Dielectrophoretic motions of a pair of particles in the vicinity of a planar wall under a direct-current electric field

Journal of Electrostatics 89 (2017) 30e41

Contents lists available at ScienceDirect

Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat

Dielectrophoretic motions of a pair of particles in the vicinity of a planar wall under a direct-current electric field Sangmo Kang Department of Mechanical Engineering, Dong-A University, Busan 49315, Republic of Korea

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 November 2016 Received in revised form 28 June 2017 Accepted 28 June 2017

This paper presents direct simulations on the two-dimensional dielectrophoretic (DEP) motion of a pair of particles in a viscous fluid, interacting with a nearby planar wall, to further understand the DEP interaction among multiple particles and a wall. Results show that, under an external direct-current electric field parallel to the wall-fluid interface, the nearby wall has significant effects on the DEP motion of both particles including their revolution, alignment and aligned movement. Regardless of their particle conductivity, the wall being less (more) conductive than the fluid pushes (draws) both particles to move away from (toward) it. © 2017 Elsevier B.V. All rights reserved.

Keywords: Immersed-boundary method Maxwell stress tensor Particle-wall dielectrophoretic interaction Sharp interface method

1. Introduction In this paper, we have performed a direct numerical simulation based study on the dielectrophoretic (DEP) motion of a pair of particles suspended freely in a viscous fluid, interacting with a nearby planar solid wall, under an external direct-current (DC) electric field to further understand the DEP interaction among multiple particles and a wall. Here, the direct numerical simulation refers to a numerical approach where one solves the Maxwell's equation or its variation for the electric potential (or the electric field) and then integrates the Maxwell stress tensor over the surface to compute the DEP force acting on each particle, while one solves the continuity and momentum equations for the flow field and the force-balance based kinetic equation for the particle motion. Since the approach does not involve any approximation, it may be very accurate compared with other existing semi-analytical approaches [1e9]. Note that two types of interactions are compositively involved in the DEP motion considered in the present study: one is the particle-particle DEP interaction and the other is the particle-wall DEP interaction. In literature, up to now, even more attention has been paid to the former than the latter out of the two types of interactions. With the purpose of understanding the pure particle-particle DEP interaction, quite a few studies based on the direct numerical

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.elstat.2017.06.008 0304-3886/© 2017 Elsevier B.V. All rights reserved.

simulation have been performed so far on the DEP motion of multiple particles suspended freely in an unbounded viscous fluid without a nearby wall under an external uniform DC electric field. First of all, Ai and Qian [1] performed direct numerical simulations on two-dimensional DEP motions of a pair of perfectly nonconducting (or insulating) particles, where the flow field, electric field and particle motion are simultaneously solved using an arbitrary Lagrangian-Eulerian method. Subsequently, Kang and Maniyeri [2] extended the same work to the DEP motion of two to five particles by applying a smoothed profile method to the solutions of the electric potential and flow field. Both early studies mentioned above drew a common conclusion that, in most cases where a uniform DC electric field is externally applied, all the particles revolve and finally get aligned in a line with the electric field. Despite such a remarkable conclusion, they assumed that all the particles involved should be non-conducting, that is they should have zero electric conductivity ðsp ¼ sp =sf ¼ 0Þ. It implies that the normal component of the electric current density or electric displacement field at each particle-fluid interface should vanish, which can be easily accessible to numerical simulations. In real applications, however, such an assumption is not general, but very exceptional. That is, each particle may usually have its own nonzero finite value of the conductivity. Some follow-up studies, therefore, for tackling such a general problem involving non-zero values of the electric conductivity have also been performed since then. Hossan et al. [3] performed direct numerical simulations on two-dimensional DEP motions of two

S. Kang / Journal of Electrostatics 89 (2017) 30e41

and three identical particles with either of two values of the electric conductivity, sp ¼ 10 and 1=1000, under an external uniform DC electric field. For the simulations, they applied an immersed interface method to the solution of the electric potential and a feedback-forcing based immersed-boundary method to that of the flow field. Subsequently, Kang [7] extended the similar work to two-dimensional DEP motions of a pair of particles with more diverse sets of the conductivity, sp ¼ 1=100, 1=2, 2 and 100, by employing a sharp interface method and a direct-forcing based immersed-boundary method to the solutions of the electric potential and flow field, respectively. Recently, Xie et al. [9] also performed similar direct numerical simulations on two-dimensional DEP motions of two and three particles using an arbitrary Lagrangian-Eulerian method. The results from the above follow-up studies indicated that the DEP motion of both particles depends significantly on the combination of their electric conductivity signs. Here, the conductivity of a particle or a wall that is smaller and larger than that of the fluid is defined to be negative- and positive-signed, respectively. When both particles have the same sign of the conductivity (negative or positive), they revolve in a variety of fashions depending on their initial configuration and finally get aligned in a line with the external electric field. Note that more multiple particles with the same sign should also lead to their final alignment in a line with the electric field. With different signs, on the other hand, they revolve in the opposite direction and finally get aligned in a line perpendicular to the electric field. As reviewed above, lots of numerical studies have been performed so far on the pure particle-particle DEP interaction. On the particle-wall interaction, only a few direct numerical simulation based studies can be found in literature. Since most of the real applications involve micro-fluidic devices or channels bounded by solid walls, rigorous understanding of the wall-induced DEP motion or the particle-wall interaction is very crucial. Very recently, Kang [10] studied two-dimensional DEP motions of a single particle suspended freely in an unbounded viscous fluid, interacting with a nearby planar wall, by performing direct numerical simulations. The results indicated that, under an external electric field parallel to the wall-fluid interface, one particle moves only in a direction normal to the interface due to the DEP force induced by the presence of a nearby wall and the motion depends strongly on the combination of the electric conductivity of the particle and wall and the separation gap between them. It was found that the direction of particle motion is determined only by the wall conductivity, irrespective of the particle conductivity: the particle is repelled to move farther away from the wall with a negative sign of the conductivity and attracted to migrate closer toward the wall with a positive sign. The intensity of particle motion is significantly influenced by the combination of their conductivity as well as the separation gap. Although obtained just from one single particle, the above results obviously imply that the wall can also play an important role even in inducing or modifying the DEP motion of multiple particles. It strongly motivates the present study. In the present study, we have numerically examined twodimensional DEP motions of a pair of particles suspended freely in an unbounded viscous fluid in contact with a planar solid wall under a uniform DC electric field applied externally in parallel with the wall-fluid interface. In addition, we also assume that any other electrokinetic effect except the DEP effect should not be involved at all. Particularly, the electrophoretic effect that may commonly occur in micro- and nano-fluidic applications is neglected under an assumption that the electric double layer (EDL) thickness should be much smaller than the particle size and the particle-wall separation gap. Since the EDL thickness is generally on the order of

