Soft Matter

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Department of Chemical and Biomolecula

College Park, Maryland 20742, USA. E-mail

Cite this: Soft Matter, 2013, 9, 4284

Received 20th November 2012Accepted 7th February 2013

DOI: 10.1039/c3sm27683j

www.rsc.org/softmatter

4284 | Soft Matter, 2013, 9, 4284–42

Deformation of an elastic capsule in a rectangularmicrofluidic channel

S. Kuriakose and P. Dimitrakopoulos*

In the present study we investigate computationally the deformation of an elastic capsule in a rectangular

microfluidic channel and compare it with that of a droplet. In contrast to the bullet or parachute shape in a

square or cylindrical channel where the capsule extends along the flow direction, in a rectangular channel

the capsule extends mainly along the less-confined lateral direction of the channel cross-section (i.e. the

channel width), obtaining a pebble-like shape. The different shape evolution in these two types of solid

channels results from the different tension development on the capsule membrane required for

interfacial stability. Furthermore, in asymmetric channel flows, capsules show a different deformation

compared to droplets with constant surface tension (which extend mainly along the flow direction) and

to vesicles which extend along the more-confined channel height. Therefore, our study highlights the

different stability dynamics associated with these three types of interfaces. Our findings suggest that the

erythrocyte deformation in asymmetric vessels (which is similar to that of capsules) results from the

erythrocyte's inner spectrin skeleton rather than from its outer lipid bilayer.

1 Introduction

The study of the interfacial dynamics of articial or physiolog-ical capsules (i.e. membrane-enclosed uid volumes) in Stokesows has seen an increased interest during the last few decadesdue to their numerous engineering and biomedical applica-tions. Articial capsules have wide applications in the phar-maceutical, food and cosmetic industries.23 In pharmaceuticalprocesses, for example, capsules are commonly used for thetransport of medical agents. In addition, the motion of redblood cells through vascular microvessels has long beenrecognized as a fundamental problem in physiology andbiomechanics, since the main function of these cells, toexchange oxygen and carbon dioxide with the tissues, occurs incapillaries.22

In the area of interest of the present paper, the study of themotion and deformation of capsules and biological cells inmicrouidic channels is motivated by a wide range of applica-tions including drug delivery, cell sorting and cell character-ization devices,1–3,5,7,15,27 fabrication of microcapsules withdesirable properties,8,17,21,26 determination of membrane prop-erties,19,24 microreactors with better mixing properties,4,32 and ofcourse its similarity to blood ow in vascular capillaries.22,23

Studying the shape of so particles in conned solid ducts,such as microuidic channels and blood microvessels, providesuseful information on the utilization of these particles inchemical, pharmaceutical and physiological processes. For

r Engineering, University of Maryland,

: dimitrak@umd.edu

96

example, understanding the stability of so particle shapesprovides helpful insight into the hydrodynamic aggregation andthe effective viscosity of suspensions.6 The deformation ofarticial capsules in microchannels is directly associated withdrug delivery, cell sorting and cell characterization.1,2 Further-more, the deformability of red blood cells plays a pivotal role inthe oxygen and carbon dioxide exchange between the micro-circulation and the body tissues,22 and helps identifying theeffects of blood disorders and diseases.3,14,27

The shape of capsules and biological cells in solid ducts isdetermined by the nonlinear coupling of the deforminghydrodynamic forces with the restoring interfacial forces of theparticle membrane. Since the latter forces depend on the type ofthe so-particle interface, this suggests that different soparticles (such as droplets, capsules, vesicles and erythrocytes)may obtain quite different shapes as they travel in a solid vessel.

In axisymmetric-like solid ducts, such as cylindrical andsquare channels, so particles commonly obtain steady-statebullet-like and parachute-like shapes, elongated along the owdirection.6,18,25,28,35 The recent investigation of Coupier et al.6

showed that in rectangular microuidic channels, vesiclesobtain an unexpected croissant shape (relatively wider in thenarrowest direction of the channel) owing to the developmentof a four-vortex pattern on the uid-incompressible vesiclemembrane.

In this paper, we show that the capsule shape in a rectan-gular microuidic channel is quite different than that in asquare or cylindrical channel. In the latter channels, the capsuleextends along the ow direction only obtaining a bullet orparachute shape while in a rectangular channel it extends

This journal is ª The Royal Society of Chemistry 2013

Paper Soft Matter

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mainly in the less-conned lateral direction of the channelcross-section, i.e. the channel width, acquiring a pebble-likeshape. The different shape evolution in these two types of solidchannels is associated with the deformation of elastic capsulesonly and results from the different tension development on thecapsule membrane needed for interfacial stability. Thus, inasymmetric channel ows, capsules show a different deforma-tion compared to droplets with constant surface tension (whichextend mainly along the ow direction) and to vesicles whichextend along the more-conned channel height.6 Our ndingsalso provide physical insight into the deformation of erythro-cytes in rectangular microuidic channels (which also extendmore along the less-conned channel width),33 and suggest thatthe particular shape results from the inner cytoskeleton of theerythrocytes rather than from their outer lipid bilayer.

2 Problem description

We consider a three-dimensional capsule (with a sphericalundisturbed shape and an elastic interface) owing along thecenterline of a straight microchannel with a (constant) rectan-gular cross-section as illustrated in Fig. 1. The capsule's interiorand exterior are Newtonian uids, with viscosities lm and m, andthe same density. The capsule size a is specied by its volumeV ¼ 4pa3/3. The height of the channel's rectangular cross-section is l z and its width l y > l z.

Far fromthe capsule, theowapproaches theundisturbedowuN in a channel characterized by an average velocity U. (The exactform of the channel's velocity eld uN and its average velocityU isgiven in Section 2 of our recent paper on capsule motion in asquare microchannel.18) We assume that the Reynolds number issmall for both the surrounding and the inner ows, and thus thecapsule deformation occurs in the Stokes regime.

In addition, we consider a slightly over-inated capsulemade of a strain-hardening membrane following the Skalaket al. constitutive law29 (and thus called Skalak capsule in thispaper) with comparable shearing and area-dilatation resistance.This capsule description represents well bioarticial capsulessuch as the capsules made of covalently linked human serumalbumin (HSA) and alginate used in the experimental study ofRisso, Colle-Pailot and Zagzoule.25

For a membrane with shearing and area-dilatation resis-tance considered in this work, the surface stress is determinedby the in-plane stresses, i.e. Df ¼ �Vs $ s which in a contra-variant form gives

Df ¼ �(sab|atb + babsabn) (1)

Fig. 1 An elastic capsule flowing at the centerline of a rectangularmicrochannel.

