Dynamical regimes and hydrodynamic lift of viscous vesicles under shear
Sebastian Meßlinger, Benjamin Schmidt, Hiroshi Noguchi, Gerhard Gompper
aa r X i v : . [ c ond - m a t . s o f t ] J un Dynamical regimes and hydrodynamic lift of viscous vesicles under shear
Sebastian Meßlinger, Benjamin Schmidt,
1, 2
Hiroshi Noguchi,
1, 3 and Gerhard Gompper
1, 4 Institut f¨ur Festk¨orperforschung, Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany Division of Engineering Science, University of Toronto, Toronto M5S 2E4, Canada Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Institute for Advanced Simulations, Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany (Dated: September 26, 2018)The dynamics of two-dimensional viscous vesicles in shear flow, with different fluid viscosities η in and η out inside and outside, respectively, is studied using mesoscale simulation techniques. Besidesthe well-known tank-treading and tumbling motions, an oscillatory swinging motion is observed inthe simulations for large shear rate. The existence of this swinging motion requires the excitationof higher-order undulation modes (beyond elliptical deformations) in two dimensions. Keller-Skalaktheory is extended to deformable two-dimensional vesicles, such that a dynamical phase diagram canbe predicted for the reduced shear rate and the viscosity contrast η in /η out . The simulation resultsare found to be in good agreement with the theoretical predictions, when thermal fluctuations areincorporated in the theory. Moreover, the hydrodynamic lift force, acting on vesicles under shearclose to a wall, is determined from simulations for various viscosity contrasts. For comparison, thelift force is calculated numerically in the absence of thermal fluctuations using the boundary-integralmethod for equal inside and outside viscosities. Both methods show that the dependence of the liftforce on the distance y cm of the vesicle center of mass from the wall is well described by an effectivepower law y − for intermediate distances 0 . R p . y cm . R p with vesicle radius R p . The boundary-integral calculation indicates that the lift force decays asymptotically as 1 / [ y cm ln( y cm )] far from thewall. PACS numbers: 87.16.D-, 82.70.-y, 47.15.G-
I. INTRODUCTION
Vesicles are fluid droplets enclosed by a fluid lipidmembrane. Typically, vesicles have sizes of the orderof 100 nanometers to 10 micrometers, whereas the thick-ness of the membrane is only of the order of a nanome-ter. Therefore, the membrane can often be regarded asa two-dimensional manifold. Vesicle shapes and fluctua-tions are then governed by the curvature elasticity. Thisdescription has been very successful to explain vesiclesbehavior in thermal equilibrium [1].The dynamics of fluid vesicles in shear flow has at-tracted much attention recently [2, 3, 4, 5, 6, 7, 8, 9,10, 11, 12, 13, 14, 15, 16, 17, 18]. Aspherical vesiclesunder shear can be found in different dynamical phases,depending on the viscosities η in and η out of inner andouter fluids, respectively, the membrane viscosity η mb ,the bending rigidity κ , the shear rate ˙ γ , the membranearea, and enclosed volume. As long as shape relaxationtimes of the vesicle are small compared to the time scaleset by the shear rate ˙ γ , the vesicle is always close to itsequilibrium shape. Under these conditions, vesicles canbe found either in tank-treading (TT) motion, or – if theviscosity contrast λ = η in /η out exceeds a critical value –in a tumbling (TB) motion. In the tank-treading regime,the vesicle shape and orientation are stationary in time,but the membrane rotates around the vesicle’s center ofmass in the same direction as the rotational part of theshear flow. Here, the orientation is characterized by theinclination angle θ with respect to the flow direction. Inthe tumbling regime, the long axis of the vesicle performsa periodic rotation. Keller and Skalak [2] developed a theory for fluid vesicles with fixed ellipsoidal shape anddifferent viscosity contrasts, which is able to explain theobserved experiments. In recent years, computer simu-lations [3, 4, 5, 6, 7] have shown that the Keller-Skalak(KS) theory provides indeed a very good description oftank-treading and tumbling.However, the vesicle dynamics is far less understoodwhen the shear rate is large enough that the vesiclecannot relax into its equilibrium shape. Only recently,it was shown that a third dynamical regime can ap-pear under these conditions, the swinging (SW) regime[11, 12, 13, 14, 15, 16, 17, 18] — also called the trem-bling [11] or vacillating-breathing regime [13]. In theswinging state, oscillations of shape and inclination an-gle together determine the vesicle dynamics. Swingingvesicles were first observed experimentally in Ref. [11].With increasing shear rate, a transition from tumbling toswinging motion was found. A perturbation theory forquasi-spherical vesicles to lowest order in the deviationfrom the spherical shape predicted swinging for a range ofviscosity contrasts [13]; however, since the shear rate ap-pears only as basic (inverse) time scale in this approach,the experimental results could not be explained. There-fore, higher-order expansions for quasi-spherical vesicles[16, 17, 18] and a generalized Keller-Skalak (KS) theoryfor ellipsoidal vesicles [15] have been developed, whichare able to predict phase transitions with varying shearrate and thereby to explain the experiments of Ref. [11].The dynamics in the TT, TB, and SW phases has beenstudied mainly for single vesicles in an unbounded fluid.However, in particular due to its physiological impor-tance, it is of high interest to study the dynamical be-havior of vesicles under shear in the presence of walls. Inthis case, vesicles are repelled from a wall due to a hydro-dynamic lift force F L . The hydrodynamic lift force playsan important role in circulatory systems of vertebrates.Since the lift force pushes red blood cells to the centerof a blood vessel, where the flow velocity is largest, it in-creases the efficiency of oxygen transport. On the otherhand, white blood cells move along the vessel walls inorder to find defects in the vascular endothelium [19, 