Eur. Phys. J. E 31, 303–310 (2010)
DOI 10.1140/epje/i2010-10577-3

THE EUROPEAN
PHYSICAL JOURNAL E

Regular Article

Drag coefficient of a liquid domain in a two-dimensional
membrane
S. Ramachandran1 , S. Komura1,a , M. Imai2 , and K. Seki3
1

Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397,
Japan
2
Department of Physics, Faculty of Science, Ochanomizu University, Tokyo 112-0012, Japan
3
National Institute of Advanced Industrial Science and Technology, Ibaraki 305-8565, Japan
Received 8 November 2009 and Received in ﬁnal form 28 January 2010c EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2010
Published online: 20 March 2010 – 
Abstract. Using a hydrodynamic theory that incorporates a momentum decay mechanism, we calculate
the drag coeﬃcient of a circular liquid domain of ﬁnite viscosity moving in a two-dimensional membrane.
We derive an analytical expression for the drag coeﬃcient which covers the whole range of domain sizes.
Several limiting expressions are discussed. The obtained drag coeﬃcient decreases as the domain viscosity
becomes smaller with respect to the outer membrane viscosity. This is because the ﬂow induced in thedomain acts to transport the ﬂuid in the surrounding matrix more eﬃciently.

1 Introduction
Membrane components such as lipids and proteins are
subject to Brownian motion, and the resulting diﬀusive
process is an important mechanism for the transport ofthese materials. Consequently, it is generally believed thatmany biochemical functions in membranes are diﬀusioncontrolled processes. Up to now, the studies of diﬀusion
in biomembranes have mainly concentrated either on protein molecules [1–5] or, more recently, on lipid domains
sometimes called as “lipid rafts” [6–10]. Lipid domains
appear due to a lateral phase separation between a liquidordered phase and a liquid-disordered phase [11], and haveattracted much attention in relation to biological functionalities [12].
In the theoretical studies, a protein molecule is assumed to be a rigid disk moving in a two-dimensional(2D) thin ﬂuid sheet under low-Reynolds-number conditions. Saﬀman and Delbrück (SD) investigated this problem by considering a membrane that is sandwiched between a three-dimensional (3D) ﬂuid medium [13, 14].
They showed that the drag coeﬃcient has only a weak logarithmic dependence on the disk radius in the small-size
limit. The SD model was further analyzed by Hughes et al.
for the whole range of protein sizes, and they also obtained
the asymptotic expression in the large-size limit [15]. The
SD model was extended by Levine et al. for a viscoelastic
ﬁlm within an inﬁnite viscous liquid, taking into accountboth in-plane and out-of-plane dynamics [16–18].
ae-mail: komura@tmu.ac.jp

On the other hand, Evans and Sackmann looked ata slightly diﬀerent situation [19], i.e., the diﬀusion of aprotein molecule moving in a supported membrane (instead of a free membrane). The presence of the solid substrate is taken into account through a friction term in
the Stokes equation. The equivalent hydrodynamic model
was independently proposed by Izuyama who suggested
that the momentum decay mechanism should generally
exist for membranes surrounded by water [20]. Since a hydrodynamic equation with a momentum decay term wasoriginally proposed by Brinkman for ﬂow in porous media [21], we call these approaches the 2D Brinkman-typemodel. One of the advantages of the Brinkman model isthat the drag coeﬃcient can be analytically obtained over
the whole range of disk sizes. In the small-size limit, theSD model and the Brinkman model give the same logarithmic dependence. In biological systems, the Brinkman
model can be more relevant because the cell membranes
are strongly anchored to the underlying cytoskeleton, or
are tightly adhered to other cells [22].
Most of the previous works have concentrated on the
problem of protein diﬀusion in a membrane for which theassumption of a rigid disk is reasonable. Liquid lipid domains, on the other hand, cannot be considered as rigid
objects. A proper description should take into account theﬂuid nature and ﬁnite viscosity of the diﬀusing domain.
It should be also noted that the viscosity of the lipid domain is diﬀerent from that of the matrix [7]. In this paper,
we derive the drag coeﬃcient of a circular liquid domain
by taking into account its ﬁnite viscosity using the 2D
Brinkman-type hydrodynamic equation. We show that thedrag coeﬃcient decreases as the domain viscosity becomes