This journal is ª The Royal Society of Chemistry 2013

where the Greek indices range over 1 and 2, while Einsteinnotation is employed for (every two) repeated indices. In thisequation, the sab|a notation denotes covariant differentiation,tb ¼ vx/vqb are the tangent vectors on the capsule surfacedescribed with arbitrary curvilinear coordinates qb, and bab isthe surface curvature tensor.12,23

The in-plane stress tensor s is described by constitutive lawsthat depend on the material composition of the membrane. Inthis work, we employ the Skalak et al. law29 which relates s'seigenvalues (or principal elastic tensions sPb, b ¼ 1, 2) with theprincipal stretch ratios lb by

sP1 ¼ Gsl1

l2

nl1

2 � 1þ Cl22hðl1l2Þ2 � 1

io(2)

Note that the reference shape of the elastic tensions is thespherical quiescent shape of the capsule while to calculate sP2reverse the lb subscripts. In eqn (2), Gs is the membrane'sshearing modulus while the dimensionless parameter C isassociated with the area-dilatation modulus Ga of themembrane (scaled with its shearing modulus).23,29

We further consider that the capsule is subjected to a posi-tive osmotic pressure difference between the interior and exte-rior uids, i.e. the capsule is (slightly) over-inated and thusprestressed. For this, we dene the prestress parameter ap suchthat all lengths in the undeformed capsule would be scaled by(1 + ap), relative to the reference shape.18 Since the capsule isinitially spherical, its membrane is initially prestressed by anisotropic elastic tension s0 ¼ sPb (t ¼ 0) which depends on theemployed constitutive law and its parameters but not onthe capsule size. For example, for a Skalak capsule with C ¼ 1and ap ¼ 0.05, the undisturbed capsule size a is 5% higher thanthat of the reference shape and the initial membrane tensionowing to prestress is s0/Gs z 0.3401. It is of interest to note thata (small) prestress also acts as a bending resistance and thusremoves the buckling instability at the capsule's lateral cross-section.18

The numerical solution of the interfacial problem is ach-ieved through our interfacial spectral boundary elementmethod for membranes, and the interested reader is referred toour earlier papers for more details on our spectral algorithmand our recent investigation on the capsule dynamics in asquare microchannel.12,18 To verify the accuracy of our results,we performed convergence runs covering the entire interfacialevolution (i.e. well past steady state) with different spacial gridsfor several capsules and ow rates. Our convergence runsshowed that our results for the interfacial shape, the capsulevelocity, the additional pressure difference and the shear stresson the solid walls (including the channel corners) were deter-mined with an accuracy of at least 3 signicant digits.

In our work we assume that the ow rate Q (or the averageundisturbed velocity U) inside the channel is xed. In addition,in Sections 3 and 4 we consider a Skalak capsule with prestressap ¼ 0.05 and membrane hardness C ¼ 1, moving in a rectan-gular channel with aspect ratio l y/l z ¼ 2. The effects ofmembrane hardness C, capsule prestress ap and channel'saspect ratio l y/l z are investigated in Section 5. The presentproblem depends on two additional dimensionless parameters:

Soft Matter, 2013, 9, 4284–4296 | 4285

Table 1 Range of parameters studied in this work. The length scale is thechannel's half-height l z, while if the channel's aspect ratio l y/l z is not specified,then our results are valid for l y/l z ¼ 2

Varying parameter Fixed parameters

Ca ¼ 0.05–0.2 a ¼ 0.6–1.1, C ¼ 1, ap ¼ 0.05a ¼ 0.1–1.1 Ca ¼ 0.1, 0.2, C ¼ 1, ap ¼ 0.05C ¼ 0.5–5 a ¼ 0.8, Ca ¼ 0.1, s0/Gs ¼ 0.3401ap ¼ 0.025–0.1 a ¼ 0.8, Ca ¼ 0.1, C ¼ 1ly/lz ¼ 1–3 a ¼ 0.8, Ca ¼ 0.1, C ¼ 1, ap ¼ 0.05

Fig. 2 Steady-state shape of a Skalak capsule with C ¼ 1, ap ¼ 0.05 and size ¼1.1 in a rectangular microchannel, for capillary number Ca ¼ 0.05. The capsuleshape is plotted as seen from (a) the negative y-axis, (b) the positive z-axis andslightly askew from it, and (c) the negative x-axis.

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the capsule size (relative to the channel height) a/lz, and thecapillary number Ca dened as

Ca ¼ mUGs

(3)

It is of interest to note that the capillary number, as denedby eqn (3), does not contain any length scale, and thus it may beconsidered as a dimensionless ow rate. In this study, if noscale is present, the channel's half-height l z is used as thelength scale, the velocity is scaled with the average undisturbedvelocity U, and thus time is scaled with l z/U, while themembrane tensions are scaled with Gs.

The range of dimensionless parameters employed in ourcomputational work (shown in Table 1) can readily be used inexperimental microuidic systems. As an example, Leclercet al.19 investigated ovalbumin microcapsules with shearmodulus Gs ¼ 0.07 N m�1 in microuidic channels with atypical height l z ¼ 50 mm. Using as external uid glycerin withviscosity m z 1 Pa s and average velocities U ¼ 1–4 cm s�1, theauthors achieved capillary numbers in the range Ca z 0.1–0.5.Considering erythrocytes with shear modulus Gs ¼ 2.5 mN m�1

(ref. 10 and 16) in microuidic channels with height l z ¼ O(10)mm and an external liquid with a viscosity similar to that ofwater, mz 1 mPa s, the same range of capillary numbers can beachieved with average velocities U ¼ O(1) mm s�1.

To facilitate description of our results, we imagine that thechannel is horizontal, as illustrated in Fig. 1. Thus, the owdirection (i.e. the x-axis) corresponds to the channel's or cap-sule's length, the z-direction will be referred to as the heightwhile the y-direction will be referred to as the width (of thechannel or the capsule). Seeing the capsule from the negativey-axis, positive z-axis or negative x-axis represents a front view, atop view or an upstream view, respectively. In addition, weadopt the standard denition of geometric shapes (e.g. poly-gons). Thus we call the capsule's rear edge as convex when theradius of curvature at the middle of the rear edge points insidethe capsule (i.e. the local curvature is positive); in the oppositecase the edge shape is concave.

