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 

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

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)

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) )

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))

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

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))

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)

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.

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

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.

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

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.

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.)

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

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)

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

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

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

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

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

2D compound vesicle Critical radius in 2D 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.

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