DOI: 10.1002/cphc.200900618

Coupled Modulated Bilayers: A Phenomenological Model
Yuichi Hirose,[b] Shigeyuki Komura,[b] and David Andelman*[a]Dedicated to Erich Sackmann on the occasion of his 75th birthday

We propose a model addressing the coupling mechanism between two spatially modulated monolayers. We obtain themean-field phase diagrams of coupled bilayers when the twomonolayers have the same preferred modulation wavelength.
Various combinations of the monolayer modulated phases are
obtained and their relative stability is calculated. Due to thecoupling, a spatial modulation in one of the monolayers induces a similar periodic structure in the second one. We havealso performed numerical simulations for the case when thetwo monolayers have different modulation wavelengths. Complex patterns may arise from the frustration between the two
incommensurate but annealed structures.

1. Introduction
Quite a number of physical, chemical, and biological systems
manifest some type of modulation in their spatial ordering.[1–3]Such structures are stripes and bubbles in two-dimensional(2D) systems, or lamellae, hexagonally packed cylinders, andcubic arrays of spheres in three-dimensional (3D) cases, as well
as more complex structures such as gyroids. Examples of such
systems include ferromagnetic layers,[4] magnetic garnet
films,[5] ferrofluids,[2, 3, 6] dipolar Langmuir films,[7] rippled phases
in lipid bilayers,[8] and block copolymers.[9, 10] Modulated phases
may also occur in systems described by two (or more) coupled
order parameters, each favoring a different equilibrium state.[11]The observed spatial patterns exhibit striking similarity evenfor systems that are very different in their nature. It is generally
understood that the modulated structures are formed spontaneously due to the competition between short- and longrange interactions.
In the case of 2D ferromagnetic layers, for example, theshort-range interaction arises from magnetic domain wallenergy, while the long-range interaction is due to magnetic
dipole–dipole interaction which induces a demagnetizing
field.[2, 3] Adding both contributions and minimizing the total
free energy with respect to the wavenumber q, one obtains
the most stable mode q* 6¼0. This description is valid in the
weak segregation limit (close to a critical point), where theequilibrium domain size is given by d * ¼ 2p=q*. In general,
this quantity depends on temperature and/or other external
fields.
Herein, we consider two modulated monolayers that are
jointly coupled. Our motivation is related to recent experiments by Collins and Keller[12] who investigated Montal–Mueller planar bilayer membranes[13] composed of lipids and cholesterol. With this technique, a bilayer is constructed by separately preparing two independent monolayers and then combining
them into one joint bilayer across a hole at the air/water interface. The experiments specifically addressed the question ofliquid domains in the two leaflets, and the mutual influence ofthe monolayers in terms of their domain phases. In the experiChemPhysChem 2009, 10, 2839 – 2846

ment, asymmetric bilayers were prepared in such a way that
one leaflet’s composition would phase-separate in a symmetricbilayer and the other’s would not. In some cases, one leaflet
may induce phase separation in the other leaflet, whereas inother cases, the second leaflet suppresses domain formation in
the original leaflet. These results imply that two-leaflet coupling is an important ingredient in determining the bilayer
phase state.
Motivated by these experiments, the coupled bilayer system
was investigated theoretically. The coupling mechanism arises
through interactions between lipid tails across the bilayer midplane, and the phase behavior of such a bilayer membrane
was computed using either regular solution theory[14] orLandau theory.[15] The theoretical results are in accord with several of the experimental observations. It should be noted thatall previous models dealt with the coupling between two
macro-phase-separated leaflets, while it is also of interest to investigate the coupling between two micro-phase-separated(modulated) leaflets. Furthermore, one might also consider theinterplay between a macro- and a micro-phase separation.
Herein, we suggest a model describing the coupling between two modulated systems, and, in particular, we analyze
the influence of this coupling on the phase behavior of two
coupled 2D monolayers. When the two monolayers have thesame preferred periodicity of modulation, we obtain meanfield phase diagrams that exhibit various combinations of
micro-phase-separated structures. In some cases, the periodic
structure in one of the monolayers induces a modulation in
the other monolayer. Interesting situations take place when[a] Prof. D. Andelman
Raymond and Beverly Sackler School of Physics and Astronomy
Tel Aviv University, Ramat Aviv, Tel Aviv 69978 (Israel)
Fax: + (972) 3-641-9529
E-mail: andelman@post.tau.ac.il[b] Y. Hirose, Prof. S. Komura
Department of Chemistry, Graduate School of Science and Engineering
Tokyo Metropolitan University, Tokyo 192-0397 (Japan)

