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PAPER www.rsc.org/softmatter | Soft Matter
Corrections to the Laplace law for vesicle aspiration in micropipettes and
other confined geometries
J.-B. Fournier and P. Galatola*
Received 18th April 2008, Accepted 26th June 2008
First published as an Advance Article on the web 1st September 2008
DOI: 10.1039/b806589f
Based on the stress tensor associated with the Helfrich Hamiltonian, we study how the Laplace law is
modified for fluid lipid membranes under small tensions or constrained within small scale devices. We
derive several exact analytical results corresponding to global or local generalized Laplace relations.
Analytical and numerical results based on standard energy minimization confirm these exact relations
and further quantify the deviations of the standard Laplace law.
PACS numbers: 87.16.D-, 68.03.Cd, 68.35.Np
1 Introduction
Nowadays, nanopipettes with aperture sizes of less than 200 nm
are commonly used 1,2 and the development of microfludics
allows on-chip fabrication and focusing of vesicles with diameters
in the 100 nm range. 3 At the same time, the technology of
solid-state nanopores is developing fast, allowing pores with
diameters ranging from a few nanometres to tens of nanometres
to be made. 4 There is also an increasing interest in the use of
individual artificial vesicles as containers suited for the isolation,
preservation and transport of small quantities of drugs or biomolecules.
5,6 It has therefore become important to understand
how aspiration pressure, membrane tension and curvature effects
are related at the nanoscale, i.e., how is the Laplace law modified.
The case of an ordinary interface between two fluids is
well-known. Calling s its surface tension and dP the pressure
difference between two phases I and II, the Laplace law gives the
local mechanical equilibrium condition,
dP ¼ s(c1 + c2) (1)
in which c1 and c2 are the interface’s principal curvatures. 7 The
latter are the minimum and maximum curvatures of the curves
obtained by cutting the surface by a plane containing the normal
vector; the directions to which they belong are perpendicular. In
eqn (1) the sign convention is such that if curvatures are
considered positive when the curvature center is inside phase I,
then dP ¼ PI PII. Lipid membranes are particular interfaces, produced by the
self assembly of lipid surfactant molecules in an aqueous
environment. They are highly flexible, fluid bilayers, of thickness
x5 nm, forming closed vesicles of sizes ranging from a few tens
of nanometres up to hundreds of mm. Vesicles are considered as
model systems for the outer walls of living cells, and they have
applications as encapsulation vectors for drug delivery. 8 In
addition to s and dP, membrane interfaces are characterized by
the Helfrich bending modulus k, defined through the energy cost
Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS
& Université Paris Diderot–Paris 7, 10 rue Alice Domon et Léonie
Duquet, F-75205 Paris Cedex 13, France
of curving a (symmetric) membrane: ½k(c1 + c2) 2 per unit area. 9
We recall that vesicles essentially have a constant volume V and
a constant membrane area A, but that the apparent, visible area
may significantly increase with s as the microscopic membrane
fluctuations unfold. 10
The equilibrium equation generalizing the Laplace law for a
membrane interface (the so-called shape equation) is a quite
complex non-linear partial differential equation. For a symmetric
membrane, it reads: 11
dP ¼ 2sH 4kH(H 2 K) 2kDsH (2)
where H ¼ ½(c 1 + c 2) is the mean curvature, K ¼ c 1c 2 the
Gaussian curvature, and Ds the covariant surface Laplacian. 12
Note that the k / 0 limit simply yields the ordinary Laplace
law. Note also that other types of soft-matter materials,
e.g., liquid crystals, exhibit different kinds of generalized
Laplace laws. 13,14 For lipid membranes, k z 1 10 19 J, while s
is an adjustable quantity that is determined by the external
forces exerted on the membrane. Typically, s ranges between
1 10 8 Jm 2 and 1 10 3 Jm 2 , the lower bound corresponding
to freely floating floppy vesicles and the upper bound
to the highest tensions that one can induce, by means of strong
aspiration or adhesion, before the lipids tear apart.
Calling R the typical curvature radius of a vesicle, or the
typical length over which the curvature varies, we see that the
surface tension term in eqn (2) is of order s/R, while the curvature
terms (those involving k) are of order k/R 3 . Hence, the
curvature corrections to the Laplace law are relevant only if R (
O(k/s). With typically s x 1 10 5 Jm 2 , curvature corrections
are found to be relevant when R ( 0.1 mm, i.e., at the nanoscale.
