This chapter is dedicated to analyzing the electrokinetic behavior of a frequency- dependent experimental study involving coplanar parallel microelectrodes. This electrode geometry is used extensively and the analysis thereof will benefit colloidal assembly study on the whole. The understanding of particle force dependence on frequency provides insight into the resulting microstructure.
A. Introduction
Bahukudumbi et al [6] reported the results of their experiments with 800nm diameter gold colloids dispersed in aqueous 0.1 M NaHCO3 (conductivity of 9 S/cm). The experimental arrangement consisted of parallel 50 nm thick gold electrodes separated by 30m on glass microscope slides. The gold colloids were confined within a thin, quasi- two-dimensional layer near the surface due to gravity and particle-surface electrostatic repulsion. The results were interpreted in terms of a competition between electrophoresis (EP), AC electro-osmosis (AC-EO), and dielectrophoresis (DP). Based on equations for the single-particle velocity as a function of frequency () for each of those three phenomena, the authors postulated that EP dominates for 3 0
10 /kHz10 , AC-EO for
0 1
10 /kHz 10 , and DP for 1 4
10 /kHz10 ). The AC-EO domain widened with increasing magnitude of the electrode potential. The experimental colloidal configurations can be seen in Fig. 13. These results consist of a matrix filled with
elements of experimental results where higher elements indicate higher voltages and elements further to the right indicate higher frequencies. Note that the lower frequencies (10 Hz) show particles settling in the gap, middle frequencies (1 kHz) show particles being ejected from the gap, and the higher frequencies (above 100 kHz) show dielectrophoretic chaining.
Fig. 13. Experimental colloidal microstructure [6]
The purpose of the research in this chapter is to further validate those conclusions by using the finite element method to model AC-EO and DP.
B. Problem Formulation
The problem domain, shown in Fig. 14, consists of an aqueous solution on two coplanar parallel electrodes. The solution is enclosed by glass except where the solution and electrodes form an interface. A two-dimensional problem was formulated by modeling the particles as cylinders infinitely extended in the direction normal to the figure.
Similarly, the electrodes were assumed to be infinitely extended in the direction normal to the figure. Note that Fig. 14 is an edge view of the domain, whereas Fig. 13 is a top view of the domain. The absolute value of the potential difference between the two electrodes was fixed at 2.5 V. The dielectric constants of the medium and particle were assigned values of 80 and 1 respectively. The conductivities of the medium and particle were 0.00573 S/m and 45.6*106 S/m respectively.
Fig. 14. 2D simulation domain where particles placed at positions 1,2, and 3
Two cases were modeled: single particles, and a 3x3 square array of particles (Fig. 15). The particle array is included to study the effects of multibody interactions. The distance between particle centers in the 3x3 array is 1.2m in the x and y directions. The three locations in Fig. 14 correspond to the locations of single particles and the center particle of the 3x3 array. The three locations were studied sequentially rather than simultaneously, i.e., no particles were present at locations 2 and 3 when particles were at location 1, etc. The forces on the single particle or central particle, in the case of the array, were computed by integrating the Maxwell and Cauchy stress tensors on the particle surface. The mesh used to obtain the solution varied between 750,000 and 900,000 quadratic Lagrangian triangles.
The formulation of Green et al. for AC-EO was used for both the electrical and fluid problems [15]. In the bulk liquid the electric potential is governed by
(( j ) ) 0
(37)
where , , and are the conductivity, permittivity, and angular frequency. The following analysis assumes that electrolysis does not occur at the electrode surface, i.e., the electrodes are considered to be perfectly polarizable, and the double layer behaves in a linear manner. The boundary condition just outside the double layer on the electrode surface is given by DL q n t (38)
where n denotes the unit normal vector and qDL is the charge per unit area in the double
layer. In this equation it is assumed that tangential currents along the double layer are negligible. If the voltage drop across the diffuse double layer is sufficiently small, there is a linear relationship between the charge and the voltage, i.e., qDLCDLDL, where
DL V
, and the equation can be written with complex amplitudes as (39).
