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Electrokinetic Motion of an Oil Droplet Attached to Water-Air Interface from Below Chengfa Wang, Yongxin Song, Xin-Xiang Pan, and Dongqing Li J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b10691 • Publication Date (Web): 09 Jan 2018 Downloaded from http://pubs.acs.org on January 9, 2018

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The Journal of Physical Chemistry

Electrokinetic Motion of an Oil Droplet Attached to Water–air Interface from Below Chengfa Wang,†, ‡ Yongxin Song,† Xinxiang Pan,† and Dongqing Li*, ‡

†

‡

Department of Marine Engineering, Dalian Maritime University, Dalian, 116026, China

Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

ACS Paragon Plus Environment

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ABSTRACT: The electrokinetic motion of a negatively charged oil droplet attached to the negatively charged water–air interface from below is numerically studied for the first time by a three–dimensional (3D) numerical model in this paper. The effects of the mobile water–air interface and the mobile water–oil interface on the electrokinetic motion of the attached oil droplet are investigated and discussed in terms of the zeta potentials at the water–air interface and the oil droplet surface, the applied electric field, the dynamic viscosity ratio of oil to water, and the droplet radius. The results show that the negatively charged oil droplet attached to the negatively charged water–air interface from below moves in the opposite direction to that of the external electric field, and its moving velocity increases with the increase of the electric field strength, the magnitudes of the zeta potentials at both the water–air interface and the water–oil interface, and the droplet size, as well as the dynamic viscosity ratio of oil to water.


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1. INTRODUCTION As an important electrokinetic phenomenon, electrophoresis is widely used to manipulate particles, biological cells or droplets in aqueous solutions for various applications ranging from biomedical science to environmental engineering.1–4 For a charged oil droplet attached to a planar water–air interface from below, when a direct current (DC) electric field is applied in water phase, the attached oil droplet will move along the water–air interface. This kind of electrokinetic motion can be used to manipulate and concentrate the oil droplets at the water–air interface, and has great and wide potential applications such as oil separation and recovery, water purification, sewage disposal, new material synthesis, and drug delivery. It is well–known that electrostatic charges on a solid–liquid interface will produce electric double layer (EDL)

5,6

. Similarly, the liquid–fluid interfaces also have electrostatic charges.7–12

For the water–air interface and water–oil interface, since only the water has ions, the EDL is formed on the water side of these interfaces. However, in comparison with the solid surfaces, the liquid–fluid interfaces are mobile. When a DC electric field is applied in water phase, the electroosmotic flow (EOF) will be generated in the EDL near the interface and pull the interface to move with the flow. Meanwhile, an electrophoretic force also exerts on the mobile interface charges by the electric field and drags the charges and consequently the interface to move. Therefore, the flow field near the liquid–fluid interfaces is much different from that near the solid surfaces. In order to describe this kind of electrokinetic flow near a liquid–fluid interface, Gao et al.

13

proposed a theoretical model with the consideration of the effects of the EDL and

surface charges (SC) at the interface (EDL+SC model). Later, Lee et al.

14,15

conducted an

experimental study on the electrokinetic flow near the liquid–fluid interfaces and found good agreement between the experimental results and the simulation results based on the EDL + SC


3


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model. For a micron–sized, charged oil droplet attached to a planar water–air interface from below, under a DC electric field applied in the water, the electroosmotic flow (EOF) of water near the mobile water–air interface will exert a hydrodynamic force on the attached oil droplet. Furthermore, the surface charges at the water–oil interface also experience an electric force from the applied electric field. Consequently, the charged oil droplet will move along the water–air interface. Regarding the electrokinetic motion of droplets, most of the relevant researches focus on the moving velocity of a charged conducting or non–conducting droplet fully submerged in an aqueous solution.16–23 The investigation on the motion of a droplet attached to a liquid–fluid interface is rare. Velev et al.

