Electric field induced instabilities at bilayer membranes
and fluid-fluid interfaces
Rochish Thaokar
Department of Chemical Engineering
IIT Bombay, Mumbai (Bombay), India
25th May 2012, KITPC, Beijing, China
1
Outline

Rayleigh Plateau Instability in Fluid Jets

Brief Introduction to Pearling in Cylindrical vesicles

Brief Introduction to Electrostatics and Electrohydrodynamics

Pearling under Uniform electric fields

Conclusions

Fluid-Fluid Electrohydrodynamics: Planar interfaces and drops
2
Apology

The talk is a little more elaborate version of my short talk

The fluid-fluid part is new, would like to have your inputs on
making connection with bio-physics
3
Rayleigh-Plateau Instability: Basic Physics
Total Energy=Surface tension*area
Long wave perturbations reduce a jet’s area
Instability happens when the wavelength of the
perturbation is larger than the circumference of
the cylinder
r=1+D ei(kz+mθ)+st)
Pin =γ(1-δ D(1-m2-k2)e i(kz+mθ)+st)
Normal mode analysis yields a simple kinematic explanation for the instability.
Stabilizing longintudinal curvature and destabilizing azimuthal curvature
4
Which wavenumber is unstable? Seen in experiments
Pin =γ(1-δ D(1-m2-k2)e i(kz+mθ)+st)
S=Ak(1-k^2)
km
Long wave instability (k<1 or λ>2 π R) are unstable
Rayleigh’s analysis Medium Air: Viscous (km=0) Inviscid (km=0.7)
Tomotika (Both fluids viscous) (km=0.56)
5
Rayleigh Plateau in cylindrical vesicles (Pearling)
In most simple cylindrical vesicular systems, the tension is identically zero.
(Tension due to thermal fluctuations too weak to induce instability).
Cylindrical vesicles do not pearl on their own.
The tension required for pearling is
c 
3 B
2a
2
Bar Ziv and Moses, 1994, PRL, first showed Laser Induced pearling
Transfer of energy of Laser results in a tension in the membrane that
causes pearling (Dielectrophoretic effect)
6
What could be issues in RP in vesicles

Fluid jets can decrease their area (area is not conserved)

In cylindrical vesicles, the membrane area has to be conserved

RP instability leads to reduction of area. A tense vesicle would
have to displace the reduced area

This is unlike planar membrane analysis where area is pulled in

The process of deflection of area can also lead to front velocity
7
A brief about cylindrical vesicles
HB 
B
2
 ( 2 H ) dA  
 dA  ( P
Bending
Area
2
e
For a sphere of radius a Minimise HB:
 Pi )  dV
volume conservation
Pi  Pe 
2
Hbend=8 πκB
a
Pressure independent of bending modulus,
Energy independent of radius
For a cylinder, minimise wrt a and L
Pi  Pe 
2 B
a
3
 
Pi  Pe 
3 B
2a
2
Pi  Pe 
Pe
Pi


a
2
a
B
2a

3
B
a
3
Pe
Pi
Negative pressure contribution by the bending term
When length does not matter (very long cylinders), Pi= Pe and at
equilibrium a=√κ/2σ, external tension
8
Stretched cylindrical vesicles
HB 
B
2
 ( 2 H ) dA  
 dA  ( P
Bending
Area
2
e
 Pi )  dV
-f L
volume conservation
f
f
For a cylinder, minimise wrt a and L, without the volume constraint (?),
infinite reservoir of fluid outside
a=√κ/2σ
f=2 π √2 κσ =2 π κ/a
In synthesis though, the radius is decided by the preparation conditions
When stretched, there might be dynamics associated reduction to
equilibrium radius a = √κ/2σ (viscosity controlled)
9
Salient observations in Bar-Ziv et. al’s work (PRL 1994)

No intrinsic curvature, no initial tension

Far from equilibrium system (Slow dynamics)