DL DL DL i q i C n (39)
where CDL is the capacitance per unit of area of the total double layer, and V is the potential applied to an electrode, and is the potential on the outer side of the diffuse layer. The capacitance per unit area can be estimated from the Debye-Huckel theory as
DL D
C
(40)
where D is the Debye length which is widely used to estimate the double layer thickness given by (41). 2 2 2 D b kT z e c (41)
where k, T, z, e and c are Boltzmann constant, absolute temperature, valence of ion, b electron charge and concentration of ions, respectively. At the interface between the electrolyte and the glass, the total normal electric displacement must be continuous.
( ) ( ) G G G i i n n (42)
where G,G and G are the electrical permittivity, conductivity and potential in the
glass. Since / / G and the conductivity of the glass is negligible, the boundary condition at the glass interface in the fluid simplifies to
0 n
(43)
The time averaged Maxwell stress tensor is given by (44).
* * 1 1 Re 2 2 E T EE E E I T (44)The fluid motion is caused by electrical body forces that are nonzero only in the diffuse double layer since the charge density in the bulk is zero. These forces result in a rapidly varying velocity profile in the diffuse double layer, changing from zero at the wall to a finite value just outside the double layer. This velocity value can be used as a boundary condition at the electrode surface to calculate the bulk motion. In the thin double layer approximation, for diffuse layers in quasiequilibrium and on a perfectly polarizable metal surface, the slip electro-osmotic velocity is given by the Helmholtz-Smoluchowski formula t DL u x (45)
where is the fluid viscosity. The Helmholtz-Smoluchoswki time averaged AC electroosmosis fluid velocity is given by
* , Re 2 t ACE DL u u x (46)
where * is the complex conjugate of . The bulk fluid is governed by Stokes flow of which the governing equations are given by (47) and (48).
2 0 u p (47) 0 u (48)
where p is the pressure. The mechanical stress tensor is given by
( )
f
T p I u u T (49)
and the traction vector is given by (50).
ˆ
f E
t T T n
(50)
The boundary conditions are (i) the tangential AC electro-osmotic velocity on the electrodes, equation (10); (ii) zero tangential velocity on the glass; (iii) zero normal velocity on every boundary; and (iv) continuous traction vector on all interfaces
0
t (51)
C. Results and Discussion
The present problem was solved, with the numerical results given in Tables 3 and 4, which contain the x and y components of the electrical force,F and ex Fey, the fluid force,
fx
F and Ffy, and the x and y components of the total force obtained by summing the electrical and fluid forces. Fig. 16 displays the typical streamline and potential distribution for the simulations where red indicates high potential and blue indicates low potential. Fig. 17 displays the force vectors for the single particle case on a logarithmic scale.
Fig. 16. Typical streamlines and potential distribution
Table 3. Single particle forces in microelectrode domain
Location Freq (Hz) Fex (N/m) Fey (N/m) Ffx (N/m) Ffy (N/m) Ftotalx (N/m) Ftotaly (N/m)
10 6.35E-15 -1.21E-14 4.11E-12 -8.44E-12 4.12E-12 -8.45E-12 1.00E+03 2.48E-11 -4.95E-11 1.15E-08 -2.36E-08 1.15E-08 -2.36E-08 1.00E+05 1.76E-10 -4.22E-10 1.80E-09 -3.82E-09 1.98E-09 -4.24E-09 10 -5.03E-13 -6.52E-13 2.87E-09 3.98E-10 2.87E-09 3.97E-10 1.00E+03 -2.30E-09 -2.53E-09 6.97E-06 1.19E-06 6.97E-06 1.19E-06 1.00E+05 -4.79E-09 -8.52E-10 5.62E-08 1.22E-08 5.14E-08 1.13E-08 10 -5.06E-15 -5.67E-13 2.84E-10 1.73E-10 2.84E-10 1.72E-10 1.00E+03 3.18E-12 -1.26E-09 3.78E-07 3.80E-07 3.78E-07 3.79E-07 1.00E+05 1.28E-10 -2.23E-10 -1.52E-09 1.17E-09 -1.39E-09 9.47E-10
1
2
Fig. 17. Normalized single particle force vectors as a function of frequency
Table 4. Particle array forces in microelectrode domain
The results in Tables 3 and 4 are supported by the experimental observations of Fig. 13. The fluid forces are largest for the intermediate frequency (1kHz) and smaller for the low (10Hz) and high (100kHz) frequencies.
The fluid force reaches its highest value at the 1 kHz frequency for all particle locations. The electric force is the highest at the 100 kHz frequency for all particle locations. However, the fluid force is greater than the electric force at all frequencies.
