pubs.acs.org/Langmuir © 2009 American Chemical Society
Influence of Streaming Potential on the Transport and Separation of Charged Spherical Solutes in Nanochannels Subjected to Particle-Wall Interactions Siddhartha Das and Suman Chakraborty* Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur-721302, India Received March 19, 2009. Revised Manuscript Received May 7, 2009 In this work, we theoretically investigate the implications of streaming potential on the transport and size-based separation of charged solutes in nanoscale confinements. By employing a regular perturbation analysis, we demonstrate that the consideration of streaming potential establishes a new paradigm of size-based separation of charged solutes in nanochannels. Depending on the sizes of the particles being handled, we establish two distinctive separation regimes. For smaller particles with significantly large electrophoretic mobilities, electrophoretic transport mechanisms predominantly influence the solutal transport characteristics, whereas for larger particles the combined pressure-driven and back electroosmotic transport mechanisms essentially dictate the resultant separation characteristics. The extent of improvement in separation characteristics, on account of the consideration of streaming effect, largely depends on the consideration of particle-wall interactions. For cases without wall effects, the streaming potential may induce dramatic enhancements in the resolution of separation for small particles by exploiting optimal combinations of zeta potential values and relative thicknesses of the electrical double layers. However, with wall effect considerations, similar combinations of zeta potential and electrical double-layer thicknesses may give rise to dramatic improvements in the separation characteristics over a much wider range of particle sizes by interacting nontrivially with the streaming potential effects. Such confluences may be exploited in practice for designing efficient nanofluidic separation systems.
1. Introduction The size-based separation of charged solutes in narrow fluidic confinements has been established as one of the most significant bioanalytical processes because of its overwhelming implications in diverse applications including biomedical, biotechnological, chemical, forensic, and so forth.1-10 Such separation strategies may broadly be classified into two categories. One class of separation methodology relies on the size dependences in the migrative tendencies of the solutes, otherwise inseparable, as a result of the hydrodynamic interactions and wall effects imposed by the confining boundaries. For instance, under the influence of Poiseuille flow, larger particles get preferentially excluded from the near-wall regions, leading to their spontaneous separation from the smaller ones. A second type of separation strategy, however, explicitly intends to exploit the size dependence of a fundamental transport characteristic of the solute. For instance, in the electrophoretic separation of charged solutes, the size-based variation of electrophoretic mobilities of the transported species is utilized to induce the differential electromigration that separates the solutes of varying radii. In nanochannels, these different strategies and their nontrivial interconnections may be suitably combined11,12 to modulate the performance of a designed separation scheme, giving rise to the possibilities in realizing hitherto unexplored separation paradigms in experimental practice. For
example, the combined consequence of wall-induced interactions and electromigration of the charged solutes may give rise to interesting transverse electrophoretic effects in narrow confinements, which may in turn augment the separation efficiency.13 The limit to which the distinctive transport characteristics of narrow confinements influence such advantageous propositions, however, depends largely on the extent to which the solutal bands are dispersed or spread (an effect called band broadening),14-17 an effect that continues until the solute distribution within the system has equilibrated. Such spreading, resulting from a combination of the diffusion and advection, may blur the steepness of the concentration peak at the channel exit to a significant extent, thereby reducing the overall separation efficiency. Several distinctive characteristics of nanochannel-based separation, mentioned above, have already been exploited by the researchers18-22 to realize very efficient separation methodologies in narrow fluidic confinements. However, in designing these separation strategies, the influences of the streaming potential gradients have largely been unexplored. An interesting implication of streaming potentials toward the separation of charged solutes in narrow confinements may intrinsically lie in the fact that despite the application of no external electrical fields, a potential gradient may implicitly be induced across two ends of the fluidic channel as a consequence of the preferential advection of mobile counterions in the diffuse part of the electrical double
*Corresponding author. E-mail:
[email protected] (1) Watson, A. Science 2000, 289, 850. (2) Griffiths, S. K.; Nilson, R. H. Anal. Chem. 2006, 78, 8134. (3) De Leebeeck, A.; Sinton, D. Electrophoresis 2006, 27, 4999. (4) Chakraborty, S. Anal. Chim. Acta 2007, 605, 175. (5) Das, S.; Chakraborty, S. AIChE J. 2007, 53, 1086. (6) Das, S.; Subramanian, K.; Chakraborty, S. Colloids Surf., B 2007, 58, 203. (7) Das, S.; Chakraborty, S. J. Appl. Phys. 2006, 100, 014098. (8) Das, S.; Chakraborty, S. Anal. Chim. Acta 2006, 559, 15. (9) Das, S.; Das, T.; Chakraborty, S. Sens. Actuators, B 2006, 114, 957. (10) Das, S.; Das, T.; Chakraborty, S. Microfluid. Nanofluid. 2006, 2, 37. (11) Das, S.; Chakraborty, S. Electrophoresis 2008, 29, 1115. (12) Das, S.; Chakraborty, S. Langmuir 2008, 24, 7704.
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(13) Xuan, X.; Li, D. Electrophoresis 2006, 27, 5020. (14) Slater, G. W. Electrophoresis 1993, 14, 1. (15) Chen, Z.; Chauhan, A. J. Colloid Interface Sci. 2005, 285, 834. (16) De Leebeeck, A.; Sinton, D. Electrophoresis 2006, 27, 4999. (17) Malone, D. M.; Anderson, J. L. Chem. Eng. Sci. 1978, 33, 1429. (18) Garcia, A. L.; Ista, L. K.; Ptsev, D. N.; O’Brien, M. J.; Bisong, P.; Mammoli, A. A.; Brueck, S. R. J.; Lopez, G. P. Lab Chip 2005, 5, 1271. (19) Baldessari, F.; Santiago, J. G. J. Nanobiotechnol. 2006, 4, 12. (20) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6772. (21) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6782. (22) Yuan, Z.; Garcia, A. L.; Lopez, G. P.; Petsev, D. N. Electrophoresis 2007, 28, 595.
