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Electric field induced instabilities at bilayer membranes and fluid-fluid interfaces. Rochish Thaokar Department of Chemical Engineering IIT Bombay, Mumbai (Bombay), India. 25 th May 2012, KITPC, Beijing, China. Outline. Rayleigh Plateau Instability in Fluid Jets - PowerPoint PPT Presentation
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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