24

designed a microfluidic system to manipulate the water or

dodecane droplets floating on a surface of fluorinated oil utilizing dielectrophoretic force evoked from arrays of electrodes arranged below the oil. In this system, the movement mechanism of droplets is the dielectrophoresis, and completely different from that studied in this paper. Recently, several papers on the electrokinetic motion of a charged solid particle at a liquid–fluid interface are published.25–27 However, in comparison with solid particle, the mobile water–oil interface and the mobile surface charge of the oil droplet as well as the internal flow inside the oil droplet also have significant impact on the droplet’s motion. Overall, the understanding of the physics of the electrokinetic behavior of an oil droplet attached to the water–air interface is very limited. Thus, it is necessary to conduct fundamental investigation on the electrokinetic motion of an oil droplet attached to a planar water–air interface from below. In this paper, a 3D numerical model based on the EDL+SC model is developed to simulate the electrokinetic motion of a charged oil droplet attached to a planar water–air interface from below. The influences of the zeta potentials at the water–air interface and the oil droplet surface,


4

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the applied electric field, the dynamic viscosity ratio of oil to water, and the droplet radius are examined. The effects of the mobile water–air interface and water–oil interface on the attached oil droplet motion are analyzed and discussed.

2. THEORETICAL MODEL Consider a micron–sided oil droplet submerged in the water. Due to the buoyancy, the oil droplet will float up and then attach to the flat water–air interface. There is a contact point between the droplet surface and the flat water–air interface, as illustrated in Figure 1. A DC electric field parallel to the flat water–air interface is applied in the water from left to right. A Cartesian coordinate (x, y, z) is used to analyze this model, and the origin of the coordinate system is set at the center of the water–air interface, as shown in Figure 1. The water phase is considered to be infinitely large in this study. The size of the computational domain is much larger than the droplet radius (a).


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Figure 1. Schematic of an oil droplet attached to the water–air interface from below: (a) 3D view; (b) 2D view. The electrostatic charges of the water–air interface and the corresponding EDL exist only on the water side. The zeta potential of the water–air interface is denoted by ζair-water. Similarly, the oil droplet–water interface also has electrostatic charges and the EDL exists only on the water side. The zeta potential of the water–oil interface is denoted by ζoil-water. The oil droplet is considered as electrically non–conducting. The DC electric field is applied in the water phase. The interaction of the applied electric field with the EDLs of the water–air interface and the oil droplet–water interface will result in electroosmotic flows on the water side of these interfaces. For simplicity, the density of the oil is assumed to be close to that of the water, so that the buoyancy of the oil droplet is small. Furthermore, the surface tension of the water–air interface, approximately 72mN/m in room temperature,28 is very strong. For example, consider an oil droplet submerged in water under the conditions of the oil density ρo = 900 kg/m3 and the droplet radius a = 5 µm, the buoyancy-corrected droplet weight is about 5.13 ×10−13 N, negligible in comparison with the surface tension of the water–air interface. Therefore, in this model the water–air interface is assumed to be flat and not affected by the oil droplet. Due to the low speed of electroosmotic flow of water around the interfaces, the capillary number (eq 1)

29

is very

small, namely, the viscous force acting on the droplet is much weaker than the interfacial tension.

Ca =

µU γ

(1)


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where µ is the dynamic viscosity of the liquid, U is the flow velocity and γ is the surface tension of the water–oil interface. In addition, under weak electric filed, the electric capillary number (eq 2) 30, which is a ratio of the electric force to the interfacial tension, is also very small.

C=

aεε w E 2

(2)

γ

where ε and εw are the dielectric constant of vacuum and the relative dielectric constant of water, respectively; E is the electric field strength. Thus, to simplify the model, the micron–sized oil droplet is assumed to be spherical and non–deformable.

