CFD-Based, Lagrangian-Eulerian coupling approach for Magnetophoretic Particle Capture

Transcription

1 CFD-Based, Lagrangian-Eulerian couling aroach for Magnetohoretic Particle Cature Saud A. KHASHAN 1*, Edward FURLANI,3 * Corresonding author: Fax: ; skhashan@uaeu.ac.ae 1: Mechanical Engineering Deartment, United Arab Emirates University, UAE : Det. of Chemical and Biological Engineering 3: Det. of Electrical Engineering, University at Buffalo, SUNY, efurlani@buffalo.edu, Abstract We study magnetohoretic cature of magnetic articles in microfluidic devices and resent a arametric characterization for the cature efficiency. We model article transort and cature using a comutational fluid dynamic, CFD-based, Lagrangian-Eulerian aroach that takes into account the dominant article forces and article-fluid couling. We introduced two dimensionless grous that characterize article cature, one that scales the magnetic and hydrodynamic forces on the article and another that scales the distance to the magnetic field source. We use the model to arameterize cature efficiency with resect to the dimensionless numbers for both one-way and two-way article-fluid couling. We demonstrate that for dilute susensions, the simlified one-way couling analysis marginally underredicts the cature efficiency comuted using the two-way fully couled analysis. Keywords: Magnetohoresis, magnetic articles, microfluidic devices, MEMS 1. Introduction Over the last several years the interest in Magnetohoretic Particle Cature in Microfluidic Systems has grown dramatically, esecially for alications in fields such as microbiology and biotechnology (Furlani 010a; Furlani 010b; Ganguly 010;Gijs 004). Magnetic articles can be functionalized to selectively bind to target biomaterials such as roteins, enzymes, nucleic acids or whole cells thereby enabling magnetohoretic control of these materials (Pankhurst et al. 009; Berry 009; Arrueboa et al. 007; Majewski and Thierry 007; Safarik and Safarikova 00). This caability is being leveraged through advances in microfluidics that enable miniature biochemical laboratories to be integrated into a single microsystem, i.e. Lab-on-a-chi and micro total analysis systems (TAS). Such microsystems tyically range in size from millimeters to the centimeters and are usually made using lanar glass-, silicon- or olymerbased substrates. Fluidic structures within these systems such as mixing chambers and flow channels range from several to hundreds of microns in size. Magnetic functionality can be integrated into these systems by embedding magnetic field source elements in the substrate, in roximity to the flow channels. These elements can be magnetically assive structures such as nickel-based microbars, or active voltage-driven conductors (Choi et. al. 001; Furlani 001; Smistru et. al. 005; Furlani 006; Furlani and Sahoo 006; Furlani et al. 007; Smistru et. al. 008). In the former case, an external field source is used to magnetize the elements. In the latter case, circuitry is required to activate the conductors. In either case, the source elements generate a magnetic field distribution that gives rise to a magnetic force on magnetically labeled material as it flows through a microchannel, thereby enabling magnetohoretic control to sort or immobilize the material. Continuous flow searation enables a relatively high throughut of a samle and it allows for real-time monitoring of searation efficiency. The latter caability enables on-line feedback for otimization, which can be - 1 -

2 achieved by adjusting arameters such as the flow rate and the induced force. In addition, the continuous flow aroach can accommodate multile inlets and outlets for the simultaneous searation of multile samle comonents. The continuous nature of the searation rocess lends itself to a high level of integration with other ustream or downstream rocess. Various mechanisms have been used to fractionate samle comonents in continuous flow alications including electric forces, standing ultrasonic waves and arrangements of obstacles in the flow ath. Magnetic searation has also been successfully used for this rocess. An early commercialized magnetic cell searation system that utilized ermanent magnets to generate the magnetic force was emloyed to searate cells labeled with biotinylated sueraramagnetic ferrit-dextran beads (Miltonic et al. 1990). Other systems using micromagnetic elements have been used for the comlete