31

nanometers, therefore, the particle radius and the separation gap should be at least on the order of micrometers in the present study. In general, the external electric field is not always parallel to the wall-fluid interface in real applications, that is they may have a non-zero included angle between them for various purposes [11,12]. As a first attempt, nevertheless, we assume in this study that the external electric field should be parallel to the interface. For the simulations, we have extended the same numerical method developed and then validated by Kang [7,10] to the present study. In other words, we employ a finite-volume based numerical approach, where a sharp interface method is adopted for the solution of the electric potential and a direct-forcing based immersed-boundary method is for that of the flow field. Then, we have also carried out parameter studies by systematically varying the combination of the electric conductivity signs of the two particles and one wall and the initial particle configuration including a separation gap between the particles and wall. 2. Numerical methods 2.1. Mathematical modeling Consider the DEP motion taken by a pair of particles (‘p’) suspended freely in an unbounded viscous fluid (‘f’) in contact with a stationary planar solid wall (‘w’) on the bottom side. Here, the particles have an equal radius, ap , and are assumed non-Brownian and neutrally buoyant, while the fluid has mass density, rf , viscosity, mf , and electric conductivity and permittivity, sf and εf . A uniform electric field, E0 , is externally applied in the horizontal, from left to right, direction (toward the positive x-direction) parallel to the wall-fluid interface. All the variables (‘’ dropped) introduced in this manuscript are normalized by the above mentioned dimensional parameters (‘’ attached), ap , rf , mf , sf , εf , and E0 .

Fig. 1. Non-dimensional schematic diagram of the flow geometry and computational domain.

32

S. Kang / Journal of Electrostatics 89 (2017) 30e41

Fig. 1 shows a non-dimensional schematic diagram of the flow geometry and computational domain, which is composed of a fluid sub-domain (Uf ) with a size of ð2L∞ Þ  ðL∞ þ G0 Þ, a wall subdomain (Uw ) with a size of ð2L∞ Þ  Hw , and two particle subdomains (Up;i , i ¼ 1; 2) (all the length scales are normalized by the particle radius, ap ). Here, the fluid sub-domain is assumed to be large enough compared with the particle size to mimic the unbounded fluid (L∞ [1), while the wall sub-domain is determined to be thick enough to have little size effect on the electric field. The origin is situated at the midpoint between both particles in an initial configuration and simultaneously away above from the wallfluid interface by G0 (y ¼ G0 ). The first and second particles are assumed initially located in the first and third quadrants, Xp;1 ¼ ðr0 ; q0 Þ and Xp;2 ¼ ðr0 ; q0 þ 180+ Þ ð0  q0  90+ Þ in a polar coordinate, respectively. Each particle has mass density, rp;i ½rp;i ¼ rp;i =rf , and electric conductivity, sp;i [sp;i ¼ sp;i =sf ]. The nearby planar wall has electric conductivity, sw , and is attached at the bottom of the fluid sub-domain, following the study of Zhao and Bau [13] that examined the effect of the wall permittivity on the DEP force acting on a single particle in induced-charge electroosmosis next to a planar solid wall. In the present study, therefore, the combination set of the electric conductivity of the two particles and one wall, ½ðsp;1 ; sp;2 Þ; sw , and the initial particle configuration including a separation gap between the origin (or both particles) and wall, ½ðr0 ; q0 Þ; G0 , are expected to have decisive effects on the DEP motion of a pair of particles in the vicinity of a nearby wall. For convenience during the explanation of the numerical method, we assume without any loss of generality that only one particle (Up and Gp ) should be involved in the DEP motion. Note that the explanation is basically identical to those in previous studies [7,10]. First of all, the electric potential, f ¼ ∪ðff ; fp ; fw Þ ½f ¼ f =ðap E0 Þ (here, the symbol, ∪, stands for the union of the fluid, particle and wall sub-domains), is obtained by solving the following Laplace equation in a non-dimensional form in each subdomain:

V2 f ¼ 0:

(1)

The Laplace equation (1) is the so-called leaky-dielectric model of Taylor-Melcher [14e16], derived from the following Maxwell equation set formulated in the electrostatic limit:

V$ðεVfÞ ¼ q;

(2)

Dt q ¼ V$ðsVfÞ;

(3)

where q is the volumetric electric charge and Dt the material derivative. In other words, the leaky dielectric model is derived by dismissing the electric potential equation (2) and neglecting the material derivative term in the electric charge equation (3). At the particle-fluid and wall-fluid interfaces (Gs , s ¼ p; w), the governing equation (1) is subjected to the following continuity of the electric potential and the electric current density:

ff ¼ fs ;

(4)

n$Vff ¼ n$ðss Vfs Þ;

(5)

After solving Eq. (1) for the electric potential, the DEP force and torque, Fe and Te , acting on the particle can be obtained by integrating the Maxwell stress tensor, se ½se ¼ se =ðεf E02 Þ, over the particle surface as follows:

Z Fe ¼

f e dS;

(7)

  x  Xp  f e dS;

(8)

Gp

Z Te ¼ Gp

f e ¼ se $n;

(9) 1 2

se ¼ EE  E2 I;

(10)

where x is the spatial coordinate [x ¼ ðx; yÞ ¼ x =ap ], dS the infinitesimal area on the particle surface, E2 ¼ E,E, and I the identity matrix. In addition, the force density (per area), f e , in Eq. (9) can be simplified through simple algebra as follows:

fe ¼

 1 2 2 En;f  Et;f n þ En;f Et;f t; 2

(11)

where En;f and Et;f are the normal and tangential components of the electric field on the fluid side at the particle-fluid interface, respectively, and t the unit vector tangential with the particle surface. The flow field is governed by the following continuity and Navier-Stokes equations in a non-dimensional form:

V$u ¼ 0;

(12)

  vu þ V$ðuuÞ ¼ Vp þ V2 u; Re vt

(13)

where

u

is

the

flow

¼ ðεf ap E02 Þ=mf , p ¼ t  ðεf E02 Þ=mf , and

velocity

½u ¼ ðu; vÞ ¼ u =u0,

where

u0

the pressure ½p ¼ p =ðεf E02 Þ, t the time

½t

Re ¼ rf u0 ap =mf the Reynolds number. For

the numerical solution of Eqs. (12) and (13), the flow velocity at the particle-fluid interface is given as follows:

  u

Gp

   ¼ Up þ Up  x

Gp

  Xp ;

(14)

where Up and Up are the translational and rotational velocities of the particle, respectively. The hydrodynamic force and torque, Fh and Th , exerted on the particle by the fluid can be calculated by integrating the Cauchy stress tensor, sh ½sh ¼ sh =ðεf E02 Þ, over the surface as follows:

Z

Fh ¼

f h dS;

(15)

  x  Xp  f h dS;

(16)

Gp

Z Th ¼ Gp

where n is the outward unit vector normal to the interfaces. Subsequently, the electric field, E ¼ ∪ðEf ; Ep ; Ew Þ ½E ¼ E =E0 , is computed from the electric potential as follows:

f h ¼ sh $n;

E ¼ Vf:

sh ¼ pI þ Vu þ ðVuÞT :

(6)

(17) h

i

(18)

S. Kang / Journal of Electrostatics 89 (2017) 30e41

For the DEP motion of the particle, the DEP force and torque have to be balanced with the hydrodynamic counterparts on the particle as follows:

Re mp

vUp vt



vUp Re Ip vt

¼ Fh þ Fe ;

dXp ¼ Up : dt

vUp ¼ Fh þ Fe ; vt

(28)

(20)

(21)

In this study, two kinds of differential equations have to be solved together with appropriate boundary conditions on their own different computational domains. That is, the Laplace equation (1) is solved simultaneously on both fluid (including the particle) and wall sub-domains ½ðHw þ G0 Þ  y  L∞ , whereas the continuity and momentum equations (12) and (13) are solved only on the fluid sub-domain ½G0  y  L∞ . Here, the conditions on the far-field boundary and wall are given as

f ¼ HL∞ at x ¼ ±L∞ ; n$Vf ¼ 0 at y ¼ ðHw þ G0 Þ; L∞

(22)

for the electric potential and

at

(27)

mp

  5 where mp ½mp ¼ mp =ðrf a3 p Þ and Ip ½Ip ¼ Ip =ðrf ap Þ are the mass and inertia moment of the particle, respectively. Finally, the trajectory taken by the particle can be traced by solving the following kinematic equation:

u¼0

du ¼ Vp þ V2 u; dt

(19)

¼ Th þ Te ;

x ¼ ±L∞ and y ¼ G0 ; L∞

33

(23)

for the flow field. Note that the unbounded fluid is replaced with a fluid filled in a very large cavity ðL∞ [1Þ during the simulations.

2.2. Numerical formulation In this study, we have performed numerical simulations on twodimensional DEP motions of a pair of particles in the vicinity of a nearby planar wall: Xp ¼ ðXp;x ; Xp;y Þ, Fe ¼ ðFe;x ; Fe;y Þ, and ðUp ; Up Þ ¼ ðUp;x ; Up;y ; Up Þ. In addition, it is also assumed that the Reynolds number should be negligibly trivial (Re≪1), which is valid in most of the micro- and nano-fluidic applications. In such a case, all the left terms in Eqs. (13), (19) and (20) may almost vanish as follows:

Vp þ V2 uz0;

(24)

Fh þ Fe z0;

(25)

Th þ Te z0:

(26)

Here, Eq. (24) corresponds to the very low Reynolds number Navier-Stokes equation or the Stokes equation. In addition, Eqs. (24)-(26) indicate that the fluid flow and particle motion always remain quasi-steady and depend only on the particle configuration besides the combination of the conductivity of the two particles and one wall. In order to solve Eqs. (12) and (24) for the flow field and Eqs. (25) and (26) for the particle motion at each real time (t), the pseudo (fictitious) unsteady terms are added as follows:

Ip

vUp ¼ Th þ Te ; vt

(29)

where t is the pseudo time. The modified equations (27e29) are iteratively solved on the pseudo time (t) in a time-marching way by continuing the iteration until the pseudo unsteady terms sufficiently vanish. For the simulations, we employ a finite-volume based numerical approach developed and then validated in previous studies [7,10], which consists of a sharp interface method for the solution of the electric potential and a direct-forcing based immersed-boundary method for that of the flow field. To the solution of the Laplace equation (1) for the electric potential, we apply the sharp interface method developed by Liu et al. [17] in order to tackle the numerical difficulty that may arise due to a large sharp jump in the coefficient (electric conductivity). For the flow field and the particle motion, we apply the direct-forcing based immersedboundary method developed by Kim et al. [18] to the solution of the continuity and momentum equations (12) and (27). In the sharp interface method [7], the jump condition on the interface is used to modify the discretization of the differential operators on the Cartesian grids in the vicinity of the interface. Since the interface is kept sharp, the jump condition on the interface can be accurately captured. In the immersed boundary method [18], the momentum forcing and mass source/sink are introduced respectively in the Navier-Stokes and continuity equations to be applied only on the body surface or inside the body for the purpose of satisfying the noslip condition on the immersed boundary and the mass conservation for the cell containing the immersed boundary. All the relevant differential equations, (1), (12) and (27), are resolved with a finite-volume approach on a staggered mesh and all the spatial derivatives are discretized with a second-order central difference scheme. A dense clustering of uniform grid points (△x ¼ △y ¼ h) is applied to the central region of the computational domain on which the particles can actually travel, whereas out of the uniform-grid region an even coarser non-uniform grid is used. Unless otherwise stated, in the present study, dual resolutions, namely ðhe ; hh Þ ¼ ð1=24; 1=12Þ, are used for the electric field and flow field. Note that the resolutions were adopted and then confirmed in previous studies [7,10]. The fluid sub-domain is set very large at L∞ ¼ 20 compared with the particle size. In addition, the wall thickness is also determined very large at Hw ¼ 10: doubling of the wall thickness causes no significant effect on the DEP force acting on the particle [10]. During the simulations, the two particles (or a particle and a wall) may approach closer toward and then penetrate into each other due to a relatively large time-step size, which cannot occur in real applications. To prevent one particle from penetrating into the other particle or the wall, we employ the so-called collision strategy suggested by Glowinski et al. [19], where Eq. (25) is replaced with

