SOFT MATERIALS
https://doi.org/10.1080/1539445X.2018.1462830

Dynamics of a bilayer membrane with membrane-solvent partial slip boundary
conditions
Kento Yasudaa, Ryuichi Okamotob, Shigeyuki Komuraa, and Jean-Baptiste FourniercaDepartment of Chemistry, Graduate School of Science, Tokyo Metropolitan University, Tokyo, Japan; bResearch Institute for Interdisciplinary
Science, Okayama University, Okayama, Japan; cLaboratoire Matiére et Systémes Complexes (MSC), Université Paris Diderot, Sorbonne Paris
Cité, Paris, France
ABSTRACTARTICLE HISTORY

We discuss the dynamics of a bilayer membrane with partial slip boundary conditions between
the monolayers and the bulk fluid. Using Onsager’s variational principle to account for the
associated dissipations, we derive the coupled dynamic equations for the membrane height
and the excess lipid density. The newly introduced friction coeﬃcients appear in the renormalized
fluid viscosities. For ordinary lipid bilayer membranes, we find that it is generally justified toignore the eﬀects of permeation and parallel slip at the membrane surface.

Received 11 February, 2018
Accepted 5 April, 2018

Much attention has been paid to artificial lipid bilayer
membranes as model systems of biological cell membranes (1). The fluidity of biomembranes is guaranteed
mainly by the lipid molecules, which are in the liquid
crystalline state at physiological temperatures.
Biomembranes exhibit a wide variety of complex phenomena, in both statics and dynamics, since lipid densities, membrane deformation and surrounding fluids
are coupled to each other. In early theoretical studies,
the relaxation rate of a lipid membrane was discussed
by regarding it as a tensionless elastic sheet undergoingout-of-plane fluctuations (2). The membrane relaxation
was then shown to be dominated by the bending rigidity and the viscosity of the surrounding bulk fluid.
Later on, Merkel et al. (3) and Seifert and Langer (4)considered both the inter-monolayer friction and the 2Dhydrodynamics of each monolayer. Importantly, theyobtained another relaxation mode which is associatedwith the density difference between the two monolayers
and is further coupled to the bending mode. Such arelaxation of the density fluctuation is dominated by theinter-monolayer friction. The existence of the predicted
compressional mode has been confirmed in the recent
experiment (5). After the work by Seifert and Langer, thebilayer nature of lipid membranes has been explicitly
considered for two-component bilayer membranes (6)and spherically closed bilayer vesicles (7–9). Morerecently, the present authors have discussed the dynamics
of a bilayer membrane coupled to a 2D cytoskeleton (10).

KEYWORDS

Dissipation; hydrodynamics;
membranes; relaxation rate

In all of these works, a partial slip boundary condition
between the two monolayers has been employed (4).
Conversely, the associated models have always assumed ano-slip boundary condition between the monolayer and theouter bulk fluid (solvent). Precisely speaking, this no-slip
boundary condition requires that the velocity of the bulk
fluid has the following properties at the monolayer surfaces:(i) the velocity component normal to the membrane coincides with the change rate of the out-of-plane membrane
displacement, so that the bulk fluid cannot permeate
through the membrane, and (ii) the lateral velocity component coincides with the fluid velocity of the monolayer.
In this paper, we discuss the dynamics of a bilayer
membrane with partial slip boundary conditions
between the monolayers and the bulk fluid, i.e., westudy the case when the above conditions (i) and (ii)
are violated. It is of importance to know which of theslipping modes (inter-monolayer or monolayer-solvent)dominates the membrane dynamics at large-wavenumber excitations. We use the framework of Onsager’s
variational principle (11) to obtain the governing hydrodynamic equations (12). In particular, the friction
between the monolayer and the bulk fluid is taken into
account through newly introduced dissipation functions.
In order to highlight the effects of the monolayer-solvent
partial slip boundary condition, we shall closely follow
the notations in Ref. (12).
As shown in Fig. 1, we describe the membrane shape bya height function hðr; tÞ, where the 2D vector r ¼ ðx; yÞ is

CONTACT Shigeyuki Komura
komura@tmu.ac.jp
Department of Chemistry, Graduate School of Science, Tokyo Metropolitan University, Tokyo 1920397, Japan.
Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lsfm.
© 2018 Taylor & Francis Group, LLC

