Dynamics of fluid vesicles in shear flow: effect of membrane viscosity and thermal fluctuations

Dynamic and rheological properties of soft biological cell suspensions

Quantifying dynamic and rheological properties of suspensions of soft biological particles such as vesicles, capsules, and red blood cells (RBCs) is fundamentally important in computational biology and biomedical engineering. In this review, recent studies on dynamic and rheological behavior of soft biological cell suspensions by computer simulations are presented, considering both unbounded and confined shear flow. Furthermore, the hemodynamic and hemorheological characteristics of RBCs in diseases such as malaria and sickle cell anemia are highlighted.

Angle of inclination of tank-treading red cells: dependence on shear rate and suspending medium

Red cells suspended in solutions much more viscous than blood plasma assume an almost steady-state orientation when sheared above a threshold value of shear rate. This orientation is a consequence of the motion of the membrane around the red cell called tank-treading. Observed along the undisturbed vorticity of the shear flow, tank-treading red cells appear as slender bodies. Their orientation can be quantified as an angle of inclination (θ) of the major axis with respect to the undisturbed flow direction. We measured θ using solution viscosities (η0) and shear rates (γ˙) covering one and three orders of magnitude, respectively. At the lower values of η0, θ was almost independent of γ˙. At the higher values of η0, θ displayed a maximum at intermediate shear rates. The respective maximal values of θ increased by ∼10° from 10.7 to 104 mPas. After accounting for the absent membrane viscosity in models by using an increased cytoplasmic viscosity, their predictions of θ agree qualitatively with our data. Comparison of the observed variation of θ at constant γ˙ with model results suggests a change in the reference configuration of the shear stiffness of the membrane.

Cooperation within von Willebrand factors enhances adsorption mechanism

von Willebrand factor (VWF) is a naturally collapsed protein that participates in primary haemostasis and coagulation events. The clotting process is triggered by the adsorption and conformational changes of the plasma VWFs localized to the collagen fibres found near the site of injury. We develop coarse-grained models to simulate the adsorption dynamics of VWF flowing near the adhesive collagen fibres at different shear rates and investigate the effect of factors such as interaction and cooperativity of VWFs on the success of adsorption events. The adsorption probability of a flowing VWF confined to the receptor field is enhanced when it encounters an adhered VWF in proximity to the collagen receptors. This enhancement is observed within a wide range of shear rates and is mostly controlled by the attractive van der Waals interactions rather than the hydrodynamic interactions among VWF monomers. The cooperativity between the VWFs acts as an effective mechanism for enhancing VWF adsorption to the collagen fibres. Additionally, this implies that the adsorption of such molecules is nonlinearly dependent on the density of flowing VWFs. These findings are important for studies of primary haemostasis as well as general adsorption dynamics processes in polymer physics.

Numerical computations of the dynamics of fluidic membranes and vesicles

Vesicles and many biological membranes are made of two monolayers of lipid molecules and form closed lipid bilayers. The dynamical behavior of vesicles is very complex and a variety of forms and shapes appear. Lipid bilayers can be considered as a surface fluid and hence the governing equations for the evolution include the surface (Navier-)Stokes equations, which in particular take the membrane viscosity into account. The evolution is driven by forces stemming from the curvature elasticity of the membrane. In addition, the surface fluid equations are coupled to bulk (Navier-)Stokes equations. We introduce a parametric finite-element method to solve this complex free boundary problem and present the first three-dimensional numerical computations based on the full (Navier-)Stokes system for several different scenarios. For example, the effects of the membrane viscosity, spontaneous curvature, and area difference elasticity (ADE) are studied. In particular, it turns out, that even in the case of no viscosity contrast between the bulk fluids, the tank treading to tumbling transition can be obtained by increasing the membrane viscosity. Besides the classical tank treading and tumbling motions, another mode (called the transition mode in this paper, but originally called the vacillating-breathing mode and subsequently also called trembling, transition, and swinging mode) separating these classical modes appears and is studied by us numerically. We also study how features of equilibrium shapes in the ADE and spontaneous curvature models, like budding behavior or starfish forms, behave in a shear flow.

Autonomous movement of a chemically powered vesicle

We investigate the diffusio-phoretic motion of a deformable vesicle. A vesicle is built from the linked catalytic and noncatalytic vertices that consumes fuel in the environment and utilize the resulting self-generated concentration gradient to exhibit propulsive motion. Under nonequilibrium conditions it is found that the self-propulsion velocity of the vesicle depends on its shape, which in turn is controlled by the bending rigidity of the membrane and solvent density around it. The self-propulsion velocity of the vesicle for different shapes has been calculated and the factors which affect the velocity are identified.

