Dynamic electrophoresis of droplet dispersions at low surface potentials
Jyh-Ping Hsu, Wei-Lun Min, Eric Lee
∗Department of Chemical Engineering, Institute of Polymer Science and Engineering, National Taiwan University, Taipei, 10617 Taiwan Received 7 September 2006; accepted 23 October 2006
Available online 28 November 2006
Abstract
The dynamic electrophoresis of a dispersion of spherical droplets under conditions of low surface potential and arbitrary double-layer thickness and droplet volume fraction is analyzed. A cell model with the Shilov–Zharkikh boundary condition for the electric potential is adopted to simulate a dispersion, and the governing equations and the associated boundary conditions are solved by a pseudo-spectral method based on Chebyshev polynomials. The influence of the frequency of the applied electric field, the volume fraction of the droplets, the thickness of the double layer, and the relative magnitude of the viscosity of the droplet fluid on the electrophoretic behavior of a dispersion is discussed.
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Keywords: Dynamic electrophoresis; Droplets; Cell model; Low surface potential
1. Introduction
The physical properties such as the charge and the linear size of a colloidal particle in a dispersion of appreciable par-ticle concentration can be measured on the basis of the effect of electrokinetic sonic amplitude[1,2]. In this case, because an alternating electric field is applied, both the magnitude and the corresponding phase angle of the electrophoretic mobility of an entity can be measured, achieving more accurate measure-ment than that when a static electric field is applied. However, compared with the analysis of the latter, that of the former is more complicated, since the time scale is involved. O’Brien[2]
investigated the dynamic electrophoresis of a spherical rigid colloid with a very thin double layer in a dilute dispersion. Since this foundation-laying work for theoretical analysis, var-ious studies on dynamic electrophoresis have been carried out in the past two decades[3–17]. Among them, Mangelsdorf and White [4] took into consideration the effect of double-layer polarization. Based on their studies, Ohshima [5] derived an analytic expression for the case of low zeta potential. These studies were all focused on the dilute dispersion, i.e., a spher-ical particle in an infinite solution. Later, Ohshima extended further to investigate the case of concentrated suspensions.
Ac-* Corresponding author. Fax: +886 2 23622530.
E-mail address:ericlee@ntu.edu.tw(E. Lee).
cording to his analysis, the dynamic electrophoretic mobility of the particle could be seriously influenced by the presence of neighboring particles[6–8]. In order to fit real dispersions more closely, various cell models were adopted and results compared [9–11]. Generally speaking, the analytical solution is confined to the limiting case of linearly electrokinetic equa-tions. When a nonlinear effect is involved, for example, the complete nonlinear Poisson–Boltzmann equation, it is difficult to analyze it analytically at arbitrary double-layer thickness and zeta potential. As a result, Lee and co-workers[12–17]applied numerical approach with a pseudo-spectral method to further release this constraint and solved the corresponding general nonlinear equations successfully. They found that the effects of double-layer polarization and overlap are crucial in the dynamic electrophoresis of concentrated dispersions of rigid colloid par-ticles.
The electrophoresis of a dispersion of nonrigid entities has many applications in practice. Emulsions, sols, and foams, for instance, belong to this category. Compared with that of rigid entities, the analysis of the electrophoresis of nonrigid entities is more complicated because the electrokinetic equations for the space inside an entity and those for the space outside an entity need to be solved simultaneously. Several attempts have been made in the literature to model the electrophoretic behavior of a dispersion of nonrigid entities[18–22]. Booth[18]first inves-tigated the electrophoretic behavior of a mercury drop. Later
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on, Levine and O’Brien[20]examined the motion of a droplet or bubble of an emulsion thoroughly and found it can be influ-enced significantly by an externally applied static electric field. They first introduced the assumption that the polarization effect dominates the electrophoresis of the mercury drop.
