Dynamic electrophoresis of concentrated droplet dispersions at arbitrary surface potentials

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www.elsevier.com/locate/ces

Dynamic electrophoresis of concentrated droplet dispersions at arbitrary

surface potentials

James Lou, Eric Lee

∗

Department of Chemical Engineering, Institute of Polymer Science and Engineering, National Taiwan University, Taipei, 10617 Taiwan

Received 30 January 2007; received in revised form 16 July 2007; accepted 16 July 2007 Available online 24 July 2007

Abstract

Dynamic electrophoresis of a dispersion of spherical droplets with arbitrary zeta potential and double layer thickness is analyzed in this study, based on Kuwabara’s unit cell model. Pseudo-spectral method with Chebyshev polynomials is adopted to solve the corresponding general electrokinetic equations. Similar to the observations of hard spheres, the droplets exhibit non-linear phenomena such as maxima or minima on both mobility magnitude and phase lag with varying frequency of input alternative electric ﬁeld, as the zeta potential of the droplets become high. Our results presented here add another dimension to the previous corresponding results of hard spheres, with the parameter of viscosity ratio,_o/_i, where_o and _i refer to the ﬂuid viscosities outside and inside of the droplets, respectively. Consistent qualitative behaviors are observed with_o/_i = 0 representing the hard spheres situation. No abrupt change is observed associating with the phase change to the droplets. The dynamic mobility is found to be faster in magnitude, as might be expected.

Keywords: Dynamic electrophoresis; Droplet; High zeta potential

1. Introduction

When an alternating voltage is applied to a colloid, the parti-cles move back and forth at a velocity that depends on its size, zeta potential, and on the frequency of the applied ﬁeld (Hunter, 1998;O’Brien et al., 1995;Mizuno et al., 2000;Dukhin et al.,

2000). As they move, the particles generate sound waves, an

effect generally referred to as the electrokinetic sonic

ampli-tude, or ESA (O’Brien, 1988). Another related electroacoustic

phenomenon is called vibration potential, or CVP, which states that a sound wave would generate an alternating electric ﬁeld as it passes through a suspension of colloids. It turns out that CVP is the converse of ESA, while the latter has become dominant in the last two decades, due to the fact that apparently it is eas-ier to measure the resulting sound waves than the tiny voltages,

considering the electric noise associated with CVP (Hunter,

1998). ESA is able to provide reliable particle size information as well as zeta potential, especially for concentrated suspen-sion of colloids. Compared with other techniques which usually

∗_{Corresponding author. Tel./fax: +886 2 23622530.}

E-mail address:ericlee@ntu.edu.tw(E. Lee).

require dilution, the ESA can measure the concentrated sample in its natural state of interest. It has been found that dilution will signiﬁcantly change the surface properties of the original suspension system (Hunter, 1998). This is the major advantage of the electroacoustic technique, such as ESA. A thorough

re-view of the experimental development was provided byHunter

(1998). As for the theoretical part,O’Brien (1988)investigated the dynamic electrophoresis of a spherical rigid colloid with very thin double layer in a dilute dispersion. He derived a for-mula relating the measurable information of the output ESA signals to the dynamic electrophoretic mobility. The surface properties were obtainable once these data were established. Various attempts were made ever since to loosen the restriction

encountered in O’Brien’s work. For example,Mangelsdorf and

White (1992)took into consideration the effect of double layer

polarization and stern layer. Based on their studies, Ohshima

(1996)derived an analytical expression for the case of low zeta potential. These studies were all focused on the dilute disper-sion, i.e., a spherical particle in an inﬁnite solution.

Later, Ohshima extended further to investigate the case of concentrated suspensions. According to his analysis, the dy-namic electrophoretic mobility of the particle could be seriously

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inﬂuenced by the presence of neighboring particles (Ohshima, 1997, 1999a,b), indicating the complexity of concentrated dispersions.

Generally speaking, analytical approach is valid for the limiting cases of linearized electrokinetic equations. When linear effect is involved, for example, the complete non-linear Poisson–Boltzmann equation, it is impossible to solve it analytically under the conditions of arbitrary zeta potential and arbitrary double layer thickness. A numerical approach is necessary under the situation. As a result, the authors have conducted a series of numerical investigations in the ﬁeld of electrokinetics, with polarization effect as the major factor of interest, which is generally not observable in traditional

analytical approaches (Lee et al., 1999). For the dynamic

electrophoresis in particular, several works (Lee et al., 2001, 2002a, 2003;Hsu et al., 2002; Lou et al., 2005) were reported by the authors regarding the rigid spheres. Typical non-linear phenomena due to polarization effect were generally observed, such as the exhibition of maxima or minima of mobility, as the frequency of the applied electric ﬁeld was varied.