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The European Physical Journal E

smaller. An analogous problem for a spherical drop of
ﬂuid was solved by Rybczyński and Hadamard about 100
years ago [23, 24]. Some authors considered the hydrodynamics of Langmuir monolayer domains at the air/waterinterface by taking into account the ﬂuid nature of thedomain [25–27]. In these works, however, the domain andthe matrix viscosities are assumed to be the same, and thedrag coeﬃcient was not obtained [28]. Although the rotational diﬀusion coeﬃcient was also calculated for a rigid
disk [13–15, 19], such a motion is irrelevant for a ﬂuid domain which has an internal ﬂow.
In the next section, we begin with a description of
the 2D Brinkman model. In sect. 3, the solutions for theinside domain are connected to those for the outer membrane matrix through the appropriate boundary conditions. Then the drag coeﬃcient is calculated for the entire
region of the domain sizes. We also obtain several limiting expressions; some of which coincide with the previous
results. Several discussions are provided in sect. 4.

2 Hydrodynamic model with momentum
decay
The usual method of obtaining the diﬀusion coeﬃcient D
from the drag coeﬃcient ζ is to use the Einstein relation
D=

kB T
,
ζ

(1)

where kB is the Boltzmann constant and T the temperature. However, for a pure 2D system in the low-Reynoldsnumber regime, a linear relation between the velocity andthe drag force cannot be obtained [29]. This fact is due tothe inability to simultaneously satisfy the boundary conditions both at the disk (cylinder) surface and at inﬁnity
within the Stokes approximation [30]. Such a problem isknown as the Stokes paradox which essentially originatesfrom the constraint of momentum conservation in a pure2D ﬂuid.
Although a lipid bilayer membrane can be treated as a
2D viscous ﬂuid, it is not an isolated system because themembrane is surrounded by a 3D ﬂuid. Due to the coupling between the 2D membrane and the 3D ﬂuid, the momentum within the membrane can leak away to the outerﬂuid. Such an eﬀect can be taken into account througha momentum decay term in the equation of motion. This
also enables the 3D ﬂuid to be discarded and we can workin a pure 2D system. A hydrodynamic equation which isconsistent with the total momentum decay is∂v+ (v · ∇)v = η∇2 v − ∇p − λv.
(2)ρ∂t
In the above, v and p are the 2D velocity and pressure,respectively, while ρ is the 2D membrane density, η themembrane 2D viscosity, and λ the momentum decay parameter. Notice that the usually reported membrane 3D
viscosity is given by η/h, where h is the membrane thickness. A more detailed derivation of eq. (2) in terms of theFig. 1. Schematic picture showing a section of an inﬁnite membrane with viscosity η and momentum decay parameter λ inwhich a circular liquid domain of radius R with viscosity η ′ andmomentum decay parameter λ′ is moving. The liquid domain
moves with a velocity −U in the x-direction.

total momentum decay is given in [31]. Since the 3D version of eq. (2) was originally used by Brinkman for ﬂows
in porous media [21], we call it as 2D Brinkman equation. But it should be kept in mind that there is no realconnection between porous media and membranes except
from the momentum decay mechanism. For a typical ﬂow
in a membrane, we can adopt the Stokes approximation
to eq. (2). Then we haveη∇2 v − ∇p − λv = 0.

