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Rheology of red blood cells under flow in highly confined microchannels. II. Effect of focusing and confinement Guillermo R. La´zaro,*a Aurora Herna´ndez-Machadoa and Ignacio Pagonabarragab We study the focusing of red blood cells and vesicles in pressure-driven flows in highly confined microchannels (10–30 mm), identifying the control parameters that dictate the cell distribution along the channel. Our results show that an increase in the flow velocity leads to a sharper cell distribution in a
Received 24th April 2014 Accepted 25th June 2014
lateral position of the channel. This position depends on the channel width, with cells flowing at outer (closer to the walls) positions in thicker channels. We also study the relevance of the object shape, exploring the different behaviour of red blood cells and different vesicles. We also analyze the
DOI: 10.1039/c4sm01382d
implications of these phenomena in the cell suspension rheology, highlighting the crucial role of the wall
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confinement in the rheological properties of the suspension.
1
Introduction
The recent development of accurate lab-on-a-chip devices with channels of roughly 10 mm offers a unique pathway for single cell tests that enable the diagnosis of certain pathologies.1 The selective treatment of the cells typically exploits their response to different external forces, including electromagnetic elds2,3 or passive lters that make use of the hydrodynamic interactions of the cell with the channel geometry.4 The focusing of cells and particles at low concentrations can be used in this direction. For instance, appropriate designing of the channel geometry (e.g. curved channels) induces an ordering in the cell distribution that allows a ne precision of cell manipulation.5 The spatial organization is controlled by different factors such as object size and rigidity,6 or rheological properties of the external medium.7 Other applications are currently under exploration, such as drug delivery based on the different behaviour of blood cells under ow.8 The ongoing improvement of the precision and applications of these devices depends on a detailed knowledge of the cell behaviour at the microscale. The spatial organization of the cells plays, additionally, a fundamental role in the rheological properties of the system.9 The Fahraeus-Lindqvist effect10 provides a good example of the non-linear rheological response caused by spatial ordering of the cells. Red blood cells (hereaer RBCs) align and concentrate on the tube core, forming an annulus of free uid ow adjacent to the wall, increasing the uidity of the blood. The effect is accentuated in narrower tubes, when the repulsion from the
a
Departament d'Estructura i Constituents de la materia, Universitat de Barcelona, Av. Diagonal 647, E08028 Barcelona, Spain. E-mail: [email protected]
b
Departament de Fisica Fonamental, Universitat de Barcelona, Av. Diagonal 647, E08028 Barcelona, Spain
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wall is stronger, giving rise to a decreased blood viscosity, until reaching the limit case of single-le channels in which RBCs ow aligned forming regular trains. The equilibrium position of a particle in the tube depends on several factors such as ow velocity or connement.11 It is well known that isolated particles owing in thick tubes move laterally (i.e. cross stream migration) towards a specic off center position, as rst observed by Segr´ e & Silberberg (1962)12 for rigid spheres in a Poiseuille ow, and extended later on to other objects,13,14 and different channel geometries.15 This effect, however, is driven by an inertial dri which forces the object to leave the center.16 The dynamics of the object is not critically determined by its specic properties and the off center focusing has been also observed for red blood cells and other so entities.5 At very narrow microchannels, when inertial effects are negligible, the absence of this dri should prevent from a lateral migration during the downstream evolution, and hard spheres and spherical vesicles do not focalize. In this regime, however, the deformability and geometry of the object arise as crucial mechanisms of symmetry breaking,17 and RBCs are known to present a much more complex behaviour than spherical objects.18 Whilst the dynamics of particles and RBCs in the presence of inertia has been extensively studied in the literature, the limit of low Reynolds number, which requires a more accurate description of the geometry and elasticity of the object, has raised interest and awareness only recently,19,20 due to its relevance in the development of new microuidic devices. In a preceding paper,21 we have studied the importance of cell elasticity in the rheological response of a dilute suspension of RBCs owing at very conned microchannels. In this regime, the interactions between cells are subdominant and the dynamics of RBCs is dictated by their interaction with the channel walls. For increasing ow velocities, the cell assumes the so-called
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discocyte, slipper and parachute morphologies. At low ow velocity, the cell behaves as a rigid object maintaining a shape close to the equilibrium discocyte, and it does not perform lateral displacements. Further increasing the incoming ow, cells are forced to migrate towards an off center position, adopting the slipper morphology. At very high ow velocity, the slipper conguration becomes unstable and cells return to an axial position where they show a symmetric parachute-like shape. Whereas parachutes present a high deformation energy, slippers represent a low energy conguration closer to the elastic energy of the equilibrium discocyte. The suspension viscosity shows a shear-thinning behaviour, determined by the interplay between the cell deformability and its lateral migration and orientation in the off center position. In this paper, we extend our study on RBC behaviour in narrow channels, driving our interest towards the spatial organization and focusing of cells. The paper is organized as follows. We rst present the physical model and briey explain the numerical methods applied in the simulations. In the Results section, we analyze the strong sensitivity of the suspension rheology to the focusing and alignment of RBCs. We then characterize the spatial organization of RBCs along the channel section and their orientation for increasing ow velocity. We focus on the relevance of the connement and the interaction of the walls with the cells. The competition between wall induced effects and hydrodynamic interactions between cells is studied in the subsequent section. In order to complete our study about the problem of particle migration, we nally identify the role of the object shape by studying the behaviour of different vesicles, i.e. varying the equilibrium shape of the cell from the discocyte to spherical geometries. We then proceed with a discussion of the coupled effect between all these parameters and compare with previous numerical and experimental studies. We nish with the main conclusions of our ndings.
2 The model The elastic properties of the RBC membrane are described by the Helfrich bending energy,22 ð ð ð k F mem ¼ c2 dS þ gdS þ xdV ; (1) 2 vU vU U where c is the total curvature of the cell surface, k is the bending rigidity of the membrane, and g and x are Lagrange multipliers which maintain constant area and volume, respectively. The constraint of constant area accounts for the in-plane incompressibility of the uid membrane. The volume is maintained constant as a consequence of the impermeability of the membrane to water exchange with the exterior of the cell. We make use of a phase-eld model for the numerical implementation of the elastic membrane energy.23 In this approach, the membrane free energy reads: ð ð 2 k* (2) F fmem ¼ f þ f3 32 V2 f dV þ gjVfj2 dV : 2 It can be shown that the rst term of eqn (2) characterizes an pffiffiffi interface with bending rigidity k ¼ ð433 =3 2Þk*. The second
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term is the constant area constraint. The requirement of constant volume is achieved by imposing a conserved dynamics, given by an advection–diffusion equation for the order parameter, vf þ v$Vf ¼ MV2 m; vt
(3)
where m ¼ dF fmem/df is the chemical potential, M is the order parameter mobility and v is the uid velocity. This equation is coupled to the Navier–Stokes equation, that describes the hydrodynamics of the uid both in the inner and outer regions of the RBC membrane, r
vv þ rðv$VÞv ¼ VP fVm þ h0 V2 v þ rf: vt
(4)
where h0 is the viscosity of the solvent, r is the density, P is the pressure and f is the external force which effectively acts as a pressure difference, rf ¼ DP/L. The coupling between eqn (3) and (4) is performed via the advective term v$Vf, that describes how the uid pushes the membrane, and the reactive term fVm which represents the elastic force exerted by the membrane.
2.1
Numerical scheme
The model comprises two coupled dynamic equations which must be solved numerically. We make use of a hybrid scheme, using a standard nite element method for integrating the order parameter equation, eqn (3), and a lattice-Boltzmann method for solving the Navier–Stokes equation (eqn (4)). The LB method is an efficient and robust numerical solver which has been extensively applied in different elds of complex uids and so matter.24–27 The LB method is based on the fact that in the hydrodynamic limit, the Boltzmann equation recovers the macroscopic dynamics given by the Navier–Stokes equation. A discrete version of the Boltzmann equation can be numerically integrated,28 fi ðx þ ci ; t þ 1Þ fi ðx; tÞ ¼
1 fi fieq þ Fi : s
(5)
where fi is the distribution function and i runs over the velocity vector ci which connects with the neighbour nodes of the lattice. Fi is related to the external force, fext. The hydrodynamic variables, density and momentum, are then recovered from the X X distribution functions by fi ¼ r and fi ci ¼ rv, respeci
i
tively. Aerwards, the velocity v is introduced in the advection term of eqn (3). The equilibrium distribution f eq is obtained from an i expansion in terms of v, 1 1 fi eq ¼ run An þ 2 v$ci þ ; (6) vv : c c þ G : c c i i i i cs 2cs 4 where we have introduced the tracelesss tensor vv ¼ vv ðv2 =3Þ1, and cs is the speed of sound. The constants uv and Av depend on the specic lattice model; we use the D3Q19 model which comprises 1 velocity of modulus c ¼ 0, 6 of velocity
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pffiffiffi c ¼ 1 and 12 of c ¼ 2. The index n identies the three velocity subsets. The force exerted by the membrane is introduced in the tensor G ¼ 9=ð2rÞP 3dTrP, where the pressure tensor is
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P ¼ ðr=3Þd þ Pmem , and Pmem must satisfy VP mem ¼ fVm. The external force term of eqn (5) reads 1 ½f ext $ci ð1 þ v$ci Þ v$f ext : Fi ¼ 4un 1 2s
(7)
We impose stick boundary conditions at the walls by means of a bounce-back method.29 The method consists of reecting the distribution functions on the solid nodes that come from the uid aer a propagation step. In this scheme, the zero velocity condition is achieved at the middle point between the uid and solid nodes. 2.2
dimensional approach it is dened as nred ¼ A/pr2, where A is the area of the object and r ¼ p/2p is the radius of the circle with the same perimeter p as that of the vesicle. For a RBC nRBC ¼ 0.48, but we explore the behaviour of other object geometries nred ¼ 0.55, 0.69 and 0.97, the later corresponding to a nearly spherical vesicle (a circle in the 2D approach). Phase-eld methods require careful control of the model parameters to enforce the appropriate membrane elastic behavior for the desired degree of discretization. The model has provided successful results in different membrane phenomena that converge towards both theoretical and numerical predictions.23,31 Further technical details of the numerical implementation and its convergence properties are explained in the preceding paper.21 The effective viscosity is computed from the mean ow vz, according to heff ¼
Channel geometry and parameter steering
The simulations are performed in 2D rectangular boxes of size Lx Lz, applying periodic boundary conditions in the direction ^z, and no-slip conditions at the walls situated at x ¼ Lx/2 and x ¼ Lx/2. The force is constant, f ¼ f0^z, so that in the absence of the RBC the imposed force gives rise to a Poiseuille ow, nz ¼ f0/ 4h0(x2 (Lz/2)2). All the geometrical parameters are expressed relative to the RBC size, a, except those referring to connement, which are expressed relative to the channel width b h Lx or half of the channel width R ¼ b/2 when referring to lateral positions in the channel (since there are two symmetric positions with respect to the channel axis). Given that we concentrate on spatial ordering, it is important to remark the 3D analogy of our simulations. It is well-known that particles owing in rectangular channels focus to two symmetric layers, whereas those owing in square-section channels focus to four symmetric points.30 In our case, the geometry of the channel might be related to a rectangular channel in which the distance between the walls in the ˆy-direction is much larger than in the ^x-direction, Ly [ Lx. Since cell dynamics is dominated by the interaction with the walls and the uid ow is effectively plane in the transverse direction ˆy, it is reasonable to assume that shape coupling with the ow is subdominant in this transverse direction. Thereby, RBCs focalize at specic positions along ^x and show a homogeneous distribution in ˆy. The characteristic viscous time is calculated as the inverse of the shear rate sh ¼ b/vz, where vz is the mean velocity in the ow direction z, and the elastic relaxational time of the cell is given by sk ¼ ha3/k. The ratio between both time scales denes the capillary number, Ck, that describes the relative effect of the viscous effects of the uid and the elastic properties of the membrane, sk h vz a2 a Ck ¼ ¼ 0 : (8) b sh k We explore the range of capillaries between 0.2 and 120. In all the simulations the Reynolds number Re ¼ rnzL/h is maintained below 0.05, within the non-inertial regime. The reduced volume denes the shape of the object, and in the two-
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f0 2 b: 12vz
(9)
This expression corresponds to the effective viscosity of a uid with mean ow velocity ^vz (at the specic channel geometry) when an external force f0 is applied. The viscosity derived from ow-curve calculations is widely extended in tube and channel geometries, especially for highly conned systems, instead of the usual shear stress/rate relationship obtained in rotational viscosimeters. Recently, a number of viscosimeters have been presented in the literature in which the rheological properties of different complex uids are studied in terms of their effective viscosity, including blood32 or suspensions of active swimmers.33 We adopt this approach to identify the effective viscosity to facilitate the comparison with experimental studies.
