Anal. Chem. 2006, 78, 8134-8141
Charged Species Transport, Separation, and Dispersion in Nanoscale Channels: Autogenous Electric Field-Flow Fractionation Stewart K. Griffiths* and Robert H. Nilson
Sandia National Laboratories, Livermore, California 94551-0969
Numerical methods are employed to examine the transport of charged species in pressure-driven and electroosmotic flow along nanoscale channels having an electric double-layer thickness comparable to the channel size. In such channels, the electric field inherent to the double layer produces transverse species distributions that depend on species charge. Flow along the channel thus yields mean axial species speeds that also depend on the species charge, enabling species separation and identification. Here we characterize field-flow separations of this type via the retention and plate height. For pressuredriven flows, we demonstrate that mean species speeds along the channel are uniquely associated with a single species charge, allowing species separation based on charge alone. In contrast, electroosmotic flows generally yield identical speeds for several values of the charge, and these speeds generally depend on both the species charge and electrophoretic mobility. Coefficients of dispersion for charged species in both planar and cylindrical geometries are presented as part of this analysis. Field-flow fractionation (FFF) is a well-established family of methods for separating and identifying charged or neutral species including particles, macromolecules, and molecular analytes.1 Pioneered by J. C. Giddings in the late 1960s,2,3 most of these methods employ pressure-driven flow along a tube or channel in conjunction with an applied field that is perpendicular to the direction of fluid motion. This applied field produces a transverse species flux that is balanced by diffusion, resulting in a quasisteady variation of species concentrations across the channel. The fluid speed in pressure-driven flow likewise varies in the transverse direction, so species travel along the channel at a mean speed that depends on the transverse species distribution. Species favoring high concentrations in the low-speed region near the tube or channel walls travel slowly along the channel; those favoring positions near the centerline travel more quickly. The overall result is fractionation of a species mixture into discrete bands much like those produced by electrophoretic or chromatographic separations. This is illustrated in Figure 1 for the species indicated by (+) and (-) symbols. * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Giddings, J. C. Science 1993, 260, 1456-1465. (2) Giddings, J. C. J. Chem. Phys. 1968, 49, 81-85. (3) Giddings, J. C. Sep. Sci. 1973, 8, 567-575.
8134 Analytical Chemistry, Vol. 78, No. 23, December 1, 2006
Figure 1. Schematic of a nanoscale tube or channel showing the electric potential, φ, species concentration, ci, and velocity profiles. Species concentrated near the centerline (+) travel faster along the channel than do those near the walls (-), enabling separation.
Members in the family of field-flow fractionation involve mainly variants in the nature of the applied transverse field. These include acceleration fields,4-10 hydrodynamic fields,11-15 thermal fields,16-20 magnetic fields,21-23 and electrical fields imposed using electrodes.24-29 Here we analyze and discuss a new variant of electric field-flow fractionation (EFFF) in which the transverse (4) Giddings, J. C.; Yang, F. J. F.; Myers, M. N. Anal. Chem. 1974, 46, 19171924. (5) Yang, F. J. F.; Myres, M. N.; Giddings, J. C. Anal. Chem. 1974, 46, 19241929. (6) Inagaki, H.; Tanaka, T. Anal. Chem. 1980, 52, 201-203. (7) Yau, W. W.; Kirkland, J. J. J. Chromatogr. 1981, 218, 217-238. (8) Schallinger, L. E.; Yau, W. W.; Kirkland, J. J. Science 1984, 225, 434-437. (9) Davis, J. M. Anal. Chem. 1986, 58, 161-164. (10) Janca, J.; Pribylova, D.; Jahnova, V. J. Liq. Chromatogr. 1987, 10, 767782. (11) Giddings, J. C.; Yang, F. J.; Myers, M. N. Anal. Chem. 1976, 48, 11261132. (12) Giddings, J. C.; Yang, F. J. F.; Myers, M. N. Science 1976, 193, 12441245. (13) Jonsson, J. A.; Carlshaf, A. Anal. Chem. 1989, 61, 11-18. (14) Litzen, A. Anal. Chem. 1993, 65, 461-470. (15) Suslov, S. A.; Roberts, A. J. Anal. Chem. 2000, 72, 43331-4345. (16) Thompson, G. H.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1969, 41, 1219-1222. (17) Hovingh, M. E.; Thompson, G. H.; Giddings, J. C. Anal. Chem. 1970, 42, 195-203. (18) Giddings, J. C.; Hovingh, M. E.; Thompson, G. H. J. Phys. Chem. 1970, 74, 4291-4294. (19) Myers, M. N.; Caldwell, K. D.; Giddings, J. C. Sep. Sci. 1974, 9, 47-70. (20) Giddings, J. C.; Smith, L. K.; Myers, M. N. Anal. Chem. 1975, 47, 23892394. (21) Latham, A. H.; Freitas, R. S.; Schiffer, P.; Williams, M. E. Anal. Chem. 2005, 77, 5055-5062. (22) Schunk, T. C.; Gorse, J.; Burke, M. F. Sep. Sci. Technol. 1984, 19, 653666. (23) Vickrey, T. M.; Garcia-Ramirez, J. A. Sep. Sci. Technol. 1980, 15, 12971304. 10.1021/ac061412e CCC: $33.50
