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Langmuir 2008, 24, 7704-7710
Transport and Separation of Charged Macromolecules under Nonlinear Electromigration in Nanochannels Siddhartha Das and Suman Chakraborty* Department of Mechanical Engineering, IIT Kharagpur-721302, India ReceiVed December 13, 2007. ReVised Manuscript ReceiVed May 5, 2008 In this work, we theoretically investigate the implications of nonlinear electrophoretic effects on the transport and size-based separation of charged macromolecules in nanoscale confinements. By employing a regular perturbation analysis, we address certain nontrivial features of interconnection among wall-induced transverse migrative fluxes, electrophoretic and electroosmotic transport, confinement-induced hindered diffusive effects, and hydrodynamic interactions in detail. We demonstrate that there occurs an optimal regime of influence of the nonlinear electrophoretic effects, within which high values of separation resolution may be achieved. This size-based optimal regime, however, can be effectively exploited only for nanochannel flows, as attributed to the strong electric double layer interactions prevalent within the same.
1. Introduction The electromigration of charged macromolecules in narrow fluidic confinements has been receiving ever-increasing attention from the research community over the past few years, primarily because of its emerging applications in biomedical, biotechnological, and forensic investigations.1–10 For example, by utilizing the size-dependent electrophoretic mobilities of the transported entities, researchers have been continuously endeavoring to exploit this mechanism toward the efficient implementation of macromolecular separation processes in laboratory-on-a-chip-based micrototal analysis systems. The electrophoretic migration of macromolecules, which is the most fundamental physical phenomenon dictating these critical applications, is inevitably accompanied by the spreading or dispersion of individual band of macromolecules11 (also known as band broadening) that continues until an equilibrium solute distribution is established within the system. Band broadening lowers the steepness of the concentration peak at the channel exit, thereby reducing the overall separation efficiency. Researchers are therefore continuously discovering new physical mechanisms for augmenting the effectiveness of electromigrative transport of charged macromolecules in microscale and nanoscale fluidic systems. The concept of nonlinear electrophoresis (also termed inducedcharge electrophoresis) has recently opened up new regimes for size-dependent electromigrative manipulation of highly polarizable charged macromolecules in confined systems. Physically, such effects stem from the fact that the influence of a local electric field drives away a flux of counterions close to the particle surface, which perturbs the equilibrium potential distribution in the electrical double layer (EDL) formed adjacent to the particle, despite conserving the net macromolecular charge. This leads to * 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) Slater, G. W. Electrophoresis 1993, 14, 1.
a nonlinear drift velocity of the macromolecules, which typically scales with the cube of the applied field. The resulting expression for the electrophoretic drift velocity thus reads12,13
Veph ) µepE + µep,3E3
(1)
The proportionality constant, µep,3, also known as the nonlinear electrophoretic mobility, represents the combined effect of the pertinent macromolecular characteristics that exhibit the nonlinear dependence (to be detailed later) of the equilibrium electrical parameters on the local field strength, E. Motivated by the degree of freedom being offered by the additional electrophoretic mobility component, the nonlinear electromigration phenomenon has been utilized in many critical applications concerning the transport of biological macromolecules (which are inherently charged in most cases). Examples of such critical applications, which have already been experimentally implemented, include the focusing of DNA molecules, the preparative isolation of DNA fragments, and the development of novel separation methods for bacterial fingerprinting.14,15 Nonlinear electromigration has also been successfully employed in achieving particle levitation,16 alignment for the electrofusion of cells,17,18 electroporation for the inclusion of molecules into cells,19,20 fabrication of dielectrophoretic field cages to trap single cells,21,22 and the separation of complexes of proteins and hydrophobic ionic detergents at pH values away from the pI of proteins.23 Furthermore, nonlinear dielectrophoretic effects have been utilized for different cell-related separation activities, including the separation of live from dead yeast cells,24 live from (12) Dukhin, A. S.; Dukhin, S. S. Electrophoresis 2005, 26, 2149. (13) Barany, S.; Shilov, V.; Madai, F. Colloids Surf., A 2007, 300, 353. (14) Frumin, L. L.; Peltek, S. E.; Zilbertstein, G. V. J. Biochem. Biophys. Methods 2001, 48, 269. (15) Frumin, L. L.; Peltek, S. E.; Zilbertstein, G. V. Phys. ReV. E 2001, 64(1-5), 021902. (16) Jones, T. B.; Kallio, G. A. J. Electrost. 1979, 6, 207. (17) Washizu, M. J. Electrost. 1990, 25, 109. (18) Wu, Y.; Sjodin, R. A.; Sowers, A. E. Biophys. J. 1994, 66, 114. (19) Andersson, H.; van den Berg, A. Curr. Opin. Biotechnol. 2004, 15, 44. (20) Mognaschi, E. R.; Savini, A. J. Phys. D: Appl. Phys. 1983, 16, 1533. (21) Schnelle, T.; Mu¨ ller, T.; Fiedler, S.; Fuhr, G. J. Electrost. 1999, 46, 13. (22) Voldman, J.; Braff, R. A.; Toner, M.; Gray, M. L.; Schmidt, M. A. Biophys. J. 2001, 80, 531. (23) Peltek, S. E.; Frumin, L. L.; Chasovskikh, V. V.; Zilberstein, G. V. Analyst 2002, 127, 337. (24) Markx, G. H.; Pethig, R. Biotechnol. Bioeng. 1995, 45, 337.