3 Effects of flow rate

In this section, we collect our steady-state results as a functionof the ow rate Ca, for capsule motion in a rectangular micro-channel with aspect ratio l y¼ 2l z, and capsule sizes comparableto the channel's height l z. In particular, we consider Skalakcapsules with size a ¼ 0.8, 0.9, 1.1 and for small-to-moderate

4286 | Soft Matter, 2013, 9, 4284–4296

capillary number Ca in the range [0, 0.2]. To obtain these steady-state results we initiate our computations from a sphericalcapsule (for a < 1) or an ellipsoidal capsule (for a > 1) at thechannel centerline using viscosity ratio l ¼ 1 and compute thecapsule dynamics until times t ¼ 10–20, i.e. well-past steadystate which usually is achieved around time t ¼ 2–4.

Owing to the specic symmetry of the channel ow and thematerial interface (i.e. the elastic-solid membrane) of thecapsule, at steady state there is no ow inside the capsule (inagreement with our computational results), and thus the steady-state capsule properties are independent of the inner viscosityor the viscosity ratio l.18,23 For the same reason, the membraneviscosity (if any), which is not accounted for in our computa-tions, does not affect the capsule's steady-state properties.

Fig. 2 shows different views of the steady-state capsule shapefor capillary number Ca¼ 0.05. Note that the three-dimensionalcapsule views presented in this paper were derived from the

This journal is ª The Royal Society of Chemistry 2013

Fig. 3 Steady-state shape of a Skalak capsule with C ¼ 1, ap ¼ 0.05 and size ¼1.1 in a rectangular microchannel, for capillary number Ca ¼ 0.2. The capsuleshape is plotted as seen from (a) the negative y-axis, (b) the positive z-axis andslightly askew from it, and (c) the negative x-axis.

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actual spectral grid by spectrally interpolating to a denser grid(with NB ¼ 25 spectral points along each direction of eachspectral element) and using orthographic projection in plotting.

At this low ow rate, the capsule appears like a bullet if seenfrom the front as in Fig. 2(a), i.e. the capsule downstream edgehas become more pointed while its upstream edge has becomeatter but remains convex. Thus, this capsule view resemblesthe bullet-like shape observed in square and circular solid ductsat low ow rates.18,25 The explanation for the common behavioris straightforward if we consider the balance of the deforminghydrodynamic forces with the restoring membrane tensionforces. That is, to balance the deforming hydrodynamic forces,the capsule tries to increase its downstream curvature anddecrease its upstream curvature so that the total restoringtension force on the membrane is increased. In essence, thiscapsule deformation results from the curvature term in themembrane traction, eqn (1), as we identied in our earlierstudies on capsule dynamics in planar extensional ows or insquare channels.11,12,18

The effects of the channel's cross-section asymmetry are seenfrom the other two directions, i.e. in Fig. 2(b) and (c). Owing tothe channel's rectangular cross-section, the capsule is extendedalong the less-conned width direction, and from the top, itsshape appears like a pebble as seen in Fig. 2(b). The capsule'swidth extension is also shown from the upstream directionwhere its shape appears like a attened ellipse, i.e. rectangular-like with rounded corners. A lateral dimple with negativecurvature has also been developed at the capsule's upstreamportion where the membrane is in proximity with the channel'stop and bottom walls (owing to the strong local normal lubri-cation forces), as shown at the right of Fig. 2(b). We emphasizethat in the present study, the membrane tensions at steady stateare always positive owing to prestress and thus negative curva-ture or dimples on the capsule interface cannot result from localnegative tensions, i.e. interfacial dimples are pure hydrody-namic effects.

Clearly, the non-axisymmetric channel's cross-section hasresulted in a highly non-axisymmetric, fully three-dimensionalcapsule shape which cannot be described from single-viewobservations as commonly happens in microuidic channels orbased on axisymmetric or two-dimensional computations.

By increasing the ow rate to Ca ¼ 0.2, the capsule appearslike a parachute from a front view as seen in Fig. 3(a). Thestronger hydrodynamic forces cause the capsule to obtain amore pointed downstream edge and a concave upstream edge(with a negative curvature). This shape increases the membraneinterfacial forces required to balance the increased hydrody-namic forces as also happens in square and circular ducts.18,25

However the capsule appears quite different if seen from thetop; the capsule shape seems circular-like as shown at the le ofFig. 3(b). In essence, the capsule has obtained now a pebbleshape attened along the channel width with two wideupstream tails. Seeing the capsule from the upstream directionin Fig. 3(c) reveals a greater width extension (compared to thatfor Ca ¼ 0.05); however this view does not provide informationon the sign of the rear curvature, i.e. if the capsule rear is convexor concave.

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The effects of the ow rate on the capsule shape is clearlyshown in Fig. 4 where we plot the capsule proles (i.e. the cross-section of the capsule surface with the planes y ¼ 0, z ¼ 0 andx ¼ 0) for capillary number Ca ¼ 0.05, 0.1, 0.2. As the ow rateincreases, the thickness h of the lubrication lm between thecapsule surface and the solid walls in the more connedchannel's height is increased as seen in Fig. 4(a). This helps inthe steady-state equilibrium by reducing the strong locallubrication forces in the ow direction which (per volume) scaleas

Fxf � m

d2ux

dz2� m

Ux

h2(4)

where Ux is the capsule velocity (see also Section 5 in ref. 18 andSection 3 in ref. 31). For the highest ow rate studied, i.e. Ca ¼0.2, the capsule's concave upstream edge is accompanied by twopointed rear tails (close to the solid walls) as shown in Fig. 4(a),and thus with the development of small local length scales.

Soft Matter, 2013, 9, 4284–4296 | 4287

Fig. 4 Steady-state profile for a Skalak capsule with C¼ 1, ap¼ 0.05 and size ¼1.1 in a rectangular microchannel, for capillary number Ca ¼ 0.05, 0.1, 0.2.Capsule profile, i.e. interface intersection with the plane (a) y¼ 0, (b) z¼ 0, and (c)x ¼ 0. All profiles are shown with centroid xc ¼ 0.

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It is of interest to note that the capsule shape is attenedalong the channel's top or bottom walls in the capsule'supstream portion (i.e. from its centroid to its rear end) as seenin Fig. 4(c). On the other hand, the capsule's downstreamportion is ellipsoidal-like especially close to its front edge. Thusthe capsule shape is fully three-dimensional mainly in itsupstream portion.

We conclude this section by stating that the ow rate effectswe have identied for the capsule size a ¼ 1.1 (i.e. slightly

4288 | Soft Matter, 2013, 9, 4284–4296

higher than the channel height l z) are identical to those formoderate-size capsules (e.g. size a ¼ 0.8, 0.9) and thus theseresults are omitted.

