THE JOURNAL OF CHEMICAL PHYSICS 124, 034906 共2006兲

Unbinding and preunbinding in surfactant solutions
S. Komura,a兲 N. Shimokawa, and T. Kato
Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan

共Received 27 September 2005; accepted 29 November 2005; published online 20 January 2006兲
We propose models for the first-order unbinding transition of lyotropic lamellae in surfactant
solutions. The coupling between the surfactant volume fraction and the elastic degree of freedom is
considered so that the net attractive interaction between the surfactant molecules is enhanced. The
elastic degree of freedom can be either 共i兲 a membrane elastic degree of freedom or 共ii兲 a bulk elastic
degree of freedom. The phase behaviors of these two models are analyzed. For both cases, theunbinding transition becomes first order when the coupling is strong enough. We determine theassociated preunbinding line which separates two lamellar phases having different repeat
distances. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2159475兴
I. INTRODUCTION

One of the simplest structures found in mixtures of waterand surfactant is the lamellar phase in which the bilayers ofamphiphile form roughly parallel layers separated by water.
In some binary systems, it is known that the lamellar phasecan be swollen almost without limit. For example, in the
mixture of water and C12E5 共n-akylpolyglycolether兲, the repeat distance of the lamellar phase can exceed 3000 Å.1 The
amount of surfactant molecules in the unbound phase is lessthan 1%. The transition from the bound lamellar phase to theunbound phase is generally called the “unbinding transition.”
The concept of the unbinding transition was theoreticallyintroduced by Lipowsky and Leibler.2 Whether this transitionis second order or first order is a subtle problem 共see alsoSec. V兲. Using the functional-renormalization-group techniques, Lipowsky and Leibler showed that two membranes
unbind continuously in the presence of thermal fluctuations.
More precisely, the mean distance between the two membranes, ᐉ, diverges as ᐉ ⬃ 共Tu − T兲−␺ with ␺ ⬇ 1 and Tu is socalled the unbinding temperature.2
For a bulk of lamellar phase, the volume fraction of
surfactant, ␾, is inversely proportional to the mean distanceᐉ. If the above prediction by Lipowsky and Leibler for two
membranes is applicable to the lamellar phase of an infinite
number of membranes, the difference in volume fraction between the bound 共lamellar兲 and the unbound phases should
decrease linearly with Tu − T.3,4 In other words, the lower
boundary of the lamellar phase in the temperatureconcentration plane should be a straight line of nonzero
slope. Several mean-field theories for a lamellar phase wereconstructed in such a way to reproduce the exact results.5–7
Unfortunately, the critical behavior is not observed in theexperimental phase diagram of surfactant solutions.1 For example, the unbinding transition in the C12E5/water system isfairly abrupt because the lower boundary of the lamellar
phase is almost horizontal.3,4 This means that ᐉ diverges discontinuously as the temperature is raised along the phaseboundary, i.e., a first-order transition. Similar behavior of thea兲Electronic mail: komura@comp.metro-u.ac.jp

0021-9606/2006/124共3兲/034906/7/$23.00

lamellar phase is observed for other types of nonionic surfactant such as C16E6 and C16E7.8,9 We show in Fig. 1 thephase diagram of the C16E7/water binary system. 共The lower
boundary of the lamellar phase in this figure is the horizontal
line at around 67 ° C兲. These experimental results suggestthat the unbinding transition in surfactant solutions may begenerally first order rather than second order.
In the present paper, we propose a possible mechanismthat leads to a discontinuous first-order unbinding transitionin surfactant solutions. Based on the mean-field theory by
Milner and Roux 共MR兲,6 we consider a phenomenologicalcoupling between the surfactant concentration and the otherelastic degree of freedom. Although the predictive power ofthe MR theory is relatively weak compared to the functionalrenormalization-group theory, it provides us a useful model
to incorporate other degrees of freedom in a simple manner.
Two possibilities are considered; 共i兲 a membrane elastic degree of freedom and 共ii兲 a bulk elastic degree of freedom. For
both cases, we assumed that the coupling affects the second
virial coefficient in such a way to enhance the net attractive
interaction. Analysis of our models shows that the unbindingtransition is second order when the coupling is weak, while it
becomes first order when the coupling is strong enough. We

