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Hydrodynamic effects on concentration fluctuations in multicomponent
membranes
Sanoop Ramachandran,a Shigeyuki Komura,*a Kazuhiko Sekib and Masayuki Imaic
Received 6th August 2010, Accepted 18th November 2010
DOI: 10.1039/c0sm00783h
We investigate the hydrodynamic effects on the dynamics of critical concentration fluctuations in
multicomponent fluid membranes. Two geometrical cases are considered; (i) confined membrane caseand (ii) supported membrane case. We numerically calculate the wavenumber dependence of theeffective diffusion coefficient by changing the temperature and/or the thickness of the bulk fluid. For
some limiting cases, the result is compared with the previously obtained analytical expression. An
analogy of the multicomponent membrane to 2D microemulsions is explored for the confined
membrane geometry.

1 Introduction
Biomembranes can be regarded as multipurpose envelopes
fundamental to the very existence of life. Composed of a hugevariety of amphiphilic molecules, this approximately 5 nm thickquasi-two-dimensional fluid sheet separates the inner and outerenvironments of the cells and organelles.1 Apart from delineating
the cell and organelle boundaries, membranes also play a significant role in a variety of physiological functions such as transmembrane transport of materials and cell signaling.1
Biomembranes have attracted renewed interest in the context
of the lipid ‘‘raft’’ hypothesis proposed a little over a decade ago.2
These 10–100 nm sized lipid domains with higher concentrationsof cholesterol were proposed to play a significant role in regulating certain cellular functions.2–4 Recent experiments using
stimulated emission depletion (STED) microscopy on plasma
membranes in vivo narrowed the size-range of the rafts to 10–20
nm.5 Despite extensive studies in this area, the details of theunderlying physical mechanisms leading to formation of rafts,
their stability, and the regulation of the finite domain size remain
elusive and controversial.6,7
Although the biological significance of rafts is still under
debate, there is no doubt about the rich physics and chemistrythat have been uncovered by the membrane studies to examine
the raft hypothesis. Numerous experiments on intact cells andartificial membranes containing saturated lipids, unsaturated
lipids and cholesterol have demonstrated the segregation of lipids
into liquid-ordered (Lo) and liquid-disordered (Ld) phases belowaDepartment of Chemistry, Graduate School of Science and Engineering,Tokyo Metropolitan University, Tokyo, 192-0397, Japan. E-mail:komura@tmu.ac.jp; Fax: +81 42 677 2525; Tel: +81 42 677 2537b
National Institute of Advanced Industrial Science and Technology,
Ibaraki, 305-8565, Japanc
Department of Physics, Faculty of Science, Ochanomizu University,
Tokyo, 112-8610, Japan

1524 | Soft Matter, 2011, 7, 1524–1531

the miscibility transition temperature.8,9 The Lo-phase is usuallyrich in saturated lipids and cholesterol. Below the transition
temperature, the domains undergo coarsening with the largest
domain limited by the system size.9 The primary driving force forthe domain coarsening is due to the positive line tension at thedomain boundaries. Studies on the diffusion of domains10 andthe dynamics of domain coarsening11–13 have been reported.
Recently, studies on multicomponent membranes above thetransition temperature have also gained much attention. As thecritical point is approached from above, one observes composition
fluctuations spanning a wide range of length and time scales.14
Veatch et al. made a notable attempt to investigate critical fluctuations in lipid mixtures.13 Deuterium NMR experiments on model
ternary membranes composed of dioleoylphosphatidylcholine(DOPC), dipalmitoylphosphatidylcholine (DPPC) and cholesterol
were used to construct the ternary phase diagram. With the determination of the line of miscibility critical points, they observedthat the NMR resonances were broadened in the vicinity of thecritical points. Such a spectral broadening was attributed to thecompositional fluctuations in the membrane having spatial dimensions less than 50 nm.
A more quantitative analysis of the critical fluctuations using
fluorescence microscopy was addressed by Honerkamp-Smithet al. for ternary mixtures of DPPC, diphytanoylphosphatidylcholine (diPhyPC) and cholesterol.15 When the criticaltemperature Tc is approached from above, the correlation length
diverges according to x z |T  Tc|n where n is the criticalexponent and T the temperature. (In order to prevent confusion
with the notations used later, the critical exponents are writtenwith a bar.) When Tc is approached from below, on the other
hand, the order parameter given by the difference in lipid
compositions vanishes as dj z (Tc  T)b. From the measure
ments of the critical exponents n and b, the authors concluded
that the critical behavior in ternary membranes is in theuniversality class of the 2D Ising model.16 Later, it was alsoThis journal is ª The Royal Society of Chemistry 2011