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D. Andelman et al.
the two monolayers have different preferred wavelengths ofmodulation. Here the frustrations between the two competing
modulated structures need to be optimized. These structures
and their dynamical behavior are examined using numericalsimulations. We note that modulated patterns of compositions
are indeed found in the experiments using giant unilamellar
vesicles formed from a ternary mixture of the lipids dioleoylphosphatidylcholine (DOPC), sphingomyelin, and cholesterol.[16, 17] In these bilayer vesicles, a micro-phase separation
occurs between a liquid-disordered phase rich in DOPC and aliquid-ordered phase rich in sphingomyelin and cholesterol. In
ref. [17], even a dynamical sequence of the striped-to-hexagonal morphological transition was observed. Although there has
been no experiment so far which directly corresponds to the
proposed model, our predictions may be verified, for example,
by constructing Montal–Mueller bilayers[13] consisting of two
lipid monolayers that exhibit a striped phase near the miscibility critical point.[18, 19]
In the next section, we present a phenomenological model
describing the coupling between two modulated lipid monolayers. In Section 3, we discuss the case when the two monolayers have the same preferred wavelength of modulation.
Monolayers having different preferred wavelengths are considered in Section 4, and some related situations are further discussed in Section 5. Although we limit our present analysis to
2D systems, the suggested model can be generalized to 3D
systems as well.

2. ModelIn order to illustrate the coupling effect between two modulated systems, we imagine a pair of lipid monolayers forming acoupled bilayer. Each of the monolayers can separately undergo a micro-phase separation. As shown in Figure 1, we assume
that each monolayer is a mixture of two lipid species, say lipid
A and lipid B. Their area fractions are defined by A ðrÞ and

Figure 1. Schematic illustration of two coupled modulated monolayers forming a bilayer membrane. Each monolayer is composed of a binary A/B lipid
mixture, which can have a spatial modulation. The relative composition of
the two lipids in the upper and the lower leaflets are defined by f and y, respectively. In general, the average composition in the two monolayers canbe different. The lipid tails interact across the bilayer midplane. The phenomenological coupling term between these two variables are assumed tobe bilinear of the form Lfy in the free-energy Equation (1).

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B ðrÞ, where r = (x,y) is the 2D positional vector. By assumingthat the monolayer is incompressible, A ðrÞ þ B ðrÞ ¼ 1, themonolayer composition can be characterized by a single-orderparameter defined by the relative A/B compositionðrÞ ¼ A ðrÞ  B ðrÞ. Let us denote this local order parameterof the upper and lower monolayers by f(r) and y(r), respectively. The coarse-grained free-energy functional for the coupled modulated bilayer is written as Equation (1):ZF ½; y ¼ Fu ½ þ F‘ ½y  L dryZt1
¼ dr 2ðr2 Þ2  2ðrÞ2 þ 2 þ 4  m 24
t1
þ2Dðr2 yÞ2  2CðryÞ2 þ y2 þ y4  my y  Ly
2
4

ð1Þ


This is a modified Ginzburg–Landau free energy expanded in
powers of the order parameters f and y and their derivatives.
The Fu ½ free energy has five terms depending only on f andits derivatives. It describes the upper monolayer and its possible modulations, while the coefficients of the Laplaciansquared, the gradient squared and the 4 terms are taken tobe numbers, for simplicity. Similarly, F‘ ½y describing the lower
monolayer contains the next five terms that are only functionsof y and its derivatives. The last term represents the coupling
between the two leaflets. The coefficients of the two gradientsquared terms are both negative (C > 0), favoring spatial modulations, whereas the coefficients of the Laplacian squared
terms are positive (D > 0) to have a stable modulation at finite
wavenumbers. The 2 , 4 , y2 and y4 terms in F are the usual
Landau expansion terms with t ¼ ðT  Tc Þ=Tc being the reduced temperature (Tc is the critical temperature). For simplicity, the two leaflets are taken to have the same critical temperature Tc (and hence the same t). Finally, the linear term coefficients, m and my , are the chemical potentials which regulate
the average values of f and y, respectively.
In the absence of the coupling term (L = 0), each of the twoleaflets can have its own modulated phase. Free-energy functionals such as Fu have been used successfully in the past to
describe a variety of modulated systems: magnetic garnet
films,[5] Langmuir films,[7] diblock copolymers,[9, 20] and amphiphilic systems.[21] Furthermore, interfacial properties between
different coexisting phases have been investigated using asimilar model.[22–24] In the above expression for the freepenergy
ﬃﬃﬃ
F, the f-leaflet has a dominantpwavenumberq* ¼ 1= 2, andﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
so has the y-leaflet with qy* ¼ C=2D. The modulation wavenumbers and amplitudes of the two monolayers coincide
when D = C = 1 and the average compositions are the same.
Next we address the physical origin of the coupling termLfy. We first note that this quadratic term is invariant underthe exchange of  $ y. When L > 0, this term can be obtained from a ð  yÞ2 term,[14, 15] which represents a local
energy penalty when the upper and lower monolayers havedifferent compositions. In the case of mixed lipid bilayers, sucha coupling may result from the conformational confinement of
the lipid chains, and hence would have entropic origin.[14] By
estimating the degree of the lipid chain interdigitation, the 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