Published on Web 07/20/2009
DOI: 10.1021/la900956k
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layer (EDL) in pressure-driven transport.23 This streaming potential field24-29 acts in a sense so as to oppose the background pressure-driven flow to which it is due and induces a back electroosmotic transport that reduces the overall net flow rate, giving rise to the so-called electroviscous effect. In addition, this induced electrical field interacts with the charges on the transported solutes and strongly influences their electromigration characteristics (both transverse as well as axial). The interactions of these additional features with the other confinement-induced interactions in nanochannels may in the end give rise to separation characteristics that elusively appear to be somewhat intuitive in nature but are by no means trivially obvious. However, to the best of our knowledge, no such study has yet been reported in the literature that investigates the contribution of streaming potential to the separation of finite-sized charged particles (including particle sizes comparable to the channel dimensions) in nanofluidic channels, experiencing different interaction forces from the wall. Possibly, the only existing study on similar lines is by Xuan and co-workers,29-31 who propose an analytical model to investigate the implications of streaming potential effects on solute transport, dispersion, and separation in nanochannels. However, in their work they do not capture the consequences of the finite size of the solute as well as the wall-solute interactions, which may have extremely significant implications on the extent to which the streaming potential can be influential in governing the separation characteristics. The aim of the present work, accordingly, is to investigate the possible implications of streaming potential on dictating the separation characteristics of finite-sized charged solutes in narrow fluidic confinements, experiencing interaction forces (such as vdW interactions, wall-analyte EDL interactions, and EDL potential induced transverse electromigration) with the wall. Toward this aim, we plan to pinpoint the possible ranges of values of nanochannel height, zeta potential, and relative sizes of the solutes (with respect to the channel height), for which the favorable influences of the streaming potential can be best exploited for augmenting the separation performance. We also emphasize how the consideration of wall influences may drastically alter the extent of influence of the streaming potential in affecting the separation performance. In executing our studies, we restrict the analysis within the broad framework of a continuum regime, which remains valid for the experimentally realizable spatiotemporal scales that are commonly addressed in the nanochannel-based separation literature.
2. Fundamental Transport Equations 2.1. Pressure-Driven Transport and Induced Streaming Potential. We consider the purely pressure-driven transport of an ionic solution of charged spherical solute particles (radius RP) through a nanoslit of height 2H (geometry shown in Figure 1). The ionic nature of the solution creates an EDL adjacent to the channel walls. The electric potential (ψ) in the EDL may be (23) Hunter, R. J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: New York 1981. (24) Yang, J.; Lu, F.; Kostiuk, L. W.; Kyok, D. Y. J. Micromech. Microeng. 2003, 13, 963. (25) Schweiss, R.; Welzel, P. B.; Werner, C.; Knoll, W. Colloids Surf., A 2001, 195, 97. (26) van der Heyden, F. H. J.; Stein, D.; Besteman, K.; Lemay, S. G.; Dekker, C. Phys. Rev. Lett. 2006, 96, 224502. (27) Alkafeef, S. F.; Alajmi, A. F. J. Colloid Interface Sci. 2007, 309, 253. (28) Daiguji, H.; Oka, Y.; Adachi, T.; Shirono, K. Electrochem. Commun. 2006, 8, 1796. (29) Xuan, X. Microfluid. Nanofluid. 2007, 3, 723. (30) Xuan, X. J. Chromatography A 2008, 1187, 289. (31) Xuan, X. Anal. Chem. 2007, 79, 7928.
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Figure 1. Flow geometry.
expressed as a function of the net charge density (Fe) as D2 ψ D2 ψ F þ 2 ¼ - e Dx2 Dy ε0 εr
ð1Þ
where Fe = e(z+n+ + z-n-), ε0 is the permittivity of free space, and εr is the relative permittivity of the medium (or the dielectric constant of the medium). Here n ( represents the number densities of the positive/negative ions and may be expressed through the Nernst equation as Dn ( þ VB 3 rn ( ¼ r 3 ðn ( μep, ( rψÞ þ D ( r2 n ( Dt
ð2Þ
where μep,( represents the ionic electrophoretic mobilities (different from the electrophoretic mobilities of the transported spherical analytes, μep, as described later) and D( represents the ionic diffusivities. Neglecting the ionic advection current of the ionic species in comparison to their electromigration and diffusion currents, eqs 1 and 2 may be collectively employed to obtain a steady-state distribution of ionic number densities and the resultant EDL potential field. (The transverse variation of the EDL potential far outweighs the axial EDL potential variation so that effectively the term ∂2ψ/∂x2 appearing in eq 1 turns out to be negligible, rendering ψ a function of y alone.) For the present analysis where the channel half height is assumed to be on the order of the Debye layer thickness λ (though always greater than λ), this induced potential distribution can be expressed as23 " ! 4kB T ezζ y -1 ψðyÞ ¼ tanhf tanh gexp ze 4kB T λ !# ezζ 2H -y -1 þ tanh ð3Þ gexp tanhf 4kB T λ where ζ is the potential at the plane of zero shear (also known as the zeta potential). The resultant velocity field established within the nanochannel may be quantitatively characterized with the sole information of the driving pressure gradient, the fluid viscosity, and the ionic concentration distributions by noting that the pressure-driven transport of the mobile part (diffuse layer) of the EDL causes an electric current, known as the streaming current (Istr), to flow in the direction of the imposed fluid motion. The resulting accumulation of the ions in the downstream section of the channel sets up its own electric field (called the streaming field), which generates a current, called a conduction current (Icond), that flows back against the direction of the pressure-driven flow. The net ionic current (Iionic) is a combination of the streaming current and the conduction current. Thus, Iionic ¼ Istr þ Icond
