/spherical colloidal particle. The nonionic surfactant C12E5 共purity⬎ 98% 兲 was purchased from Nikko
Chemicals Inc. and used without further purification. The
phase behavior of the C12E5/water system is well characterized and we can control the interlamellar distance by asimple dilution law.22 As the spherical colloidal particles, weused the polystyrene latex purchased from Magsphere Inc. as
a 10% suspension in H2O. The purchased latex particles ofradius Rc = 80 Å have surface charge ⬃0.1 e / nm2 but nosteric layer covering colloid surface. In order to remove theexcess surfactants and impurities, the latex solutions weredialyzed using cellulose tubes 共Viskase Companies Inc, poresize 40– 50 Å兲 against purified water for about two weeks.
The completeness of the dialysis was verified by adsorption
spectrum measurements in the region of 200– 400 nm. We
confirmed by the SANS profiles that the dialysis did not
induce the aggregation of the colloidal particles.
In this study we performed two types of SANS experiments to make clear the structures of surfactant assemblies,
one is a standard SANS measurement and the other is thecontrast matching measurement. The standard SANS experiments for surfactant and colloid mixture systems were performed using D2O / H2O mixture solvent with the composition of 85/ 15, except for the sample with ⌽c = 1.3% 共⌽c:volume fraction of colloidal particles兲. For the mixtures with⌽c = 1.3%, we used the solvent with the composition of
D2O / H2O = 48/ 52. On the other hand, for the contrast
matching experiments, the D2O / H2O mixture solvent had amatching composition of 28.5/ 71.5. Details of the contrast
matching experiment will be described in a contrast matching technique of SANS section.
The ternary mixtures composed of water/C12E5/colloidal
particle were prepared by the following procedure. The required amounts of surfactants 共C12E5兲 were dissolved in colloidal suspensions with prescribed colloid concentrations andthe mixtures were homogenized using a vortex mixer atroom temperature 共micellar phase兲. We increased the sample
temperature 共T兲 to the measurement temperature 共57 ° C:lamellar phase兲 and then equilibrated the sample for several
minutes after the homogenization. During the scattering
measurements at 57 ° C, we regulated the temperature within⫾0.1 ° C of accuracy. In this study we prepared two series ofsamples. In the first series we fixed the volume fraction of
surfactant to ⌽s = 3.1% and varied the volume fractions of
colloidal particles in the range of 0 % ⬍ ⌽c ⬍ 3.0%;
on the other hand, in the second series, we fixed ⌽c

C. Contrast matching technique of SANS

Generally scattering function from the surfactant assembly and colloidal particle mixture is given by
I共q兲 = 共␳c − ␳w兲2Scc共q兲 + 2共␳c − ␳w兲共␳s − ␳w兲Scs共q兲+ 共␳s − ␳w兲2Sss共q兲,

共1兲

冕

共2兲

Sij共q兲 =具␾i共ri兲␾ j共r j兲典exp关− iq · 共ri − r j兲兴dr,

where ␳i and ␾i共r兲 are the scattering length density and volume fraction of component i 共i denotes the components c, s,
and w for colloid, surfactant, and solvent, respectively兲. In
order to obtain the scattering function from the surfactant
assemblies without the interference effects, the scattering
length density of the solvent is chosen to be the same as thatof the colloidal particles. The contrast matching point wascalculated by the following expression:

␳D2O ⫻ ␾0 + ␳H2O ⫻ 共1 − ␾0兲 = ␳c ,共3兲where ␾0 is the volume fraction of D2O in the solvent, and␳D2O and ␳H2O are the scattering length densities of D2O
and H2O, respectively. Using ␳D2O = 6.36⫻ 10−6 Å−2,
␳H2O = −0.56⫻ 10−6 Å−2, and ␳c = 1.41⫻ 10−6 Å−2, we obtained the D2O volume fraction at the matching point as
␾0 = 0.285. Actually the SANS profile of the colloidal suspension at the matching point showed no significant scattering from the colloidal particles. Under this matching condition, the scattering function is expressed by

134903-3

J. Chem. Phys. 129, 134903 共2008兲

SANS studies of lamellar to micelle transitions

I共q兲 = 共␳s − ␳w兲2Ns Ps共q兲Ss共q兲,

共4兲

where Ns is the number density of the surfactant assemblies,
and Ps共q兲 and Ss共q兲 are the form factor and the partial structure factor of the surfactant assembly.