A Comparison with droplet dynamics

Owing to the rather unexpected effect of the ow rate on thecapsule shape in a rectangular microchannel (i.e. to increase thecapsule extension in the less-conned lateral direction), weproceed now by investigating the corresponding effects on thedroplet deformation. A droplet is also a deformable object but,in contrast to a capsule, its surface tension g remains constant.Thus, our goal here is to clarify the similarities and differencesof droplet and capsule dynamics in rectangular microchannels.

To determine the droplet dynamics, we utilized both ourexplicit and the fully implicit time-integration spectral algo-rithms for droplets.9,34 Our convergence runs showed that ourdroplet dynamics results were determined with at least 3signicant-digit accuracy. We emphasize that our spectralboundary element method has also been used to study dropletmotion in solid channels of different cross-sections and we havemade successful comparisons with published experimental andcomputational results.20

For droplet motion, the capillary number is dened as Ca ¼mU/g where g is the droplet's surface tension. In addition, theviscosity ratio l does affect the droplet shape even at steady stateowing to the non-zero inner circulation. As the viscosity ratio l

increases, the droplet deformation increases owing to thehigher inner hydrodynamic forces. Since our droplet results forl ¼ 1, 5 are qualitatively similar, we only present our results forl ¼ 1 droplets.

As seen in Fig. 5(a) the droplet's y ¼ 0 prole changes frombullet-like to parachute-like as the ow rate increases, similarlyto what happens for a capsule shown earlier in Fig. 4(a). Thus athigh enough Ca, the droplet's rear edge also becomes concavewith a negative curvature. However, the droplet does notdevelop very pointed tails at its upstream section in proximity tothe solid walls. Furthermore, the ow rate increase causes asignicant droplet extension along the ow direction whichresults in a different z ¼ 0 prole (compared to that for acapsule) as shown in Fig. 5(b). Even the droplet's x ¼ 0 prolealong the channel cross-section is not so attened but it is moreelliptical as seen in Fig. 5(c).

The signicant differences in the shape of capsules anddroplets are clearly shown in Fig. 6 where we plot three-dimensional views of the droplet shape for capillary numberCa ¼ 0.3. This ow rate causes the same rear concavity (i.e. thesame negative curvature at the rear edge) as for the capsule withCa ¼ 0.2 shown earlier in Fig. 3.

Our capsule-vs.-droplet conclusions are quantied in Fig. 7and 8 where we plot the steady-state lengths and the tailcurvature as a function of the capillary number Ca for these twodeformable particles.

Fig. 7 shows that as the ow rate increases, both deformableparticles are restricted similarly along the channel's moreconned height direction, i.e. their height Lz is reduced simi-larly. However, to accommodate the higher hydrodynamic

This journal is ª The Royal Society of Chemistry 2013

Fig. 5 Steady-state profile for a droplet with viscosity ratio l¼ 1 and size ¼ 1.1in a rectangular microchannel, for capillary number Ca ¼ 0.05, 0.1, 0.2, 0.3.Droplet profile, i.e. interface intersection with the plane (a) y¼ 0, (b) z¼ 0, and (c)x ¼ 0. All profiles are shown with centroid xc ¼ 0.

Fig. 6 Steady-state shape of a droplet with viscosity ratio l ¼ 1 and size ¼ 1.1in a rectangular microchannel, for capillary number Ca¼ 0.3. The droplet shape isplotted as seen from (a) the negative y-axis, and (b) the positive z-axis and slightlyaskew from it.

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forces, the droplet extends mainly along the ow direction (i.e.higher length Lx) while the capsule extends rather unexpectedlyalong the channel's less conned width direction (i.e. higherwidth Ly).

Fig. 8 shows the steady-state maximum curvature Cmaxxz as a

function of the ow rate for the two so particles. Observe thatCmaxxz is a line curvature, determined along the interfacial cross-

section with the y¼ 0 plane, and takes on positive values for thespherical quiescent shape where Cmax

xz ¼ a�1. Thus, for theproblem studied in this paper, this curvature represents thelocal curvature at the interfacial tail developed in the particleupstream portion near the channel walls, as seen in Fig. 4(a)

This journal is ª The Royal Society of Chemistry 2013

and 5(a). Clearly, the tail curvature is much higher for a capsulethan for a droplet, and thus at their rear tails capsules developpointed local areas characterized by a local length scale (orradius of curvature) which is of O(30) smaller than the capsulesize.

B Reasoning for the capsule deformation

The explanation for the different shape development between adroplet and a capsule in a rectangular channel lies on thetension development on the capsule membrane. Excluding theprestress tension s0, a capsule does not have membranetensions at the quiescent spherical condition. Membranetensions develop as the capsule deforms due to the externalow; they may grow locally under extension dynamics or decayunder local compression.

In a rectangular channel, a so particle (with a size compa-rable to the channel's height) causes a signicant ow blockingalong the channel height; however its blocking is much smalleralong the channel width. As the ow rate increases, the particleheight Lz is decreased owing to the stronger normal hydrody-namic forces in the small gap between the particle surface andthe channel's top and bottom walls.

For a droplet with constant surface tension g, the increasedow rate and the height reduction Lz cause a signicant exten-sion Lx along the ow direction, i.e. both the particle's heightreduction and length extension are pure hydrodynamic effects.However, the droplet's constant surface tension creates stronglateral interfacial forces which are able to balance the lateralow forces; thus the droplet is not extended much along itswidth.

For the case of a capsule, the increased ow rate and theheight reduction cause also the capsule to extend along the owdirection. Thus, along its lateral cross-section, the membrane

Soft Matter, 2013, 9, 4284–4296 | 4289

Fig. 7 Steady-state projection lengths, Lx, Ly and Lz (scaled with the length 2 ofthe undisturbed spherical shape) of (a) a Skalak capsule with C¼ 1 and ap ¼ 0.05,and (b) a droplet with viscosity ratio l¼ 1, as a function of the capillary number Cafor size ¼ 1.1 in a rectangular microchannel. These lengths are determined asthe maximum distance of the deformable interface in the x, y and z directions.

Fig. 8 Steady-state maximum curvature Cmaxxz along the y¼ 0 profile, for a Skalak

capsule with C ¼ 1 and ap ¼ 0.05, and for a droplet with viscosity ratio l ¼ 1, as afunction of the capillary number Ca for size ¼ 1.1 in a rectangular microchannel.The curvatures are scaled with the curvature of the undisturbed spherical shape.

