Anal. Chem. 2008, 80, 9542–9550
Electropreconcentration with Charge-Selective Nanochannels Adrien Plecis,* Cle ´ ment Nanteuil, Anne-Marie Haghiri-Gosnet, and Yong Chen Laboratoire de Photonique et Nanostructures, CNRS, 91460 Marcoussis, France We report on the systematic investigation of electropreconcentration phenomena in hybrid micro/nanofluidic devices. The competition between the electroosmotic dragging force and the highly nonlinear electrophoretic forces induced by the polarization effect is responsible for four preconcentration regimes within such structures that can arise at both cathodic and anodic sides of the nanochannel. Numerical calculations on the spatiotemporal concentration of charged molecules confirm such a classification, showing a general agreement with the reported experimental data at low and moderate ionic strengths. The results also suggest that both the mobility and the valence of the species of interest are important parameters in the determination of the preconcentration rates. Recently there has been increasing interest in using nanochannels for charge-selective ionic transportation.1-3 Indeed, when the size of a microfluidic channel is down to a few tens of nanometers, the surface charge induced an exclusion enrichment effect; i.e., co-ions of the surface charges are excluded from the nanochannel while counterions are enriched. When electric fields are applied, the enhanced counterionic transport can be used for electropreconcentration of charged biomolecules. The first attempts to use nanochannels in microfluidic networks were based on the integration of charge-selective nanoporous membranes.4,5 In particular, Foote et al. have used porous silica membranes to preconcentrate protein samples prior to electrophoretic separations.6 The preconcentration rate in such devices, however, could not be easily predicted because of the problems of membrane clogging and aging. More robust nanofluidic channels could be obtained using advanced nanofabrication techniques, allowing more systematic investigations.7-16 While some groups have demonstrated the stacking of fluorescein molecules at the cathodic nanochannel entrances,5,15,16 some others have shown preconcentration effects where proteins17 and * To whom correspondence should be addressed. E-mail: adrien.plecis@ lpn.cnrs.fr. Phone: +33 1 69 63 62 30. (1) Peng, Y. Y.; Pallandre, A.; Tran, N. T.; Taverna, M. Electrophoresis 2008, 29, 157–178. (2) Han, J. Y.; Fu, J. P.; Schoch, R. B. Lab Chip 2008, 8, 23–33. (3) Yuan, Z.; Garcia, A. L.; Lopez, G. P.; Petsev, D. N. Electrophoresis 2007, 28, 595–610. (4) Kuo, T. C.; Cannon, D. M.; Chen, Y. N.; Tulock, J. J.; Shannon, M. A.; Sweedler, J. V.; Bohn, P. W. Anal. Chem. 2003, 75, 1861–1867. (5) Kuo, T. C.; Sloan, L. A.; Sweedler, J. V.; Bohn, P. W. Langmuir 2001, 17, 6298–6303. (6) Foote, R. S.; Khandurina, J.; Jacobson, S. C.; Ramsey, J. M. Anal. Chem. 2005, 77, 57–63.
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ss-DNA18 were accumulated at a given distance from the nanochannel entrance in the cathodic microchannel. On the other hand, the best preconcentration rates were actually reached at the anodic side of the nanochannel. Han et al. reported, for example, millionfold preconcentrations of GFP in hybrid micronanofluidic devices.13 Hasselbrink et al. also showed impressive preconcentration by using field-induced nanochannels.19 Both of these results were obtained at moderate ionic strengths where the exclusion effect is low and reported a detachment of the preconcentration front from the nanochannel. Despite the rapid advance in nanochannel-based preconcentration devices, it is still difficult to quantify the whole preconcentration process because of a large amount of experimental parameters that have to be controlled. Nevertheless, important differences in preconcentration rate and dynamics could be observed for different species within the same experimental conditions, which suggests that the preconcentration process is dependent on the molecule properties. The purpose of this work was to propose a theoretical qualitative and quantitative picture of all these cathodic and anodic preconcentration processes and evaluate the efficiency of the electropreconcentration as a function of the electrokinetic properties of the molecules of interest. Few theoretical works have been dedicated to the nanochannelbased preconcentration problems. Most of them20-22 were carried out on nanochannels of very simple geometry without taking into account the influence of the embedding microchannel parts. The concentration polarization induced by the charge selectivity of the nanochannels is then systematically omitted. For this reason, the most relevant works are usually found in the field of (7) Datta, A.; Gangopadhyay, S.; Temkin, H.; Pu, Q. S.; Liu, S. R. Talanta 2006, 