III. DATA ANALYSIS

Pneut共q兲 =

N−1
n=1

冋

In order to estimate the intermembrane interaction potential, we extract the Caillé parameter ␩ of fluctuating membranes from the SANS profiles. The Caillé parameter ␩ 共Ref.
25兲 is defined by
q20kBT冑

8␲ KB̄

共5兲

,

where K is the bulk bending modulus 共K = ␬ / d, where ␬ isthe bending modulus for a single membrane兲 and q0 is the
position of the first Bragg peak. In the case of the lamellar
phase containing guests, the layer compressibility B̄ is related to the interlamellar interaction potential per unit bilayer
area Fl−l byB̄ = d冦⳵2Fl−l
⳵d̄

2

−

冋

1 ⳵Fl−l−⳵2Fl−ld̄ ⳵⌽̄c ⳵⌽̄c⳵d̄⳵2Fl−l

⳵⌽̄2c

册

2

冧

,

共6兲

B. Analysis of SANS profiles of lamellar phase

The Caillé parameter ␩ is a parameter to describe thecorrelation function of the fluctuating lamellar membranes
and can be extracted from the static structure factor of the
SANS function. Nallet et al.27 proposed a model scatteringfunction for the randomly oriented lamellar membrane
system2␲ P共q兲S共q兲,
dq2

with a form factor冉 冊 冉⫻exp −

qdnncos
N
1 + 2⌬q2d2␹共n兲

2qz2d2␣共n兲 + ⌬q2d2n
2共1 + 2⌬q2d2␹共n兲兲

册冑

冊

1

1 + 2⌬q2d2␹共n兲

,

共9兲
where N is the number of layers. The correlation function offluctuating membranes ␹共n兲 is given by␹共n兲 =

␩关ln共␲n兲 + ␥兴,
4␲2

共10兲

where ␥ is Euler’s constant. The width of the resolutionfunction is expressed by ⌬q and we adopted the following
expression:⌬q2 = q2

冉 冊

冋

册

R22 共⌬R兲2
1⌬␭ 2 共2␲兲2 R21++,
2 3 2 +3
8 ln 2 ␭
12␭
L ⬘2
L1
L22
共11兲

where ⌽̄c is the volume fraction of the colloidal particles inwater layers.15,26 Thus, we can evaluate the interlamellar interaction potential from the compression modulus B̄. In theprevious paper, we showed by means of neutron spin echomeasurements that the presence of the colloidal particles
does not affect the bending modulus.19 Then the modificationof interlamellar interaction induced by the presence of colloidal particles can be evaluated through the Caillé parameter␩. In the following, we explain the procedures to obtain ␩from the SANS profiles.

I共q兲 =

共8兲

where ␦neut is the thickness of membrane determined by the
SANS, ␴ is the measure of membrane thickness distribution
fixed at ␦neut / 4, and ⌬␳ is the scattering contrast between themembrane and the solvent. The structure factor S共q兲 is givenbyS共q兲 = 1 + 2 兺 1 −A. Estimation of intermembrane interaction␩=

2共⌬␳兲22 2
关1 − cos共q␦neut兲e−q ␴ /2兴,
2
q

共7兲

L⬘ =

L 1L 2,L1 + L2

共12兲

where R1 is the radius of the source aperture 共10 mm兲, R2 isthe radius of the sample aperture 共2.5 mm兲, L1 is the incident
flight path length, L2 is the scattered flight path length, and⌬R is the detector resolution 共5 mm兲.28 For the data analysis,
we used the isotropic 2D scattering patterns indicating randomly oriented lamellar systems.
IV. RESULTS AND DISCUSSIONS
A. Phase behavior of lamellar membrane and colloidal
particle mixtures

In the previous paper,19 we reported that the addition of
the neutral polymer into the flexible lamellar membranes induces an attractive interlamellar interaction and finally destabilizes the lamellar phase. The attractive interaction can beinterpreted by the depletion interaction for the membraneand polymer mixture systems. In contrast, the spherical colloidal particles confined between the lamellar membranes
suppress lamellar fluctuations, which results in the effective
repulsive interlamellar interaction. Figure 1共a兲 shows a seriesof SANS profiles as a function of volume fraction of thecolloidal particle ⌽c at a constant volume fraction of thenonionic surfactant C12E5, ⌽s = 3.1%. Here we shifted the
SANS profiles vertically to avoid the overlapping of the profiles. The standard profile is the top profile in the figure andeach profile is multiplied by a factor of 0.1. Without thecolloidal particles, the scattering profile shows monotoni-

134903-4

Suganuma et al.

FIG. 1. 共a兲 Change of SANS profiles by addition of colloidal particles into
the lamellar phase as a function of ⌽c at constant ⌽s = 3.1% and T = 57 ° C.
For the mixtures with 0 % ⬍ ⌽c ⬍ 0.13%, the solvent has the composition of
D2O / H2O = 85/ 15 and for the mixture with ⌽c = 1.3% the solvent has thecomposition of D2O / H2O = 48/ 52. 共b兲 Change of SANS profiles by additionof colloidal particles into the micellar phase at 16 ° C 共⌽s = 3.1%, ⌽c= 0.0% in D2O / H2O = 85/ 15 solvent and ⌽s = 3.1%, ⌽c = 1.3% in
D2O / H2O = 48/ 52 solvent兲. For both figures, the standard profile is the top
profile and each profile is multiplied by a factor of 0.1 to avoid overlapping.

cally decreasing profile having no characteristic lamellar
peaks, which is due to the large membrane fluctuations. The
addition of colloidal particles causes the emergence of thefirst Bragg peak. With an increase of ⌽c, the first Bragg peak
becomes sharper and the second and higher order Bragg
peaks begin to appear, indicating that the membrane fluctua-