20].This is achieved by special ligands, which are located atthe outside of white blood cells and bind to receptorson the vessel wall to resist the hydrodynamic lift force[21, 22].The existence of a hydrodynamic lift force was firstreported by Poiseuille in 1836 [23], who observed thiseffect on blood cells. In recent years, the hydrody-namic lift force was studied intensively, both theoretically[24, 25, 26, 27, 28] and experimentally [29, 30, 31, 32].Abkarian et al. [30, 31] observed the unbinding of a heavyvesicle, which was pulled by gravity towards a wall, withincreasing shear rate. For vesicles which are not in directcontact with the wall, only studies in three dimensionswith equal viscosities of inside and outside fluids exist.Both, boundary-integral simulations [28] as well as the-oretical studies [24, 25] show that the lift force decayswith a power law 1 /y with increasing distance betweenthe vesicle’s center of mass and the wall. For vesicles intwo dimensions, there are only theoretical and numeri-cal studies which focus on adhering vesicles bound to thewall by a short-ranged attractive potential [26, 27].In this paper, we study the dynamics of a two-dimensional (2D) vesicle as a function of viscosity con-trast λ and shear rate ˙ γ , both in the bulk and near a wall.The advantage of simulations of a vesicle in two dimen-sions is (i) the reduced numerical effort of hydrodynamicssimulations, which allows for larger system sizes, longeraccessible time scales, and better statistics, and (ii) thesimpler form of the equations of the KS theory, whereno integrals remain in the geometric factors – unlike inthe 3D version (see App. A). This facilitates a detailedcomparison of the results of theory and simulations. Us-ing a mesoscopic hydrodynamics approach, we first showthat the SW mode also exists in two dimensions, and de-termine the dynamical phase diagram. The simulationresults are compared with the predictions of a general-ized KS theory. Second, we study the lift force F L of 2Dvesicles, by covering the full range of wall distances y cm ,and investigate the effects of viscosity contrast λ . More-over, we investigate the effect of a wall on the TT-TBbehavior. For comparison with the results of mesoscopichydrodynamics simulations, we also determine F L andthe inclination angle θ by the boundary-integral methodfor tank-treading vesicles with λ = 1. II. THEORY AND METHODSA. Dimensionless Parameters
In a 2D vesicle, the perimeter L p and the enclosed area A are kept constant (analogously to the constant mem-brane surface and the enclosed volume of 3D vesicles). Itis useful to combine these two parameters into a dimen-sionless quantity, the reduced area A ∗ := 4 πAL p2 = (cid:16) R A R p (cid:17) . (1)Here R p = L p / π and R A = p A/π are the radii ofcircles with the same L p and A as those of the vesicle,respectively. A ∗ is the ratio between the enclosed area A and the area of a circle with the same perimeter L p .We focus here on a reduced area of A ∗ = 0 . α and inclination angle θ basedon the gyration tensor of the vesicle membrane. WhenΛ max and Λ min are the two eigenvalues of the gyrationtensor (Λ max ≥ Λ min ), and ˆe max and ˆe min the corre-sponding eigenvectors, the “asphericity” is described by α = (Λ max − Λ min ) / (Λ max + Λ min ) and the vesicle orien-tation by the inclination angle θ = ∡ ( ˆx , ˆe max ), where ˆx is the shear and ˆy the gradient direction.The stability of dynamical phases mainly depends ontwo parameters, the viscosity contrast λ and the reducedshear rate ˙ γ ∗ := ˙ γη out R κ . (2)The time η out R /κ is the characteristic relaxation timein thermal equilibrium, where κ is the bending rigidity.Thus ˙ γ ∗ expresses the interplay between the perturbationby the external field ˙ γ and the ability of the vesicle torestore its equilibrium shape. B. Generalized Keller-Skalak Theory in TwoDimensions
Keller-Skalak (KS) theory [2] is based on the assump-tion that vesicles have a fixed ellipsoidal shape. There-fore, it cannot describe the swinging state with oscillatingvesicle shapes. Therefore, KS theory has been general-ized to include shape deformation in three dimensions[15]. This theory is applicable to ellipsoidal vesicles overa wide range of reduced volumes, while higher-order per-turbation theory [16, 17, 18] is limited to quasi-sphericalvesicles. Here, we employ the two-dimensional version ofthe generalized KS theory. The differential equations forthe asphericity α and inclination angle θ are given by1˙ γ dαdt = − b A ∗ ˙ γ ∗ R p κ ∂F∂α + b sin(2 θ ) , (3)1˙ γ dθdt = 12 [ − B ( α ) cos(2 θ )] , (4)with prefactors b = 34 π ( λ + 1) and b = 32( λ + 1) . (5)There are no adjustable parameters. An explicit expres-sion for B ( α ) and its derivation are described in App. A.The time evolution of θ is described by Eq. (4), whichhas the same form as in two-dimensional KS theory.However, B ( α ) is now not constant but depends on thetime-dependent vesicle shape α ( t ). The time evolutionof α (see Eq. (3)) is derived based on the perturbationtheory of quasi-circular vesicles [33]. Here, F is the freeenergy of the vesicle shape at constant A ∗ . F attains itsminimum for an elliptical vesicle shape in equilibrium.Thus, the first term on the right hand side of Eq. (3)causes a relaxation of α towards its equilibrium value.The second term represents the change of α due to theexternal flow field. Eqs. (3) and (4) are solved numeri-cally using a fourth-order Runge-Kutta method.Thermal fluctuations can be incorporated in this ap-proach by adding Gaussian white noises g α ( t ) and g θ ( t )to Eqs. (3) and (4), respectively. The noise terms obeythe fluctuation-dissipation theorem, such that h g i ( t ) i = 0and h g i ( t ) g j ( t ′ ) i = (2 k B T /ζ i ) δ i,j δ ( t − t ′ ) with i, j ∈{ α, θ } , where k B T is the thermal energy. As a reasonableapproximation, we employ the rotational friction coeffi-cients of a circle, ζ α = 4 π η out R A2 ( λ + 1) and ζ θ = 4 πη out R p2 . (6) C. Mesoscale Hydrodynamics Simulation Method
1. Membrane Model
The membrane is modeled by a closed chain of n monomers of mass M . For a monomer with index i (with1 ≤ i ≤ n ), we introduce the notation i − = ( i −