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develops stabilizing tensions pointing in the ow direction. Atthe same time, the ow forces also extend the capsule along theless-conned width direction owing to the initial weak lateraltensions. This lateral extension results in tension development(pointing in the lateral direction) while the width extensionceases when the lateral membrane tensions grow strong enoughto withstand the deforming hydrodynamic forces.

Therefore, both the capsule's length and width extensionscontribute to increased membrane tensions and thus to theinterfacial stability. The greater width extension compared tothe length extension reveals the membrane preference indeveloping strong lateral tensions and clearly shows that thecapsule dynamics is a three-dimensional phenomenon. Inessence, the capsule's width extension resembles the lateralextension of capsules in planar extensional ows which alsobuilds strong lateral tensions.11,12

To support our reasoning, in Fig. 9(a) we plot the steady-statedistribution of the two principal tensions, sPF and sPL, on the

4290 | Soft Matter, 2013, 9, 4284–4296

membrane cross-section with the plane x ¼ xc, i.e. the x-coor-dinate of the capsule centroid. Observe that the tensions sPFpoint mainly towards the ow direction while the tensions sPLpoint mainly in the lateral direction. Owing to the capsule's owextension, the associated tensions sPF have been increasedsignicantly with respect to the prestress tension s0 of thespherical shape. However, the capsule's width extension alsocreates local extension on the membrane in the lateral direc-tion. Thus the lateral tensions sPL are also increased everywherealong the capsule lateral cross-section and especially at thelocations of the highest width extension, as seen in Fig. 9(a).

Even though the ow-oriented tensions sPF are higher thanthe laterally oriented tensions sPL, both of them contributesignicantly to the capsule stability since the membrane forcesalso include the associated interfacial curvatures. That is, assuggested by the curvature term in eqn (1), the restoringmembrane normal stress f nm along the capsule's lateral cross-section can be approximated by

f nm � sPFCF + sPLCL (5)

where CF and CL are the interfacial curvatures along the owand lateral directions, respectively. In general, CL > CF and thusboth normal stress components contribute to the capsulestability along its cross-section.

It is of interest to note that in a square or cylindrical channel,the required tension development results only from the capsuleextension along theowdirection, i.e. the only directionwhich isnot conned by the solid walls. In particular, the lateralmembrane tensions are decreased owing to the compressiondynamics associated with the reduction of the capsule's lateraldimensions in these solidducts. Both effects on the cross-sectionprincipal tensions are shown in Fig. 9(b) based on the results ofour earlier work for capsule motion in a square channel.18 Thisconstitutes a major difference in the membrane stability for acapsule moving in an asymmetric or symmetric solid duct.

This journal is ª The Royal Society of Chemistry 2013

Fig. 9 Steady-state distribution of the two principal tensions, sPF and sPL, amongthe spectral discretization points, on the membrane cross-section with the planex ¼ xc, as a function of the azimuthal angle f (measured in degrees), with respectto the positive y-direction. For a Skalak capsule with C ¼ 1, ap ¼ 0.05 and size ¼1.1 moving in (a) a rectangular channel with aspect ratio l y/l z ¼ 2 and Ca ¼ 0.2,and (b) a square channel with Ca ¼ 0.1. Also included as a horizontal line is the(constant) tension distribution s0 due to prestress.

Fig. 10 Steady-state capsule lengths, Lx, Ly and Lz, as a function of the capsule'ssize for a Skalak capsule with C¼ 1, ap¼ 0.05 and capillary number Ca¼ 0.2, in arectangular microchannel. The lengths are scaled with the length 2 of theundisturbed spherical shape.

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Note that in Fig. 9, the sinusoidal-like variation of themembrane principal tensions, sPF and sPL, with the azimuthalangle f results from the fact that, in a square or rectangularchannel, the local ow stresses near the middle of the channelwalls are different than those at the channel corners. Thiscreates varying local extension on the capsule membrane andthus varying tensions with the azimuthal angle f. Owing to thecross-section asymmetry, the period of this variation in a rect-angular channel is obviously twice that in a square channel.

The membrane's tension evolution also explains the muchmore pointed tails developed at the upstream portion of thecapsule compared to the droplet tails. For both so particles,the restoring tension forces (which are required to balance thedeforming ow forces) result from both the local tension andthe local curvature, e.g. the curvature term in eqn (1) or (5). Ourcomputations show that at the capsule tail, local compression

This journal is ª The Royal Society of Chemistry 2013

occurs and thus there the membrane tensions become weakeras the ow rate increases. Therefore, the capsule needs toincrease the tail curvature signicantly to produce a strongenough local interfacial force. On the other hand, on the dropletinterface, the surface tension g remains constant and thus asufficiently strong local surface tension force results from asmaller local curvature.

4 Effects of capsule size

In this section, we collect our steady-state results as a functionof the capsule size a, for capsule motion in a rectangularmicrochannel with aspect ratio l y ¼ 2l z and capillary numberCa ¼ 0.2. In particular, we consider Skalak capsules with C ¼ 1and ap ¼ 0.05, for size a ¼ 0.1, 0.2, ., 1.1, i.e.much smaller upto slightly higher than the channel height l z.

The effects of the capsule size a on the steady-state capsuledimensions and proles are shown in Fig. 10 and 11, respec-tively. For small capsule sizes a ( 0.4 the hydrodynamic forcesassociated with the ow rate Ca ¼ 0.2 are weak and causeminimal deformation. Thus the capsule remains nearly spher-ical with Lx z Ly z Lz z 2a and circular proles. For moderatecapsule sizes 0.5 ( a ( 0.7, the length Lx and the height Lz ofthe capsule are practically equal to their undisturbed value 2a.However, now the hydrodynamic forces are stronger owing tothe smaller gap between the capsule interface and the solidwalls, and cause the capsule to deform into a bullet-like shapewith a pointed downstream edge and a attened, convex rear asshown in Fig. 11(a). Most important, the capsule extends in theless-conned channel width and thus the capsule width Lybecomes larger than its other two dimensions as seen in Fig. 10.For higher capsule sizes a T 0.8, the capsule surface is inproximity to the channel's top and bottom walls. Thus, thecapsule height Lz is restricted owing to the strong normallubrication forces in the narrow gap between the capsule

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Fig. 11 Steady-state profile for a Skalak capsule with C ¼ 1, ap ¼ 0.05 andcapillary number Ca ¼ 0.2, in a rectangular microchannel, for capsule size ¼ 0.4,0.6, 0.7, 0.8, 0.9, 1, 1.1. Capsule profile, i.e. interface intersection with the plane (a)y ¼ 0, (b) z ¼ 0, and (c) x ¼ 0. All profiles are shown with centroid xc ¼ 0.