3 Results 3.1
Effective viscosity and focusing
Our results show that heff depends on the initial condition of the RBC, especially at low Ck. The viscosity dispersion Dheff ¼ (heff eff)/ eff is the average viscosity, can be as high as h heff, where h 25%. This dispersion is shown in Fig. 1A, for a number of different congurations. In experiments, the effect of thermal noise, that allows random exploration of different orientations, may lead to a more uniform measurement. Therefore, in order to obtain a robust measurement of the viscosity, we average over a set of 7 different inclinations and 3 different lateral positions. The dispersion of each conguration is shown in Fig. 1B (inclinations) and 1C (lateral positions). According to their particular position and alignment with the ow, RBCs will offer variable resistance to ow. RBCs aligned parallel to the incoming ow and owing close to the walls (where the shear stress decreases) present smaller contribution to the viscosity than those centered and with normal orientation, which induce a severe perturbation of the surrounding ow. The results show that RBCs oriented symmetrically with respect to the normal ow direction (e.g., orientation of q ¼ 45 and 135 ) show a similar viscosity. This symmetry suggests that the relevant
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Fig. 1 Effective viscosity of a suspension of RBCs as a function of the capillary number, Ck. The coloured regions represent the three main morphological regimes: discocytes, slippers and parachutes. Examples of each morphology, as well as the steady position and orientation for different initial configurations, are shown in the snapshots. For further information about the morphological sequence see the preceding paper.21 At low capillaries, RBCs flow showing a certain dependence on the initial conditions. RBCs deform and accommodate their shape to the flow profile, without rotating towards a specific orientation, and maintaining their initial position along the tube. At intermediate Ck, RBCs rotate orienting their axis with the flow, but still maintain their initial position. Only RBCs placed very close to the wall migrate to more centered positions – hence, focusing in this regime is very weak. RBCs inclination therefore depends on their particular position. At high Ck, the focusing implies that regardless of the initial configuration RBCs flow at a fixed lateral position, assuming the slipper shape with their inclination aligned with the flow profile. Finally, RBCs exhibit parachute shapes for typically Ck > 105. (A) The viscosity presents a certain sensitivity to the initial conditions of the RBC, especially at low capillary numbers. Each dashed line represents the viscosity for a specific initial configuration. The mean eff (red line), is therefore averaged over the different configurations, covering: (B) initial inclinations (with respect to the effective viscosity, h channel axis), with normal RBCs presenting a higher resistance to flow compared to aligned RBCs; (C) initial heights in the channel, with centered RBCs opposing more resistance than RBCs flowing close to the walls. The viscosity dispersion is defined as Dheff ¼ (heff h eff)/ heff, where heff is the effective viscosity for each particular initial configuration. In all cases, within the slipper regime (V) the final configuration is nearly independent from the initial one. The dependence is stronger at intermediate (,) and especially low (B) capillaries. The schemes on the right represent the RBC inclination q and center of mass height hcm.
parameter controlling the value of the viscosity, within this rigid limit, is the ratio between the section occupied by the cell with respect to the total section of the channel, regardless of the cell orientation. The dispersion reduces for intermediate Ck and it is negligible at high values. The memory of the RBC of its initial conguration can be separated into two different contributing phenomena: alignment of the RBC with the ow, and focusing to a nal position. Due to the channel symmetry with respect to its axis, RBCs focus at two symmetric lateral bands, depending on which channel region was the cell located with respect to the axis at the initial condition. Migration of initially centered RBCs is determined by numerical noise. Henceforth, we concentrate on one of the channel regions (i.e. from the axis to one of the walls), bearing in mind that symmetric processes occur in the opposite region. At very low Ck, RBCs do not orient their axis with the ow, and they ow maintaining the initial position, without showing migration across streamlines. The increasing external ow forces the cell to rotate and align with the ow, but still showing a dependence on the initial position along the channel section. The dispersion of the viscosity decreases since the range of cell inclinations is reduced, but still maintain the contribution due to the different position along the tube. Further increase of Ck
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induces a migration towards an off center region, and the RBC distribution narrows forming a thin band. The viscosity dispersion decreases with the focusing and, for sufficiently high ow velocities, the nal position and orientation of the RBC is unique regardless of the initial condition, leading to a welldened value of the effective viscosity.
3.2
RBC focusing and alignment
RBC focusing has been extensively studied in the inertial regime, but it is not well understood in the viscous one. In this section we study the focusing of RBCs to a dened, off-center position, and how their lateral position can be controlled by tuning the capillary number and the distance between the channel walls. 3.2.1 Focusing. At low Ck, RBCs ow occupying the entire channel section, without any favoured position. If the external force increases, however, RBCs concentrate on a narrow band off the channel axis, as shown in Fig. 2. In Fig. 2A, at Ck ¼ 10, we show the height along the channel section of RBCs placed at different initial positions during the downstream ow. RBCs initially placed close to the wall experience a repulsion and migrate towards the center, stabilizing at hcm/R ¼ 0.5. RBCs
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Fig. 2 (A) Evolution during the downstream flow of the center of mass height, hcm, of RBCs initially placed at different lateral positions, within the slipper regime, Ck ¼ 13.2. RBCs close to the wall are repulsed and migrate towards the center; likewise, centered RBCs migrate towards the walls. All RBCs stabilize at intermediate positions, forming a band of stable trajectories. In this case the confinement has been increased with respect to the standard one, with a/b ¼ 0.49 in this plot, in order to observe larger and more perceptible migrations. (B) Width of the band of stable trajectories as a function of Ck. The width of the band can be controlled by increasing the flow velocity (confinement a/b ¼ 0.71). High velocities induce a narrowing of the band, until the RBC eventually occupies a unique position in the channel. Within the parachute regime, RBCs can develop parachute and slipper morphologies at fixed flow conditions depending on their initial position: RBCs initially placed close to the wall are repulsed and assume a slipper shape, whereas cells placed at the channel core deform into a parachute.
initially placed in the channel axis also migrate from the center and reach an equilibrium position at hcm/R ¼ 0.25. The rest of the RBCs occupy intermediate positions, forming a band of stable trajectories. The channel width has been increased to a/b ¼ 0.49, in order to allow larger and more perceptible migrations. The time scale of the migration is typically half of the deformation time sk. The increase in Ck induces a narrowing of the band of stable trajectories, as shown in Fig. 2B, and eventually RBCs are found in a unique lateral position regardless of their initial condition, for Ck > 40. In the parachute regime, we obtain that parachute and slipper shapes can be observed at the same conditions depending on the initial conguration: cells initially placed close to the walls assume asymmetric shapes, whilst those placed close to the center adopt a parachute shape. 3.2.2 Alignment. RBCs at very low capillary Ck < 2 ow maintaining their initial orientation, as shown in Fig. 3A.
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Fig. 3 (A) Orientation of the RBC as a function of the capillary number for different initial conditions, q0. Within the discocyte regime, the RBC maintains its initial orientation, with slight deformation. For Ck > 2, all RBCs rotate and orient their axis with the imposed flow profile. (B) Temporal evolution of the inclination of a RBC at Ck ¼ 4.5. The rotation towards the equilibrium inclination occurs on a fast time scale compared to the migration and relaxation time of the cell. In the inset some snapshots of the evolution are shown.
However, beyond the critical value Ck ¼ 2, RBCs lose this dependence and rotate towards a xed orientation, which in turn depends on the lateral position along the channel. RBCs owing close to the wall assume slightly more horizontal inclinations than those more centered, as a result of the nonuniform curvature of the ow prole. The focusing of RBCs takes place for higher capillaries. The sensitivity to the initial conditions of the orientation and the lateral position is seemingly uncoupled. Accordingly, the dispersion on the nal congurations can be separated into three different situations: (i) for Ck < 2, RBCs ow maintaining their initial distribution and orientation, deforming their shape to the local ow prole; (ii) for 2 < Ck < 7, RBCs show a xed orientation aligned with the ow, but they still retain their memory of the initial position along the channel section; and (iii) Ck > 7 RBCs focalize at two symmetric lateral positions in the channel, aligned with the ow. The temporal characterization of the rotation of a RBC towards its stable orientation is shown in Fig. 3B. RBC rotation occurs on a much shorter time scale than migration, typically 0.1sk. In the inset, some snapshots of the process are depicted. 3.2.3 Effect of wall connement. The focusing of RBCs can be controlled by both increasing the ow velocity or the degree Soft Matter, 2014, 10, 7207–7217 | 7211
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of connement of the channel. By varying the distance between _ we identify the walls b, but maintaining a constant shear rate g, how the closer presence of the walls affects the RBC morphology. We focus on intermediate and high capillaries in which the presence of the walls is more relevant. In Fig. 4A, a phase diagram for different capillaries and connements is shown. Moving along a row of constant capillary numbers, the increasing connement induces higher migrations and lower inclinations. This effect is especially noticeable at intermediate capillaries, in which highly conned RBCs have migrated and adopted the slipper morphology, whereas less conned RBCs still remain centered and retain a discocyte shape. The relative distance to the wall arises as an important factor to trigger the migration, and at thick channels the capillary required to observe slippers and parachutes might be extremely high. Interestingly, the diagram highlights a similar effect of moving along the vertical and horizontal axes. This suggests a coupling between the effect of connement and capillary, implying that increasing shear rates induce stronger interactions between the cell and the walls, leading to a larger repulsive force from the axis, and this effect can be emphasized by a closer distance of the RBC to the walls. The lateral position and focusing of RBCs for different wall connements is shown in Fig. 5. The connement positively affects focusing, and for all capillaries Ck > 20 RBCs are localized at a dened lateral position if a/b > 0.5. The distance from the axis of this position increases for thicker channels, from hcm ¼ 0.05R to 0.3R in the range of channels studied. At the narrowest channels, the RBC is placed close to the center, likely due to the geometrical constraints but also to the strong wall repulsion. The increase in Ck reinforces the focusing effects, and especially for Ck ¼ 109.5 focusing has extended to thicker channels and RBCs are localized at
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Focusing of RBCs as a function of the wall confinement and the capillary number. For each Ck, the top and bottom values correspond, respectively, to the outermost (i.e. closest to the wall) and innermost (closest to the center) equilibrium positions; all the rest stable positions are found in between these two. The wall confinement favours the focusing and typically the effect is reduced below a/b < 0.5. The lateral position of the RBC is found further away from the axis in thicker channels, and in the limit in which the width of the channel is equal to the cell diameter, RBCs flow nearly centered. The capillary number also induces an increased focusing (especially in the case Ck ¼ 109.5), though its effect is weaker. Fig. 5
positions further from the channel axis. Note, however, that the effect is relatively weak for the two other cases Ck ¼ 18.5 and 43.2. The larger inclination and migration observed in RBCs owing along the narrowest channels are accompanied by lower values of the effective viscosity, as shown in Fig. 4B. In the case
Fig. 4 Effect of channel confinement on the RBC behaviour at different Ck. All the RBCs are initially placed at the channel axis. (A) RBC morphologies found at different confined channels; cells shown here are initially placed at the channel axis, normal to the flow direction. The shear rate is maintained constant for each value of Ck. The effect of the walls is important to induce lateral migration, as confined RBCs exhibit the characteristic slipper morphology, whereas the RBC flowing at the thicker tube still retains a discocytic shape. Less confined RBCs, which may present weaker interactions with the walls, require higher Ck to deform and migrate. (B) Effective viscosity heff for a suspension of RBCs as a function of the channel confinement and capillary number, at constant volume fraction. heff, especially at high capillaries, strongly depends on the confinement. For confined RBCs which have migrated, the viscosity is low and uniform. However, if the channel is thick enough, the RBC does not migrate and its center position and normal orientation imply a higher viscosity.