Bending elasticity of vesicle membranes studied by Monte Carlo simulations of vesicle thermal shape fluctuations

The membrane bending stiffness of nearly spherical lipid vesicles can be deduced from the analysis of their thermal shape fluctuations. The theoretical basis of this analysis [Milner and Safran, Phys. Rev. A: At., Mol., Opt. Phys., 1987, 36, 4371-4379] uses the mean field approximation. In this work we apply Monte Carlo simulations and estimate the error in the determination of the bending stiffness due to the approximations applied in the theory. It is less than 10%. The method presented in this work can also be used to determine the changes of the bending stiffness of biological membranes due to their chemical and/or structural modifications.

Multi-particle collision dynamics algorithm for nematic fluids

Research on transport, self-assembly and defect dynamics within confined, flowing liquid crystals requires versatile and computationally efficient mesoscopic algorithms to account for fluctuating nematohydrodynamic interactions. We present a multi-particle collision dynamics (MPCD) based algorithm to simulate liquid-crystal hydrodynamic and director fields in two and three dimensions. The nematic-MPCD method is shown to successfully reproduce the features of a nematic liquid crystal, including a nematic-isotropic phase transition with hysteresis in 3D, defect dynamics, isotropic Frank elastic coefficients, tumbling and shear alignment regimes and boundary condition-dependent order parameter fields.

Shape transformations of toroidal vesicles

Morphologies of genus-1 and 2 toroidal vesicles are studied numerically by dynamically triangulated membrane models and experimentally by confocal laser microscopy. Our simulation results reproduce shape transformations observed in our experiments well. At large reduced volumes of the genus-1 vesicles, obtained vesicle shapes agree with the previous theoretical prediction, in which axisymmetric shapes are assumed: double-necked stomatocyte, discoidal toroid, and circular toroid. However, for small reduced volumes, it is revealed that a non-axisymmetric discoidal toroid and handled discocyte exist in thermal equilibrium in the parameter range, in which the previous theory predicts axisymmetric discoidal shapes. Polygonal toroidal vesicles and subsequent budding transitions are also found. The entropy caused by shape fluctuations slightly modifies the stability of the vesicle shapes.

Statistical analysis of vesicle morphology dynamics based on a free energy landscape

We here present a method to reconstruct effective free energy landscapes (FELs) of lipid vesicles from the statistical analysis of a large number of microscope images. This method, not only allows us to define possible energy landscapes, but also highlights minority vesicle shapes that were otherwise hidden in the majority. When compared with temporal evolution of deforming lipid vesicles, it was found that the trajectory of deforming vesicles was in accordance with the reconstructed landscape, in which the minority shapes play a key role. When compared with theoretical models, it revealed that the vesicle shapes characterised in the reconstructed FELs were consistent with the theoretically predicted shapes. These results suggest that the FEL analysis can be a useful tool to investigate the morphological dynamics of lipid vesicles, in conjunction with other analytical methods.

Fluid vesicles in flow

We review the dynamical behavior of giant fluid vesicles in various types of external hydrodynamic flow. The interplay between stresses arising from membrane elasticity, hydrodynamic flows, and the ever present thermal fluctuations leads to a rich phenomenology. In linear flows with both rotational and elongational components, the properties of the tank-treading and tumbling motions are now well described by theoretical and numerical models. At the transition between these two regimes, strong shape deformations and amplification of thermal fluctuations generate a new regime called trembling. In this regime, the vesicle orientation oscillates quasi-periodically around the flow direction while asymmetric deformations occur. For strong enough flows, small-wavelength deformations like wrinkles are observed, similar to what happens in a suddenly reversed elongational flow. In steady elongational flow, vesicles with large excess areas deform into dumbbells at large flow rates and pearling occurs for even stronger flows. In capillary flows with parabolic flow profile, single vesicles migrate towards the center of the channel, where they adopt symmetric shapes, for two reasons. First, walls exert a hydrodynamic lift force which pushes them away. Second, shear stresses are minimal at the tip of the flow. However, symmetry is broken for vesicles with large excess areas, which flow off-center and deform asymmetrically. In suspensions, hydrodynamic interactions between vesicles add up to these two effects, making it challenging to deduce rheological properties from the dynamics of individual vesicles. Further investigations of vesicles and similar objects and their suspensions in steady or time-dependent flow will shed light on phenomena such as blood flow.

Morphological variation of a lipid vesicle confined in a spherical vesicle

Morphologies of a double-bilayer vesicle were explored experimentally by fast confocal laser microscopy and numerically by a dynamically triangulated membrane model with area-difference elasticity. The confinement was found to induce several shapes of the inner vesicles that had not been observed in unilamellar vesicles: double and quadruple stomatocytes, slit vesicle, and vesicles of two or three compartments with various shapes. The simulations reproduced the experimental results very well and some of the shape transitions can be understood by a simple theoretical model for axisymmetric shapes.