(3)

On the other hand, the incompressibility condition is givenby∇ · v = 0.
(4)

Due to the presence of the momentum decay term, theStokes paradox is now eliminated and eqs. (3) and (4) canbe solved analytically.
For a supported membrane, Evans and Sackmann assumed that a thin lubricating layer of bulk ﬂuid (thickness
H and 3D viscosity ηf ) exists between the membrane andthe substrate [19]. In this case, they speciﬁcally identiﬁed the friction parameter as λ = ηf /H, provided that
H is small enough. Here we consider that the parameter
λ represents all kinds of frictional interactions between
the lipid head group and the surrounding ﬂuid even for a
free membrane, and hence has more general signiﬁcance.
Previously, the above Brinkman model was used to calculate velocity autocorrelation function of a disk [31], diffusion coeﬃcient of a polymer chain in membranes [32],or dynamics of concentration ﬂuctuations in binary ﬂuid
membranes [33]. As pointed out in ref. [34], the translational symmetry along the membrane surface is broken
for eq. (3) due to the friction term. We have implicitly
assumed here that the velocity at inﬁnity vanishes so thatthe friction term is proportional to v itself. We also notethat eqs. (3) and (4) describe a purely 2D model so thatthe 3D hydrodynamic nature of the bulk ﬂuid is not taken
into account.
As schematically presented in ﬁg. 1, we consider a circular liquid domain of radius R and viscosity η ′ moving
with a velocity −U = (−U, 0) in a thin membrane sheetof viscosity η. As a result of the domain motion, a velocity ﬁeld is induced around it as well as inside the domain.
Our purpose is to calculate the drag force experienced by
the domain. To generalize our treatment, we allow for theS. Ramachandran et al.: Drag coeﬃcient of a liquid domain in a two-dimensional membranesituation where the momentum decay parameters are different between the matrix (λ) and the domain (λ′ ). Hereafter, quantities without prime correspond to those of thematrix, while quantities with prime refer to those of thedomain.
The ﬂuid velocity can generally be expressed as a
gradient of some potential ϕ and curl of some vector
A = (0, 0, A) which has only a single component [30].
Then the velocities in the matrix and the domain regions
are expressed as
v = −∇ϕ + ∇ × A,

(5)v′ = −∇ϕ′ + ∇ × A′ ,

(6)

and
respectively. Substituting eqs. (5) and (6) into eq. (4), we
obtain∇2 ϕ = 0,∇2 ϕ′ = 0,(7)which are the Laplace equations. With the use of eqs. (5)and (6), one can show that eq. (3) can be satisﬁed if thepressures are given byp = λϕ,p ′ = λ ′ ϕ′ ,(8)

while A and A obey the following equations:(∇2 − κ2 )A = 0,(∇2 − κ′2 )A′ = 0.

where we have used the recursion relations among themodiﬁed Bessel functions (see appendix A). Then thecomponents of the stress tensor can be obtained as∂vr
σrr = −p + 2η ∂r2κ
4+ 3= −η C1
r
r


4
2κ
+C2
K0 (κr) cos θ,
K1 (κr) +
r2
r


1 ∂vr
∂vθ
vθ
+
−
σrθ = η
r ∂θ
∂r
r


4
4C1
+ C2
K1 (κr) + κ2 K1 (κr)
= −η
r3
r2

2κ
+ K0 (κr) sin θ.
r(9)Here we have deﬁned the (inverse) hydrodynamic screening lengths for the matrix and the domain as κ ≡ (λ/η)1/2
and κ′ ≡ (λ′ /η ′ )1/2 , respectively. We shall next seek forthe solutions to eqs. (7) and (9) subject to the appropriate boundary conditions for the translational motion of
the domain.