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of high capillary, the viscosity is similar for all the channels in which the RBC has been forced to migrate and align. However, if the connement is low and the RBC still ows along the centerline, it will displace a larger amount of external uid and therefore the suspension viscosity will be also larger. For decreasing connements the effective viscosity increases, until it eventually stabilizes when the wall inuence is negligible and the RBC behaviour is similar to that in an unbounded ow. The connement necessary to recover this unbounded behaviour also depends on the capillary number, as expected given the existing coupling between connement and the capillary number. At lower values of the capillary, all the RBCs exhibit a less marked migration and alignment, and the differences between the viscosity of each initial condition are less accentuated.
Effectively, the system reproduces a one-dimensional array of identical RBCs separated by the domain length distance, Lz, which reminds the ordered structures observed experimentally. Although we cannot address the stability of these regular arrays and the possibility that RBCs reorder and form clusters, the method allows us to explore systematically the competition between wall-induced effects and RBC interactions. The membrane stiffness dictates the ow disruption induced by the RBC. Rigid cells induce stronger perturbations of the incoming ow than soer ones, as seen in Fig. 6 A. We dene the ow amplitude .Xh i X AðzÞ ¼ ðan a0n þ bn b0n Þ ða0n Þ2 þ ðb0n Þ2 of the imposed
3.3
ow amplitude to a Gaussian decay A(z) ¼ A0 exp(((z zcm)/ lp)2), a typical distance of the ow distortion is obtained, the penetration length lp. Fig. 6 B displays the penetration length for different Ck, at very long tubes Lz [ a when RBCs do not interact. The results show that the ow disruption strongly decays far from the cell and for Ck > 10 the ow deviates from the reference one only at very close distances from the RBC, 0.1b. This could suggest that the coupling between RBCs is only relevant if they are placed extremely close. However, even if deviations from the imposed ow are small when RBCs are distant, interactions are strengthened for lower
Hydrodynamic interactions between RBCs
While owing along conned channels at high concentrations, RBCs oen order in regular trains, as rst noted by Gaehtgens et al. (1980).34 The formation of these ordered congurations is important in the designing of microuidic devices as it increases the control of cell manipulation. From the theoretical point of view, the organization in trains also offers an interesting way to study the hydrodynamic interactions between neighbouring cells, and how it affects the RBC dynamics. We take advantage of the periodic boundary conditions in the ^z direction35 to study the hydrodynamic interactions between the cell and its images.
Poiseuille, where (an, bn) and (a0n, b0n) are the coefficients of the Fourier decomposition of the actual vz (x, z0) and imposed vz0 (x, z0) velocity proles, respectively, for the rst n ¼ 20 modes, X vðx; z0 Þ ¼ ½an sinð2pnx=bÞ þ bn cosð2pnx=bÞ. By tting the
Fig. 6 (A) Flow disruption induced by a RBC in a Poiseuille flow. The color map represents the value of the velocity component vz. Ck affects the flow disturbance that the cell induces: rigid RBCs (up, Ck ¼ 0.8) disturb a larger region of the surrounding flow than soft RBCs (bottom, Ck ¼ 15.1). (B) Penetration length, lp of the flow disruption caused by the RBC for different Ck. lp is the typical length of the flow distortion caused by the cell in the surrounding flow with respect to the unperturbed Poiseuille, but see the main text for a formal definition. lp decreases rapidly with Ck. For Ck > 10 it falls down to lp < 0.1a, meaning that RBCs induce a limited perturbation of the flow at intermediate and high capillaries. (C) Penetration distance lp for closely placed RBCs, as a function of the distance between cells L. Even if the flow disruption shows a strong decay at high capillaries, RBC behaviour is highly sensitive to the presence of neighbouring cells. The dashed line separates the single-cell behaviour (when RBCs are able to migrate off center and assume slipper morphologies) from the array configuration (when they order in regular arrays of centered, symmetric cells). For Ck ¼ 19.5, RBCs only recover the single-cell behaviour if the distance between them is higher than 1.5a (inset). For lower values, hydrodynamic interactions between cells dominate and migration is inhibited. The penetration length lp shows that the coupling between RBCs is large if the distance is much lower than the length of the channel (wall confinement here is a/b ¼ 0.71).
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distances between cells, favouring the collective behaviour. If RBCs are initially placed closer to each other, even at high Ck, they do not migrate towards the wall but ow maintaining a centered position, forming an ordered array of RBCs aligned normal to the channel axis, as shown in Fig. 6C (inset, RBCs on the right) for Ck ¼ 19.5. RBCs bend, coupling their surface to the ow prole. If the distance between cells is increased, eventually slippers are recovered, initially showing a slight distortion and then fully exhibiting the single-cell behaviour. The distance necessary to separate this array-conguration from the singlecell behaviour depends on the capillary number, requiring larger cell-to-cell distances at low Ck. For Ck ¼ 19.5, the RBC behaves as hydrodynamically isolated for lp > 1.4a, but the critical distance decays to 0.5 for Ck ¼ 97. The formation of regular arrays originates from the transversal connement that the RBC feels due to the presence of its neighbours. The ow disturbance generated by the RBC is symmetrically compressed by both the preceding and the rear cells of the array, inhibiting the symmetry breaking of the ows generated around the RBC that preludes the migration in the case of isolated cells. Thereby, the transversal connement constrains the RBC shape to a centered, symmetric morphology. Fig. 6C displays the dependence of the penetration length on the separation between RBCs for a given channel width, a/b ¼ 0.71, and for an intermediate Ck value, for which lp is typically of the order of the channel width. For distances L much smaller than the channel width the wall effect is subdominant and the RBC behaviour is controlled by the transversal connement, forming regular arrays of RBCs. Only in sufficiently long channels, L/a > 2, RBCs behave as hydrodynamically isolated, recovering the singlecell behaviour, exhibiting lateral migration and assuming slipper shapes. This transition from the array-conguration to the singlecell behaviour determines the range in which wall effects are dominant with respect to cell-to-cell interactions. The formation of arrays of RBCs has important implications in the rheology of the suspension. The normal orientation of the cells implies a larger resistance to ow than in the case of aligned RBCs (i.e. slippers), and the solvent is repelled from the channel core towards the walls, where it ows free of cell disruption. This effect, however, implies that the effective viscosity of the suspension increases, as shown in Fig. 7A, where the effective viscosity of the suspensions shown in Fig. 6C, at constant Ck, is plotted. A sweep in Ck reveals that the differences in the viscosity are accentuated at lower values, as a direct consequence of the stronger interactions between more rigid cells. This effect is shown in Fig. 7B, where the effective viscosity as a function of Ck is shown for three different distances between RBCs. At high Ck, the distance between RBCs in all the cases is larger than the critical distance in the single-cell regime. For lower values of Ck, the differences of the viscosity are primarily due to the different volume fraction. At some point, marked by the dashed line, the distance between RBCs is lower than the increasing critical length and RBCs switch to the train conguration, inducing a substantial increase of the effective viscosity. Accordingly, the presence of hydrodynamic interactions between RBCs critically determines both the morphological and rheological behaviour of the suspension.
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Fig. 7 (A) Effective viscosity for a regular array as a function of the intracellular distance, a/L, at a constant capillary number Ck ¼ 18. Each value of the viscosity corresponds here to the shapes shown in Fig. 6C, for initially centered RBCs. The increase in the viscosity can be decomposed into two different contributions: (i) the change in the volume fraction, from 0.40 to 0.13, (ii) the effect of the interactions between RBCs at low L, which allow the formation of trains of cells at the channel core that present higher flow resistance. (B) Effective viscosity as a function of Ck for different cell-to-cell distances. If the channel is not sufficiently long, at some point, marked by a dashed line, the penetration length lp exceeds the distance between cells L and the RBC interacts with its image (forming an array of RBCs due to the periodic boundary conditions) and this fact triggers a considerable increase in the effective viscosity. The change between the single-cell and train like configurations is marked by a dashed line, and the specific cell shapes are shown in the inset. At low distances between RBCs, they maintain a centered position but bending its surface. Only for large distances between cells slippers are observed, RBCs on the left.
3.4
Vesicle shape
We explore the relevance of the object geometry by studying the case of three vesicles with different reduced volumes and compare with the RBC case. The relaxed shapes of the vesicles are obtained from minimizing several initial ellipsoids with different areas and perimeters, nred ¼ 0.97, 0.69, 0.55, 0.48, corresponding to the equilibrium shapes from a nearly circular to a discocyte shape. This minimization is performed in the absence of uid. The vesicles are then placed in the channel and the uid is switched on. Due to their symmetric shape, almost circular vesicles present less degrees of freedom to adapt and orient with the external ow, showing slight deformations even at high forcements. They do not migrate out of the axis,
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Fig. 8 Effect of the reduced volume on vesicles and RBC behaviour and suspension rheology. (A) Vesicle morphologies for different reduced volumes. Circular vesicles, which are symmetric, do not migrate and remain centered flowing at the channel center line. The increasing deflation of the vesicles (i.e., lower reduced volumes) induces a higher degree of migration and asymmetry. Discocytes exhibit the largest degree of migration towards the wall and orientation with the flow profile. (B) Effective viscosity as a function of the capillary number for different reduced volumes. The capability of deflated vesicles to migrate and orient enables them to reduce their resistance to flow, and this property implies a sharper shear-thinning behaviour than in the case of circular vesicles.
remaining at the center line, and the low area excess allows only a slight deformation. The vesicle center of mass moves to a more forward position at high capillaries, acquiring at intermediate capillaries a more triangular shape which could remind us of a parachute, the so-called bullet, as shown in Fig. 8A. We check the evolution of a circular vesicle when initially placed close to the wall, nding that it migrates towards the center recovering the symmetric shape, as opposed to discocytes. Deated vesicles present an intermediate behaviour between the circular and the discocyte vesicles. For all the cases studied, the equilibrium position is strictly asymmetric and therefore vesicles eventually deform and migrate, but lower values of the capillary require more deated vesicles to acquire asymmetric shapes. In the inertial regime, the off center position of objects does not depend on their specic geometry, and hard spheres, so beads and cells behave similarly, migrating towards roughly the same equilibrium position. The repulsive force from the axis does not manifest a critical dependence on the specic properties of the object. The sensitivity shown by different objects in the viscous regime might be explained by asymmetric distribution of normal stresses on deated vesicles, implying an effective repulsive dri, which is on the contrary balanced in circular vesicles because of their symmetry. In the rigid limit, discocytes present a considerably higher resistance to ow than circular vesicles, in spite of their lower volume. This may be due to the smoother streamlines of the ow when passing round the object, whereas the ater face of the discocyte imposes a sharper change in the ow direction. However, the capability to deform, migrate and orient implies that discocytes present a larger gap in the effective viscosity (Fig. 8B). The mechanisms of migration and orientation permit a sharp relaxation, and at high shear rates the circular vesicle presents higher effective viscosity than the discocyte. All the
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deated vesicles behave very similarly, especially at high capillaries, with a marked shear-thinning behaviour which increases for lower reduced volumes. Conversely, the viscosity of the circular vesicle is rather constant at high capillaries, when the vesicle has adopted the so-called bullet shape (i.e. the equivalent of the parachute for circular shapes).
