March 2010
EPL, 89 (2010) 56001
doi: 10.1209/0295-5075/89/56001

www.epljournal.org

Eﬀects of an embedding bulk fluid on phase separation dynamics
in a thin liquid film
S. Ramachandran1 , S. Komura1(a) and G. Gompper2
1

Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University
Tokyo 192-0397, Japan
2
Institut für Festkörperforschung, Forschungszentrum Jülich - D-52425 Jülich, Germany, EU
received 5 December 2009; accepted in final form 23 February 2010
published online 25 March 201068.05.-n – Liquid-liquid interfaces
87.16.D- – Membranes, bilayers, and vesicles
PACS 87.16.dp – Transport, including channels, pores, and lateral diﬀusion
PACS
PACS

Abstract – Using dissipative-particle-dynamics simulations, we study the eﬀects of an embedding
bulk fluid on the phase separation dynamics in a thin planar liquid film. The domain growth
exponent is altered from 2D to 3D behavior upon the addition of a bulk fluid, even though thephase separation occurs in 2D geometry. Correlated diﬀusion measurements in the film show thatthe presence of bulk fluid changes the nature of the longitudinal coupling diﬀusion coeﬃcient fromlogarithmic to algebraic dependence of 1/s, where s is the distance between the two particles. This
result, along with the scaling exponents, suggests that the phase separation takes place throughthe Brownian coagulation process.
c EPLA, 2010
Copyright Introduction. – Lipid molecules constituting themembranes of biological cells play a major role in
the regulation of various cellular processes. About adecade ago Simons and Ikonen proposed a hypothesis
which suggests that the lipids organize themselves intosub-micron–sized domains termed “rafts” [1]. The rafts
serve as platforms for proteins, which in turn attributes
a certain functionality to each domain. Although there
have been extensive studies in this area, the details of
the underlying physical mechanisms leading to formationof rafts, their stability, and the regulation of their finite
domain size remain elusive. Numerous experiments on
intact cells and artificial membranes containing mixturesof saturated lipids, unsaturated lipids and cholesterol,
have demonstrated the segregation of the lipids into
liquid-ordered and liquid-disordered phases [2]. Recent
experimental observations of critical ﬂuctuations point
towards the idea that the cell maintains its membraneslightly above the critical point [3,4]. Below the transition
temperature, there have been studies on the dynamics
in multicomponent membranes such as diﬀusion of
domains [5] and domain coarsening [6]. A clear understanding of phase separation may contribute towards a
better explanation of the dynamics of lipid organization(a) E-mail: komura@tmu.ac.jp

in cell membranes. Apart from biological membranes, it
is of relevance to understand the dynamics of Langmuirmonolayer systems which are also thin ﬂuid films.
Phase separation of binary ﬂuids following a quench has
been under study for over forty years [7]. The dynamic
scaling hypothesis assumes that there exists a scaling
regime characterized by the average domain size R thatgrows with time t as R ∼ tα with a universal exponent
α. For three-dimensional (3D) oﬀ-critical binary ﬂuids,
there is an initial growth by the Brownian coagulation process [8], followed by the Lifshitz-Slyozov (LS)evaporation-condensation process [9]; both mechanisms
show a growth exponent α = 1/3. This is followed by a latetime inertial regime of α = 2/3 [10]. For critical mixtures,
there is an intermediate α = 1 regime owing to interface
diﬀusion [11]. The scenario is slightly diﬀerent for pure
two-dimensional (2D) systems [12]. For an oﬀ-criticalmixture, it was predicted that after the initial formationof domains, they grow by the Brownian coagulation
mechanism with a diﬀerent exponent α = 1/2 (as will be
explained later), followed by a crossover to the LS mechanism which gives α = 1/3 even in 2D. For critical mixtures,on the other hand, the initial quench produces an interconnected structure which coarsens and then breaks up dueto the interface diﬀusion with an exponent α = 1/2. Afterthe breakup processes, coarsening takes place through56001-p1