Micropipette aspiration experiment
In this classical experiment (Fig. 1), used to measure k through
the expansion of the visible area of the vesicle, 15,16 the standard
Laplace law, eqn (1), is commonly used instead of eqn (2). This is
indeed reasonable, since the typical radius of the micropipettes is
a x 3 mm, yielding a [ O(k/s) as soon as s [ 1 10 8 Jm 2
(which is almost instantly attained as soon as the vesicle is
pressurized).
This journal is ª The Royal Society of Chemistry 2008 Soft Matter, 2008, 4, 2463–2470 | 2463

Fig. 1 Geometry of the vesicle aspiration experiment. The pipette radius
is a, the aspiration pressure is DP ¼ P0 P1 > 0, where P0 is the pressure
outside the channel and P1 the pressure inside the channel; the pressure
inside the vesicle is P. The outer part of the vesicle is quasi-spherical. The
regions M0, M and M1 corresponding to the outer, tubular (length L)
and inner parts, respectively, are indicated. The region M is pressed onto
the channel, while the other two are free.
The free part of the membrane inside the micropipette is
usually described as a perfect spherical cap of radius r ¼ a. 15–17 As
we shall see, this is inexact. Indeed, in the absence of adhesion, it
is well-known 18 that the equilibrium curvature at the detachment
point in the plane of symmetry must be equal to the curvature of
the substrate in that direction, i.e., must vanish in the present
case. Therefore, the curvature cannot be constant. Nevertheless,
in the ordinary case [a [ O(k/s)], eqn (1) has to be almost
perfectly satisfied—except possibly in a boundary layer. Hence
the curvature radius r must be constant except very close to the
micropipette wall. As we shall see, in the absence of adhesion it
turns out that this constant curvature radius is indeed r x a.
Therefore the standard picture is correct in the ordinary case, i.e.,
when there is no adhesion and a [ O(k/s).
As for the outer free part, it is usually almost a spherical cap of
radius R [ a [ O(k/s). Then, applying, far from the boundary
layers, eqn (1) twice, once to the free part of the membrane
inside the micropipette and once to the outer free part, one gets
P P1 ¼ 2s/a and P P0 ¼ 2s/R, which yields the relationship
between s and DP ¼ P0 P1 used in the literature: 15
DP ¼ 2s 1
a
Note that the fluctuation corrections to s are neglected here. 17,19
Scope and organization of the paper
In this work, we discuss several forms of the generalized Laplace
law for membranes, in situations where the standard Laplace law
does not hold—essentially in nanochannels or in the presence of
adhesion between the membrane and the channel. We consider
both tubular and conical channels. We discuss global relationships
of the type shown in eqn (3), that take into account the
corrections coming from the curvature energy and from the
adhesion energy. We also discuss local forms, simpler than eqn
(2), valid on the axis of a revolution symmetric shape. We obtain
these relationships from a straightforward force equilibrium
analysis, which is completely equivalent to the standard Helfrich
energy approach. 9 We then recover these results within a more
traditional energy minimization analysis. In the last part of the
paper, we present numerical results showing the departure of
the free membrane part from a spherical cap and quantifying
the violation of the standard Laplace law.
1
R
(3)
2 Force equilibrium analysis
In the Helfrich model, 9 the free energy of a vesicle of volume V
and area A within an aspiration channel can be expressed as:
F ¼ sA GAc þ Wpr þ k
ð
dAðc1 þ c2Þ
2
2
(4)
where Ac represents the portion of the vesicle area in contact with
the channel, G the adhesion energy per unit surface (possibly
negligible), Wpr the work of the pressure forces, and the other
quantities have been previously defined. In the present case,
Wpr ¼ PV + P0Vext + P1Vint, where Vint is the volume of the
vesicle portion aspirated in the channel (where the pressure is P1),
Vext ¼ V Vint is the remaining volume of the vesicle outside the
channel (where the pressure is P 0), and P is the pressure inside the
vesicle. Note that s and P can be interpreted as Lagrange
multipliers fixing A and V. We have not included the Gaussian
rigidity k, which does not contribute in the absence of topology
changes 9 and we have assumed that the membrane is symmetric
(no spontaneous curvature).
