electrophoresis of a spherical particle normal to an air–water interface

Research Article

Electrophoresis of a spherical particlenormal to an air–water interface

Electrophoresis of a spherical particle normal to an air–water interface is considered theo-

retically in this study. The presence of the air–water interface is found to reduce the particle

mobility in general, especially when the double layer is very thick. This boundary effect

diminishes as the double layer gets very thin. The higher the surface potential, the more

significant the reduction of mobility due to the polarization effect from the double layer

deformation when the particle is in motion. Local extrema are observed in the mobility

profiles with varying double layer thickness as a result. Comparison with a solid planar

boundary is made. It is found that the particle mobility near an air–water interface is smaller

than that near a solid one when the double layer is thick, and vice versa when the double layer

is thin, with a critical threshold value of double layer thickness corresponding roughly to the

touch of the interface. The reason behind it is clearly explained as the buildup of electric

potential at the air–water interface, which reduces the driving force as a result.

Keywords:

Air–water interface / Boundary effect / Polarization effect / Spherical particleDOI 10.1002/elps.201000012

1 Introduction

Electrophoresis is probably the most well-known electro-

kinetic phenomenon which relates the particle hydrody-

namic velocity with the electric properties on the colloid

surface. It has been used extensively in the colloid science as

a power tool to explore the colloid surface properties, as well

as a versatile separation technique due to its simplicity to set

up and so forth. Recently, it also finds tremendous potential

applications in nanotechnology such as a biosensor or a lab-

on-a-chip device [1, 2].

While uniform suspensions, either dilute or concentrated,

are normally used to investigate the electrophoretic behavior of

colloidal particles in fundamental colloid science, systems

involving a nearby physical boundary are often encountered in

practical applications [3–7]. Among them, the electrophoresis of

a colloidal particle near a planar wall has been studied exten-

sively with theoretical approaches. The presence of a planar

wall in general was found to retard the particle motion due to

the hydrodynamic drag and the electrostatic interaction with

the double layer surrounding the particle. In particular, Lou

and Lee [8] found that whether the double layer touches the

plane serves as a criterion predicting the overall impact on the

magnitude and direction of the particle motion. Corresponding

studies for a free surface, however, has been quite limited.

There is a fundamental discrepancy between a solid

plane and a free surface. For example, the electrolyte solu-

tion in contact with a solid wall is immobile, whereas it is

mobile at a free interface. The motion of the electrolyte

nearby greatly affects the interfacial electric potential; hence,

the particle motion. Moreover, perhaps the fundamental

difference between the two boundaries in terms of electro-

static interaction can be best illustrated by the image-charge

analogue in electrostatics [9]. As a result, the conventional

boundary conditions and analyses regarding a solid–liquid

interface are not directly applicable to a free surface.

Among various types of free surfaces, the air–water

interface is probably the most important and widely

encountered one in practice. Colloidal crystallization [10–12]

is a typical example of motion of particles near an air–water

interface. Aggregation of particles and adsorption of

proteins at an air–water interface find wide applications in

denaturation and stabilization of products in the biological

and food industries [12]. In corresponding theoretical

analyses, Terao and Nakayama [10] studied the suspension

of colloids trapped at an air–water interface by Monte Carlo

simulations. Mbamala and von Grunberg [11] explored

further the electrostatic interaction between a charged

particle and an air–water interface when the particle is

approaching or trapped at the interface. According to their

results, the electrical double layer around the particle forms

a significant barrier when the particle moves to the

air–water interface.

These studies reviewed above have focused on mechan-

isms controlling aggregation or the particle–interface electro-

static interaction. As for the electrokinetics involving particle

motion near an air–water interface, Huddeson and Smith [13]

and Usui and Healy [14] developed a technique based on

particle electrophoresis to determine the zeta potential of

Peter TsaiJames LouYan-Ying HeEric Lee

Department of ChemicalEngineering, National TaiwanUniversity, Taipei, Taiwan

Received January 8, 2010Revised June 15, 2010Accepted July 23, 2010

Correspondence: Professor Eric Lee, Department of ChemicalEngineering, National Taiwan University, Taipei 10617, TaiwanE-mail: [email protected]: 1886-2-23622530

& 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.electrophoresis-journal.com

Electrophoresis 2010, 31, 3363–3371 3363

an air–water interface experimentally. Gao and Li [15] recently

studied the electrophoresis of a particle parallel to a

liquid–fluid planar interface, motivated by its potential appli-

cation in microfluidic operations.

