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Simulation study of dielectrophoretic particle sorters
Sensors and Actuators B 103 (2004) 331–338
Simulation study of dielectrophoretic particle sorters J.H. Nieuwenhuis∗ , M.J. Vellekoop Institute of IEMW, Vienna University of Technology, Gusshausstrasse 27-29, A-1040 Vienna, Austria Available online 1 June 2004
Abstract In this paper, the performance of five electrode configurations for particle sorters based on negative dielectrophoresis is compared using a novel analysis method. By equating the formulas for dielectrophoresis, drag and buoyancy, the velocity vectors for the particles with respect to the liquid can be obtained. This information is combined with the liquid velocity vectors to calculate three-dimensional particle trajectories. The maximum allowed flow-speed at which the sorters still work properly is the criterion applied to compare their performance. For the five different electrode configurations analysed, the performance varied as much as a factor five, where configurations with electrodes on both sides of the channel proved to have the best performance. © 2004 Elsevier B.V. All rights reserved. Keywords: Dielectrophoresis; Microfluidics; Particle sorter; Simulation
1. Introduction Currently, the biosensor field is rapidly growing in importance and numerous lab-on-a-chip based systems are emerging. Many of them are equipped with some kind of integrated sensors. The functionality of such systems can be greatly extended by adding a flow sorter to sort out rare bio-particles (e.g. cells) based on the readings of the sensors. In literature, integrated particle sorters have been presented based on different principles such as thermal [1,2], magnetic [3,4], mechanical [3,5–10], electrical [11–16] and optical [17] actuation. In this paper, we will focus on flow-through particle sorters with negative dielectrophoresis actuation. Dielectrophoresis is defined as the lateral motion imparted on uncharged particles as a result of polarisation induced by non-uniform electric fields. Dielectrophoresis is an attractive actuation principle for microfluidics because the actuators are simple electrodes and it requires no moving parts, no labelling of the sample and no high voltages. Although for most macro-systems, the forces generated by dielectrophoresis are not significant, the situation is quite different for micro-systems. Thanks to IC-technology very small electrode geometries can be realised with dimensions comparable to those of the particles. With these ∗ Corresponding author. Tel.: +43-1-58801-36677; fax: +43-1-58801-36699. E-mail address: [email protected] (J.H. Nieuwenhuis). URL: http://iss.iemw.tuwien.ac.at.
0925-4005/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.snb.2004.04.062
micro-electrodes, very strong field gradients can be realised, and therefore, large dielectrophoretic forces. In this paper, we compare the performance of five different electrode configurations for a dielectrophoretic particle sorter by calculating the three-dimensional trajectories of the particles. Performance is here defined as the maximum flow-speed the sorters can handle. The sorters considered here consist of a gradually sloped dielectrophoretic actuator that is used to divert a particle from its straight trajectory to one side of the flow-channel so that it can be sorted out with a simple y-junction configuration (see Fig. 1). The gradual sloped shape of the actuators was selected to maximise the time the particles are exposed to the dielectrophoretic force.
2. Theory The force on a particle generated by the non-uniform electric field can be calculated with the following formula [18]: 02 FDEP = 2πεl r 3 KCM ∇E
(1)
where εl is the permittivity of the liquid, r the radius of the particle, KCM the Clausius–Mossotti function and E0 the electric field. In the lossless case, KCM can be written as: εp − εl KCM = (2) εp + 2εl where εp is the permittivity of the particle. Frequently, formula 1 is combined with that of the buoyancy force for static analyses of for example the levitation height of a particle
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What is not considered in this paper is the influence of Joule heating, which could slightly reduce the force. However in [20], it is demonstrated that this effect is only of importance in highly conductive media (>1 S/m) or at very low flow-speeds; this is a consequence of the micro-scaling. Since distilled water is considered and the flow-speed is significant, Joule heating can be neglected here.
3. Sorter variables
Fig. 1. The operation principle of a gradually sloped particle sorter based on negative dielectrophoresis that allows to send the particles either into the left branch (left) or into the right branch (right) of the outlet channel.
[19]. To analyse the dynamic behaviour of a particle also the drag force is important, which is related to the particle speed, size, shape and surface characteristics. For a smooth sphere, the stokes approximation for this force is: FDRAG = 6πηrv
(3)
where η is the viscosity of the fluid and v the speed of the particle. The final force considered here is the buoyancy force: FB = 43 πr 3 (ρl − ρp )g
(4)
where ρl and ρp are the densities of the liquid and the particle respectively and g is the gravitational constant. When these forces are constant a steady state will be reached (after an initial transient period) where the settling-speed of the particle is given by: vsettle =
2 εl r 2 KCM ∇E 2r 2 (ρl − ρp )g 0 + 3η 9η
(5)
The first part of this equation is caused by FDEP and the second part is determined by FB . Of course, FB only plays a role in the vertical direction. The time-constant for this process is: τ=
2ρp r 2 . 9η
(6)
In this article, we analyse the dielectrophoretic force on polystyrene particles with a radius of 5 m in purified water. For this situation, the time-constant is only 5.6 s. Because of this short time-constant, the particles can be considered to travel at settling-speed constantly, as long a there are no sudden changes in the electric field. With the gradually sloping sorter configurations analysed here, this seems justified.
