VOLUME 92, N UMBER 5

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PHYSICA L R EVIEW LET T ERS

Temperature and Pressure Effects on the Bending Modulus of Monolayers
in a Ternary Microemulsion
Y. Kawabata,1,2,* M. Nagao,3 H. Seto,4 S. Komura,1 T. Takeda,5 D. Schwahn,6 N. L. Yamada,2 and H. Nobutou2
1

Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan
Graduate School of Bio-Sphere Science, Hiroshima University, Higashi-Hiroshima 739-8521, Japan
3
Institute for Solid State Physics, The University of Tokyo, Tokai 319-1106, Japan
4
Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
5
Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan
6
Institut für Festkörperforschung des Forschungszentrums Jülich GmbH, D-52425 Jülich, Germany(Received 21 April 2003; published 6 February 2004)
2

We performed small-angle neutron scattering and neutron spin echo experiments on a ternary
microemulsion composed of ionic surfactant AOT, water, and decane. Thermal fluctuations of monolayers have been investigated as a function of temperature and pressure. The amphiphilic monolayers
become more flexible with increasing temperature and more rigid with increasing pressure. These
results are consistent with the microscopic picture that the head-head repulsion of the AOT molecules isenhanced at high temperature while an attractive interaction between the hydrophobic tails of the AOT
molecules increases at high pressure.
DOI: 10.1103/PhysRevLett.92.056103

PACS numbers: 68.05.Gh, 61.12.Ex, 62.50.+p

energy [8],Electronic, magnetic, or structural properties of condensed matters react to variations in temperature andpressure. In order to investigate the physical propertiesof condensed matters, the temperature and pressure
dependencies of structures and physical quantities areimportant. In many materials such as simple liquids,
gas, or crystalline solids, an increase of temperature has
the same effect as a decrease of pressure because
the Clausius-Clapeyron relation dT=dP > 0 is usual.
For a soft condensed matter, however, there are systems
such as surfactant solutions in which increasing pressureand decreasing temperature have different effects(dT=dP < 0). In a mixture composed of water, decane,and AOT (dioctyl sulfosuccinate sodium salt), a water-inoil droplet phase is formed over a wide range of composition at ambient temperature and pressure [1]. By raisingtemperature, a single droplet phase undergoes a phaseseparation [2,3]. A similar phenomenon is observed
when pressure is increased [4,5]. Hence increasing temperature and increasing pressure play a similar role in thephase behavior of this system at least on a phase behavior.
However, such pressure effects on the structural formation [6] have not been investigated in much detail.
Especially, only very few studies exist on dynamical
properties of microemulsion systems under the variationof pressure and temperature [7].
In this Letter, we present the results of small-angle
neutron scattering (SANS) and neutron spin echo (NSE)experiments in the dilute droplet regime of the ternary
microemulsion composed of AOT, water, and decane. We
discuss the temperature and pressure dependencies of thethermal fluctuations of AOT monolayers. Furthermore,
we discuss a microscopic picture of the temperature andpressure effects on AOT monolayers based on the bending

where R1 and R2 are the principal curvature radii, Rs the
spontaneous curvature radius,  the bending modulus, the saddle-splay modulus, and dS an area element. Thereexist a number of studies to obtain  and  in a dilute
droplet system by means of NSE [9–11]. We improve here
the procedure used in the above studies in order to explore
the temperature and pressure dependencies of .
We prepared a dilute droplet sample as described below. AOT (purity, 99%) was purchased from Sigma
Chemical and purified by extraction with benzene.
Deuterated water (D2 O) and deuterated decane (C10 D22 )were purchased from Isotec Inc. with isotopic purities of
99.9%-D for D2 O and C10 D22 . These ingredients weremixed at 5.4 vol. % of AOT, 4.6 vol. % of D2 O, and90 vol. % of C10 D22 , so that the molar ratio of D2 O to
AOT was 18:4, and the volume fraction of droplets was 0:1. Because the neutron scattering length densitiesof D2 O and C10 D22 are almost the same, only the AOT
layers can be observed (shell contrast). The SANS experiments were carried out at the SANS-U and KWS1
diffractometers in JRR3M, JAERI, Japan, and FRJ2, FZJülich, Germany, respectively. The NSE experiments
were performed at the ISSP-NSE spectrometer in
JAERI [12]. The conditions for each experiment are listed
in Table I.
Typical SANS intensities Iq as a function of momentum transfer q are shown in Fig. 1. The critical scattering
due to the droplet density fluctuations in the one-phaseregion is enhanced with increasing temperature as the056103-1