shown that giant plasma membrane vesicles extracted from thatof living rat basophil leukemia cells exhibit a critical behavior.17
These experimental observations suggest that lateral heterogeneity present in real cell membranes at physiological conditions
correspond to critical fluctuations.15,17 In other words, concentration fluctuations above the transition temperature (rather
than below) can also be responsible for raft structures in cell
membranes.
There have also been several theoretical works on concentration fluctuations in multicomponent membranes. Using renormalization group techniques, Tserkovnyak and Nelson
calculated protein diffusion in a multicomponent membraneclose to a rigid substrate.18 They pointed out that, in the vicinityof the critical point, the effective protein diffusion coefficientacquires a power-law behavior. Seki et al. reported equivalent
results with the use of a two-dimensional (2D) hydrodynamic
model involving a phenomenological momentum decay mechanism.19 Later Haataja showed that the effective diffusion coefficient exhibits a crossover from a logarithmic behavior to an
algebraic dependence for larger length scales.20 In his theory, anapproximate empirical relation for the diffusion coefficient ofa moving object was employed.21 A rigorous hydrodynamic
calculation performed by Inaura and Fujitani arrived at similar
results.22
The present article uses the idea of critical phenomena tocalculate the effective diffusion coefficient in multicomponent
lipid membranes. Based on the time-dependent Ginzburg–
Landau approach with full hydrodynamics, we calculate inparticular the decay rate of the concentration fluctuations
occurring in membranes. We deal with the case where themembrane is surrounded by a bulk solvent and two walls as
depicted in Fig. 1. Such a situation is worth considering because
biological membranes interact strongly with other cells,
substrates or even the underlying cytoskeleton which affects the
structural and transport properties of the membrane.23 We notethat effects of the substrates on the diffusion coefficient ofproteins have been investigated before.24,25 Two surrounding

geometries of the membrane are discussed; (i) confined
membrane and (ii) supported membrane. We also study thesituation when the multicomponent membranes form 2D
microemulsions.26 This interesting viewpoint is motivated bya recent work which predicts the reduction of the line tension inmembranes containing saturated, unsaturated and hybrid lipids(one tail saturated and the other unsaturated).27,28 Based on
chain entropy arguments, they proposed that hybrid lipids in 2D
play an equivalent role to surfactant molecules in three-dimensions (3D). Hence we shall explore the concentration fluctuationsin 2D microemulsions.
This paper is organized as follows. In Section 2, we first obtainthe membrane mobility tensors that are used in the subsequentcalculations. In Section 3, the decay rates of concentration fluctuations are analyzed for confined and supported membranecases. Section 4 deals with the microemulsion picture of themulticomponent membranes. Finally we close with somediscussions in Section 5.

2

Membrane hydrodynamics

In this section, we start with the membrane equations of motion
as well as the requisite membrane mobility tensors for the twogeometries. The membrane is assumed to be an infinite planar
sheet of liquid and its out-of-plane fluctuations are totallyneglected, which is justified for typical bending rigidities ofbilayers. The liquid membrane, fixed in the xy-plane at z ¼ 0, isembedded in a bulk fluid such as water or solvent that is further
bounded by hard walls as shown in Fig. 1. The upper (z > 0) andthe lower (z < 0) fluid regions are denoted by ‘‘+’’ and ‘‘’’,respectively. Since the 3D viscosities of the upper and the lower
solvent can be different, we denote them as hs . Consider thesituation in which impenetrable walls are located at z ¼ h,
where h+ and h can also be different in general.
Let v(r) be the 2D velocity of the membrane fluid. The 2D
vector r ¼ (x,y) represents a point in the plane of the membrane
which is assumed to be incompressible
V$v ¼ 0.

