Dynamics of Fluid Vesicles in Flow through Structured Microchannels

0.9 MB·English

by  Hiroshi Noguchi

epl draft
Dynamics of Fluid Vesicles in Flow through Structured Microchan-
nels
Hiroshi Noguchi1,2 (a), Gerhard Gompper1 (b), Lothar Schmid3 (c), Achim Wixforth3 , and Thomas
0 Franke3 (d)
1
0 1 Institut fu¨r Festk¨orperforschung, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany,
2 2 Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan,
n 3 Experimentalphysik I, Microfluidics Group, Universita¨t Augsburg, 86159 Augsburg, Germany
a
J

3
1 PACS 87.16.D-–Membranes, bilayers, and vesicles
PACS 83.50.Ha–Flow in channels

] PACS 82.70.Uv–Surfactants, micellar solutions, vesicles, lamellae, amphiphilic systems
t
f
o Abstract. - The dynamics of fluid vesicles is studied under flow in microchannels, in which the
s
widthvariesperiodicallyalongthechannel. Threetypesofflowinstabilitiesofprolatevesiclesare
.
t found. Forsmallquasi-sphericalvesicles–compared totheaveragechannelwidth–perturbation
a
m theorypredictsatransitionfrom astatewithorientationaloscillations ofafixedprolateshapeto
astatewithshapeoscillations ofsymmetricalellipsoidalorbullet-likeshapeswithincreasingflow
-
d velocity. Experimentally, such orientational oscillations are observed during the slow migration
n of a vesicle towards the centerline of the channel. For larger vesicles, mesoscale hydrodynamics
o simulations and experiments show similar symmetric shape oscillation at reduced volumes V∗ &
c 0.9. However,fornon-sphericalvesicleswith V∗ .0.9, shapesarefoundwithtwosymmetricora
[ single asymmetric tail.

2
v
2
6
8
0 Introduction. – Soft deformable objects such as liq- cells and vesicles, e.g. the measurement of the dynamic
. uid droplets, vesicles, and cells show a complex behavior pressure drop for single-cell deformation [12], the separa-
1
1 under flow. For example, in simple shear flow, fluid vesi- tion of RBCs from the suspending plasma [13,14], and
8 cles exhibit tank-treading, tumbling, and swinging (also the control of oxygen concentration to investigate sickle
0 called vacillating-breathing, or trembling) motions, de- cells [10]. There are many potential applications such as
:
v pending on parameters such as shear rate, viscosity con- blood diagnosis on chip. Under steady flows in homo-
i trast, and internal volume [1–9]. Understanding the flow geneous glass capillaries and rectangular microchannels,
X
behavior of lipid vesicles and red blood cells (RBCs) is vesicles [15] and RBCs [14,16–18] deform into a bullet
r
not only an interesting problem of the hydrodynamics of (with a flattened rear end) or parachute (with an inside
a
deformable, thermally fluctuating membranes, but is also bulge at the rear end) shape. Compared to these steady-
important for medical applications. In microcirculation, flow conditions, the vesicle dynamics in time-dependent
the deformation of RBCs reduces the flow resistance of flow is much less explored. Only very recently, phenom-
microvessels. In diseases such as sickle cell anemia, RBCs enalikethewrinklingofvesiclesafterinversionofanelon-
havereduceddeformability and often block microvascular gational flow [19,20] or shape oscillation of RBCs [21,22]
flow [10]. Lipid vesicles are considered as a simple model and fluid vesicles [23] under oscillatory shear flow have
of RBCs and also have applications as drug-delivery sys- been discovered.
tems.