10.1021/la703892q CCC: $40.75 2008 American Chemical Society Published on Web 07/12/2008
Transport and Separation of Charged Macromolecules
dead bacteria cells,25 malaria-infected cells from healthy cells,26 and human leukemia cells from healthy blood cells.27 A plethora of these applications is fundamentally based on the contradiction that unlike its linear counterpart (µep) µep,3 increases with an increase in radius, imposing counter-balancing influences on the resultant macromolecular drift. A simple example of an application of this principle involves the migration of macromolecules of different sizes in opposite directions in pulsed-field gel electrophoresis.14 A somewhat more complicated arrangement may involve the interaction of an unbalanced alternating transverse electrical field with the nonlinear electromigration of charged particles, resulting in their simultaneous stacking and separation.12 The successful implementation of most of these applications has been augmented by the favorably high surface conductivity and high polarizability inherent to different types of DNA,28,29 proteins,28,30,31 and nanoparticles28,32–35 that are commonly employed. Despite being attributed to several advantageous features mentioned above, the successful exploitation of nonlinear electromigrative influences characteristically suffers from a critical bottleneck in the sense that it necessitates the employment of large transverse electric fields for the prominent manifestation of the pertinent desirable features. Such constraints, in effect, may lead to several technological drawbacks. For instance, persistently large globally imposed electric fields may lead to the serious degradation of thermally labile biological entities such as DNA, as attributed to significant Joule heating effects. In addition, such fields may induce large global ionic gradients in the flow medium, which may in turn impose adverse physiological changes on the biological cells being handled and manipulated. Although these adversities may be arrested to some extent through the imposition of time periodic electrical fields, the corresponding technological complications (especially those demanding the employment of alternating transverse electric fields) are not far from becoming ominous. Scientific intuition offers insight into the idea that the interactions of nonlinear electrophoretic effects and confinementinduced migrative fluxes in nanochannels may indeed turn out to be of utmost significance in terms of dictating the overall particle transport characteristics because additional surface effects may be potentially exploited over reduced length scales. In addition, implicitly induced high electrical energy densities may be potentially realized over the system length scales when the characteristic EDL thicknesses are comparable to the channel dimensions, which might obviate the necessity of employing externally imposed transverse electric fields. However, despite such interesting implications, no study has yet been reported in the literature to elucidate the interactions between the wall-induced transverse fluxes and the nonlinear electrophoretic influences, in terms of dictating the macromolecular transport characteristics in nanofluidic devices and systems. (25) Lapizco-Encinas, B. H.; Simmons, B. A.; Cummings, E. B.; Fintschenko, Y. Anal. Chem. 2004, 76, 1571. (26) Gascoyne, P.; Mahidol, C.; Ruchirawat, M.; Satayavivad, J.; Watcharasit, P.; Becker, F. F. Lab Chip 2002, 2, 70. (27) Becker, F. F.; Wang, X.-B.; Huang, Y.; Pethig, R.; Vykoukal, J.; Gascoyne, P. J. Phys. D: Appl. Phys. 1994, 27, 2659. (28) Zheng, L.; Brody, J. P.; Burke, P. J. Biosens. Bioelectron. 2004, 20, 606. (29) Regtmeier, J.; Duong, T. T.; Eichhorn, R.; Anselmetti, D.; Ros, A. Anal. Chem. 2007, 79, 3925. (30) Nayeem, A., Jr.; Stouch, T. Biopolymers 2003, 70, 201. (31) Quillin, M. L.; Breyer, W. A.; Griswold, I. J.; Matthews, B. W. J. Mol. Biol. 2000, 302, 955. (32) Chang, R.; Chiang, H. -P.; Leung, P. T.; Tse, W. S. Opt. Commun. 2003, 225, 353. (33) Alu, A.; Engheta, N. J. Appl. Phys. 2005, 97, 094310. (34) Murray, D. B.; Netting, C. H.; Mercer, R. D.; Saviot, L. J. Raman Spectrosc. 2007, 38, 770. (35) Baldessari, F.; Santiago, J. G. J. Nanobiotech. 2006, 4, 12.
