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Research Article
Dynamic Electrophoresis of a Droplet in a Spherical Cavity
Eric Lee, Wei-Lun Min, and Jyh-Ping Hsu
Langmuir, 2006, 22 (8), 3920-3928 • DOI: 10.1021/la0533821 Downloaded from http://pubs.acs.org on November 21, 2008
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Dynamic Electrophoresis of a Droplet in a Spherical Cavity
Eric Lee, Wei-Lun Min, and Jyh-Ping Hsu*
Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617 ReceiVed December 14, 2005. In Final Form: February 16, 2006
The electrophoretic behavior of a droplet in a spherical cavity subject to an alternating electric field is analyzed theoretically under the conditions of an arbitrary level of surface potential and double-layer thickness. The influences of the thickness of the double layer, the level of surface potential, the size of a droplet, the viscosity of the droplet fluid, and the frequency of the applied electric field on the electrophoretic behavior of a droplet are examined through numerical simulations. We show that, because of the effect of double-layer deformation, the magnitude of the electrophoretic mobility of a droplet could have a local maximum and the phase angle could have a negative (phase lead) local minimum as the frequency of the applied electric field varies. In general, the lower the surface potential, the thicker the double layer and the larger the viscosity of the droplet fluid, and the more significant the boundary effect, the smaller the magnitude of the electrophoretic mobility of a droplet.
Introduction
Electroacoustic phenomena involve two basic effects: the colloid vibration potential and the electrokinetic sonic amplitude.1 The former is a macroscopically measurable potential difference when a supersonic wave is applied to a colloidal dispersion. The latter is the reverse effect of the former, that is, the application of an alternating electric field to a colloidal dispersion yields a measurable supersonic wave. Several attempts have been made to model the electroacoustic phenomena. O’Brien,2for example, considered the dynamic electrophoresis of a spherical dispersion for the case of a thin double layer. The governing equations were solved numerically, and the results were used to evaluate the dynamic electrophoretic mobility. Sawatzky and Babchin3derived an approximate expression for the dynamic electrophoretic mobility of a spherical dispersion. Applying the numerical method of O’Brien, Magelsdorf and White4solved the electroacoustic equations for a spherical dispersion at low surface potential. Under the conditions of low surface potential and arbitrary double-layer thickness, Ohshima5derived an analytical expression for the dynamic mobility of a spherical dispersion. All these studies focused on the electrophoresis of rigid entities. For the case of nonrigid entities such as droplets, since the electric and the flow fields inside an entity may also need to be considered, solving the governing equations is usually more difficult than solving the corresponding equations for rigid entities.6Nonrigid dispersion is one of basic categories of colloidal dispersion, and is of practical significance in modern technology. Microemulsion systems, for instance, are often adopted to produce nanosized particles. Compared with the results for the electrophoresis of rigid entities, those for nonrigid entities are very limited.6-13
In practice, electrophoresis is often conducted in a confined space where boundary effect can play a significant role. A typical example includes electrophoresis in a porous medium where the influence of the porosity of a medium is usually not negligible. Capillary electrophoresis is another example where the presence of a capillary wall should be considered. Although the boundary effect on electrophoresis has been studied by many authors for the case in which a static electric field is applied, relevant analyses on cases in which a dynamic electric field is applied are very limited, especially when an entity is of a nonrigid nature. In this study, the boundary effect on the dynamic electrophoresis of a nonrigid entity is investigated by considering a droplet in a spherical cavity. A pseudospectral method based on Chebyshev polynomials is adopted to solve the governing electroacoustic equations and the associated boundary conditions. The influences of the key parameters of the system under consideration on the electrophoretic behavior of a droplet are examined. These include the frequency of the applied electric field, the thickness of the double layer surrounding a droplet, the relative size of a cavity, and the physical properties of a droplet, such as its surface potential and viscosity.