68, 659–665. (8) Karnik, R.; Castelino, K.; Fan, R.; Yang, P.; Majumdar, A. Nano Lett. 2005, 5, 1638–1642. (9) Kutchoukov, V. G.; Pakula, L.; Parikesit, G. O. F.; Garini, Y.; Nanver, L. K.; Bossche, A. Sens. Actuators, A 2005, 123-24, 602–607. (10) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6782–6789. (11) Plecis, A.; Schoch, R. B.; Renaud, P. Nano Lett. 2005, 5, 1147–1155. (12) van Delft, K. M.; Eijkel, J. C. T.; Mijatovic, D.; Druzhinina, T. S.; Rathgen, H.; Tas, N. R.; van den Berg, A.; Mugele, F. Nano Lett. 2007, 7, 345–350. (13) Wang, Y. C.; Stevens, A. L.; Han, J. Y. Anal. Chem. 2005, 77, 4293–4299. (14) Yang, B.; Pang, S. W. J. Vac. Sci, Technol. B. 2006, 24, 2984–2987. (15) van der Heyden, F. H. J.; Stein, D.; Dekker, C. Phys. Rev. Lett. 2005, 95. (16) Schoch, R. B.; Renaud, P. Appl. Phys. Lett. 2005, 86. (17) Schoch, R. B.; Caprioli, L.; Bertsch, A.; Renaud, P. Nanotechnology 2005, (18) Dai, J. H.; Ito, T.; Sun, L.; Crooks, R. M. J. Am. Chem. Soc. 2003, 125, 13026–13027. (19) Kim, S. M.; Burns, M. A.; Hasselbrink, E. F. Anal. Chem. 2006, 78, 4779– 4785. (20) Conlisk, A. T. Electrophoresis 2005, 26, 1896–1912. (21) De Leebeeck, A.; Sinton, D. Electrophoresis 2006, 27, 4999–5008. (22) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6772–6781. 10.1021/ac8017907 CCC: $40.75 2008 American Chemical Society Published on Web 11/12/2008
permselective membranes23 and ionic channels,24 where Poisson-Nernst-Planck (PNP) equations are solved to predict ionic fluxes. The electrical properties of asymmetric nanofluidic channels were investigated recently based on the work on ionic channels,25 which requires coupling the PNP equations with Navier Stockes equations in order to take the convective flux correctly into account. Two-dimensional (2D) modeling of a micronanofluidic interconnect device was then proposed,26 but the preconcentration effect of the ionic distribution on a “third-party” molecule was not taken into account. Moreover, the computational time required for 2D simulations prevent any investigation of the long-time dynamics, which is necessary in the case of electropreconcentration processes. Dhopeshwarkar et al. have recently proposed a 1D modeling of the transient preconcentration of anionic molecules using charge-selective hydrogels.27 The cathodic preconcentration of ss-DNA was qualitatively explained by the competition between the electromigration of molecules and the opposite induced electroosmotic flow (EOF). The treatment of the Navier-Stockes equations is avoided in such hydrogels as electroosmosis cannot be simply modeled and a constant arbitrary flow is chosen through the structure. However, the flow velocity is one of the most important components in electropreconcentrations, and its numerical treatment remains necessary to explain the diversity of the preconcentration regimes observed experimentally. In this paper, the 1D modeling of electropreconcentration processes through a typical micronanofluidic channel is proposed. First, a phenomenological description is given and the variety of preconcentration regimes is classified into four categories. Then, the theoretical background is exposed together with the approximations necessitated by our 1D approach. The finite element calculation procedure is also provided in order to facilitate its implementation for further studies. Finally, the numerical system was applied to a number of cases that are representative of most the previously reported experiments. In particular, cathodic preconcentration of fluorescein dyes diluted in low ionic strength buffers will be studied as a function of the surface charge. Then, the anodic preconcentration of proteins in salted buffers will be explained, and its dependence on proteins mobility and charge will be quantified in order to answer the question of the applicability of electropreconcentration to the detection of lowconcentration samples. PRINCIPLE Electropreconcentration of proteins and anionic dyes was shown to highly depend on electrolyte composition. In the present paper, KCl was chosen as the background buffer because of the symmetric mobility of the cation K+ and the anion Cl-. Proton and hydroxide ions concentrations will be neglected, considering a constant pH 7 of the solution, and only the effect of ionic concentration gradient will be considered. This approximation is obviously very restrictive, and future work will have to consider (23) Sokalski, T.; Lingenfelter, P.; Lewenstam, A. J. Phys. Chem. B 2003, 107, 2443–2452. (24) Corry, B.; Kuyucak, S.; Chung, S. H. Biophys. J. 2000, 78, 2364–2381. (25) Constantin, D.; Siwy, Z. S. Phys. Rev. E 2007, 76. (26) Jin, X. Z.; Joseph, S.; Gatimu, E. N.; Bohn, P. W.; Aluru, N. R. Langmuir 2007, 23, 13209–13222. (27) Dhopeshwarkar, R.; Crooks, R. M.; Hlushkou, D.; Tallarek, U. Anal. Chem. 2008, 80, 1039–1048.