p

Fh þ Fe þ Fr þ Fw r z0:

(30)

Here, the short-range repulsive force, Fsr (s ¼ p; w), exerted on the particle by the other particle or the wall can be given as

34

S. Kang / Journal of Electrostatics 89 (2017) 30e41

( Fsr

¼

dp > 2 þ l   2 ; ð1=gÞ Xp  X0s 2 þ l  dp dp  2 þ l 0

(31)

0

where X s represents the center of the other particle (s ¼ p) or the imaginary particle located on the other side of the wall-fluid  interface (s ¼ w). In addition, dp ¼ Xp  X0s  is the distance between the centers of the two particles, l the force range and g a very small positive stiffness parameter. In the present study, we determine to set the force range at r ¼ 1:5hh and the stiffness parameter at g ¼ 2  104 . 3. Results and discussion In order to further understand the DEP interaction arising between multiple particles and a wall, we perform direct numerical simulations on two-dimensional DEP motions of a pair of particles, interacting with a nearby planar wall, and then discuss their results. Since the DEP interaction between both particles without involving a nearby wall was fully discussed in a previous study [7], we want to concentrate mainly on the effect of the nearby wall on the particle-particle DEP interaction. For the discussion, we adopt two extreme values of the electric conductivity, ss ¼ ss =sf ¼ 1=100 and 100 (s ¼ p; w), for the two particles and one wall, which are representative of the negative and positive signs of the conductivity, respectively. That is, both particles may have four different combinations of the conductivity, ðsp;1 ; sp;2 Þ ¼ ð1=100; 1=100Þ, ð100; 100Þ, ð1=100; 100Þ, and ð100; 1=100Þ, while the wall may have two different values of the conductivity, sw ¼ 1=100 and 100, giving rise to a total of eight different combinations of the conductivity among the particles and wall, ½ðsp;1 ; sp;2 Þ; sw . In addition, we adopt two initial separation gaps between the two particles (or the origin) and wall, G0 ¼ 3:5 and 7, to examine the gap effect on the DEP interaction. 3.1. DEP interactions among two particles and one wall To see the effect of the nearby wall, we want to scrutinize the instantaneous electric and flow fields formed around a pair of particles located at ðr0 ; q0 Þ ¼ ð2; 45+ Þ and G0 ¼ 3:5 and the corresponding DEP forces acting on them. Toward this end, we perform numerical simulations for a total of eight different combinations of the conductivity, ½ðsp;1 ; sp;2 Þ; sw , and then present their

Fig. 2. Contours of the electric potential (△f ¼ 0:2) for ðsp;1 ; sp;2 Þ ¼ (a) ð1=100; 100Þ and (b) ð100; 1=100Þ at sw ¼ 1=100. Here, the particles are located at ðr0 ; q0 Þ ¼ ð2; 45+ Þ and G0 ¼ 3:5.

representative results in Figs. 2e7. Figs. 2 and 3 show contours of the electric potential typically at ðsp;1 ; sp;2 Þ ¼ ð1=100; 100Þ and ð100; 1=100Þ for the walls at sw ¼ 1=100 (colored in red) and 100 (in blue), respectively. It is observed that the distribution of the electric potential around each particle is determined mostly by the conductivity that the particle has. In case of the particle at sp ¼ 1=100 (colored in red) where the particle is even less conductive than the fluid (sp ≪1 or sp ≪sf ), the contour lines of the electric potential meet the particle-fluid interface on the fluid side   nearly at a right angle (En;f  z0). From Eq. (11), therefore, the Gp

force density reduces to

1 2 f e z  Et;f n: 2

(32)

It implies that the force density should always act normally inwards and thus be purely compressive on the whole surface. In case of the particle at sp ¼ 100 (in blue) where the particle is even more conductive than the fluid (sp [1 or sp [sf ), the contour lines are nearly  parallel or tangential with the interface on the fluid side  (Et;f  z0). The force density, therefore, simply becomes Gp

1 2 f e z En;f n: 2

(33)

Similarly, it implies that the force density should always act normally outwards from the particle and thus be purely tensile. On both cases, the tangential component is negligibly trivial (fe;t z0). Figs. 4 and 5 show profiles of the normal component of the force density, fe;n , acting along the particle-fluid interfaces on both particles for the walls at sw ¼ 1=100 and 100, respectively. To understand the effect of the nearby wall, the present results (designated by symbols) are also compared with those (by lines) for a pair of particles without a nearby wall [7]. Here, a is the counterclockwise circumferential angle along the interface starting from each caseby-case axis. As defined in Fig. 1, the angle starts from the positive and negative x axes for the first and second particles, respectively. Since the second particle is much closer to the wall than the first one ½ðGp;1 ; Gp;2 Þ ¼ ð2:8; 4:2Þ, the nearby wall should exert a significant effect on the particle-particle DEP interaction mainly through the second particle. Here, Gp;i is the distance of the i-th particle center from the wall. During the discussion of the particlewall interaction, therefore, the second particle has to be more concentrically examined.

Fig. 3. Contours of the electric potential (△f ¼ 0:2) for ðsp;1 ; sp;2 Þ ¼ (a) ð1=100; 100Þ and (b) ð100; 1=100Þ at sw ¼ 100. Here, the particles are located at ðr0 ; q0 Þ ¼ ð2; 45+ Þ and G0 ¼ 3:5.