The electrophoresis of a dispersion of non-rigid entities has many applications in practice. Emulsion, sol, and foam, for instance, belong to this category. Compared with that of rigid entities, the analysis of non-rigid entities is more complicated because the electrokinetic equations for the space inside an en-tity and outside an enen-tity need to be solved simultaneously. Sev-eral attempts have been made in the literature to model the static electrophoretic behavior of a dispersion of non-rigid entities. Booth (1951)ﬁrst investigated the electrophoretic behavior of a

mercury drop. Later on,Levine and O’Brien (1973)examined

thoroughly the motion of a mercury drop and found that it can be influenced significantly by an externally applied static elec-tric field. They found that the polarization effect is dominant in

the electrophoresis of the mercury drop.Baygents and Saville

(1991a,b)analyzed the electrophoretic mobility as a function of the zeta potential for conducting and non-conducting drops. Ohshima (1999c) also extended the study to concentrated dispersion of charged spherical mercury drops and derived a formula for it.

It should be noted that while there were many published reports on dynamic electrophoresis, essentially all were focused on the rigid particle. And while there are many studies on the electrophoresis of droplet dispersions, few if none of them are focused on the dynamic electrophoresis; essentially all of them are confined to the static electrophoresis. As a result, the au-thors studied the dynamic electrophoresis of a spherical droplet within a solid cavity (Lee et al., 2006). Indeed it is possible that the tiny electroacoustic vibration is able to penetrate the droplet interface and set the inner fluid into an internal flow, thus changing the overall dynamic behavior of the emulsions. It modeled the electrophoresis in a porous medium with spe-cial emphasis on the boundary effect. Polarization effect of double layer was found to be important. As for the dynamic electrophoresis of concentrated dispersion of droplets, the au-thors also presented a study confined to low zeta potential sit-uation (Hsu et al., 2007). It was found there that the mobility is larger than the corresponding rigid particles as the frequency

gets higher. No polarization effect is involved, however, since the zeta potential is low.

As a conclusive generalization of the previous work, we consider here the dynamic electrophoresis of spherical droplets with arbitrary zeta potential and double layer thickness, as well as arbitrary volume fraction of droplets. The double layer and polarization overlapping effect are taken into account.

Kuwabara’s (1959)unit cell model is adopted with the bound-ary conditions on cell surface proposed byShilov et al. (1981), which has been found to give better experimental agreement in many other related works on electrokinetic phenomena, as con-centrated suspension is considered. A pseudo-spectral method based on Chebyshev polynomials is adopted to solve the re-sultant governing equations subject to the associated boundary conditions. The method has proved to be both powerful and accurate in solving the general electrokinetic equations, static

or dynamic (Lee et al., 1999, 2001). Moreover, it also

pre-vailed in the analyses of polymeric ﬂuid ﬂow, as published

by the authors (Lee et al., 2002b, 2004). Various important

factors are examined, such as the frequency of the applied electric field, the volume fraction of droplets, the thickness of the double layer, and the relative viscosity of the droplet fluid. Its influences on the dynamic electrophoretic behavior of the dispersion are thoroughly explored and discussed in this study.

2. Theory

Consider the electrophoretic problem illustrated in Fig. 1

where a dispersion of electrolyte-free drops of radius a move

with a velocity U= Ue−itez under an applied electric ﬁeld

E= EZe−itez back and forth in the z-direction. Note that

both U and E are complex quantities with real and imaginary

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parts to represent its oscillating nature. The dispersion medium contains z1: z2 electrolyte, z1 and z2 being, respectively, the

valences of cations and anions. Kuwabara’s (1959) unit cell model is adopted, where a representative droplet is surrounded by a concentric liquid shell of radius b. The spherical coordi-nates (r,, ) are used with its origin located at the center of the droplet. The physical properties of the electrolyte solution and those of the drop ﬂuid are assumed to maintain constant, and the shape of the drop remains spherical, which will be elaborated later again in this section. Moreover, creeping ﬂow isassumed for the system under consideration.