While there are many published reports on dynamic elec-trophoresis, none of them is for nonrigid droplet dispersions; they are essentially all focused on rigid hard particles, to the best of our knowledge. And while there are many published studies on the electrophoresis of droplet dispersions, none of them is focused on dynamic electrophoresis; essentially all of them are confined to static electrophoresis. The only related work in the field of dynamic electrophoresis of nonrigid spheres was published recently by Lee et al.[22], who studied the dy-namic electrophoresis of a spherical droplet within a solid cav-ity. They modeled the electrophoresis in a porous medium with special emphasis on the boundary effect. The polarization effect of a double layer was found to be important.
In this study, we consider dynamic electrophoresis in a con-centrated dispersion of spherical droplets when an alternating electric field is applied under conditions of low surface potential and an arbitrary volume fraction of droplets and double-layer thickness. Here, the dispersion is simulated by Kuwabara’s unit cell model[23]with the boundary conditions on the cell sur-face proposed by Shilov and Zharkikh[11]. A pseudo-spectral method based on Chebyshev polynomials used previously for the analysis of electrophoresis under a static applied electric field[24,25]is adopted to solve the resultant governing equa-tions and the associated boundary condiequa-tions. The influence of the frequency of the applied electric field, the volume fraction of droplets, the thickness of the double layer, and the relative viscosity of the droplet fluid on the electrophoretic behavior of the dispersion is discussed.
2. Theory
Let us consider the electrophoresis of an aqueous dispersion of droplets subject to an alternating electric field. The fluid of a droplet is electrolyte-free, and the dispersion medium contains
z1:z2electrolyte, where z1and z2are respectively the valences
of cations and anions. We adopt Kuwabara’s unit cell model
[23], illustrated inFig. 1, where a dispersion is simulated by a representative droplet of radius a surrounded by a concentric liquid shell of radius b. a:b is chosen to be 1:2 in the current calculations. The spherical coordinates (r, θ, ϕ) are used with origin located at the center of the droplet. E= EZe−iωtezand
U= Ue−iωtezare respectively the applied electric field and the
corresponding electrophoretic velocity of a droplet, where ezis
the unit vector in the Z-direction. Let U= UR+ iUI, where UR
and UI are respectively the real and the imaginary parts of U
and i=√−1.
Since a droplet contains no electrolyte, only the flow field inside it needs to be considered. However, the electric, concen-tration, and flow fields have to be solved simultaneously for the dispersion medium. In the present problem, the electric field
Fig. 1. Kuwabara’s unit cell model for a spherical dispersion of droplets of radius a, where b is the radius of a concentric liquid shell, Eze−iωtez is an applied alternating electric field in the z-direction, U e−iωtezis the elec-trophoretic velocity of a droplet, and (r, θ, ϕ) are the spherical coordinates with origin located at the center of the droplet.
can be derived based on Gauss’s law to give
(1) ∇2φ= −ρe ε = − 2 j=1 zjˆenj ε ,
where∇2 is the Laplace operator, φ the electrical potential,
ρe the space charge density, ε the dielectric constant of the
dispersion medium, ˆe the elementary charge, and nj the
num-ber concentration of ionic species j . The conservation of ionic species j yields (2) ∂nj ∂t = ∇ · Dj ∇nj+ njˆezj kT ∇φ + nju ,
where∇ is the gradient operator, Dj the diffusivity of ionic
species j , and u the velocity of the dispersion medium. Since the Reynolds number is very small, the flow field is described by (3) ∇ · u = 0, 0 < r < b, (4) ρi ∂u ∂t = −∇p + ηi∇ 2u, 0 < r < a, (5) ρ0 ∂u ∂t = −∇p + η0∇ 2 u− ρe∇φ, a < r < b.
In these expressions, p is the pressure, ηi and η0are
respec-tively the viscosity of the droplet fluid and that of the disper-sion medium, and ρi and ρ0are respectively the density of the
droplet fluid and that of the dispersion medium.