searation of 1µm magnetic articles (Smistru et al. 006). In this aer, we develo a scaled fullycouled article-fluid model for redicting the transort and cature of magnetic articles in microfluidic systems. The model involves a CFD-based Eulerian-Lagrangian analysis that includes the dominant forces on the articles and two-way momentum transfer between the moving articles and the flow field. To our knowledge, this is one of the first studies of article searation in microfluidic system that is based on fully-couled article-fluid CFD analysis. Our model takes into account key magnetic variables including the size and magnetic roerties of the articles and the magnetic field arameters, i.e. the magnitude and gradient of the field as well as the distance of the field source to the microchannel. It also accounts for fluidic arameters including the dimensions of the microchannel and the fluid viscosity. Several simlifying assumtions are made in the analysis. Secifically, we assume the fluid is incomressible and that the flow is laminar, which is usually the case in microfluidic devices. We further assume that the article susension is sufficiently dilute so that article-article interactions can be neglected. Particle coagulation as well as build-u and subsequent channel blockage are also neglected. The article magnetism is assumed to be in the linear range, below saturation. We also assume that the articles enter the channel with a uniform satial distribution. In the develoment of the model we introduce two dimensionless grous that characterize article cature, one that scales the magnetic and hydrodynamic forces on the article and another that scales the distance to the magnetic field source. We use the model to arameterize cature efficiency with resect to the dimensionless numbers for both one-way and two-way article-fluid couling. For one-way couling in which the article motion does not alter the flow momentum, we develo correlations that rovide insight into system erformance towards otimization. We quantify for the first time the difference in cature efficiency redicted using one-way versus two-way couling analysis. Our work demonstrates that one-way couling rovides a conservative estimate of cature efficiency. Secifically, it over redicts the magnetic force needed for article cature comared to the more rigorous fully-couled analysis. As such, one-way couling can be used for raid arametric screening of article cature erformance. However, more accurate redictions require two-way article-fluid couling analysis. This is esecially true when considering higher cature efficiencies and/or higher article concentrations.. Mathematical Formulation Broadly seaking, two aroaches are commonly used to model article transort in microfluidic systems, the Eulerian and the Lagrangian aroaches. In the Eulerian aroach articles are modeled collectively in terms of a time-deendent satially-varying concentration. The concentration is governed by a PDE that accounts for both force-induced drift and Brownian diffusivity of the articles (Khashan et al. 011; Furlani and Ng 008). - -

3 The Eulerian method is used to study the behavior of sub-micron articles when the Brownian diffusivity, as modeled by a Stokes- Einstein-like equation, has a significant imact on the article transort. In this case, the articles are assumed to resond instantaneously (with a negligibly short acceleration) to a balance between the fluidic drag and magnetic forces. The article moves at a constant terminal velocity relative to the base fluid, which is commonly referred to as the magnetohoretic velocity. This velocity adds to fluid velocity to define the overall article motion that drives its convective flux in the Eulerian frame work. The PDE that governs the article concentration has the same form as the fluidic momentum equations and can be solved simultaneously with these equations using almost the same numeric algorithm. Thus, an Eulerian-based article transort model can be readily integrated into Eulerian-based CFD codes (Khashan et al. 011; Khashan and Haik 006). In the Lagrangian aroach, articles are treated as discrete entities and the trajectory of each article is determined by integrating the Newtonian equations of motion. The fluid is governed by the Navier-Stokes equations and the flow field is usually considered to be incomressible. This aroach is aroriate for larger articles when Brownian motion is negligible. In this