Figure 1. Schematic of the simulated device. (a) typical lengths in the system. (b) volumic surface charge model and boundary conditions.
the implication of a pH gradient on the preconcentration rate of charged molecules. However, the general use of buffered solutions in experimental studies also suggests that the preconcentration effect is mainly due to ionic concentration effects as will be discussed in the following section. Here, a qualitative description of the mechanism of preconcentration for charged species P (valence zp) will be presented in the simplest geometry composed of a microchannel interrupted by a central nanochannel: the micro-nano-micro (MNM) channel (Figure 1). It will be shown that four types of preconcentration can actually be achieved in such micro/nanofluidic networks. In standard capillary electrophoresis, both electric field and liquid velocity are homogeneous along the capillary, which results in a constant velocity of the molecules. In our case, as the section of the MNM is not constant, the velocity of the liquid as well as the electric field will vary, and it is more appropriate to consider a transport coefficient defined as the product of the particle velocity with the section area: Jp ) Svp ) (µpElS + Db)
(1)
where Db is the liquid flow rate across the channel, µp is the particle electrophoretic mobility, El is the local electric field, and S is the local section in the MNM structure. Preconcentration will arise from variations of Jp along the networks. For a noncompressive liquid, Db is constant, which means that the variation in Jp is mainly due to the variation of the electric field, while the liquid flow rate is responsible for a translation of the transport coefficient. Figure 2 summarizes the preconcentration scenarios that can occur in negatively charged MNM structures. The variation of Jp due to the electric fields is explained in detail in Supporting Information. Typically, the variation of the electrophoretic flux along the MNM structure can be explained by two phenomena: (i) at nanochannel entrances, a sharp increase of the potential is due to the exclusion enrichment effect;11 (ii) in the microchannels, a concentration polarization occurs that increases (decreases) the ionic concentration at the cathodic (anodic) side of the nanochannel. This results in a smooth decrease (increase) of the electrophoretic flux in the cathodic (anodic) microchannels when the molecules try to cross the nanochannel. Depending on the liquid flow rate, the transport coefficient of anionic molecule P cuts the zero flux axe at different positions along the micro/nanochannel. There will be systematically two Analytical Chemistry, Vol. 80, No. 24, December 15, 2008
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Figure 2. Preconcentration regimes as a function of electrophoretic vs electroosmotic competition. Anodic stacking (AS) and counter gradient focusing (ACGF) regimes on the left are EOF-dominated regimes while cathodic stacking (CS) and counter gradient focusing (CCGF) on the right are electrophoretic dominated preconcentration regimes.
equilibrium points where the net velocity of the molecule will be zero: an unstable front (positive derivative of the local flux) situated within the nanochannel and a stable preconcentration front within the microchannel (negative derivative of the flux) where preconcentration will occurs. When liquid flow rate is increased, the first preconcentration regime comes out because of the electrostatic potential barrier at the nanochannel cathodic side. This is the CS regime. For higher Dv, the equilibrium point migrates toward the cathode and is then a consequence of the induced electric field gradient inherited from the polarization phenomenon. This regime can be assimilated to a CCGF phenomenon. Finally, there is a value of Dv for which convective transport becomes more important than electrophoretic migration. This happens when the electroosmotic mobility of the surface is higher than the electrophoretic mobility of the analyte. In that case, the anionic molecules will globally migrate from the anode to the cathode and preconcentration will occurs at the anodic side of the nanochannel. There are also two regimes that can be observed for anodic preconcentrations. The one based on the ionic gradient, that we name the ACGF regime and the one resulting from the electrostatic barrier at nanochannel entrance that we name AS regime. Therefore, preconcentration of anionic particles can happen on both cathodic and anodic sides of the nanochannel and can result from two different mechanisms (electrostatic exclusion or ionic gradient focusing). For cationic particles, it is important to notice that the stable equilibrium point would be found in the nanochannel. However, as electroosmotic flows and electrophoretic velocities have the same direction in negatively charged MNM channels, it is not possible to find a zero velocity. In order to preconcentrate cationic species, positively charged MNM structures will be more appropriate, as it will inverse the ionic gradient polarization and results in the four preconcentration regimes described before. As this case was experimentally not reported by previous work, we will now focus on the quantitative modeling of anionic particles preconcentration in negatively charged MNM structures and will try to describe more precisely the features of the four abovementioned regime.
