Proceedings of Meetings on Acoustics Volue 19, 2013 htt://acousticalsociety.org/ ICA 2013 Montreal Montreal, Canada 2-7 June 2013 Physical Acoustics Session 1PAb: Acoustics in Microfluidics and for Particle Searation III: Biological Alications 1PAb4. Nuerical odeling for analyzing icrofluidic acoustohoretic otion of cells and articles with alication to identification of vibro-acoustic roerties Zhongzheng Liu*, Han Wang, Aru Han and Yong-Joe Ki *Corresonding author's address: Mechanical Engineering, Texas A&M University, College Station, Texas 77843, liuzz008@tau.edu Microfluidic acoustohoretic cell/article searation has gained significant interest recently. The otion of cells/articles in acoustohoretic searation is coonly analyzed by using a one-diensional (1-D) analytical odel in a "static" fluid ediu, while the effects of acoustic streaing, viscous boundary layers, and 2-D/3-D geoetries are usually not considered. This akes it challenging to accurately redict the otion of cell/articles. Here a nuerical odeling ethod for accurately analyzing the acoustohoretic otion is resented by including the aforeentioned effects in the odel. The first-order ressure and the second-order streaing velocity are first calculated by using a higherorder finite difference ethod. Then, acoustohoretic force is calculated based on the force equation roosed by Gorkov and is alied to the Newton's second law to calculate the acoustohoretic otion. The effects of acoustic streaing, viscous boundary layers, and 2-D geoetry on the otion of cells/articles are studied by coaring the to 1-D odeling results. Since the acoustohoretic otion deends on the vibroacoustic roerties (e.g., density, coressibility, and size) of articles/cells, these roerties can be estiated by otially fitting the exeriental and siulated trajectories. The roerties of olystyrene beads obtained fro exeriental results through the resented nuerical analysis show good agreeent with data reorted in literature. Published by the Acoustical Society of Aerica through the Aerican Institute of Physics 2013 Acoustical Society of Aerica [DOI: 10.1121/1.4799390] Received 23 Jan 2013; ublished 2 Jun 2013 Proceedings of Meetings on Acoustics, Vol. 19, 045015 (2013) Page 1
INTRODUCTION Microfluidic, acoustohoretic searation of cells/icroarticles can have relatively high energy efficiency and high throughut when coared with other searation ethods using dieletrohoretic, agnetohoretic, and inertia forces [1]. In the revious work on icrofluidic acoustohoresis, the tie-averaged acoustohoretic force is derived based on the first- and second-order acoustic ressure and article velocities. It is then reresented in ters of the first-order acoustic ressure and article velocities only [2, 3, 4]. In articular, the acoustohoretic force equation, roosed by Gorkov [4] in Ref 4, is reresented in ters of the derivatives of the first-order acoustic ressure and article velocities, that can be alicable to arbitrary 2-D or 3-D acoustic fields in an inviscid, static fluid ediu. In the acoustohoretic force reresentations, the effects of oving fluid edia, viscous boundary layers, and acoustic streaing are not considered. In this aer, a nuerical ethod including the aforeentioned effects is resented to accurately redict the acoustohoretic otion of cells/icroarticles. In the roosed ethod, zeroth-order fluid flow fields and first- and second-order acoustic fields are firstly calculated by using a erturbation ethod and a six-order finite difference ethod. Here, the zeroth-order fluid flow is obtained to consider the effects of the oving fluid ediu and the viscous boundary layers on the first-order and second-order acoustic fields. In addition, the second-order, tie-indeendent acoustic article velocity is calculated as the acoustic streaing. Then, the Gorkov s equation [4] is used to calculate the acoustohoretic force in ters of the