© 2006 American Chemical Society Published on Web 10/31/2006
field is not applied by external means. Instead, this field is provided by the electric field naturally present in the double layer of a nanoscale tube or channel. As the field is produced without external influence, we refer to this technique as autogenous electrical field-flow fractionation (AEFFF). Pennathur and Santiago30-32 recently described a new technique relying in part on one variant of AEFFF. In their approach, referred to as EKSIV, separations are performed using electroosmotic flow in both microscale and nanoscale channels. The two results are then combined to extract simultaneously both the species charge and electrophoretic mobility. This of course requires correlating each peak in one of these separations with its counterpart in the other. As part of this work, Pennathur and Santiago computed convective and electrophoretic contributions to the average species speeds for electroosmotic flow over a small range of the species charge. They also examined diffusive spreading of the species bands but did not consider hydrodynamic dispersion. The results of these theoretical and experimental studies clearly indicate that the mean species speed was influenced by the double-layer electric field when the Debye layer thickness was comparable to the channel height. Garcia et al.33 have also examined charged-species transport and separation in nanoscale channels via electroosmotic flow. Using both a neutral and charged species (zi ) -2) and channel dimensions from 35 to 200 nm, they demonstrated successful separation into two species bands for a single case in which the product of the charge and surface potential is positive. They also demonstrated good agreement between their experimental and theoretical results when surface adsorption of the neutral species was taken into account through a semiempirical correction to their model. Their theoretical treatment was based on a simplifying assumption that overlap of the potential fields from opposing channel walls was not large. A similar assumption was employed by Petsev34 to develop a closed-form, explicit expression for the mean speed of charged species in terms of Euler and Lerch functions. In the present paper, we use numerical methods to more broadly examine the motion and AEFFF separation of charged nonadsorbing species along nanoscale channels for cases in which the species size is negligible compared to the smallest channel dimension. Both pressure-driven and electroosmotic flows are considered. AEFFF separation performance is characterized by retention and plate height, taking into account both diffusive and dispersive spreading of species bands. Resolution and optimum design will be examined in a subsequent paper. (24) Caldwell, K. D.; Giddings, J. C.; Myers, M. N.; Kesner, L. F. Science 1972, 176, 296-298. (25) Caldwell, K. D.; Gao, Y. S. Anal. Chem. 1993, 65, 1764-1772. (26) Palkar, S. A.; Schure, M. R. Anal. Chem. 1997, 69, 3223-3229. (27) Palkar, S. A.; Schure, M. R. Anal. Chem. 1997, 69, 3230-3238. (28) Tri, N.; Caldwell, K.; Beckett, R. Anal. Chem. 2000, 72, 1823-1829. (29) Gale, B. K.; Caldwell, K. D.; Frazier, A. B. Anal. Chem. 2001, 73, 23452352. (30) Pennathur S.; Santiago, J. G. Proc. IMECE 2004, ASME International Mechanical Engineering Congress and Exposition, Anaheim, CA, November 13-29, 2004. (31) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6772-6781. (32) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6782-6789. (33) Garcia, A. L.; Ista, L. K.; Petsev, D. N.; O’Brien, M. J.; Bisong, P.; Mammoli, A. A.; Brueck, S. R. J.; Lopez, G. P. Lab Chip 2005, 5, 1271-1276. (34) Petsev, D. N. J. Chem. Phys. 2005, 123, 244907.
