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JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 19 (2007) 072101 (8pp)

doi:10.1088/0953-8984/19/7/072101

FAST TRACK COMMUNICATION

Concentration fluctuations in binary fluid membranes
K Seki1 , S Komura2,4 and M Imai3
1

National Institute of Advanced Industrial Science and Technology (AIST), Ibaraki 305-8565,
Japan
2 Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397,
Japan
3 Department of Physics, Faculty of Science, Ochanomizu University, Tokyo 112-0012, Japan
E-mail: komura@comp.metro-u.ac.jp

Received 30 October 2006, in final form 12 January 2007
Published 2 February 2007
Online at stacks.iop.org/JPhysCM/19/072101
Abstract
We investigate the dynamics of critical fluctuations in binary fluid membranes
using a two-dimensional hydrodynamic model with momentum decay to thesurrounding water. In particular, the decay rate of concentration fluctuations is
obtained analytically. In the limit of small wavenumber q with respect to thecorrelation length, the decay rate is proportional to q 2 , as usual. In the large-q
limit, however, the effective diffusion coefficient increases only logarithmicallywith q .

Biological membranes typically contain various components such as lipid mixtures, sterols, andproteins that are indispensable to cell functions [1]. Rather than being uniformly distributedin the membrane, there are growing evidences that some intra-membrane components areincorporated in domains arising from lateral lipid segregation in membranes. This phenomenon
has attracted great interest in the context of ‘rafts’ [2], i.e., liquid domains rich in cholesterol,saturated lipids (typically sphingomyelin lipids), and in some cases particular proteins [3].
Cholesterol-rich domains have been directly observed in model membranes composed of lipid
mixtures and cholesterol, using advanced fluorescence microscopy [4–6].
Recent researches have focused on the dynamical aspect of lateral phase separation in
mixed membranes. For example, Veatch and Keller observed domain formation in giant
vesicles consisting of dipalmitoyl-phosphatidylcholine (DPPC), dioleoyl-phosphatidylcholine(DOPC), and cholesterol [7]. Depending on the composition, phase separation occurs througheither domain ripening or spinodal decomposition. Saeki et al reported that the average domain
size develops with time as ∼t 0.15 , which is a very slow process [8]. In the experiment by
Yanagisawa et al, two different types of domain coarsening were observed, i.e., ‘normalcoarsening’ and the ‘trapped coarsening’ [9]. In the former case, the growth law is found to be∼t 2/3 , whereas in the latter case, domain coarsening is suppressed at a certain size. Moreover,
4 Author to whom any correspondence should be addressed.

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J. Phys.: Condens. Matter 19 (2007) 072101

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domain growth and budding in phase-separating fluid vesicles were investigated by meansof several simulation methods such as the time-dependent Ginzburg–Landau approach [10],Monte Carlo simulation [11], or dissipative particle dynamics [12].
In this letter, we investigate the dynamics of two-component fluid membranes from thetheoretical point of view. Taking into account the effect of hydrodynamic interaction within themembrane, we calculate the decay rate of the time-correlation function of the concentration
fluctuations. Since fluid membranes are sandwiched by surrounding water, we use a twodimensional hydrodynamic equation with momentum decay to this water. Such an equation has
previously been used to calculate the diffusion constant of proteins [13–16] or polymers [17]moving in membranes. Our formulation closely follows that in [18], which deals with the
dynamics of microemulsions. An effective diffusion coefficient is calculated in the entire rangeof wavenumber. In the large-wavenumber limit, the effective diffusion coefficient depends
logarithmically on the wavenumber, which is in contrast to the case for three-dimensional
critical fluids. Although there have been no experimental reports on concentration fluctuations
in membranes, they could be measured in the near future since the critical point has been
determined in various phase diagrams of multi-component membranes [4, 6, 7].
We consider a two-component fluid membrane composed of lipid A and lipid B (orcholesterol) whose local area fractions are denoted by φA (r) and φB (r), respectively (r is thetwo-dimensional vector). Since the relation φA (r) + φB (r) = 1 holds, we introduce a variable
defined by ψ(r) ≡ φA (r) − φB (r). The simplest form of the free-energy functional F{ψ}describing the fluctuation around the homogeneous state is



a 2 c
2
F{ψ} = dr ψ + (∇ψ) − μψ ,(1)22
where a > 0 is proportional to the temperature difference with respect to the criticaltemperature, c > 0 is related to the line tension, and μ is the chemical potential.
Next we discuss the hydrodynamics of the fluid membrane. Introducing the twodimensional local velocities vA and vB for each component, we define the average local velocity
by v = φA vA + φB vB , which satisfies the incompressibility condition:∇ · v = 0.