Even within the fixed configuration of a straight diagonal dielectrophoretic sorter element, such as depicted in Fig. 1, there are still many design variables that have significant influence on the performance that can be obtained. In this paper, the focus is on the influence of the electrode configuration on the performance of the dielectrophoretic sorter. The influence of both the dimensions and the positions of the electrodes will be investigated. The other properties of the sorter are kept constant during the simulation experiments. Below, the influence of these other sorter properties is briefly discussed so that the performance of a sorter configuration different from the one analysed in this paper can be estimated. Firstly, there are the channel dimensions. The height of the channel has a strong influence on the maximum force that can be generated. When electrodes are applied both on top of the channel and on the bottom of the channel the maximum field strength is inversely proportional to the height of the channel, and therefore, the dielectrophoretic force is inversely proportional to the square of the channel height (see formula 1). When electrodes are applied only on the bottom of the channel, the maximum force also changes with the channel height but now the relation is more complicated and depends on the electrode geometry. However, in both cases the following is clear—the smaller the channel height, the larger the maximum attainable forces. The minimum channel height is governed by clogging problems for relatively small channel dimensions. So, there is a trade-off between performance and reliability. The width of the channel has in first order not much influence on the performance when the flow-rate is kept constant, because the two main effects from widening the channel play a compensating role. When the channel is wider, a particle has to travel a longer distance from one side of the channel to the other side. At the same time, the flow-speed of the liquid suspending the particle goes down proportional to the width-increase of the channel. When buoyancy effects are neglected, these effects exactly balance out. Buoyancy forces add an additional vertical motion to the particles, which can bring them closer to the electrodes into a region where the electric field gradient is larger. At lower flow-speeds, this buoyancy effect is more dominant, which might lead to a small performance increase. A second small performance increase for wider channels might be observed as a consequence of a change in the velocity
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profile that becomes less parabolic and more uniform for large aspect ratio channels. This reduces the relative liquid velocity maximum in the centre of the channel somewhat. The length of the sorter element is directly proportional to the maximum flow-rate the sorter can handle. When the sorter element is longer (and the angle of the electrode with the direction of flow is less), the particles will experience a force from the dielectrophoretic element for a longer period of time. Therefore, less displacement per unit time is required for successful sorting operation in a longer sorter, so, the flow-rate can be higher. Besides the geometry of the channel also the size and electrical properties of the particle and liquid have influence on the performance of the sorter. The dielectrophoretic force is proportional to the third power of the particle radius (see formula 1), while the drag force is proportional to the radius (see formula 3). As a consequence, larger particles can be sorted faster than smaller ones. Furthermore, the combination of the liquid and the particles should be chosen such that their permittivity is quite different, which results in a Clausius–Mossotti factor close to −0.5 (the maximum negative dielectrophoretic effect, see formula 2). Using a liquid with a low viscosity also improves performance, since it reduces the drag force on the particle (see formula 3). Finally, the applied voltage has direct influence on the sorter performance—the electric field strength increases proportional to the voltage which leads to a quadratic increase in the dielectrophoretic force (see formula 1). The voltage is limited by Joule heating effects [21,22]. Firstly, Joule heating may lead to temperatures that are too high, especially in highly conductive media, for the (bio)particles that are sorted. Secondly, the Joule heating causes temperature gradients in the liquid, and therefore, gradients in density, permittivity and conductivity. The changes in density result in liquid flows due to natural convection. Furthermore, the gradients in permittivity and conductivity cause interactions with the electrical field which lead to electric forces on the liquid—at frequencies below 0.5 MHz AC electro-osmosis at the electrode surface can induce unwanted forces on the liquid [21].
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bottom of the channel extend to the side-wall on one side (see Fig. 2 left). Using finite element simulations, the electric field distribution for each electrode configuration was calculated in a 2D plane for a channel with a height of 50 m and a width of 200 m. The logarithm of the electric field intensity squared is plotted in the left column of Fig. 2 (for clarity reasons, the vertical dimension of the plots in Fig. 2 was extended; therefore, the aspect ratio is not to scale). By applying formula 5, the steady state velocity vectors for a particle with respect to the liquid can be calculated for a cross-section of the sorter module. The streamlines for this velocity-vector field are plotted in the middle column of Fig. 2 for polystyrene particles with a radius of 5 m in purified water and with voltages of ± 3 V applied to the two electrodes, respectively. The maximum velocity that can be obtained with each electrode configuration over the full height of the channel is plotted in the right column of Fig. 2. The graphs for each configuration will now be briefly discussed. 4.1. Configuration I The main part of the electric field distribution of configuration I is concentrated in the area around the electrodes and drops rapidly in all other directions. This means that there is a strong field gradient, which can generate a large force in this area, but in the rest of the channel cross-section a particle will only experience a weak force. From the streamline plot, it becomes clear that besides the wanted horizontal motion, there also is a large vertical component that pushes the particle upwards as they move along the actuator into the region where the electric field intensity drops rapidly. It can be seen that gravity plays a minor role—only near the sides of the channel, the particles tend to sediment to the bottom of the channel. In the right graph that depicts the maximum horizontal speed that can be realised with this actuator, it can be seen that the maximum achievable horizontal speed drops dramatically at larger heights. While at a height of 10 m, a speed of almost 2.5 mm/s can still be achieved, the maximum speed for a height of 20 m is already five times less. 4.2. Configuration II
4. Electrode configurations Five different electrode configurations are analysed, the first three of them only have electrodes located on the bottom of the channel and the last two have electrodes both on the top and on the bottom of the channel. The first configuration (I) has two equally sized electrodes located at the bottom of the channel. Configuration II is similar but now the left electrode extends up to the channel side-wall and in configuration III, both electrodes extend to the side-walls of the channel. The fourth configuration (IV) has one electrode on the bottom of the channel and one electrode at the top of the channel. In configuration V, the electrodes on top and
The graphs for configuration II look quite similar to those of configuration I, but there are some differences. For configuration II, the electric field extends higher into the channel because of the larger width of the left electrode. On the left side, the field intensity is now quite a bit larger, but the largest field gradient for this configuration is found next to the narrow electrode, which can be seen in the density of the field lines. This is the side of the sorter that will be used to push the particle sideways. The streamline plot is no longer symmetrical and on the left side, above the wide electrode, the field lines are more horizontal. The maximum velocity plot shows that at larger heights also this
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Fig. 2. The distributions of the electric field (left) and the direction vectors for a particle (middle) in a cross-section of the channel (for clarity, the vertical dimension has been stretched, and therefore, the aspect ratio of the plot differs from the channel aspect ratio); in the right graphs the maximum horizontal speed generated by the actuator (horizontal axis in mm/s) versus the position in the channel (vertical axis in m) are displayed for the five electrode configurations analysed (from top to bottom).
