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The role of electrode impedance and electrode geometry in the design of microelectrode systems
Journal of Colloid and Interface Science 297 (2006) 819–831 www.elsevier.com/locate/jcis
The role of electrode impedance and electrode geometry in the design of microelectrode systems Hao Zhou a , Robert D. Tilton a,b , Lee R. White a,∗ a Department of Chemical Engineering, Center for Complex Fluids Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA b Department of Biomedical Engineering, Center for Complex Fluids Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Received 11 August 2005; accepted 9 November 2005 Available online 5 December 2005
Abstract Microelectromechanical systems (MEMS) employing spatially and/or temporally nonuniform electric fields have been extensively employed to control the motion of suspended particles or fluid flow. Design and control of microelectromechanical processes require accurate calculations of the electric field distribution under varying electrolyte conditions. Polarization of electrodes under the application of an oscillating voltage difference produces dynamic electrical double layers. The capacitive nature of the double layers significantly inhibits the penetration of the electric field through the double layer and into the surrounding bulk electrolyte at low frequencies. This paper quantitatively discusses the effect of electrode impedance on the electric field distribution as a function of field frequency, electrolyte composition, and electrode zeta potential in microelectrode systems. The design principles for the electrode geometry and configuration are also discussed in terms of their effects on the electric field magnitude and nonuniformity. © 2005 Elsevier Inc. All rights reserved. Keywords: MEMS; Electrode impedance; Microelectrode arrays; AC electric field
1. Introduction Applications of microelectromechanical systems (MEMS) to microfluidic devices, drug delivery devices, biosensors, bioreactors, and cell separation and arraying schemes have been extensively explored in recent years [1–7]. MEMS have the ability to control the positioning of colloidal particles and/or to create electrohydrodynamic flows using AC electric fields under two dominant mechanisms, dielectrophoresis (DEP) and AC electroosmosis (EO) [8–12]. DEP refers to the migration of a polarizable particle in a nonuniform electric field. The dielectrophoretic force on a particle scales as ∇E 2 , causing particles to migrate along gradients of electric field intensity. Particles undergoing positive dielectrophoresis (pDEP) migrate toward regions of greatest electric field intensity, while particles undergoing negative dielectrophoresis (nDEP) migrate toward regions of lowest field intensity [7]. AC EO flow arises from the * Corresponding author.
E-mail address: [email protected] (L.R. White). 0021-9797/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.11.024
interaction of the nonuniform electric field with the free charge generated in the diffuse double layer above the electrodes in a microelectrode array [8–15]. As with conventional electroosmosis, the electromotive force on the ions is transferred through viscous interactions to a fluid flow. Because both the sign of the free charge and the electric field direction reverse when the electrode polarity reverses in an AC circuit, the direction of the electromotive force and resulting fluid flow is unchanged [8–15]. This fluid flow results in a viscous drag on any particle residing near the microelectrode. In order to calculate either the DEP force or the AC EO drag force on a particle in the vicinity of a microelectrode in a MEMS device, it is necessary to first accurately calculate the electric field distribution. Such is the purpose of this paper. To date the calculation of the electric field distribution in a microelectrode system is limited by the use of approximate boundary conditions or consideration of only specific geometries [12–18]. The effect of electrode geometry on microelectrode systems has been discussed qualitatively in the literature [13–15]. In this paper, we calculate the electric field distribution using electrode double layer boundary conditions at the bound-
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ing electrode and dielectric surfaces, which allows us to obtain the electric field distribution for the full field frequency range in any specified geometries. When an oscillating voltage difference is applied across the electrolyte domain, the electrodes are polarized, forming dynamic electrical double layers. At low frequencies (1 MHz), the capacitive nature of the electric double layers significantly inhibits the penetration of the electric field through the double layer and into the surrounding bulk electrolyte. Therefore, the calculation of the electric field distribution above patterned electrodes depends strongly on the double layer impedance (Zel ). The electrode double layer impedance, Zel , used in this paper is that of an ideally polarizable plane calculated using the standard model of colloidal electrokinetics [19]. This approach incorporates the classic retardation and relaxation effects on the double layer but excludes the case where ion current penetrates the electrode surface via electrochemical reactions. The double layer impedance is a function of the zeta potential on the electrode surface, the frequency of the applied voltage and the electrolyte concentration, valency and limiting conductance. The electric field distribution within a MEMS microelectrode array is discussed as a function of field frequency, electrode surface zeta potential, and electrolyte composition. The effect of the electrode geometry on the field distribution is also discussed and compared between two common microelectrode configurations ( and ⊥), as described below. In this paper, symbol represents an electrode array in which voltage is applied between adjacent co-planar parallel electrode strips, shown in Fig. 1a, in order to produce highly nonuniform, high intensity fields. This microelectrode configuration is the most common MEMS design for DEP processes. The characteristic separation width hw for this configuration
(a)
(b) Fig. 1. (a) Schematic view of the microelectrode system. The top plate is a dielectric substrate, the bottom plate is a dielectric substrate patterned with parallel microelectrode strips. The electric potential is applied co-planarly to adjacent parallel strips. The characteristic separation width (hw ) is defined using the dielectric spacing between the edges of the electrode strips. (b) Schematic view of the ⊥ microelectrode system. The top plate is a uniform planar electrode. The bottom plate is a dielectric substrate patterned with parallel microelectrode strips. The electric voltage is applied normally to the top and bottom electrode. The characteristic separation width (hw ) for this ⊥ microelectrode system is defined using the normal distance between the connected top and bottom electrodes.
refers to the spacing (the dielectric strip width) between the edges of two adjacent strips. Rather than using the typical design in our previous experimental work, we employed a vertical configuration of the electrodes, denoted ⊥ as shown in Fig. 1b [20]. The electric field was applied between a uniform top electrode and a bottom, multi-strip electrode array. The characteristic separation width hw for this microelectrode configuration refers to the normal separation distance between the top and bottom electrodes. It has been demonstrated experimentally that the lateral positioning of colloidal particles and cells strongly depends on the frequency and strength of the applied electric field [20]. In that work an approximate electric field model was used to calculate the high frequency electric field distribution. In this paper, we reexamine the electric field distribution and the lateral DEP force on colloidal particles in this ⊥ microelectrode system, by including the Zel effect over the full frequency range. The particle patterning mechanisms in a ⊥ microelectrode system, originally proposed in Ref. [20], are further elaborated here. By comparing the effects of electrode geometry on the electric field distribution in the two microelectrode systems ( and ⊥), desirable design features are suggested for various DEP or microfluidic application purposes. The results are necessary for a quantitative understanding and control of microelectromechanical processes. 2. Theoretical modeling The electric field distribution in microelectrode systems and ⊥ are calculated using the models shown in Fig. 2. The bounding surfaces of the model domain consist of two half electrode strip surfaces (L and R), dielectric surface (D), two symmetry planes S, and a top bounding surface (T). In the configuration, T is a dielectric surface, shown in Fig. 2a. In the ⊥ configuration, T is a uniform electrode surface as seen in Fig. 2b. The characteristic separation width, hw , is defined by the separation distance between the two connected electrodes. Hence hw = lD is the gap width between two adjacent electrode strips in the configuration, and hw is an arbitrary electrode cell thickness in the ⊥ configuration. In order to be consistent with the electrode dimensions in our previous experimental work [20], all dimensions in this paper are scaled by the half electrode strip width l (l = 15 µm). The geometric dimensions of each bounding surface are lL = lR = l = 15 µm, lD = 8l = 120 µm, and lT = lL + lD + lR = 10l = 150 µm; the electrolyte is 0.1 mM NaHCO3 ; the electrode surface zeta potential, ζ , is −80 mV; and the dielectric surface zeta potential is −100 mV, unless otherwise stated. The electric field E exp(−iωt) above the electrode array is obtained by solving ∇ 2 Φ = 0,
(1)
E(x, y) = −∇Φ(x, y)
(2)
in the bulk electrolyte domain together with appropriate boundary conditions on the bounding surfaces.
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iting conductance Λ∞ j by λj =
NA e2 |zj | , Λ∞ j
(4)
where NA is the Avogadro number. The boundary condition on the outer solution Φ(x, y) at an electrode surface is then Φ(x, Y − ) − Φ(x, Y + ) = −K ∞ (ω)Zel (ω)nˆ · E(x, Y + ), where
Φ(x, Y − )
(5)
is the potential of the electrode surface.
(a)
2.2. Boundary conditions on the dielectric surface At the dielectric boundary, y = Y , the current density is just the displacement current (no ion penetration or electrode electrochemistry) and we should have iωε< ε0 nˆ · E(x, Y − ) = −K ∞ (ω)nˆ · E(x, Y + ),
(b) Fig. 2. Electric field calculation model used for the microelectrode systems. (a) Calculation model for configuration, shown in Fig. 1a. Electric potentials are applied to two parallel electrode strip surfaces, left (L) and right (R), which are positioned at y = 0. The lengths lL and lR are the half strip widths of L and R electrode strips, respectively. S stands for the symmetry plane. The characteristic separation width between two connected electrode strips, hw , equals the dielectric strip width lD . The top dielectric surface is positioned at y = 1 mm. (b) Calculation model for ⊥ configuration, shown in Fig. 1b. Electric potentials are applied normally to the electrode strip surfaces, T, L, and R. The L and R electrode strips are positioned at y = 0. The lengths lL and lR are the half strip widths of L and R electrode strips, respectively. S strands for the symmetry plane. The edges of the strips L and R are separated by a dielectric surface of length lD . The length of the top electrode lT equals lL + lR + lD . The characteristic separation width between two connected electrode strips, hw , equals the microelectrode cell thickness.