(1)

Here V is a 2D differential operator. We work in the lowReynolds number regime of the membrane hydrodynamics sothat the inertial effects can be neglected. This allows us to use the2D Stokes equation given by
hV2v  Vp + fs + F ¼ 0,

(2)

where h is the 2D membrane viscosity, p the 2D in-plane pressure, fs the force exerted on the membrane by the surrounding
fluid (‘‘s’’ stands for the solvent), and F is any other force acting
on the membrane.
Once we know fs, the membrane velocity can be obtained fromeqn (2) asv[q] ¼ G[q]$F[q],Fig. 1 Schematic picture showing a planar liquid membrane having 2D
viscosity h located at z ¼ 0. It is sandwiched by a solvent of 3D viscosityhs . Two impenetrable walls are located at z ¼ h bounding the solvent.

This journal is ª The Royal Society of Chemistry 2011(3)

where v[q], G[q] and F[q] (q ¼ (qx,qy)) are the Fourier componentsof v(r), G(r) and F(r) defined bySoft Matter, 2011, 7, 1524–1531 | 1525

vðrÞ ¼GðrÞ ¼

ð

dq

ð2pÞ2
ð

dq

ð2pÞ2

v½qexpðiq,rÞ;(4)G½qexpðiq,rÞ;(5)F½qexpðiq,rÞ;

(6)

and
FðrÞ ¼

ð

dq

ð2pÞ2

respectively. Using the stick boundary conditions at z ¼ 0 andz ¼ h, we can solve the hydrodynamic equations to obtain fs.29
Its perpendicular component with respect to q is given by
fst[q] ¼ h+svtq coth(qh+)  hs vtq coth(qh),(7)

where q ¼ |q| and vt is the perpendicular component of v withrespect to q. On the other hand, one can show that the parallel
component of fs is zero. After some calculations, the componentsof the mobility tensor G[q] finally becomes
Gab ½q ¼

1





þ þ h coth qh
hq2 þ q hþ
coth
qh
s
s


qa qb
 dab  2 ;q(8)with a,b ¼ x,y. The full details of the calculation are given in theseparate article by Ramachandran et al.29
As in ref. 22, we first consider the case when the two walls arelocated at equal distances from the membrane, i.e., h+ ¼ h ¼ h.
Then the above mobility tensor simplifies to


1
qa qb
Gab ½q ¼
;
(9)
d

ab
h½q2 þ nq cothðqhÞq2
where n ^ 2hs/h with hs ¼ (h+s + hs )/2. This expression, taken in
the limit of large h, has been used previously in calculating thecorrelated diffusion coefficients of particles embedded ina membrane.30
For supported membranes, h+ is infinitely large while h isfinite when compared to the membrane thickness. In this case,eqn (8) reduces to29


1
qa qb
  dab  2 :(10)
Gab ½q ¼ q
h q2 þ nq 1 þ coth qh2This equation has been used for the investigation of the correlated dynamics of inclusions in a supported membrane.31

3 Dynamics of concentration fluctuations
In order to discuss the dynamics of concentration fluctuations
above the transition temperature, we closely follow theformalism used in ref. 32. Here we extend our previous work in
ref. 19 for membranes embedded in a solvent confined by twowalls (confined membrane) or supported on a substrate (supported membrane).
Consider a two-component fluid membrane composed of lipid
A and lipid B whose local area fractions are denoted by fA(r) andfB(r), respectively. Since the relation fA(r) + fB(r) ¼ 1 holds, we
introduce a new variable defined by j(r) ^ fA(r)  fB(r). The
1526 | Soft Matter, 2011, 7, 1524–1531

simplest form of the free energy functional F {j} describing thefluctuation around the homogeneous state is
ð h
i
a
c
F fjg ¼ dr j2 þ ðVjÞ2 mj ;(11)
2
2
where a > 0 is proportional to the temperature difference fromthe critical temperature, c > 0 is related to the line tension and m isthe chemical potential.
The time evolution of concentration in the presence ofhydrodynamic flow is given by the time-dependent Ginzburg–
Landau equation for a conserved order parameter32
vjdF
þ V,ðvjÞ ¼ LV2 ;
vt
dj

