Eur. Phys. J. E 13, 391–400 (2004)
DOI 10.1140/epje/i2003-10085-7

THE EUROPEAN
PHYSICAL JOURNAL EInter-lamellar interactions modulated by addition of guest
components
M. Imai1,a , R. Mawatari1 , K. Nakaya1 , and S. Komura2
1
2

Department of Physics, Faculty of Science, Ochanomizu University, 2-1-1 Otsuka, Bunkyo, Tokyo 112-0012, Japan
Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji,
Tokyo 192-0397, Japan
Received 9 September 2003 and Received in ﬁnal form 26 March 2004 /c EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2004
Published online: 4 May 2004 – 
Abstract. We have investigated the eﬀects of a guest component (polymer or spherical colloidal particle)
conﬁned between ﬂexible lamellar slits on the inter-lamellar interaction by means of a small-angle X-ray
scattering technique and a neutron spin echo technique. The dominant interaction between ﬂexible lamellar
membranes without guest components is the Helfrich mechanism. The addition of a neutral polymer into thelamellar phase induces an attractive inter-lamellar interaction and ﬁnally destabilizes the lamellar phase.
On the other hand, spherical colloidal particles conﬁned between ﬂexible lamellar membranes reduce theundulational ﬂuctuations of lamellae and bring a repulsive inter-lamellar interaction. The behavior of thelayer compression modulus of the lamellar membrane containing colloidal particles is well described by theentropical repulsive inter-lamellar interaction driven by steric hindrance.
PACS. 82.70.Uv Surfactants, micellar solutions, vesicles, lamellae, amphiphilic systems, (hydrophilic andhydrophobic interactions) – 61.25.Hq Macromolecular and polymer solutions; polymer melts; swelling –83.80.Hj Suspensions, dispersions, pastes, slurries, colloids – 89.75.Fb Structures and organization incomplex systems

1 Introduction
Complex systems consisting of surfactant membranes andguest components such as polymers or colloidal particles are widely recognized in the ﬁelds of biological membranes, drug carrier systems, lubricants, cosmetic, etc. andhave acquired great interest from the scientiﬁc and industrial points of view. These systems show a characteristicphase behavior as a result of the modiﬁcation of the intermembrane interaction induced by the guest components.
The lamellar phase consisting of stacks of surfactant bilayers separated by solvent layers is a suitable system toinvestigate the inter-membrane interaction induced by theguest components, since behaviors of the inter-lamellar interactions are well characterized by recent theoretical andexperimental studies [1–4]. In this context, there are extensive studies on the ternary systems of surfactant, water,
and polymers [5–17].
The role of added polymers between the lamellar bilayers was investigated theoretically by Daoud and deGennes [5] and revisited by Brooks and Cates [7]. They
classiﬁed the situations of the polymer conﬁnement into
the following four regimes and derived the correspondingae-mail: imai@phys.ocha.ac.jp

free energy for each regime: 1) three-dimensional dilute
solution of polymer chains (3D D), 2) three-dimensionalsemi-dilute solution of polymer chains (3D S-D), 3) twodimensional semi-dilute solution of polymer chains (2DS-D) and 4) two-dimensional dilute solution of polymer chains (2D D). On the basis of these theoretical works, Ligoure et al. [11–13,15,16] have investigatedthe lyotropic lamellar system of cetyl pyridium chloride(CPCL)/hexanol/water in which a non-adsorbing watersoluble polymer polyvinylpyrolidone (PVP) is introduced.
The CPCL lamellar structure without polymer is stabilized by the electrostatic interaction indicating that theﬂuctuations of the bilayer are strongly suppressed by thelong-range electrostatic interaction [2,18]. The additionof polymer chains into the lamellar membranes induces adecrease of the smectic compressibility in the above four
regimes, which is well described by the theoretical predictions [11]. Furthermore, when the electrostatic interaction is screened by salt, the addition of polymer bringsa lamellar-lamellar phase separation [11,12]. Thus, the effective inter-lamellar interaction mediated by the presenceof polymer chains is attractive.
On the other hand, if the guest component has a hardcore potential, the ﬂuctuations of lamellar membranes
might be strongly aﬀected by the excluded-volume eﬀect.