nuerically-redicted, first-order acoustic ressure and article velocities. The acoustohoretic force, the zeroth-order fluid velocities, and the acoustic velocities are then alied to the Newton s equation of otion to obtain the acoustohoretic otion of cells/icroarticles. Finally, the nuerically-redicted trajectories of a olystyrene bead in a one-diensional (1-D) static icrochannel are otially fitted to exeriental trajectories to identify the roerties (e.g., density and coressibility) of the olystyrene bead. THEORY Acoustic fields generated by an ultrasonic excitation in a icrofluidic channel can be obtained fro the Mass and Moentu Conservation Equations and the State Equation. These three governing equations in a coressible, isotroic, viscous Newtonian fluid ediu can be then reresented as v t v t v v 2 v v 2 2. 0 2 0 0 0 (1a) (1b) (1c) By using a erturbation ethod, the above equations can be decoosed into zeroth-, first- and second-order governing equations. Based on the satial discretization and the sixth-order difference oerators defined in Ref 5, the decoosed governing equations are then exressed in atrix fors. The zeroth-, first- and second-order governing equations can be solved algebraically in sequence with the aroriate boundary conditions. This rocedure is described in detail in Ref 5. By substituting the calculated first-order acoustic ressure and article velocities into the equation roosed by Govkov [4], the acoustohoretic force can be calculated. In the roosed odeling rocedure, the viscous drag force and the acoustohoretic force are only considered since the effects of other forces such as buoyance, gravity, and hydrodynaic focusing forces on the acoustohoretic otion are uch ore insignificant than the viscous and acoustohoretic forces. Fro the Newton s second law of otion, the cell or icroarticle s equation of otion is reresented as [5] U QU FU, t (2) t where Proceedings of Meetings on Acoustics, Vol. 19, 045015 (2013) Page 2
6a Fax 6avx 0 0 0 6a Fay 6av y U = [v x v y r x r y ], Q 0 0 0, and F. 1 0 0 0 0 0 1 0 0 0 In Eq. (2), v j is the cell or icroarticle s velocity in the j-direction (j = x or y), r j is the cell or icroarticle s osition, is the dynaic viscosity of the fluid ediu, a is the radius of the cell or icroarticle, is the ass of the cell or icroarticle, F aj is the acoustohoretic forces in the j-direction, v j is the total tie-indeendent fluid ediu velocity in the j-direction including the zeroth-order fluid flow velocities and the acoustic velocities. In this case, the acoustic streaing is included in the v j ters. Eq. (2) is solved for the cell or icroarticle s location and velocity vectors by using the fourth-order Runge-Kutta ethod. SIMULATION RESULTS AND DISCUSSION The siulation setu is shown in Fig. 1. As shown in Fig. 1, an acoustic excitation is alied on both the sidewalls. The fluid ediu is water and the fluid flow at the inlet of the icrofluidic channel is set to be arabolic with a satially-averaged velocity of 14 /s at the ustrea boundary (see Fig. 1). The excitation frequency is deterined at the first half-wavelength resonance frequency in the y-direction (i.e., f = 2.117 MHz). The nuerically-redicted, first-order acoustic ressure and acoustic streaing velocities are shown in Fig. 2. Figure 1 Siulation setu of the icrofluidic channel with acoustic excitation. Figure 2 Nuerically-redicted acoustic fields: (a) Alitude of first-order, linear acoustic ressure, and (b) Tie-indeendent, second-order acoustic streaing velocity. In Fig. 2(b), the grid nubers are 5714 in the x-direction and 50 in the y-direction. In Fig. (2b), the four vortices of the acoustic streaing, can be observed, whose the axiu velocity alitude is significantly sall. For a siulation of icroarticle s otion in the icrofluidic channel in Fig. 1, the density, coressibility and diaeter of the icroarticle are set to be 1050 kg/ 3, 2.25 10-10 Pa -1, and 10 µ, resectively. In order to resent the effects of the acoustic streaing, the icroarticle s trajectories are calculated in two cases including and excluding the acoustic streaing velocities. These two cases are resented in Fig. 3. It is shown that in Fig. 3, the acoustic streaing has an insignificant effect on the icroarticle s otion and thus it can be ignored in analyzing the acoustohoretic otion in the icrofluidic channel. Proceedings of Meetings on Acoustics, Vol. 19, 045015 (2013) Page 3