2.1. EDL field Due to the existence of ions in the water, the EDL fields exist only on the water side of the interfaces. The electric potential (ψ) distribution of the EDL field is determined by the Poisson– Boltzmann equation 31:

∇ 2ψ = −

ρe εε w

(3)

where ρ e is the local net charge density and given by

ρ e = −2 zen ∞ sinh(

ze ψ ) kbT

(4)

In the above equations, z and e are the valence of ions and the elementary charge, respectively;

n∞ is the bulk ionic number concentration ( n∞ = 1000NaM , Na = 6.022×1023/mol). In this study, the ionic concentration in the water phase is chosen as 1×10−6M. kb is the Boltzmann constant


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and T is the absolute temperature. The EDL potential (ψ) is equal to the zeta potential at the water–air interface and the water–oil interface. That is, the boundary conditions are: ψ = ζ air − water

at the water–air interface

(5a)

ψ = ζ oil − water

at the water–oil interface

(5b)

2.2. Applied electric field When the DC electric field is applied, the electric potential (V) field in water phase is described by Laplace’s equation:

∇ 2V = 0

(6)

The local applied electric field strength ( E ) is given by E = −∇ V

(7)

The boundary conditions of the electric field are

n⋅ E = 0

(8a)

at the bottom and side boundaries of the computation domain, at the water–air interface and the water–oil interface, where n is the unit normal vector of the boundary surfaces or the interfaces.

V = V0

(8b)

at the left boundary of the computation domain (Figure 1b).

V =0

(8c)

at the right boundary of the computation domain.


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2.3. Flow field The incompressible laminar flow under steady state is governed by the Navier–Stokes equation (eq 9) and the continuity equation (eq 10).

ρ U ⋅ ∇ U = −∇ p + µ∇ 2U + F

(9)

∇ ⋅U = 0

(10)

where ρ is the density of the liquid, µ is the dynamic viscosity of the liquid, U is the velocity vector, ∇p is the local pressure gradient, F is the body force. For electroosmotic flow, the velocity and the Reynolds number generally are very small, therefore, the inertia term ( ρU ⋅∇U ) is very small and can be neglected. As there are EDLs in the water phase, the electrical body force needs to be considered. The Navier–Stokes equation in the water phase can be reduced to ∇p = µ w∇ 2U w + ρ e E

(11)

where ρe, µw and U w are the local net charge density, viscosity and velocity vector of water, respectively. As oil is considered non–conducting, there is no electric field in the oil droplet; therefore, the item representing the electrical body force can be neglected and the Navier–Stokes equation in the oil phase can be reduced to ∇p = µo ∇ 2U o

(12)

where µo and U o are the viscosity and velocity vector of the oil, respectively.


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In this system, there is no applied pressure difference. Thus, the pressure and viscous stress are all set as zero at the inlet and outlet of the computational domain:

P=0

(13a)

n ⋅∇Uw = 0

(13b)

In comparison with the solid surface, the water–air interface and the oil droplet surface are mobile. After a DC electric field is applied in water phase, the electroosmotic flows (EOFs) generated on the water side of the water–air interface and the water–oil interface will exert shear stress on the two interfaces. At the same time, due to the existence of surface charges at the water–air interface and the water–oil interface, the external electric field will exert electric force on these mobile interface charges. The two factors will contribute to the movement of the two interfaces. At the oil droplet–water interface, the continuity of velocity (eq 14a) and the shear stress balance (eq 14b) must be obeyed13,15,32

Uw = Uo

(14a)

∂U w ∂U o σ oil − water E⊥ −β = ∂n ∂n µw

(14b)

where β = µo µ w is the dynamic viscosity ratio of oil to water, E⊥ is the tangential electric field strength on the droplet surface, σ oil− water is the surface charge density of the water–oil interface, and can be calculated by31

σ oil − water =

4 zen∞ ze ζ oil − water sinh( ) k 2 k bT

(15)


10

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2 2 where k = 2n∞ z e εε wkbT is the Debye–Hückel parameter.