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In the present article, we demonstrate the possibility of unveiling a new paradigm of nonlinear electrophoretic migration in which the requirement of imposing large transverse electric fields (either direct or alternating) is completely eliminated, thereby overcoming the associated adverse consequences. We achieve this improvement by exploiting nontrivial physical chemistry associated with the pertinent interfacial phenomena over nanoscopic scales, which is the central theme behind the present investigation. In particular, we focus our attention on specific nanoscale electromigrative regimes, over which the fundamental physicochemical mechanisms dictating the solute transport characteristics are nontrivially influenced by the complex interplay among the EDL interactions, van der Waals forces, and the confinement-induced hydrodynamic interactions.35–40 In effect, the van der Waals forces, which decay rapidly with distances away from the wall, may induce strong wall-directed attractive velocities (∼1 mm/s) on particles of radii of the order of 20 nm (typically, within 10 nm of the separation distance from the wall).38 The particle-wall EDL interactions, however, create repelling drifts of similar orders of magnitude but spread over larger separation distances.38 Another contrasting feature is that the van der Waals force increases with macromolecular radius whereas the force due to EDL interaction may exhibit a reverse trend.38 On the basis of a judicious exploitation of these interaction mechanisms, we establish a yet unexplored operating regime for optimal electromigration of charged macromolecules in nanofluidic systems. A critical feature that works favorably in this respect, as compared to its scaled-up microfluidic counterpart, is the fact that strong EDL interactions (substantially aggravated by extremely short separation distances between the electrically interacting surfaces that are typical of nanochannels) are likely to establish large induced local potential gradients. Because of distinctive particle sizes and particle positions, this confluence is implicitly subjected to spatiotemporal variations that mimic the effects of externally imposed time-varying electrical fields, to a broader extent in a notional sense. It is also important to mention here that although the linear component of the resultant local variations in electrophoretic mobility is governed solely by the particle charge and radius its nonlinear counterpart has an additional constraint of large particle polarizability. In our previous work,38 we established that the mutually competitive particle-wall interaction forces (ensuring a definite transverse location of a macromolecule of a given size, which in turn decides its rate of axial transport), coupled with the linear electrophoretic effects, hydrodynamic forces (due to finite macromolecular sizes), and hindered diffusivity constraints (which originated from hindrance on the transverse or longitudinal migration of particles in confinements having dimensions comparable to the particle sizes) may be judiciously exploited for achieving an extremely high resolution of separation (∼10), which is far from being obtained in any other standard separation methodology. In this article, we extend this analysis to show that complex nonlinear interactions between the induced particle-wall EDL field and the electromigrative mechanisms, operating on highly polarizable macromolecules, may result in their extremely efficient and selective size-based transport, with the synergetic aid of other confinement-induced hydrodynamic interactions. This, clearly, is of twofold technological advantage. First, a more stringent handle on the particle migration features as a function of their (36) Xuan, X.; Li, D. Electrophoresis 2006, 27, 5020. (37) 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. (38) Das, S.; Chakraborty, S. Electrophoresis 2008, 29, 1115. (39) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6772. (40) Paul, D.; Chakraborty, S. J. Appl. Phys. 2007, 102, 074921.
7706 Langmuir, Vol. 24, No. 15, 2008
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expressed through the Nernst equation as
∂n( +b V · ∇ n( ) ∇ (n(z(νF ∇ φ) + D∇2n( ∂t
Figure 1. Flow geometry.
relative sizes enables one to achieve extremely efficient macromolecular transport characteristics, which grossly outweigh the corresponding limits attainable through the employment of other alternative arrangements such as alternating electric fields (typical of examples involving nonlinear dielectrophoresis). More importantly, the present proposition completely eliminates the need to employ any externally imposed transverse electric field, which simplifies the underlying technological arrangements to a large extent. The only limitations associated with this method are the narrow-confinement-induced constraints on the ranges of macromolecular sizes being handled and the requirements of high particle polarizabilities (which are otherwise satisfied by typical biological macromolecules such as DNA). It is also important to emphasize here that the present analysis is by no means restricted to the applications of macromolecular separation but is designed to be somewhat more generic in nature so as to address the fundamental scientific issues pertinent to the interaction mechanisms between nanoscale interfacial phenomena and nonlinear transverse electromigration in narrow confinements. Rather, we take macromolecular separation to be a concrete and particular example through which the practicalities of this scientific principle may be best illustrated. The organization of the remaining part of this article is as follows. First, we execute a regular perturbation analysis, similar in principle to that of Chen and Chauhan,41 to obtain closedform expressions quantifying the macromolecular band velocity and dispersion coefficient. As an example of the effects of nonlinear electromigration on macromolecular transport, we parametrically calculate the resolution of separation as a function of the significant nondimensional operating parameters. From these analyses, we highlight certain nontrivial features of the resultant transport characteristics, which can be potentially exploited over specific regimes of macromolecular sizes so as to achieve dramatically augmented separation performances.
2. Fundamental Transport Equations 2.1. Electroosmotic Transport and the Solvent Velocity Field. We consider the combined electroosmotic and electrophoretic transport of an ionic solution of charged spherical nondeformable macromolecules (radius RP) through a nanoslit of height 2H (Figure 1). The ionic nature of the solution induces an EDL adjacent to the channel walls. The electric potential in the EDL (ψ) is a function of the net charge density distribution, Fe, and is expressed through the Poisson equation as
Fe ∂2ψ ∂2ψ + 2 )2 ε ∂x ∂y
(2)
with Fe ) e(z+n+ + z-n-), where n ( is the ionic number density (41) Chen, Z.; Chauhan, A. J. Colloid Interface Sci. 2005, 285, 834.