In this paper, we propose a structured microchannel
The recent development of microfluidic techniques [11]
system to study vesicle dynamics. The width of a mi-
allows the investigation and manipulation of individual
crochannel is spatially modulated along the channel, so
that a flowing vesicle is exposed to an oscillatory elonga-
(a)E-mail:[email protected]
tional flow. We have fabricated such microchannels and
(b)E-mail:[email protected]
(c)E-mail:[email protected] observe the time-dependent vesicle deformation via opti-
(d)E-mail:[email protected] cal microscopy. In parallel, we use perturbation theory
p-1
H. Noguchi et al.
for quasi-spherical vesicles and mesoscale-hydrodynamics
simulations for non-spherical vesicles to predict several
flowinstabilities. Inparticular,wepredictforwiderchan-
nels(comparedtothevesiclesize)atransitionfromshape
oscillations of symmetrical bullet-like shapes to orienta-
tional oscillations with decreasing flow velocity; for nar-
rower channels, we observe stable shapes with two sym-
metric or with a single asymmetric tail. Our results
demonstrate that structured channels are well suited to
investigate the dynamical behavior of vesicles.
Experimental Systems and Methods. – Lipid
vesicles, consisting of 1,2-Dioleoyl-sn-Glycero-3- Fig. 1: (Color online) Side view of the channel setup with
Phosphocholine (DOPC, Avanti Polar Lipids) with the reservoirs. The inset shows a top view of the modulated
0.1mol% fluorescently labeled T-Red DHPE (Texas Red channel. The length of themodulation is Lx =100µm.
1,2-dihexadecanoyl-sn-glycero-3-phosphoethanolamine,
triethylammonium salt, Invitrogen, USA), were prepared
Theory for quasi-spherical vesicles in modulated
inaqueoussucrosesolution(200mM)byelectroformation,
channels. – We consider vesicles which have the same
as described elsewhere [24]. Using this method, many
viscosity η of the outer and inner fluids. Vesicles have
giantunilamellarvesicleswith diameterlargerthan10µm
constant volume V and constant surface area S, so that
wereproduced. Priortotheexperiments,thevesicleswere thereducedvolumeV∗ andtheexcessarea∆ aredefined
introduced into an slightly hypertonic aqueous glucose byV∗ =(R /R )3 =(1+∆ /4π)−3/2 and∆S =S/R2
or sucrose solution. This procedure enables control of V S S S V−
4π, where R =(3V/4π)1/3 and R =(S/4π)1/2.
the volume-to-surface ratio by inflation and deflation of V S
First,wederiveananalyticaldescriptionoftheflowbe-
the vesicles. The vesicles in sucrose solution are perfectly
haviorofsmallquasi-sphericalvesicleswith∆ 1,based
density matched. The density difference between sucrose S ≪
on the theory for linear flows [2,5–7,20]. The microchan-
and glucose solutions at 200mM is ∆ρ/ρ = 0.012, which
nelhasaconstantheightL andaperiodicallymodulated
implies a small buoyancy force. z
width with walls at h (x), where
y
Microchannels were fabricated by soft lithography [30]. ±
The structure of an AutoCAD designed mask was Ly 2πx
h (x)= 1+a cos . (1)
transferred by illumination to a photoresist (SU-8, mi- y 2 n y (cid:16) L (cid:17)o
x
croresist, Berlin), which serves as a master to cast
Here, L is the periodicity length along the channel (see
poly(dimethylsiloxane)(PDMS)replicas. After treatment x
Fig.1). Theflowfieldv(r)forweaklymodulatedchannels
with an O -plasma, the PDMS mold was bonded onto a
2
(a L /L ) is obtained from lubrication theory,
thin coverslip. The assembly was connectedto a reservoir y ≪ x y
containing the vesicle solution (inlet) and a water reser- v = (9v L /8h ) 1 (y/h )2 1 (2z/L )2 ,
x m y y y z
voir (outlet) to adjust the hydrostatic pressure difference, { − }{ − }
v = πv yL a sin(2πx/L )/h L , (2)
which drives the fluid through the structured microchan- y − x y y x y x
nel (see Fig. 1). All channels have a height Lz = 50µm vz = 0,
and a periodic length L =100µm.
