IOP PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 25 (2013) 195105 (8pp)

doi:10.1088/0953-8984/25/19/195105

Growth kinetics of circular liquiddomains on vesicles bydiffusion-controlled coalescence
Kazuhiko Seki1 , Shigeyuki Komura2 and Sanoop Ramachandran3
1

National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 5,
Higashi 1-1-1, Tsukuba, Ibaraki 305-8565, Japan2
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan
University, Tokyo 192-0397, Japan3
Physique des Polymères, Université Libre de Bruxelles, Campus Plaine, CP 223, 1050 Brussels,
Belgium
E-mail: k-seki@aist.go.jp and komura@tmu.ac.jp

Received 15 October 2012, in final form 19 February 2013
Published 19 April 2013
Online at stacks.iop.org/JPhysCM/25/195105
Abstract
Motivated by recent experiments on multicomponent membranes, the growth kinetics ofdomains on vesicles are theoretically studied. It is known that the steady-state rate ofcoalescence cannot be obtained by taking the long-time limit of the coalescence rate when themembrane is regarded as an infinite two-dimensional (2D) system. The steady-state rate ofcoalescence is obtained by explicitly taking into account the spherical vesicle shape. Using theexpression for the 2D diffusion coefficient obtained in the limit of small domain size, ananalytical expression for domain growth kinetics is obtained when a circular shape is always
maintained. For large domains, the growth kinetics are discussed by investigating the size
dependence of the coalescence rate, using the expression for the diffusion coefficient ofarbitrary domain size.

1. Introduction

In recent experiments, the growth kinetics of circulardomains on ternary fluid vesicles have been observed byfluorescence microscopy [11, 13, 14]. In these experiments,
liquid domains are formed in giant vesicles by phaseseparation into the liquid-ordered phase and the liquiddisordered phase, on lowering the temperature from theone-phase region. Yanagisawa et al found two different
types of domain coalescence kinetics [14]. In one of the
coalescence processes, the domains grew by collision andcoalescing while keeping their circular shape until the large
domains of vesicle size appeared. These growth kinetics due
to diffusion-controlled coalescence (DCC) were describedby a power-law. In the other coalescence process, domain
growth was suppressed by membrane-mediated repulsive
inter-domain interactions. Recently, it was pointed out thatdomain coalescence could be prevented by the membranemediated interactions between liquid domains associated with
deformations of the membrane [17–21]. When the liquiddomain size exceeds a critical value, the boundary line energy

Lipid bilayer membranes can be regarded as two-dimensional(2D) systems embedded in a three-dimensional (3D) solvent.
The membranes are coupled to the solvent since thelipids composing the membrane interact with the solvent
surrounding them and momentum can be exchanged between
the membrane and the solvent [1–10]. In this sense,
membranes can be regarded as quasi-2D systems. With
recent advances in experiments, domains formed by phaseseparation in multicomponent membranes are visualized, andthe lipid spatial organization and its dynamics have beenstudied [11–15]. The phase separation kinetics and coarsening
are influenced by the dimensionality, the domain shapes andthe hydrodynamics in the systems [16]. The phase separation
in a multicomponent lipid bilayer exhibits rich dynamics,
partly due to momentum dissipation to third dimensions while
the motion is confined to 2D [7–9].
0953-8984/13/195105+08$33.00

1

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J. Phys.: Condens. Matter 25 (2013) 195105

K Seki et al

Smoluchowski theory in pure 3D and 2D infinite systems
are reviewed. In section 4, the coalescence rate is obtainedby taking into account the vesicle shape. In section 5, ananalytical expression representing the growth of mean domain
size is obtained when the circular shape and the area of theliquid domains are kept the same before and after coalescence.
The size dependence of the coalescence rate is investigatedusing an analytical expression for the diffusion coefficientfor the liquid domain of arbitrary size in section 6. In
section 7, the theoretical results are discussed in relation to
those obtained by recent experiments [14].