2.1. Boundary conditions on the electrode surfaces
K ∞ (ω) = e2
N n∞ z ∞ j j j =1
λj
− iωεε0 ,
where ε< is the dielectric constant of the dielectric substrate. Equation (5) also applies at the dielectric boundary but Φ(x, Y − ) is not specified. In principle, Φ(x, Y − ) and nˆ · E(x, Y − ) are obtained from solving Eq. (1) in the dielectric and matching to the outer solution in the electrolyte using (5) and (6). However, the left-hand side of (6) is negligible up to frequencies ωc ∼ κ 2 kT
N βj j =1
(7)
λj
(ωc = 1.4 × 106 for 0.1 mM NaHCO3 ), where zj2 n∞ j βj = N , 2 ∞ z j =1 j nj
N
βj = 1,
(3)
and ε, ε0 , e, zj , nj , λj are the electrolyte dielectric constant, vacuum permittivity, the electron charge, and the valency, number density, and ionic drag coefficient of the j th ion type, respectively. The drag coefficient λj is related to the ionic lim-
(8)
j =1
and these dielectric boundary conditions for ω ωc can be replaced by nˆ · E(x, Y + ) = 0
On an electrode at y = Y (Y = 0 or Y = hw ), where voltage is applied, Φ(x, Y − ) − Φ(x, Y + ) is the potential drop across the double layer at position x on the surface, and is equal to the double layer impedance Zel (ω) times the normal current density evaluated just outside the charge-bearing region as −K ∞ (ω)nˆ · E(x, Y + ), where nˆ is the surface normal pointing into the electrolyte. Here, K ∞ (ω) is the bulk electrolyte conductivity,
(6)
(9)
with negligible error. For the systems, the dielectric boundary conditions (5) and (6) are applied to bounding dielectric surfaces D and T. For the ⊥ systems, the dielectric boundary conditions (5) and (6) are applied to the bounding dielectric surface D only. 2.3. Boundary conditions on surfaces of symmetry On the bounding surfaces of symmetry S, the appropriate boundary condition is nˆ · E(x, y)|S = 0
(10)
The magnitude of the outer electrical potential and electric field distribution is calculated by using the commercial finite element software package FEMLAB (The Comsol Group).
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3. Results and discussion 3.1. Electrode impedance effects in the ⊥ configuration When the top plate electrode voltage is V exp(−iωt) and the bottom electrode strip electrodes are earthed, the electric field E can be written as E(x, y) = V E0 (x, y) exp −i(ωt + ϕ) , (11) where E0 (x, y) exp(−iϕ) = −∇Φ(x, y)
(12)
and ϕ is the phase angle between the electric field and the applied potential due to the complex electrode impedance Zel . Equation (1) is solved for Φ(x, y) with boundary conditions (5), (6), and (10) at appropriate surfaces. For the ⊥ configuration, the inner boundary potentials are Φ(x, h− w )|T = 1, −
(13) −
Φ(x, 0 )|L = Φ(x, 0 )|R = 0.
(14)
A suitable scale for E0 in this geometry is 1/ hw and we will use this scaling where convenient in the plots that follow. Unless otherwise stated, the electric field distribution curves are plotted for y = 3 µm from the bottom electrode surfaces. 3.1.1. Frequency variation of the electric field distribution In Fig. 3, the scaled electric field intensity distribution is plotted as a function of x at various frequencies in the ⊥ microelectrode system (hw = 66.7l = 1 mm). We note that the Zel = 0 result is almost identical to the 20 kHz results, but is quite different from the lower frequency plots where electrode impedance is an important effect. The capacitive nature of the electrode double layer at low frequencies hinders the penetration of the electric fields into the electrolyte region. When the frequency increases, the electrode double layer impedance decreases [19], which leads to an increase in electric field intensity. Zero electrode impedance has been used previously to
Fig. 3. The electric field intensity distribution calculated at y = 0.2l = 3 µm from the electrode surface without electrode impedance (Zel = 0) and with the electrode double layer impedance at 5, 10, 20, 60, 300 Hz, and 20 kHz in the ⊥ configuration (hw = 66.7l = 1 mm). The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
calculate approximately the electric field intensity in a microelectrode system [20]. From the results shown in Fig. 3, it can be seen that this zero electrode impedance approximation only generates valid results at frequencies higher than 20 kHz. The electric field intensity is vanishingly small in the center of the dielectric surface D, where the global electric field minimum can be found. Local minima occur over the electrode strip centers for all frequencies examined above 5 Hz. The electric field maxima are located just inside the electrode strip edges when the frequency is lower than 300 Hz, and are located at the electrode strip edges at high frequency (20 kHz, y = 3 µm). It can be seen that the overall electric field intensity increases as the frequency is increased from 5 to 60 Hz and Zel decreases. From 300 Hz to 20 kHz the electric field intensity increase is observed at the electrode strip edges and above the dielectric surfaces. The electric field intensity in the electrode strip centers, however, decreases when the frequency increases, and a stronger electric field gradient is generated on the electrode strips. Thus for a pure DEP process, high frequency should generate stronger traps and sharper patterning adjacent to both the dielectric surfaces and the electrode surfaces due to the high electric field intensity and high electric field gradient. 3.1.2. Electric field distribution above the electrode surface The scaled electric field distribution at various heights (y) above the bottom electrode surface is plotted in Fig. 4 for the ⊥ configured system (hw = 66.7l = 1 mm) at a frequency of 20 Hz. Note that the electric field decreases as the normal distance from the bottom electrode increases. The field maxima shift from the electrode strip edges toward the electrode strip centers as the normal distance increases, which predicts a strong correlation between the lateral DEP positioning and the vertical elevation in a colloidal patterning process. Correlation between the vertical elevation and lateral colloidal positioning has been observed, although the effects of AC EO flows become important in this frequency range [20]. Also from this result, it can be seen that the initial elevation of colloidal particles can be important to a patterning/separation process. If the electric poten-
Fig. 4. The electric field intensity distribution calculated at various heights, y = 0.1l, 0.2l, 0.4l, 0.6l, 0.8l, and 1l, above the electrode surface at 20 Hz in the ⊥ configuration (hw = 66.7l = 1 mm). The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
H. Zhou et al. / Journal of Colloid and Interface Science 297 (2006) 819–831
tial is applied while nDEP colloidal particles are sedimenting onto the bottom surface, the particles can be easily collected on the dielectric surface where the global minimum resides. As the particles sediment, they would not likely be trapped in the local field minimum located at low elevations over the electrode strip center, since there is no local minimum there at high elevations. This field distribution would guide sedimenting particles around the electrode strips to finally locate adjacent to the dielectric region. If instead the particles sedimented in the absence of an electric field, then some fraction of nDEP particles would likely be trapped in the local field minima above the electrode strip centers. The relative number of nDEP particles trapped along electrode centers or the dielectric region would depend on the overall field strength and the particle buoyancy. In a DEP process to separate nDEP and pDEP colloids, application of the electric field during sedimentation may generate more efficient separation. At low frequencies, where the EO effect may be significant, the positioning of the colloidal particles can be very sensitive to the particle polarizability, particle size, and particle mass density due to the strong correlation between ∇E20 and the vertical elevation [20]. 3.1.3. Electrode zeta potential effect on the electric field distribution The electrode double layer impedance is a function of electrode zeta potential, ζ , the electrostatic potential at the start of the diffuse double layer for the resting electrode. In Fig. 5, the electrode ζ potential effect on the electric field distribution is illustrated for various frequencies and ζ values in the ⊥ configured system (h = 66.7l = 1 mm). In Fig. 5a, the right electrode zeta potential is systematically changed from −80 to −140 mV, where the left electrode and the top electrode remain at −80 mV. The frequency is set at 5 Hz, where electrode impedance is significant. It can be seen that the electric field intensity above the right electrode strip increases dramatically when the magnitude of its zeta potential increases. The electric field intensity above the left electrode, however, decreases moderately as the right electrode zeta potential increases. The double layer capacitance (Cel ) increases with |ζ | and the electrode impedance (dominated by the capacitive contribution 1/ωCel at 5 Hz) decreases. Thus, less potential drop occurs across the double layer, resulting in an increased outer electric field strength (see Eq. (5)). When the right electrode impedance drops, its share of the total current carried increases. Consequently, the current and hence outer electric field at the left electrode decreases. Therefore the electric field intensity difference between the left and right electrode strips can be controlled by the zeta potentials of the electrode strips. In Fig. 5b we show the effect of changing frequency in a ⊥ configuration, where the right electrode zeta potential is ζR = −100 mV, and the top and left electrode zeta potentials are ζT = ζL = −80 mV. At 20 kHz, electrode impedance is negligible and the field distributions on the left and right electrodes are identical. For lower frequencies, as the electrode impedance on all electrodes increases, the asymmetry of the left and right electrodes becomes apparent. Between 20 and 60 Hz, the electric field intensities on both left and right
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(a)
(b)
(c) Fig. 5. The electric field intensity distribution calculated at y = 0.2l = 3 µm above the electrode surface for various electrode zeta potentials in the ⊥ configuration (hw = 66.7l = 1 mm). The electrolyte is 0.1 mM NaHCO3 . (a) ζT = ζL = −80 mV, ζR = −80, −100, −120, and −140 mV at 5 Hz. (b) ζT = ζL = −80 mV, ζR = −100 mV at 5, 10, 20, 60, 300 Hz, and 20 kHz. (c) ζT = ζL = ζR = −80 mV; ζT = ζL = −80 mV, ζR = −140 mV; ζT = −80 mV, ζL = ζR = −140 mV; ζT = −140 mV, ζL = ζR = −80 mV; ζT = ζL = −140 mV, ζR = −80 mV; and ζT = ζL = ζR = −140 mV at 5 Hz.