2.1 Membrane stresses
We have introduced the free energy in eqn (4) in order to fully
specify the system under study. Until Sec. 3, however, we shall
not use it . Indeed, the Helfrich model is equivalent to the
following description in terms of stresses. 20,21 The force S M dlthat
a piece of membrane M exerts onto an adjacent piece M 0
through a cut dlhas a very simple expression in the case where
the cut is parallel to a direction of principal curvature (otherwise
a tensorial description must be used). Let t M denote the unit
vector tangent to the membrane, perpendicular to the cut, and
directed toward the interior of M (Fig. 2). Let n denote the
normal to the membrane, with the convention 22 that curvatures
are considered positive if the membrane is curved towards n,
then: 20,21
S M ¼ S M t t M + S M n n (5)
S M
t
k 2
¼ s G þ ck 2
S M
n
¼ 2k vH
vt M
k 2
ct 2
where c || and c t are the curvatures parallel and perpendicular to
the cut, respectively, H ¼ ½(c || + c t) is the mean curvature, and
t M is the coordinate in the direction of t M . The term G arises
when the membrane is in contact with a substrate with adhesion
energy G. 21 For the free membrane parts, G ¼ 0.
Fig. 2 The geometry for the definition of the force S M dlexerted by the
region M onto the region M 0 through a cut dlin the plane tangent to the
membrane.
2464 | Soft Matter, 2008, 4, 2463–2470 This journal is ª The Royal Society of Chemistry 2008
(6)
(7)

Note that eqn (5)–(7) satisfy the action–reaction principle
SM0 ¼ SM , where SM0 is the force per unit length that M0 exerts
onto M. Indeed, SM0 t ¼ SM t , since eqn (6) depends only on the
local curvatures. On the contrary, SM0 n ¼ SM n , since tM0 ¼ tM .
Then, since tM0 ¼ tM while n is the same both for M and M0 , the
action–reaction principle follows.
2.2 Continuity relations
Let us comment on the continuity of the components of the stress
along the membrane. At equilibrium, everywhere along the free
part of the membrane, the continuity of S must obviously be
satisfied. But what happens at the detachment point between the
membrane and the pipette? The part of the membrane in contact
with the pipette substrate is free to glide, therefore the continuity
of St must be satisfied—even though the curvatures are discontinuous.
However, in general, there will be a discontinuity of S n.
This is due to the fact that there is actually a narrow region,
intermediate between the free and the bound part, where the
membrane feels a potential W(z) depending on the distance z to
the substrate. This intermediate region is subject to a density of
normal forces dW(z)/dz. The latter results in an apparent
discontinuity of Sn in a theory neglecting the width of the
intermediate region.
2.3 Aspiration in a cylindrical nanochannel
Let us determine the balance along ^z of the forces acting on
region M1 (Fig. 1). They have two origins: the pressure forces
acting all along the surface of M1 and the membrane elastic
forces acting through the contour separating M from M1.
Hence, at equilibrium, we have:
Ð
M1 (P P1) n$^z dA+ 2pa SM $^z ¼ 0 (8)
Taking into account that Ð M 1 n$^z dA ¼ pa 2 and S M $^z ¼ S M t ,
while S M t ¼ s G +½k/a 2 (within the tube c|| ¼ 1/a and ct ¼ 0),
we obtain:
s G k
P P1 ¼ 2 þ
a a3 (9)
which is an exact global Laplace relation that takes into account
the curvature and adhesion corrections. Note that this equation
can be rewritten as:
P P1 þ 2G 2s
¼
a a
1 þ 1
2
l
a
2
(10)
where l ¼ O(k/s), which shows that the adhesion energy acts as
a suction pressure that adds up to the pressure difference P P1,
and that the curvature correction is important for a ( l, i.e.,
essentially for nanopipettes at ordinary tensions. In the latter
case, the curvature corrections are important everywhere in M1,
and M 1 departs significantly from a spherical cap. Nevertheless,
eqn (9) holds exactly.
Note now the following subtlety that will be important in the
following. In eqn (8), we have used for S M t the expression s G +
½k/a 2 , which is actually valid inside the tubular region, while we
should use the stress exerted immediately after the detachment
point. We are safe, however, because the tangential component of
the stress is continuous. This is why eqn (9) holds independently
of the fact that the shape of the cap is not a half-sphere of radius
a; indeed, the size a that enters eqn (9) is the radius of the
cylinder.
