Multi-Physics Analysis of Microfluidic Devices with Hydrodynamic Focusing and Dielectrophoresis

Transcription

1 Multi-Physics Analysis of Microfluidic Devices with Hydrodynamic Focusing and Dielectrophoresis M. Kathryn Thompson Mechanical Engineering Dept, MIT John M. Thompson, PhD Consulting Engineer Abstract Among the applications for MEMS devices is that of sorting biological specimens. Closed form solutions fail to provide the necessary means for evaluating these devices, particularly if more than one Physics environment is involve. This paper provides an overview of some of the lessons learned in the fluidic, electric and thermal domains. Introduction As microfluidic devices become smaller and more complex, fewer options are available for modeling the behavior these devices. Analytic solutions that provided scoping results when the devices were simple fail to provide adequate assurance when the devices become complex. And if more than one physics environment is involved, complexity will be present for the degrees of freedom as well as the geometry. As a result, the only viable alternative is to use finite element simulation. One of the strengths of the ANSYS finite element program is its ability to simulate multi-physics systems in both the macro and micro scale domains. This paper presents some preliminary models for a microfluidic device to demonstrate some of these abilities for a micromachined dielectrophoretic flow cytometer. The behavior of the cells as they flow through the system undergoing hydrodynamic focusing can be determined using Flotran for a fluidic model of the system. The dielectrophoretic behavior of cells present in the fluid can be found as a result the electric fields generated between the submerged electrodes in the flow. The Joule heating created by those electrodes and the subsequent temperature rise can be determined by modeling the thermal behavior of the system. Optical detection methods, device fabrication and packaging, and the performance of the electrodes and of the device as a whole will not be discussed. The Phenomena Flow Cytometry Flow cytometry is defined as a technique for identifying and sorting cells and their components (as DNA) by staining with a fluorescent dye and detecting the fluorescence usually by laser beam illumination. 1 In the simplest type of device, cells flow past an optical detector in a single file line and a count of cells is made. If the device is a cell sorter, the output signal from the detector may be used to decide what direction the cell should take. A secondary signal is then be sent to the sorting mechanism to change the path of the cell, if necessary. Finally, cells flow out of the device through one of the two outlet channels. In this example, hydrodynamic focusing uses the addition of a sheath fluid as the pipe narrows to focus the sample flow into a single file line of cells before they enter the detection region. Dielectrophoresis Dielectrophoresis is the movement of dipole particles as a result of the mechanical forces exerted by a field on a suspended particle, through interaction with its induced dipole moment. 2 This can be useful for manipulating polymer particles like latex spheres, as well as with living cells. Cells can experience positive or negative dielectrophoresis depending on the properties of the cells and their surrounding medium. Those

2 experiencing negative electrophoresis migrate towards regions of low field strength. Depending on the strength of the field, the cells are then either trapped at the field minima or they are deflected slightly from their original course of travel. In flow cytometers, electrodes can be placed near or in a fluid stream to exert a force on the cells as they pass the electrodes. If the dielectrophoretic force is sufficient to overcome the drag force on the cell, the cell will be pushed away from the electrodes. In this way, cells can be moved from the center of the channel to one side or the other, effectively sorting them into separate output channels (figure 1). Figure 1. Device Schematic Cytological Geometric Constraints and Assumptions It is assumed that the device is made from a moldable polymer like PDMS or SU-8 which requires that device geometry be approximately a two dimensional extrusion instead of being fully three dimensional. The manufacturing technology requires that the channels must also be square instead of cylindrical and concentric fluid flows cannot be achieved. The velocity of the cells flowing through the detection region was given to be 1 m/s. The scale for the electrode geometry was given and the electrodes are actuated at +/- 5V. Cytological Constraints It is important that the living cells in the fluid sample are not permanently damaged or killed in the process of being manipulated. Mammalian cells should be subjected to no more than 1 Pa shear stress, no more than 70 mv across the cell membrane and a maximum temperature of no more than 39 degrees C. (Body temperature is 37 C.) It is also preferable to keep the cells in a biologically compatible medium like a physiological saline solution or a well regulated sucrose solution. Modeling in the Fluid Domain Hydrodynamic Focusing The use of a sheath flow for hydrodynamic focusing is described in detail by Lee et al. 3 Since the geometry of the focusing region was not given, it was chosen using the velocity and shear stress requirements. The behavior of the fluid in the focusing region of the microfluidic device is dependent on shear stress, velocity profile and conversation of mass. The velocity profile at a radius r for a single fluid flowing through a cylindrical pipe is given by:

3 2 r vr () = vmax (1 ) Eq 1 2 R where r is the radial distance from the centerline of a pipe of inner radius R with a maximum fluid velocity of v max found at the center of the pipe. The shear stress on a cell in the flow is given by: du τ = µ Eq 2 dy where τ is the shear stress, µ is the viscosity of the liquid, and du is the change in fluid velocity over the distance dy. The allowable shear stress over the cell, cell diameter and the velocity of the cells are givens, so equation (2) is solved for du which is then substituted into equation (1) to solve for the minimum inner radius of the pipe in the detection region. The geometry of the sheath flow region is entirely dictated by the principle of conservation of mass which says that the total mass flow rate of all fluid streams entering a control volume must equal the total mass flow rate of all fluid streams exiting the control volume. Since concentric sheath flow cannot be achieved, two dimension sheath flow is used. In two dimensional sheath flow, the sample fluid enters the system with a channel of sheath fluid on either side. These three streams converge and the total width of the channel is decreased. It is assumed that the same incompressible fluid will be used for the sheath and sample fluids. Using conservation of mass, the following relationship can be found between the required diameter of the focused core (which must be the same as the diameter of the cells) and the geometry and initial velocities of the core and sheath fluids as they enter the system 4 : d v 2 = Ds = vcore D a vd 1 1 vd ( + ) vd 2 2 vd 2 2 Eq. 3 where d is the diameter of the focused core, v c is the velocity of the focused core (and also the maximum velocity of the focused stream), D a is the diameter of the focused stream, v 1 is the initial velocity of the top sheath fluid, v 2 is the initial velocity of the sample fluid, v 3 is the initial velocity of the bottom sheath fluid and D 1, D 2 and D 3 are their respective diameters (figure 2). This method of focusing works because the flow is in the laminar regime (Reynold s number ~ 45). The streams will converge but will not mix since diffusion across those boundaries is very slow. Therefore, the cells in the focused stream will stay in their separate streamlines through the device. Figure 2. Hydrodynamic Focusing Schematic For simplicity, the diameters and velocities of the sheath fluids were set equal and the diameters of all of the inlet fluid channels were set equal to the diameter of the focused stream. This allowed the initial velocities of the system to be solved analytically. Since the channels are square instead of cylindrical, the minimum inner diameter of the pipe set to the width and depth of the flow channels. Horizontal Focusing A fully parameterized two dimensional finite element model was created using Flotran to demonstrate the fluid focusing in the horizontal direction. Units for all models in this paper are in the micro MKS system and the fluid is assumed to be physiological saline. The L2TAN command was used to create the narrowing region between the inlet and outlet channels of the horizontal focuser; rectangles formed the other areas of

4 the focuser. FLUID141 elements were used with no slip boundary conditions specified on all fluid channel walls. The velocities of all three fluid inlets were set as initial conditions and the pressure was set equal to zero at the outlet. Meshing was done with the automatic meshing tool using a SMRTSIZE of 1. A velocity vector plot of the horizontal focusing region is shown in Figure 3. The flow becomes fully developed very quickly after the streams converge and a parabolic velocity profile can be seen at the outlet. The maximum velocity is approximately 0.06 m/s. This is lower than the expected maximum velocity of the system because the flow requires one more focusing step (vertical focusing) to achieve a maximum velocity of 0.1 m/s before entering the detection region. Figure 3. Vector Plot of Two-Dimensional Horizontal Focusing Figure 4 shows a plot of the particle trace of the sample fluid flow. Both the initial sample flow channel and the final flow channel in this model are 35 microns x 35 microns. The width of the focused sample flow is 10 microns as desired. Figure 4. Particle Trace of Two-Dimensional Horizontal Focusing