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The European Physical Journal E

Ponsinet et al. [19,20] found that colloidal magnetic particles stabilizes the lamellar phase due to steric hindrance.
From this standpoint, we systematically investigate themodulation of the inter-lamellar interactions between theﬂuctuating lamellar bilayers by the addition of guest components, namely, polymer chains having ﬂexible entity or
spherical colloidal particles having rigid entity. In thisstudy we deal with the 3D D regime, thus the mean interlamellar distance is fairly larger than the size of the guest
component and its volume fraction is quite low. We examine the non-ionic surfactant lamellar system, where
the inter-lamellar interactions are governed by the entropic repulsive interaction, i.e., Helfrich interaction [4].
The changes of the statical and dynamical structure of
ﬂuctuating lamellar bilayers induced by the guest components were measured by small-angle X-ray scattering(SAXS), small-angle neutron scattering (SANS) and neutron spin echo (NSE) techniques. From the SAXS, SANS
and NSE proﬁles we can extract the elastic properties ofthe lamellar structure, that is the bending modulus andthe layer compression modulus [2,3,21,22]. Especially, thebehavior of the layer compression modulus is examined interms of the modulation of the inter-lamellar interaction
potential by the addition of the guest components.

2 Experimental section
We investigate ternary mixtures of water, C12 E5 plus
polymer or spherical colloid particle. The non-ionic surfactant C12 E5 (purity > 98%) was purchased from Nikko
Chemicals Inc., and used without further puriﬁcation.
The phase behavior of the C12 E5 /water system is wellcharacterized and we can control the inter-lamellar distance by a simple dilution law [23]. First, we investigatedthe phase behavior of the water, C12 E5 and polyethylene oxide (PEO) mixture system. This polymer is nonionic and easily soluble in the lamellar phase of the
C12 E5 /water system. The PEO sample has molecularweight ∼ 70000, which corresponds to a radius of gyration, Rg , of 128 Å [24], and was obtained from Wako
Pure Chemical Industries, Ltd. The volume fraction of
PEO, ΦPEO , was calculated using the following densities:
1.00 g cm−3 (H2 O); 1.104 g cm−3 (D2 O); 0.967 g cm−3
(C12 E5 ); 1.21 g cm−3 (PEO). The phase behavior of thewater, C12 E5 and colloidal-particle system was investigated using a polystyrene latex having diameter 200 Å
(±80 Å), which was obtained from Magsphere Inc. The
polystyrene latex has surface charge ∼ 0.1e/nm2 , which
is high enough to prevent it from aggregation. The ionicstrength I of the colloidal suspension is expressed by the
relation I(mM) = 24ρcol ( wt%), where ρcol is the weightfraction of colloidal particles. Thus the colloidal suspension with ρcol = 0.01% has an ionic strength of 0.24 mM.
We conﬁrmed that the scattering proﬁle of the lamellar
structure was not aﬀected by the presence of the small
amount of the ionic species. The volume fraction of thepolystyrene colloidal particle Φcol was calculated using adensity of 1.05 g/cm3 .

The ternary-mixture samples for the scattering measurements were prepared by the following procedure. The
surfactant and water mixtures (C12 E5 /D2 O for SANS andC12 E5 /H2 O for SAXS) were homogenized and then theguest components were added to the surfactant solution
at room temperature (isotropic micelle phase). The sample was transferred to a cell for scattering measurements
and then annealed at a measurement temperature of 70 ◦ C(lamellar phase). The standard thickness of the sample
cell for SANS measurements was 4.0 mm or above. In
the standard sample cells, the lamellar phase prefers polycrystalline arrangements. During the scattering measurements, we regulated the temperature within ±0.01 ◦ C.
SAXS measurements were performed using theBL-15A instrument at the photon factory (PF) in thehigh-energy accelerator research organization (KEK) [25].
In this experiment we used a 1.5 Å wavelength X-ray beam
having 0.8 × 0.8 mm2 square cross-section. The scattered
beam was recorded using the CCD area detector coveringa scattering vector q (q = 4π sin θ/λ) range from 0.03 to
0.33 Å−1 . The obtained scattering patterns were corrected
for the non-uniformity, image distortion and backgroundscattering.
SANS measurements were carried out using the
SANS-U instrument of the Institute of Solid State Physics,
University of Tokyo at the JRR3-M reactor of the Japan
Atomic Energy Research Institute in Tokai [26]. In thisexperiment, we used a neutron beam having a wavelength(λ) of 7.0 Å with 10% resolution. The diameter of theirradiated neutron beam was 5 mm and the scatteredneutrons were detected by a two-dimensional positionsensitive detector having 65 × 65 cm2 (128 × 128 pixels)
area. By changing the sample-to-detector distance, we covered the q range from 0.002 to 0.3 Å−1 . The obtained twodimensional isotropic scattering patterns were corrected
for background scattering and then circularly averaged toobtain one-dimensional scattering proﬁles.
The intermediate scattering functions of the samples
were obtained using the NSE spectrometer at the C2-2
port of the JRR-3M reactor [27]. The incident neutron
wavelength was 7.14 Å with 18% resolution, which covers
a q range from 0.04 to 0.16 Å−1 . The Fourier time rangeof the spectrometer is 0.08 < t < 15 ns. The obtained intermediate scattering function S(q, t) was corrected usinga resolution standard sample of Grafoil.