1) mod n and i + = ( i + 1) mod n (7)for the indices of its two neighboring monomers. Thereby,the ring topology is taken into account correctly. Themonomers are connected by a harmonic spring potential U sp = k sp n X i =1 ( | R i | − l ) , (8)where R i := r i + − r i are the bond vectors, and l is the re-laxed bond length. The curvature elasticity of the mem-brane is described by the bending potential U bend = κl n X i =1 (cid:18) − R i + · R i | R i + k R i | (cid:19) . (9) An area potential U A = k A A − A ) . (10)is introduced to control the deviations of the area A fromits target value A . Here, the enclosed area A in Eq. (10)is obtained from the monomer positions by A = 12 ˆz · n X i =1 r i × r i + . (11)
2. Multi-Particle Collision Dynamics
For the solvent hydrodynamics, we employ multi-particle collision dynamics (MPC), a particle-basedmesoscopic simulation technique [34, 35, 36]. The dy-namics of an MPC fluid evolves in two alternating steps.In the “streaming step”, particles move ballistically for atime ∆ t , the collision time, according to their current ve-locities. For the “collision step”, solvent particles are firstsorted into the cells of linear size a of a regular square lat-tice; all particles in a cell then exchange momenta suchthat the total translational momentum is conserved ineach collision cell.Several modifications of the original MPC algorithmhave been introduced recently [37], which differ in theway the collision step is executed. We employ the MPC-AT+ a version of multi-particle collision dynamics, whichuses an Anderson thermostat (AT) and locally conservesangular momentum (+ a ) in addition to translational mo-mentum. In MPC-AT, new particle velocities relative tothe center-of-mass velocity are chosen from a Maxwell-Boltzmann distribution with temperature T . This ther-mostat avoids any heating due to energy dissipation insheared system. For details of the MPC-AT+ a algo-rithm, see Refs. [37, 38]. We use this algorithm, sincelocal angular-momentum conservation is crucial in binaryfluid systems with different viscosities [39].Simulations are performed with a rectangular simula-tion box with linear sizes L x and L y , periodic boundaryconditions in the x direction, and no-slip wall bound-ary conditions in the y direction. Linear shear flow withshear rate ˙ γ is realized by moving the upper wall with avelocity ˙ γL y ˆx , whereas the lower wall is held at rest.Many properties of the MPC-AT+ a solvent can be ad-justed by the simulation parameters collision time ∆ t ,the particle number density n s , and the particle mass m .The solvent viscosity η = η kin + η coll is a sum of a kinetic η kin and a collisional contribution η coll , which have beencalculated analytically [38], η kin = n s k B T ∆ ta (cid:20) n s n s − − (cid:21) , (12) η col = m ( n s − / t . (13)The viscosity η out of the fluid outside of the vesicle is ad-justed by varying the collision time ∆ t in the range from∆ t = 0 . a p m/k B T to ∆ t = 0 . a p m/k B T . Since forthese collision times the mean free path is much smallerthan the cell size a , the total shear viscosity η is domi-nated by η coll (see Eqs. (12) and (13)). Since the colli-sional viscosity η coll ∝ m , the viscosity contrast λ can bevaried by using different masses m in and m of the innerand outer fluid particles, respectively, which implies λ = η in η out ≈ m in m . (14)In our simulations, the viscosity contrast is varied from λ = 1 to λ = 10 (with m ≤ m in ≤ m ), while all theother MPC parameters are the same for the fluid on bothsides of the membrane.
3. Membrane Interactions
In order to describe an impermeable membrane in flow,it has to be ensured that MPC particles stay on the cor-rect side of the membrane ( i.e. inside or outside of thevesicle). For numerical efficiency, it is advantageous torelax this condition for short length and time scales, asit was done in previous 3D vesicle simulations [6]. Thestreaming and collision steps for the fluid particles arecarried as in the absence of the membrane. This impliesthat after each streaming step, some MPC particles havecrossed the membrane. For the (few) particles which arenow located on the wrong side of the membrane, with adirection of their velocity which would bring them awayeven further away from the membrane, the velocities haveto be modified such that they move towards the mem-brane instead, in order to cross back to their correct side.We denote this velocity update a “membrane collision”.It has to be constructed such that the translational andangular momentum as well as the kinetic energy of thefluid particles and membrane monomers are conservedlocally. Our procedure for membrane collisions is a gen-eralization of the standard bounce-back rule for no-slipboundary conditions. A detailed description of this pro-cedure is provided in App. B.In order to prevent the membrane from crossing thewalls, a purely repulsive Lennard-Jones potential U w ( y ) = ε (cid:20)(cid:16) σy (cid:17) − (cid:16) σy (cid:17) (cid:21) + ε, ≤ y ≤ √ σ , otherwiseis employed, which depends only on the distance y of amonomer from a wall.For the determination of hydrodynamic lift forces, weemploy a gravitational body force f G = − ˆy g ∆ ̺ , whichacts on the internal fluid of the vesicle. Here, g denotesthe strength of the gravitational field, and ∆ ̺ is the mass-density difference between the inner and outer fluids. Thegravitational body force f G acting on the inner fluid canbe expressed as a potential U G , which only depends on the monomer positions, U G = F G A X i (cid:0) y i + y i + (cid:1) (cid:0) r i × r i + (cid:1) · ˆz . (15)Here, y i and y i + are the y components of the monomerpositions r i and r i + , respectively, and F G = (cid:12)(cid:12)R A f G dA (cid:12)(cid:12) isthe total gravitational force acting on the vesicle. F G hasa constant value and is used as a simulation parameter.As long as not specified otherwise, the parameters usedin our vesicle simulations are n = 50, l = a = √ σ , n s = 10 a − , M = 10 m , ε = 10 k B T , and κ/l = 50 k B T .For the reduced area, we require that it deviates lessthan 1% from its target value of A ∗ = 0 .