is reduced by budding, at the expense of some bending
energy. The significant slowing down of domain growth canbe observed for large budded domains [14, 21]. In the steadystate, the domain patterns and the membrane shapes can bestabilized by coupling between the local membrane curvatureand the line tension [12, 15, 22–24].
As briefly summarized above, the liquid domains coarsen
under the influence of several competing processes. Even
without budding, the observed growth kinetics were different
from those obtained from the scaling hypothesis [13, 14].
Motivated by the experiments, we study theoretically thegrowth kinetics of domains on vesicles by DCC.
The study of domain growth on vesicle surfaces is still
limited compared to that in 3D [16]. According to the scaling
hypothesis, the domain growth exponent due to DCC in 2D
is 1/2, in contrast to that in 3D given by 1/3 [25, 26].
However, it should be noted that, in the scaling hypothesis,
the coalescence process is not explicitly taken into account.
Moreover, it is known that a steady-state rate of coalescencecannot be obtained for an infinitely large 2D system [27, 28].
In contrast to diffusion in 3D space, in order to obtain the
coalescence rate when diffusion is restricted in 2D, boundary
conditions are crucial. This point is missing in the scaling
hypothesis.
In addition to the above argument for pure 2D systems,
the coalescence of liquid domains can be influenced bycoupling between the membrane and the solvent. The
diffusion of large domains is influenced more by coupling
compared to that of small domains. As the domain grows,
the influence of coupling between the membrane and thesolvent increases. The domain growth kinetics have beenstudied by dissipative particle dynamics simulations andcontinuum simulations [7–9, 29]. The simulation results
suggest the slowing down of domain growth by DCC due tothe coupling [7].
In this paper, we study the growth of liquid domains
immersed in a 2D membrane using an analytical theory
which goes beyond the scaling hypothesis. We note that a
steady-state rate of coalescence can be obtained by taking
into account explicitly the vesicle shape [30, 31]. Using the
diffusion coefficient for domains obtained by taking into
account coupling between the membrane and the solvent [1–6,10], we show that the coalescence rate is independent of
domain size in the limit of small domain size. For such acase, it is known that the size distribution is described by
the Smoluchowski theory of coalescence processes [32]. By
further assuming that the circular shape of liquid domains isalways maintained, the time evolution equation of the mean
domain size is obtained from the size distribution using theconservation of domain area upon coalescence. The mean
domain growth is expressed by a single function for thewhole time regime, starting from the initial induction periodof coalescence to the final asymptotic regime given by thepower-law. When the domain size is large, we discuss theinfluence of coupling between the membrane and the solventon domain growth by analyzing the size dependence of the
coalescence rate.
In section 2, we present the results obtained fromthe scaling hypothesis. In section 3, the results from

2. Scaling theory
In this section, we summarize the results obtained from
scaling theory. Scaling theory is the simplest way to derive
the power-law growth of domain size. Obviously, one cannot
obtain both the transient growth leading to asymptotic
power-law kinetics and the magnitude of the power-law
growth.
Scaling theory is based on the hypothesis that the mean
domain radius ha(t)i in d-dimension is related to time t byha(t)i2 ∼ Dd t,(1)where Dd is the diffusion coefficient and subscript d denotes
the dimensionality. In 3D, the diffusion coefficient forthe domain is inversely proportional to ha(t)i from the
Stokes–Einstein relation, D3 ∼ 1/ha(t)i. Substituting thisrelation in the scaling relation, we obtain ha(t)i ∼ t1/3 [16,25, 26]. For 2D, we have D2 ∼ kB T/η from the dimensional
argument, where kB is the Boltzmann constant, T thetemperature, and η the 2D membrane viscosity surrounding
the domain. Using the above expression for D2 , the scaling
hypothesis leads to ha(t)i ∼ t1/2 [16, 25, 26]. It should bepointed out that a logarithmically time-dependent diffusion
coefficient has been derived in pure 2D by hydrodynamic
theory [2]. When the membrane couples to the solvent, aconstant diffusion coefficient is derived, and D2 depends
logarithmically on the domain size in the weak coupling
limit [1, 2]. The logarithmic size dependence and thenumerical factors are ignored in the dimensional argument.
We also note that the time given by ha(t)i2 /Dd is notequal to the coalescence time for the domain of size ha(t)i.
The coalescence time should be given by the mean first timethat domains coalesce by diffusion from a random initial
distribution. Below, we show that the latter time is different
from the one given by ha(t)i2 /Dd .
The scaling hypothesis merely relates a single lengthscale given by the mean domain size at time t withthe diffusion coefficient as shown in equation (1) andit should be justified. In subsequent sections, we pointout that the results for the scaling hypothesis cannot be
obtained for domain growth by diffusion coalescence in 2D
infinite systems. We show then that the scaling hypothesis
is consistent with domain growth kinetics via diffusion
coalescence when the available diffusion area is finite in 2D.
The whole kinetics, including transient growth and the final
power-law growth, can be obtained by taking into account the
diffusion-coalescent process and the vesicle shape explicitly.
2