4 Discussion and conclusions The results obtained describe the extensive phenomenology of RBC circulation in microchannels, and highlight the subtle dependence of the cell dynamics on the ow velocity, wall connement and distance between cells. The focusing and alignment of RBCs play an important role in the rheological behaviour of the suspension. At low Ck, when RBCs ow with variable orientation and position, the effective viscosity of the suspension shows high sensitivity to the particular conguration. RBCs owing normal to the ow direction oppose higher resistance than those owing parallel to the ow. Additionally, RBCs owing close to the walls are also characterized by lower contributions to the viscosity. We have veried that the dependence of the RBC morphology on its initial condition is not due either to a small relaxation process or to an underlying nite size correlation between cells. Fig. 9 shows the effective deformation time, teff k , computed as the effective relaxation time of the membrane energy,† as a function of Ck. The relaxation time increases when decreasing Ck, but it does not diverge in the regime of small Ck where we observe strong memory effects. We have carried out † We have veried that this relaxation time is comparable to the characteristic time in which the RBC migrates laterally in the channel or reorients, corresponding to the quantities plotted in Fig. 3A and 2B, respectively.
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Effective deformation time seff k , measured from the evolution of the deformation energy, as a function of Ck. We consider that the RBC has reached a steady shape when the energy achieves a value at 1% of the final value. The curve is restricted to the regime in which RBCs rotate and orient with the flow, not presenting dependence on the initial inclination. The evolution of the typical time scale of evolution is smooth and unlikely to diverge at lower Ck. Deviations between the theoretical deformation time sk and the effective value measured may respond to the definition of the relevant length of the cell, which we have fixed for simplicity as the cell diameter a. Fig. 9
simulations 20 times longer than the highest teff k measured, thus largely exceeding the expected relaxation time scale, and observed the same dependence on the RBC initial conguration. Section 3.3 has shown that the correlation length lp increases with decreasing Ck. However, we have checked that with increasing the channel length from L ¼ 4a to 12a we do not observe any change in the nal RBC morphology at small Ck, ruling out that the memory reported is due to nite size effects. The origin of the sensitivity of RBCs to their initial conguration at small Ck remains unclear. It could be due to the elastic nature of the RBC and its ability to slightly deform, adapting to the position dependent ow. The focusing of RBCs in two narrow bands at symmetric lateral positions of the channel is primarily controlled by the capillary number. At low Ck, RBCs maintain their initial height in the channel during the ow, and thus cross stream migration is not observed. Supposing a suspension in which at the initial condition RBCs are uniformly distributed along the channel section, the downstream evolution does not change the cell distribution. For Ck > 10, RBCs exhibit a marked migration until they reach an equilibrium off center position. In terms of the RBCs spatial organization, the ow gives rise to a converging distribution of cells, an effect oen known as tubular pinch. The off-center migration of particles and cells in the inertial regime results from a balance between the wall repulsion and a dri from the axis towards the wall, as demonstrated analytically for rigid spheres.36,37 This dri is caused by inertial effects, so that the physical origin of the migration observed in the viscous regime must be different, and it might be found in the asymmetry and deformable properties of the object, explaining the
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diverging behaviour observed for RBCs and spherical vesicles. These results highlight the relevance of the specic properties of the deformable object, as opposed to the inertial regime in which RBCs, vesicles and particles are known to share a similar behaviour.5 The channel connement determines the off center equilibrium position of the RBC within the slipper regime: cells owing in thick channels (e.g. a/b 0.5) migrate towards a position 0.3R, whereas vesicles owing at the narrowest channels (e.g. a/b 0.9) are constrained to positions closer to the axis 0.1R. This lateral position is considerably lower than the classic Segr´ e & Silverberg inertial result of 0.6R, but the difference with our results might not relate to the hydrodynamic regime but it is likely explained by the geometric constraints imposed by the large size of the cell in comparison with the channel width, since recent experimental results at large connements in the inertial regime also found equilibrium positions in the range of 0.2–0.4R.38 The wall effect on the RBC is, additionally, coupled to the ow velocity. Less conned cells require higher Ck (even if the shear rate is maintained constant) to undergo lateral migration. Therefore, focusing at thick channels is achieved for increasingly higher Ck, implying that slippers and especially parachutes are rarely found at low connement conditions. The coupling between capillary and connement has been experimentally observed for single RBCs. Abkarian et al. (2008)18 presented a phase-diagram in which advanced parachutes are obtained at the highest connements and ow velocities, whilst slippers are restricted to lower values of connement and velocity. For a xed thick channel, they also observe more sparse positions of the RBC at low ow velocities. Their results might be interpreted as a transition from parachutes to slippers when the channel width is increased up to a/b < 0.4, suggesting that they are in the limit of high Ck. The RBC dynamics is very sensitive to the ow disruptions caused by neighbouring cells. At narrow channels, if the RBC concentration is high enough to force low separations between cells, they order in regular arrays. The transversal connement induced between vesicles forces a symmetric conguration, inhibiting the typical lateral position and deformation of slippers, and modifying the suspension rheology. These congurations of centered RBCs present much larger resistance to ow than slippers. For the range of Ck studied, minimum distances between RBCs required to reproduce the single-cell behaviour are 1–2 cells diameters, which represent volume fractions of typically fv ¼ 0.04–0.08. The critical distance separating the single and array congurations, however, reduces with increasing Ck. At thick tubes, the inertially driven lateral migration is robust for higher concentrations up to fv ¼ 0.2,14 but the much higher Reynolds and capillary numbers might attenuate the interactions between RBCs. Future work should study the importance of connement for high concentration suspensions, and how it determines the arrangement and organization of the RBCs when several of them can occupy the channel section (i.e. transition from the single-le ow, studied here, to multi-le ow). In this case, the wall connement and the interactions between cells (which
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effectively act as a transversal connement) act in the same direction, and thus the interplay between both effects will dictate the spatial cell distribution and consequently the suspension rheology.
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Acknowledgements We acknowledge the nancial support from MICINN (Spain) under project FIS 2009-12964-C05-02 and DURSI project SGR 2009-00014, and the Direcci´ on General de Investigaci´ on (Spain) and DURSI project for nancial support under projects FIS 2011-22603 and 2009SGR-634, respectively. G.R.L. also acknowledges Generalitat de Catalunya for support under grant FI-DGR2011.
References 1 M. J. Rosenbluth, W. A. Lam and D. A. Fletcher, Lab Chip, 2008, 8, 1062–1070. 2 X. Y. Hu, P. H. Bessette, J. R. Qian, C. D. Meinhart, P. S. Daugherty and H. T. Soh, Proc. Natl. Acad. Sci. U. S. A., 2005, 102, 15757–15761. 3 Y. W. Kim and J. Y. Yoo, Biosens. Bioelectron., 2009, 24, 3677– 3682. 4 S. Yang, A. Undar and J. D. Zahn, Lab Chip, 2006, 6, 871–880. 5 D. D. Carlo, D. Irimia, R. G. Tompkins and M. Toner, Proc. Natl. Acad. Sci. U. S. A., 2007, 104, 18892–18897. 6 A. Kumar and M. D. Graham, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2011, 84, 066316. 7 A. M. Leshansky, A. Bransky, N. Korin and U. Dinnar, Phys. Rev. Lett., 2007, 98, 234501. 8 A. Kumar and M. D. Graham, So Matter, 2012, 8, 10536. 9 A. R. Pries and T. W. Secomb, Clin. Microrheol. Circ., 2003, 29, 143–148. 10 R. Fahraeus and T. Lindqvist, Am. J. Physiol., 1931, 96, 562– 568. 11 J.-P. Matas, J. F. Morris and E. Guazzelli, J. Fluid Mech., 2004, 515, 171–195. 12 G. Segr´ e and A. Silberberg, J. Fluid Mech., 1962, 14, 136–157. 13 R. C. Jeffrey and J. R. A. Pearson, J. Fluid Mech., 1965, 22, 721– 735. 14 M. Han, C. Kim, M. Kim and S. Lee, J. Rheol., 1999, 43, 1157– 1174. 15 M. Tachibana, Rheol. Acta, 1973, 12, 58–69.
This journal is © The Royal Society of Chemistry 2014
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16 H. Wang and R. Skalak, J. Fluid Mech., 1969, 38, 75–96. 17 P. Olla, Phys. Rev. Lett., 1999, 82, 453. 18 M. Abkarian, M. Faivre, R. Horton, K. Smistrup, C. A. BestPopescu and H. A. Stone, Biomed. Mater., 2008, 3, 034011. 19 J. B. Freund, Annu. Rev. Fluid Mech., 2014, 46, 67–95. 20 D. A. Fedosov, H. Noguchi and G. Gompper, Biomech. Model. Mechanobiol., 2013, 239–258. 21 G. R. L´ azaro, A. Hern´ andez-Machado and I. Pagonabarraga, So Matter, 2014, DOI: 10.1039/C4SM00894D. 22 W. Helfrich, Z. Naturforsch., C: Biochem., Biophys., Biol., Virol., 1973, 28, 693–703. 23 F. Campelo and A. Hern´ andez-Machado, Eur. Phys. J. E, 2006, 2, 37–45. 24 A. J. C. Ladd and R. Verberg, J. Stat. Phys., 2001, 104, 1191– 1251. 25 M. E. Cates, J.-C. Desplat, P. Stansell, A. J. Wagner, K. Stratford and I. Pagonabarraga, Philos. Trans. R. Soc., A, 2005, 363, 1917–1935. 26 R. Benzi, M. Sbragaglia, S. Succi, M. Bernaschi and S. Chibbaro, J. Chem. Phys., 2009, 131, 104903. 27 G. Gonnella, E. Orlandini and J. M. Yeomans, Phys. Rev. Lett., 1997, 78, 1695. 28 S. Succi, The Lattice Boltzmann Equation: for Fluid Dynamics and beyond, Oxford University Press, Norfolk, 2001. 29 A. J. C. Ladd, J. Fluid Mech., 1994, 271, 285–309. 30 Y. W. Kim and J. Y. Yoo, J. Micromech. Microeng., 2008, 18, 065015. 31 F. Campelo and A. Hern´ andez-Machado, Phys. Rev. Lett., 2007, 99, 088101. 32 N. Srivastava, R. D. Davenport and M. A. Burns, Anal. Chem., 2005, 77, 383–392. 33 J. Gachelin, G. Mino, H. Berthet, A. Lindner, A. Rousselet and E. Cl´ ement, Phys. Rev. Lett., 2013, 110, 268103. 34 P. Gaehtgens, C. Duhrssen and K. H. Albrecht, Blood Cells, 1980, 6, 799–812. 35 J. L. McWhirter, H. Noguchi and G. Gompper, So Matter, 2011, 7, 10967. 36 B. P. Ho and L. G. Leal, J. Fluid Mech., 1974, 65, 365–400. 37 E. S. Asmolov, J. Fluid Mech., 1999, 381, 63–87. 38 D. D. Carlo, J. F. Edd, K. J. Humphry, H. A. Stone and M. Toner, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2009, 102, 094503.