7. Since A ∗ is afunction of the perimeter L p and the enclosed area A (seeEq. (1)), the parameters k sp and k A for the potentials U sp and U A , respectively, have to be sufficiently large.We chose k sp = 10 k B T /a and k A = 80 k B T /a . Withthese parameters, the effective vesicle radius is obtainedto be R p = 7 . l . The size of the simulation box is L x = L y = 80 a . Gravitational forces F G are only applied insimulations for the hydrodynamic lift force, where valuesin the range k B T /a ≤ F G ≤ k B T /a are investigated.In simulations, different reduced shear rates ˙ γ ∗ can beachieved, according to Eq. (2), by varying ˙ γ , η out , R p ,or κ . Since equilibrium properties like the undulationspectrum depend on R p and κ , we vary ˙ γ ∗ by adjusting˙ γ and η out . In order to avoid inertial effects, we restrictthe shear rates to obtain low Reynolds numbers Re =˙ γρR /η out , where ρ is the density of the outer fluid. Themaximum Reynolds number is Re= 0 . D. Boundary-Integral Method
For comparison with our MPC simulation results of thelift force, we also perform numerical boundary-integralcalculations. The hydrodynamic lift force in 2D has beenstudied previously with the boundary-integral approachfor vesicles in direct contact with the wall [26, 27]. Thismethod has the advantage that it can be used to cal-culate lift forces on vesicles even for very large distances y cm from the wall and for reduced areas A ∗ close to unity,which are not easily accessible by MPC simulations. Onthe other hand, our boundary-integral calculation is re-stricted to elliptical shapes and ignores thermal fluctu-ations, which give rise, e.g. , to undulation-induced re-pulsion near a wall. We focus on tank-treading ellipticalvesicles without viscosity contrast, i.e. λ = 1. In thesteady tank-treading state the lift force can calculatedfrom a single, time-independent vesicle shape. Whereasin MPC simulations, the wall distance y cm is calculatedfor a given strength of the gravitational force, we fol-low the opposite procedure with the boundary-integralapproach, by calculating the lift force for a given walldistance y cm .For ellipse half axes a and a , wall distance y cm , andinclination angle θ , the location r of the vesicle membrane (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) PSfrag replacements a a ˆu i l i r i ϕ θ y cm FIG. 1: For the boundary-integral calculation, the vesicle isdiscretized into segments i of length l i , which have the orien-tation ˆu i and a center-of-mass position r i . is uniquely defined (with x cm ≡ ϕ , it is r ( ϕ ) = y cm ˆy + (cid:18) cos θ − sin θ sin θ cos θ (cid:19) r ′ ( ϕ ) (16)where r ′ ( ϕ ) is the membrane position in the principal-axis system of the ellipse, r ′ ( ϕ ) = (cid:18) a cos ϕa sin ϕ (cid:19) . (17)In the steady tank-treading state, the center-of-massvelocity v cm = ˆx v cm of the vesicle has only a non-vanishing component in shear direction. If v cm and thetank-treading angular velocity ω are known, the velocityof the tank-treading membrane is given by v ( ϕ ) = ˆx v cm + R p ω ˆt ( ϕ ) . (18)with the tangent vector ˆt ( ϕ ) = (cid:18) a a sin θ a a cos θ − a a cos θ a a sin θ (cid:19) r ′ ( ϕ ) | r ′ ( ϕ ) | . (19)In a tank-treading membrane in shear flow, forces arisingfrom pressure and viscous stress have to be balanced inorder to maintain a steady motion. The force distribution f ( r ′ ) along the membrane ∂A is related to the velocityfield at position r by v ( r ) − ˙ γy ˆx = Z ∂A G ( r , r ′ ( s )) f ( r ′ ( s )) ds. (20)Here, ds is a line element of the membrane ∂A , and thesecond-order tensor G ( r , r ′ ( s )) is the Greens function ofthe Stokes equation which satisfies the boundary condi-tions. For vesicles in an unbounded fluid, G ( r , r ′ ( s )) isthe Oseen tensor. In our case of a vesicle near a wall, thehalf-space Oseen tensor – also known as Blake tensor –is convenient, as it realizes no-slip boundary conditionsat the wall. The full expression for the two-dimensionalBlake tensor can be found in Ref. [40]. The difficulty is that the force distribution f ( r ′ ) alongthe membrane is a priori unknown. Instead, we know thevelocities v ( r ) at each site of the membrane. Eq. (20)is thereby a Fredholm integral equation of the first type.This integral equation is solved numerically. For this pur-pose, we discretize the membrane in N straight segments,which have to be small enough such that the differencein velocities between two neighboring segments is smalland the force distribution can be assumed to be constantalong the segment. A segment with index i has a velocity v i , center-of-mass position r i , length l i and orientation ˆu i (see Fig. 1). The discretized form of Eq. (20) is v i − ˙ γy i ˆx = N X j =1 Z l j / − l j / G ( r i , r j + ˆu j s ) f ( r j + ˆu j s ) ds. (21)Since the force distribution f j = f ( r j + ˆu j s ) is assumedto be constant over the whole segment j , it can be movedoutside of the integral, v i − ˙ γy i ˆx = N X j =1 "Z l j / − l j / G ( r i , r j + ˆu j s ) ds f j = N X j =1 H ij f j . (22)The calculation of H ij can be performed analytically,both for the free-space and the half-space Oseen tensor.Thus, the integral equation (20) is reduced to a set oflinear algebraic equations which can be easily solved nu-merically.The segment velocities v i depend linearly on ω and v cm (see Eq. (18)). Therefore, we can extend the linearsystem of equations (20) by two additional conditions,which determine ω and v cm self-consistently in the steadystate. For the first condition, we require that the sumof tangential forces along the membrane vanishes. Thesecond condition is that the vesicle does not experiencea net force in shear direction. The total system of linearequations finally reads − ˙ γy i ˆx = N X j =1 H ij f j − v cm ˆx − ˆt i R p ω (23)0 = N X i =1 ˆx · f i l i (24)0 = N X i =1 ˆt i · f i l i . (25)This set of equations is solved numerically with up to N = 600 segments. Once, ω , v cm and the force distri-bution are known, quantities like the lift force F L andthe torque M on the vesicle, as well as the velocity v ( r )and pressure fields p ( r ) in the surrounding fluid can becalculated as F L = N X j =1 f j l j , (26) M = N X j =1 ( r j − y cm ˆy ) × f j l j , (27) v ( r ) = ˙ γy ˆx + N X j =1 "Z l j / − l j / G ( r , r j + ˆu j s ) ds f j , (28) p ( r ) = N X j =1 "Z l j / − l j / g ( r , r j + ˆu j s ) ds · f j . (29)Here, g ( r , r ′ ) is the half-space pressure vector (seeRef. [40]). The lift force F L and the torque M (seeEqs. (26) and (27)), are thereby functions of the four pa-rameters θ , y cm , a , and a , which define the membranelocation uniquely. Using a numerical root finder (Brent’smethod), the stable inclination angle θ , for which thetorque vanishes, is determined while keeping the otherparameters y cm , a , and a fixed. III. DYNAMICAL REGIMES OF VISCOUSVESICLES IN UNBOUNDED SHEAR FLOWA. Phase Diagram
We consider first the dynamics of vesicles in shear flow,far from walls and in the absence of a gravitational field.The 2D generalized KS theory predicts a phase diagram,see Fig. 2, which shows the qualitatively the same fea-tures as a function of ˙ γ ∗ and λ as the 3D version [15]. Atsmall and large λ , a vesicle exhibits tank-treading (TT)and tumbling (TB) motion, respectively. At large ˙ γ ∗ andintermediate λ , the swinging (SW) phase appears. As in3D generalized KS theory, TT with negative inclinationangles θ < θ < θ >
0) or of a TT and a SW states arealso seen. In 2D, TT with θ < swinging tumbling t an k - t r ead i ng PSfrag replacements ˙ γ ∗ λ FIG. 2: (Color online) Phase diagram at A ∗ = 0 .