The continuity of the tangential stress at the detachment point
gives us an important equilibrium condition. Calling c0 the
boundary curvature at the detachment point in the direction
parallel to the tube axis, we obtain S M1
t
Then S M t ¼ S M1
t yields:
c0 ¼
rffiffiffiffiffiffi
2G
k
1
¼ s þ
2 k=a2 1
2
kc 2
0 .
(11)
This corresponds to the well-known Seifert–Lipowsky adhesion
boundary condition, 18 which is valid also on a curved wall
when the curvature of the wall in the direction perpendicular
to the contact line vanishes. 23,24 Note that when this latter
condition is not satisfied, the full equilibrium condition requires
also the torque balance. 25 In the absence of adhesion, eqn (11)
yields:
c 0 ¼ 0 (12)
Therefore, as mentioned in the introduction, M 1 is never an
exactly perfect spherical cap. 26
Note that relation shown in eqn (9) may also be obtained by
a careful minimization of the total free energy shown in eqn (4)
with respect to the tube length L. Indeed, the curvature term in
eqn (4), proportional to k, is equal to Ec ¼ pkL/a + Ec 1 + Ec 0 ,
where the first term corresponds to the cylindrical part and the
other two contributions correspond to regions M1 and M0.
Thanks to the Lagrange multiplier P, V can be varied independently
of the other parameters; thus the same holds for L.
Therefore, since E c1 and E c0 are independent of L, we obtain
0 ¼ svA/vL GvA c/vL PvV/vL + P 0vV ext/vL + P 1vV int/vL +
vE c/vL ¼ 2pas 2paG pa 2 P + pa 2 P 1 + kp/a, yielding
eqn (9).
Generalized global Laplace relation
Let us now assume that the vesicle outside the nanochannel is
much larger than O(k/s), which allows us to use the approximate
Laplace relation eqn (1) instead of the exact one eqn (2). Calling
R the curvature radius where the membrane meets the revolution
axis, we get P P0 x 2s/R. Combining this equation with eqn
(9), we obtain the generalized global Laplace relation:
DP x 2s 1
a
1
R
G
sa
þ k
a 3
(13)
which generalizes eqn (3). This is one of our central results. It
holds most generally for vesicle aspiration in any cylindrical
nanochannel (even when the free membrane cap M1 is far from
being spherical).
Macroscopic channels
Note that for channels large enough such that R [ a [ O(k/s),
the term involving k is negligible in eqn (9) and we obtain P P1
x 2(s G)/a. Now, since in this case the ordinary Laplace law
This journal is ª The Royal Society of Chemistry 2008 Soft Matter, 2008, 4, 2463–2470 | 2465

(1) holds everywhere except possibly in a boundary layer (cf. the
discussion in the Introduction), region M 1 must be a spherical
cap almost within the whole gap. Therefore, the radius of the
spherical cap M1 is not r x a, but:
r x
a
1 G=s
(14)
This is our second central result. For G ¼ 0, we recover the
standard approximation that M 1 is a half-sphere matching the
tube radius.
2.4 Aspiration in a conical nanochannel
The problem is more complex for a conical channel (see Fig. 3),
because of the discontinuity of the normal stress at the detachment
point. The force balance along ^z, for region M1, can be
written as:
pa 2 (P P1) 2pa(S M 1
t cosa + S M 1
n sina) ¼ 0 (15)
Note that instead of using S M as previously, we have used
S M 1 , which is correct, because S M 1 is indeed the force
exerted on M1 by the rest of the membrane (action–reaction
principle). Thanks to the continuity of the tangential stress,
S M1
t ¼ SM t
¼ s G þ 1
2 kðcos aÞ2 =a 2 , because of the conical
shape. We now discuss S M 1
n . The symmetry of revolution implies
H ¼ 1
2 ðck þ ctÞ ¼ 1
ðcos q=r þ cÞ, where r is the distance to the
2
revolution axis, q is the inclination of the tangent with respect to
this axis, and c is the curvature of the membrane section in
the plane containing the revolution axis. Thus, according to
eqn (7),
S M !