(2)

Although a lipid membrane can be regarded as a two-dimensional viscous fluid, it is not an
isolated system since lipids are coupled to the adjacent water. Accordingly, the momentum
within the membrane can leak to the outer fluid. Such an effect can be phenomenologicallytaken into account through a momentum relaxation term in the two-dimensional hydrodynamic
equation [13–17]. Moreover, a thermodynamic force due to the concentration fluctuations
should also be included [19]. Then the hydrodynamic equation for a mixed fluid membrane
can be written as


∂v
δF
+ (v · ∇)v = η∇ 2 v − ∇ p − λv − ψ∇
,
ρ
(3)
∂t
δψ
where all the vector operators are two dimensional. In the above, p is the pressure, and theconstants ρ and η are the density and dynamic viscosity of the lipid membrane, respectively5 .
The phenomenological parameter λ is inversely proportional to the momentum relaxation time
which reflects the coupling strength between the membrane and the surrounding water. Within
the present treatment, thermal fluctuations of the membrane shape are ignored. Details of thephysical meaning of λ will be discussed later.
5

2

The dimension of η is that of three-dimensional viscosity times length such as the membrane thickness.

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The time evolution of concentration in the presence of hydrodynamic flow is given by thetime-dependent Ginzburg–Landau equation for a conserved order parameter [18]:
∂ψ
δF
(4)
+ ∇ · (vψ) = L∇ 2,∂tδψwhere L is the kinetic coefficient. Equations (1)–(4) provide the complete set of differential
equations to be solved.
In the following, we assume that the relaxation of velocity v is much faster than that ofconcentration ψ , so the left-hand side of equation (3) can be neglected. In this case, the velocity
can be formally solved from equations (2) and (3) asδF,v α (r, t) = dr T αβ (r, r )(∇β ψ)(5)δψ(r )
where T αβ (r, r ) is the two-dimensional Oseen tensor which takes into account the extra
dissipation due to the surrounding fluid. Defining the Fourier transform bydq
αβ
T (r) =T αβ eiq·r ,(6)(2π)2 qwe obtain


1
qαqβ
αβ
δαβ − 2 ,Tq =(7)
η(q 2 + κ 2 )qwhere q = |q| and κ −1 ≡ (η/λ)1/2 is the hydrodynamic screening length. Equation (7) reduces
to the usual Oseen tensor when κ → 0. Since we are interested in the concentration fluctuations
around the homogeneous state, we define δψ(r, t) = ψ(r, t) − ψ̄ , where the bar indicates aspatial average. Then the free-energy functional (1) is expanded in powers of δψ as



a
c
2
2
F{δψ} = dr (δψ) + (∇δψ) .
(8)22Then the static correlation function is given bykB T
χq = δψq δψ−q  =(9),2c(q + ξ −2 )where kB is the Boltzmann constant, T the temperature, and ξ ≡ (c/a)1/2 the correlation
length. Substituting equation (5) into (4), we obtain∂δψ(r, t)
δF
2 δF
= L∇
− dr (∇α δψ(r))T αβ (r, r )(∇β δψ(r ))
,
(10)
∂t
δ(δψ)
δ(δψ(r ))for the dynamics of the concentration fluctuations.
Using the above equation, we now consider the dynamics of the time-correlation function
defined by

S(r, t) = δψ(r1 , t)δψ(r2 , 0),(11)

where r = r2 − r1 . With the use of the factorization approximation6 , the spatial Fouriertransform of S(r, t) satisfies the equation∂ Sq (t)= −(q(1) + q(2) )Sq (t).
(12)∂t
The first part of the relaxation rate q(1) corresponds to the so-called van Hove part, which issimply given byq(1) = LkB T q 2 χq−1 = Lcq 2 (q 2 + ξ −2 ).

(13)More precisely, the fourth-order correlations are approximated as (∇α δψ(r1 , t))(∇β δψ(r2 , t))δψ(r3 , t)δψ(r4 , 0) ≈(∇α δψ(r1 , t))(∇β δψ(r2 , t))δψ(r3 , t)δψ(r4 , 0). See [18] for details.