Theory
Referring to Figure 1, we consider the case where an electrolyte-free, Newtonian droplet of radius a is located at the center of a spherical cavity of radius b. Let H ) (a/b)3. An alternating electric field Eze-iωtezis applied along the z-axis, where t is time, Ezand
ω are respectively the strength of the field and the frequency of
the applied electric field, ezis the unit vector in the z-direction, and i )x-1. Let Ue-iωtezbe the electrophoretic velocity of the droplet. The spherical coordinates (r,θ,φ) are adopted, with its
origin located at the center of the droplet. We consider the case where the liquid phase in the cavity is a Newtonian fluid containing z1/z2electrolyte, with z1and z2respectively being the valences of cations and anions with z2) -Rz1. For convenience, U is expressed as U ) UR+ iUI, in which URand UIare respectively the real and imaginary parts of U. We assume that the strength of the applied electric field is relatively weak compared with that * To whom correspondence should be addressed. Tel: 886-2-23637448.
Fax: 886-2-23623040. E-mail: jphsu@ntu.edu.tw. (1) Hunter, R. J. Colloids Surf., A 1998, 141, 37. (2) O’Brien, R. W. J. Fluid Mech. 1988, 190, 71.
(3) Sawatzky, R. P.; Babchin, A. J. J. Fluid Mech. 1993, 246, 321. (4) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1992, 88, 3567.
(5) Ohshima, H. J. Colloid Interface Sci. 1996, 179, 431.
(6) Baygents, J. C.; Saville, D. A. J. Chem. Soc., Faraday Trans. 1991, 87, 1883.
(7) Booth, F. J. Chem. Phys. 1951, 19, 1331.
(8) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962.
(9) Levine, S.; O’Brien, R. N. J. Colloid Interface Sci. 1973, 43, 616. (10) Levine, S. Materials Proceeding in the Reduced GraVity EnVironment of
Space; Rindone, G. E., Ed.; North-Holland: Amsterdam, 1981; Vol. 9, p 241.
(11) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans.
2 1984, 80, 1643.
(12) Lee, E.; Kao, J. D.; Hsu, J. P. J. Phys. Chem. B 2002, 106, 8790. (13) Lee, E.; Fu, C. H.; Hsu, J. P. Langmuir 2003, 19, 3035. 10.1021/la0533821 CCC: $33.50 © 2006 American Chemical Society
established by the droplet. The latter is on the order of surface potential/Debye length. Because the surface potential ranges from 25 to 150 mV and the Debye length ranges from 10 to 100 nm, the strength of the electric field established by the droplet is on the order of 2500 kV/m. In practice, the strength of the applied electric field is much weaker than this value. It is known that, if the Reynolds number (Re) is small, the shape of a drop remains spherical.14,15In our case, because the typical value of Re is on the order of 10-10 - 10-8, the deformation of droplet can be neglected. Consequently, the distribution of charge on the droplet surface is uniform, and the surface potential is position independent. The surface conductivity of the droplet is neglected, that is, the surface potential is represented by theζ potential.
Both the droplet fluid and the fluid in the cavity are incompressible and have constant physical properties.
Governing Equations
The governing equations for the present problem include those for the electric, concentration, and flow fields. For the fluid outside the droplet, these equations are
In these expressions,∇ and ∇2are respectively the gradient operator and the Laplace operator; φ is the electrical potential;
Feis the space charge density; is the permittivity of the fluid
in the cavity; k and T are respectively the Boltzmann constant and the absolute temperature; eˆ is the elementary charge; nj, fj, and Djare respectively the number concentration, the flux, and the diffusivity of the jth ionic species; u is the velocity of fluid;
p is the pressure; Foandηoare respectively the density and the viscosity of the fluid in the cavity.
For the domain inside the droplet, since the fluid contains no electrolyte, only the flow field needs be solved. Because the Reynolds number is small we have
where Fiandηiare respectively the density and the viscosity of the droplet fluid.
For convenience, all the dependent variables are partitioned into an equilibrium term (i.e., when the applied electric field is absent), and a perturbed term arises from the applied electric field. The symmetric nature of the present problem implies that the dependent variables are functions of r,θ, and t only. We have
In these expressions, the subscript and the superscript “e” denote the equilibrium properties, andδ denotes the perturbed properties.
Note that, for the equilibrium case, since the droplet remains fixed when the applied electric field is absent, the velocity vanishes.