J. Phys.: Condens. Matter 25 (2013) 195105

K Seki et al

3. Smoluchowski theory in infinite systems

3.2. Two dimensions (2D)
Applying the boundary condition at σ , we obtain

In Smoluchowski theory, domain motion is assumed to be
independent of others and coalescence between a pair of
domains is considered. When both domains can move, it isdifficult to solve the problem analytically. We assume that
one of the domains is fixed and the other diffuses with amutual diffusion coefficient Dd , which is the sum of the
diffusion coefficients for two spherical domains of equal
size [33]. Coalescence takes place immediately when two
domains come in contact at the encounter distance, which isthe sum of the radii of the two domains. In the theoreticalformulation, the spatial distribution of domains satisfies the
diffusion equation and the spatial domain density should
vanish at the encounter distance. The domain size increases
immediately after coalescence and the spatial distribution of
the new domain size should be zero at the encounter distancefor the increased domain size.
The density of the mobile domain around the immobile
domain satisfies the diffusion equation∂ρd (r, t) = Dd ∇ 2 ρd (r, t),∂tρ2 (r) = C ln(σ/r),

where C is a constant determined from the other boundary
condition at r → ∞. However, it is impossible to fix C
because ρ2 (r) diverges for r → ∞ as long as C is finite.
Unlike 3D infinite systems, the density cannot satisfy bothboundary conditions, equations (4) and (5), simultaneously.

4. Smoluchowski theory in spherical surface
The difficulty mentioned in the previous section for 2D can beovercome if the available diffusion area is finite. The density
profile depends crucially on the shape of the 2D region.
In earlier experiments [14], DCC of circular domains wasobserved on the vesicle surfaces. In principle, the coalescence
rate can be obtained by modifying the method shown in theprevious section to the spherical region, but the calculation
is rather complicated. In this paper, we employ an alternative
method.
Here, we investigate the life time of the density of mobile
domains which survive from collision with the immobile
domain. In a confined region, the density decay can be wellapproximated by a single exponential. The time constant of
the exponential decay can be reasonably obtained from themean first-passage time of a mobile domain to the peripheryof the immobile domain assuming uniform distribution for thestarting point. For coalescence, the mean first-passage timeis the mean coalescence time corresponding to the encounter
time between two domains. When the initial position of themobile reactant is z = cos θ (see figure 1 for the geometry), themean coalescence time τ (z) satisfies the following equation(see the appendix for the derivation) [30](2)where r is the distance to the center of the immobile domain.
We assume a random initial condition, given byρd (r, t = 0) = 1.

(3)

The boundary condition applied at the encounter distance σ isgiven byρd (r = σ, t) = 0.

(4)

We should set another boundary condition such that thedensity at infinite separation is unity, i.e.,lim ρd (r, t) = 1.

r→∞

(7)

D2 ∂
∂
(1 − z2 ) τ (z) = −1.
∂z
R2 ∂z
The boundary conditions are∂τ (z)
τ (z = − cos α) = 0,
= 0,∂z z=1(5)

3.1. Three dimensions (3D)
Before investigating domain growth on the 2D sphericalsurface, we shall briefly present the known results for 3D
infinite systems and show that the corresponding results do
not hold for 2D cases.
For 3D infinite systems, the density profile which satisfies
both of the boundary conditions is ρ3 (r) = 1−(σ/r) [33]. The
mean coalescence rate is given by the inward flow of domains
across the surface at σdρ3 (r)2= 4π σ D3 .
(6)
k3 = 4π σ D3
dr
r=σ

(8)

(9)

where α is the angle between domain centers at collision(see figure 1). Notice that a simple geometric argument givessin(α/2) = σ/(2R) and cos α = 1−σ 2 /(2R2 ) [30]. Equations(8) and (9) can be easily solved, and we obtain
 2

R2
2R
τ (z) =
ln
(1
+
z)
.
(10)
D2
σ2
Averaging over the random initial distribution, we obtain [30]R1dz τ (z)τav = − cos α22 − σ /(2R2 )

 

R2
2
2R
=
ln
−1 .
(11)
D2 1 − (σ/2R)2σ

In a 3D fluid of viscosity ηs , the diffusion coefficient isinversely proportional to the size of the diffusing object, D3 =kB T/3π ηs σ . Using this Stokes–Einstein relation, the mean
coalescence rate can be expressed by k3 = 4kB T/3ηs . Noticethat σ/2 is the radius of the spherical domain [32].

The probability that the mobile domain has not reached theimmobile domain up to time t is given by exp (−t/τav ) when3

J. Phys.: Condens. Matter 25 (2013) 195105

K Seki et alunder the condition of νa  1. Strictly speaking, the SD result
was also obtained for 2D flat membranes. Recently, it was
shown that the SD result was applicable for spherical vesicles
when a < R < 1/ν [36].
Substituting equation (15) into equation (13), we obtain
k2 ≈

(12)
(13)

5. Growth kinetics of liquid domains

domains are initially distributed uniformly in the sphericalsurface. The same probability can be expressed using thebulk bimolecular rate as exp (−kd t/A), where the surface area
available for diffusion is A = 2π R2 [2−σ 2 /(2R2 )]. Comparing
these expressions, we find that the bulk bimolecular rate
can be calculated from the mean coalescence time by k2 =A/τav [30, 31, 34].
Using the bimolecular rate, the mean coalescence rate is
obtained as [30, 31, 34]
4π [1 − (a/R)2 ]D2 (a)2 ln(R/a) − 1 + (a/R)2
4π D2 (a)',2 ln(R/a) − 1(16)when a/R  1. The coalescence rate is independent of ain this limit. k2 t then approximately represents the area
explored by a diffusive object of radius σ during timet [34, 35]. Hence k2 t is given by D2 (a)t times the effective
collision cross-section. The size independence is the result of
the two opposing effects; with increasing domain size, thediffusion coefficient decreases while the effective collisioncross-section given by 4π/[2 ln(R/a) − 1] in equation (13)increases.
Strictly speaking, the coalescence rate depends on thedomain size if the coalescence occurs between domains ofdifferent sizes. However, such a size dependence is small
for 2D because of the weak logarithmic dependence inequations (13) and (15), and will be ignored hereafter. In fact,
the size dependence was not taken into account, even to study
the coalescence processes in 3D [32].
Equation (12) is valid even if the condition a/R  1 is notsatisfied. The full size dependence of the coalescence rate will
be studied by substituting equation (14) into equation (12) as
we shall discuss in section 6.