2

K. YASUDA ET AL.

V+ (R)+v (r)ρ+ (r)eρ− (r)zyv− (r)V− (R)xh(r)Figure 1. Schematic picture of a laterally compressible bilayer
membrane. The deformation of the membrane is described by
the height variable hðrÞ and the excess lipid mass densities of
the upper and the lower monolayers by ρ ðrÞ. The lateralvelocities of the upper monolayer and the lower monolayer
are v ðr; tÞ. The velocities of the bulk fluid in the upper and thelower side of the membrane are V ðR; tÞ.

a generic point in a reference plane. We use ρþ ðr; tÞ andρ ðr; tÞ to describe relative excess mass densities of theupper and the lower monolayers, respectively. The two
lipid monolayers are regarded as 2D fluid sheets with ashear viscosity η2 and a dilatational viscosity λ2 . The lateralvelocity of each monolayer is denoted vi ðr; tÞ, withi 2 fx; yg. The outer bulk fluids are assumed identical on
both sides of the membrane and are characterized by thebulk fluid viscosity η, the pressure P ðR; tÞ, and thevelocity Vα ðR; tÞ, where α 2 fx; y; zg and R ¼ ðr; zÞ is a
3D vector.
The dynamical equations can be obtained without any
ambiguity from the extremalization of the Rayleighian(11) of the whole system, i.e., the bilayer membrane andthe bulk fluid. In general, the Rayleighian consists of adissipation function plus the time derivative of a freeenergy. The conservation laws and the boundary conditions are taken into account by introducing Lagrange
multipliers. Regarding the bulk solvent as an incompressible fluid, we require @α Vα ¼ 0, whereas the massconservation is expressed by ρ_  þ @i vi ¼ 0, where ρ_ 
indicates the time derivative of ρ . In this paper, we shalllater introduce new friction coefficients between themonolayers and the bulk fluid.
Concerning the boundary condition of the normalcomponent of the velocities, we require that the normal
velocities of the upper and lower monolayers coincide.
However, since we shall use partial slip boundary conditions between the monolayers and the bulk fluid, thevelocities of the membrane and the bulk fluid should be
different in general. Hence we only require the following boundary conditions regarding the bulk fluid
velocities:

Vzþ jz¼0 ¼ Vz jz¼0 ;(1)which ensures that no water is stocked by themembrane.
Next, we consider the dissipation functions of thesystem. The dissipation functions corresponding to thebulk fluid P b and the 2D fluid monolayers P s are
given byðP b ¼ d3 R ηD
(2)αβ Dαβ ;where Dαβ ¼ ð@α Vβ þ @β Vα Þ=2, andð 
λ2  
2
 
P s ¼ d r η2 dij dij þ dii djj ;
2

(3)


where dij ¼ ð@i v
j þ @j vi Þ=2. Furthermore, the dissipation due to the inter-monolayer friction is given byðb(4)
P i ¼ d2 r ðviþ  vi Þ2 ;2

in which the friction coefficient b between the twomonolayers appears.
In contrast to Ref. (12) we shall consider here the dissipation which occurs at the boundaries between the monolayers and the bulk fluid. The friction in the x; y-direction is
proportional to the relative velocity vi  Vi jz¼0 , whereasthat in the z-direction is determined by h_  Vzþ jz¼0 : Hencethe corresponding dissipation functions can be written as
ð
bk

(5)
P k ¼ d2 r ðvi  Vi jz¼0 Þ2 ;
2
ð
b?
(6)
P ? ¼ d2 r ðh_  Vzþ jz¼0 Þ2 ;2
where bk is the friction coefficient in the parallel direction, and b? is that in the normal direction. These are
the new dissipation functions that we consider in thispaper. Notice that the inverse of b? is known as themembrane permeation coefficient (13).
Next we briefly discuss the free energy of a laterally compressible bilayer membrane with finite thickness. The elasticity of a flexible membrane is
generally characterized by a surface tension σ and abending rigidity κ. Using these quantities, the freeenergy per unit area of a membrane is given by
2
2
1
1 2
2 σðÑhÞ þ 2 κc where c  Ñ h is twice the mean
curvature. We also consider the coupling between
the membrane curvature and the lipid density inthe monolayers (4). When the membrane has a positive mean curvature, the upper monolayer isstretched while the lower one is compressed. The
amount of stretching or compression is simply
given by ec, where e is the distance between the