Laser 50mW with 0.3 microns radius, generates tension of
1.8 10-4 mN/m. Lipid molecules sucked into the laser
(akin negative dielectrophoresis)

A wavenumber k=2πR/λ=0.8-1.0 of the instability is observed

Significantly different from the fluid-jet analysis
10
Salient observations in Bar-Ziv et. al’s work

The reduced area during RP instability is
absorbed in the laser trap

Leads to a propogating instability from the
laser trap

A front seen to propogate at around 30
microns/s

This velocity increases with laser power,
tension
11
Late stage pearling

The reduced volume in a cylinder in the large L limit is
v=3/21/3 (R/L)1/2
•
Large L leads to small v can have variety of equilibrium
shapes!!
•
Late stage pearling!! Volume conservation leads to,
Rp=1.806 Ro
Rneck= √κ/2σ=470 nm
12
Different techniques for inducing Pearling Instability
in cylindrical vesicles
Optical Tweezers
(Bar-Ziv et al., 1994)
Polymer anchoring
(Tsafrir, 2001)
Magnetic field
(Menager et al., 2002)
Elongational flow
(Kanstler, 2008)
Nanoparticle encapsulation
(Yan Yu and Steve ,2009)
Application of tweezers on membrane
creates surface tension by drawing lipid
molecules into the tweezed area
Spontaneous curvature because of the
amphiphilic polymer backbone induces
tension in the outer leaflet of bilayer
membrane tube
Deformation of magnetoliposome
takes place under applied magnetic
field leading to tension in the cylinder
Stretching of a tubular vesicles with
initial length to dia. ratio L/D0 > 4.2
by an elongational induces shape
transformation from dumbbell to a
transient pearling state
Encapsulation
of
excess
of
nanoparticles within GUVs induces
13
shape transformation
Synthesis of cylindrical vesicles
Spin coating (1kRPM,
10sec) of microscopic glass
slide with DMPC lipid
SS-electrodes (Thickness
0.45mm) at a spacing of
3mm
DMPC Lipid
Conc.
1,2Dimyristoylsn-glycero-3phosphocholin
e
10 mg/ml
(CHCl3:CH3
OH = 2:1)
Spin coating
1000 RPM
for 10 sec
Sucrose
solution
conc.
0.1M
Electrode
thickness
0.45mm
(Stainless
steel)
Electrode
spacing
3mm
Conductivity
(0.1M
Sucrose
solution)
5.7µS/cm
Sucrose
solution
injection
rate
3ml/min
Fixing upper glass slide to
the bottom one
Sealing from four sides to
form a closed chamber
Dry lipid layer hydration by
sucrose solution injection
(3ml/min)
All the experiments conducted at 26 oC above Tg(23 oC)
14
Images of Cylindrical vesicles
Variety of sizes of cylindrical vesicles observed
Vesicles appear as single cylinders or a bunch
They are free at both ends or connected to lipid reservoirs
Myelin and multi-lamellar cylindrical vesicles also observed
15
Electric field Experimental setup
Computer
CCD Camera
Inverted
microscope
Experimental cell
Figure: Electric field setup
Function generator
High frequency
amplifier
16
2.5 mm spacing
DC experiments without amplifier (Voltages around 1.5 V)
AC experiments: 500kHz to 2 MHz (Voltages around 60 V)
Oscilloscope
Some important experimental observations
Pearling
Late Pearls
Some important experimental observations
Flutter
Budding
18
Some important experimental observations
Pearling seen on increasing the field
Seems to start at one end of the cylinder
Late pearls in some cases show bimodal distribution
Simultaneous stretching is observed but is remarkably reversible
Flutter at strong fields.
A fluttered vesicle often pearled on removal of field: Tension is dissipated much slowly
μea/σ
19
Effect of electric field
In most systems, the tension is almost zero: Cylindrical vesicles do not
pearl on their own!!
The tension required for pearling is
c 
3 B
2a
2
How does electric field induce tension in a cylindrical vesicle?
Problem complicated by end-caps. What is the field distribution around end caps?
Axial part
End Caps
No Simple base state on which stability analysis
can be conducted
Normal mode analysis difficult if ends are considered
20
Maxwell’s stress (Origin)
E
+
+
+ +
-
Net Maxwells force due to difference
+
+
in Dielectric constants