3 Results and discussion
In this section we describe the features of lamellar + guest
systems in the following way. First, we show the diﬀerencein the phase behavior between the lamellar + polymer
system and the lamellar + colloid-particle system. Before
doing a detailed analysis for lamellar + guests systems, we
examine the dominant interactions between the non-ionicsurfactant membranes without guests using experimental
results of SANS and NSE measurements. Then, the phasebehavior of the lamellar + polymer system will be discussed based on Ligoure’s model. A characteristic phaseM. Imai et al.: Inter-lamellar interactions modulated by addition of guest components393

the ρC12 E5 -ρPEO phase diagram and the ρC12 E5 -ρcol phase
diagram of the ternary system at T = 57 ◦ C (ρC12 E5 :weight fraction of C12 E5 , ρPEO : weight fraction of PEO).
We waited several weeks to examine the stability ofthe observed phases, because in the case of lamellar +guest component systems it took extremely long time toequilibrate the phases [11]. The addition of PEO chains
into the lamellar phase brings the phase separation forρPEO > 0.02 wt%, which is presumably the coexistenceof surfactant-rich lamellar phase and polymer-rich dilute
lamellar phase [28]. This observation is consistent with theprevious results [8,12]. The phase separation concentration was independent of the surfactant concentration. On
the other hand, by the addition of spherical colloidal particles, the lamellar phase transformed to the micellar phasethrough the coexistence of lamellar and micellar phases.
The phase transition concentration ρcol increased with theincrease of the surfactant concentration ρC12 E5 . The samples in the lamellar phase without the guest components
are monophasic and slightly cloudy. By the addition of
the colloidal particles, the samples became transparent
and were a clear ﬂuid in the L1 -phase, whereas by theaddition of PEO the samples became cloudy.
The diﬀerence between the polymer chains and thecolloidal particles becomes apparent by measuring SANS
proﬁles. Figure 2 shows a series of SANS proﬁles as a function of (a) ρPEO and (b) ρcol at constant ΦC12 E5 = 0.045.
Unfortunately, there is no visible Bragg peak in the SANS
proﬁle of the ΦC12 E5 = 0.045 sample without guest species,
because of the large membrane ﬂuctuations. The membrane ﬂuctuations cause the excess scattering observed in
the low-q region [29]. In this case, we evaluate the lamellar repeat distance d from the following relationship in theliterature [23]:


δ
kB T
d=
ln ΦC12 E5 ,1−
(1)
ΦC12 E54πκ0Fig. 1. Phase diagrams at T = 57 ◦ C of the ternary
C12 E5 /water/PEO system (a) and C12 E5 /water/colloidalparticle system (b). The vertical axis is the weight fractionof the surfactant ρC12 E5 and the horizontal axis is the weightfraction of the guest components, PEO (ρPEO ) (a) and colloidal particle (ρcol ) (b). Lα : lamellar phase, L1 : micelle phase,
Lα /Lα : lamellar and lamellar two-phase region having diﬀerent
spacing, and Lα /L1 : lamellar and micelle two-phase region.