J. Chem. Phys. 129, 134903 共2008兲

tions are suppressed by the colloidal particles. When ⌽creaches ⬃0.1%, the scattering peaks shift toward higher qside, indicating a phase separation. Further increase of⌽c causes a drastic change of the scattering profile at⌽c ⬃ 1.3%, as shown in Fig. 1共a兲. The Bragg peaks characteristics of the lamellar phase disappear and the scatteringprofile in low q region becomes almost flat.
In Fig. 1共b兲 we show the change of SANS profiles by theaddition of colloidal particles into the micellar phase at
16 ° C 共⌽s = 3.1%, ⌽c = 0.0% in D2O / H2O = 85/ 15 solutionand ⌽s = 3.1%, ⌽c = 1.3% in D2O / H2O = 48/ 52 solution兲.
The scattering profile of pure C12E5 micellar solution at
16 ° C is well described by the prolate core-shell model asshown later. By the addition of colloidal particles into themicellar phase, the scattering profiles are modified. The
SANS profile for micelle+ colloid system is very close tothat obtained by the addition of colloidal particles into thelamellar phase shown in Fig. 1共a兲. These profiles suggest thatthe lamellar assembly is transformed to micellar structure by
the doping of the colloidal particles.
We show the phase diagrams of the C12E5 and colloidal
particle mixture system determined by eye observation in
Fig. 2, 共a兲 ⌽c − T phase diagram at ⌽s = 3.1% and 共b兲
⌽c − ⌽s phase diagram at T = 57 ° C. In the figures, open
circles, crosses, and filled circles indicate lamellar phase
共L␣兲, coexisting phase 共L␣ / L1兲, and micellar phase 共L1兲, respectively. In order to construct these phase diagrams, we
needed to equilibrate the samples for several days.15 The⌽c − T phase diagram indicates that with the increase of ⌽c,the micellar phase expands to higher temperature region, andaccordingly the lamellar phase disappears. This phase behavior is consistent with the ⌽c − ⌽s phase diagram, Fig. 2共b兲.
Thus SANS profile change shown in Fig. 1共a兲 corresponds tothe lamellar to micellar phase transition via the coexistence
region.
Here we mention the nature of colloidal particles in the
C12E5/water system. When we added a very small number ofcolloidal particles 共⬃0.004% 兲 into the dilute lamellar phase共⌽s = 3.1% 兲, the SANS profiles showed a lamellar peak, asshown in Fig. 1共a兲, indicating suppression of lamellar membrane fluctuations. Whereas when we added the same number of colloidal particles into the micellar phase, the scattering profiles did not show any change 关not shown in Fig.
1共b兲兴, indicating that the micelle structure and intermicellarinteraction are not affected by the addition of this very small
number of colloidal particles. Thus the observed suppressionof lamellar membrane fluctuations is caused by the geometryof surfactant assemblies and not by the specific interaction
between the colloids and the surfactants. Next we consider
whether the colloidal particles are confined between thelamellar sheets or not. As shown in Figs. 2共a兲 and 2共b兲, when
we added a small number of colloids to the lamellar phase,the system kept the homogeneous one phase appearance, indicating that the colloidal particles are homogeneously distributed in the sample. In the one phase region, the SANS
profiles of lamellar membrane and colloid mixture 关Fig. 1共a兲兴showed that the lamellar peak position keeps constant134903-5

J. Chem. Phys. 129, 134903 共2008兲

SANS studies of lamellar to micelle transitions

FIG. 2. Phase diagrams of C12E5 + colloidal particle mixture system; 共a兲⌽c − T phase diagram at ⌽s = 3.1% and 共b兲 ⌽c − ⌽s phase diagram at T
= 57 ° C. For both figures, open circles, crosses, and filled circles indicate
lamellar phase 共L␣兲, coexisting phase 共L␣ / L1兲, and micellar phase 共L1兲,
respectively.

against ⌽c. These results are strong evidences for theconfinement of colloidal particles between the lamellar
membranes.
B. Interlamellar interactions mediated by additionof colloidal particles