sure) are uniform along the channel height. The choice of this 1D simplification was made in order to get a sufficiently realistic picture of the preconcentration mechanism while keeping the calculation time minimum. As the number of parameters is very important, it is crucial to be able to vary as many factors as possible to predict the general trends of the preconcentration regime. Ion Flux Equations. Ionic transport is governed by the Nernst-Planck flux equations where diffusion is no more neglected as in eq 1:
NUMERICAL MODELING We propose a one-dimensional modeling of the electropreconcentration process inspired by the field of permselective membrane science23 in which systems are explicitly 1D. The attempt to reduce micronanofluidic networks to 1D model systems necessitates a few approximations. First, the channel width of our MNM systems is generally much larger that the channel height. The second approximation consists of considering that all the variables (i.e., ionic concentrations, electric potential, and pres-
∂2φ(x) F(x) )εoεr ∂x2
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[
Ji(x) ) h(x)Di ∇ci(x)
]
ziF ∇ φ(x)ci(x) + Dbci(x) RT
(2)
Ji is the ionic flux for the ith ion at x across the section of height h. The three terms in the right side of the equation represent respectively the following: (i) the diffusion flux, which is proportional to the gradient of concentration ∇ci and the diffusion coefficient Di; (ii) the electrophoretic flux, which is proportional to the local concentration and electric field -∇Φ; DiziF/RT corresponds to the electrophoretic mobility of the ionic specie where zi is the valence of the ion; F is the Faraday constant; R is the molar gas constant; and T is the temperature (in kelvin); and (iii) the convective transport related flux, which is proportional to the flow rate Db. In our model system, i stands for for K+, Cl-, and Pzp ions. In order to follow the dynamics of ionic concentrations, this transport equation must be related to the material balance equation: ∂h(x)ci(x) ) - ∇ Ji(x) ∂t
(3)
In addition, the potential Φ and the liquid flow rate Db can be described as follows. Potential Determination and Volumic Surface Charge. The potential variations are governed by the Poisson equation:
(4)
where F is the net charge density and ε0εr is the permittivity of the medium (water). Within 1D simulations, the question of the potential is related to the question of modeling the surface charge. Indeed, we know that surface charge is implicated both in the electrostatic barrier at nanochannel entrance and in the polarization process responsible for the ionic gradient creation.
which is not the case in our system. In order to correctly consider the electroosmotic contribution, we have to consider an apparent zeta potential ζap for veof that also depends on the local Debye length λ(x) and the height of the channel h(x):
veof(x) ) -
Figure 3. Apparent ζ potential as a function of the ionic strength for the micro and the nanochannels.
The surface charge was actually modeled via a volumic surface charge cs (see Figure 1) consisting in immobile ions homogeneously distributed along the micro/nanochannel section. Then, the volumic net charge density depends on the surface charge through F(x) ) cK+(x) - cCl-(x) + zpcp-(x) cs(x)cs(x) )
σx(x) h(x)
(5)
Here, the exclusion enrichment effect in the nanochannel comes from the fact that this volumic surface charge is no more negligible for smaller sections. Surface ions are actually treated as immobile ions (a zero diffusion coefficient). Liquid Flow Rate and EOF Component. The liquid flow rate is governed by the incompressibility of the liquid, which tends to maintain a constant flow rate:
(
Db ) vb(x)h(x) ) veof(x) +
)
h(x)2 ∂P(x) h(x) 12η ∂x
(6)
Here, P is the pressure that arises in the system in order to maintain a constant liquid flow rate. η is the viscosity of the fluid, and the factor 12 results from the parabolic flow profile generated by a pressure gradient between two infinite parallel planes and that can be superimposed to the global electroosmotic flow veof induced by the electric field. The osmotic pressure that can arise from ionic concentration gradients was neglected in eq 6 as it results in negligible velocities compared to the electroosmotic flow and unnecessary additional computing time. veof has to be evaluated in order to both solve the pressure gradient and evaluate the total flow rate Db. Usually, veof is described as proportional to the local electric field and the surface ζ potential through the Helmholtz-Smoluchowski equation: veof(x) ) -
ηζ(x) ∂φ(x) ε0εr ∂x
