Rochish Thaokar

Department of Chemical EngineeringIIT Bombay, Mumbai (Bombay), India

25th May 2012, KITPC, Beijing, China

Electric field induced instabilities at bilayer membranes and fluid-fluid interfaces

1

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

Outline

2

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

Apology

3

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

Rayleigh-Plateau Instability: Basic Physics

4

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

Which wavenumber is unstable? Seen in experiments

5

Pin =γ(1-δ D(1-m2-k2)e i(kz+mθ)+st)

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)

S=Ak(1-k^2)

km

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

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)

22

3

aB

c

6

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

What could be issues in RP in vesicles

7

8

Bending Area volume conservation

dVPPdAdAHH ieB

B )()2(2

2

A brief about cylindrical vesicles

For a sphere of radius a Minimise HB: a

PP ei

2

Pressure independent of bending modulus, Energy independent of radius

For a cylinder, minimise wrt a and L

3

2

aPP Bei

22

3

aB

32aaPP Bei

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

PePi

PePi

Hbend=8 πκB

3

2

aaPP Bei

9

Bending Area volume conservation

dVPPdAdAHH ieB

B )()2(2

2

Stretched cylindrical vesicles

For a cylinder, minimise wrt a and L, without the volume constraint (?), infinite reservoir of fluid outside

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)

-f L

a=√κ/2σ f=2 π √2 κσ =2 π κ/a

ff

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

Salient observations in Bar-Ziv et. al’s work (PRL 1994)

10

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

Salient observations in Bar-Ziv et. al’s work

11

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

Late stage pearling

12

Different techniques for inducing Pearling Instability

in cylindrical vesiclesOptical Tweezers

Polymer anchoring

Magnetic field

Elongational flow

Nanoparticle encapsulation

(Bar-Ziv et al., 1994)

(Tsafrir, 2001)

(Menager et al., 2002)

(Kanstler, 2008)

(Yan Yu and Steve ,2009)

Application of tweezers on membrane creates surface tension by drawing lipid molecules into the tweezed area

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

Deformation of magnetoliposome takes place under applied magnetic field leading to tension in the cylinder

Spontaneous curvature because of the amphiphilic polymer backbone induces tension in the outer leaflet of bilayer membrane tube

Encapsulation of excess of nanoparticles within GUVs induces shape transformation 13

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

Sealing from four sides to form a closed chamber

Dry lipid layer hydration by sucrose solution injection (3ml/min)

Fixing upper glass slide to the bottom one

14All the experiments conducted at 26 oC above Tg(23 oC)

DMPC Lipid Conc.1,2-

Dimyristoyl-sn-glycero-3-phosphochol

ine

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

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

Oscilloscope

High frequencyamplifierFunction generator

Computer CCD CameraInverted

microscope

Experimental cell

Figure: Electric field setup

16

2.5 mm spacing

DC experiments without amplifier (Voltages around 1.5 V)AC experiments: 500kHz to 2 MHz (Voltages around 60 V)

Some important experimental observations

Pearling

Late Pearls

Some important experimental observations

Flutter

18Budding

Some important experimental observations

19

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

Effect of electric fieldIn most systems, the tension is almost zero: Cylindrical vesicles do not pearl on their own!!

The tension required for pearling is 22

3

aB

c

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

20

No Simple base state on which stability analysiscan be conducted

Normal mode analysis difficult if ends are considered

Maxwell’s stress (Origin)

+E + +

- -+

++

F/Vol=ρ E+P.grad E-1/2 grad εo(ε -1) E.E I

=ρ E-1/2 εo E.E grad ε =del.T Τ=F/Area=ε εo (EE-1/2 I E2)

E

Net Maxwells force due to difference

in Dielectric constants

21

+-

+-

+-

+-

+-

+-

E cos(ωt)

+

- -

+ +

-Air

Net Maxwells force due to difference

in conductivities

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.

Axial part

Compressive axial electric stress on the walls

Eo

nE

nE

EEnE

EEn ooenoinoee 2

)2

()2

(.222

22

What about the caps?

Solve for electrostatics on a spherical vesicle, and consider one half of the same

23

0,2 em

End Caps: Electrostatics for a spherical vesicle

•Assume spherical vesicle as a dielectric drop in a conducting liquid (Helfrich 1983)

•Continuity of potential at interface of membrane inner and outer medium•Normal field zero at the conductor-dielectric interface

)(

)(0.

ar

arEn

me

e

2 a=6 microns

em

•Compressive stresses

16

9 22oeoE

DC

aEF

How is tension generated by compressive electric stresses?

Axial part

Compressive axial electric stress

Eo

nE

nE

EEnE

EEn ooenoinoee 2

)2

()2

(.222

24

16

9 22oeoE

DC

aEF

22

3

aB

c

Can axial and end-cap compressive forces generate tension?