strips increases with frequency but the field intensity asymmetry between the strips stays relatively constant. From 60 to 300 Hz, the electric field intensity on the right strip de-
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creases slightly and the field intensity on the left electrode increases. In Fig. 5c, the electric field intensity distribution for more complicated electrode zeta potential combinations is displayed. When the electrode zeta potentials are identical, ζT = ζL = ζR , increase of |ζ | decreases the electrode double layer impedance [19] and, hence, increases the outer electric field intensity. Increasing |ζT | increases the electric field intensity on both L and R electrode strips. Increasing either |ζL | or |ζR | increases the electric field intensity on that electrode strip. By exploiting different zeta potentials on the T, L, and R electrode surfaces, more complicated electric field intensity distributions (magnitude and gradient) can be achieved. It can be seen that the electrode zeta potential effect plays a significant role on the electric field distribution in low frequency DEP/EO processes. The zeta potential asymmetry between the electrode strips creates field maxima and minima of varying strength in different regions of the microelectrode system. This impacts directly on the lateral positioning of the colloidal particles. In practice, the zeta potential of the electrode strip surface can be varied by modifying the electrode surface, perhaps by adsorption of polymers or polyelectrolytes, or by chemisorption of self-assembled monolayers. Conversely, contamination of the electrode strips will also alter the electrode ζ potential and may significantly skew the behavior of a MEMS process when operated at the moderate frequencies where the zeta potential effect is strongest. 3.1.4. Electrolyte concentration effect on the electric field distribution In Figs. 6a and 6b, the electrolyte concentration effect is plotted for 5 and 300 Hz for the ⊥ configuration (h = 66.7l = 1 mm). In these calculations, the electrode zeta potentials ζT = ζL = ζR = −80 mV, although in practice the ζ potential changes with increasing electrolyte concentration [21]. It can be seen that the electrolyte concentration effect is more pronounced at low frequencies where its effect on the electrode impedance is significant. At low frequencies, smaller electrolyte concentrations generate stronger field intensities. At moderate frequency (∼300 Hz), low concentration electrolyte serves better in a DEP process where stronger field gradients adjacent to the electrode are achieved. 3.1.5. Electrolyte ion type effect on the electric field distribution The electrolyte ion transport properties affect the electric field intensity distribution, as illustrated in Figs. 7a and 7b, for 5 and 300 Hz in the ⊥ configuration (h = 66.7l = 1 mm). As expected, the electrolyte ion type effect is strongest at lower frequencies. The ionic limiting conductivities used in this paper are listed in Table 1. Variation in the electrolyte ion type changes the impedance of the electrode double layer and the bulk electrolyte impedance simultaneously. At low frequencies, decreasing the ion limiting conductivities has a very small effect on the electrode double layer impedance [19], but it decreases the bulk electrolyte conductance K ∞ (see Eq. (3)).
(a)
(b) Fig. 6. The electric field intensity distribution calculated at y = 0.2l = 3 µm above the electrode surface for 0.1, 1, and 10 mM NaHCO3 at (a) 5 Hz and (b) 300 Hz in the ⊥ configuration (hw = 66.7l = 1 mm). The electrode zeta potentials are ζT = ζL = ζR = −80 mV.
We note from Eq. (5) that the electrode boundary condition involves the product of Zel and K ∞ , and the effect of decreasing ion conductance is to decrease the effect of electrode impedance, with a consequent increase in the outer electric field. Hence in Fig. 7a, the least conducting salt NaHCO3 produces the smallest electrode impedance effect and the largest outer fields. The most conducting salt HCl produces the smallest fields. The electrode impedance at 300 Hz is small, and we see in Fig. 7b that the effect of ion type is accordingly weak. 3.2. The effect of electrode geometry 3.2.1. Characteristic length h effect on the field distribution In Figs. 8a and 8b, we plot E02 at y = 3 µm as a function of x for 5 and 300 Hz for variations in the normal electrode spacing hw in the ⊥ configuration. At 300 Hz (Fig. 8b) we see that increasing the electrode spacing decreases the field in an unsurprising way. At this frequency, the effects of electrode impedance are small. In contrast at 5 Hz, where the electrode impedance is large due to the capacitive nature of the double
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(a)
(a)
(b)
(b)
Fig. 7. The electric field intensity distribution calculated at y = 0.2l = 3 µm from the electrode surface for 0.1 mM NaHCO3 , NaCl, KCl, and HCl at (a) 5 Hz and (b) 300 Hz in the ⊥ configuration (hw = 66.7l = 1 mm). The electrode zeta potentials are ζT = ζL = ζR = −80 mV.
Fig. 8. The electric field intensity distributions calculated at y = 0.2l = 3 µm above the electrode surface for characteristic separation width hw = 6.67l = 0.1 mm, hw = 33.33l = 0.5 mm, and hw = 66.67l = 1 mm at (a) 5 Hz and (b) 300 Hz in the ⊥ configuration. The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
Table 1 Parameters used in the calculations in this paper Temperature Medium viscosity Medium density Dielectric constant of medium Limiting conductance of Na+ Limiting conductance of H+ Limiting conductance of K+ Limiting conductance of Cl− Limiting conductance of HCO− 3
T = 298 K η = 0.8904 × 10−3 kg/(m s) ρ0 = 997.04 kg/m3 ε = 78.54688 Λ∞ = 50.11 × 10−4 m2 / equiv Λ∞ = 349.82 × 10−4 m2 / equiv Λ∞ = 73.5 × 10−4 m2 / equiv Λ∞ = 76.35 × 10−4 m2 / equiv Λ∞ = 41.5 × 10−4 m2 / equiv
layer, we observed a surprisingly weak sensitivity of the field intensity to the normal electrode spacing (Fig. 8a). This somewhat counterintuitive result can be rationalized as follows. Near the bottom electrodes the transverse fields are determined mainly by the electrode strip geometry and are consequently relatively insensitive to normal spacing hw . However, our numerical studies show that the normal outer electric field is of a comparable magnitude. Thus the insensitivity of nˆ · E must be explained to understand these results. The outer normal field is determined by the outer potential difference between top and
bottom electrodes divided by the spacing hw . The outer potential difference is the voltage difference applied at the electrodes minus the potential drops across the double layers on top and bottom electrodes. These potential drops are (from Eq. (5)) a product of the electrode impedance and the outer normal field. Thus nˆ · E ∼
T + Z B )n V − K ∞ (ω)(Zel el ˆ · E hw
(15)
and, rearranging, nˆ · E ∼
V . T + ZB ) hw + K(ω)(Zel el
(16)
When electrode impedance is large (at 5 Hz), nˆ · E is insensitive to hw . When electrode impedance is small (at 300 Hz), nˆ · E varies inversely with hw . 3.2.2. ⊥ and configuration comparison For the configured systems, the left and right electrode strips are connected to the AC power source. The boundary condition (5) applies on electrodes L and R, and the inner boundary
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(a)
(b)
(c) Fig. 9. (a) The electric field intensity distributions, (b) lateral field intensity, and (c) normal field intensity calculated at y = 0.2l = 3 µm above the electrode surface in the ⊥ configuration and the configuration with characteristic interelectrode spacing width hw = 8l = 120 µm at 300 Hz in the ⊥ configuration. The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
potentials on L and R are just the applied electric potentials, Φ(x, 0− )|L = 0, −
Φ(x, 0 )|R = 1.
(17) (18)
In Fig. 9a, the electric field distribution is compared between ⊥ and configurations at 300 Hz (hw = 120 µm for both configurations). It can be seen from Fig. 9a that the ⊥ configuration generates stronger electric field intensity than the configuration when the same voltage is applied. In Figs. 9b and 9c, we separate the normal and transverse field components. The configuration produces stronger electric field intensity on the dielectric surface D due to its strong lateral electric field component (see Fig. 9b). The broader minimum in E02 can result in DEP positioning on the dielectric surface in the configuration being less well defined than in the ⊥ configuration. 3.2.3. Voltage configuration effect on the field distribution By varying the voltages applied at the electrodes, a variety of electric field distributions can be generated. In Fig. 10, the electric field distribution is plotted for a number of different voltage applications in a ⊥ configured system. Strong electric field bias on one bottom electrode is obtained by applying the
Fig. 10. The electric field intensity distributions calculated at y = 0.2l = 3 µm above the electrode surface for various voltage configurations, ΦL = 0, ΦR = 1 in the configuration (where ΦT is not applicable as there is no top electrode); ΦT = ΦL = 0, ΦR = 1; ΦT = 1, ΦL = ΦR = 0; and ΦT = ΦR = 1, ΦL = 0 in the ⊥ configuration. All at 300 Hz. The electrode zeta potentials are ζT = ζL = ζR = −80 mV and hw = 8l = 120 µm. The electrolyte is 0.1 mM NaHCO3 .
same voltage to the top electrode and the other strip. By varying the applied voltage at the top electrode, high electric field
H. Zhou et al. / Journal of Colloid and Interface Science 297 (2006) 819–831
(a)
827
(a)
(b) Fig. 11. The electric field intensity distributions calculated at y = 0.2l = 3 µm above the electrode surface for various dielectric spacings lD = l, 2l, 4l, 6l, and 8l between two electrode strips (a) in the ⊥ configuration (hw = 66.7l = 1 mm) and (b) in the configuration; all at 60 Hz. The electrode zeta potentials are ζT = ζL = ζR = −80 mV and the electrode widths are fixed at lL = lR = l. The electrolyte is 0.1 mM NaHCO3 .
bias can be easily switched from left to right electrode. Such strongly biased switchable electric field intensities may have interesting novel MEMS applications. 3.2.4. Dielectric strip width effect on the field distribution In Figs. 11a and 11b, the dielectric strip width (the spacing between adjacent electrode strips) is systematically varied for microelectrode system in either the ⊥ or configuration. In a ⊥ configured system, the global minimum is always at the center of the dielectric strip. Increasing the dielectric strip width increases both the electric field intensity and the electric field gradient on the electrode strips. In contrast, in a configured system, increasing the dielectric strip width decreases the overall electric field intensity and gradient. For DEP processes, where both strong field intensity and field gradient are preferable, a larger separation between the electrode strips in the ⊥ configuration is more efficient. For the configured microelectrode system, however, small interelectrode strip separations are preferable.
