비균일 전기장에서의 유체 방울의 이동에 대한 계면활성제의 영향

하종욱, 양승만

한국과학기술원 화학공학과

Effect of surfactant on the motions of a fluid particle

in a nonuniform electric field

Jong-Wook Ha, Seung-Man Yang

Department of Chemical Engineering,

Korea Advanced Institute of Science and Technology

INTRODUCTION

In a nonuniform electric field uncharged small dielectric particles suspended in a fluid experience a net force which impels them away from or toward regions of high field. The resultant motion was defined as dielectrophoresis. The dielectrophoresis of particles is important in a variety of applications including biophysics, bioengineering, multiphase separation, etc.

Dielectrophoretic phenomena have been traditionally described as systems made of ideal dielectrics and perfect conducting particles. However, actually, all dielectrics encountered in practice exhibit a conduction mechanism to some extent in addition to a polarization mechanism. This dielectric is referred to as leaky dielectrics.

Theoretical analyses if the dielectrophoretic motion of uncharged drops and bubbles in nonuniform fields have been poorly understood relative to that of rigid spheroidal particles. Recently, Feng considered the primary effects of electrohydrodynamic flow in the dielectrophoresis of deformable fluid particles on the basis of leaky dielectric theory [1]. In that study, he showed that the electrohydrodynamic flow either enhances or hinders the dielectrophoretic motion of fluid particles and obtained the first correction of interface shape.

Here, we extend the Feng's results to the case of insoluble surfactant molecules are resided in the interface. It is well-known experimentally and theoretically that a surfactant present even at very low concentration reduces both the terminal and interfacial velocities of a droplet translating in another fluid. In many practical applications, surface-active agents are added deliberately in order to lower interfacial tension so that achieve fine morphology of liquid-liquid dispersion. In addition, inevitable impurities such as dust act like surface-active agents.

THE PROBLEM

The nonuniform electric field is produced by a point charge in the presence of a leaky dielectric drop in this problem. Such a field configuration is chosen because it can be treated analytically in precise way and also because it is applicable to numerous practical situations, for example, when point electrodes occurs. The governinig equations for the electrostatic potential inside and outside the drop are Laplace's equations and after applying appropriate boundary conditions,

[eqn: phi =- SUM from { { n}=1} to inf LAMBDA ^{n-1} LEFT [ r^n -{n(1-R) OVER {n+R(1+n)}} {1 OVER r^{n+1} }RIGHT ] P_n (eta)],                       (1)

[eqn: {TILDE phi} =- SUM from { { n}=1} to inf LAMBDA ^{n-1} LEFT [ {R(2n+1) OVER {n+R(1+n)}} r^n RIGHT ] P_n (eta)],                               (2)

where, R is the resistivity ratio. The parameter [eqn: LAMBDA] is a measure of the relative magnitude of the field nonuniformity. Since the electric field is disturbed due to the presence of the drop, the electric traction is exerted on the interface. The electric traction outside the drop can be expressed in terms of Maxwell stress tensor.

To formulate the problem for velocity field induced by the electric stress, we assume that the inertia is negligible, and thus, fluid motion can be described by Stokes equation and continuity equation inside and outside the drop. The dielectrophoretic motion of the drop is equivalently represented by a uniform flow velocity at infinity. Moreover, the buoyant force due to gravity is assumed to be aligned with the applied electric field so that axisymmetry of the problem is preserved. Under the assumptions made previously, the motion of fluidoutside the drop can be determined by solving a linear fourth-order equation for the stream function.

[eqn: LEFT [ { PARTIAL ^2 OVER {PARTIAL r^2}}+{{sin theta} Over r^2 } { PARTIAL OVER {sin theta} } LEFT ( {1 OVER {sin theta}}{PARTIAL OVER {PARTIAL theta}} RIGHT ) RIGHT ]^2 PSI =0]

that yields

[eqn: PSI = -r^2 Q_1 (eta) + SUM from { { n}=1} to inf LEFT [ LEFT ( C_n + {C^*}_n RIGHT ) r^{2-n} + LEFT ( D_n + {D^*}_n RIGHT )r^{-n} RIGHT ] Q_n (eta)].          (3)

Similarly, for the inside the drop

[eqn: {TILDE PSI} = SUM from { { n}=1} to inf LEFT [ LEFT ( A_n + {A^*}_n RIGHT ) r^{n+3} + LEFT ( B_n + {B^*}_n RIGHT )r^{n+1} RIGHT ] Q_n (eta)].                    (4)

In which asterisk marks denote the velocity fields responsible for the translation of the drop.