behavior observed in the lamellar + colloidal-particle system will be investigated in terms of an entropical repulsive
interaction model.
3.1 Phase behaviors of lamellar + polymer andlamellar + colloid-particle systems
First, we brieﬂy compare phase behaviors of the lyotropic
lamellar membrane + polymer system and lyotropic lamellar membrane + colloidal-particle system. Figure 1 showswhere δ is the thickness of the membrane and κ0 is thebare elastic bending modulus of a single membrane. Using δ = 30 Å and κ0 = 1.3kB T , we obtained d(ΦC12 E5 =0.045) = 794 Å. Thus, the size of the guest components
is much smaller than the period, and we are in the 3D D
region. By the addition of PEO chains, the scattering functions keep the initial proﬁle until ρPEO ∼ 0.03% and thenthe scattering intensity in the low-q region increases withincrease of ρPEO , which is due to the phase separation. On
the other hand, the addition of colloidal particles brings
the sharpening of the ﬁrst Bragg peak and even the emergence of the second harmonic as shown in Figure 2(b).
Thus, the inter-lamellar interactions induced by the addition of the guest components strongly depend on thenature of the guest species.
3.2 Non-ionic surfactant lamellar membrane without
guestsFirst, we examine the dominant inter-lamellar interaction of a non-ionic surfactant membrane. Generally, it is394

The European Physical Journal Epressed by d¯ = d − δ (δ: bilayer thickness), kB is Boltzmann’s constant, and T is the temperature. Using the local¯ m is expresseddisplacements of the membrane u(r) and d,byu2 (r) = md¯2 .
(3)The expression of equation (2) is diﬀerent from theusual expression for the Helfrich interaction with m =4/(3π 2 ), because the numerical prefactor is still under controversy [30]. Hereafter we use the prefactor 1/(32m) for
the Helfrich interaction. When the inter-lamellar forces
are dominated by the Helfrich interaction given by equation (2), we can expect to follow the simple geometricalexpression [1]
2

2π
δ
η=
.
(4)
1−
d
3/m
Here the Caillé parameter η is related to the elastic
nature of the membrane by the following expression [31]:η=

q02 kB T√,
8π K B̄(5)

where K is the bulk bending modulus having a relationship K = κ/d, B̄ is the layer compression modulus, and
q0 is the position of the ﬁrst Bragg singularity, q0 = 2π/d.
The Caillé parameter can be estimated from the SAXS orSANS proﬁles of the lamellar phase.
Figure 3 shows the SAXS proﬁles of the C12 E5 /water
system as a function of ΦC12 E5 at 70 ◦ C (lamellar phase) inan I(q)q 4 -q plot. In order to obtain the Caillé parameter,
the scattering proﬁles were ﬁtted by the model scatteringfunction for the undulating lamellar structure proposed
by Nallet et al. [3]:Nl
P (q)S(q) ,(6)q2
4P (q) = 2 {∆ρH [sin[q(δH + δT )] − sin(qδT )]q+∆ρT sin(qδT )}2 ,
(7)




N
−1
l

qdn
n
S(q) = 1 + 2
1−cos
2 d2 α(n)
N
1
+
2∆q
l
n=1


2q 2 d2 α(n) + ∆q 2 d2 n2
× exp − z
2(1 + 2∆q 2 d2 α(n))
1
×
,
(8)
1 + 2∆q 2 d2 α(n)Ipowd (q) ∼Fig. 2. SANS proﬁles of the lamellar phase conﬁning guest
components, PEO (a) and colloidal particle (b), at constantsurfactant volume fraction ΦC12 E5 = 0.045 with varying theguest components fraction.

considered that the inter-lamellar interaction of the nonionic surfactant membrane system is governed by the Helfrich mechanism. For the Helfrich interaction [4], the potential energy is expressed by
FHel =(kB T )2,32mκd¯2(2)

where m is the measure of the mean amplitude of thermal undulations of the membrane restricted by steric hindrance due to the adjacent bilayers, d¯ is the average thickness of the water layer between two adjacent bilayers ex-

where Nl is the number of lamellar layers, ∆q is the widthof the resolution function assuming a Gaussian proﬁle, and∆ρH and ∆ρT are the contrast between the hydrophilic
layer (thickness δH ) and the solvent (H2 O), and between
the hydrophobic part (thickness δT ) and the solvent, respectively. In this model the correlation function for undulating lamellae α(n) = (un − u0 )2 /2d2 in equation (8)is expressed by
α(n) =

η[ln(πn) + γ]d2
4d2 π 2(9)

M. Imai et al.: Inter-lamellar interactions modulated by addition of guest components

395

Fig. 3. SAXS proﬁles of the lamellar phase for the pure C12 E5 /water system as a function of ΦC12 E5 in an I(q)q 4 -versus-q plot.
The solid lines are least-square ﬁtting curves using the model scattering function proposed by Nallet et al.