(b) Fig. 12. The electric field intensity distributions calculated at y = 0.2l = 3 µm above the electrode surface for various electrode strip widths lL = lR = l, 2l, 3l, and 4l and fixed dielectric spacing lD = 2l (a) in the ⊥ configuration (hw = 66.7l = 1 mm) and (b) in the configuration; all at 60 Hz. The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
3.2.5. Electrode strip width effect on the electric field distribution In Figs. 12a and 12b, the electrode strip width is systematically varied for ⊥ and microelectrode systems. Decreasing the electrode strip width increases both the electric field intensity and the electric field gradient on the dielectric surfaces for the ⊥ configuration (Fig. 12a). For an efficient DEP process, smaller electrode strip widths are more appropriate in the ⊥ configuration. By contrast again, for a configured microelectrode system (Fig. 12b), wider electrode strips significantly increase the electric field intensity at fixed dielectric strip spacing. Note that the global electric field intensity minimum switches from the dielectric center to the electrode strip center as the electrode strip width increases. For the configuration, a wider electrode strip will achieve more precise DEP patterning results on both dielectric and electrode strip surfaces.
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(a)
(b)
(c)
(d)
Fig. 13. The electric field intensity distributions calculated at y = 0.2l = 3 µm above the electrode surface for various left electrode strip widths lL = l, 2l, 3l, 4l, 5l, 6l, 7l, and 8l with fixed lR = l and lT = 10l (a) in the ⊥ configuration (hw = 66.7l = 1 mm) at 5 Hz, (b) in the configuration at 5 Hz, (c) in the ⊥ configuration (hw = 66.7l = 1 mm) at 60 Hz, and (d) in the configuration at 60 Hz. The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
3.2.6. Electrode/dielectric strip width ratio effect on the electric field distribution When the dimension of the microelectrode cell is fixed (i.e., constant working area), varying one of the electrode strip widths would simultaneously vary the dielectric strip spacing. This variation affects the electric field distribution, as shown in Figs. 13a and 13b (for 5 Hz) and Figs. 13c and 13d (for 60 Hz). The total electrode center-to-center spacing and the right electrode width are fixed at lT = lL + lD + lR = 10l and lR = l, respectively. For the ⊥ configuration, increasing the left electrode strip width decreases the electric field intensity in the centers of both electrode strips, as shown in Figs. 13a and 13c. The global minimum is always in the center of the dielectric strips as shown previously. In the 5 Hz case (Fig. 13a), there exists an optimum electrode strip and dielectric strip spacing combination where the strongest local maximum electric field on the left strip is obtained. For the configuration at 5 Hz, however, a systematic shifting of the local field maximum on the left strip toward the right electrode and a strong electric field intensity bias on the right electric strip is observed when the left electrode width increases. There exists a combination of the left
electrode strip width and dielectric strip width where the local maximum on the left electrode strip width reaches a minimum value (Fig. 13b). A merging of the electric field peak on the left electrode into the electric field peak on the right electrode eventually occurs when increasing the left electrode strip width. At 60 Hz, the optimal spacing observed in the ⊥ configuration at 5 Hz does not occur (Fig. 13c). The configuration at 60 Hz (Fig. 13d) is qualitatively similar to the 5 Hz case. From Fig. 13, it can be seen that the configuration can be used to generate a strong electric field bias on one electrode strip by increasing the width of the other electrode strip. 3.3. Experimental patterning of polystyrene (PS) colloids in ⊥ configuration In our previous publication [20], we systematically examined the patterning behavior of a variety of PS colloids in 0.1 mM NaHCO3 over the frequency range from 60 Hz to 1 MHz in the ⊥ configuration. Here we discuss observations for the 9.6 µm diameter particles with −44 mV zeta potential, hereafter referred to as PS 9.6/44. The trends are representative of the other PS colloids we examined as well. The key
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(a)
(b)
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In this paper, we reexamine the lateral dielectrophoretic force calculation for PS 9.6/44 particles using the full solution that accounts for the electrode double layer. The results are shown in Fig. 15. Note that the direction and the strength of the lateral DEP force that a colloidal particle experiences above the electrode strip surface is strongly dependent on the vertical elevation and the field frequency. At 60 Hz (Fig. 15a), the lateral DEP force on PS 9.6/44 is inward toward the electrode strip center at low elevations (y 3 µm). At y = 6 and 9 µm, a distance comparable to the particle sizes used in this thesis research, however, FDEP_x for PS 9.6/44 points outward toward the strip edges. This indicates that particles at higher elevations can experience a lateral DEP force that acts in the opposite direction as the force on particles at low elevation, or particles with smaller sizes. At higher frequencies, 1 kHz, the direction of the lateral DEP force remains inward at y = 3 µm. At y = 6 µm, however, the lateral DEP force remains inward, unlike what occurs at 60 Hz. The lateral DEP force points outward toward the edges only at y = 9 µm or higher. The lateral DEP force is less sensitive to particle elevation at higher frequencies. These calculations further demonstrate the importance of the particle elevation for the lateral positioning of colloidal particles in nonuniform fields at low frequencies (<1 kHz). Not only is the direction of the lateral AC EO drag force dependent on the vertical elevation, but the direction of the lateral DEP force acting on the colloidal particle is also sensitive to the particle elevation. This effect on the DEP force was not apparent in the zero impedance approximation. Both the DEP force and the AC EO drag force influence the vertical elevation and therefore the preferred lateral positioning. 4. Summary
(c) Fig. 14. Representative patterning results for PS 9.6/44 colloids obtained after 10 min of applying an electric field. (a) 60 Hz, 0.17 V. (b) 60 Hz, 0.51 V. (c) 1 kHz, where 0.17 or 0.51 V produced similar results.
observation was that these particles were patterned at 60 Hz along the electrode edges for a 0.51 V applied voltage, but in the electrode centers for 0.17 V. See Figs. 14a and 14b. While the particle patterning behavior was sensitive to the field strength at 60 Hz, at 1 kHz, however, particles were only patterned in the electrode centers at both voltages. See Fig. 14c. Using the electric field distribution calculated with zero double layer impedance [20], we estimated the dielectrophoretic forces exerted on a PS 9.6/44 particle, and we proposed that the patterning mechanism at low frequency was attributed to a coupling between dielectrophoretic forces and AC electroosmotic drag forces on particles near the microelectrode strips and the relative direction of these two types of forces depended on the particle elevation [20]. As discussed above, the zero impedance approximation is only strictly valid for high frequencies.
This study quantitatively showed the effect of electrode impedance on the electric field distribution as a function of field frequency, electrolyte composition, and electrode zeta potential in microelectrode systems. The design principles for the electrode geometry and configuration are also discussed in terms of the effect on desired electric field properties—magnitude and nonuniformity. At low frequencies, the capacitive electrode impedance weakens the electric field intensity and its nonuniformity adjacent to microelectrode surfaces. This offers the possibility of controlling the electric field intensity profile by engineering the electrode zeta potential and tuning the electric field frequency. Field asymmetry can be generated and controlled by varying the electrode zeta potential, electrode/dielectric strip width, voltage configuration and field frequency in the low frequency range, suggesting possible novel MEMS strategies. The ability to create a desired electric field nonuniformity, magnitude and asymmetry provides opportunities to precisely control the motions of particles or fluid flows in MEMS processes. Accounting for the double layer impedance provides a more accurate calculation of the dynamic electric field distribution, and thus a more accurate predictive capability for the low frequency DEP behavior of colloidal particles in nonuniform fields.
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(a)
(b) Fig. 15. The DEP force that a PS 9.6/44 particle experiences near a single strip electrode of width 2l at an applied voltage of 0.51 V in the ⊥ configuration. A lateral position of zero indicates the electrode strip center. Lateral DEP force FDEP_x as a function of normalized lateral distance from a single strip center at elevations of 3, 6, 9, and 12 µm above the surface at (a) 60 Hz and (b) 1 kHz.
References [1] C.L. Asbury, Biophys. J. 74 (1998) 1024–1030. [2] A. Docoslis, N. Kalogerakis, L.A. Behie, K.V.I.S. Kaler, Biotechnol. Bioeng. 54 (1997) 239–250. [3] R. Bashir, Adv. Drug Deliv. Rev. 56 (2004) 1565–1586. [4] P.R.C. Gascoyne, J. Vykoukal, Electrophoresis 23 (2002) 1973–1983. [5] A.C. Richards Grayson, R.S. Shawgo, A.M. Johnson, N.T. Flynn, Y. Li, M.J. Cima, R. Langer, Proc. IEEE 92 (2004) 6–21. [6] P.R.C. Gascoyne, X.-B. Wang, Y. Huang, F.F. Becker, IEEE Trans. Ind. Appl. 33 (1997) 670–678. [7] T.B. Jones, Electromechanics of Particles, Cambridge Univ. Press, New York, 1995. [8] A. Castellanos, A. Ramos, A. Gonzalez, N.G. Green, H. Morgan, J. Appl. Phys. D 36 (2003) 2584–2597.
[9] N.G. Green, H. Morgan, J. Phys. Chem. B 103 (1999) 41–50. [10] N.G. Green, A. Ramos, A. Gonzalez, H. Morgan, A. Castellanos, Phys. Rev. E 61 (4-B) (2000) 4011–4018. [11] A. Gonzalez, A. Ramos, N.G. Green, A. Castellanos, H. Morgan, Phys. Rev. E 61 (2000) 4019–4028. [12] N.G. Green, A. Ramos, A. Gonzalez, H. Morgan, A. Castellanos, Phys. Rev. E 66 (2002) 0263011–0263051. [13] A. Ajdari, Phys. Rev. E 61 (2000) R45–R48. [14] F. Nadal, F. Argoul, P. Kestener, B. Pouligny, C. Ybert, A. Ajdari, Eur. Phys. J. E 9 (4) (2002) 387–399. [15] M.Z. Bazant, T.M. Squires, Phys. Rev. Lett. 92 (2004) 0661011–0661014. [16] X.-B. Wang, Y. Huang, P.R.C. Gascoyne, F.F. Becker, IEEE Trans. Ind. Appl. 33 (1997) 660–669. [17] D.R. Albrecht, R.L. Sah, S.N. Bhatia, Biophys. J. 87 (2004) 2131– 2147.
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[18] D. Lastochkin, R. Zhou, P. Wang, Y. Ben, H.-C. Chang, J. Appl. Phys. 96 (2004) 1730–1733. [19] H. Zhou, M.A. Preston, R.D. Tilton, L.R. White, J. Colloid Interface Sci., in press.
