Physica A 192 (1993) 27-46
North-Holland

Dynamical fluctuations of spherically closed fluid
membranes
Shigeyuki Komura 1
Department of Applied Physics, Faculty of Science, Tokyo Institute of Technology,
Meguro-ku, Tokyo I52, Japan
Kazuhiko Seki 2
FacultO des Sciences, UniversitO Libre de Bruxelles, Campus Plaine C.P. 231,
Boulevard du Triomphe, 1050 Brussels, Belgium
Received 31 March 1992

Properties of dynamical shape fluctuations of spherically closed fluid membranes such as
vesicles or microemulsion droplets are discussed. As a boundary condition at the interface, we
employ the generalized Laplace's formula obtained by Zhong-can and Helfrich. We calculate
the oscillation frequencies and the relaxation times of the membranes for a small deformation
under the constraint of either constant area or constant volume. Furthermore, the diffusion
coefficient of the droplet is estimated from the translational sideways mode. Our result does
not depend on the form of the shape energy, in agreement with the recent prediction by
Edwards and Schwartz.

I. Introduction
Properties of membranes with high flexibility are of current interest inconnection with statistical mechanics of fluctuating surfaces, biophysics ofmembranes and also high-energy physics [1,2]. Such membranes are typicallyrealized in microemulsion systems being homogeneous mixtures of oil, waterand surfactants [3]. Since the surface tension of the membranes is zero orpractically zero, the system can contain large internal interracial areas separating oil and water. Alternatively, the deformation of the surface is governed
mainly by the elastic bending energy [4]. Since the associated bending rigidity
is known to be the order of kBT, membranes fluctuate due to thermal~Address after 1 April 1992: Department of Physics, Faculty of Science, Kyoto University,
Kyoto 606, Japan.
2 Address after 1 January 1993: Department of Energy Sciences, Faculty of Science andEngineering, Tokyo Institute of Technology, Meguro-ku, Tokyo 152, Japan.
0378-4371/93/$06.00 (~) 1993- Elsevier Science Publishers B.V. All rights reserved

28

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

excitations and hence one should handle this problem from the point of view ofstatistical mechanics.
A similar system is also realized in spherically closed membranes of lipid
bilayers called "vesicles" which are typically exemplified by red blood cells.
This type of thin-walled fluid vesicles have received great attention as models
of cell membranes. Shape transformation among various conformations can becaused by changing, e.g., the osmotic conditions, the temperature or thecomposition of the lipids [5-7]. These aspects of vesicles might be closely
related to the physiological functions of biomembranes. Although vesicles
differ from microemulsion droplets by several decades in length scale [8], theruling physics behind them is expected to be the same.
There have been several dynamical measurements of microemulsion droplets(neutron scattering [9]) and vesicles (fluorescence microscopy [10], video
microscopy [11,12], reflection interference contrast microscopy [13]). Alongwith these experiments, some authors calculated the time correlation functionof the out-of-plane displacement for spherically closed fluid membranes.
Schneider, Jenkins and Webb were the first ones who obtained this quantity byusing the stationary solutions of the Stokes equations which describe thesurrounding incompressible fluids [10,14]. They required the so called "stick"boundary conditions and the balance of forces on the surface. Milner andSafran extended their results to the case of non-zero spontaneous curvature [8].
The important assumption in their calculations is that the total area is aconserved quantity as well as the total volume, both for vesicles and microemulsion droplets. In order to incorporate these two constraints simultaneously, they had to introduce the idea of "constant excess area" together withthe unknown Lagrange multiplier.
In fact, the appropriate constraint on these systems is still controversial. Van
der Linden, Bedeaux and Borkovec insisted that only the area should be kept
constant for vesicles, whereas only the volume constraint is necessary for
microemulsion droplets since the supply and the loss of surfactants from thebulk phase would take place in a time scale short enough compared to thedeformation of droplets [15]. Having focused on the microemulsion droplet
case, namely, on the constant volume case, they obtained a dynamical correlation function different from previously mentioned results. Their calculation canbe regarded as a generalization of the work by Mellema, Blom and Beekwilder
who considered only finite surface tension but took into account the different
viscosity inside and outside the droplet [16].
In this paper, we calculate the oscillation frequencies and the relaxation
times of a spherically closed fluid membrane whose equilibrium shape isdetermined by the minimization of the following shape energy [5,7,17,18]:

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

Hf= ½K~dA(Cl+C2-Co)2+Z~dA+~P f dV.

