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### Electrophoresis of a non-conducting Newtonian drop of low electrical

### potential normal to a plane

### Eric Lee, Shih-Han Lou, Jyh-Ping Hsu

∗*Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan*

Received 24 November 2005; received in revised form 21 February 2006; accepted 22 February 2006 Available online 28 February 2006

**Abstract**

The electrophoretic behavior of a non-conducting, Newtonian drop of low surface potential normal to a plane is investigated theoretically under the conditions of weak applied electric ﬁeld and arbitrary thickness of double layer. The governing equations and the associated boundary conditions expressed in terms of bipolar spherical coordinates are solved by an orthogonal collocation method. In general, the thinner the double layer surrounding a drop and/or the longer its distance from a planar surface, the larger its mobility, and if a drop is sufﬁciently close to a plane, its mobility may change sign. These results are similar to the case of a rigid particle. The mobility of a drop decreases with the increase in the ratio (viscosity of drop ﬂuid/viscosity of dispersion medium). Under the conditions assumed, a drop can be treated as a rigid sphere if the viscosity ratio exceeds about 100, and as a bubble if it is smaller than about 0.01.

䉷 2006 Elsevier Ltd. All rights reserved.

*Keywords: Electrophoresis; Boundary effect; Drop normal to plane; Non-conducting Newtonian drop*

**1. Introduction**

Electrokinetic phenomena, which include electrophoresis, electroosmosis, sedimentation potential, streaming potential, and electric conductivity, play a key role in colloidal and in-terfacial science. These phenomena are closely related to the charged conditions of a colloidal entity or a surface. Among the electrokinetic phenomena, electrophoresis is not only a power-ful analytical tool for the characterization of the surface prop-erties of a colloidal entity but also one of the key techniques in nanotechnology. While electrophoresis is studied extensively in the past decades, most of the relevant results in the literature focused on rigid entities. In contrast, available results for non-rigid dispersions such as emulsions or micro-emulsions, which are important dispersed systems in practice, are much more limited. Solving the electrokinetic equations for a non-rigid dis-persion is usually more complicated than solving those for a rigid dispersion since both the ﬂow and the electric ﬁelds in-side a non-rigid entity need be conin-sidered. The electrophoretic

∗_{Corresponding author. Tel.: +886 2 23637448; fax: +886 2 23623040.}

*E-mail address:*jphsu@ntu.edu.tw(J.-P. Hsu).

0009-2509/$ - see front matter䉷2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2006.02.030

behaviors of non-rigid particles was investigated for an iso-lated mercury drop (Craxford et al., 1937; Booth, 1951; Levich, 1962; Levine and O’Brien, 1973), an isolated drop or gas bub-ble (Baygents and Saville, 1991a,b;Ohshima, 2003), a disper-sion of mercury drops (Ohshima et al., 1984; Ohshima, 1997, 1999;Lee et al., 2003a), a dispersion of general non-rigid par-ticles (Kelsall et al., 1996;Lee et al., 2003b), and a dispersion of non-Newtonian drops (Lee et al., 2005a).

One of the important factors in conducting electrophoresis is the presence of a boundary. This factor needs to be considered, for instance, when a particle is sufﬁciently close to the wall of electrophoresis cell. Another typical example is in electrodepo-sition where charged particles are driven by an applied electric ﬁeld toward electrodes. A thorough review of literature reveals that while many attempts have been made in the past to ex-plore the boundary effect on electrophoresis, most of them are focused on rigid entities (e.g.,Keh and Anderson, 1985; Ennis and Anderson, 1997). Relevant results on non-rigid entities, al-though of both fundamental and practical signiﬁcance, are very limited. The application of capillary electrophoresis in char-acterization and separation of vesicles and multi-components liquid separation, for instance, all involve this problem (Breyer et al., 2003; Ashok and Joykrishna, 2005). The analysis of the

electrophoresis of non-rigid particles is more complicated than that of rigid ones because both the electrokinetic equations in-side a particle and those outin-side a particle need to be conin-sidered simultaneously.Lee et al. (2002)analyzed the electrophoresis of a non-rigid particle in a spherical cavity. The electrophoresis (Lee et al., 2004) and the sedimentation (Lee et al., 2005b) of a composite particle in a spherical cavity were studied recently. In general, because the particle–dispersion medium interface is not non-slip, the electrophoretic mobility of a non-rigid particle is larger than that of the corresponding rigid particle.