The applied electric fields are all parallel to the free

surface in the above electrophoresis analyses though. No

corresponding studies for vertically applied electric field

have been carried out, at least in the theoretical front, to the

best of our knowledge. Very little is known about the elec-

trophoretic motion of a particle normal to an air–water

interface. It is very important to understand how the

air–water interface affects the electrokinetic transport of a

colloidal particle, judging from its already proven significant

impact in the horizontal situation.

In research fields other than electrophoresis, the appli-

cation of a vertical electric field across an air–water interface

has been explored by various researchers with different

motivations [16–18]. Among them, Hayes [16] investigated

experimentally the variation of surface tension of an

air–water interface in the presence of an applied vertical

electric field. Arisawa [17] explored the feasibility of

controlling molecular orientations by a pulsating vertical

electric field at the air–water interface, which has promising

potential in the fabrication of 2-D colloidal crystal. Sugiyama

et al. [18] tried to achieve the same goal with similar idea

but using direct electric field instead. Although the appli-

cation of a vertical electric field across an air–water interface

takes place frequently in experimental studies involving

colloids, there have been few theoretical treatments, if any.

As a result, we present here a study of the electrophoretic

behavior of a spherical particle normal to an air–water inter-

face in response to a vertically applied electric field. It should

be noted that an extremely thin, about half the radius of an

electrolyte ion-free layer may take place near the interface

[19–21]. Its effect is usually negligible, however, unless the

particle is extremely close to the interface, or the particle itself

is of the comparable dimension of this layer. Either situation

is excluded from the scope of the current analysis. A pseudo-

spectral method adopted by Lee et al. [7–9] is applied here

again for the solution of the resulted governing electrokinetic

equations. The method has proven to be a very powerful tool

treating electrophoresis problems and other electrokinetic

systems. Key parameters are examined for their effects on the

mobility such as the distance between the particle and the

air–water interface, the thickness of double layer surrounding

the particle, the zeta potential of the particle, and so on. In

particular, a detailed comparison of boundary effects between

that of an air–water interface and a solid planar wall are

extensively discussed. The analysis and results are presented

in the subsequent sections.

2 Theory

2.1 General electrokinetic equations with major

assumptions

Consider a charged particle of radius a moving with a

velocity U near an air–water interface containing a general

electrolyte in an applied electric field E, as illustrated in

Fig. 1. The distance between the center of the particle and

the interface is denoted as h. The symmetric electrolyte

solution contains two ionic species of valences zj (j 5 1, 2),

and nj0 be the bulk number concentration of the jth ionic

species in the electroneutral solution.

The bipolar coordinates (x, Z, j) are adopted to describe

the system. The bipolar coordinates and the Cartesian

coordinates are related by [22]

z ¼ csinhZ

cosh Z� cos x; ð1Þ

$ ¼ csin x

cosh Z� cos x; ð2Þ

where c is the focal length. For convenience, we define

Z0 5�cosh�1(h/a). Z5 0 and Z5�Z0 represent, respec-

tively, the air–water interface and the particle surface.

The main assumptions in our analysis are as follows.

(i) The Reynolds numbers of the liquid flow is small enough

to ignore inertial terms in the Navier–Stokes equations and

the liquid can be regarded as incompressible. (ii) The applied

field E is weak so that the particle velocity U is proportional to

E and terms of higher order in E may be neglected. In practice

this means that E is small compared with the fields that occur

in the double layer, with |E|oozak, the characteristic electric

field measured by the zeta potential divided by the double

layer thickness. (iii) The air phase can be treated as a non-

conducting medium, as the dielectric constant of air is much

smaller than that of water (1:80). (iv) The air phase is

considered to be inviscid, as the viscosity of air is much less

than that of water. (v) The air–water interface remains planar

when the particle approaches it [22]. (vi) The particle surface is

ion-impenetrable. (vii) There is no intrinsic charge at the

air–water interface. (viii) The system is at quasi-steady state

ϖ

z

a

h

air

U

Ez

Colloid

electrolyte solution

Figure 1. Geometric configuration of the system in this study.

