pubs.acs.org/Langmuir © 2010 American Chemical Society
Acoustic-Wave-Induced Analyte Separation in Narrow Fluidic Confinements in the Presence of Interfacial Interactions Bharath Bhat and Suman Chakraborty* Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur-721302, India Received May 18, 2010. Revised Manuscript Received July 19, 2010 In the present work, we attempt to analyze the influences of acoustic forces, in conjunction with the intrinsic electrokinetic effects as well as the near-wall attractive and repulsive forces, on the transport and size-based separation of charged analytes in a background-pressure-driven flow through a narrow fluidic confinement. By executing a regular perturbation analysis, we establish that the speed of traverse and the extent of spreading (dispersion) of the analyte bands is effectively determined by the ratio of particle to channel heights, channel height relative to the Debye length, and all other significant acoustic and nonacoustic parameters. These factors in tandem may dictate the analyte separation characteristics (quantified by the resolution of separation), in tune with the particular harmonic of the acoustic wave, the strengths of the induced electrical double layer fields, and the van der Waals interaction mechanisms. We quantitatively pinpoint the relationship between the harmonics to be employed and particle sizes to be separated. Our study reveals that there is a critical channel height beyond which the acoustic effects may effectively mask the near-wall interactions and below which the transverse migrative influences induced by the walls may influence the separation characteristics in a rather profound manner. The results implicate an interesting high-efficiency separation regime that can be obtained with a judicious combination of the background flow, energy intensity of the acoustic effects, and induced electrical double layer interactions.
1. Introduction Microfluidic devices of multifarious functionalities have attracted considerable research attention over the past few years, primarily motivated by the necessity to develop more efficient procedures for the reaction, separation, detection, and synthesis of a wide range of species at a very rapid rate. Following this trend, several separation techniques have been systematically attempted and investigated by the concerned research community over the years for efficient handling of biological and nonbiological entities. Many of these separation techniques are essentially based on disparities in the sizes of the analytes that are targeted to be separated off. On the basis of the relative size distribution of the analytes in the sample, there may be distinct preferential speeds with which their corresponding bands tend to move. When an analyte is transported through a fluidic conduit, its center of mass is excluded from the low-velocity region near the surface because of its finite size. Thus, the average speed of an analyte becomes greater than that of the fluid when an ensemble is taken over a sufficiently long separation time. As is obvious, larger analytes are excluded from the low-velocity regions, which are otherwise accessible to smaller analytes. Accordingly, in the absence of other counteracting effects, the larger analytes acquire greater average velocities than the smaller ones, thereby establishing a fundamental basis for size-based separation techniques in narrow confinements.
*Corresponding author. E-mail:
[email protected]. (1) Han, J.; Turner, S. W.; Craighead, H. G. Phys. Rev. Lett. 2001, 86, 1394. (2) Das, S.; Chakraborty, S. Electrophoresis 2008, 29, 1115. (3) Clicq, D.; Vervoort, N.; Vounckx, R.; Ottevaere, H.; Buijs, J. B.; Gooijer, C.; Ariese, F.; Baron, G. V.; Desmet, G. J. Chromatogr., A 2002, 979, 33. (4) Lin, Y. C.; C. P. Jen, C. P. Lab Chip 2002, 2, 164. (5) Garcia, A. L.; Ista, L. K.; Petsev, D. N.; O’Brien, M. J.; Bisong, P.; Mammoli, A. A.; Brueck, S. R. J.; Lo0 pez, G. P. Lab Chip 2005, 5, 1271. (6) Datta, R.; Kotamarthi, V. R. AIChE J. 1990, 36, 916. (7) Griffiths, S. K.; Nilson, R. H. Anal. Chem. 2000, 72, 4767. (8) Datta, B.; McEldoon, J. P. Anal. Chem. 2002, 64, 227.