 2004 The American Physical Society

0031-9007=04=92(5)=056103(4)$22.50

HZ   1
2 R1



1
2

R2 Rs

2




1
dS;
R1 R2

(1)

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PHYSICA L R EVIEW LET T ERSVOLUME 92, N UMBER 5

TABLE I. The experimental settings for SANS-U, KWS1, and ISSP-NSE.  is the neutron wavelength for each spectrometer, and= is its resolution (FWHM). Trange and Prange are the range of measured temperature and pressure. T and P are temperatureand pressure variation steps, respectively. t is the range of NSE time. The pressure variation and the temperature variation in the
SANS-U and NSE experiments were done at fixed temperature (T  298 K) and at fixed pressure (P  0:1 MPa), respectively.
 1 ]q [A (=%) [A]SANS-UKWS1
ISSP-NSE

7:0 (10)
7:0 (20)
7:14 (18)

0:008
0:006
0:04

Trange (T) [K]

0:3
0:14
0:13

284
286
284

phase separation line is approached from below. The
scattering intensity is suppressed at higher temperaturein the two-phase region. Similar behavior is observed bychanging pressure across the phase separation line. From
this observation, we estimated the phase separation temperature to be Tps  332 K for P  0:1 MPa (ambientpressure), and the phase separation pressure to be Pps 
43 MPa for T  298 K (room temperature).
The SANS profiles were analyzed using the procedure
by Gradzielski et al. [13], and we obtained the mean
radius of thedroplets R0 (hRi) and the polydispersityp
index p  hR2 i=hRi2  1. The polydispersity index isgiven by
1kB T
R0 3kB Tp ;
62     8 
f 
Rs
4
2
2

(2)

where kB is the Boltzmann constant and f  the mixing
entropy per droplet as a function of the volume fraction[14]. For T  298 K and P  0:1 MPa, we obtained p  from the SANS data. When tempera0:16 and R0  32 A
ture is increased to 329 K, p  0:15, and R0 decreased to On the other hand, R0 did not depend on pressure
28 A.
very much.

Prange (P) [MPa]353 (3)298 (2)338 (5)

0:1
0:1
0:1

t [ns]

66:5 (5)30 (10)60 (20)

0:15

15

From the NSE experiment, we obtained the normalizedintermediate scattering function Iq; t=Iq; 0 which weanalyzed following the procedure by Huang et al. [9,10].
In their treatment, Iq; t=Iq; 0 is given byIq; t=Iq; 0  exp Deff qq2 t ;(3)Deff q  Dtr  Ddef q;(4)

where Deff q is the effective diffusion coefficient, with
Dtr  Deff q  0 the translational diffusion coefficientof droplets, and Ddef the diffusion coefficient in terms ofdroplet shape deformations. We found that the hydrodynamic radius obtained from Dtr with the Stokes’s law wasabout 2 times larger than the droplet radius R0 determined by SANS. The values from the fitting of Dtr at
each condition are shown by the arrows in Fig. 2.
Figure 2 shows Deff q as a function of q. It is clearlyseen that Deff q becomes larger with increasing temperature and smaller with increasing pressure. Hence theincrease of temperature and pressure give rise to opposite
effects on the diffusion coefficient of droplets.

12
T=328K

T=298K
P=22 MPap=0.13
R 0=32.3Å

10

10

1

8

T=322K-7-1

I(q) [cm ]2

Deff [10 cm /s]

100

T=329K
P=0.1MPap=0.15
R 0=27.7Å

T=308K

T=298K
P=0.1MPap=0.16
R0=32.5Å
2

0.01

6

3

4

4

5 6 7 89

-1

2

0.1

q [Å ]

2

T=292K
T=298K
P=21MPa
P=40MPa
P=60MPa
0.00

0.04

0.08

-1

0.12

q [Å ]

FIG. 1. SANS profiles observed at three different conditions.
The data at T  298 K (room temperature) and P  0:1 MPa(ambient pressure) is shown in the absolute unit. Two othercurves are multiplied by factors of 2 and 4, respectively.

056103-2

FIG. 2. Deff obtained from NSE data at various conditions.
Lines are fitting results to Eq. (4). The values of Deff at q  0
correspond to Dtr , which is shown by the arrows.