1   2
+
=ρ E-1/2 εo E.E grad ε =del.T
Τ=F/Area=ε εo (EE-1/2 I E2)

E
-- ++ +
+
F/Vol=ρ E+P.grad E-1/2 grad εo(ε -1) E.E I
+
Net Maxwells force due to difference
in conductivities

+
Air
- -

+
E cos(ωt)
+
21
What are the axial and end-cap forces?
Consider the vesicle to be a dielectric in a conductor medium Helfrich (1983) in
DC fields.
Eo
Axial part
n . e   e  o ( EE n 
E
2
2
n )   i  o ( EE n 
E
2
2
n )    e o
Eo
2
n
2
Compressive axial electric stress on the walls
What about the caps?
Solve for electrostatics on a spherical vesicle, and consider one half of the same
22
End Caps: Electrostatics for a spherical vesicle
  m ,e  0
2
•Assume spherical vesicle as a dielectric drop in a
conducting liquid (Helfrich 1983)
2 a=6 microns
m
•Continuity of potential at interface of membrane inner and outer medium
•Normal field zero at the conductor-dielectric interface
n.E e  0 ( r  a )
e  m (r  a )
•Compressive stresses
9 E o  a  e o
2
F
E
DC

2
16
23
e
Can axial and end-cap compressive forces generate tension?
How is tension generated by compressive electric stresses?
Eo
Axial part
n . e   e  o ( EE n 
E
2
2
n )   i  o ( EE n 
E
2
2
n )    e o
Eo
2
n
2
Compressive axial electric stress
F
c 
9 E o  a  e o
2
E
DC

2
16
3 B
2a
2
24
Effect of membrane thickness on electric field
Electric field calculations assuming the vesicle to be a dielectric drop
incorrect, although one can still predict generation of tension
The membrane is just a thin layer of dielectric. The inner core is a
conductor and although the field inside is zero, the charges at the inner
core would be substantial
A detailed model to describe the electrostatics should be suggested. The
net electric traction is
f E  n . e ( r  a  d )  n . m ( r  a  d )  n . m ( r  a )  n . i ( r  a )
d=5nm
2 a=6 microns
i
e
m
25
Dielectrics, Leaky dielectrics and Conductors

Perfect Dielectrics
+
1   2
-- ++ +

Layered Dielectrics (PD-PD): Net
bound charge at the interface
E
- -
+
+ +
+-
+-
+
+
Leaky Dielectrics
+

+ +
-- - + + ++

+
- ++ +
E
Layered Dielectrics (LD-LD): Accumulation
of charges at the interface. The charge
relaxation time is given by tc=ε/σ
 1 E1   2 E 2
Conductors
Steady state assumption in most cases
Charges accumulate at the interface
Assumption that charge relaxation time
Equi-potential assumption
tR=ε/σ(t is faster than other time-scales
(Low frequency)  E   E
1 1
2 2
Current continuity condition
Is realised when the conductivity is very
large
High frequency: Dielectrophoretic behaviour
26
 1 E1   2 E 2
Electrostatics equations
  i ,m ,e  0
2
•No free charge, conductors, perfect or leaky dielectrics
2 a=6 microns
•The boundary conditions are important
i m
•Continuity of potential at interface of membrane inner and outer medium
 o ( e E e   m E m )  q e ( r  a  d )
 m Em   eEe 
 o ( m E m   i E i )  q i ( r  a )
 i Ei   m Em 
qe
t
qi
t
(r  a  d )
(σi=σe=5 10-5 S/m σm=0
εe=εi =80 εm =2)
(r  a )
Typically, we assume σm=0
tMW=εe εo /σ