5 Although two dimensional modeling gives a good approximation of the behavior of the fluid in the system, three dimensional models were created for the system to bring the models closer to reality. The horizontal focuser model was created by extruding the 2D geometry with the VEXT command. FLUID142 elements were used with the same initial and boundary conditions. Figure 5 shows the velocity vector plot for the 3D horizontal focuser. The three dimensional model predicts a maximum velocity of m/s which is significantly higher than the value predicted by the 2D model justifying the decision to create the 3D models. Figure 6 shows the particle trace for the 3D horizontal focuser. Careful examination of the particle trace near the outlet shows that the sample flow has been successfully focused in the horizontal direction but the sample flow still extends over almost the entire width in the vertical direction. The zero fluid velocity at the walls prevents the sample fluid from extending all the way to the walls. Figure 5. Vector Plot of Three-Dimensional Vertical Focusing Figure 6. Particle Trace of Three-Dimensional Vertical Focusing

6 Vertical Focusing A fully parameterized two dimensional finite element model was created using Flotran to demonstrate the fluid focusing in the vertical direction. In this case, the sheath flow enters the channel through square holes in the top and bottom of the channel. Instead of converging with the sample flow and entering a narrowed channel, the sheath flow is entrained by the horizontal flow. The added mass flow rate is sufficient to cause focusing without the need to reduce the total volume of the system. The construction of the vertical focusing model was almost identical to the horizontal focusing model. FLUID141 elements were again used with the no slip boundary condition. The average velocity from the previous model was used as the inlet velocity of the horizontal stream. The inlet velocity of the vertical sheath flow was applied to the inlet lines in the model and the pressure was again set to zero at the outlet. Figure 7 shows a velocity vector plot for the two dimensional vertical focuser. Again, the flow becomes fully developed very quickly and predicts a maximum velocity of m/s. Figure 8 shows a plot of the particle trace of the sample fluid flow. Again, the sample flow is reduced to a height of 10 microns centered in the channel as expected. Figure 7. Vector Plot of Three-Dimensional Horizontal Focusing Figure 8. Particle Trace of Three-Dimensional Vertical Focusing

7 The 3D focuser model was created by extending the 3D horizontal focuser model to include the vertical focusing features. By putting the two models in series, the error of averaging and rounding the velocity of the sample flow as it leaves the horizontal focuser is eliminated. Figure 9 shows the velocity vector plot of the 3D focuser. The maximum predicted velocity is 0.12 m/s or just above the desired maximum. Figure 10 shows the particle trace for the 3D focuser. The sample flow is focused in two dimensions and can be seen in the middle of the channel. Figure 9. Vector Plot of Three-Dimensional Vertical Focusing Figure 10. Particle Trace of Three-Dimensional Vertical Focusing Fluid Flow around the Electrodes If the analysis is to include dielectrophoresis, electrode must be added to the model. A three dimensional model was created to visualize the flow past the electrodes. In this model, two pairs of electrodes (one on each side of the channel) were set inside a rectangular flow channel. The no-slip boundary condition was applied to the electrode surfaces as well as the channel walls. The inlet velocity was determined by the outlet velocity of the 3D focuser. Figure 11 shows the velocity vector plot of the flow past the electrodes

8 and figure 12 shows the particle trace past the electrodes. This clearly shows that the cells in the focuser flow will no longer be perfectly centered in the channel. The cells move 2 to 3.5 microns away from the electrodes depending on their initial position in the stream although they return to their initial position down stream of the electrodes. Since the force that the electrodes exert on the cells is a function of the distance between the cells and the electrodes, this will affect the electrode design and performance. Figure 11. Vector Plot Around Electrodes in Flow Figure 12. Particle Trace Around Electrodes in Flow Full Three-Dimensional Model Finally, a three dimensional fluid model was created that combined all of the elements of the cell sorter including the two outlet channels. With all of the factors affecting the flow combined in one model, the inlet velocities of the sample flow and the sheath flows could be fine tuned to produce the desired outlet cell velocity. Figure 13 shows a velocity vector plot of the entire sorting system which predicts the target

9 maximum velocity of 0.1 m/s. Figure 14 shows a particle trace of the sample fluid as it travels through the sorting system. Figure 13. Vector Plot of Full Three-Dimensional Device Figure 14. Particle Trace of Full Three-Dimensional Device Modeling in the Electrical Domain Sorting in a microfluidic device can be done in a variety of ways, but dielectrophoresis (DEP) is one of the more promising methods available. Dielectric particles inside an inhomogeneous electric field will move towards regions of high field strength (positive DEP) or low strength (negative DEP) depending on the