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The separation techniques that have been most commonly employed on microscales and nanoscales1-13 have primarily been based on electrokinetic principles. This has been essentially motivated by the fact that with a favorable exploitation of electrokinetic interfacial phenomena over reduced length scales, efficient separation of analytes might be possible over much shorter distances than those demanded by larger separation systems. Motivated by this challenging proposition, several researchers in the recent past have studied various aspects of species separation in microscale and nanoscale fluidic systems. However, it needs to be mentioned in this context that the employment of an electric field is often complicated by phenomena such as Joule heating, which may adversely affect thermally labile samples that are being separated. An alternative approach, considering such restraints in purview, may be to utilize other bulk forces for achieving continuous flow separation and size sorting by exploiting pertinent physical properties of dispersed particles. Such a separation principle may also have the advantage of working inline as a flowthrough separator and may turn out to be suitable for a wide range of materials without any biomolecular specificity. For example, one may cite the use of centrifugal forces. Nevertheless, centrifugation also has its own drawbacks. Such drawbacks primarily stem from the relatively high expenses of centrifugal separation systems, their relatively low throughput, and possible losses of control on account of an imbalanced spinning of the rotor. Considering these constraints, the use of ultrasonic waves to exert a radiation force proportional to the volume of dispersed particles to be transported through a fluidic conduit indeed offers an appealing concept. Laurell and colleagues14 first described a (9) Martin, C. R.; Nishizawa, M.; Jirage, K.; Kang, M. J. Phys. Chem. B 2001, 105, 1925. (10) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6772. (11) Guo, L. J.; Cheng, X.; Chou, C. F. Nano Lett. 2004, 4, 69. (12) Stein, D.; Kruithof, M.; Dekker, C. Phys. Rev. Lett. 2004, 93, 035901. (13) Das, S.; Chakraborty, S. Langmuir 2008, 24, 7704.
Published on Web 08/30/2010
DOI: 10.1021/la101993g
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separation methodology based on acoustic forces that works on a continuous flow of sample in a miniaturized fluidic device. The method has been successfully employed for separating erythrocytes from lipid microemboli in human blood (thus, operative on particles typically 1 to 5 μm in size). This study has been a logical extension of their earlier work on the use of acoustic forces for moving particles from one laminar stream to another in a microfluidic channel with applications in postoperative blood cleaning.15 It may be of further interest to investigate whether this method may be applied for separating a wider range of particle sizes. Additional features that may complicate a comprehensive assessment of such implications may include the other transverse migrative influences, over and above the acoustic effects. However, no theoretical studies in the literature have yet been addressed to analyze the interesting interplay between the acoustic separation of analytes and the influence of other transverse migrative fluxes in narrow fluidic confinements as well as the influences of electrokinetic effects intrinsic to the separation system. Such investigations, nevertheless, may turn out to be of immense consequence in designing efficient separation mechanisms that work over microscales and nanoscales. This is primarily due to the fact that other than transverse migrative influences induced by the acoustic forces, the near-wall interaction mechanisms originating from van der Waals forces, the particle-wall electric double layer (EDL) interactions, and electrophoretic effects due to the EDL field established within the fluidic confinement are likely to interact with each other in an intricate manner to establishing the solutal concentration field in such systems. The ability of ultrasonic sound waves to manipulate microscopic particles and cells has been exploited in several experimental studies on separation and size sorting.16-20 These are based on the fact that suspended particles exposed to an ultrasonic standing wave are acted upon by an acoustic radiation force. This force moves the particles toward either the pressure antinodes or the nodes depending on the density and compressibility of the particles and the medium. Studies have so far concentrated on the development of transverse separation techniques based on this concept. The aim of the present study is to determine the effect of acoustic forces on axial separation techniques in narrow fluidic confinements, as modulated by the electrophoretic forces as the intrinsic electrokinetic effects and the near-wall attractive and repulsive influences. The method considers the action of an actuator (such as a piezoceramic one) positioned farther downstream from the channel through the inlet of which the mixture to be separated is introduced with the aid of pressure-driven flow. The actuator generates an acoustic standing wave inside the channel, which is orthogonal to the fluid flow. As the mixture travels through the acoustic wave, its components separate into their individual laminar streams. We aim to establish regimes wherein a combination of acoustic forces and near-wall forces such as van der Waals forces and EDL forces can be used to obtain these interesting separation characteristics in such physical arrangements. We further explore the effects of using (14) Petersson, F.; A˚berg, L.; Sw€ard-Nilsson, A.-M.; Laurell, T. Anal. Chem. 2007, 79, 5117. (15) Petersson, F.; Nilsson, A.; J€onsson, H.; Laurell, T. Anal. Chem. 2005, 77, 1216. (16) Townsend, R. J.; Hill, M.; Harris, N. R.; White, N. M. Ultrasonics 2004, 42, 319. (17) Hawkes, J. J.; Coakley, W. T. Sens. Actuators, B 2001, 75, 213. (18) Gupta, S.; Feke, D. L.; Zloczower, I. Chem. Eng. Sci. 1995, 50, 3275. (19) Nilsson, A.; Petersson, F.; Jonsson, H.; Laurell, T. Lab Chip 2004, 4, 131. (20) Yasuda, K.; Umemura, S.; Takeda, K. Jpn. J. App. Phy. 1995, 34, 2715.