configuration shows a steep drop in performance, although settling at a slightly higher minimum. 4.3. Configuration III In configuration III, both electrodes extend to the side-walls of the channel. In the electric field plot, it can be seen that as a consequence the electric field is stronger in the higher parts of the channel. A second effect is that the maximum horizontal gradient in the electric field is somewhat less than in configurations I and II. The streamline plot for this configuration, as expected, looks much like a symmetrical version of the left part of the stream-
line plot for configuration II. The maximum velocity plot reflects the observations that were made in the plot of the electric field distribution. Although for this configuration, there still is a large drop in the maximum velocity higher in the channel, the absolute value at the top of the channel is still 50%, 14% higher than in configuration I and II, respectively. This relates to the higher field strength near the top of the channel. The lower maximum horizontal gradient in the electric field leads to somewhat lower flow-speeds at smaller heights. This, however, is not an issue since the overall maximum sorter speed is constrained by its (substantially lower) performance at higher vertical positions.
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4.4. Configuration IV For configuration IV, the electric field distribution looks completely different. The electrodes at top and bottom create an electric field that is concentrated around the middle of channel and that is relatively constant over its full height. Such a field distribution is very suitable to push a particle sideways, because in the horizontal plane there is a large field gradient over the full height of the channel. This is reflected in the horizontal motion expressed in the streamline plot. The velocity streamlines further show that particles will be concentrated in a horizontal plane near the middle of the channel. The maximum obtainable horizontal flow-speeds are quite different for a configuration with electrodes on both top and bottom of the channel. The maximum speed plot shows that the maximum speed drops less and that the minimum occurs at the middle of the channel. But this minimum is substantially larger than for the configurations with electrodes located on only the bottom of the channel. 4.5. Configuration V In configuration V, the electrodes on top and bottom of the channel extend to the left side-wall. In the plot of the electric field distribution, two effects can be observed. Firstly, on the left side of the channel, a mainly constant strong electric field is present. Secondly, more interestingly,
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the horizontal gradient in the electric field has increased for the right side of the channel. This means that a particle can be pushed sideways more strongly. In the streamline plot, the strongly non-symmetrical behaviour of this configuration is very clear—particles will only be pushed to the right with this configuration. The maximum horizontal speed plot shows the result of the increased gradient most clearly by an increase in the larger speeds that can be achieved in the middle of the channel.
5. Particle trajectories In the previous paragraph, it was demonstrated that the qualitative flow behaviour can be deduced from a two-dimensional analysis. To analyse the performance of the different configurations in a quantitative way, the simulation model will now be extended to three dimensions. The method applied in this paper is based on the orthogonality of the dielectrophoretic force and buoyancy force in a cross-sectional plane of the flow-channel with respect to the velocity of the liquid in a direction along the flow-channel. The velocity profile of the liquid under pressure driven actuation was calculated for the flow-channel using finite element simulation (Coventorware 2001.3). By adding the velocity vector components of the liquid itself to the vectors that describe the particle motion with respect to the liquid, the three-dimensional vectors are obtained that describe the
Fig. 3. A 3D view (top) and a top view (bottom) of the simulation of the particle trajectories for a sloped dielectrophoretic actuator with electrode configuration V, operated at its maximum flow-speed (for clarity reasons, only a limited number of trajectories in the middle plane of the sorter are depicted).