Electrophoresis 2010, 31, 3363–33713364 P. Tsai et al.


when the particle is in motion. (ix) The particle is noncon-

ducting and assumes a constant surface potential, or zeta

potential za. Note also that that the scope of the current

theoretical analysis assuming chargeless interface has to be

confined to systems with no ions adsorbed to the air–water

interface, as observations of surface charges were made in

recent years for some specific electrolytes [23].

The fundamental electrokinetic equations are as follows

[7]:

H2f ¼ � re¼ �

X2

j¼1

zjenj

e; ð3Þ

� Hp1mH2v� rHf ¼ 0; ð4Þ

H � v ¼ 0; ð5Þ

H � f j ¼ 0; ð6aÞ

with

f j ¼ �Dj Hnj1zjenj

kBTHf

� �1nj v; ð6bÞ

where e is the permittivity of the solution, r the space charge

density, e the elementary charge, m the viscosity of the solu-

tion, Dj the diffusivity of the ionic species j, kB the Boltzmann

constant, and T the absolute temperature. In Eqs. (3)–(6), f, v,

and nj are, respectively, the electrical potential, the liquid

velocity, and number concentration of the jth ionic species.

Note that Eq. (3) is the Poisson equation. Eq. (4) is the Stokes

equation with an extra electric body force term, whereas

Eq. (5) is the continuity equation, indicating that the fluid is

incompressible. Eq. (6a) stands for the conservation law of

ion flux at quasi-steady state and Eq. (6b) is a modified

Nernst–Planck equation taking account of the convection

contribution.

Taking curl of Eq. (4) to get rid of the pressure term,

introducing a stream function c so defined that Eq. (5) is

satisfied automatically, and substituting Eqs. (6b) to (6a), we

end up with:

E4c ¼ x

mc

qrqx

qfqZ� qrqZ

qfqx

� �sinx; ð7Þ

H2nj1zje

kBTðHnj � Hf1njH2fÞ � 1

Djv � Hnj ¼ 0; ð8Þ

where E4 5 E2E2, x 5 cosh Z–cos x, and

H2 ¼ x2

c2

�q2

qZ2þ q2

qx2� sinh Z

x

qqZþ cosxcosh Z� 1

xsinxqqx

�; ð9Þ

E2 ¼ x2

c2

�q2

qZ2þ q2

qx2þ sinh Z

x

qqZþ 1� cosxcosh Z

xsinxqqx

�: ð10Þ

The velocity field is related to the stream function c as:

nx ¼ �ðcosh Z� cos xÞ2

c2sinxqcqZ

;

nZ ¼ðcosh Z� cos xÞ2

c2sinxqcqx:

ð11Þ

Eqs. (7) and (8) plus Eq. (3) are the governing equations in

the current analysis. Three coupled nonlinear equations in

terms of three independent variables: f, c, and nj. Note that

for a symmetric electrolyte solution, only one nj is inde-

pendent, with the other determined by the electro-neutrality

constraint in the bulk liquid phase.

2.2 Equilibrium state

At equilibrium state, no external electric field is applied, and

the system is motionless. Corresponding electric field and

number concentration can be obtained by solving Eqs. (3) and

(8) together, which is simply the famous Poisson–Boltzmann

equation, subject to appropriate boundary conditions:

H2fe ¼ �X2

j¼1

zjenj0

eexp � zjefe

kBT

� �; ð12Þ

where subscripts e denotes the equilibrium state variables.

The two boundary conditions needed for fe are fe 5 za at

Z5�Z0 and qfe/qZ5 0 at Z5 0, which corresponds,

respectively, the assumption of a constant zeta potential on the

particle surface and the air–water interface is non-conducting.

In addition, the axial symmetry condition requires that qfe/

qx5 0 at x5 0, p. The resulting non-linear, 2-D equilibrium

state solution can thus be calculated numerically.