S. Kang / Journal of Electrostatics 89 (2017) 30e41

35

Fig. 4. Profiles of the normal component of the force density along the particle-fluid interface for ðsp;1 ; sp;2 Þ ¼ (a) ð1=100; 1=100Þ, (b) ð100; 100Þ, (c) ð1=100; 100Þ, and (d) ð100; 1=100Þ at sw ¼ 1=100. Here, the particles are located at ðr0 ; q0 Þ ¼ ð2; 45+ Þ and G0 ¼ 3:5.

Fig. 5. Profiles of the normal component of the force density along the particle-fluid interface for ðsp;1 ; sp;2 Þ ¼ (a) ð1=100; 1=100Þ, (b) ð100; 100Þ, (c) ð1=100; 100Þ, and (d) ð100; 1=100Þ at sw ¼ 100. Here, the particles are located at ðr0 ; q0 Þ ¼ ð2; 45+ Þ and G0 ¼ 3:5.

For the second particle at sp;2 ¼ 1=100, the contour lines of the electric potential become more densely clustered and more sparsely scattered in the gap between the particle and wall, or on the lower side of the particle (a ¼ 0  180+ ), by the nearby walls at sw ¼ 1=100 [see Figs. 2(b) and 4(d)] and 100 [Figs. 3(b) and 5(d)], respectively, compared with the no-wall cases [7]. That is, the electric field on the lower side of the second particle grows strong when the particle and wall have the same sign of the conductivity, whereas it becomes weak when they have different signs [10]. The

force density on the lower side, therefore, which should face upwards because it is always compressive, increases for the wall at sw ¼ 1=100 and decreases for the wall at sw ¼ 100. Such changes can be obviously observed over the range of a ¼ 0  180+ , particularly around at az90+ , in the graphs of Figs. 4(a,d) and 5(a,d). As a result, the upward force is added onto the second particle at sp;2 ¼ 1=100 by the wall at sw ¼ 1=100, whereas the downward force is added onto the same particle by the wall at sw ¼ 100. A very similar explanation can also be applied to the second

36

S. Kang / Journal of Electrostatics 89 (2017) 30e41

Fig. 6. Streamlines (△j ¼ 0:002, where j is the stream function) for ðsp;1 ; sp;2 Þ ¼ (a) ð1=100; 1=100Þ and (b) ð1=100; 100Þ at sw ¼ 1=100. Here, the particles are located at ðr0 ; q0 Þ ¼ ð2; 45+ Þ and G0 ¼ 3:5 and revolve clockwise in (a) and counterclockwise in (b).

particle at sp;2 ¼ 100. Due to the particle-wall interaction, the walls at sw ¼ 1=100 [see Fig. 2(a)] and 100 [Fig. 3(a)] make the electric field weaker and stronger on the lower side of the particle, respectively, compared with the no-wall cases. The force density on the lower side, therefore, which should face downwards because it

Fig. 7. Streamlines (△j ¼ 0:002, where j is the stream function) for ðsp;1 ; sp;2 Þ ¼ (a) ð100; 100Þ and (b) ð100; 1=100Þ at sw ¼ 100. Here, the particles are located at ðr0 ; q0 Þ ¼ ð2; 45+ Þ and G0 ¼ 3:5 and revolve clockwise in (a) and counterclockwise in (b).

is always tensile, decreases for the wall at sw ¼ 1=100 and increases for the wall at sw ¼ 100. Such changes can also be obviously observed over the range of a ¼ 0  180+ , particularly around at az0+ and 180+ , in the graphs of Figs. 4(b,c) and 5(b,c), although

Fig. 8. Trajectories taken by both particles at ðsp;1 ; sp;2 Þ ¼ (a) ð1=100; 1=100Þ, (b) ð100; 100Þ, (c) ð1=100; 100Þ, and (d) ð100; 1=100Þ, interacting with a nearby wall at sw ¼ 1=100 for G0 ¼ 3:5. The motion starts at r0 ¼ 2 and q0 ¼ 0+ , 10+ , 30+ , 60+ , 80+ and 90+ in sequence (open circles).

S. Kang / Journal of Electrostatics 89 (2017) 30e41

37

Fig. 9. Trajectories taken by both particles at ðsp;1 ; sp;2 Þ ¼ (a) ð1=100; 1=100Þ, (b) ð100; 100Þ, (c) ð1=100; 100Þ, and (d) ð100; 1=100Þ, interacting with a nearby wall at sw ¼ 100 for G0 ¼ 3:5. The motion starts at r0 ¼ 2 and q0 ¼ 0+ , 10+ , 30+ , 60+ , 80+ and 90+ in sequence (open circles).

they are very delicate. After all, the wall at sw ¼ 1=100 adds an additional upward force on the second particle so as to push it farther away from the wall, whereas the wall at sw ¼ 100 adds a downward force on the particle so as to draw it closer toward the wall. Figs. 6 and 7 show streamlines at ðsp;1 ; sp;2 Þ ¼ ð1=100; 1=100Þ and ð1=100; 100) for the wall at sw ¼ 1=100 and at ðsp;1 ; sp;2 Þ ¼ ð100; 100Þ and ð100; 1=100) for the wall at sw ¼ 100, respectively. The streamlines indicate that the revolution of both particles still occurs despite the presence of a nearby wall (negative or positive). That is, both particles with the same sign of the conductivity revolve clockwise [see Figs. 6(a) and 7(a)], whereas those with different signs revolve counterclockwise [Figs. 6(b) and 7(b)]. Although the particles receive additional upward or downward forces, their translational velocities become more or less reduced due to the blockage and friction effects of the wall. The aforementioned effect by the nearby wall on the DEP motion of a pair of particles can also be explained by the so-called Clausius-Mossotti (CM) factor. The DEP force on a spherical particle can be formulated as