3 Calculation of drag coefficient

(13)

(14)

Inside the liquid domain where r < R, the proper solutions to eqs. (7) and (9) subject to the condition thatthey do not diverge as r → 0 are
ϕ′ = C1′ r cos θ,′

305

A′ = C2′ I1 (κ′ r) sin θ,(15)Here, C1′ and C2′ are unknown coeﬃcients, and I1 (z) is
the modiﬁed Bessel function of the ﬁrst kind of order one.
Since the corresponding radial and tangential componentsof the velocity are now


C2′
′
′
′
I1 (κ r) cos θ,(16)
vr = −C1 +
r


C′
(17)
vθ′ = C1′ − C2′ κ′ I0 (κ′ r) + 2 I1 (κ′ r) sin θ,r
the components of the stress tensor can be obtained as

3.1 Velocity and stress tensor
It is convenient to work in the cylindrical polar coordinates (r, θ) with the origin at the center of the circulardomain. First we consider the matrix region where r > R.
Under the condition that the velocity and pressure must
approach zero at large distances, we write down the solutions to eqs. (7) and (9) as follows:C1
cos θ,A = C2 K1 (κr) sin θ.
(10)rHere, C1 and C2 are unknown coeﬃcients which will be
determined from the boundary conditions, and K1 (z) is
the modiﬁed Bessel function of the second kind of orderone. Although the general solutions for ϕ and A can be
expressed as series expansions in terms of r, we have kept
only the least number of terms satisfying the requisitepressure and velocity conditions. From eq. (5), the radial
and tangential components of the velocity are given by


C1
C2
K1 (κr) cos θ,
+
vr =
(11)
r2
r


C1
C2
K1 (κr) sin θ,+ C2 κK0 (κr) +
(12)
vθ =
r2
r

′
σrr
=



4
2κ′
′ ′2
′
′
′
′
−η C1 κ r + C2
I0 (κ r) cos θ, (18)I1 (κ r) −
r2
r
′
σrθ
=


4
2κ′
−η ′ C2′ 2 I1 (κ′ r) + κ′2 I1 (κ′ r) −I0 (κ′ r) sin θ. (19)rr

ϕ=

3.2 Boundary conditions
Next we assume that the no-slip condition is satisﬁed at
the domain boundary. This means that, at r = R, theradial component of the ﬂuid velocities should be equal tothe domain velocity −U cos θ, the tangential components
should be continuous, and so should the components ofthe stress tensor [29]. These conditions are written as
vr = −U cos θ,
vr′ = −U cos θ,vθ = vθ′ ,′σrθ = σrθ
.

(20)
(21)
(22)
(23)

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Since we have kept only the lowest-order terms in r forthe solutions to eqs. (7) and (9) (see eqs. (10) and (15)),
we are allowed to neglect the shape deformation of thecircular domain while it moves. In other words, we are
implicitly assuming that the line tension at the domainboundary exceeds a typical viscous force. A more quantitative argument to justify this condition will be given
later.
Using the above four boundary conditions, we can obtain the four unknown coeﬃcients C1 , C2 , C1′ and C2′
which are listed in appendix B. In the following, we introduce the dimensionless parameters to measure the relative viscosities E ≡ η/η ′ and the relative decay param′
eters L ≡ λ/λ. We also use the notations ǫ ≡ κR and′′ǫ ≡ κ R = ǫ E/L to measure the domain radius. Furthermore, the arguments of the modiﬁed Bessel functions
will be omitted as Kn ≡ Kn (ǫ) and In ≡ In (ǫ′ ) in orderto keep the notation compact.
3.3 Force exerted on a domain
The force exerted on the domain is given by the integralof the stress tensor over the boundary,

125

100

¦"1"6rj

306

75

50

25

0

10

5

20

15

"g
Fig. 2. The dimensionless drag coeﬃcient ζ as a function ofdimensionless domain radius ǫ = κR when L = 1. The diﬀerent
curves are for E = 0.1 (solid), E = 1 (dashed), and E = 10
(dotted).

1.8
F =R

 2π
0

dθ (σrr cos θ − σrθ sin θ).