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Fig. 4. The performance of the different electrode configurations clearly show that sorters with electrodes on both top and bottom of the channel (IV and V) clearly perform better than sorters that only have electrodes on the bottom of the channel.
three-dimensional particle motion in a single plane of the device. By repeating this procedure with a slight horizontal offset for the electric field, the 3D particle velocity-vector field for the sloped sorter layout is obtained. This can be used to calculate particle trajectories (see Fig. 3). The maximum average flow-speed (since the flow-speed is not uniform across the channel) of the liquid for which the particles could be sorted (or in other words where the trajectories remained on the proper side of the actuator) was determined for each electrode configuration. The situation depicted in Fig. 3 shows a sorter of configuration V operated at maximum flow-speed—all of the particle trajectories just end up in the first 50 m of the flow-channel, following the outline of the actuator. The conditions for the simulations were as follows. The channel has a length, width and height of 1200, 200 and 50 m, respectively, with an effective sorter length of 750 m. The liquid used is deionised water and the particles consist of polystyrene spheres with a radius of 5 m. Voltages of ± 3 V are applied to the two electrodes. While the particle moves along the channel, it experiences a maximum displacement of 150 m (when it started near the wall on the ‘wrong’ side of the channel). The gradual slope of the sorter (1:5) justifies the orthogonality approximation made earlier. The particle trajectories start evenly distributed over the whole channel cross-section (see Fig. 3) and should all end within the right 50 m of the channel. The results for the different sorters are depicted in Fig. 4. The absolute values are not so important, since they vary depending on the sorter layout, but the relative performance of the electrode configurations is the most important result here. 6. Discussion The most striking result is the variation in maximum performance for the different sorter configurations—the
performance varies by a factor of five when only changes are made to the electrode configuration. Furthermore, as predicted from the graphs in Fig. 2, there is a large performance difference between the single-sided configuration with electrodes only on the bottom of the channel (I, II and III) compared to configurations with electrodes on both top and bottom of the channel (IV and V). This has to do with the much lower electric field density of the single-sided electrode configurations near the top of the channel. Since the overall maximum flow-speed that can be handled is determined by the minimum performance anywhere along the cross-section of the channel, this causes the single-sided configurations to perform substantially less. For the single-sided electrode configurations extending one electrode all the way to the channel side-wall in configuration II, improved the electric field intensity near the top of the channel quite significantly. The simulation results show an overall performance increase of 33% for configuration II compared to configuration I. Extending also the second electrode to the channel wall in configuration III added only 8% extra performance. Although the performance obtained with these configurations is considerably lower than with the double side electrode configurations, their performance is still very usable. Furthermore, they have the advantage that they are easier to fabricate than the double-sided configurations. So, if in a certain application performance is not critical, these configurations can be the right choice. A second situation, where the application of single-sided sorters may be favourable, can be found in devices that have a flow-profile in which the particles are focused to the bottom of the flow-channel [23]. Near the bottom of the flow-channel the single-sided configurations have a maximum field gradient is even higher than is the case with the double-sided electrode configurations. Finally, the performance of the single-sided sorters can be increased, if necessary, by increasing the length of the sorter or increasing the applied voltages. Under the same conditions, the double-sided configurations IV and V show much higher performance. Configuration IV already outperforms the best single-sided configuration by 100% and configuration V almost doubles the performance of configuration IV. This can be understood when one considers that for the double-sided electrode configurations, the electric field intensity fans out over only half the channel height after which its density increases again thanks to the opposite electrode. The higher performance of configuration V with respect to configuration IV can also be explained. Configuration V has a higher maximum horizontal field gradient, which is located near the edge of the electrodes where the relatively uniform field on the left side goes over into the more non-uniform field on the right side (see Fig. 2 bottom left). Because the performance increases with the square of the electric field intensity, moderate changes in the electric field can result in substantial performance differences. Besides their added complexity, another drawback of the double-sided electrode configurations can be that with
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common electrode materials (gold, platinum), they partially block the view inside a channel. This can be solved by using a top electrode that is made of a transparent conducting material, such as for example indium tin oxide (ITO).
7. Conclusions In this paper, the performance of dielectrophoretic particle sorters was studied using a novel analysis method that is based on the orthogonality of the dielectrophoretic and buoyancy forces with respect to the direction of flow. By calculating the particle trajectories, the performance of different electrode configurations was compared quantitatively. It was found that electrode configurations with electrodes both on top and on the bottom of the flow-channel performed best by a large margin. This is due to the relative steep drop-off in electric field intensity in the higher part of the channel for the single-sided configurations. Although the field intensity at higher positions inside the channel can be improved by extending the electrodes to the sides of the channel, the double-sided electrode configurations still perform 100–250% better. The main advantages of single-sided sorter configurations over the double-sided configurations are that they are easier to fabricate and that they do not obstruct visual inspection of the channel.
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Biographies Jeroen H. Nieuwenhuis was born in Bergen op Zoom, The Netherlands, in 1977. In 2000, he received the MS degree in electrical engineering with honours from Delft University of Technology, The Netherlands. In 2000, he did a 6-months internship at IC-Sensors, Milpitas, California, where he worked on accelerometers and pressure sensors. Since January 2001, he is a PhD student, currently at the Vienna University of Technology, Austria. His main research interests are microfluidics and in particular the design of integrated sensors for particle analysis. Michael J. Vellekoop was born in Amsterdam in 1960. He received the BSc degree in Physics in 1982 and the PhD degree in Electrical
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Engineering in 1994. In 1988, he co-founded Xensor Integration B.V., where he was managing director until 1996. In that year, he initiated a new group on the topic of physical chemosensors at the DIMES Electronic Instrumentation Laboratory of the Delft University of Technology, where in 1997 he became an associated professor. Since 2001, he is a full professor of Industrial Sensor Systems at the Institute of
Industrial Electronics and Material Science at the Vienna University of Technology, Austria. Keywords of research are physical chemosensors, physical biosensors, sensor systems, lab on a chip, micro and nanofluidics, micro and nanotechnology. He authored and co-authored more than 120 publications in peer reviewed journals and international conferences.