ð4Þ
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The streaming current, resulting from the advection of the ionic charges, may be expressed as Z 2H Z 2H Istr ¼ Fe u dy ¼ ez ðn þ -n - Þu dy 0
0
ðassuming a z=z symmetric electrolyteÞ
ð5Þ
The velocity field, u, as appearing in eq 5, needs to be considered to be a result of the imposed pressure-driven transport and the opposing streaming-field-induced electroosmotic transport. This electroosmotic component of the streaming current in pure pressure-driven transport is often neglected in the theoretical analysis of microchannel-based separation systems. However, as demonstrated in a recent study by Chakraborty and Das,32 neglecting this electroosmotic component may significantly overestimate the streaming field, particularly in nanochannels, giving rise to an energetically absurd situation where the average flow strength of the induced electroosmotic transport may become greater than the pressure-driven transport. Hence, for generality, we consider the velocity field to be a combined consequence of the imposed pressure gradient and the induced streaming field, obtained from the Stokes flow equation, as 1 dp ε0 εr ζES ψ ð6Þ ð2Hy -y2 Þ 1u ¼ up þ uES ¼ 2μ dx ζ μ where dp/dx is the pressure gradient, μ is the dynamic viscosity of the fluid, and ES is the induced streaming potential field. In arriving at eq 6, we consider that for the present case the channel width (i.e., the dimension along the transverse direction) is much larger than the other dimensions so that there is no flow component in the transverse (z) direction (Figure 1), rendering the flow field to be effectively 2D. Further, we assume the flow to be fully developed, which implies u = u(y) alone. In addition, we assume the thermophysical properties dictating the velocity field to be constants in order to capture the essential physics within the constraints of a simplistic mathematical tractability. The streaming potential field, as appearing in eq 6, may be specified explicitly by considering the net flow of ionic current across each section of the fluidic channel, which is the sum of the streaming current and the conduction current. The conduction current (Icond) results from the transport of the ions in response to the established streaming potential so that one may write Z 2H ðn þ z þ ucond, þ þ n - z - ucond, - Þ dy ð7Þ Icond ¼ e 0
where ucond,( represents the electromigration velocities of the cations/anions in response to the streaming potential and may be expressed as ucond, ( ¼
z ( eES f(
ð8Þ
Here f ( represents the friction coefficients of the ions. Using eq 8 in eq 7 and considering identical cationic and anionic friction coefficients (f+ = f- = f = 6πμrion, where rion is the ionic radius; here we assume, with some degree of approximation, that the cationic and anionic radii are approximately equal to around 1.5 A˚), we get Icond ¼
z2 e2 ES f
Z
2H
0
ðn þ þ n - Þ dy
(32) Chakraborty, S.; Das, S. Phys. Rev. E 2008, 77, 037303.
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ð9Þ
For pure pressure-driven background transport, there is no net ionic current at steady state, which implies Iionic = 0. Using this condition along with eqs 4, 5, 6, and 9 and considering the Boltzmann distribution of ionic charges (i.e., n( = n0 exp(ezψ/kBT) (where n0 is the bulk ionic conductivity), we obtain the following expression for the streaming field: ezψ 1 dp 2 ½ - 2μ dx ð2Hy -y Þsinh kB T dy ES ¼ 2 2 R 2H R ezψ n0 ezε0 εr ζ 2H n0 e z 1 - ψζ sinh kezψ dy 0 cosh kB T dy þ 0 μ f BT n0 ez
R 2H 0
ð10Þ 2.2. Solute Transport. To study the solute transport, we assume that a pulse of charged solutes with a mean velocity of u has been introduced into the channel at t = 0. In a reference frame having a velocity u, the generalized species transport equation reads Dc D D þ ððuP -uÞcÞ þ ðvP cÞ Dt Dx Dy D Dc D Dc D Dφ Dx þ Dy þ μep c ¼ Dx Dx Dy Dy Dx Dx D Dφ μ c þ Dy ep Dy
ð11Þ
Equation 11 considers the following important assumptions: (a) The analytes may be idealized as finite-sized nondeformable charged spherical particles. (b) Finite size effects of the particles introduce Steric exclusion effects, size-dependent transverse velocity variations (where this velocity depends on the wallanalyte interactions, namely, the EDL and van der Waals interaction, which in turn depend on the particle sizes), size-dependent electrophoretic effects (by introducing a size-dependent electrophoretic mobility), and size-dependent reduction of the diffusivity values. These effects modify the values of the various transport coefficients appearing in eq 11. (c) The electrophoretic mobility of the analyte particles depends on their effective charges as μep = zMe/ 6πμRP. Strictly speaking, this expression holds true only for particle sizes that are smaller than the Debye layer thickness. However, the concerned approximations turn out to be inconsequential for the ranges of particle sizes considered in this work. (d) The values of the flow field and the EDL potential field at the particle centers are unaffected by the finite size effects of the particles. In eq 11, φ denotes the potential developed within the system so that one can write Dψ rφ ¼ -ES i^ þ j^ Dy
ð12aÞ
In eq 11, we consider the particle velocities, uP and vP, to be33 uP = u + urel,x ≈ u (urel,x, which is the x component of the particle velocity relative to the flow velocity, is virtually negligible because the rotational effects of the particles are not considered to be important in the present context) and νP = ν + νrel,y = νrel,y (expressions for vrel,y, the transverse velocity of analytes relative to (33) Lin, Y. C.; Jen, C. P. Lab Chip 2002, 2, 164.