In this study, the boundary effect on the electrophoresis of a non-rigid particle is analyzed by considering the electrophoresis of a non-conducting Newtonian drop of low surface potential normal to a planar surface. An orthogonal collocation method is adopted for the resolution of the governing electrokinetic equations expressed in terms of bipolar spherical coordinates and the associated boundary conditions. The inﬂuences of the thickness of double layer, the distance between a drop and a plane, and relative magnitude of the viscosity of a drop on its electrophoretic behavior are examined.

**2. Theory**

Let us consider the electrophoresis problem illustrated in

Fig. 1*where an electrolyte-free drop of radius a moves with*
**velocity U normal to a planar surface as a response to a uniform**
**applied electric ﬁeld Ez** of strength*Ezin the z-direction. The*
*distance between the center of the drop and the surface is h.*
The dispersion liquid contains*z*1:*z*2electrolytes,*z*1and*z*2are,

respectively, the valences of cations and anions. Let*n*10and*n*20

be, respectively, the bulk concentrations of cations and anions
with *n*10 *= n*20. The bipolar spherical coordinates *(, , )*

are adopted where* = 0 and = *0 represents, respectively,

the planar surface and the drop–dispersion liquid interface, and
*∞ > > *0 denotes the domain inside the drop. The bipolar

coordinates and the Cartesian coordinates*(x, y, z) are related*
by (Happel and Brenner, 1983)

*z = c* sinh

cosh* − cos *, (1)

*y = c* sin

cosh* − cos *, (2)

*where c is the focal length, 0 < ∞, and 0. The *
phys-ical properties of the dispersion liquid and those of the drop
ﬂuid are assumed to maintain at constant values, and the shape
of a drop remains spherical.

The electrical potential of the system under consideration,
*, can be described by the Poisson equation*

∇2* _{ = −}*
= −
2

*j=1*

*zjenj*, (3)

where* and are, respectively, the space charge density and the*
*permittivity of the dispersion liquid, e is the elementary charge,*
and*n _{j}*

*is the number concentration of ionic species j. If the*de-formation of double layer surrounding a drop is insigniﬁcant,

the spatial distribution of ions follows the Boltzmann
distribu-tion
*nj= nj0*exp
−*zje*
*kBT*
, (4)

*where T is the absolute temperature and* *k _{B}* is the Boltzmann
constant. For convenience,

*is decomposed into an equilibrium*potential

*and a perturbed potential arising from the applied electric ﬁeld,*

_{e}*(*Tang et al., 2001).

For the case* _{e}*is low, it can be shown that in the dispersion
liquid

∇2* _{}*∗

*e= (a)*2∗*e, *0*0,* (5)

where ∗_{e}*= _{e}/_{a}*,

*is the potential on drop surface and*

_{a}*n*∗

*j* *= nj/n*10. The reciprocal Debye length * and the scaled*

gradient operator∇2are deﬁned, respectively, by

* =*
⎡
⎣2
*j=1*
*nj0(ezj)*2*/kBT*
⎤
⎦
1*/2*
, (6)
∇2_{=} *x*2
*c*∗2
j2
*j*2 +
j2
*j*2 −
sinh
*x*
j
*j* +
cos* cosh − 1*
*x sin *
j
*j*
(7)
with *c*∗ *= c/a and x = cosh − cos . Since the drop is*
electrolyte-free, the corresponding governing equation for∗* _{e}*is
∇2

*∗*

_{}*e= 0, ∞ > > *0. (8)