with Euler’s constant γ = 0.5772. The results of ﬁtting
are shown in Figure 3 by the solid lines. The experimentally obtained η is plotted against (1 − δ/d)2 in Figure 4.
Here we used δ = 30 Å [23]. According to equation (4),this plot should give a straight line throughthe origin
and its slope corresponds to the prefactor 2π/ 3/m. The
expected value for the slope is 4/3√ by the Helfrich prediction (m = 4/(3π 2 )) [4], and 4 2/3 by the computer
simulations (m = 8/(3π 2 )) [30]. The solid line in Figure 4
shows the best ﬁt of experimental√ data with equation (4)and gives a slope of 1.85 ∼ 4 2/3, which corresponds
to m = 0.26 ∼ 8/(3π 2 ). Hence we concluded that theinter-lamellar interaction of the lyotropic lamellar phaseconsisting of the non-ionic surfactant C12 E5 is governedby the Helfrich interaction.
Next, we estimate the bending modulus κ from the intermediate scattering function I(q,t) of a ﬂuctuating membrane obtained by the NSE measurement [21,22]. Generally, the intermediate scattering function is expressed byI(q, t) = Ns−1

Ns Ns
i=1

Fig. 4. The Caillé parameter η versus the (1 − δ/d)2 plot forthe pure C12 E5 /water system.

exp{−iqri (0)} exp{iqrj (t)} ,

j=1

(10)
where Ns is the number of scattering particles. Zilman
and Granek [21] derived the expression for the normalizedintermediate scattering function of the ﬂuctuating membranes:I(q, t)= exp{−(Γz t)2/3 } ,I(q, 0)(11)Γz = Dz q 3 ,(12)

Dz = 0.0056γκ (kB T )3/2 κ−1/2 ηκ−1 ,(13)where ηκ is the viscosity of water. γκ is given byγκ = 1 − 3ln(qξ)kB T
,
4πκ

(14)

where ξ is the typical size of the mesoscopic structureof the system. Thus, we can obtain κ from the normalized intermediate scattering function. The normalized intermediate scattering functions for the C12 E5 /water system (ΦC12 E5 = 0.045) obtained from the NSE experiments are shown in Figure 5 with the ﬁtting curves using the stretched exponential form of equation (11). The

396

The European Physical Journal Ewith m = 0.26 κ = 3.1kB T , d = 794 Å and δ = 30 Å,
the evaluated value of the compression modulus for thepure C12 E5 /water (ΦC12 E5 = 0.045) system was B̄ Hel ∼25 dyn/cm2 .
3.3 Lamellar + polymer system

Fig. 5. Normalized intermediate scattering functions of thelamellar phase for the pure C12 E5 /water system at 57 ◦ C(ΦC12 E5 = 0.045). The solid lines are least-square ﬁtting curves
using the Zilman-Granek theory.

relaxation rate Γz obtained by the ﬁtting shows the powerlaw of q 3 in agreement with the Zilman and Granek theory. However, it is still a controversy for the value of numerical parameters to evaluate the bending modulus fromequations (11–14), because when the macroscopic viscosity of the solvent ηκ is used, one obtains a fairly large
value of the bending modulus compared with the bending modulus estimated from other techniques. In order to
ﬁll up this gap, some adjustment techniques have been
proposed [22,32,33]. A familiar technique for the adjustment is to use an eﬀective viscosity ηeﬀ instead of themacroscopic viscosity ηκ . The eﬀective viscosity corrects
the local dissipation at the interface between the membrane and the solvent, although the value of ηeﬀ is determined by the phenomenological considerations. It is interesting to note that the intermediate scattering functions
obtained from a dynamical light scattering (DLS) method
give consistent bending moduli without the viscosity correction [34,35]. Then, we estimated κ of the standard
lamellar sample (C12 E5 /D2 O: ΦC12 E5 = 0.015 at 57 ◦ C)
by the NSE and DLS measurements. The NSE and the
DLS measurements gave values of κNSE = 9.05kB T andκDLS = 2.01kB T , respectively, and the value of κDLS isclose to the reported values for κ obtained from othermethods [36]. From these experimental results, we estimated the value of ηeﬀ = 9.05/2.01 ∼= 2.1ηκ , which isslightly smaller than the value used in reference [22]. In
this study, we used this eﬀective viscosity to obtain κ fromthe NSE measurements. For the pure C12 E5 /D2 O lamellar sample of ΦC12 E5 = 0.045, we obtained κ = 3.1kB T
from κNSE = 14kB T (at ΦC12 E5 = 0.045) using ξ = 794 Å(∼ d) [37].
When the inter-lamellar interaction is governed by theHelfrich mechanism, the layer compression modulus is expressed by [4]B̄Hel