10 properties of the particles, liquid and electrode frequency. 5 Mammalian cells can be approximated as spherical dipoles that move towards areas of low field strength or away from the source electrodes. The time averaged dielectrophoretic force on a spherical particle of radius r in a field E is given by the equation 6 : where u* is the Clausius-Mossotti (CM) ratio: F = r E u Eq * 2πε m Re( ) u * * * ( ε particle εmedium ) * * ε particle + εmedium = ( 2 ) Eq 5 * and the complex permittivity is defined asε = ε jσ / w. Since the CM ratio for the example system is negative, the cells are pushed away from fields of high concentration. A two dimensional model of the electrodes in the fluid flow was created using PLANE67 (2D Thermal Electric Solid) elements to model the behavior of the system in the electrical domain. The electrodes were assumed to be gold rectangular boxes 3 microns wide and 15 microns deep with a 6 micron gap between the top and bottom electrode (figure 15). A voltage of +5 V was applied to the nodes of the top line of the top electrode. A voltage of -5 V was applied to the nodes of the bottom line of the bottom electrode. A convection to bulk temperature was also applied at the right side of the model to allow the solution to converge. (In the actual system, the dominant thermal initial condition is the temperature of the fluid as it enters the electrode region while the thermal resistances of the walls and the device packaging will have almost no impact. For this reason, the two dimensional model cannot be used for calculating temperature increases and is presented only to demonstrate the shape of the electric fields generated.) In this model, DC current was used for simplicity. Either AC or DC may used, but DC has the disadvantage of generating electrolysis, which may be a concern for applications with biological specimens. Figure 15. Material Plot of Two-Dimensional Electrode Geometry Figure 16 shows the distribution of voltage through the electrodes and the surrounding fluid. Since gold is an excellent electrical conductor with low resistance, both electrodes appear to have constant voltages while the fluid shows significant gradients. Figure 17 shows the sum of the electric fields in the system. This electric field can be exported and used to calculate the dielectrophoretic force applied to the cells and evaluate the performance of the sorter, however it cannot be calculated directly in ANSYS.

11 Figure 16. Voltage Distribution of Two-Dimensional Electrode Geometry Figure 17. Electric Field Sum for Two-Dimensional Electrode Geometry Equation 4 can be reduced to a constant multiplied by the gradient of the electric field squared, meaning that the dielectrophoretic forces may be calculated once the electric field is calculated. However, ANSYS does not calculate the gradient of the electric field (or the electric field squared) and user calculation of the gradient of the electric field is not convenient at the current release of ANSYS (Release 7.1). This presents several logistic problems. 1. The calculation of the gradient requires that the nodes be sorted by coordinate in insure correct calculation. (To understand this, imagine an ANSYS area meshed in the standard manner (lines first and then the interior of the area). If the electric field distribution is cylindrical with a variation in the X direction and no variation in the Y direction, calculation in the ANSYS vector along an end line would result in a zero value when the actual value would be very steep.) 2. ANSYS does not permit the sorting of one vector based upon a second vector.

12 3. ANSYS does not permit calculating the DER1 for vector operations when the denominator vector is not monotonically increasing. For 2 and 3 dimensional models, the denominator values are likely to loop from minimum to maximum repeatedly. This will require a check on the denominator values. 4. User calculation of the gradient will require *DO and *IF commands. The values for electric field and for the node coordinates can be loaded into an ANSYS vector and the necessary calculations performed. This calculation will be time consuming for large models and it may be more convenient to export the data to another program, perform the calculations and return the force values as loads for the subsequent calculation. The best options to directly calculate the DEP forces on the cells in the fluid flow would be to export the electric field and do the calculations in a separate program that was intended to do large scale linear algebra or to write a custom macro that would calculate the gradient of the element electric energy (ETABLE, SENE) distribution. Joule Heating One of the outputs of the electric domain analysis is the Joule heating that occurs as a result of the electric current. The fluid between the electrodes acts as an electrical resistor. Some of the energy carried by the current is passed from one electrode to the other, and some of it is dissipated as heat. The joule heating can be combined with the fluid model to calculate the temperature rise and distribution in the fluid flow to ensure that the cells will not go into thermal shock or be killed by the sorting process. Since electrical conductivity (the reciprocal of resistivity) is a function of frequency, the joule heating is also a function of frequency. Electrical conductivity is given by σ = σ0 + jwε where σ o is the base conductivity, j is equal to the square root of -1, w is electrode frequency, and ε is the permittivity of the medium. The frequency dependence of the conductivity of saline at 1 MHz was found to be at least two orders of magnitude smaller than the base conductivity, so for this example, the joule heating will be approximately the same for AC and DC currents. A three dimensional model of two electrode pairs in the fluid was created to display the Joule heating between the electrodes. The Joule heating model uses SOLID69 elements and the same geometry and initial conditions as the electric field model. Figure 18 shows a cut away view of the model. The area of highest heat dissipation is in the area between the electrodes as expected but has a very limited range away from the electrodes. Figure 18. Joule Heating Distribution in Saline