The electrical potential of the system under consideration,

, can be described by the Poisson equation:

∇2_{= −}e = − 2 j=1 zjenj , (1)

where∇2_{is the Laplace operator,}_{the electrical potential, e}

the space charge density, the dielectric constant of dispersion medium, e the elementary charge, and nj the number

concen-tration of ionic species j. The conservation of ionic species j yields jnj jt = ∇Q Dj ∇nj + njˆezj kT ∇ + nju , (2)

where∇ is the gradient operator, Dj the diffusivity of ionic

species j, k the Boltzmann constant, and u the velocity of dis-persion medium. Since the Reynolds number is very small, the ﬂow ﬁeld is described by

∇ · u = 0, (3)

ju_jt = −∇p + ∇2

u− e∇, (4)

where p is the pressure, and and , respectively, the viscosity and the density of the ﬂuid.

Following the same treatment as before (Lee et al., 1999), the variables, nj, u, and p can be decomposed into an equilibrium

state plus a perturbed state arising from the applied electric ﬁeld. The strength of the applied electric ﬁeld is assumed very weak in comparison with the equilibrium state value established by the charge on the droplet surface. Therefore,

(r, , t) = e(r)+ (r, )EZe−it, (5)

nj(r,, t) = nej(r,) + nj(r,)EZe−it, (6)

u(r,, t) = 0 + u(r, )EZe−it, (7)

p(r,, t) = pe(r,) + p(r, )EZe−it, (8) where the subscript e denotes equilibrium property and the symbol represents a perturbed quantity arising from the ap-plied electric ﬁeld. Note that, for the equilibrium case, since the droplet remains stagnant when the applied electric ﬁeld is absent, the velocity vanishes. The governing equations and its associated boundary conditions are grouped into two sub-problems: the equilibrium system and the perturbed system as follows.

2.1. Equilibrium system

Since the distribution of ionic species at equilibrium follows Boltzmann distribution, the governing equation fore is ∇2_e_{= −} N j=1 zjˆene_j0 exp −zjˆee kT , a < r < b, (9) ∇2_e_{= 0, 0 < r < a.} ₍₁₀₎

Eq. (9) is the well-known Poisson–Boltzmann equation. 0 < r < a denotes the region inside the drop, and a < r < b represents the dispersion medium. ne_j₀ is the bulk concentra-tion of ionic species j. Suppose that the potential on the droplet surface is kept constant, and no net electric current across the outer virtual surface, a logic claim reﬂecting the fact that the bulk system is electrically neutral as a whole. Then the boundary conditions associated with Eqs. (9) and (10) are

e= a at r= a, (11) de dr = 0 at r = b. (12) Moreover, je jr = 0 at r = 0 (13)

considering the spherical symmetry of the system under con-sideration.

2.2. Perturbed system

A modiﬁed Boltzmann distribution is presented here to ac-count for the polarization effect of the double layer as the droplet is set to motion:

nj = nej0exp −zjˆe kT (e+ Eze −it_{+ g} jEze−it) , (14)

where gj represents the equivalent perturbed potential arising

from double layer polarization (Lee et al., 1999). Substituting Eqs. (5)–(8) and (14) into Eqs. (1)–(4), after some mathematical

treatments available elsewhere (Lee et al., 2006), we end up

with the governing equations for the perturbed system, which are further divided as

(A) Inside the droplet:

−ii uEZe−it= − ∇ pEZe−it

+ i∇

2_uEZ

e−it, 0 < r < a, (15) where Eq. (15) is the Navier–Stokes equation in complex form. Note that u is the perturbed velocity and is a complex quantity with both real and imaginary parts to represent its oscillating nature. Moreover, since Eq. (15) is of complex nature involving complex quantities, there are actually two corresponding equa-tions associated with it, representing real and imaginary parts separately. The associated boundary conditions are

urEZe−iwt= 0, r = 0, (16)

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which are obtained from the symmetric nature of the present problem. Note that there is no electrolyte inside the drop; thus,

nj is zero when 0 < r < a.