2.3 Linearized perturbed state

As the applied field is weak, the ion cloud surrounding the

particle is only slightly distorted and so is the subsequent

particle motion. In other words, the electric potential field in

electrophoresis can be decomposed into the sum of an

equilibrium electrical potential fe and a perturbed electrical

potential df arising from the applied electric field, i.e.,

f ¼ fe1df; ð13Þ

with dfoofe. Similarly,

nj ¼ nje1dnj; ð14Þ

The symbol d represents the disturbed state, which is

assumed very small. Following the analysis by O’Brien and

White [24], the effect of double layer polarization is

introduced at this stage by defining a position-dependent

perturbation function gj as follows:

nj ¼ nj0exp � zje

kBTðfe1df1gjÞ

� �; ð15Þ

where gj represents an equivalent perturbed potential to be

determined that measures the magnitude of the double

layer polarization effect. Note that gj refers to the extra

deformation of the double layer due to the flow field. In the

original analysis by O’Brien and White [24] for a single

sphere suspended in an infinite medium, the polarization

effect refers to the extra effect in addition to the

concentrically symmetric distribution of ions predicted by

the Poisson–Boltzmann equation. It should be noted,

Electrophoresis 2010, 31, 3363–3371 General 3365


however, that the prediction by the Poisson–Boltzmann

equation is generally not concentrically symmetric for 2-D or

3-D systems. For instance, the equilibrium state electric

potential profile here is computed as a 2-D system. With the

presence of a nearby planar air–water interface, the double

layer is compressed somehow as shown later. As a result,

dnj ¼ �njezje

kBTðdf1gjÞ

� �: ð16Þ

Substituting the forms for f and nj in the Eqs. (13)–(16) into

the original governing Eqs. (3)–(8), and using the fact that

the equilibrium state quantities feand nje must satisfy these

equations too, we obtain the approximate linear equations:

H2df ¼ �X2

j¼1

zjenj0

eexpð� zje

kBTðfe1df1gjÞ

��

�exp � zjefe

kBT

� �; ð17Þ

H2g1 �z1e

kBT

qfe

qxqg1

qx1qfe

qZqg1

qZ

� �

¼ 1

D1

x

c

� �3 1

sinxqfe

qxqcqZ

1qfe

qZqcqx

� �; ð18Þ

H2g2 �z2e

kBT

qfe

qxqg2

qx1qfe

qZqg2

qZ

� �

¼ 1

D2

x

c

� �3 1

sinxqfe

qxqcqZ

1qfe

qZqcqx

� �; ð19Þ

E4c ¼ x

mc

�qre

qxqdfqZ

1qdrqx

qfe

qZ� qre

qZqdfqx� qdr

qZqfe

qx

�: ð20Þ

In deriving these equations we have neglected the products

of the small quantities such as (dnj)2 or df � dnj.

Corresponding boundary conditions for df, gj, and cfollowing the physical assumptions we mentioned earlier

are given in Eqs. (21)–(30). On the particle surface, we have:

qdfqZ¼ 0; Z ¼ �Z0; ð21Þ

qgj

qZ¼ 0; Z ¼ �Z0; ð22Þ

c ¼ � 1

2

csinxx

� �2

U; Z ¼ �Z0; ð23Þ

qcqZ¼ c2

x3sin2 xsinh ZU; Z ¼ �Z0: ð24Þ

Eqs. (22)–(24) reflect the fact that the particle surface is non-

conducting, ion-impenetrable, and moves with a constant

velocity U.

At the air–water interface, on the other hand, we have:

qdfqZ¼ �E; Z ¼ 0; ð25Þ

where E is the magnitude of electric field at the ‘‘liquid

phase’’ side of the interface. The air phase is treated here in

this analysis as a non-conducting medium as the dielectric

constant of air is much smaller than that of water (1:80), as

mentioned earlier in the main assumptions, i.e. eair _¼0.

Moreover, the possible accumulation of electric charges at

the interface is taken to be zero as well by the order of

magnitude analysis, to be consistent with the earlier

assumption in electrostatics that the air–water interface is

chargeless, or the scope is limited to this situation. As a

result, the corresponding physical boundary condition

considering interfacial polarization, or the Maxwell-Wagner

effect [25–27], yields Eq. (25).

There is an extremely thin ion-free layer near the air–water

interface at equilibrium state, as inferred by the ‘‘image charge

repulsion’’ theory in electrostatics [11, 19–21]. Assuming the

essential physical situation remains the same in electrophor-

esis, as the applied electric field df is much smaller than the

characteristic equilibrium state value fe, we have:

gj ¼ �df; Z ¼ 0; ð26Þ

or the ion concentration at the interface remains intact in

electrophoresis.