ηζap(λ(x), h(x)) ∂φ(x) λ(x) ) ε0εr ∂x
ε0εrRT
2
F
2 i i
i
ζap was calculated for a surface charge of 5 mC/m2 in Figure 3 for the two main sections of our system (2.5 µm and 50 nm in height). This calculation was achieved numerically by solving Gouy-Chapman equations (see Supporting Information for mathematical developments). It is clear from this figure that the flow rate induced in the nanochannel by electroosmosis is much lower than the EOF that develops in the microchannels because of the proximity of the lateral walls. It is also possible to fit this apparent ζ potential by
(
m⁄n ζap (x) ) a1,m⁄n 1 - tanh(
)
-log(λ(x)) + a3,m⁄n a2,m⁄n
(9)
where a1,m/n, a2,m/n, and a3,m/n only depends on surface charge and channel height (m/n refers respectively to micro/nanochannel sections) and can be determined before solving the main equations sytem. Numerical Implementation with Initial and Boundary Conditions. The typical dimensions of our model system are presented in Figure 1. A transition region between the micro- and nanochannel was modeled by a continuous change of section height, which preserves the system of abrupt change in height resulting in singularities in the electric field and instabilities in the finite element calculations. First, the fitting coefficients of the apparent ζ potential expression were calculated for each subregion. A Matlab function (bvp4c) based on a finite difference code that implements the three-stage Lobatto IIIa formula was used for the potential determination across the section. The apparent ζ potential was then deduced for different KCl concentrations (Figure 3) and the resulting apparent ζ potentials were fitted with a Nelder-Mead Simplex Method (fminsearch function, Matlab) in order to determine a1, a2, and a3 in eq 13. This analytical expression of the apparent ζ potential was then used in the five coupled equations system that was solved using a commercial finite element program (Comsol AB). These equations consisted in (i) The three mass equations for each ci resulting from eqs 2 and 3
∂h(x)ci(x) )∇ ∂t
((
[
]
ziF ∇ φ(x)ci(x) RT hx2 ∂P(x) ηζap(x) ∂φ(x) h(x)ci(x) 12η ∂x ε0εr ∂x h(x)Di ∇ci(x) -
)
(7)
This equation is nevertheless only accurate as long as the Debye length λ is small compared to the microchannel section,
(8)
∑ z c (x)
)
(10)
where i stand for the K+, Cl-, and Pzp concentrations. (ii) The Poisson equation for the potential φsequations 4 and 5 Analytical Chemistry, Vol. 80, No. 24, December 15, 2008
9545
∂2φ(x) )∂x2
cK+(x) - cCl-(x) - zPcP-(x) +
σs(x) h(x)
εoεr
(11)
(iii) The liquid conservation equation for the pressure P (eq 6 derived along x)
(
(
ηζap(x) ∂φ(x) ∂P(x) ∂ h(x) ε0εr ∂x ∂x ) 12η ∂x ∂x
)
∂ hx3
)
(12)
Initial conditions for the three ionic concentration ci were first calculated using the same strategy as for the apparent ζ potential determination. In each subregion, the cross-section potential was first calculated for the background KCl concentration using the Gouy-Chapman model. The local ion concentrations were then deduced from the potential and averaged over the section to give 0 the initial concentration ci,m/n . The subscript m/n still stands for, respectively, the microchannel and nanochannel regions. The initial potential Φm/n in each subregion was also determined averaging the cross-sectional potential (see Supporting Information for initial conditions calculation details). The initial pressure was set to zero across the whole system. In the transition regions between the microchannels and the nanochannel, the apparent ζ potential of the microchannel region was used, which clearly overestimates the EOF contribution in these sections. Indeed, these regions should have an apparent ζ potential that is intermediate between the micro- and nanochannel’s one. However, the contribution of these regions to the total induced electrokinetic flow can be neglected as they represent less than 0.1% of the total system length. This was confirmed by applying the nanochannel apparent ζ potential to these sections instead of the microchannel’s one, which did not result in a significant change in the total liquid flow rate. As for the apparent ζ potential, the initial conditions in the intermediate regions between micro- and nanochannels were assimilated to microchannel initial conditions. This obviously led to inconsistent initial conditions that were corrected by the internal backward Euler step of the finite element simulation. The boundary conditions for the system are also given in Figure 1. In order to avoid singularities at initial time, Φleft was applied exponentially with a short time constant (typically 0.1 s). Dirichlet boundary values for the ionic concentrations were used and the initial concentrations calculated in the microchannels 0 ci,m/n were imposed at each extremities of the MNM channel. Pressure was maintained at zero at both extremities of the MNM channel. Lagrange quadratic elements were used as a base for the five variables. The typical mesh had 3776 elements with size ranging from 6 µm (near the reservoirs) to 60 nm (at nanochannel entrances).The calculation time to solve the system over 180 s was under 5 min using a Pentium M 1.5 Ghz, 1.5mB RAM