MATHEMATICAL MODEL Consider an electrolytic fluid in a tube or planar channel as shown in Figure 1. A surface charge builds at the boundary of the fluid in contact with the tube or channel walls, and this gives rise to an electric potential φ and associated net electric charge distributed in the fluid near these walls. A uniform electric field Ex is applied along the tube or channel axis using electrodes in reservoirs far removed from the region shown. This applied field acts on the charged fluid, inducing an electroosmotic flow. The transverse profile of this electroosmotic velocity field depends on fluid properties, the transverse tube or channel dimension and the electric ζ potential at the walls. Alternatively, a pressure gradient ∇p may be applied along the channel. This pressure gradient likewise produces flow in the channel, but in this case, the velocity profile is always parabolic so long as the lateral dimension is more than ∼10 nm. For a permittivity that does not vary with position, the Poisson equation governing the electric potential across this tube or channel is
∇2φ ) -Fe
(1)
where the net local charge density Fe is related to the electric potential through the Boltzmann distribution. For a symmetric electrolyte this can be written as Fe ) -2zFc0 sinh(zFφ/RT), where F is the Faraday constant, z is the charge number, c0 ) c( is the bulk-fluid ion concentration, R is the universal gas constant, and T is the temperature. Assuming the fluid is incompressible, the flow is steady and the viscosity is uniform, fluid motion is governed by
∇‚u ) 0
(2)
µ∇2u ) ∇p - FeE
(3)
and
Here u is the local fluid velocity, µ is the fluid viscosity, p is the local pressure, and E is the applied electric field. Now consider a dilute analyte species suspended in this flow. The overall flux of this species is given by
j ) uci - Di∇ci + νiziciF(E - ∇φ)
(4)
where Di is the species diffusivity, νi is its mobility, zi is the charge number, and F is Faraday’s constant. Because both the fluid velocity u and the applied field E possess only axial components u and Ex for a long straight tube or channel, eq 4 in the absence of an axial concentration gradient can be rewritten as
jx ) uci + νiziciFEx ) uici
(5)
where ui ) u + νiziFEx is the local species speed, and
jy ) -Di
∂ci ∂φ - νiziciF ∂y ∂y
Analytical Chemistry, Vol. 78, No. 23, December 1, 2006
(6) 8135
Here x and y denote the axial and transverse directions, respectively. Note that eq 6 can be integrated analytically to within a constant for the equilibrium condition jy ) 0. Using the NernstEinstein relation, νi ) Di/RT, the result is
()
ziFφ ci ln )c0i RT
(7)
where c0i is the bulk concentration at zero potential outside the tube or channel. This is the familiar Boltzmann distribution.35 The mean species concentration, spatially averaged over the tube or channel cross section, is
jci )
n+1 an+1
∫
a
ciyn dy
0
(8)
where a is the tube radius or channel half-height and the parameter n ) 0 denotes a planar channel; n ) 1 denotes a tube of circular cross section. Using this definition of the mean concentration, the mean axial speed (zone velocity) can be expressed as
u ji )
n+1 jcian+1
∫
a
0
jxyn dy )
n+1 jcian+1
∫
a
0
(u + νiziFEx)ciyn dy (9)
ψ)-
n+1 an+1
∫
a
0
u yn dy
γ)
(13)
2 ψ 1 a ∇xp )3 + 5n 3 + 5n µ
(14)
(10) for the special case of purely pressure-driven flow, ψ f (∞. For purely electroosmotic flow, ψ f 0, the result is
2
(aλ) ) 2z RT Fac
(11)
2 2 2
0
characterizing the tube or channel dimension. The fifth parameter characterizes operating conditions. It is a normalized pressure gradient, (35) Probstein, R. F. Physicochemical Hydrodynamics; John Wiley & Sons: New York, 1995.