(3)
where ∇φ is the combination of the gradient of the EDL potential (∇ψ) and the applied axial electric field. Neglecting the ionic convection current in comparison to the conduction current induced by the electrical field developed within the EDL, eqs 2 and 3 may be collectively employed to obtain a steady-state distribution of the buffer ionic charge concentration and the resultant induced potential. 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 (EDL) potential distribution can be expressed as42
ψ(y) )
[ ( { } ( )) ( { } (
4kBT ezζ y tanh-1 tanh exp + ze 4kBT λ ezζ 2H - y tanh-1 tanh exp 4kBT λ
))] (4)
where ζ is the wall zeta potential. The flow field is assumed to be steady, axial, hydrodynamically fully developed, and purely electroosmotically driven (as the result of an imposed axial electric field) and is expressed as37
b V)uiˆ ) -
εζEx ψ 1 - ˆi µ ζ
(
)
(5)
It is important to mention here that we consider the no-slip hydrodynamic boundary condition in this work, although the hydrodynamics within the EDL may induce certain nontrivial apparent slip43 effects, to be attributable to the confinementinduced hydrophobic interactions. In addition, we neglect any pressure-driven flow component. In practice, however, there may be interesting and nonintuitive coupling between the induced streaming electrical fields and the imposed pressure gradients for nanochannel flows, the details of which are reported elsewhere.44 2.2. Solute Transport. To study the solute transport, let us assume that a pulse of macromolecules with a mean velocity of uj has been introduced into the channel at t ) 0. In a reference frame moving with a velocity uj, the generalized species transport equation reads
∂c ∂c ∂c ∂ ∂ ∂ ∂ D + D + + ((u -u¯)c) + (VPc) ) ∂t ∂x P ∂y ∂x x ∂x ∂y y ∂y ∂ ∂φ ∂ ∂φ ∂ ∂φ µ c + µ c + µ c( )3 (6) ∂x ep ∂x ∂y ep ∂y ∂y ep,3 ∂y
(
)
( ) ( ) ) ( ) (
In eq 6, uP is the axial nonelectrophoretic macromolecular velocity, VP is the transverse nonelectrophoretic part of macromolecular velocity, µep (µep ) zMe/6πµRP) is the linear electrophoretic mobility, and µep,3 is the nonlinear electrophoretic mobility given as12
µep,3 ) RP2z+z-
Du eε 3kBT 1 + 2Du
(7)
In eq 7, RP is the macromolecular radius,and Du (Du ) Kσ/ KmRP) is the Dukhin number, where Kσ and Km are the surface conductivity of the macromolecule and the bulk conductivity of (42) Hunter, R. J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: New York, 1981. (43) Chakraborty, S. Phys. ReV. Lett. 2008, 100, 097801. (44) Chakraborty, S.; Das, S. Phys. ReV. E 2008, 77, 037303.
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Langmuir, Vol. 24, No. 15, 2008 7707
the medium, respectively. The surface conductivity can be expressed as45
Kσ ) Kσd + Kσi
(8)
where Kσd and Kσi are the conductivities due to the diffuse and stagnant parts of the macromolecular double layer, respectively. For a symmetric electrolyte (|z+| ) |z-| ) z), the expression for Kσd becomes45 2 2
Kσd )
{ [ (
) ]( ) [ ( ) ]( )}
2e z n0λ ezζM 3m+ D+ exp -1 1+ 2 + kBT 2kBT z 3mezζM D- exp -1 1+ 2 2kBT z
eFRpz2n0 K ) kBTNA
[
z-δ-Dz+δ+D+ + z+eζM z-eζM z exp z exp kBT kBT
( )
( )
]
(8a)
(8b)
where δ+(δ-) is the surface conductivity parameter of the cations (anions), given as
( )( ) [ ( ) ( )]
DSL ez( σ ( exp S kBT CSL RPK( D( δ( ) n0 ez+ σ 1+ exp ζ K+ kBT M CSL
∂c ∂c ∂ ∂ ∂ D + + ((u - u¯ + µepEx)c) + (Vrel,yc) ) ∂t ∂x ∂y ∂x x ∂x ∂ ∂c ∂ψ 3 ∂ ∂ψ ∂ D + µ c + µ c (10) ∂y y ∂y ∂y ep ∂y ∂y ep,3 ∂y
( )
jy ) -Dy
n0 ezσ exp ζ KkBT M CSL
)] )]
(8c)
∂c ∂ψ 3 ∂ψ ) 0 at + Vrel,yc - µepc - µep,3c ∂y ∂y ∂y y ) 0, 2H (10a)
( )