Fig. 12 Steady-state curvatures as a function of the capsule size for a Skalakcapsule with C ¼ 1, ap ¼ 0.05 and capillary number Ca ¼ 0.2, in a rectangularmicrochannel. (a) Edge curvature along the y ¼ 0 and z ¼ 0 profile. Downstreamedge: Cdxz and Cdxy; upstream edge: Cuxz and Cuxy. (b) Maximum curvature Cmax

xz alongthe y ¼ 0 profile. The curvatures are scaled with the curvature of the undisturbedspherical shape.

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surface and the solid walls. For these capsules, the capsulelength Lx starts to increase in order to accommodate the cap-sule's larger volume; however it always remains smaller than thecapsule width Ly as seen in Fig. 10.

Looking at the x¼ 0 prole of the different capsules includedin Fig. 11(c), we observe that the capsule remains axisymmetricuntil a z 0.4. At higher sizes, the increase of the capsule widthLy causes the capsule's x ¼ 0 prole to become rectangular-like

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with rounded corners, especially for the larger capsules westudied where the capsule interface becomes parallel to thechannel's top and bottom walls, as seen in Fig. 11(c). Therefore,for moderate and especially for large sizes, the capsule defor-mation is highly affected by the duct's rectangular cross-sectionand thus the capsule shape deviates signicantly from that in asquare or cylindrical channel.18,25

As the capsule size increases, the smaller gap between thecapsule and the channel's top and bottom walls creates strongerhydrodynamic forces. To increase the downstream membraneforces needed for interfacial balance, the capsule front becomesmore pointed along the y ¼ 0 prole and thus the downstreamedge curvature Cd

xz is increased, as shown in Fig. 11(a) and 12(a).However, the edge curvature Cd

xy along the z ¼ 0 prole remainspractically unchanged as seen in Fig. 12(a) since the capsulefront does not cause signicant ow blocking along the (wider)channel width.

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The three deformation regimes identied based on thecapsule dimensions result in three distinct regimes with respectto the shape of the capsule rear. For small capsule sizes a( 0.4,the capsule rear is spherical-like as shown in Fig. 11(a) and (b).For moderate capsule sizes 0.5 ( a ( 0.7, the increasedhydrodynamic forces make the capsule rear atter but stillconvex in both the front and top view, i.e. the y ¼ 0 and z ¼0 proles, as seen in Fig. 11(a) and (b). Therefore, both rearcurvatures Cu

xz and Cuxy decrease but remain positive; see

Fig. 12(a). This increases the net membrane forces on thecapsule by reducing the adverse upstream membrane force. Inessence, the capsule shape is a attened bullet, extended alongthe channel width.

For higher capsule sizes a T 0.8, to account for the strongerlubrication forces in the narrow gap along the channel height,the capsule obtains a parachute-like front view (or y ¼ 0 prole)with a concave upstream shape which produces a positive

Fig. 13 Transient evolution of the lengths, Lx, Ly and Lz, for a Skalak capsule withsize ¼ 0.8 and capillary number Ca ¼ 0.1 in a rectangular microchannel. Themembrane hardness is C¼ 0.5, 1, 2, 5 and the prestress ap ¼ 0.0732, 0.05, 0.0311,0.0147, respectively, so that all capsules have the same initial prestress tension s0/Gs z 0.3401. Capsule lengths: (a) Lx, and (b) Ly and Lz.

Fig. 14 Transient evolution of the lengths, Lx, Ly and Lz, for a Skalak capsule withC ¼ 1, size ¼ 0.8 and capillary number Ca ¼ 0.1 in a rectangular microchannel.The capsule prestress is ap ¼ 0.025, 0.03, 0.05, 0.1 and thus the initial prestresstension is s0/Gs z 0.1597, 0.1941, 0.3401, 0.7716. Capsule lengths: (a) Lx, and (b)Ly and Lz. For ap ¼ 0.025 our computations cover up to time t ¼ 15 where thesteady state is well established.

This journal is ª The Royal Society of Chemistry 2013

upstream membrane force (see Fig. 11(a) and 12(a)). However,along the less-conned channel width, the capsule rear stillappears like a bullet, with a convex shape and positive curvatureCuxy, owing to the smaller deforming hydrodynamic forces along

this direction. Therefore for these sizes, the capsule has obtaineda attened pebble shape with two pointed upstream tails.

For the larger capsules studied in this work, the stronglubrication forces in the narrow gap between the capsulesurface and the channel's top and bottom walls, cause thedevelopment of upstream pointed tails (in proximity to the solidwalls) as shown in the proles included in Fig. 11(a). Thus thetail curvature, represented by the curvature Cmax

xz in Fig. 12(b),increases signicantly with the capsule size. As explained inSection 3B, these pointed tails result from the decrease of thelocal membrane tensions, and thus the need for the capsule toincrease the tail curvature signicantly to produce strongenough local interfacial forces.

Soft Matter, 2013, 9, 4284–4296 | 4293

Fig. 15 Transient evolution of the lengths, Lx, Ly and Lz, for a Skalak capsule withC ¼ 1, ap ¼ 0.05 and size ¼ 0.8, for capillary number Ca ¼ 0.1 in a rectangularmicrochannel with aspect ratio l y/l z¼ 1.1, 1.25, 1.5, 2. Capsule lengths: (a) Lx, and(b) Ly and Lz.

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5 Effects of membrane hardness, prestressand channel asymmetry

In this section we consider briey the effects of the membranehardness, capsule prestress and channel's aspect ratio on thedeformation of the elastic capsule. The goal of this section is toinvestigate the effects of these parameters on the mainconclusion of this work, i.e. the fact that a capsule in a rectan-gular channel extends mainly along the less-conned channelwidth.

Fig. 13 focuses on the deformation of strain-hardeningSkalak capsules with increasing membrane hardness (i.e. C ¼0.5, 1, 2, 5) and the same initial prestress tension s0/Gs z0.3401. As the membrane hardness increases, its increasedarea-dilatation modulus creates stronger locally isotropictensions and thus tries to make the deformed capsule shapemore axisymmetric-like. (See the isotropic term containing themembrane hardness C at the right-hand-side of the Skalak et al.law in eqn (2).) Thus, as seen in Fig. 13(b), at steady state both

4294 | Soft Matter, 2013, 9, 4284–4296

the reduction of the capsule height Lz and the increase of itswidth Ly are decreased with C, and thus the capsule length Lx isincreased to accommodate the same capsule volume. Never-theless, even the most strain-hardening Skalak capsule westudied shows mainly an extension along the less-connedchannel width and thus an increased width Ly.