Equilibrium System
Since the distribution of ionic species at equilibrium follows Boltzmann distribution, the governing equation for φeis
where nj0e is the bulk number concentration of ionic species j. Suppose that both the potential on the droplet surface and that on the cavity surface are kept constant. Then the boundary conditions associated with eq 12 are
Perturbed System
Taking∇2on both sides of eq 8 gives
The governing equation for φ is assumed to take the form
where the effect of double-layer deformation is taken into account by the function gjEze-iωt. Substituting eq 8 into this expression yields
(14) Saito, S. Sci. Rep. Tohoku Imp. UniV. (Sendai, Jpn.) 1913, 2, 179. (15) Taylor, T. D.; Acrivos, A. J. Fluid Mech. 1964, 18, 466.
Figure 1. Illustration of the problem considered in which an electrolyte-free, Newtonian droplet of radius a is located at the center of a spherical cavity of radius b. An alternating electric field Eze-iωtez is applied along the z-axis and Ue-iωtezis the electrophoretic velocity of the droplet. The spherical coordinates (r,θ,φ) are adopted with
its origin at the center of droplet.
∇2 φ ) -F e ) -
∑
j)1 2 z jeˆnj (1) ∂nj ∂t ) -∇‚fj (2) fj) -Dj(
∇nj+njeˆzj kT∇φ)
+ nju (3) ∇‚u ) 0 (4) F o ∂u ∂t ) -∇p + ηo∇ 2 u - Fe∇φ (5) ∇‚u ) 0 (6) F i ∂u ∂t ) -∇p + ηi∇ 2u (7) φ(r,θ,t) ) φe(r) +δφ(r,θ)Eze -iωt (8) nj(r,θ,t) ) nj e (r,θ) + δnj(r,θ)Eze -iωt (9) u(r,θ,t) ) 0 + δu(r,θ)Eze -iωt (10) p(r,θ,t) ) pe(r,θ) + δp(r,θ)EZe -iωt (11) ∇2 φe(r) ) -∑
j)1 2 z jeˆnj0 e exp(
-zjeˆφe kT)
(12) φe) ζa, r ) a (13) φe) ζb, r ) b (14) ∇2 (δφ(r,θ)Eze -iωt ) )∇2φ(r,θ,t) -∇2φe(r) (15) ∇2 φ ) -∑
j)1 2 z jeˆnj0 e exp(
-zjeˆ(φ + gjEze-iωt) kT)
(16)Equations 1 and 17 lead to
Combining eqs 12, 15, and 17 yields the governing equation for
δφEze-iωt:
The function gjEze-iωtcan be determined by first combing eqs 2 and 3 to give
Substituting eqs 8-10 into this expression and dividing both sides of the resultant expression by Eze-iωt, we obtain
The equilibrium ionic concentration is nje) nj0e exp(-zjeˆφe/kT), and the corresponding perturbed concentration can be determined from eqs 9 and 18. We have
Substituting eqs 18 and 22 into eq 21 yields the governing equation for gj. For a simpler treatment, gj, instead ofδnj, is solved in subsequent discussions.
Note that gjneeds to be solved simultaneously with the flow field. The fluid is stagnant in the absence of the applied electric field, and therefore,∇pe) 0 and ue) 0, and only the perturbed flow field needs to be solved. Equations 4-8 and 10 yield
The mathematical treatments can be made simpler by employing a stream function representationψ. In terms of ψ, the r and the θ components of δu, δurandδuθ, can be expressed respectively
asδur) -(1/r2sinθ)(∂ψ/∂θ) and δuθ) (1/r sin θ)(∂ψ/∂r). We have
Note that eq 23 is satisfied automatically byδurandδuθ. Taking curl on both sides of eqs 24 and 25 to eliminate the pressure term and introducingψ, we obtain
