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 bilayer 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
•Vesicle in shear flow
conserved as well
 s  u s  I  n  n : u  0,
• For red blood cell mimic, the
vesicle membrane also has a   u  0.
(J. Fluid Mech. Submitted (2010) )
finite shear elasticity
p(in, out)  in, out 2 uin, out    T in, out  0,   uin, out  0,
 in
T
out
ˆ
T  pI   u  u , n  T  T   m .


 m is the membrane surface forces with elastic component.
Biological mimic: Capsule (cont.)
• Small-deformation theory is
employed to understand the
dynamics of capsule in shear
flow
j

rs  1 f ,   1   f jmY jm ,
j 2 m j
where Y jm is the scalar spherical harmonics.
•Capsule in shear flow
(J. Fluid Mech. Submitted (2010))
T he leading- order equat ions for amplit ude
f 2m  R(t) eim t 
1
1

S sin 2  2 ,
Ý  

cos2   0


2 2R
2R
1


1
RÝ 1 1 R 2 sin2   S 0 1 R 2 cos(2  2 )  R 1 R 2 1 02  ,
1
2
Ý  . 0 : asphericity,S -1 

2
8 30
Ca1, 1 
,  : excess area.
15
23  32 
ÝR0
 out 
A
Ca 
,   02  4  ,  ~  .

R0
Biological mimic: Capsule (cont.)
• Three types of capsule
dynamics in shear flow: tanktreading (TT), swinging (SW),
and tumbling (TB)
•Capsule in shear flow
•0=0.5, Ca=0.2 and
=0.02
ÝR0
 out 
Capillary number: Ca 
.

• Transition from
SW to TB as a
function of
a outin, and
b 0

Biological mimic: Capsule (cont.)
• SW-TB transition at the
limit 0 <<1 and R~ 0
1
RÝ~ 1 sin2   S 0 cost  2   R,
Assuming 0 S 1 ~ O(1), at leading order
R ~ 0 1
0 S sin2   cost  2 ,
1
cos2   S 10 sint  2 

Ý~

.
2S sin2   S 10 cost  2 
P eriodic solution for is possible forS 10  1.
T B occurs whenS 10  1 
15 1

0  Reproduces t he t ransition
2
from comput ations for
0  0  0.5.
Ca1
c 
•Capsule in shear flow
• SW-TB transition for
0 1 and R 1
1 1
S sin t  2
Ý
 ~  
cos2  


2 2
2
1 1
=- 
Bcos2   B ,
2 2
boundary
 S 1 cost  
2
1
1
B  1 S  2S sint , B  tan 
.
1
1 S sint 
1
2
Ýd t   2    0  0,
For periodic solutions 
0
(J. Fluid Mech. Submitted (2010))
23  322 
15 
this is possible only whenCa 
1 
 
2 

 
8
30



1
c
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
 s  v s  I  n  n : v  0,
conserved as well.
• A vesicle is placed in a
  v  0.
linear (planar) shear flow.
Formulation
• The system contains three
dimensionless parameters:
Excess area , Viscosity
ratio Capillary number 
Excess Area 
A0
 4 ,
2
R0

3
2
  
Reduced Volume V  1
 .
 4  
*
R0  3 3V0 /4  ,
Ý
inside
a 3

, 
.
outside

 0
1  0
1 


Ý
v  E  r , E = 1 
0
0,   1 for linear shear flow.
2 


0
0
 0
outside 2v outside  p outside  0,   v outside  0.
outside 2v inside  p inside  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: m j
r  r0  f ()  r0    f jmY jm ,
j 2 m j
 
jm
( j  2)( j 1)
*
f jm f jm
 h.o.t.,
2
y jm0  j  j  1
1/ 2
y jm1  irˆ  y jm0,
y jm2  rˆY jm .
r Y jm ,
•The rigid sphere is assumed
to be concentric with the vesicle.
 f jm ~ 
Small-deformation theory
• Velocity field inside and outside vesicle




v   c u  u   c jmq u jmq   X jm q | q'u jmq' 




jmq
jmq
q'


in


v   c jmq u jmq   X(q | q')u jmq ' 




jmq
q'
out

jmq

jmq

jmq
•Singular at origin u

jm0

ujm1 
ujm2 
•Singular at infinity
1
r j 1
u

jm2
r
 Y jm
j( j  1)
y jm1  irˆ  y jm0
y jm0 
y jm2  rˆY jm
 1  
j  j  11 2 y jm2 
 r  