 2   FDEP ¼ 2pfCM VE ;

(34)

where fCM is the CM factor [20]. Here, the CM factor depends solely on the electric conductivity of the particle and fluid and is expressed as

fCM ¼

sp  1 sp þ 2

(35)

under an external DC electric field or AC field at a low frequency. For the two values of the particle conductivity, sp ¼ 1=100 and 100, adopted in the present study, the CM factors are calculated to be 0:49 (fCM < 0) and 0.97 (fCM > 0), respectively. Eq. (34) implies that the particle with a negative CM factor (sp < 1 or sp < sf ) is forced from a region of strong electric field to that of weak electric field, whereas the particle with a positive factor (sp > 1 or sp > sf ) is forced in the opposite direction. These effects are well known as negative and positive DEPs, respectively. Consider the wall with a negative-signed conductivity (sw ¼ 1=100). The presence of the particle at sp ¼ 1=100 near the wall makes the electric field strengthened between the particle and wall because of the same sign (negative) and is pushed to drift away from the wall due to the negative DEP. The particle at sp ¼ 100, on the other hand, makes the electric field weakened between them because of the different signs and is also pushed to drift away from the wall due to the positive DEP. Such a similar explanation can be applied to the wall with a positive-signed conductivity (sw ¼ 100). The particles at sp ¼ 1=100 and 100 make the electric field weakened and strengthened between them because of the different signs and the same sign (positive) and thus are pulled to move toward the wall due to the negative and positive DEPs, respectively. The analytical interpretation is in exact agreement with the

38

S. Kang / Journal of Electrostatics 89 (2017) 30e41

Fig. 10. Trajectories taken by both particles at ðsp;1 ; sp;2 Þ ¼ (a) ð1=100; 1=100Þ, (b) ð100; 100Þ, (c) ð1=100; 100Þ, and (d) ð100; 1=100Þ, interacting with a nearby wall at sw ¼ 1=100 for G0 ¼ 7. The motion starts at r0 ¼ 2 and q0 ¼ 0+ , 10+ , 30+ , 60+ , 80+ and 90+ in sequence (open circles).

numerical observations in that the direction of particle motion is determined only by the wall conductivity, irrespective of the particle conductivity: the particle is repelled to move farther away from the wall being less conductive than the fluid and attracted to migrate closer toward the wall being more conductive. 3.2. Trajectories of both particles in the vicinity of a nearby wall Up to now, we have observed that the nearby wall has a significant effect on the DEP interaction between both particles. To actually confirm such an observation, we also perform timeevolving numerical simulations on the DEP motion and present the long trajectories taken by them in Figs. 8e11. Here, both particles and one wall have either of two values of the electric conductivity, ss ¼ ss =sf ¼ 1=100 and 100 (s ¼ p; w), while the particles are initially located away from the wall by G0 ¼ 3:5 (Figs. 8 and 9) and 7 (Figs. 10 and 11). Under each circumstance, the motion starts at r0 ¼ 2 and q0 ¼ 0+ (horizontal arrangement), 10+ , 30+ , 60+ and 80+ (inclined arrangements), and 90+ (vertical arrangement) in sequence. Fig. 8 shows the trajectories taken by both particles, interacting with the wall at sw ¼ 1=100, which are initially located at G0 ¼ 3:5. Results indicate that, for all the inclined arrangements considered (0+ < q0 < 90+ ), both particles revolve and then align themselves parallel or perpendicular to the electric field although their DEP motion is significantly influenced by the nearby wall. In other words, their trajectories are very similar to the no-wall cases except

for some degree of course variations due to the presence of a nearby wall: both particles with the same sign of the conductivity revolve clockwise and then get aligned in a line parallel with the electric field or the wall-fluid interface (horizontal alignment) [Fig. 8(a,b)], whereas those with different signs revolve counterclockwise and then get aligned in a line perpendicular to the electric field (vertical alignment) [Fig. 8(c,d)]. During the revolution, both particles are pushed upward so as to drift farther away from the wall in addition to the pure DEP interaction between them. At the same time, the second particle located closer to the wall is more strongly forced than the first particle. Such observations are obviously attributed to the repulsive force acting on each particle induced by the same wall. It is also found that, even after their in-line alignment, both particles continue to take the DEP motion while sustaining their alignment. That is, the horizontally or vertically aligned particles are pushed to move upward due to the repulsive force induced by the same wall. Here, it is seen that the horizontally aligned particles move upward much faster than the vertically aligned ones: see Figs. 12 and 13 in advance. Note that, in the no-wall cases, no movement occurs longer after the alignment [7]. For the horizontal arrangement (q0 ¼ 0+ ), both particles with the same sign of the conductivity approaches each other and then attain their horizontal in-line alignment while they are pushed to move upward. With different signs, to the contrary, both particles drift apart from each other while also pushed to move upward. At a much later time, they should attain their vertical in-line alignment, which cannot be verified because of the computational limitation

S. Kang / Journal of Electrostatics 89 (2017) 30e41

39

Fig. 11. Trajectories taken by both particles at ðsp;1 ; sp;2 Þ ¼ (a) ð1=100; 1=100Þ, (b) ð100; 100Þ, (c) ð1=100; 100Þ, and (d) ð100; 1=100Þ, interacting with a nearby wall at sw ¼ 100 for G0 ¼ 7. The motion starts at r0 ¼ 2 and q0 ¼ 0+ , 10+ , 30+ , 60+ , 80+ and 90+ in sequence (open circles).