e
 e o
(qe   m E m ) 
 ( q e   m E m )  t MW
qe
t
qe
t
Conductor Behavior ω>t-1MW
Dielectric behavior ω<t-1MW
Water (5 10-5 S/m) , t-1MW =70 kHz
27
e
Model 2
f E  n . e ( r  a  d )  n . m ( r  a  d )  n . m ( r  a )  n . i ( r  a )
DC Case ω<<tmw-1
High frequency ω>tmw-1
Model 1
9 a E o  m  o
3
F
E
DC

2
8d

Tensile Axial stress
2 2 a   F E 
Ec obtained by requiring
F
E
AC

 /2
0
adE
2
o
 ( e   m )  o
2
2 m
d  f E a sin 
c 
2
3 B
2a
2
28
Critical Electric field for pearling
Vesicles turn in the direction of field
The frequency dependent tension, when
exceeds the critical tension, onset of
Pearling is observed
Both AC and DC experiments are reported
Low DC voltage and fields to prevent
electroporation (<1kV/cm, DC) and
electrolysis
29
Governing equations and Boundary conditions
Variables
Scalings
X
a
T
μea3/κB
For Hydrodynamics, membrane acts as an interface
V
κB /μea2
Electrostatics solved for internal and external fluids
and the membrane phase
P,τ
κB / a3
Φ,E
√κB /aεo, √κB /a3εo
ω
κB/μea3
Linear stability analysis is conducted
Stokes equations for Hydrodynamics
30
Stability Analysis
Put normal mode perturbations for all the variables
Get dispersion relation and determine the value of s
m=0 is the symmetric mode
m=1,2 are the non-axisymmetric modes
Low wave number instability is often seen
Floquet analysis is conducted for time-periodic
voltages
31
Rayleigh Plateau instability in liquid-liquid jets
For εi > εe, the Maxwell’s stress is out of
phase with the displacement D by Π/2,
stabilizing action of the electric field.
At B the field is obstructed so +ve free charges
-ve perturbation charges at position A.
Axial perturbation electric field e is in phase with
the interface displacement D.
E-Field stabilizes RP instability in liquid-liquid jets
Base state stress at interface is
-(εe -εi)/2 E2 and is compressive
The normal perturbation stress is
-(εe -εi)/2 e E and is directed
inwards at the crests and outwards
at the trough, leading to
stabilization.
e.g electrospinning
32
Rayleigh Plateau instability in cylindrical vesicles
Governing equations and Boundary conditions
Normal stress BC has a tension term
Intrinsic tension due to electric field (Maxwell’s
stresses, in the base state)
Perturbed stress, incompressibility condition leads
to a tension (a Lagrange parameter)
Compare with fluid-fluid model (No tension,
tangential stress continuity) or immobile interface
(zero tangential velocity)
33
Effect of electric field on wavelength
Dual Role of Electric field: It generates tension needed to induce the instability (σ>
σc). But also suppresses RP instability in jets
Balance of electric field induced tension and stabilizing effect of E yeilds an E
independent plateau km
This results in increase in km with E and plateaus to a value less than 0.56 (fluids)
DC fields: Two possible cases (Ee =Eo , Em =Ei=0) and (Ee =Em =Ei=Eo)
The plateau value of km decreases with an increase in the frequency
34
Effect of electric field on wavelength: Comparisons with
Experiments
DC Experiments
AC Experiments
Laser Tweezing Data
(BarZiv and Moses, PRL 1994)
Issues:
Issues:
Significant
scatter
in the than
km theory much
smaller
experimental
Experiments data
The
fields
DC
Either
MSCrequired
effect orfor
some
instability
much smaller than AC
Physics missing?
Weak dependence of electric
field is seen unlike fluid jets
35
Conclusions
Late stage Pearl size
(When unimodal distribution)
Rp=1.806 Ro
Completely reversible Pearling instability observed and explained
Dual behavior of Electric field: Induces tension as well as stabilizes the
instability
Experimental km values significantly higher than the theory
Analytical theory does not predict flutter if membrane is non-conducting
36
Research in my group