3(kB T )2=d,
16mκd¯4(15)

Here we discuss the observed polymer-induced phase separation using the lamellar-lamellar phase separation model
proposed by Ligoure et al. [11,15]. They found that theaddition of the polymer into the smectic bilayer decreases
the inter-lamellar repulsive force and ﬁnally brings thelamellar-lamellar phase separation when the layer compression modulus vanishes (B̄ = 0). In what follows, we
brieﬂy show an explanation for the observed phase behavior of the polymer-containing lamellar system basedon Ligoure’s scenario. In the case of the 3D D conﬁnement system, the additional inter-lamellar interaction energy per unit bilayer area due to the polymer conﬁnement
is given by
kB T Φ̄eﬀlog(Φ̄eﬀ ) ,Fpol ∼= 3 d¯aN(16)

where a is the monomer length of the polymer, and N isthe degree of polymerization. The eﬀective volume fraction of polymers in the inter-lamellar water layer Φ̄eﬀ iscorrected due to the depletion zone covering the interface
between water and bilayer,d¯Φ̄eﬀ = ΦPEO ¯,
d − 2Rg(17)

where Rg is the radius of gyration of a polymer coil. Thisadditional potential energy modiﬁes the compression modulus of the lamellar phase which is accessible from thescattering experiments. In the case of the lamellar phase
containing guest components, the layer compression modulus is expressed by [38]


2

1 ∂F
∂2F
 ∂2F

− ∂ Φ̄g ∂ d¯ 
d¯ ∂ Φ̄g,(18)B̄µ = d−∂2F2 ∂ d¯∂ Φ̄2gwhere Φ̄g is the volume fraction of the guest component
in the water layer. Using this expression, the contributionof the conﬁned polymer to the layer compression modulus
is given by [11]

 ¯
2
4kB T Rg2 dΦ̄dΦ̄
B̄µpol = −
.
(19)
1
+
log
a3 N d¯3
d¯ − 2Rg
Here we assume that the total inter-lamellar interaction for the lamellar membrane + polymer system is expressed by the sum of the two terms [11,15]:
Ftotal = Fpol + FHel

(20)

and the total layer compressibility is given by
total
B̄pol
= B̄µpol + B̄ Hel .

(21)

M. Imai et al.: Inter-lamellar interactions modulated by addition of guest components

Fig. 6. Variation of the theoretical total layer compression
total
modulus B̄polfor the lamellar + PEO system as a functionof ΦPEO . The experimentally observed phase separation concentration Φ∗e
PEO is indicated by an arrow.

It is important to note that the polymer contributionB̄µpol is negative and induces destabilization of the lamellar phase. Experimentally, Ligoure et al. reported thatby the addition of polymer in the lamellar system, thelamellar phase is destabilized by the additional attractive
inter-lamellar interaction and ﬁnally undergoes a lamellartotal
lamellar phase separation at B̄pol= B̄µpol + B̄ Hel = 0. The
polvalue of B̄µ can be estimated by equation (19) using thestatistic chain parameters a and N . For PEO chain, we
determined the values of a and N from the experimental
relationship
0.635
Rg (Å) = 0.1078 × MPEO,

(22)

where MPEO is the molecular weight of PEO. The calculated values a and N for the PEO chain with MPEO =70000 were a = 5.6 Å [24] and N = 1070. Using thesevalues we obtained the B̄µpol as a function of ΦPEO :B̄µpol (dyn/cm2 ) = −2.85 × 104 ΦPEO (1 + log[1.47ΦPEO ])2 .
(23)
total
In Figure 6 we plotted B̄pol= B̄µpol + B̄ Hel as a functotaltion of ΦPEO . Figure 6 shows that the B̄poldecreases∗twith increase of ΦPEO and crosses zero at ΦPEO ∼ 0.001%,
where the phase separation starts. Ligoure’s model explain
the phase separation behavior only qualitatively but quantitatively there is a large diﬀerence between the theoretical prediction and the experimental data Φ∗e
PEO ∼ 0.02%.
The discrepancy between theoretical prediction and experimental result may be originated from the ambiguityof the estimation of κ or B̄ Hel .
3.4 Lamellar + colloid-particle system
In the case of the non-ionic surfactant/water and colloidalparticle mixture system, the phase behavior is completely

