Shear thinning and shear thickening of a confined suspension of vesicles.

Widely regarded as an interesting model system for studying flow properties of blood, vesicles are closed membranes of phospholipids that mimic the cytoplasmic membranes of red blood cells. In this study we analyze the rheology of a suspension of vesicles in a confined geometry: the suspension, bound by two planar rigid walls on each side, is subject to a shear flow. Flow properties are then analyzed as a function of shear rate γ[over ̇], the concentration of the suspension ϕ, and the viscosity contrast λ=η_{in}/η_{out}, where η_{in} and η_{out} are the fluid viscosities of the inner and outer fluids, respectively. We find that the apparent (or effective viscosity) of the suspension exhibits both shear thinning (decreasing viscosity with shear rate) or shear thickening (increasing viscosity with shear rate) in the same concentration range. The shear thinning or thickening behaviors appear as subtle phenomena, dependant on viscosity contrast λ. We provide physical arguments on the origins of these behaviors.

Amoeboid motion in confined geometry.

Many eukaryotic cells undergo frequent shape changes (described as amoeboid motion) that enable them to move forward. We investigate the effect of confinement on a minimal model of amoeboid swimmer. A complex picture emerges: (i) The swimmer's nature (i.e., either pusher or puller) can be modified by confinement, thus suggesting that this is not an intrinsic property of the swimmer. This swimming nature transition stems from intricate internal degrees of freedom of membrane deformation. (ii) The swimming speed might increase with increasing confinement before decreasing again for stronger confinements. (iii) A straight amoeoboid swimmer's trajectory in the channel can become unstable, and ample lateral excursions of the swimmer prevail. This happens for both pusher- and puller-type swimmers. For weak confinement, these excursions are symmetric, while they become asymmetric at stronger confinement, whereby the swimmer is located closer to one of the two walls. In this study, we combine numerical and theoretical analyses.

Rheological properties of a vesicle suspension.

The rheological behavior of a dilute suspension of vesicles in linear shear flow at a finite concentration is analytically examined. In the quasispherical limit, two coupled nonlinear equations that describe the vesicle orientation in the flow and its shape evolution were derived [Phys. Rev. Lett. 96, 028104 (2006)PRLTAO0031-900710.1103/PhysRevLett.96.028104] and serve here as a starting point. Of special interest is to provide, for the first time, an exact analytical prediction of the time-dependent effective viscosity η_{eff} and normal stress differences N_{1} and N_{2}. Our results shed light on the effect of the viscosity ratio λ (defined as the inner over the outer fluid viscosities) as the main controlling parameter. It is shown that η_{eff},N_{1}, and N_{2} either tend to a steady state or describe a periodic time-dependent rheological response, previously reported numerically and experimentally. In particular, the shear viscosity minimum and the cusp singularities of η_{eff},N_{1}, and N_{2} at the tumbling threshold are brought to light. We also report on rheology properties for an arbitrary linear flow. We were able to obtain a constitutive law in a closed form relating the stress tensor to the strain rate tensor. It is found that the resulting constitutive markedly contrasts with classical laws known for other complex fluids, such as emulsions, capsule suspensions, and dilute polymer solutions (Oldroyd B model). We highlight the main differences between our law and classical laws.

The plasma protein fibrinogen stabilizes clusters of red blood cells in microcapillary flows.

The supply of oxygen and nutrients and the disposal of metabolic waste in the organs depend strongly on how blood, especially red blood cells, flow through the microvascular network. Macromolecular plasma proteins such as fibrinogen cause red blood cells to form large aggregates, called rouleaux, which are usually assumed to be disaggregated in the circulation due to the shear forces present in bulk flow. This leads to the assumption that rouleaux formation is only relevant in the venule network and in arterioles at low shear rates or stasis. Thanks to an excellent agreement between combined experimental and numerical approaches, we show that despite the large shear rates present in microcapillaries, the presence of either fibrinogen or the synthetic polymer dextran leads to an enhanced formation of robust clusters of red blood cells, even at haematocrits as low as 1%. Robust aggregates are shown to exist in microcapillaries even for fibrinogen concentrations within the healthy physiological range. These persistent aggregates should strongly affect cell distribution and blood perfusion in the microvasculature, with putative implications for blood disorders even within apparently asymptomatic subjects.

On the problem of slipper shapes of red blood cells in the microvasculature.