3. Regular Perturbation Analysis We employ regular perturbation analysis to solve eq 10, with the respective choices of the following time scales and length scales:15 t0 ) L/〈u〉 and l ≈ (DyL/〈u〉)1/2, where L is the channel length and 〈u〉 is the mean fluid velocity. Accordingly, eq 10 may be nondimensionalized by employing the following normalization parameters
T)
νrel,y t 〈u〉t u j u¯ c ) ,U) ,U) ,V) ,C) ,X) t0 L 〈u〉 〈u〉 〈u〉 c0 x Ψ y 〈u〉2H ,Ψ) ,Y) , Pe ) l 2H Dy ζ
to yield 2 ∂C Pe ∂ ¯ + Uep,x)C) + Pe ∂ (VC) ) R ∂ C + + ((U - U 2 ∂T ε ∂X ε ∂Y ∂X2 3 (Pe)ep,3 ∂ ∂ψ 1 ∂2C (Pe)ep ∂ ∂ψ + C + C (11) ∂Y ε2 ∂Y2 ε2 ∂Y ∂Y ε2 ∂Y
+
( )
S N(
In eq 8c, is the number of sites available per unit area for the adsorption of z( ions, K( is the dissociation constant of those SL sites, D( is the diffusivity of the ions in the stagnant layer, σ(σ˜ ε/4πζM/λ) is the surface charge density of the macromolecule based on a given ζM, and CSL is the capacitance of the stagnant ionic layer. Furthermore, the bulk conductivity may be expressed as46
Km ) Km0(1 + 3φvfdM)
)
Expressions for Vrel,y (transverse velocity of macromolecules relative to the background flow) may be derived by considering the van der Waals and EDL interaction potentials, as detailed elsewhere.38,40,47,48 It is also important to mention here that the diffusion coefficients appearing in eq 10 may need to be modified to account for the confinement-induced hindered diffusive effects, which are not considered in this work.
NS(
( ) [ ( )( ( ) [ ( )(
(
( ) ( ( ))
Equation 10 is subjected to a no-flux boundary condition at the walls, which reads
In eq 8a, n0 is the bulk ionic number density, ζM is the macromolecular surface zeta potential, D+(D-) is the diffusivity of the cations (anions), and m+(m-) is the dimensionless ionic mobility of the cations (anions). To calculate Kσi, we use the finite stagnant layer conductivity (SLC) approach. Unlike the classical approach in which ions of any kind located within the stagnant layer of the macromolecular EDL is assumed to be immobile (thereby rendering Kσi ) 0), only the liquid in the stagnant layer is assumed to be static in the finite SLC method. Thus, ions may possess their own diffusive and electromigrative velocities within the EDL, thereby giving rise to a nonzero value of Kσi. If we follow this approach, then Kσi may be expressed as45 σi
velocity and is virtually negligible because the rotational effects of the macromolecules are not considered to be important), VP ) V + Vrel,y ) Vrel,y, ∂φ/∂x ) -Ex, and ∂φ/∂y ) ∂ψ/∂y, eq 6 simplifies to38
(9)
where Km0 is the conductivity without considering the presence of the macromolecules, φvf is the volume fraction of the macromolecules, and dM is the dipolar coefficient of the macromolecules (∼1). For the present problem, we consider a very dilute solution of the macromolecules, ensuring φvf < 0.1, so that Km ≈ Km0. Considering that uP ) u + urel,x ≈ u (urel,x, which is the x component of the macromolecular velocity relative to the flow (45) Carrique, F.; Arroyo, F. J.; Shilov, V. N.; Cuqejo, J.; Jimenez, M. L.; Delgado, A. V. J. Chem. Phys. 2007, 126, 104903. (46) Grosse, C.; Pedrosa, S.; Shilov, V. N. J. Colloid Interface Sci. 2002, 251, 304.
( ( ))
In eq 11, Uep,x ) µepEx/〈u〉, R ) Dx/Dy, ε ≡ 2H/l(2H/L)1/2 , 1, (Pe)ep ) ζµep/Dy, and (Pe)ep,3 ) [µep,3ζ3/(2H)3]2H/Dy. The dimensionless form of the boundary condition given by eq 10a is
-
∂C ∂ψ ∂ψ + Pe(VC) - (Pe)epC - (Pe)ep,3C( )3 ) 0 at Y ) ∂Y ∂Y ∂Y RP ⁄ 2H and (1 - RP ⁄ 2H) (11a)
To solve eq 11, one may utilize a regular asymptotic expansion for C as
C ) C0 + εC1 + ε2C2 + ε3C3 + ...
(12)
We substitute eq 12 in eqs 11 and 11a 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. (47) Adamczyk, Z.; Van De Ven, T. G. M. J. Colloid Interface Sci. 1981, 81, 497. (48) Lin, Y. C.; Jen, C. P. Lab Chip 2002, 2, 164.