397

Fig. 7. Variation of the Caillé parameter η of the lamellar +colloid-particle system as a function of Φ̄col at ΦC12 E5 = 0.045.
Inset: scattering proﬁle for the lamellar + colloid-particle system (ΦC12 E5 = 0.045 and Φ̄col = 0.028%) with the ﬁtting result(solid line) using equations (6–8).

diﬀerent from the non-ionic surfactant/water and polymer
mixture system. The addition of colloidal particles brings
the sharpening of the ﬁrst Bragg peak and the emergenceof the second harmonic as shown in Figure 2(b). This
result is consistent with the behavior of the lamellar +colloidal-magnetic-particle system [19,20]. Here we consider the dominant inter-lamellar interaction in the presence of the spherical colloidal particles. In order to examine the inter-lamellar interaction, the scattering proﬁlesfor the C12 E5 + colloidal-particle/water system were ﬁtted using the scattering function proposed by Nallet etal., equations (6–8). The variation of the obtained η asa function of Φ̄col at ΦC12 E5 = 0.045 is plotted in Figure 7. For reference, we display a typical ﬁtting resultof the scattering proﬁle for the system (ΦC12 E5 = 0.045
and Φ̄col = 0.028%) in the ﬁgure. The Caillé parameter η decreases sharply as Φcol increases, until it reaches
a plateau value of η ∼ 0.4. Thus, the addition of hardsphere colloidal particles brings restriction of undulation
ﬂuctuations of lamellar membranes at extremely low volume fraction of the colloid particle. An interesting featureof the inter-lamellar interaction induced by the conﬁnement of colloidal particles is the dependence of the Caillé
parameter η on the lamellar spacing d as shown in Figure 8. When the inter-lamellar distance is governed by
the Helfrich interaction, η obeys the simple geometricalexpression η = 1.85(1 − δ/d)2 . Thus, η determined by theHelfrich interaction increases with increase of d as shownby the solid line in Figure 8. On the contrary, in the presence of colloidal particles, η decreases monotonically withincrease of d. Interestingly, the restriction of membrane
ﬂuctuations is remarkable in the dilute lamellar region.
In order to elucidate this characteristic inter-lamellarinteraction induced by the addition of colloidal particle,

398

The European Physical Journal E

Fig. 8. Variation of the Caillé parameter η of the lamellar +colloid-particle system as a function of d at Φ̄col = 0.01%. The
solid line is the prediction of the Helfrich model.

Fig. 9. Normalized intermediate scattering functions of thelamellar system and the lamellar + colloidal-particle systemat ΦC12 E5 = 0.045.

Fig. 10. Variation of the experimentally obtained layer compression modulus B̄ for the lamellar + colloid-particle system
as a function of Φ̄col at ΦC12 E5 = 0.045. The theoretical prediction B̄µcol ∼ Φ̄4col is indicated in the ﬁgure.

from ∼ 25 dyn/cm2 (Φcol = 0) with increase of Φcol andreaches an equilibrium value ∼ 600 dyn/cm2 . Thus the
addition of hard-sphere colloids brings strong repulsive
force between adjacent lamellar layers and this eﬀect isremarkable at low concentrations of the non-ionic surfactant. This observation is hardly explained by the previous
model for the 3D D region, because in this model the
addition of guest components necessarily brings negative
contribution to the layer compression modulus as shownin equation (19).
In order to explain the observed experimental results,
we took into account the suppression of membrane ﬂuctuations due to the excluded-volume eﬀect of the colloidal
particles, which gives the entropic repulsive inter-lamellainteraction. Here we derive the expression of the steric
repulsive interaction potential. First, we introduce a dimensionless surface coverage parameter deﬁned byχ = πR2 ncol = πR2

we estimated the layer compression modulus B̄. For thispurpose we obtained the bending modulus of the membrane of the C12 E5 /water + colloidal-particle systems by
the NSE experiments. The comparison of intermediate
scattering functions between lamellar system and lamellar+ colloidal-particle system is shown in Figure 9. The intermediate scattering functions were independent of the addition of the colloidal particles and obeyed the stretched exponential low of equation (11). The Zilman-Granek analysis of the observed intermediated scattering functions gives
the bending modulus κ. The value of κ was independentof Φcol in the range 0 < Φcol < 0.02% and kept the original value ∼ 3kB T . Using the values of η and κ, we obtained the layer compression modulus B̄ as a function ofΦcol (Fig. 10). The layer compression modulus B̄ increases