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Rheology of red blood cells under flow in highly confined microchannels. II. Effect of focusing and confinement Guillermo R. La´zaro,*a Aurora Herna´ndez-Machadoa and Ignacio Pagonabarragab We study the focusing of red blood cells and vesicles in pressure-driven flows in highly confined microchannels (10–30 mm), identifying the control parameters that dictate the cell distribution along the channel. Our results show that an increase in the flow velocity leads to a sharper cell distribution in a
Received 24th April 2014 Accepted 25th June 2014
lateral position of the channel. This position depends on the channel width, with cells flowing at outer (closer to the walls) positions in thicker channels. We also study the relevance of the object shape, exploring the different behaviour of red blood cells and different vesicles. We also analyze the
DOI: 10.1039/c4sm01382d
implications of these phenomena in the cell suspension rheology, highlighting the crucial role of the wall
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confinement in the rheological properties of the suspension.
1
Introduction
The recent development of accurate lab-on-a-chip devices with channels of roughly 10 mm offers a unique pathway for single cell tests that enable the diagnosis of certain pathologies.1 The selective treatment of the cells typically exploits their response to different external forces, including electromagnetic elds2,3 or passive lters that make use of the hydrodynamic interactions of the cell with the channel geometry.4 The focusing of cells and particles at low concentrations can be used in this direction. For instance, appropriate designing of the channel geometry (e.g. curved channels) induces an ordering in the cell distribution that allows a ne precision of cell manipulation.5 The spatial organization is controlled by different factors such as object size and rigidity,6 or rheological properties of the external medium.7 Other applications are currently under exploration, such as drug delivery based on the different behaviour of blood cells under ow.8 The ongoing improvement of the precision and applications of these devices depends on a detailed knowledge of the cell behaviour at the microscale. The spatial organization of the cells plays, additionally, a fundamental role in the rheological properties of the system.9 The Fahraeus-Lindqvist effect10 provides a good example of the non-linear rheological response caused by spatial ordering of the cells. Red blood cells (hereaer RBCs) align and concentrate on the tube core, forming an annulus of free uid ow adjacent to the wall, increasing the uidity of the blood. The effect is accentuated in narrower tubes, when the repulsion from the
a
Departament d'Estructura i Constituents de la materia, Universitat de Barcelona, Av. Diagonal 647, E08028 Barcelona, Spain. E-mail: [email protected]
b
Departament de Fisica Fonamental, Universitat de Barcelona, Av. Diagonal 647, E08028 Barcelona, Spain
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wall is stronger, giving rise to a decreased blood viscosity, until reaching the limit case of single-le channels in which RBCs ow aligned forming regular trains. The equilibrium position of a particle in the tube depends on several factors such as ow velocity or connement.11 It is well known that isolated particles owing in thick tubes move laterally (i.e. cross stream migration) towards a specic off center position, as rst observed by Segr´ e & Silberberg (1962)12 for rigid spheres in a Poiseuille ow, and extended later on to other objects,13,14 and different channel geometries.15 This effect, however, is driven by an inertial dri which forces the object to leave the center.16 The dynamics of the object is not critically determined by its specic properties and the off center focusing has been also observed for red blood cells and other so entities.5 At very narrow microchannels, when inertial effects are negligible, the absence of this dri should prevent from a lateral migration during the downstream evolution, and hard spheres and spherical vesicles do not focalize. In this regime, however, the deformability and geometry of the object arise as crucial mechanisms of symmetry breaking,17 and RBCs are known to present a much more complex behaviour than spherical objects.18 Whilst the dynamics of particles and RBCs in the presence of inertia has been extensively studied in the literature, the limit of low Reynolds number, which requires a more accurate description of the geometry and elasticity of the object, has raised interest and awareness only recently,19,20 due to its relevance in the development of new microuidic devices. In a preceding paper,21 we have studied the importance of cell elasticity in the rheological response of a dilute suspension of RBCs owing at very conned microchannels. In this regime, the interactions between cells are subdominant and the dynamics of RBCs is dictated by their interaction with the channel walls. For increasing ow velocities, the cell assumes the so-called
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discocyte, slipper and parachute morphologies. At low ow velocity, the cell behaves as a rigid object maintaining a shape close to the equilibrium discocyte, and it does not perform lateral displacements. Further increasing the incoming ow, cells are forced to migrate towards an off center position, adopting the slipper morphology. At very high ow velocity, the slipper conguration becomes unstable and cells return to an axial position where they show a symmetric parachute-like shape. Whereas parachutes present a high deformation energy, slippers represent a low energy conguration closer to the elastic energy of the equilibrium discocyte. The suspension viscosity shows a shear-thinning behaviour, determined by the interplay between the cell deformability and its lateral migration and orientation in the off center position. In this paper, we extend our study on RBC behaviour in narrow channels, driving our interest towards the spatial organization and focusing of cells. The paper is organized as follows. We rst present the physical model and briey explain the numerical methods applied in the simulations. In the Results section, we analyze the strong sensitivity of the suspension rheology to the focusing and alignment of RBCs. We then characterize the spatial organization of RBCs along the channel section and their orientation for increasing ow velocity. We focus on the relevance of the connement and the interaction of the walls with the cells. The competition between wall induced effects and hydrodynamic interactions between cells is studied in the subsequent section. In order to complete our study about the problem of particle migration, we nally identify the role of the object shape by studying the behaviour of different vesicles, i.e. varying the equilibrium shape of the cell from the discocyte to spherical geometries. We then proceed with a discussion of the coupled effect between all these parameters and compare with previous numerical and experimental studies. We nish with the main conclusions of our ndings.
2 The model The elastic properties of the RBC membrane are described by the Helfrich bending energy,22 ð ð ð k F mem ¼ c2 dS þ gdS þ xdV ; (1) 2 vU vU U where c is the total curvature of the cell surface, k is the bending rigidity of the membrane, and g and x are Lagrange multipliers which maintain constant area and volume, respectively. The constraint of constant area accounts for the in-plane incompressibility of the uid membrane. The volume is maintained constant as a consequence of the impermeability of the membrane to water exchange with the exterior of the cell. We make use of a phase-eld model for the numerical implementation of the elastic membrane energy.23 In this approach, the membrane free energy reads: ð ð 2 k* (2) F fmem ¼ f þ f3 32 V2 f dV þ gjVfj2 dV : 2 It can be shown that the rst term of eqn (2) characterizes an pffiffiffi interface with bending rigidity k ¼ ð433 =3 2Þk*. The second
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term is the constant area constraint. The requirement of constant volume is achieved by imposing a conserved dynamics, given by an advection–diffusion equation for the order parameter, vf þ v$Vf ¼ MV2 m; vt
(3)
where m ¼ dF fmem/df is the chemical potential, M is the order parameter mobility and v is the uid velocity. This equation is coupled to the Navier–Stokes equation, that describes the hydrodynamics of the uid both in the inner and outer regions of the RBC membrane, r
vv þ rðv$VÞv ¼ VP fVm þ h0 V2 v þ rf: vt
(4)
where h0 is the viscosity of the solvent, r is the density, P is the pressure and f is the external force which effectively acts as a pressure difference, rf ¼ DP/L. The coupling between eqn (3) and (4) is performed via the advective term v$Vf, that describes how the uid pushes the membrane, and the reactive term fVm which represents the elastic force exerted by the membrane.
2.1
Numerical scheme
The model comprises two coupled dynamic equations which must be solved numerically. We make use of a hybrid scheme, using a standard nite element method for integrating the order parameter equation, eqn (3), and a lattice-Boltzmann method for solving the Navier–Stokes equation (eqn (4)). The LB method is an efficient and robust numerical solver which has been extensively applied in different elds of complex uids and so matter.24–27 The LB method is based on the fact that in the hydrodynamic limit, the Boltzmann equation recovers the macroscopic dynamics given by the Navier–Stokes equation. A discrete version of the Boltzmann equation can be numerically integrated,28 fi ðx þ ci ; t þ 1Þ fi ðx; tÞ ¼
1 fi fieq þ Fi : s
(5)
where fi is the distribution function and i runs over the velocity vector ci which connects with the neighbour nodes of the lattice. Fi is related to the external force, fext. The hydrodynamic variables, density and momentum, are then recovered from the X X distribution functions by fi ¼ r and fi ci ¼ rv, respeci
i
tively. Aerwards, the velocity v is introduced in the advection term of eqn (3). The equilibrium distribution f eq is obtained from an i expansion in terms of v, 1 1 fi eq ¼ run An þ 2 v$ci þ ; (6) vv : c c þ G : c c i i i i cs 2cs 4 where we have introduced the tracelesss tensor vv ¼ vv ðv2 =3Þ1, and cs is the speed of sound. The constants uv and Av depend on the specic lattice model; we use the D3Q19 model which comprises 1 velocity of modulus c ¼ 0, 6 of velocity
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pffiffiffi c ¼ 1 and 12 of c ¼ 2. The index n identies the three velocity subsets. The force exerted by the membrane is introduced in the tensor G ¼ 9=ð2rÞP 3dTrP, where the pressure tensor is
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P ¼ ðr=3Þd þ Pmem , and Pmem must satisfy VP mem ¼ fVm. The external force term of eqn (5) reads 1 ½f ext $ci ð1 þ v$ci Þ v$f ext : Fi ¼ 4un 1 2s
(7)
We impose stick boundary conditions at the walls by means of a bounce-back method.29 The method consists of reecting the distribution functions on the solid nodes that come from the uid aer a propagation step. In this scheme, the zero velocity condition is achieved at the middle point between the uid and solid nodes. 2.2
dimensional approach it is dened as nred ¼ A/pr2, where A is the area of the object and r ¼ p/2p is the radius of the circle with the same perimeter p as that of the vesicle. For a RBC nRBC ¼ 0.48, but we explore the behaviour of other object geometries nred ¼ 0.55, 0.69 and 0.97, the later corresponding to a nearly spherical vesicle (a circle in the 2D approach). Phase-eld methods require careful control of the model parameters to enforce the appropriate membrane elastic behavior for the desired degree of discretization. The model has provided successful results in different membrane phenomena that converge towards both theoretical and numerical predictions.23,31 Further technical details of the numerical implementation and its convergence properties are explained in the preceding paper.21 The effective viscosity is computed from the mean ow vz, according to heff ¼
Channel geometry and parameter steering
The simulations are performed in 2D rectangular boxes of size Lx Lz, applying periodic boundary conditions in the direction ^z, and no-slip conditions at the walls situated at x ¼ Lx/2 and x ¼ Lx/2. The force is constant, f ¼ f0^z, so that in the absence of the RBC the imposed force gives rise to a Poiseuille ow, nz ¼ f0/ 4h0(x2 (Lz/2)2). All the geometrical parameters are expressed relative to the RBC size, a, except those referring to connement, which are expressed relative to the channel width b h Lx or half of the channel width R ¼ b/2 when referring to lateral positions in the channel (since there are two symmetric positions with respect to the channel axis). Given that we concentrate on spatial ordering, it is important to remark the 3D analogy of our simulations. It is well-known that particles owing in rectangular channels focus to two symmetric layers, whereas those owing in square-section channels focus to four symmetric points.30 In our case, the geometry of the channel might be related to a rectangular channel in which the distance between the walls in the ˆy-direction is much larger than in the ^x-direction, Ly [ Lx. Since cell dynamics is dominated by the interaction with the walls and the uid ow is effectively plane in the transverse direction ˆy, it is reasonable to assume that shape coupling with the ow is subdominant in this transverse direction. Thereby, RBCs focalize at specic positions along ^x and show a homogeneous distribution in ˆy. The characteristic viscous time is calculated as the inverse of the shear rate sh ¼ b/vz, where vz is the mean velocity in the ow direction z, and the elastic relaxational time of the cell is given by sk ¼ ha3/k. The ratio between both time scales denes the capillary number, Ck, that describes the relative effect of the viscous effects of the uid and the elastic properties of the membrane, sk h vz a2 a Ck ¼ ¼ 0 : (8) b sh k We explore the range of capillaries between 0.2 and 120. In all the simulations the Reynolds number Re ¼ rnzL/h is maintained below 0.05, within the non-inertial regime. The reduced volume denes the shape of the object, and in the two-
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f0 2 b: 12vz
(9)
This expression corresponds to the effective viscosity of a uid with mean ow velocity ^vz (at the specic channel geometry) when an external force f0 is applied. The viscosity derived from ow-curve calculations is widely extended in tube and channel geometries, especially for highly conned systems, instead of the usual shear stress/rate relationship obtained in rotational viscosimeters. Recently, a number of viscosimeters have been presented in the literature in which the rheological properties of different complex uids are studied in terms of their effective viscosity, including blood32 or suspensions of active swimmers.33 We adopt this approach to identify the effective viscosity to facilitate the comparison with experimental studies.