(24)

After some calculation, the drag coeﬃcient obtained from
ζ = F/U becomes

An alternate form can be obtained by using the following
recurrence relation:
I0 (ǫ′ ) =2I1 (ǫ′ )+ I2 (ǫ′ ),
ǫ′

(26)

so that we haveǫ2
ǫK1 (ǫ′ I1 − 2I2 + 2EI2 )ζ(ǫ; E, L)
=
+
.
4πη
4
K0 (ǫ′ I1 − 2I2 ) + E(2K0 + ǫK1 )I2(27)
This is the main result of the paper.
In ﬁg. 2, we plot the dimensionless drag coeﬃcient ζ
as a function of dimensionless domain size ǫ = κR for
E = 0.1, 1, 10 when L = 1. In all these cases, the drag coeﬃcient increases with the domain radius R as it should
be. For ﬁxed values of ǫ, on the other hand, the drag coeﬃcient is smaller when the domain viscosity η ′ becomes
smaller (larger E). Fixing the domain size to ǫ = 1, we
have plotted in ﬁgs. 3 and 4 the drag coeﬃcient ζ as afunction of E and L, respectively. In the former, we chosevarious values of L ranging from L = 0.1 to 10, while thevalues of E were changed from E = 0.1 to 10 in the latter. Note that these graphs are both semi-log plots. From¦"1"6rjζ(ǫ; E, L)
ǫ2
=
4πη
4
ǫK1 [(4 + ǫ′2 )I1 − 2ǫ′ I0 + 2E(ǫ′ I0 − 2I1 )]. (25)+
K0 [(4+ǫ′2 )I1 −2ǫ′ I0 ]+E(2K0 +ǫK1 )(ǫ′ I0 −2I1 )

1.6

1.4

1.2

1

-2

10

0

2

10

10

4

10

E
Fig. 3. The dimensionless drag coeﬃcient ζ as a function of
the relative viscosity ratio E = η/η ′ when ǫ = 1. The diﬀerent
curves are for L = 0.1 (solid), L = 1 (dashed) and L = 10
(dotted).

ﬁg. 3, we see that the drag coeﬃcient monotonically decreases with increasing E (smaller domain viscosity). Thiscan be attributed to the internal ﬂows generated in the domain because they are more eﬃcient in transporting theﬂuid around it. As a result, a domain with ﬁnite viscosity
feels a smaller drag force than a rigid disk. The asymptotic
values of ζ for E → 0 and E → ∞ converge to respective
constants. In ﬁg. 4, the drag coeﬃcient is a decreasing
function of L. The large L values of ζ are dependent on
E as discussed below.

1.8

10

1.6

10

4

1.4

1.2

1 -6
10

307

6

¦"1"6rj¦"1"6rjS. Ramachandran et al.: Drag coeﬃcient of a liquid domain in a two-dimensional membrane

2

10

0

10

-2

10

-4

10

-2

0

10

10

2

10

10

-3

-1

10

10

1

3

10

"g

L
Fig. 4. The dimensionless drag coeﬃcient ζ as a function of
the relative momentum decay parameter ratio L = λ/λ′ when
ǫ = 1. The diﬀerent curves are for E = 0.1 (solid), E = 1(dashed) and E = 10 (dotted).

Fig. 5. The limiting dimensionless drag coeﬃcient ζ0 (solid,eq. (30)) and ζ∞ (dashed, eq. (31)) as a function of dimensionless domain radius ǫ = κR when L = 1.