In this study, the air above the bulk water surface is considered stationary. The water under the water–air interface experiences electroosmotic flow (EOF) originated in the EDL beneath the water–air interface. The boundary condition at the water–air interface is given by:

µw

∂U w = σ air − water E x ∂n

(16)

where E x is the applied electric field strength in the x–direction, σ air−water is the surface charge density of the water–air interface, and can be calculated by31

σ air − water =

4 zen ∞ ze ζ air − water sinh( ) k 2 k bT

(17)

In this study, we consider the micron–sized oil droplet is in an infinite water domain, and the height of the computation domain is set to be sufficiently large so that the zero velocity boundary condition (eq 18) is applied at the bottom boundary of the computation domain. Uw = 0

(18)

Considering the fact that the EOF is only in the x–direction in any position sufficiently far away from the droplet, the boundary condition for the two side boundary surfaces of the computation domain should be no viscous stress in the tangential direction (eq 13b) and no flow across these boundaries (eq 19). Uw ⋅ n = 0

(19)

2.4. Oil droplet velocity


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Due to the existence of the electrostatic charges at the oil droplet–water interface, when the external electric filed is applied in the water, an electric force ( Fe ) will exert on the oil droplet. Meanwhile, there is also hydrodynamic force ( Fh ) acting on the oil droplet by the flow filed around the droplet. Thus, the net force acting on the oil droplet is given by 33,34

Fnet = Fe + Fh

(20)

where the hydrodynamic force ( Fh ) contains two parts:

Fh = Fhin + Fho

(21)

where Fhin is the hydrodynamic force acting on the oil droplet surface by the flowing water in the EDL of the water–oil interface, and Fho is the hydrodynamic force generated by the flow of water outside the EDL of the water–oil interface. When the electroosmotic flow around the droplet is fully developed, the Fe is balanced by

Fhin 33,34, and the net force exerting on the droplet becomes33–36

Fnet = Fho = ∫ σ w ⋅ ndS

(22)

where S is the area of the outer surface of the EDL around the droplet, σ w is the hydrodynamic stress tensor in water, denoted by

σ w = − PI + µ w [∇U w + (∇U w )T ] (23) where I is the second order unit tensor. When the net force exerting on the oil droplet becomes zero (eq 24), the droplet motion reaches the steady state, and the steady velocity of the oil droplet (Ud) can be obtained.


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Fnet = 0

(24)

3. NUMERICAL SIMULATION COMSOL MULTIPHYSICS 5.2a® was employed to solve this 3D numerical model in this study. The droplet is put at the geometric center plane of the computation domain, as shown in Figure 1a. In order to reduce the computation, only half of the computation domain is calculated because of symmetry. The mesh elements at the water–air interface and the oil droplet–water interface have a huge impact on the accuracy of the calculated droplet velocity in the model; therefore, extremely fine meshes were used at these interfaces. Meanwhile, the dependence of the simulation results on the mesh number was also tested. Test results show that the difference of the moving velocities of the attached droplet is less than 0.5% when the mesh number is between 437,791 and 461,203. Thus, the mesh number of 437,791 was used in the simulations. Table 1 lists the parameter values used in the simulation. In all the simulations reported in this paper, the radius of droplet (a) is taken as the characteristic length and the reference velocity is Uref = 100 µm/s. Table1. Parameter values used in the numerical simulations Parameters

Values

Relative dielectric constant of water, εw

80

Dielectric permittivity in vacuum, ε (F/m)

8. 85×10−12

Viscosity of the water, µw (Pa.s)

1.0×10−3

Density of the water, ρw (kg/m3)

1.0×103

Density of the oil, ρo (kg/m3)

0.9×103

valence of ions, z

1


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Elementary charge, e (C)

1.602×10−19

Boltzmann constant, kb (J/K)

1.3806488×10−23

Absolute temperature, T (K)

298

In the numerical simulations, the droplet is kept stationary. Instead, a slip velocity condition (−Ud) is given at the bottom boundary of the computational domain, namely, the bottom boundary moves with the velocity of −Ud. During calculating the velocity of the oil droplet, different values of Ud are used. The chosen Ud which satisfies eq 24 is the steady velocity of the oil droplet. In order to evaluate the accuracy of the numerical methods used in this study, the model verification is also conducted, as shown in the Supporting Information.

4. RESULTS AND DISCUSSIONS 4.1. Flow field of the water–air interface system with an attached oil droplet Figure 2 shows the flow field around the oil droplet attached to the flat water–air interface under the following conditions: The dynamic viscosity ratio of oil to water β = 1, the radius of the oil droplet a = 5 µm, the zeta potential at the water–oil interface ζoil-water = −20 mV and the zeta potential of the water–air interface ζair-water = −50 mV, and the electric field strength E = 50 V/cm. In Figure 2, U*w is the dimensionless water velocity (U*w = Uw / Uref, Uref = 100 µm/s), and z* is the dimensionless z–coordinate (z* = z / a).