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[20] H. Zhou, L.R. White, R.D. Tilton, J. Colloid Interface Sci. 285 (2005) 179–191. [21] R.J. Hunter, Zeta Potential in Colloid Science, Academic Press, San Diego, 1981.
The role of electrode impedance and electrode geometry in the design of microelectrode systems Hao Zhou a , Robert D. Tilton a,b , Lee R. White a,∗ a Department of Chemical Engineering, Center for Complex Fluids Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA b Department of Biomedical Engineering, Center for Complex Fluids Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Received 11 August 2005; accepted 9 November 2005 Available online 5 December 2005
Abstract Microelectromechanical systems (MEMS) employing spatially and/or temporally nonuniform electric fields have been extensively employed to control the motion of suspended particles or fluid flow. Design and control of microelectromechanical processes require accurate calculations of the electric field distribution under varying electrolyte conditions. Polarization of electrodes under the application of an oscillating voltage difference produces dynamic electrical double layers. The capacitive nature of the double layers significantly inhibits the penetration of the electric field through the double layer and into the surrounding bulk electrolyte at low frequencies. This paper quantitatively discusses the effect of electrode impedance on the electric field distribution as a function of field frequency, electrolyte composition, and electrode zeta potential in microelectrode systems. The design principles for the electrode geometry and configuration are also discussed in terms of their effects on the electric field magnitude and nonuniformity. © 2005 Elsevier Inc. All rights reserved. Keywords: MEMS; Electrode impedance; Microelectrode arrays; AC electric field
1. Introduction Applications of microelectromechanical systems (MEMS) to microfluidic devices, drug delivery devices, biosensors, bioreactors, and cell separation and arraying schemes have been extensively explored in recent years [1–7]. MEMS have the ability to control the positioning of colloidal particles and/or to create electrohydrodynamic flows using AC electric fields under two dominant mechanisms, dielectrophoresis (DEP) and AC electroosmosis (EO) [8–12]. DEP refers to the migration of a polarizable particle in a nonuniform electric field. The dielectrophoretic force on a particle scales as ∇E 2 , causing particles to migrate along gradients of electric field intensity. Particles undergoing positive dielectrophoresis (pDEP) migrate toward regions of greatest electric field intensity, while particles undergoing negative dielectrophoresis (nDEP) migrate toward regions of lowest field intensity [7]. AC EO flow arises from the * Corresponding author.
E-mail address: [email protected] (L.R. White). 0021-9797/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.11.024
interaction of the nonuniform electric field with the free charge generated in the diffuse double layer above the electrodes in a microelectrode array [8–15]. As with conventional electroosmosis, the electromotive force on the ions is transferred through viscous interactions to a fluid flow. Because both the sign of the free charge and the electric field direction reverse when the electrode polarity reverses in an AC circuit, the direction of the electromotive force and resulting fluid flow is unchanged [8–15]. This fluid flow results in a viscous drag on any particle residing near the microelectrode. In order to calculate either the DEP force or the AC EO drag force on a particle in the vicinity of a microelectrode in a MEMS device, it is necessary to first accurately calculate the electric field distribution. Such is the purpose of this paper. To date the calculation of the electric field distribution in a microelectrode system is limited by the use of approximate boundary conditions or consideration of only specific geometries [12–18]. The effect of electrode geometry on microelectrode systems has been discussed qualitatively in the literature [13–15]. In this paper, we calculate the electric field distribution using electrode double layer boundary conditions at the bound-
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ing electrode and dielectric surfaces, which allows us to obtain the electric field distribution for the full field frequency range in any specified geometries. When an oscillating voltage difference is applied across the electrolyte domain, the electrodes are polarized, forming dynamic electrical double layers. At low frequencies (1 MHz), the capacitive nature of the electric double layers significantly inhibits the penetration of the electric field through the double layer and into the surrounding bulk electrolyte. Therefore, the calculation of the electric field distribution above patterned electrodes depends strongly on the double layer impedance (Zel ). The electrode double layer impedance, Zel , used in this paper is that of an ideally polarizable plane calculated using the standard model of colloidal electrokinetics [19]. This approach incorporates the classic retardation and relaxation effects on the double layer but excludes the case where ion current penetrates the electrode surface via electrochemical reactions. The double layer impedance is a function of the zeta potential on the electrode surface, the frequency of the applied voltage and the electrolyte concentration, valency and limiting conductance. The electric field distribution within a MEMS microelectrode array is discussed as a function of field frequency, electrode surface zeta potential, and electrolyte composition. The effect of the electrode geometry on the field distribution is also discussed and compared between two common microelectrode configurations ( and ⊥), as described below. In this paper, symbol represents an electrode array in which voltage is applied between adjacent co-planar parallel electrode strips, shown in Fig. 1a, in order to produce highly nonuniform, high intensity fields. This microelectrode configuration is the most common MEMS design for DEP processes. The characteristic separation width hw for this configuration
(a)
(b) Fig. 1. (a) Schematic view of the microelectrode system. The top plate is a dielectric substrate, the bottom plate is a dielectric substrate patterned with parallel microelectrode strips. The electric potential is applied co-planarly to adjacent parallel strips. The characteristic separation width (hw ) is defined using the dielectric spacing between the edges of the electrode strips. (b) Schematic view of the ⊥ microelectrode system. The top plate is a uniform planar electrode. The bottom plate is a dielectric substrate patterned with parallel microelectrode strips. The electric voltage is applied normally to the top and bottom electrode. The characteristic separation width (hw ) for this ⊥ microelectrode system is defined using the normal distance between the connected top and bottom electrodes.
refers to the spacing (the dielectric strip width) between the edges of two adjacent strips. Rather than using the typical design in our previous experimental work, we employed a vertical configuration of the electrodes, denoted ⊥ as shown in Fig. 1b [20]. The electric field was applied between a uniform top electrode and a bottom, multi-strip electrode array. The characteristic separation width hw for this microelectrode configuration refers to the normal separation distance between the top and bottom electrodes. It has been demonstrated experimentally that the lateral positioning of colloidal particles and cells strongly depends on the frequency and strength of the applied electric field [20]. In that work an approximate electric field model was used to calculate the high frequency electric field distribution. In this paper, we reexamine the electric field distribution and the lateral DEP force on colloidal particles in this ⊥ microelectrode system, by including the Zel effect over the full frequency range. The particle patterning mechanisms in a ⊥ microelectrode system, originally proposed in Ref. [20], are further elaborated here. By comparing the effects of electrode geometry on the electric field distribution in the two microelectrode systems ( and ⊥), desirable design features are suggested for various DEP or microfluidic application purposes. The results are necessary for a quantitative understanding and control of microelectromechanical processes. 2. Theoretical modeling The electric field distribution in microelectrode systems and ⊥ are calculated using the models shown in Fig. 2. The bounding surfaces of the model domain consist of two half electrode strip surfaces (L and R), dielectric surface (D), two symmetry planes S, and a top bounding surface (T). In the configuration, T is a dielectric surface, shown in Fig. 2a. In the ⊥ configuration, T is a uniform electrode surface as seen in Fig. 2b. The characteristic separation width, hw , is defined by the separation distance between the two connected electrodes. Hence hw = lD is the gap width between two adjacent electrode strips in the configuration, and hw is an arbitrary electrode cell thickness in the ⊥ configuration. In order to be consistent with the electrode dimensions in our previous experimental work [20], all dimensions in this paper are scaled by the half electrode strip width l (l = 15 µm). The geometric dimensions of each bounding surface are lL = lR = l = 15 µm, lD = 8l = 120 µm, and lT = lL + lD + lR = 10l = 150 µm; the electrolyte is 0.1 mM NaHCO3 ; the electrode surface zeta potential, ζ , is −80 mV; and the dielectric surface zeta potential is −100 mV, unless otherwise stated. The electric field E exp(−iωt) above the electrode array is obtained by solving ∇ 2 Φ = 0,
(1)
E(x, y) = −∇Φ(x, y)
(2)
in the bulk electrolyte domain together with appropriate boundary conditions on the bounding surfaces.
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iting conductance Λ∞ j by λj =
NA e2 |zj | , Λ∞ j
(4)
where NA is the Avogadro number. The boundary condition on the outer solution Φ(x, y) at an electrode surface is then Φ(x, Y − ) − Φ(x, Y + ) = −K ∞ (ω)Zel (ω)nˆ · E(x, Y + ), where
Φ(x, Y − )
(5)
is the potential of the electrode surface.
(a)
2.2. Boundary conditions on the dielectric surface At the dielectric boundary, y = Y , the current density is just the displacement current (no ion penetration or electrode electrochemistry) and we should have iωε< ε0 nˆ · E(x, Y − ) = −K ∞ (ω)nˆ · E(x, Y + ),
(b) Fig. 2. Electric field calculation model used for the microelectrode systems. (a) Calculation model for configuration, shown in Fig. 1a. Electric potentials are applied to two parallel electrode strip surfaces, left (L) and right (R), which are positioned at y = 0. The lengths lL and lR are the half strip widths of L and R electrode strips, respectively. S stands for the symmetry plane. The characteristic separation width between two connected electrode strips, hw , equals the dielectric strip width lD . The top dielectric surface is positioned at y = 1 mm. (b) Calculation model for ⊥ configuration, shown in Fig. 1b. Electric potentials are applied normally to the electrode strip surfaces, T, L, and R. The L and R electrode strips are positioned at y = 0. The lengths lL and lR are the half strip widths of L and R electrode strips, respectively. S strands for the symmetry plane. The edges of the strips L and R are separated by a dielectric surface of length lD . The length of the top electrode lT equals lL + lR + lD . The characteristic separation width between two connected electrode strips, hw , equals the microelectrode cell thickness.