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To assess the efficiency of macromolecular transport, we first expand variables C0 and C1 as separable functions of (X, t) and Y and execute an analysis similar to that outlined in our previous work.38,40 Following that approach, we express the dimensionless dispersion coefficient, D*, in the following form RP
Pe
dG - f1(Y)G ) 0 dY where
P
2H
D )R-
RP 12H RP
∫
(13) F dY
where RP
∫R1- 2H (UF) dY
〈C0 〉 )
P
2H RP 12H RP
∫
F dY
The function F(Y) can be expressed as -
-
(
F(Y) ) exp(-(Pe)[φW - φW,C] + (Pe)Fg Y -
1 2
)
-
(Pe)ep[ψ - ψC] - I1 + I2 (15) where
φ¯ W )
φW 6πµRP(2H)〈u〉 I1 )
∫1⁄2Y
)
6πµRP(2H)〈u〉
∂ψ ∂Y
and
I2 )
φDL + φVdW
( ( ) Pe)ep,3
µ
∫1⁄2Y 〈u〉(Zmep,3 e)(2H)
3
dY
( ) ∂φ¯ DL ∂Y
(Zm is the macromolecular valence and
(
φ¯ DL )
n0M c0wh
(14)
2H
-
( )
Evaluating the dispersion coefficient finally leads to the concentration profile at the channel exit (x ) L), as given by the following Gaussian distribution
2H
U ) Uep,x +
3
dψ dψ - PeV + (Pe)ep,3 dY dY 3 d2ψ d dψ dV - Pe f2 ) (Pe)ep 2 + (Pe)ep,3 dY dY dY dY ¯ + Uep,x)F) (16b) f3 ) Pe((U - U
f1 ) (Pe)ep
∫R1- 2H (U - U¯ + Uep,x)G dY
/
( )
(16a)
φDL 6πµRP(2H)〈u〉
3
dY
)
is the dimensionless EDL (particle)-EDL (wall) interaction potential.41) We evaluate the closed-form expressions for I1 and I2 with the aid of a symbolic mathematics package. We abstain from reproducing those tedious and lengthy expressions here for the sake of brevity. We obtain the function G(Y), as appearing in eq 13, from the numerical solution of the following ordinary differential equation
d2G dG + f1(Y) + f2(Y)G ) f3(Y) 2 dY dY
(16)
subject to the following boundary conditions at Y ) RP/2H and (1 - RP/2H):
1
√4πDyD/t
exp[-
¯ 〈u〉t)2 (L - U ] 4DyD/t
(17)
where n0M is the number of moles of macromolecules introduced into the channel. On evaluating the above expressions, we first note that the dispersion coefficient depends almost solely on the diffusion coefficient ratio (as also predicted by Pennathur and Santiago,32 although following a somewhat different approach). This may be attributed to the fact that the contribution of the velocitydependent term (or the Taylor contribution to the overall dispersion) turns out to be negligible in the description of the effective dispersion coefficient (as governed by eq 13), as attributed to the characteristically low Peclet number values in typical nanochannel flows. Furthermore, noting that with F(Y) j -ψ j C]), we also assess the special case of ) exp(-(Pe)ep[ψ linear electrophoresis without involving any wall-induced transverse migration influences, for which we obtain the same form of the band velocity expression (in dimensionless form) as given by eq 19 of Pennathur and Santiago.39 (Pennathur and Santiago miss one dividing factor of 〈exp(-zSe(ψ(y) - ψC)/ kBT)〉 in the first term on the RHS of eq 19 in ref 39). On the basis of this preliminary validation with the reported benchmark literature mentioned above, we extend our analysis further to demonstrate the implications of nonlinear electromigration of charged macromolecules on their separation performance in nanoscale confinements. Toward that end, we first note that an effective measure of the separation efficiency, namely, resolution (Rs), may be derived directly from eq 17 by utilizing the following definition
Rs )
(tR1 - tR2) (wb1 + wb2) ⁄ 2
(18)
where tR1 and tR2 are the times required by 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 can all be predicted from a plot of eq 17 for two distinctive radii RP1 and RP2). The higher the value of the resolution, corresponding to a
Table 1. Summary of Perturbation Analysis order
governing equation
boundary conditionat Y ) RP/2H and (1 - RP/2H)
1/ε2
j /∂Y) + (Pe)ep,3(∂/ Pe(∂/∂Y)(VC0) ) ∂ C0/∂Y + (Pe)ep(∂/∂Y)(C0 ∂ψ j /∂Y)3) ∂Y)(C0(∂ψ j ep,x)C0)+Pe(∂/∂Y)(VC1) ) ∂2C1/∂Y2 + (Pe)ep∂/∂Y(C1 Pe(∂/∂X)((U -U j /∂Y) + (Pe)ep,3∂/∂Y(C1(∂ψ j /∂Y)3) ∂ψ j - U + Uep,x)C1) + Pe(∂/∂Y)(VC2) ) R(∂2C0/ ∂C0/∂T + Pe(∂/∂X)((U j /∂Y) +(Pe)ep,3(∂/∂Y)(C2(∂ψ j /∂Y)3) ∂X2) + ∂2C2/∂Y2 + (Pe)ep∂/∂Y(C2 ∂ψ
j /∂Y) - (Pe)ep,3(C0(∂ψ j /∂Y)3) ) 0 Pe(VC0) - ∂C0/∂Y - (Pe)ep(C0 ∂ψ
1/ε ε0
2
2
j /∂Y)- (Pe)ep,3(C1(∂ψ j /∂Y)3) ) 0 Pe(VC1) - ∂C1/∂Y - (Pe)ep(C1 ∂ψ j /∂Y)- (Pe)ep,3(C2(∂ψ j /∂Y)3) ) 0 Pe(VC2) - ∂C2/∂Y - (Pe)ep(C2 ∂ψ
Transport and Separation of Charged Macromolecules
Figure 2. Variation of the linear and nonlinear components of the transverse electrophoretic velocity field (resulting from the interaction of wall EDLs) as functions of the dimensionless macromolecular position (yCP/2H) for different H/λ values corresponding to RP/H ) 0.3.
given pair of macromolecules, the more efficiently they can be separated.