Red blood cells (RBC) are known to exhibit non symmetric (slipper) shapes in the microvasculature. Vesicles have been recently used as a model for RBC and numerical simulations proved the existence of slipper shapes under Poiseuille flow (both in unconfined and confined geometry). However, in our recent numerical simulations the transition from symmetric (parachute) shape to the slipper one was found to take place upon decreasing the flow strength, while experiments on RBCs showed the contrary. In this work we show that if the viscosity contrast (ratio between the internal over external fluid viscosities) is different from unity, as is the case with RBCs, the transition from parachute to slipper shape occurs upon increasing the flow strength, in agreement with experiments. We provide the phase diagram of shapes in the microcirculation. The slipper is found to have a higher speed than the parachute (for the same parameters), that we believe to be the basic reason for its prevailing in the microvasculature. We provide a simple geometrical picture that explains the slipper flow efficiency over the parachute one. Finally, we show that there exists in parameter space regions of co-existence of slipper/parachute shapes and suggest simple experimental protocols to test these findings. The coexistence of shapes seems to be supported by experiments, though a systematic experimental study is lacking. A potential application of this work is to guide designing flow-based experiments in order to link the shape of RBCs to pathologies affecting cell deformability, such as sickle cell diseases, malaria, and those affecting blood hematocrit, as in polycythemia vera disease.

Dynamic modes of quasispherical vesicles: exact analytical solutions.

In this paper we introduce a simple mathematical analysis to reexamine vesicle dynamics in the quasispherical limit (small deformation) under a shear flow. In this context, a recent paper [Misbah, Phys. Rev. Lett. 96, 028104 (2006)] revealed a dynamic referred to as the vacillating-breathing (VB) mode where the vesicle main axis oscillates about the flow direction and the shape undergoes a breathinglike motion, as well as the tank-treading and tumbling (TB) regimes. Our goal here is to identify these three modes by obtaining explicit analytical expressions of the vesicle inclination angle and the shape deformation. In particular, the VB regime is put in evidence and the transition dynamics is discussed. Not surprisingly, our finding confirms the Keller-Skalak solutions (for rigid particles) and shows that the VB and TB modes coexist, and whether one prevails over the other depends on the initial conditions. An interesting additional element in the discussion is the prediction of the TB and VB modes as functions of a control parameter Γ, which can be identified as a TB-VB parameter.

Hydrodynamic interaction between two vesicles in a linear shear flow: asymptotic study.

Interactions between two vesicles in an imposed linear shear flow are studied theoretically, in the limit of almost spherical vesicles, with a large intervesicle distance, in a strong flow, with a large inner to outer viscosity ratio. This allows to derive a system of ordinary equations describing the dynamics of the two vesicles. We provide an analytic expression for the interaction law. We find that when the vesicles are in the same shear plane, the hydrodynamic interaction leads to a repulsion. When they are not, the interaction may turn into attraction instead. The interaction law is discussed and analyzed as a function of relevant parameters.

Complexity of vesicle microcirculation.

This study focuses numerically on dynamics in two dimensions of vesicles in microcirculation. The method used is based on boundary integral formulation. This study is inspired by the behavior of red blood cells (RBCs) in the microvasculature. Red RBCs carry oxygen from the lungs and deliver it through the microvasculature. The shape adopted by RBCs can affect blood flow and influence oxygen delivery. Our simulation using vesicles (a simple model for RBC) reveals unexpected complexity as compared to the case where a purely unbounded Poiseuille flow is considered [Kaoui, Biros, and Misbah, Phys. Rev. Lett. 103, 188101 (2009)]. In sufficiently large channels (in the range of 100 µm; the vesicle size and its reduced volume are taken in the range of those of a human RBC), such as arterioles, a slipperlike (asymmetric) shape prevails. A parachutelike (symmetric) shape is adopted in smaller channels (in the range of 20 µm, as in venules), but this shape loses stability and again changes to a pronounced slipperlike morphology in channels having a size typical of capillaries (5-10 µm). Stiff membranes, mimicking malaria infection, for example, adopt a centered or off-centered snakelike locomotion instead (the denomination snaking is used for this regime). A general scenario of how and why vesicles adopt their morphologies and dynamics among several distinct possibilities is provided. This finding potentially points to nontrivial RBCs dynamics in the microvasculature.

Rheology of particulate suspensions in a Poiseuille flow.

Particulate dense suspensions behave as complex fluids. They do not lend themselves easily to analytical solution. We propose an analytical model to mimic this problem. Namely, we consider arrays of long parallel plates which represent a simplification of arrays of chains of spherical particles. This simplified model can be solved analytically. The effect of effective rotation of the spherical particles is taken into account by attributing different velocities on each side of the plate that mimics the fact that particles are subject to shear. This work is an extension of a previous study where particle rotation was disregarded. The flow rate, the dissipation and the apparent viscosity are studied as a function of the underlying structure. For a single plate placed out of the flow center, the viscosity is lower when rotation is taken into account. For two plates, the minimal viscosity corresponds to the situation where the particles are as close as possible to the center and arranged symmetrically with respect to the center. We compute the rheological properties for arbitrary plate positions, and exploit them for a periodic arrangement. For N plates, and in a confined geometry, the viscosity is about twice as small as compared to the situation where rotation is ignored. We have conducted a numerical study of a suspension of spherical particles, and linear chains of spherical particles. The numerical study is in good qualitative and semiquantitative agreement with the analytical theory considering long plates. This agreement highlights the fact that our analytical model captures the essential features of a real suspension. The numerical study is based on a fluid dynamic particle method where the particles are represented by a scalar field having high viscosity inside.