Figure 1. A schematic picture of the liquid domains embedded in a
fluid vesicle (left). The radius of the liquid domains is a while thevesicle radius is R. Liquid domains undergo Brownian motion on
the vesicle surface. Shown on the right is the geometricalparametrization. θ denotes the azimuthal angle and α is the angle
between domain centers at collision. σ represents the encounter
distance.

k2 =

kB T
,
η

where a  R is used to obtain the second equality. The
mutual diffusion coefficient is now expressed by D2 (a) since
it depends on the domain radius a as shown below.
Unfortunately, the full size dependence of the diffusion
coefficient is not known for spherical vesicles. However, ananalytical expression is known for a circular liquid domain
which has the same viscosity as the outside of the domain
for 2D flat membranes. For this case, the mutual diffusion
coefficient was obtained by De Koker as [3]Z
J 2 (z)2kB T ∞
D2 (a) =dz 2 1,(14)πη 0
z (z + νa)

Now we consider the formation of m-fold domains frominitial domains of the same size. When the coalescence rateis independent of the domain size as discussed above, thenumber density of an m-fold domain nm (t) at time t is given
by [32]

when the two circular domains have the same radius a. In
the above, J1 (z) is the Bessel function of the first kind, η isthe 2D membrane viscosity, and ν = 2ηs /η with ηs being theviscosity of the outer fluids. Here we have assumed that theviscosities of the liquid inside and outside the vesicle are thesame. The analytical expression after the integration can be
expressed using Meijer G-functions [5, 6]. For the case ofνa  1, the above expression reduces to [5, 6]
  

kB T
2
1
ln
−γ +
,
(15)
D2 (a) ≈πηνa4

n1 is the initial number density of the primary domains, andkd is the coalescence rate in d-dimensional space.
In general, if aggregates of large size are in the solidstate, there are many possible shapes for them. Shapecharacterization and size distribution have been studied in theconventional theory of Brownian coagulation. Here, we study
Brownian coalescence of liquid domains under the condition
that a circular shape is always maintained. For comparison,
we also consider Brownian coalescence of liquid domains
in 3D infinite systems, where a spherical shape is always
maintained. Under this assumption, the time evolution of thedomain size can be obtained analytically.
Assuming that the domain shape is immediately restored
after coalescence, the radius of the m-fold domain am

nm (t)(t/τ )m−1=,
n1(1 + t/τ )m

(17)

where
τ=

where γ = 0.5772 · · · is Euler’s constant. Equation (15)is slightly larger than the mutual diffusion coefficient ofSaffman–Delbrück (SD) theory derived for solid domains
4

1
,
kd n1

(18)

J. Phys.: Condens. Matter 25 (2013) 195105

K Seki et al

Figure 2. The average domain size ha(t)i/a1 as a function ofdimensionless time t/τ when domains are confined in the sphericalsurface. The solid line is obtained from equation (20) and the dashedline represents the asymptotic behavior given by equation (22).

Figure 3. k2 η/ (kB T) as a function of the dimensionless domain
size νa. The solid line is obtained from equation (12) using an
expression for the diffusion coefficient of a circular liquid domain
which has the same viscosity as the outside of the domain. Rν = 5,
10 and 50 from top to bottom, respectively. The dashed lines are
obtained by substituting equation (15) into equation (12). The dots
are obtained by substituting equation (24) into equation (12).

is determined by the area (d = 2) or volume (d = 3)
conservation relation,
adm = mad1 .

(19)

account explicitly the vesicle shape, the finite coalescence rate
is obtained from Smoluchowski theory and the results indicate
the induction period before asymptotic growth.