(B) Outside the droplet:

The governing equations for the perturbed electric ﬁeld , perturbed ionic concentration nj, perturbed pressure p, and perturbed velocity ﬁeld u are, respectively,

∇2₍_EZ e−it) = 2 j=1 zjêne_j₀ exp −zjê kT(e+ EZe −it +gjEZe−it) − exp −zjê kTe , a < r < b, (18) −i nj = − Dj(∇2( nj)+ zjê kT(∇n e j· ∇( nj) + ∇( nj)· ∇e + ∇( nj)· ∇( EZe−it)+ nej∇ 2₍₎ + ( nj)∇2e + ( nj)∇2( EZe−it))) + ( u)∇(ne j+ njEZe−it), a < r < b, (19) ∇ · ( uEZe−it)= 0, a < r < b, (20) and

−io uEZe−it= − ∇ pEZe−it+ o∇2 uEZe−it

− e∇, a < r < b, (21)

where_i and_oare, respectively, the viscosity of droplet ﬂuid and that of dispersion medium, and_i and_o are the density of droplet ﬂuid and that of dispersion medium. The associated boundary conditions are

∇( EZe−it)· ˆn = 0 at r = a. (22)

Eq. (22) indicates that the droplet is non-conducting. At the outer cell surface, Shilov–Zharkikh boundary condition is adopted in the present study, which is derived from thermo-dynamics consideration relating macroscopic and microscopic electric ﬁelds appropriately (Shilov et al., 1981). This model has been successfully applied in concentrated suspensions and yields better agreement with experimental results than the

tra-ditional Levine and Neal boundary conditions (Dukhin et al.,

1999). Based on the Shilov–Zharkikh boundary condition, we

have, after simple arithmetic cancellations,

( ) = b cos at r = b. (23)

Since the bulk liquid cannot penetrate a droplet and both the velocity and the shear stress are continuous across the droplet surface, the following conditions are assumed:

urEZe−iwt|r=a+= urEZe−iwt|r=a−= 0 at r = a, (24)

uEZe−iwt|r=a+= uEZe−iwt|r=a− at r= a, (25) (n· r) × r|r=a+= (n· r) × r|r=a− at r= a, (26)

where n _{is the shear stress tensor on the droplet surface.}

Here, we assume that the cell surface moves with a velocity −UEZe−itrelative to a droplet. Also, according toKuwabara (1959), the vorticity should vanish on the cell surface. Based on these assumptions, the following conditions are assumed on the cell surface:

∇ × uEZe−it= 0, r = b (27) urEZe−it= −(UR+ iUI)EZe−itcos, r = b. (28)

Here, UR and UI are the real and the imaginary parts of the

drop velocity.

Moreover, we assume that the surface of the droplet is im-permeable to ions; hence, the associated boundary condition of

nj is

jnj

jr = 0 at r = a. (29)

Besides, no disturbance of concentration of ionic species is allowed at the outer virtual surface; thus, the corresponding boundary conditions for the polarization effect are

gj+ = 0 at r = b. (30)

The governing equations, Eqs. (9), (10), (15), (18)–(21), and its associated boundary conditions are then solved by a pseudo-spectral method based on Chebyshev polynomials, which is found to be a powerful and accurate algorithm for the prob-lems of the present type. Note that we have assumed that the droplet remains spherical under the applied oscillating electric field. This is justified as the Weber numbers, both inertial and electrostatic as defined below, of the system are much less than unity (Taylor and Acrivos, 1964; Eow et al., 2003):

W e= U2a/, (31)

W ee=

2E2a

, (32)

where is the surface tension of the droplet interface and other variables are deﬁned in the Notations. The dimensionless We-ber numWe-bers, We and W ee, are, respectively, measurements of

relative signiﬁcance of inertia force and electric force in com-parison with the surface tension, which is the shape-keeping force for the droplet to remain spherical. In the cases under our investigation, which are typical in dynamic electrophore-sis, both We and W eeare of the order around 10−10, deﬁnitely

small enough to justify the spherical-shape assumption for the droplet. Moreover, it also indicates the trivialness of Maxwell stress tensor over the shape of the droplet as far as the defor-mation of liquid droplet is concerned. However, it would al-ways be a good idea to check the validity of trivial electrostatic Weber number whenever facing an electrophoretic system of liquid droplets.

