Dielectrophoresis and deformation of a liquid drop in a non-uniform, axisymmetric AC electric field.

Analytical theory for the dielectrophoresis and deformation of a leaky dielectric drop, suspended in a leaky dielectric medium, subjected to non-uniform, axisymmetric Alternating Current (AC) fields is presented in the small deformation limit. The applied field is assumed to be a combination of a uniform part and a quadrupole component. The analysis shows that the magnitude and the sign of the steady and time-periodic dielectrophoretic velocity depend upon the frequency of the applied voltage. The frequency of oscillatory motion is twice that of the applied frequency and the phase lag is a consequence of charge dynamics. A deformed drop under non-uniform axisymmetric AC fields admits Legendre modes l = 2, 3, 4 . The deformation has a frequency-dependent steady and time-periodic parts due to charge and interface dynamics. The steady deformation can be zero at a certain critical frequency in leaky dielectric systems. The time-periodic deformation also has a frequency which is twice the frequency of the applied voltage. In perfect dielectric systems, unlike the steady state deformation which is a balance of Maxwell and curvature stresses, the time-periodic deformation additionally includes viscous stresses associated with the oscillatory shape changes of the drop. A consequence of this effect is a phase lag that is dependent on the charge and interface hydrodynamics and a lag of π/2 at high frequencies. It also results in vanishing amplitude of the oscillatory deformation at high frequencies. The study should lead to a better understanding of dielectrophoresis under non-uniform axisymmetric AC fields and better electrode design to affect drop breakup.

Stability of a charged drop near a conductor wall.

The effect of conductor boundaries on the deformation and stability of a charged drop is presented. The motivation for such a study is the occurrence of a charged conductor drop near a conductor wall in experiments (Millikan-like set-up in studies on Rayleigh break-up) and applications (such as electrospraying, ink-jet printing and ion mass spectroscopy). In the present work, analytical (linear stability analysis (LSA)) and numerical methods (boundary element method (BEM)) are used to understand the instability. Two kinds of boundaries are studied: a spherical, conducting, grounded enclosure (similar to a spherical capacitor) and a planar conducting wall. The LSA of a charged drop placed at the center of a spherical cavity shows that the Rayleigh critical charge (corresponding to the most unstable l = 2 Legendre mode) is reduced as the non-dimensional distance ̂d = (b - a)/a decreases, where a and b are the radii of the drop and spherical cavity, respectively. The critical charge is independent of the assumptions of constant charge or constant potential conditions. The trans-critical bifurcation diagram, constructed using BEM, shows that the prolate shapes are subcritically unstable over a much wider range of charge as [Formula: see text] decreases. The study is then extended to the stability of a charged conductor drop near a flat conductor wall. Analytical theory for this case is difficult and the stability as well as the bifurcation diagram are constructed using BEM. Moreover, the induced charges in the conductor wall lead to attraction of the drop to the wall, thereby making it difficult to conduct a systematic analysis. The drop is therefore assumed to be held at its position by an external force such as the electric field. The case when the applied field is much smaller than the field due to inherent charge on the drop ((a(3)ρg)/(3ε(0)Ψ(2)) ≪ 1 is considered. The wall breaks the fore-aft symmetry in the problem, and equilibrium, predominantly prolate shapes corresponding to the legendre mode, l = 2 , are observed. The deformation increases with increasing charge on the drop. The breakup of the prolate equilibrium shapes is independent of the legendre modes of the initial perturbations. The prolate perturbations are subcritically unstable. Since the equilibrium prolate shapes cannot continuously exchange instability with equilibrium oblate shapes, an imperfect transcritical bifurcation is observed. A variety of highly deformed equilibrium oblate shapes are predicted by the BEM calculations.

Linear stability analysis of electrohydrodynamic instabilities at fluid interfaces in the "small feature" limit.