When we confine a small amount of colloidal particles
between the lamellar sheets, the scattering functions from thelamellar structure show the characteristic changes. In Fig.
3共a兲 we show detailed scattering profile change as a functionof ⌽c at ⌽s = 3.1%. The addition of colloidal particles causes
the sharpening of the first Bragg peak and the emergence ofthe second and higher order Bragg peaks. It is noteworthy
that a very small number of the colloidal particles⌽c ⬃ 0.003% brings the change of scattering profile. On theother hand, when we increase ⌽s while keeping ⌽c = 0.01%,
the lamellar peak position shifts toward high q side withdecreasing peak intensity, as shown in Fig. 3共b兲. Thus theregularity of lamellar membranes decreases with decreasing
the interlamellar distance. This is quite different from thebehavior of pure nonionic surfactant/water systems. In thecase of the pure nonionic surfactant/water systems, the amplitude of membrane fluctuations increases with increasing
interlamellar distance, which makes the lamellar peak obscure. These results indicate that the confined colloidal particles modify the original interlamellar interaction 共Helfrichinteraction兲.
The mediated interlamellar interaction is examinedthrough the layer compression modulus, which can be calculated from the Caillé parameter ␩ and the bending modulus
␬, as expressed in Eq. 共5兲. The Caillé parameter was obtainedby fitting of the observed SANS profiles with the model
scattering function for lamellar membranes described in Sec.
III. In Fig. 3共a兲 we show the results of fitting by the solidlines and the extracted parameters are tabulated in Table I.
Here for pure lamellar structure 共⌽s = 3.1% 兲 in the absenceof colloidal particles, we show a predicted profile assumingthat the interlamellar interaction is governed by the Helfrichinteraction.29
The variation of obtained ␩ as a function of ⌽c is plottedin Fig. 4共a兲. The Caillé parameter decreases monotonicallywith an increase of ⌽c, indicating the membrane fluctuations
are suppressed by the colloidal particles. In order to obtainthe layer compression modulus B̄ from ␩, we have to estimate the value of the bending modulus ␬. In the previous
paper, we estimated the bending modulus from the dynamical structure factor of fluctuating membranes obtained byneutron spin echo measurements.19 Using the values of ␩ and␬, we calculated the layer compression modulus B̄, as a function of ⌽c, as shown in Fig. 5共a兲. The layer compression
modulus B̄ increases with an increase of ⌽c in the range of
0.002% ⬍ ⌽c ⬍ 0.02%. Thus the addition of hard sphere colloids brings strong repulsive force between the adjacentlamellar membranes.
On the other hand, the scattering profiles from the dilute
lamellar phase showed a characteristic ⌽s dependence, as
shown in Fig. 3共b兲. In order to elucidate the characteristic
behavior, we extracted the Caillé parameters from the SANS
profiles. The results of fitting are shown by the solid lines inFig. 3共b兲 and the obtained parameters are tabulated in Table
I. The obtained Caillé parameters at ⌽c = 0.01% and 0.02%
are plotted as a function of interlamellar distance d̄ in Fig.
4共b兲. The Caillé parameter decreases monotonically with increasing d̄ for the both cases. When the interlamellar interaction is governed by the Helfrich interaction, the Caillé parameter is expressed by␩ = 2␲

冑冉

冊 冑冉冊

␮
␦Hel 2
1−
= 2␲
3
d

␮ d̄
3 d

2

,

共13兲

where ␦Hel is the effective membrane thickness based on thesteric hindrance and ␮ is related to the local displacement of
membranes u共r兲 by the following geometrical expression:

134903-6

J. Chem. Phys. 129, 134903 共2008兲

Suganuma et al.

TABLE I. Obtained parameters by fitting of SANS profiles of lamellar
membrane+ colloid systems.
⌽s / %⌽c / %

d/Å

␩

3.1
3.1
3.1
3.1
3.1
3.1
3.1
3.1
3.1
3.1
3.1
3.1
4.1
4.1
5.2
5.2
6.2
6.2
7.2
7.2

0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.010
0.010
0.012
0.014
0.019
0.010
0.020
0.010
0.020
0.010
0.020
0.010
0.020

995
946
1003
979
1004
980
1014
992
1006
997
974
975
755
760
573
593
499
507
424
432

1.01
1.17
0.85
0.82
0.63
0.68
0.54
0.51
0.52
0.50
0.50
0.46
0.50
0.43
0.63
0.49
0.63
0.61
1.01
0.72

具u2共r兲典 = ␮d̄2 .

共14兲

In the previous paper we confirmed that the pure C12E5/water
system satisfies the simple relation expressed by Eq. 共13兲,
which gives the values of ␮ = 0.284 and ␦Hel = 36 Å.29 Thus,
Caillé parameter increases with increasing d̄, which is incontrast to the experimental results. By combining the bending modulus ␬ and the Caillé parameter ␩, we obtained thed̄ dependence of the layer compression modulus at⌽c = 0.01% and 0.02%, as shown in Fig. 5共b兲. When thelamellar membranes confine colloidal particles, B̄ decreases
with an increase of d̄.
The interlamellar interaction in the presence of colloidal
particles shows two characteristic behaviors, B̄ increases
with an increase of ⌽c and decreases with an increase of d̄.
To describe these behaviors, we already proposed a model inwhich the colloidal particles suppress the fluctuations ofmembranes.19,30 Following the Helfrich formulation,21 thefree energy of the interlamellar interaction in the presence of
colloidal particles per unit area of the membrane is given by

Fl−c =
FIG. 3. A series of SANS profiles of lamellar membrane+ colloidal particle
mixture system as a function of ⌽c at ⌽s = 3.1% 共a兲 and ⌽s at ⌽c = 0.01%
共b兲. Both measurements were performed at 57 ° C. For both figures, the solidlines indicate the results of fitting using model scattering function proposed
by Nallet et al. For pure lamellar structure 共⌽s = 3.1% 兲 in the absence of
colloidal particles, we show a predicted profile assuming that the interlamellar interaction is governed by the Helfrich interaction. For both figures, thestandard profile is the top profile and each profile is multiplied by a factor of
0.1 to avoid overlapping.