d cos q
1
n ¼ k c þ ¼ k
ds r 1
dc cos a c1a
þk
ds 1 a2 sin a (16)
where the index ‘‘1’’ means evaluation at the detachment point of
region M1, and where s is the curvilinear coordinate oriented in
the same direction as ^z. The tangential stress continuity implies
again
rffiffiffiffiffiffi
2G
c1 ¼
(17)
k
at the detachment point. Hence, we obtain:
Fig. 3 Geometry of vesicle aspiration in a conical nanochannel of
aperture 2a. The regions M 0, M and M 1 corresponding to the outer,
conical and inner parts, respectively, are indicated. The pressure within
the channel is P 1, the pressure outside is P 0, and the pressure inside the
vesicle is P. The radius of the channel at the contact point between region
M 1 and region M is a.
s G
P P1 ¼ 2 cos a þ
a
k
a3 cos a 1 þ sin2a 2 ffiffiffiffiffiffiffiffi p
2Gk
a2 sin 2 a þ 2 k dc
sin a
a
ds 1
(18)
This equation generalizes the global Laplace relation, eqn (9);
unfortunately (dc/ds)|1 is unknown and can only be determined
by solving for the complete shape of region M1. More results
concerning the conical channel will be given in Sec. 3.2
2.5 Generalized local Laplace law at an umbilical point
For a shape with revolution symmetry, the point where the
membrane meets the revolution axis has its two principal
curvatures equal. At such a point, called an umbilical point,
the generalized Laplace law, eqn (2), takes a simpler form:
indeed, the contribution proportional to H2 K vanishes,
because H2 ¼ c2 ¼ K, and the covariant surface laplacian Ds
equals 2v2 /vx2 , where x is a cartesian coordinate in the tangent
plane. Identifying, at lowest order, the coordinate x in the
tangent plane with the curvilinear coordinate s in a symmetry
plane, we obtain DsH ¼ 2 v2H/vs2 ¼ 2d2c/ds2 , where the last
equality easily follows from the symmetry of revolution and
some differential geometry. Hence, the generalized Laplace law,
eqn (2), becomes:
dP ¼ 2sc 4k d2c ds2 (19)
To verify this relation, we write the balance of the normal
forces acting on an elementary cap of radius r / 0 centered on
an umbilical point O (see Fig. 4):
dPpr2 h
¼ 2pr
s þ k 2
ck 2
2k vH
cos f
vs
k
2 c2 sin f
(20)
where dP is the pressure difference across the membrane and one
recognizes the expressions of St and Sn, as expressed in eqn (6)
and (7), with ct ¼ c, G ¼ 0 (free membrane), and s the curvilinear
coordinate, measured from O. Note that for r s 0, we have
c|| s c. The limit r / 0 yields eqn (19), since s þ k
2 c 2 k
k
2 c2 /s,
(sinf)/r / c, cosf / 1 and (1/r)vH/vs / v 2 H/vs 2 |0.
Eqn (19) is an exact generalized Laplace relation including the
curvature corrections. Note that it can be applied to the tip of
region M1 in the nanopipette problem (Fig. 1), but the values of c
and d 2 c/ds 2 are unknown unless we solve the whole problem. The
Fig. 4 Elementary cap of radius r around an umbilical point for the
determination of the local generalized Laplace law at the tip of
a symmetry axis.
2466 | Soft Matter, 2008, 4, 2463–2470 This journal is ª The Royal Society of Chemistry 2008

method we used above is therefore more efficient. Note, however,
that this equation can be used at the tip of region M 0, implying
that eqn (13) takes the exact form:
DP ¼ 2s 1
a
1
R
G
sa
þ k
a 3
4k d2 c
ds 2
where d 2 c/ds 2 is evaluated at the tip of region M0.
3 Energy minimization analysis
(21)
In this section we give the differential equations and the
boundary conditions governing the shape of the free membrane
inside the cylindrical or conical nanochannels of Fig. 1 and 3. As
a first application, we discuss numerically the features of the
solution that cannot be predicted analytically. We then redemonstrate
analytically the detachment condition eqn (17) and the
generalized Laplace eqn (18) and (19) that we have obtained from
the stress tensor analysis.