As the capsule prestress parameter ap increases, theincreased prestress tension s0 acts as an additional restoringinterfacial force and thus reduces the deformation of thecapsule for a given ow rate Ca, as shown in Fig. 14. However,the capsule asymmetry along the two channel's lateral direc-tions is preserved while the capsule is mainly extended alongthe less-conned channel width. It is of interest to note that thereduction (or complete elimination) of the capsule prestressincreases signicantly the capsule's asymmetric deformationand thus the growth of the capsule width Ly (relative to itsheight Lz). Via this lateral extension, the capsule builds stronglateral membrane tensions necessary for interfacial stability, asdiscussed in Section 3B.

We conclude this session by stating that the capsule's lateralextension is also present in rectangular channels with varyingaspect ratio l y/l z, as shown in Fig. 15. Therefore, this propertyrepresents a general feature of the deformation of elastic strain-hardening capsules in rectangular channels with comparablesizes, a and l z. In essence, in these duct ows, capsules show adifferent deformation compared to droplets with constantsurface tension (which extend mainly along the ow direction)and the vesicles which extend along the more-conned channelheight.6

6 Conclusions

In this paper we have investigated computationally the defor-mation of an elastic capsule in a rectangular microuidicchannel and compared it with that of a droplet. In particular, wehave considered an elastic capsule made of a strain-hardeningmembrane (following the Skalak et al. constitutive law) withcomparable shearing and area-dilatation resistance. Ourinvestigation involves low-to-moderate ow rates with capillarynumber Ca¼ O(0.1) and capsule size a comparable or smaller tothe channel's narrowest direction (height). This study is moti-vated by a wide range of applications including drug delivery,cell sorting and cell characterization devices, microcapsulefabrication, determination of membrane properties, and ofcourse its similarity to blood ow in vascular capillaries.

Our investigation shows that the capsule shape in a rectan-gularmicrouidic channel is quite different than that in a squareor cylindrical channel where the capsule extends along the owdirection obtaining a bullet-like or parachute-like shape.18,25 Bycontrast, in a rectangular channel the capsule extends mainlyalong the less-conned lateral direction of the channel cross-section (i.e. its width) obtaining a pebble-like shape. Thedifferent shape evolution in these two types of solid channelsresults from the different tension development on the capsulemembrane required for interfacial stability. In a square orcylindrical channel, the required tension development resultsonly from the capsule extension along the ow direction, i.e. the

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only direction which is not conned by the solid walls. However,in a rectangularmicrochannel, the capsule lateral extension alsoresults in tension development, and thus both the capsule's owand lateral extensions contribute to the interfacial stability. Thegreater width-to-length extension of the capsule in a rectangularchannel reveals the membrane preference in developing stronglateral tensions and clearly shows that the capsule dynamics is athree-dimensional phenomenon.12

Therefore, the channel's cross-section asymmetry results in ahighly non-axisymmetric, fully three-dimensional capsuleshape (as shown in Fig. 2 and 3), which cannot be describedfrom single-view observations as commonly happens in micro-uidic experiments or based on axisymmetric or two-dimen-sional computations. It is of interest to note that seeing thecapsule only from the channel height (as it may happen in amicrouidic experiment) provides misleading informationsince from this view the capsule still resembles a bullet or aparachute. To overcome this issue, experimental studies maywork with two sets of rectangular microuidic channels so thatthey are able to observe the so particles from both the chan-nel's width and height.

In addition, our work shows that the capsule deformation ina rectangular microchannel is different from that of dropletswith constant surface tension (which extend mainly along theow direction) and of vesicles which extend along the moreconned channel height owing to the development of a four-vortex pattern on the uid-incompressible vesicle membrane.6

In essence, our study highlights the different stability dynamicsassociated with these three types of interfaces, i.e. constanttension interface of a droplet, capsule membrane (withcomparable shearing and area-dilatation resistance), and thelocal-incompressible vesicle membrane with no shearingresistance.

It is of interest to note that our conclusions are not restrictedto articial capsules but should also represent physiologicalcapsules, such as erythrocytes, for the same physical reasons.For example, in their experimental study on the viscoelasticityof the erythrocyte's membrane, Tomaiuolo et al.33 reported thatin rectangular microuidic PDMS channels, erythrocytes obtaina more attened shape elongated along the less-connedchannel's width. (See also the experimental photos in Fig. 6(a)of the earlier study.) We note that the erythrocyte membraneconsists of an outer lipid bilayer (which is essentially a two-dimensional incompressible uid with no shearing resistanceas in vesicles) and an underlying spectrin skeleton (whichexhibits shearing and area-dilatation resistance like the elasticmembrane of common articial capsules).13,16,30 Therefore, ourndings suggest that the similar erythrocyte deformation inrectangular microchannels (and more general in asymmetricvessels) results from the erythrocyte's inner spectrin skeletonrather than from its outer lipid bilayer.

Acknowledgements

This work was supported in part by the National ScienceFoundation and the National Institutes of Health. Mostcomputations were performed on multiprocessor computers

This journal is ª The Royal Society of Chemistry 2013

provided by the Extreme Science and Engineering DiscoveryEnvironment (XSEDE) which is supported by the NationalScience Foundation.

References

1 M. Abkarian, M. Faivre, R. Horton, K. Smistrup, C. A. Best-Popescu and H. A. Stone, Cellular-scale hydrodynamics,Biomed. Mater., 2008, 3, 034011.

2 A. Alexeev and A. C. Balazs, Designing smart systems toselectively entrap and burst microcapsules, So Matter,2007, 3, 1500–1505.

3 M. Antia, T. Herricks and P. K. Rathod, Microuidicmodeling of cell–cell interactions in malaria pathogenesis,PLoS Pathog., 2007, 3, 0939–0948.

4 C. N. Baroud, F. Gallaire and R. Dangla, Dynamics ofmicrouidic droplets, Lab Chip, 2010, 10, 2032–2045.

5 M. Chabert and J.-L. Viovy Microuidic, High-throughputencapsulation and hydrodynamic self-sorting of singlecells, Proc. Natl. Acad. Sci. U. S. A., 2008, 105, 3191–3196.

6 G. Coupier, A. Farutin, C. Minetti, T. Podgorski andC. Misbah, Shape diagram of vesicles in Poiseuille ow,Phys. Rev. Lett., 2012, 108, 178106.