8136
RT RT ) Λµ z F νµ 2 2
where Λ is average molar conductivity of the two equivalent and equimobile ions. This parameter, referred to here as the Levine number,36-38 exhibits a highly restricted range. The value of Λ for aqueous solutions varies only between about 4 and 20 mS‚ m2/mol,35 so realistic values of the Levine number will likely span only the very small range 0.1 < γ < 0.5 at 20 °C. A typical value is γ ) 0.3. Introducing these normalized variables into the primitive governing equations yields the dimensionless governing equations provided in Supporting Information. These are solved using a shooting technique.39 Additional details and a discussion of the accuracy are also provided in Supporting Information. Finally, the dimensional mean fluid speed is important in the design and evaluation of an AEFFF system. From eq 10 (or eq S3), this speed can be written as
u j)-U
Note that the subscripted speeds ui and uj i refer to species “i”, while those without the subscript refer to the fluid. We now introduce a set of dimensionless variables. The new normalized dependent variables are taken as u* ) u/U and φ* ) φ/ζ, where U ) -ζEx/µ is the Helmholtz-Smoluchowski speed for flow past a charged planar surface and ζ is the electric potential at the tube or channel wall. The new independent variables are x* ) x/a and y* ) y/a, where again x and y are the axial and the transverse coordinates. This normalization leads to six dimensionless parameters. The first three are the normalized zeta potential, ζ* ) zFζ/RT, characterizing the fluid-solid interaction, a dimensionless species mobility, νi* ) νi/ν, and the normalized species charge, zi* ) zi/z. The fourth parameter is a normalized Debye length,
λ*2 )
(12)
This parameter is defined such that flow in the positive x direction occurs for ψ < 0. The last of these parameters characterizes the electrolytic fluid and is given by
The corresponding average axial fluid speed is
u j)
a2∇xp ζEx
Analytical Chemistry, Vol. 78, No. 23, December 1, 2006
u j ) Uβ ) -
ζEx β µ
(15)
where
β ) (n + 1)
∫
1
0
(1 - φ*)y*n dy*
(16)
is the normalized contribution to the mean fluid speed from electrokinetic flow. Note that the limit ψ f (∞ for pressure-driven flow does not account for the streaming potential produced by charge transport along the channel. Under open-circuit conditions, this potential induces an electric field that may influence fluid motion through the electroviscous effect and may additionally contribute an electrophoretic component to the transport of charged species. If no current passes along the channel, then ψ cannot be infinite but is instead given by (36) Griffiths, S. K.; Nilson, R. H. Electrophoresis 2005, 26, 351-361. (37) Levine, S.; Marriott, J. R.; Robinson, K. J. Chem. Soc., Faraday Trans. 2 1975, 71, 1-11. (38) Levine, S.; Marriott, J. R.; Neale, G.; Epstein, N. J. Colloid Interface Sci. 1975, 52, 136-149. (39) Griffiths, S. K.; Nilson, R. H. Anal. Chem. 2005, 77, 6772-6781.
ψ)
(n + 1) 2
γβλ* ζ*
2
∫
1
0
[cosh(ζ*φ*) + γζ*(1 - φ*) sinh(ζ*φ*)]y*n dy* (17)
Values of ψ computed from this expression are indeed finite, but they are very large for most conditions of practical importance. These include all conditions for which λ* is small, ζ* is small, or the product γβλ*2ζ*2 is small. Alternatively, the condition ψ f (∞ can be realized exactly for any pressure-driven flow simply by grounding the reservoirs at the channel ends. In this case, there is a current along the channel, but no electric field. Separation Performance. Separations using FFF are typically characterized by retention, plate height, and peak resolution. Retention in this case is simply the ratio of the species mean speed to the average speed of the fluid.
Ri ) u j i/u j)u j i*/u j*
(18)
Note that the mean fluid speed is the same as the mean speed of any neutral species, so the retention is defined in all cases such that Ri ) 1 for zi ) 0. The plate height, Hi, is the spatial variance of the species peak distribution, σi2, divided by the distance of travel or channel length, L. This is often expressed in the dimensionless form of a reduced plate height, hi ) Hi/a, where again a is the tube radius or channel half-height. The reduced plate height can therefore be written as
Hi σi2 2D'iti 2D'i hi ) ) ) ) a aL aL au ji
(19)
where D′i is the effective species diffusivity due to a combination of dispersion and axial diffusion. This effective diffusivity is given by40
D′i ) Di(1 + RiPei2)
(20)
where Di is the species diffusivity and Pei ) uj ia/Di is the species Peclet number based on the mean species speed. The coefficient of dispersion, Ri, appearing in eq 20 can be expressed as2,17,41,42
∫
1
ci*ωi2y*n dy*
(21)
ci*(ui*/u j i* - 1)y*n dy*
(22)
Ri ) (n + 1)
0
in which
ωi )
1 ci*y*n
∫
y*
0
Combining eqs 19 and 20, the reduced plate height can be (40) Aris, R. Proc. R. Soc. A 1956, 235, 67-77. (41) Brenner, H.; Edwards, D. A. Macrotransport Processes; Butterworth-Heinemann: Boston, MA, 1993. (42) Martin, M.; Giddings, C. J. J. Chem. Phys. 1981, 85, 727-733.