Averaging am over the distribution of m-fold aggregates (seeequation (17)), the mean domain radius ha(t)i at time t is
obtained as∞X
ha(t)iτt/τ
nm (t)=
m1/d= Li−1/d. (20)
a1
n1
t
1 + t/τ
m=1

6. Size dependence of the coalescence rate
In the previous section, growth kinetics for the circularliquid domain via diffusion coalescence are obtained when
the coalescence rate is independent of the size of the liquiddomain. The result for domain growth on the vesicle isobtained in the limit of νa  1. Below, we study the size
dependence of the coalescence rate using equations (12) and(14). The diffusion coefficient, equation (14), is obtained for acircular liquid domain of arbitrary size. In the limit of νa  1,the domain size dependence of the diffusion coefficient islogarithmic, as shown by equation (15). The coalescence rate
is given by kB T/η, independent of the domain size because of
the cancellation of the logarithmic domain size dependence
both in equations (12) and (15). For the opposite limit ofνa  1, the diffusion coefficient is obtained as [5, 6]

In the above, we have used the polylogarithm function definedby [37]
Lis (x) =

∞ k
X
x
k=1

ks

.

(21)

In the asymptotic limit of t/τ  1, equation (20) reduces
to either√  1/2
ha(t)i
π t
≈
(22)
a1
2 τ
for coalescence in 2D spherical surfaces, or   1/3
ha(t)i
4
t
≈0
a1
3
τ

(23)

D2 (a) ≈for coalescence in 3D infinite systems. Here 0(x) is
the gamma function. The time evolution of ha(t)i/a1 isshown in figure 2 for domain coalescence on the vesicle.
The asymptotic time dependence is well approximated byequation (22) when t/τ  1. We can see the induction
period of coalescence when t/τ < 1. The induction period
is characterized by the inverse of the apparent coalescence
rate given by the coalescence rate times the initial number
density for the primary domains, equation (18). Taking into

8kB T,
3π 2 ηs a(24)which is inversely proportional to the domain radius a.
The coalescence rate obtained by substituting equation (14) into equation (12) is indicated by the solid lines
in figure 3. For a relatively small vesicle, say νR = 5,
the coalescence rate is almost independent of the domain
size and is approximately given by kB T/η. The situation isvery different for large vesicles. When νR ≥ 10, the size
dependence of the coalescence rate can be ignored only when5

J. Phys.: Condens. Matter 25 (2013) 195105

K Seki et alνa < 0.1. In the same figure, we indicate the results obtainedby substituting equation (15) into equation (12) by dashedlines. As long as the domain size dependence of the diffusion
coefficient can be approximated by equation (15), the size
dependence of the coalescence rate is weak.
When νR = 5, the size dependence is weak, even beyond
the regime given by νa  1. However, when νR ≥ 10 andνa > 0.1 the coalescence rate decreases with increasing
domain size. In the figure, we indicate the results obtainedby substituting equation (24) into equation (12) by dots. The
coalescence rates with a full size dependence of the diffusion
coefficients (equation (14)) approach the results indicated bydots as the size increases. The dots rapidly decrease withincreasing domain size because of the power-law dependenceof the domain size in equation (24). When νR ≥ 10 andνa > 0.1, the coalescence rate decreases with increasing
domain size due to the strong size dependence of the diffusion
coefficient. If the coalescence rate decreases with increasing
size, the growth kinetics of the domain size should be
suppressed. The growth law given by equation (20) and theresult shown in figure 2 can be applicable when νR = 5, butshould be applicable only when νa  1 if the vesicle radius
is large, such as νR ≥ 10.
We also note that the coalescence rate increases withincreasing domain size if the domain size approaches close
to the vesicle radius. This dependence originates from thedenominator of equation (12) and can be interpreted as thefinite size effect of the domains confined in the sphericalsurface.
The diffusion coefficient, equation (14), shows alogarithmic size dependence (equation (15)) and a power-law
size dependence (equation (24)) as νa is varied. The
dimensionless quantity νa characterizes the coupling between
the embedded bulk fluid and the membrane for a given size.
The coupling is small when νa  1. The domain growthand its asymptotic limit given by equations (20) and (22),respectively, are obtained in the weak coupling limit. In thestrong coupling limit νa  1, domain growth is suppressed
as a result of the small coalescence rate compared to that in
the weak coupling limit, as shown in figure 3. The results are
consistent with the simulation results that the growth law issuppressed with increasing hydrodynamic coupling when thegoverning mechanism is DCC in 2D [7, 8].
The above conclusion is not altered if we use the diffusion
coefficient for the solid domains instead of equation (14).
Recently, the diffusion coefficient for the solid domain in the2D flat membrane has been approximated by a closed-form
empirical expression [10]. The interpolation formula forthe diffusion coefficient reproduces the logarithmic size
dependence obtained from SD theory when νa  1. Whenthe interpolation formula is introduced, the coalescence
rate equation (12) is slightly smaller but the overall size
dependence is similar to that obtained using equation (14).
The coalescence rate decreases with decreasing diffusion
coefficient for the solid domain compared to that for the liquiddomain of the same size. In general, the diffusion coefficientfor the solid domain is smaller than that for the liquid domainof the same size since the friction between the membrane andthe solid edge is larger than that between the membrane andthe liquid domain.
As shown above, the coalescence rate is constant intime when domains are confined on a spherical vesicle. The
finite coalescence rate constant can also be calculated byconsidering a circular flat sheet of radius L and setting theabsorbing boundary condition at r = a and the reflecting
boundary condition at r = L (a < L). When a circular domainof radius a is placed at the center of a circular flat sheet, thecoalescence rate is given by equation (13), by replacing R with
L [38]. Within this substitution, the domain size dependenceof the coalescence rate is the same as that in a sphericalvesicle. The diffusion coefficient of a domain placed at thecenter of a circular sheet of radius L was obtained, taking
into account the viscosity of a circular flat membrane [1].
However, the result was limited for a  L and obtained byignoring the hydrodynamic coupling between the membrane
and the solvent.