共kBT兲232␮␬

1d̄2 tanhn

冉 冊
1

,

共15兲

w⌽̄effcwhere n and w are the constants describing the degree ofsuppression of membrane fluctuations by the colloidal
particles.30 In this expression we introduced a phenomenological volume fraction of colloidal particles between lamellar sheets ⌽̄effc given by

134903-7

J. Chem. Phys. 129, 134903 共2008兲

SANS studies of lamellar to micelle transitions

FIG. 4. Caillé parameters ␩ of lamellar membrane+ colloidal particle mixture. 共a兲 ⌽c dependence at ⌽s = 3.1% with theoretical prediction based on
Eq. 共20兲 indicated by the solid line and 共b兲 d̄ dependence at ⌽c = 0.01% and0.02% with theoretical predictions based on Eq. 共20兲 indicated by the solidline for ⌽c = 0.01% and the dotted line for ⌽c = 0.02%.

⌽̄eff
c = ⌽̄c

冉冊
d̄
Rc

m

共16兲

,

where the exponent m is an unknown parameter and should
be determined by the comparison with experimental results.
The layer compression modulus for the lamellar membranes
confining colloidal particles can be calculated using Eqs. 共6兲
and 共15兲 as
B̄l−c =

共kBT兲2 g共⌽̄effc 兲

d,32␮ ␬d̄4 h共⌽̄eff兲
c

冉 冊

冉 冊

冉 冊
冉 冊

2Rm
2
c
cosh
+ 2共m − 1兲mnw
w⌽̄eff
c

冉 冊

Rm
2
c
d̄m⌽̄c sinh
共3 − 4m − m2兲
eff
w
w⌽̄c

冉 冊
冉 冊
冉 冊 冉 冊冉 冊冋 冉 冊
冉 冊 冉 冊
冉 冊册

−3

冉 冊

Rm
4
c
d̄m⌽̄c sinhww⌽̄effc
4+ 3d̄2m⌽̄2c coshmh共⌽̄eff
c 兲 = 2d̄ ⌽̄c sinh

共17兲

2
Rm
Rm
c
c
2 2
2
2 2
兲
=
−
3d̄
⌽̄
−
2mn
+
2m
n
g共⌽̄eff
c
c
w
w

+n

FIG. 5. Layer compression modulus B̄ of lamellar membrane+ colloidal
particle mixture. 共a兲 ⌽c dependence at ⌽s = 3.1% with theoretical prediction
based on Eq. 共17兲 indicated by the solid line and 共b兲 d̄ dependence at ⌽c= 0.01% and 0.02% with theoretical predictions based on Eq. 共17兲 indicatedby the solid line for ⌽c = 0.01% and the dotted line for ⌽c = 0.02%.

⫻tanh−n−w⌽̄effc

1w⌽̄effc

1w⌽̄eff
c

cosh

−n

共18兲

,

1

w⌽̄eff
c

Rm
c
w

Rm
2
2
c
+ d̄m⌽̄c sinh
cosh
eff
w
w⌽̄c
w⌽̄effc.

共19兲
By substituting Eqs. 共17兲–共19兲 to Eq. 共5兲, the Caillé parameter for lamellar membrane and colloid mixture, ␩l−c is expressed by

134903-8

J. Chem. Phys. 129, 134903 共2008兲

Suganuma et al.

␩l−c = 2␲冑2␮

冉 冊冑
d̄
d

2

h共⌽̄effc 兲
g共⌽̄eff
c 兲

.

共20兲

In Figs. 4共a兲 and 4共b兲 we show the experimentally obtained⌽̄c and d̄ dependences of ␩ with the fitting results using Eq.
共20兲, respectively. In both cases our model describes the observed data using the numerical parameters of m = 2.3,
n = 0.75, and w = 1200. Furthermore we calculated the layer
compression modulus B̄ using Eq. 共17兲 with the same numerical parameters. The theoretical predictions of ⌽̄c and d̄dependences of B̄l−c agree with the experimentally obtainedB̄ behaviors, as shown in Figs. 5共a兲 and 5共b兲. Thus we consider that our free energy model is suitable to describe theinterlamellar interaction in the presence of colloidal particles. By confining the colloidal particles between the lamellar sheets, the particles suppress the membrane fluctuations
due to the steric hindrance, which brings entropic repulsive
interaction.