The endeavour to effectively harness interfacial electrohydrodynamic instabilities, to create small patterns, involves reducing the wavelength of the instability. This can be accomplished by decreasing the separation between the electrodes which may not always be possible. One may therefore have to reduce the surface tension or increase the applied voltage at a fixed electrode spacing. This can result in the wavelength of the pattern becoming of the same order as the electrode separation. Pease and Russel (J. Chem. Phys. 118, 3790 (2003)) were the first to argue that the commonly used Thin-Film Approximation (TFA) that involves an asymptotic expansion in the small parameter δ = (ε(0) φ(0)(2)/(γh(0)))(1/2) (where ε (0) is the permittivity of vacuum, φ (0) is the root mean square value of the applied potential, γ is the surface tension and h (0) is the thickness of the thin film) need not always be valid and γ may not be small in experiments. Higher-order corrections to the TFA might therefore be necessary. We extend the Direct Current (DC) field analysis of Pease and Russel to an Alternating Current (AC) field. AC field has been suggested as an effective way of controlling the wavelength of electrohydrodynamic instabilities at fluid-fluid interfaces. Infact, the perfect and leaky dielectric limits can be realised in the same fluid at very high and very low electric field frequencies, respectively. Recently, Roberts and Kumar (J. Fluid Mech. 631, 255 (2009)) carried out an analysis using TFA to investigate AC-field-induced instabilities at air-polymer interfaces. We propose a Generalized Model (GM), without the lubrication approximation, and carry out detailed comparison with the TFA. We consider the top fluid to be air, a perfect dielectric, and the bottom fluid to be a perfect or a leaky dielectric. The analysis is carried out for both DC and AC fields, and the deviation from TFA is expressed in terms of the parameter B = γh(0)/(ε(0) φ(0)(2) ) = δ(-2). We discuss variation of the wavelength of the fastest growing mode with frequency of the applied field for any arbitrary value of B, unlike the analysis of Roberts and Kumar which is restricted to B ≫ 1(δ ≪ 1) . We also revisit the analysis of Pease and Russel for instabilities under DC field and present the results in terms of the single parameter, B.

Effect of counterions on the Rayleigh-Plateau instability of a charged cylinder.

The effect of counterions on the instability of a charged cylinder is investigated. Both axisymmetric and asymmetric perturbations are considered. The analysis shows that the Rayleigh-Plateau instability is modified for a charged cylinder in the presence of counterions. For the axisymmetric instability, the counterions have a stabilizing effect at low values of kappa, the inverse Debye layer thickness. However, the effect is destabilizing at higher values of kappa . The asymmetric modes which are stable for an uncharged cylinder are rendered unstable at high values of kappa . The analysis should be important in pearling instability of charged cylindrical vesicles. The expression for the correlation time of thermally induced shape fluctuations of charged cylindrical vesicles is also derived.

Weakly nonlinear analysis of the electrohydrodynamic instability of a charged membrane.

The effect of nonlinear interactions on the linear instability of shape fluctuations of a flat charged membrane immersed in a fluid is analyzed using a weakly nonlinear stability analysis. There is a linear instability when the surface tension reduces below a critical value for a given charge density, because a displacement of the membrane surface causes a fluctuation in the counterion density at the surface, resulting in an additional Maxwell normal stress at the surface which is opposite in direction to the stress caused by surface tension. The nonlinear analysis shows that at low surface charge densities, the nonlinear interactions saturate the growth of perturbations resulting in a new steady state with a fluctuation amplitude determined by the balance between the destabilizing electrodynamic force and surface tension. As the surface charge density is increased, the nonlinear terms destabilize the perturbations, and the bifurcation is subcritical. There is also a significant difference in the predictions of the approximate Debye-Huckel and more exact Poisson-Boltzmann equations at high charge densities, with the former erroneously predicting that the bifurcation is supercritical at all charge densities.

Role of conductivity in the electrohydrodynamic patterning of air-liquid interfaces.

The effect of electrical conductivity on the wavelength of an electrohydrodynamic instability of a leaky dielectric-perfect dielectric (LD-PD) fluid interface is investigated. For instabilities induced by dc fields, two models, namely the PD-PD model, which is independent of the conductivity, and the LD-PD model, which shows very weak dependence on the conductivity of the LD fluid, have been previously suggested. In the past, experiments have been compared with either of these two models. In the present work, experiments, analytical theory, and simulations are used to elucidate the dependence of the wavelength obtained under dc fields on the ratio of the instability time (τs=1/smax) and the charge relaxation time (τc=εε0/σ, where ε0 is the permittivity of vacuum, ε is the dielectric constant, and σ is the electrical conductivity). Sensitive dependence of the wavelength on the nondimensional conductivity S2=σ2µ2h0(2)/(ε0(2)φ0(2)δ2) (where σ2 is the electrical conductivity, µ2 is the viscosity, h0 is the thickness of the thin liquid film, φ0 is the rms value of the applied field, and δ is a small parameter) is observed and the PD-PD and the LD-PD cases are observed only as limiting behaviors at very low and very high values of S2, respectively. Under an alternating field, the frequency of the applied voltage can be altered to realize several regimes of relative magnitudes of the three time scales inherent to the system, namely τc, τs, and the time period of the applied field, τf. The wavelength in the various regimes that result from a systematic variation of these three time scales is studied. It is observed that the linear Floquet theory is invalid in most of these regimes and nonlinear analysis is used to complement it. Systematic dependence of the wavelength of the instability on the frequency of the applied field is presented and it is demonstrated that nonlinear simulations are necessary to explain the experimental results.