aer we use a combined Lagrangian- Eulerian CFD-based aroach to model the fully-couled article-fluid behavior of magnetic articles in a microfluidic channel under the influence of a magnetic force. We use a Lagrangian analysis to track the motion of individual articles, and we coule the article motion to the fluid by introducing a article force sink into the Navier Stokes momentum equations, which are solved using an Eulerian-based CFD analysis. The alication of two-way couling is a key result of this work and distinguishes it from most other studies that consider only one-way article-fluid couling. We quantify for the first time the difference in cature efficiency redicted using one-way versus two-way couling analysis as described below. The magnetic force as alied on the magnetic article susended in a state of dilute susension in a non-conducting carrier fluid can be exressed as (Furlani EP 010c) 1 Fmag o mvh (1) where is the free-sace magnetic 0 ermeability and m (dimensionless) is the effective volume-averaged suscetibility of the linearly magnetic article, which is estimated based on its magnetically active volume and relative to the suscetibility of the carrier fluid, i.e. the magnetic moments of the olymericbased microarticles are derived comletely from the magnetic nanocrystals encasulated or embedded in the olymer matrix. Here H H H 1/ is the magnitude of the alied external magnetic field (A/m), which can be related to the magnetic field induction B (in T) of a article susended in a non-magnetic or weakly diamagnetic fluid by B H. o Both H and B can be determined from Maxwell s field equations under magnetostatic conditions. With the article s mobility defined as 3d 1 the adoted article motion takes the form: du 1 m b uu V g dx () 1 o mvh The variables u, and are the instantaneous local velocity vector, density and the molecular viscosity, resectively, of the carrier fluid, a and V are the density, radius and volume of the article, and g is the gravitational acceleration. The trajectory equations are solved by stewise integration over discrete time stes. Integration of time in article motion equation yields the velocity of the article at each oint along the trajectory as du 1 uu a (3) dx - 3 -

4 where 1 V g omvh a (4) m and mb. The trajectory can be redicted using dx dt u (5) Here resents the time required for a article to resond to changes in the base fluid motion, also, referred to as the article relaxation time. Note that Eqs. (3) and (5) are a set of couled ordinary differential equations. These can be integrated using a number of different numerical techniques. We have used a fourth-order Runge-Kutta method for our analysis (Furlani and Sahoo 006). We model the fluid hase using a Eulerian aroach. The fluid velocity field is described by the Eulerian incomressible Navier-Stokes equations u 0 (6) u uu P u t (7) N cell, 1 f fj Vcell j1 In this study, the article volume fraction is sufficiently low so that its effect on the continuity equation and on the inertia and stress flux terms of the momentum equation is neglected. The two-way couling is accounted for through the last sink term that accounts for the drag, buoyancy and magnetic forces (the right-hand side of the Lagrangian equation) for all articles in each fluidic comutational cell, where V cell is the volume of a comutational cell, N is the number cell, of articles in that cell and f fj is the fluid force on the j'th article in that cell. In other words, force f simulates the interhase fj momentum transfer from the continuous fluid to a single j'th article. This force is equal but oosite to the force that a article exerts on the fluid ( f ), which is thus the sink term for fj continuous-hase equation. It is imortant to note that for this aroach to work, the article size should be much smaller than the size of the cells that make u the fluidic comutational domain. 