S. Ramachandran et al.
Brownian coagulation that is again characterized by the
α = 1/2 scaling [8]. These predictions were confirmed bymolecular-dynamics simulations in 2D [13]. The exponent
α = 1/2 was also observed in 2D lattice-Boltzmann
simulations in the presence of thermal noise for a criticalmixture [14].
Although biomembranes composed of lipid bilayers canbe regarded as 2D viscous ﬂuids, they are not isolated
pure 2D systems since lipids are coupled to the adjacentﬂuid. Hence it is of great interest to investigate the phaseseparation dynamics in such a quasi-2D liquid film in thepresence of hydrodynamic interaction. (We use the word
“quasi-2D” whenever the film is coupled to the bulk ﬂuid.)
To address this problem, we consider a 2D binary viscous
ﬂuid in contact with a bulk ﬂuid. Our approximation of themembrane as a planar 2D liquid film is valid in the limit oflarge bending rigidity (common in biological membranes)
or in the presence of a lateral tension, which both act
to suppress membrane undulations. We employ a simple
model in which the film is confined to a plane with thebulk ﬂuid particles added above and below. In our modelusing a dissipative-particle-dynamics (DPD) simulation
technique, the exchange of momentum between the film
and the bulk ﬂuid is naturally taken into account. We
particularly focus on the eﬀect of bulk ﬂuid on the quasi2D phase separation. We show that the presence of a bulk
ﬂuid will alter the domain growth exponent from thatof 2D to 3D indicating the significant role played by thefilm-solvent coupling. In order to elucidate the underlying
physical mechanism of this eﬀect, we have looked into thediﬀusion properties in the film by measuring two-particle
correlated diﬀusion. Our result suggests that quasi-2D
phase separation proceeds by the Brownian coagulation
mechanism which reﬂects the 3D nature of the bulk ﬂuid.
Such a behavior is universal as long as the domain size
exceeds the Saﬀman-Delbrück length [15].
Model and simulation technique. – For the purposeof our study, we use a structureless model of the 2D liquidfilm within the DPD framework [16,17]. As shown in fig. 1,
the 2D film is represented by a single layer of particles
confined to a plane. In order to study phase separation,
we introduce two species of particles, A and B. The bulk
ﬂuid which we call as “solvent” (S) is also represented bysingle particles of same size as that of the film particles.
All particles have the same mass m. We avoid using theexisting DPD models for a self-assembling bilayer [18,19]
as they inherently include bending and protrusion modes,
which makes it diﬃcult to separate hydrodynamic eﬀects
from the eﬀect of membrane shape deformations.
In DPD, the interaction between any two particles,
within a range r0 , is linearly repulsive. The pairwiseinteraction leads to full momentum conservation, which
in turn brings out the correct ﬂuid hydrodynamics. The
force on a particle i is given bym
dvi   CR=
Fij (rij ) + FD
ij (rij , vij ) + Fij (rij ) ,dt(1)Fig. 1: (Colour on-line) Image of the model film with the bulk
fluid called solvent. The yellow and red particles represent thetwo components constituting the model film, while blue ones
represent the solvent. For clarity, only a fraction of the solvent
particles are shown.

where rij = ri − rj and vij = vi − vj . Of the three typesof forces acting on the particles, the conservative force
on particle i due to j is FC
ij = aij ω(rij )r̂ij , where aij
is an interaction strength and r̂ij = rij /rij with rij =|rij |. The second type of force is the dissipative force
2FD
ij = −Γij ω (rij )(r̂ij · vij )r̂ij , where Γij is the dissipative
strength for the pair (i, j). The last is the random force−1/2FR
ω(rij )ζij r̂ij , where σij is the amplitude
ij = σij (Δt)
of the random noise for the pair (i, j), and ζij is arandom variable with zero mean and unit variance whichis uncorrelated for diﬀerent pairs of particles and diﬀerent
time steps. The dissipative and random forces act as athermostat, provided the ﬂuctuation-dissipation theorem2= 2Γij kB T is satisfied (kB is Boltzmann constant andσij
T is the thermostat temperature). The weight factor ischosen as ω(rij ) = 1 − rij /r0 up to the cutoﬀ radius r0
and zero thereafter. The particle trajectories are obtainedby solving eq. (1) using the velocity-Verlet integrator.
In the simulation, r0 and m set the scales for lengthand mass, respectively, while kB T sets the energy scale.
The time is measured in units of τ = (mr02 /kB T )1/2 . The
numerical value of the amplitude of the random force
is assumed to be the same for all pairs such that σij =3.0[(kB T )3 m/r02 ]1/4 , and the ﬂuid density is set as ρ = 3.0.
We set kB T = 1 and the integration time step is chosen to
be Δt = 0.01τ .
The thin film is constructed by placing particles inthe xy-plane in the middle of the simulation box (seefig. 1). Owing to the structureless representation of theconstituent particles, we apply an external potential so as
to maintain the film integrity. This is done by fixing thez-coordinates of all the film particles. It is known that
confinement of simulation particles between walls leads toa reduction in the solvent temperature near the wall [20].
However, since we allow for the in-plane motion of the film
particles, the solvent temperature is found to be only 2%
less than the bulk near the 2D film. Hence we consider that
this eﬀect is negligible. Our work involves the systematicj=i56001-p2