Under conditions of practical significance, the strength of the applied electric field is relatively weaker than that estab-lished by the charge on the droplet surface. For convenience, each dependent variable is expressed as the sum of its equilib-rium value, that is, the value when the external electric field is absent, and a perturbed value, which arises from the applied
electric field. Also, because the Reynolds number is very small for the present problem, the droplet remains spherical, and the dependent variables are functions of r, θ , and t only. Therefore, (6)
φ(r, θ, t)= φe(r)+ δφ(r, θ)EZe−iωt,
(7)
nj(r, θ, t )= nej(r, θ )+ δnj(r, θ )EZe−iωt,
(8)
u(r, θ, t)= 0 + δu(r, θ)EZe−iωt,
(9)
p(r, θ, t )= pe(r, θ )+ δp(r, θ)EZe−iωt.
In these expressions, the subscript e and the superscript e denote equilibrium property, and a symbol with δ represents a perturbed quantity arising from the applied electric field. Note that since a droplet remains stationary when the external electric field is absent, u= 0. Also, φeis a function of r only.
3. Equilibrium system
In the absence of the applied electric field, droplets remain stationary, and therefore, only the electric and the concentra-tion fields need to be considered. Since the surface potential is low, the equilibrium potential satisfies the linearized Poisson– Boltzmann equation, (10) ∇2φ e= N j=1 zjˆenej0 ε zjˆeφe kT ,
where nej0is the bulk concentration of ionic species j . Suppose that the electric potential remains constant on the droplet sur-face, and there is no net electric current across the outer virtual cell surface. Then
(11) φe= ζa, r= a, (12) dφe dr = 0, r = b. 4. Perturbed system
Taking∇2on Eq.(6)gives
(13) ∇2δφ(r, θ )E Ze−iωt = ∇2φ(r, θ, t)− ∇2φ e(r).
Under the condition of low surface potential the perturbed ionic concentration is negligible, δnjEe−iwt = 0, which
im-plies that (14) ∇2φ(r, θ, t)= ∇2φ e(r). Therefore (15) ∇2δφ(r, θ )Ee−iwt= 0.
In the absence of E the fluid is stagnant,∇pe= 0, and ue=
0, where peand ue are respectively the pressure and the fluid
velocity at equilibrium. Therefore, only the perturbed flow field needs to be considered. Equations(3)–(6) and (8)yield
(16) ∇ ·δuEZe−iωt
= 0, 0 < r < b,
(17) −iωρ0δuEZe−iωt= −∇δpEZe−iωt+ η0∇2δuEZe−iωt
− ρe∇φ, a < r < b,
(18) − iωρiδuEZe−iωt= −∇δpEZe−iωt+ ηi∇2δuEZe−iωt,
0 < r < a.
For an easier treatment, the stream function ψ is adopted to describe the flow field. In terms of ψ , we have
δuEZe−iωt=
δurˆr + δuθˆθEZe−iωt
(19) = − 1 r2sin θ ∂ψ ∂θ ˆr + 1 rsin θ ∂ψ ∂r ˆθ EZe−iωt,
where δur = −(1/r2sin θ )(∂ψ/∂θ ) and δuθ = (1/r sin θ) ×
(∂ψ/∂r) are respectively the r- and the θ -components of δu. Note that Eq. (16)is satisfied automatically by these two ex-pressions. Taking the curl on both sides of Eqs.(17) and (18)
and introducing the stream function, we obtain
η0E4ψ+ iωρ0E2ψ= ∂ρe ∂r ∂φ ∂θ − ∂ρe ∂θ ∂φ ∂r sin θ, (20) a < r < b, (21) ηiE4ψ+ iωρiE2ψ= 0, z 0 < r < a, where E2= (∂r∂22)+ sin θ r2 ∂ ∂θ( 1 sin θ ∂ ∂θ)and E 4= E2E2.
For the electric field, because the fluid inside a droplet is electrolyte-free and the surface conductivity is negligible, we assume that the droplet surface is impenetrable to ions; that is,
(22) ∇δφEZe−iωt
· ˆn = 0, r = a.
On the cell surface, the boundary condition proposed by Shilov and Zharkikh[11]is adopted; that is,
(23)
δφEZe−iωt
= EZe−iωtbcos θ, r= b.