3 Flow-Magnetic Configurations We aly the model described above to the analysis of article transort and cature in a Newtonian fluid flowing in a two-dimensional microchannel as shown in Figure 1. We erform a arametric analysis of this system about a base configuration in which the microchannel has a height h = 100 m and a length L = 1000 m. A magnetic diole field source in the form of a air of anti-arallel current carrying conductors is ositioned at a distance ymag 100m beneath the lower wall of the microchannel, midway along its length ( x 500m ). mag Figure 1: Microfluidic System with magnetic diole source. In the following analysis the width of the channel (into the age) is assumed to be sufficiently large to justify a two-dimensional flow analysis, i.e. we ignore flow variation in that direction. The channel height is large enough to justify a continuum Newtonian fluid analysis and we ignore article-wall hydrodynamic interactions. The length of the channel is sufficiently long so that hydrodynamic effects at the inlet and outlet do not imact the magnetic field-directed article motion near the mid-length of the channel. In our model, incomressible fluid enters the channel at the left side (inlet) with a fully develoed laminar flow rofile in which the average velocity is u i = 00 m /s. The outlet ressure is set to zero. Solid sherical magnetic articles are injected into the - 4 -

5 comutational domain at the inlet with a uniform distribution over the entrance lane. We assume that the fluid is water, which is essentially nonmagnetic. The viscosity and density are Ns/m 3 and f 1000 kg/m, resectively. The roerties of the magnetic articles are chosen to be comatible with the MyOne beads ( The article has a radius a 0.55 m, density kg/m, saturation magnetization 4 M s A/m and an effective suscetibility 1.4. The magnetization of the articles will be below saturation as long as B M / where B B (Furlani et o s al. 007). This will occur as long as B does not exceed 38 mt, which was the case in all simulations. The article volume fraction for our base analysis is %. It is imortant to note that the article size is much smaller that the grid size in the comutational domain so that many articles can exist within a comutational cell. This condition is needed when comuting the momentum transfer from the articles to the fluid. Lastly, the channel walls are assumed to be nonmagnetic and therefore are treated as boundaries in the simulation model. 3 Results and Analysis The erformance criterion for magnetic searation is defined in terms of collection efficiency, which is the ercentage of incoming articles that are catured (traed) in the microchannel by the magnetic force. First, we sought a dimensionless grou that scales both the magnetic and drag force on the articles. Accordingly, we resent the following two dimensionless grous 1 V / 6auh 5 (8) o i and ymag / h. (9) where Id /, I is the current in each conductor and d is the distance between the conductors It should be noted that is indeendent of the location of the magnetic source, while deends on the osition of the source and therefore the field gradient that it generates within the microchannel. Note that scaling the magnetic and fluidic forces with directly accounts for the increased magnetic force due to larger effective article suscetibility and size. It also account for the article s mobility, defined as 6a 1, in different viscous base fluids. = 64 µam = 19 µam = 15 µam (critical cature) Figure : Comarison of redicted article trajectories using one-way couling (left) and two-way couling (right) for the base roblem (h=100 µm, L=1000 µm, yc = ymag=100 µm, xmag=500 µm, ui=00 µm/s, d= 1.05 µm, = 1.4, and = 10-3 N.s/m). The trajectories (samled over a selected section of the microchannel ) show an increased article cature efficiency with increased diole field strength. From this figure, we find that the one-way couling analysis under-redicts article cature relative to the two-way fully-couled analysis. That is, the magnetic force required to achieve a given cature efficiency is redicted to be higher using one-way couling as comared to the fully-couled analysis. The =15 µam calculations are illustrative of this effect. Secifically, for one-way couling, this field strength renders critial article cature, i.e. all articles are catured, albeit some at locations beyond the diole field source. However, it is obvious that the critial article cature occurs for at a lower field strength (i.e