SOFT MATERIALS

monolayer neutral surface and the membrane midsurface (see Figure 1). Introducing the monolayer
stretching coefficient k, one can write down the
total free energy as
ð h
σ
κ
2
H ¼ d2 r ðÑhÞ2 þ ðÑ2 hÞ
2
2

k þ
k 
2 2
2 2
þ ðρ þ eÑ hÞ þ ðρ  eÑ hÞ :(7)
2
2
The whole set of dynamical equations can beobtained by extremalizing the Rayleighian with respect
to all of the dynamical variables (11). The Rayleighianis given by the sum of all the dissipation functions andthe time derivative of the free energy, i.e., R ¼ P þ
b þ
þ

þ

_
P
þ
P
þ
P
þ
P
þ
P
þ
P
þ
P
þ
H:
The
i
?
b
s
s
jj
jj
incompressibility condition and the mass conservation
condition, as shown before, are taken into account by
the method of Lagrange multiplier. Extremalizing R_ ρ_  and thewith respect to the fields v , V  , h,Lagrange multiplier fields, we finally obtain the dynamical equations for a compressible bilayer membranewith partial slip boundary conditions between themonolayers and the bulk fluids.
The equations for the outer bulk fluid are the
Stokes equation and the incompressibility condition
which are given by  ηÑ2 Vα þ @α P ¼ 0 and@α Vα ¼ 0, respectively. Solving these equations withthe use of a 2D Fourier transform of the variables
such asðXðq; zÞ ¼ d2 r XðRÞeiqr ;(8)
we obtain the bulk fluid velocity asVz ðq; zÞ ¼ ðA þ B zÞeqz ;(9)Vk ðq; zÞ ¼ iðB =q  A  B zÞeqz ;(10)

where Vk ðq; zÞ is the component of the bulk fluid
velocity parallel to the direction of q, while A andB are coefficients yet to be determined. In the above,
we have assumed that Vα ! 0 for z ! 1. In addition, the pressure is given by P ðq; zÞ ¼ 2ηB eqz .
Notice that the component of the bulk fluid velocity
perpendicular to q, denoted by V? ðq; zÞ, is completely
decoupled from the other components.
The boundary conditions at the membrane, z ¼ 0,
are Eq. (1) and ηð@z Vi þ @i Vz Þ  bk ðvi  Vi Þ ¼ 0;

(11)

 2ηð@z Vzþ  @z Vz Þ þ Pþ  P  b? ðh_  Vzþ Þ ¼ 0; (12)3which are the force balance equations in the perpendicular and parallel directions to the membrane, respectively, in the presence of partial slip effects. Theseboundary conditions are now used to obtain theunknown coefficients as
b?
_
hðqÞ;
b? þ 4ηq

(13)

bk
b?
_
v ðqÞ:hðqÞ iqb? þ 4ηqbk þ 2ηq k

(14)

A ¼
B ¼ q

Note that vk ðqÞ in Eq. (14) is the component of themembrane velocity parallel to q.
The force balance equations for the membrane itself
are given by~κÑ4 h  σÑ2 h þ keÑ2 ðρþ  ρ Þ þ Pþ  2η@z Vzþ P þ 2η@z Vz ¼ 0;

(15)
where ~κ ¼ k þ 2ke and2 η2 Ñ2 vi  ðη2 þ λ2 Þ@i Ñ  v þ bðvi  vi Þþ k@i ðρ  eÑ2 hÞ  ηð@z Vi þ @i Vz Þ ¼ 0:

(16)

Using the solutions for the bulk fluid as calculated
above, we finally obtain the dynamical equations forthe membrane height h and the density difference ρ ¼ρþ  ρ as




_ tÞ
hðq; tÞ
hðq;;(17)¼ MðqÞ
ρðq; tÞ
_ tÞ
ρðq;
where
0
B
B
MðqÞ ¼ B@

σq þ ~κq3
4η? ðqÞ
keq4b þ ηk ðqÞq þ ηs q2

1
keq
C
4η? ðqÞ
C
C:
2
kq
A
2
2½b þ ηk ðqÞq þ ηs q (18)