7. Dashed(red) and solid (blue) lines represent the results of the gener-alized Keller-Skalak theory with b = 3 / [2( λ +1)] (see Eq. (5))and b = 1 / ( λ +1), respectively. The black circles ( • ) indicatethe location of the simulation which are shown in Fig. 5. flow causes an elongation of the vesicle shape (increasing α ). For a large shape parameter α , the vesicle is tem-porarily in the tumbling regime, until a negative inclina-tion angle θ is reached (for 0 ≤ t ˙ γ ≤
10 in Fig. 3(c)). Fornegative θ , the elongational component of the flow actsto reduce α . Due to the constraint of fixed perimeter L p and fixed enclosed area A , the vesicle assumes a potato-like shape, such that α decreases (for 10 ≤ t ˙ γ ≤
15 inFig. 3(c)). The vesicle is then stretched again by theelongational flow leading to a positive inclination angle θ and increasing shape parameters α (for t ˙ γ & B. TT-TB Transition
The generalized KS theory predicts that for small shearrates, with ˙ γ ∗ .
6, the TT-TB transition at λ ≃ . γ ∗ (see Fig. 2). In this regime, shapedeformations are very small, and the behavior can be welldescribed by the original KS theory. We choose a shearrate ˙ γ = 0 . p k B T /ma in our simulation, correspond-ing to a small reduced shear rate ˙ γ ∗ = 3 . θ on the viscosity contrast λ is shown. Our MPC sim- (a) Tank-treading: PSfrag replacements t ˙ γ PSfrag replacements t ˙ γ (b) Swinging: PSfrag replacements t ˙ γ PSfrag replacements t ˙ γ (c) Tumbling: PSfrag replacements t ˙ γ PSfrag replacements t ˙ γ PSfrag replacements t ˙ γ FIG. 3: (Color online) Sequences of vesicle snapshots for eachof the dynamical regimes, shown (a: TT, b: SW, c: TB).A (red) bullet marks one fixed membrane element to indi-cate the membrane motion. All systems share the param-eters κ/l = 50 k B T and A ∗ = 0 .
7. Further parameters are(a) η out = 36 √ k B T m/a , λ = 1, ˙ γ = 0 . p k B T /m/a cor-responding to ˙ γ ∗ = 3 .
6; (b) η out = 120 √ k B T m/a , λ = 4,˙ γ = 0 . p k B T /m/a corresponding to ˙ γ ∗ = 38; and (c) η out = 36 √ k B T m/a , λ = 10, ˙ γ = 0 . p k B T /m/a corre-sponding to ˙ γ ∗ = 3 . PSfrag replacements Beaucourt:This work:KS theory: λ θ [ ◦ ] -50510152025 0 2 4 6 8 10 FIG. 4: (Color online) Inclination angle θ as a function of vis-cosity contrast λ for simulations with ˙ γ ∗ = 3 .
6. For compari-son the results of the boundary-integral calculation of Beau-court et al. [4] (without thermal fluctuations) as well as thecurve of KS theory [2] (see Eq. (30)) are shown. ulations well reproduce the results of previous boundary-integral calculations by Beaucourt et al. [4]. Deviationsclose to the TT-TB transition at λ ∗ ≃ λ = 3, and also simulations with λ > λ ∗ ≃ & θ decreases more gradually with increasing λ , andno tumbling motion was observed at viscosity contrastsas large as λ = 10. Thus, we conclude that inertial effectsenhance the TT-membrane rotation.Fig. 4 also shows that KS theory [2] provides a gooddescription of the λ dependence of θ and the TT-TBtransition. This transition is explained by the KS theoryas follows. The stationary inclination angle θ in the tank-treading regime is determined by Eq. (4) with fixed α as θ = −
12 arccos (cid:18) − B (cid:19) . (30)For small λ , the inclination angle θ decreases mono-tonically up to a critical viscosity contrast λ ∗ , where θ = 0. For larger viscosity contrasts λ > λ ∗ , the tum-bling regime, there is no real solution of Eq. (30), i.e. nostationary inclination angle exists, and the vesicle per-manently rotates. C. TB-SW Transition
To investigate the TB-SW transition, we consider afixed viscosity contrast of λ = 4, and perform simulationsfor four different reduced shear rates ˙ γ ∗ = 5 , ,
22, and38. The locations of these four shear rates in the dynam-ical phase diagram are indicated in Fig. 2. The resultingtrajectories in the θ - α plane are shown on the left-handside of Fig. 5. In this representation, closed cycles indi-cate swinging events, whereas trajectories spanning thefull [ − π/ , + π/
2] range of θ are tumbling events. Obvi-ously, thermal noise has a large impact on the vesicle dy-namics. In particular, at small inclination angles θ ≃ γ ∗ = 17 and 22. Thisis about a factor 2 larger than the prediction ˙ γ ∗ = 9 ofgeneralized KS theory. In the generalized KS theory, apossible source of error can be found in the estimate of b and b , which have both been calculated in the cir-cular limit, see Eq. (5). Therefore, we also calculate thephase diagram with b reduced by a factor 2 / i.e. with b = 1 / ( λ + 1), which gives a better agreement with oursimulations for non-circular vesicles with A ∗ = 0 . b and b analytically for non-circular shapes. We conclude that generalized KS theory Simulations: Theory:
PSfrag replacements α θ [ ◦ ] ˙ γ ∗ = 5 ˙ γ ∗ = 17 (a)(b)(c)(d)(e)(f )(g)(h) -90 -45 0 45 9000.20.40.6 PSfrag replacements α θ [ ◦ ] ˙ γ ∗ = 5 ˙ γ ∗ = 17 (a) (b)(c)(d)(e)(f )(g)(h) -90 -45 0 45 9000.20.40.6 PSfrag replacements α θ [ ◦ ] ˙ γ ∗ = 5 ˙ γ ∗ = 17 (a)(b) (c)(d)(e)(f )(g)(h) -90 -45 0 45 9000.20.40.6 PSfrag replacements α θ [ ◦ ] ˙ γ ∗ = 5 ˙ γ ∗ = 17 (a)(b)(c) (d)(e)(f )(g)(h) -90 -45 0 45 9000.20.40.6 PSfrag replacements α θ [ ◦ ] ˙ γ ∗ = 5 ˙ γ ∗ = 17 (a)(b)(c)(d) (e)(f )(g)(h) -90 -45 0 45 9000.20.40.6 PSfrag replacements α θ [ ◦ ] ˙ γ ∗ = 5 ˙ γ ∗ = 17 (a)(b)(c)(d)(e) (f )(g)(h) -90 -45 0 45 9000.20.40.6 PSfrag replacements α θ [ ◦ ] ˙ γ ∗ = 5 ˙ γ ∗ = 17 (a)(b)(c)(d)(e)(f ) (g)(h) -90 -45 0 45 9000.20.40.6 PSfrag replacements α θ [ ◦ ] ˙ γ ∗ = 5 ˙ γ ∗ = 17 (a)(b)(c)(d)(e)(f )(g) (h) -90 -45 0 45 9000.20.40.6 FIG. 5: (Color online) Trajectories in the θ - α plane for λ =4, both with (thin red lines) and without (thick blue lines)thermal noise. The trajectories in (a), (c), (d), and (g) areobtained from simulations, whereas the curves in (b), (d), (f),and (h) are calculated from the generalized KS theory withnoise and b = 1 / ( λ + 1). The reduced shear rates are ˙ γ ∗ = 5for (a) and (b), ˙ γ ∗ = 17 for (c) and (d), ˙ γ ∗ = 22 for (e) and (f),and ˙ γ ∗ = 38 for (g) and (h). In all plots, the correspondingtheoretical trajectory according the generalized Keller-Skalaktheory without thermal noise is shown as a thick blue (dark)line. provides a good description of vesicle dynamics in shearflow in both two and three spatial dimensions.Elastic capsules [44, 45] and red blood cells [46] canalso exhibit a swinging motion. However, the angle θ ( t ) isalways positive during these oscillations — unlike SW offluid vesicles. The physical mechanism is an energy bar-rier for the TT rotation caused by the membrane shearelasticity and the anisotropic shape of the spectrin net-work [46, 47, 48]. Although, a vesicle in 2D (a closedstring) does not have membrane shear elasticity, an en-ergy barrier for the TT rotation can be introduced by in- (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) PSfrag replacements r cm y cm F G (a) (b) FIG. 6: (Color) (a) Shape and (b) pressure field of a tank-treading vesicle under shear flow close to a wall, in steadystate with viscosity contrast λ = 3. The color code is ex-pressed in units of n s k B T . The hydrodynamic lift force isbalanced by an external gravitational force F G = 14 k B T /a ,where the average distance from the wall is y cm = 7 . a . Seealso movie [50]. homogeneities in the spontaneous curvature [49]. In thefuture, it will be interesting to investigate the coupling ofdifferent swinging mechanisms in composite membranes. IV. LIFT FORCE
We now consider a vesicle under the combined effect ofa shear flow and a gravitational force F G , see Fig. 6(a).The vesicle moves towards or away from the wall un-til gravitational F G and lift forces F L ( y cm ) balance eachother (see also movie [50]). In this steady state, the liftforce F L ( y cm ) equals the gravitational force in magni-tude.Fig. 6(b) shows the pressure field in the outer fluid forthe steady-state configuration of a tank-treading vesicle.The hydrodynamic lift force is the integral of the pressureforces over the membrane contour. The higher pressurein the gap between the vesicle and the wall is responsi-ble for the lift force. Fig. 6(b) also nicely demonstratesthat there is a lower pressure at the two caps of the vesi-cle, which is the origin of vesicle elongation. The hydro-dynamic lift force is a pressure force which is of purelyviscous nature – in contrast to e.g. aerodynamic forcesacting on the wings of an airplane, which are caused byinertial forces.The dependence of the hydrodynamic lift force on thewall distance is shown in Fig. 7(a) — calculated as h y cm i for fixed gravitational force in the simulations, and aslift force at fixed y cm in the Oseen calculations, as ex-plained in detail in Sec. II C 3 and Sec. II D above. Liftforces of vesicles with λ ≤ F L ∝ y − for F L ≤ . k B T /R p , correspond-ing to y cm & R p . For these distances, the vesicle is notin direct contact with the wall. At applied gravitationalforces larger than 2 . k B T /R p , the vesicle touches the wall( y cm . R p ). However, the distance y cm between the cen-ter of mass and the wall can be reduced even further PSfrag replacements (b)(a) λ = 1 : λ = 2 : λ = 3 : λ = 4 : λ = 5 : λ = 7 : λ = 10 :Oseen: θ [ ◦ ] y cm /R p y − : F L R p / ( k B T ) ✍✌✎☞ ✍✌✎☞✍✌✎☞ ✍✌✎☞✍✌✎☞ ✍✌✎☞✍✌✎☞✍✌✎☞✍✌✎☞✍✌✎☞✍✌✎☞✍✌✎☞ PSfrag replacements (b)(a) λ = 1 : λ = 2 : λ = 3 : λ = 4 : λ = 5 : λ = 7 : λ = 10 :Oseen: θ [ ◦ ] y cm /R p y − : F L R p / ( k B T )
100 100.5 1 2 3 4 55 6-5 001520250.5123456 -50152025 ✍✌✎☞ ✍✌✎☞✍✌✎☞✍✌✎☞ ✍✌✎☞✍✌✎☞ ✍✌✎☞✍✌✎☞ ✍✌✎☞✍✌✎☞✍✌✎☞✍✌✎☞✍✌✎☞✍✌✎☞ ✍✌✎☞✍✌✎☞
FIG. 7: (Color online) (a) Lift forces and (b) average incli-nation angle as a function of the average wall distance y cm ofthe vesicle, from MPC simulations with ˙ γ ∗ = 3 . y − is plotted in (a). Theright-most data points in (b) correspond to F G = 0, so thatthey cannot be shown in the double-logarithmic presentationin (a). by vesicle deformation. The 1 /y dependence does notapply in this regime. Finally, the constraints of fixedenclosed area A and fixed perimeter L p keep the walldistance larger than y cm & . R p .Fig. 7(a) shows that the lift forces decrease with in-creasing viscosity ratio λ for a fixed wall distance y cm .This behavior is analyzed in more detail in Fig. 8, wherethe amplitude F L y of the lift force is plotted as afunction of the viscosity contrast λ . Although solid col-loidal particles of elliptical shape experience no net liftforce [51], tumbling vesicles with finite λ obtain lift forcedue to an asymmetry of its shape deformations and asmall tank-treading component (compare Fig. 3(c)).For vesicles in three dimensions, both boundary-integral simulations [28] as well as theoretical studies[24, 25] show a 1 /y dependence of the lift force forvesicles far from the wall. The theory of Olla [24, 25] as- PSfrag replacements F L y c m / ( k B T R p ) λ SimulationOseen
00 2 4 6 8 1020406080100120140160180
FIG. 8: (Color online) Amplitude of the lift force, F L y / ( k B T R p ), as a function of the viscosity contrast λ .The amplitudes are fits to the curves in Fig. 7(a) for whichvesicles are not in direct contact with the wall. sumes an ellipsoidal shape for the vesicles with half axes a , a , a ≪ y cm . It is not possible in this case to de-rive expressions for two dimensions by taking the limit a → ∞ — as done in App. A for KS theory of vesi-cles in unbounded flows — because this limit is incon-sistent with the assumption a , a , a ≪ y cm . Therefore,instead of an analytical theory, we perform boundary-integral calculations of 2D elliptical vesicles with λ = 1in the presence of a wall, as described in Sec. II D. Forthe results in Fig. 7(a), the effect of the opposite wallat L y = 10 R p is also taken into account by plotting F L ( y cm ) − F L ( L y − y cm ), where F L ( y cm ) and F L ( L y − y cm )are obtained from two independent boundary-integralcalculations. Of course, a more precise calculation wouldrequire the use the two-wall Oseen tensor [40] instead ofthe half-space Greens function. However, as long as thedistance L y between the two walls is sufficiently large, F L ( y cm ) − F L ( L y − y cm ) is a good approximation of thetwo-wall lift force. Fig. 7(a) as well as Fig. 8 show thatthe boundary-integral calculation indeed agrees nicelywith the corresponding MPC simulation for λ = 1. Theamplitudes F L y cm2 only differ by about 25%. Reasons forthis deviation are that the boundary-integral calculationis done for elliptical shapes, whereas vesicles in simula-tions are closer to the equilibrium shape (compare Fig. 1and Fig. 6). Moreover thermal fluctuations in the MPCsimulations may cause differences.Fig. 7(a) shows that tumbling is suppressed when thegravitational force exceeds a threshold value, dependingon the viscosity contrast λ . In order to perform a tum-bling motion, the center-of-mass distance y cm has to beon the order of or larger than the long vesicle axis a .However, for larger gravitational forces, the center-of-mass distance y cm becomes smaller than a , such thateven vesicles with high viscosity contrasts do not tumble.Even if y cm is slightly larger than a , the vesicle cap hasto come so close to the wall that the resulting pressure0 F L R p / ( k B T ) λ TTTB
FIG. 9: (Color online) Dynamic phase diagram of tank-treading and tumbling states as a function of the gravita-tional force F G and the viscosity contrast λ . The reducedshear rate is ˙ γ ∗ = 3 .
6. Circles ( • ) indicate tumbling motion(TB), squares ( (cid:4) ) tank-treading motion (TT). The line forthe phase boundary is a guide to the eye. forces prevent the inclination angle to reach π/
2. Thisleads to the dynamical phase diagram shown in Fig. 9.When the gravitational force is large, tumbling is sup-pressed and the vesicle displays a tank-treading motionat the wall. With increasing λ , the gravitational forcenecessary to prevent tumbling increases.The dependence of the inclination angle θ on the walldistance y cm is shown in Fig. 7(b). Without a gravita-tional force, the lift force caused by the upper wall at y = 10 R p compensates the lift force of the lower wallwhen the vesicle is in the center, at y cm = 5 R p . Sincethe lift forces are very small nearby, strong fluctuationsare observed in the wall distances for small F G .As long as a vesicle is tank-treading, its inclination an-gle θ decreases when it approaches the wall. Even if thevesicle does not touch the wall, the pressure at the lowestpart of the membrane is highest (see Fig. 6(b)) such thatit causes a torque which lowers θ . For very small walldistances, the vesicle comes into direct contact with thewall, where the repulsive wall potential causes an addi-tional torque, which decreases θ even further, until thevesicle is finally completely parallel to the wall.Vesicles with λ ≥ λ = 3and λ = 4 are still tank-treading most of the time andonly occasionally perform a tumbling motion, their in-clination angles are non-zero, whereas for λ = 10 and F G ≤
1, the average inclination angle θ essentially van-ishes (see Fig. 7(b)).We employ the boundary-integral approach to calcu-late the dependence of the hydrodynamic lift force ofvesicles with λ = 1 on the reduced area A ∗ in the ab-sence of thermal fluctuations. Also, since this method isnot restricted by the system size (as simulations), the liftforces can be calculated even for very large wall distances.Fig. 10 shows the hydrodynamic lift force as a functionof the wall distance y cm for different reduced areas A ∗ inthe range 0 . ≤ A ∗ ≤ .
99. This plot shows that the
PSfrag replacements y cm /R p F L R p / ( k B T ) y − y − . A ∗ = 0 . . . . . . . ∝ / { ( y cm /R p ) ln( y cm /R p ) } FIG. 10: (Color online) Hydrodynamic lift force F L on vesi-cles with varying A ∗ vs. wall distance y cm obtained fromboundary-integral calculations. PSfrag replacements F L y c m l n ( y c m / R p ) / ( k B T ) − A ∗ Oseen ∝ √ − A ∗ FIG. 11: (Color online) Amplitude of the lift force F L as afunction of 1 − A ∗ . The data points are fits to the curves inFig. 10 for y cm ≥ R p . These data points (Oseen) are fittedto a √ − A ∗ dependence for small values of 1 − A ∗ . y − dependence of the hydrodynamic lift force F L onlyholds for distances y cm . R p from the wall. For largerdistances y cm , a crossover to a 1 / [( y cm /R p ) ln ( y cm /R p )]dependence is obtained. This power law with a loga-rithmic correction fits perfectly the numerical data ofthe boundary-integral calculation for y cm & R p for allconsidered reduced areas. Fig. 11 shows the amplitudes K = F L y cm ln( y cm /R p ) of the lift forces F L vs. 1 − A ∗ inthe far-field limit. These amplitudes K are determinedby fitting the expression F L = K/ [( y cm /R p ) ln ( y cm /R p )]to all data with y cm ≥ R p . For 1 − A ∗ = 0, i.e. fora circular vesicle, the lift force vanishes, which directlyfollows from the time-inversion symmetry of the Stokesequation. For small deviations from the circular shape,the lift force rapidly increases with 1 − A ∗ , and followsa power-law dependence K ∼ √ − A ∗ ≃ p ∆ p /π , withthe excess length ∆ p = L p /R A − π , for 1 − A ∗ . . − A ∗ over the whole range of1reduced areas A ∗ .Since boundary-integral calculations do not take intoaccount thermal fluctuations, lift forces can be deter-mined for large y cm (only limited by numerical accuracy).However in simulations as well as in real systems, liftforces at large y cm have a vanishing effect compared tothermal noise. Moreover, with very large wall distance y cm ∼ p η out /ρ ˙ γ , inertial effects are not negligible, sothat the Stokes approximation becomes less reliable [25]. V. SUMMARY AND CONCLUSION