6 for between 19 µam 15 µam) in the twoway couling analysis. One reason for this is that two-way couled articles act to decrease the fluid velocity in roximity to the magnet, which in turn romotes article cature in this region. Two-way couled articles in effect increase the bulk effective viscosity of the fluid, whereas one-way couled articles have no imact on the momentum or viscoity or fluid flow. Next, we study the cature efficiency CE as a function of the dimensionless number, where Figure 5 resents the critical cr corresondence at which comlete cature CE 1 can be achieved under the assumtion of one-way couling. A lower article magnetization is needed when the magnetic source is brought closer to the microchannel, a condition that obviously corresonds to a higher gradient and, therefore, a higher magnetic force. Over the considered range, the critical can be exressed in the cr following analytical form number of articles catured CE (10) total number of articles Figure 3 shows the cature efficiency CE vs. with fixed ( 1 ) for one-way couling. Note that a single curve is obtained for all variations of the inlet velocity, article diameter, article magnetic suscetibility and fluid viscosity as long as is held constant. In our study, articles are assumed to be catured once they contact the lower late (at y y mag ). In many cases, they are catured at locations well before the outlet as shown in Fig. c, and it might be more aroriate to define cature efficiency with resect to a desired target site rather than the whole length of the lower wall. This is esecially the case when confined article focusing or searation is desired. The CE corresondence at fixed is also illustrated in Fig. 3 for different values of the hydraulic channel diameter h. It is evident from the figure that the unique corresondence holds even though the inlet mass flow rate of the base fluid ( uh i ) is doubled for each cases. Figure 4 below shows two distinct CE relations for equal to 1.0 and 0.5, resectively. These lots indicate that at smaller (i.e. the magnetic source closer to the microchannel), a higher cature efficiency can be obtained at smaller. For instance, at =0.5, the cature efficiency is about 40% and 80%, for =1 and 0.5, resectively. Figure 3: The cature efficiency CE verses at =1.0 (one-way couling analysis). The base roblem corresonds to h=100 µm, L=1000 µm, y c = y mag =100 µm, x mag =500 µm, u i =00 µm/s, d = 1.05 µm, = 1.4, and = 10-3 N.s/m. Figure 4: The cature efficiency CE verses at =1.0. and 0.5 (one-way couling analysis). The arameter, as defined by equation 0 can be any value. cr 3 cr (11) 5 h - 6 -

7 This equation rovides a direct means to relate the following arameters: (a grouing of hydrodynamics and magnetization arameters), h (the microchannel height) and (the required critical diole strength) to the arameter. Thus, we can use this exression to correlate the critical and values at which we have comlete cature. Furthermore, the definition of indicates that this equation may be used to relate a variation in any of, cr or h with resect to at comlete cature. Another imortant oint is that this correlation, which is based on oneway couling, can be used as a conservative design arameter for the more rigorous twoway couling. Figure 6 shows a comarison of the redicted cature efficiency using one-way versus a twoway couling analysis for two different values of. Note that for both values of the oneway and two-way couled simulations yield aroximately the same functional rofile for values of that render CE 0.6. However, the former under redicts the latter for larger vales of. This last result is imortant, and is a key finding of this study. Secifically, we quantify for the first time the difference in cature efficiency redicted using one-way versus two-way couling analysis. Secifically, we show that one-way couling over redicts the magnetic force needed for article cature as comared to the more rigorous fully-couled analysis, esecially at higher cature efficiencies and/or higher article concentrations. 4 Conclusion There are many advantages to using magneticbased biomaniulation and it has been successfully imlemented in microfluidic devices for alications that include biosearation and continuous cell sorting. However, desite the versatility and otential imact of this aroach, the develoment of rigorous transort models that account for fully-couled article-fluid interactions are lacking. In this aer, we have resented a model for redicting field-directed transort of colloidal magnetic articles in microfluidic systems taking into account two-way articlefluid couling. The method involves the use a CFD-based Lagrangian-Eulerian aroach to redict the transort and cature of articles and the imact of their motion on the flow field. We have demonstrated the method via alication to article transort and cature in a two-dimensional laminar flow field within a microfluidic channel. The magnetic