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2.3. Dynamic electrophoretic mobility

The deﬁnition of the dynamic electrophoretic mobilitymis

m= R+ iI =

U

E, (33)

where E= EZe−it and U= (UR+ iUI)EZe−it,RandI

being, respectively, the real and the imaginary parts ofm. To obtainm, a force balance exerted on the drop is required. The expression of the force balance is shown as

Fh+ Fe= 4 3 a 3₍_i₋ o) dU dt, (34)

where Fhand Feare, respectively, the hydrodynamic force and

the electric force acting on the drop. Feand Fhcan be calculated

as follows (Happel and Brenner, 1983):

Fe= S e(−∇)s· ezdS = 2 2 a 0 j jr r=a j jr cos −1 r j jsin r=a r2sin d, (35) Fh= − o 0 r2sin2j jt( uEZe−it) r=a d + 0 [r2

sin2(∇2 uEZe−it)]r=ad

−

0

[r2

sin2e(∇)]_r_=ad, (36) where S denotes the droplet surface andethe surface charge density, determined by Gauss law. For an easier treatment, we iterate the velocity of the droplet under some applied electric ﬁeld EZe−it until Eq. (34) is satisﬁed. Since the

dynamic electrophoretic mobility is a complex number, it can be characterized by its magnitude m =

2

R+

2

I and

phase angle  = tan−1(I/R). Thus, E= EZe−itez and

U=

U_R2+ U_I2EZe−it+ez, < 0 implies that U leads E,

and the reverse is true if > 0.

3. Results and discussion

Since a variety of important factors might inﬂuence the electrophoretic behavior of the system under considera-tion, we group them in dimensionless form to simplify the analysis. They include: the scaled dynamic electrophoretic mobility ∗_m = /(a/), the scaled frequency ∗ = oa2_/_o_{, the scaled double layer thickness} _{a, where}

−1_{=(kT /}2

j=1nej0(ezj)2)1/2, the viscosity ratio (o/i), the

scaled surface potentialr = z1ea/kT, and the volume

frac-tion of droplets, measured by H=(a/b)3. The followingvalues are also assumed in numerical simulations: T = 298.15 K,

z1= −z2= 1, o= 8.904 × 10−3g/cm s,o/i= 0.909, and  = 8.854 × 10−12× 78.54688 F/m, typical for aqueous KCl solution. ω* 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 ω* 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 0.2 0.4 0.6 1.0 φr=5.0

a

2.0 3.0 4.0 Θ (Degrees) 0 20 40 60 80 φr=1.0 5.0

b

μm *

Fig. 2. Variation in the scaled magnitude of dynamic electrophoretic mobility

∗

m(a), and phase angle(b), as a function of the scaled frequency∗for various values ofr ata= 1, H = 0.1, ando/i= 1. Dash line: low zeta potential.

3.1. Effect of frequency and surface potential

Figs. 2–5 show the variations in the scaled magnitude of electrophoretic mobility∗_m, and the corresponding phase angle

, as functions of the scaled frequency of the applied electric

ﬁeld∗at various values ofr. For the purpose of comparison,

corresponding results of low zeta potential presented by Hsu

et al. (2007)are also shown in the graphs. Clearly our results approach the limiting case asr → 0. Actually as r= 1, they

still almost coincide with each other in lots of cases shown.

With even higher_r, however, the outcome is quite different.

For low surface potential, the mobility ∗_m monotonically

de-creases with increasing frequency. Examining Eqs. (15) and (21) closely, as∗gets larger, the ﬂow ﬁeld will be dominated

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ω* 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 ω* 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.0 2.0 3.0 4.0 φr= 5.0

a

Θ (Degrees) 0 20 40 60 80 φr= 1.0 5.0

b

μm *

∗

m(a), and phase angle(b), as a function of the scaled frequency∗ for various values of_rata= 3, H = 0.1, ando/i= 1. Dash line: low zeta potential.

by the inertia term, and the influences of the other terms di-minish. Moreover, as the direction of the applied electric field changes back and forth, the direction of the drop motion will change accordingly. And eventually the drop simply fails to keep up with the rapid pace of the applied AC field as∗gets large, as we reported and explained in a related paper earlier (Hsu et al., 2007). However, the mobilities corresponding to high surface potentials here increase first, and then decreases

rapidly with increasing ∗, exhibiting maximums at medium

to high frequencies. In the case of static electric ﬁeld, double layer polarization induces an internal electric ﬁeld in the op-posite direction of the externally applied one, hence retarding