We have studied the dynamics of vesicles in shear flowin a two-dimensional model system. This system showsa variety of interesting dynamical phenomena.First, we have investigated the effect of the viscositycontrast λ , i.e. the ratio between the inner and outer vis-cosities of a vesicle, on the dynamics in unbounded flows.With increasing λ , the sequence from “tank-treading”over “swinging” to “tumbling” motion is generically ob-served — except for small shear rates ˙ γ , where the in-termediate swinging phase is absent. Thus, the swingingphase appears in the phase diagram of 2D vesicles un-der shear in the same way as it was found previouslyfor 3D ellipsoidal vesicles. However, the mechanism ofswinging is different in two and three dimensions. Whilein 3D, ellipsoidal deformations are sufficient to obtainswinging, in 2D higher-order undulation modes are re-quired. Thermal fluctuations play an important role;they lead to a smooth crossover between the dynami-cal states, with intermittent tumbling and tank-treadingmotions. Our simulations are in semi-quantitative agree-ment with a theoretical description based on a general-ized Keller-Skalak approach.Second, we have investigated the behavior of vesiclesnear walls. Close to a wall, tumbling is strongly sup-pressed. Furthermore, the vesicle is repelled from thewall by the hydrodynamical lift force. We have foundby boundary-integral calculations that the hydrodynamiclift force decays with increasing wall distance y cm like1 / ( y cm ln y cm ). However, for small wall distances – inparticular in the regime of the MPC simulations – aneffective y − dependence is observed. With increasingviscosity contrast, the lift force becomes weaker, as thevesicle becomes less deformable. The lift force also de-creases with increasing reduced area A ∗ , and vanishes inthe circular limit. We find that our numerical data arewell described by a √ − A ∗ dependence.Our results show that there is a different behavior ofthe lift force at intermediate and large distances from thewall, and that the lift force decreases significantly withincreasing viscosity contrast. This may shed some lighton the behavior in three dimensions, where experimentsshow a dependence of the lift force on the wall-membranedistance h , which decays as h − for distances smallerthan the vesicle radius [30], whereas a y − decay has beenfound theoretically in a small range 1 . . y cm /R p . .
25 of wall distances [28]. Thus, we hope that our resultswill stimulate new experiments and simulations in 3Dover a wider range of wall distances, reduced volumes,and viscosity contrasts.
Acknowledgments
Sebastian Messlinger acknowledges a fellowship ofthe International Helmholtz Research School “BioSoft”.Benjamin Schmidt thanks the DAAD for financial sup-port through the RISE (Research Internships in Scienceand Engineering) program and for giving him the oppor-tunity of a visit at the Research Center J¨ulich.
APPENDIX A: DERIVATION OF 2DGENERALIZED KELLER-SKALAK THEORY1. Keller-Skalak Theory in Two Dimensions
Keller and Skalak [2] derived analytical expressions forthe inclination angle θ and the average angular velocity ω for 3D vesicles of fixed ellipsoidal shape, with ( x/a ) +( y/a ) + ( z/a ) = 1, based on the Jeffery theory [52].Although the KS theory is formulated for vesicles inthree dimensions, it is straightforward to transfer it totwo-dimensional systems by simply taking the limit a →∞ . The resulting cylindrical three-dimensional geometryis equivalent to a 2D vesicle with the shape of an ellipse,( x/a ) + ( y/a ) = 1 with a ≥ a . Let S ′ be the framewhich has its origin at the center of the ellipse, and the x ′ direction points into the direction of the long axis. Thenthe local velocity v ′ of an element of the tank-treadingmembrane is assumed to be( v ′ x , v ′ y ) = ω ( − a a x ′ , a a x ′ ) . (A1)in the frame S ′ . We define the auxiliary variables f := 1 − α D2 α D2 , f := 1 − α D2 α D , f := 1 + α D2 , where α D = ( a + a ) / ( a − a ). The balance of torques onthe membrane and the assumption that the work done onthe vesicles by the shear flow is dissipated in the interiorof the vesicle leads to the non-linear differential equation dθdt = ˙ γ − B ( α D , λ ) cos(2 θ )] (A2) B ( α D , λ ) = f (cid:26) f + f − f ( λ − (cid:27) . (A3)Furthermore, the average angular velocity ω is found tobe ω ˙ γ = cos(2 θ )2 f { f ( λ − } (A4)2 F / k B T α FIG. 12: (Color online) Free energy F ( α ) of the two-dimensional vesicle relative to the prolate shape for A ∗ = 0 . κ/l = 50 k B T , k sp = 10 k B T /a , and k A = 80 k B T /a . Thesolid (red) and dashed (blue) lines represent the simulationdata and fitted curve, respectively.
2. Shape Equation in Two Dimensions
The vesicle shape is expanded in Fourier modes withpolar angle φ as r ( φ ) = R A e r { P u m exp( imφ/ √ π ) } .Based on the Stokes approximation and perturbation the-ory, the dynamics of a quasi-circular vesicle is describedby [33] ∂u m ∂t = i ˙ γm u m − κ Γ m E m η out R u m ∓ ih ˙ γδ m, ± , (A5)where Γ m = | m | / λ + 1)( m − E m = ( m − m − / σ ), and h = √ π/ λ + 1). A Lagrange multi-plier σ keeps the perimeter L p constant. Following theprocedure for 3D [15], we decompose u ± into ampli-tude and phase, u ± = r exp( ∓ iθ ), and replace theforce 2 κE m r by ∂F/∂r . Then, Eq. (3) is obtained with α = 3 r/ √ π + O ( r ).The free energy F ( α ) for the same simulation parame-ters calculated with a version of the generalized-ensembleMonte Carlo method [6] (see Fig. 12): The vesicle con-formations are sampled under the uniform distributionof α , and then the canonical distribution is obtained bythe reweighing. Instead of an interpolation [5, 6, 15],we use fit functions here to obtain smooth functions. The force is fitted as a function − (1 /k B T ) ∂F/∂α =9 + 180 α − α − exp(80 α − α D = 2 α/ . α by fitting for the ellipse of A ∗ = 0 . APPENDIX B: MEMBRANE COLLISIONS WITHTHE SOLVENT
For the modeling of an impermeable membrane, inte-rior and exterior solvent particles have to stay on theappropriate side of the membrane. Depending on theposition of the MPC particle with respect to membranebonds, either one or two membrane monomers partici-pate in a membrane collision. The new velocities of the 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