29

(1.1)

In the above, dA and dV are surface and volume elements, respectively, K thebending rigidity, c 1 and c 2 are two principal curvatures, c o the spontaneous
curvature, ~: the surface tension and Ap the (osmotic) pressure differenceAP--= e - P',(1.2)

measured between outside and inside. ( H e r e and below we shall use the prime
in order to distinguish the quantities of the fluid inside from the corresponding
quantities of the fluid outside.) The second and third terms either represent theconstraints of constant area and volume or actual work.
T h e first point we would like to stress in this paper is that our results holdregardless of whether area or volume is kept constant. This fact is based on thecalculation of shape energy for fluid vesicles by Zhong-can and Helfrich [18].
H e n c e the present results can be applied both to vesicles and microemulsion
droplets in the sense that one of the constraints is incorporated. We consider
that simultaneous inclusion of the both constraints needs separate considerations. Secondly, as conditions of balance of forces at the boundary, we employ
the generalized Laplace's formula given also by Zhong-can and Helfrich [18].
We believe that this choice is much simpler and straightforward as compared tothe previously used boundary conditions [8,10,14,15]. Our calculation differsfrom that by van der Linden et al. at this stage. Thirdly, we estimate thediffusion coefficient of such a deformable spherically closed membrane according to the translational sideways excitation mode. In contrast to the result byvan der Linden et al., our expression does not depend on the surface tension
nor on the bending rigidity. This is consistent with the recent papers on thestochastic dynamics of a deformable membrane by Edwards and Schwartz[19,20].
The outline of this article is as follows. First, after some mathematicaldefinitions, the generalized Laplace's formula is extended to the case where theviscous forces are also taken into account. In section 3, we list the expressionsof the variations of the shape energy (1.1) up the second order in terms of theout-of-plane displacement. In section 4, simple fluctuating hydrodynamics
describing the surrounding fluids are summarized. In section 5, the expressionof the oscillation frequencies in the absence of viscous dissipation are derived
for small deformations. Although our result is identical to that given by van der
Linden et al., we have derived it in a somewhat different manner. In section 6,
using the boundary conditions of section 1, we derive the formulae of theS. K o m u r a , K. Seki / Fluctuations o f spherically closed fluid membranes

30

relaxation times which are one of the main and new results in this paper.
Brownian motion of a droplet is discussed in the last section.

2. Shape equation and boundary condition
First, we will collect some formulas from differential geometry. One can, in
general, parameterize a 2-dimensional membrane in 3-dimensional space by
two real inner coordinates s = (s 1, s2). The shape of the membrane is thendescribed by a 3-dimensional vector r = r(s). At each point on the membrane,
there are two tangent vectors r / = 8 r / O s i with i = 1, 2. The outward unit normal
vector fi is perpendicular to these tangent vectors, i.e., fi = (r~ x r 2 ) / [ r 1 x rzl.
All properties related to the intrinsic geometry of the membrane areexpressed in terms of the metric tensor defined by(2.1)

gij = ri " ri •Two important quantities are the determinant and the inverse of the metric
which will be denoted byg = det(gq)andgq = ( g 6 )

-1

(2.2)

In addition, one has to consider the (extrinsic) curvature tensor given by
h~ = f i . r ~ = - f i i ' ~ ,(2.3)

with rij 02r/OsiOs j. Similar to (2.2), the determinant and the inverse of thecurvature tensor are denoted by=

h = det(h/i )andh ii =(hq) -1(2.4)

T h e mean curvature H and the Gaussian curvature K are calculated accordingto/ 4 = _ 1 ( ¢ , + ¢2) =½g"hij,

(2.5)

and
K=

ClC 2 = h / g ,respectively.

(2.6)

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

31

At zero temperature, the membrane is supposed to be in the (undeformed)
reference state described by r = R . Any deformed state of the membranewithout any overhangs can then be parameterized by using the normal vector
in the reference state, i.e., lq = (R 1 × R 2 ) / I R ~ × R21, in the following way:r = R + l(s ], s 2, t) 1~.