3 Results 3.1
Effective viscosity and focusing
Our results show that heff depends on the initial condition of the RBC, especially at low Ck. The viscosity dispersion Dheff ¼ (heff eff)/ eff is the average viscosity, can be as high as h heff, where h 25%. This dispersion is shown in Fig. 1A, for a number of different congurations. In experiments, the effect of thermal noise, that allows random exploration of different orientations, may lead to a more uniform measurement. Therefore, in order to obtain a robust measurement of the viscosity, we average over a set of 7 different inclinations and 3 different lateral positions. The dispersion of each conguration is shown in Fig. 1B (inclinations) and 1C (lateral positions). According to their particular position and alignment with the ow, RBCs will offer variable resistance to ow. RBCs aligned parallel to the incoming ow and owing close to the walls (where the shear stress decreases) present smaller contribution to the viscosity than those centered and with normal orientation, which induce a severe perturbation of the surrounding ow. The results show that RBCs oriented symmetrically with respect to the normal ow direction (e.g., orientation of q ¼ 45 and 135 ) show a similar viscosity. This symmetry suggests that the relevant
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Fig. 1 Effective viscosity of a suspension of RBCs as a function of the capillary number, Ck. The coloured regions represent the three main morphological regimes: discocytes, slippers and parachutes. Examples of each morphology, as well as the steady position and orientation for different initial configurations, are shown in the snapshots. For further information about the morphological sequence see the preceding paper.21 At low capillaries, RBCs flow showing a certain dependence on the initial conditions. RBCs deform and accommodate their shape to the flow profile, without rotating towards a specific orientation, and maintaining their initial position along the tube. At intermediate Ck, RBCs rotate orienting their axis with the flow, but still maintain their initial position. Only RBCs placed very close to the wall migrate to more centered positions – hence, focusing in this regime is very weak. RBCs inclination therefore depends on their particular position. At high Ck, the focusing implies that regardless of the initial configuration RBCs flow at a fixed lateral position, assuming the slipper shape with their inclination aligned with the flow profile. Finally, RBCs exhibit parachute shapes for typically Ck > 105. (A) The viscosity presents a certain sensitivity to the initial conditions of the RBC, especially at low capillary numbers. Each dashed line represents the viscosity for a specific initial configuration. The mean eff (red line), is therefore averaged over the different configurations, covering: (B) initial inclinations (with respect to the effective viscosity, h channel axis), with normal RBCs presenting a higher resistance to flow compared to aligned RBCs; (C) initial heights in the channel, with centered RBCs opposing more resistance than RBCs flowing close to the walls. The viscosity dispersion is defined as Dheff ¼ (heff h eff)/ heff, where heff is the effective viscosity for each particular initial configuration. In all cases, within the slipper regime (V) the final configuration is nearly independent from the initial one. The dependence is stronger at intermediate (,) and especially low (B) capillaries. The schemes on the right represent the RBC inclination q and center of mass height hcm.
parameter controlling the value of the viscosity, within this rigid limit, is the ratio between the section occupied by the cell with respect to the total section of the channel, regardless of the cell orientation. The dispersion reduces for intermediate Ck and it is negligible at high values. The memory of the RBC of its initial conguration can be separated into two different contributing phenomena: alignment of the RBC with the ow, and focusing to a nal position. Due to the channel symmetry with respect to its axis, RBCs focus at two symmetric lateral bands, depending on which channel region was the cell located with respect to the axis at the initial condition. Migration of initially centered RBCs is determined by numerical noise. Henceforth, we concentrate on one of the channel regions (i.e. from the axis to one of the walls), bearing in mind that symmetric processes occur in the opposite region. At very low Ck, RBCs do not orient their axis with the ow, and they ow maintaining the initial position, without showing migration across streamlines. The increasing external ow forces the cell to rotate and align with the ow, but still showing a dependence on the initial position along the channel section. The dispersion of the viscosity decreases since the range of cell inclinations is reduced, but still maintain the contribution due to the different position along the tube. Further increase of Ck
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induces a migration towards an off center region, and the RBC distribution narrows forming a thin band. The viscosity dispersion decreases with the focusing and, for sufficiently high ow velocities, the nal position and orientation of the RBC is unique regardless of the initial condition, leading to a welldened value of the effective viscosity.
3.2
RBC focusing and alignment
RBC focusing has been extensively studied in the inertial regime, but it is not well understood in the viscous one. In this section we study the focusing of RBCs to a dened, off-center position, and how their lateral position can be controlled by tuning the capillary number and the distance between the channel walls. 3.2.1 Focusing. At low Ck, RBCs ow occupying the entire channel section, without any favoured position. If the external force increases, however, RBCs concentrate on a narrow band off the channel axis, as shown in Fig. 2. In Fig. 2A, at Ck ¼ 10, we show the height along the channel section of RBCs placed at different initial positions during the downstream ow. RBCs initially placed close to the wall experience a repulsion and migrate towards the center, stabilizing at hcm/R ¼ 0.5. RBCs
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Fig. 2 (A) Evolution during the downstream flow of the center of mass height, hcm, of RBCs initially placed at different lateral positions, within the slipper regime, Ck ¼ 13.2. RBCs close to the wall are repulsed and migrate towards the center; likewise, centered RBCs migrate towards the walls. All RBCs stabilize at intermediate positions, forming a band of stable trajectories. In this case the confinement has been increased with respect to the standard one, with a/b ¼ 0.49 in this plot, in order to observe larger and more perceptible migrations. (B) Width of the band of stable trajectories as a function of Ck. The width of the band can be controlled by increasing the flow velocity (confinement a/b ¼ 0.71). High velocities induce a narrowing of the band, until the RBC eventually occupies a unique position in the channel. Within the parachute regime, RBCs can develop parachute and slipper morphologies at fixed flow conditions depending on their initial position: RBCs initially placed close to the wall are repulsed and assume a slipper shape, whereas cells placed at the channel core deform into a parachute.
initially placed in the channel axis also migrate from the center and reach an equilibrium position at hcm/R ¼ 0.25. The rest of the RBCs occupy intermediate positions, forming a band of stable trajectories. The channel width has been increased to a/b ¼ 0.49, in order to allow larger and more perceptible migrations. The time scale of the migration is typically half of the deformation time sk. The increase in Ck induces a narrowing of the band of stable trajectories, as shown in Fig. 2B, and eventually RBCs are found in a unique lateral position regardless of their initial condition, for Ck > 40. In the parachute regime, we obtain that parachute and slipper shapes can be observed at the same conditions depending on the initial conguration: cells initially placed close to the walls assume asymmetric shapes, whilst those placed close to the center adopt a parachute shape. 3.2.2 Alignment. RBCs at very low capillary Ck < 2 ow maintaining their initial orientation, as shown in Fig. 3A.
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Fig. 3 (A) Orientation of the RBC as a function of the capillary number for different initial conditions, q0. Within the discocyte regime, the RBC maintains its initial orientation, with slight deformation. For Ck > 2, all RBCs rotate and orient their axis with the imposed flow profile. (B) Temporal evolution of the inclination of a RBC at Ck ¼ 4.5. The rotation towards the equilibrium inclination occurs on a fast time scale compared to the migration and relaxation time of the cell. In the inset some snapshots of the evolution are shown.
However, beyond the critical value Ck ¼ 2, RBCs lose this dependence and rotate towards a xed orientation, which in turn depends on the lateral position along the channel. RBCs owing close to the wall assume slightly more horizontal inclinations than those more centered, as a result of the nonuniform curvature of the ow prole. The focusing of RBCs takes place for higher capillaries. The sensitivity to the initial conditions of the orientation and the lateral position is seemingly uncoupled. Accordingly, the dispersion on the nal congurations can be separated into three different situations: (i) for Ck < 2, RBCs ow maintaining their initial distribution and orientation, deforming their shape to the local ow prole; (ii) for 2 < Ck < 7, RBCs show a xed orientation aligned with the ow, but they still retain their memory of the initial position along the channel section; and (iii) Ck > 7 RBCs focalize at two symmetric lateral positions in the channel, aligned with the ow. The temporal characterization of the rotation of a RBC towards its stable orientation is shown in Fig. 3B. RBC rotation occurs on a much shorter time scale than migration, typically 0.1sk. In the inset, some snapshots of the process are depicted. 3.2.3 Effect of wall connement. The focusing of RBCs can be controlled by both increasing the ow velocity or the degree Soft Matter, 2014, 10, 7207–7217 | 7211
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of connement of the channel. By varying the distance between _ we identify the walls b, but maintaining a constant shear rate g, how the closer presence of the walls affects the RBC morphology. We focus on intermediate and high capillaries in which the presence of the walls is more relevant. In Fig. 4A, a phase diagram for different capillaries and connements is shown. Moving along a row of constant capillary numbers, the increasing connement induces higher migrations and lower inclinations. This effect is especially noticeable at intermediate capillaries, in which highly conned RBCs have migrated and adopted the slipper morphology, whereas less conned RBCs still remain centered and retain a discocyte shape. The relative distance to the wall arises as an important factor to trigger the migration, and at thick channels the capillary required to observe slippers and parachutes might be extremely high. Interestingly, the diagram highlights a similar effect of moving along the vertical and horizontal axes. This suggests a coupling between the effect of connement and capillary, implying that increasing shear rates induce stronger interactions between the cell and the walls, leading to a larger repulsive force from the axis, and this effect can be emphasized by a closer distance of the RBC to the walls. The lateral position and focusing of RBCs for different wall connements is shown in Fig. 5. The connement positively affects focusing, and for all capillaries Ck > 20 RBCs are localized at a dened lateral position if a/b > 0.5. The distance from the axis of this position increases for thicker channels, from hcm ¼ 0.05R to 0.3R in the range of channels studied. At the narrowest channels, the RBC is placed close to the center, likely due to the geometrical constraints but also to the strong wall repulsion. The increase in Ck reinforces the focusing effects, and especially for Ck ¼ 109.5 focusing has extended to thicker channels and RBCs are localized at
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Focusing of RBCs as a function of the wall confinement and the capillary number. For each Ck, the top and bottom values correspond, respectively, to the outermost (i.e. closest to the wall) and innermost (closest to the center) equilibrium positions; all the rest stable positions are found in between these two. The wall confinement favours the focusing and typically the effect is reduced below a/b < 0.5. The lateral position of the RBC is found further away from the axis in thicker channels, and in the limit in which the width of the channel is equal to the cell diameter, RBCs flow nearly centered. The capillary number also induces an increased focusing (especially in the case Ck ¼ 109.5), though its effect is weaker. Fig. 5
positions further from the channel axis. Note, however, that the effect is relatively weak for the two other cases Ck ¼ 18.5 and 43.2. The larger inclination and migration observed in RBCs owing along the narrowest channels are accompanied by lower values of the effective viscosity, as shown in Fig. 4B. In the case
Fig. 4 Effect of channel confinement on the RBC behaviour at different Ck. All the RBCs are initially placed at the channel axis. (A) RBC morphologies found at different confined channels; cells shown here are initially placed at the channel axis, normal to the flow direction. The shear rate is maintained constant for each value of Ck. The effect of the walls is important to induce lateral migration, as confined RBCs exhibit the characteristic slipper morphology, whereas the RBC flowing at the thicker tube still retains a discocytic shape. Less confined RBCs, which may present weaker interactions with the walls, require higher Ck to deform and migrate. (B) Effective viscosity heff for a suspension of RBCs as a function of the channel confinement and capillary number, at constant volume fraction. heff, especially at high capillaries, strongly depends on the confinement. For confined RBCs which have migrated, the viscosity is low and uniform. However, if the channel is thick enough, the RBC does not migrate and its center position and normal orientation imply a higher viscosity.
