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)

(J. Fluid Mech. Submitted (2010) )

•Vesicle in shear flow

• 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

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∇s ⋅r u

s≡ I − n ⊗ n( ) :∇

r u = 0,

∇ ⋅r u = 0.

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−∇p( in, out) + μ in, out( )∇ 2 r u in, out( ) =∇ ⋅T in, out( ) = 0, ∇ ⋅

r u in, out( ) = 0,

T = −pI + μ ∇r u + ∇

r u ( )

T

[ ], ˆ n ⋅ T out − T in( ) = τ m .

τ m is the membrane surface forces with elastic component.

Biological mimic: Capsule (cont.)

(J. Fluid Mech. Submitted (2010))

•Capsule in shear flow

• Small-deformation theory is employed to understand the dynamics of capsule in shear flow

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rs =1+ εf θ,ϕ( ) =1+ ε f jmY jm

m=− j

j

∑j= 2

∞

∑ ,

where Y jm is the scalar spherical harmonics.

€

The leading - order equations for amplitude f2m = R(t) e−imψ t( )

˙ ψ = −1

2+

Λ−1

2Rcos 2ψ( ) + ε0

SΛ( )−1

2Rsin 2φ − 2ψ( ),

˙ R = Λ−1 1− R2( )sin 2ψ( ) + SΛ( )

−1ε0 1− R2

( )cos(2φ − 2ψ ) − R 1− R2( ) 1−ε0

2( ){ },

˙ φ = −1

2. ε0 : asphericity, S -1 =

2Δ

15πCa−1, Λ−1 =

8 30π

23λ + 32( ) Δ, Δ : excess area.

Ca =μ out ˙ γ R0

η, Δ ≡

A0

R02 − 4π , ε ~ Δ.

Biological mimic: Capsule (cont.)

•Capsule in shear flow

• Three types of capsule dynamics in shear flow: tank-treading (TT), swinging (SW), and tumbling (TB)

•0=0.5, Ca=0.2 and =0.02

• Transition from SW to TB as a function of (a)outin, and(b) 0

€

Capillary number : Ca ≡μ out ˙ γ R0

η.

Biological mimic: Capsule (cont.)

•Capsule in shear flow

• SW-TB transition at the

limit 0 <<1 and R~ 0

• SW-TB transition for

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˙ R ~ Λ−1 sin 2ψ( ) + SΛ( )−1

ε0 cos t + 2ψ( ) − R[ ],

Assuming ε0S−1 ~ O(1), at leading order

R ~ ε0 ε0−1S sin 2ψ( ) + cos t + 2ψ( )[ ],

˙ ψ ~Λ−1

2S

cos 2ψ( ) − S−1ε0 sin t + 2ψ( )sin 2ψ( ) + S−1ε0 cos t + 2ψ( )

.

Periodic solution for ψ is possible for S−1ε0 ≤1.

TB occurs when S−1ε0 >1⇒

Cac−1 =

15π

2Δε0

−1 ⇒ Reproduces the transition boundary

from computations for 0 ≤ ε0 ≤ 0.5.

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0 →1 and R →1

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˙ ψ ~ −1

2+

Λ−1

2cos 2ψ( ) −

SΛ( )−1

2sin t + 2ψ( )

= -1

2+

Λ−1

2Bcos 2ψ −θB( ),

B ≡ 1+ S−2 − 2S−1 sin t( ),θB ≡ tan−1 S−1 cos t( )1− S−1 sin t( )

⎛

⎝ ⎜

⎞

⎠ ⎟.

For periodic solutions ˙ ψ dt0

2π

∫ =ψ 2π( ) −ψ 0( ) = 0,

this is possible only when Cac−1 ≤

15π

2Δ1− Δ

23λ + 32

8 30π

⎛

⎝ ⎜

⎞

⎠ ⎟2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

(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.