(2.7)

Here the variable l(s 1, s 2, t) represents the transverse (out-of-plane) displacement field which can generally depend on time t. (Here we did not include thein-plane displacements which are irrelevant in the bending energy up to thesecond order in terms of displacement fields. In-plane displacements should betaken into account in describing polymerized elastic membranes [21].)
By requiring that the first variation of the shape energy (1.1) vanishes forany infinitesimal displacement 1, Zhong-can and Helfrich obtained the mechanical equilibrium condition of the membrane vesicle such that [18]A P - - 2 ~ H + K(2H + Co)(2H 2 - 2 K - Coil ) + 2KV2BH= 0 ,(2.8)

where V2~ is the Laplace-Beltrami operator on the surface defined by
VZLB = V 1~ Oi(g ij~

0/).

(2.9)

(2.8) represents the balance of normal forces per unit membrane area andreduces to the well-known Laplace's formula when K = 0. (Recently a moregeneral case has been obtained by Onuki and Kawasaki [22].)
In order to construct the boundary condition that must be satisfied at theinterface between two viscous fluids in motion, it is natural and straightforward
to extend (2.8) in accordance with Landau and Lifshitz [23]. Hence in thepresence of both surface tension and bending rigidity, the equality of the forceson the surface of each fluid can be written as(P - P ' ) n , - (%8 - °",t3)n~+ [ - 2 ~ H + K(2H + Co)(2H 2 - 2 K - Coil ) + 2KVZsH]n,~ = 0,(2.10)
where n~ are components of the normal vector and o-~t~ is the viscous stresstensor given in section 4. (We shall use Greek indices for the range 1, 2 and 3.)(2.10) will be used in sections 5 and 6.

S. Komura, K. Seki / Fluctuations o f spherically closed fluid membranes

32

3. S p h e r i c a l l yclosed fluid membranes

Now, consider a spherically closed fluid membrane of radius r o with internal
coordinates (s 1, s 2) = (0, ok). As a local basis of 3-dimensional space, we
introduce three unit vectors:(s,nOcos:)er =sin 0 sin
cos0,'cosO :),eo = ~ cos 0 sin\ -sin0,~+ =cos ~b ] .
0 /

(3.1)

With these notations, the reference state is described by(3.2)

R = roe r ,and a slightly deformed sphere can be represented by (see (2.7))
r = R + l(0, dp, t) N = [r o + l(0, oh, t)]~ r .

(3.3)

The undeformed reference sphere (3.2) is always a solution of the equilibrium
shape condition (2.8) if the next relation holds among the parameters [18]:A P r 3 + 2 ~ r ~ + KCoro(cor o - 2) = 0 .

(3.4)

This relation is called the "capillarity condition" [15].
When a spherical membrane deforms slightly without any overhangs according to (3.3), a straightforward calculation up to the first order in terms of theout-of-plane displacement l yields the following expression for the normal
vector:10l

fi~er-----e0
r o 0013l

r 0sin0 0~b

~,

(3.5)

and for the mean curvature H and the Gaussian curvature K:
2 H ~ _ _2 + - 51 ( 2 + V ± 2 ) l ( 0 , ~b, t)
ro

(3.6)

r0

and
K ~- 12ro

13 (2 + V])/(0, ¢, t)r0

respectively, where(3.7)

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

V2 -

33

1
O (/ s i n 0 0 \ )1
02
sin 0 00
0-0 + sin2~ 04) 2"(3.8)

It is convenient to expand the function l(O, 4), t) in terms of the sphericalharmonics Y,m(O, oh),l(O, d), t)= ~ l,m(t ) Y,m(O, 4>).

(3.9)

n,rtl

As usual, we have l,*m(t) = (--1)mQ_m(t) in order to ensure that the displacement field is real (the asterisk denotes the complex conjugate value) and thesummation is over n = 0, 1, 2 , . . . and Iml ~< n. Hereafter, the well-known
relation(3.10)vZ Y, m(O, 4>) = - n ( n + 1)Y~m(O, 4))will be used frequently.
So far, the variations of the bending energy Hb, the area A and the volume
V for spherical parameterization have been calculated by several authors[8,10,18,24,25]. Up to the second order in terms of l,m, they can be summarized as

8Hb -~ ~KCoro(Coro -- 2) l°-~°r o+ ½K ~'~ {[n(n + 1)12 - (2 + 2Coro - ½Coro)n(n2 22 2[/n,~[+ 1) + Coro}
- 2,
n,m
r0
(3.11)
8 ~ d A = 2X/-4--~ro/oo + ~ [1 + ln(n + 1)]]l.m] 2 ,(3.12)

n,m

and

8f dV~- X/-4--~roloo
2 + ro ~, II.ml 2 ,(3.13)

n.m

respectively. U p o n adding all these expressions in accordance with the shapeenergy (1.1), the total second variation turns out to be