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Coupled Modulated Bilayers
magnitude of the coupling parameter L was recently estimated by May.[25] In general, the coupling constant L can also be
negative depending on the specific coupling mechanism.[25]However, it is explained later that the phase diagram for L < 0
can easily be obtained from the L > 0 one. Hence, it is sufficient to consider only the L > 0 case without loss of generality.
Although the microscopic origin of the coupling may differ between systems, we will regard L as a phenomenological parameter and investigate its role on the structure, phase behavior and dynamics of coupled modulated bilayers.
The phase behavior for uncoupled case, L = 0, can be obtained from the analysis of Fu ½[11] and is only briefly reviewed
here (see also Figure 2). For a 2D system, the mean-field phase
diagram can be constructed by comparing the free energies ofstriped (S) and hexagonal (H) phases. In terms of the f orderparameter, the stripe phase is described by Equation (2):S ðrÞ ¼ 0 þ 2q cosðq xÞ

ð2Þwhere 0 ¼ hi is the spatially averaged composition (imposedby the chemical potential m ), and q is the amplitude of the
q*-mode in the x-direction. Similarly, the composition of thehexagonal phase is given by a superposition of three 2Dmodes of equal magnitude, jqi j ¼ q* as shown in Equation (3):32q X
H ðrÞ ¼ 0 þ pﬃﬃﬃcosðqi  rÞ
3 i¼1

ð3Þ

t
1
3
fS ð0 ; q Þ ¼ 20 þ 40 þ ðt  1 þ 320 Þ2q þ 4q
2
4
2
t
1
4
5
fH ð0 ; q Þ ¼ 20 þ 40 þ ðt  1 þ 320 Þ2q þ pﬃﬃﬃ 0 3q þ 4q
2
4
2
3

ð6Þ
t
1
fD ð0 Þ ¼ 20 þ 40
2
4

ð7Þ

In Figure 2 a, we reproduce the original phase diagram ofrefs. [7, 11]. The striped, hexagonal, and disordered phases are
separated by first-order phase-transition lines. Regions of twophase coexistence do exist, but are omitted from the Figure
for clarity’s sake.[26] Thus, the transition lines indicate the locus
of points at which the free energies of two different phasescross each other, and are not the proper phase boundaries (binodals). The critical point (*) is located at ð0 ; tÞ = (0,1).

3. Two Coupled Leaflets with the Same q*

ð4Þ

3.1. Free-Energy Densities
First we consider the case when D = C = 1 so that the preferred
wavenumbers are the same forbothpﬃﬃmonolayers,ﬃ
q* ¼ qy* ¼ q* ¼ 1= 2.
Themean-field phase diagram is calculated within the single-mode
approximation. Various combinations of 2D modulated structuresappearing in the two monolayers are possible. The first example is the striped–striped (SS)phase, in which both monolayers exhibit the striped phase.
This can be expressed by Equations (8) and (9):Figure 2. a) Mean-field phase diagram of a single monolayer with a modulated structure in the vicinity of the critical temperature, computed using a model as in Equations (5)–(7). 0 is the average composition and t is the reduced temperature. The three phases are: striped (S), hexagonal (H), and disordered (D). These phases are separated by first-order transition lines, while for clarity we omit showing coexistence regions. (*) located atð0 ; tÞ ¼ ð0; 1Þ indicates the critical point. Note the shift of the critical temperature from zero to unity when themodulated phases are considered. b) Mean-field phase diagram of decoupled (L = 0) modulated monolayers at
t = 0.8. 0 and y0 are the average compositions in the two leaflets. The notations of the different phases are described in Section 3.1. All the phases are separated by first-order transition lines.