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where jw is the wall-analyte interaction potential primarily contributed by the EDL and vdW interactions between the analytes and the channel walls. Thus, ð12cÞ jW ¼ jDL þ jvdW Appropriate expressions for these potentials are as follows33-36 kB T 2 eζ eζm " tanh tanh hs e 4kB T 4kB T exp " #1=2 λ 2RP eζm þ1 2 λ 1 þ 1 - R 2 tanh P þ1 4kB T λ # 2H -2Rp -hs ð12dÞ þexp λ
32ε0 εr RP
jDL ¼
where hs is the minimum separation distance of the analyte surface from the channel bottom wall, which can be expressed in terms of the coordinate of the analyte center (yCP) as hs = yCP - RP. In eq 12d, ζm is the zeta potential at the surface of the analytes (which is different from the wall zeta potential) given as23 q ζm ¼ ð12eÞ 4πRP ε0 εr 1 þ RλP where q = zMe is the total charge on the analyte. The expression for jvdW (appearing in eq 12c) may be written as !" # A λ hs þ 2RP 2RP hs þ RP ln jvdW 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 ð12fÞ 2H -2RP -hs 2H -hs where A is the Hamaker constant for interaction between the analytes and the nanochannel wall with an intervening liquid medium, λh is the London characteristic wavelength, often assumed to be on the order of 100 nm, and hs is a constant that, in this study, is taken as 11.116.37 It is also important to mention here that the diffusion coefficients appearing in eq 11 get modified to account for the confinement-induced hindered diffusive effects as R fS, ¥ jW ðyCP Þ exp dA ACS f ðy Þ kB T Dr S, r CP ð12gÞ ¼ R D0 jW ðyCP Þ dA ACS exp kB T where ACS is the channel cross sectional area, D0 is the unrestricted bulk diffusivity of analytes given as D0 = kBT/6πμRP, jw is (34) (35) 335. (36) 17. (37)
given by eq 12c, fS,¥ (= 6πμRP) is the unrestricted Stokes coefficient for the analyte, fS,r(yCP) is the enhanced friction coefficient for the analyte (due to restrictions imposed by the channel walls) whose center is at a distance of yCP from the bottom wall and translates parallel and/or perpendicular to the confining walls. Expressions for fS,¥/fS,r, (yCP) (describing the particle motion parallel to the two confining walls and hence dictating the axial diffusivity Dx) are taken from Grassel and Lobry38 and Happel and Brenner,39 whereas those for fS,¥/fS,r, ^(yCP) (describing the particle motion perpendicular to the two confining walls and hence dictating the transverse diffusivity Dy) are taken from Brenner40 and Happel and Brenner.39 The same expressions are not repeated here for the sake of brevity. The above considerations simplify the solutal transport equation (eq 11) as )
the background flow, may be derived by considering the van der Waals (vdW) and EDL interaction potentials).11,33-35 Accordingly, we may write Dj - W Dy ð12bÞ vrel, y ¼ 6πμRP
Dc D D þ ððu -u þ μep ES ÞcÞ þ ðvrel, y cÞ Dt Dx Dy D Dc D Dc D Dψ Dx þ Dy þ μ c ¼ Dx Dx Dy Dy Dy ep Dy
ð13Þ
Equation 13 is solved in the presence of a no-flux boundary condition at the walls, which reads jy ¼ -Dy
Dc Dψ þ vrel, y c -μep c ¼ 0 at y Dy Dy
¼ RP , 2H -RP
ð13aÞ
It needs to be noted here that the particle-wall interaction mechanisms get effectively manifested in solutal transport through two terms appearing in eq 13. These are due to the nonelectrophoretic interactions (the term ∂/∂y(vrel,yc)) and the electrophoretic interactions (the term ∂/∂y(μepc(∂ψ/∂y))).
3. Regular Perturbation Analysis We employ a regular perturbation analysis to solve eq 13, with the respective choices of the √ following time scale and length scale11 : t0 = L/Æuæ and l ≈ (DyL/Æuæ), where L is the channel length, Æuæ is the mean fluid velocity, and l is the axial spread of the analyte pulse in time t0. Accordingly, eq 13 may be nondimensionalized by employing the following normalization parameters: T = t/t0 = Æuæt/L, U = u/Æuæ, U = u/Æuæ, V = vrel,y/Æuæ, C = c/ c0, X = x/l, Y = y/2H, Pe = Æuæ2H/Dy, ψh = ψ/ζ to yield DC Pe D Pe D þ ððU -U þ Uep, x ÞCÞ þ 2 ðVCÞ DT ε DX ε DY ! D2 C 1 D2 C ðPeÞep D CDψ ¼R 2 þ 2 2 þ 2 DX ε DY ε DY DY
ð14Þ
√ In eq 14, Uep,x = μepES/Æuæ, R = Dx/Dy, ε 2H/l (2H/L) , 1, and (Pe)ep = ζμep/Dy.The dimensionless form of the boundary condition given by eq 13a reads as -
DC Dψ þ PeðVCÞ -ðPeÞep C ¼ 0; DY DY RP RP and 1 at Y ¼ 2H 2H
ð14aÞ
Paul, D.; Chakraborty, S. J. Appl. Phys. 2007, 102, 074921. Bell, G. M.; Levine, S.; McCartney, L. N. J. Colloid Interface Sci. 1970, 33, Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 90, Adamczyk, Z.; Van De Ven, T. G. M. J. Colloid Interface Sci. 1981, 81, 497.
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(38) Grasselli, Y.; Lobry, L. Phys. Fluids 1997, 9, 3929. (39) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; PrenticeHall: Englewood Cliffs, NJ, 1965. (40) Brenner, H. Chem. Eng. Sci. 1961, 16, 242.