(12)

where L is the kinetic coefficient. In the membrane hydrodynamic equation (2), on the other hand, we need to incorporate
the thermodynamic force due to the concentration fluctuations.
Hence we havedF
F ¼ jV :dj(13)

We implicitly assume that the relaxation of the velocity v ismuch faster than that of concentration j.19 The membrane
velocity can be formally solved using the appropriate 2D
mobility tensor Gab(r,r ) derived in the previous section,ð 0  0dF
0va ðr; tÞ ¼ dr Gab r; r Vb j:
(14)djðr0 Þ0

Since our interest is in the concentration fluctuations around the where the bar
homogeneous state, we define dj(r,t) ¼ j(r,t)  j,indicates the spatial average. The free energy functional
expanded in powers of dj becomes,
ð h
i
a
c
F fdjg ¼ dr ðdjÞ2 þ ðVdjÞ2 :(15)
2
2
Substituting eqn (14) into eqn (12), we getvdjðr; tÞ
dF
¼ LV2
vt
dðdjÞ
ð
 0  0  0 0 dr ðVa djðrÞÞGab r; r Vb dj r

dF:dðdjðr0 ÞÞ(16)

We now consider the dynamics of the time-correlation function defined byS(r,t) ¼ hdj(r1,t)dj(r2,0)i,(17)

where r ¼ r2  r1. Within the factorization approximation,32 thespatial Fourier transform of S(r,t) defined byðdq
S½q; texpðiq,rÞ;(18)
Sðr; tÞ ¼
ð2pÞ2
satisfies the following equation

vS½q; t
¼  Gð1Þ ½q þ Gð2Þ ½q S½q; t:
vt

(19)

In the above, the first term G(1)[q] denotes the van Hove part of
the relaxation rate given by
G(1)[q] ¼ LkBTq2c1[q].

(20)This journal is ª The Royal Society of Chemistry 2011

Here the static correlation function is defined bykB T;c½q ¼ hdj½qdj½qi ¼  2
c q þ x2(21)

where x ^ (c/a)1/2 is the correlation length, kB the Boltzmann
constant, and T the temperature.
As for the second term in eqn (19), G(2)[q] denotes the hydrodynamic part of the decay rate
ð
1
dp
ð2Þ
G ½q ¼
qa Gab ½pqb c½q þ p;(22)
c½q ð2pÞ2
where either eqn (9) or eqn (10) will be used for the mobility
tensor Gab.

3.1 Confined membraneFirst we consider the situation in which the membrane is confinedby two walls which are located at equidistant h from themembrane. The appropriate form of the mobility tensor is given
by eqn (9). Substituting it into eqn (22), the hydrodynamic partof the decay rate is expressed asðkB T dpGð2Þ ½q ¼
hc½q ð2pÞ2
(23)
c½p
q2 p2  ðq,pÞ2
:jp  qj2 þ njp  qj cothðjp  qjhÞjp  qj2
We introduce an effective diffusion coefficient D[q] (due only tothe hydrodynamic effect) defined by
G(2)[q] ¼ q2D[q].

(24)

In order to deal with dimensionless quantities, we rescale all thelengths by the hydrodynamic screening length n1 ¼ h/(2hs) suchthat P ^ p/n, Q ^ q/n, X ^ xn and H ^ hn. Then D[q] can be
rewritten as

kB T 
1 þ Q2 X 2
4p2 hð N ð 2pP3 sin2 q
pﬃﬃﬃﬃ ;dP dq 

1 þ P2 X 2 G2 þ G 3=2 coth GH
00

Fig. 2 Scaled effective diffusion coefficient D as a function of Q for
H ¼ 0.01,1,100 when X ¼ 1 for the confined membrane case. The
numerically calculated points are joined by lines for clarity. The redlines are from the analytical expression given in eqn (26) obtained in thelimit of small H.

from numerical fitting. We note that the above logarithmic
behavior for Q [ 1 is in contrast to that of 3D critical fluids asgiven by the Kawasaki function which increases linearly with q.33
In Fig. 4, we explore the effect of the correlation length X on
D for different values of H when Q ¼ 103. The quantity X
measures an effective size of the correlated region formed transiently in the membrane due to thermal fluctuations. When X 
1, the diffusion coefficient D decreases only logarithmically,
which is typical for a pure 2D system.34–36 When X [ 1, on the