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Figure 2. Flow field of water–air interface system with an attached oil droplet under the conditions of ζair-water = −50 mV, ζoil-water = −20 mV, E = 50 V/cm, a = 5 µm and β = 1(U*w = Uw / Uref, z* = z / a, Uref = 100 µm/s): (a) water velocity profile along the negative z–coordinate direction and (b) distribution of velocity field around and inside the droplet at the cut plane of y =


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0. The insert in Figure 2a shows the plotting position of the flow velocity profile, and the arrow is the liquid velocity vector. The water velocity profile along a cut line parallel to z–coordinate far away from the oil droplet is displayed in Figure 2a. Different from a solid–liquid interface, the water–air interface is mobile. Thus, once an external electric field E is applied in water phase, the electroosmotic flow (EOF) generated on the water side of the negatively charged water–air interface (negative zeta potential) is in the same direction as E (positive x–direction) and exerts positive hydrodynamic stress on the water–air interface, which tends to drag the interface to move in the same direction as E. Meanwhile, the external electric field will exert electric force on the negative charges of the interface and tends to drag the interface to move in the opposite direction of E. Additionally, the external electric field will exert electric force on the negatively charged oil droplet that is attached to the water–air interface. This force will drag the oil droplet and hence the attached water–air interface to move in the opposite direction of E. In other words, the net motion of the water–air interface depends on the motion of the EOF formed near the water– air interface, the electrophoretic motion of the negatively charged water–air interface and the motion of the negatively charged oil droplet attached to the water–air interface. As indicated by eq 16, the shear stress generated by the EOF on the water–air interface ( µ w

∂U w ) is balanced by ∂n

the electrophoretic force on the negative surface charges of the water–air interface ( σ air − water E x ). Therefore, the net motion of the water–air interface is caused by the electrophoretic motion of the negatively charged oil droplet attached to the water–air interface, and moves in opposite direction to the applied electric field E. The velocity at point A in Figure 2a is the net velocity of


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the water–air interface, and the velocity profile from point A to point B is the detailed distribution of velocity field in the EDL near the water–air interface. Figure 3 shows the flow field around and inside a negatively charged oil droplet that is fully submerged in water and far away from any boundaries, under a DC electric field. The net motion of the mobile water–oil interface is the combination of the motion dragged by the EOF formed around the droplet surface and the electrophoretic motion of the negatively charged water–oil interface. The net motion of the mobile water–oil interface determines the flow field inside the droplet. Two same–sized vortices are generated inside the droplet because of symmetry. However, when the same oil droplet is attached to the flat water–air interface, the flow field is much different due to the effect of the water–air interface, as displayed in Figure 2b. Firstly, the moving water–air interface with negative moving velocity (Figure 2a) will drag the water–oil interface near the contact point to move faster in opposite direction to E. More importantly, near the contact point the local electric field becomes much stronger than that at other areas, which greatly increases the electrophoretic motion of the negatively charged water–oil interface near the contact point. Thus, in Figure 2b it is not difficult to understand that inside the droplet the flow velocity near the contact point to the water–air interface is large and only one visible vortex rotating counter–clockwise is formed near the contact point.