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of high capillary, the viscosity is similar for all the channels in which the RBC has been forced to migrate and align. However, if the connement is low and the RBC still ows along the centerline, it will displace a larger amount of external uid and therefore the suspension viscosity will be also larger. For decreasing connements the effective viscosity increases, until it eventually stabilizes when the wall inuence is negligible and the RBC behaviour is similar to that in an unbounded ow. The connement necessary to recover this unbounded behaviour also depends on the capillary number, as expected given the existing coupling between connement and the capillary number. At lower values of the capillary, all the RBCs exhibit a less marked migration and alignment, and the differences between the viscosity of each initial condition are less accentuated.
Effectively, the system reproduces a one-dimensional array of identical RBCs separated by the domain length distance, Lz, which reminds the ordered structures observed experimentally. Although we cannot address the stability of these regular arrays and the possibility that RBCs reorder and form clusters, the method allows us to explore systematically the competition between wall-induced effects and RBC interactions. The membrane stiffness dictates the ow disruption induced by the RBC. Rigid cells induce stronger perturbations of the incoming ow than soer ones, as seen in Fig. 6 A. We dene the ow amplitude .Xh i X AðzÞ ¼ ðan a0n þ bn b0n Þ ða0n Þ2 þ ðb0n Þ2 of the imposed
3.3
ow amplitude to a Gaussian decay A(z) ¼ A0 exp(((z zcm)/ lp)2), a typical distance of the ow distortion is obtained, the penetration length lp. Fig. 6 B displays the penetration length for different Ck, at very long tubes Lz [ a when RBCs do not interact. The results show that the ow disruption strongly decays far from the cell and for Ck > 10 the ow deviates from the reference one only at very close distances from the RBC, 0.1b. This could suggest that the coupling between RBCs is only relevant if they are placed extremely close. However, even if deviations from the imposed ow are small when RBCs are distant, interactions are strengthened for lower
Hydrodynamic interactions between RBCs
While owing along conned channels at high concentrations, RBCs oen order in regular trains, as rst noted by Gaehtgens et al. (1980).34 The formation of these ordered congurations is important in the designing of microuidic devices as it increases the control of cell manipulation. From the theoretical point of view, the organization in trains also offers an interesting way to study the hydrodynamic interactions between neighbouring cells, and how it affects the RBC dynamics. We take advantage of the periodic boundary conditions in the ^z direction35 to study the hydrodynamic interactions between the cell and its images.
Poiseuille, where (an, bn) and (a0n, b0n) are the coefficients of the Fourier decomposition of the actual vz (x, z0) and imposed vz0 (x, z0) velocity proles, respectively, for the rst n ¼ 20 modes, X vðx; z0 Þ ¼ ½an sinð2pnx=bÞ þ bn cosð2pnx=bÞ. By tting the
Fig. 6 (A) Flow disruption induced by a RBC in a Poiseuille flow. The color map represents the value of the velocity component vz. Ck affects the flow disturbance that the cell induces: rigid RBCs (up, Ck ¼ 0.8) disturb a larger region of the surrounding flow than soft RBCs (bottom, Ck ¼ 15.1). (B) Penetration length, lp of the flow disruption caused by the RBC for different Ck. lp is the typical length of the flow distortion caused by the cell in the surrounding flow with respect to the unperturbed Poiseuille, but see the main text for a formal definition. lp decreases rapidly with Ck. For Ck > 10 it falls down to lp < 0.1a, meaning that RBCs induce a limited perturbation of the flow at intermediate and high capillaries. (C) Penetration distance lp for closely placed RBCs, as a function of the distance between cells L. Even if the flow disruption shows a strong decay at high capillaries, RBC behaviour is highly sensitive to the presence of neighbouring cells. The dashed line separates the single-cell behaviour (when RBCs are able to migrate off center and assume slipper morphologies) from the array configuration (when they order in regular arrays of centered, symmetric cells). For Ck ¼ 19.5, RBCs only recover the single-cell behaviour if the distance between them is higher than 1.5a (inset). For lower values, hydrodynamic interactions between cells dominate and migration is inhibited. The penetration length lp shows that the coupling between RBCs is large if the distance is much lower than the length of the channel (wall confinement here is a/b ¼ 0.71).
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distances between cells, favouring the collective behaviour. If RBCs are initially placed closer to each other, even at high Ck, they do not migrate towards the wall but ow maintaining a centered position, forming an ordered array of RBCs aligned normal to the channel axis, as shown in Fig. 6C (inset, RBCs on the right) for Ck ¼ 19.5. RBCs bend, coupling their surface to the ow prole. If the distance between cells is increased, eventually slippers are recovered, initially showing a slight distortion and then fully exhibiting the single-cell behaviour. The distance necessary to separate this array-conguration from the singlecell behaviour depends on the capillary number, requiring larger cell-to-cell distances at low Ck. For Ck ¼ 19.5, the RBC behaves as hydrodynamically isolated for lp > 1.4a, but the critical distance decays to 0.5 for Ck ¼ 97. The formation of regular arrays originates from the transversal connement that the RBC feels due to the presence of its neighbours. The ow disturbance generated by the RBC is symmetrically compressed by both the preceding and the rear cells of the array, inhibiting the symmetry breaking of the ows generated around the RBC that preludes the migration in the case of isolated cells. Thereby, the transversal connement constrains the RBC shape to a centered, symmetric morphology. Fig. 6C displays the dependence of the penetration length on the separation between RBCs for a given channel width, a/b ¼ 0.71, and for an intermediate Ck value, for which lp is typically of the order of the channel width. For distances L much smaller than the channel width the wall effect is subdominant and the RBC behaviour is controlled by the transversal connement, forming regular arrays of RBCs. Only in sufficiently long channels, L/a > 2, RBCs behave as hydrodynamically isolated, recovering the singlecell behaviour, exhibiting lateral migration and assuming slipper shapes. This transition from the array-conguration to the singlecell behaviour determines the range in which wall effects are dominant with respect to cell-to-cell interactions. The formation of arrays of RBCs has important implications in the rheology of the suspension. The normal orientation of the cells implies a larger resistance to ow than in the case of aligned RBCs (i.e. slippers), and the solvent is repelled from the channel core towards the walls, where it ows free of cell disruption. This effect, however, implies that the effective viscosity of the suspension increases, as shown in Fig. 7A, where the effective viscosity of the suspensions shown in Fig. 6C, at constant Ck, is plotted. A sweep in Ck reveals that the differences in the viscosity are accentuated at lower values, as a direct consequence of the stronger interactions between more rigid cells. This effect is shown in Fig. 7B, where the effective viscosity as a function of Ck is shown for three different distances between RBCs. At high Ck, the distance between RBCs in all the cases is larger than the critical distance in the single-cell regime. For lower values of Ck, the differences of the viscosity are primarily due to the different volume fraction. At some point, marked by the dashed line, the distance between RBCs is lower than the increasing critical length and RBCs switch to the train conguration, inducing a substantial increase of the effective viscosity. Accordingly, the presence of hydrodynamic interactions between RBCs critically determines both the morphological and rheological behaviour of the suspension.
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Fig. 7 (A) Effective viscosity for a regular array as a function of the intracellular distance, a/L, at a constant capillary number Ck ¼ 18. Each value of the viscosity corresponds here to the shapes shown in Fig. 6C, for initially centered RBCs. The increase in the viscosity can be decomposed into two different contributions: (i) the change in the volume fraction, from 0.40 to 0.13, (ii) the effect of the interactions between RBCs at low L, which allow the formation of trains of cells at the channel core that present higher flow resistance. (B) Effective viscosity as a function of Ck for different cell-to-cell distances. If the channel is not sufficiently long, at some point, marked by a dashed line, the penetration length lp exceeds the distance between cells L and the RBC interacts with its image (forming an array of RBCs due to the periodic boundary conditions) and this fact triggers a considerable increase in the effective viscosity. The change between the single-cell and train like configurations is marked by a dashed line, and the specific cell shapes are shown in the inset. At low distances between RBCs, they maintain a centered position but bending its surface. Only for large distances between cells slippers are observed, RBCs on the left.
3.4
Vesicle shape
We explore the relevance of the object geometry by studying the case of three vesicles with different reduced volumes and compare with the RBC case. The relaxed shapes of the vesicles are obtained from minimizing several initial ellipsoids with different areas and perimeters, nred ¼ 0.97, 0.69, 0.55, 0.48, corresponding to the equilibrium shapes from a nearly circular to a discocyte shape. This minimization is performed in the absence of uid. The vesicles are then placed in the channel and the uid is switched on. Due to their symmetric shape, almost circular vesicles present less degrees of freedom to adapt and orient with the external ow, showing slight deformations even at high forcements. They do not migrate out of the axis,
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Fig. 8 Effect of the reduced volume on vesicles and RBC behaviour and suspension rheology. (A) Vesicle morphologies for different reduced volumes. Circular vesicles, which are symmetric, do not migrate and remain centered flowing at the channel center line. The increasing deflation of the vesicles (i.e., lower reduced volumes) induces a higher degree of migration and asymmetry. Discocytes exhibit the largest degree of migration towards the wall and orientation with the flow profile. (B) Effective viscosity as a function of the capillary number for different reduced volumes. The capability of deflated vesicles to migrate and orient enables them to reduce their resistance to flow, and this property implies a sharper shear-thinning behaviour than in the case of circular vesicles.
remaining at the center line, and the low area excess allows only a slight deformation. The vesicle center of mass moves to a more forward position at high capillaries, acquiring at intermediate capillaries a more triangular shape which could remind us of a parachute, the so-called bullet, as shown in Fig. 8A. We check the evolution of a circular vesicle when initially placed close to the wall, nding that it migrates towards the center recovering the symmetric shape, as opposed to discocytes. Deated vesicles present an intermediate behaviour between the circular and the discocyte vesicles. For all the cases studied, the equilibrium position is strictly asymmetric and therefore vesicles eventually deform and migrate, but lower values of the capillary require more deated vesicles to acquire asymmetric shapes. In the inertial regime, the off center position of objects does not depend on their specic geometry, and hard spheres, so beads and cells behave similarly, migrating towards roughly the same equilibrium position. The repulsive force from the axis does not manifest a critical dependence on the specic properties of the object. The sensitivity shown by different objects in the viscous regime might be explained by asymmetric distribution of normal stresses on deated vesicles, implying an effective repulsive dri, which is on the contrary balanced in circular vesicles because of their symmetry. In the rigid limit, discocytes present a considerably higher resistance to ow than circular vesicles, in spite of their lower volume. This may be due to the smoother streamlines of the ow when passing round the object, whereas the ater face of the discocyte imposes a sharper change in the ow direction. However, the capability to deform, migrate and orient implies that discocytes present a larger gap in the effective viscosity (Fig. 8B). The mechanisms of migration and orientation permit a sharp relaxation, and at high shear rates the circular vesicle presents higher effective viscosity than the discocyte. All the
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deated vesicles behave very similarly, especially at high capillaries, with a marked shear-thinning behaviour which increases for lower reduced volumes. Conversely, the viscosity of the circular vesicle is rather constant at high capillaries, when the vesicle has adopted the so-called bullet shape (i.e. the equivalent of the parachute for circular shapes).