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different harmonics of the acoustic standing wave on separation characteristics.
2. Theory 2.1. Acoustic Radiation Forces. When an acoustic standing wave is set up in a fluid across a channel, the radiation force acting on a particle of radius r, where r is much smaller than the sound wavelength, is given by21 F ¼
2πðkRp 3 Þ2E ΦðΛ, σÞ sinð2kxÞ k2
ð1Þ
where k is the sound wavenumber, Rp is the particle radius, E is the energy density of sound waves, Λ = (Fp/F) is the ratio of particle density to fluid density, σ = (cp/c) is the ratio of the sound velocity of a particle to the sound velocity in a fluid and Φ ¼
1 5Λ - 2 1 - 2 3 2Λ þ 1 Λσ
ð1aÞ
According to this theory, particles accumulate in the nodes of the acoustic force field. As is evident from eq 1, the force on the particles strongly depends on the radius of the particles and hence can be used to bring about the transverse migration of particles in the channel. Hence, the major effect of the acoustic force field is to bring about the differential transverse movement of particles to certain positions across the channel. By changing the wavelength of the acoustic standing wave, the position of the nodes and hence the region of particle accumulation can be conveniently changed to achieve the desired separation characteristics. 2.2. Fluid Velocity Field. We consider the transport of a dilute solution of spherical particles through a slit-like channel of height 2H, length l, and width w (w . l, H) under the action of an applied pressure gradient of -dP/dx. The axial and transverse coordinates are given by x and y, respectively, in the plane of flow, with the origin of the y axis being located on the bottom plate. The parabolic fully developed fluid velocity profile is given by u ¼ -
1 dP 2 ðy - 2yHÞ 2μ dx
ð2Þ
In addition, the flow due to acoustic streaming is given by the expression22 Vs ¼ -
3 dVðxÞ VðxÞ 4ω dx
where ω is the acoustic frequency. This, on simplification, leads to the following order-of-magnitude expression Vs
V2 C0
ð2aÞ
where V is the particle velocity due to the acoustic standing wave and C0 is the wave speed. Assuming the energy density of the wave to be on the order of 100 J/m3, the streaming velocity Vs turns out to be on the order of 10-4 m/s. This is at least 2 orders of magnitude less than typical background-pressure-driven flow velocities and hence can be conveniently neglected for further calculations. (21) Nyborg, W. L. In Ultrasound: Its Application in Medicine and Biology; Fry, F. J., Ed.; Elsevier: New York, 1978; Vol. 1, pp 1-76. (22) Laurell, T.; Bengtsson, M. Anal. Bioannal. Chem. 2004, 378, 1716.
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2.3. Solutal Transport. Particle transport occurs in the channel under the action of several transverse and longitudinal migrative influences in the presence of the background flow field. For convenience in the theoretical analysis, one may split these effects into two parts, namely, electrokinetic and nonelectrokinetic ones. It is important to mention here that despite the application of no external electrical field, electrokinetic effects may still be pertinent in the present context because of the fact that the channel substrates may become spontaneously charged when in contact with an electrolyte. The solution being transported may thus acquire a net charge within an electrical boundary layer adjacent to the solid boundaries. Within that walladjacent layer, the liquid bears a charge equal in amplitude and opposite in sign to the bound charge on the solid substrate. This charged layer is commonly known as the electric double layer (EDL). Immediately next to the solid boundary, the EDL simply contains a layer of immobilized counterions, also known