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The second part of the relaxation rate, q(2) , is the hydrodynamic part that is given by an integral

q(2) =

kB T
ηχq



dp
1
q 2 p2 − (q · p)2χ.
p(2π)2 (p − q)2 + κ 2(p − q)2

We first carry out the integral over the angle between p and q:
 2π
p 2 q 2 sin2 θdθ 2( p + q 2 − 2 pq cos θ )( p 2 + q 2 − 2 pq cos θ + κ 2 )
0



π
2
2
2
2
2
2
2
=
−1 + [( p + q) /κ + 1][( p − q) /κ + 1] − |q − p |/κ .
2

(14)

(15)

Introducing the dimensionless quantities Q ≡ qξ and  ≡ κξ , we further perform the integral
over the magnitude of p to obtain the resultq(2) = q 2 D Q (),(16)with an effective diffusion coefficient



 4

Q + + Q 2− + Q 2+ 
kB T Q 2 + 1 Q 2 + 1
− ln  +.
D Q () =ln
ln
8πη Q 22
Q2 + 122
 − Q 2− − 1(17)Here we have used the following abbreviations: = (Q 2 + 2 − 1)2 + 4 Q 2 ,

Q± =



Q 2 ± 2 .

(18)

For three-dimensional binary critical fluids, an analogous calculation yields the famous
Kawasaki function [20]. Its validity has been successfully confirmed by several dynamic lightscattering experiments [21]. Equation (17) is the two-dimensional analogue of the Kawasaki
function, provided that the momentum can leak to the surrounding fluid.
In figure 1, we plot D Q () (scaled by kB T /4πη) as a function of Q = qξ for various
values of . Since q(2) can be generally expanded in terms of Q 2 in the isotropic state,
D Q () is constant in the limit of Q
1. The effective diffusion coefficient starts to increase
when Q ≈ 1. All the curves are generally S-shaped, and an inflection point appears in theintermediate Q -region. In the limit of Q
1, the asymptotic form of equation (17) can beapproximately given by

 2

  2
kB T
Q + 2Q2
Q + 2
D Q () ≈
ln
ln
(19)
+
.
8πη
2
2
Q2
If we plot this equation in figure 1 for different  values (not shown), they all coincide with theexact result as long as Q > 1. Equation (19) can be further approximated as

D Q () ≈when QkB Tln(Q/),
4πη

(20)

kB T
(Q/)2 ,
4πη

(21)

, or
D Q () ≈when Q (except the logarithmic correction). Equation (20) shows that D Q () depends
only logarithmically on Q in the large- Q limit, i.e., Q
1 and Q
. This is an importantresult of this letter. Such a weak wavenumber dependence should be contrasted with that inthree-dimensional critical fluids for which the effective diffusion coefficient increases linearlywith q (or the relaxation rate is proportional to q 3 ) in the large-q limit [20].
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101

4πηDQ(Λ)/kBT

Λ=0.1
100

1

10-1

10
10-2
10-2

10-1

100

101

102

Q
Figure 1. Scaled effective diffusion coefficient D Q () as a function of the dimensionlesswavenumber Q = qξ for  = 0.1, 1, 10 from top to bottom.

Next we argue the dependence of D Q () on , which measures the coupling strength
between the membrane and the water. In figure 2, we plot D Q () against  = κξ for various
values of Q . When  is large, the momentum transfer from the membrane to the water becomes
faster. Under such conditions, the concentration fluctuations decay slowly because the velocity
field in the membrane is damped out quickly by the momentum transfer. Therefore, the effective
diffusion coefficient D Q () should be smaller for larger coupling strength, and D Q () is
a monotonically decreasing function of . However, its -dependence has two asymptotic
regimes, as obtained by equations (20) and (21): D Q () decreases only logarithmically with 
when Q, while it decreases algebraically (∼−2 ) when Q
. According to figure 2,
the logarithmically dependent regime increases for larger Q (compare Q = 10 and 100). Thisis because the length scale of our investigation (q −1 ) is much smaller than the screening lengthκ −1 , and D Q () is hardly affected by the coupling effect.
From equation (17) the limiting value of D Q () for Q → 0 becomesD0 () =kB T ln ,
4πη 2 − 1(22)which is called the cooperative diffusion constant. This equation is plotted in figure 2 for
Q = 0. We note that equation (22) is not singular at  → 1 and gives a finite value
D0 () = kB T /(8πη). Two further limits of equation (22) are

D0 () ≈
for 

kB T
ln(1/),
4πη

(23)

kB T
(1/)2 ,4πη

(24)

1, andD0 () ≈1. These logarithmic and algebraic dependences on  are analogous to those givenfor 
in equations (20) and (21), respectively. It should be noticed that  is proportional to thecorrelation length. The cooperative diffusion constant D0 () is smaller for larger correlation
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101