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Article Table 1. Summary of Perturbation Analysis
order 1/ε2 1/ε ε0
boundary condition at Y = RP/2H and (1 - (RP/2H))
governing equation Pe(∂/∂Y)(VC0) = (∂2C0/∂Y2) + (Pe)ep(∂/∂Y)(C0(∂ψ h/∂Y) Pe(∂/∂X)((U - U + Uep,x)C0) + Pe(∂/∂Y) (VC1) = (∂2C1/∂Y 2) + (Pe)ep(∂/∂Y)(C1(∂ψh/∂Y) (∂C0/∂T) + Pe(∂/∂X)((U - U + Uep,x)C1) + Pe(∂/∂Y) (VC2) = R(∂2C0/∂X 2) + ∂2C2/∂Y 2 + (Pe)ep(∂/∂Y)(C2(∂ψh/∂Y)
To solve eq 14, one may utilize a regular asymptotic expansion for C as C ¼ C0 þ εC1 þ ε2 C2 þ ε3 C3 þ ::::::
ð15Þ
We substitute eq 15 into eqs 14 and 14a and match terms pertaining to different orders of ε. This leads to a coupled system of partial differential equations with the closing boundary conditions, as summarized in Table 1. To assess the efficiency of analyte transport, we first expand the variables C0 and C1 as separable functions of (X, T) and Y and execute an analysis similar to that outlined in our previous work.11,34 Following that approach, we express the dimensionless dispersion coefficient, D*, in the following form D ¼ R -
Pe
R 1 -RP =2H RP =2H
ðU -U þ Uep, x ÞG dY R 1 -RP =2H F dY RP =2H
ð16Þ
where U is the dimensionless band velocity given as R 1 -RP =2H
ðUFÞ dY R =2H U ¼ Uep, x þ RP1 -R =2H P F dY RP =2H
ð17Þ
In the above expressions the function F(Y) can be expressed as (neglecting gravitational effects) FðYÞ ¼ expð -ðPeÞ½j W -j W, C -ðPeÞep ½ψ -ψ C Þ
ð18Þ
In the present perturbation analysis, F(Y) is defined as C0 = A(X, T) F(Y), where A(X, T) is the value of C0 at the channel centerline; for more details, see ref 11. In eq 18, jhW is the dimensionless wallanalyte interaction potential expressed as jW ¼
jW ðj þ jvdW Þ ¼ DL 6πμRP ð2HÞÆuæ 6πμRP ð2HÞÆuæ
jhW,C is the dimensionless wall-analyte interaction potential evaluated at the channel centerline (i.e., y = H), and ψhC is the dimensionless EDL potential evaluated at the channel centerline (and hence ψhC = 0 where the EDLs are assumed to be nonoverlapping). Furthermore, we obtain the function G(Y), as appearing in eq 16, from the numerical solution of the following ordinary differential equation. (It needs to be noted that in the perturbation analysis G(Y) is defined by the relationship C1 = [∂/∂X A(X, T)](G(Y); for more details see ref 11.) d2 G dG þf2 ðYÞG ¼ f3 ðYÞ þf1 ðYÞ dY 2 dY
ð19Þ
Equation 19 is subject to the following boundary conditions at Y= RP/2H and (1 - RP/2H) dG -f1 ðYÞG ¼ 0 dY Langmuir 2009, 25(17), 9863–9872
ð19aÞ
Pe(VC0) - (∂C0/∂Y) - (Pe)ep(C0(∂ψh/∂Y) = 0 Pe(VC1) - (∂C1/∂Y) - (Pe)ep(C1(∂ψh/∂Y) = 0 Pe(VC2) - (∂C2/∂Y) - (Pe)ep(C2(∂ψh/∂Y) = 0
where dψ d2 ψ dV -PeV ; f2 ¼ ðPeÞep 2 -Pe , dY dY dY f3 ¼ PeððU -UþUep, x ÞFÞ ð19bÞ f1 ¼ ðPeÞep
An evaluation of the dispersion coefficient finally leads to the concentration profile at the channel exit (x = L), as given by the following Gaussian distribution " # nM 1 ðL -UÆuætÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ð20Þ ÆC0 æ ¼ c0 wh 4πDy D t 4Dy D t where nM is the number concentration of solute particles introduced into the channel. We extend our analysis further to demonstrate the implications of the streaming potential on the separation performance of charged macromolecules in nanoscale confinements. Toward that, we first note that an effective measure of the separation efficiency, namely, resolution (Rs), may be derived directly from eq 20 by utilizing the following definition ðtR1 -tR2 Þ ð21Þ Rs ¼ ðwb1 þwb2 Þ=2 where tR1 and tR2 are the times required by the macromolecules of radii RP1 and RP2 to attain the corresponding maximum concentration values at the nanochannel exit and wb1 and wb2 are the average widths of the corresponding concentration profiles at half-maximum. (Parameters tR1, tR2, wb1, and wb2 may all be predicted from a plot of eq 20 for two distinctive radii RP1 and RP2). The higher the value of the resolution, corresponding to a given pair of particles, the more efficiently they can be separated.
4. Results and Discussions We study the implications of the streaming potential on the separation characteristics of the negatively charged solute particles (zM = -1) with the help of the different parameters enlisted in Table 2. In Figure 2, we plot the variation of the streaming field with zeta potential for different H/λ ratios. For smaller ranges of the zeta potential values, increments in the zeta potential augment the strength of the streaming field. However, the trend is reversed for higher ranges of the zeta potential. Increments of the zeta potential increase the concentration of counterions in the EDL. The larger the advection velocity of these counterions, the greater the value of the streaming potential. When the number density of counterions is small (i.e., for low zeta potential values), even relatively weaker pressure-driven transport is sufficient to impart substantial advective strength to the counterions, thereby ensuring that the streaming field increases with the zeta potential over that regime. However, for substantially higher zeta potential values, the number density of counterions becomes prohibitively large to permit rapid advective transport of those with the aid of a weak driving pressure gradient. The net effect is that the streaming field decreases with increasing zeta potential over this regime. DOI: 10.1021/la900956k
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Das and Chakraborty Table 2. Parameter Values
ε0 εr L kB T H μ dp/dx A zM
8.8 10-12 C V-1 m-1 79.8 10 mm 1.38 10-23 J K-1 300 K 50 nm 10-3 Pa s -10-9 Pa m-1 10-20 J -1
Figure 3. Variation of the fluid velocity across the channel height from pure pressure-driven flow considerations as well as from the considerations of combined pressure-driven and streaming-potential-induced electroosmotic transport, corresponding to H/λ = 2.
Figure 2. Variation of the streaming potential with the wall zeta potential for different values of the relative EDL thickness. In each plot, we have specified the corresponding maximum values of the streaming potential and the zeta potentials (ζc) at which the respective peaks are attained.