4. Results and Discussions We study the implications of nonlinear electrophoresis on the separation characteristics of negatively charged macromolecules (zM ) -1) with the aid of the following parameters: L ) 10 mm, ε ) 6.65 × 10-10 C/Vm, kB ) 1.38 × 10-23 J/K, µ ) 10-3 Pa s, T ) 300 K, ζ ) -100 mV, n0M ) 0.1 moles, A ) 10-20 J, S λ ) 40 nm, Ex ) 104 V/m, z ) 1, ζM ) -100 mV, N+ ) 5.0/nm2, S SL SL 2 N- ) 0.0, C ) 130 µF/m , K+ ) 0.1, K- ) 0.01, D+/D+ ) SL 1, and D-/D- ) 1. In Figure 2, we separately plot the variation of the linear and nonlinear components of the transverse electrophoretic velocity fields (resulting from interaction of wall EDLs) with the dimensionless macromolecular position (yCP/2H) for different values of H/λ corresponding to RP/H ) 0.3. There are some common characteristics for both the linear and nonlinear components of electrophoretic velocity, as evident in Figure 2. In general, the negatively charged macromolecules are repelled from the walls toward the bulk. However, a closer examination of the Figure reveals that the nonlinear electrophoretic effects are significantly more pronounced in the walladjacent layers as compared to those in the outer region because of strong local EDL potential gradients established adjacent to the walls. Figure 2 shows that the nonlinear component of the transverse electrophoretic velocity possesses significant strength only at locations close to the walls, whereas the linear counterpart exhibits its existence well into the bulk. The larger the values of H/λ, the less prominent are these effects because of weakened EDL interactions. For a fixed RP/H ratio, a larger value of H/λ also implies a larger absolute value of RP, which in turn reduces the linear counterpart of the electrophoretic mobility(µep ∝ RP-1). However, the nonlinear counterpart, µep,3, increases in proportion to RP2, which compensates for the reduction in the transverse electric field to a considerable extent. Thus, the extent of lowering of the nonlinear velocity component, caused by an increase in H/λ, is somewhat small as compared to the linear part (Figure 2). Figure 3 depicts the variation of the band velocity Uj〈u〉 as a function of the relative macromolecular size (RP/H) for cases with and without the consideration of nonlinear effects for different values of H/λ. As has been reported elsewhere,38 the macromolecular band velocity is a consequence of the wall forces, electrophoresis in different directions, and hydrodynamic effects
Langmuir, Vol. 24, No. 15, 2008 7709
j 〈u〉) as a function Figure 3. Variation of the band velocity Uband () U of the relative macromolecular size (RP/H) for cases with and without the consideration of nonlinear effects for different H/λ values.
(i.e., the effects by which macromolecules with larger sizes are excluded from near-wall regions) in nanochannel flows. Nonlinear effects are introduced into the transverse migrative fluxes primarily because of the interactions between the nonlinearities in the transverse electrophoretic mobilities with the EDL potential gradients. For the different H/λ values mentioned in Figure 3, the nonlinear effects are most prominent over an intermediate range of relative macromolecular radii (RP/H), exhibited by j 〈u〉 values between the linear and significant differences in U nonlinear cases. The specific extent of this intermediate range depends on the particular value of H/λ considered for the pertinent simulation. For example, the intermediate regime spans an RP/H range of 0.1 to 0.15 (approximately) for H/λ ) 5, whereas the same spans an RP/H range of 0.1 to 0.2 (approximately) for H/λ ) 4, as evidenced from Figure 3. For smaller values of RP/H, the nonlinear effects are rather weak to have any influence, whereas the hydrodynamic effects completely overweigh the nonlinear electrophoretic effects for larger values of the same. Nonlinear transverse electrophoresis imparts an additional repelling push on the macromolecules (which are charged similarly to the walls), forcing them to migrate further away from the walls. This allows them to access the streamlines closer to the bulk more easily, thereby resulting in higher values of j 〈u〉 over an intermediate RP/H range, as compared to those that U may be obtained for the cases of linear electrophoresis. j 〈u〉 within this Interestingly, there also occurs a minimum in U regime, and the corresponding value of RP/H is reminiscent of the situation in which the transverse wall influences (linear and nonlinear) turn out to be the most dominating ones for a given value of H/λ. Furthermore, as mentioned earlier, repulsive EDL interactions are primarily responsible for transverse velocity perturbations directed away from the solid boundaries. This, in turn, imparts a consequent perturbation to the axial component of the concentration-averaged macromolecular velocity and tends to slow it down. The larger the value of H/λ, the weaker the transverse electrophoretic influences become so that the hydrodynamic effects start to dominate at a much lower RP/H values. j 〈u〉 shifts more toward Thus for a larger H/λ, the minimum in U lower values of RP/H. The separation times may be estimated by utilizing the j 〈u〉 and D*Dy (eq 17) and subsequently through variations in U an evaluation of the time at which the peak of the concentration profile is attained at the channel exit. Because the axial diffusion effects (which are virtually independent of any nonlinear electrophoretic influences) outweigh the Taylor dispersion effects
7710 Langmuir, Vol. 24, No. 15, 2008
Das and Chakraborty
tromigrative influences may be dominant, the difference in resolution for the linear and the nonlinear cases tends to be reduced with value increments of H/λ. Therefore, there occurs an intermediate optimal range of H/λ over which a reasonably large peak of resolution may be attained, with most significant contributions from the nonlinear electromigration effects.