056103-2

According to Ref. [14], Ddef q is given by
Ddef q 

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PHYSICA L R EVIEW LET T ERSVOLUME 92, N UMBER 5

52 f2 qR0 hja2 j2 i2q 4 j0 qR0  2  5f2 qR0 hja2 j2 i;

f2 qR0   4j2 qR0   qR0 j3 qR0  2 :

(5)

(6)

Here, jn x is the nth spherical Bessel function, 2 thedamping frequency of the droplet deformation, andhja2 j2 i the mean square displacement of the 2nd-mode
spherical harmonic (n  2). We fitted the data by Eq. (4)as shown in Fig. 2, and obtained 2 and hja2 j2 i.
In Fig. 3, we present the temperature and pressure
dependencies of hja2 j2 i. We find that the shape fluctuations are enhanced for higher temperature while they aresuppressed for higher pressure.
We estimated the bending modulus  of the monolayers. For higher pressure, the viscosity of deuterated
decane increases more than that of water. Therefore, wemust consider the difference between 0 and  which are
the viscosities inside and outside a droplet. Thus 2 isexpressed as [15]R0 3kB T
24

f

2 
: (7)

3
4
3
 4
230  32
R0 RsCombining Eqs. (2) and (7), we get the expression of thebending modulus  [16] with the observable parameters,
p2 , R0 , and 2 ,
0
1 kB T
3 23  32
;
(8)


R

2
0
48 p23
where  and 0 are the bulk viscosities of D2 O anddecane, respectively.

Figure 4 shows the temperature and pressure dependencies of  obtained from Eq. (8). It is clear that 
becomes smaller with increasing temperature while it
becomes larger with increasing pressure. For example,
 at T  312 K was estimated to be 0:26 kB T, which wasslightly smaller than that at room temperature. On thecontrary,  at P  43 MPa becomes 1.3 times larger thanthat at ambient pressure (P  0:1 MPa).
From the results of hja2 j2 i and  shown in Figs. 3 and 4,
we conclude that the AOT monolayers become more
flexible with increasing temperature, while they become
more rigid with increasing pressure. These tendencies are
consistent with the NSE results obtained by Nagao et al.
in the dense droplet regime (0:6) [18]. Using a model
describing single membrane undulations [19], they
showed that  at higher temperature was smaller thanthat at room temperature, and  at higher pressure waslarger than that at ambient pressure. Thus, all these results
indicate that the temperature effect on the dynamics ofthe monolayers is opposite to the pressure effect.
The values of  and hja2 j2 i were experimentally obtained as  ’ 0:3 kB T and hja2 j2 i ’ 0:7 at ambient temperature and pressure, which are smaller and larger thanthose in literature [  0:9 kB T, hja2 j2 i  0:01] [11]. The
origin of the small value of  and the too large value ofhja2 j2 i could be the same; the value of Dtr was ambiguous
because it was treated as a fit parameter. Hellweg et al.
suggested that a combination of dynamic light scattering(DLS) and NSE gives more appropriate value [11]. With
our preliminary DLS experiment at ambient temperatureand pressure, the values of  and hja2 j2 i were reasonable;  0:8kB T and hja2 j2 i  0:07. However, the tendency
should be correct from the above results (Figs. 3 and 4)
Temperature [K]290Temperature [K]
290

300

310

320

330

2.0

<|a2| > (pressure)κ [kBT]

2

0.45

2

<|a2| > (temperature)

320

330

340

κ (temperature)κ (pressure)

0.50

2.5

310

340

3.0

<|a2| >

300

2

0.40
0.35

1.5
0.30

1.0
0.25

0.5
0

0

20

40

Pressure [MPa]

40

60

60

FIG. 3. The temperature and pressure dependencies of hja2 j2 i
obtained from Eq. (5). The left dotted line indicates the room
temperature and ambient pressure, whereas the right one gives
the phase separation temperature and pressure. The shapefluctuation of a droplet becomes larger with increasing temperature, while it becomes smaller with increasing pressure.
Curves are drawn to guide the eye.

056103-3

20

Pressure [MPa]FIG. 4. Temperature and pressure dependencies of  obtained
from Eq. (8). The left dotted line indicates the room temperature and ambient pressure, whereas the right one gives thephase separation temperature and pressure. AOT monolayers
become flexible with increasing temperature, while they become rigid with increasing pressure. Curves represent thebending modulus cal obtained from the calculation according
to the model proposed by Würger [17]. The value of cal ismultiplied by a factor of 0:38.