ω* 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 1.0 2.0 3.0 4.0 φr= 5.0

a

Θ (Degrees) 0 20 40 60 80 φr= 1.0 5.0

b

μm * ω*

∗

the movement of drops. Generally speaking, with the increase in r, the polarization effect becomes even more significant. However, if a dynamic electric field is applied, its direction changes with time and the corresponding induced electric field cannot respond instantly. A phase lag is generally observed as a result. It is interesting to note that the associated phase angle

 observed here becomes even negative at medium ∗_,

indi-cating the response of the drop velocity leads that of the applied electric ﬁeld. This is certainly impossible from the viewpoint of cause and consequence: How can a response take place be-fore the input signal is yet introduced? The apparent phase lead by the dynamic electrophoresis signal over the input AC ﬁeld

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10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1.0 2.0 3.0 4.0 φr= 5.0

a

Θ (Degrees) 0 20 40 60 80 φr= 1.0 5.0

b

μm * ω* ω*

∗

is simply an indication of a way too slow response that actu-ally lag an entire period. Thus, the induced electric ﬁeld due to polarization may actually accelerate the movement of the drops and result in an ascending trend of the mobility curves. And similar to the situation at low zeta potential, as∗ fur-ther increases, eventually the accelerating time in each individ-ual period becomes so short, due to the fast offset by the back and forth rapid changing of input AC ﬁeld, that the mobility actually goes down again, and the polarization effect, hence, is not important anymore. The combined effects mentioned above generate the maxima as observed inFigs. 2(a)–5(a). The peaks

κa 10-2 ₁₀-1 ₁₀0 ₁₀1 0 1 2 3 4 5 6 7 φr = 5.0 4.0 3.0 2.0 1.0 μm *

Fig. 6. Variation in the scaled magnitude of electrophoretic mobility∗_mas a function ofafor various values of_r at∗= 1,_o/_i= 1, and H = 0.1.

κa 10-2 ₁₀-1 ₁₀0 ₁₀1 0 1 2 3 4 φr= 5.0 4.0 3.0 2.0 1.0 φr= 5.0 1.0 φr= 5.0 1.0 μm *

Fig. 7. Variation in the scaled magnitude of electrophoretic mobility∗mfor various values ofr at∗= 0.002,o/i= 1, and H = 0.1.

shown inFigs. 2(a)–5(a)also become more pronounced as_r

increases. Also, as ∗ gets large, the time for drops to react

by altering its direction of movement becomes short; hence, the phase lag increases accordingly.Figs. 2(b)–5(b)reveal this phenomenon in the present problem.

3.2. Effect of double layer thickness

The effect of double layer thickness on the dynamic elec-trophoretic mobility is different for high or low frequency, as

shown inFigs. 6 and 7. InFig. 6,∗= 1, which corresponds

to a dimensional frequency of about 10 MHz (a= 300 nm)

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μm * ω* 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 ω* 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 0.2 0.4 0.6 0.8

a

H = 0.01 0.05 0.1 0.2 0.3 0.5 Θ (Degrees) 0 20 40 60 80

b

H = 0.01 0.5

∗

m(a), and phase angle(b), as a function of the scaled frequency∗ for various values of H ata= 1,r= 3, ando/i= 1.

dynamic electrophoresis. The dynamic electrophoretic mobil-ity increases monotonically witha in the range of 1–10. This is because the polarization effect, i.e., the distortion of ionic cloud within the double layer, is dampened due to phase lag, as explained in the last paragraph, thus no longer a signiﬁ-cant factor. The mobility is mainly determined by such fac-tors as surface potential and double layer thickness. Asa in-creases, indicating the double layer gets thinner, thus less “hin-drance effect” from double layer overlapping with neighboring droplets, the mobility hence increases monotonically as shown inFig. 6. InFig. 7,∗= 0.002, which corresponds to a dimen-sional frequency of about 20 kHz, a low frequency in typical dynamic electrophoresis. The situation here is kind of

compli-cated, and we see a reverse of _r dependence of ∗_m as a

κa 10-2 ₁₀-1 ₁₀0 ₁₀1 κa 10-2 10-1 100 101 0 1 2 3 4 H = 0.5 0.3 0.2 0.1 0.05 0.01

a

Θ (Degrees) 5 10 15 20 25 H = 0.5 0.3 0.2 0.1 0.05 0.01

b

μm *

Fig. 9. Variation in the scaled magnitude of electrophoretic mobility∗_m(a), and phase angle(b), as a function ofafor various values of H at∗=1,

o/i= 1, andr= 3.