D½Q; X ; H ¼

(25)

with G ¼ P2 + Q2  2PQ cos q. Since this integral cannot be
performed analytically, we evaluate it numerically. We explore
the dependencies of D on the variable Q, and the parameters X
and H.
In Fig. 2, we plot the diffusion coefficient D (scaled by kBT/4ph) as a function of dimensionless wavenumber Q for different
solvent thickness H while the correlation length is fixed to X ¼ 1(i.e., fixed temperature). In the limit of Q  1, D is almosta constant. The calculated D starts to increase at around Q z 1
and a logarithmic behavior (extracted via numerical fitting) isseen for Q [ 1. This trend has been previously reported by
Seki et al.19 or Inaura and Fujitani.22 When H is small such as
H ¼ 0.01, the value of D decreases by one order of magnitude
compared to H ¼ 100. Figure 3 shows the diffusion coefficient
D as a function of wave number Q for different X (i.e., different
temperatures) while the solvent height is fixed to H ¼ 1. Again,
D is nearly constant for Q  1, and follows an S-shaped curvewith increasing Q. Finally, a logarithmic dependence is observed
This journal is ª The Royal Society of Chemistry 2011

Fig. 3 Scaled effective diffusion coefficient D as a function of Q for
X ¼ 0.01,1,100 when H ¼ 1 for the confined membrane case.

Soft Matter, 2011, 7, 1524–1531 | 1527

Fig. 4 Scaled effective diffusion coefficient D as a function of X for H ¼
0.01,1,100 when Q ¼ 103 for the confined membrane case. The red lines
are from the analytical expression given in eqn (26) obtained in the limitof small H.

other hand, the behavior of D depends on the value of H. The
proximity to the walls results in a loss of momentum from themembrane.19,29,37 This leads to a rapid suppression of thevelocity field within the membrane such that the concentration
fluctuations decay slowly. Consequently, the values of D are
lower for smaller H. The flattening of the curves for large X isdue to the dominance of the X2 terms in the numerator anddenominator of eqn (25). In Fig. 5, we plot D as a function of H
for different values of X when Q ¼ 103. In general, there isa monotonic increase in D with larger H followed bya saturation to a constant value. We see that the effect of H is
most prominent for large X (close to the critical point), while
there is only a weak dependence for X  1 (far from the criticalpoint). The former reflects the fact that the membrane fluid isaffected by the outer environment when the correlation length x
is larger than the hydrodynamic screening length n1, i.e., X [1.29 For correlation lengths smaller than n1, the outer environment surrounding the membrane is less important. In thissituation, the system behaves essentially as a pure 2D one.
When the correlation length becomes larger than n1, as is thecase near the critical point, the outer environment significantly
affects membrane dynamics.
For small h, the nqcoth(qh) term in the mobility tensor eqn (9)can be replaced by a constant n/h. In this case, Seki et al. obtained
an analytical expression for the effective diffusion coefficient.19
Their result in terms of the dimensionless quantities Q, X and H
can be reproduced as
kB T 1 þ Q 2 X 2
X
p
ﬃﬃﬃﬃ
ﬃ
D½Q; X ; H ¼
ln
4ph 2Q2 X 2
H
!

H
X
2 2

þ 2 1 þ Q X ln pﬃﬃﬃﬃﬃ
(26)X
H 1 þ Q2 X 2!#
Q4þ þ Q2 þ Q2þ U
HU
;
ln
þ
2
2X
U  Q2  1
where
U¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
Q2 X 2 þ X 2 =H  1 þ4Q2 X 2 ;(27)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Q2 X 2  X 2 =H :

(28)

and
Q ¼

Equation (26) is plotted using red curves in Fig. 2 for H ¼ 0.01
and 1 with X ¼ 1. For H ¼ 0.01, the analytical and numerical
data coincide giving credence to accuracy of the numericalsolutions. It is seen that even for H ¼ 1 the agreement is still
acceptable. For H ¼ 100, however, a significant deviation isobserved (not shown), which is expected as this limit is beyond
the valid range of eqn (26).
The red curves in Fig. 4 also represent the analytical result ofeqn (26). It is seen that the analytical and the numerical data
points almost coincide for H ¼ 0.01 and H ¼ 1. For H ¼ 100,
there is significant deviation from the numerical data (notshown). However, the agreement between the numerical result
and the analytical expression is beyond the expected range ofH  1 and reaches up to H z 1, as pointed out by Stone andAjdari.25 Hence eqn (26) can be useful in analyzing the experimental data in many situations.

