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Shape and stability of two-dimensional lipid domains with dipole-dipoleShape and stability of two-dimensional lipid domains with dipole-dipole

interactionsinteractions

Mitsumasa Iwamoto, Fei Liu, and Zhong-can Ou-Yang

Citation: J. Chem. Phys. 125125, 224701 (2006); doi: 10.1063/1.2402160

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Shape and stability of two-dimensional lipid domains

with dipole-dipole interactions

Mitsumasa Iwamotoa?

Department of Physical Electronics, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku,

Tokyo 152-8552, Japan

Fei Liu

Center for Advanced Study, Tsinghua University, Beijing 100084, China

Zhong-can Ou-Yang

Institute of Theoretical Physics, The Chinese Academy of Sciences, P.O. Box 2735 Beijing 100080, China

?Received 14 June 2006; accepted 30 October 2006; published online 8 December 2006?

We study the general energy and shape of the two-dimensional solid monolayer domains with the

dipole-dipole interactions. Compared with the domain energy without tilted dipole moments ?M.

Iwamoto and Z. C. Ou-Yang, Phys. Rev. Lett. 93, 206101 ?2004??, the general dipolar energy is not

only shape and size but also boundary orientation dependent. The general shape equation derived by

this energy using variational approach predicts a circular solution and an equilibrium shape grown

from this circle. In particular, the latter is composed of two branches: a translation-induced growth

of all odd harmonic modes and a pressure-induced cooperative deformation by all even harmonic

modes. The good qualitative agreement between our prediction and the experimental observations

shows the validity of the present theory. © 2006 American Institute of Physics.

?DOI: 10.1063/1.2402160?

I. INTRODUCTION

As an extension of the order parameters in liquid crystals

?LCs?,1–4the orientational state of the floating monolayer

composed of rod polar molecules can be characterized by

three nonzero orientational order parameters Sn?n=1,2,3?.

These parameters are defined by the thermodynamic average

of the Legendre polynomials, Pn?cos ?? ?n=1, 2, and 3?, of

the orientational angle ? from the normal direction to the

surface.Among them, S1and S3do not appear in LCs and are

the specific parameters of the monolayer.5–8The two param-

eters connect to symmetry breaking and naturally relate to

the spontaneous and nonlinear polarization which have been

probed by Maxwell-displacement current ?MDC? and second

harmonic generation ?SHG?.8The presence of nonzero S1

and S3also implies that the dipole-dipole interaction as well

as the dipolar energy are inevitable when we discuss the

pattern formation of the monolayers and the domain shapes

of the monolayers. Many sophisticated microscopic tech-

niques, such as florescence microscopy9?FM? and Brewster

angle microscopy ?BAM?, were developed to detect various

monolayer domains under different physical or chemical cir-

cumstances. The simplest system should be pure single-

component phospholipid at the air-water interface.10–12

On the theoretical side, during the past two decades,

great theoretical efforts have been devoted to quantitatively

understanding the two-dimensional ?2D? domain shapes and

shape transitions in this simplest system.13–16They were al-

most based on the idea firstly given by Andelman et al.17the

shape and shape transition of the domains are determined by

a competition between the line tension at the domain bound-

ary between fluid and the domain region and the long range

dipole-dipole electrostatic force between the lipid molecules,

namely, the minimum of the energy13,14

F = ??dl + F?+ F?,

?1?

where ? is the unit line tension, F?is the energy of the

dipole-dipole interaction between the dipoles oriented nor-

mally to the water surface,

2??

F?= −??

2

t?l? · t?s?

?r?l? − r?s??dlds,

?2?

F?is the energy of the dipole-dipole interaction between the

dipoles oriented in parallel to the water surface,

2??

?r?l? − r?s??

F?=??

2

?t?l? · y ˆ0??t?s? · y ˆ0?

dlds,

?3?

r?s? describes the domain boundary curve, s is the arc length,

t=dr/ds is the unit tangential vector of the boundary, and y0

is the unit vector of the y axis; see Fig. 1. Here all dipoles are

assumed to uniformly tilt along the x direction with the same

tilted angle ?, i.e.,

? = ???,0,??? = ?0?sin ?,0,cos ??.

?4?