7. Discussion and conclusion
We discuss the relevant length scales in the recent experimenton domain growth kinetics by diffusion coalescence [14].
In the experiments, giant vesicles with a diameter of about
20 µm undergo phase separation at 30 ◦ C after a temperaturedrop from the one-phase region (42–43 ◦ C). Circular domains
were observed when the size exceeded an optical resolution of
the microscope (roughly 0.8 µm). In one of the coalescenceprocesses, the large domains of vesicle size appeared withinseveral minutes. Domain growth by collision and coalescence
was observed. In the other coalescence process, domain
growth was suppressed for several tens of minutes. The former
process can be theoretically studied by assuming DCC.
The value for ν = 2ηs /η can be estimated using typicalvalues of ηs = 10−3 Pa s, with η given by 0.1 Pa s times
the membrane thickness 5 nm as ν = 4.0 × 106 m−1 . We
estimate νR ≈ 40 by introducing a typical radius for thevesicles of 10 µm. When νR = 50, the coalescence rateis almost independent of the domain size as long as a <25 nm, obtained from the condition νa < 0.1. The domain
size is much smaller than the optical resolution of the
microscope, typically 0.8 µm. Therefore, when the domain
size grows and reaches the optical resolution, the coalescence
rate decreases with increasing domain size and domain growth
can be suppressed compared to that given by equation (22).
According to figure 3, the coalescence rate is almost constantover the wide range of the domain size when νR = 5. Thisvalue corresponds to a vesicle radius close to 1 µm, which
may be the maximum vesicle radius at which to observe thepower-law growth given by equation (22).
It should be remembered that the full size dependence ofthe diffusion coefficient for spherical vesicles is not known.
In this paper, the size dependence of the coalescence rate
has been discussed by substituting the known expression forthe diffusion coefficient for 2D flat membranes. The fullsize dependence of the diffusion coefficient for the sphericalvesicle is needed to further develop the coagulation theory forlarge domains within the current limit of optical resolutions.
6

J. Phys.: Condens. Matter 25 (2013) 195105

K Seki et alpair of domains at a relative position r at time t if their initial
relative position was ri and this was uniformly distributedZρd (r, t) = dri pd (r, t|ri , 0).
(A.1)