For the fluid flow at the interface, the kinematic

condition describing the impenetrability of the air–water

interface requires that the normal velocity must vanish

there:

qcqx¼ 0; Z ¼ 0: ð27Þ

Furthermore, the shear stress exerted on the tangential

plane at the interface must vanish there too, as inferred by

the assumption that the air phase is inviscid [8, 22]. Hence

we have:

eðEE� 12 E2IÞ1m½Hv1ðHvÞT�g � iZ � ix ¼ 0; Z ¼ 0; ð28Þ

where E 5�HF and iZ, ix are the unit vectors along the Zand x direction, respectively. Or:

eqdfqZ

@fe

@x� m

x

csinx@2c@Z2¼ 0;Z ¼ 0; ð29Þ

In addition, all variables must satisfy the axial symmetry

condition at x5 0 and x5 p which yields

qdfqx¼ qgj

qx¼ qc

qx¼ c ¼ 0; x ¼ 0;p; ð30Þ

where c5 0 is a reference value for stream function.

To simplify the analysis, dimensionless variables are

introduced as follows: r�5 r/a, n�j ¼ nj=n10, E�5 E/xa/a,

v�5 v/UE, f�5f/za, g�j ¼ gj=za, and c�5c/UEa2, where

the superscript � denotes dimensionless quantities. Note

that n10 is the bulk number concentration of the ionic

species 1, and UE is the particle velocity predicted by

Smoluchowski’s theory when an electric field za/a is applied,

i.e., UE ¼ ez2a=Za.

Following the analysis of O’Brien and White [24], the

current electrophoresis problem can be decomposed into

two subproblems to facilitate subsequent mathematical

treatments. The mobility of a particle can thus be calculated



based on the requirement that the net force acting on the

particle surface, the summation of the electric driving force

and the retarding hydrodynamic force, must vanish at the

quasi-steady state in electrophoresis. The details of the

numerical procedure can be found elsewhere [6–8].

3 Results and discussion

Standard procedure of mesh refinement is carried out

to ensure the convergence of the numerical scheme

adopted in terms of grid independence. A mesh with

40� 40 nodal points, in (x, Z) domain is found to be

sufficiently accurate, and is used to carry out the calculations

in this study. Figure 2 depicts the scaled mobility

m�m(defined as m�m ¼ mm=ðeza=mÞ) as a function of ka for

h�5 74.2. At such a distance from the free surface, the

presence of the surface is hardly ‘‘felt’’ by the colloid [11].

The system is essentially like a single particle suspended in

an infinite medium.

Corresponding theoretical predictions based on Henry’s

classic analysis valid for the low zeta potential situation as

well as those by Lee et al. for high potentials based on a

separate cell model approach [28] are shown in Fig. 2 for

verification purpose. As can be seen in Fig. 2, our results

reduce successfully to either case for the range of a exam-

ined. In addition, the results of Wiersema et al. [29] are

recovered successfully at high zeta potential situations as

well with the separate cell model approach, which is not

shown here due to different parameter settings. Note that

the almost invariant mobility observed in Fig. 2 is the

asymptotically approached Huckel’s [30] prediction

m�m ¼ 23 � 0:67, which is valid when ka is very small. More-

over, all mobility profiles asymptotically converge to

Smoluchowski’s [31] results with thin double layer when agets large, as they should be.

As for the effect of double layer thickness, a thinner

double layer (higher a) tends to enhance the particle mobi-

lity in general due to a steeper gradient of electric potential

near the particle surface as well as more space charges

nearby. For the system considered here, however, as the

double layer can be compressed by the presence of a nearby

boundary (boundary effect) and further distorted due to the

particle motion (polarization effect), the overall impact on

the particle mobility is the outcome of the combination of all

these three factors. With this general understanding in

mind, we go on to examine the specific impact of each factor

for various situations of interest, as shown in Figs. 2–4, and

Figs. 6–8.

3.1 Influence of the polarization effect and double

layer thickness with the presence of air–water

interface

Figure 3 depicts the scaled mobility m�m as a function of ka at

various scaled surface potentials at Z0 5�2, or equivalently

h�5 h/a 5 3.76. As discussed above, a thinner double layer

(higher ka) tends to enhance the particle mobility except

when fr (defined as fr 5 za/(kT/z1e)) is high at the same

time, as a steeper gradient of electrical potential near the

particle surface is expected, which in turn generates higher

electric driving force. Moreover, more space charges reside

near the particle at a thinner double layer, which also

contributes to the enhancement of particle mobility. At

higher zeta potential, however, polarization effect sets in as

well which tends to decrease the mobility. When a particle is

in motion, the ion clouds tends to cluster around the rear

end of the particle due to the convection fluid flow, which

κa10-1 100 101

0.5

0.6

0.7

0.8

φr=4

φr=3

φr=2

φr=1

φr→0

Henry function

μm*

Figure 2. Scaled mobility as a function of ka with Z0 5�5.Dashed lines: results based on a cell model approach with verydilute volume fraction. Dashed dot line: results based on Henryfunction.