Electrohydrodynamics in fluid-fluid systems

Recently started working in the area of “Effect of fields on
bilayers and vesicles (Spherical and cylindrical)”

Apologies for the fluid-fluid systems. Would be keen to know if
there are similar problems of biological importance
37
Electrohydrodynamic instabilities at fluid-fluid
interfaces in low conductivity, low frequency limit
Possible Mechanism of Electroformation
System
GOAL
Shimanouchi, Langmuir, 25(9),2009
Y=βho
external
fluid (e)
internal
fluid (i)
Bilayer (m)
Late stage swelling under osmosis and
maxwell stresses
Y=0
Y=- ho
Fluid 1
Fluid 2
ho
λ
WE ARE HERE
39
Introduction
Soft

Pattern
Lithography
replication is an important tool in many industries.
 More challenging for lower length scales.
 Methods
Photolithography
Advantages
Sizes accessible: 50nm
Electronbeam
lithography
Disadvantages: Complex
chemical treatments, needs a
Direct
positive
replicaclean
of the
mask.
mask,
needs
room.
Sizes
accessible:
<10nm
Soft
Lithography
Inexpensive
Complex
and
expensive
Fig:Disadvantages:
a) PET preform mould
b) Pressure
driven
microfluidic
instrumentation
bioreactor
40
No sophisticated devices/ expertise required
Schaffer
al, 2000, Morariu et al, 2002, Deshpande
et al, 2004
Website: et
blowmolding-machine.en.made-in-china.com
and loac-hsg-imt.de
Soft Lithography: The story so far
1999
Chou et al
2001
Schaffer et al, Lin et al
2002-04
Morariu et al, Leach et
al
2006-08
Dickey et al, Voicu et
al
2009-10
41
Heier et
al, Sharma et
al
• LISA (no electric fields), Pattern
depends on mol weight of the
polymer
• Electric fields , hexagonal pattern,
rosette structure, linear mask
replicated, bilayers
• Patterned electrodes, polymerpolymer-air trilayers
• Novel 3d cages, high aspect
ratio pillars, time evolution of
instability
• Spatially Modulated fields
• Viscous to elastic behaviour
Experiments (Protocol)
Scotch tape is stuck on the
lower slide as spacer
A drop of Poly dimethyl siloxane
30000cSTk fluid is placed
The fluid is spin coated at 3k rpm for 3 mins to get a ~37 micron film
The second slide is placed against the
first and it is connected to AC supply
Undulations form and grow at the fluid
interface to form columns which touch the
top slide. The mean spacing is then
measured
a
Pattern formation: Simple theory
2
 e   o (V / H )   s   H / 
a
 
b1 / 2 1 / 2
a
c
E

H
3/ 2
V
2
1
b
c500 Hz 1 kV/cm
d
c
500 Hz 0.5 kV/cm
DC 0.5 kV/cm
d
DC 0.5 1 kV/cm
Ω
43
Mechanism of the instability
44
1   2
ε1
E
+
- - + + +
+
+
+
+
+
+
+
+
+
ε2
1   2
ε1
High pressure in fluid 1
E
+
- - + + +
+
+
+
+
+
+
+
+
+
ε2
Net negative bound
charge

Base State
1   2
 m1
ε1
y
E
ε2
 m 2
y


1   2
ε1
~
~
 2
E
 1
Point of equal potential
~
 1
~
 2

ε2

1   2
+
ε1
+
- - - - + + + + +
E
Net negative bound
perturbation
charge
ε2
- + +
+
+