C. Morphology transition of surfactant assembly
induced by addition of colloidal particles

Further addition of colloidal particles brings the lamellar
to micellar transition. In order to confirm the lamellar to
micelle transition, we performed the contrast matching
SANS measurements for pure micelle phase 共⌽s = 3.1% 兲 at
T = 16 ° C, the micelle plus the colloidal particle mixture
共⌽s = 3.1% and ⌽c = 1.3%兲 at T = 16 ° C and the lamellar
membrane plus the colloidal particle mixture 共⌽s = 3.1% and⌽c = 1.3%兲 at T = 57 ° C. The obtained scattering profiles areshown in Fig. 6. We fitted the observed scattering profiles
using the scattering function for prolate core-shell model31,32
given by
I共q兲 = Nm冕⬘ 共q兲,
g共r兲Pm共q,r兲drSm

共21兲

where Nm is the number density of micelles. For the form
factor Pm共q兲, we adopted a prolate core-shell model expressed by
具Pm共q兲典 ⬅

冕

1

兩H共q,x兲兩2dx,

共22兲

FIG. 6. SANS profiles for pure micellar phase 共⌽s = 3.1% and T = 16 ° C兲and the C12E5 + colloidal particle mixture 共⌽s = 3.1% and ⌽c = 1.3%兲 at T
= 16 and 57 ° C at contrast matching condition. The solid lines indicate theresults of fitting using prolate core-shell model. In the figure, the standard
profile is a top profile and each profile is multiplied by factor of 0.1 to avoid
overlapping.

structure factor Sm⬘ 共q兲 共Refs. 31 and 32兲 is related to the true
structure factor Sm共q兲 by⬘ 共q兲 = 1 + ␤共q兲关Sm共q兲 − 1兴,
Sm

共25兲

␤共q兲 = 兩具H共q兲典兩2/具兩H共q兲兩2典.

共26兲

The true structure factor S共q兲 for prolate micelles was calculated by assuming that a prolate micelle can be treated as arigid sphere having equivalent volume. The polydispersity of
micelle core radius is expressed by the Gaussian distributionfunction g共r兲 in Eq. 共21兲,

0

冋 册
冋 册

g共r兲 =

4
j共ut兲
H共q,x兲 = 共␳t − ␳h兲 ␲atR3t 3
3
ut

4
j共u兲
+ 共bh − bs兲 ␲aR3 3,3uut = qRt冑a2t x2 + 共1 − x2兲,u = qR冑a2x2 + 共1 − x2兲.

共23兲
共24兲

Here, R and Rt, are the outer and inner radii of the prolate
minor axis, a and at are the ratios of major to minor axes forthe outer and inner radii, respectively, and j共x兲 is the firstorder spherical Bessel function. The scattering length densities bt, bk, and bs correspond to the hydrophobic, the hydrophilic, and the solvent regions, respectively. The apparent1冑2␲␯ exp关− 共r − Rt 兲 /2␯ 兴,
m 2

2

共27兲

where ␯ is the variance and Rm
t is the mean micelle inner
radius of the prolate minor axis.
Using the hard core intermicellar interaction potential
共Percus–Yevick approximation33兲, we fitted the observed
scattering function at the matching point and the result offitting is shown in Fig. 6 by solid lines. The scattering profile
of pure C12E5 micellar solution without colloidal particles at
16 ° C 共⌽s = 3.1%, ⌽c = 0.0%兲 was well described by themodel scattering functions with the fitting parameters of efeff= 5%, R = 28 Å, Rm
fective volume fraction of micelles ⌽mict
= 17 Å, a = 18 共micelle aggregation number, Nagg = 2730兲, bt= −4.72⫻ 10−6 Å−2, bh = 1.0⫻ 10−7 Å−2, bw = 1.41⫻ 10−6 Å−2,

134903-9

J. Chem. Phys. 129, 134903 共2008兲

SANS studies of lamellar to micelle transitions

␯ = 2.6 Å, and Nm = 1.4⫻ 1015 micelles. Although the thicknesses of the hydrophobic part and hydrophilic part are consistent with the molecular structure, we found a very large
axis ratio a = 18 corresponding to the total micelle length ofL ⬃ 1000 Å. This long rod length is beyond the Kuhn lengthof the C12E5 micelle of ⬃200 Å.34 Thus the micelles may
have a semiflexible rodlike shape or a banana shape, which isconsistent with the transmission electron microscopy observation of the C12E5 micelles35 and the hydrodynamic radiusof C12E5 micelles obtained by dynamical light scattering.34
The large polydispersity of micelle radius 共␯ / Rm
t = 0.15兲 obtained by the fitting may be due to the semiflexible nature.
By the addition of a small amount of colloid in the micelle phase, the scattering profile showed a characteristic
broad peak even at low colloid volume fraction, ⌽c = 1.3%.
The fitting of the SANS profile gives the geometricaleff= 26%, R = 28 Å, Rm
parameters of ⌽mic
t = 17 Å, a = 12, andNagg = 1830. Thus by the addition of colloidal particles, thelength of prolate micelle decreases. This observation is consistent with our previous reports.36 We interpret the observed
decrease of the prolate micelle length induced by the addition of colloidal particles using the depletion concept.1–3,37 It
is well known that in a colloid and polymer mixture, thecenters of mass of polymer chains cannot approach closer
than the radius of gyration of polymer chain from colloid
surface, i.e., excluded volume called depletion zone. We applied this depletion concept to the colloid+ elongated micellesystem. The centers of mass of the semiflexible prolate micelle are excluded from “depletion zone” surrounding thesurface of a colloidal particle. Then, in order to increase thefree volume of micelles, the size of micelles may be decreased, which is consistent with our experimental results. It
should be noted that the surfactant assembly has an ability tochange its size depending on the external conditions.
In addition we observed large effective volume fraction,eff= 26%, estimated by the fitting of the scattering profile.
⌽mic
In this fitting we assumed that 6.6 water molecules hydrate
one ethylene oxide group of the hydrophilic chains, which
results in the scattering length density of bh = 1.0⫻ 10−7 Å−2.
This hydration of the hydrophilic chain gives the volume
fraction of prolate micelles, ⌽mic = 7.6% 共⌽s = 3.1% 兲,which is still smaller than the effective volume fraction ofeff⌽mic= 26%. Then we applied the depletion concept. According to the depletion concept, the centers of mass of micelles
cannot penetrate into the depletion zone, which increases theeffective volume fraction of micelles. The contribution of thedepletion zone to the effective volume fraction is expressedby a free volume fraction given by␣ = Vfree/V,