In this study, we consider the asymptotic case in which the deformed drop shape deviates slightly from sphericity. As a consequence, the concentration distribution of surfactant on the drop surface also must deviates from an equlibrium state. In this case, the distribution of surfactant concentration can be expressed with the same form as the shape function which will be obtained later as a part of solution.

[eqn: GAMMA =1+omega = 1+ Ca_0 SUM from { { n}=1} to inf G_n P_n (eta) + O({Ca_0}^2 )].                              (5)

The coefficient [eqn: G_n] can be determined from mass conservation equation. Here, [eqn: Ca_0] is the modified capillary number considering the effect of surfactant, defined by

[eqn: Ca_0 = a mu U^INF / gamma_0],                                                        (6)

in which, [eqn: gamma_0] is the interfacial tension at the initial equilibrium state. Since we assume that the surfactant is insoluble and the convection and diffusion of surfactant through the interface are balanced, the dimensionless convective-diffusion equation is

[eqn: TRIANGLED _s CDOT LEFT ( GAMMA {BOLD u}_s RIGHT )={1 OVER Pe_s} {TRIANGLED ^2 }_s GAMMA],                                              (7)

in which, [eqn: Pe_s] is the surface Peclet number.

STRESS BALANCES

The tengential stress balance at the drop interface in this case is

[eqn: epsilon {E_0}^2 {(1-SR) OVER R} {{PARTIAL {TILDE phi}} OVER {PARTIAL r}}{{PARTIAL {TILDE phi}} OVER {PARTIAL theta}} +{PARTIAL OVER {PARTIAL r}} LEFT ( {TILDE u_theta} OVER r RIGHT ) -lambda {PARTIAL OVER {PARTIAL r}} LEFT ( {TILDE u_theta} OVER r RIGHT )=beta LEFT ( {{PARTIAL varpi} OVER {PARTIAL theta}} RIGHT )],       (8)

where, [eqn: beta LEFT ( =E_s / LEFT ( 1-E_s RIGHT ) RIGHT )] is a parameter defined in terms of Gibbs elasticity [eqn: E_s]. After applying no-slip condition and kinematic condition as well as (8) at r = 1, we obtain

[eqn: {A_1}^* = -{B_1}^* = -{3 OVER {6(1+lambda)+2beta XI }}], [eqn: {C_1}^* = {A_1}^* +{3 OVER 2}], [eqn: {D_1}^* = -{A_1}^* - {1 OVER 2}],

[eqn: {A_n}^* = {B_n}^* = {C_n}^* = {D_n}^* = 0] for [eqn: (n >= 2)],

[eqn: {A_1} = -{B_1} =C_1 = -D_1 = {xi OVER 5} LEFT ( {{SR-1} OVER R} RIGHT ) {{LAMBDA d_1 d_2} OVER {3(1+lambda)+beta XI}}],

[eqn: {A_2} = -{B_2} =C_2 = -D_2 = -{xi OVER 7} LEFT ( {{SR-1} OVER R} RIGHT ) {{6LAMBDA^2 {d_2}^2 +3LAMBDA^2 d_1 d_3 + {d_1}^2} OVER {5(1+lambda)+beta XI}}],

[eqn: {A_3} = -{B_3} =C_3 = -D_3 = {24xi OVER 5} LEFT ( {{SR-1} OVER R} RIGHT ) {{LAMBDA d_1 d_2} OVER {7(1+lambda)+beta XI}}],

[eqn: {A_4} = -{B_4} =C_4 = -D_4 = {4xi OVER 7} LEFT ( {{SR-1} OVER R} RIGHT ) {{9LAMBDA^2 {d_2}^2 +15LAMBDA^2 d_1 d_3} OVER {9(1+lambda)+beta XI}}],

where, [eqn: d_1 = {3R OVER {2R+1}}], [eqn: d_2 = {5R OVER {3R+2}}], and [eqn: d_3 = {7R OVER {4R+3}}]. Moreover, □is the ratio of capillary number and the modified electrical Weber number defined by [eqn: We_0 = a epsilon {E_0}^2 / gamma_0]. Thus, velocity field corresponding to the translating motion and electric-field-induced flow is determined.