Three-dimensional dynamics of oblate and prolate capsules in shear flow

We study computationally the dynamics of oblate and prolate spheroidal capsules in simple shear flow with small inertia for a range of dimensionless shear rates. The capsule is modelled as a liquid droplet enclosed by a hyperelastic membrane, and its equatorial plane is initially tilted out of the plane of shear. We find, at low shear rates, the well-accepted tumbling motion is not always stable for both oblate and prolate capsules. For an oblate capsule, the dominant stable modes for increasing dimensionless shear rate are as follows: rolling with the equatorial plane staying in the plane of shear, precessing following Jeffery's orbit [Proc. R. Soc. London A 102, 161 (1922)], and tumbling. Interestingly, the order of modes is reversed for a prolate capsule: tumbling, precessing, and rolling with increasing dimensionless shear rate. At transitional regimes, we find the stable motion of a capsule can depend on its initial titled angle, even at the same shear rate. At high dimensionless shear rates, a spheroidal capsule undergoes a complicated oscillating-swinging motion: Its major axis oscillates about the plane of shear in addition to the swinging about a mean angle with flow direction found previously, and the amplitudes of both oscillations decrease when increasing the dimensionless shear rate towards a steady tank treading motion asymptotically. We summarize the results in phase diagrams and discuss the reorientation of both oblate and prolate capsules in a wide range of dimensionless shear rates.

Computational biorheology of human blood flow in health and disease

Hematologic disorders arising from infectious diseases, hereditary factors and environmental influences can lead to, and can be influenced by, significant changes in the shape, mechanical and physical properties of red blood cells (RBCs), and the biorheology of blood flow. Hence, modeling of hematologic disorders should take into account the multiphase nature of blood flow, especially in arterioles and capillaries. We present here an overview of a general computational framework based on dissipative particle dynamics (DPD) which has broad applicability in cell biophysics with implications for diagnostics, therapeutics and drug efficacy assessments for a wide variety of human diseases. This computational approach, validated by independent experimental results, is capable of modeling the biorheology of whole blood and its individual components during blood flow so as to investigate cell mechanistic processes in health and disease. DPD is a Lagrangian method that can be derived from systematic coarse-graining of molecular dynamics but can scale efficiently up to arterioles and can also be used to model RBCs down to the spectrin level. We start from experimental measurements of a single RBC to extract the relevant biophysical parameters, using single-cell measurements involving such methods as optical tweezers, atomic force microscopy and micropipette aspiration, and cell-population experiments involving microfluidic devices. We then use these validated RBC models to predict the biorheological behavior of whole blood in healthy or pathological states, and compare the simulations with experimental results involving apparent viscosity and other relevant parameters. While the approach discussed here is sufficiently general to address a broad spectrum of hematologic disorders including certain types of cancer, this paper specifically deals with results obtained using this computational framework for blood flow in malaria and sickle cell anemia.

Dynamics of self-assembly of model viral capsids in the presence of a fluctuating membrane

A coarse-grained computational model is used to investigate the effect of a fluctuating fluid membrane on the dynamics of patchy-particle assembly into virus capsid-like cores. Results from simulations for a broad range of parameters are presented, showing the effect of varying interaction strength, membrane stiffness, and membrane viscosity. Furthermore, the effect of hydrodynamic interactions is investigated. Attraction to a membrane may promote assembly, including for subunit interaction strengths for which it does not occur in the bulk, and may also decrease single-core assembly time. The membrane budding rate is strongly increased by hydrodynamic interactions. The membrane deformation rate is important in determining the finite-time yield. Higher rates may decrease the entropic penalty for assembly and help guide subunits toward each other but may also block partial cores from being completed. For increasing subunit interaction strength, three regimes with different effects of the membrane are identified.

Noisy nonlinear dynamics of vesicles in flow

We present a model for the dynamics of fluid vesicles in linear flow which consistently includes thermal fluctuations and nonlinear coupling between different modes. At the transition between tank treading and tumbling, we predict a trembling motion which is at odds with the known deterministic motions and for which thermal noise is strongly amplified. In particular, highly asymmetric shapes are observed even though the deterministic flow only allows for axisymmetric ones. Our results explain quantitatively recent experimental observations [Levant and Steinberg, Phys. Rev. Lett. 109, 268103 (2012)].

Multiscale modeling of blood flow: from single cells to blood rheology

Mesoscale simulations of blood flow, where the red blood cells are described as deformable closed shells with a membrane characterized by bending rigidity and stretching elasticity, have made much progress in recent years to predict the flow behavior of blood cells and other components in various flows. To numerically investigate blood flow and blood-related processes in complex geometries, a highly efficient simulation technique for the plasma and solutes is essential. In this review, we focus on the behavior of single and several cells in shear and microcapillary flows, the shear-thinning behavior of blood and its relation to the blood cell structure and interactions, margination of white blood cells and platelets, and modeling hematologic diseases and disorders. Comparisons of the simulation predictions with existing experimental results are made whenever possible, and generally very satisfactory agreement is obtained.