FIG. 1. Phase diagram of the C16E7 / water system, redrawn from Fig. 1 inRef. 9. L␣, lamellar phase; L1, isotropic micellar phase; V1, cubic phase; H1,
hexagonal phase; and Nc, nematic phase.
124, 034906-1© 2006 American Institute of Physics

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J. Chem. Phys. 124, 034906 共2006兲

Komura, Shimokawa, and Katoalso determine the associated preunbinding transition line
which separates the two lamellar phases of different
spacing.10 In particular, the coupling to the bulk elastic degree of freedom leads to a three-phase coexistence.
In the next section, we briefly review the theories ofunbinding transitions. We put an emphasis on the MR theory
for a stack of lamellae in a surfactant solution, and confirm
that it predicts a second-order unbinding transition. In Secs.
III and IV, we extend the MR theory to take into account theabove mentioned coupling mechanism. We show that ourmodel exhibits a first-order unbinding transition when the
coupling is strong enough. Finally, we give some discussions
in the last section.

II. UNBINDING TRANSITION

Fluid bilayer membranes experience steric repulsion
arising from their reduced undulation entropy. The corresponding interaction energy per unit area of the membrane
was considered by Helfrich,11␷s共ᐉ兲 =

b共kBT兲2,
␬共ᐉ – ␦兲2

共1兲

where kB is the Boltzmann constant, T the temperature, ␬ thebending rigidity, ␦ the membrane thickness, and ᐉ the average distance between bilayers. Note that ᐉ – ␦ represents theintermembrane distance within which membranes can fluctuate. The numerical prefactor b was calculated to be b
= 3␲2 / 128 in the original paper,11 but subsequent various
simulation studies revealed that the value of b is smaller than
the above one by a factor of 2.12–14
In general, the combination of the above steric repulsion
and other direct microscopic interactions, such as longranged van der Waals attraction and short-ranged hydration
repulsion, determines whether two membranes bind eachother or unbind between them.15 Lipowsky and Leibler
pointed out that solving a simple superposition of Eq. 共1兲 andother direct interactions at a mean-field level gives incorrect
共first-order兲 description of the unbinding transition.2 An appropriate treatment of this problem using a functional
renormalization-group method predicted that the unbindingtransition should be second order 共critical unbinding transition兲.
Later, Milner and Roux proposed a theory for the unbinding transition in a bulk of the lamellar phase following
the spirit of mean-field theory for polymers.6 In their treatment, the Helfrich estimate of the entropy is taken into account accurately, whereas the other direct microscopic interactions are approximately incorporated as a correction to thehard-wall result for the second virial coefficient. The suggested free energy per unit volume of a lamellar stack isĝMR共␾兲 = − kBT␹ˆ ␾2 +b共kBT兲2 ␾3ˆ ␾,−␮␬␦3 共1 − ␾兲2共2兲

where the membrane volume fraction ␾ = ␦ / ᐉ 艌 0 cannot be
negative. In the above, ␹ˆ is the correction to the hard-wallresult, and is calculated by

FIG. 2. The phase diagrams of the Milner and Roux model as a function of
共a兲 surfactant volume fraction ␾ and virial coefficient ␹, and 共b兲 surfactant
chemical-potential ␮ and virial coefficient ␹. The solid line is a first-orderline, whereas the dashed line is a second-order one. The horizontal lines andthe dot-dashed line in 共a兲 are tielines and the spinodal line, respectively. The
filled circle denotes the tricritical point 共tcp兲.

␹ˆ = −

12␯2

冕dr共1 − exp关− U␯共r兲/kBT兴兲,

共3兲

where U␯共r兲 is the interaction between bits of membrane ofvolume ␯. All the direct microscopic 共van der Waals, electrostatic, and hydration兲 interactions are taken into accountˆ , is needed for thethrough U␯共r兲. The chemical potential, ␮conservation of the membrane volume fraction ␾. The factorof 共1 − ␾兲−2 in the second term comes from the finite membrane thickness 关see Eq. 共1兲兴.16
It is convenient to rescale all the energy densities by
2b共kBT兲2 / 共␬␦3兲. Then Eq. 共2兲 can be presented asgMR共␾兲 = − ␹␾2 +␾3− ␮␾ .
2共1 − ␾兲2

共4兲The corresponding phase diagrams in the 共␹ , ␾兲 and 共␹ , ␮兲
planes are shown in Fig. 2. The volume fractions and chemical potentials for the binodal and the spinodal satisfy the
relations16