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Figure 3. Flow field around and inside an oil droplet submerged in stationary water (E = 50 V/cm, ζoil-water = −20 mV, β = 1, a = 10 µm). The color legend shows the dimensionless velocity magnitude (Uref = 100 µm/s). Regarding the effect of the water–air interface on the flow field inside the oil droplet, it can be seen more clearly in Figure 4. Figure 2b shows the velocity field on the cut plane of symmetry (y = 0). It should be realized that the flow field is different on different cut planes. For example, on the cut plane of y = − 0.3a, the surface of the oil droplet is no longer in contact with the water–air interface, as shown in Figure 4. As a result, the electric field strength between the water–air interface and the oil droplet is smaller significantly in comparison with that at the contact point. Hence, the velocities of the EOFs formed near the two interfaces and the electric forces acting on the surface charges at the two interfaces all decline. Therefore, it is easily understood that the flow velocity in Figure 4a is much smaller than that in Figure 2b. In addition, in Figure 4a the magnitude of ζair-water is larger than that of ζoil-water, the electrophoretic velocity (in the opposite direction to E) of the negatively charged water–air interface are larger than EOF velocity (in the same direction as E) around the oil droplet surface. Therefore, in Figure 4a we can see the net velocity of water between the droplet and the water–air interface is negative (in the opposite direction to E). By contrast, in Figure 4b the magnitude of ζair-water is smaller than that of ζoil-water, the flow between the two interfaces is in same direction as E; hence the flow tends to drag the water–oil interface on the top of the droplet to move in the same direction as E. However, in comparison with Figure 4a, the magnitude of ζoil-water is much larger in Figure 4b, accordingly, the electrophoretic motion of the negatively charged water–oil interface increases greatly. Overall, in Figure 4b, the water–oil interface on the top of the droplet still moves in opposite direction to E and its velocity is much larger than that at other areas.


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Figure 4. Distribution of velocity field on the cut plane of y = − 0.3a (E = 50 V/cm, β = 1, a = 5 µm): (a) ζair-water = −50 mV, ζoil-water = −20 mV; (b) ζair-water = −20 mV, ζoil-water = −50 mV. Based on the interface boundary conditions, Eq. (14b) and Eq. (16), the applied electric field strength (E), the zeta potentials of the water–air interface (ζair-water) and the oil droplet–water interface (ζoil-water), and the dynamic viscosity ratio (β) of oil to water affect the flow field in both water and oil droplet. The droplet zeta potential (ζoil-water) determines the electric force acting on the droplet by the external electric field. Additionally, the surface area of the oil droplet in the


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EDL region of the water–air interface is different under different droplet diameters (d). All these factors can influence the electrokinetic velocity of the oil droplet contacting the water–air interface, and will be discussed in the following sections.

4.2. Effects of electric field and droplet size Under different electric fields, the electric forces exerting on the surface charges of the water– air interface and the water–oil interface are different. Meanwhile, the EOFs produced near these two interfaces are also different. Thus, the moving velocity of an oil droplet attached to the water–air interface varies with the electric field strength. Furthermore, the droplet diameter determines the surface area of the water–oil interface interacting with the flowing water near the water–air interface, and consequently affects the droplet motion. Figure 5 shows the effect of the electric field and the oil droplet size on the moving velocity of the oil droplet attached to the water–air interface from below. In this section, the zeta potentials at the water–air interface (ζairwater)

and the oil droplet–water interface (ζoil-water) are set to −50 mV and −20 mV, respectively;

the dynamic viscosity ratio of oil to water is β = 1. The dimensionless velocity of the oil droplet is defined by U*d = Ud / Uref (Uref = 100 µm/s).


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Figure 5. Effects of electric field and droplet size on droplet velocity (ζair-water = −50 mV, ζoil-water = −20 mV, β = 1). From Figure 5, it can be seen that the moving direction of the negatively charged oil droplet attached to the negatively charged water–air interface is opposite to the direction of the applied electric field E, and the droplet velocity increases with the increase of Ε. In addition, under the same electric field, the droplet velocity increases with the increase of the droplet radius a. Furthermore, Figure 5 compares the electrokinetic velocity of an oil droplet of 10 µm in radius between two cases: the droplet is fully submerged in stationary and infinitely large water, and the droplet is attached to a flat water–air surface from below. It can be clearly seen that the velocity of the same droplet fully submerged in stationary water is much smaller than that attached to the water–air interface. The reasons are as follows. For a negatively charged oil droplet attached to the water–air interface, after applying a DC electric field in water phase, the electric field will exert negative electric force (in negative x– direction) on the oil droplet, making the negatively charged oil droplet move towards the positive electrode. Meanwhile, due to the interaction of the applied electrical field with the negative surface charges of the water–air interface, the water–air interface moves towards the positive electrode, i.e., to the negative x–direction. Thus the motion of the water–air interface will generate a hydrodynamic force on the droplet in the negative x–direction. Although the hydrodynamic force produced by the electroosmotic flow underneath the water–air interface is in the same direction as the electric field E, i.e., in the positive x–direction, the net force on the oil droplet is dominated by the electrophoretic forces acting on the oil droplet and on the surface charges of the water–air interface. The net force dictates that the oil droplet moves in the opposite direction of E. With the increase in the applied electric field Ε, the net hydrodynamic