force on the articles is generated by a diole field source, which is in the form of two antiarallel current carrying wires located beneath the channel and midway along its length. We have introduced two dimensionless grous for this analysis that characterize article cature. Figure 5: CE ma with resect to and (one-way couling analysis). The shaded area below CE=1.0 corresonds to comlete article cature. One grou scales the magnetic and hydrodynamic forces on the article and the other scales the distance to the magnetic field source. We show that one-way couling over redicts the magnetic force needed for article cature as comared to two-way couling analysis. As such, it can be used for raid arametric screening of novel magnetic article cature systems

8 Figure 6: The cature efficiency CE verses at =1.0 (one-way couling analysis). The base roblem corresonds to h=100 µm, L=1000 µm, y c = y mag =100 µm, x mag =500 µm, u i =00 µm/s, d = 1.05 µm, = 1.4, and = 10-3 N.s/m. ACKNOWLEDGEMENT The first author acknowledges the financial suort received from the Research Affairs at the UAE University under contract number /10. References Arrueboa, M., Fernández-Pachecoa, R., Ibarraa, R.M. et al., 007, Magnetic nanoarticles for drug delivery. Nanotoday (3), -3. Berry, C. C., 009, Progress in functionalization of magnetic nanoarticles for alications in biomedicine. J. Phys. D: Al. Phys. 4, 4003 (9). Choi, J-W., Liakooulos, T. M. and Ahn, C. H., 001, An on-chi magnetic bead searator using siral electromagnets with semi-encasulated ermalloy. Biosensors & Bioelectronics. 16, Furlani, E. P., 010a, Magnetic Biotransort: Analysis and Alications. Materials, 3(4), Furlani, E. P., 010b, Particle Transort in Magnetohoretic Microsystems. In: Kumar C. S. S. R. (ed), Microfluidic Devices in Nanotechnology: Fundamental Concets, John Wiley, NY, : Furlani, E. P., 010c, Nanoscale Magnetic Biotransort. In: Sattler K (ed) Handbook of Nanohysics, CRC Press. Furlani, E. P., 006, Analysis of Particle Transort in a Magnetohoretic Microsystem. J. Al. Phys. 99 (), 0491 (11). Furlani, E. P., 001, Permanent Magnet and Electromechanical Devices; Materials, Analysis and Alications. Academic Press, NY. Furlani, E. P., and Ng, K. C., 008, Nanoscale magnetic biotransort with alication to magnetofection. Phys. Rev. E 77, (8 ages). Furlani, E. P., and Sahoo, Y., 006, Analytical model for the magnetic field and force in a magnetohoretic microsystem. J. Phys. D - Al. Phys. 39, Furlani, E. P., Sahoo, Y., Ng, K. C., Wortman, J. C., and Monk, T. E., 007, A model for redicting magnetic article cature in a microfluidic biosearator. Biomedical Microdevices 9 (4), Ganguly, R., and Puri, I. K., 010, Microfluidic transort in magnetic MEMS and biomems. Wiley Interdiscilinary Reviews: Nanomedicine and Nanobiotechnology (4), Gijs, M. A. M., 004, Magnetic bead handling onchi: new oortunities for analytical alications. Microfluidics and Nanofluidics 1 (1), 40. Khashan, S. A., Elnajjar, E., and Haik, Y., 011, Numerical Simulation of the Continuous Biomagnetic Searation in a Two-dimensional Channel, International Journal of Multihase Flow, In-ress. Khashan, S. A., and Haik, Y., 006, Numerical Simulation of Bio-magnetic Fluid Downstream an Eccentric Stenotic Orifice, Physics of Fluids. 18 (11), Majewski, P., and Thierry, B., 007, Functionalized magnetite nanoarticles - synthesis, roerties, and bio- Alications. Critical Reviews in Solid State and Materials Sciences 3, Miltenyi, S., Muller, W., Weichel, W., Radbruch, A., 1990, High gradient magnetic cell searation with MACS. Cytometry 11, Pankhurst, Q. A., Thanh. N. K. T., Jones, S. K., and Dobson, J., 009, Progress in alications of magnetic nanoarticles in biomedicine. J. Phys. D: Al. Phys 4, Safarik, I., and Safarikova, M., 00, Magnetic nanoarticles and biosciences. Monatshefte fur Chemie 133, Smistru, K., Hansen, O., Bruus, H., and Hansen, M. F., 005, Magnetic searation in microfluidic systems using microfabricated electromagnets: exeriments and simulations. J. of Magnetism and Magnetic Materials 93, Smistru, K., Bu, M. Q., Wolff, A., Bruus, H., Hansen, M. F., 008, Theoretical analysis of a new, efficient microfluidic magnetic bead searator based on magnetic structures on multile length scales. Microfluidics and Nanofluidics 4 (6), Smistru, K., Lund-Olesen, T., Hansen, M. F., and Tang, P. T., 006, Microfluidic magnetic searator using an array of soft magnetic elements. J. Al. Phys. 99, 08P