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∇s ⋅v s≡ I − n ⊗ n( ) :∇v = 0,

∇ ⋅v = 0.

Formulation• The system contains three

dimensionless parameters: Excess area , Viscosity ratio Capillary number

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Excess Area Δ ≡A0

R02 − 4π ,

Reduced Volume V * ≡ 1+Δ

4π

⎛

⎝ ⎜

⎞

⎠ ⎟−

3

2.

R0 = 3V0 /4π3 ,

λ ≡η inside

η outside

,χ ≡ληa3 ˙ γ

κ.

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v∞ = ˙ γ E ⋅r r , E =

1

2

0 1+ ω 0

1−ω 0 0

0 0 0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟, ω =1 for linear shear flow.

η outside∇2v outside −∇poutside = 0, ∇ ⋅v outside = 0.

λη outside∇2v inside −∇pinside = 0, ∇ ⋅v inside = 0.

∇ ⋅v = 0, in - extensible membrane ⇒ ∇ s ⋅v s= 0.

Formulation (cont.)• The compound vesicle

encloses a particle (sphere of radius a < R0)

• Small-deformation theory is employed:

€

r = r0 + f (Ω) = r0 + f jmY jm,m=− j

m= j

∑j= 2

∞

∑

Δ =( j + 2)( j −1)

2f jm

jm

∑ f jm* + h.o.t.,

r y jm 0 = j j +1( )[ ]

−1/ 2r∇ΩY jm,

r y jm1 = −iˆ r ×

r y jm 0,

r y jm 2 = ˆ r Y jm .

•The rigid sphere is assumed to be concentric with the vesicle.

€

• f jm ~ ε

Small-deformation theory• Velocity field inside and outside vesicle

•Singular at origin

•Singular at infinity

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vout = c jmq∞

jmq

∑ u jmq+ − u jmq

−( ) + c jmq u jmq

− + X jm q | q'( )q'

∑ u jmq '−

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥jmq

∑

vin = c jmq u jmq− + X(q | q')u jmq '

−

q'

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥jmq

∑

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u jm 0− =

1

2r j 2 − j +j

r2

⎛

⎝ ⎜

⎞

⎠ ⎟r y jm 0 + j j +1( ) 1−

1

r2

⎛

⎝ ⎜

⎞

⎠ ⎟r y jm 2

⎡

⎣ ⎢

⎤

⎦ ⎥

u jm1− =

1

r j +1

r y jm1

u jm 2− =

1

2r j2 − j( )

j

j +11−

1

r2

⎛

⎝ ⎜

⎞

⎠ ⎟r y jm 0 + j +

2 − j

r2

⎛

⎝ ⎜

⎞

⎠ ⎟r y jm 2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

u jm 0+ =

r j−1

2− j +1( ) + j + 3( )r2

[ ]r y jm 0 − j j +1( ) 1− r2

( )r y jm 2

u jm1+ = r j r y jm1

u jm 2+ =

r j−1

23+ j( )

j +1

j1− r2( )

r y jm 0 + j + 3− j +1( )r2

( )r y jm 2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

Vector Spherical Harmonics

r y jm 0 =

r

j( j +1)∇ΩY jm

r y jm1 = −iˆ r ×

r y jm 0

r y jm 2 = ˆ r Y jm

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.

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vS inc + vS scat = α mu1m1+

m

∑ at r = a,

vS inc = c jmqu jmq+ , vS scat = c jmq X jm q | q'( )u jmq '

−

qq '

∑ .jm

∑jmq

∑

Scattering matrix Xjm(q|q’) (cont’d)

• For any coefficients c2mq the following equations have to be satisfied

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X11 = −a5, X00 = −1

2a5 −3+ 5a2

( ),

X20 = −5 6

4a5 1− a2

( ),X02 =5 6

4a3 1− a2

( )2,

X22 =a3

4−15a4 + 36a2 − 25( ).

X01 = X10 = X12 = X21 = 0.