3.2 Supported membrane

Fig. 5 Scaled effective diffusion coefficient D as a function of H for X ¼
0.01,1,100 when Q ¼ 103 for the confined membrane case.

1528 | Soft Matter, 2011, 7, 1524–1531

Apart from studying membrane dynamics on vesicles, experiments are also conducted on supported membranes which havea benefit of avoiding curvature effects.38 We proceed now to
calculate concentration fluctuations for the supported membranecase. As given by eqn (10), the mobility tensor of a supported
membrane is slightly modified from that of a confined
membrane. By substituting eqn (10) into eqn (22), we obtain theexpression for the effective diffusion coefficient asThis journal is ª The Royal Society of Chemistry 2011

ð
ð 2p

 kB T  N
D Q; X ; H  ¼ 2 1 þ Q2 X 2
dP dq
4p h
0
0
P3 sin2 qpﬃﬃﬃﬃ  ;
2
1 þ P2 X 2 G 2 þ G 3=2 1 þ coth G H (29)where H ^ hn and G ¼ P2 + Q2  2PQcosq as before.
Performing numerical integrations, we plot in Fig. 6 theeffective diffusion coefficient D as a function of Q for different
values of H when X ¼ 1 (closed symbols). In general, thebehavior of D is similar to that of the confined membrane case.
When H  1, the calculated D is slightly larger than theconfined membrane compared with the same value of H. Thisis because the supported membrane is subjected to only one
wall, while the confined membrane is sandwiched by two walls on
both sides. The confined and the supported membranes show analmost identical behavior when both H and H are large enough,as expected. The other dependencies of D are similar to theconfined membrane case except for a slight increase at smallH values.

4 Membrane as a 2D microemulsion
The role of surfactant molecules in 3D microemulsions is to
reduce the surface tension at the interface between oil and water.
In an analogy to 3D microemulsions, hybrid lipids (one chain
unsaturated and the other saturated) act as lineactant molecules
which stabilize finite sized domains in 2D. In other words, hybrid
lipids play a similar role to surfactant molecules at the interface
between the Lo and Ld domains. It should be also noticed thathybrid lipids form a major percentage of all naturally existing
lipids.39,40 Based on a simple model of hybrid lipids, Brewster
et al. showed that finite sized domains can be formed in equilibrium.27,28 A subsequent model predicted stabilized domains evenin a system of saturated/hybrid/cholesterol lipid membranes.41

Being motivated by this idea, we calculate the decay rate ofconcentration fluctuations when the free energy of the multicomponent membrane has the form of a 2D microemulsion. Here we
consider only the confined membrane geometry, and use eqn (9)for the mobility tensor.
The free energy functional for a microemulsion includes ahigher order derivative term and is expressed in terms of dj as26
ð h
2 i
a
c
g
F ME fdjg ¼ dr ðdjÞ2 þ ðVdjÞ2 þ V2 dj ;
2
2
2

(30)

with a, g > 0 and c < 0. The negative value of c creates 2Dinterfaces, while the term with positive g is a stabilizing term.
This form of the free energy has been used previously to study
coupled modulated bilayers.42 As in the previous section, thedecay rate of the correlation function can be split into two parts.
First, the van Hove part becomes now
2 1
G(1)
ME[q] ¼ LkBTq cME [q],(31)

where L is the kinetic coefficient assumed to be the same as
before, and the static correlation function cME[q] is43
kB T:
gq4 þ cq2 þ a

(32)

c
;
2g

(33)

 2
a
c

;
g
2g

(34)

cME ½q ¼
By defining

q20 ¼ s4 ¼

the static correlation function can be also written askB T
i:cME ½q ¼ h
2
2g ðq  q20 Þ þs4(35)

On plotting cME as a function of q, a peak appears at q ¼ q0
followed by a q4-decay. The width of the peak is given by s, anda lamellar phase appears when s ¼ 0. Notice that c ¼ 0 is called
the Lifshitz point at which the peak occurs for q ¼ 0.44 Using theform of eqn (35), we can rewrite eqn (31) as
2
2
2 2
4
G(1)
ME [q] ¼ Lgq [(q  q0) + s ].