共28兲

where V is the system volume and Vfree is the free volume
available for the micelles in the system. For the sake of simplicity, we regard a flexible prolate micelle as a spherical coileffwith a radius Rmof 320 Å and estimate the free volume
fraction using a scaled particle theory38 expressed by

FIG. 7. Interlamellar interaction free energy in the presence of colloidal
particles per unit volume as a function of ⌽c.

冋 冉 冊 冉 冊
冉 冊册

␣ = 共1 − ⌽c兲exp − A1
− A3

⌽c 3
,
1 − ⌽c

⌽c
⌽c 2
− A2
1 − ⌽c1 − ⌽c共29兲

with
A1 = 3␰ + 3␰2 + ␰3 ,共30兲

A2 = 4.5␰2 + 3␰3 ,共31兲

A3 = 3␰3 ,共32兲

where ␰ = Reffm / Rc = 4.0. Using this expression, we obtained
␣ = 0.276, which gives ⌽mic = 0.275. This value is close to theeffeffective volume fraction of ⌽mic
= 0.26. Then we consider
that the large effective volume fraction is due to the depletion zone surrounding each colloidal particle.
When we add the colloidal particles into the lamellar
phase 共⌽s = 3.1%, ⌽c = 1.3%, and T = 57 ° C兲, the scatteringprofile changes to the characteristic profile for the micellar
phase with colloidal particles, as shown in Fig. 6. The fittingof obtained profile using the prolate core-shell model gaveeff
= 0.28, R = 28 Å,
the following parameters of ⌽micm
Rt = 17 Å, a = 14.8, and Nagg ⬃ 2240. Thus the length of theprolate micelle increases with the increase of temperature.
In the case of nonionic surfactant/water systems, the lengthof elongated micelles increases with the increase of
temperature,39 which is consistent with our observation. Thus
the observed scattering profiles show that by the addition ofcolloidal particles into the lamellar phase, the lamellar membranes transform to the prolate micelles.
Finally, we consider a scenario of the lamellar to micellar transition induced by the addition of colloidal particles. In
Fig. 7 we plot the interlamellar interaction free energy in thepresence of colloidal particles per unit volume, n1Fl-c, as afunction of ⌽c, where n1 is the number of lamellar membranes per unit volume 共n1 = 1.01⫻ 107 for ⌽s = 3.1%兲. The
frustration energy increases with the increase of ⌽c and134903-10

reaches 103 – 104 J / m3 at ⌽c = 1.0% – 10%. When this additional potential surpasses the free energy difference between
the lamellar assembly and the micellar assembly, the morphology transition will take place. Thus, in order to release
the frustration due to the restriction of membrane fluctuations, the surfactant assembly transforms the morphology
from the two dimensional sheet structure to one dimensional
flexible prolate structure which does not have strong confinement effect for colloidal particles.
V. CONCLUSIONS

The depletion interaction is one of the central interactions governing the stability of a binary mixture of hard particles having different sizes or shapes, where the phase behavior is determined by a balance between the mixing
entropy and the translational entropy. This concept can beextended to surfactant membrane and polymer mixture systems, where the attractive interlamellar interaction is induced
so as to obtain the large free volume for guest polymer
chains. In the case of the surfactant membrane and hard particle mixture systems, the free volume of fluctuating membranes is restricted by the presence of colloidal particles,
which causes strong repulsive colloid-membrane interaction.
Further addition of the colloidal particles brings the lamellar
membrane to prolate micelle transition. Thus the surfactant
assemblies reduce the entropic frustration by changing themorphology instead of the overlapping of the depletion zone.
This may be a unique feature of the surfactant assembly andhard particle mixtures.
ACKNOWLEDGMENTS

This work is supported by KAKENHI 共Grant-in-Aid for
Scientific Research兲 on Priority Area “Soft Matter Physics”
and Grant-in-Aid Scientific Research 共B兲 共Grant No.
19340118兲 from the Ministry of Education, Culture, Sports,
Science and Technology of Japan. The SANS experiments
were done under the approval of the Neutron Scattering Program Advisory Committee.
S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 共1954兲.
A. Vrij, Pure Appl. Chem. 48, 471 共1976兲.
V. J. Anderson and H. N. W. Lekkerkerker, Nature 共London兲 416, 811
共2002兲.
4
P. B. Warren, J. Phys. I 4, 237 共1994兲.
1
2
3

J. Chem. Phys. 129, 134903 共2008兲

Suganuma et al.