x
where v is the mean flow velocity. Furthermore,we con-
m
Due to the lift force from a wall [25–27] and the mi-
sider small vesicles with R L ,L , and neglect back-
V y z
gration in a parabolic Poiseuille flow [28,29], vesicles are ≪
flow effects due to the channel wall.
aligned close to the center of the channel. In the glu-
The vesicle shape is expanded in spherical coordinates
cose solutions, the buoyancy force leads to a displace-
asr =R (1+ u Y ),withthesphericalharmon-
ment of the vesicles from the center of the channel in V Pl,m l,m l,m
ics Y (θ,ϕ) and the polar axis in the z direction. We
l,m
the z-direction (see Fig. 1). The balance of the buoy-
consider a vesicle shape which is mirror symmetric in the
ancy force and the lift force [25,27] implies a distance
z direction, which implies, e.g., u2,±1 =u3,0 =u3,±2 =0.
ℓ (ηγ˙R /(∆ρg))1/2 between membrane and wall, where
0 Thecurvatureenergyofthemembranewithbendingrigid-
≃
γ˙ is an effective shear rate, R a mean vesicle radius, η
0 ity κ and reduced surface tension σ is given by
the fluid viscosity, and g the gravitational acceleration.
FveolrocRit0y=of1a0bµomu,tγ˙20=µm1s/−s1)(acnodrrethspeovnidscinogsittyooafmweaatner,flowwe F =(κ/2)XEl|ul,m|2+κ(12+2σ/3)u(23), (3)
l,m
obtaintheorder-of-magnitudeestimateofℓ=10µm,com-
parabletothechannelsizeLz. Thedisplacementfromthe whereEl =(l+2)(l−1){l(l+1)+σ}. Here,thethirdorder
center implies an asymmetric deformation of the vesicle term u(3) = ( 5/π/7)(u 3 6u u 2) for l = 2 is
2 p 2,0 − 2,0| 2,2|
shape in glucose solution under flow. necessarytoobtainaprolateshapeasthermal-equilibrium
p-2
Dynamics of Fluid Vesicles in Flow through Structured Microchannels
state[7,31]. WiththeStokesapproximationandtheLamb
k *=0.01
solutionfor the flowfield, the dynamics ofthe amplitudes
0.4 (a)
ul,mofavesiclemovingalongthecenterline(y =z =0)of k *=100
the channel with velocity v(xG,0,0) = (9vmLy/8hy,0,0) >
)
is governed by [2,7,20]
q2
2n (0.2
∂ul,m ∗ si
=sl,m(xG) κ Γlfl,m (4) <
∂t − lx
0
with reduced bending rigidity κ∗ = κL /(ηR 3v ) = 1 k * 10 100
y V m
L /(v τ) and Γ = l(l+1)/(2l+1)(2l2+2l 1). Here,
y m l
−
τ =ηR3/κ is the characteristic relaxation time of a vesi- ly
V
cle. Two kinds of forces determine the membrane defor- 2 (b)
mation in Eq. (4), the curvature force fl,m = Elul,m V
(3) − R
t(h12e +elo2nσg/a3t)i∂onua2l /fl∂ouwl,.m,Tahnedfothrceefosrce(xsl,m)(ixsGg)ivdeune btyo l / lz
l,m G 1.8
sl,m(xG)=(al,m+Blbl,m)Ly/RVvm [2,7,20]. Here, Bl = lx
1/(2l2+2l 1),anda andb arethesphericalharmon-
l,m l,m
−
ircesspeexcptaivnesliyo,nwohfevrre(rv)(arn)disRtVh∂evrra(rd)i/a∂lrcoamtp|ro−nernGts|=ofRthVe, -0.5 0 x G / Lx 0.5
r
flow field v(r) in the absence of the vesicle. The tension 4 4
a y = 0.5
σ is determined by the area constraint ∂∆S/∂t = 0 with *c (c)
v∆eSsi=clePwli,tmh(la+sm2)a(lll−ex1)c|eusls,ma|r2e/a2∆−2u=(230)/.13.(cWorerceospnosinddeirnag *ck2 0.4 8
S 0.3
to V∗ =0.988), where it is sufficient to take into account k
2 0.1
l = 2,3 modes, and a channel with L = L = 20R and 0
y z V
0 5 10 15
various Lx and ay. L y / Lx 16