as the Stern layer or the Helmholtz layer. The sublayer of EDL subsequent to this, known as the Gouy-Chapman layer (or the diffuse part of the EDL), consists of mobile counterions and coions. The Stern layer and the Gouy-Chapman layer, in a simplified representation, are assumed to be separated by a plane known as the shear plane. The electrical potential at the shear plane is known as the zeta potential (ζ), which for all practical purposes may be approximated as the electrical potential of the adjacent solid boundary. One may characterize the span of the EDL by considering the distance from the shear plane over which the EDL potential reduces to 1/e times ζ, also known as the Debye length (λ). For simplicity, we consider a symmetric electrolyte (zþ = -z- = z). The potential distribution within the EDL may be obtained by employing the Poisson-Boltzmann equation as d2 ψ F ¼ - e dy2 ε
ð3Þ
where Fe = ze(nþ - n-) is the net volumetric charge density and n( = n0 exp(-zeψ/kBT), with ψ being the potential distribution within the EDL, kB being the Boltzmann constant, ε being the permittivity of the flow medium, and T being the absolute temperature. Considering specified ζ and the symmetry boundary condition at the channel centerline, one may analytically estimate the distribution of ψ as23 2 ! 4kB T 4 ezζ y -1 ψðyÞ ¼ tanh tanh exp ze 4kB T λ !# ezζ 2H - y -1 þ tanh exp tanh 4kB T λ
where uP and vP are the axial and transverse components of the nonelectrophoretic part of the analyte velocity and μep (μep = zMe/6πμRP) is the electrophoretic mobility of the analytes. The particle velocities, appearing in eq 5, may be expressed as uP ¼ u þ urel;x
ð6aÞ
vP ¼ v þ vrel;y
ð6bÞ
where urel,x and vrel,y are the velocity components of the particles relative to the flow along the x and y directions, respectively, without taking the electrophoretic effects into account. (Electrophoretic effects are represented by separate source terms; see eq 5.) To simplify eq 5, we make a further assumption that the particle and fluid velocity components along the x direction are virtually identical4 so that urel,x=0. On the basis of these considerations, eq 5 gets reduced to ∂c ∂ ∂ ∂ ∂c ∂ ∂c þ ððu -uÞcÞ þ ðvrel, y cÞ ¼ D þ D ∂t ∂x ∂y ∂x ∂x ∂y ∂y ∂ ∂ψ μ c ð7Þ þ ∂y ep ∂y subject to no-flux boundary conditions at the channel walls: jy ¼ - D
∂c ∂ψ þ vrel, y c - μep c ¼ 0 at y ¼ 0, 2H ð7aÞ ∂y ∂y
2.4. Transverse Migration of Analytes. The nonelectrophoretic transverse migration flux of the particles relative to the background flow, manifested by the term vrel,y, is a combined consequence of the other surface forces and body forces acting on each particle. The effect due to surface forces can be expressed as vrel, y, surface force ¼
Fsurface 6πμRP
ð8Þ
where Fsurface is the force due to the interaction between the particles and the channel walls. Fsurface can be obtained from the potential energy of interaction between the analytes and the channel wall, φPE, given as Fsurface ¼ -
∂φPE ∂y
ð9Þ
This potential energy is primarily contributed by the EDL and van der Waals interactions between the particles and the channel walls. Accordingly, ð4Þ
φPE ¼ φDL þ φVDW
ð10Þ
∂c ∂ ∂ ∂ ∂c ∂ ∂c þ ððuP - uÞcÞ þ ðvP cÞ ¼ Dx Dy þ ∂t ∂x ∂y ∂x ∂x ∂y ∂y ∂ ∂ψ ∂ ∂ψ þ ð5Þ þ μ c μ c ∂x ep ∂x ∂y ep ∂y
In the present study, the following expressions for the abovementioned potential are employed:4,24,25 ! eζp kB T 2 eζ " tanh 32εRP tanh e 4kB T 4kB T hs φDL ¼ 2 31=2 exp KD 2RP ! þ1 6 7 eζp 7 KD 2 1þ6 2 tanh 61 - 7 4k 4 BT 5 RP þ1 KD 2H - 2Rp - hs ð11Þ þ exp KD
(23) Hunter, R. J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: London, 1981.
(24) Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 90, 17. (25) Bell, G. M.; Levine, S.; McCartney, L. N. J. Colloid Interface Sci. 1970, 33, 335.
