Phase Diagram of Single Vesicle Dynamical States in Shear Flow J. Deschamps, V. Kantsler,and V. Steinberg Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, 76100 Israel (Received 30 October 2008; published 17 March 2009) 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. DOI: 10.1103/PhysRevLett.102.118105 PACS numbers: 87.16.D, 82.70.Uv, 83.50.v Understanding rheology of biofluids remains a great challenge and its progress is based on detail studies of dynamics of a single cell. Vesicle is a model system to study dynamical behavior of biological cells, and its dy- namics in a shear flow was a subject of intensive theoretical [1–6], numerical [7–10] and experimental [11–16] re- search for the past decade. A vesicle is a droplet of viscous fluid surrounded by a phospholipid bilayer membrane suspended in a fluid of either the same or different viscosity as the inner one. Both the volume and the surface area of the vesicle are conserved. The former means that the vesicle membrane is considered to be impermeable, at least on the time scale of the experiment, and the latter means that the membrane dilatation is neglected [1,2]. Already the first model of vesicle dynamics in a shear flow [17] reveals tank-treading (TT) and tumbling (TU) motions and transition between them, where the regions of existence of TT and TU depend on two control parameters: the excess area,  ¼ A=R 2  4%, and the viscosity contrast, ! ¼  in = out . Here R is the effective vesicle radius related to its volume via V ¼ 4 3 %R 3 , A is the vesicle surface area,  in and  out are the dy- namic viscosities of the inner and outer fluids, respectively. At sufficiently low !<! c ðÞ, a vesicle preserves , the inclination angle between its long axis and the shear flow direction, and its shape besides thermal fluctuations, while the membrane implements TT motion [1,7,12,13,17]. At !>! c ðÞ, according to theoretical predictions [2,17] and recent experiments [14], the transition to TU occurs, when a vesicle axis rotates with respect to the flow direction. It is remarkable that both TT and the transition line are inde- pendent of the shear rate, _ , and described by a single equation for  [1,17]. The next key observation is a new type of an unsteady motion, coined by us trembling (TR) [14]. This dynamical regime is distinguished by the incli- nation angle oscillation jðtÞj < %=2 around the flow di- rection accompanied by strong deformations of the vesicle shape. The latter is also revealed in the TU but in much less degree. A crucial aspect of TR is dependence of its exis- tence region, which is separate from the TU region, on _ . This is a distinct signature of the TR. Vesicle dynamics qualitatively similar to experimentally observed TR was independently predicted theoretically, though the distinc- tive feature was missing (there TR was coined as vacillat- ing breathing mode coexisted with TU) [3]. The discovery of TR lead to reconsideration of the basic theoretical model. Recently three different theoretical models, which take into account coupling between the shape deformation and  of a vesicle and dependence of TR dynamics on _ , were suggested. They describe the regions of existence and transitions between TT, TR, and TU [5,10,18]. The quali- tatively new result of the model presented in Ref. [5] is the self-similar solution obtained at   1, which reduces five parameters in the problem, namely  out , !, _ , R and , just to two dimensionless ones [5]: S ¼ 7% 3 ffiffiffi 3 p _  out R 3  ; (1)  ¼ 4 ffiffiffiffiffiffiffiffiffi 30% p  1 þ 23 32 !  ffiffiffiffi  p ; (2) where  is the bending modulus. It means that all vesicles with different geometrical characteristics, R and , can be presented on the 2D phase diagram, contrary to the 3D phase diagrams suggested in Ref. [10] with  as the third parameter. On the other hand, the authors of Ref. [18] claim that a hydrodynamic response of the the next order term in the Helfrich force, which is significant and breaks the self-similarity, is not taking into account in Ref. [5]. However, the additional terms suggested in Ref. [18] com- pared with Ref. [5] are of the next order in ffiffiffiffi  p in the equation for the shape deformation parameter and of the same order in the equation for . It means that theory of Ref. [18] suggests a significant correction even for TT compared with the older [1,2,7,17] and recent theories [6] that are rather well established and tested by the ex- periment [13]. To test these theories and verify, which of them proper describe experimental data, significant improvement of the existing techniques of vesicle characterization and control of _  is required. Indeed, the determination of R and  of a 3D object extrapolated from a 2D vesicle contour leads up to about 20% error. The deviation of ! in each vesicle from PRL 102, 118105 (2009) PHYSICAL REVIEW LETTERS week ending 20 MARCH 2009 0031-9007= 09=102(11)=118105(4) 118105-1 Ó 2009 The American Physical Society