The different phases and their coexistence in the 2D do-

main were expected to have an analogy with the three-

dimensional ?3D? case.18Such prediction has been directly

a?Electronic mail: iwamoto@pe.titech.ac.jp

THE JOURNAL OF CHEMICAL PHYSICS 125, 224701 ?2006?

0021-9606/2006/125?22?/224701/9/$23.00 © 2006 American Institute of Physics

125, 224701-1

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visualized by the FM.19,20Helfrich suggested that the shape

energy of a 3D vesicle is21

F = ?P?dV + ??dA +kc

2??2H + c0?2dA,

?5?

where ?P is the osmotic pressure, ? is the tensile stress

acting on the membrane, dA and dV are the surface and

volume elements, respectively, kcis the bending rigidity, H is

the mean curvature, and c0is the spontaneous curvature. The

last term represents the curvature-elastic energy Fc. Hence

we asked whether the 2D domain energy F can also be ex-

pressed geometrically like Eq. ?5?. If so, the previous knowl-

edge about Eq. ?5? ?Refs. 22–24? can then be mapped onto

the 2D domain case. Recently the present authors indeed

found that25when the dipole moment is perpendicular to the

water plane, the double-line integral in the energy ?Eq. ?2?? is

approximated by a sum of an additionally negative line ten-

sion and a curvature-elastic energy of the domain boundary.

Many important physical properties and experimental phe-

nomena about the 2D monolayer domain were understood

analytically and quantitatively.

In the present paper, we follow the same approximation

approach to consider the general domain energy having tilted

dipole moments. We think that such an extension is not only

for a more general formalism to treat the domain shape prob-

lem of over 20 years but also very essential to consider in

recent experiments. Radhakrishnan et al. found that an in-

plane applied electric field can destabilize the condensed

complex and induce micron-scale domain formation around

the electrode.26This observation implies that the in-plane

dipole moments of lipids might have an important effect in

the formation of the 2D monolayer domains. On the other

hand, McConnell and Moy also studied the thin shape tran-

sition of a rectangular domain induced by increasing size.

They found that a small tilted angle ? can completely oblit-

erate the sharp transition observed in the perpendicular di-

pole case.14In particular, they observed a circle to clover-

leaf-shaped

?CLS?

domain

dipalmitoylphosphatidylcholine27?DPPC? induced by com-

pression. A remarkable feature of the CLS domains is that it

is not of exact C3symmetry but a distorted triangle with C2

symmetry. Although these intriguing phenomena in the latter

experiment were attributed to the presence of the tilted di-

growthina racemic

pole moments, to the best of our knowledge, they lack quan-

titative explanation. We will show that the analytical formal-

ism developed for the tilted dipole moments in the present

work can deal with the problem.

The paper is organized as follows. In Sec. II some defi-

nitions and basic formulas are given. In Sec. III, we apply

Taylor’s expansion to approximate the general domain en-

ergy as a sum of general line tensions and curvature-elastic

energy. Then the general shape equation of the energy is

yielded by variation calculation in Sec. IV. The circular so-

lutions and their instabilities are treated in Secs. V and VI,

respectively. Section VII concludes the paper.

II. DEFINITIONS AND BASIC FORMULAS

Compared with the model given by McConnell,15we

additionally introduce a Lagrange multiplier ?P in analogy

with the Helfrich curvature-elastic energy. This multiplier is

viewed as the constraint of constant domain area and/or has

real physical meaning, i.e., ?P=?−g0, where ? is the 2D

surface pressure of the monolayer and g0is the difference in

the Gibbs free energy density between the outer ?e.g., fluid?

and inner ?solid? phases. Because the solid phase is more

stable than the fluid one, g0is always positive. Then the 2D

domain energy is

F = ?P?dA + ??dl + F?+ F?.

?6?

We describe the boundary of a domain as a closed curve

r?s? in a 2D plane. The unit tangential vector t?s? and the

outward unit normal vector m?s? of the curve at s are related

to r?s? through the Frenet formulas in the plane:28

dr?s?

ds

= t?s?,

t · t = 1,

dt?s?

ds

= ??s?m?s?,

m · m = 1,

?7?

dm?s?

ds

= − ??s?t?s?,

m · t = 0,

where ??s? is the curvature of the curve at s. Another conve-

nient representation of the unit vectors is as follows:

t?s? = ?cos ??s?,sin ??s??,

?8?

m?s? = ?sin ??s?,− cos ??s??,

where ??s? is the boundary orientational angle at s and is

related to the curvature by

??s? = −d??s?

ds

.