2.1. Boundary conditions on the electrode surfaces
K ∞ (ω) = e2
N n∞ z ∞ j j j =1
λj
− iωεε0 ,
where ε< is the dielectric constant of the dielectric substrate. Equation (5) also applies at the dielectric boundary but Φ(x, Y − ) is not specified. In principle, Φ(x, Y − ) and nˆ · E(x, Y − ) are obtained from solving Eq. (1) in the dielectric and matching to the outer solution in the electrolyte using (5) and (6). However, the left-hand side of (6) is negligible up to frequencies ωc ∼ κ 2 kT
N βj j =1
(7)
λj
(ωc = 1.4 × 106 for 0.1 mM NaHCO3 ), where zj2 n∞ j βj = N , 2 ∞ z j =1 j nj
N
βj = 1,
(3)
and ε, ε0 , e, zj , nj , λj are the electrolyte dielectric constant, vacuum permittivity, the electron charge, and the valency, number density, and ionic drag coefficient of the j th ion type, respectively. The drag coefficient λj is related to the ionic lim-
(8)
j =1
and these dielectric boundary conditions for ω ωc can be replaced by nˆ · E(x, Y + ) = 0
On an electrode at y = Y (Y = 0 or Y = hw ), where voltage is applied, Φ(x, Y − ) − Φ(x, Y + ) is the potential drop across the double layer at position x on the surface, and is equal to the double layer impedance Zel (ω) times the normal current density evaluated just outside the charge-bearing region as −K ∞ (ω)nˆ · E(x, Y + ), where nˆ is the surface normal pointing into the electrolyte. Here, K ∞ (ω) is the bulk electrolyte conductivity,
(6)
(9)
with negligible error. For the systems, the dielectric boundary conditions (5) and (6) are applied to bounding dielectric surfaces D and T. For the ⊥ systems, the dielectric boundary conditions (5) and (6) are applied to the bounding dielectric surface D only. 2.3. Boundary conditions on surfaces of symmetry On the bounding surfaces of symmetry S, the appropriate boundary condition is nˆ · E(x, y)|S = 0
(10)
The magnitude of the outer electrical potential and electric field distribution is calculated by using the commercial finite element software package FEMLAB (The Comsol Group).
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3. Results and discussion 3.1. Electrode impedance effects in the ⊥ configuration When the top plate electrode voltage is V exp(−iωt) and the bottom electrode strip electrodes are earthed, the electric field E can be written as E(x, y) = V E0 (x, y) exp −i(ωt + ϕ) , (11) where E0 (x, y) exp(−iϕ) = −∇Φ(x, y)
(12)
and ϕ is the phase angle between the electric field and the applied potential due to the complex electrode impedance Zel . Equation (1) is solved for Φ(x, y) with boundary conditions (5), (6), and (10) at appropriate surfaces. For the ⊥ configuration, the inner boundary potentials are Φ(x, h− w )|T = 1, −
(13) −
Φ(x, 0 )|L = Φ(x, 0 )|R = 0.
(14)
A suitable scale for E0 in this geometry is 1/ hw and we will use this scaling where convenient in the plots that follow. Unless otherwise stated, the electric field distribution curves are plotted for y = 3 µm from the bottom electrode surfaces. 3.1.1. Frequency variation of the electric field distribution In Fig. 3, the scaled electric field intensity distribution is plotted as a function of x at various frequencies in the ⊥ microelectrode system (hw = 66.7l = 1 mm). We note that the Zel = 0 result is almost identical to the 20 kHz results, but is quite different from the lower frequency plots where electrode impedance is an important effect. The capacitive nature of the electrode double layer at low frequencies hinders the penetration of the electric fields into the electrolyte region. When the frequency increases, the electrode double layer impedance decreases [19], which leads to an increase in electric field intensity. Zero electrode impedance has been used previously to
Fig. 3. The electric field intensity distribution calculated at y = 0.2l = 3 µm from the electrode surface without electrode impedance (Zel = 0) and with the electrode double layer impedance at 5, 10, 20, 60, 300 Hz, and 20 kHz in the ⊥ configuration (hw = 66.7l = 1 mm). The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
calculate approximately the electric field intensity in a microelectrode system [20]. From the results shown in Fig. 3, it can be seen that this zero electrode impedance approximation only generates valid results at frequencies higher than 20 kHz. The electric field intensity is vanishingly small in the center of the dielectric surface D, where the global electric field minimum can be found. Local minima occur over the electrode strip centers for all frequencies examined above 5 Hz. The electric field maxima are located just inside the electrode strip edges when the frequency is lower than 300 Hz, and are located at the electrode strip edges at high frequency (20 kHz, y = 3 µm). It can be seen that the overall electric field intensity increases as the frequency is increased from 5 to 60 Hz and Zel decreases. From 300 Hz to 20 kHz the electric field intensity increase is observed at the electrode strip edges and above the dielectric surfaces. The electric field intensity in the electrode strip centers, however, decreases when the frequency increases, and a stronger electric field gradient is generated on the electrode strips. Thus for a pure DEP process, high frequency should generate stronger traps and sharper patterning adjacent to both the dielectric surfaces and the electrode surfaces due to the high electric field intensity and high electric field gradient. 3.1.2. Electric field distribution above the electrode surface The scaled electric field distribution at various heights (y) above the bottom electrode surface is plotted in Fig. 4 for the ⊥ configured system (hw = 66.7l = 1 mm) at a frequency of 20 Hz. Note that the electric field decreases as the normal distance from the bottom electrode increases. The field maxima shift from the electrode strip edges toward the electrode strip centers as the normal distance increases, which predicts a strong correlation between the lateral DEP positioning and the vertical elevation in a colloidal patterning process. Correlation between the vertical elevation and lateral colloidal positioning has been observed, although the effects of AC EO flows become important in this frequency range [20]. Also from this result, it can be seen that the initial elevation of colloidal particles can be important to a patterning/separation process. If the electric poten-
Fig. 4. The electric field intensity distribution calculated at various heights, y = 0.1l, 0.2l, 0.4l, 0.6l, 0.8l, and 1l, above the electrode surface at 20 Hz in the ⊥ configuration (hw = 66.7l = 1 mm). The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
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tial is applied while nDEP colloidal particles are sedimenting onto the bottom surface, the particles can be easily collected on the dielectric surface where the global minimum resides. As the particles sediment, they would not likely be trapped in the local field minimum located at low elevations over the electrode strip center, since there is no local minimum there at high elevations. This field distribution would guide sedimenting particles around the electrode strips to finally locate adjacent to the dielectric region. If instead the particles sedimented in the absence of an electric field, then some fraction of nDEP particles would likely be trapped in the local field minima above the electrode strip centers. The relative number of nDEP particles trapped along electrode centers or the dielectric region would depend on the overall field strength and the particle buoyancy. In a DEP process to separate nDEP and pDEP colloids, application of the electric field during sedimentation may generate more efficient separation. At low frequencies, where the EO effect may be significant, the positioning of the colloidal particles can be very sensitive to the particle polarizability, particle size, and particle mass density due to the strong correlation between ∇E20 and the vertical elevation [20]. 3.1.3. Electrode zeta potential effect on the electric field distribution The electrode double layer impedance is a function of electrode zeta potential, ζ , the electrostatic potential at the start of the diffuse double layer for the resting electrode. In Fig. 5, the electrode ζ potential effect on the electric field distribution is illustrated for various frequencies and ζ values in the ⊥ configured system (h = 66.7l = 1 mm). In Fig. 5a, the right electrode zeta potential is systematically changed from −80 to −140 mV, where the left electrode and the top electrode remain at −80 mV. The frequency is set at 5 Hz, where electrode impedance is significant. It can be seen that the electric field intensity above the right electrode strip increases dramatically when the magnitude of its zeta potential increases. The electric field intensity above the left electrode, however, decreases moderately as the right electrode zeta potential increases. The double layer capacitance (Cel ) increases with |ζ | and the electrode impedance (dominated by the capacitive contribution 1/ωCel at 5 Hz) decreases. Thus, less potential drop occurs across the double layer, resulting in an increased outer electric field strength (see Eq. (5)). When the right electrode impedance drops, its share of the total current carried increases. Consequently, the current and hence outer electric field at the left electrode decreases. Therefore the electric field intensity difference between the left and right electrode strips can be controlled by the zeta potentials of the electrode strips. In Fig. 5b we show the effect of changing frequency in a ⊥ configuration, where the right electrode zeta potential is ζR = −100 mV, and the top and left electrode zeta potentials are ζT = ζL = −80 mV. At 20 kHz, electrode impedance is negligible and the field distributions on the left and right electrodes are identical. For lower frequencies, as the electrode impedance on all electrodes increases, the asymmetry of the left and right electrodes becomes apparent. Between 20 and 60 Hz, the electric field intensities on both left and right
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(a)
(b)
(c) Fig. 5. The electric field intensity distribution calculated at y = 0.2l = 3 µm above the electrode surface for various electrode zeta potentials in the ⊥ configuration (hw = 66.7l = 1 mm). The electrolyte is 0.1 mM NaHCO3 . (a) ζT = ζL = −80 mV, ζR = −80, −100, −120, and −140 mV at 5 Hz. (b) ζT = ζL = −80 mV, ζR = −100 mV at 5, 10, 20, 60, 300 Hz, and 20 kHz. (c) ζT = ζL = ζR = −80 mV; ζT = ζL = −80 mV, ζR = −140 mV; ζT = −80 mV, ζL = ζR = −140 mV; ζT = −140 mV, ζL = ζR = −80 mV; ζT = ζL = −140 mV, ζR = −80 mV; and ζT = ζL = ζR = −140 mV at 5 Hz.
