Orientation and Dynamics of a Vesicle in Tank-Treading Motion in Shear Flow Vasiliy Kantsler and Victor Steinberg Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, 76100 Israel (Received 8 June 2005; published 12 December 2005) Experimental results on mean inclination angle and its fluctuation due to thermal noise in tank-treading motion of a vesicle in shear flow as a function of vesicle excess area, normalized shear rate, viscosity, and viscosity contrast between inner and outer fluids, , are presented. Good quantitative agreement with theory made for   1 was found. At > 1 the dependence is altered significantly. Dependence of the vesicle shape on shear rate is consistent with theory. A tank-treading velocity of the vesicle membrane is found to be a periodic function close to that predicted by theory. DOI: 10.1103/PhysRevLett.95.258101 PACS numbers: 87.16.Dg, 47.60.+i, 87.17.Jj Dynamics of deformable mesoscopic objects under hydrodynamic stresses determine rheology of many com- plex fluids, such as emulsions, suspensions of droplets or bubbles, solutions of vesicles, blood, biological flu- ids, etc. From a theoretical point of view this nonequilib- rium problem is rather challenging due to the coupling between the object deformations and the flow that leads to a free-boundary hydrodynamic problem, where the ob- ject shape is not given a priori but determined by an inter- play between flow, bending energy, and various physical constraints. A vesicle, an example of such deformable objects, is a liquid droplet bounded by a closed phospholipid bilayer membrane (usually unilamellar) suspended in a fluid that can either be the same solvent as an inner one or different. Both the volume and the total surface area of the vesicle are conserved, and the latter means that the membrane dilata- tion is neglected. Vesicle surface undulations due to ther- mal fluctuations and an external flow were analyzed in Ref. [1]. In this Letter, we report experimental studies of average inclination angle and its fluctuations, membrane tank- treading velocity and shape of a single vesicle as a function of excess area, normalized shear rate, viscosity, and vis- cosity contrast,    in = out . Here  in and  out are the dynamic viscosities of inner and outer fluids, respectively. Dynamics of a vesicle subjected to a shear flow is governed by three main control parameters: the total excess area (dimensionless),   S=R 2  4, the viscosity con- trast, , and the dimensionless shear rate,   _  out R 3 =, that characterizes the deformability of the vesicle under the shear rate, _ . Here R is the effective vesicle radius related to its volume via V  4 3 R 3 , S is the vesicle surface area that exceeds that of a sphere with the same volume by , and  is the bending rigidity. As followed from theory [1] and confirmed by numerical simulations [2] at   1, a vesicle subjected to a shear flow acquires a stationary mean inclination angle at a given  with the membrane that undergoes a tank-treading motion. The stationary inclination angle results from the balance of the following torques: torque associated with the rotational component of a shear velocity field, torque due to its elongation part, and torque resulting from the tank- treading motion of the membrane [3]. As the rotational torque component pushes the inclination angle away from =4 towards lower values, the vesicle membrane acquires a tank-treading motion, through which the elongation com- ponent of the torque is introduced into the torque balance [4]. A theory based on a simple model of a fixed shape membrane [3] as well as a more intricate theory that takes into account the shape deformations due to flow [5], and numerical simulations [4,6], predict two types of a vesicle motion. For < c , where the value of  c depends on the excess area [4,6], a vesicle achieves a fixed inclination angle with the membrane in a tank-treading motion. At >  c a transition to a tumbling motion is predicted. The vesicle axis then rotates with respect to the flow direction. In this Letter, we will concentrate just on the first type of the vesicle dynamics. In the case of   1, the detailed calculations that take into account both the vesicle shape deformations due to a shear flow and thermal fluctuations lead to the following expression for the mean inclination angle [1] in a quasi- spherical approximation at    1 [see Fig. 1(a) for defi- nition of ]:  0  1 2 arctan  a       1=2  =4    1=2 2a 1=2 : (1) Here a  412=11 2 2=15’ 1:994;   is the systematic part of the excess area that shows up due to the shear flow and is stored in the second harmonic of the shape defor- mations, where the last expression is valid at  =a  1. We would like to point out that Eq. (1) is valid for all values of , and this expression is relevant to experimental measure- ments of the vesicle shape, since only   can be measured. A shear flow pulls out the hidden membrane area from the thermal fluctuations. The more excess area is available, the larger deviations of the mean inclination angle from the equilibrium one, =4, obtained for small shear rates. It was reported in Ref. [7] that measurements of  0 as a function PRL 95, 258101 (2005) PHYSICAL REVIEW LETTERS week ending 16 DECEMBER 2005 0031-9007= 05=95(25)=258101(4)$23.00 258101-1  2005 The American Physical Society