?9?

Given that r?s? is the boundary curve of a domain in

equilibrium, consider a slightly distorted curve,

r??s? = r?s? + ??s?m?s?,

?10?

where ??s? is a sufficiently small smooth variation function.

With the help of the formulas given in the Appendix, we

FIG. 1. Schematic illustration of the shape of a monolayer domain, whose

dipoles ? uniformly tilt along the x direction with the same tilted angle ?.

The geometric quantities describing its boundary curve are also shown, and

h is the thickness of the monolayer.

224701-2 Iwamoto, Liu, and Ou-YangJ. Chem. Phys. 125, 224701 ?2006?

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calculate the variations of the boundary length L and area A

of the domain:

?L =???ds? =??− ?? +1

2?s

2+ o??3??ds

?11?

and

?A =1

2???r · mds? =??? −1

2??2+ o??3??ds.

?12?

In the derivations the following two relations are useful:

?f?s??sds = −?fs?ds

?13?

and

?f?s??ssds =?fss?ds.

?14?

Here f?s? is any smooth continuous function.

III. ENERGY FORMULA OF THE 2D DOMAIN

The key step to approximate the double-line integrals is

to rewrite the energies of the two components of the dipole

moments as

2???

2???

?r?s + x? − r?s??

F?+ F?= −??

2

t?s? · t?s + x?

?r?s + x? − r?s??dx?ds

+??

2

?t?s + x? · y ˆ0??t?s? · y ˆ0?

dx?ds,

?15?

where the range of the arc variable x?l−s is ?h,L?; h rep-

resents a nonzero monolayer thickness as a cutoff; see Fig. 1.

Substituting Eqs. ?A3?–?A5? into the above equation, the in-

ner integrals ?the brackets? are then the sum of ci?s?xi, i=

−1,0,1,..., where ci?s? are the functions of the curvature

??s?, its derivatives with different orders, and the angle ??s?.

If we consider a somewhat smooth curve and only take into

account the terms of x till the first order, after integrating we

obtain an approximation given by

2?ds?

h

h?ds +11

F?= −??

2

L

dx?1

x−11

24??s?2x + o?x2??

2L2???s?2ds

? −??

2

2

lnL

96??

?16?

and

F?=??

2

2?ds?

−?1

h

L

dx?1

xsin2??s? −1

24??s?2sin2??s??x + o?x2??

h? sin2??s?ds −

???11+ 13 cos 2??s????s?2ds.

2??s?sin 2??s?

4?s?s?sin 2??s? +11

???

2

2

lnL

1

192??

2L2

?17?

In the derivations Eq. ?9? has been invoked to remove the

derivatives of ??s?.

The validity of the above approximation is as follows: If

we consider the limit h→0 the divergences of Eqs. ?16? and

?17? depend only on the integral of the first integrand term of

1/x; it is a good approximation to neglect the terms of o?x2?.

As we pointed out in our Letter,25for the exact calculation

for a circle of radius ?0, the logarithmic divergence is

ln?8?0/e1/2h??ln?5?0/h? ?see Eq. ?2.11? in Ref. 16? and this

nearly equals ln?2??0/h? calculated by our Taylor’s expan-

sion. In fact, such an ansatz approach by expanding the line

integral was also proposed by Langer et al. ?see Eqs. ?4.11?

and ?4.12? in Ref. 16? early.

We see that F?is approximated by a sum of an addition-

ally negative line tension and a curvature-elastic energy of

the domain boundary. This indeed has a Helfrich-type form

in the 2D case. Generally, the total energy F of the domain

depends on the size ?L?, the orientation ???, and the shape

??? ?SOS?.

IV. SHAPE EQUATION

To obtain the equilibrium-shape equation, we derive the

first variation of the general energy ?Eq. ?6??, which includes

four parts:

??1?F = ?P??1??dA + ???1??ds + ??1?F?+ ??1?F?. ?18?