where
We assume that the surface of a droplet is impenetrable to ionic species, and therefore,
On the cavity surface, the electric field is that which arises from the applied electric field, and the ionic concentration reaches the equilibrium value, that is,
The droplet surface is impenetrable to dispersion medium, and both the velocity and the shear stress are continuous on that surface. Therefore,
whereτnis the shear stress tensor on the droplet surface. The
relative velocity between the cavity and the droplet is -UEze-iωt, that is,
At the center of the droplet, we must have
For a simpler treatment, dimensionless quantities are used in subsequent discussions. On the basis of the scaling factors a,ζa,
∇2 φ ) -
∑
j)1 2 zjeˆnj0e exp(
-zjeˆ(φe+ δφEze -iωt+ g jEze -iωt) kT)
(17)nj) nj0e exp
(
-zjeˆ(φe+ δφEze-iωt+ g jEze -iωt) kT
)
(18) ∇2 (δφEze-iωt) )∑
j)1 2 z jeˆnj0 e(
exp(
-zjeˆ kT(φe+ δφEze -iωt+ g jEze -iωt ))
-exp(
-zjeˆ kTφe))
(19) ∂nj ∂t ) -Dj(
∇ 2n j+ zjeˆ kT(∇nj‚∇φ + nj∇ 2 φ))
+ u‚∇nj (20) -iωδnj) -Dj{
∇2 (δnj) + zjeˆ kT[∇nj e‚∇(δn j) +∇(δnj)‚∇φe+ ∇(δnj)‚∇(δφEze -iωt) + n j e∇2 (δφ) + (δnj)∇ 2 φe+(δnj)∇2(δφEze-iωt)]
}
+ (δu)‚∇(nje+ δnjEze-iωt) (21)δnjEze-iωt)
nj0e
[
exp(
-zjeˆ(φe+ δφEze-iωt+ g jEze -iωt) kT
)
- exp(
-zjeˆφe kT)
]
(22) ∇‚(δuEze -iωt) ) 0, 0 < r < b (23)-iωFoδuEze-iωt) - ∇δpEze-iωt+ ηo∇2δuEze-iωt
-F e∇φ, a < r < b (24) -iωFiδuEze -iωt) -∇δpEze -iωt+ η i∇ 2 δuEze-iωt, 0 < r < a (25)
δuEze-iωt) (δurrˆ +δuθθˆ)Eze
-iωt (26) ηoL4ψ + iωFoL2ψ )
(
∂Fe ∂r ∂φ ∂θ -∂Fe ∂θ ∂φ ∂r)
sinθ, a < r < b (27) ηiL4ψ + iωFiL2ψ ) 0, 0 < r < a (28) L2)(
∂ 2 ∂r2)
+ sin θ r2 ∂ ∂θ(
1 sinθ ∂ ∂θ)
(29) L4) L2L2)[
(
∂ 2 ∂r2)
+ sin θ r2 ∂ ∂θ(
1 sinθ ∂ ∂θ)
]
2 (30) ∇(δφEze -iωt)‚nˆ ) 0, r ) a (31) (fj- nju)‚nˆ ) 0, r ) a (32) ∂(δφEze -iωt ) ∂r ) -Eze -iωtcos θ, r ) b (33) nj) nj0e exp(
-zjeˆ kTφe)
, r ) b (34)δurEze-iωt|r)a+) δurEze-iωt|r)a-) 0, r ) a (35)
δuθEze
-iωt|
r)a+) δuθEze-iωt|r)a-, r ) a (36)
(τn‚nˆ) × nˆ|r)a+) (τn‚ nˆ) × nˆ|r)a-, r ) a (37)
δurEze-iωt) -(UR+ iUI)Eze-iωtcosθ, r ) b (38)
δurEze
-iωt) 0, r ) 0 (39)
n10e , and UE) ζa2/ηoa, we define the following dimensionless quantities: Ez/) Ez/(ζa/a); U*Ez
/ e-iωt) UEze-iωt/UE; φe / ) φe/ ζa; δφ*Ez /
e-iωt ) δφEze- iωt/ζa; gj
/
Ez/e-iωt ) gjEze-iωt/ζa;
ψ*Ez/e-iωt) ψEze-iωt/UEa; nj
/ ) nj/n10
e; and r* ) r/a. Since the applied electric field is relatively weak compared to that established by the droplet,|Eze-iωt| , (ζa/a), which implies that
|δφEe-iωt| , φ
eand|gjEe-iωt| , φe. Therefore, eq 18 can be approximated by
and
where φr) z1eζa/kT is the scaled surface potential. Applying the method of separation of variables, we have
where subscripts R and I respectively denote the real and the imaginary parts of a variable. Substituting these expressions into the corresponding governing equations yields
In these expressions, L12) (d2/dr*2) + (2/r*2)(d/dr*) - (2/r*2), L24) L