κa10-2 10-1 100 101 102

0.4

0.5

0.6

0.7

0.8

0.9

1

φr→ 0

φr=1

2

3

4

Henry functionμm

*

Figure 3. Scaled mobility as a function of ka at different fr withZ0 5�2.0.



results in an uneven distribution of ionic concentrations, or

equivalently, an uneven distortion of double layer. An

induced electric field opposite to the applied field is

generated as a result, which slows down particle motion

in general, and is referred to as the polarization effect

here [29]. The competition between the accelerating

factor of a and the decelerating polarization effect yields a

clear local maximum at ka 5 0.7 for fr 5 4, approximately.

In general the polarization effect is most significant for

ka�1. As the double layer gets thinner even further, the

driving force becomes dominant and the mobility increases

eventually, yielding a local minimum around ka 5 3 for

fr 5 4. The polarization effect at lower zeta potential is still

significant, but not strong enough to yield local extrema as

fr 5 4 does.

The above observations of ka and fr effects are consis-

tent with the corresponding case of an isolated sphere. The

boundary effect of an air–water interface is analyzed as

follows by direct comparison with Henry function [32]. The

effect of an air–water interface is hardly felt when the double

layer is very thin, whereas profoundly significant when

double layer is thick. The deformation and even collapse of

the double layer drastically influence the particle motion.

The deviation of mobility readings with or without the

presence of the air–water interface is a good measurement

of the boundary effect. Regardless of the zeta potential, the

boundary effect amounts to a 30% reduction of mobility at

h�5 3.76, which appears at kaffi 10�2. This is consistent

with the observation that the thicker the double layer, the

more severely it can be distorted in shape by the interface.

Note that both the polarization effect and the boundary

effect slow down the particle motion once the double layer of

the particle touches the interface. The thicker the double

layer, the more profound the boundary effect in general, as

the degree of deformation gets more severe accordingly. The

only difference is that the boundary effect depends on the

location of the particle while the polarization effect does not.

As a result, the boundary effect further slows down the

particle motion as the double layer starts to touch the

interface.

3.2 Influence of the distance to the air–water

interface with varying ja

Figure 4 shows the scaled mobility as a function of a at

various values of Z0 for fr 5 3. The larger the distance away

from the interface, the larger the mobility is, as the boundary

effect diminishes with increasing distance in general.

As can be measured in Fig. 4, at Z0 5�1 boundary

effect amounts to a reduction of mobility roughly by 70% for

a very thick double layer (ka 5 10�2) to around 15% for

relatively thin double layer (ka 5 3). In general, the thicker

the double layer and the closer the particle to the interface,

the more significant the boundary effect is in terms of the

mobility reduction, as might be expected. As fr 5 3 is a

representative example for high zeta potential situation,

Fig. 4 serves as a very useful chart to estimate the boundary

effect for real systems of interest.

3.3 Characteristics of an air–water interface in terms

of electric potential distribution

The contour plots of the equilibrium electric potential

around the particle as well as its distribution profile at the

interface are illustrated in Fig. 5A for fr 5 3, ka 5 0.1, and

Z0 5�2. The contour plots represent the actual shape of the

double layer in physical sense. Note that at such a distance

the double layer has touched the interface.

In corresponding case of a solid boundary with constant

potential, such as a grounded metal plane, the double layer

surrounding the particle is simply squeezed in the front end

of the particle as it approaches the plane, hence yields closed

contour plots as shown in Fig. 5B. With an air–water

interface, however, the contour plots ‘‘collapse’’ in shape

here due to the outward horizontal fluid flow resulted from

the electric potential buildup at the air–water interface as

shown in Fig. 5C. Unlike a grounded metal plane where the

electric potential is kept as a constant, the electric potential

at an air–water interface builds up as shown in Fig. 5C.