Simulation study of dielectrophoretic particle sorters J.H. Nieuwenhuis∗ , M.J. Vellekoop Institute of IEMW, Vienna University of Technology, Gusshausstrasse 27-29, A-1040 Vienna, Austria Available online 1 June 2004
Abstract In this paper, the performance of five electrode configurations for particle sorters based on negative dielectrophoresis is compared using a novel analysis method. By equating the formulas for dielectrophoresis, drag and buoyancy, the velocity vectors for the particles with respect to the liquid can be obtained. This information is combined with the liquid velocity vectors to calculate three-dimensional particle trajectories. The maximum allowed flow-speed at which the sorters still work properly is the criterion applied to compare their performance. For the five different electrode configurations analysed, the performance varied as much as a factor five, where configurations with electrodes on both sides of the channel proved to have the best performance. © 2004 Elsevier B.V. All rights reserved. Keywords: Dielectrophoresis; Microfluidics; Particle sorter; Simulation
1. Introduction Currently, the biosensor field is rapidly growing in importance and numerous lab-on-a-chip based systems are emerging. Many of them are equipped with some kind of integrated sensors. The functionality of such systems can be greatly extended by adding a flow sorter to sort out rare bio-particles (e.g. cells) based on the readings of the sensors. In literature, integrated particle sorters have been presented based on different principles such as thermal [1,2], magnetic [3,4], mechanical [3,5–10], electrical [11–16] and optical [17] actuation. In this paper, we will focus on flow-through particle sorters with negative dielectrophoresis actuation. Dielectrophoresis is defined as the lateral motion imparted on uncharged particles as a result of polarisation induced by non-uniform electric fields. Dielectrophoresis is an attractive actuation principle for microfluidics because the actuators are simple electrodes and it requires no moving parts, no labelling of the sample and no high voltages. Although for most macro-systems, the forces generated by dielectrophoresis are not significant, the situation is quite different for micro-systems. Thanks to IC-technology very small electrode geometries can be realised with dimensions comparable to those of the particles. With these ∗ Corresponding author. Tel.: +43-1-58801-36677; fax: +43-1-58801-36699. E-mail address: [email protected] (J.H. Nieuwenhuis). URL: http://iss.iemw.tuwien.ac.at.
0925-4005/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.snb.2004.04.062
micro-electrodes, very strong field gradients can be realised, and therefore, large dielectrophoretic forces. In this paper, we compare the performance of five different electrode configurations for a dielectrophoretic particle sorter by calculating the three-dimensional trajectories of the particles. Performance is here defined as the maximum flow-speed the sorters can handle. The sorters considered here consist of a gradually sloped dielectrophoretic actuator that is used to divert a particle from its straight trajectory to one side of the flow-channel so that it can be sorted out with a simple y-junction configuration (see Fig. 1). The gradual sloped shape of the actuators was selected to maximise the time the particles are exposed to the dielectrophoretic force.
2. Theory The force on a particle generated by the non-uniform electric field can be calculated with the following formula [18]: 02 FDEP = 2πεl r 3 KCM ∇E
(1)
where εl is the permittivity of the liquid, r the radius of the particle, KCM the Clausius–Mossotti function and E0 the electric field. In the lossless case, KCM can be written as: εp − εl KCM = (2) εp + 2εl where εp is the permittivity of the particle. Frequently, formula 1 is combined with that of the buoyancy force for static analyses of for example the levitation height of a particle
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What is not considered in this paper is the influence of Joule heating, which could slightly reduce the force. However in [20], it is demonstrated that this effect is only of importance in highly conductive media (>1 S/m) or at very low flow-speeds; this is a consequence of the micro-scaling. Since distilled water is considered and the flow-speed is significant, Joule heating can be neglected here.
3. Sorter variables
Fig. 1. The operation principle of a gradually sloped particle sorter based on negative dielectrophoresis that allows to send the particles either into the left branch (left) or into the right branch (right) of the outlet channel.
[19]. To analyse the dynamic behaviour of a particle also the drag force is important, which is related to the particle speed, size, shape and surface characteristics. For a smooth sphere, the stokes approximation for this force is: FDRAG = 6πηrv
(3)
where η is the viscosity of the fluid and v the speed of the particle. The final force considered here is the buoyancy force: FB = 43 πr 3 (ρl − ρp )g
(4)
where ρl and ρp are the densities of the liquid and the particle respectively and g is the gravitational constant. When these forces are constant a steady state will be reached (after an initial transient period) where the settling-speed of the particle is given by: vsettle =
2 εl r 2 KCM ∇E 2r 2 (ρl − ρp )g 0 + 3η 9η
(5)
The first part of this equation is caused by FDEP and the second part is determined by FB . Of course, FB only plays a role in the vertical direction. The time-constant for this process is: τ=
2ρp r 2 . 9η
(6)
In this article, we analyse the dielectrophoretic force on polystyrene particles with a radius of 5 m in purified water. For this situation, the time-constant is only 5.6 s. Because of this short time-constant, the particles can be considered to travel at settling-speed constantly, as long a there are no sudden changes in the electric field. With the gradually sloping sorter configurations analysed here, this seems justified.