Since∗*= *∗* _{e}+ *∗, Eqs. (3) and (5) lead to
∇2

*∗*

_{ }_{= ∇}2

*∗*

_{}_{− ∇}2

*∗*

_{}*e*= 0. (9)

Suppose that both the drop liquid and the dispersion medium are Newtonian, and the ﬂow ﬁeld can be described by the conti-nuity equation and the Navier–Stokes equation in the creeping ﬂow regime,

**∇ · v = 0,** (10)

*∇*2_{v}_{− ∇p − ∇ = 0,}_{(11)}

*where p and are, respectively, the pressure and the viscosity.*
Taking curl on both sides of Eq. (11) and substituting Eqs. (4)–
(13) into the resultant expression yields

*E*4_{
}_{∗}*(e) _{= −(a)}*2

*x sin*

*c*∗

*j*∗

_{e}*j*

*j*∗

*j*−

*j*∗

_{e}*j*

*j*∗

*j*, (12)

where*
*∗*=
/(k _{B}T /z*1

*e)*2

*a/ is the scaled stream function,*

*is the stream function, the superscript e denotes the exterior*of a drop,

*E*4

*= E*2

*E*2, and

*E*2

_{=}

*x*2

*c*∗2 j2

*j*2 + j2

*j*2 + sinh

*x*j

*j*+ 1

*− cos cosh*

*x sin*j

*j*. (13)

**Fig. 1. The electrophoresis of a liquid drop of radius a moves with velocity U normal to a plane as a response to a uniform applied electric ﬁeld E****z**in the
*z-direction where h is the drop–plane distance.*

In terms of*
and the present bipolar spherical coordinate, the*
*- and the -components of the ﬂuid velocity, v* and*v _{}*, can
be expressed, respectively, as

*v*=

*x*2

*c*2

_{sin}

_{}*j*

*j*, (14)

*v*=

_{}*−x*2

*c*2

_{sin}

_{}*j*

*j*. (15)

Following the same procedure as that employed in the deriva-tion of Eq. (12), it can be shown that for the domain inside a drop

*E*4_{
}∗(i)_{= 0,}

(16)
*where the superscript i denotes the interior of a drop.*

We assume that the drop remains spherical and has a uniform
equilibrium surface potential. Also, there is no net charge on
the plane, and the perturbed potential arising from the applied
electric ﬁeld is normal to the plane. The ﬂow ﬁeld is assumed
to be continuous across the drop–dispersion medium interface,
and the planar surface is no-slip. Based on these assumptions,
the boundary conditions associated with Eqs. (5), (8), (9), (12),
(16) can be expressed as
∗* _{e}= 1, = *0, (17)
∗

*(18)*

_{e}= 0, = 0,*j*∗

_{e}*j*

*= 0, = 0 and = ,*(19) ∗

*(20)*

_{e}= 0 as → ∞,*∗*

**= −E**∗

**z· r**∗

*, = 0,*(21)

*j*∗

*j*

*= 0, =*0, (22)

*∗*

*= 0, >*0, (23)

*∗(e)= 0, = 0,*(24)

*j ∗(e)*

*j*

*= 0, = 0,*(25)

*j ∗(e)*

*j*−

*j ∗(i)*

*j*=

*c*∗2

*x*3 sin 2

*∗*

_{ sinh U}*0, (26)*

_{, = }*∗(e)*0, (27)

_{= }∗(i)_{, = }*∗(e)*

*r*2

*→ 0, = 0 and = 0,*(28)

*
∗(e)*= 0 and *j
_{j}∗(e)*

*= 0, = 0 and = ,*(29)

*∗(i)*

_{}*= ∗(e)*

_{}*, =*0. (30)

In these expressions,∗* _{}*is the dimensionless shear stress on

**the drop surface, E**∗

**z= Ez**

*/(*

_{a}/a) is the scaled electric ﬁeld,**r**∗**= (r/a)i****r**is the normal vector pointing from the center of a

drop to its interface, and*U*∗*= U/U _{E}*, where

*U*2

_{E}=*/*The solution procedure used by O’Brien and White (1978)