T. Koda, M. Numajiri, and S. Ikeda, J. Phys. Soc. Jpn. 65, 3551 共1996兲.
M. Adams, Z. Dogic, S. L. Keller, and S. Fraden, Nature 共London兲 393,
349 共1998兲.
7
G. H. Koenderink, G. A. Vliegenthart, S. G. J. M. Kluijtmans, A. vanBlaaderen, A. P. Philipse, and H. N. W. Lekkerkerker, Langmuir 15,4693 共1999兲.
8
Z. Dogic, D. Frenkel, and S. Fraden, Phys. Rev. E 62, 3925 共2000兲.
9K.-H. Lin, J. C. Crocker, A. C. Zeri, and A. G. Yodh, Phys. Rev. Lett. 87,
088301 共2001兲.
10
A. Matsuyama and T. Kato, Eur. Phys. J. E 6, 15 共2001兲.
11
M. Imai, N. Urakami, A. Nakamura, R. Takada, R. Oikawa, and Y. Sano,
Langmuir 18, 9918 共2002兲.
12
S. V. Savenko and M. Dijkstra, J. Chem. Phys. 124, 234902 共2006兲.
13
M. Daoud and P. G. de Gennes, J. Phys. 共France兲 38, 85 共1977兲.
14
J. T. Brooks and M. E. Cates, J. Chem. Phys. 99, 5467 共1993兲.
15
C. Ligoure, G. Bouglet, G. Porte, and O. Diat, J. Phys. II 7, 473 共1997兲.
16
L. Porcar, C. Ligoure, and J. Marignan, J. Phys. II 7, 493 共1997兲.
17
G. Bouglet, C. Ligoure, A.-M. Bellocq, E. Dufoure, and G. Mosser, Phys.
Rev. E 57, 834 共1998兲.
18
V. Ponsinet and P. Fabre, J. Phys. II 6, 955 共1996兲.
19
M. Imai, R. Mawatari, K. Nakaya, and S. Komura, Eur. Phys. J. E 13,
391 共2004兲.
20
P. Sens, M. S. Tumor, and P. Pincus, Phys. Rev. E 55, 4394 共1997兲.
21
W. Helfrich, Z. Naturforsch. A 33a, 305 共1978兲.
22
R. Strey, R. Schomäcker, D. Roux, F. Nallet, and U. Olsson, J. Chem.
Soc., Faraday Trans. 86, 2253 共1990兲.
23
Y. Ito, M. Imai, and S. Takahashi, Physica B 213–214, 889 共1995兲.
24
S. Okabe, M. Nagao, T. Karino, S. Watanabe, T. Adachi, H. Shimizu, and
M. Shibayama, J. Appl. Crystallogr. 38, 1035 共2005兲.
25
A. Caillé, Comptes Rendus des Seances de l’Academie des Science, Serie
B: Sciences Physique 274, 891 共1972兲.
26
F. Nallet, D. Roux, C. Quilliet, P. Fabre, and S. T. Milner, J. Phys. II 4,1477 共1994兲.
27
F. Nallet, R. Laversanne, and D. Roux, J. Phys. II 3, 487 共1993兲.
28
C. J. Glinka, J. M. Rowe, and J. G. LaRock, J. Appl. Crystallogr. 19, 427
共1986兲.
29
T. Masui, M. Imai, K. Nakaya, and T. Taniguchi, J. Chem. Phys. 124,
074904 共2006兲.
30
Y. Suganuma, M. Imai, and K. Nakaya, J. Appl. Crystallogr. 40, s303
共2007兲.
31
M. Kotlarchyk and S.-H. Chen, J. Chem. Phys. 79, 2461 共1983兲.
32
J. B. Hayter and J. Penfold, Colloid Polym. Sci. 261, 1022 共1983兲.
33
J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids, 3rd ed.
共Academic, Amsterdam, 2006兲.
34
S. Shirai, S. Yoshimura, and Y. Einaga, Polym. J. 共Tokyo, Jpn.兲 38, 37
共2006兲.
35
A. Bernheim-Groswasser, E. Wachtel, and Y. Talmon, Langmuir 16,
4131 共2000兲.
36
M. Imai, Y. Suganuma, K. Nakaya, and S. Komura, J. Phys.: Condens.
Matter 17, S2929 共2005兲.
37
H. N. W. Lekkerkerker, W. C.-K. Poon, P. N. Pusey, A. Stroobants, and P.
B. Warren, Europhys. Lett. 20, 559 共1992兲.
38
P. G. Bolhuis and H. N. W. Lekkerkerker, Physica A 196, 369 共1993兲.
39
T. Kato, S. Anzai, and T. Seimiya, J. Phys. Chem. 94, 7255 共1990兲.
56