increases, which is absent in the previous high frequency case

(∗= 1). Again, polarization effect is the determining factor

behind it. Taking a closer look atFig. 7, we found that if the

double layer becomes thinner (a increases) but still

compara-ble with the droplet radius, the additional electrohydrodynamic interactions due to double layer overlapping from neighbor-ing drops decrease, and the distortion of ionic cloud near the charged surface becomes more signiﬁcant. As a result, the in-duced electric force mentioned on the last paragraph will have

a signiﬁcant retarding effect on the movement of drops. Asa

increases further, however, the double layer thickness gets so thin that eventually the distortion of the ionic cloud becomes negligible again, and the dynamic electrophoretic mobility re-stores to its previous behavior, i.e., to increase accordingly with

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ω* 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 ω* 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 0.1 0.2 0.3 0.4 ηo/ηi=100 ηo/ηi=100 0.1 1.0 10.0

a

0.01 Θ (Degrees) 0 20 40 60 80 0.1 1.0 10.0

b

0.01 μm *

Fig. 10. Variation in the scaled magnitude of electrophoretic mobility∗_m(a), and phase angle(b), as a function of∗ for various values of_o/_i at

H= 0.1,a= 0.5, andr= 3. Dash line: hard spherical particle.

increasing surface potential. Similar behaviors are also ob-served in the case of an isolated spherical particle (Preston et al., 2005) and concentrated dispersion of spherical particles (Hsu et al., 2002), both at high surface potential.

3.3. Effect of volume fraction

The effect of volume fraction H is presented inFigs. 8 and

9. It is obvious that ∗m and decrease with the increase in

volume fraction H. This is because as the volume fraction of drops gets large, the distance between droplets becomes nar-row. For example, if the volume fraction of drops changes from

H= 0.1 to 0.2, the dimensionless distance between droplets,

2(b− a), would be shortened approximately from 2.3 to 1.42.

Double layer overlapping may occur in the current analysis if

ω* 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 ₁ ω* 10-3 10-2 10-1 100 101 102 02 0.2 0.4 0.6 0.8 ηo/ηi= 100 ηo/ηi= 100 0.1 1.0 10.0

a

0.01 Θ (Degrees) 0 20 40 60 80 0.1 1.0 10.0

b

0.01 μm *

Fig. 11. Variation in the scaled magnitude of electrophoretic mobility∗_m(a), and phase angle(b), as a function of∗for various values of _o/i at H= 0.1,a= 1.0, andr= 3. Dash line: hard spherical particle.

the electrolyte concentration is low (smalla) or the drops be-come concentrated (large H ). The inﬂuence of the double lay-erthickness surrounding a drop on its electrophoretic behavior is also illustrated inFig. 9. In the limit asa → 0, the magni-tude of the dynamic mobility always decreases to zero, as

ex-pected. This is because asa → 0, the thickness of the double

layer surrounding the drop surface gets inﬁnitely thick, hence inﬂuencing all the other drops throughout the entire dispersion system. Due to the double layer overlapping effect, it actually yields a uniform distribution of ions and thus a uniform electric potential in the solution. As a result, no electric force is gen-erated upon the drop and it remains stationary. However, the effect of double layer overlapping gradually disappears with

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ω* 10-3 10-2 10-1 100 101 102 ω* 10-3 10-2 10-1 100 101 102 0.5 1 1.5 2 2.5 3 0.1 1.0 10.0

a

0.01 Θ (Degrees) 0 20 40 60 80 0.1 1.0 10.0

b

0.01 μm * ηo/ηi=100 ηo/ηi=100

Fig. 12. Variation in the scaled magnitude of electrophoretic mobility∗_m(a), and phase angle(b), as a function of∗ for various values of_o/i at H= 0.1,a= 5.0, andr= 3. Dash line: hard spherical particle.

a non-uniform distribution of ions, which sets up the motion eventually. The higher thea, the less signiﬁcant the double layer overlapping from neighboring drops, and the mobility in-creases accordingly. The drop is also easier to respond to the variation in the applied electric ﬁeld with less entanglement from the double layer overlapping. Thus, the phase lag  in

Fig. 9(b) decreases with increasinga. This is because the elec-trical double layer is conﬁned to a narrow space, and the time necessary to adjust the ionicdistribution inside is short, and the phase lag is getting smaller.