_{r}a.is used where an electrophoresis problem is decomposed into two subproblems. In the ﬁrst subproblem, a drop moves in the absence of the applied electric ﬁeld, and in the second problem, it is held ﬁxed when the electric ﬁeld is applied. The mobility of a drop can be evaluated based on the condition that the to-tal force acting on it vanishes at the steady state (O’Brien and White, 1978;Baygents and Saville, 1991a). A pseudo-spectral method based on Chebyshev polynomials (Finlayson, 1980), which is readily applicable to the present problem, is adopted for the resolution of the governing equations and the associ-ated boundary conditions. Convergence test is conduced, and using 30× 30 nodal points inside droplet and 30 × 45 outside droplet is found to be sufﬁciently accurate;Fig. 2shows the re-sultant mesh system. The applicability of the present numerical scheme is also justiﬁed by comparing the results ofWacholder and Weihs (1972), where a pure hydrodynamic problem was analyzed for the case of a droplet moving normal to a plane, with those evaluated by our method.Tables 1and2summarize the result of comparison. These tables indicate that the perfor-mance of the present numerical scheme is satisfactory.

y Z 0 1 2 3 4 5 0 1 2 3 4

Fig. 2. The system mesh for a liquid drop normal to the plane (_{0}= 1.0;
zoom-in).

Table 1

Dimensionless drag on a droplet for the case of a pure hydrodynamic problem
for various values of0 at*= 1.0*

0 Exact analytical Present numerical Percentage

result result deviation

0.5 28.9070 28.90628 0.00249075 1 12.14115 12.14712 0.04921115 1.5 8.236236 8.238643 0.02922451 2 6.6335 6.686423 0.79781413 2.5 5.8963 5.9568 1.0260719 3 5.5121 5.56999 1.05023494

The exact analytical result is reported byWacholder and Weihs (1972).

Table 2

Dimensionless drag on a rigid plane for the case of a pure hydrodynamic
problem for various values ofat_{0}*= 1.0*

Exact analytical Present numerical Percentage

result result deviation

0 8.22184 8.68154 5.59120586

0.5 10.66268 10.69016 0.25770257

1 12.14115 12.14712 0.04921115

10 16.86964 16.87936 0.05762042

100 18.06295 18.10583 0.23739978

The exact analytical result is reported byWacholder and Weihs (1972).

**3. Results and discussion**

The inﬂuences of the key parameters of the present
prob-lem, including viscosity of the drop and boundary effect, on the
electrophoretic behavior of a drop are examined through
nu-merical simulation. For illustration, we assume that*P e*1*=0.01*

and*P e*2*=0.01. The contours of the scaled equilibrium electric*

**y**
**Z**
0 1 2 3 4 5
0
1
2
3
4
5
(a)
(b) **y**
**Z**
0 1 2 3 4 5
0
1
2
3
4
5
Level FI1
15 0.9375
14 0.875
13 0.8125
12 0.75
11 0.6875
10 0.625
9 0.5625
8
7 0.4375
6
5 0.3125
4
3 0.1875
2
1 0.0624999
0.5
0.375
0.25
0.125
Level FI1
15 0.9375
14 0.875
13 0.8125
12 0.75
11 0.6875
10 0.625
9 0.5625
8
7 0.4375
6
5 0.3125
4
3 0.1875
2
1 0.0625
0.5
0.375
0.25
0.125

Fig. 3. Contours of scaled equilibrium potential∗* _{e}* for two levels of

*a at*0

*= 1.0, and= 1.0: (a)a = 0.01, (b)a = 7.943.*

potential∗* _{e}* at two levels of

*a are presented in*Fig. 3, and the corresponding contours of the scaled stream function

*∗ are shown inFig. 4.Fig. 3indicates that a larger value of*

*a leads*to a narrower distribution in∗

*and a steeper gradient of∗*

_{e}*near the surface of a drop. Since the drop is non-conductive, the latter implies that the larger the value of*

_{e}*a, the higher the*ion concentration in the solution, the higher the charge den-sity on its surface, and therefore, a greater electric force acting on it, which yields a larger mobility. The ﬂow pattern shown inFig. 4justiﬁes this inference. Note that the streamlines are continuous across the drop–dispersion liquid interface, which results from the boundary conditions assumed.