100
10

4πηDQ(Λ)/kBT

100

1
Q=0
10-1

10-2

10-3
10-2

10-1

100

Λ

101

102

Figure 2. Scaled effective diffusion coefficient D Q () as a function of  = κξ for Q = 0, 1,
10 and 100 from bottom to top. The line Q = 0 corresponds to the scaled cooperative diffusion
constant given by equation (22).

length ξ , as it should be. However, its dependence on the correlation length is different fromthat of the three-dimensional cooperative diffusion constant, which is inversely proportional tothe correlation length ξ [20]. Moreover, equation (24) shows that the membrane viscosity η inthe denominator of D0 () drops out since (1/)2 = η/λξ 2 . The fact that D0 () is sensitive
to the coupling to the environment (represented by λ) has been pointed out recently for lipid
mixtures close to a critical point [22].
It is worthwhile comparing the obtained cooperative diffusion constant with the tracer
diffusion constant of a disc-like molecule with radius R moving in a two-dimensional fluidwith momentum decay. Based on the same hydrodynamic equation as in equation (3) (without
the thermodynamic force), the tracer diffusion constant was previously obtained as [13, 16]
kB T z 2
z K 1 (z) −1
D=
+
,
(25)
4πη 4
K 0 (z)with z = κ R . In the above, K 0 (z) and K 1 (z) are the modified Bessel functions of the second
kind of order zero and one, respectively. In the weak coupling limit, κ R
1, equation (25)becomes


kB T
2
−γ ,
D≈
ln
(26)
4πη
κR
where γ = 0.5772 · · · is Euler’s constant. Such a logarithmic dependence is consistent withthe result of Saffman and Delbrück [23, 24]. In the strong coupling limit, κ R
1, we have2
kB T 2D≈.
(27)
4πη κ R
Comparing these equations with equations (23) and (24), we can easily confirm that thedependences on the correlation length ξ and the radius R are the same. Notice that η inequation (27) drops out, as in equation (24). Indeed the validity of equation (25) was recently
confirmed by the measurement of diffusion constant of phase-separated domains in mixed
membranes [25].
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Finally, we discuss the physical meaning of the parameter λ in equation (3). When thelipid membrane is placed on a solid substrate, the term −λv represents the friction, which is
proportional to the local membrane velocity relative to the substrate. Under the presence ofa thin liquid layer of thickness  between the membrane and the substrate, the laminar flow
gives rise to the friction η v/, where η is the fluid viscosity. This means that λ = η / h for amembrane with thickness h [13].
When  is infinitely large, the flow of the fluid surrounding the membrane deviates fromlaminar flow. In this case, Saffman and Delbrück obtained the diffusion constant in the weakcoupling limit by taking into account the surrounding flow [23, 24]. By comparing their resultand equation (26), we find that λ = η(2η / hη)2 [17]. In the strong coupling limit, on theother hand, a small deviation was found between equation (25) and the extended result of the
Saffman and Delbrück theory [13, 26]. In the strong coupling limit of Saffman and Delbrück
theory, the drag force comes not only from the cylinder walls of molecules but also fromthe parts directly interacting with surrounding liquids [26]. Although the membrane-boundmolecules are assumed to be flush with the membrane interface, the results depend on whether
the molecules protrude into surrounding liquids or not in the limit [26]. Since equation (25)correlates with the extended results of Saffman and Delbrück theory over a wide range of λvalues, our phenomenological approach is justified at least for the molecules that are flush withthe membrane interface. It should also be pointed out that the size of diffusing molecules should
be larger than that of the lipids composing the membrane in the Saffman and Delbrück theory.
Therefore, the applicability of equation (3) to a mixed membrane is rationalized for the laminarflow of surrounding liquids but not justified from the model of Saffman and Delbrück theory.
In summary, we have calculated the decay rate of concentration fluctuations in a binary
fluid membrane which is coupled to the adjacent water. In contrast to the case of threedimensional critical fluids, the effective diffusion coefficient increases logarithmically withwavenumber. Our prediction should be checked experimentally.
We thank T Kato for useful discussions. This work is supported by KAKENHI (Grant-in-Aid
for Scientific Research) on Priority Areas ‘Soft Matter Physics’ from the Ministry of Education,
Culture, Sports, Science and Technology of Japan.
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J. Phys.: Condens. Matter 19 (2007) 072101

Fast Track Communication[19] Chaikin P M and Lubensky T C 1995 Principles of Condensed Matter Physics (New York: Cambridge University
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