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force acting on the droplet increases. Therefore, it is not difficult to understand that the droplet velocity increases with the increase of the applied electric field, Ε, as shown in Figure 5. From Figures 2b and 4, one can see that the strongest local electric field and flow field occur in the region near the point of oil droplet contacting the water–air interface. Apparently, the droplet experiences the strongest electrophoretic force and the strongest hydrodynamic force from the moving water–air interface in this area. When the droplet size is larger, the surface area of the water–oil interface near the contact point region is larger. Accordingly, the droplet will experience higher hydrodynamic force from the flowing water near the water–air interface. Although larger droplet will experience a larger resistance from the water, the electric force acting on the droplet by the applied electric field also increases. Therefore, it is not difficult to understand that the larger droplet has a higher velocity as shown in Figure 5. For a negatively charged oil droplet submerged in an aqueous solution and far away from all boundaries, its motion is only determined by the electrophoretic force acting on the droplet under an applied electric field. However, when the same droplet is attached to the water–air interface from below, another force generated by the electrophoretic motion of the negatively charged water–air interface also exerts on the droplet. Therefore, the velocity of the oil droplet attached to the water–air interface is larger than the velocity of the same–sized oil droplet fully submerged in bulk water, as shown in Figure 5. Again, this indicates that the mobile and charged water–air interface has a great influence on the velocity of the attached oil droplet at the water– air interface. In addition, different from the solid particle, the oil droplet surface is mobile. As displayed in Figure 3, because of the electrokinetic effect of the mobile and negatively charged water–oil interface, the oil droplet surface moves in the opposite direction to the electroosmotic flow in the


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EDL around the oil droplet. This reduces the hydrodynamic force ( F ho ) acting on the droplet, hence the electrophoretic velocity of the droplet. Thus, from Figure 5 it can be seen that under same conditions the droplet electrophoretic velocity is smaller than the theoretical Smoluchowski velocity (namely, the electrophoretic velocity of a solid particle in a stationary electrolyte solution).6 When the negatively charged oil droplet is attached to the water–air interface from below, the moving water–oil interface may have an impact on the velocity of the attached oil droplet. The dynamic viscosity ratio of oil to water (β) is a key parameter in Eq. (14b) to reflect the influences of the water–oil interface. When the dynamic viscosity ratio of oil to water (β) is smaller, the water–oil interface will move faster. Accordingly, the effect of internal flow of the droplet is stronger; hence the velocity of the attached droplet is smaller. As shown in Figure 6, the droplet velocity increases with β. However, when the viscosity ratio β is larger than 10, the oil droplet approaches a rigid particle20, and the influence of internal flow becomes very weak and the oil– water interface gradually loses its mobility. Therefore, in Figure 6 one can see that, when the dynamic viscosity ratio β is larger than 10, the droplet velocity is essentially unchanged.


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Figure 6. Effect of dynamic viscosity ratio of oil to water on droplet velocity (E = 50V/cm, ζairwater

= −50 mV, ζoil-water = −50 mV, a = 5 µm).