3.1 Differential equations and boundary conditions
We consider generically the region M1 in the conical nanochannel
of Fig. 3. Indeed, the nanopipette geometry of Fig. 1 is
a particular case where a ¼ 0, while the region M 0 of the conical
nanochannel of Fig. 3 corresponds to the case a < 0. The minimization
of the free energy shown ineqn (4) for axisymmetric
shapes yields the set of differential equations for the profile of the
free part M1: 27,28
j 00 ¼
g sin j
kr þ ðP1 PÞr cos j
2k
g 0
¼ 1
2 k j 0 2 sin2 j
r 2
þðP1 PÞr sin j
j 0 cos j sin 2j
þ
r 2r2 þ s
(22)
(23)
r 0 ¼ cosj (24)
z 0 ¼ sinj (25)
where j is the angle that the tangent to the profile forms with
the radial direction r and a prime indicates derivation with
respect to the arclength s. We set s ¼ 0 at the detachment
point of M1 and s ¼ s1 at the revolution axis r ¼ 0; s1 is thus
the length of the profile. The function g(s) is a Lagrange
multiplier field that enforces the condition that s is everywhere
the arclength of the profile. The boundary conditions are
obtained by setting to zero the boundary terms of the free energy
variation: 27,28
jð0Þ ¼ p
þ a; rð0Þ ¼a; zð0Þ ¼0 (26)
2
j(s1) ¼ p, r(s1) ¼ 0 (27)
P
2 gð0Þtan a
P1 ¼
a2 2 s k cos a
þ þ
a cos a a3 k
a cos a j 02 ð0Þ
(28)
P P1 ¼
2 gð0Þtan a
a2 2ðs GÞ k cos a
þ þ
a cos a a3 (29)
Eqn (26) and (27) reflect the geometry of the problem, eqn (28)
is obtained by varying the length of the axisymmetric contour
(taking into account the existence of a conserved Hamiltonian29 ),
and eqn (29) is obtained by displacing the detachment point of
region M1, taking into account the free energy FM of the
adhering part M:
FM ¼ 2p
ð z1
z0
k cos a
2rðzÞ
ðs GÞrðzÞ
þ þ
cos a
ðP1 PÞr2ðzÞ 2
dz (30)
Here, r(z) ¼ a ztana is the equation of the cone, z0 is the z
coordinate of the detachment point of M 0 and z 1 ¼ 0 is the z
coordinate of the detachment point of M 1, at which r ¼ a.
Eqn (22)–(25) with the seven boundary conditions (26)–(29)
fully determine the problem. Indeed, we have five first-order
partial differential equations, plus the two unknowns s1 and
P P1.
3.2 Numerical analysis
We solve numerically the shape eqn (22)–(25) with the boundary
conditions shown in eqn (26)–(29) by using a standard finite
difference scheme with deferred correction and Newtonian iteration
30 . We introduce the normalized quantities:
z ¼ z r s
; r ¼ ; s ¼
a a a ; s1 ¼ s1
a
s ¼ sa2
3 ðP P1Þa
; p ¼
k k
(31)
(32)
G ¼ Ga2 ga
; g ¼ (33)
k k
To cope with the unknown parameters s1 and p, we write
the differential equations in terms of the independent parameter
t ¼ s/s1, with 0 # t # 1, and we treat s1 and p1 as two extra
functions of the parameter t, obeying the differential equations
ds1/dt ¼ dp/dt ¼ 0. Our problem is thus transformed into
a standard set of seven first-order differential equations obeying
seven boundary conditions: five at the left boundary t ¼ 0(s¼ 0)
and two at the right boundary t ¼ 1, (s ¼ s1).
Because of the singularity of the differential equations at r ¼ 0,
we replace the two right boundary conditions, eqn (27), with the
following conditions at s ¼ s1 3 (with 0 < 3 s1): r (s1 3) ¼ 3 (34)
jðs1 3Þ ¼p 3 1 þ 3
3 gðs1 3Þ dj
ds s1 3
(35)
that are obtained by solving the shape equations up to order 3 2
around s ¼ s1.