22L22, and L22) (d2/dr*2) - (2/r*2); κ-1) (kT/Σj)1 2 nj0e(ezj)2)1/2is the Debye length; and Pej) UEa/Djis the electric Peclet number of ionic species j. (Foωa2/ηo) is a measure of the relative significance between the dynamic inertial force and the viscous force, (ωa2/D
j) is a measure of the relative significance between the rate of transport of ionic species j arising from the applied electric field and that due to diffusion. The associated boundary conditions are
L24ΨR -F oωa 2 ηo L2 2 ΨI) -(κa)2 (1 + R)(G1Rexp(-φrφe / ) + RG2Rexp(Rφrφe/))dφe / dr*, a < r < b (50) L24ΨI+ F oωa 2 ηo L2 2 ΨR) -(κa)2 (1 + R)(G1Iexp(-φrφe / ) + RG2Iexp(Rφrφe/))dφe / dr*, a < r < b (51) L24ΨR -F oωa 2 ηo
(
ηo ηi)(
F i F o)
L22ΨI) 0, 0 < r < a (52) L24ΨI+ F oωa 2 ηo(
ηo ηi)(
F i F o)
L22ΨR) 0, 0 < r < a (53){
∂ΦR ∂r* ) 0, r* ) 1 ∂ΦR ∂r* ) -Ez / , r* )b a (54){
∂ΦI ∂r*) 0, r* ) 1 ∂ΦI ∂r*) 0, r* ) b a (55){
dG1R dr* ) 0, r* ) 1 G1R) -ΦR, r* )b a (56){
dG1I dr* ) 0, r* ) 1 G1I) -ΦI, r* )b a (57){
dG2R dr* ) 0, r* ) 1 G2R) -ΦR, r* )b a (58){
dG2I dr* ) 0, r* ) 1 G2I) -ΦI, r* )b a (59) n1/) exp(-φrφe/)[1 - φr(δφ* + g1/)Ez/e-iωt] (41) n2/) R exp(Rφrφe/)[1 + Rφr(δφ* + g2/)Ez/e-iωt] (42){
δφ*(r,θ) ) (ΦR(r) + iΦI(r)) cosθ g1/(r,θ) ) (G1R(r) + iG1I(r)) cosθ g2/(r,θ) ) (G2R(r) + iG2I(r)) cosθ ψ*(r,θ) ) (ΨR(r) + iΨI(r)) sin 2 θ (43) L12ΦR- (κa) 2 1 + R[exp(-φrφe / ) + R exp(Rφrφe/)]ΦR) (κa)2 1 + R[exp(-φrφe / )G1R+ R exp(Rφrφe/)G2R], a < r < b (44) L12ΦI- (κa) 2 1 + R[exp(-φrφe / ) + R exp(Rφrφe/)]ΦI) (κa)2 1 + R[exp(-φrφe / )G1I+ R exp(Rφrφe/)G2I], a < r < b (45) L12G1R- φrdφe / dr* dG1R dr* + Pe1 1 r*2ΨR dφe/ dr*- ωa 2 D1(ΦI+ G1I) ) 0, a < r < b (46) L12G1I- φrdφe / dr* dG1I dr* + Pe1 1 r*2ΨI dφe/ dr*+ ωa 2 D1(ΦR+ G1R) ) 0, a < r < b (47) L12G2R+ Rφrdφe / dr* dG2R dr* + Pe2 1 r*2ΨR dφe/ dr*- ωa 2 D2(ΦI+ G2I) ) 0, a < r < b (48) L12G2I+ Rφrdφe / dr* dG2I dr* + Pe2 1 r*2ΨI dφe/ dr*+ ωa 2 D2(ΦR+ G2R) ) 0, a < r < b (49)The scaled governing equations for the equilibrium potential and the associated boundary conditions are
For the present problem, the dynamic electrophoretic mobility of dropletµ is defined as
where E ) Eze-iωtand U ) (UR+ iUI)Eze-iωt, withµRandµI respectively being the real and imaginary parts ofµ, which can
be determined from a force balance on a droplet. The z components of the forces acting on a droplet include the electric force Feand the hydrodynamic drag Fh. According to the Newtonian second law of motion,
In terms of scaled quantities, it can be shown that this expression can be rewritten as
Fecan be evaluated by
where the surface charge densityσ can be determined from Gauss
law as
where q is the total amount of charge enclosed by surface S, and
E ) -∇φ. Substituting eq 68 into eq 67 and using the relation
dS ) 2πr2sinθdθ, it can be shown that
This expression implies that knowing φe/,ΦR, andΦIsuffices the determination of Fe/.