The inﬂuence of the nature of a drop, measured by * =*

*(i) _{/}(e)*

_{, is illustrated in}

_{Figs. 5}

_{and}

_{6}

_{. For comparison, the}

corresponding results for the case of a rigid sphere (Ennis and Anderson, 1997; Tang et al., 2001) are also presented in these

**P**
**Z**
0 1 2 3 4
0
1
2
3
4
Level FI5
10 -0.0036187
9 -0.014559
8 -0.0322256
7 -0.0498921
6 -0.0675586
5 -0.0852251
4 -0.102892
3 -0.120558
2 -0.138225
1 -0.148762
**P**
**Z**
0 1 2 3 4 5
0
1
2
3
4
5
Level FI5
17 0.136973
16 0.113062
15 0.0881361
14 0.0392991
13 0.00927523
12 -0.00953793
11 -0.0583749
10 -0.107212
9 -0.156049
8 -0.204886
7 -0.30256
6 -0.364627
5 -0.449071
4 -0.546745
3 -0.644419
2 -0.742093
1 -0.839767
(a)
(b)

Fig. 4. Contours of stream function for the case ofFig. 3.

ﬁgures. Note that as* → ∞, the behavior of a drop approaches*
to that of a rigid sphere, and as* → 0 the behavior of the drop*
approaches to that of a bubble.Figs. 5and6 indicate that the
mobility of a drop decreases with the increase in*, which is*
expected since the larger the value of*, the greater the viscous*
drag acting on a drop, as is illustrated inFigs. 7and8, where
the drag on a rigid particle is based on the result ofWacholder
and Weihs (1972). The negative value of the scaled drag force
in these ﬁgures implies that its direction is always opposite to
that of the movement of a drop, that is, it is a retardation force.

Figs. 5and6suggest that a drop can be treated as a rigid sphere
if* exceeds about 100, and as a bubble if is smaller than about*
0.01. In general, the larger the value of*a, the larger is the *
mo-bility of a drop. A comparison betweenFigs. 6and8 reveals
that for a ﬁxed value of*a the qualitative behavior of the *
mobil-ity is the same as that of the drag. According to Eq. (12), if*a*
is very small, the present electrophoresis problem is essentially

κa U * / E z * 10-2 10-1 100 101 0 1 2 3 4 5 σ = 0.01 0.1 1.0 10 100 rigid sphere gas bubble

Fig. 5. Variation of scaled electrophoretic mobility*(U*∗*/E*∗*) as a function*
of*a for various values of*at_{0}*= 1.0, dashed line represents result for a*
rigid sphere (Ennis and Anderson, 1997; Tang et al., 2001).

σ
U
* / E
z
*
10-2 _{10}-1 _{10}0 _{10}1 _{10}2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
rigid sphere
gas bubble

Fig. 6. Variation of scaled electrophoretic mobility*(U*∗*/E*∗*) as a function*
offor the case when0*= 1.0 anda = 1.0, dashed line represents result*
for a rigid sphere (Ennis and Anderson, 1997; Tang et al., 2001).

the same as a pure hydrodynamic problem. In this case, the
mo-bility of a drop depends mainly upon the viscosity ratio*. For*
instance, if*a = 0.01, the mobility of rigid sphere/mobility of*
drop with* = 1.0 is equal to 0.672, which is about the same as*
the ratio (drag on rigid sphere/drag on drop). However, as *a*
increases, the ratio (mobility of rigid sphere/mobility of drop)
declines; for example, it becomes 0.325 at *a = 10. In other*
words, the rate of increase in the mobility increases with the
increase in*a, as is seen in*Fig. 5. As mentioned previously,
this is because the larger the value of*a, the greater the electric*
driving force acting on a drop.