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

25

)()()()( .... ariarmdarmdareE nnnnf

d=5nm

2 a=6 microns

ei m

Dielectrics, Leaky dielectrics and Conductors

Perfect Dielectrics Leaky Dielectrics

26

+- E

+-

+-

+-

Layered Dielectrics (PD-PD): Net bound charge at the interface

+-+-

+-

+-+-

21 +-

+-

+-

+- E

+- +

-+-

Layered Dielectrics (LD-LD): Accumulation of charges at the interface. The charge relaxation time is given by tc=ε/σ

+-

+-

+-

+-

+- +- +

-+-

++-

-

-

-+

+

-+- -++ +

2211 EE

Steady state assumption in most cases

Assumption that charge relaxation time

tR=ε/σ(t is faster than other time-scales (Low frequency)

Current continuity condition

High frequency: Dielectrophoretic behaviour

2211 EE

Conductors

Charges accumulate at the interface

Equi-potential assumption

Is realised when the conductivity is very large

2211 EE

27

0,,2 emi

Electrostatics equations

•No free charge, conductors, perfect or leaky dielectrics

•The boundary conditions are important

•Continuity of potential at interface of membrane inner and outer medium

)()(

)()(

arqEE

darqEE

iiimmo

emmeeo

)(

)(

art

qEE

dart

qEE

immii

eeemm

2 a=6 microns

ei m

t

qtEq

t

qEq

eMWmme

emme

oe

e

)(

)(

tMW=εe εo /σ

Conductor Behavior ω>t-1MW

Dielectric behavior ω<t-1MW

Water (5 10-5 S/m) , t-1MW =70 kHz

(σi=σe=5 10-5 S/m σm=0 εe=εi =80 εm =2)

Typically, we assume σm=0

Model 2

)()()()( .... ariarmdarmdareE nnnnf

sin2 22/

0afdFa EE

d

EaF omoE

DC 8

9 23

m

omeoEAC

adEF

2

)( 22

28

Tensile Axial stress

DC Case ω<<tmw-1 High frequency ω>tmw

-1

22

3

aB

c

Ec obtained by requiring

Model 1

2

Critical Electric field for pearling

29

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

Governing equations and Boundary conditions

30

Variables Scalings

X a

T μea3/κB

V κB /μea2

P,τ κB / a3

Φ,E √κB /aεo, √κB /a3εo

ω κB/μea3

Linear stability analysis is conducted

Stokes equations for Hydrodynamics

For Hydrodynamics, membrane acts as an interface

Electrostatics solved for internal and external fluids and the membrane phase

Stability Analysis

31

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

32

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

33

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)

34

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

35

Effect of electric field on wavelength: Comparisons with Experiments

Issues:

Significant scatter in the experimental data

The fields required for DC instability much smaller than AC

Weak dependence of electric field is seen unlike fluid jets

Issues:

km theory much smaller thanExperiments

Either MSC effect or some Physics missing?

DC Experiments

AC Experiments

Laser Tweezing Data (BarZiv and Moses, PRL 1994)

36

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

Conclusions

Late stage Pearl size(When unimodal distribution)

Rp=1.806 Ro

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

Research in my group

37

Electrohydrodynamic instabilities at fluid-fluid interfaces in low conductivity, low frequency

limit

System

39

Y=βho

Y=- ho

Y=0Fluid 1

Fluid 2

ho

λBilayer (m)

external fluid (e)

internal fluid (i)

Late stage swelling under osmosis and maxwell stresses

Shimanouchi, Langmuir, 25(9),2009

Possible Mechanism of Electroformation

GOAL

WE ARE HERE

Introduction

40

Pattern replication is an important tool in many industries. More challenging for lower length scales.

Methods

PhotolithographySizes accessible: 50nmDisadvantages: Complex chemical treatments, needs a mask, needs clean room.

Electronbeam lithographySizes accessible: <10nmDisadvantages: Complex and expensive instrumentation

Soft Lithography

Soft Lithography

Schaffer et al, 2000, Morariu et al, 2002, Deshpande et al, 2004

Advantages

No sophisticated devices/ expertise required

Direct positive replica of the mask.

InexpensiveFig: a) PET preform mould b) Pressure driven microfluidic bioreactor

Website: blowmolding-machine.en.made-in-china.com and loac-hsg-imt.de

Soft Lithography: The story so far

41

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

Experiments (Protocol)

Pattern formation: Simple theorya

a b

c

43Ω

cdE

b

c d

a

500 Hz 1 kV/cm DC 0.5 kV/cm

500 Hz 0.5 kV/cm DC 0.5 1 kV/cm

22 /)/( HHV soe

12/32/12/1 VH

44

Mechanism of the instability

ε1

ε2

21

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

E

ε1

ε2

21

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

E

Net negative bound charge

High pressure in fluid 1

ε1

ε2

E

ym

1

ym

2

21

Base State

ε1

ε2

1

~2

~E

21

Point of equal potential

1

~

2

~

ε1

ε2

E

21

+-

+-

+-

+-

+-

+-

+-

+- +

-+-

+-

+-

+-

Net positive bound perturbation charges

Net negative bound perturbation charge

ε1

ε2

E

21

Leads to attraction

ε1

ε2

+-

+-+-

+-

+-+-

21

21

+-

+-

-+

-

+

-+

-

+-

+-

+

-+

- +-+

-+

-+

-+

Mechanism of the Instability (leaky dielectrics)