␹bin =and␾bin,
共1 − ␾bin兲3␮bin = −2
␾bin
共1 + ␾bin兲,
2共1 − ␾bin兲3

共5兲

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␹sp =J. Chem. Phys. 124, 034906 共2006兲

Unbinding and preunbinding in surfactant solutions3␾sp,
2共1 − ␾sp兲4

␮sp =

2
2
␾sp
共␾sp
− 4␾sp − 3兲,
2共1 − ␾sp兲4

共6兲

respectively. When ␹ ⬎ 0 the binodal line separates between
the bound phase and the unbound phases where ␾ = 0. For␹ ⬍ 0, on the other hand, the complete unbinding of thelamellae occurs at ␮ = 0 upon swelling with excess water.17
These first-order and second-order lines are continuously
connected at the tricritical point 共tcp兲. If we denote ␹u as thevalue of ␹ at which the lamellae first unbind as a single
phase, we have here ␹u = ␹tcp = 0. As we go down in ␹ along
the binodal line in Fig. 2共a兲 关more precisely, infinitesimallyon the lamellar side of coexistence in Fig. 2共b兲兴 and approach␹u from above, the repeat distance ᐉ increases continuously,
and the critical unbinding transition takes place. This is consistent with the prediction by Lipowsky and Leibler.2
So far, the unbinding transition in a bulk of lamellar
phase has been discussed within the MR theory. Next, we
extend the MR theory by considering another degree of freedom which is coupled to the volume fraction of the surfactant. Two models are proposed, 共i兲 the coupling to the membrane elastic degree of freedom 共model A兲, and 共ii兲 thecoupling to the bulk elastic degree of freedom 共model B兲.

III. MODEL A
A. Free energy

As an example of the membrane elastic degree of freedom, we consider the interfacial area S per surfactant molecule. This quantity can fluctuate around the so-called saturated state S = S* at which the effective surface tension
vanishes.18 If we define the fluctuation of the area by s = 共S
− S*兲 / S*, the free energy per surfactant molecule is simply
given by19
f s共s兲 =

A 2
s ,
2

共7兲

where A ⬎ 0 has the dimension of energy.
The proposed scaled free energy per unit volume of thesystem isgA共␾,s兲 = − ␹␾2 +

␾3
a 2
2
2 − ␮␾ − ␥␾ s + ␾s ,2共1 − ␾兲2

共8兲with the constraint ␾ 艌 0 as before and a = 共A / S ␦兲⫻关␬␦3 / 2b共kBT兲2兴. The first three terms depend only on ␾,
and are identical to the right-hand side of Eq. 共4兲. The term
−␥␾2s represents a phenomenological coupling between ␾and s with a dimensionless coefficient ␥ ⬎ 0. It is assumed
here that the area fluctuation affects the bare virial coefficientso that the binary interaction between the surfactant molecules is shifted. Note that ␹eff = ␹ + ␥s can be regarded as aneffective virial coefficient of the system. Moreover, a constant shift in s can always be adsorbed by redefining ␹. A
lower-order coupling term such as ␾s simply renormalizes ␹after the minimization with respect to s, and is neglected
here. A somewhat similar coupling was proposed in Ref. 20
where the unbinding of a single mixed membrane was discussed.

Minimizing the free energy in Eq. 共8兲 with respect to sgives

␥s = ␾.
a共9兲

By substituting back this relation into Eq. 共8兲, the free energy
as a function of ␾ only becomesg A共 ␾ 兲 = − ␹ ␾ 2 +␾3
␥2 3
␾ − ␮␾ .
2 −
2共1 − ␾兲2a共10兲We then minimize gA共␾兲 with respect to ␾ to get the equation of state. The two-phase region is obtained by the standard Maxwell construction.
The variable s can represent other physical quantities.
For example, s can be the nematic order parameter magnitude of the surfactant molecules. In liquid-crystal systems,
the microscopic interaction between two rodlike molecules
depends not only on their relative distance but also on theirrelative orientation. Maier and Saupe considered that the interaction potential is lowered as the molecules align eachother when they are located at a fixed distance.21 Such aneffect would lead to a virial coefficient which depends on theorientational order, as expressed in Eq. 共8兲. In the lyotropic
systems, however, the orientational order should vanish
when ␾ → 0, and the last term in Eq. 共8兲 is necessary. In fact,a linear relation between the orientational order parameter s
and the volume fraction ␾ 关see Eq. 共9兲兴 has been indicated insome NMR studies of nonionic surfactant solutions.22,23
B. Phase behavior