Shapes in the cylindrical nanochannel
In the absence of adhesion (G ¼ 0), we find that the vesicle has
a prolate shape at small tensions (see top curves of Fig. 5). As the
This journal is ª The Royal Society of Chemistry 2008 Soft Matter, 2008, 4, 2463–2470 | 2467

Fig. 5 Calculated shapes of the vesicle inside the cylindrical nanochannel
in a symmetry plane (x,z). Top: G ¼ 0 (no adhesion) for the
reduced tensions s ¼ 0, s ¼ 10, s ¼ 1 10 2 , s ¼ 1 10 3 , s ¼ 1 10 5 (from
top to bottom). Bottom: G ¼ 2 for the same reduced tensions, however
from bottom to top. In both cases, for s ¼ 1 10 5 the shape is almost
a half-sphere.
tension increases, the vesicle tends to a sphere of radius equal to
the radius of the channel—except within a small region of size
xO(k/s) close the channel walls.
The deviation of the shape from a spherical cap of radius
equal to the radius of the channel can be characterized by the
normalized height h ¼ z(r ¼ 0)/a of the top of the vesicle with
respect to the detachment point (see Fig. 6). Increasing the
adhesion energy at fixed tension reduces the height of the
vesicle. When the adhesion energy G equals k/(2a 2 )(G ¼ 1/2),
the vesicle becomes exactly a half-sphere with radius equal to
the channel radius, whatever the tension. Indeed, according to
eqn (11), for G ¼ k/(2a 2 ) the curvature at the detachment point is
1/a and a perfect sphere is always a solution of the shape
equations. In this case, the pressure drop is related to the
tension by the ordinary Laplace law; indeed, the curvature
terms in eqn (2) vanish. At adhesion energies higher than k/(2a 2 )
(G > 1/2), the vesicle develops an oblate shape that tends to
a sphere as the tension increases (see Fig. 6 and bottom curves
of Fig. 5).
Fig. 6 The normalized height h of the top of the vesicle (with respect to
the detachment point) in the cylindrical nanochannel, as a function of the
reduced tension s for different normalized adhesion energies G (G ¼ 0, G
¼ 0.2, G ¼ 0.5, G ¼ 2, G ¼ 10). For G ¼ 1/2, the vesicle is a perfect halfsphere
of radius a whatever the tension.
Fig. 7 The normalized curvature c 0 of the top of the vesicle in a cylindrical
channel, as a function of the normalized tension s, for G ¼ 10.
Continuous line: numerical result; dashed line: analytical approximation
c0 ¼ 1 G/s according to eqn (14). Inset, continuous line: shape of the
vesicle, in a symmetry plane (x,z), for s ¼ 40. The dashed line corresponds
to a spherical cap having the same curvature as the top of the
vesicle.
High tension and strong adhesion in the cylindrical nanochannel
As predicted above, at tensions s [ k/a 2 the vesicle should have
an almost spherical shape of radius given by eqn (14). We have
checked this numerically in the strong adhesion case (G x s), for
which the predicted radius differs significantly from the radius
a of the channel (see Fig. 7). When G > s, we find numerically
that the curvature of the top of the vesicle becomes negative (the
membrane invaginates), in agreement with eqn (14).
Deviation from the naive prediction using the ordinary Laplace
law
We consider the general conical channel case of Fig. 3. The
ordinary Laplace law, that neglects the curvature energy, would
predict a pressure drop dPnaive ¼ (2scosa)/a, corresponding to
a spherical cap of radius a/cosa matching tangentially the conical
channel. In our normalized units, this yields pnaive ¼ 2scosa.
Curvature and adhesion energies introduce the corrections displayed
in eqn (18), and cannot be evaluated analytically for a s
0. Fig. 8 shows the deviation p p naive, in the absence of adhesion,
for different cone angles a. For a cylindrical channel
(a ¼ 0), according to eqn (9), the deviation is p p naive ¼ 1,
independent of the tension. On the other hand, for a conical
nanochannel (a s 0), the deviation p pnaive increases as the
tension increases. Asymptotically, for s [ 1, we find that p
pnaive f s 0.5 ; therefore, the relative error (p pnaive)/pnaive x (p
pnaive)/s tends to zero as 1/Os. At small tensions (s ( 1), the
deviation p pnaive is almost independent of the tension; the inset
of Fig. 8 shows the extrapolated deviation in the limit s / 0,
which is maximum for a x 48 .
3.3 Validation of the force equilibrium analysis
Here, we show that the relations previously deduced from the
force equilibrium analysis can be recovered directly from the
shape eqn (22)–(25) and the boundary conditions shown in eqn
(26)–(29).