Eﬀects of an embedding bulk ﬂuid on phase separation dynamics in a thin liquid film

1
0.9

0.8

0.7

α = 1/2

log10 R

0.5

0.6
0.3
2

3

2.5

0.4
α = 1/3
0.2

70:30
0

1

1.5

2

2.5

3

log10 tFig. 3: (Colour on-line) The average domain size R as a functionof time t for a 70 : 30 oﬀ-critical mixture. The upper black curveis the pure 2D case showing an α = 1/2 scaling, and the lower
red curve is the quasi-2D case when Lz = 40 showing a distinct
α = 1/3 scaling. The inset shows the zoomed in portion with
diﬀerent box heights in the z-direction, i.e., Lz = 0, 5, 20 and
40 starting from the top black curve.

Fig. 2: (Colour on-line) The snapshots for a 70 : 30 mixture(yellow and red) undergoing phase separation at t = 0, 150 and1000 (top to bottom) for a pure 2D (left column) and quasi-2D
system with Lz = 40 (right column). The above snapshots arefrom one of the ten independent trials that were conducted.

variation of the height of the simulation box starting fromthe pure 2D case. In the absence of solvent, we workwith a 2D-box of dimensions Lx × Ly = 80 × 80 with 19200
particles constituting the film. For the quasi-2D studies,
we add solvent particles S above and below the model film,
and increase the height of the box as Lz = 5, 20 and 40.
For all the cases there are 19200 film particles. The largest
box size (Lz = 40) has 748800 solvent particles. The box
with height Lz = 40 is found to be suﬃciently large enoughto prevent the finite-size eﬀect which aﬀects the solventfilm interaction. The system is then subject to periodic
boundary conditions in all the three directions. For phaseseparation simulations, we introduce two species of film
particles A and B. The interaction parameter between
various particles are given by aAA = aBB = aSS = aAS =aBS = 25 and aAB = 50. In order to do a quench, the film isfirst equilibrated with a single component, following whicha fraction of the particles are instantaneously changed to
the B type.

Phase separation. – First we describe the results ofthe phase separation dynamics. The snapshots for theA : B composition set to 70 : 30 (oﬀ-critical mixture) are
shown in fig. 2 for both the pure 2D case (left column)
and the quasi-2D case with Lz = 40 (right column). Qualitatively, it is seen that the domains for the quasi-2D caseare smaller in size when compared at the same time step.
We also monitor the average domain size R(t) which canbe obtained from the total interface length L(t) between
the two components. This is because R(t) and L(t) are
related by L(t) = 2πN (t)R(t), where N (t) is the numberof domains. The area occupied by the B-component isgiven by A = πN (t)R2 (t) which is a conserved quantity.
Then we haveR(t) = 2A/L(t).
(2)
When the domain size grows as R ∼ tα , one has L ∼ t−α
and N ∼ t−2α . The domain size R(t) for a 70 : 30 mixture
is shown in fig. 3. In this plot, the average over 10
independent trials has been taken. It can be seen that thepure 2D case has a growth exponent α = 1/2. Upon the
addition of solvent, we observe that the exponent shifts toa lower value of α = 1/3. This exponent is reminiscent of
the phase separation dynamics of an oﬀ-critical mixture
in 3D. Upon systematically increasing the amount ofsolvent in the system by changing the height Lz , we cansee a clear deviation from the pure 2D behavior (seethe inset of fig. 3). There is no further change if Lz is
increased beyond 40. We note that the scaling regime