3.4. Effect of viscosity ratio

Probably the most important factor affecting the dynamic electrophoresis of a droplet is its viscosity. After all, that is what

set it apart from the conventional studies on the rigid sphere. The inﬂuence of the viscosity of the droplet ﬂuid at several

a on its electrophoretic behavior is illustrated inFigs. 10–12. These ﬁgures reveal that the larger the ratio of (o/i), the

larger the∗m. This is because asi gets small, the ﬂuid inside

the droplet is easier to flow; hence, the shear rate at the droplet surface decreases and results in a smaller retarding drag force. The mobility of the droplets thus increases with declining_i. In comparison, corresponding results of rigid spherical particles (Hsu et al., 2002) are also presented inFigs. 10–12. As can be clearly seen from these figures, they serve as the limiting cases as (_o/_i)approaches zero. The phase lag is also found to increase with larger (_o/_i), since longer time is needed for the “lighter” (less viscous) internal fluid to react accordingly. The influence of (o/i)on the phase lag is not as substantial as

that on the mobility magnitude∗_m, though.Figs. 10–12serve as an extension of the traditional dynamic electrophoresis data for dispersions of rigid colloids.

4. Conclusion

The dynamic electrophoretic behavior of a concentrated dis-persion of drops with arbitrary surface potential is investigated. The effect of volume fraction of drops and double layer thick-ness is examined in particular. In summary, we conclude the following:

(i) If the zeta potential on droplet surface is high, the effect of double layer polarization cannot be neglected. The dy-namic electrophoretic mobility exhibits a maximum as a result and then decreases rapidly with increasing frequency of AC ﬁeld.

(ii) At medium frequency, the induced electric force coming from the distortion of ionic cloud may actually accelerate the movement of drops. However, with further increase in the frequency, the time for drops to alter its direction of movement becomes short, which slows down the mobility. The phase lag increases accordingly.

(iii) The magnitude of electrophoretic mobility declines with the volume fraction of droplets due to the hindrance effect and electrostatic interaction from neighboring drops. (iv) The effect of double layer overlapping is very signiﬁcant

in concentrated dispersion if the volume fraction of drops is high and the double layer is thick.

(v) The smaller the viscosity of the droplet ﬂuid, the larger the magnitude of dynamic electrophoretic mobility and the phase lag. As the viscosity of the droplet ﬂuid increases, the electrophoretic behavior of a droplet approaches that of a rigid particle eventually.

Notation

a radius of droplets, m

b radius of the liquid shell, m

Dj the diffusivity of ionic species j , m2/s e charge of electron (1.6× 10−19Coul)

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E applied AC electric ﬁeld, V/m

Ez strength of the AC electric ﬁeld, V/m

Fe electrical force, N

Fh hydrodynamic force, N

gj the potential function for the double layer polarization, V

H volume fraction of droplets, dimensionless

I i=√−1

k Boltzmann constant (1.38× 10−23J/K)

nj number density of electrolytes j, number/m3

nj0 bulk concentration of electrolytes j, number/m3

p pressure, N/m2

r r-component of spherical coordinates

T absolute temperature, K

t time, s

u velocity of dispersion medium, m/s

U terminal velocity of the droplets, m/s

We inertia Weber number, dimensionless

W eeelectrostatic Weber number, dimensionless zi valence of ionic species i, dimensionless Greek letters

dielectric constant of dispersion medium, C/V/m

a surface potential on the droplets, V

ﬂuid viscosity, kg/m/s

i ﬂuid viscosity inside of the droplets, kg/m/s

o ﬂuid viscosity outside the droplets, kg/m/s

-component of spherical coordinates

 phase angle between velocity and electric ﬁeld, deg reciprocal Debye length, m−1

m dynamic electrophoretic mobility, m2/V/s ﬂuid density, kg/m3

e space charge density in dispersion medium, C/m3 i ﬂuid density inside the droplets, kg/m3

o ﬂuid density outside the droplets, kg/m3

surface tension, kg/s2

e surface charge density, C/m3

n _{shear stress tensor, kg/m/s}2

electrical potential, V

r scaled surface potential on the droplets, dimensionless -component of spherical coordinates

dynamic frequency of the AC electric ﬁeld, s−1

Superscripts

* scaled symbols

Subscript

e equilibrium properties

S properties on the droplet surface

perturbed properties

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