Dynamics of a compound vesicle*

1. Dynamics of a compound vesicle* Yuan-Nan Young New Jersey Institute of Technology Shravan Veerapaneni, New York University Petia Vlahovska, Brown University Jerzy Blawzdziewicz, Texas Technological University *Submitted to Phys. Rev. Lett., 2010 Funding from NSF-CBET, NSF-DMS

2. Biological motivation: Red blood cell (RBC) (Alison Forsyth and Howard Stone, Princeton University)

3. RBC dynamics, ATP release, and shear viscosity • Correlation between RBC dynamics and ATP release • Correlation between RBC dynamics and shear viscosity (Alison Forsyth and Howard Stone, Princeton University)

4. Biological mimic: Elastic membrane (vesicle) • A vesicle is a closed lipid bi-layer membrane, and the total area is conserved because the number of lipids in a monolayer and the area per lipid are fixed • The enclosed volume is conserved as well • For red blood cell mimic, the vesicle membrane also has a finite shear elasticity • Vesicle in shear flow (J. Fluid Mech. Submitted (2010) )

5. Biological mimic: Capsule (cont.) • Small-deformation theory is employed to understand the dynamics of capsule in shear flow • Capsule in shear flow (J. Fluid Mech. Submitted (2010))

6. Biological mimic: Capsule (cont.) • Three types of capsule dynamics in shear flow: tank-treading (TT), swinging (SW), and tumbling (TB) • Capsule in shear flow • 0=0.5, Ca=0.2 and =0.02 • Transition from SW to TB as a function of • outin, and • 0

7. Biological mimic: Capsule (cont.) • SW-TB transition at the limit 0 <<1 and R~ 0 • Capsule in shear flow • SW-TB transition for (J. Fluid Mech. Submitted (2010))

8. Introduction • Enclosing lipid membranes with sizes ranging from 100 nm to 10 m • Vesicle as a multi-functional platform for drug delivery (Park et al., Small 2010)

9. Configuration • A vesicle is a closed lipid bi-layer membrane, and the total area is conserved because the number of lipids in a monolayer and the area per lipid are fixed. • The enclosed volume is conserved as well. • A vesicle is placed in a linear (planar) shear flow.

10. Formulation • The system contains three dimensionless parameters: Excess area , Viscosity ratio Capillary number 

11. Formulation (cont.) • The compound vesicle encloses a particle (sphere of radius a < R0) • Small-deformation theory is employed: • The rigid sphere is assumed to be concentric with the vesicle.

12. Small-deformation theory • Velocity field inside and outside vesicle • Singular at origin • Singular at infinity

13. Scattering matrix Xjm(q|q’) • The enclosed rigid sphere (of radius a <1) is concentric with the vesicle. Thus the sphere can only rotate inside the vesicle in a shear flow. This means the velocity must be the rigid-body rotation at r=a.

14. Scattering matrix Xjm(q|q’) (cont’d) • For any coefficients c2mq the following equations have to be satisfied • Velocity continuity at r=a gives (Young et al., to be submitted to J. Fluid Mech.)

15. Amplitude equations • Surface incompressibility gives • Balance of stresses on the vesicle membrane gives the tension and cjm2. Combining everything, we obtain

16. Tank-treading to tumbling:  >1 • In a planar shear flow, vesicle tank-treads at a steady inclination angle  for small excess area . • Vesicle tumbles if • In experiments (3D) and direct numerical simulations (2D), vesicle in a shear flow does not tumble without viscosity mismatch even at large . • Inclination angle (Vlahovska and Gracia, PRE, 2007)

17. Tumbling of a compound vesicle:  • The vesicle rotates as a rigid particle as This is because • The inclination angle is a function of enclosed particle radius a and excess area  • Compound vesicle tumbles when the inclusion size is greater than the critical particle radius ac. • Effectively the interior fluid becomes more viscous due to the rigid particle, and we can quantitatively describe the effective interior viscosity by the transition to tumbling dynamics. • Geometric factor vs radius • Inclination angle vs excess area =2 =0 • Critical radius vs reduced volume

18. Effective interior fluid viscosity • The compound vesicle can be viewed as a membrane enclosing a homogeneous fluid with an effective viscosity, estimated as Rheology of c-vesicles • Effective interior fluid viscosity • First normal stress for the dilute suspension • Effective shear viscosity for the dilute suspension

19. Conclusion • Compound vesicle can tumble in shear flow without viscosity mismatch • Effective interior viscosity is quantified as a function of particle radius a • Rheology of the dilute compound vesicle suspension depends on the “internal dynamics” of compound vesicles

20. Compound Capsule • A pure fluid bi-layer membrane is infinitely shear-able. • Polymer network lining the bi-layer gives rise to finite shear elasticity. • Assuming linear elastic behavior, the elastic tractions are

21. Compound Capsule (cont.) • Extra parameter  for shear elasticity • Starting from the tank-treading unstressed “reference” membrane • For deformation of a membrane with fixed ellipticity, the transition between trank-treading (swinging) and tumbling can be found using min-max principle

22. Critical radius in 2D 2D compound vesicle • Following Rioual et al. (PRE 2004) the critical particle radius can be found as a function of the swelling ratio (reduced area)  in two-dimensional system. • Rigorous small-deformation for the 2D compound vesicle is conducted. • Comparison with boundary integral simulation results is consistent.

23. Effective interior fluid viscosity • Effective interior fluid viscosity • The compound vesicle can be viewed as a membrane enclosing a homogeneous fluid with an effective viscosity, estimated as Dilute Suspension of c-vesicles • Effective shear viscosity