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ð5Þ

Having introduced the free energy and explained the phasebehavior of the uncoupled case, we now explore the equilibrium and non-equilibrium properties of two coupled modulatedmonolayers, L ¼
6 0.

where [Eq. (4)]:q1 ¼ q* x̂,pﬃﬃﬃ 
q* 
q2 ¼
x̂ þ 3ŷ ,2pﬃﬃﬃ q* 
x̂  3ŷ
q3 ¼
2

P3
and
i¼1 qi ¼ 0. In the above, only the most stable wavenumber q* is used within the single-mode approximation. This canbe justified for the weak segregation region close to the critical point.[5]Averaging over one spatial period, we obtain the freeenergy densities of the striped, hexagonal, and disorderedphases, respectively, shown in Equations (5)–(7): 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

S ðrÞ ¼ 0 þ 2q cosðq* xÞð8Þ

yS ðrÞ ¼ y0 þ 2yq cosðq* xÞð9Þ

where 0 ¼ hi and y0 ¼ hyi
are the average compositions, q

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D. Andelman et al.
and yq are the respective amplitudes. These composition profiles are substituted into the free energy of Equation (1). Averaging over one spatial period, we obtain the free-energy density of the SS phase, given by Equation (10):fSS ¼ fS ð0 ; q Þ þ fS ðy0 ; yq Þ  Lð0 y0 þ 2q yq Þ



3
2yq X
2p
cos qi  r þ
yH* ðrÞ ¼ y0 þ pﬃﬃﬃ
3
3 i¼1

The free energy density of the HH* phase is then obtained as
Equation (17):ð10Þ
fHH* ¼ fH ð0 ; q Þ þ fH ðy0 ; yq Þ  Lð0 y0  q yq Þ

where fS is defined in Equation (5). We then minimize fSS withrespect to both q and yq for given 0 , y0 , t and L. Wheneither q or yq vanishes, the corresponding monolayer is in its
disordered phase and the mixed bilayer state will be called thestriped–disordered (SD) or the disordered–striped (DS) phase.
Note that we use the convention that the first index is of the
f-leaflet and the second of the y-one. When both q and yq
are zero, the free energy density of the disordered–disordered(DD) phase is given by Equation (11):fDD ¼ fD ð0 Þ þ fD ðy0 Þ  L0 y0

ð11Þ

where fD is defined in Equation (7). This free energy fDD was analyzed in ref. [15] in order to investigate the macro-phase separation of a bilayer membrane with coupled monolayers. It was
shown that the bilayer can exist in four different phases, andcan also exhibit a three-phase coexistence.
Similar to the stripe case, the order parameters of the hexagonal–hexagonal (HH) phase can be represented by Equations (12) and (13):32q X
H ðrÞ ¼ 0 þ pﬃﬃﬃcosðqi  rÞ
3 i¼1

ð12Þ

3
2yq X
yH ðrÞ ¼ y0 þ pﬃﬃﬃcosðqi  rÞ
3 i¼1Another combination which should be considered in thepresent model is the asymmetric case where one monolayer
exhibits the striped phase and the other the hexagonal phase.
This striped–hexagonal (SH) phase is expressed as Equations (18) and (19):S ðrÞ ¼ 0 þ 2q cosðq* xÞ

ð18Þ

3
2yq X
cosðqi  rÞyH ðrÞ ¼ y0 þ pﬃﬃﬃ
3 i¼1

ð19Þ

The free energy density of this SH phase is calculated to be[Eq. (20)]:
2
fSH ¼ fS ð0 ; q Þ þ fH ðy0 ; yq Þ  L 0 y0 þ pﬃﬃﬃ q yq
3

ð14Þ

where fH is defined in Equation (6). When either q or yq vanishes, one of the monolayers is in the disordered phase andthe bilayer will be called the hexagonal–disordered (HD) phaseor the disordered–hexagonal (DH) phase.
When the normal hexagonal phase in one leaflet is coupled
to the inverted hexagonal phase in the other leaflet, it is energetically favorable to have a particular phase shift of 2p/3 between the two hexagonal structures. The order parameterswhich represent such a different type of hexagonal–hexagonal(HH*) phase can be written as Equations (15) and (16):www.chemphyschem.org

ð20Þ

The phase in which S and yH in Equations (18) and (19) are
interchanged with H , yS is called the hexagonal–striped (HS)phase, and its free energy is obtained from the SH phase bynoting the  $ y symmetry. In addition to these phases, wehave also taken into account the square–square (QQ) phase expressed by Equations (21) and (22):2qQ ðrÞ ¼ 0 þ pﬃﬃﬃ ½cosðq* xÞ þ cosðq* yÞ
2