Thermodynamically consistent picture of the phase-field model of vesicles: elimination of the surface tension.

In two recent papers [D. Jamet and C. Misbah, Phys. Rev. E 76, 051907 (2007); 78, 031902 (2008)], we considered a thermodynamically consistent model for vesicles and membranes, where we dealt, in the first paper, with the membrane local incompressibility condition, while in the second one with the bending energy and the derivation of a constitutive law of the composite fluid: ambient fluid+membrane . This is the last paper of this series and focuses on the elimination of surface tension (inherent in phase-field models), retaining the thermodynamically consistent model. We write down the complete set of equations and the full constitutive law for membranes embedded in a Newtonian fluid.

Toward a thermodynamically consistent picture of the phase-field model of vesicles: curvature energy.

We extend our recent work on phase-field model for vesicles [D. Jamet and C. Misbah, Phys. Rev. E 76, 051907 (2007)]-where only the membrane local incompressibility was treated-to the situation where the bending forces and spontaneous curvature are included. We show how the general phase-field equations can be derived within a thermodynamic consistent picture. We analyze a general form of the bending energy, where the Helfrich bending force is treated as a special case. The dynamical evolution equation derived here for the velocity field allows one to write down a constitutive law of the composite fluid: The ambient fluid plus the membrane. This constitutive law has a viscoelastic form, the viscous part arises from the fluid, while the elastic one represents the action of the membrane. It is shown that the elastic stress tensor is not symmetric, owing to bending torque, inherent to a diffuse membrane model.

Lateral migration of a two-dimensional vesicle in unbounded Poiseuille flow.

The migration of a suspended vesicle in an unbounded Poiseuille flow is investigated numerically in the low Reynolds number limit. We consider the situation without viscosity contrast between the interior of the vesicle and the exterior. Using the boundary integral method we solve the corresponding hydrodynamic flow equations and track explicitly the vesicle dynamics in two dimensions. We find that the interplay between the nonlinear character of the Poiseuille flow and the vesicle deformation causes a cross-streamline migration of vesicles toward the center of the Poiseuille flow. This is in a marked contrast with a result [L. G. Leal, Annu. Rev. Fluid Mech. 12, 435 (1980)] according to which the droplet moves away from the center (provided there is no viscosity contrast between the internal and the external fluids). The migration velocity is found to increase with the local capillary number (defined by the time scale of the vesicle relaxation toward its equilibrium shape times the local shear rate), but reaches a plateau above a certain value of the capillary number. This plateau value increases with the curvature of the parabolic flow profile. We present scaling laws for the migration velocity.

On coupling between the orientation and the shape of a vesicle under a shear flow.

A simple 2D model of deformable vesicles tumbling in a shear under flow is introduced in order to account for the main qualitative features observed experimentally as shear rates are increased. The simplicity of the model allows for a full analytical tractability while retaining the essential physical ingredients. The model reveals that the main axes of the vesicle undergo oscillations which are coupled to the vesicle orientation in the flow. The model reproduces and sheds light on the main novel features reported in recent experiments [M. Mader et al., Eur. Phys. J. E. 19, 389 (2006)], namely that both coefficients A and B that enter the Keller-Skalak equation, d psi/dt = A+B cos(2 psi) (psi is the vesicle orientation angle in the shear flow), undergo a collapse upon increasing shear rate.

Towards a thermodynamically consistent picture of the phase-field model of vesicles: local membrane incompressibility.

A phase-field model for vesicles including hydrodynamics was presented in two and three dimensions [T. Biben and C. Misbah, Phys. Rev. E 67, 031908 (2003); T. Biben, K. Kassner, and C. Misbah, Phys. Rev. E 72, 041921 (2005)]. A particularly important feature for vesicles is that their membrane is locally incompressible. In these works a tension field defined everywhere in the bulk was introduced in order to fulfill local membrane inextensibility. Here we reconsider the original model by treating the phase field as a thermodynamic variable and develop a picture which is consistent with the second law of thermodynamics. This enables us to write the phase-field evolution equations in terms of a thermodynamical potential. This potential acquires, at global equilibrium, a Lyapunov functional character. The goal of this paper is twofold: (i) The first and primary goal is purely conceptual, in that we can write down a first and second principle for membranes, from which the evolution equations follow, thanks to the evaluation of the entropy production and the use of concepts of irreversible thermodynamics. (ii) Due to the monotonous character of the evolution of the functional (at global equilibrium), we expect this formulation to be more appropriate for numerical studies. The formalism developed to account for the local incompressibility of the membrane is believed to offer a systematic framework in order to include naturally other physical ingredients, as briefly discussed here and demonstrated in future works.