4 Discussion and conclusions The results obtained describe the extensive phenomenology of RBC circulation in microchannels, and highlight the subtle dependence of the cell dynamics on the ow velocity, wall connement and distance between cells. The focusing and alignment of RBCs play an important role in the rheological behaviour of the suspension. At low Ck, when RBCs ow with variable orientation and position, the effective viscosity of the suspension shows high sensitivity to the particular conguration. RBCs owing normal to the ow direction oppose higher resistance than those owing parallel to the ow. Additionally, RBCs owing close to the walls are also characterized by lower contributions to the viscosity. We have veried that the dependence of the RBC morphology on its initial condition is not due either to a small relaxation process or to an underlying nite size correlation between cells. Fig. 9 shows the effective deformation time, teff k , computed as the effective relaxation time of the membrane energy,† as a function of Ck. The relaxation time increases when decreasing Ck, but it does not diverge in the regime of small Ck where we observe strong memory effects. We have carried out † We have veried that this relaxation time is comparable to the characteristic time in which the RBC migrates laterally in the channel or reorients, corresponding to the quantities plotted in Fig. 3A and 2B, respectively.
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Effective deformation time seff k , measured from the evolution of the deformation energy, as a function of Ck. We consider that the RBC has reached a steady shape when the energy achieves a value at 1% of the final value. The curve is restricted to the regime in which RBCs rotate and orient with the flow, not presenting dependence on the initial inclination. The evolution of the typical time scale of evolution is smooth and unlikely to diverge at lower Ck. Deviations between the theoretical deformation time sk and the effective value measured may respond to the definition of the relevant length of the cell, which we have fixed for simplicity as the cell diameter a. Fig. 9
simulations 20 times longer than the highest teff k measured, thus largely exceeding the expected relaxation time scale, and observed the same dependence on the RBC initial conguration. Section 3.3 has shown that the correlation length lp increases with decreasing Ck. However, we have checked that with increasing the channel length from L ¼ 4a to 12a we do not observe any change in the nal RBC morphology at small Ck, ruling out that the memory reported is due to nite size effects. The origin of the sensitivity of RBCs to their initial conguration at small Ck remains unclear. It could be due to the elastic nature of the RBC and its ability to slightly deform, adapting to the position dependent ow. The focusing of RBCs in two narrow bands at symmetric lateral positions of the channel is primarily controlled by the capillary number. At low Ck, RBCs maintain their initial height in the channel during the ow, and thus cross stream migration is not observed. Supposing a suspension in which at the initial condition RBCs are uniformly distributed along the channel section, the downstream evolution does not change the cell distribution. For Ck > 10, RBCs exhibit a marked migration until they reach an equilibrium off center position. In terms of the RBCs spatial organization, the ow gives rise to a converging distribution of cells, an effect oen known as tubular pinch. The off-center migration of particles and cells in the inertial regime results from a balance between the wall repulsion and a dri from the axis towards the wall, as demonstrated analytically for rigid spheres.36,37 This dri is caused by inertial effects, so that the physical origin of the migration observed in the viscous regime must be different, and it might be found in the asymmetry and deformable properties of the object, explaining the
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diverging behaviour observed for RBCs and spherical vesicles. These results highlight the relevance of the specic properties of the deformable object, as opposed to the inertial regime in which RBCs, vesicles and particles are known to share a similar behaviour.5 The channel connement determines the off center equilibrium position of the RBC within the slipper regime: cells owing in thick channels (e.g. a/b 0.5) migrate towards a position 0.3R, whereas vesicles owing at the narrowest channels (e.g. a/b 0.9) are constrained to positions closer to the axis 0.1R. This lateral position is considerably lower than the classic Segr´ e & Silverberg inertial result of 0.6R, but the difference with our results might not relate to the hydrodynamic regime but it is likely explained by the geometric constraints imposed by the large size of the cell in comparison with the channel width, since recent experimental results at large connements in the inertial regime also found equilibrium positions in the range of 0.2–0.4R.38 The wall effect on the RBC is, additionally, coupled to the ow velocity. Less conned cells require higher Ck (even if the shear rate is maintained constant) to undergo lateral migration. Therefore, focusing at thick channels is achieved for increasingly higher Ck, implying that slippers and especially parachutes are rarely found at low connement conditions. The coupling between capillary and connement has been experimentally observed for single RBCs. Abkarian et al. (2008)18 presented a phase-diagram in which advanced parachutes are obtained at the highest connements and ow velocities, whilst slippers are restricted to lower values of connement and velocity. For a xed thick channel, they also observe more sparse positions of the RBC at low ow velocities. Their results might be interpreted as a transition from parachutes to slippers when the channel width is increased up to a/b < 0.4, suggesting that they are in the limit of high Ck. The RBC dynamics is very sensitive to the ow disruptions caused by neighbouring cells. At narrow channels, if the RBC concentration is high enough to force low separations between cells, they order in regular arrays. The transversal connement induced between vesicles forces a symmetric conguration, inhibiting the typical lateral position and deformation of slippers, and modifying the suspension rheology. These congurations of centered RBCs present much larger resistance to ow than slippers. For the range of Ck studied, minimum distances between RBCs required to reproduce the single-cell behaviour are 1–2 cells diameters, which represent volume fractions of typically fv ¼ 0.04–0.08. The critical distance separating the single and array congurations, however, reduces with increasing Ck. At thick tubes, the inertially driven lateral migration is robust for higher concentrations up to fv ¼ 0.2,14 but the much higher Reynolds and capillary numbers might attenuate the interactions between RBCs. Future work should study the importance of connement for high concentration suspensions, and how it determines the arrangement and organization of the RBCs when several of them can occupy the channel section (i.e. transition from the single-le ow, studied here, to multi-le ow). In this case, the wall connement and the interactions between cells (which
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effectively act as a transversal connement) act in the same direction, and thus the interplay between both effects will dictate the spatial cell distribution and consequently the suspension rheology.
Published on 25 June 2014. Downloaded by Purdue University on 31/08/2014 10:05:09.
Acknowledgements We acknowledge the nancial support from MICINN (Spain) under project FIS 2009-12964-C05-02 and DURSI project SGR 2009-00014, and the Direcci´ on General de Investigaci´ on (Spain) and DURSI project for nancial support under projects FIS 2011-22603 and 2009SGR-634, respectively. G.R.L. also acknowledges Generalitat de Catalunya for support under grant FI-DGR2011.
References 1 M. J. Rosenbluth, W. A. Lam and D. A. Fletcher, Lab Chip, 2008, 8, 1062–1070. 2 X. Y. Hu, P. H. Bessette, J. R. Qian, C. D. Meinhart, P. S. Daugherty and H. T. Soh, Proc. Natl. Acad. Sci. U. S. A., 2005, 102, 15757–15761. 3 Y. W. Kim and J. Y. Yoo, Biosens. Bioelectron., 2009, 24, 3677– 3682. 4 S. Yang, A. Undar and J. D. Zahn, Lab Chip, 2006, 6, 871–880. 5 D. D. Carlo, D. Irimia, R. G. Tompkins and M. Toner, Proc. Natl. Acad. Sci. U. S. A., 2007, 104, 18892–18897. 6 A. Kumar and M. D. Graham, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2011, 84, 066316. 7 A. M. Leshansky, A. Bransky, N. Korin and U. Dinnar, Phys. Rev. Lett., 2007, 98, 234501. 8 A. Kumar and M. D. Graham, So Matter, 2012, 8, 10536. 9 A. R. Pries and T. W. Secomb, Clin. Microrheol. Circ., 2003, 29, 143–148. 10 R. Fahraeus and T. Lindqvist, Am. J. Physiol., 1931, 96, 562– 568. 11 J.-P. Matas, J. F. Morris and E. Guazzelli, J. Fluid Mech., 2004, 515, 171–195. 12 G. Segr´ e and A. Silberberg, J. Fluid Mech., 1962, 14, 136–157. 13 R. C. Jeffrey and J. R. A. Pearson, J. Fluid Mech., 1965, 22, 721– 735. 14 M. Han, C. Kim, M. Kim and S. Lee, J. Rheol., 1999, 43, 1157– 1174. 15 M. Tachibana, Rheol. Acta, 1973, 12, 58–69.
This journal is © The Royal Society of Chemistry 2014
Soft Matter
16 H. Wang and R. Skalak, J. Fluid Mech., 1969, 38, 75–96. 17 P. Olla, Phys. Rev. Lett., 1999, 82, 453. 18 M. Abkarian, M. Faivre, R. Horton, K. Smistrup, C. A. BestPopescu and H. A. Stone, Biomed. Mater., 2008, 3, 034011. 19 J. B. Freund, Annu. Rev. Fluid Mech., 2014, 46, 67–95. 20 D. A. Fedosov, H. Noguchi and G. Gompper, Biomech. Model. Mechanobiol., 2013, 239–258. 21 G. R. L´ azaro, A. Hern´ andez-Machado and I. Pagonabarraga, So Matter, 2014, DOI: 10.1039/C4SM00894D. 22 W. Helfrich, Z. Naturforsch., C: Biochem., Biophys., Biol., Virol., 1973, 28, 693–703. 23 F. Campelo and A. Hern´ andez-Machado, Eur. Phys. J. E, 2006, 2, 37–45. 24 A. J. C. Ladd and R. Verberg, J. Stat. Phys., 2001, 104, 1191– 1251. 25 M. E. Cates, J.-C. Desplat, P. Stansell, A. J. Wagner, K. Stratford and I. Pagonabarraga, Philos. Trans. R. Soc., A, 2005, 363, 1917–1935. 26 R. Benzi, M. Sbragaglia, S. Succi, M. Bernaschi and S. Chibbaro, J. Chem. Phys., 2009, 131, 104903. 27 G. Gonnella, E. Orlandini and J. M. Yeomans, Phys. Rev. Lett., 1997, 78, 1695. 28 S. Succi, The Lattice Boltzmann Equation: for Fluid Dynamics and beyond, Oxford University Press, Norfolk, 2001. 29 A. J. C. Ladd, J. Fluid Mech., 1994, 271, 285–309. 30 Y. W. Kim and J. Y. Yoo, J. Micromech. Microeng., 2008, 18, 065015. 31 F. Campelo and A. Hern´ andez-Machado, Phys. Rev. Lett., 2007, 99, 088101. 32 N. Srivastava, R. D. Davenport and M. A. Burns, Anal. Chem., 2005, 77, 383–392. 33 J. Gachelin, G. Mino, H. Berthet, A. Lindner, A. Rousselet and E. Cl´ ement, Phys. Rev. Lett., 2013, 110, 268103. 34 P. Gaehtgens, C. Duhrssen and K. H. Albrecht, Blood Cells, 1980, 6, 799–812. 35 J. L. McWhirter, H. Noguchi and G. Gompper, So Matter, 2011, 7, 10967. 36 B. P. Ho and L. G. Leal, J. Fluid Mech., 1974, 65, 365–400. 37 E. S. Asmolov, J. Fluid Mech., 1999, 381, 63–87. 38 D. D. Carlo, J. F. Edd, K. J. Humphry, H. A. Stone and M. Toner, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2009, 102, 094503.
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