Figure 2. Distribution of species concentrations. Concentrations are high in low-speed regions near the wall when zi*ζ* < 0, producing a low mean species speed. High concentrations near the centerline yield a high mean speed for zi*ζ* > 0.
rewritten in terms of just the coefficient of dispersion and the species Peclet number,
hi )
2 (1 + RiPei2) Pei
(23)
Note that this expression differs from the common form based on the fluid Peclet number. That alternate form can be recovered, however, using the identity Pei ) uj aRi/Di ) PeRi/Di* ≈ PeRi/ νi*, where Di* ) Di/D ≈ νi/ν ) νi*, and Pe ) uj a/D is the Peclet number based on the average fluid speed uj and a diffusivity D characteristic of the electrolyte ions. Also note that eq 23 exhibits a minimum given by
hi,min ) 4xRi at Pei,opt )
1
xRi
(24)
This is the smallest possible reduced plate height. It is obtained only at the optimum operating condition given by the optimum species Peclet number. Species Distribution and Retention. As discussed shortly, the case of small ζ* is of special interest for obtaining sensitive separations over a broad range of species charge. In this limit, the normalized electric potential, φ* is independent of ζ*. Because of this, the normalized fluid speed, u*, is also independent of ζ*, and the species distribution, ci*, depends on ζ* only through the product zi*ζ*. As a result, the retention, coefficient of dispersion, and reduced plate height also depend only on the product zi*ζ* for fixed λ*, ψ, γ, νi*, and Pei. Computed species distributions are shown in Figure 2 for the cylindrical geometry, n ) 1, and the conditions λ* ) 0.2 and ζ* f 0. We see as expected that the distribution of a neutral species, zi*ζ* ) 0, is uniform across the tube. For zi*ζ* < 0, the charged species are attracted to the wall, y* ) 1, producing a high concentration adjacent to the surface and a low concentration near the centerline, y* ) 0. This trend is simply reversed for Analytical Chemistry, Vol. 78, No. 23, December 1, 2006
8137
Figure 3. Species retention for pressure-driven (black) and electroosmotic flow (gray) as a function of normalized Debye layer thickness. Retentions exceed unity for zi*ζ* > 0 and fall below unity for zi*ζ* < 0. They always approach unity for small and large λ*.
Figure 4. Species retention for the geometry of a wide planar channel using pressure-driven (black) and electroosmotic flow (gray). Maximum possible retention zi*ζ* > 0 is Ri ) 3/2 corresponding to the ratio of the maximum to mean fluid speeds.
zi*ζ* > 0. Figure 2 also shows the electrical potential for these conditions, as well the normalized electroosmotic fluid speed, uEK*, and the parabolic profile of the fluid speed for pressure-driven / flow in a tube, -4u∆p /ψ. These speeds exhibit a maximum at the centerline and fall to zero at the tube wall. Species for which zi*ζ* < 0 are thus concentrated in a region of low fluid speed and so will exhibit mean speeds that are less than the mean fluid speed. Conversely, species for which zi*ζ* > 0 appear predominantly in the region of high fluid speed and so travel with a mean speed that is less than that of the fluid. Since neither the distribution of the fluid speed nor the species distribution depends on the species mobility, we can see from this that AEFFF is capable of separation based on species charge alone. This is always the case for pressure-driven flows. It is also the case for electroosmotic flows under some special conditions. The ratio of the mean species speed to the mean fluid speed is the retention given by eq 18. Retention as a function of the normalized Debye layer thickness is illustrated in Figure 3, again for the case of small ζ*. Results are presented for both pressuredriven flow (ψ f -∞) and electroosmotic flow (ψ f 0). In the case of electroosmotic flow, these solutions are further restricted to conditions for which the electrophoretic speed is very small. Since a single electric field drives both the electroosmotic flow and any electrophoretic motion, these conditions are satisfied only when the term νi*zi*/γζ* is very small, as apparent from eq S7 (Supporting Information). This entire term is identical to the parameter “β” used by Pennathur and Santiago.31 The solutions in Figure 3 for electroosmotic flow thus carry the additional constraint that the normalized species mobility is very small, νi* f 0. As we will see, however, this assumption is not usually justified. High sensitivity, dRi/dzi*, is important to obtaining high resolution in AEFFF separation. At both small and large λ*, however, the retentions of Figure 3 for all species approach unity, and the sensitivity vanishes. Further, high sensitivities for positive zi*ζ* occur only when the electric field from opposing tube walls overlap just slightly, and such overlap arises only for 0.01 < λ* < 2. In the limit zi*ζ* f 0, maximum sensitivity occurs at λ* ) 0.20