•Velocity continuity at r=a gives

(Young et al., to be submitted to J. Fluid Mech.)

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c2m 0

a

2−3+ 5a2

( ) +1

a4X00

⎡ ⎣ ⎢

⎤ ⎦ ⎥+ c2m2 5a

3

81− a2( ) +

X20

a4

⎡

⎣ ⎢

⎤

⎦ ⎥= 0,

c2m 0

3

2− 1− a2

( ) +a2 −1

2a4 X00 +1

a4

2

3X02

⎡

⎣ ⎢

⎤

⎦ ⎥+ c2m2

a 5 − 3a2( )

2+

3

2

a2 −1

a4 X20 +X22

a2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥= 0.

Amplitude equations• Surface incompressibility gives

• Balance of stresses on the vesicle membrane gives the tension and cjm2. Combining everything, we obtain

€

c jm2 =j( j +1) 1+ X00( ) − 2X02

2 − j j +1( )X20 + 2X22

c jm0 ≡ αc jm0.

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D'=D

1+X02

α+ X22

, D =1

α−

2 j +1

j( j +1)+

3(2 j +1)

j( j +1)− 2(2 j +1)

⎡

⎣ ⎢

⎤

⎦ ⎥ when λ =1.

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df jm

dt= iω

m

2f jm −

1

D'c jm 0

∞ 2 j +1

j( j +1)+ c jm2

∞ −3(2 j +1)

j( j +1)+ 2 2 j +1( )

⎛

⎝ ⎜

⎞

⎠ ⎟−

Ef jm

χ

⎡

⎣ ⎢

⎤

⎦ ⎥

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 .

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> c ≡ −9

23+

120

23

2π

15Δ

(Vlahovska and Gracia, PRE, 2007)

€

˜ Δ = Δ1/ 2 9 + 23λ

16π 3 / 2 30

•Inclination angle

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

€

a →1

€

1+X02

α+ X22 → 0 as a →1.

•Inclination angle vs excess area

•Geometric factor vs radius

=2 =0•Critical radius vs reduced volume

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V ≡ 1+Δ

4π

⎛

⎝ ⎜

⎞

⎠ ⎟−3 / 2

• 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.

Effective interior fluid viscosity• The compound vesicle can be

viewed as a membrane enclosing a homogeneous fluid with an effective viscosity, estimated as

•Effective interior fluid viscosityRheology of c-vesicles

•Effective shear viscosity for the dilute suspension

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ηeff acritical( )η out

≡ −9

23+

120

23

2π

15Δ

•First normal stress 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

€

t μ = −2 KA − μ( )(∇ s ⋅d)H ˆ n + KA − μ( )∇ s∇ s ⋅d + μ∇ s ⋅ ∇ sd ⋅ Is + Is ⋅ ∇ sd( )T

[ ]

d is the displacement of a material particle

KA stretch modulus( ) ~ 200N /m,μ shear elastic modulus( ) ~ 10−6 N /m

∇ s ⋅d = 0 for an inextensible capsule.

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

• 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.

•Critical radius in 2D

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≡A

πR2

2D compound vesicle

Effective interior fluid viscosity• The compound vesicle can be

viewed as a membrane enclosing a homogeneous fluid with an effective viscosity, estimated as

•Effective interior fluid viscosity

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Txy ≡5

2−

f (a)

D'−

D'2

E 2 + D'2 ⎛

⎝ ⎜

⎞

⎠ ⎟, f (a) =

5 6 1+ X00 + X20α( ) +10αζ

αζD'.

Dilute Suspension of c-vesicles•Effective shear viscosity

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ηeff acritical( )η out

≡ −9

23+

120

23

2π

15Δ

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N1 ≡3Δ

4πEf (a) ≥ 0, E =

160π

3Δ− D'2 ,ζ =

6 1+ X00( ) − 2X02

2 − 6X20 + 2X22

.