Considering the electrophoretic interactions on account of charged particles and the induced EDL field, one may write the species conservation equation as a pulse of analytes introduced into the channel at t = 0 and having a mean velocity of u in a reference frame moving with a velocity u to obtain
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Bhat and Chakraborty Table 1. Summary of Perturbation Analysis
order
governing equation 2 ∂ψ ∂ ∂ Pe ∂Y ðVC0 Þ ¼ ∂∂YC20 þ ðPeÞep ∂Y C0 ∂Y
1/ε2 1/ε
∂ ∂ Pe ∂X ððU - U ÞC0 Þ þ Pe ∂Y ðVC1 Þ ¼
ε0
∂C0 ∂T
∂ 2 C1 ∂Y 2
∂ψ ∂ þ ðPeÞep ∂Y C1 ∂Y
∂ ∂ þ Pe ∂X ððU - U ÞC1 Þ þ Pe ∂Y ðVC2 Þ ¼
∂ 2 C0 ∂X 2
2 ∂ψ ∂ þ ∂∂YC22 þ ðPeÞep ∂Y C2 ∂Y
Here hs is the separation distance of the particle surface from the channel bottom wall, which can be expressed in terms of the coordinate of the particle center as hs = yCP - RP. In eq 11, κD is the particle Debye length and ζp is the zeta potential at the particle surface (which is conceptually different from the zeta potential, ζ, due to EDL formation at the channel wall) given as23 ζm ¼
Q p RP 4πRP ε 1 þ KD
ð12Þ
where Qp is the total charge on the particle. The expression for φVDW is given by4,26 φVDW
!" # A λ hs þ 2RP 2RP hs þ RP ln ¼ 6 λ þ shs hs hs hs þ 2RP
!" A λ 2H - hs ln þ 6 λ þ sð2H - 2RP - hs Þ 2H - 2RP - hs 2RP 2H - RP - hs 2H - 2RP - hs 2H - hs
ð13Þ
ð14Þ
Therefore, the final expression for vrel, y can be written as dðφVDW þ φDL þ φbody Þ 1 6πμRp dy
!
ð16Þ
In eq 16, (Pe)ep = ζμep/D = (ζμep/h)(2H/D) = uep,y(2H/D) and ε (2H/l)(2H/L)1/2 , 1. The nondimensional form of the boundary condition (eq 7a) reduces to -
∂C ∂ψ þ PeðVCÞ - ðPeÞep C ∂Y ∂Y RP RP and 1 at Y ¼ 2H 2H
ð17Þ
ð18Þ
Substituting this expression into eqs 15 and 16, we obtain systems of partial differential equations by separating terms containing different orders of ε (refer to Table 1). We further consider leading-order terms C0 and C1 to be variable-separable and write their expressions in the following form: ∂ AðX, TÞ GðYÞ ð19Þ C0 ¼ AðX, TÞ FðYÞ, C1 ¼ ∂X Following this approach, the governing differential equation for the cross sectionally averaged solutal concentration (ÆC0æ) is obtained in the form of 2 3 R 1 - ðR =2HÞ Pe RP =2HP ðU - UÞG dY ∂ÆC0 æ ∂ 2 ÆC0 æ4 5 ¼ 1R 1 - ðRP =2HÞ ∂T ∂X 2 F dY
ð20Þ
RP =2H
ð15Þ
2.5. Solution by Perturbation Analysis: An Outline. We execute a regular perturbation analysis as outlined in the literature to solve for the particle concentration profile.2 A brief outline of the analysis is as follows. First, a nondimensional form of the species conservation equation is obtained by using the following length parameters: timescale t0=L/Æuæ, where L is the channel R 2H and R Æuæ is the mean fluid velocity given as Æuæ = ( 2H 0 u dy/ 0 dy), length scale in the x direction l ≈ (DL/Æuæ)1/2, which is the width that the analyte pulse will spread to in time t0. Using normalized variables T = t/t0 = Æuæt/L, U = u/Æuæ, U = u/Æu, V = vrel,y/Æu, C = c/c0, X = x/l, Y = y/2H, Pe = Æuæ2H/Dy, and ψh = ψ/ζ, (26) Hamaker, H. C. Physica 1937, 4, 1058. (27) Adamczyk, Z.; Van De Ven, T. G. M. J. Colloid Interface Sci. 1981, 84, 497.