The equation for the normal stress balance at the fluid interface takes the form

[eqn: {1 OVER 2} LEFT [ {1 OVER R^2 }(1-SR^2 ) {{PARTIAL {TILDE phi}} OVER {PARTIAL r}}{{PARTIAL {TILDE phi}} OVER {PARTIAL r}}-(1-S) {{PARTIAL {TILDE phi}} OVER {PARTIAL theta}}{{PARTIAL {TILDE phi}} OVER {PARTIAL theta}} RIGHT ]]+

[eqn: + (p-2{{PARTIAL u_r} OVER {PARTIAL r}})-lambda({TILDE p}-2{{PARTIAL {TILDE u}_r} OVER {PARTIAL r}}) = { 1} over {We_0 } TRIANGLED CDOT n].    (9)

Each term in (9) can be written as an expansion in [eqn: P_n (eta)] where n ranges from 0 to 4, with the terms on the left side approximated by the results for the spherical particles and the capillary force term on the right side retaining the first-order effects of interface deformation. Only the terms associated with n = 1 in (9) contribute to a net force, which is of primary concern in dielectrophoresis. By gathering the terms associated with [eqn: P_1 (eta)] in (9), the equation describing net force balance in the direction of [eqn: theta = 0] can be obtained as

[eqn: { 4 pi a^3 epsilon {E_0}^2 LAMBDA} over {(2R+1)(3R+2) } LEFT [ 2(2-3R^2 +SR^2 ) +15 R(1-SR) { 2+lambda} over {15(1+lambda)+3beta XI } RIGHT ]]

[eqn: = { 4} over {3 } pi a^3 g ({TILDE rho}-rho)+4 pi mu a U^INF { 3(2+3lambda)} over {6(1+lambda)+2beta XI }]                      (10)

where g is the acceleration of gravity, and [eqn: U^INF] denotes the dielectrophoretic velocity. The term accosiated with [eqn: U^INF] represents the viscous drag on a creeping spherical particle.

DISCUSSION

The intensity of electrohydrodynamic flow decreases as [eqn: R~ ->~ 0] and vanishes as [eqn: R ~->~ 1/S] because of the disapprearance of surface charge accumulation. Actually, electrohydrodynamic flow also vanishes as [eqn: R~ ->~ 0]. In this case, the fluid inside the particle is much more conductive than the surrounding fluid and the tangential component of electric field becomes zero, leading to zero tangential electric stress despite a substantial amount of surface charge existing at the fluid interface.

When [eqn: beta = 0] or [eqn: XI = 0], the force balance equation (10) is reduced to the previous result [1] since the interfacial tension is uniform through the interface and consequently no additional fluid motion associated Marangoni effect. In addition, if the nonuniformity of the electric field is negligible ([eqn: LAMBDA~->~0]), relative translation of the drop caused by electric field is not produced and the force balance corresponds to Hadamard-Rybczynski problem. However, in this case, the shape of the drop is deformed by the uniform electric field even at leading order.

CONCLUSION

Bahavior of a leaky dielectric drop in a nonuniform electric field is considered. It is assumed that the interface is contaminated by the insoluble surfactant. In order to study the effect of surfactant on the dielectrophoretic motion and electrohydrohynamic deformation of the drop, small deformation theory is utilized. When the surfactant distribution on the surface is uniform, Feng's results are recovered. The electrohydrodynamic flow is shown to either enhance or hinder the dielectrophoretic motion of fluid particle depending on the fluid properties.

REFERENCE

[1] Feng, J. Q., Phys. Rev. E, 54, 4438 (1996)