56001-p3

S. Ramachandran et al.

1.5

gives rise to an exponent α = 1/3. (In general, the exponent is α = 1/d, where d is the space dimension.) The
change in the exponent from α = 1/2 to 1/3 due to theaddition of solvent implies the crossover from 2D to 3D
behaviors of the phase separation dynamics even though
the lateral coarsening takes place only within the 2D
geometry1 . Therefore it is necessary to examine the size
dependence of the domain diﬀusion coeﬃcient in quasi-2D
systems. This can be calculated by tracking the meansquared displacement of domains of various radii. The
equivalent information can be more eﬃciently obtainedby calculating the two-particle longitudinal coupling diﬀusion coeﬃcient in a single component film rather than ina binary system. This is described in the next section.

1.21

α = 1/2

log10 R

1

0.8

0.6

2

2.2

2.4

2.6

α = 1/3
0.5

Correlated diﬀusion. – Consider a pair of particles
separated by a 2D vector s, undergoing diﬀusion in theliquid film. The two-particle mean squared displacement
is given by [21]

50:50
0

1

1.5

2

2.5

log10 tkl(s)t,Δski Δslj  = 2Dij

Fig. 4: (Colour on-line) The average domain size R as a functionof time t for a 50 : 50 critical mixture. The upper black curveis the pure 2D case showing an α = 1/2 scaling, and the lower
red curve is the quasi-2D case when Lz = 40 showing a distinct
α = 1/3 scaling. The inset shows the zoomed in portion with
diﬀerent box heights in the z-direction as in fig. 3.

covers about one decade in time, which is similar to that
previously shown in the literature [18]. A larger system size
Lx × Ly = 200 × 200 also produced the same scaling for thepure 2D case, which demonstrates that finite-size eﬀects
are small. However, we present here only the results forthe 80 × 80 system in 2D, because this is the system sizestudied for the quasi-2D case with a bulk ﬂuid, which
requires a large amount of particles in 3D.
In fig. 4, we show the result for a component ratioof 50 : 50 (critical mixture). In this case, the growth
exponent for the pure 2D case is less obvious owing torapid coarsening of the domains. However, by simulatinga bigger system 200 × 200 with the same areal density, an
α = 1/2 exponent is indeed obtained. Similar to the oﬀcritical case, the growth of the domains is slowed downby the addition of solvent and the exponent is reduced to
α = 1/3. These results indicate that solvent is responsible
for slowing down the growth dynamics.
The observed exponent α = 1/2 in pure 2D systems canbe explained in terms of the Brownian coagulation mechanism [12]. From dimensional analysis, the 2D diﬀusion
coeﬃcient of the domain is given by D2 ∼ kB T /η, where ηis the film 2D viscosity. Using the relation
2R ∼ D2 t ∼ (kB T /η)t,(4)

where Δski is the displacement of the particle k (= 1, 2)
klalong the axis i (= x, y), Dij
is the diﬀusion tensor giving
self-diﬀusion when k = l and the coupling between themwhen k = l. The x-axis is defined along the line connecting
a pair of particles 1 and 2, i.e., s = sx̂. Hence, we have12
= 0 by symmetry. The longitudinal coupling diﬀusion
Dxy12(sx̂), gives the coupled diﬀusion
coeﬃcient, DLc (s) = Dxx
along the line of centers of the particles. We first describe
the analytical expression of DLc (s) based on the Saﬀman
and Delbrück (SD) theory which was originally developed
for protein diﬀusion in membranes [15].
Since the calculation of the diﬀusion coeﬃcient in apure 2D system is intractable due to Stokes paradox, SD
circumvented this problem by taking into account thepresence of the solvent with 3D viscosity ηs above andbelow the membrane. Suppose a point force f directedalong the plane of the film lying in the xy-plane acts atthe origin. Then we seek for the velocity v(s) induced at
the position s. According
to the SD theory, it is given inFourier space, v[q] = ds e−iq·s v(s), as [15,21,22]