Modeling cell proliferation for simulating three-dimensional tissue morphogenesis based on a reversible network reconnection framework

Tissue morphogenesis in multicellular organisms is accompanied by proliferative cell behaviors: cell division (increase in cell number after each cell cycle) and cell growth (increase in cell volume during each cell cycle). These proliferative cell behaviors can be regulated by multicellular dynamics to achieve proper tissue sizes and shapes in three-dimensional (3D) space. To analyze multicellular dynamics, a reversible network reconnection (RNR) model has been suggested, in which each cell shape is expressed by a single polyhedron. In this study, to apply the RNR model to simulate tissue morphogenesis involving proliferative cell behaviors, we model cell proliferation based on a RNR model framework. In this model, cell division was expressed by dividing a polyhedron at a planar surface for which cell division behaviors were characterized by three quantities: timing, intracellular position, and normal direction of the dividing plane. In addition, cell growth was expressed by volume growth as a function of individual cell times within their respective cell cycles. Numerical simulations using the proposed model showed that tissues grew during successive cell divisions with several cell cycle times. During these processes, the cell number in tissues increased while maintaining individual cell size and shape. Furthermore, tissue morphology dramatically changed based on different regulations of cell division directions. Thus, the proposed model successfully provided a basis for expressing proliferative cell behaviors during morphogenesis based on a RNR model framework.

Scattering intensity of bicontinuous microemulsions and sponge phases

Monte Carlo simulations of dynamically triangulated surfaces of variable topology are used to investigate the scattering intensities of bicontinuous microemulsions. The bulk scattering intensity is shown to follow the Teubner-Strey expression. The domain size and the correlation length are extracted from the scattering peaks as a function of the bending rigidity, saddle-splay modulus, and surfactant density. The results are compared to earlier theories based on Ginzburg-Landau and Gaussian random field models. The ratio of the two length scales is shown to be well described by a linear combination of logarithmically renormalized bending rigidity and saddle-splay modulus with universal prefactors. This is in contrast to earlier theoretical predictions in which the scattering intensity is independent of the saddle-splay modulus. The equation of state, and the asymptotics of the bulk and film scattering intensities for high and low wave vectors are determined from simulations and compared with theoretical results.

Effect of thermal noise on vesicles and capsules in shear flow

We add thermal noise consistently to reduced models of undeformable vesicles and capsules in shear flow and derive analytically the corresponding stochastic equations of motion. We calculate the steady-state probability distribution function and construct the corresponding phase diagrams for the different dynamical regimes. For fluid vesicles, we predict that at small shear rates thermal fluctuations induce a tumbling motion for any viscosity contrast. For elastic capsules, due to thermal mixing, an intermittent regime appears in regions where deterministic models predict only pure tank treading or tumbling.

Three-dimensional numerical simulation of vesicle dynamics using a front-tracking method

Three-dimensional numerical simulation using the front-tracking method is presented on the dynamics of a vesicle in a linear shear flow. The focus here is to elucidate the parametric dependence and the self-similarity of the vesicle dynamics, quantification of vesicle deformation, and the analysis of shape dynamics. A detailed comparison of the numerical results is made with various theoretical models and experiments. It is found that the applicability of the theoretical models is limited despite some general agreement with the simulations and experiments. The deviations between the perturbative results and the simulation results occur even in the absence of thermal noise. Specifically, we find that the vesicle dynamics does not follow a self-similar behavior in a two-parameter phase space, as proposed in a theoretical model. Rather, the dynamics is governed by three controlling parameters, namely, the excess area, viscosity ratio, and dimensionless shear rate. Additionally, we find that a linear scaling of the tank-treading angle, as proposed in the theoretical model, is possible only for nearly spherical vesicles. The breakdown of the scaling occurs at higher values of the excess area even in the absence of thermal noise. We find that the vesicle deformation saturates at large shear rates, and the asymptotic deformation matches well with a theoretical prediction for nearly spherical vesicles. The dependence of the critical viscosity ratio associated with the onset of unsteady dynamics on the vesicle excess area is in excellent agreement with the experimental observation. We show that near the transition between the tank-treading and tumbling dynamics, both the vacillating-breathing-like motion characterized by a smooth ellipsoidal shape and the trembling-like motion characterized by a highly deformed shape are possible. For the trembling-like motion, the shape is highly three-dimensional with concavities and lobes, and the vesicle deforms more in the vorticity direction than in the shear plane. A Fourier spectral analysis of the vesicle shape shows the presence of the odd harmonics and higher order modes beyond fourth order.