13 Modeling in the Thermal Domain Finally, the results of the Joule heating model must be transferred to the thermal domain to calculate the temperature increase in the fluid flow. The thermal model uses SOLID70 (Thermal Solid) elements with the same geometry as the electric models. One of the key options of the SOLID70 elements allows the specification of a velocity in the solid elements. In this way, the velocity of the fluid flow within the channels was added to the model. Power graphics for all thermal models was turned off to ensure that the true maximum temperature of the system would be shown. First, an approximate model of the temperature distribution was created using an average value of the joule heating. To apply the boundary conditions, the element table from the electric model was generated, summed and then averaged over the area of the model which was affected by the Joule heating. This average Joule heating value was then applied to the same affected area in the thermal model to give a close approximation of the temperature rise in the fluid. The temperature distribution from this approximation can be seen in Figure 19. Figure 19. Approximate Temperature Distribution in Saline To test the validity of the approximation, a second model was created for the temperature distribution. In the second case, the exact joule heating distribution was read into the thermal model using the LDREAD,HGEN command. Figure 20 shows the temperature distribution created by reading the Joule heating element table directly into the thermal model, thus bypassing the approximation and leading to a better solution. The exact solution predicts a maximum temperature of degrees C while the approximate solution predicts a maximum temperature of 28 degrees C. Thus, the exact solution is necessary to calculate secondary effects like electrohydrodynamic and density driven flows. However, in both cases the temperature distribution is approximately the same and both predict that the temperature of the sample fluid will rise only a few degrees, indicating that the approximation holds for fluid far away from the electrodes.

14 Figure 20. Exact Temperature Distribution in Saline The high temperature increase in the electrode region prompted the creation of one last model which assumed the use of an isotonic sucrose solution with a conductivity of 0.12 S/m based on the work of Gimsa and Wachner. 7 The electrical conductivity of the sucrose solution in much higher than the conductivity of the physiological saline solution which results is much lower temperature increases in the fluid. Figure 21 shows the joule heating of the electrodes submerged in a sucrose solution. Figure 22 shows the temperature distribution in the sucrose solution using the exact joule heating distribution. The temperature profile looks identical to those generated in the previous models, however the maximum temperature in this system is only C. Figure 21. Approximate Temperature Distribution in Sucrose

15 Figure 22. Exact Temperature Distribution in Sucrose Conclusion It has been shown that the behavior of microfluidic systems that operate in a number of physical domains can be modeled and optimized using finite element methods. However, dielectrophoretic forces cannot be modeled in ANSYS at this time without writing a custom macro to perform a direct calculation. The creation of this macro will be the focus of future work. Acknowledgements The authors would like to thank the MIT Dept. of Chemical Engineering MicroChemical Systems Technology Center for its support of this work, Prof. Alexander Slocum, Jason Kralj, Eddy Karat, Elena Antonova and Dave Looman for their feedback, and the MIT class for the Spring, 2003 for their contributions to a similar project. M. K. Thompson would like to thank the National Science Foundation for its support of her research through a Graduate Research Fellowship. The authors would also like to thank ANSYS, Inc. for providing the software to make this analysis possible. References Fiedler et al. Anal. Chem. 1998, 70, Lee et al. Journal of Fluids Engineering, 2001, (123) Lee et al. Journal of Fluids Engineering, 2001, (123) Fiedler et al. Anal. Chem. 1998, 70, Foster et al. Biophys. J. 1992,63, Gimsa and Wachner. Biophys J, August 1998, 75,