for pressure-driven flow in a tube; for electroosmotic flow, this optimum occurs for λ* ) 0.25. The sensitivities for these cases are dRi/dzi* ) 0.15ζ* and 0.12ζ*, respectively. At zi*ζ* ) 16, the optimums yielding maximum sensitivity are λ* ) 0.93 and 0.96 for pressure-driven and electroosmotic flow; these yield dRi/dzi* ) 0.021ζ* in both cases. In contrast, high sensitivities for negative zi*ζ* do not depend on this overlap of the electric field, so good separation performance can be obtained even for very small λ*. For example, the optimums yielding maximum sensitivity for pressure-driven and electroosmotic flow are λ* ) 0.0011 and 0.0012 at zi*ζ* ) -8, and these both yield very large sensitivities of 0.21ζ* and 0.19ζ*, respectively. We therefore expect that negative zi*ζ* will yield higher resolution than do positive values, especially given that retentions for negative zi*ζ* are also small. Retentions for the geometry of a wide planar channel are shown in Figure 4. These are clearly similar to the previous results, except in the magnitude of the retention especially for zi*ζ* > 0. The results of Figure 4 for electroosmotic flow agree well with the results in Figure 1 of the theoretical paper by Pennathur and Santiago31 for -2 < zi < 2 and ζ* ) -2. Note that the inset in their Figure 1 labeled ζ* 0; it is increased for zi*ζ* < 0. Taylor result Ri ) 1/48 is obtained for both flows and all charges in the limit of large λ*.
just -2 < zi* < 1 even for νi* ) 0.1. AEFFF using electroosmotic flow thus seems fundamentally problematic, and this cannot be remedied by altering the sign of ζ*, the applied field, the channel size, or any other simple means. Separations using pressure-driven flow of course do not suffer from these problems because there is no applied axial electric field and so no electrophoretic contribution to species motion. Dispersion and Plate Height. Both the plate height and resolution for AEFFF depend on the coefficient of dispersion as given by eqs 21 and 22. Computed values of this coefficient for the cylindrical geometry are shown in Figure 7 as a function of the normalized Debye layer thickness for various values of zi*ζ*. Results for both pressure-driven and electroosmotic flow are shown. For pressure-driven flow in a planar channel, the coefficient of dispersion of a neutral species is independent of λ* and equal to the well-known Taylor value of 1/48 ≈ 0.0208 for large and small λ*.43 For charged species, the coefficient of dispersion is smaller than this value when zi*ζ* is positive and larger than this value when zi*ζ* is negative. This is because species concentrations are high near the centerline for zi*ζ* > 0, and velocity gradients in this region are small. Concentrations are high near the surface for zi*ζ* < 0, and here velocity gradients are large. As a result, dispersion coefficients for zi*ζ* < 0 are larger than the neutral-species value. However, the coefficients of dispersion do approach the neutral-species value for all zi*ζ* when λ* is either very small or very large. Again this is because the electric field is small over most of the domain in both of these limits, so the species distribution becomes relatively uniform across the tube or channel. As with pressure-driven flows, the coefficients of dispersion for charged species in electroosmotic flow exceed the neutralspecies value for zi*ζ* < 0 and fall below this value for zi*ζ* > 0. For electroosmosis, however, the coefficient of dispersion approaches the Taylor value only for large λ*. In this limit, the electric potential is uniform across the tube or channel, the transverse electric field is negligible, species concentrations are nearly uniform, and the velocity profile is parabolic. And, these 8140 Analytical Chemistry, Vol. 78, No. 23, December 1, 2006
Figure 8. Minimum reduced plate heights at optimum Pei for pressure-driven and electroosmotic flows. Smaller plate heights offer the potential for improved resolution, but only for cases in which the retention varies strongly with charge.