Dynamics of semi-flexible tethered sheets : a simulation study using stochastic rotation dynamics

The dynamics of a semi-flexible sheet or tethered membrane in a solvent is studied using the method of stochastic rotation dynamics. Hydrodynamic interactions between different parts of the sheet are naturally included in this method. We confirm the scaling law for the radius of gyration versus sheet size predicted for a self-avoiding tethered membrane. The mean-square displacement shows both sub-diffusive and diffusive behavior similar to linear polymers. In the intermediate scattering function the sub-diffusive behavior appears as stretched exponential which we reproduce in our simulations. Thereby, we confirm an early prediction between the roughness and the sub-diffusion exponent derived from Zimm dynamics (E. Frey, D.R. Nelson, J. Phys. I 1, 1715 (1991)). Finally, we show that the diffusion coefficient of the square sheet is inversely proportional to the edge length of the sheet again in good agreement with theoretical predictions.

Orientation and internal flow of a vesicle in tank-treading motion in shear flow

Deformation, orientation and internal flow of lipid bilayer vesicles in linear shear flows are investigated using phase contrast microscopy. We construct a rotating-cylinder apparatus, which can generate a linear shear flow with constant shear rates. Vesicles are prepared from 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) by the gentle hydration method. When visualizing internal flows, polystyrene tracer particles are mixed with the hydration water solution. In our observation, vesicles deform to steady ellipsoidal shapes and show constant orientations given by θ(i), which is the angle between the major axis and the flow direction. The tracer particles inside a vesicle rotate around the center of the vesicle along ellipsoidal orbits, which are homothetic to the shape of the vesicle. It is shown that the relationship between θ(i) and the swelling ratio (volume/surface ratio) S(w) agrees quantitatively with the experimental result of Abkarian et al. [Biophys. J.89, 1055 (2005)], which was obtained with vesicles in wall-bounded shear flows. It also agrees with a theoretical analysis of Keller and Skalak [J. Fluid Mech.120, 27 (1982)] and other numerical simulations. It is also shown that angular velocities of the particles near the membrane change periodically and agree quantitatively with the experimental result for the motion of a particle adhering to the membrane of a tank-treading vesicle [Kantsler and Steinberg, Phys. Rev. Lett.95, 258101 (2005)]. A statistical analysis indicates that the velocity of the internal fluid close to the membrane is not constant along the circumference, which implies the possibility of a three-dimensional flow field of the lipid molecules or an apparent stretching motion of the membrane by the effect of hidden surface area due to thermal fluctuation.

Micro-capsules in shear flow

This paper deals with flow-induced shape changes of elastic capsules. The state of the art concerning both theory and experiments is briefly reviewed starting with dynamically induced small deformation of initially spherical capsules and the formation of wrinkles on polymerized membranes. Initially non-spherical capsules show tumbling and tank-treading motion in shear flow. Theoretical descriptions of the transition between these two types of motion assuming a fixed shape are at odds with the full capsule dynamics obtained numerically. To resolve the discrepancy, we expand the exact equations of motion for small deformations and find that shape changes play a dominant role. We classify the dynamical phase transitions and obtain numerical and analytical results for the phase boundaries as a function of viscosity contrast, shear and elongational flow rate. We conclude with perspectives on time-dependent flow, on shear-induced unbinding from surfaces, on the role of thermal fluctuations and on applying the concepts of stochastic thermodynamics to these systems.

Tank-treading and tumbling frequencies of capsules and red blood cells

This study is motivated in part by the discrepancy that exists in the literature with regard to the dependence of the tank-treading frequency of red blood cells on the shear rate and suspending medium viscosity. Here we consider three-dimensional numerical simulations of deformable capsules of initially spherical and oblate spheroidal shapes and biconcave discoid representing the red blood cell resting shape. By considering a much broader range of the viscosity ratio (ratio of capsule or cell interior to suspending fluid viscosity), shear rate, and aspect ratio (ratio of minor to major axes) than that considered in the previous experiments, we find several new characteristics of the tank-treading and tumbling frequencies that have not been reported earlier. These new characteristics are the result of the large shape deformation and the coupling between shape and angular oscillations of the capsules or cells. For the spherical and oblate spheroidal capsules, the tank-treading frequency shows a nonmonotonic trend that is characterized by an initial decrease leading to a minimum followed by an increase with increasing viscosity ratio. For red blood cells, we find two regimes of the viscosity dependence of the tank-treading frequency: an exponential regime in which the tank-treading frequency decreases at a slower rate with increasing viscosity ratio, and a logarithmic range in which it decreases at a much faster rate. While this trend agrees well with different theoretical models of shape-preserving capsules, it was not evident in previous experimental results. When the shear rate dependence is considered, the tank-treading frequency of red blood cells and capsules of highly elongated initial shapes exhibits a nonmonotonic trend that is characterized by an initial increase leading to a maximum followed by a sharp decrease with decreasing shear rate. This anomalous behavior of the tank-treading frequency is shown to be due to a breathing-like dynamics of the capsule or cell that is characterized by a repeated emergence and absence of deep, crater-like dimples, and a large swinging motion. We further observe that the tumbling frequency exhibits a decreasing trend with increasing viscosity ratio that is in contrast to the theoretical result for the shape-preserving capsules and is due to the periodic deformation and preferential alignment of the capsules in the extensional quadrant of the flow.