(36)

As in the previous section, we next write the hydrodynamic
part of the decay rate in terms of the effective diffusion coefficient
DME[q] as
2G(2)ME [q] ¼ q DME [q].

(37)

Using eqn (9) for the mobility tensor, we obtain DME as
DME ½Q; Q0 ; S; H ¼
Fig. 6 Scaled effective diffusion coefficient D as a function of Q forH ¼ 0.01,1,100 when X ¼ 1 for the supported membrane case plottedwith filled symbols. For comparison, the data in Fig. 2 are also plottedwith open symbols.

This journal is ª The Royal Society of Chemistry 2011

ð 2p

ið N

kB T h 2
4
2 2
Q

Q
þS
dP
0
4p2 h
0

P3 sin2 qi
dq hpﬃﬃﬃﬃ ;
20
P2  Q20 þS4 G 2 þ G3=2 coth GH(38)

where P ^ p/n, Q ^ q/n, Q0 ^ q0/n, S ^ s/n, H ^ hn, and G ¼ P2
+ Q2  2PQcosq.
Soft Matter, 2011, 7, 1524–1531 | 1529

Fig. 7 Scaled effective diffusion coefficient DME as a function of Q for
H ¼ 0.01,1,100 when Q0 ¼ S ¼ 1 for the confined membrane case.

In Fig. 7, we plot DME as a function of Q for different values of
H when Q0 ¼ S ¼ 1 are fixed. When Q  1, D shows a constantvalue. We also observe the 2D characteristic of logarithmic
behavior of D for Q [ 1. For Q  1, the effect of the outerenvironment is felt with the suppression of the diffusion coefficient with smaller H. The curves almost overlap when Q [ 1indicating the negligible effect of the outer environment at large
wave numbers. An interesting feature of DME is the dip occurring
at Q z Q0 which does not exist for binary critical fluids. This canbe attributed to the peak at q ¼ q0 in cME[q].32,43,45 For 3D
microemulsions, however, it is known that the effective diffusion
coefficient varies linearly with q for large wave numbers.32
Although the analogy between a 3D and 2D microemulsion
has been invoked, it should be pointed out that a 3D microemulsion formed from an oil/water/surfactant mixture arises
predominantly due to the differences in relative affinity between
the components. For the 2D case, it is the physical interactions
from the hydrocarbon chain packing requirements at the Lo/Ldinterface that gives rise to the lineactant properties of the hybrid
lipid.

5 Discussion
In summary, we have calculated the decay rate of concentration
fluctuations in a multicomponent fluid membrane for two
geometries. First we considered the membrane surrounded bysolvent of finite depth which is further bounded by two walls. In
the second geometry, we allow the solvent depth on one side of
the membrane to be infinitely large. This is equivalent to a supported membrane in experimental situations. The resulting
integrals for the effective diffusion coefficient of the concentration fluctuations are calculated numerically to determine thevarious dependencies. We have also explored a possibility ofconsidering the multicomponent membrane as a 2D microemulsion.
1530 | Soft Matter, 2011, 7, 1524–1531