conditions yield Taylor’s result. For small λ*, in contrast, the velocity profile in electroosmotic flow becomes very flat, giving rise to a coefficient of dispersion that grows with the square of λ*.39 We see in Figure 7 that this functional dependence for small λ* is independent of the species charge; only the multiplicative constant varies with charge. Coefficients of dispersion for the planar channel geometry are qualitatively very similar to those shown in Figure 7. In this case, however, the asymptototic values at large or small λ* are the Wooding value 2/105 ≈ 0.0190.44 These results are shown in Figure S1 of the Supporting Information. At the optimum Peclet number, reduced plate heights depend only on the coefficient of dispersion as described by eq 24. This is the minimum possible plate height for any given λ*, ζ*, zi*, and Pei. However, Pei cannot be optimum for more than one species charge in a separation process, so this minimum plate height provides only guidance as to the maximum performance over a small range of charges in the vicinity of the specified charge. Such minimum plate heights are shown in Figure 8 for pressure-driven and electroosmotic flow in a tube. As expected, these resemble the coefficients of dispersion. The reduced plate heights for pressure-driven flow range only from about 0.1 to 10 for all λ* and all -8 e zi*ζ* e 16. This indicates that submicrometer plate heights should be attainable by AEFFF over a broad range of conditions for tube or channel dimensions less than 100 nm. Such small dimensions will be required in any case to obtain acceptable values of λ*. Figure 8 also indicates that reduced plate heights for electroosmotic flow may be considerably smaller than those for pressure-driven flow at equivalent conditions, but these results are again based on the approximation νi* f 0. As already discussed, conditions for which this approximation remains valid will be very difficult to obtain. SUMMARY The transport of charged analyte species by flow along nanoscale channels exhibits mean species speeds that depend on (43) Taylor, G. I. Proc. R. Soc. A 1953, 186-203. (44) Wooding, R. A. J. Fluid Mech. 1960, 7, 501-515.
the species charge. This results from the nonuniform distribution of species across the channel due to the intrinsic electric field of the Debye layer. Such dependence of mean speed on species charge enables the separation of species of mixed charge into species bands by the mechanism referred to here as AEFFF. For pressure-driven flows, the mean species speed always exceeds that of the fluid when the product of the normalized ζ potential and species charge is positive, zi*ζ* > 0, and is less than that of the fluid when this product is negative. Further, the ratio of the mean species speed to that of the fluid depends only on the charge when all other parameters are fixed, and this retention varies monotonically with species charge. Pressure-driven flow thus enables separation based on charge alone. Electroosmotic flows, in contrast, exhibit retention that generally decreases with increasing species charge when the charge is large and negative, zi*ζ* > 0. Moreover, retention for electroosmotic flow generally depends on both the charge and electrophoretic mobility. Interpreting peak arrival times using electroosmotic flow is thus highly problematic unless the electrophoretic mobilities of all analyte species are extremely small. Plate heights for nonadsorbing species depend only on the coefficient of dispersion and the species Peclet number. For both pressure-driven and electroosmotic flow, dispersion coefficients for charged species in nanoscale channels are reduced from that of a neutral species when the product zi*ζ* is positive and are increased relative to a neutral species when this product is negative. As a result, plate heights for negative zi*ζ* are always larger than those for positive values of this product. AEFFF using pressure-driven flow offers several advantages (as well as new challenges) relative to conventional EFFF and
other established separation techniques. One key benefit is that embedded electrodes are not required to produce the transverse electric field, so bubble formation due to hydrolysis is of no concern. AEFFF using pressure-driven flow also permits concurrent separation of species of mixed charge sign in a single channel. The challenges of course are that nanoscale channels are still difficult to fabricate, the required driving pressures for optimum performance may be very high due to the small channel dimensions, and surface adsorption of analyte species may be a problem owing to the extremely high surface to volume ratio of nanoscale channels. Even when adsorption is not an issue, steric exclusion may affect mean species speeds for large macromolecules since these may be comparable in size to the smallest nanoscale channels of interest. Finally, in most cases of practical importance, plate heights are much less that 1 µm, so injection and detection volumes (lengths) must be extremely small to realize the full capability of the technique. ACKNOWLEDGMENT This work was funded by a Sandia Engineering Sciences LDRD. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.
Received for review July 31, 2006. Accepted September 14, 2006. AC061412E
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