3Φ̄col d3Φ̄col d
,
=
3
4πR
4R

(24)

where ncol is the number of colloidal particles per unit
bilayer area. We assume that the restriction of the membrane ﬂuctuations is expressed by a function of the surface
coverage χ, that is¯ (χ) .
|u(r)|  df

(25)

The function f (χ) has the following features: i) f (χ) →1 for χ → 0, and ii) f (χ) → 0 for χ → ∞. In order tosatisfy these conditions, we adopted the function given byf (χ) = tanh2 (1/αχ) ,(26)

where α is a coeﬃcient to correct eﬀective surface coverage. Using this function, the restriction of the membraneM. Imai et al.: Inter-lamellar interactions modulated by addition of guest components

ﬂuctuations are expressed byu2 (r) = mc d¯2 tanh44R3αΦ̄col d


,

(27)

where the factor mc must be smaller than unity. Following
the Helfrich formalism [4], the free energy of the stericinteraction per unit area of a membrane is given by
Fcol =

1(kB T )2.
32mc κ d¯2 tanh4 ( 3α4R
)
Φ̄ d

(28)

col

) approaches unity. ThenFor αχ
1, tanh4 ( 3α4R
Φ̄col dequation (28) agrees with the expression for the usual Helfrich interaction free energy, equation (2). On the other
hand, in the high surface coverage region αχ  1, thefree energy is expressed by4 4d(kB T )2 3αΦ̄colFcol =
for αχ  1 .
(29)
32mc κ
4R
d¯2
Using this free-energy expression and equation (18),the layer compression modulus is given by4 5
d(kB T )2 3αΦ̄colB̄µcol =for αχ  1 . (30)3mc κ
16Rcold¯4
In Figure 10 we compared the theoretical predictionB̄µcol with the experimentally obtained Φ̄col -dependence ofB̄. The steep increase of B̄ with increase of Φ̄col is welldescribed by equation (30). Here we assume that the coeﬃcient α takes a large value (> 103 ). This large α meansthat the presence of one colloidal particle aﬀects ﬂuctuations of a large membrane area. The experimentally obtained B̄ deviates from the theoretical prediction and levels oﬀ at high Φ̄col . This deviation may be originated fromthe fact that the form of f (χ) given by equations (24) and(26) is not valid at high Φ̄col . Thus, the layer compression
modulus is independent of Φ̄col at χ  1, because the Φ̄colreaches a saturated value. Using equation (30), the Caillé
parameter of this system is expressed by√2  ¯21
π 3mc 16Rdcol
η =for αχ  1. (31)
2
2Φcol
3α
d
d2
In Figure 8 we compared this theoretical prediction¯ ∼with the experimental results. Taking into account d/d=
1, the characteristic d-dependence of η is well describedby equation (31). Thus, the colloidal particles introduced
between lamellar membranes restrict the membrane ﬂuctuations due to the hard-sphere nature, which brings steric
repulsive inter-lamellar interactions.

4 Conclusions
The inter-lamellar interaction induced by the additionof the guests depends on the rigidity of the added materials. The addition of neutral polymer into the lamellar phase induces an attractive inter-lamellar interaction399

and ﬁnally destabilizes the single lamellar phase. This behavior is qualitatively described by a free-energy model
for polymer-containing lyotropic lamellar phases developed by Ligoure et al. On the other hand, the spherical colloidal particles conﬁned between the ﬂexible lamellar membranes suppress lamellar ﬂuctuations and bringa repulsive inter-lamellar interaction. The behavior of thelayer compression modulus of the lamellar membrane containing colloidal particles is well described by the entropical repulsive inter-lamellar interaction driven by steric hindrance. The diﬀerence between polymer and colloid particle is probably attributed to the fact that the polymer
chains do not suppress the membrane ﬂuctuations due totheir large internal degrees of freedom.
This work is supported by a Grant-in-Aid for Scientiﬁc Research (B) (No. 15340138) and (C) (No. 15540395) from the
Ministry of Education, Science, Sports and Culture of Japan.
The SANS and NSE experiments were done under the approvalof the Neutron Scattering Program Advisory Committee (Proposal No. 3541).

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