κa
DF
10-2 _{10}-1 _{10}0 _{10}1
-20
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
-9
-8
rigid sphere
100.0
10.0
1.0
0.1
σ = 0.01
gas bubble

Fig. 7. Variation of scaled drag force DF as a function of*a for various*
values ofat_{0}*= 1.0. Dash-dot line represents result for a rigid sphere*
(Wacholder and Weihs, 1972).

σ
DF
10-2 _{10}-1 _{10}0 _{10}1 _{10}2
-19
-18
-17
-16
-15
-14
-13
-12
-11
-10
-9
-8
rigid sphere DF = -18.21638
gas bubble DF = -8.2953

Fig. 8. Variation of scaled drag force DF on the drop surface as a function
offor the case when0*= 1.0 anda = 1.0, dashed line represents result*
for a rigid sphere (Wacholder and Weihs, 1972).

The inﬂuence of the distance between drop and surface,
mea-sured by0= cosh−1*(h/a), on the electrophoretic behavior of*

a drop is presented inFig. 9. This ﬁgure reveals that for a ﬁxed
value of*a the mobility of a drop increases with its distance*
from a plane; similar behavior was also observed in the case of
a rigid sphere (Ennis and Anderson, 1997;Tang et al., 2001). It
is expected that as_{0}→ ∞, the behavior of a drop approaches
to that of an isolated drop in an inﬁnite liquid. Note that if a
drop is sufﬁciently close to a plane, its mobility may change

κa
U
* / E
z
*
10-3 _{10}-2 _{10}-1 _{10}0 _{10}1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
0.5
0.75
1.25
1.0
η0=2.0

Fig. 9. Variation of scaled electrophoretic mobility*(U*∗*/E*∗*) as a function*
of*a for various values of*0 at= 1.0.

κa
Fin
v
10-3 _{10}-2 _{10}-1 _{10}0 _{10}1
-10
-5
0
0.5
0.75
1.25
1.0
η0=2.0

Fig. 10. Variation of scaled induced electric force*F*invas a function of*a*
for various values of_{0}at*= 1.0.*

sign, that is, it moves in the opposite direction as that of the applied electric ﬁeld. This phenomenon was also observed in the case of a rigid sphere (Tang et al., 2001; Chih et al., 2002), and can be explained by the effect of the deformation of the double layer surrounding a drop. As illustrated inFig. 3(a), if a drop is sufﬁciently close to a plane, the double layer sur-rounding the former will be distorted by the latter, that is, the equilibrium electric ﬁeld constructed by the drop will be com-pressed by the plane. In this case, an electric force, which is in the opposite direction as that arises from the applied electric ﬁeld, is induced. Fig. 10illustrates the variation of the scaled

magnitude of the electric force acting on a drop arising from
this induced electric ﬁeld,*F*inv , as a function of*a. This *

ﬁg-ure shows that*F*invincreases with the decrease in the distance

between the drop and the plane. Note that if0is smaller than

about 0.5,*|F*inv| becomes greater than the scaled electric force

arising from the applied electric ﬁeld, which is about 7.051 at
0*= 0.5 for the present case.*

Although we consider the case when only one plate is present,
the result obtained is also applicable to the case when a drop
is moving normal to one of the two parallel plates, if it can
be considered to be sufﬁciently far away from one of those
plates. The electrophoresis of droplet between two electrode
surfaces is a typical example. The criterion for the critical
dis-tance can be inferred from the result ofWacholder and Weihs
(1972). For a system consisting of a drop and a plate they
con-cluded that if 0*5.5, or the ratio (drop–plate distance/drop*

radius)*122.348, the drag on the drop is essentially the same*
as that when the plate is absent. The deformation of droplet
is neglected in the present analysis. According toTaylor and
Acrivos (1964), the deformation of a drop is proportional to

*Re*2* _{, or to the Weber number We}_{= U}*2

_{a/}1*, Re and*1being,

respectively, the Reynolds number and the surface tension of
*drop liquid. In our case, because typical value of Re is on the*
order of 10−10–10−8*and that of We is on the order of 10*−17,
the deformation of a drop can be neglected.