ð21Þ

2yq
yQ ðrÞ ¼ y0 þ pﬃﬃﬃ ½cosðq* xÞ þ cosðq* yÞ2ð22ÞThen its free energy density is given by Equation (23):fQQ ¼ fQ ð0 ; q Þ þ fQ ðy0 ; yq Þ  Lð0 y0 þ 2q yq ÞfHH ¼ fH ð0 ; q Þ þ fH ðy0 ; yq Þ  Lð0 y0 þ 2q yq Þ

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ð17Þ

ð13Þ

where the basis of the three qi is defined in Equation (4). By repeating the same procedure as for the SS phase, the freeenergy density of the HH phase is obtained as Equation (14):32q X
H ðrÞ ¼ 0 þ pﬃﬃﬃcosðqi  rÞ
3 i¼1

ð16Þ

ð15Þ

ð23Þ

where [Eq. (24)]:t1
9fQ ð0 ; q Þ ¼ 20 þ 40 þ ðt  1 þ 320 Þ2q þ 4q
2
4
4

ð24Þ

However, we show below that this QQ phase cannot be morestable than the other phases.

3.2. Bilayer Phase Diagrams
Minimizing Equations (10), (14), (17), (20) and (23) with respect
to both q and yq , we obtain the phase diagram for the coupled bilayer. As a reference, we first show in Figure 2 b thephase diagram in the decoupled case (L = 0) for t = 0.8. Thiscan easily be obtained from Figure 2 a by combining its twocross-sections (one for 0 and one for y0 ) at t = 0.8. Figure 3

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Coupled Modulated Bilayers
gions is almost unaffected by
the coupling. Even when thetemperature is lowered by decreasing t, only the phases located
close
to
the
origin(0 ¼ y0 ¼ 0) would expand,
and the global topology does
not change substantially.
For a larger value of the coupling parameter (L = 0.2), thefive regions of the SS phasemerge together forming one
single continuous SS region. The
Figure 3. Mean-field phase diagram of coupled modulated bilayers for t = 0.8. 0 and y0 are the average compofour HH regions are still distinct
sitions in the two leaflets. The coupling parameter is chosen to be a) L = 0.02 and b) L = 0.2. The notations of theand separate the SS region fromdifferent phases are described in Section 3.1. All the phases are separated by first-order transition lines. The phasefour DD phase regions. Notediagram is symmetric with respect to the two principal diagonals 0 ¼ y0 and 0 ¼ y0 .
that in Figure 3 b, all phases
have a symmetric combinationof phase modulation such as SS or HH. The asymmetric combigives the phase diagram for a coupled bilayer when a) L =nation such as the SH phase does not appear, because the0.02 and b) L = 0.2, while the temperature is fixed to t = 0.8 aslarge coupling parameter strongly prefers symmetric phases ofbefore. On the ð0 ; y0 Þ-plane, we have identified the phaseequal modulations in the two monolayers, although the qwhich has the lowest energy, whereas possible phase coexisand yq amplitudes of the two modulated monolayers are nottence regions between different phases were ignored. All thethe same in the stripe SS phase (or the hexagonal HH phase).
boundary lines indicate first-order transitions. Since the freeAs the value of L increases from 0.02 to 0.2, first the SH phaseenergy Equation (1) is invariant under the exchange of  $ y,disappears, followed by the disappearance of the HH* phase.
the phase diagrams are symmetric about the diagonal line
When the value of L increases further, the regions of the SS
0 ¼ y0 as the upper and lower leaflets have been chosen arand HH phases expand at the expense of the DD phase rebitrarily. These phase diagrams are also symmetric under rotagions. This means that the coupling between the monolayers
tion of 180 degrees around the origin because Equation (1) iscauses more structural order in the bilayer. Finally we remark
invariant (except for the linear terms) under the simultaneousthat the QQ phase was never found to be more stable thantransformations of  !  and y ! y. This is reasonable as
any of the other phases considered above.
the labels of “A” or “B” for the two lipids were assigned arbitraAlthough we have so far assumed that L is positive, therily. As a consequence, the phase diagrams are also symmetricphase diagrams for L < 0 can be easily obtained from thoseabout the diagonal line 0 ¼ y0 . The symmetries with refor L > 0 by rotating them by 90 degrees around the origin.
spect to both 0 ¼ 0 and y0 ¼ 0 in Figure 2 b for L = 0 are
This is because the free-energy Equation (1) is invariant under
now broken because of the coupling between the two leaflets.
the simultaneous transformations of either  !  andWhen the coupling parameter is small (L = 0.02), the global
L ! L , or y ! y and L ! L.
topology of the phase diagram resembles that of the uncoupled case presented in Figure 2 b. Close to the origin,0 ¼ y0 ¼ 0, there is a region of SS phase surrounded byeight other phases: two SH, two HS, two HH, and two HH*
3.3. Modulated Bilayer Dynamics
phases. The HH phase appearing in the region of 0 < 0 and
In order to check the validity of the obtained phase diagramy0 < 0 is a combination of the two inverted hexagonal strucand to investigate the dynamics of coupled modulated bilaytures on each monolayer. One sees that the HH* phase apers, we consider now the time evolution of the coupled equapears in the regions of 0 y0 < 0, where the hexagonal and thetions for f and y, as shown in Equation (25):inverted hexagonal structures are coupled to each other.
A remarkable feature of this phase diagram is the existence
@
dF
@y
dF
of the SS and HH phases in the regions where either j0 j orð25Þ
¼ L r 2;¼ Ly r 2
@t
@t
d
dy
jy0 j are large. These outer SS and HH phases extend up to the
maximum or the minimum values of the compositions. These
Here we have assumed that both f and y are conserved orderregions of the SS and HH phases with L > 0 roughly correparameters in each of the monolayers (model B in the Hohenspond to those of the SD (DS) and HD (DH) phases, respectiveberg–Halperin classification[27]). For simplicity, the kinetic coeffily, in Figure 2 b with L = 0. Hence, the modulated structure incients L and Ly are taken to be unity, and both the hydrodyone of the monolayers induces the same modulated phase innamic effect and thermal fluctuations are neglected. We solve
the other monolayer due to the coupling term. Notice that thethe above equations numerically in 2D using the periodic
SD (DS) phase and HD (DH) phase do not exist in Figure 3 a.
boundary condition. Each simulation starts from a disordered
We further remark that the extent of the four DD phase restate with a small random noise around the average composiChemPhysChem 2009, 10, 2839 – 2846