strips increases with frequency but the field intensity asymmetry between the strips stays relatively constant. From 60 to 300 Hz, the electric field intensity on the right strip de-
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creases slightly and the field intensity on the left electrode increases. In Fig. 5c, the electric field intensity distribution for more complicated electrode zeta potential combinations is displayed. When the electrode zeta potentials are identical, ζT = ζL = ζR , increase of |ζ | decreases the electrode double layer impedance [19] and, hence, increases the outer electric field intensity. Increasing |ζT | increases the electric field intensity on both L and R electrode strips. Increasing either |ζL | or |ζR | increases the electric field intensity on that electrode strip. By exploiting different zeta potentials on the T, L, and R electrode surfaces, more complicated electric field intensity distributions (magnitude and gradient) can be achieved. It can be seen that the electrode zeta potential effect plays a significant role on the electric field distribution in low frequency DEP/EO processes. The zeta potential asymmetry between the electrode strips creates field maxima and minima of varying strength in different regions of the microelectrode system. This impacts directly on the lateral positioning of the colloidal particles. In practice, the zeta potential of the electrode strip surface can be varied by modifying the electrode surface, perhaps by adsorption of polymers or polyelectrolytes, or by chemisorption of self-assembled monolayers. Conversely, contamination of the electrode strips will also alter the electrode ζ potential and may significantly skew the behavior of a MEMS process when operated at the moderate frequencies where the zeta potential effect is strongest. 3.1.4. Electrolyte concentration effect on the electric field distribution In Figs. 6a and 6b, the electrolyte concentration effect is plotted for 5 and 300 Hz for the ⊥ configuration (h = 66.7l = 1 mm). In these calculations, the electrode zeta potentials ζT = ζL = ζR = −80 mV, although in practice the ζ potential changes with increasing electrolyte concentration [21]. It can be seen that the electrolyte concentration effect is more pronounced at low frequencies where its effect on the electrode impedance is significant. At low frequencies, smaller electrolyte concentrations generate stronger field intensities. At moderate frequency (∼300 Hz), low concentration electrolyte serves better in a DEP process where stronger field gradients adjacent to the electrode are achieved. 3.1.5. Electrolyte ion type effect on the electric field distribution The electrolyte ion transport properties affect the electric field intensity distribution, as illustrated in Figs. 7a and 7b, for 5 and 300 Hz in the ⊥ configuration (h = 66.7l = 1 mm). As expected, the electrolyte ion type effect is strongest at lower frequencies. The ionic limiting conductivities used in this paper are listed in Table 1. Variation in the electrolyte ion type changes the impedance of the electrode double layer and the bulk electrolyte impedance simultaneously. At low frequencies, decreasing the ion limiting conductivities has a very small effect on the electrode double layer impedance [19], but it decreases the bulk electrolyte conductance K ∞ (see Eq. (3)).
(a)
(b) Fig. 6. The electric field intensity distribution calculated at y = 0.2l = 3 µm above the electrode surface for 0.1, 1, and 10 mM NaHCO3 at (a) 5 Hz and (b) 300 Hz in the ⊥ configuration (hw = 66.7l = 1 mm). The electrode zeta potentials are ζT = ζL = ζR = −80 mV.
We note from Eq. (5) that the electrode boundary condition involves the product of Zel and K ∞ , and the effect of decreasing ion conductance is to decrease the effect of electrode impedance, with a consequent increase in the outer electric field. Hence in Fig. 7a, the least conducting salt NaHCO3 produces the smallest electrode impedance effect and the largest outer fields. The most conducting salt HCl produces the smallest fields. The electrode impedance at 300 Hz is small, and we see in Fig. 7b that the effect of ion type is accordingly weak. 3.2. The effect of electrode geometry 3.2.1. Characteristic length h effect on the field distribution In Figs. 8a and 8b, we plot E02 at y = 3 µm as a function of x for 5 and 300 Hz for variations in the normal electrode spacing hw in the ⊥ configuration. At 300 Hz (Fig. 8b) we see that increasing the electrode spacing decreases the field in an unsurprising way. At this frequency, the effects of electrode impedance are small. In contrast at 5 Hz, where the electrode impedance is large due to the capacitive nature of the double
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Fig. 7. The electric field intensity distribution calculated at y = 0.2l = 3 µm from the electrode surface for 0.1 mM NaHCO3 , NaCl, KCl, and HCl at (a) 5 Hz and (b) 300 Hz in the ⊥ configuration (hw = 66.7l = 1 mm). The electrode zeta potentials are ζT = ζL = ζR = −80 mV.
Fig. 8. The electric field intensity distributions calculated at y = 0.2l = 3 µm above the electrode surface for characteristic separation width hw = 6.67l = 0.1 mm, hw = 33.33l = 0.5 mm, and hw = 66.67l = 1 mm at (a) 5 Hz and (b) 300 Hz in the ⊥ configuration. The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
Table 1 Parameters used in the calculations in this paper Temperature Medium viscosity Medium density Dielectric constant of medium Limiting conductance of Na+ Limiting conductance of H+ Limiting conductance of K+ Limiting conductance of Cl− Limiting conductance of HCO− 3
T = 298 K η = 0.8904 × 10−3 kg/(m s) ρ0 = 997.04 kg/m3 ε = 78.54688 Λ∞ = 50.11 × 10−4 m2 / equiv Λ∞ = 349.82 × 10−4 m2 / equiv Λ∞ = 73.5 × 10−4 m2 / equiv Λ∞ = 76.35 × 10−4 m2 / equiv Λ∞ = 41.5 × 10−4 m2 / equiv
layer, we observed a surprisingly weak sensitivity of the field intensity to the normal electrode spacing (Fig. 8a). This somewhat counterintuitive result can be rationalized as follows. Near the bottom electrodes the transverse fields are determined mainly by the electrode strip geometry and are consequently relatively insensitive to normal spacing hw . However, our numerical studies show that the normal outer electric field is of a comparable magnitude. Thus the insensitivity of nˆ · E must be explained to understand these results. The outer normal field is determined by the outer potential difference between top and
bottom electrodes divided by the spacing hw . The outer potential difference is the voltage difference applied at the electrodes minus the potential drops across the double layers on top and bottom electrodes. These potential drops are (from Eq. (5)) a product of the electrode impedance and the outer normal field. Thus nˆ · E ∼
T + Z B )n V − K ∞ (ω)(Zel el ˆ · E hw
(15)
and, rearranging, nˆ · E ∼
V . T + ZB ) hw + K(ω)(Zel el
(16)
When electrode impedance is large (at 5 Hz), nˆ · E is insensitive to hw . When electrode impedance is small (at 300 Hz), nˆ · E varies inversely with hw . 3.2.2. ⊥ and configuration comparison For the configured systems, the left and right electrode strips are connected to the AC power source. The boundary condition (5) applies on electrodes L and R, and the inner boundary
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(c) Fig. 9. (a) The electric field intensity distributions, (b) lateral field intensity, and (c) normal field intensity calculated at y = 0.2l = 3 µm above the electrode surface in the ⊥ configuration and the configuration with characteristic interelectrode spacing width hw = 8l = 120 µm at 300 Hz in the ⊥ configuration. The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
potentials on L and R are just the applied electric potentials, Φ(x, 0− )|L = 0, −
Φ(x, 0 )|R = 1.
(17) (18)
In Fig. 9a, the electric field distribution is compared between ⊥ and configurations at 300 Hz (hw = 120 µm for both configurations). It can be seen from Fig. 9a that the ⊥ configuration generates stronger electric field intensity than the configuration when the same voltage is applied. In Figs. 9b and 9c, we separate the normal and transverse field components. The configuration produces stronger electric field intensity on the dielectric surface D due to its strong lateral electric field component (see Fig. 9b). The broader minimum in E02 can result in DEP positioning on the dielectric surface in the configuration being less well defined than in the ⊥ configuration. 3.2.3. Voltage configuration effect on the field distribution By varying the voltages applied at the electrodes, a variety of electric field distributions can be generated. In Fig. 10, the electric field distribution is plotted for a number of different voltage applications in a ⊥ configured system. Strong electric field bias on one bottom electrode is obtained by applying the
Fig. 10. The electric field intensity distributions calculated at y = 0.2l = 3 µm above the electrode surface for various voltage configurations, ΦL = 0, ΦR = 1 in the configuration (where ΦT is not applicable as there is no top electrode); ΦT = ΦL = 0, ΦR = 1; ΦT = 1, ΦL = ΦR = 0; and ΦT = ΦR = 1, ΦL = 0 in the ⊥ configuration. All at 300 Hz. The electrode zeta potentials are ζT = ζL = ζR = −80 mV and hw = 8l = 120 µm. The electrolyte is 0.1 mM NaHCO3 .
same voltage to the top electrode and the other strip. By varying the applied voltage at the top electrode, high electric field
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(b) Fig. 11. The electric field intensity distributions calculated at y = 0.2l = 3 µm above the electrode surface for various dielectric spacings lD = l, 2l, 4l, 6l, and 8l between two electrode strips (a) in the ⊥ configuration (hw = 66.7l = 1 mm) and (b) in the configuration; all at 60 Hz. The electrode zeta potentials are ζT = ζL = ζR = −80 mV and the electrode widths are fixed at lL = lR = l. The electrolyte is 0.1 mM NaHCO3 .
bias can be easily switched from left to right electrode. Such strongly biased switchable electric field intensities may have interesting novel MEMS applications. 3.2.4. Dielectric strip width effect on the field distribution In Figs. 11a and 11b, the dielectric strip width (the spacing between adjacent electrode strips) is systematically varied for microelectrode system in either the ⊥ or configuration. In a ⊥ configured system, the global minimum is always at the center of the dielectric strip. Increasing the dielectric strip width increases both the electric field intensity and the electric field gradient on the electrode strips. In contrast, in a configured system, increasing the dielectric strip width decreases the overall electric field intensity and gradient. For DEP processes, where both strong field intensity and field gradient are preferable, a larger separation between the electrode strips in the ⊥ configuration is more efficient. For the configured microelectrode system, however, small interelectrode strip separations are preferable.
(b) Fig. 12. The electric field intensity distributions calculated at y = 0.2l = 3 µm above the electrode surface for various electrode strip widths lL = lR = l, 2l, 3l, and 4l and fixed dielectric spacing lD = 2l (a) in the ⊥ configuration (hw = 66.7l = 1 mm) and (b) in the configuration; all at 60 Hz. The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
3.2.5. Electrode strip width effect on the electric field distribution In Figs. 12a and 12b, the electrode strip width is systematically varied for ⊥ and microelectrode systems. Decreasing the electrode strip width increases both the electric field intensity and the electric field gradient on the dielectric surfaces for the ⊥ configuration (Fig. 12a). For an efficient DEP process, smaller electrode strip widths are more appropriate in the ⊥ configuration. By contrast again, for a configured microelectrode system (Fig. 12b), wider electrode strips significantly increase the electric field intensity at fixed dielectric strip spacing. Note that the global electric field intensity minimum switches from the dielectric center to the electrode strip center as the electrode strip width increases. For the configuration, a wider electrode strip will achieve more precise DEP patterning results on both dielectric and electrode strip surfaces.