**4. Conclusions**

The boundary effect on the electrophoresis of a non-rigid particle is investigated by considering a non-conducting New-tonian drop moving normal to a plane. The present boundary-valued problem is solved numerically by an orthogonal collo-cation method and the inﬂuences of the key parameters of the system under consideration are investigated through numerical simulation. We found that, in general, the thinner the double layer surrounding a drop and/or the longer its distance from a planar surface, the larger its mobility. Also, if a drop is sufﬁ-ciently close to a plane, its mobility may change sign. These results are similar to the case of a rigid particle. The mobility of a drop decreases with the increase in the ratio (viscosity of drop ﬂuid/viscosity of dispersion medium). Under the condi-tions assumed, a drop can be treated as a rigid sphere if the vis-cosity ratio exceeds about 100, and as a bubble if it is smaller than about 0.01.

**Notation**

*a* radius of particle, m

*c* focus length of bipolar coordination, m

DF hydrodynamic drag force, N

*e* charge of electron, 1*.6 × 10*−19C

**Ez** applied electric ﬁeld, V/m

*F*inv electrical force induced by the imbalance charge

distribution, N

*F* force, N

*h* distance between center of particle and plane, m

*kB* Boltzmann constant, 1*.38 × 10*−23J*/K*
*nj* *number density of electrolytes j, number/m*3
*nj0* *bulk density of electrolytes j, number/m*3

*p* pressure, N/m2

*Re* Reynolds number, dimensionless

*T* absolute temperature, K

*U* terminal velocity, m/s

**v** velocity vector, m/s

*We* *Weber number, We= U*2*a/*1, dimensionless
*x* *x = cosh − cos , dimensionless*

*y* *y-component of Cartesian coordinates, m*
*z* *z-component of Cartesian coordinates*
*zi* *valence of ionic species i, dimensionless*
*Greek letters*

*n*10/*n*20, dimensionless

permittivity, C/V/m

*a* zeta potential, V

*-component of bipolar coordinates, m*
0 position of the liquid drop, m

reciprocal Debye length, 1/m

viscosity of dispersion medium, kg/m/s
*-component of bipolar coordinates, m*

space charge density in dispersion medium,

C*/m*3

viscosity ratio*(i)/(e)*, dimensionless
1 surface tension, kg*/s*2

electrical potential, V

*-component of bipolar coordinates, deg*

*
* stream function, m3/s
*Superscripts*
* scaled symbols
*e* exterior of a drop
*i* interior of a drop
*Subscripts*
*e* equilibrium properties
* * perturbed properties
**Acknowledgment**

This work is supported by the National Science Council of the Republic of China.

References

Ashok, M., Joykrishna, D., 2005. Vesicles as pseudostationary phase for enantiomer separation by capillary electrophoresis. Journal of Chromatography A 1070, 185–192.

Baygents, J.C., Saville, D.A., 1991a. Electrophoresis of drops and bubbles. Journal of the Chemical Society—Faraday Transactions 87, 1883–1898. Baygents, J.C., Saville, D.A., 1991b. Electrophoresis of small particles and

ﬂuid globules in weak electrolytes. Journal of Colloid and Interface Science 146, 9–37.

Booth, F., 1951. The cataphoresis of spherical ﬂuid droplets in electrolytes. Journal of Chemical Physics 19, 1331–1342.

Breyer, E.D., Howard, S., Raje, N., Allison, S., Apkarian, R., Brown, W.V., Strasters, J.K., 2003. Study of lipid and apolipoprotein binding interactions

using vesicle afﬁnity capillary electrophoresis. Analytical Chemistry 75, 5160–5169.

Chih, M.H., Lee, E., Hsu, J.P., 2002. Electrophoresis of a sphere normal to a plane at arbitrary electrical potential and double layer thickness. Journal of Colloid and Interface Science 248, 383–388.