8Hf--SHb +.~ 8 ~ d A + b P S f dV
_ 1 ~ (n - 1)(n + 2){K[n(n + 1) -- Cor0] -- ~Ap1 ro}3 II, 2[ 2,,,mro
= ½ Z ( n - 1)(n + 2){•ro 2 + K[n(n + 1 ) - 2 C o r o + ½c2r2]}
n,m

r 0

(3.14)

S. Komura, K. Seki / Fluctuations o f spherically closed fluid membranes

34

Here the capillarity condition (3.4) has been used to obtain the last identityand all the first order terms have cancelled each other as they should. Since(3.14) depends only on n but not on m, the shape energy has (2n + 1)-folddegeneracy. Notice that we have not incorporated any constraints at this stage.
Next we consider the case where either the area or the volume is kept
constant. The constraint of constant volume can be easily incorporated byusing the expression (3.13) for the volume. It then follows from 8V = 0 thatV~

100 ~ - ~] II"ml2n,m(3.15)

r0

Hence the constant volume constraint leads to the elimination of the/00-terms.
Inserting this into (3.11) and (3.12) and forming the variation~Hr = ~H b + Z ~ d A ,(3.16)

we obtain

1 2 21a(rt -- 1)(n + 2){•ro z + K[n(n + 1) - 2c0r o + ~Cor0l~~Hf ~ ½n,mr0

(3.17)
where the only change compared to (3.14) is the prime on the summation
indicating that (n, m) = (0, 0) is excluded. This expression is identical to thatby van der Linden et al. [15] (our spontaneous curvature differs in sign).
Likewise, the constraint of constant area can be incorporated by requiring
6A = 0, namely,

V~-~ roloo ~- -½ E [1 + ½n(n + 1)]ll.ml

2 .

(3.18)

n,m

By eliminating the 100-terms, one finds~Hf = ~H b + AP ~ f dV

(3.19)

is identical to (3.17) again. Therefore (3.17) holds generally regardless ofwhether volume or area is kept constant. In the following sections, we shallconsider the case (3.17) where one of the constraints is included. Although we
use the word "microemulsion droplets" in the following, the readers areadvised to replace it also by "vesicles".

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

35

4. Hydrodynamic equations
In describing the motion of the surrounding fluids, we assumed that both theinside and the outside fluids are incompressible. The basic hydrodynamical
equations of mass conservation (continuity equation) and momentum conservation (Navier-Stokes equation) are
div v = 0 ,

(0o

(4.1)

)

P -~- + (v. grad)v -- - g r a d P + rlVZe,(4.2)

respectively. Here p denotes the density of the fluid, v the velocity, rI thedynamic viscosity. In (4.2), we have used the fact that the viscous stress tensor
for an incompressible fluid is{ Ov,~ Ovt3 ~°"~t~ = ~7~,Oxt3 + Ox~,/(4.3)(4.3) will be used later to apply the boundary condition (2.10).
When we are interested only in the fluctuating properties, the above
equations can be linearized by putting
P = Po + ~P,

P = Po + ~ P ,v = ~v,(4.4)

where Po, Po stand for the values in equilibrium and ~p, ~P, 8v are theirvariations. Retaining in (4.2) only up to the first order terms, we obtainau

Po ~ - = - g r a d ~P + ~TVzv.

(4.5)

In the above equations, it is convenient to decompose the velocity field into
irrotational (longitudinal) and rotational (transverse) parts according to
V =

V~ +

(4.6)

Vt ,

div Ut = O,

rot v e = O.

(4.7)

Obviously, only the longitudinal part remains in (4.1);
div o e = 0 ,while (4.5) splits into two equations:(4.8)

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

36

Ovt

'0 V2vt,

3t

P0

(4.9)

Ov e

Po Ot -

gradgP.

(4.10)

Hereafter we will be concerned only with the longitudinal part of the velocity
o e which will be denoted simply by v.
Like any vector field having zero rotation, it is advantageous to introduce thevelocity potential 6 which provides the velocity o throughv = grad 6 -(4.11)

Substituting (4.11) into (4.8) and (4.10) yields

V21//= 0,
8P=-Po(4.12)
06
Ot'(4.13)

respectively, where the constant term has been neglected in (4.13). Upon
neglecting the inertial term in (4.2), the incompressible condition (4.1) implies
that the pressure field also satisfies Laplace's equation
V 2 8P = O.