Infastflow,whereflowforcesdominateandthereduced
bending rigidity is small, κ∗ 1, the analysis of Eq. (4) 4
shows that the shape is mir≪ror symmetric with respect 2
0
to both xz and xy planes, and that the vesicle lengths 0 0.2 a 0.4
y
l and l oscillate (see Fig. 2). The flow elongates the
x y
vesicle in the y and x direction for L /2 < x < 0 and
0 < x < Lx/2, respectively. In the−regxime κ∗ ≪ 1, the Fclieg.a2t:V∗(C=olor0.9o8n8li.ne)(aD)ynDaempiecnsdeonfcae qoufatshi-esphtieltricaanlgvleesiθ-
extremalelongationsaredeterminedbythebalanceofthe ∗
of a vesicle on κ , for Lx/Ly = 4 and ay = 0.5. Here,
flow forces, sl,m(xG), and the area constraint. The forces hsin2(2θ)i = h(Im(u22)/|u22|)2i describes the deviation from
s2,2(xG) and s3,3(xG) have maxima at xG = 0.38Lx symmetric shape (with respect to the xz plane). The insets
|and xG =| Lx/|2 for ay|=0.5, respectively. The r±esulting show sliced snapshots of vesicles in the xy plane for κ∗ =0.01
positions o±f extremal elongations are found to be close to and κ∗ = 50. Solid and dashed lines indicate shapes of ex-
x =0 and x =L /2 (see Fig. 2(b)). tremal elongation or tilt. (b) Maximum vesicle lengths in the
G G x
In contrast, in slow flow with κ∗ 1, the vesicle is x,y,z directions as a function of the center-of-mass position
∗
predictednottochangeitsshape,butt≫odisplayaperiodic xG, for κ = 0.01 and ay = 0.5. Solid and dashed lines rep-
oscillation of its orientation. The tilt angle θ oscillates resent results for Lx/Ly = 4 and Lx/L∗y = 16, respectively.
(c) Critical reduced bending rigidity κ as a function of the
c
around π/4 (or π/4 depending on initial positions), see
− corrugation amplitudes ay for channel geometry Lx/Ly = 2,
snapshot in Fig. 2(a). Thus, a symmetry breaking occurs 4, 8, and 16. The inset shows κ∗c as a function of Lx/Ly for
with decreasing flow velocity. ay =0.1, 0.3, 0.4, and 0.5.
Both types of vesicle motions are limit cycles. A sharp
but continuous transition between these two motions oc-
curs at a critical reduced bending rigidity κ∗, where the Mesoscale Hydrodynamics Simulations of Vesi-
c
tilt angle θ vanishes (see Fig. 2(a)). The critical value clesunderFlow.– Weemploymesoscalehydrodynam-
κ∗ increases with increasing corrugation a of the chan- icssimulationstostudytheflowbehaviorofvesiclesinnar-
c y
nel, but is almost independent of L for L /L > 2 (see row channels. A dynamically-triangulated surface model
x x y
Fig.2(c)). Thesymmetricvesicledeformationreducesthe for the membrane [32] is combined with a particle-based
disturbance of the original flow but increases the free en- mesoscale simulation technique — multi-particle collision
ergyofavesicle. Atsmallorlargeκ∗,theformerorlatter dynamics (MPC) [33–35] — for the embedding fluid. A
contribution dominates, respectively. detaileddescriptionofthisapproachtomodelvesiclesdy-
p-3
H. Noguchi et al.
0.5 lx
y ly
L
l / lz
0.4
0.3
-0.5 0 x / L 0.5
G x
Fig. 4: (Color online) Maximum lengthslx, ly, and lz of vesi-
clesinexperiments(solidlines) andsimulations(dashedlines)
from thesame data as shown in Fig. 3.