3.4 Limiting expressions

small compared to the ﬁrst terms in eqs. (30) and (31).
Upon taking the ǫ → 0 limit of eqs. (30) and (31), we
obtainNext we examine several asymptotic limits of eq. (25) oreq. (27). Some useful formulae are listed in appendix A.
First we consider arbitrary values of E and L. In thelimit of ǫ → 0, eq. (25) gives a logarithmic behavior,
ζ(ǫ → 0)1+E≈
, (28)4πη
ln (2/ǫ) − γ + E[ln (2/ǫ) − γ + (1/2)]
where γ = 0.5772 . . . is Euler’s constant. The above Edependent logarithmic behavior is a new outcome of ourresult. In the opposite ǫ → ∞ limit, we have an algebraic
dependenceζ(ǫ → ∞)
ǫ2
≈
,
(29)
4πη
4
which does not depend on E. We also note that the above
asymptotic expressions for small and large domain size
limits do not depend on L.
For arbitrary ǫ, we next consider the rigid disk case(η ′ → ∞) where the limit of E → 0 is taken. Then we
obtainζ(ǫ; E → 0)ǫ2
ǫK1
ζ0 (ǫ)
=
+
,
(30)
≡
4πη
4
K0
4πη
which coincides with the result by Evans and Sackmann [19]. The opposite limit of E → ∞ corresponds to a
2D gas bubble (η ′ → 0). In this case, we obtain a diﬀerent
expression:ζ(ǫ; E → ∞)
ζ∞ (ǫ)
ǫ2
2ǫK1
=
+
.
≡
4πη
4
2K0 + ǫK1
4πη

(31)

In ﬁg. 5, we plot both ζ0 and ζ∞ as a function of ǫ in loglog scales. In the limit of ǫ → ∞, the diﬀerence between
the two curves is of the order of ǫ which becomes negligiblyζ0 (ǫ → 0)
1
≈
,
4πη
ln (2/ǫ) − γ
ζ∞ (ǫ → 0)
1
≈
.
4πη
ln (2/ǫ) − γ + (1/2)

(32)
(33)

These are nothing but the E → 0 and E → ∞ limits ofeq. (28). We note that small-ǫ behavior of ζ0 and ζ∞ does
not coincide in ﬁg. 5. The limiting expressions of eqs. (30)and (31) for ǫ → ∞ coincide with eq. (29).
In the limit of L → 0, which is the case of high momentum dissipation in the domain region, we recover eq. (30)again,
ζ(ǫ; L → 0)
ǫ2
ǫK1
=
+
.
(34)
4πη
4
K0
In the opposite L → ∞ limit, we obtainζ(ǫ; L → ∞)
ǫ2
2ǫ(1 + E)K1
=
+
,
4πη
4
2(1 + E)K0 + ǫEK1(35)

which depends on E as observed in ﬁg. 4. Equation (35)
reduces to eqs. (30) and (31) for E → 0 and E → ∞,respectively.

4 Conclusion and discussion
In summary, we have obtained the drag coeﬃcient of a circular liquid domain which has a ﬁnite viscosity using a 2Dhydrodynamic equation with momentum decay or the 2D
Brinkman equation. We showed that the drag coeﬃcientdecreases as the domain viscosity becomes smaller withrespect to the matrix viscosity. This is because the ﬂow in

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The European Physical Journal E