Such contrasting trends in the streaming potential gradient for two distinctive regimes of the zeta potential force the streaming field to attain its maximum at an intermediate zeta potential value. Regarding the role of the relative EDL thickness toward influencing this physical scenario, it may be inferred that the total number of ions in the EDL for a given zeta potential value is reduced for thinner EDLs (when considering the nanochannel under study to be connected to ionic reservoirs). Consequently, the zeta potential value at which the streaming field attains a maximum is larger for larger H/λ values. The presence of the streaming potential field may significantly reduce the resultant flow velocities by inducing an electrokinetic body force that opposes the pressure-driven transport, as depicted in Figure 3. Implications of such reductions in the flow strength, leading to equivalent decrements in the axial velocities of the analytes, may bear far-ranging consequences on the separation characteristics, as illustrated later. In a subsequent part of this article, we illustrate the implications of streaming potential on the size-based migration and separation of particles for cases with and without particle-wall interaction considerations (which include the particle-wall EDL potential, vdW potential, and wall EDL potential-induced transverse electrophoresis of the particles) so as to isolate the nontrivial coupling between the particle-wall interaction mechanisms and the streaming potential effects. 4.1. Migration and Separation in the Absence of Wall Effects. Figure 4 depicts the size-based band velocity variations with and without the streaming potential consideration (for cases without wall effects). For the cases without streaming effects, larger particles tend to amplify the extent of hydrodynamic influences (by virtue of which larger particles get more preferentially excluded from the near-wall regions because of their center 9868 DOI: 10.1021/la900956k
Figure 4. Variation of dimensional band velocity (referred to as UÆuæ in the text) with dimensionless particle sizes with (different zeta values) and without streaming potential effects for H/λ = 2 for cases without wall effects. In the inset, we separately demonstrate the contributions of the streaming-field-induced electrophoretic and electroosmotic influences in altering the band velocity for the case with ζ = -50 mV.
of masses being in greater proximity to the bulk than to the walls and hence being more effectively advected by the pressure-driven flow that strengthens its effects more prominently as one approaches the channel centerline away from the wall) so that there is a monotonic increment in the band velocity with the particle size. On the contrary, for the case with streaming potential considerations, the streaming-field-induced size-dependent axial electrophoresis introduces a size-dependent variation of the band velocity, which is a distinctive characteristic feature that is not manifested with pure pressure-driven flow considerations alone. For small enough particles, the electrophoretic mobility values are substantially large, which tends to increase the band velocity (because the induced streaming field and the electrophoretic mobilities are identical in the sense that the axial electrophoresis augments the pressure-driven transport). However, there is a simultaneous retarding influence of the streaming-field-induced back electroosmosis. Depending on the strength of the zeta potential, either of these effects may have a larger contribution, resulting in the band velocity being either larger or smaller than in the cases without steaming effect considerations. However, the significant decrements in electrophoretic mobility with increasing Langmuir 2009, 25(17), 9863–9872
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Figure 5. Temporal variation of analyte concentration at the channel exit for different RP/H values, considering streaming potential effects (but no wall effects), corresponding to ζ = -25 mV and H/λ = 2.
particle size effectively ensure that for larger particles the streaming effect contribution is primarily due to the electroosmotic backflow, which reduces the strength of the band velocity. (In the inset of Figure 4, we illustrate the contributions of varying streaming fields on the band velocity variations.) We may thus infer that the consideration of the streaming field induces two distinct size-based separation paradigms; for small enough particles, the electrophoretic transport mechanisms have a strong role to play toward dictating the separation characteristics, whereas for larger particles the scenario is primarily dictated by a combined consequence of the pressure-driven and induced back electroosmotic transport. Such a shift in paradigm, in turn, implicates that significant differences in the band velocity values (and hence separation time, a factor that dictates the separation resolution) may be obtained for particles that fall into these two distinctive separation regimes. Knowledge of UÆuæ and D*Dy helps us to evaluate the time at which the peak of the concentration profile is attained at the channel exit (eq 20). Because the axial diffusion effects (which are independent of the streaming potential considerations) outweigh the Taylor dispersion effects for nanochannel flow, the implications of streaming potential on separation characteristics are predominantly manifested through the variations in UÆuæ alone. To understand the effect of the streaming potential on the separation characteristics, we first plot the temporal variations in the concentration profiles at the channel exit for different analyte sizes corresponding to H/λ=2 and ζ = -25 mV (Figure 5). For very small analytes, the streaming-potential-induced electrophoresis predominantly dictates the analyte band velocities. For such cases, the time to attain the peak of the concentration profiles increases with increments in the analyte radii through a reduction in the effective electrophoretic mobilities. This implies that increments in the analyte radii enhance the differences in separation time between two different analytes, thereby augmenting the resolution of separation. However, beyond a certain radius, electrophoretic effects tend to get significantly outweighed by back-electroosmotic effects, and the size dependence is accounted for primarily through the hydrodynamic influences. Thus, for such cases, increments in analyte radii increase the corresponding band velocities, leading to a shorter separation time. Depending on whether this reduced separation time is greater or less than the separation time for the smallest analyte, the resolution may increase or decrease with the increment in the radii ratio. Langmuir 2009, 25(17), 9863–9872
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Figure 6. Variation of resolution of separation with RP/RP,min (RP,min = 0.01 H) for cases with (for different zeta potential values) and without streaming potential effects, for cases without considering wall interactions.
In Figure 6, we plot the corresponding trends in resolution (eq 21) as a function of RP/RP,min (note that two distinctive particle radii are required to obtain the resolution of separation, out of which we fix one value as RP,min = 0.01H) for different zeta potential values, corresponding to a representative case of H/λ = 2. Commensurate with the band velocity variations, the resolution of separation monotonically increases with increments in RP/ RP,min (i.e., with enhanced disparities in the particle sizes) for cases in which streaming potential effects are not considered. However, with a consideration of streaming potential effects, the resolution of separation shows three distinctive trends depending on the RP/ RP,min values. For relatively small values of RP/RP,min, the separation time monotonically increases with increments in RP/ RP,min (as shown in Figure 5), causing enhancements in the corresponding resolution values. This regime is essentially governed by the electrophoretic transport due to the induced streaming potential field. Interestingly, the extent of this increment as well as the range of values of RP/RP,min over which this augmented resolution value (in comparison to the case without streaming potential effects) persists depends on the strength of the zeta potential. We may accordingly distinguish the operating zeta potential values as favorable and adverse. Favorable zeta potential values are essentially the ones in the proximity of ζc (zeta potential value for which the streaming potential is a maximum; see Figure 2), corresponding to a given H/λ. Such values ensure the imposition of strong electrophoretic effects as induced by the streaming potential field, leading to the establishment of large differences in size-based band velocity values and hence an augmented resolution of separation for a given pair of analytes. For the results depicted in Figure 6, favorable zeta potential values are the ones that are close to -50 mV. Similarly, adverse zeta potential values are the ones that are much larger or smaller in magnitude than ζc so that the corresponding streaming potential values are significantly smaller. (For the results shown in Figure 6, adverse zeta potential values, for example, may be -25 and -100 mV.) Beyond a threshold RP/RP,min value, however, the size dependence is predominantly controlled by the hydrodynamic interactions so that increments in the band velocities cause a monotonic reduction (with RP/RP,min) in the separation time (Figure 5). This reduces the difference in separation time between analyte particles of sizes RP and RP,min, causing a decrement in the resolution of separation. However, beyond a threshold value of RP (this value again depends on the zeta DOI: 10.1021/la900956k
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Figure 7. Variation of resolution of separation with zeta potential for RP/RP,min=10 for different H/λ values, with a consideration of streaming potential effects but without considering particle-wall interactions.