4.3. Effects of zeta potentials of water–air interface and water–oil interface According to the interface boundary conditions (eqs 14 and 16), the zeta potential of water–air interface affects the EOF velocity near the water–air interface and, consequently, influences the motion of the oil droplet attached to the water–air interface from below by affecting the hydrodynamic force acting on the oil droplet and the motion of the water–oil interface. The zeta potential of water–oil interface determines the electric force acting on the oil droplet by the external electric field, the electrophoretic motion of the mobile water–oil interface, and the EOF around the water–oil interface, and then impacts the motion of the attached oil droplet. Figure 7 shows the effects of the dimensionless zeta potentials (the reference zeta potential is ζref = −50 mV) of the water–air interface and the water–oil interface with the conditions of β = 1, a = 5 µm, and E = 50 V/cm. In Figure 7, when examining the effect of the zeta potential of one

interface, the zeta potential of another interface is fixed. It is obvious from Figure 7 that the magnitude of the negative droplet velocity all increase with the increases of the dimensionless zeta potentials of water–air interface (ζ *air−water ) and water–oil interface (ζ *oil−water ). This may be understood as follows: With the increase in the dimensionless zeta potential of the water–air interface (namely, the increase in the absolute value of the negative zeta potential of the water– air interface), due to the electrokinetic effect of the mobile water–air interface, the net flow of water near the water–air interface is faster in the opposite direction to the applied electric field E, as shown in Figure 4. Consequently, the oil droplet will be subjected to a higher hydrodynamic force in that direction. Furthermore, with the increase in the absolute value of the negative zeta potential of the droplet surface, the droplet will experience larger electrophoretic force by the


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electric field owing to the larger surface charge density at the oil droplet surface (Eq. (15)). Therefore, it is not difficult to understand the droplet velocity all increase with the increase in the dimensionless zeta potentials of water–air interface and water–oil interface in Figure 7.

Figure 7. Effects of zeta potentials of water–air interface and water–oil interface on oil droplet velocity (β = 1, E = 50 V/cm, a = 5 µm, ζref = −50 mV). Generally, in the same aqueous solution, the zeta potentials of the water–air interface and the water–oil interface have the same sign. They are all negative under a wide range of pH conditions, but become positive under strong acidic conditions (pH < 5)

37–39

. From Figure 7 it

can be seen that the negatively charged oil droplet attached to a negatively charged water–air interface will move in opposite direction of the applied electric field. When the zeta potentials of the two interfaces all become positive under strong acidic conditions, it is not difficult to predict that the droplet will move in the same direction of the applied electric field.

5. CONCLUSIONS


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This paper presents a numerical investigation on the electrokinetic motion of an oil droplet attached to the water–air interface from blow. Just as the most solid surfaces, the liquid–fluid interfaces also have electrostatic charges. For the water–air interface and the water–oil interface, since the ions only exist in water phase, the EDL forms only on the water side of these interfaces. However, different from the solid surfaces, the liquid–fluid interfaces are mobile. Therefore, under an applied electric field, the EOF near the interfaces and the electrophoretic force acting on the mobile charges of these interfaces will influence significantly the flow field near the liquid–fluid interfaces, and consequently the motion of the oil droplet attached to the water–air interface. The simulation results show that the moving direction of the negatively charged oil droplet attached to the negatively charged water–air interface is opposite to that of the applied electric field. The velocity of the oil droplet increases with the applied electric field and the zeta potentials of the interfaces. While the droplet velocity increases also with the oil–water viscosity ratio and the droplet size, the effects of these two parameters are small. The fundamental understanding of electrokinetic motion of the charged oil droplet attached to the water–air interface from below developed in this paper has great potential applications in oil droplet concentration and separation.

ASSOCIATED CONTENT Supporting Information. Model verification on the accuracy of the numerical methods used in this study (PDF)

AUTHOR INFORMATION


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Corresponding Author *E-mail: [email protected]. Tel: +1-519-888-4567 x38682. ORCID Dongqing Li: 0000-0001-6133-4459 Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENT The authors wish to thank the financial support of the Chinese Scholarship Council to Chengfa Wang, the National Natural Science Foundation of China (51679023) to Yongxin Song, and the National Natural Science Foundation of China (51479020) to Xinxiang Pan. The Natural Sciences and Engineering Research Council of Canada through a research grant to D. Li are greatly appreciated.

ABBREVIATIONS 3D, three–dimensional; DC, direct current; EDL, electric double layer; EOF, electroosmotic flow SC, surface charges.


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