15038 DOI: 10.1021/la101993g
∂C Pe ∂ Pe ∂ þ ððU - UÞCÞ þ 2 ðVCÞ ∂T ε ∂X ε ∂Y ! ∂ 2 C 1 ∂ 2 C ðPeÞep ∂ ∂ψ ¼ C þ þ 2 ∂X 2 ε2 ∂Y 2 ∂Y ∂Y ε
C ¼ C0 þ εC1 þ ε2 C2 þ :::::
In eq 13, A is the Hamaker constant for interaction between a particle and the channel wall with an intervening liquid medium, λh is the London characteristic wavelength, which is often assumed to be around 100 nm, and s is a constant that is taken as 11.116.27 The acoustic body force on the particles is given by eq 1. For convenience in mathematical analysis, we express this force also in terms of an equivalent interaction potential, which may be obtained by integrating eq 1, and we obtain
vrel, y ¼
a nondimensional form of eq 7 is written as
Next, we assume an asymptotic expansion for C in ε as
φbody ¼ 2πRp 3 ðEΦÞ cosðπy=HÞ
boundary condition at Y = RP/2H and (1 - (RP/2H)) ∂ψ 0 PeðVC0 Þ - ∂C ¼ 0 ∂Y - ðPeÞep C0 ∂Y ∂ψ ∂C1 PeðVC1 Þ - ∂Y - ðPeÞep C1 ∂Y ¼ 0 ∂ψ 2 PeðVC2 Þ - ∂C ¼ 0 ∂Y - ðPeÞep C2 ∂Y
R P/2H) C0 dY. where ÆC0æ = R1-(R P/2H From eq 20, it is clear that the dimensionless dispersion coefficient can be expressed as D ¼ 1 -
Pe
R 1 - ðRP =2HÞ
ðU - UÞG RP =2H R 1 - ðRP =2HÞ F dY RP =2H
dY
ð21Þ
where U is the dimensionless band velocity given by R 1 - ðRP =2HÞ U ¼
ðUFÞ dY RP =2H R 1 - ðRP =2HÞ F dY RP =2H
ð22Þ
The expressions for F(Y) and G(Y) are similar to the ones obtained in previous work.2 However, it is to be noted that the Langmuir 2010, 26(18), 15035–15043
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Gaussian macromolecular concentration profile increase the resolution, indicating an augmented separation efficiency.
3. Results and Discussions
Figure 1. Acoustic potential distribution across the channel height for the two harmonics considered in this work.
major difference lies in the expression for the interaction potential, which now includes the effects of the acoustic potential as well. (See Figure 1 for a schematic illustration.) Through the definitions of the band velocity (eq 22) and the effective dispersion coefficient (eq 21), one may reduce the species transport equation to an equivalent dispersion equation, which is expressed solely in terms of C0 as ∂ÆC0 æ ∂ÆC0 æ ∂ 2 ÆC0 æ þ Æuæ U ¼ DD ∂t ∂x ∂x2
ð23Þ
From eq 23, the concentration profile at the channel exit (x = L) is given by the following Gaussian distribution " # n0 1 ðL - ÆuæUtÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ð24Þ ÆC0 æ ¼ c0 wh 4πDDt 4DDt where n0 is the number of moles of the analytes introduced into the channel. From eq 24, it is clear that a bell-shaped (Gaussian-type) distribution of analytes is formed at the channel exit at different times for particles of different radii. Consequently, it is possible to define the time at which the concentration of particles of a given radius attains a maximum value at the channel exit. This time, designated as the separation time, varies according to the particle radii, the nature of the flow field, and the forces involved and can be looked upon as a measure of the separation performance. A more precise measure of the separation efficiency, namely, the resolution (Rs), may also be derived directly from eq 24. The resolution predicts the effectiveness of a particular flow field in separating particles with two different radii (e.g., RP1 and RP2). Mathematically, we define the resolution as Rs ¼
ðtR1 - tR2 Þ ðwb1 þ wb2 Þ=2
ð25Þ
where tR1 and tR2 are the times required by the particles of radii RP1 and RP2 to attain the corresponding maximum concentration values at the channel exit and wb1 and wb2 are the average halfwidths of the corresponding maximum concentration profiles. The higher the value of the resolution, corresponding to a given pair of particle sizes, the more efficient the separation between them. Equation 25, in addition, implicates that an increase in the difference of the separation times for a pair of particles and an increase in the steepness (i.e., reduction in base width) of the Langmuir 2010, 26(18), 15035–15043
From the theoretical analysis presented in section 2, it is apparent that a combination of the band velocity and the effective dispersion coefficient in effect dictates the size-based analyte concentration distribution within the separation system. The interesting interplay between the various physical parameters that give rise to the resultant separation performance, in turn, is a combined consequence of several forcing parameters that act on the system, which we discuss through the representative case studies in the foregoing discussions. Various parameters used for the present simulations are given in Table 2. It is important to mention in this context that we explore two harmonics of the acoustic wave in the current study because our aim is to establish conditions wherein the acoustic forces can be advantageously used in conjunction with nearwall effects to achieve the desired separation characteristics. Figure 1 represents the variation of acoustic potential for a given analyte radius across the cross section for the