(4.14)

5. Undamped oscillations

In this section, we determine the characteristic oscillation frequency of aspherically closed membrane in the absence of any viscous damping effect. The
system is regarded as an infinite set of uncoupled harmonic oscillators; each
oscillator identified by the set of (n, m). Then the equation of motion of theout-of-plane displacement lnm(t ) becomes [15]=2--09 nmlnm(t ) ,(5.1)

where the dot indicates the derivative with respect to time. Our purpose is toexpress the oscillation frequency OOnmin terms of surface tension and bending
rigidity. In view of (5.1), we write lnm(t) = lnm e i%mt in (3.9), yieldingl(O, eh, t) = 2 lnmYnm(O, qb) e i~°m' .
n,m

(5.2)

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

37(As mentioned in section 3, we exclude (n, m) = (0, 0) due to the presence ofthe constraint.)
Since both inside and outside fluids are assumed to be incompressible, therespective velocity potentials satisfy Laplace's equation (4.12). Hence, as given
by Lamb [15,26], they can be expanded in terms of the solid sphericalharmonics:
tp'(r, t) =~dnm

Ynm(O, ~t~)e it°hint ,(5.3)~nm(rO) n+l Ynm(O, ~)) e it°nmt .

(5.4)

n,m

~O(r,,) = ~
n,m

\ r /

In the absence of a dissipation mechanism due to the viscous damping effect,
the so called "slip" boundary condition is employed here [15]. This condition
requires the continuity of the radial component of the velocity, i.e., v r =O~/Or = Ol/Ot at the boundary. Since the velocity field is linear in the displacement amplitude l, one can impose the boundary condition at r = r 0. Accordingly, the first boundary condition follows from (5.3) and (5.4) asor(r = r° ) = __1 ~ Ht~tlnmYnrn(O, ~)) eiOJnmt
r 0

_

n,m

1 ~ (n + 1)~nmYnm(O , tip) e i°~mt
r 0 n,m= ~i°)nmlnmYnm(O, 6 ) e i % m t

n,m

,

(5.5)

where we took the time derivative of (5.2) to obtain the last equation. By
comparing the coefficients of the spherical harmonics, one findsn~rnm = - - ( n + 1)t~n m = irotOnmlnm .

(5.6)

The second boundary conditions stems from the balance of forces on thesurface. In the absence of viscous forces, the boundary condition (2.10)
reduces to the generalized Laplace's formula (2.8). By noticing that the
Laplace-Beltrami operator is now the usual Laplacian operator on the sphereV~s - V~
2,
r0

(5.7)

and substituting (3.6) and (3.7) into (2.8), the force balance for the radial part

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

38

up to the first o r d e r in terms o f l(0, 4~, t) turns out to be~p_~p,__l [ x r 2 _ K ( v Z+2coro _ ~Coro)](2122+ VZ)l(O, qa, t ) = O ,

(5.8)

ro

at r = r o. In the a b o v e , we used the fact that n r ~ 1 and all the constant terms
h a v e cancelled each o t h e r because of the capillarity condition (3.4). If wereplace ~P and ~ P ' with the use of (4.13), one obtains

Po -~- - P ;+ -1- [~r2° _ K(V2 + 2Coro - ~Cor0)](2+1 2 272)1(0, qS, t) = 0 ,
F0

(5.9)
at r = r o. By substituting ( 5 . 2 ) - ( 5 . 4 ) into (5.9) and c o m p a r i n g the coefficientso f the spherical harmonics as before, we have•¢t

iPo O0,mOn,~ -- lP0%m 0,m1= ~ (n-1

2 21)(n + 2 ) { X r ~ + K[n(n + 1 ) - 2 C o r o + 5Corol}l.m .

(5.10)

FO(5.6) and (5.10) are sufficient to obtain the oscillation f r e q u e n c y O.)nmwhichis2 -O)nm

12
21a
1 5 (n - 1)(n + 2 ) { ~ r ~ + K[n(n + 1) - 2Cor 0 + ~CoroD,(5.11)

P, ro

with
!