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PHYSICA L R EVIEW LET T ERSVOLUME 92, N UMBER 5

Temperature [K]
290

300

310

320

330

340

64
aH (temperature)aH (pressure)62
2

aH [ ]

60
58
56
54
52
0

20

40

Pressure [MPa]60

FIG. 5. The temperature and pressure dependencies of area
per AOT molecule aH . The left dotted line indicates the room
temperature and ambient pressure, whereas the right one gives
the phase separation temperature and pressure.

although the absolute values of  and hja2 j2 i might not bevery correct.
The microscopic origin of the temperature and pressure
dependencies of  has been interpreted as follows [6,18].
With increasing temperature, a counterion dissociationfrom a head group of AOT is enhanced, and an electrostatic head-head repulsive force increases. On the other
hand, at high pressure, an attractive interaction betweenhydrophobic tails of the AOT molecules increases. Thus it
is expected that the head area per AOT molecule aH
becomes larger at higher temperature and smaller athigher pressure with the result of the change of monolayer
flexibility.
The area per AOT molecule aH was obtained from the
SANS data following the relation, aH  4R20 =ns , where
ns is the number of surfactant per droplet. Figure 5 shows
the temperature and pressure dependencies of aH . We findthat aH at higher temperature is larger than that at higher
pressure. It is clear that the temperature and pressure
dependencies of aH and  are consistent with the microscopic picture described above.
This picture is consistent with a microscopic model
describing the bending rigidity [17]. In the model, theinteraction between hydrocarbon chains of surfactants is
assumed to be a Lennard-Jones potential with the potential minimum  at an intermolecular distance . As
shown by the dashed line in Fig. 4, the temperaturedependence of  was reproduced by using this modeland the temperature dependence of aH . Note that theabsolute value of the calculated cal was about 0:8 kB T
and the line in the figure was multiplied by the factor of
about 0:38.
The solid line in Fig. 4, representing the pressure
dependence of , is a fitting result to the experimental
data according to the model. In this fitting procedure,linear pressure dependencies of  and were assumed.
The fit result reproduced the pressure dependence of ,
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and the potential depth  increased linearly with P, while
the potential minimum
was decreased with P. This
result is also consistent with the microscopic origin ofpressure- induced phase transition, in which the attractive
interaction between hydrocarbon chains of AOT molecules increases at high pressure. The details of this analysis will be discussed in our following paper.
In summary, we measured the temperature and pressure dependencies of the bending modulus in the ternary
microemulsion system for the first time. The result clearly
showed that the AOT monolayer becomes more flexible
with increasing temperature while more rigid with increasing pressure although the phase transitions induced
by temperature and pressure seem to be the same. Thisevidence was consistent with the microscopic picture of
the origins of the phase transitions.
We appreciate Mr. M. Koizumi at Hikari HighPressure Machinery Co. Ltd. for developing the highpressure cell for the NSE experiments. One of the authors(M. N.) was supported by the Grant-in-Aid for Encouragement of Young Scientists (No. 13740234) and(H. S) and (T. T) were also supported by a Grant-in-Aid
for Scientific Research (No. 12640377, No. 09640466,
No. 13440119) from the Japanese Ministry of Education, Culture, Sports, Science and Technology. Our
SANS and NSE experiments at JAERI were done
under approval of the Neutron Scattering Program
Advisory Committee (Proposal No. 00-199, No. 00200, No. 00-201).

*Electronic address: youheik@comp.metro-u.ac.jp[1] M. Nagao et al., J. Appl. Crystallogr. 36, 602 (2003).
[2] M. Kotlarchyk et al., Phys. Rev. A 29, 2054 (1984).
[3] H. Seto et al., Phys. Rev. E 54, 629 (1996).
[4] M.W. Kim et al., Phys. Rev. Lett. 54, 46 (1985).
[5] M. Nagao and H. Seto, Phys. Rev. E 59, 3169 (1999).
[6] H. Seto et al., J. Chem. Phys. 112, 10 608 (2000).
[7] Y. Kawabata et al., Appl. Phys. A 74 (Suppl.), S534
(2002).
[8] W. Helfrich, Z. Naturforsch. 28C, 693 (1973).
[9] J. S. Huang et al., Phys. Rev. Lett. 59, 2600 (1987).
[10] B. Farago et al., Phys. Rev. Lett. 65, 3348 (1990).
[11] T. Hellweg and D. Langevin, Physica (Amsterdam)
264A, 370 (1999).
[12] Y. Kawabata et al., J. Phys. Soc. Jpn. Suppl. A 70, 445(2001).
[13] M. Gradzielski et al., J. Phys. Chem. 99, 13232 (1995).
[14] S. T. Milner and S. A. Safran, Phys. Rev. A 36, 4371
(1987).
[15] K. Seki and S. Komura, Physica (Amsterdam) 219A, 253(1995).
[16] Y. Kawabata et al., J. Neutron Res. 10, 131 (2002).
[17] A. Würger, Phys. Rev. Lett. 85, 337 (2000).
[18] M. Nagao et al., J. Chem. Phys. 115, 10 036 (2001).
[19] A. G. Zilman and R. Granek, Phys. Rev. Lett. 77, 4788
(1996).

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