5. Conclusions
Figure 4. Variation of the resolution (RS) as a function of the ratio of macromolecular size (RP/RP,max)((RP,max/H)H/λ)4 ) 0.20, (RP,max/H)H/λ)5 ) 0.16, and (RP,max/H)H/λ)7 ) 0.12 for cases with and without a consideration of nonlinear effects for different H/λ values.
for nanochannel flows (as has already been explained), the implications of nonlinear electrophoretic effects on separation characteristics are predominantly manifested through the variaj 〈u〉. We plot the resultant trends in resolution (eq 18) tions in U in Figure 4 as a function of RP/RP,max. (Note that two distinctive macromolecular radii are required to obtain the resolution of separation, out of which we fix one value as RP,max). In ascertaining the normalizing macromolecular radius, RP,max, we first numerically explore the full regime of RP values over which distinctive and improved resolutions of separation may be obtained by exploiting the nonlinear electromigrative effects for a given choice of H/λ. We choose RP,max as the largest possible value of RP falling in this regime. We find that the smaller the value of H/λ, the larger the corresponding value of RP,max for a specified nanochannel dimension. This may be attributed to the fact that lower values of H/λ result in stronger EDL interactions, thereby giving rise to strong induced transverse electric fields penetrating more prominently into the bulk. This, in turn, amplifies the nonlinear electromigrative fluxes to a more considerable extent (Figure 2). However, if one is solely concerned with the peak of the resolution (rather than the range over which the advantageous aspects of nonlinear electrophoresis may be best exploited), then the situation is altered more dramatically because higher values of H/λ may be characterized by higher-resolution maxima. This may be attributed to the fact that the reduction of transverse migration velocities induced by the van der Waals forces is more prominent for larger channels, corresponding to a given RP/H ratio (which corresponds to the decreasing nature of the band velocity characteristics; see Figure 3). However, because of a relative thinning of the EDL and a consequent decrement of the relative extent over which nonlinear elec-
We have carried out detailed theoretical investigations to assess the consequences of nonlinear electrophoretic effects on the sizebased transport and separation of charged macromolecules in nanochannels. Toward this end, we have demonstrated that implicitly induced large potential gradients due to strong EDL interactions may be favorably exploited to generate significant nonlinear electromigrative drift velocities, giving rise to immense technological benefits. Accordingly, the significance of the present proposition lies in the fact that this dramatic improvement in the electromigrative transport characteristics of charged macromolecules is predicted without necessitating the employment of any external influences such as large transverse fractionizing electric fields or very low pH buffer solutions, which might otherwise impose serious practical constraints on implementing the principle. The resultant separation characteristics exhibit substantial improvements in resolution over an intermediate range of relative macromolecular radii, with optimal effects clustered around a peak. Thus, in effect we have demonstrated that an optimal regime of augmented macromolecular transport characteristics may be realized as a parametric function of the relative macromolecular radii and the relative Debye layer thickness. Accordingly, nonlinear electrophoretic effects may be judiciously exploited toward improving the separation performance by choosing appropriate values of H/λ corresponding to the range of RP values encountered in the particular separation system. Because these effects are not manifested as a result of weakened EDL interactions in microchannel flows, nanochannel-based separation paradigms indeed open up a new window for the possible exploitation of nonlinear electrophoretic effects to the best possible consequence. By exploiting these effects, we have demonstrated that a value resolution of even more than 15 may be attainable in practice, which is significantly larger than the characteristic resolution values that have been experimentally realized until now (typically less than 5). Incidentally, typical biological macromolecules possess reasonably high polarizibilities, which also act as an advantage to the present proposition. Thus, findings from this study may act as precursors to focused experimental investigations involving generic transport features of charged macromolecules in nanoscale confinements, which may optimally exploit the benefits associated with the nonlinear transverse electromigrative influences in the most effective manner, on the basis of the specific applications. LA703892Q