7 H. A. Cranston, C. W. Boylan, G. L. Carroll, S. P. Sutera,J. R. Williamson, I. Y. Gluzman and D. J. Krogstad,Plasmodium falciparum maturation abolishes physiologicred cell deformability, Science, 1984, 223, 400–403.

8 D. Dendukuri, T. A. Hatton and P. S. Doyle, Synthesis andself-assembly of amphiphilic polymeric microparticles,Langmuir, 2007, 23, 4669–4674.

9 P. Dimitrakopoulos, Interfacial dynamics in Stokes ow via athree-dimensional fully implicit interfacial spectralboundary element algorithm, J. Comput. Phys., 2007, 225,408–426.

10 P. Dimitrakopoulos, Analysis of the variation in thedetermination of the shear modulus of the erythrocytemembrane: effects of the constitutive law and membranemodeling, Phys. Rev. E: Stat., Nonlinear, So Matter Phys.,2012, 85, 041917.

11 W. R. Dodson III and P. Dimitrakopoulos, Spindles, cuspsand bifurcation for capsules in Stokes ow, Phys. Rev. Lett.,2008, 101, 208102.

12 W. R. Dodson III and P. Dimitrakopoulos, Dynamics ofstrain-hardening and strain-soening capsules in strongplanar extensional ows via an interfacial spectralboundary element algorithm for elastic membranes, J.Fluid Mech., 2009, 641, 263–296.

13 W. R. Dodson III and P. Dimitrakopoulos, Tank-treading oferythrocytes in strong shear ows via a non-stiffcytoskeleton-based continuum computational modeling,Biophys. J., 2010, 99, 2906–2916.

14 D. A. Fedosov, H. Lei, B. Caswell, S. Suresh andG. E. Karniadakis, Multiscale modeling of red blood cellmechanics and blood ow in malaria, PLoS Comput. Biol.,2011, 7, e1002270.

15 L. K. Fiddes, E. W. K. Young, E. Kumacheva andA. R. Wheeler, Flow of microgel capsules through

Soft Matter, 2013, 9, 4284–4296 | 4295

Soft Matter Paper

Dow

nloa

ded

by U

nive

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of

Mar

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Col

lege

Par

k on

27

Mar

ch 2

013

Publ

ishe

d on

12

Mar

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013

on h

ttp://

pubs

.rsc

.org

| do

i:10.

1039

/C3S

M27

683J

View Article Online

topographically patterned microchannels, Lab Chip, 2007, 7,863–867.

16 S. Henon, G. Lenormand, A. Richert and F. Gallet, A newdetermination of the shear modulus of the humanerythrocyte membrane using optical tweezers, Biophys. J.,1999, 76, 1145–1151.

17 K. Jiang, C. Xue, C. Arya, C. Shao, E. O. George, D. L. DeVoeand S. R. Raghavan, A new approach to in situ“micromanufacturing”: microuidic fabrication ofmagnetic and uorescent chains using chitosanmicroparticles as building blocks, Small, 2011, 7, 2470–2476.

18 S. Kuriakose and P. Dimitrakopoulos, Motion of an elasticcapsule in a square microuidic channel, Phys. Rev. E:Stat., Nonlinear, So Matter Phys., 2011, 84, 011906.

19 E. Leclerc, H. Kinoshita, T. Fujii and D. Barthes-Biesel,Transient ow of microcapsules through convergent–divergent microchannels, Microuid. Nanouid., 2012, 12,761–770.

20 M. S. Lee, Computational studies of droplet motion anddeformation in a microuidic channel with a constriction,MS thesis, University of Maryland, 2010.

21 D. Lensen, K. V. Breukelen, D. M. Vriezema andJ. C. M. V. Hest, Preparation of biodegradable liquid corePLLA microcapsules and hollow PLLA microcapsules usingmicrouidics, Macromol. Biosci., 2010, 10, 475–480.

22 A. S. Popel and P. C. Johnson, Microcirculation andHemorheology, Annu. Rev. Fluid Mech., 2005, 37, 43–69.

23 Modeling and Simulation of Capsules and Biological Cells, ed.C. Pozrikidis, Chapman and Hall, London, 2003.

24 M. Prevot, A. L. Cordeiro, G. B. Sukhorukov, Y. Lvov,R. S. Besser and H. Mohwald, Design of a microuidicsystem to investigate the mechanical properties of layer-by-

4296 | Soft Matter, 2013, 9, 4284–4296

layer fabricated capsules, Macromol. Mater. Eng., 2003, 288,915–919.

25 F. Risso, F. Colle-Pailot and M. Zagzoule, Experimentalinvestigation of a bioarticial capsule owing in a narrowtube, J. Fluid Mech., 2006, 547, 149–173.

26 S. Seiffert, J. Thiele, A. R. Abate and D. A. Weitz, Smartmicrogel capsules from macromolecular precursors, J. Am.Chem. Soc., 2010, 132, 6606–6609.

27 J. P. Shelby, J. White, K. Ganesan, P. K. Rathod andD. T. Chiu, A microuidic model for single-cell capillaryobstruction by Plasmodium falciparum infectederythrocytes, Proc. Natl. Acad. Sci. U. S. A., 2003, 100,14618–14622.

28 R. Skalak and P.-I. Branemark, Deformation of red bloodcells in capillaries, Science, 1969, 164, 717–719.

29 R. Skalak, A. Tozeren, R. P. Zarda and S. Chien, Strain energyfunction of red blood cell membranes, Biophys. J., 1973, 13,245–264.

30 R. Skalak, N. Ozkaya and T. C. Skalak, Biouid mechanics,Annu. Rev. Fluid Mech., 1989, 21, 167–204.

31 H. A. Stone, Interfaces: in uid mechanics and acrossdisciplines, J. Fluid Mech., 2010, 645, 1–25.

32 S.-Y. Teh, R. Lin, L.-H. Hung and A. P. Lee, Dropletmicrouidics, Lab Chip, 2008, 8, 198–220.

33 G. Tomaiuolo, M. Barra, V. Preziosi, A. Cassinese, B. Rotolidand S. Guido, Microuidics analysis of red blood cellmembrane viscoelasticity, Lab Chip, 2011, 11, 449–454.

34 Y. Wang and P. Dimitrakopoulos, A three-dimensionalspectral boundary element algorithm for interfacialdynamics in Stokes ow, Phys. Fluids, 2006, 18, 082106.

35 Y. Wang and P. Dimitrakopoulos, Low-Reynolds-numberdroplet motion in a square microuidic channel, Theor.Comput. Fluid Dyn., 2012, 26, 361–379.

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