Applying the result of Happel and Brenner,16 Fh can be expressed as
The first term on the right-hand side of this expression is the force arising from the change of the momentum of fluid, the second term is the drag on a droplet, and the third term is the electric force arising from the flow of electrolyte solution. Applying the relation ∂φe//∂θ ) 0, it can be shown that
This expression implies that knowing φe/,ΦR,ΦI,ΨR, andΨI suffices the determination of Fh
/
.
To avoid tedious recursive procedure in the evaluation of dynamic mobility, the present problem is divided into two subproblems. In the first one, a droplet moves with a scaled velocity (UR/+ iUI/)Ez/e-iωtin the absence of the applied electric field, and, in the second one, a scaled electric field Ez
/
e-iωtis applied, but a droplet is fixed in space. Note that the partition of the present problem is for a simpler mathematical treatment; the two subproblems have no actual physical meaning. The linear nature of the present problem implies that the scaled total force (16) Happel, J.; Brenner, H. Low-Reynolds Number Hydrodynamics; Nijhoff: Boston, MA, 1983. Fh/+ Fe/) -i
(
4 3πζ 2)(
Foωa 2 ηo)(
F i- Fo F o)
(UR/ + iUI/)Ez/e-iωt (66) Fe)∫∫
r)aσ(-∇φ)s‚ ezdS (67) q )∫∫
(E‚nˆ)dS )∫∫
σdS (68) Fe/) 8 3πζa 2[
r*dFe / φe/ dr* (ΦR+ iΦI)Ez / e-iωt]
r*)1 (69) Fh) -πFo∫
0 π[
r2sinθ∂ ∂t(
∂(ψEze-iωt) ∂r)
]
sdθ + ηπ∫
0π[
r4sin3∂ ∂r(
E2ψEze -iωt r2sin2θ)
]
s dθ - π∫
0π[
r2sin2θFe ∂φ ∂θ]
s dθ (70) Fh/) πζa2i(
ωFoa 2 ηo)
∫
0 π[
r*2sinθ∂(ψ*Ez / e-iωt) ∂r*]
r*)1dθ + πζa 2∫
0 π[
r*4sin3θ ∂ ∂r*(
E*2ψ*Ez/e-iωt r*2sin2θ)
]
r*)1 dθ -πζa 2 (κa) 2 (1 + R)φr∫
0 π[
r*2sin2θ(exp(-φrφe / ) -exp(Rφrφe/))∂δφ* ∂θ Ez / e-iωt]
r*)1 dθ (71){
ΨR) 0, r* ) 0 dΨR dr* ) 0 r* ) 0 ΨR|r*)1+) ΨR|r*)1-) 0, r* ) 1 dΨR dr*|
r*)1+) dΨR dr*|
r*)1- r* ) 1 d dr*(
1 r*2 dΨR dr*)
|
r*)1+ )ηi ηo d dr*(
1 r*2 dΨR dr*)
|
r*)1 -, r* ) 1 ΨR) 1 2URr*, r* ) b a ∂ΨR ∂r* ) URr*, r* ) b a (60){
ΨI) 0, r* ) 0 dΨI dr*) 0 r* ) 0 ΨI|r*)1+) ΨI|r*)1-) 0 r* ) 1 dΨI dr*|
r*)1+) dΨI dr*|
r*)1- r* ) 1 d dr*(
1 r*2 dΨI dr*)
|
r*)1+ )ηi ηo d dr*(
1 r*2 dΨI dr*)
|
r*)1- r* ) 1 ΨI) 1 2UIr* r* ) b a ∂ΨI ∂r*) UIr* r* ) b a (61) 1 r*2 d dr*(
r* 2dφe / dr*)
) - (κa) 2 (1 + R)φr(exp(-φrφe / ) - exp(Rφrφe/)) (62){
φe/) 1, r* ) 1 φe/)ζb ζa , r* )b a (63) µ ) µR+ iµI) U E (64) Fh+ Fe) 4 3πa 3 (Fi- Fo)dU dt (65)acting on a droplet in the first subproblem, F1/, is proportional to (UR/+ iUI/)Ez/e-iωt, and that in the second subproblem, F2/, is proportional to Ez/e-iωt. That is,
whereχ and β are complex constants. Therefore, eq 65 can be
rewritten as
Equations 64 and 72-74 yield
whereµ* ) µ/(ζa/ηo). Therefore, knowing the values ofχ and
β suffices the determination of the dynamic mobility, and no
recursive procedure is necessary. Note that these complex constants are independent of the applied electric field and the velocity of a droplet. For the present problem, the electrophoretic velocity is complex, and we define the scaled magnitude of the mobility µm/ asµm/ )
x
µR2+µI2and the corresponding phase angle Θ as Θ ) tan-1(µI/µR), that is, µ* )µm/
eiΘ. Here, Θ
measures the phase difference between U and E; a negativeΘ
implies that U leads E, and the reverse is true ifΘ is positive.