Figure 3 Coarison of icroarticle s otion with/without acoustic streaing. By coaring exerientally-easured and nuerically-redicted icroarticle trajectories, the density and coressibility of a icroarticle and the acoustic excitation alitude P 1 can be otially deterined. In this aer, the acoustohoretic otion of olystyrene beads is exerientally recorded in a icrofluidic channel. Since the excitation is alied in a large area on one of the icrochannel s sidewalls, it is assued that 1-D lane-wavelike acoustic fields are generated in the icrochannel for the nuerical rediction. The density of the olystyrene bead is reorted as 1050 kg/ 3 and the coressibility, in the range of 2.1~2.4 10-10 Pa -1 [6]. The radius of the olystyrene bead is 5 µ that is easured fro the recorded icroscoic iages. By fitting the exeriental trajectories of the olystyrene bead to the nuerically-redicted ones, the otial araeters are identified. The resulting acoustic ressure alitude P 1 is 1.12 10 5 Pa that can be considered as the first-order acoustic ressure alitude in the icrofluidic channel. The otial density is identified to be 1057.87 kg/ 3 with the axiu variation of 0.52%. The otial coressibility is 2.4 10-10 Pa -1 that is in line with the reorted range of 2.1~2.4 10-10 Pa -1 [6]. CONCLUSION In this aer, the nuerical ethod including the effects of oving fluid edia, viscous boundary layers, and acoustic streaing is roosed to accurately redict the acoustohoretic otion of cells/icroarticles. Through the nuerically-redicted trajectories of a icroarticle, it is shown that the agnitude of the acoustic streaing velocities in the siulation setu in Fig. 1 is significantly sall and thus does not affect the acoustohoretic otion of the icroarticle significantly. Additionally, the density and coressibility of the olystyrene bead are identified by otially fitting the nuerically-redicted and exerientally-easured olystyrene bead s trajectories. The identified otial roerties of the olystyrene bead are close to the reorted values. ACKNOWLEDGMENT This research has been sonsored in art by research grants of the National Science Foundation (Grant No.: ECCS-1232251) and the U.S. Ary Engineering Research and Develoent Center (Grant No.: W9132T-12-2- 0022). REFERENCE 1. H. Tsutsui and C.-M. Ho, Cell searation by non-inertial force fields in icrofluidic systes, Mechanics Research Counications, 36, 92-103 (2009). 2. L.V. King, On the Acoustic Radiation Pressure on Sheres, Proceedings of the Royal Society of London, Series A, 147, 212-240 (1934). 3. K.Yosioka and Y.Kawasia, Acoustic Radiation Pressure on a Coressible Shere, Acoustica, 5, 167-173 (1955). Proceedings of Meetings on Acoustics, Vol. 19, 045015 (2013) Page 4
4. L. P. Gorkov, On the forces acting on a sall article in an acoustical field in an ideal fluid, Soviet Physics-Doklady, 6, 773-775 (1962). 5. Z. Z. Liu, A. Han and Y.-J. Ki, Two-Diensional Nuerical Analyses of Acosutohoresis Phenoena in Microfluidic Channel with Microarticle-Susended Viscous, Moving Fluid Mediu, Proceedings of the ASME 2012 International Mechanical Engineering Congress & Exosition, IMECE2012-89912, Houston, USA, 2012. 6. K. Hartono, Y. Liu, P.L. Tan, X. Y. S. Then, L.-Y. L. Yung and K.-M. Li, On-chi easureents of cell coressibility via acoustic radiation, Lab on a Chi, 11, 4072-4080 (2011). Proceedings of Meetings on Acoustics, Vol. 19, 045015 (2013) Page 5