Vesicles in shear flow

H. Noguchi and G. Gompper, Dynamics of fluid vesicles in shear flow effect of membrane viscosity and thermal fluctuations, Phys. Rev. E 72, 011901 (2005). 

Figure 7.6 Calculated shape transformation of an oblate vesicle in shear flow — yz. The numbers in each frame refer to time in seconds after the shear flow is turned on. They hold for a vesicle with i> = 0.8, shear rate y = 1/s", bending rigidity k— 10 erg, and size R = 10 pm. In each frame, the top picture shows the view onto the (x, z) plane. The bottom one shows the view onto the (x, y) plane. The arrows give the local velocity of the membrane.

Fig. 25 Snapshot of a discocyte vesicle in shear flow with reduced shear rate y = 0.92, reduced volume V = 0.59, membrane viscosity = 0, and viscosity contrast rjin/rjo = 1. The arrows represent the velocity field in the xz-plane. From [180]...

Fig. 29 Dynamical phase diagram of a vesicle in shear flow for reduced volume V = 0.59. Symbols correspond to simulated parameter values, and indicate tank-treading discocyte and tank-treading prolate (circles), tank-treading prolate and unstable discocyte (triangles), tank-treading discocyte and tumbling (transient) prolate (open squares), tumbling with shape oscillation (diamonds), unstable stomatocyte (pluses), stable stomatocyte (crosses), and near transition (filled squares). The dashed lines are guides to the eye. From [180]...

Abkarian, M. and Viallat, A. (2008) Vesicles and red blood cells in shear flow. Soji Matter, 4 (4), 553-557. 

The imposition of shear flow can have quite dramatic consequences on the structure and phase behavior of complex fluids. Steady shearing of binary amphiphilic systems can lead to a completely new phase of densely packed onionlike vesicles [140]. Shear flow also strongly affects the stability of the lamellar phase [141-145]. We want to discuss here the role of shear in the microemulsion-to-lamellar transition. 

Noguchi H, Gompper G (2004) Fluid vesicles with viscous membranes in shear flow. Phys Rev Lett 93(25) 258102... 

Hoffmann, H. and Ulbricht, W., Vesicle phases of surfactants and their behaviour in shear flow, Tenside Surf. Det., 35, 421-438 (1998). 

Vesicles were swollen from 1,2-dioleoyl-s/i-glycero-3-phosphocholine (Avanti Polar Lipids, USA) and V- (6-(biotinoyl)amino)hexanoyl)-l,2-dihexadecanoyl-sn-glycero-3-phosphoethanolamine (biotin-X DHPE, Molecular Probes, The Netherlands) at a ratio of 99 1. Tubes were grown out of lipid globules in shear flow and cut to the desired length with a pipette. Streptavidin-coated beads (Dynal, Germany) were used... 

Linden [49]. To our knowledge the theory has not yet been applied to experimental results. E. v. d. Linden assumes that multilamellar vesicles (droplets) are deformed in shear flow from a spherical to an elliptical shape. Turning into the deformed state the energy of closed shells is shifted because their curvature as well as their interlamellar distance D are changed. Due to the interaction of the bilayers, expressed by the bulk compression modulus B, the inner shells are deformed and the total deformation energy of the lamellar droplet gets minimized. Assuming that the volume of a droplet is not modified by the deformation, the surface A must increase. One can define an effective surface tension (Tef[=ElAA. E. v. d. Linden obtains ... 

The electron micrograph in Fig. 11.12 shows the existence of thermodynamically stable vesicles in the L i phase which can be formed as small unilamellar vesicles besides large multilamellar vesicles (liposomes). Depending on the composition of the ternary system the vesicles can also show structural faults like holes (perforated vesicles) [37]. The vesicle phases show a significantly reduced electric conductivity because a part of the water phase together with the ions is included in the interior of the vesicles. With stopped-flow experiments and optical or conductivity readout it is thus possible to determine the permeability of the vesicle membranes for dissolved compoimds [99]. Vesicle phases show often high viscosities and viscoelasticity due to the mutual hindrance of the vesicles in sheared solu-... 

The theory of Keller and Skalak [194] describes the hydrodynamic behavior of vesicles of fixed ellipsoidal shape in shear flow, with the viscosities t)in and qo of the internal and external fluids, respectively. Despite of the tqtproximations needed to derive the equation of motion for the inclinalion angle 6, which measures the deviation of the symmetry axis of the ellipsoid from the flow direction, this theory describes vesicles in flow surprisingly well. It has been generalized later [197] to describe the effects of a membrane viscosity Tjmb-... 

For B < 1, no stationary solution exists, and the vesicle shows a tumbling motion, very similar to a solid rod-like colloidal particle in shear flow. 