Net positive
bound
perturbation
charges

1   2

ε1
E
ε2
Leads to attraction


Mechanism of the Instability (leaky dielectrics)
1   2
+
12
+
-
-
-
+
+
-
-
+
+
+
-
+
-
-
ε1
+
-
+
-
++
-
+
+
+
-
+
-+
ε2
1   2
 m1
12
σ1
y
Different conductivities of the two
fluids results in this base state
For the case of
equal ε
 m 2
y
σ2
1   2
σ1
+
12
-
-
+
+
-q
+
-
-
+
+-
+
σ2
Leads to a base state charge given by
q=ε1E1-ε2E2
+
+
+
-
+
-+

1   2
σ1
12
~
~
 2
E
 1
Point of equal potential
σ2


1   2
σ1
12
-
+ +- -
- - +-
-
+
E
Net negative
perturbation
charge
σ2


1   2

12
ε1
E
ε2
Leads to attraction


Model
Y=βho
Fluid 1
ho
Y=0
Y=- ho
Fluid 2
λ
57
Model
Boundary Conditions
Governing Equations
Continuity of velocities u 1  u 2
Equation of continuity
Balance of normal and tangential stresses
 .v  0
n.( 1   2 ). n  n.(
Momentum Balance
v
t
e
  2 ). n      gh ( x , t )
e
1
n .( 1   2 ). t  n .(
~0
Re(
 v . v )   p    v
2
Gauss’s law for electrostatic
potential
  0
2
No free charges and
electric body force in the
bulk
v1  v 2
Kinematic condition
e
1

h
t
e
2
). t  0
 V . s h  v 2
Continuity of potential in normal and
tangential direction
1   2
  0 1
 1
y
  0 2
 2
y
q
Charge conservation equation
q
t
 u . s q  qn .( n . ) u   1
 1
y
2
58
 2
y
Electric stress
Scaling
Parameter
General model
Thin Film approx.
Length
h0
h0
 0 0
Velocity
2
 2 h0
 0 0 2
Pressure
h0
Time
 2 h0
 0 0
Interfacial
Charge
Conductivity
2
2
or
for AC

 2 h0
2
 0 0 
2
h0
h0
 2 h0
2
2
1
or
2
 0 0

2
2
 h0
2
  h0
 
  2
 0 0
for AC




1
2
A small parameter
  0 0  
 2 h0
s 
3
2
2
lateral length scale
 0 0 2
 0 0
  0 0 

2
2
h0
e




Stress due to surface
tension
 0 0  2
 2 h0
 0 0 
 2 h0
2
1
h0
 0
 0
 h
 0
2
 
h0
  0 0

  h
0

2




1
2
Tools used
 LSA for AC and DC systems
 Nonlinear simulations using thin film approximation
 Comparison with experiments
60
Effect of conductivity in DC
experiments
Linear Stability Analysis
Perturbations are expressed as
ikx  st
~
m  m0  me
k- wavenumber of the instability
s- growth rate of the instability
Instability is characterized by a
fastest growing mode (kmax)
kmax grows
=
k1
+
+
k2
k3
The inverse of kmax gives the wavelength
of the pattern obtained experimentally.
62
Background
 Two time scales of interest
 Time for growth of instability (τs=1/smax)
 Time for charge migration/relaxation (τc =ε/σ)
 If τc >> τs ---- PD-PD
 If τc << τs ---- PD-Conductor
 If τc ~ τs ---- PD-LD
 Linear Stability analysis assumes a well defined base state
(A conductor)
 What do you compare experiments? SIMULATIONS
Background
Case : Both
Charge
Instability
happen
relaxes
grows
simulatneously
faster
beforethan
charge
the instability
migrates grows
A leaky-leaky interface c cc
s s  ?