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D. Andelman et al.

Figure 4. Equilibrium patterns of coupled modulated monolayers with t = 0.8. Setting D = C = 1 implies the same
q* in both monolayers. The patterns of f, y, f+ y, and fy at t = 5000 are presented. The other parameters arechosen to be a) 0 ¼ 0:2, y0 ¼ 0:2, L = 0.02, and b) 0 ¼ 0:5, y0 ¼ 0, L = 0.2.

tern of f. This corresponds to
the SS phase shown in Figure 3 b. The periodicities of thetwo striped structures are thesame, although their amplitudes
differ. Notice that the modulation phase of f+ y is shifted byp relatively to that of fy. Since
the patterns in Figure 4 would
correspond to the equilibrium
configurations, they can be compared with the phase diagramsin Figure 3. We conclude thatthese simulation results indeed
reproduce the predicted equilibrium modulated structures.

4. Coupled Monolayers with Two Different q*

tions 0 and y0 . In Figure 4, we show typical equilibrium patterns of f, y, f+ y and fy for two choices of parameters.
We consider next the more general case in which the preferred
The f+ y pattern is presented here because this quantity canwavelengths of modulation in the two uncoupled leaflets are
be directly observed in the experiment on Montal–Mueller bidifferent, q* 6¼q*y . The free-energy densities cannot be obtained
layers using fluorescence microscopy.[12] The quantity fyanalytically as was done in Section 3.1, because there is not ameasures the concentration contrast between the f and ysingle periodicity on which one can average f(r) and y(r). Dueleaflets. Time is measured in discrete time steps, and t = 5000
to such a difficulty in the analytical treatment, we present
corresponds to a well-equilibrated system. In all the simulabelow the results of numerical simulations, relying on Equations below, the temperature is fixed to be t = 0.8, correspondtion (25) for the time evolution of the two coupled-order paing to the weak segregation regime. Notice that all the patrameters.
terns in Figure 4 are presented
with the same gray scale.
Figure 4 a illustrates the coupling between a hexagonalphase with 0 ¼ 0:2 and an inverted hexagonal phase withy0 ¼ 0:2 in the weak coupling
regime (L = 0.02). Being consistent with the phase diagram of
Figure 3 a, this parameter choiceyields the HH* phase as seenfrom the pattern of f+ y, where
the two hexagonal structures aresuperimposed. We note that thedifference in the order parameter fy also exhibits a hexagonal structure.
Figure 4 b shows the equilibrium patterns for 0 ¼ 0:5 andy0 ¼ 0 in the strong coupling
regime (L = 0.2). If there were
no coupling, the f-monolayer
would not exhibit any modulation (as it is in its own disordered phase), whereas the ymonolayer is in the striped
Figure 5. Patterns of coupled modulated monolayers with t = 0.8. The patterns of f, y, f+ y, and fy are prephase. We clearly see that, due
sented for simulation time, t = 5000, and the average compositions are set to be 0 ¼ y0 ¼ 0. The other parameto the coupling effect, the stripe ters are chosen to be a) D = C = 1, L = 0.02, b) D = 0.1296, C = 0.36, L = 0.02, c) D = 0.1296, C = 0.36, L = 0.2, andstructure is induced in the pat- d) D = 0.1296, C = 0.36, L = 0.4. In all cases but (a), D ¼
6 C and the two periodicities are nonequal, q* 6¼q*y .