With this positive potential buildup right in front, the

particle motion is expected to be slower as compared with a

constant potential solid plane. This is consistent with the

observation of an image-charge analogue in electrostatics [9]

as well. A charged colloidal particle at a distance z 5 h from

the air–water interface behaves as if there were another

colloidal particle of the same charge at z 5 h. The presence

of a solid plane, on the other hand, is equivalent to an image

particle with opposite charge. In terms of electrostatic force,

free surface poses as a repulsive force while solid plane an

attractive one. However, in pure electrostatics there is no

κa10-2 10-1 100

0.2

0.4

0.6

η0 =-1 (h*=1.54)

-1.5 (h*=2.35)

-2 (h*=3.76)

-2.5 (h*= 6.13)

-4 (h*=27.31)

-3 (h*=10.06)

h*→∞

μm*

Figure 4. Scaled mobility as a function of ka at different Z0 atfr 5 3. Dashed line: corresponding result for a single particle inan infinite medium of electrolyte solution based on a cell model.



externally applied electric field and the particle is assumed

stationary somehow. But in electrophoresis an external

electric field is applied which sets up the particle in

motion. Additional consideration of hydrodynamic

aspect is thus necessary. Therefore, strictly speaking, there

is no corresponding image charge analogue in terms of

boundary effect, but the reasoning with the electric potential

buildup is still applicable. The a dependence of the

electric potential profile at the free surface is presented

in Fig. 5C. The thinner the double layer, the less

buildup of the electric potential at the interface, hence the

flatter the profile is. This is because the counterions are

sparse outside the double layer. When the double layer gets

thinner, lesser counterions exist near the interface to enable

a buildup of electric potential above the reference value

corresponding to a neutral bulk solution. As double layer

gets very thin, the electric potential profile will be very close

to that of a solid plane.

3.4 Comparison of the boundary effect between an

air–water interface and a solid planar wall

We compare the effects on electrophoresis of two funda-

mental boundaries: the air–water interface and the solid

planar wall plane with constant surface potential.

Figure 6 illustrates the scaled mobility as a function of

ka. Dashed lines are mobility profiles corresponding to a solid

planar wall, whereas the solid ones an air–water interface.

Figure 6 reveals that for a fixed value of ka, the mobility

increases with its distance from either boundary in general.

As ka is low, the particle mobility toward an air–water

ϖ*

ϖ*

ϖ*

0 0 2 4 61 2 3 4 5 6 7

0 2 4 6 8 10 12

-7

-6

-5

-4

-3

-2

-1

0

18 0.9517 0.916 0.8515 0.814 0.7513 0.712 0.6511 0.610 0.559 0.58 0.457 0.46 0.355 0.34 0.253 0.22 0.151 0.05

z* z*

-7

-6

-5

-4

-3

-2

-1

0

18 0.9517 0.916 0.8515 0.814 0.7513 0.712 0.6511 0.610 0.559 0.58 0.457 0.46 0.355 0.34 0.253 0.22 0.151 0.1

0

0.1

0.2

0.3

0.4

κa=0.05

φ*e

1

0.1

0.5

0.3

A

C

B

Figure 5. (A) Contours of scaled equili-brium electric potential f�e with ka 5 0.1,Z0 5�2, and fr 5 3. The top edge corre-sponds to the location of an air–waterinterface. (B) Contours of scaled equili-brium electric potential f�e with ka 5 0.1,Z0 5�2, and fr 5 3. The top edge corre-sponds to the location of a solid planarwall. (C) f�e distribution along theair–water interface at different ka withZ0 5�2 and Fr 5 3.



interface is smaller than that toward a solid planar wall. And

the smaller the ka is, the larger the discrepancy between the

two boundaries. This can be attributed to the different

situations of the electrostatic interaction between the particle

and the two planar boundaries. If the planar boundary is an

electrically grounded metal, the electric potential on the plane

is maintained at zero at all time. Hence the electric potential

difference between the front end of the particle and the plane

remains unchanged as shown in Fig. 5B. However, at the

air–water interface there will be an electric potential buildup

as we have discussed earlier, hence the potential difference

now is lower in comparison, yielding a lower driving force.

When ka gets large, however, the double layer ceases to touch

the boundary and the buildup of electric potential at the

interface is negligible. As a result, the electric boundary

condition will be similar to that of a solid planar wall.