In the above matrix, we have defined the effective surface viscosity asηs ¼ η2 þ

λ2
;
2

(19)

whereas the wavenumber dependent renormalized viscosities areη? ðqÞ ¼

ηb?
;b? þ 4ηqηk ðqÞ ¼

ηbk:bk þ 2ηq(20)(In Ref. (4), the density difference was defined asρ ¼ ðρþ  ρ Þ=2. This leads to a somewhat different
appearance of the corresponding relaxation matrix.) We
do not discuss here the dynamics of the density sum ρ ¼
ρþ þ ρ since it is completely decoupled from the othervariables (4). First we note that our result generalizes that of

K. YASUDA ET AL.

Seifert and Langer (4), because the bulk viscosity η is now
replaced either by η? ðqÞ or ηk ðqÞ. In other words, both η?
and ηk reduce to η when b?  4ηq and bk  2ηq, respectively, corresponding to the previous no-slip boundary
condition. Equations (17) and (18) are the main result of
this paper.
Next we discuss the wavenumber dependence of therelaxation rates, which are the two eigenvalues of thematrix in Eq. (18). First of all, one can demonstrate thatthe two eigenvalues are real and positive in all situations.
This property is ensured by the positivity of all thedissipation functions as well as the positivity of the staticparameters such as κ and k. In order to see clearly theeffects of the partial slip boundary conditions, we consider here the tensionless case and set σ ¼ 0. The asymptotic analysis of the relaxation matrix yields the following
wavenumber dependences of the two eigenvalues:8qq1 ;< ðk=2bÞq2 ;γ1  ð~κ=4ηÞq3 ; q1
(21)
q
q2 ;
:
q2
q;
ð~κ=b? Þq4 ;8qq1 ;< ðκ=4ηÞq3 ;γ2  ðk=2bÞðκ=~κÞq2 ; q1
q
q3 ;
(22)
:
ðk=2ηs Þðκ=~κÞ;q3
q:Here the three crossover wavenumbers are
pﬃﬃﬃﬃﬃﬃﬃﬃﬃ
q1 ¼ 2ηk=ðb~κÞ, q2 ¼ b? =ð4ηÞ, and q3 ¼ b=ηs .
Among these, q1 and q3 were discussed before (4), while
q2 introduces a new length scale.
The parameters of ordinary lipid membranes havethe following order of magnitudes: e  109 m, k 
0:1 J/m2 , κ  1019 J, b  109 J  s/m4 , ηs  109 J  s/
m2 , and η  103 J  s/m3 (pure water) (12). Note thatthese values are consistent with the relation b  ηs =e2 .
Likewise, it seems reasonable to estimate the friction
coefficient bk between the monolayer and the bulk
fluid by using the water viscosity η and the watermolecular size a  3 1010 m through bk  η=a 
106 J  s/m4 . As mentioned before, the friction coefficient b? is the inverse of the membrane permeation
coefficient; it is known to be b?  1012 J  s/m4 (13),which is far larger than bk . With these numericalvalues, the three crossover wavenumbers can beroughly estimated as q1  106 m1 , q2  1014 m1 ,and q3  109 m1 . Therefore, we recognize that q2 isbeyond the appropriate range for the present theory
and that the scaling behavior γ1 ,q4 for q  q2 should
not be observable.
As for the scale dependent viscosities in Eq. (20), we
expect essentially η?  η because b? is very large.
Besides, ηk is generally dependent on q and becomesηk  bk =ð2qÞ when q  q where q ¼ bk =ð2ηÞ  108m1 for the above parameter values. Hence, forqqq3 , the monolayer friction coefficient isrenormalized as b ! b þ bk . However, since bk =b 
103 for the above typical parameter values, the modification of b due to the partial slip boundary condition
may not be observable. We also note that q is alreadyclose to q3 . From these results, one can conclude theeffects of partial slip boundary conditions do not show
up in the relaxation dynamics of a compressible bilayer
membrane. In other words, is it justified to neglect both
permeation and parallel slip at the membrane surface
for ordinary lipid membranes.
Using the above parameter values, we numericallycalculate the two relaxation rates γ1 and γ2 as shownin Fig. 2 (solid lines). Although the partial slip boundary
conditions are included here, the result is essentially thesame as that by Seifert and Langer (4) because of theabove-mentioned reasons. The wavenumber dependencies of the relaxation rates are in accordance with theasymptotic behaviors in Eqs. (21) and (22), although we
do not see the dependence γ1 ,q4 for q  q2 since q2 is
too large. Just for comparison, we have also plotted inFig. 2 the relaxation rates for b?  106 J  s/m4 (dashedline) which corresponds to a very permeable membrane.
Here γ1 increases as γ1 ,q4 for q  108 m1 . In Fig. 3