The present work follows Seki et al. who were able to use
analytical means to obtain the effective diffusion coefficient.19
Although their result should be valid only in the limit of verysmall h, the agreement between the numerical result and theanalytical expression is fairly well as long as h  n1.
The opposite limit of very large h was studied by Inaura andFujitani through numerical methods.22 Using the generalmobility tensor given by eqn (8), we are able to probe all theintermediate situations of finite h. We have verified that our
results properly interpolates between these previous works in thelimits of H / 0 and H / N.
In our case, the measure of the size of the transient structures isgiven by the correlation length x. Close to the critical point, x
diverges and the hydrodynamic effects of the outer fluid playa significant role in altering the diffusion coefficient. It has beenpreviously shown through explicit calculations that the diffusion
coefficient has a logarithmic behavior D z ln(1/R) when the sizeof the diffusing object R is less than the hydrodynamic screening
length, i.e., R  n1.34,35 When R [ n1, on the other hand, thefluid flow in the bulk leads to the diffusion coefficient to show
1/R-behavior (this condition occurs when there are no walls).36 In
the presenceof walls or a substrate, the screening length ispﬃﬃﬃﬃﬃﬃﬃ
altered to h=n, where h p
is ﬃﬃﬃﬃﬃﬃﬃ
the distance of the walls from themembrane.25 When R[ h=n, the diffusion coefficient now
shows an algebraic decay D z 1/R2.19 This change is attributed to
the loss of momentum from the membrane to the walls whereas
momentum is conserved otherwise.37
In the present study, we have kept the model as simple aspossible. The bending stiffness of typical membranes is of theorder of 10kBT which is sufficiently large enough to neglect theout-of-plane displacements of the membrane itself. The dynamics
of the out-of-plane fluctuations have been previously studied by
Levine and MacKintosh.46 In our treatment, we have also
neglected the effects of membrane curvature which can besignificant when the radius of curvature becomes close to thehydrodynamic screening length. Recent calculations by Henleet al. considered the diffusion of a point object on a sphericallyclosed membrane.47,48 The extension of this work to finite sized
objects is a particularly difficult proposition.
From the experiments on model multicomponent vesicles, thestatic critical exponents for the order parameter and correlation
 ¼ 1/8 and n ¼ 1,
length were found to have values close to b
15
respectively.
Furthermore, experiments on giant plasma
 ¼ 7/4 which
membrane vesicles measured the critical exponent g
characterizes the critical behavior of the osmotic compressibility.17 These static exponents coincide with the exact results ofthe 2D Ising model.49,50 The description presented in this paper
uses the mean-field approach and therefore the corresponding
 ¼ 1/2, n ¼ 1/2, and g
 ¼ 1, respectively.
static exponents are b
However, attributing biological relevance to these critical fluctuations should be exercised with caution. This is because theplasma membranes do not show phase separation phenomena
without chemical treatment.51 Extraction of membranes fromreal cells also ruptures the association with the underlying
cytoskeleton and other active cellular processes like vesiculartrafficking.
As mentioned earlier, experiments on real plasma membranes
have suggested that the cell maintains the membranes at a criticalcomposition.17 This leads to nanometre-sized composition
This journal is ª The Royal Society of Chemistry 2011

fluctuations at the physiological temperatures, although thesestructures are much smaller than what can be resolved throughoptical microscopy. We therefore speculate that there is somebiological relevance in studying concentration fluctuations.
However, this article has mainly concentrated on the dynamics
towards the equilibrium state of lipid bilayer membranes. In realcells, there are many active non-equilibrium cellular processes
that are involved in the proper functioning of the cells. It has
indeed been proposed that nano-domain formation may berelated to the underlying cytoskeleton.52 There have been several
other models which make use of the active non-equilibrium
phenomena to explain the existence of finite sized domains in
multicomponent membranes. A review of the non-equilibrium
models can be found in ref. 53.
Another experimental system which can be used to verify ourmodel is the Langmuir monolayer setup. The upper ‘‘+’’ region ina Langmuir monolayer system is occupied by air and hence h+s
should be set to zero in eqn (8). In this case, the mobility tensor isgiven by eqn (9), but the definition should be replaced with n ¼hs /h. The diffusion coefficient as a function of wave vector can beobtained via light scattering techniques.

Acknowledgements
We thank H. Diamant, Y. Fujitani, T. Kato and N. Oppenheimer for useful discussions. This work was supported by
KAKENHI (Grant-in-Aid for Scientific Research) on Priority
Area ‘‘Soft Matter Physics’’ and Grant No. 21540420 from the
Ministry of Education, Culture, Sports, Science and Technologyof Japan.

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