Electrophoresis of a non-conducting Newtonian drop of low electrical potential normal to a plane

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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.

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).

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

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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. 1where 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 strengthEzin the z-direction. The distance between the center of the drop and the surface is h. The dispersion liquid containsz1:z2electrolytes,z1andz2are,

respectively, the valences of cations and anions. Letn10andn20

be, respectively, the bulk concentrations of cations and anions with n10 = n20. 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, andn_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= nj0exp −zje kBT , (4)

where T is the absolute temperature and k_B is the Boltzmann constant. For convenience, is decomposed into an equilibrium potential_eand a perturbed potential arising from the applied electric ﬁeld,  (Tang et al., 2001).

For the case_eis low, it can be shown that in the dispersion liquid

∇2∗

e= (a)2∗e, 00, (5)

where ∗_e = _e/_a, _a is the potential on drop surface and n∗

j = nj/n10. The reciprocal Debye length and the scaled

gradient operator∇2are deﬁned, respectively, by

= ⎡ ⎣2 j=1 nj0(ezj)2/kBT ⎤ ⎦ 1/2 , (6) ∇2₌ x2 c∗2 j2 j2 + j2 j2 − 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∗_eis ∇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

E4_∗(e)_{= −(a)}2x sin c∗ j∗_e j j ∗ j − j∗_e j j ∗ j , (12)

where ∗= /(k_BT /z1e)2a/ is the scaled stream function, is the stream function, the superscript e denotes the exterior of a drop,E4= E2E2, and E2₌ x2 c∗2 j2 j2 + j2 j2 + sinh x j j + 1− cos cosh x sin j j . (13)

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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 Ezin 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 andv, can be expressed, respectively, as v= x 2 c2_sin j j, (14) v= −x2 c2_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

E4∗(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) ∗_e= 0, = 0, (18) j∗_e j = 0, = 0 and = , (19) ∗_e= 0 as → ∞, (20) ∗= −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 x3 sin 2_{sinh U}∗_{, =} 0, (26) ∗(e)₌∗(i)_{, =} 0, (27) ∗(e) r2 → 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)iris the normal vector pointing from the center of a

drop to its interface, andU∗= U/U_E, whereU_E= 2/_ra. The solution procedure used by O’Brien and White (1978)

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.

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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 ofat₀= 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 thatP e1=0.01

andP e2=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 ofa at 0= 1.0, and= 1.0: (a)a = 0.01, (b)a = 7.943.

potential∗_e at two levels ofa are presented inFig. 3, and the corresponding contours of the scaled stream function ∗ are shown inFig. 4.Fig. 3indicates that a larger value ofa leads to a narrower distribution in∗_e 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 ofa, 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} _and₆_{. 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

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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 ofa, the larger is the mo-bility of a drop. A comparison betweenFigs. 6and8 reveals that for a ﬁxed value ofa the qualitative behavior of the mobil-ity is the same as that of the drag. According to Eq. (12), ifa 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 ofa for various values ofat₀= 1.0, dashed line represents result for a rigid sphere (Ennis and Anderson, 1997; Tang et al., 2001).

σ U * / E z * 10-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 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, ifa = 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 ina, as is seen inFig. 5. As mentioned previously, this is because the larger the value ofa, the greater the electric driving force acting on a drop.

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κa DF 10-2 ₁₀-1 ₁₀0 ₁₀1 -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 ofa for various values ofat₀= 1.0. Dash-dot line represents result for a rigid sphere (Wacholder and Weihs, 1972).

σ DF 10-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 -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 ofa 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 as₀→ ∞, 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 ₁₀-1 ₁₀0 ₁₀1 -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 ofa for various values of0 at= 1.0.

κa Fin v 10-3 ₁₀-2 ₁₀-1 ₁₀0 ₁₀1 -10 -5 0 0.5 0.75 1.25 1.0 η0=2.0

Fig. 10. Variation of scaled induced electric forceFinvas a function ofa for various values of₀at= 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

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magnitude of the electric force acting on a drop arising from this induced electric ﬁeld,Finv , as a function ofa. This

ﬁg-ure shows thatFinvincreases with the decrease in the distance

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

about 0.5,|Finv| 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 05.5, 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

Re2_{, or to the Weber number We}_{= U}2_a/

1, Re and1being,

respectively, the Reynolds number and the surface tension of drop liquid. In our case, because typical value of Re is on the order of 10−10–10−8and 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

Finv 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/m3 nj0 bulk density of electrolytes j, number/m3

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= U2a/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

n10/n20, 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/m3

viscosity ratio(i)/(e), dimensionless 1 surface tension, kg/s2

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.

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