Hydrodynamic fluctuations in thermostatted multiparticle collision dynamics

In this work we study the behavior of mesoscopic fluctuations of a fluid simulated by Multiparticle Collision Dynamics when this is applied together with a local thermostatting procedure that constrains the strength of temperature fluctuations. We consider procedures in which the thermostat interacts with the fluid at every simulation step as well as cases in which the thermostat is applied only at regular time intervals. Due to the application of the thermostat temperature fluctuations are forced to relax to equilibrium faster than they do in the nonthermostatted, constant-energy case. Depending on the interval of application of the thermostat, it is demonstrated that the thermodynamic state changes gradually from isothermal to adiabatic conditions. In order to exhibit this effect we compute from simulations diverse correlation functions of the hydrodynamic fluctuating fields. These correlation functions are compared with those predicted by a linearized hydrodynamic theory of a simple fluid in which a thermostat is applied locally. We find a good agreement between the model and the numerical results, which confirms that hydrodynamic fluctuations in Multiparticle Collision Dynamics in the presence of the thermostat have the properties expected for spontaneous fluctuations in fluids in contact with a heat reservoir.

Three-dimensional vesicles under shear flow: numerical study of dynamics and phase diagram

The study of vesicles under flow, a model system for red blood cells (RBCs), is an essential step in understanding various intricate dynamics exhibited by RBCs in vivo and in vitro. Quantitative three-dimensional analyses of vesicles under flow are presented. The regions of parameters to produce tumbling (TB), tank-treating, vacillating-breathing (VB), and even kayaking (or spinning) modes are determined. New qualitative features are found: (i) a significant widening of the VB mode region in parameter space upon increasing shear rate γ and (ii) a robustness of normalized period of TB and VB with γ. Analytical support is also provided. We make a comparison with existing experimental results. In particular, we find that the phase diagram of the various dynamics depends on three dimensionless control parameters, while a recent experimental work reported that only two are sufficient.

Dynamic modes of red blood cells in oscillatory shear flow

The dynamics of red blood cells (RBCs) in oscillatory shear flow was studied using differential equations of three variables: a shape parameter, the inclination angle θ, and phase angle ϕ of the membrane rotation. In steady shear flow, three types of dynamics occur depending on the shear rate and viscosity ratio. (i) tank-treading (TT): ϕ rotates while the shape and θ oscillate. (ii) tumbling (TB): θ rotates while the shape and ϕ oscillate. (iii) intermediate motion: both ϕ and θ rotate synchronously or intermittently. In oscillatory shear flow, RBCs show various dynamics based on these three motions. For a low shear frequency with zero mean shear rate, a limit-cycle oscillation occurs, based on the TT or TB rotation at a high or low shear amplitude, respectively. This TT-based oscillation well explains recent experiments. In the middle shear amplitude, RBCs show an intermittent or synchronized oscillation. As shear frequency increases, the vesicle oscillation becomes delayed with respect to the shear oscillation. At a high frequency, multiple limit-cycle oscillations coexist. The thermal fluctuations can induce transitions between two orbits at very low shear amplitudes. For a high mean shear rate with small shear oscillation, the shape and θ oscillate in the TT motion but only one attractor exists even at high shear frequencies. The measurement of these oscillatory modes is a promising tool for quantifying the viscoelasticity of RBCs, synthetic capsules, and lipid vesicles.

Dynamic modes of microcapsules in steady shear flow: effects of bending and shear elasticities

The dynamics of microcapsules in steady shear flow were studied using a theoretical approach based on three variables: the Taylor deformation parameter αD , the inclination angle θ , and the phase angle ϕ of the membrane rotation. It is found that the dynamic phase diagram shows a remarkable change with an increase in the ratio of the membrane shear and bending elasticities. A fluid vesicle (no shear elasticity) exhibits three dynamic modes: (i) tank treading at low viscosity ηin of internal fluid (αD and θ relaxes to constant values), (ii) tumbling (TB) at high ηin (θ rotates), and (iii) swinging (SW) at middle ηin and high shear rates γ (θ oscillates). All of three modes are accompanied by a membrane (ϕ) rotation. For microcapsules with low shear elasticity, the TB phase with no ϕ rotation and the coexistence phase of SW and TB motions are induced by the energy barrier of ϕ rotation. Synchronization of ϕ rotation with TB rotation or SW oscillation occurs with integer ratios of rotational frequencies. At high shear elasticity, where a saddle point in the energy potential disappears, intermediate phases vanish and either ϕ or θ rotation occurs. This phase behavior agrees with recent simulation results of microcapsules with low bending elasticity.