potential), hydrodynamic influences render the velocity of particles of size RP to be more than that of size RP,min, making the separation time for analytes of size RP much less than that for analytes of size RP,min. This, in turn, implies that the resolution of separation increases with increments in RP/RP,min. Evidently, beyond the RP value for which the streaming-potential-induced electrophoretic influence ceases to have any perceptible effect on the band velocity variation, the separation characteristics (for the corresponding RP/RP,min values) exhibit no further improvement with the consideration of streaming potential. Thus, for a favorable utilization of the streaming effects for enhanced separation efficiency, an optimal choice of RP/RP,min, consistent with the chosen zeta potential and H/λ values, becomes essential. In Figure 7, we demonstrate the separation characteristics corresponding to such an optimally chosen RP/RP,min value (RP/RP,min = 10) as a function of the zeta potential for different H /λ values. In similar lines to our previous arguments, we find that the resolution of separation is maximum for the favorable zeta potential values (or zeta potential values that are close to ζc corresponding to a given H/λ). Lowered resolution of separation for higher H/λ values may be attributed to reduced strengths of the streaming potential field (Figure 2) with lowered relative EDL thicknesses as compared to the channel height. 4.2. Migration and Separation in the Presence of Wall Effects. Wall effects (vdW interactions, particle-wall EDL interaction,s and EDL-potential-induced transverse electrophoresis) induce strong size dependences of the particle migration characteristics, typically for analytes that are small enough for hydrodynamic effects to be influential. In Figure 8, we illustrate the size-based influences of these different effects in altering the transverse velocity of the analytes. One may clearly observe from this Figure that the TEP and wall-analyte EDL interactions impart repelling influences on the analytes (directing those away from the wall), enabling those to access higher velocity streamlines. However, vdW influences push the analytes toward the wall, leading to a decrement in the band velocity. The band velocity variation (as a function of RP/H) is depicted in Figure 9 with and without streaming potential effects. Under no streaming potential influences, the band velocity variation with analyte size carries the signature of distinctive walls effects over different regimes of RP/H values. For example, for small enough RP, particle-wall EDL interactions and EDL potential induced transverse electrophoresis outweigh the vdW interaction forces so that band velocity increases with RP/H (for reasons 9870 DOI: 10.1021/la900956k
Figure 8. Variation of the transverse velocity (for the channel bottom half) of the analytes with their dimensionless centerline coordinates (yCP/H) considering (a) wall-analyte vdW interaction, (b) wall-analyte EDL interaction (for H/λ = 4 and ζ = -50 mV), and (c) EDL-potential-induced transverse electrophoresis (for H/λ=4 and ζ=-50 mV) for different values of RP/H.
already discussed). However, beyond a threshold RP/H, vdW effect starts to be influential so that the band velocity first comes to a plateau (the vdW effect balances the EDL effects) and then rapidly decreases with RP/H. Eventually, for even larger RP/H values, hydrodynamic influences take over and the band velocity variation exhibits a monotonic increase with macromolecular size. When streaming effects are included, similar to the no-wall case, we observe a rapid decrement in the band velocity with increments in RP/H (for small enough RP/H values), implicating the effect of streaming-potential-induced electrophoresis. Once this electrophoretic effect dies down and the streaming effect is manifested primarily through the retarding influence of induced back electroosmosis, the band velocity follows the same qualitative size dependence of the no-streaming case, but with a reduced magnitude. The nature of this relative variation, however, is such that even for situations where streaming-potentialinduced electrophoretic effects are rather weak, the band velocity difference between particles of sizes RP and RP,min is always greater with the involvement of streaming potential effects, which Langmuir 2009, 25(17), 9863–9872
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Figure 9. Variation of dimensional band velocity (referred to as UÆuæ in the text) with dimensionless particle sizes with and without streaming potential effects for H/λ = 2 and ζ = -50 mV for cases with particle-wall interaction considerations.
implies improved resolution for the entire studied range of RP/RP,min. The temporal variation of the concentration of analytes of different sizes is depicted in Figure 10. Increments in the separation time for smaller analytes with increasing radius occur as a result of the decrements in the band velocity on account of the reduced strength of the streaming-field-induced electrophoretic effects. In the intermediate radii range, vdW influences become dominant, causing a steep reduction in the band velocity (Figure 9) and hence an increased separation time. Beyond a threshold radius, however, the wall interactions as well as electrophoretic influences tend to become somewhat inconsequential, as the hydrodynamic effects start influencing the separation characteristics in a more prolific manner, resulting in monotonic increments in the band velocity (Figure 9) and accordingly a reduction in the separation time. Figure 11 depicts the variations in the resolution of separation as a function of the analyte radius. As dictated by the concentration variations, for smaller RP/RP,min values, there is a monotonic increment in the resolution of separation. This is followed by a steep rise in the resolution of separation over the ranges of RP in which the vdW effects become dominant. However, once the RP values become large enough for the hydrodynamic influences to dominate, the resolution monotonically decreases, for reasons already mentioned in the context of Figure 6. It is also important to mention in this context that the electrophoretic effects induced by the streaming potential lead to significant enhancements in the resolution of separation in the lower ranges of RP/RP,min values. Interestingly, the enhanced resolution persists for the entire range of RP/RP,min values, signifying that the effect of streaming potential in improving the separation efficiency is amplified in the presence of wall interactions. 4.3. Possible Experimental Realization of the Present Design. Over the past few years, a number of significant experimental studies have been reported in the literature on nanochannel-based analyte transport and separation characteristics.18,21,22 Using the methodologies described in these studies, one may experimentally design efficient test benches for assessing the separation characteristics of charged analytes subjected to streaming potential effects. A critical issue in conducting such (41) van der Heyden, F. H. J; Stein, D; Dekker, C. Phys. Rev. Lett. 2005, 95, 116104.