Here we discuss how the shape of the phase diagrams
changes as the value of the coupling strength ␥2 / a ischanged. 共We do not change ␥ and a independently.兲 Depending on the coupling strength, there are two different
types of phase behavior; 共i兲 weak coupling regime 共␥2 / a⬍ 1兲 and 共ii兲 strong coupling regime 共␥2 / a ⬎ 1兲. The corresponding phase diagrams are shown in Figs. 3 and 4 for␥2 / a = 0.5 and 5, respectively.
In the weak coupling case ␥2 / a ⬍ 1 共see Fig. 3兲, thephase behavior is basically unchanged compared to that of
the MR model 共see Fig. 2兲. Here the binodal and the spinodal
are given by␹bin =

*

␾bin
␥2
−
␾bin ,
共1 − ␾bin兲3 a

␾2 共1 + ␾bin兲 ␥2 2
␮bin = − bin+ ␾bin ,2共1 − ␾bin兲3
2a

共11兲

and

␹sp =

3␾sp
3␥2
−
␾sp ,
2共1 − ␾sp兲4 2a

␾2 共␾2 − 4␾sp − 3兲 3␥2 2
␮sp = sp sp␾ ,+2共1 − ␾sp兲4
2a sp

共12兲

respectively. We immediately see that these relations reduce
to Eqs. 共5兲 and 共6兲 in the limit of ␥2 / a → 0. The binodal

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Komura, Shimokawa, and Kato

FIG. 3. The phase diagrams for ␥2 / a = 0.5 in model A as a function of 共a兲surfactant volume fraction ␾ and virial coefficient ␹, and 共b兲 surfactant
chemical-potential ␮ and virial coefficient ␹. The same notation of different
lines is used as in Fig. 2. The filled circle denotes the tricritical point 共tcp兲.

FIG. 4. The phase diagrams for ␥2 / a = 5 in model A. The same notation ofdifferent lines is used as in Fig. 2. The filled square and diamond indicate
the critical end point 共cep兲 and critical point 共cp兲, respectively.

␹u = −
separates between the lamellar phase and the excess water.
The tricritical point appears at ␹tcp = 0 which again coincides
with the unbinding transition point; i.e., ␹u = ␹tcp = 0. It should
be noted, however, that the two-phase region in Fig. 3共a兲 iswider compared to that in Fig. 2共a兲. This is due to the enhanced attractive interaction between the surfactant molecules.
In the strong coupling regime ␥2 / a ⬎ 1 共see Fig. 4兲, there
exists a first-order line separating two lamellar phases of different spacing ᐉ. This first-order line corresponds to the preunbinding line and ends at an ordinary critical point 共cp兲.
The locus of the critical point can be calculated from theconditions that both the second and the third derivatives ofgA共␾兲 vanish; gA⬙ 共␾兲 = gA⵮共␾兲 = 0. This yields

␹cp = −

26␾cp,
共1 − ␾cp兲5

␥2 1 + 3␾cp=.
a 共1 − ␾cp兲5

␾2u,
共1 − ␾u兲3

1 + ␾u␥2=.
a 共1 − ␾u兲3

共14兲The behavior of ␹u as a function of ␥2 / a is also plotted inFig. 5.
IV. MODEL B
A. Free energy

In the model B, we consider the bulk elastic degree of
freedom ␴. The suggested free energy per unit volume of thesystem is

共13兲

In Fig. 5, we have plotted ␹cp as a function of ␥2 / a.
As shown in Fig. 4共b兲, the second-order line ends at thecritical end point 共cep兲 on the first-order line. It is important
to note that the value of ␹ at the critical end point is negative;
␹cep ⬍ 0. Since ␹u = ␹cep in the strong coupling regime, theunbinding transition point is determined by the conditions
f A共 ␾ 兲 = f A⬘ 共␾兲 = 0, where f A共␾兲 = gA共␾兲 + ␮␾. Then we have

FIG. 5. The values of ␹ at the critical point ␹cp 共solid line兲 and the unbinding transition point ␹u 共dashed line兲 as a function of the coupling strength␥2 / a for model A.