= P_20+ PoP"nn +-----]-"(5.12)

W h e n K = 0, (5.11) reduces to the old expression first given by R a y l e i g h a longtime ago [27]. (5.11) is also identical to the result by van der L i n d e n , B e d e a u x
and B o r k o v e c [15].
Since the expression (5.11) vanishes for n = 1, the smallest possible freq u e n c y o f oscillations c o r r e s p o n d s to n = 2, and is=¢'Omin4. 2z[ X r 2 + K(6--2COr o + 1 C0ro)](O~/2 + po/3)r~(5.13)

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

39

Furthermore, the total variation of the shape energy (3.17) can be simply
expressed in terms of OJ,m ast~nf = 1 ~n,m3 z ,tnm ,z •Dnro°Jnm(5.14)

With the use of equipartition theorem (or straightforward Gaussian integrations), the fluctuation amplitudes are easily estimated:(it°ml 2) -kBT
3 2PnrOO)nm__

2 2( n - 1)(n + 2){Zr 2 + K[n(n + 1 / - 2 c o r o + Coro/2]}(5.15)

where k B is the Boltzmann constant and T is the temperature. It is important torealize that (5.15) is valid only for n/> 2, since n = 1 corresponds to the simple
translational sideways displacement of the droplet as a whole requiring noenergy. This mode is essentially related to the Brownian motion of the droplet
and will be discussed in detail in the last section.

6. Overdamped oscillations
In the previous section, the frictional forces due to the viscous damping
effect have been neglected. The inclusion of viscous forces introduces thedamping term in the equation of motion (5.1) in such a way that [15]~mn(t) = __Tnmlnm(t)-2tOnmlnm(l).

(6.1)

In this section, we consider the case where the motion of the membranesurface is overdamped. Then the restoring forces of the membrane balancewith the viscous resistance forces due to the surrounding fluid after a shortinitial period of motion. In such a case, (6.1) reduces to]nm(t)_

1rnm

l.m(t) '

(6.2)

where
1

2
£Onm

,l.nm

)tnm

(6.3)

40

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

From (6.2), the time dependence of l,m(t ) is now given by l,m(t ) = l,m e -'/'°'.
This leads to the expansion of l(O, 49, t) in the form of(6.4)

l(O} 49, t) = ~ lnmYnm(O , ~)) e -'/*'m .
tl,mBy using the results of the fluctuation amplitudes (5.15), the time correlation
function is simply provided by(lnrn(t) lnm(O)* ) --k B3 Z 2 e --t/.rnm
pn?'OOJ nm

k Bzr~ e t/,,,.
2 2( n - 1)(n + 2){2r 2 + K[n(n + 1 ) - 2cor o + Coro/2]}(6.5)
Upon taking the viscous effect into account, the motion of the surrounding
fluid is described by the "creeping flow" neglecting the inertial term at low
Reynolds number (Stokes equation),r/V2v = grad P ,(6.6)

together with the incompressibility condition
div v = 0.

(6.7)

As has been already discussed by several authors [8,10,15], we are interested in
the irrotational solutions, which simplifies the problem to some extent. The
solution of (6.6) and (6.7) in spherical coordinates can be constructed by thetwo scalar functions describing inside the fluid, q/(r, t), P'(r, t), and by thoseoutside the fluid, ~b(r, t), P(r, t). Since all of these functions satisfy Laplace's
equation (see (4.12) and (4.14)), they can be expanded in terms of solid
spherical harmonics as before:
tp'(r, t) = ,~m q &, ' ~/ ) r ~" Y"m(O' 49) e_,/,,,m ,

(6.8)

°

P'(r, t) =

r
P,,~

r

Y,m(O, 49) e_t/~, m ,(6.9)

Y.m(O, 49) e -'/~"r" ,(6.1o)

and~b(r, t) = .,,. ~b.,. r41

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranesr0) n+lP(r, t) = • P.., rn,mY.m(O, 6) e -'/""(6.11)(For the pressure fields P' and P, we include (n, m) = (0, 0) terms because of
the presence of the non-fluctuating hydrostatic pressure.)
According to the general stationary solution of the Stokes equation given by
Lamb [26], the velocity field inside of the droplet is written as [28]~(vt(r, t) = n,m_