first and second harmonics of the acoustic wave considered. For the first harmonic, the acoustic wavelength is equal to twice the channel height, which leads to particles being concentrated at the single node formed at the center of the channel. For the second harmonic, the acoustic wavelength is equal to the channel height, leading to two nodes, one on each half of the channel. Particles are acted upon by a force proportional to their volume, moving those to either of the two nodes. This, in combination with the steric effects and near-wall forces, gives rise to interesting paradigms that offer insight into the optimum conditions for achieving efficient separation. Physically, an interesting interplay of these effects determines the transport characteristics of the particles relative to the flow field for given strengths of the imposed pressure gradient. The steric component of the particle-wall interaction arises by virtue of the fact that the particles are restricted from approaching the wall to a distance equivalent to their respective radii, thereby making them accessible to higher fluid flow velocities. In the absence of other interaction forces, this gives rise to a monotonically increasing particle transport velocity with increases in particle size. This may be attributed to the fact that center lines of larger particles may effectively access streamlines corresponding to faster-moving fluid elements. The van der Waals interaction arises out of dipole-dipole interactions and is attractive in nature, thereby tending to “pull” the molecules toward the channel walls, thus hindering their access to higher flow rates. However, the particle-wall EDL interactions, as considered in the present study, turn out to be repulsive in nature, thereby tending to “push” the analytes toward the center of the channel. The interaction between these forces, acoustic interactions, as well as the externally applied pressure gradient determines the transport characteristics of particles of a given size. In the foregoing discussions, we consider three parameters, namely, the band velocity (U), the effective dimensionless dispersion coefficient (D*), and the resolution of separation (Rs). We illustrate the variations of these parameters with particle radius, channel height, and the two harmonics of the acoustic wave. We compare the resolution of separation for the two harmonics with each other and in the case where no acoustic forces are used and indicate regions wherein each can be used advantageously. DOI: 10.1021/la101993g
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Bhat and Chakraborty Table 2. Physical Properties and Problem Data
parameter
value
L ε kB μ T ζ, ζm n0 (bulk inlet ionic concentration)
10 mm 6.5 10-10 C/V 3 m 1.38 10-23 J/K 10-3 Pa 3 s 300 K -50 mV 6.023 1022 /m3 (This number concentration is equivalent to a concentration of 100 μM.) 1100 kg/m3 1000 kg/m3 -1 10-20 J 10 nm -109 Pa/m 100 J/m3 2 10-8 m
FP Ff zM A (Hamaker constant) κD dP/dx Eac σ (Cp/C) κd
3.1. Dimensionless Band Velocity. In Figure 3a-c, we plot the variation of the dimensionless band velocity as a function of the dimensionless particle size Rp/H. The curves are plotted for three different channel heights and for three cases, namely, without acoustic effects and for the two harmonics considered. The influence of the near-wall forces is clearly seen in the case of smaller particles that see a marked decrease in band velocity due to the predominance of van der Waals attractive forces. These particles are thus kept near the wall where the background velocity is low, and hence they have a greatly reduced band velocity. Larger particles cannot approach the near-wall regions and thus are exposed to higher-velocity regions, leading to the band velocity increase. In general, a monotonic increase in band velocity may be observed even without acoustic forces, as a combined consequence of the background parabolic flow and steric effects. Beyond a certain particle size (Rp/H > 0.2), the effect of acoustic forces is more conspicuously felt. In the case of the first harmonic of the acoustic effects, all particles are moved toward the channel center and hence we see a general increase in the band velocity as compared to the case without acoustic effects as particles are exposed to higher background flow regions. Most particles tend to accumulate near the central region, and hence the band velocity profile flattens out for Rp/H > 0.4. This effect is prominently seen in wider channels where the acoustic forces become predominant compared to other forces. In the case of the second harmonic, particles are moved toward the two nodes formed in the two halves of the channel. However, beyond a certain radius, that is, beyond Rp/H > 0.5, particle centers cannot reach this nodal point and hence follow an increase in band velocity with size dictated by the background flow. Particles smaller than this size, however, are mostly aggregated at the nodal points and hence the flat profile in the region is 0.2 < Rp/H