Location Freq (Hz) Fex (N/m) Fey (N/m) Ffx (N/m) Ffy (N/m) Ftotalx (N/m) Ftotaly (N/m)
10 2.04E-13 2.25E-13 1.34E-11 -3.18E-11 1.36E-11 -3.16E-11 1.00E+03 8.18E-10 6.73E-10 3.77E-08 -8.92E-08 3.85E-08 -8.85E-08 1.00E+05 5.99E-09 4.79E-11 6.65E-09 -1.46E-08 1.26E-08 -1.46E-08 10 -4.67E-13 -3.04E-13 3.94E-10 7.51E-11 3.94E-10 7.48E-11 1.00E+03 -1.77E-09 -1.48E-09 9.58E-07 2.21E-07 9.56E-07 2.20E-07 1.00E+05 -2.76E-09 -4.19E-09 6.30E-09 -2.54E-09 3.54E-09 -6.73E-09
10 2.96E-15 -3.06E-13 3.58E-11 2.88E-11 3.58E-11 2.85E-11 1.00E+03 1.10E-11 -5.87E-10 4.51E-08 5.81E-08 4.51E-08 5.75E-08 1.00E+05 6.11E-11 -1.15E-09 -1.94E-10 -1.19E-09 -1.33E-10 -2.34E-09
1
2
For particle locations 1 and 2, the force Ftotalxis weakest, strongest, and intermediate for
10 Hz, 1kHz, and 100 kHz, respectively. For particle location 3, the force increases from 10Hz to 1kHz and then reverses direction for 100kHz. At 1kHz, the particles are forced to the right at all frequencies, in agreement with Fig. 13 that shows the particles accumulating on the right half of the electrode.
Fig. 13 demonstrates that at 10Hz the particles accumulate near x=0. Thus, Ftotalx at location 1 should be to the left. In contrast, the current simulation predicts that Ftotalx at location 1 is to the right, but the force is very small, indicating stagnation and particle accumulation. At 100kHz, at particle location 2, Ftotalxis to the right, whereas Ftotalx is to the left at location 3. This is supported by the accumulation of particles at the innermost electrode edge in Fig. 13.
The force on the center particle of the array is reversed, in some cases, with respect to the force on a single particle. For example, at location 1 at 1kHz and 100kHz, Feyis upward for the center particle and downward for the single particle. However Ffy, which is the dominate term, is downward at location 1 in all cases. A more significant difference between the center particle and the single particle results occurs at locations 2 and 3 for the case of 100kHz, in which Ffy , a significant force, is downward for the center particle and upward for the single particle.
Finally, it is noted that the sedimentation force Fsed , which is in the y direction, is equal to: 3 -14 4 ( ) 4.81x10 / 3
sed gold solute
F g R N m (52)
which is negligible compared toFtotaly for all cases considered.
The electric and fluid forces calculated by integrating the Maxwell and Cauchy tensors on the surface of a particle provide insight into the equilibrium configuration of the particles. The results support the conclusions of Bahukudumbi et al [6] concerning their experiments with 800nm diameter gold colloids dispersed in aqueous 0.1mM NaHCO3: for 2.5V magnitude, AC-EO dominates for intermediate frequencies and DP for higher frequencies. The fluid forces calculated from FEA are largest for the intermediate frequency (1kHz) and smaller for the low (10Hz) and high (100kHz) frequencies. The electric forces are largest for the high (100 kHz) frequency. Also, the sedimentation force is much smaller than the electric and fluid forces.
D. Key Insights and Contributions
The simulation results show good agreement with the experimental results. The particles experience larger forces pointing away from the gap at the intermediate frequency (1 kHz) indicating larger AC electroosmotic force at this frequency. The particles start to experience smaller forces pointing away from the gap and even some forces pointing toward the gap indicating a weakening in the AC electroosmotic force at the high (100
kHz) frequency. The electric force also reaches its largest value at the high frequency indicating a transfer from AC electroosmosis dominance to dielectrophoresis dominance. The frequency regimes dominated by AC electroosmosis will form colloidal systems with particles ejected from the gap, while those dominated by dielectrophoresis will form particle chains between the electrodes. This study has demonstrated the simulation method’s ability for capturing colloidal microstructure behavior in AC fields.
CHAPTER IV