personnal computer (30 s for the prior apparent ζ potential and initial conditions calculation). RESULTS AND DISCUSSION We applied numeric calculations to systems that were previously investigated experimentally, including cathodic preconcentration of fluorescein dyes at low ionic strengths (exp 1) and preconcentration of slower particles such as proteins in moderate ionic strengths electrolytes (exp 2). In the second case, the preconcentration efficiency was studied as a function of particle mobility. Finally, we also focused our study on the ACGF preconcentration rate as a function of the charge of the preconcentrated molecule (exp 3), showing a charge dependence of the preconcentration rate. The simulation parameters for the three types of experiments are given in Table 1. Preconcentration of Fluorescein at Low Ionic Strengthes: The Electrophoretically Dominated Regimes. Numeric calculations were first performed to explain our experimental results obtained with dianionic fluorescein dyes at low KCl concentrations.11,28 The concentration of KCl was set to 10 µM so that exclusion enrichment in the nanochannel is maximal. Since the glass surface charge is strongly dependent on the type of glass, buffer composition, and pretreatments, determining its influence on the preconcentration is relevant. Figure 4 (from left to right) shows the variation of fluorescein concentration profiles in the cathodic microchannel as a function of the density of the surface charge. The voltage was also varied in order to evaluate how the competition between electrophoretic forces and electroosmosis can influence the preconcentration effect. Let us now describe the three different regimes that can be clearly observed in Figure 4a. At low surface charge and intermediate voltage (1 mC and 40 V, for example), an accumulation of fluorescein rapidly happens at the nanochannel cathodic entrance. As the fluid counterflow is not sufficient, fluorescein dyes are only rejected by the electrostatic barrier induced by the surface charges of the nanochannel. Only a small fraction of dianions is accepted in the nanochannel, so fluorescein concentration increases until the incoming flux of fluorescein equalizes the outflux through the nanochannel. This saturating regime corresponds to a CS regime (Figure 2), that was already reported by Pu29 and Datta.7 When surface charge is increased (Figure 4a from left to right), the EOF is enhanced, which results in a translation of the transport coefficient toward positive values. As predicted in Figure 2, the preconcentration front line is shifted toward cathodic reservoirs in the microchannel. The preconcentration rate of this CCGF regime is much higher than for the CS regime as can be observed in Figure 4b where the maximum concentration after 1 min is reported to the initial concentration. This CCGF regime has been previously observed in experiments.17,30 The preconcentration front line location is slowly displaced when EOF is increased with
Table 1. Parameters of the Numerical Simulations exp 1 2 3
9546
cKCl (M) -5
10 5 × 10-2 5 × 10-2
cP (M) -9
10 10-9 10-9
σs (mC · m-2)
∆Φ (V)
DP (m2 · s-1)
zP (valence)
[0.5->10] 25 25
[5->100] 25 25
0.485 × 10-9 [2.34->0.01] × 10-9 [0.65->0.07] × 10-9
-2 -2 [-1->-10]
Analytical Chemistry, Vol. 80, No. 24, December 15, 2008
Figure 4. Cathodic preconcentration of fluorescein for a 10 µM KCl background electrolyte solution. (a) Preconcentration profiles as a function of surface charge and electric field in the cathodic microchannel as a function of time. Each curve corresponds to an additional time of 6 s (the lighter corresponds to 60 s). (b) Preconcentration rate (ratio of the final maximum concentration on the initial fluorescein concentration) as a function of surface charge and applied voltage. (d) Peak shift velocity measured from the displacement of the maximum preconcentration position between 55 and 60 s. (c) Cartography of the CS and stable and unstable CCGF regimes. The best preconcentration occurs in the stable CGF regime.