Vesicles in sawtooth-shaped channels in fast
flows. – We investigate the dynamics of large prolate
vesicles in fast flows, corresponding to small κ∗, both by
experiments and simulations. In this case, hydrodynamic
interactions between the vesicle and wall are not negligi-
ble. Weconsidernowaperiodically-patternedmicrochan-
nel with a saw-tooth shape (see Figs. 1 and 3), for which
the walls are located at
L 4x
y
h (x)= 1+a 1 (5)
y 2 n y(cid:16) ± L (cid:17)o
x
for L /2 < x 0 and 0 < x L /2, respectively.
x x
− ≤ ≤
Two types of channels are used, which are characterized
by L = 50µm, a = 0.2 and L = 75µm, a = 0.33
y y y y
(as well as L = 50µm and L = 100µm). The glucose
Fig. 3: (Color online) Sequence of snapshots (at equal time z x
and sucrose solutions are used for the experiments with
intervals)ofvesicleswithV∗=0.96andRS/Ly =0.21moving
the narrower and wider channels, respectively. Typical
through a microchannel, experimentally observed by optical
microscopy(insucrosesolution)forκ∗ =0.04(vm =38µm/sec, experimental flow velocities are in the range vm = 10 –
Ly = 75µm) (top), and in simulation for κ∗ = 0.08 (bottom). 100µm/s.
Fortheexperimentaldata,thetimeintervalbetweenimagesis First, we describe the dynamics in the wider channel
0.97 sec. (L = 75µm). Experimental and simulation results for
y
the shape deformation are shown in Figs. 3 and 4. As ex-
pected, the vesicles change their shapes periodically. For
namics under flow can be found in Refs. [4,18]. In MPC, V∗ &0.9, vesicle shapes vary continuously between an al-
the fluid is described by N point-like particles of mass most ellipsoidal(in the narrowpart of the channel) and a
s
m . After free streaming of every particle for a time cone-like(inthewidepartofthechannel)bulletshape,see
s
step ∆t, the fluid particles collide in cubic boxes of lat- Fig. 3. This is in contrast to the behavior for unbounded
tice constant a = L /16, which is also the mean length steadyPoiseuilleflow,whereperturbationtheorypredicts
x
between membrane vertices. We use the fluid viscosity acoexistencebetweenbullet-andparachute-likeshapesat
η = 550√m k T/a2 (with number density n = 100a−3 the center line [29]. The amplitudes andphases ofthe de-
s B
and the time step ∆t = 0.01a m /k T), corresponding formationsinthex-andy-directions(seeFig.4)agreewell
p s B
tolowReynoldsnumbers,tosimulateexperimentalcondi- betweensimulationsandexperiments,whileverylittlede-
tions. We study vesicles with bending rigidity κ=20k T formation is seen in the z-direction. The small difference
B
where k T is the thermal energy. The flow velocity is of amplitudes between the experiment and simulation is
B
chosen to be v τ/L = 170(R /L )3 for a channel with probably due to the somewhat different values of the re-
m y V y
L = L = L /2, corresponding to a slow vesicle relax- duced volume and radius of the vesicle, since an accurate
y z x
ationcomparedtothepassagetimethroughachannelseg- experimentalestimation is difficult (because the length in
ment. For vesicles with R /L = 0.36 and R /L = 0.46 the z direction could not be measured). The dynamics
S y S y
at V∗ = 0.9, the reduced bending rigidity is κ∗ = 0.14 also agrees qualitatively with the theoretical predictions
and κ∗ = 0.07, respectively. In this regime, the vesicle for quasi-spherical vesicles, compare Fig. 2, although the
dynamics is not very sensitive to κ∗. parameters are outside the range of validity of the theo-
p-4
Dynamics of Fluid Vesicles in Flow through Structured Microchannels
0.3 e1 ly s4
e4
s1 e3
s2
0.2 e2 s3
>
<l e1
l /
D0.1
e2
l
x
0
0.8 0.9 1
V*
0.25
p/
q
0
0 1 x / L 2 3
G x
Fig. 6: (Color online) Oriental oscillation of a vesicle (in su-
crose solution) in a wide structured channel with Ly =75µm.