Since the bulk liquid cannot penetrate a droplet and both the velocity and the shear stress are continuous across the droplet surface, the following conditions are assumed,
(24)
δurEZe−iwt r=a+= δurEZe−iwt r=a−= 0, r = a,
(25)
δuθEZe−iwt r=a+= δuθEZe−iwt r=a−, r= a,
(26)
(τn· ˆn) × ˆn r=a+= (τn· ˆn) × ˆn r=a−, r= a,
where τn is the shear stress tensor on the droplet surface. Here, we assume that the cell surface moves with a velocity −UEZe−iωtrelative to a droplet. Also, according to Kuwabara
[23], the vorticity should vanish on the cell surface. Therefore the following conditions are assumed on the cell surface:
(27) ∇ × δuEZe−iωt= 0, r = b,
(28)
δurEZe−iωt= −(UR+ iUI)EZe−iωtcos θ, r= b.
Furthermore, the symmetric nature of the present problem requires that
(29)
δurEZe−iwt= 0, r = 0,
(30)
5. Electrophoretic mobility
For the present case, the electrophoretic mobility μ is de-fined by
(31)
μ= μR+ iμI=
U E,
where E= EZe−iωtand U = (UR+ iUI)EZe−iωt, μRand μI
being respectively the real and the imaginary parts of μ. To evaluate μ, a force balance on a droplet is conducted to give
(32) Fh+ Fe= 4 3π a 3(ρ i− ρ0) dU dt ,
where Fh and Fe are respectively the hydrodynamic force and
the electric force acting on a droplet. For the present problem, only the z-components of these forces need to be considered. Since the charge on the droplet surface is distributed uniformly and the droplet fluid is electrolyte-free, the z-component of Fe,
Fe, can be calculated as Fe= x S σ (−∇φ)s· ezdS = 2πεζ2 a π 0 ∂φ ∂r r=a ∂φ ∂r cos θ− 1 r ∂φ ∂θ sin θ r=a (33) × r2 sin θ dθ,
where S denotes the droplet surface and σ is the surface charge density, which can be determined by Gauss’s law. Under a creeping flow condition the z-component of Fh, Fh, can be
eval-uated by[26] Fh= −πρ0 π 0 r2sin θ ∂ ∂t ∂(ψ EZe−iωt) ∂r r=a dθ + ηπ π 0 r4sin3 ∂ ∂r E2ψ EZe−iωt r2sin2θ r=a dθ (34) − π π 0 r2sin2θρe ∂φ ∂θ r=a dθ.
For an easier treatment, the problem under consideration is decomposed into two subproblems. In the first subproblem a droplet moves with a scaled velocity (UR∗+ iUI∗)E∗Ze−iωt in the absence of E, and in the second subproblem E is applied but the droplet remains stationary in space. EZ∗ represents the scaled electric field. Let F1∗ and F2∗ be respectively the scaled total forces in subproblems 1 and 2. Since the influence of the perturbed electric field is of a linear nature, F1∗is proportional to (UR∗+ iUI∗)EZ∗e−iωt, and F2∗ is proportional to EZ∗e−iωt EZ∗e−iωt; that is,
(35)
F1∗= χUR∗+ iUI∗EZ∗e−iωt,
(36)
F2∗= βEZ∗e−iωt,
where χ and β are complex constants. In scaled symbols, Eq.(32)can be rewritten as
(37) Fh∗+ Fe∗= F1∗+ F2∗= − i 4 3π εζ 2 a ρ0ωa2 η0 ρi− ρ0 ρ0 ×UR∗+ iUI∗EZ∗e−i t.
Based on Eqs.(31) and (35)–(37), a scaled mobility μ∗=
μ/(εζa/η0)can be expressed as (38) μ∗= −β χ+ i(ρ0ωa2 η0 )( ρi−ρ0 ρ0 ) .
Here, μ∗ is a complex number, which can be characterized by its magnitude μ∗m =
μ2R+ μ2I and phase angle Θ = tan−1(μI/μR); that is,
(39)
μ∗= μ∗meiΘ.