Even within the fixed configuration of a straight diagonal dielectrophoretic sorter element, such as depicted in Fig. 1, there are still many design variables that have significant influence on the performance that can be obtained. In this paper, the focus is on the influence of the electrode configuration on the performance of the dielectrophoretic sorter. The influence of both the dimensions and the positions of the electrodes will be investigated. The other properties of the sorter are kept constant during the simulation experiments. Below, the influence of these other sorter properties is briefly discussed so that the performance of a sorter configuration different from the one analysed in this paper can be estimated. Firstly, there are the channel dimensions. The height of the channel has a strong influence on the maximum force that can be generated. When electrodes are applied both on top of the channel and on the bottom of the channel the maximum field strength is inversely proportional to the height of the channel, and therefore, the dielectrophoretic force is inversely proportional to the square of the channel height (see formula 1). When electrodes are applied only on the bottom of the channel, the maximum force also changes with the channel height but now the relation is more complicated and depends on the electrode geometry. However, in both cases the following is clear—the smaller the channel height, the larger the maximum attainable forces. The minimum channel height is governed by clogging problems for relatively small channel dimensions. So, there is a trade-off between performance and reliability. The width of the channel has in first order not much influence on the performance when the flow-rate is kept constant, because the two main effects from widening the channel play a compensating role. When the channel is wider, a particle has to travel a longer distance from one side of the channel to the other side. At the same time, the flow-speed of the liquid suspending the particle goes down proportional to the width-increase of the channel. When buoyancy effects are neglected, these effects exactly balance out. Buoyancy forces add an additional vertical motion to the particles, which can bring them closer to the electrodes into a region where the electric field gradient is larger. At lower flow-speeds, this buoyancy effect is more dominant, which might lead to a small performance increase. A second small performance increase for wider channels might be observed as a consequence of a change in the velocity
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profile that becomes less parabolic and more uniform for large aspect ratio channels. This reduces the relative liquid velocity maximum in the centre of the channel somewhat. The length of the sorter element is directly proportional to the maximum flow-rate the sorter can handle. When the sorter element is longer (and the angle of the electrode with the direction of flow is less), the particles will experience a force from the dielectrophoretic element for a longer period of time. Therefore, less displacement per unit time is required for successful sorting operation in a longer sorter, so, the flow-rate can be higher. Besides the geometry of the channel also the size and electrical properties of the particle and liquid have influence on the performance of the sorter. The dielectrophoretic force is proportional to the third power of the particle radius (see formula 1), while the drag force is proportional to the radius (see formula 3). As a consequence, larger particles can be sorted faster than smaller ones. Furthermore, the combination of the liquid and the particles should be chosen such that their permittivity is quite different, which results in a Clausius–Mossotti factor close to −0.5 (the maximum negative dielectrophoretic effect, see formula 2). Using a liquid with a low viscosity also improves performance, since it reduces the drag force on the particle (see formula 3). Finally, the applied voltage has direct influence on the sorter performance—the electric field strength increases proportional to the voltage which leads to a quadratic increase in the dielectrophoretic force (see formula 1). The voltage is limited by Joule heating effects [21,22]. Firstly, Joule heating may lead to temperatures that are too high, especially in highly conductive media, for the (bio)particles that are sorted. Secondly, the Joule heating causes temperature gradients in the liquid, and therefore, gradients in density, permittivity and conductivity. The changes in density result in liquid flows due to natural convection. Furthermore, the gradients in permittivity and conductivity cause interactions with the electrical field which lead to electric forces on the liquid—at frequencies below 0.5 MHz AC electro-osmosis at the electrode surface can induce unwanted forces on the liquid [21].
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bottom of the channel extend to the side-wall on one side (see Fig. 2 left). Using finite element simulations, the electric field distribution for each electrode configuration was calculated in a 2D plane for a channel with a height of 50 m and a width of 200 m. The logarithm of the electric field intensity squared is plotted in the left column of Fig. 2 (for clarity reasons, the vertical dimension of the plots in Fig. 2 was extended; therefore, the aspect ratio is not to scale). By applying formula 5, the steady state velocity vectors for a particle with respect to the liquid can be calculated for a cross-section of the sorter module. The streamlines for this velocity-vector field are plotted in the middle column of Fig. 2 for polystyrene particles with a radius of 5 m in purified water and with voltages of ± 3 V applied to the two electrodes, respectively. The maximum velocity that can be obtained with each electrode configuration over the full height of the channel is plotted in the right column of Fig. 2. The graphs for each configuration will now be briefly discussed. 4.1. Configuration I The main part of the electric field distribution of configuration I is concentrated in the area around the electrodes and drops rapidly in all other directions. This means that there is a strong field gradient, which can generate a large force in this area, but in the rest of the channel cross-section a particle will only experience a weak force. From the streamline plot, it becomes clear that besides the wanted horizontal motion, there also is a large vertical component that pushes the particle upwards as they move along the actuator into the region where the electric field intensity drops rapidly. It can be seen that gravity plays a minor role—only near the sides of the channel, the particles tend to sediment to the bottom of the channel. In the right graph that depicts the maximum horizontal speed that can be realised with this actuator, it can be seen that the maximum achievable horizontal speed drops dramatically at larger heights. While at a height of 10 m, a speed of almost 2.5 mm/s can still be achieved, the maximum speed for a height of 20 m is already five times less. 4.2. Configuration II
4. Electrode configurations Five different electrode configurations are analysed, the first three of them only have electrodes located on the bottom of the channel and the last two have electrodes both on the top and on the bottom of the channel. The first configuration (I) has two equally sized electrodes located at the bottom of the channel. Configuration II is similar but now the left electrode extends up to the channel side-wall and in configuration III, both electrodes extend to the side-walls of the channel. The fourth configuration (IV) has one electrode on the bottom of the channel and one electrode at the top of the channel. In configuration V, the electrodes on top and
The graphs for configuration II look quite similar to those of configuration I, but there are some differences. For configuration II, the electric field extends higher into the channel because of the larger width of the left electrode. On the left side, the field intensity is now quite a bit larger, but the largest field gradient for this configuration is found next to the narrow electrode, which can be seen in the density of the field lines. This is the side of the sorter that will be used to push the particle sideways. The streamline plot is no longer symmetrical and on the left side, above the wide electrode, the field lines are more horizontal. The maximum velocity plot shows that at larger heights also this
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Fig. 2. The distributions of the electric field (left) and the direction vectors for a particle (middle) in a cross-section of the channel (for clarity, the vertical dimension has been stretched, and therefore, the aspect ratio of the plot differs from the channel aspect ratio); in the right graphs the maximum horizontal speed generated by the actuator (horizontal axis in mm/s) versus the position in the channel (vertical axis in m) are displayed for the five electrode configurations analysed (from top to bottom).