surface charge (Figure 4a right). While its location remains within a few tens of micrometers for moderate surface charges, a rapid shift of this preconcentration front suddently appears above a critical surface charge. A comparison of the preconcentration profiles at 5 and 5.5 mC at 40 V illustrates this transition. The regime at a concentration above this critical surface corresponds to an unstable CCFG. To determine the limit between those two stable and unstable CCGF regimes, the shift velocity of the preconcentration peak is reported in Figure 4d that shows an abrupt increase at high surface charges. Preconcentration rates for these unstable CCGF regimes are still higher than for the CS regime (Figure 4b). However, as the preconcentration front line is rapidly pushed back to the reservoirs, only the stable CCGF regime is recommended for preconcentration applications. Conditions of an optimal preconcentration are summarized in Figure 4c. To determine the stable CCGF region, panels b and d in Figure 4 were superimposed. It can be observed that the applied voltage drop can be optimized as a function of the surface charge of the MNM structure to obtain a high-efficiency stable preconcentration. When the electric field is increased, both convective (28) Plecis, A. S. P.; Chen, Y. µTAS 2007, 2, 1583–1585. (29) Pu, Q. S.; Yun, J. S.; Temkin, H.; Liu, S. R. Nano Lett. 2004, 4, 1099–1103. (30) Plecis, A.; Schoch, R. B.; Renaud, P. µTAS 2005, 1038–1041.
flux due to EOF and electrophoretic flux are increased. However, it seems that the electrophoretic flux increases more slowly than the EOF. Indeed, when focusing on a constant surface charge of 5 mC and increasing linearly the anodic potential, preconcentration switches from an unstable CCGF (at 10 and 20 V) to a stable CCGF regime (at 40 V) and finally reaches a CS regime at higher electric fields (80 V), thus indicating an electrophoretic dominated preconcentration at high voltages. This electrophoretic domination at higher electric fields was only observed for low ionic strength buffers and can be explained by the apparent ζ potential saturation at low ionic strength (plateau in Figure 3), which limits the EOF. On the contrary, numerical calculations achieved with moderate and high ionic strength buffers revealed a saturation of the electrophoretic flux at high electric fields due to the polarization effect,31 while EOF remains proportional to the electric field. These results could thus explain why cathodic preconcentrations (electrophoretically dominated regimes) were usually observed with low ionic strength buffers while anodic preconcentrations (electroosmotically dominated regime) were always found at higher ionic strengths. We will now (31) Kim, S. J.; Wang, Y. C.; Lee, J. H.; Jang, H.; Han, J. Phys. Rev. Lett. 2007, 99.
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Figure 5. (a) Preconcentration peak location and maximum preconcentration rate after 1 min as a function protein mobility (center). Preconcentration profiles in a-d correspond to decreasing electrophoretic mobilities (respectively, 7.1, 5.4, 1, and 0.54 × 10-8 m2 · s-1 · V-1).
focus on anodic preconcentration regimes in order to evaluate their efficiency for biopreconcentration applications. Preconcentration of Proteins at Moderate Ionic Strengths: Electroosmotically Dominated Regime. Impressive preconcentration rates have been obtained with proteins.13,19 In these cases, the background electrolyte had a moderate ionic strength (∼10 mM) and the applied voltage difference varied from 10 to 200 V. One of the questions raised by these experiments is the dependence of the preconcentration factor as a function of the protein charge and mobility. The preconcentration rate after 1 min is plotted versus protein mobility in Figure 5 (center bottom). In these experiments, the net charge of the protein was maintained to -2e, while the diffusion coefficient was varied. Preconcentration factor remains small for faster proteins, but it increases dramatically for slower proteins. Below a minimum electrophoretic mobility (-1 10-8 m2 · S1- · V-1), the preconcentration rate falls abruptly. From the maximal concentration peak position after 1 min (Figure 5, center up), it is possible to determine the kind of preconcentration regime charged molecules undergo. When the mobility of the particle is sufficiently high ( -7 10-8 m2 · s-1 · V-1), the maximum concentration peak slowly moves toward the cathode that corresponds to the CCGF regime (Figure 5b). The inset in this figure shows that the CGF regime results in a better preconcentration rate than the S regime. However, the maximum of preconcentration at moderate ionic strengths is clearly obtained for the ACGF regime when µp > -1.2 × 10-8 m2 · s-1 · V-1. When the electrophoretic transport is further decreased (µp > -1 × 10-8 m2 · s-1 · V-1), the preconcen9548
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tration peak returns to the nanochannel entrance, which results in an anodic S regime exhibiting negligible preconcentration rates. We have shown previously that it was possible to vary the transport coefficient Jp and the kind of preconcentration regime by changing the total liquid flow rate through variation of the surface charge and EOF velocity. However, it is clear from eq 1 that one could also maintain the convective transport constant and decrease the electrophoretic migration by deceasing µp. In that case, the preconcentration of molecules switches from cathodic regimes to anodic regimes following the scenario depicted in Figure 2. The total flux increase due to the diminution of electrophoretic mobilities explains why most of the experimental studies involving fast migrating dyes observed preferentially