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Rochish Thaokar Department of Chemical Engineering IIT Bombay, Mumbai (Bombay), India

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

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Rochish Thaokar Department of Chemical Engineering IIT Bombay, Mumbai (Bombay), India

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  1. Electric field induced instabilities at bilayer membranes and fluid-fluid interfaces Rochish ThaokarDepartment of Chemical EngineeringIIT Bombay, Mumbai (Bombay), India 25th May 2012, KITPC, Beijing, China

  2. 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

  3. 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

  4. 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

  5. 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)

  6. 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)

  7. 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

  8. A brief about cylindrical vesicles Bending Area volume conservation For a sphere of radius a Minimise HB: Hbend=8 πκB Pe Pi Pressure independent of bending modulus, Energy independent of radius For a cylinder, minimise wrt a and L 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

  9. Stretched cylindrical vesicles -f L Bending Area volume conservation f f a=√κ/2σf=2 π √2 κσ =2 πκ/a 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)

  10. 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

  11. 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

  12. 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

  13. Different techniques for inducing Pearling Instability in cylindrical vesicles Optical Tweezers Application of tweezers on membrane creates surface tension by drawing lipid molecules into the tweezed area (Bar-Ziv et al., 1994) Spontaneous curvature because of the amphiphilic polymer backbone induces tension in the outer leaflet of bilayer membrane tube Polymer anchoring (Tsafrir, 2001) Magnetic field Deformation of magnetoliposome takes place under applied magnetic field leading to tension in the cylinder (Menager et al., 2002) 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 Elongational flow (Kanstler, 2008) Nanoparticle encapsulation Encapsulation of excess of nanoparticles within GUVs induces shape transformation (Yan Yu and Steve ,2009)

  14. 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 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)

  15. 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

  16. Electric field Experimental setup Inverted microscope Computer CCD Camera Experimental cell Figure: Electric field setup High frequency amplifier Function generator 2.5 mm spacing DC experiments without amplifier (Voltages around 1.5 V) AC experiments: 500kHz to 2 MHz (Voltages around 60 V) Oscilloscope

  17. Some important experimental observations Pearling Late Pearls

  18. Some important experimental observations Flutter Budding

  19. 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

  20. 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 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

  21. - - - - - - + + + + + + - - - + + + Maxwell’s stress (Origin) Net Maxwells force due to difference in Dielectric constants + + + E + + - - E + F/Vol=ρE+P.grad E-1/2 grad εo(ε -1)E.E I =ρE-1/2 εoE.E grad ε =del.T Net Maxwells force due to difference in conductivities Τ=F/Area=εεo (EE-1/2 I E2) Air E cos(ωt)

  22. 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 What about the caps? Solve for electrostatics on a spherical vesicle, and consider one half of the same

  23. 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 2 a=6 microns e m • Compressive stresses

  24. Can axial and end-cap compressive forces generate tension? How is tension generated by compressive electric stresses? Eo Axial part Compressive axial electric stress

  25. 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 d=5nm 2 a=6 microns e i m

  26. Dielectrics, Leaky dielectrics and Conductors + + + + + + + + + + + + - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + E E Layered Dielectrics (PD-PD): Net bound charge at the interface Layered Dielectrics (LD-LD): Accumulation of charges at the interface. The charge relaxation time is given by tc=ε/σ - - - - - - - + + + + + + + + Perfect Dielectrics Leaky Dielectrics 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 Conductors Charges accumulate at the interface Equi-potential assumption Is realised when the conductivity is very large

  27. 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 (σi=σe=5 10-5 S/m σm=0 εe=εi =80 εm =2) Typically, we assume σm=0 tMW=εeεo/σ Conductor Behavior ω>t-1MW Dielectric behavior ω<t-1MW Water (5 10-5 S/m) , t-1MW =70 kHz 2 a=6 microns e i m

  28. Model 2 DC Case ω<<tmw-1 High frequency ω>tmw-1 Model 1 Tensile Axial stress 2 Ec obtained by requiring

  29. 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

  30. Governing equations and Boundary conditions 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

  31. 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

  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 DC Experiments Laser Tweezing Data (BarZiv and Moses, PRL 1994) 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 than Experiments Either MSC effect or some Physics missing? AC Experiments

  36. 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

  37. 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

  38. Electrohydrodynamic instabilities at fluid-fluid interfaces in low conductivity, low frequency limit

  39. Possible Mechanism of Electroformation System GOAL external fluid (e) Bilayer (m) internal fluid (i) Shimanouchi, Langmuir, 25(9),2009 Y=βho Fluid 1 ho Y=0 λ Fluid 2 Y=- ho WE ARE HERE Late stage swelling under osmosis and maxwell stresses

  40. Introduction • Pattern replication is an important tool in many industries. • More challenging for lower length scales. • Soft Lithography • Methods • Photolithography • Advantages • No sophisticated devices/ expertise required • Direct positive replica of the mask. • Inexpensive • Sizes accessible: 50nm • Disadvantages: Complex chemical treatments, needs a mask, needs clean room. • Electronbeam lithography • Sizes accessible: <10nm • Disadvantages: Complex and expensive instrumentation • Soft Lithography Fig: a) PET preform mould b) Pressure driven microfluidic bioreactor Schaffer et al, 2000, Morariu et al, 2002, Deshpande et al, 2004 Website: blowmolding-machine.en.made-in-china.com and loac-hsg-imt.de

  41. Soft Lithography: The story so far

  42. Experiments (Protocol) A drop of Poly dimethyl siloxane 30000cSTk fluid is placed Scotch tape is stuck on the lower slide as spacer The fluid is spin coated at 3k rpm for 3 mins to get a ~37 micron film Undulations form and grow at the fluid interface to form columns which touch the top slide. The mean spacing is then measured The second slide is placed against the first and it is connected to AC supply

  43. Pattern formation: Simple theory a a a b b c 500 Hz 1 kV/cm DC 0.5 kV/cm E c c d d 500 Hz 0.5 kV/cm DC 0.5 1 kV/cm Ω

  44. Mechanism of the instability

  45. ε1 E - - - - - - - - - - - - - + + + + + + + + + + + + + ε2

  46. ε1 High pressure in fluid 1 E - - - - - - - - - - - - - + + + + + + + + + + + + + ε2 Net negative bound charge

  47. Base State ε1 E ε2

  48. ε1 E Point of equal potential ε2

  49. ε1 E Net negative bound perturbation charge - - - - - - - - - - - - - + + + + + + + + + + + + + ε2 Net positive bound perturbation charges

  50. ε1 E ε2 Leads to attraction

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