Moreover, under this situation the electric driving force will

increase due to a steeper potential gradient near the particle

surface. The hydrodynamic drag force becomes the crucial

factor to reckon with as the particle mobility is concerned. An

air–water interface exerts less retardation impact upon the

particle motion than a solid planar wall from the hydro-

dynamic point of view, due to the ability of fluid flow in the

tangential direction at the surface. Therefore, the mobility

will be larger now instead. Between these two limiting cases

exists a threshold value of a where both boundaries yield the

same mobility, as shown in Fig. 6 by the symbol ‘‘& ’’. The

locations of these threshold values of ka correspond roughly

to the onset of touching the boundaries.

Figure 7 presents the scaled mobility as a function of ka at

even closer distance (Z0 5�1, or equivalently h�5 1.54) to

the air–water interface, as compared with h�5 3.76. Both the

air–water interface and solid planar wall are considered.

Boundary effects in either cases, as well as polarization effects

are presented nicely in this figure. Corresponding results at a

distance h�5 3.76 are summarized in Fig. 8 as well.

The polarization effect tends to decrease the particle

mobility in general, as shown in Figs. 7 and 8, where higher

fr yields lower mobility in both figures. As discussed earlier,

this is due to the opposite electric field generated by the

uneven distortion of the double layer when the particle is in

motion. In general, the polarization effect is found to be

significant around ka 5 1, and the higher the zeta potential

of the particle, the more profound this effect is. However,

the qualitative behavior of mobility profile appears to be

different in Figs. 7 and 8, which are analyzed below.

In Fig. 7, the mobility increases with increasing kamonotonously for fr 5 4; whereas in Fig. 8, the mobility

exhibits a local maximum. The involvement of the addi-

tional boundary effect explains this seemingly contradictory

behavior. In Fig. 8, the double layer does not touch the

κa10-2 10-1 100 1010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φr=1

Solid Planar Wall

Air-Water Interface η =-1

h =1.54

Henry function

μm*

23

4

Figure 7. Scaled mobility as a function of ka at different fr withZ0 5�1.0. Dashed lines: results of solid planar wall.

κa10-2 10-1 100 101 1020

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φr → 0

12

3

4Solid planar wall

Air-water interface

η =-2

h =3.76

Henry function

μm*

Figure 8. Scaled mobility as a function of ka at different fr withZ0 5�2.0. Dashed lines: results of solid planar wall.

κa10-2 10-1 100

0.2

0.4

0.6

0.8

-2.5

η0=-1

-1.5

-2

φr=3

-3

Single particle in an infinite electrolyte solution

μm*

Figure 6. Scaled mobility as a function of ka at different Z0 withfr 5 3. Dashed lines: results of solid planar wall.



interface in the range of ka when the polarization effect is

significant, indicated by the local maxima observed. In

Fig. 7, however, the extra boundary effect further suppresses

particle motion as the double layer touches the interface in

the a range corresponding roughly to the left side of the

local maximum for fr 5 4, which results in a seeming

monotonous behavior.

4 Concluding remarks

Electrophoresis of a spherical particle normal to an

air–water interface is investigated theoretically. Key para-

meters of electrokinetic interest are examined to explore

their respective effect on the electrophoretic mobility.

Moreover, a detailed comparison with the solid planar wall

is carried out to examine respective boundary effects and the

unique phenomena associated with the air–water interface.

Classic results of Henry for a single particle valid at low

potential are recovered excellently when the particle is far away

from the interface. The presence of the air–water interface is

found to reduce the particle mobility in general, especially

when the double layer is very thick. Up to 70% mobility

reduction is observed for Z0 5�1 and fr 5 3. This boundary

effect diminishes as the double layer gets very thin. It is also

found that the higher the surface potential, the more signifi-

cant the reduction of mobility due to the additional polariza-

tion effect. Local extrema are observed in the mobility profiles

with varying double layer thickness as a result. Finally,

comparison with a solid planar boundary is made. The particle

mobility near an air–water interface is smaller than that near a

solid one when the double layer is thick, and vice versa when

the double layer is thin, with a critical threshold value of

double layer thickness corresponding roughly to the onset of

touching of the interface. The reason behind it is clearly

explained as the buildup of electric potential at the air–water

interface, which reduces the driving force as a result.

This work is finically supported by the National ScienceCouncil of the Republic of China.

The authors have declared no conflict of interest.

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electrophoresis of a spherical particle normal to an air–water interface

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