configuration shows a steep drop in performance, although settling at a slightly higher minimum. 4.3. Configuration III In configuration III, both electrodes extend to the side-walls of the channel. In the electric field plot, it can be seen that as a consequence the electric field is stronger in the higher parts of the channel. A second effect is that the maximum horizontal gradient in the electric field is somewhat less than in configurations I and II. The streamline plot for this configuration, as expected, looks much like a symmetrical version of the left part of the stream-
line plot for configuration II. The maximum velocity plot reflects the observations that were made in the plot of the electric field distribution. Although for this configuration, there still is a large drop in the maximum velocity higher in the channel, the absolute value at the top of the channel is still 50%, 14% higher than in configuration I and II, respectively. This relates to the higher field strength near the top of the channel. The lower maximum horizontal gradient in the electric field leads to somewhat lower flow-speeds at smaller heights. This, however, is not an issue since the overall maximum sorter speed is constrained by its (substantially lower) performance at higher vertical positions.
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4.4. Configuration IV For configuration IV, the electric field distribution looks completely different. The electrodes at top and bottom create an electric field that is concentrated around the middle of channel and that is relatively constant over its full height. Such a field distribution is very suitable to push a particle sideways, because in the horizontal plane there is a large field gradient over the full height of the channel. This is reflected in the horizontal motion expressed in the streamline plot. The velocity streamlines further show that particles will be concentrated in a horizontal plane near the middle of the channel. The maximum obtainable horizontal flow-speeds are quite different for a configuration with electrodes on both top and bottom of the channel. The maximum speed plot shows that the maximum speed drops less and that the minimum occurs at the middle of the channel. But this minimum is substantially larger than for the configurations with electrodes located on only the bottom of the channel. 4.5. Configuration V In configuration V, the electrodes on top and bottom of the channel extend to the left side-wall. In the plot of the electric field distribution, two effects can be observed. Firstly, on the left side of the channel, a mainly constant strong electric field is present. Secondly, more interestingly,
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the horizontal gradient in the electric field has increased for the right side of the channel. This means that a particle can be pushed sideways more strongly. In the streamline plot, the strongly non-symmetrical behaviour of this configuration is very clear—particles will only be pushed to the right with this configuration. The maximum horizontal speed plot shows the result of the increased gradient most clearly by an increase in the larger speeds that can be achieved in the middle of the channel.
5. Particle trajectories In the previous paragraph, it was demonstrated that the qualitative flow behaviour can be deduced from a two-dimensional analysis. To analyse the performance of the different configurations in a quantitative way, the simulation model will now be extended to three dimensions. The method applied in this paper is based on the orthogonality of the dielectrophoretic force and buoyancy force in a cross-sectional plane of the flow-channel with respect to the velocity of the liquid in a direction along the flow-channel. The velocity profile of the liquid under pressure driven actuation was calculated for the flow-channel using finite element simulation (Coventorware 2001.3). By adding the velocity vector components of the liquid itself to the vectors that describe the particle motion with respect to the liquid, the three-dimensional vectors are obtained that describe the
Fig. 3. A 3D view (top) and a top view (bottom) of the simulation of the particle trajectories for a sloped dielectrophoretic actuator with electrode configuration V, operated at its maximum flow-speed (for clarity reasons, only a limited number of trajectories in the middle plane of the sorter are depicted).