1
qi qj
vi [q] = GSD[q]f=−fj , (5)δjijijηq(q + ν)
q2
where GSD is the 2D film analog of the Oseen tensor. In
the above, the SD length is defined by ν −1 = η/2ηs .
For over damped dynamics, we can use the Einstein
relation to relate the diﬀusion tensor DLc to GSD
ij [21]. AfterSD
converting Gij [q] into real space, we obtain(3)

DLc (s) = kB T GSD
(sx̂)
xx
2
kB T πH1 (νs) πY1 (νs)
=
−
−
,
4πη
νs
νs
(νs)2

(6)
we find R ∼ t1/2 . For 3D systems, on the other hand, thediﬀusion coeﬃcient of the droplet is inversely proportional
1 We note that the LS mechanism shows an exponent of α = 1/3
to its size, D3 ∼ 1/R, a well-known Stokes-Einstein rela- for both 2D and 3D. Thus our simulations are still in the early stagetion. Hence the Brownian coagulation mechanism in 3D of the coarsening dynamics.
56001-p4

Eﬀects of an embedding bulk ﬂuid on phase separation dynamics in a thin liquid film
-1

10

-2

DL

c

10

10

-3

1/s
-4

10 1

10

s

100

Fig. 5: (Colour on-line) Longitudinal coupling diﬀusion DLc as
a function of particle separation s. The upper circles are data
for the pure 2D case. The lower squares correspond to the casewith solvent when Lz = 40. The upper solid line is the fit byeq. (7), and the lower solid line is the fit by eq. (6). The dashedline shows the 1/s-dependence.

where H1 and Y1 are the Struve function and theBessel function of the second kind, respectively. At shortdistances s ≪ ν −1 , the asymptotic form of the above
expression becomes
  
2
kB T
1
DLc (s) ≈
ln
−γ +
,
(7)
4πη
νs
2
where γ = 0.5772 · ·· is Euler’s constant. At large interparticle separations s ≫ ν −1 , on the other hand, eq. (6)reduces to
kB T
kB T
=
,
(8)
DLc (s) ≈2πηνs 4πηs sshowing the asymptotic 1/s decay which reﬂects the 3D
nature of this limit. Notice that eq. (8) depends only on
the solvent viscosity ηs but not on the film viscosity η any
more.
In fig. 5, we plot the measured longitudinal coupling
diﬀusion coeﬃcient DLc as a function of the 2D distances. In these simulations, we have worked only with singlecomponent films with the same system sizes and numberof particles as those used for the phase separation simulations. We have also taken the average over 20 independent trials. In the pure 2D case without any solvent,DLc shows a logarithmic dependence on s. This is consistent with eq. (7) obtained when the coupling between thefilm and the solvent is very weak so that the film canbe regarded almost as a pure 2D system. Using eq. (7)as an approximate expression, we get from the fitting
kB T /4πη ≈ 0.89 × 10−2 and ν −1 ≈ 20. In an ideal case, theSD length should diverge due to the absence of solvent.