Otnm grad +n+3,22r/'(n + 1)(2n + 3) P~mr grad

n

,

r

~'(n + 1)(2n + 3) e.mr

L~(O, 4))e-'/"~,(6.12)

whereas the corresponding solution for the outside is provided byo ( r , t) = ~t/,m(0nm grad
n + 1

n-2

2

2r/n(2n - 1) Pnmr grad
\ [ r o \"+1

q- T/n(-2n -~ 1) e n m r ) ~ r )Ynm(O'q~) e-t/'~"m"(6.13)In (6.12) and (6.13), the gradient operators also act on the solid sphericalharmonics outside the square bracket. For the purpose of writing down theboundary conditions explicitly, we should keep in mind that the gradient
operator in spherical coordinates takes the form
O
1 0
1
0
grad = er ~rr + e0 -r ~ + e6 r sin 0 O~b(6.14)

In contrast to the previous section, the so called "stick" boundary condition
is used here as in van der Linden et al. [15]. This condition requires that boththe velocity of the membrane and the velocity of the fluid on each side of themembrane are equal to Or/Ot (see (3.3)). The first two boundary conditions
come from the continuity of the velocity field. In the ~r-direction, this is
written as~ ( n6nm, + 2r/'(2nn + 3) r°Pnm
2 , )ynm(O, qb) e-t/~,,,
vr(r = r° ) = lro n,m_1 ~(r0 ,.m (n + 1)~0,,,

=~

n+l
2)
e--t/rn,n2r/~n Z 1) r°Pnm Ynm(O' 6)--1 lnmYnm(O' q~)e_,,¢,,, 'n,m "l"nm(6.15)

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

42

where the last equation has been obtained by taking the time derivative of(6.4). Likewise, the continuity of the velocity in the ~0-direction yields
n+32 t )OYnm
V o ( r = r o ) = 1- ~ ( ~'nm + 2~/'(n + l)(2n + 3) r°Pnm - - ~ e-t/~'.mr 0 n,mr 0 n,mmn- 22 ) 3 Ynm --t/~.m
2r~n-(-2-nZ 1) r°P"m ~e(6.16)

The continuity condition in the ~6-direction results in the same condition as(6.16). From (6.15) and (6.16), we obtain the following relations among
coefficients of the spherical harmonics:
n+l2, + 2r/'(2nn + 3) r°P"'~
2 , = - ( n + 1)~,, m + 2T/~n ----1) r°Pnm
n6"m
r° ln~%m-(6.17)

and
,

n+3

2

,

n-2

2

(6.18)~ltnm -~- 2~7'(n + 1)(2n + 3) r°Pnm = &n,n 2~Tn(2n- 1) r°Pnm 'a t r -- r 0.

Additional sets of boundary conditions follow from the balance of forces on
the membrane. Up to the first order in terms of the amplitudes I, the force
balance in the ~r-direction is given by( S p _ 8 p , ) _ (2r/ OG0v'~
0 7 - 2rf O r /1 [,Vr 2 - K ( V 2 + 2 c 0 r ° - ~Cor0)l(2, 22+ V2)/(0, 6, t) = 0 ,(6.19)

r0

at r = r 0. Like in eq. (5.8), the fact that nr ~ 1 and the capillarity condition(3.4) has been used. In fact, (6.19) differs from (5.8) only due to the viscousterms. Likewise, the force balance in the ~0-direction yields(Po-/'~)no+(2 Co

n r O0 + Or)y ( 2 - coro) n o = 0 ,

-~7

r-~

+ Or
(6.20)

roat r = r o. Notice that n o ~ -(1/ro)(Ol/O0) which is linear in I. Again the terms

S. Komura, K. Seki / Fluctuations of spherically closed fluid membranes

43

proportional to n o cancel each other because of the capillarity condition (3.4).
After substituting the expressions for the velocity and the pressure fields into
(6.19) and (6.20), the relations between the coefficients for n/> 1 are given by

P',,-

--

2rfn(n-1)'
6~+n(n + 1) , )2 n + 3 P,mPnm -~ (2"q(n + 1)(n + 2) qj"'~2ron(n + 1) P,m)2n - 11 (n - 1)(n + 2){.Vro2 + K[n(n + 1) - 2Cor0 + ~1c2r21~= --~o o,, l ,m,(6.2l)r0andt2r/'(n - 1) ~b2 +n ( n + 2)p,ro(n + 1)(2n + 3)nm

+ 2r/(n + 2) ff~''
r2(n - 1)(n + 1)
n ( 2 n - 1)
Pnm = 0.