The governing equations and the associated boundary condi-tions are solved numerically by a pseudospectral method based on Chebyshev polynomials. This approach has been justified to be accurate and efficient for the electrokinetic phenomenon of the present type.17,18
Results and Discussion
Numerical simulations are conducted to examine the influences of the key parameters of the system under consideration on the (17) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65. (18) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1999, 209, 240.
Figure 2. Variation in the scaled magnitude of electrophoretic mobilityµm
/
(a) and phase angle Θ (b) as a function of scaled
frequencyω* for various φrat κa ) 1, H ) 0.1, andηo/ηi ) 1.
F1/) χ(UR/+ iUI/)Ez/e-iωt (72) F2/) βEz/e-iωt (73) Fh/+ Fe/) F1/+ F2/) -i
(
4 3πζa 2)(
Foωa 2 ηo)(
F i- Fo F o)
(UR/ + iUI/)Ez/e-iωt (74) µ* ) -β χ + i(Foωa 2 /ηo)[(Fi- Fo)/Fo] (75)Figure 3. Variation in the scaled magnitude of electrophoretic mobility µm
/
(a) and phase angle Θ (b) as a function of scaled
frequencyω* for various ηo/ηiat κa ) 1, H ) 0.1, and φr ) 1.
electrophoretic behavior of a droplet. These include (ωa2/D 1), (ωa2/D
2), Pe1, Pe2, κa, (Fi/Fo), (ηo/ηi), φr, H ) (a/b)3, and the scaled frequency of the applied electric fieldω* ) Foωa2/ηo. The following values, which are based on an aqueous KCl solution at 298.15 K, are chosen: ) 8.854× 10-12× 78.54688 F m-1;
ηo) 0.8904 cp; Fo) 0.99704 g/cm3, Z1) 1; Z2) -1; D1) 1.962297× 10- 5cm2/s; and D
2) 2.037051 × 10- 5cm2/s. These values yieldωa2/D
1) 454.88(Foωa2/ηo) andωa2/D2) 472.21(Foωa2/ηo). Also, we assume (Fi- Fo)/Fo) 0.1 and Pe1
) Pe2) 0.01.
Influence of Surface Potential. The variations of the scaled
magnitude of the dynamic mobility of a droplet µm/ and the corresponding phase angleΘ as a function of the scaled frequency
of the applied electric fieldω* at various levels of the scaled
surface potential φrare presented in Figure 2. For the case in which φris low,µm
/
decreases andΘ, which is positive, increases
with the increase inω*. The former is because, as ω* increases,
the time interval available for the acceleration of a droplet becomes short. The latter is because, if φris low, the deformation of the double layer surrounding a droplet is negligible; therefore, the higher the frequency, the greater its inertia and the slower its
response to the applied electric field. As φrincreases and the thickness of double layer is on the order of the radius of the droplet (κa ) 1), µm/ shows a local maximum and Θ has a
negative (phase lead) local minimum asω* varies. These arise
from the effect of double-layer deformation, which yields an induced electric field near the droplet. When an external electric field is applied, since this induced electric field is in the opposite direction of that of the applied electric field, it has the effect of retarding the movement of a droplet. However, when the applied electric field changes its direction, if the redistribution of the ionic species in the double layer is not fast enough, the induced electric field becomes advantageous to the movement of a droplet. The presence of the local maximum inµm/ and the phase lead become inappreciable ifω* is sufficiently high. This is because,
ifω* is high, the change in the direction of the movement of the
droplet is fast, and the deformation of the double layer becomes inappreciable.
