Migration behaviors of leaky dielectric droplets with electric and hydrodynamic forces

The external electric field enables separation and transport of droplets effectively in microfluidic devices. Herein, a volume-of-fluid (VOF) + level-set (LS) + smoothed physical parameters (SPP) method associated with the dynamically adaptive grid technique is extended to simulate three-dimensional leaky dielectric droplets in the electric field. The effects of electric and hydrodynamic forces on droplet behaviors are investigated. It is demonstrated that the electric force could act toward the outside or inside of a droplet and produce different droplet deformations. For the dielectrophoretic migration of droplets in the nonuniform electric field, the electric force has a dominant effect. It is found that when the electric conductivity ratio is greater than 1, an unbalanced electric force toward a stronger electric field is generated, bringing about the migration toward a stronger electric field. In contrast, when the electric conductivity ratio is smaller than 1, the unbalanced electric force direction is reversed and the droplet migrates toward a weaker electric field. The hydrodynamic force produces little promotion or hindrance to droplet migration. A greater permittivity ratio usually produces greater electric force and migration velocity. The droplet migrates along one direction in a symmetric nonuniform electric field but tends to migrate along the normal direction of electric potential profiles in an asymmetric nonuniform electric field.

Figure 1

Physical problem of droplet deformation in the electric field: (a) schematic diagram and (b) grid system.

Figure 2

Schematic of droplet deformation.

Figure 3

Evolution of the droplet deformation rate with time for three different grid systems (${\lambda }_{\sigma }=0.5, {\lambda }_{\varepsilon }=2, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$).

Figure 4

Droplet deformation rates: (a) ${\lambda }_{\sigma }=2.5, {\lambda }_{\varepsilon }=2, {\lambda }_{\mu }=1$; (b) ${\lambda }_{\sigma }=100, {\lambda }_{\varepsilon }=17.54, {\lambda }_{\mu }=0.00071$; (c) ${\lambda }_{\sigma }=5, {\lambda }_{\mu }=16, \mathrm{C}{\mathrm{a}}_E=0.2$; and (d) ${\lambda }_{\varepsilon }=2, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 5

Velocity vectors (left) and stream functions (right) around the droplet at $y=5R$: (a) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=0.5, {\lambda }_{\mu }=16, \mathrm{C}{\mathrm{a}}_E=0.2, D=0.092$; (b) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=8, {\lambda }_{\mu }=16, \mathrm{C}{\mathrm{a}}_E=0.2, D=0.014$; and (c) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=30, {\lambda }_{\mu }=16, \mathrm{C}{\mathrm{a}}_E=0.2, D=-0.123$.

Figure 6

(a) Velocity magnitude distributions around the droplet at $y=5R$ when ${\lambda }_{\sigma }=0.2, {\lambda }_{\varepsilon }=2, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$. The red color corresponds to high velocity while the blue color corresponds to low velocity. (b)–(d) The dimensionless maximum velocity magnitude near the droplet interface. (b) ${\lambda }_{\sigma }=2.5, {\lambda }_{\varepsilon }=2, {\lambda }_{\mu }=1$; (c) ${\lambda }_{\sigma }=5, {\lambda }_{\mu }=16, \mathrm{C}{\mathrm{a}}_E=0.2$; and (d) ${\lambda }_{\varepsilon }=2, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 7

Distribution of volume density of local free charges $q$ at $y=5R$: (a) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=0.5, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$; and (b) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=30, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 8

Electric force distribution around the droplet interface at $y=5R$: (a) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=0.5, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$. $D>0$; (b) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=10, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2, D\approx 0$; (c) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=30, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2, D<0$; (d) ${\lambda }_{\sigma }=0.5, {\lambda }_{\varepsilon }=0.2, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2, D>0$; (e) ${\lambda }_{\sigma }=0.5, {\lambda }_{\varepsilon }=1, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2, D<0$; and (f) ${\lambda }_{\sigma }=0.5, {\lambda }_{\varepsilon }=2, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2, D<0$.

Figure 9

Transient development of droplet position along $z$ direction when ${\lambda }_{\sigma }=100, {\lambda }_{\varepsilon }=17.02, {\lambda }_{\rho }=1.04$, and ${\lambda }_{\mu }=0.0013$.

Figure 10

Electric potential distribution at $y=5R$ when the droplet does not exist.

Figure 11

Transient development of droplet position along $z$ direction: (a) ${\lambda }_{\varepsilon }=2, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$ and (b) ${\lambda }_{\varepsilon }=10, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 12

Transient development of droplet position along $z$ direction when ${\lambda }_{\sigma }=5, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 13

Droplet deformation and electric force distribution at $y=5R$ in the moment of $T=14$: (a) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=0.5, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$; (b) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=10, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$; and (c) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=30, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 14

Droplet velocity fields at $y=5R$ in the moment of $T=14$: (a) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=0.5, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$; (b) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=10, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$; and (c) ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=30, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 15

Droplet deformation, electric force distribution (left), and the corresponding velocity fields (right) at $y=5R$ in the moment of $T=14$ when ${\lambda }_{\sigma }=2, {\lambda }_{\varepsilon }=10, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 16

Transient development of droplet position along $z$ direction when ${\lambda }_{\sigma }=0.5, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 17

Droplet deformation and electric force distribution at $y=5R$ in the moment of $T=14$: (a) ${\lambda }_{\sigma }=0.5, {\lambda }_{\varepsilon }=0.2, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$; (b) ${\lambda }_{\sigma }=0.5, {\lambda }_{\varepsilon }=5, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$; and (c) ${\lambda }_{\sigma }=0.5, {\lambda }_{\varepsilon }=20, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 18

Droplet velocity fields at $y=5R$ in the moment of $T=14$: (a) ${\lambda }_{\sigma }=0.5, {\lambda }_{\sigma }=0.2, \mathrm{C}{\mathrm{a}}_E=0.2$; (b) ${\lambda }_{\sigma }=0.5, {\lambda }_{\sigma }=5, \mathrm{C}{\mathrm{a}}_E=0.2$; and (c) ${\lambda }_{\sigma }=0.5, {\lambda }_{\varepsilon }=20, \mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 19

Droplet deformation, electric force distribution (left), and the corresponding velocity fields (right) at $y=5R$ in the moment of $T=14$ when ${\lambda }_{\sigma }=0.8, {\lambda }_{\varepsilon }=0.2, {\lambda }_{\mu }=1$, and $\mathrm{C}{\mathrm{a}}_E=0.2$.

Figure 20

Transient development of droplet position along $z$ direction when ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=2, {\lambda }_{\mu }=1$.

Figure 21

Evolution of droplet positions and corresponding electric force distributions at $y=5R$ when ${\lambda }_{\sigma }=5, {\lambda }_{\varepsilon }=2, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$: (a) $T=0$, (b) $T=6$, (c) $T=12$, and (d) $T=18$.

Figure 22

Evolution of droplet positions and corresponding electric force distributions at $y=5R$ when ${\lambda }_{\sigma }=0.5, {\lambda }_{\varepsilon }=2, {\lambda }_{\mu }=1, \mathrm{C}{\mathrm{a}}_E=0.2$: (a) $T=0$, (b) $T=120$, (c) $T=240$, and (d) $T=480$.