However, the vesicle shape in shear flow is often not as constant as assnmed by Keller and Skalak. In these situations, it is very helpful to compare simulation results with a generalized KeUer-Skalak theory, in which shape deformation and thermal fluctuations are taken into account. Therefore, a phenomenological model has been suggested in [180], in which in addition to the inclination angle 9 a second parameter is introduced to characterize the vesicle shape and deformation, the asphericity [207]... 

Fig. 32 Snapshots of vesicles in capillary flow, with bending rigidity K/k T = 20 and capillary radius / cap = 1-4/fo- a Fluid vesicle with discoidal shape at the mean fluid velocity v T/ffcap =41, both in side and top views, b Elastic vesicle (RBC model) with parachute shape at t m r/Rcap — 218 (with shear modulus nRl/ksT = 110). The blue arrows represent the velocity field of the solvent, c Elastic vesicle with shpper-like shape at v r/Rcap = 80 (with iiRl/k T = 110). The inside and outside of the membrane are depicted in red and green, respectively. The upper front quarter of the vesicle in (b) and the front half of the vesicle in (c) are removed to allow for a look into the interior, the black circles indicate the lines where the membrane has been cut in this procedure. Thick black lines indicate the walls of the cylindrical capillary. From [187]...

Experiments by Muller et al. [17] on the lamellar phase of a lyotropic system (an LMW surfactant) under shear suggest that multilamellar vesicles develop via an intermediate state for which one finds a distribution of director orientations in the plane perpendicular to the flow direction. These results are compatible with an undulation instability of the type proposed here, since undulations lead to such a distribution of director orientations. Furthermore, Noirez [25] found in shear experiment on a smectic A liquid crystalline polymer in a cone-plate geometry that the layer thickness reduces slightly with increasing shear. This result is compatible with the model presented here as well. 

It is easy to understand that these solutions must exhibit viscoelastic properties. Under shear flow the vesicles have to pass each other and, hence, they have to be deformed. On deformation, the distance of the lamellae is changed against the electrostatic forces between them and the lamellae leave their natural curvature. The macroscopic consequence is an elastic restoring force. If a small shear stress below the yield stress ery is applied, the vesicles cannot pass each other at all. The solution is only deformed elastically and behaves like Bingham s solid. This rheological behaviour is shown in Figure 3.35. which clearly reveals the yield stress value, beyond which the sample shows a quite low viscosity. 

Small-angle neutron scattering (SANS) [Cates and Candau, 1990 Rehage and Hoffmann, 1991] and cryo-TEM studies [Hoffmann et al., 1985] have shown that surfactant solutions with vesicle structures in the quiescent state can transform to TLMs under shear, accounting for the rare and unexpected observations that solutions with vesicle structures can be DR in turbulent flow flelds [Zheng et al., 2000]. 

The dynamical behavior of fluid vesicles in simple shear flow has been stodied experimentally [190-193], theoretically [194-201], numerically with the boundary-integral technique [202,203] or the phase-field method [203,204], and with meso-scale solvents [37,180,205]. The vesicle shape is now determined by the competition of the curvature elasticity of the membrane, the constraints of constant volume V and constant surface area S, and the external hydrodynamic forces. 

Shear flow is characterized (in the absence of vesicles or cells) by the flow field V = jyCx, where Cx is a unit vector, compare Sect. 10.4. The control parameter of shear flow is the shear rate f, which has the dimension of an inverse time. Thus, a dimensionless, scaled shear rate y = ft can be defined, where T is a characteristic relaxation time of a vesicle. Here, t = rjoRpkBT is used, where rjo is the solvent viscosity, Rq the average radius [206]. For < 1, the internal vesicle dynamics is... 

The form of the stochastic equations (105) and (106) is motivated by the following considerations. The first term in (105), dF/da, is the thermodynamic force due to bending energy and volume constraints it is calculated from the free energy F a). The second term of (105) is the deformation force due to the shear flow. Since the hydrodynamic forces elongate the vesicle for 0 < 0 < r/2 but push to reduce the elongation for - r/2 < 0 < 0, the flow forces should be proportional to sin(20) to leading order. The amplitude A is assumed to be independent of the asphericity a. C,a and A can be estimated [205] from the results of a perturbation theory [199] in the quasi-spherical limit. Equation (106) is adapted from KeUer-Skalak theory. While B is a constant in KeUer-Skalak theory, it is now a function of the (time-dependent) asphericity a in (106). 

Shear flow does not only induce different dynamical modes of prolate and oblate fluid vesicles, it can also induce phase transformations. The simplest case is a oblate fluid vesicle with Tjn,b = 0 and viscosity contrast rjin/rjo = 1- When the reduced shear rate reaches 1, the discocyte vesicles are stretched by the flow forces into a prolate shape [37,180,202], A similar transition is found for stomatocyte vesicles, except that in this case a larger shear rate 3 is required. In the case of non-zero membrane viscosity, a rich phase behavior appears, see Fig. 29. 

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