Birth and morphological evolution of step bunches under electromigration.

We analyze the dynamics of electromigration-induced step bunching in the absence of desorption. We show that, even when the instability occurs at long wavelength, hinting to a smooth morphology, the surface suddenly splits into bunches escorted with wide terraces, in agreement with several observations. As the size of the bunches increases, a nonstandard regime is exhibited, namely, the bunches do not match tangentially to the facet, as would the classical Pokrosvky-Talapov shape dictate. This Letter presents a complete scenario of evolution of bunches from their birth up to their ultimate stage.

Fluctuations and instability of a biological membrane induced by interaction with macromolecules.

This paper studies the dynamics and fluctuations of a phospholipidic membrane in the presence of a diffusion field of foreign molecules, such as polymers, proteins, etc., which have the ability to adsorb on, and to desorb from, the membrane. We develop a model that includes, besides hydrodynamics, molecular diffusion in the surrounding fluid and lateral diffusion (i.e., diffusion along the membrane) as well as the kinetics of attachment and detachment to and from the membrane. This model is exploited here for the case of a free membrane which is globally at equilibrium while the nonequilibrium part will be presented in the future. We show that if the coupling between the membrane and the molecules is strong enough, the flat membrane can suffer a morphological instability. The numerical calculation in the nonlinear regime reveals budding, and an initial stage of spontaneous vesicle emission. When the condition of stability is satisfied we show how kinetic fluctuations lead to a rich variety of dynamical behaviors expressing the dominant dissipation mechanisms. We show that in the limit of well separated length scales related to the physical mechanisms that enter into play, the width of the membrane fluctuations exhibits various dynamical scalings with universal scaling exponents. It is shown that the usual behavior with time w approximately t(1/3) is altered in various time and length scales of interest. For example, we find that , w approximately t(1/4), w approximately t(1/2), and w approximately [t ln(t)](1/2), depending on which physical mechanism limits the membrane fluctuations on the time scale under consideration. The experimental study of the fluctuation spectrum can be viewed as a precious tool in order to have access to the underlying microscopic physical mechanisms.

Peculiar effects of anisotropic diffusion on dynamics of vicinal surfaces.

We report on peculiar behaviors due to anisotropic terrace diffusion on step meandering on a vicinal surface. We find that anisotropy triggers tilted ripples. In addition, if the fast diffusion direction is perpendicular to the steps, the instability is moderate and coarsening is absent, while in the opposite case the instability is promoted, and interrupted coarsening may be observed. Strong enough anisotropy restabilizes the step for almost all step orientations. These findings point to the nontrivial effect of anisotropy and open promising lines of inquiries in the design of surface architectures.

Analytical analysis of a vesicle tumbling under a shear flow.

Vesicles under a shear flow exhibit a tank-treading motion of their membrane, while their long axis points with an angle

Steady to unsteady dynamics of a vesicle in a flow.

We investigate the dynamics of a vesicle in a shear flow on the basis of the newly proposed advected field (AF) method [T. Biben and C. Misbah, Eur. Phys. J. E 67, 031908 (2003)]. We also solve the same problem with the boundary integral formulation for the sake of comparison. We find that the AF results presented previously overestimated the tumbling threshold due to the finite size of the membrane, inherent to the AF model. A comparison between the two methods shows that only in the sharp interface limit (extrapolating the results to a vanishing width) the AF method leads to accurate quantitative results. We extensively investigate the tank-treading to tumbling transition, and compare our numerical results to the theory of Keller and Skalak which assumes a fixed ellipsoidal shape for the vesicle. We show that this theory describes correctly the two regimes, at least in two dimensions, even for the quite elongated non-convex shapes corresponding to red blood cells (and therefore far from ellipsoidal), This theory is, however, not fully quantitative. Finally we investigate the effect of a confinement on the tank-treading to tumbling transition, and show that the tumbling regime becomes unfavorable in a capillary vessel, which should have strong effects on blood rheology in confined geometries.

Interrupted coarsening of anisotropic step meander.

We report on the effect of anisotropy on the step meandering instability on vicinal surfaces during molecular beam epitaxy growth. A scenario of interrupted coarsening is found: the lateral length scale of the structure first significantly increases with time and then freezes at a larger length scale. The wavelength selection mechanism results from a nontrivial nonlinear effect of anisotropy. Anisotropy also leads to solutions which drift sideways, resulting from the loss of the back-front symmetry of the meander and the nonvariational character of dynamics.