2468 | Soft Matter, 2008, 4, 2463–2470 This journal is ª The Royal Society of Chemistry 2008

Fig. 8 The deviation p 2scosa of the normalized pressure difference p
from the naive Laplace prediction pnaive ¼ 2scosa, in the absence of
adhesion (G ¼ 0) for different cone angles a (from bottom to top, a ¼ 0 ,
a ¼ 3 , a ¼ 10 , a ¼ 30 , a ¼ 45 ). Above the dashed line, the relative
violation (p pnaive)/s is larger than 10%. Inset: normalized pressure p
extrapolated at zero tension as a function of the cone angle a.
Curvature at the detachment point
The detachment condition, eqn (17), can be deduced from the
boundary conditions shown in eqn (28) and (29), since c1 ¼ j 0 (0).
Global general Laplace relation
Eqn (18) can be deduced from the shape eqn (22), using the
boundary conditions shown in eqn (26) and (29), eliminating
g(0).
Local generalized Laplace law at an umbilical point
Eqn (19) can be recovered in the following way. Eqn (22), with
the help of eqn (24), can be rewritten as:
k
r
d
ds ck
ðsin jÞg
þ c ¼
r2 þ ðP1 PÞcos j
2
(36)
since c || ¼ (sinj)/r and c ¼ j 0 . The local generalized Laplace law
(19) is then the limit for r / 0(i.e., s / s 1) of eqn (36). Indeed,
using the L’Hôpital rule, we have:
lim
r/0
1
r
d
ds ck þ c ¼
1
r 0 ðs1Þ
d 2 ðck þ cÞ
ds 2
s¼s 1
¼ 2 H 00 ðs1Þ (37)
sin j
lim
r/0 r ¼ j0 ðs1Þ cos jðs1Þ
r 0 ¼ j
ðs1Þ
0 ðs1Þ ¼cðs1Þ (38)
g
lim
r/0 r ¼ g0 ðs1Þ
r 0 ¼ s (39)
ðs1Þ
where H ¼ ½(c|| + c) is the mean curvature. Here, we have taken
into account that r 0 (s 1) ¼ cos[j(s 1)] ¼ 1 [see eqn (24) and (27)]
and g 0 (s 1) ¼ s, according to eqn (23) with r(s 1) ¼ 0 and c(s 1) ¼
c ||(s 1).
4 Conclusions
In micropipette experiments, the vesicle tension is usually
determined by means of the ordinary Laplace relation (3). Here
we have shown that the correct generalized relation, that takes
into account the curvature and adhesion corrections, is eqn (13).
The curvature correction is relevant when the pressure drop DP is
comparable with k/a 3 , as it can occur in nanochannel. For
instance, taking a ¼ 100 nm and DP ¼ 1.1 10 2 Jm 3 , with k ¼ 1
10 19 J and a radius of the free part R [ a, the naive relation
(3) would give s ¼ 5.5 10 6 Jm 2 , while the correct value given
by eqn (13) is s ¼ 5 10 7 Jm 2 , i.e., ten times smaller.
Eqn (13) also allows us to determine the tension s in the
presence of an adhesion energy G. Indeed, we have shown that
for tensions s [ k/a 2 the vesicle profile inside the micropipette is
a spherical cap of radius depending on the adhesion energy G
according to eqn (14). This relation can be used to determine the
ratio G/s, then s and G can be determined separately with the
help of eqn (13).
Note that eqn (13) is an approximation holding when R [
O(k/s), which is usually the case. We have also determined the
general relation shown in eqn (21), that is exact for cylindrical
micropipettes whatever the value of R, using an exact generalization
of the Laplace law valid locally at an umbilical point [see
eqn (19)].
These analytical results have been confirmed by numerical
calculations, that also allow us to determine the exact profile of
the vesicle and the relation between the tension and the pressure
drop in conical geometries (see Fig. 8). In particular, we have
shown that the relative error on the Laplace pressure is significantly
larger in the conical case. For instance, for a channel of
aperture a ¼ 45 with size a ¼ 1 mm, with k ¼ 1 10 19 J and s ¼
1 10 6 Jm 2 , the true pressure drop is dP ¼ 1.9 J m 3 , almost
40% larger than the value predicted by the ordinary Laplace law.
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