in the present study. For the vertical arrangement (q0 ¼ 90+ ), both particles with the same sign move upward, but at different speeds, so that they cannot attain their in-line alignment forever because the separation gap between them becomes widening. The first particle moves faster because the upward forces due to the particleparticle interaction and particle-wall interaction are additive, whereas the second particle moves more slowly because the upward force due to the particle-wall interaction is partially cancelled out by the downward force due to the particle-particle interaction. With different signs, to the contrary, both particles approach each other at different speeds so that they can attain their vertical in-line alignment. The first particle moves downward more slowly because the downward force due to the particle-particle interaction is partially cancelled out by the upward force due to the particle-wall interaction, whereas the second particle moves upward faster because the upward forces due to the particle-particle interaction and particle-wall interaction are additive. After the vertical in-line alignment, the aligned particles are pushed to move upward while sustaining their alignment. Fig. 9 shows the trajectories taken by both particles, interacting with the wall at sw ¼ 100, which are initially located at G0 ¼ 3:5. The particle trajectories traced for the wall at sw ¼ 100 are very similar to those for the wall at sw ¼ 1=100 except that both particles are drawn to move closer toward the same wall so that either of them can come in close proximity to the wall. Consider, first of all, the cases with the same sign of the particle conductivity as shown in Fig. 9(a,b). For the horizontal arrangement (q0 ¼ 0+ ), both

particles approach each other and then attain their horizontal inline alignment while they are pushed to move upward unlike our expectation. After the alignment, the aligned particles are drawn to move downward so that they can come in near-contact with the wall and then become stationary. For the inclined arrangements (0+ < q0 < 90+ ), they revolve clockwise, then get aligned in a line parallel with the electric field and finally move in a state of alignment while attracted to move closer toward the same wall. When the particles are near-vertically located (for example, q0 ¼ 60+ and 80+ ), particularly, the second particle located much closer to the wall may first come in near-contact with the wall during the revolution before the alignment. For the vertical arrangement (q0 ¼ 90+ ), the first particle moves upward more slowly without stoppage because the upward force acting on it due to the particleparticle interaction is partially canceled out by the downward force due to the particle-wall interaction, whereas the second particle moves downward faster to come in near-contact with the wall because the downward forces due to the particle-particle and particle-wall interactions are additive. Consider, secondly, the cases with different signs as shown in Fig. 9(c,d). For the horizontal arrangement (q0 ¼ 0+ ), both particles drift apart from each other and then independently get in close proximity to the wall at sw ¼ 100 while attracted to move downward. For the inclined arrangements (0+ < q0 < 90+ ), they revolve counterclockwise and then get aligned in a line perpendicular to the electric field. For most arrangements, however, since the second particle moves downward due to the counterclockwise revolution,

40

S. Kang / Journal of Electrostatics 89 (2017) 30e41

Fig. 12. DEP forces on and translational velocity of the aligned particles according to the separation gap between the particles and wall at ðsp;1 ; sp;2 Þ ¼ (a) ð1=100; 1=100Þ, (b) ð100; 100Þ, (c) ð1=100; 100Þ, and (d) ð100; 1=100Þ for sw ¼ 1=100. Here, the particles are assumed to be located at r0 ¼ 1 þ ε=2 (ε ¼ 0:115) and q0 ¼ (a,b) 0+ and (c,d) 90+ and move in a single body. F e;y ¼ ðFe;y;1  Fe;y;2 Þ=2.

Fig. 13. DEP forces on and translational velocity of the aligned particles according to the separation gap between the particles and wall at ðsp;1 ; sp;2 Þ ¼ (a) ð1=100; 1=100Þ, (b) ð100; 100Þ, (c) ð1=100; 100Þ, and (d) ð100; 1=100Þ for sw ¼ 100. Here, the particles are assumed to be located at r0 ¼ 1 þ ε=2 (ε ¼ 0:115) and q0 ¼ (a,b) 0+ and (c,d) 90+ and move in a single body. F e;y ¼ ðFe;y;1  Fe;y;2 Þ=2.

it may first come in near-contact with the wall and then get aligned in a line with the first particle. For the vertical arrangement (q0 ¼ 90+ ), the first particle moves downward faster because the downward forces acting on it due to the particle-particle and particle-wall interactions are additive, whereas the second particle moves downward more slowly to come in near-contact with the wall because the downward force due to the particle-wall interaction is partially canceled out by the upward force due to the particle-particle interaction. At a later time, both particles attain their vertical in-line alignment. Subsequently, we continue to perform the same numerical simulations on the DEP motion of both particles by increasing the initial separation gap to G0 ¼ 7 and then present their results in Figs. 10 and 11. The figures indicate that the trajectories are very similar to those for G0 ¼ 3:5, but the wall effect grows significantly weak because of the increased separation gap. Particularly, both

particles with different signs move in a state of vertical alignment very slowly, compared with those with the same sign moving in a state of horizontal alignment. As seen in Figs. 10(c,d) and 11(c,d), therefore, we do not perform numerical simulations after the alignment any longer: see Figs. 12 and 13 in advance. So far, we have found that, even after the in-line alignment, both particles continues to take the DEP motion without stoppage as if they were in a single body (aligned movement). Depending on the sign of the wall conductivity, they move upward or downward perpendicular to the electric field or the wall-fluid interface while sustaining their horizontal or vertical in-line alignment without a revolution. To see the wall effect on the aligned movement, therefore, we perform numerical simulations by varying the separation gap between the origin (or the particles) and wall up to G0 ¼ 12 in sequence and then present their results in Figs. 12 and 13. During the simulations, we assume that both particles should be in