Since E = EZe−iωtez and U =
UR2+ UI2EZe−iωt+Θez,
Θ <0 implies that U leads E, and the reverse is true if Θ > 0. The governing equations, Eqs.(10) and (13)–(21), and the associated boundary conditions, Eqs.(11), (12), and (22)–(30), are solved by a pseudo-spectral method based on Chebyshev polynomials, which is found to be efficient and accurate for problems of the present type[24,25].
6. Results and discussion
The factors that might influence the electrophoretic behavior of the system under consideration include the scaled frequency
ω∗= ρ0ωa2/η0, the scaled double-layer thickness κa, where
κ−1= (εkT / 2j=1nej0(ezj)2)1/2, the viscosity ratio (η0/ηi),
the scaled surface potential φr = z1eζa/kT, and the volume
fraction of droplets, which can be measured by H = (a/b)3. An aqueous KCl solution is chosen to illustrate the typical behavior of a dispersion. The following values are assumed in numerical simulations: T = 298.15 K, z1= −z2= 1 F/m,
η0= 8.904 × 10−3 g/(cm s), ρ0/ρi = 0.909, ζa= 25.6 mV,
and ε= 8.854 × 10−12× 78.54688.
Fig. 2shows the variations of the scaled magnitude of elec-trophoretic mobility μ∗mand the corresponding phase angle Θ as a function of the scaled frequency of the applied electric field ω∗ for various values of H , and those as a function of
κa are presented inFig. 3. H represents the volume fraction of droplets. For example, if H = 0.1, the dimensionless dis-tance between droplets, 2(b− a), would be approximately 2.3. Double-layer overlapping is allowed in the current analysis. For the configuration under discussion (a:b = 1:2), κa smaller than 1 indicates the occurrence of overlapping, where 1/κ is chosen as the index of the double-layer thickness. As can be seen in
Fig. 2, for a fixed value of H , an increase in ω∗has the effect of reducing μ∗m. This is because in Eqs.(20) and (21), as ω∗ gets large, the flow field will be dominated by the inertia term, and the influences of the other terms become less important. Moreover, as the direction of the applied electric field changes back and forth, that of the movement of a droplet varies ac-cordingly. Therefore, the higher the ω∗ the shorter the time
(a)
(b)
Fig. 2. Variation of scaled magnitude of electrophoretic mobility μ∗m(a), and phase angle Θ (b), as a function of scaled frequency ω∗for various values of Hat κa= 1 and η0/ηi= 1.
available for the acceleration of a droplet, and hence the smaller the mobility. It is expected that as ω∗→ ∞, μ∗m→ 0; that is,
a droplet remains stationary.Fig. 2also shows that for a fixed level of ω∗, μ∗mdecreases with the increase in H . This is mainly due to the hindrance effect between neighboring droplets. The higher the volume fraction ratio H , the more significant this ef-fect becomes. Fig. 2b reveals that Θ > 0 (phase lag) for the present problem. This is because a droplet needs some time to adjust its surrounding ionic cloud so that they can move si-multaneously. If ω∗is large, the time for a droplet to react by altering its direction of movement becomes short, and hence the phase lag increases accordingly. The droplets simply run out of time to catch up with the fast-alternating input elec-tric field. The apparent outcome is increasing phase lag, like
(a)
(b)
Fig. 3. Variation of scaled magnitude of electrophoretic mobility μ∗m(a), and phase angle Θ (b), as a function of κa for various values of H at ω∗= 1 and η0/ηi= 1.
the increasing distance between two runners on a round race-track.