Equations ?11? and ?12? have given the first variations of the

boundary length L and the area of the domain A,

??1?L =???1?ds =?− ???s?ds

?19?

and

??1?A =???1?dA =???s?ds.

?20?

According to Eq. ?A12?, the first variation of F?is

2?11

2L2???1????s?2ds?

2??11

2L2???3??s?ds + 2??ss??s?ds?. ?21?

??1?F?=??

2

24L???s?2ds − lnLe

h????1?ds

+11

96??

= −??

2

24L???x?2dx − lnLe

h????s?ds

+11

96??

For F?, its first variation is

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??1?F?=??

2

2?1

L? sin2??s?ds −11

h???1??sin2??s?ds? +13

2??1

2L2?

192

48L???s?2ds −13

48L? cos 2???s?2ds????1?ds −12

2L2???1??sin2???s?2ds?

48L???x?2dx −13

2L2? sin 2???s??s?ds − ??

96??

2L2???1????s?2ds?

+??

2

2

lnL

96??

= −??

2

L? sin2??x?dx −11

48L? cos 2???x?2dx +1 + 3 cos 2??s?

2

lnL

h????s?ds

+ ??

39 cos 2??s? − 11

?3??s?ds −13

24??

2L2?

39 cos ??s? + 11

96

?ss??s?ds.

If the curve r?s? describes a boundary of an equilibrium do-

main, it must satisfy ??1?F=0 for any infinitesimal function

??s?. Combining Eqs. ?19?–?22?, we get the shape equation

of an equilibrium domain,

?P − ?? + ??3+ ??ss+ ???s= 0,

?22?

where the definitions of the coefficients are

? = ? −??

2

2

lnLe

h

+??

2?1 + 3 cos 2??

4

2L? sin2??s?ds

lnL

h

+11

48??

2L???s?2ds +??

2L??11+ 13 cos 2????s?2ds,

2

−

1

96??

? =11

96??

2L2+ ??

2L239 cos 2? − 11

192

,

?23?

? =11

48??

2L2− ??

2L211+ 13 cos 2?

96

,

? = −13

24??

2L2sin 2?,

respectively. The readers are reminded that the coefficients

are also the functions of the domain shapes. Obviously, com-

pared to the specific shape ?Eq. ?9? in Ref. 25? ???=0?, the

current equation is more complex. In addition to the more

complicated coefficients of ??s? and its derivatives, ??sis

involved.

V. THE CIRCULAR SOLUTIONS

A. The perpendicular dipole case

We rewrite the general shape equation in the absence of

??as follows:

?P − ?0? + ?0?3+ 2?0?ss= 0,

?24?

where

?0=11

96??

2L2,

?25?

?0= ? −??

2

2

lneL

h

+11

48L??

2???s?2ds.

Because the above equation is derived by defining the nor-

mal direction to be outward, ?=−1/?cfor a circle with ra-

dius ?c. According to this definition, ?=1/?tis then the cur-

vature of the inner circle of a torus with radius ?t. Hence it is

understandable that outer and inner circles can coexist if

?t??c. In the case of the planar circles ?i, i=c,t, Eq. ?24? is

simplified as

±k2?e?i

h

+ r = ln2?e?i

h

,

?26?

where k=?Ph/?e??

positive and negative signs before k correspond to the circu-

lar and torus cases, respectively. According to the values and

signs of the two constants k and r, which are also the linear

functions of the pressure ?P and the line tension ?, the so-

lutions of Eq. ?26? are divided into four cases; see Fig. 2.

2

and r=2?/??

2+11?2/12, and the

?1?

Only one circle exists for 0?r?k or 0?k?−r

?regions I and II in Fig. 2?.

Two circles of different sizes can coexist for 0?−r

?k and r?−?1+ln k? or 0?k?r and r?−?1+ln k?

?regions III, IV, and VI in the figure?.

Two circles and two tori of different sizes can coexist

for 0?r?k and r?−?1+ln k? ?region V in the figure?.

?2?

?3?

FIG. 2. Distribution of the circular and torus solutions of the equilibrium-

shape equation Eq. ?26?. The lines and the curve are k=r, k=−r, and

r=−?1+ln k?, respectively.

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