In original work by Yanagisawa et al, the best
fitted exponent 2/3 was obtained for power-law domain
growth [14]. However, the mechanism which leads tothis large exponent is not well understood. An attractive
interaction between domains seems to be present due
to hydrodynamic flow around the domains, which would
accelerate domain growth [39].
The power-law growth of circular domains can be
induced by transport of molecules from one domain toanother through the medium [16, 40–42]. The growth of large
domains is associated with evaporation of small domains,which is known as Ostwald ripening. According to the
Lifshitz–Slyozov–Wagner theory of Ostwald ripening, thepower-law exponent is 1/3 [40]. In this paper, we studied
domain growth using the DCC mechanism and did notconsider the evaporation and condensation mechanism in 2D.
We just remark that the exponent 1/3 is independent of thedimensionality and holds also in 2D [8, 41, 42].
In conclusion, we have investigated the DCC mechanism
for the growth kinetics of liquid domains on fluid vesicles.
Applying bimolecular reaction theory in the spherical surface
and using a 2D diffusion coefficient, the 2D coalescence rate
is found to be independent of the liquid domain size if it issmall enough. As a result, the domain size distribution is given
by classical Smoluchowski theory. When a circular shape isalways maintained, we have obtained the mean domain sizefor the whole time range (equation (20)) using the domain
size distribution and the area conservation relation. In theasymptotic long-time limit, we expect a power-law behavior
with the exponent 1/2 (equation (22)).
Domain growth kinetics have been derived under thecondition that the 2D coalescence rate is independent of thedomain size. This condition is investigated using a recently
obtained analytical expression for the diffusion coefficient ofarbitrary domain size. When the vesicle radius is small, thecoalescence rate can be approximated well as a constant over
the wide range of the domain size. When the vesicle radius
is large, the coalescence rate becomes independent of thedomain size only in the limit of small domain size. In general,the coalescence rate decreases with increasing domain size up
to a certain size where the finite size effect dominates. The
results are discussed in relation to the recent experimental
observations of DCC in vesicles.

We introduce the survival probability that the pair has notcoalesced up to time t if their initial relative position was rZ
wd (r, t) = drf pd (rf , t|r, 0).
(A.2)
Because pd (rf , 0|r, −t) satisfies the backward Kolmogorov
equation and hence pd (rf , t|r, 0) = pd (rf , 0|r, −t), we obtain [43–45]∂
wd (r, t) = Dd ∇ 2 wd (r, t),∂twith the initial condition
wd (r, t = 0) = 1,

(A.3)

(A.4)

and the boundary conditions
wd (r = σ, t) = 0,

lim wd (r, t) = 1.

r→∞

(A.5)

Since 1 − wd (r, t) is the probability that the pair coalesce
at time t, the mean coalescence time τ (r) is given by
Z ∞
Z ∞
∂
τ (r) =
dt t [1 − wd (r, t)] =dt wd (r, t). (A.6)∂t00
In general, the mean coalescence time is called the mean
first-passage time. Integrating equation (A.3) over time andusing equation (A.4), we find that the mean coalescence
time satisfies equation (8) when domains are confined in thespherical surface. Note that in equation (A.3), the diffusion
equation in the spherical surface can be written assuming
azimuthal symmetry such that


1
∂
∂w2
2
D2 ∇ w2 = D2 2sin θ∂θR sin θ ∂θ
=

D2 ∂
∂w2
(1 − z2 )
,
2
∂z
R ∂z

(A.7)

where z = cos θ .

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Acknowledgments
We would like to thank M Imai and M Yanagisawa
for valuable discussions. KS and SK are supported by aGrant-in-Aid for Scientific Research (grant No. 24540439)
from the MEXT of Japan. SK also acknowledges support
from the JSPS Core-to-Core Program ‘International research
network for non-equilibrium dynamics of soft matter’.

Appendix. The mean coalescence time
In this appendix, we briefly present the derivation ofequation (8) when the density satisfies equations (2)–(5). The
density can be expressed using the probability of finding a
7

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