Rheology of a dilute suspension of liquid-filled elastic capsules

Rheology of a dilute suspension of liquid-filled elastic capsules in linear shear flow is studied by three-dimensional numerical simulations using a front-tracking method. This study is motivated by a recent discovery that a suspension of viscous vesicles exhibits a shear viscosity minimum when the vesicles undergo an unsteady vacillating-breathing dynamics at the threshold of a transition between the tank-treading and tumbling motions. Here we consider capsules of spherical resting shape for which only a steady tank-treading motion is observed. A comprehensive analysis of the suspension rheology is presented over a broad range of viscosity ratio (ratio of internal-to-external fluid viscosity), shear rate (or, capillary number), and capsule surface-area dilatation. We find a result that the capsule suspension exhibits a shear viscosity minimum at moderate values of the viscosity ratio, and high capillary numbers, even when the capsules are in a steady tank-treading motion. It is further observed that the shear viscosity minimum exists for capsules with area-dilating membranes but not for those with nearly incompressible membranes. Nontrivial results are also observed for the normal stress differences which are shown to decrease with increasing capillary number at high viscosity ratios. Such nontrivial results neither can be predicted by the small-deformation theory nor can be explained by the capsule geometry alone. Physical mechanisms underlying these results are studied by decomposing the particle stress tensor into a contribution due to the elastic stresses in the capsule membrane and a contribution due to the viscosity differences between the internal and suspending fluids. It is shown that the elastic contribution is shear-thinning, but the viscous contribution is shear thickening. The coupling between the capsule geometry and the elastic and viscous contributions is analyzed to explain the observed trends in the bulk rheology.

Swinging and synchronized rotations of red blood cells in simple shear flow

The dynamics of red blood cells (RBCs) in simple shear flow was studied using a theoretical approach based on three variables: a shape parameter, the inclination angle theta, and phase angle phi of the membrane rotation. At high shear rate and low viscosity contrast of internal fluid, RBCs exhibit tank-treading motion, where phi rotates with swinging oscillation of shape and theta . At low shear rate, tumbling motion occurs and theta rotates. In the middle region between these two phases, it is found that synchronized rotation of phi and theta with integer ratios of the frequencies occurs in addition to intermittent rotation. These dynamics are robust to the modification of the potential of the RBC shape and membrane rotation. Our results agree well with recent experiments.

Dynamics of nonspherical capsules in shear flow

Three-dimensional numerical simulations using a front-tracking method are presented on the dynamics of oblate shape capsules in linear shear flow by considering a broad range of viscosity contrast (ratio of internal-to-external fluid viscosity), shear rate (or capillary number), and aspect ratio. We focus specifically on the coupling between the shape deformation and orientation dynamics of capsules, and show how this coupling influences the transition from the tank-treading to tumbling motion. At low capillary numbers, three distinct modes of motion are identified: a swinging or oscillatory (OS) mode at a low viscosity contrast in which the inclination angle theta(t) oscillates but always remains positive; a vacillating-breathing (VB) mode at a moderate viscosity contrast in which theta(t) periodically becomes positive and negative, but a full tumbling does not occur; and a pure tumbling mode (TU) at a higher viscosity contrast. At higher capillary numbers, three types of transient motions occur, in addition to the OS and TU modes, during which the capsule switches from one mode to the other as (i) VB to OS, (ii) TU to VB to OS, and (iii) TU to VB. Phase diagrams showing various regimes of capsule dynamics are presented. For all modes of motion (OS, VB, and TU), a large-amplitude oscillation in capsule shape and a strong coupling between the shape deformation and orientation dynamics are observed. It is shown that the coupling between the shape deformation and orientation is the strongest in the VB mode, and hence at a moderate viscosity contrast, for which the amplitude of shape deformation reaches its maximum. The numerical results are compared with the theories of Keller and Skalak, and Skotheim and Secomb. Significant departures from the two theories are discussed and related to the strong coupling between the shape deformation, inclination, and transition dynamics.

Dynamical regimes and hydrodynamic lift of viscous vesicles under shear

The dynamics of two-dimensional viscous vesicles in shear flow, with different fluid viscosities etain and etaout inside and outside, respectively, is studied using mesoscale simulation techniques. Besides the well-known tank-treading and tumbling motions, an oscillatory swinging motion is observed in the simulations for large shear rate. The existence of this swinging motion requires the excitation of higher-order undulation modes (beyond elliptical deformations) in two dimensions. Keller-Skalak theory is extended to deformable two-dimensional vesicles, such that a dynamical phase diagram can be predicted for the reduced shear rate and the viscosity contrast etain/etaout. The simulation results are found to be in good agreement with the theoretical predictions, when thermal fluctuations are incorporated in the theory. Moreover, the hydrodynamic lift force, acting on vesicles under shear close to a wall, is determined from simulations for various viscosity contrasts. For comparison, the lift force is calculated numerically in the absence of thermal fluctuations using the boundary-integral method for equal inside and outside viscosities. Both methods show that the dependence of the lift force on the distance ycm of the vesicle center of mass from the wall is well described by an effective power law ycm(-2) for intermediate distances 0.8Rp< approximately ycm< approximately 3Rp with vesicle radius Rp. The boundary-integral calculation indicates that the lift force decays asymptotically as 1/[ycm ln(ycm)] far from the wall.