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Coupled Modulated Bilayers
In Figure 5, we show the patterns for t = 0.8 and0 ¼ y0 ¼ 0, when both monolayers exhibit the striped phasewithout the coupling. As a reference, we show in Figure 5 a
the case when D = C = 1 and L = 0.02 corresponding to the SS
phase in Figure 3 a. The patterns of f and y match each otheras the composition difference fy vanishes throughout thesystem. In Figure 5 b, the parameters are chosen to be D =0.1296, C = 0.36 and L = 0.02. The preferred wavenumbers of
the two monolayers are different: q*y ¼ 1:67q* for uncoupledleaflets. The above set of parameters, especially D and C, ischosen in such a way that the amplitudes of the two stripes
are nearly equal. As long as the coupling parameter is small,the two stripes of different periodicities are formed rather independently. The superposition of the two striped structures
produces an interference pattern resulting in a new modulation as seen from the pattern of f+ y. The striped modulations of f and y are almost parallel or perpendicular to eachother.
When L increases, up to L = 0.2 as in Figure 5 c, the y-field
exhibits a complex pattern in which two different length scales
coexist (reflecting q* and qy* ), whereas the pattern of f is characterized by a single mode (reflecting q* ). The patterns of f
and y almost match each other when L = 0.4, as seen in Figure 5 d. In this case, the modulation with a longer wavelength(q* ) dominates both monolayers. Figures 5 b–d provide a typical sequence of morphological changes, that is, interference
pattern!two-mode pattern!single-mode pattern, as the coupling constant L is increased.
To further analyze the temporal correlations of the twoorder parameters, f and y, we have plotted in Figure 6 thetime evolution of the quantity shown in Equation (26):DðtÞ ¼1L2Z

dr½ðr; tÞ  yðr; tÞ2

ð26Þ

where L = 128 is the linear system size in the simulations.
(c), (a), (g) and (d) correspond to the time evolutions of D(t) in Figures 5 a, b, c, and d respectively. The (c, a)first increases and then approaches zero, as the patterns of thetwo stripes coincide in the late stage. In (b), (a) increases in
two separate stages. At first it increases because the growthrate of the modulation with the smaller wavelength (corresponding to y) is faster than that with larger periodicity (corresponding to f) as is also revealed from the linear stability analysis of Equation (25) which will be published elsewhere. Later,
the value of D remains large for small coupling parameter(L = 0.02). When the coupling becomes even stronger (L =0.2) as for (g) in (c), the value of D is suppressed comparedto (a, b), because f and y tend to have more overlap forlarger L. The same applies for (d, d) with L = 0.4 as compared to (g, c).

5. Discussion and Conclusions
We propose a minimal model describing the coupling phenomena between two modulated bilayers. Considering 2D
case, we obtain the mean-field phase diagram when the two
coupled and spatially modulated monolayers have the same
preferred periodicity. Various combinations of modulated
phases can exist, such as the SS, HH, HH* and SH (HS) phases
as described in Section 3. We have seen that modulations in
one of the monolayers induce similar modulations in the other.
The region of the induced modulated phase expands as thecoupling parameter becomes larger.
When the two monolayers have different inherent wavelengths in the decoupled case, we have conducted numericalsimulations to investigate the morphologies and dynamics ofthe coupled system. We obtain several complex patterns arising from the frustration induced by the two incommensurate
structures. As the coupling constant L increases, the two different modes start to interfere with each other and eventuallycoincide. The time evolution of the striped structures can takeplace in two steps, reflecting the different growth rate of thetwo modulations.
It is instructive to rewrite the free-energy Equation (1) interms of the sum and the difference of f and y, that is,h ¼   y. When D = C = 1, we obtain Equation (27):11dr ðr2 hþ Þ2  ðrhþ Þ2 þ ðt  LÞh2þ þ h4þ  mþ hþ
4
32

1
1 4
3 2 2
2
2
2
2
þðr h Þ  ðrh Þ þ ðt þ LÞh þ h  m h þ hþ h
432
16
F hþ; h ¼

Z

ð27Þ

Figure 6. Time evolutions of D defined by Equation (26). (c), (a),(g)and(d) correspond to the parameters of Figures 5a, b, c and d, respectively.