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Fig. 13. The electric field intensity distributions calculated at y = 0.2l = 3 µm above the electrode surface for various left electrode strip widths lL = l, 2l, 3l, 4l, 5l, 6l, 7l, and 8l with fixed lR = l and lT = 10l (a) in the ⊥ configuration (hw = 66.7l = 1 mm) at 5 Hz, (b) in the configuration at 5 Hz, (c) in the ⊥ configuration (hw = 66.7l = 1 mm) at 60 Hz, and (d) in the configuration at 60 Hz. The electrode zeta potentials are ζT = ζL = ζR = −80 mV. The electrolyte is 0.1 mM NaHCO3 .
3.2.6. Electrode/dielectric strip width ratio effect on the electric field distribution When the dimension of the microelectrode cell is fixed (i.e., constant working area), varying one of the electrode strip widths would simultaneously vary the dielectric strip spacing. This variation affects the electric field distribution, as shown in Figs. 13a and 13b (for 5 Hz) and Figs. 13c and 13d (for 60 Hz). The total electrode center-to-center spacing and the right electrode width are fixed at lT = lL + lD + lR = 10l and lR = l, respectively. For the ⊥ configuration, increasing the left electrode strip width decreases the electric field intensity in the centers of both electrode strips, as shown in Figs. 13a and 13c. The global minimum is always in the center of the dielectric strips as shown previously. In the 5 Hz case (Fig. 13a), there exists an optimum electrode strip and dielectric strip spacing combination where the strongest local maximum electric field on the left strip is obtained. For the configuration at 5 Hz, however, a systematic shifting of the local field maximum on the left strip toward the right electrode and a strong electric field intensity bias on the right electric strip is observed when the left electrode width increases. There exists a combination of the left
electrode strip width and dielectric strip width where the local maximum on the left electrode strip width reaches a minimum value (Fig. 13b). A merging of the electric field peak on the left electrode into the electric field peak on the right electrode eventually occurs when increasing the left electrode strip width. At 60 Hz, the optimal spacing observed in the ⊥ configuration at 5 Hz does not occur (Fig. 13c). The configuration at 60 Hz (Fig. 13d) is qualitatively similar to the 5 Hz case. From Fig. 13, it can be seen that the configuration can be used to generate a strong electric field bias on one electrode strip by increasing the width of the other electrode strip. 3.3. Experimental patterning of polystyrene (PS) colloids in ⊥ configuration In our previous publication [20], we systematically examined the patterning behavior of a variety of PS colloids in 0.1 mM NaHCO3 over the frequency range from 60 Hz to 1 MHz in the ⊥ configuration. Here we discuss observations for the 9.6 µm diameter particles with −44 mV zeta potential, hereafter referred to as PS 9.6/44. The trends are representative of the other PS colloids we examined as well. The key
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In this paper, we reexamine the lateral dielectrophoretic force calculation for PS 9.6/44 particles using the full solution that accounts for the electrode double layer. The results are shown in Fig. 15. Note that the direction and the strength of the lateral DEP force that a colloidal particle experiences above the electrode strip surface is strongly dependent on the vertical elevation and the field frequency. At 60 Hz (Fig. 15a), the lateral DEP force on PS 9.6/44 is inward toward the electrode strip center at low elevations (y 3 µm). At y = 6 and 9 µm, a distance comparable to the particle sizes used in this thesis research, however, FDEP_x for PS 9.6/44 points outward toward the strip edges. This indicates that particles at higher elevations can experience a lateral DEP force that acts in the opposite direction as the force on particles at low elevation, or particles with smaller sizes. At higher frequencies, 1 kHz, the direction of the lateral DEP force remains inward at y = 3 µm. At y = 6 µm, however, the lateral DEP force remains inward, unlike what occurs at 60 Hz. The lateral DEP force points outward toward the edges only at y = 9 µm or higher. The lateral DEP force is less sensitive to particle elevation at higher frequencies. These calculations further demonstrate the importance of the particle elevation for the lateral positioning of colloidal particles in nonuniform fields at low frequencies (<1 kHz). Not only is the direction of the lateral AC EO drag force dependent on the vertical elevation, but the direction of the lateral DEP force acting on the colloidal particle is also sensitive to the particle elevation. This effect on the DEP force was not apparent in the zero impedance approximation. Both the DEP force and the AC EO drag force influence the vertical elevation and therefore the preferred lateral positioning. 4. Summary
(c) Fig. 14. Representative patterning results for PS 9.6/44 colloids obtained after 10 min of applying an electric field. (a) 60 Hz, 0.17 V. (b) 60 Hz, 0.51 V. (c) 1 kHz, where 0.17 or 0.51 V produced similar results.
observation was that these particles were patterned at 60 Hz along the electrode edges for a 0.51 V applied voltage, but in the electrode centers for 0.17 V. See Figs. 14a and 14b. While the particle patterning behavior was sensitive to the field strength at 60 Hz, at 1 kHz, however, particles were only patterned in the electrode centers at both voltages. See Fig. 14c. Using the electric field distribution calculated with zero double layer impedance [20], we estimated the dielectrophoretic forces exerted on a PS 9.6/44 particle, and we proposed that the patterning mechanism at low frequency was attributed to a coupling between dielectrophoretic forces and AC electroosmotic drag forces on particles near the microelectrode strips and the relative direction of these two types of forces depended on the particle elevation [20]. As discussed above, the zero impedance approximation is only strictly valid for high frequencies.
This study quantitatively showed the effect of electrode impedance on the electric field distribution as a function of field frequency, electrolyte composition, and electrode zeta potential in microelectrode systems. The design principles for the electrode geometry and configuration are also discussed in terms of the effect on desired electric field properties—magnitude and nonuniformity. At low frequencies, the capacitive electrode impedance weakens the electric field intensity and its nonuniformity adjacent to microelectrode surfaces. This offers the possibility of controlling the electric field intensity profile by engineering the electrode zeta potential and tuning the electric field frequency. Field asymmetry can be generated and controlled by varying the electrode zeta potential, electrode/dielectric strip width, voltage configuration and field frequency in the low frequency range, suggesting possible novel MEMS strategies. The ability to create a desired electric field nonuniformity, magnitude and asymmetry provides opportunities to precisely control the motions of particles or fluid flows in MEMS processes. Accounting for the double layer impedance provides a more accurate calculation of the dynamic electric field distribution, and thus a more accurate predictive capability for the low frequency DEP behavior of colloidal particles in nonuniform fields.
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(b) Fig. 15. The DEP force that a PS 9.6/44 particle experiences near a single strip electrode of width 2l at an applied voltage of 0.51 V in the ⊥ configuration. A lateral position of zero indicates the electrode strip center. Lateral DEP force FDEP_x as a function of normalized lateral distance from a single strip center at elevations of 3, 6, 9, and 12 µm above the surface at (a) 60 Hz and (b) 1 kHz.
References [1] C.L. Asbury, Biophys. J. 74 (1998) 1024–1030. [2] A. Docoslis, N. Kalogerakis, L.A. Behie, K.V.I.S. Kaler, Biotechnol. Bioeng. 54 (1997) 239–250. [3] R. Bashir, Adv. Drug Deliv. Rev. 56 (2004) 1565–1586. [4] P.R.C. Gascoyne, J. Vykoukal, Electrophoresis 23 (2002) 1973–1983. [5] A.C. Richards Grayson, R.S. Shawgo, A.M. Johnson, N.T. Flynn, Y. Li, M.J. Cima, R. Langer, Proc. IEEE 92 (2004) 6–21. [6] P.R.C. Gascoyne, X.-B. Wang, Y. Huang, F.F. Becker, IEEE Trans. Ind. Appl. 33 (1997) 670–678. [7] T.B. Jones, Electromechanics of Particles, Cambridge Univ. Press, New York, 1995. [8] A. Castellanos, A. Ramos, A. Gonzalez, N.G. Green, H. Morgan, J. Appl. Phys. D 36 (2003) 2584–2597.
[9] N.G. Green, H. Morgan, J. Phys. Chem. B 103 (1999) 41–50. [10] N.G. Green, A. Ramos, A. Gonzalez, H. Morgan, A. Castellanos, Phys. Rev. E 61 (4-B) (2000) 4011–4018. [11] A. Gonzalez, A. Ramos, N.G. Green, A. Castellanos, H. Morgan, Phys. Rev. E 61 (2000) 4019–4028. [12] N.G. Green, A. Ramos, A. Gonzalez, H. Morgan, A. Castellanos, Phys. Rev. E 66 (2002) 0263011–0263051. [13] A. Ajdari, Phys. Rev. E 61 (2000) R45–R48. [14] F. Nadal, F. Argoul, P. Kestener, B. Pouligny, C. Ybert, A. Ajdari, Eur. Phys. J. E 9 (4) (2002) 387–399. [15] M.Z. Bazant, T.M. Squires, Phys. Rev. Lett. 92 (2004) 0661011–0661014. [16] X.-B. Wang, Y. Huang, P.R.C. Gascoyne, F.F. Becker, IEEE Trans. Ind. Appl. 33 (1997) 660–669. [17] D.R. Albrecht, R.L. Sah, S.N. Bhatia, Biophys. J. 87 (2004) 2131– 2147.
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[18] D. Lastochkin, R. Zhou, P. Wang, Y. Ben, H.-C. Chang, J. Appl. Phys. 96 (2004) 1730–1733. [19] H. Zhou, M.A. Preston, R.D. Tilton, L.R. White, J. Colloid Interface Sci., in press.
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[20] H. Zhou, L.R. White, R.D. Tilton, J. Colloid Interface Sci. 285 (2005) 179–191. [21] R.J. Hunter, Zeta Potential in Colloid Science, Academic Press, San Diego, 1981.