034906-5

J. Chem. Phys. 124, 034906 共2006兲

Unbinding and preunbinding in surfactant solutions

g B共 ␾ , ␴ 兲 = − ␹ ␾ 2 +

␾3
c 2
2
2 − ␮␾ − ␭␾ ␴ + ␴ , 共15兲
2共1 − ␾兲2

where ␭ is a new dimensionless coupling constant and c⬎ 0. The physical meaning of the coupling term −␭␾2␴ issimilar to that in Eq. 共8兲. The only difference compared to
Eq. 共8兲 is that the last term c␴2 / 2 does not vanish in the limitof ␾ → 0. This is the reason why ␴ is called the bulk elastic
degree of freedom. Such a variable could play a role, forexample, when not only the surfactant molecules but also thesolvent molecules can sustain a local orientational orderthroughout the system. Although the above model may notdirectly describe the binary surfactant solutions, it is worth
considering because of the conceptual difference and significance.
By minimizing Eq. 共15兲 first with respect to ␴, we obtain␭␴ = ␾2 .
c

共16兲Then the minimized free energy becomesg B共 ␾ 兲 = − ␹ ␾ 2 +

␾3
␭2 4
−
␾ − ␮␾ .
2共1 − ␾兲2 2c

共17兲

which contains a quartic term in ␾ with a negative coefficient. This should be contrasted with the cubic term in Eq.
共10兲, and leads to a somewhat different phase behavior.
B. Phase behavior

For the model B, there are three different regimes; 共i兲weak coupling regime 共␭2 / c ⬍ 5.36兲, 共ii兲 intermediate coupling regime 共5.36⬍ ␭2 / c ⬍ 6.75兲, and 共iii兲 strong coupling
regime 共␭2 / c ⬎ 6.75兲.
In the weak coupling regime ␭2 / c ⬍ 5.36, the phase diagrams are similar to those obtained for the weak coupling
regime in the model A 共see Fig. 3兲. The binodal and thespinodal are now given by␹bin =

3␭2 2
␾bin
−
␾ ,
共1 − ␾bin兲3 2c bin

␾2 共1 + ␾bin兲 ␭2 3
␮bin = − bin+ ␾bin ,2共1 − ␾bin兲3
c

共18兲

and

␹sp =

3␾sp
3␭2 2
−
␾ ,
2共1 − ␾sp兲4c sp

␾2 共␾2 − 4␾sp − 3兲 4␭2 3
␮sp = sp sp␾ ,+2共1 − ␾sp兲4
c sp

共19兲

respectively.
In the intermediate coupling regime 5.36⬍ ␭2 / c ⬍ 6.75,
the phase behavior turns out to be qualitatively different, asshown in Fig. 6 for ␭2 / c = 6. In this regime, a lamellarlamellar coexisting region starts to appear. In addition, there
is another coexistence region between the lamellar phasewith a larger spacing 共smaller ␾兲 and the excess water phase.
As seen in Fig. 6共b兲, the corresponding first-order line, ends

FIG. 6. The phase diagrams for ␭2 / c = 6 in model B. The same notation ofdifferent lines is used as in Fig. 2. The filled circle, triangle, diamond indicate the tricritical point 共tcp兲, triple point 共tr兲, critical point 共cp兲,
respectively.

at a tricritical point 共␹tcp = ␮tcp = 0兲 from which a secondorder line emerges. Hence the unbinding transition takes
place at ␹u = ␹tcp = 0 in this regime. As ␹ is increased, wereach a triple point 共tr兲 at which the two lamellar phases andthe unbound phase coexist. Above the triple point, the firstorder line separates between the lamellar phase and the excess water. The locus of the critical point which appears for␭2 / c ⬎ 5.36 is␹cp =3␾cp共1 − 5␾cp兲
,
4共1 − ␾cp兲5

1 + 3␾cp␭2=.
c 4␾cp共1 − ␾cp兲5

共20兲

In Fig. 7, we have plotted ␹cp as a function of ␭2 / c. We seethat ␹cp is a decreasing function of ␭2 / c, and is limited to␹cp ⬍ 0.068.
In the strong coupling regime ␭2 / c ⬎ 6.75, the phase diagrams are similar to Fig. 4 of model A. The unbinding transition point is given by␹u =␾u共1 − 3␾u兲,
4共1 − ␾u兲3

1 + ␾u␭2=.
c 2␾u共1 − ␾u兲3

共21兲The behavior of ␹u as a function of ␭2 / c is plotted in Fig. 7
as well.
V. DISCUSSION