In this case, interfacial charge doesn’t reach
its steady
state value
and the
A perfect
dielectric
leaky
dielectric
behaviour
assumption of the linear theory becomes invalid. Non-linear simulations are
requiredbehaviour
64
Results for DC fields
simulation
The actual wavelength seen in experiments decreases considerably
with decrease in the conductivity as predicted by the non-linear
65
analysis.
AC Conditions
Important questions?
 Effect of frequency of AC fields on wavelength of patterns: Theory
and Experiments? (Not reported yet)
 Frequency as a tool (identical to rheometry) to probe different time
scales
 Can simulations reveal more about instabilities under AC fields?
 τc << τs and τω< τs (Fully charged surface)
 τω<< τc and τω >> τc
 For τc << τs and τω> τs
 For τc =τs and τω> τs and τω< τs
67
Experiments
 c   s and     s
 c   s and     s
ND conductivities are 20, 1 and 0.05
 c   s and     s
Experiments (Scale frequency with conductivity)
69
Observations
 No fitting parameters
 Reasons for disagreement :
 Errors in conductivity measurement
 High polydispersity in the pattern, especially at high frequencies (low
growth rates)
 Beta, the ratio of heights of air and liquid is not the same in all the
experiments.
70
Conclusions
 The non-dimensional conductivity does matter in electrohydrodynamic
instabilities for both AC and DC fields
 Simulations might be necessary to make accurate predictions
 The experimental observations support the above two
71
Acknowledgements
 DST
 Priya Gambhire, PhD student
72
Drop deformation and translation in non-uniform
fields
Why nonuniform elecric field

Drops deform and break or coalesce under electric fields

What is the best electrode configuration?

Dielectrophoresis is movement of a particle in non-uniform fields

Dielectrophoresis has several applications biophysics, bioengineering,
multiphase separation
To investigate drop deformation, breakup and motion in the
simplest non-uniform electric field
74
Why nonuniform elecric field
Drop Transport by Dielectrophoresis
Breakup of water drop in castor oil
Nature, 426(2003) 515
Separation of Tobacco Mosaic Virus(+ve
dielectrophoresis) and Herpes Simplex
Virus(-ve dielectrophoresis)
(Kua C. H. et al 2004)
Deformation, dielectrophoresis and
oscillation of water drop in castor oil
75
Drop breakup under P1 Electric field
Large CaE LD systems showing breakup with prolate deformation
Q=10, R=0.1,CaE=0.342
Ca E 
R 


Large CaE LD systems showing breakup with oblate deformation
(Q=10, R=10,CaE=0.342)
 0 E
2

a

Q 


Summary of Leaky dielectric results
R Q<1
No deformation for
quadrupole field
No deformation for
uniform field
No
electrohydrodynamic flow
λ = 0.01
R Q>1
λ =1
λ = 100
···
—
---
Salient features:
Boundaries for prolate and oblate changed
Possibly the drop shapes are also more complex
77
Acknowledgements
Shivraj Deshmukh
Sameer Mhatre
Thank You
78
Planar Membranes
HB 
B
2
 (2 H )
2
dA  
 dA  ( P
e
h
 Pi )  dV
Shape equation
•
For the planar case, the height-height correlation of a fluctuating membra
assumes no area conservation!! Membrane can be drawn from the sides.
•
One can still enforce local lipid conservation.
•
In 1-D it means the tangential velocity is zero
•
In linearised theory, the tension is O(ε)
•
•
The tension γ is always externally imposed
( hq hq * ) 
kT
A ( q   q 4 )
2
It is not the tension (Lagrange Multiplier) to conserve area
79
Planar Membranes: Questions
•
In an ideal planar bilayer, there is no concept
h
of reduced volume, so area can be drawn from
edges, unless pinned which would generate tension by area conservation
What is the energy required for pulling the excess energy from the edges
Can one conduct studies with pinned flat bilayer as a reference state, defining
an excess area and doing a systematic analysis. The excess area will popup or down the base state of the membrane. Ve=L/Lpin
Would you get
or
and when
80