ChemPhysChem 2009, 10, 2839 – 2846

where m ¼ ðm  my Þ=2. Hence the coupling term between hþand h in the free energy takes the form of an h2þ h2 term witha numerical positive coefficient. The original coupling parameter L enters in the coefficients of the h2þ and h2 terms (butnot in the h2þ h2 coupling term). It shifts the respective transition temperatures of hþ and h in opposite directions. Whenthe gradient terms are absent and m ¼ 0, a similar model was
considered by MacKintosh and Safran, who studied transitions
between lamellar and vesicle phases in two-component fluid
bilayers.[28] 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

www.chemphyschem.org

2845

D. Andelman et al.
The proposed free energy Equation (1) has some analogies
to the previous model for the rippled phase in lipid bilayers.[29]It was argued that the coupling between the membrane curvature and the asymmetry in the area per molecule between thetwo monolayers would induce a structural modulation of a bilayer. By considering a similar mechanism, Kumar et al.[30] investigated various modulated phases in two-component bilayer
membranes. They claimed that the phase behavior of twocomponent bilayers resembles that of three-component monolayer. This is because the three different local combinations ofupper/lower composition in bilayers (A/B, B/A, and A/A for
excess of A), would correspond to three different types of molecules for the monolayer. One of the new aspects in ourmodel is that the preferred wavelengths of the two monolayers can, in general, be different from one another leading to afrustrated bilayer state.
We also point out that there are some similarities between
coupled modulated structures and the problem of atoms adsorbed on a periodic solid substrate. The latter topic has been
extensively studied within the Frenkel–Kontorova (FK) model
which provides a simple description of the commensurate–incommensurate transition.[31] Our model and the FK model are
analogous in the sense that there are two natural length scales
whose ratio changes as a function of other model parameters.
In the FK model, however, these length scales are quenched,
whereas in our model they are annealed.
Another related experimental system can be seen for surface-induced ordering in thin film of diblock copolymers.[32]When the surface is periodically patterned, a tilt of the lamellae
is induced in order to match the surface periodicity. The situation becomes more complex if a copolymer melt is confined
between two surfaces. An interesting case arises when thespacing between the two surfaces is incommensurate with thelamellar periodicity.[32]For systems out of equilibrium, spatial resonances and superposition patterns combining stripes and/or hexagons were
investigated in a reaction–diffusion model with interacting
Turing modes of different wavelengths.[33] These models weresuccessful in reproducing hexagonal superlattice patterns
which are known as “black-eyes”. Although the mechanism ofpattern formation is different than in our model, we observe
similar superposition patterns as reported in ref. [33], such as
hexagons on stripes or hexagons on hexagons (not shownherein).
A more detailed study of the present model and several interesting extensions will be published elsewhere. One possible
extension is to consider vector order parameters describing,for example, the molecular tilt for coupled bilayers.[34, 35] Whenthe two order parameters are vectors, the nature of the transitions between different phases can be different, and even thesquare phase may exist in thermodynamic equilibrium.[36]Scientists No. 215097 from the Japan Society for the Promotionof Science (JSPS). SK acknowledges support by KAKENHI (Grantin-Aid for Scientific Research) on Priority Areas “Soft Matter Physics” and Grant No. 21540420 from the Ministry of Education, Culture, Sports, Science and Technology of Japan. DA acknowledges
support from the Israel Science Foundation (ISF) under grant No.
231/08 and the US–Israel Binational Foundation (BSF) under
grant No. 2006/055.
Keywords: lipids · membranes · phase transitions · selfassembly · statistical thermodynamics

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We thank T. Kato, S. L. Keller, M. Schick, and K. Yamada for useful
discussions. YH acknowledges a Research Fellowship for Young

Received: August 5, 2009
Published online on October 21, 20092846

www.chemphyschem.org

 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

ChemPhysChem 2009, 10, 2839 – 2846