Craxford, S.R., Gatty, O., McKay, H.A.C., 1937. The theory of electrocapillarity—Part VI. A note on electrophoresis. Philosophical Magazine 25, 172.

Ennis, J., Anderson, J.L., 1997. Boundary effects on electrophoretic motion of spherical particles for thick double layers and low zeta potential. Journal of Colloid and Interface Science 185, 497–514.

Finlayson, B.A., 1980. Nonlinear Analysis in Chemical Engineering. McGraw-Hill, New York.

Happel, J., Brenner, H., 1983. Low-Reynolds Number Hydrodynamics. Martinus Nijhoff, Dordrecht.

Keh, H.J., Anderson, J.L., 1985. Boundary effects on electrophoretic motion of colloidal spheres. Journal of Fluid Mechanism 153, 417–439. Kelsall, G.H., Tang, S.Y., Yurdakul, S., Smith, A.L., 1996. Electrophoretic

behaviour of bubbles in aqueous electrolytes. Journal of the Chemical Society—Faraday Transactions 2 92, 3887–3893.

Lee, E., Kao, J.D., Hsu, J.P., 2002. Electrophoresis of a nonrigid entity in a spherical cavity. Journal of Physical Chemistry B 106, 8790–8795. Lee, E., Hu, J.K., Hsu, J.P., 2003a. Electrophoresis of concentrated mercury

drops. Journal of Colloid and Interface Science 257, 250–257.

Lee, E., Fu, C.H., Hsu, J.P., 2003b. Electrophoresis of a concentrated dispersion of nonrigid particles. Langmuir 19, 3035–3040.

Lee, E., Tang, Y.P., Hsu, J.P., 2004. Electrophoresis of a membrane-coated sphere in a spherical cavity. Langmuir 20, 9415–9421.

Lee, E., Chang, C.J., Hsu, J.P., 2005a. Electrophoresis of a concentrated aqueous dispersion of non-Newtonian drops. Journal of Colloid and Interface Science 282, 486–492.

Lee, E., Huang, T.H., Hsu, J.P., 2005b. Sedimentation of a composite particle in a spherical cavity. Langmuir 21, 1729–1737.

Levich, V.G., 1962. Physicochemical Hydrodynamics. ﬁrst ed. Prentice-Hall, New York. (Chapter IX).

Levine, S., O’Brien, R.N., 1973. Theory of electrophoresis of charged mercury drops in aqueous-electrolyte. Journal of Colloid and Interface Science 43, 616–629.

O’Brien, R.W., White, L.R., 1978. Electrophoretic mobility of a spherical colloidal particle. Journal of the Chemical Society—Faraday Transactions 2 74, 1607–1626.

Ohshima, H., 1997. A simple expression for the electrophoretic mobility of charged mercury drops. Journal of Colloid and Interface Science 189, 376 –378.

Ohshima, H., 1999. Electrokinetic phenomena in a concentrated dispersion of mercury drops. Journal of Colloid and Interface Science 218, 533–544. Ohshima, H., 2003. Electrophoretic mobility of a liquid drop in a salt-free

medium. Journal of Colloid and Interface Science 263, 333–336. Ohshima, H., Healy, T.W., White, L.R., 1984. Electrokinetic phenomena in

a dilute suspension of charged mercury drops. Journal of the Chemical Society—Faraday Transactions 2 80, 1643–1667.

Tang, Y.P., Chih, M.H., Lee, E., Hsu, J.P., 2001. Electrophoretic motion of a charge-regulated sphere normal to a plane. Journal of Colloid and Interface Science 242, 121–126.

Taylor, T.D., Acrivos, A., 1964. On the deformation and drag of a falling viscous drop at low Reynolds number. Journal of Fluid Mechanics 18, 466–476.

Wacholder, E., Weihs, D., 1972. Slow motion of a ﬂuid sphere in the vicinity of another sphere or a plane boundary. Chemical Engineering Science 27, 1817–1828.