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Figure 10. Temporal variation of the concentration at the channel exit for analytes with different values of RP/H considering streaming potential effects and wall interactions, corresponding to ζ = 50 mV and H/λ = 2.
Figure 11. Variation of the resolution of RP/RP,min (RP,min = 0.01 H) with and without streaming effects for H/λ = 2 and ζ = -50 mV for cases with particle-wall interaction considerations.
experiments with quantitative precision involves the accurate measurement of streaming potentials, the protocol of which has been well documented in the works of van der Heyden et al.26,41 The real challenge in the design, however, may originate from the requirements of handling analyte particles of sizes comparable to the channel height without clogging the flow passage. Nevertheless, the transport of nanometer-sized beads in narrow confinement has been successfully demonstrated and controlled by several researchers in the recent past. An important design challenge, however, may lie in coming up with optimal sizes of the charged analytes that are large enough for wall effects to be consequential and yet small enough to prevent any clogging of the flow passage. Once this size regime has been identified through experimental trials, one may vary the relative EDL thickness (by altering the ionic concentration of the buffer) or may even alter the strength of the particle-wall interactions by engineering the physicochemical characteristics of the nanofluidic substrate so as to examine the cases with the best possible experimentally realizable separation performance.
5. Conclusions We have carried out detailed theoretical investigations to assess the consequences of streaming potential on the size-based DOI: 10.1021/la900956k
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ε0 εr ψ Fe n( VB μep,( D( kB T ζ λ Istr Icond Iionic H u uP uES μ z(
permittivity of free space (C V-1 m-1) relative permittivity of the medium EDL potential field (V) charge density due to free charges within EDL (C m-3) number density of the cations and anions (m-3) velocity vector (m s-1) electrophoretic mobility of the cations and anions (m2 V-1 s-1) diffusivities of the cations and anions (m2 s-1) Boltzmann constant (J K-1) absolute temperature (K) zeta potential (V) Debye length (m) streaming current per unit length (C m-1 s-1) conduction current per unit length (C m-1 s-1) ionic current per unit length (C m-1 s-1) channel half height (m) axial velocity field (m s-1) pressure-driven component of the axial velocity field (m s-1) streaming-potential-induced component of the axial velocity field (m s-1) dynamic viscosity of the fluid (Pa s) ionic valence
9872 DOI: 10.1021/la900956k
vrel,y zM RP φ Dx Dy jW jDL jvdW ζm hs A λh yCP μep D0 Dr fS,¥ fS,r fS,r,
)
List of Symbols
f ES p ucond,( n0 uP vP urel,x
fS,r,
)
transport and separation of charged solutes in pressure-driven nanochannel flows. The additional size dependence (over and above the hydrodynamic influences) introduced by the consideration of streaming effects results in distinctive separation characteristics that exhibit substantial improvements in the resolution of separation for pairs of particles having slight differences in size. In effect, we have demonstrated that an optimal regime of particle-separation characteristics may be obtained far beyond the constraints imposed by the hydrodynamic interaction limits for small particles, on the basis of the relative EDL thicknesses and the zeta potential values, for a given particle-size ratio. When wall effects are introduced, the extent of this improvement, on account of streaming potential considerations, is dramatically enhanced in a sense that even for extremely large particles the resolution is significantly augmented. Thus, findings from this study may act as precursors to focused experimental investigations demonstrating the exploitation of streaming-potential-induced electromigration effects to the most favorable limits for improving the separation performance of charged particles in all size limits by exploiting their nontrivial interconnection with particle-wall interactions.
u Æuæ Pe l (Pe)ep ε U Uep,x D* nM w Rs
ionic friction coefficient (N s m-1) streaming potential gradient (V m-1) pressure (Pa) ionic velocity due to the conduction current (m s-1) bulk ionic number density (m-3) axial velocity of the analyte (m s-1) transverse velocity of the analyte (m s-1) axial velocity of the analyte relative to the flow (m s-1) transverse velocity of the analyte relative to the flow (m s-1) analyte valence analyte radius (m) resultant electrical potential (V) axial diffusivity (m2 s-1) transverse diffusivity (m2 s-1) wall interaction potential (J) double-layer interaction potential (J) van der Waals interaction potential (J) zeta potential at the analyte surface (V) separation distance between the analyte and the channel bottom wall (m) Hamaker constant (J) London characteristic wavelength (m) coordinate of the analyte center of mass (m) electrophoretic mobility of the solute (m2 V-1 s-1) bulk diffusivity (m2 s-1) reduced diffusivity due to confinement effects (m2 s-1) unrestricted Stokes friction coefficient for the analyte (N s m-1) enhanced friction coefficient for the analyte (N s m-1) enhanced friction coefficient due to analyte motion parallel to the wall (N s m-1) enhanced friction coefficient due to analyte motion perpendicular to the wall (N s m-1) analyte band velocity (m s-1) average flow velocity (m s-1) Peclet number based on channel height and average flow velocity Æuæ width of the sample plug (m) Peclet number based on the channel height and electrophoretic velocity perturbation oparameter dimensionless band velocity dimensionless axial electrophoretic velocity dimensionless dispersion coefficient number of moles of analyte width of the channel (m) resolution of separation
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