The obtained finite value for ν −1 is roughly set by half ofthe system size in the simulation.
When we add solvent (Lz = 40), the DLc is decreased
and no longer behaves logarithmically. In this case, we use
the full expression eq. (6) for the fitting, and obtained
kB T /4πη ≈ 1.35 × 10−2 and ν −1 ≈ 1. In the above twofits we have neglected the first two points as they lie
outside the range of validity, s ≫ 1, of eq. (6) [21]. Sinceν −1 ≈ 1 when the solvent is present, the data shown infig. 5 are in the crossover region, s  ν −1 , showing anapproach towards the asymptotic 1/s-dependence as ineq. (8). Hence we conclude that the solvent brings in the
3D hydrodynamic property to the diﬀusion in films. Thisis the reason for the 3D exponent α = 1/3 in the phaseseparation dynamics, and justifies that it is mainly driven
by the Brownian coagulation mechanism.
In our simulations the film and the solvent have very
similar viscosities. This sets the SD length scale to beof the order of unity, which is consistent with the valueν −1 ≈ 1 obtained from the fitting. As explained above, thefit also provides the 2D film viscosity as η ≈ 6, and hence
we obtain as ηs ≈ 3. This value is in reasonable agreementwith the value ηs ≈ 1 calculated in ref. [23] by using thereverse Poiseuille ﬂow method. The reason for the slightly
higher value of ηs in our simulations is that the tracer
particles are of the same size as the film particles. This
may lead to an underestimation of the correlated diﬀusion
coeﬃcient. In real biomembranes sandwiched by water,
however, the value of the SD length is much larger than thelipid size, and is in the order of micron scale [15]. Hence the3D nature of hydrodynamics can be observed only for large
enough domains observed under optical microscopes [5].
Discussion. – Several points merit further discussion.
The growth exponents obtained from our simulations
for critical mixtures are the same for the oﬀ-criticalcase, namely α = 1/2 without solvent and α = 1/3 withsolvent. A previous DPD study by Laradji and Kumar onphase separation dynamics of two-component membranes(both critical and oﬀ-critical cases) used a coarse-grainedmodel for the membrane lipids [18]. In their model, theself-assembly of the bilayer in the presence of solvent
is naturally taken into account. The exponent for theoﬀ-critical case α = 1/3 is the same as that obtained inour study, although they attributed this value to the
LS mechanism. For critical mixtures in the presence ofsolvent, they obtained a diﬀerent value α = 1/2. A suitable
explanation for this exponent was not given in their paper.
As a related experimental work, the diﬀusion of
tracer particles embedded in a soap film was recently
reported [24]. When the diameter of the tracer particles
is close to the thickness of the soap film, the systemshows a 2D behavior. On the other hand, if the particle
diameter is much smaller than the soap film thickness, it
executes a 3D motion. On systematically increasing thesoap film thickness, they identified a transition from 2D
to 3D nature. In this paper, we have demonstrated the56001-p5

S. Ramachandran et al.
analogue for a 2D liquid film-solvent system using DPD
simulations.
In the SD theory, the bulk ﬂuid is assumed to occupy
an infinite space above and below the membrane. The
situation is altered when the solvent and the membrane
are confined between two solid walls [22]. If the distance
H between the membrane and the wall is small enough,we are in a situation similar to that described by Evans
and Sackmann [25]. The Oseen tensor GES in this case isdefined through the relation [26],


1
qi qj
[q]f
=
−
fj , (9)vi [q] = GES
δ
j
ij
ij
η(q 2 + κ2 )q2
where the new length scale is defined as κ−1 = ηH/2ηs .
Notice that κ−1 is the geometric mean of ν −1 andH [27]. Following the same procedure as in the previous
section, the longitudinal coupling diﬀusion coeﬃcient canbe obtained as
1kB T
K1 (κs)c−
DL (s) =,(10)2πη (κs)2κs
where K1 is a modified Bessel function of the second kind.
At short distances s ≪ κ−1 , we have
  
2
kB T
1
c
DL (s) ≈
ln
−γ +
,
(11)
4πη
κs
2
which is almost identical to eq. (7) except ν is replaced now
by κ. At long distances s ≫ κ−1 , on the other hand, we get
DLc (s) ≈kB T
kB T H=,2πηκ2 s2 4πηs s2(12)which exhibits a 1/s2 -dependence. This is in contrast toeq. (8). Following a similar scaling argument as before, we
predict that, in the presence of solid walls, the domain
growth exponent should be α = 1/4 within the Brownian coagulation mechanism. In biological systems, theabove model with solid walls can be relevant because thecell membranes are strongly anchored to the underlying
cytoskeleton, or are tightly adhered to other cells.
In summary, we have shown that the bulk ﬂuid has asignificant eﬀect on the phase separation dynamics of thinliquid films through a simple quasi-2D simulation model.
We have demonstrated the change in the growth exponents
from 2D to 3D by the addition of bulk ﬂuid. This is further
justified by the two-particle correlation studies, whichshowed that the longitudinal coupling diﬀusion coeﬃcientchanges from a logarithmic dependence for the pure 2D
case to an algebraic one for the quasi-2D case. Future
directions include investigating the role of out-of-plane
ﬂuctuations and the eﬀect of boundary walls on the phaseseparation.
∗∗∗
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 Technology ofJapan.
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