Influence of Droplet Viscosity. The influence of the relative
viscosity of the droplet fluid, measured by the ratioηo/ηi, on its electrophoretic behavior for the case in which φr is low is illustrated in Figure 3. Note that, asηo/ηif 0, the behavior of
a droplet approaches that of a rigid particle, and, asηo/ηif∞, Figure 4. Variation in the scaled magnitude of electrophoretic
mobilityµm /
(a) and phase angle Θ (b) as a function of scaled
frequencyω* for various κa at φr ) 1, H ) 0.1, and ηo/ηi ) 1.
Figure 5. Variation in the scaled magnitude of electrophoretic mobilityµm/(a) and phase angleΘ (b) as a function scaled frequency
its behavior approaches that of a gas bubble. Figure 3 reveals that, in general,µm/ increases with the increase inηo/ηi, so it is the phase lag. This is expected because the larger the value of
ηo/ηi, the smaller the hydrodynamic drag acting on the droplet.
Influence of Double-Layer Thickness. Figures 4 and 5 show
the influence of the thickness of the double layer surrounding a droplet, measured by κa, on its electrophoretic behavior at various levels of φr. In general, for a fixed value ofω*, µm
/
increases with the increase in κa. This is because the thinner the double layer surrounding a droplet, the greater the absolute value of the potential gradient on its surface, and, therefore, the greater the electrical driving force for its movement. Figures 4 and 5 indicate that, if φris sufficiently high and the double layer is not thick,µm/has a local maximum andΘ has a negative (phase lead)
local minimum asω* varies. As pointed out previously, these
arise from the effect of double-layer deformation. Note that the phase lag occurring at a medium to high value ofω* decreases
with the increase in κa. This is because, as the double layer becomes thinner, it is easier for a droplet to catch up to the variation of the applied electric field.
Influence of Cavity Size. The influences of the relative size
of a cavity, measured by H, on the electrophoretic behavior of a droplet at various levels of φrare illustrated in Figures 6 and 7. These figures indicate thatµ/mdeclines with the increase in H. This is because the larger the value of H, the closer a droplet is to a cavity, and the greater the hydrodynamic retardation. Figure 7 reveals that, as H increases, the phase lead occurring at a low to medium value ofω* becomes less appreciable. This
is because the larger the value of H, the smaller the space available for double-layer deformation, and, therefore, its influence on the electrophoretic behavior of a droplet is confined. The behavior of the phase lag occurring at a medium to high value ofω* can
be interpreted by similar reasoning.
Conclusions
The boundary effect on the dynamic electrophoresis of a droplet is investigated by considering an electrolyte-free droplet in a spherical cavity. The influences of the frequency of the applied electric field, the thickness of the double layer surrounding a droplet, the relative size of a cavity, and the physical properties of a droplet on its electrophoretic behavior are discussed. We
Figure 6. Variation in the scaled magnitude of electrophoretic mobilityµm
/
(a) and phase angle Θ (b) as a function of scaled
frequencyω* for various H at φr) 1, κa ) 1, and ηo/ηi) 1.
Figure 7. Variation in the scaled magnitude of electrophoretic mobility µm/ (a) and phase angle Θ (b) as a function of scaled
frequencyω* for various H at φr) 5, κa ) 1, and ηo/ηi) 1.
show that, if the deformation of the double layer is significant, the magnitude of the electrophoretic mobility of a droplet has a local maximum and the phase angle has a negative (phase lead) local minimum as the frequency of the applied electric field varies. The more important the boundary effect, the less significant the effect of double-layer deformation. In general, the smaller
the viscosity of the fluid of a droplet, the larger the magnitude of its mobility.
Acknowledgment. This work is supported by the National
Science Council of the Republic of China. LA0533821