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Fig. 4. The performance of the different electrode configurations clearly show that sorters with electrodes on both top and bottom of the channel (IV and V) clearly perform better than sorters that only have electrodes on the bottom of the channel.
three-dimensional particle motion in a single plane of the device. By repeating this procedure with a slight horizontal offset for the electric field, the 3D particle velocity-vector field for the sloped sorter layout is obtained. This can be used to calculate particle trajectories (see Fig. 3). The maximum average flow-speed (since the flow-speed is not uniform across the channel) of the liquid for which the particles could be sorted (or in other words where the trajectories remained on the proper side of the actuator) was determined for each electrode configuration. The situation depicted in Fig. 3 shows a sorter of configuration V operated at maximum flow-speed—all of the particle trajectories just end up in the first 50 m of the flow-channel, following the outline of the actuator. The conditions for the simulations were as follows. The channel has a length, width and height of 1200, 200 and 50 m, respectively, with an effective sorter length of 750 m. The liquid used is deionised water and the particles consist of polystyrene spheres with a radius of 5 m. Voltages of ± 3 V are applied to the two electrodes. While the particle moves along the channel, it experiences a maximum displacement of 150 m (when it started near the wall on the ‘wrong’ side of the channel). The gradual slope of the sorter (1:5) justifies the orthogonality approximation made earlier. The particle trajectories start evenly distributed over the whole channel cross-section (see Fig. 3) and should all end within the right 50 m of the channel. The results for the different sorters are depicted in Fig. 4. The absolute values are not so important, since they vary depending on the sorter layout, but the relative performance of the electrode configurations is the most important result here. 6. Discussion The most striking result is the variation in maximum performance for the different sorter configurations—the
performance varies by a factor of five when only changes are made to the electrode configuration. Furthermore, as predicted from the graphs in Fig. 2, there is a large performance difference between the single-sided configuration with electrodes only on the bottom of the channel (I, II and III) compared to configurations with electrodes on both top and bottom of the channel (IV and V). This has to do with the much lower electric field density of the single-sided electrode configurations near the top of the channel. Since the overall maximum flow-speed that can be handled is determined by the minimum performance anywhere along the cross-section of the channel, this causes the single-sided configurations to perform substantially less. For the single-sided electrode configurations extending one electrode all the way to the channel side-wall in configuration II, improved the electric field intensity near the top of the channel quite significantly. The simulation results show an overall performance increase of 33% for configuration II compared to configuration I. Extending also the second electrode to the channel wall in configuration III added only 8% extra performance. Although the performance obtained with these configurations is considerably lower than with the double side electrode configurations, their performance is still very usable. Furthermore, they have the advantage that they are easier to fabricate than the double-sided configurations. So, if in a certain application performance is not critical, these configurations can be the right choice. A second situation, where the application of single-sided sorters may be favourable, can be found in devices that have a flow-profile in which the particles are focused to the bottom of the flow-channel [23]. Near the bottom of the flow-channel the single-sided configurations have a maximum field gradient is even higher than is the case with the double-sided electrode configurations. Finally, the performance of the single-sided sorters can be increased, if necessary, by increasing the length of the sorter or increasing the applied voltages. Under the same conditions, the double-sided configurations IV and V show much higher performance. Configuration IV already outperforms the best single-sided configuration by 100% and configuration V almost doubles the performance of configuration IV. This can be understood when one considers that for the double-sided electrode configurations, the electric field intensity fans out over only half the channel height after which its density increases again thanks to the opposite electrode. The higher performance of configuration V with respect to configuration IV can also be explained. Configuration V has a higher maximum horizontal field gradient, which is located near the edge of the electrodes where the relatively uniform field on the left side goes over into the more non-uniform field on the right side (see Fig. 2 bottom left). Because the performance increases with the square of the electric field intensity, moderate changes in the electric field can result in substantial performance differences. Besides their added complexity, another drawback of the double-sided electrode configurations can be that with
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common electrode materials (gold, platinum), they partially block the view inside a channel. This can be solved by using a top electrode that is made of a transparent conducting material, such as for example indium tin oxide (ITO).
7. Conclusions In this paper, the performance of dielectrophoretic particle sorters was studied using a novel analysis method that is based on the orthogonality of the dielectrophoretic and buoyancy forces with respect to the direction of flow. By calculating the particle trajectories, the performance of different electrode configurations was compared quantitatively. It was found that electrode configurations with electrodes both on top and on the bottom of the flow-channel performed best by a large margin. This is due to the relative steep drop-off in electric field intensity in the higher part of the channel for the single-sided configurations. Although the field intensity at higher positions inside the channel can be improved by extending the electrodes to the sides of the channel, the double-sided electrode configurations still perform 100–250% better. The main advantages of single-sided sorter configurations over the double-sided configurations are that they are easier to fabricate and that they do not obstruct visual inspection of the channel.
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Biographies Jeroen H. Nieuwenhuis was born in Bergen op Zoom, The Netherlands, in 1977. In 2000, he received the MS degree in electrical engineering with honours from Delft University of Technology, The Netherlands. In 2000, he did a 6-months internship at IC-Sensors, Milpitas, California, where he worked on accelerometers and pressure sensors. Since January 2001, he is a PhD student, currently at the Vienna University of Technology, Austria. His main research interests are microfluidics and in particular the design of integrated sensors for particle analysis. Michael J. Vellekoop was born in Amsterdam in 1960. He received the BSc degree in Physics in 1982 and the PhD degree in Electrical
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Engineering in 1994. In 1988, he co-founded Xensor Integration B.V., where he was managing director until 1996. In that year, he initiated a new group on the topic of physical chemosensors at the DIMES Electronic Instrumentation Laboratory of the Delft University of Technology, where in 1997 he became an associated professor. Since 2001, he is a full professor of Industrial Sensor Systems at the Institute of
Industrial Electronics and Material Science at the Vienna University of Technology, Austria. Keywords of research are physical chemosensors, physical biosensors, sensor systems, lab on a chip, micro and nanofluidics, micro and nanotechnology. He authored and co-authored more than 120 publications in peer reviewed journals and international conferences.