S. Kang / Journal of Electrostatics 89 (2017) 30e41

near-contact with each other so as to move in a single body. That is, the particles are set apart from each other at ε ¼ 0:115 on the basis of surface to surface to mimic the aligned particles. Fig. 12 shows the DEP forces acting on and translational velocity of the aligned particles, interacting with the wall at sw ¼ 1=100, according to the separation gap. The figure indicates again that the wall with a negative-signed conductivity pushes the aligned particles upward so as to drift farther away from the wall and their DEP motion depends strongly on the particle alignment (arrangement) and the separation gap. For the horizontal alignment obtained with the same sign of the particle conductivity, the DEP motion exhibits symmetry and thus both particles have the same upward DEP force and translational velocity: see Fig. 12(a,b). With increasing separation gap, the DEP force exponentially decreases to near-zero, whereas the translational velocity increases to a certain value near the wall and then also decreases to near-zero. For the vertical alignment with different signs, the first (upper) and second (lower) particles receive downward and upward DEP forces, respectively, resulting into a slight upward net force on the aligned particles: see Fig. 12(c,d). Note that the DEP motion for the vertical alignment is driven by the difference in the DEP forces acting on both particles, whereas the motion for the horizontal alignment is driven by the DEP forces themselves. With increasing separation gap, the DEP force on the first particle stays nearly constant and that on the second particle decreases, thus also decreasing the net driving force. In addition, the translational velocity of the aligned particles exponentially decreases. Fig. 13 shows the DEP forces acting on and translational velocity of the aligned particles, interacting with the wall at sw ¼ 100. It is found again that the wall with a positive-signed conductivity draws the aligned particles downward so as to move closer to the wall. In addition, the dependency of the DEP forces and translational velocity on the particle alignment and the separation gap is very similar to that for the wall at sw ¼ 1=100 except for the direction of the DEP motion. At G0 ¼ 7, for all the vertical alignments considered in Figs. 12(c,d) and 13(c,d), the velocity of the aligned particles reads a near-zero value regardless of the sign of the wall conductivity. It implies that the aligned particles should move negligibly slowly or be nearly stationary. This is a main reason why we do not continue to perform numerical simulations after the in-line alignment in Figs. 10(c,d) and 11(c,d). 4. Conclusions To understand the dielectrophoretic (DEP) interaction among multiple particles and a wall, we have numerically studied twodimensional DEP motions of a pair of particles suspended freely in a viscous fluid, interacting with a nearby planar wall, under an external direct-current electric field. For the study, we solve the leaky dielectric model of Taylor-Melcher (Laplace equation) with a large sharp jump in the electric conductivity at the particle-fluid and wall-fluid interfaces for the electric potential and then integrate the Maxwell stress tensor to compute the DEP force acting on each particle, while we solve the continuity and Stokes equations for the flow field. For the simulations, we employ a finite-volume based numerical approach, where a sharp interface method is adopted for the solution of the electric potential and a directforcing based immersed-boundary method is for that of the flow field. Results show that, under a uniform electric field parallel to the wall-fluid interface, the motion depends strongly on the combination of the conductivity signs of the two particles and one wall

41

and the initial particle configuration including a separation gap between the particles and wall. When both particles have the same sign (positive or negative), regardless of the wall conductivity sign, they revolve and then get aligned in a line with the electric field despite the presence of a nearby wall. With different signs, they revolve in the opposite direction and then get aligned perpendicular to the field. In the courses of their revolution, alignment and aligned movement, regardless of their conductivity signs, both particles are repelled to move farther away from the wall being less conductive than the fluid, whereas they are attracted to move closer toward the wall being more conductive. Finally, it is also found that, as both particles drift apart from the wall, the wall effect gets weak and thus the particle-particle DEP interaction stands out as being more dominant than the particle-wall interaction. Acknowledgments This work has been supported by the Dong-A University research fund. References [1] Y. Ai, S. Qian, DC dielectrophoretic particle-particle interactions and their relative motions, J. Colloid Interf. Sci. 346 (2010) 448e454. [2] S. Kang, R. Maniyeri, Dielectrophoretic motions of multiple particles and their analogy with the magnetic counterparts, J. Mech. Sci. Technol. 26 (2012) 3503e3513. [3] M.R. Hossan, R. Dillon, A.K. Roy, P. Dutta, Modeling and simulation of dielectrophoretic particle-particle interactions and assembly, J. Colloid Interf. Sci. 394 (2013) 619e629. [4] M.R. Hossan, R. Dillon, A.K. Roy, P. Dutta, Hybrid immersed interfaceimmersed boundary methods for AC dielectrophoresis, J. Comput. Phys. 270 (2014) 640e659. [5] Y. Ai, Z. Zeng, S. Qian, Direct numerical simulation of AC dielectrophoretic particle-particle interactive motions, J. Colloid Interface Sci. 417 (2014) 72e79. [6] A.P.S. Bhalla, R. Bale, B.E. Griffith, N.A. Patankar, Fully resolved immersed electrohydrodynamics for particle motion, electrolocation, and self-propulsion, J. Comput. Phys. 256 (2014) 88e108. [7] S. Kang, Dielectrophoretic motion of two particles with diverse sets of the electric conductivity under a uniform electric field, Comput. Fluids 105 (2014) 231e243. [8] S. Kang, Dielectrophoretic motions of multiple particles under an alternatingcurrent electric field, Eur. J. Mecha. B-Fluid 54 (2015) 53e68. [9] C. Xie, B. Chen, C.-O. Ng, X. Zhou, J. Wu, Numerical study of interactive motion of dielectrophoretic particles, Eur. J. Mecha. B-Fluid 49 (2015) 208e216. [10] S. Kang, Dielectrophoretic motions of a single particle in the vicinity of a planar wall under a direct-current electric field, J. Electrostat 76 (2015) 159e170. [11] C.-T. Huang, C.-H. Weng, C.-P. Jen, Three-dimensional cellular focusing utilizing a combination of insulator-based and metallic dielectrophoresis, Biomicrofluidics 5 (2011) 044101. [12] M. Camarda, S. Scalese, A.L. Magna, Analysis of the role of the particle-wall interaction on the separation efficiencies of field flow fractionation dielectrophoretic devices, Electrophoresis 36 (2015) 1396e1404. [13] H. Zhao, H.H. Bau, On the effect of induced electro-osmosis on a cylindrical particle next to a surface, Langmuir 23 (2007) 4053e4063. [14] J.R. Melcher, G.I. Taylor, Electrohydrodynamics: a review of the role of interfacial shear stresses, Annu. Rev. Fluid Mech. 1 (1969) 111e146. [15] J.R. Melcher, Continuum Electromechanics, MIT Press, Cambridge, MA, USA, 1982. [16] D.A. Saville, Electrohydrodynamics: the Taylor-Melcher leaky dielectric model, Annu. Rev. Fluid Mech. 29 (1997) 27e64. [17] X.-D. Liu, R.P. Fedkiw, M. Kang, A boundary condition capturing method for Poisson's equation on irregular domains, J. Comput. Phys. 160 (2000) 151e178. [18] J. Kim, D. Kim, H. Choi, An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comput. Phys. 171 (2001) 132e150. [19] R. Glowinski, T.W. Pan, T.I. Hesla, D.D. Joseph, A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. Multiphas. Flow. 25 (1999) 755e794. [20] B. Cetin, D. Li, Dielectrophoresis in microfluidics technology, Electrophoresis 32 (2011) 2410e2427.