Dynamics of a vesicle in general flow

An approach to quantitatively study vesicle dynamics as well as biologically-related micro-objects in a fluid flow, which is based on the combination of a dynamical trap and a control parameter, the ratio of the vorticity to the strain rate, is suggested. The flow is continuously varied between rotational, shearing, and elongational in a microfluidic 4-roll mill device, the dynamical trap, that allows scanning of the entire phase diagram of motions, i.e., tank-treading (TT), tumbling (TU), and trembling (TR), using a single vesicle even at lambda = eta(in)/eta(out) = 1, where eta(in) and eta(out) are the viscosities of the inner and outer fluids. This cannot be achieved in pure shear flow, where the transition between TT and either TU or TR is attained only at lambda>1. As a result, it is found that the vesicle dynamical states in a general are presented by the phase diagram in a space of only 2 dimensionless control parameters. The findings are in semiquantitative accord with the recent theory made for a quasi-spherical vesicle, although vesicles with large deviations from spherical shape were studied experimentally. The physics of TR is also uncovered.

Modeling morphological instabilities in lipid membranes with anchored amphiphilic polymers

Anchoring molecules, like amphiphilic polymers, are able to dynamically regulate membrane morphology. Such molecules insert their hydrophobic groups into the bilayer, generating a local membrane curvature. In order to minimize the elastic energy penalty, a dynamic shape instability may occur, as in the case of the curvature-driven pearling instability or the polymer-induced tubulation of lipid vesicles. We review recent works on modeling of such instabilities by means of a mesoscopic dynamic model of the phase-field kind, which take into account the bending energy of lipid bilayers.

Flow-induced clustering and alignment of vesicles and red blood cells in microcapillaries

The recent development of microfluidic devices allows the investigation and manipulation of individual liquid microdroplets, capsules, and cells. The collective behavior of several red blood cells (RBCs) or microcapsules in narrow capillaries determines their flow-induced morphology, arrangement, and effective viscosity. Of fundamental interest here is the relation between the flow behavior and the elasticity and deformability of these objects, their long-range hydrodynamic interactions in microchannels, and thermal membrane undulations. We study these mechanisms in an in silico model, which combines a particle-based mesoscale simulation technique for the fluid hydrodynamics with a triangulated-membrane model. The 2 essential control parameters are the volume fraction of RBCs (the tube hematocrit, H(T)), and the flow velocity. Our simulations show that already at very low H(T), the deformability of RBCs implies a flow-induced cluster formation above a threshold flow velocity. At higher H(T) values, we predict 3 distinct phases: one consisting of disordered biconcave-disk-shaped RBCs, another with parachute-shaped RBCs aligned in a single file, and a third with slipper-shaped RBCs arranged as 2 parallel interdigitated rows. The deformation-mediated clustering and the arrangements of RBCs and microcapsules are relevant for many potential applications in physics, biology, and medicine, such as blood diagnosis and cell sorting in microfluidic devices.

Phase diagram of single vesicle dynamical states in shear flow

We report the first experimental phase diagram of vesicle dynamical states in a shear flow presented in a space of two dimensionless parameters suggested recently by V. Lebedev et al. To reduce errors in the control parameters, 3D geometrical reconstruction and determination of the viscosity contrast of a vesicle in situ in a plane Couette flow device prior to the experiment are developed. Our results are in accord with the theory predicting three distinctly separating regions of vesicle dynamical states in the plane of just two self-similar parameters.

Relaxation dynamics of fluid membranes

We study the effect of membrane viscosity in the dynamics of liquid membranes-possibly with free or internal boundaries-driven by conservative forces (curvature elasticity and line tension) and dragged by the bulk dissipation of the ambient fluid and the friction occurring when the amphiphilic molecules move relative to each other. To this end, we formulate a continuum model which includes a form of the governing equations for a two-dimensional viscous fluid moving on a curved, time-evolving surface. The effect of membrane viscosity has received very limited attention in previous continuum studies of the dynamics of fluid membranes, although recent coarse-grained discrete simulations suggest its importance. By applying our model to the study of vesiculation and membrane fusion in a simplified geometry, we conclude that membrane viscosity plays a dominant role in the relaxation dynamics of fluid membranes of sizes comparable to those found in eukaryotic cells, and is not negligible in many large synthetic systems of current interest.

Vesicles and red blood cells in shear flow

We describe the similarities and the specificities of the behaviour of individual soft particles, namely, drops, lipid vesicles and red blood cells subjected to a shear flow. We highlight that their motion depends in a non-trivial way on the particle mechanical properties. We detail the effect of the presence of a wall with or without wall-particle attractive interaction from a biological perspective.