the domain helps to transport the ﬂuid in the surrounding
matrix more eﬃciently. Our result shows an E-dependence
in the small-domain-size limit (eq. (28)). New limiting expressions for the E → ∞ (eq. (31)) and L → ∞ (eq. (35))
are also obtained.
Several points merit further discussion. We ﬁrst discuss
the realistic value of E = η/η ′ in the case of lipid domains.
Lipid domains are known to arise from a phase separation
between a liquid-ordered (Lo ) phase rich in saturated lipid
such as sphingomyelin and a liquid-disordered (Ld ) phaserich in unsaturated lipid [11]. Applying the pulsed ﬁeld
gradient NMR spectroscopy, Orädd et al. measured lateral
diﬀusion coeﬃcients of a single lipid molecule both in the
Lo and Ld phases [7]. With the use of the SD logarithmic
expression for the diﬀusion coeﬃcient, the 2D viscositiesof these phases are estimated as ηo ≈ 1.6×10−9 N s/m andηd ≈ 0.4 × 10−9 N s/m at 293 K (the membrane thickness
is chosen to be h ≈ 3.8 × 10−9 m). When the Lo phaseforms a domain in the matrix of the Ld phase, the typical viscosity ratio would be E ≈ 0.2. In the opposite casewhere the Ld phase forms a domain, the ratio tends tobe E ≈ 4. Hence, it is necessary to take into account theﬁnite viscosity ratio between the domain and the matrix
rather than regarding the domain as a rigid disk. Incidentally, sphingomyelin constituting the Lo phases has alarge head group which can protrude out into the 3D ﬂuid,
whereas the Ld phase are devoid of such structures [12].
It is therefore reasonable to assume that the momentum
decay parameter λ experienced by the Lo and Ld domains
is diﬀerent.
When imposing the boundary conditions as given fromeq. (20) to (23), we have assumed that the domain does
not undergo any shape deformation. This implies that theline tension should be large enough to overcome the viscous force of deformation. In the case of lipid domains,the line tension was measured to be σ ∼ 10−12 N using
domain boundary ﬂicker spectroscopy [35]. For a ﬂow of
the order of U ∼ 10−6 m/s, a typical viscous force turns
out to be ηU ∼ 10−15 N, which is much smaller than σ.
Hence it is reasonable to assume that the line tension islarge enough to maintain the circular shape of the domains unless the temperature is very close to the criticalpoint. Diﬀerent boundary conditions were used in order toconsider the relaxation of deformed domains in a polymer
monolayer [36].
The crossover from the logarithmic to algebraic behaviors of the drag coeﬃcient takes place when ǫ ∼ 1 orR ∼ κ−1 . A typical domain size which corresponds to thiscondition is roughly estimated below. As stated in sect. 2,
the momentum decay parameter for a supported membrane was originally identiﬁed as λ = ηf /H, where ηf is the3D viscosity in the outer ﬂuid and H is the thickness of athin lubricating water which exists between the membraneand substrate [19]. In this situation,
 we have the hydrodynamic screening length κ−1 = ηH/ηf . Although we
do not assume the presence of a solid substrate, the hydrodynamic screening length in the Brinkman approach
would typically be κ−1 ∼ 10−7 m when H = 10−8 m,
ηf ∼ 10−3 N s/m2 and η ∼ 10−9 N s/m. Domain size inthe order of R ∼ 10−7 m is accessible in the experiments.

Within the present Brinkman model, the drag coefﬁcient has a R2 dependence in the large-size limit as ineq. (29). Such a behavior was indeed observed experimentally for Brownian motion of phase-separated domains on
stacked-supported membranes [37]. However, this dependence diﬀers from the asymptotic analysis of the SD modelaccording to which the drag coeﬃcient increases linearlywith R (similar to a 3D sphere) [15]. For ternary vesicles,
Cicuta et al. have measured the drag coeﬃcient of a domain which is proportional to R [8], supporting the SDmodel. The large-scale behavior of the Brinkman modeland the SD model diﬀers because the former is essentiallya 2D model while the latter has a 3D character. It shouldbe also noted that the hydrodynamic screening length in
the SD theory is given by ν −1 ≡ η/ηf , which is diﬀerent
from κ−1 . In the present Brinkman approach, the hydrodynamic screening length is the geometric mean of ν −1
and H.
At this stage we mention that, based on the SD theory,
an integral form of the drag coeﬃcient for a 2D liquiddomain with E = 1 was obtained by De Koker as [38]
ζDK
1
=
4πη
4

 ∞
0

dz

J12 (z)2z (z + δ)

−1

,

(36)

where δ ≡ νR and J1 (z) is the Bessel function of the ﬁrst
kind. We have estimated the above integral numerically. In
the limit of δ → 0, the above expression gives a logarithmic
dependence on R, in agreement with our result for small
ǫ as seen in eq. (28). In the opposite limit of δ → ∞,however, ζDK is proportional to R due to the 3D nature of
the SD theory. As described above, this is diﬀerent fromeq. (29) which gives the R2 dependence.
Several authors considered a more general scenario in
which a disk or a domain moves in a membrane being
located at a ﬁnite distance H from a ﬁxed substrate [26,39–42]. Among these, Stone and Ajdari numerically solved
the coupled dual integral equations which are obtained assolutions for the Stokes equations [39]. They showed thatthe Brinkman model gives a qualitatively good approximation over a wide range of membrane-substrate separations. One can explicitly show that the Brinkman and
the SD models are recovered in the limit of H → 0 andH → ∞, respectively.
For comparison, we ﬁnally write down the analogous
expression for the drag coeﬃcient of a spherical drop having 3D viscosity η3′ and radius R moving in a ﬂuid of 3D
viscosity η3 [23, 24, 29]
2E3 + 3ζ3=,
2πη3 R
E3 + 1(37)