Figs. 3a and 3breveal that μ∗mand Θ decrease with the in-crease in H . This is because if the volume fraction of droplets is large, the space between droplets becomes narrow. In this case, since the electric double layer is confined to a narrow space, the time necessary to adjust the ionic distribution inside is short, and the phase lag is small. The influence of the thickness of the double layer surrounding a droplet on its electrophoretic be-havior is also illustrated inFig. 3. In the limit, as κa→ 0, the magnitude of the dynamic mobility always decrease to zero. This is because as κa→ 0, indicating an infinitely thick double layer, it influences all the other droplets throughout the entire
(a)
(b)
Fig. 4. Variation of scaled magnitude of electrophoretic mobility μ∗m(a), and
phase angle Θ (b), as a function of ω∗for various values of (η0/ηi)at κa= 1
and H= 0.1.
electrolyte dispersion system. Due to the double-layer over-lap effect, it actually yields a uniform distribution of ions and thus electric potential throughout. As a result, no electric force is generated upon the droplet and it remains stationary. How-ever, the effect of the double-layer overlap gradually disappears with the increase of κa; that is, the decrease of double-layer thickness yield a nonuniform distribution of ions, which sets up the motion eventually. The higher the κa, the less signif-icant the double-layer overlap from neighboring droplets, and the mobility increases accordingly. The phase lag, Θ, decreases with increasing κa, because now it is easier to react to the ap-plied alternating electric field, with less entanglement from the double-layer overlap. This phenomenon gets less significant as
H is increased, because of the associated retarding hindrance
(a)
(b)
Fig. 5. Variation of scaled magnitude of electrophoretic mobility μ∗m(a), and phase angle Θ (b), as a function of κa for various values of (η0/ηi)at ω∗= 1
and H= 0.1.
effect, which appears to be more dominant in concentrated dis-persions.
The influence of the viscosity of the droplet fluid on its elec-trophoretic behavior is illustrated inFigs. 4 and 5. These figures reveal that the larger the ratio (η0/ηi), that is, the more viscous
the bulk fluid is relative to the droplet fluid, the larger the μ∗m. This is because the smaller the viscosity of the droplet fluid, the smaller the drag acting on a droplet. For comparison, the corresponding result for hard spherical particles[13]is also pre-sented inFigs. 4 and 5. As can be seen from the two figures, they serve as limiting cases. If (η0/ηi)is equal to 0.01, we can
see that the dynamic electrophoretic mobility of liquid drops is very close to that of hard spheres. The phase lag is found to increase with the decrease in the viscosity of the droplet fluid
(a)
(b)
Fig. 6. Variation of scaled magnitude of electrophoretic mobility μ∗m(a), and
phase angle Θ (b), as a function of ω∗for various values of κa at H= 0.1 and η0/ηi= 1.
(larger (η0/ηi)). This is because a droplet with a smaller
vis-cosity has a larger μmand inertia, and a longer time is needed
for it to alter its direction of movement. The variations of μm
and Θ with κa at various viscosity ratios inFig. 5are mainly due to the double-layer overlap effect, as we deduced before in
Fig. 3.
The influence of the thickness of the double layer surround-ing a droplet on its electrophoretic behavior is also illustrated in
Fig. 6. This figure indicates that μ∗mincreases with the increase in κa for a fixed frequency. As we said before, the overlap of neighboring double layers has the effect of restraining the
movement of droplets. When κa= 0.1, for example, the over-lap effect is so severe that μ∗m approaches zero, which results
in a nearly horizontal line along varying ω∗ inFig. 6a. Simi-larly, the phase lag declines with the increase in κa because the thinner the double layer, the more easily it moves accordingly.
In summary, the dynamic electrophoretic behavior of a droplet dispersion is analyzed under conditions of low surface potential and an arbitrary volume fraction of droplets and dou-ble layer thickness. We conclude the following: (i) If the effect of double-layer polarization is neglected, as in the current case of low zeta potential, the phase of the electrophoretic velocity of a droplet, and consequently its mobility, is behind that of the applied electric field. (ii) The magnitude of electrophoretic mo-bility declines with the volume fraction of droplets. (iii) The smaller the viscosity of the droplet fluid the larger the magni-tudes of the electrophoretic mobility and the phase lag. As the viscosity of the droplet fluid increases, the electrophoretic be-havior of a droplet approaches that of a rigid particle. (iv) The magnitude of electrophoretic mobility increases, but the corre-sponding phase lag decreases, with the decline in the thickness of the double layer.
Acknowledgment
This work is supported by the National Science Council of the Republic of China.
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