σ1

21

21

σ2

For the case of equal ε

Different conductivities of the two fluids results in this base state

ym

1

ym

2

σ1

21

21

σ2

-+

-

+

-

+

-

+-

+-

+

-+

- +-+

-+-

+

-+

-q

Leads to a base state charge given by q=ε1E1-ε2E2

1

~2

~E

Point of equal potential

σ1

σ2

21

21

E

Net negative perturbation charge

σ1

σ2

- ++ + +- - -

--- -

-

21

21

ε1

ε2

E

Leads to attraction

21

21

Model

57

Y=βho

Y=- ho

Y=0

Fluid 1

Fluid 2

ho

λ

Model

58

21 vv 21 uu

21 qyy

220

110

Continuity of velocities

Continuity of potential in normal and tangential direction

Balance of normal and tangential stresses

),()..()..( 2121 txghnnnn ee

0)..()..( 2121 tntn ee

yyunqnqu

t

qs

2

21

1)..(.

2. vhVt

hs

Kinematic condition

Charge conservation equation

Equation of continuity

0. v

02

vpvvt

v 2).Re( ~0

Gauss’s law for electrostatic potential

Momentum Balance

Governing EquationsBoundary Conditions

No free charges and electric body force in the bulk

ScalingParameter General model Thin Film approx.

Length

Velocity

Pressure

Time

Interfacial Charge

Conductivity

20 0

20h

22 0

20 0

1 or for AC

h

0 0

0h

2

0 0

22 0h

0h

20 0

2 0h

0h

20 0

20h

22 0

20 0

1 or for AC

h

0 0

0h

2

0 0

22 0h

0h

20 0

2 0h

2

0 0

2 0h

2

2

2

2

0

00

he

Electric stress

20

h

s

Stress due to surface tension

21

200

30

h

lateral length scale

21

0

2000

hh

A small parameter

Tools used

60

LSA for AC and DC systems

Nonlinear simulations using thin film approximation

Comparison with experiments

Effect of conductivity in DC experiments

Instability is characterized by a fastest growing mode (kmax)

stikxemmm ~0

Perturbations are expressed as k- wavenumber of the

instability

s- growth rate of the

instability

The inverse of kmax gives the wavelength of the pattern obtained experimentally.

= + +

k1 k2 k3kmax

grows

Linear Stability Analysis

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

sc sc ? sc

In this case, interfacial charge doesn’t reach its steady state value and the assumption of the linear theory becomes invalid. Non-linear

simulations are required

A leaky-leaky interface

Case : Charge relaxes faster than the instability growsCase : Instability grows before charge migratesCase : Both happen simulatneously

A leaky dielectric behaviourA perfect dielectric behaviour

64

Results for DC fields

The actual wavelength seen in experiments decreases considerably with decrease in the conductivity as predicted by the non-linear analysis. 65

simulation

AC Conditions

67

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

Important questions?

Experiments

ssc and

ssc and ssc and

ND conductivities are 20, 1 and 0.05

69

Experiments (Scale frequency with conductivity)

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

Separation of Tobacco Mosaic Virus(+vedielectrophoresis) and Herpes Simplex Virus(-ve dielectrophoresis) (Kua C. H. et al 2004)

Drop Transport by Dielectrophoresis Nature, 426(2003) 515

Breakup of water drop in castor oil

Deformation, dielectrophoresis and oscillation of water drop in castor oil

75

Large CaE LD systems showing breakup with prolate deformation

Drop breakup under P1 Electric field

(Q=10, R=10,CaE=0.342)

Q=10, R=0.1,CaE=0.342

Large CaE LD systems showing breakup with oblate deformation

aE

CaE

2

0

QR

No deformation for quadrupole field

No deformation for uniform field

No electrohydrodynamic flow

λ = 0.01 · · ·λ = 1 —λ = 100 - - -

Summary of Leaky dielectric results

R Q>1

R Q<1

Salient features:

Boundaries for prolate and oblate changed

Possibly the drop shapes are also more complex77

Acknowledgements

Shivraj DeshmukhSameer Mhatre

Thank You

78

Planar Membranes

79

dVPPdAdAHH ieB

B )()2(2

2

Shape equation

• For the planar case, the height-height correlation of a fluctuating membrane 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

• It is not the tension (Lagrange Multiplier) to conserve area

h

)()(

42* qqA

kThh qq

Planar Membranes: Questions

80

• In an ideal planar bilayer, there is no concept

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 pop-up or down the base

state of the membrane. Ve=L/Lpin

Would you get or and when

h