(6.22)

Combining (6.17), (6.18), (6.21) and (6.22), one can express q/rim, P'm, qtnm
and P,m in terms of l,m. After some calculations, the relaxation time Tnm in(6.2) turns out to be1"l'nm(n - 1)n(n + 1)(n + 2)(2n + 1)(E + 1)
r/r 3o [(2n 2 + 4 n + 3 ) E + 2 n ( n + 2 ) l [ 2 ( n2-1)E+2n 2+1]
~Tr
1

1 2 2

× {2fr 2 + K[n(n + 1 ) - 2Cor0 + ~c0r0]},(6.23)

with E---r/'hT. For K = 0, the result (6.23) reduces to that given by Mellma,
Blom and Beekwilder [16]. This relaxation time is quite similar to thatobtained by Lisy [29], at least concerning the dependence on n, but it still
differs in the prefactor.
Since OOnm has already been obtained in the previous section, one canimmediately obtain the frictional coefficient %,, via (6.3) as2~ n m = "i'nm tO n mr/[ ( 2 n 2 + 4 n + 3 ) E + 2n(n + 2 ) ] [ 2 ( n 2 - 1 ) E + 2 n 2 + 11

2
Pn r0

n ( n + 1)(2n + 1)(E + 1)

(6.24)

It is important to note that our result for the frictional coefficient depends only
on the properties of the surrounding fluids but not on the membrane prop-

S. Komura, K. Seki / Fluctuations o f spherically closed fluid membranes

44

erties, such as the surface tension or the bending rigidity. This point is in sharp
contrast to the result by van der Linden et al. [15].

7. Diffusion coefficient

In this section, we estimate the diffusion coefficient of a deformable droplet.
The simplest way is to use the expression of the frictional coefficient withrespect to the translational motion identified with the n = 1 mode. Since (6.24)
corresponds to the frictional coefficient per unit mass, the total frictionalcoefficient F can be effectively obtained by [15]3E+2F = ( 4 3"rrPn=lr°)Y"=l'm3= 2"rr~?r° E + 1(7.1)

H e n c e the diffusion coefficient D can be calculated through Einstein's relation
as

kBT
D-

F

kBT

rl + 7l'- 2rrr/r 0 277 + 3 r / ' '(7.2)

which coincides with the old result by Hadamard [30] and Rybczynski [31],
who did not take into account the deformation of the droplet. When r / ' = r/,(7.2) reduces to

D-

kBT

(7.3)

57r.0r° •As discussed by Landau and Lifshitz [23], in the case of r/'--~ ~ (corresponding
to a solid sphere), (7.2) becomes the well-known Stokes formula

D-

kBT

(7.4)

6~.qr ° "In the opposite limit T/'---~0 (corresponding to a gas bubble), (7.2) reduces to

D-

kBT

(7.5)

4~rr/r0
In view of (7.2), one can conclude that the form of the shape energy isirrelevant to the diffusion coefficient as far as the deformation of the droplet issmall enough. A more general case has been recently discussed by Edwards
and Schwartz in their theory of stochastic dynamics of a deformable membrane

s. Komura, K. Seki / Fluctuations of spherically closedfluid membranes

45[19,20]. In fact, (7.2) can be also derived by following their arguments. In theabsence of thermal agitation, the equation for [nm should be of the following
form:

o nf
inm=-Kn

aln_m= -K,(n-I 2 2 , .)l n =2 .
1)(n + 2 ) { ~ r ~ + K[n(n + 1 ) - 2Cor o + ~Corol
ro

(7.6)

Combining this result with our result of Tnm (see (6.23)), one concludes that1n(n + 1)(2n + 1)(E + 1)
K, = -nro [(2n 2 + 4n + 3 ) E + 2n(n + 2)][2(n 2 - 1)E + 2n 2 + 1]

(7.7)

When E = 1 (~7' = ~7), (7.7) reduces toK, = 2n(n + 1)
r/r o (2n - 1)(2n + 1)(2n + 3) '(7.8)

as given by Edwards and Schwartz [20] who considered only the surface tension
in the shape energy. One of their important results is that the diffusion
coefficient of the deformable droplet is identified with the n = 1 mode as
D=kBT(

3

K,=1).

(7.9)

Substituting (7.7) into (7.9) yields (7.2).

Acknowledgements
We would like to thank Prof. T. Izuyama and Prof. K. Kitahara for theirinterest and useful comments. We are also grateful to Miss. A. Tezuka for
critical reading of the manuscript. One of us (K.S.) wishes to thank Prof. R.
Kutner for his kind hospitality at University of Warsaw.

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46

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