10-4

2

10-6

γi/q2 [m2/s]

4

1
10-8

10-10

γ1
γ2

10-12
104

−2

1
106

108

1010

q [1/m]Figure 2. The relaxation rates γ1 and γ2 as a function of thewavenumber q. Both are normalized by q2 . The parameter values,
given in the text, are those of a tensionless ordinary lipid membrane. The solid and dashed lines correspond to b? ¼ 1012 J  s/m4
(ordinary membrane) and b? ¼ 106 J  s/m4 (very permeable
membrane), respectively. The different numbers indicate the slope
representing the exponent of the power-law behaviors.

SOFT MATERIALS

γi/q2 [m2/s]

10-6

1

10-8

γ1

10-10

γ2
10-12
104

1

−2
106

108

1010

q [1/m]Figure 3. The relaxation rates γ1 and γ2 as a function of thewavenumber q. The solid lines show the same data as in Fig. 2.
The dashed line corresponds to the same parameters except fora finite membrane tension σ ¼ 106 J/m2 . The latter affects
only the small q behaviors. Although the partial slip boundary
conditions have been taken into account they also make novisible difference here.

we show the effect of a finite membrane tension of therelaxation rates, as presented by the dashed line.
It is instructive to compare here the different dissipation mechanisms introduced in the Rayleighian.
The dissipation due to the inter-monolayer friction is
given by Eq. (4). In Fourier space, using the massconservation law, it can be rewritten as
ð 2
dq b
2
_
Pi ¼
jρðqÞj:(23)ð2πÞ2 2q2
Similarly, one can Fourier transform the dissipation P k[see Eq. (6)] due to the friction between the monolayer
and the bulk fluid:2
ð 2
d q bkηq2_
P¼
jρðqÞj:(24)k
ð2πÞ2 q2 bk þ 2ηq2_
in theLooking at the q-dependent coefficient of jρðqÞj
2
integrand, we see that it decays as q in Eq. (23),whereas in Eq. (24), it is independent of q whenqq , while it also decays as q2 for q  q . Hence
P i always dominates for q
q , whereas the sum b þ
bk contributes to the dissipation for q  q . This isconsistent with the previous argument on the renormalized friction coefficient, but we note again that bk istypically much smaller than b for ordinary membranes.
In Figs. 2 and 3, we have plotted up to q ¼ 1010 m1 .
Although directly detecting molecular scale dynamics

5

may not be so easy, we note that a long-wavelength
deformation can excite a collection of modes withmuch shorter wavelengths, e.g., when a bilayer membrane is coupled with a cytoskeleton (10). This isbecause the lattice structure of a cytoskeleton breaks
lateral continuous translational symmetry and couples
Fourier modes with different wave vectors.
Some years ago, Müller and Müller-Plathe investigated shear viscosity of a lipid bilayer system by using
reverse non-equilibrium molecular dynamics simulations (14). They showed that water molecules are lessmobile near the lipid head groups than in the bulk water,
and the local viscosity of water close to the head groupinterface is several times larger than the bulk waterviscosity. This means that the parallel friction coefficientbk  η=a can be even larger than our estimate (assuming a is the same) because we have used the bulk waterviscosity value for η. In our theory, the increase of bk
leads to the increase of the crossover wavelength q .
In conclusion, we have studied the effect on membrane dynamics of partial slip boundary conditions at
the monolayers-solvent interface. We found that a new
regime may appear in the spectrum of the relaxation
rates and that the new friction coefficients associatedwith the partial slip boundary conditions renormalize
the solvent viscosity. For ordinary lipid bilayer membranes, however, these effects should not be detectable
and it is plainly justified to ignore them. It is nonetheless possible that exotic membranes may someday
display the new regimes that we have calculated.

Funding
S.K. and R.O. acknowledge support by Grant-in-Aid for
Scientific Research on Innovative Areas “Fluctuation and
Structure” (grant no. 25103010) from the Ministry of
Education, Culture, Sports, Science, and Technology(MEXT) of Japan, and by Grant-in-Aid for Scientific
Research (C) (Grant No. 15K05250) from the Japan Society
for the Promotion of Science (JSPS).

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K. YASUDA ET AL.

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