y jm1
 2  j  
1 
j  1 
2

j
1
y


 jm0 j  2 y jm2 



2r j 
j  1  r 2 
r  
r j1
u 
 j  1   j  3r 2 y jm0 

2
u jm1  r j y jm1

jm0

1 
j 
2

j


y jm0 

2r j 
r 2 
Vector Spherical Harmonics
j  j  11 r 2 y jm2

r j1 
j 1
2
2

3  j 
1 r y jm0  j  3   j  1r y jm2
2 
j

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.

vS inc  vS scat  a mu1m1
at r  a,
m
vS inc   c jmq u jmq , vS scat    c jmq X jm q | q'ujmq' .
jmq
jm qq'
Scattering matrix Xjm(q|q’) (cont’d)
• For any coefficients c2mq the following
equations have to be satisfied

a

1
3
X 20 
2
2
c 2m 0  3  5a  4 X 00  c 2m 2 5a 1 a  4  0,
2

a
a 
 8
a5  3a 2 
3 
a 2 1
1 2 
3 a 2 1
X 22 
2
c 2m 0 1 a 
X 00  4
X 02  c 2m 2 

X 20  2  0.
4
4
2 
2a
a 3 
2
2 a
a 



•Velocity continuity at r=a gives
(Young et al., to be submitted to
J. Fluid Mech.)
1
X11  a 5 , X 00   a 5 3  5a 2 ,
2
2
5 6 5
5 6 3
X 20  
a 1 a 2 , X 02 
a 1 a 2  ,
4
4
a3
X 22  15a 4  36a 2  25.
4
X 01  X10  X12  X 21  0.
Amplitude equations
• Surface incompressibility gives
c jm2 

j( j  1) 1 X 00   2X 02
2  j  j  1X 20  2X 22
c jm0  ac jm0 .
• Balance of stresses on the vesicle
membrane gives the tension and cjm2.
Combining everything, we obtain
3(2 j  1)
 Ef jm 
m
1  
2 j 1

 i f jm  c jm0
 c jm2 
 22 j  1

dt
2
D' 
j( j  1)
 j( j 1)
  
df jm
D
D'
1
X 02
a

1  2 j  1
3(2 j  1)
, D  

 2(2 j  1)when  = 1.
a  j( j  1)
j( j  1)

 X 22
Tank-treading to tumbling:  >1
• In a planar shear flow, vesicle tank-treads at
a steady inclination angle  for small excess
area .
•Inclination angle
• Vesicle tumbles if
9 120 2
  c   
23 23 15
(Vlahovska and Gracia, PRE, 2007)
• In experiments (3D) and
9  23
˜ 

direct numerical simulations (2D),
16
30
vesicle in a shear flow does not tumble
 large .
without viscosity mismatch even at
1/ 2
3/2
Tumbling of a compound vesicle: 
• The vesicle rotates as a rigid particle
as a 1
X
This is because 1 a02  X22  0 as a 1.
• The inclination angle is a
function of enclosed particle radius a

and excess
 area 
•Inclination angle vs 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
3 / 2

 
V  1

 4 
=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  a    9  120 2
eff
critical
out
23
23 15
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

t   2K A  ( s  d)Hnˆ  K A   s s  d   s   sd  Is  Is   sd
d is the displacement of a material particle
K A stretch modulus ~ 200N /m, shear elastic modulus
 ~ 106 N /m
 s  d  0 for an inextensible capsule.
T

Compound Capsule (cont.)
• Extra parameter  for shear
elasticity
• Starting from the tanktreading 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
• 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


A
R2
Effective interior fluid viscosity
• The compound vesicle can be
viewed as a membrane
enclosing a homogeneous fluid
with an effective viscosity,
estimated as  a    9  120 2
eff
•Effective interior fluid viscosity
critical
out
23
23 15
Dilute Suspension of c-vesicles

•Effective shear viscosity
5 6 1 X 00  X 20a  10a
5 f (a) 
D'2 
Txy  
,
f
(a)

.
 2

2
D'  E  D'2 
aD'
N1 
6 1 X 00   2X 02
3
160
Ef (a)  0, E 
 D'2 , 
.
4
3
2  6X 20  2X 22