A few points merit further discussion. There are someother possibilities which may cause the unbinding transition
to become a first-order one.15 First, Lipowsky argued that,034906-6

J. Chem. Phys. 124, 034906 共2006兲

Komura, Shimokawa, and Kato

FIG. 7. The values of ␹ at the critical point ␹cp 共solid line兲 and the unbinding transition point ␹u 共dashed line兲 as a function of the coupling strength␭2 / c for model B.

even in the presence of fluctuations, the unbinding transitionof two membranes can be first order when the interaction
potential exhibits a sufficiently high potential barrier.24 We
encounter such a potential barrier when the membranes carryelectric charges. In this case, the direct interaction exhibits
an attractive potential well at small separations and a repulsive potential barrier at intermediate separations. For nonionic surfactant systems, however, the direct interaction potential exhibits only a single minimum resulting from a
repulsive hydration and an attractive van der Waals interaction.
Second, Lipowsky discussed that, for a stack of N membranes, the amplitude of the critical singularities have astrong N dependence so that the critical region for thesecond-order transitions become very small.25 As a result, theunbinding transition may be very abrupt in the large N limit.
This conjecture could explain, at least in part, the discontinuous nature of the unbinding transition in the surfactant systems. However, the quantitative phase behavior for an infinite stack has not yet been considered prior to our work.
Next, we refer to some other works which are related tothe present paper. Some years ago, a direct observation of adiscontinuous unbinding transition was reported for hydrated
multilamellar phospholipid membranes.26 Using x-ray reflectivity techniques, they measured the repeat distance as wellas the fluctuation spectra. But since they did not observe an
increase in the repeat distance as the temperature is raised,they concluded that the steric repulsion was not responsible
for the unbinding. Later the same systems were reinvestigated by another group using grazing angle neutronscattering techniques.27 In contrast with the previous experiment, they detected the increase of the repeat distance, which
was attributed to the enhancement of thermal fluctuations.
On the other hand, they did not observe the unbinding transition, and even questioned the results of Ref. 26. At the
moment, it is not clear whether multilamellar phospholipid
membranes exhibit the unbinding transition or not. However,
we expect that the coupling between the local lipid concentration and the elastic degree of freedom will drive the unbinding transition to be first order.
From theoretical viewpoints, Matsen showed that a
blend of asymmetric diblock copolymers and homopolymers

exhibits an unbinding transition by using a self-consistentfield theory.28,29 With the use of the same method, Janert andSchick investigated a mixture of symmetric diblock copolymers and homopolymers which have identical chain lengths.3
They demonstrated that the unbinding in this system is firstorder, and located the preunbinding transition line. In theirlater work, the channel of unbinding is shown to close assoon as the homopolymer is made slightly larger than theblock.4 It should be noted, however, that these calculations
do not include the attractive long-ranged van der Waals interaction, nor do they include the effect of fluctuations.
In this paper, we did not consider the presence of theisotropic micellar phase, and argued only the phase separation between the bound and unbound phases. However, in theexperimental phase diagram shown in Fig. 1, the micellar
phase 共denoted by L1兲 exists in the lower surfactant concentration region. This micellar phase not only coexists with thelamellar phase 共denoted by L␣兲, but also exhibits the phaseseparation between a dilute and a dense micellar phases. In
order to explain all of these phase behaviors, one has to take
into account the free energy of the micellar phase and compare it with that of the lamellar phase. Such an analysis as
well as other extensions of the present study are currently
under progress.
In summary, we have proposed a phenomenologicalmodel which exhibits a first-order unbinding transition oflamellae in surfactant solutions. The main idea is to take into
account the coupling effect between the surfactant volume
fraction and the other elastic degree of freedom. For the latter, we considered two cases, 共i兲 a membrane elastic degreeof freedom or 共ii兲 a bulk elastic degree of freedom. For bothmodels, the unbinding transition becomes discontinuous
when the coupling is strong enough. We also determined theassociated preunbinding line which separates the two lamellar phases having different repeat distances.
ACKNOWLEDGMENTS

This research was partly supported by the Grant-in-Aid
for Scientific Research 共C兲, Japan Society for the Promotionof Science under Grant No. 15540395.
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Riste 共Plenum, New York, 1989兲.
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S. T. Milner and D. Roux, J. Phys. I 2, 1741 共1992兲.
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