(a) Micrograph of an oscillating vesicle with R = 24µm,
V
v = 34 µm/s, corresponding to κ∗ = 0.0132. The image is
a superposition of video frames taken at time intervals of 1s.
(b)Orientationalangle asafunction of thecenter-of-mass po-
Fig.5: (Coloronline)Normalizedoscillationamplitudes∆l/hli
sition x . The (red) circles (◦) represent the same data as in
inthex(red)andy (blue)directions. Opensymbolsrepresent G
(a). Thevesicleoscillatesbetweentheextremaofθ =0.07π
simulation data for RS/Ly = 0.36 (◦, (cid:3)) and RS/Ly = 0.46 min
and θ =0.3π. The (blue) triangles represent another vesi-
(△, ⋄) at ay = 0.2, respectively. Closed symbols represent max
cle (with R =18µm, v=±55µm/s). Here, we have inverted
experimental data for Ly = 50µm with ay = 0.2 in glucose V
the direction of the fluid flow; symbols indicate angles before
solution(•,(cid:4))andLy =75µm withay =0.33insucrosesolu-
(△) and after (N) flow inversion.
tion (N, (cid:7)), respectively. The solid lines are guide to the eye.
The lower part shows snapshots of simulations (s1, s2, s3, s4)
andexperiments(e1,e2,e3,e4)forLy =50µm. Allsnapshots
the tail is found in the z direction. Recently, Abkarian et
are displayed with thesame length scale. The picturese1 and
e2 show the same vesicle before and after the transformation al. [14] found a similar asymmetric tail for RBCs in very
from ellipsoid to tailed shape, respectively. All images (e1-e4) fast flows in homogeneous capillaries (with flow velocity
are vesicles in glucose solution. v = 10 35 cm/s and capillary diameter 10µm). Such
m
−
a long tail is much more difficult to form in RBCs due
to the shear resistance of the spectrin network, and may
retical approximations. evenrequirealocalseparationofthelipidbilayerfromthe
At reduced volumes V∗ .0.9, we find that the vesicles spectrin network. Thus, the formation of an asymmetric
deformintonovelshapes,whilethey arestillellipsoidalin longtailis ageneric feature incapillaryflows. Structured
the absence of flow. At V∗ 0.85, the simulated vesicles channels promote tail formation already at smaller flow
∼
developtwotailsinthexy plane(seeFig.5(s3)). Thistail velocities. The low contrastandthe blurrycontourin the
position is very stable; the tails quickly return to the xy rear part of the vesicles in the experimental micrographs
plane,evenfromaninitial state withthe tails in xz plane of Fig. 5 (e3, e4) indicates that vesicles exhibit enhanced
(obtained by π/2 rotation). At V∗ 0.80, the symmetric shape fluctuations at the rear part, due to a locally re-
∼
tailsbecomeunstable;theyarereplacedbyastableshape duced membrane tension; the low contrast regions in the
withasingleasymmetrictail(seeFig.5(s2)). Inthiscase, shapes of Fig. 5 (e1, e2) is probably due to a tilt of the
the configurationwith a tail in the xz plane is also stable vesicleaxisandavesicleasymmetryinducedbyoff-center
within the accessible simulation time (see Fig. 5(s1)). A motion with a buoyancy force.
single-tail shape is also observed in our experiments with Ourexperimentalandsimulationresults for the oscilla-
vesicles in glucose solution (see Fig. 5(e2)); here, the off- tion amplitude ∆l, displayed in top part of Fig. 5, show
center motioninduced by the buoyancy forceimplies that that the difference between maximum and minimum of
p-5
H. Noguchi et al.
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