where E3 = η3 /η3′ . In the solid sphere limit (η3′ → ∞),we obtain ζ3 = 6πη3 R (the Stokes result). In the opposite
bubble limit (η3′ → 0), we have ζ3 = 4πη3 R. The reductionin the drag coeﬃcient with the decrease in the drop viscosity η3′ is similar to that seen for a liquid domain in 2D
as discussed before. It is also important to point out thatthe eﬀect of the viscosity ratio E3 exists for all the drop
sizes R because there is no typical length scale for a 3D

S. Ramachandran et al.: Drag coeﬃcient of a liquid domain in a two-dimensional membrane

ﬂuid. This is in sharp contrast to our model in which thehydrodynamic screening length κ−1 plays a crucial role.
We thank S.L. Keller and Y. Sakuma for useful discussions.
This work was supported by KAKENHI (Grant-in-Aid for Scientiﬁc Research) on Priority Area “Soft Matter Physics” andGrant No. 21540420 from the Ministry of Education, Culture,
Sports, Science and Technology of Japan.

309

with
(B.5)
M1 = 2I0 K0 R(η − η ′ )κ′ + I0 K1 R2 ηκκ′ ,′2 ′2
M2 = −2I1 K1 Rηκ − 4I1 K0 η + I1 K0 η (4 + R κ ). (B.6)
In the above, Kn = Kn (κR) = Kn (ǫ) and In = In (κ′ R) =In (ǫ′ ) as denoted in the text.

References
Appendix A. Useful formulae
According to ref. [43], the following recursion relations
hold:
K0′ (z) = −K1 (z),K1 (z),
K1′ (z) = −K0 (z) −zI0′ (z) = I1 (z),I1 (z).
I1′ (z) = I0 (z) −
z

(A.1)
(A.2)
(A.3)
(A.4)

Here the primes indicate the derivative with respect to z.
In the limit of z → 0, we have
 

  
2
2
1
K0 (z) ≈ ln
−γ + 1+ln−γ z 2 + . . . , (A.5)
z
4
z
 


2
1 1 1
+ ln
K1 (z) ≈ −− γ z + ...,
(A.6)
z
2 2
z
z2
+ ...,(A.7)
I0 (z) ≈ 1 +4z3z
I1 (z) ≈ ++ ...,
(A.8)
2 16
z4
z2
I2 (z) ≈++ ...,(A.9)
896
where γ = 0.5772 . . . is the Euler constant. In the limit ofz → ∞, we have



π −z
4n2 − 1Kn (z) ≈e+ ... ,
(A.10)
1+
2z
8z


ez
4n2 − 1+ ... .
(A.11)In (z) ≈ √
1−
8z
2πz

Appendix B. Coefficients C1 , C2 , C′1 and C′2
The coeﬃcients C1 , C2 , C1′ and C2′ are given by
C1 = −R2 U − 2K1 RU/(κK0 )+2K12 R2 ηU (−2I1 +I0 Rκ′ )/[K0 (M1 +M2 )], (B.1)
C2 = 4I0 R(η − η ′ )κ′ U/[κ(M1 + M2 )]−2I1 U [4η − η ′ (4 + R2 κ′2 )]/[κ(M1 + M2 )], (B.2)′
C1 = U + 2I1 K1 RηκU/(M1 + M2 ),
(B.3)
′
2
C2 = 2K1 R ηκU/(M1 + M2 ),(B.4)

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