cathodic preconcentration regimes, while studies based on larger molecules, like ovalbumin or GFP, reported anodic regimes. It is nevertheless obvious that only the ACGF regime is really fitted for intensive preconcentration purposes. At high ionic strength, the electrostatic barrier at nanochannel is too small to result in an efficient stacking, which explains why both anodic and cathodic stacking regimes result in poor preconcentration rates. On the contrary, the charge selectivity of the nanochannel still enables the creation of an ionic gradient polarization. At nanochannel cathodic side, this ionic gradient is spread out by the electroosmotic flow (see .Supporting Information), which results in a rapid shift of the preconcentration line and limit the preconcentration process at longer time (no stable CCGF regimes). On the contrary, the anodic gradient is compressed at nanochannel entrance by the liquid flow rate, which results in a very sharp and stable ionic gradient (Figure 5c). This ACGF regime produces the best preconcentration rates at moderate ionic strengths. Preconcentration Efficiency as a Function of Biomolecules Charge. The preconcentration of biomolecules through ACGF regimes can greatly enhance the detection limit of a number of biodetection applications, when it is coupled with electrophoretic separation columns19 or directly to on-chip enzyme
µp )
zpFDp ) constant RT
(13)
Figure 6a shows preconcentration rates and profiles obtained for a set of molecules having the same electrophoretic mobility but different total net charges. The higher is the valence of the molecule; the smaller is its diffusion coefficient. Neglecting diffusion effects should result in a constant preconcentration rate (same incoming flux at anodic microchannel extremity). However, it is clear that the lower the diffusion coefficient of the preconcentrated molecule, the stronger is the preconcentration. In Figure 6b, one can clearly observe the effect of smaller diffusion coefficients on the sharpness of the preconcentration front, which can partially explain the increase of preconcentration efficiency with the change of the molecule. On the other hand, when the concentration of the molecule is integrated along the whole anodic microchannel (Peak area instead of peak height), the total number of molecules trapped by the flux barrier also increases with the valence of the molecule. This result suggests that the diffusion of larger molecules toward the second unstable front located inside the nanochannel (see Figure 2) is decreased, which results in a smaller probability for the molecule to cross the energy barrier created by the polarization effect. This analysis was supported by the observation of smaller “leakage” concentrations beyond the nanochannel when the diffusion coefficient was lowered. Figure 6. Influence of molecule charge at constant electrophoretic mobility on the preconcentration rate and the profile of the preconcentration region.
assays.32 For these applications, the understanding of preconcentration dynamic is essential and Figure 5 shows that this dynamic also depends on the nature of the preconcentrated molecules. Since the incoming flux from the anodic reservoir is translated toward higher values when the electrophoretic mobility is decreased (eq 1), a high preconcentration rate is observed for slower molecules until an electrophoretic mobility cut off. If we consider that all molecules are stopped at the preconcentration front, the incoming particle flux and thus the preconcentration rate should vary linearly with the mobility of the particle up to an optimal value, after which ACGF regime is no more initiated. It can be observed in Figure 6 that this dependence is actually not linear, which affects the quantitative analysis of biological samples. This nonlinearity can be explained by diffusion effects that are not taken into account in the phenomenological approach of Figure 2 and eq 1. Indeed, in numerical experiments reported in Figure 5, the mobility of the preconcentrated molecule was varied by changing its diffusion coefficient while keeping its net charge constant. When molecules accumulate at the preconcentration front line, an important concentration occurs that is responsible for a spreading of the front line. The diffusion coefficient of the molecules is then involved in both the sharpness of the preconcentration region and the “leakage” of molecules across the counter flux barrier. This diffusion contribution can clearly be observed when the mobility of the molecule is kept constant, while its diffusion coefficient is varied together with the valence according (32) Lee, J. H.; Song, Y. A.; Tannenbaum, S. R.; Han, J. Anal. Chem. 2008, 80, 3198–3204.
CONCLUSION The phenomenological description of the electropreconcentration process using charge-selective micro/nano/microchannels led us to consider four classes of preconcentration regimes. Electrostatic exclusion is responsible for the “so-called” stacking (S) regimes, while concentration polarization effects are responsible for the “so-called” Counter Gradient Focusing (CGF) regimes. Both of them can occur either at the cathodic side (electrophoretically dominated regimes) or at the anodic side (electroosmotically dominated regimes) of the nanochannel. In order to exactly take into account the competition of both EOF and electrophoresis in the preconcentration process, a 1D model and numerical algorithm has been developed. The introduction of a volumic surface charge associated to an apparent ζ potential allows modeling of the surface effects within 1D system, while keeping the computing time to a reasonable value. The implementation of this model within a commercial finite element code enabled us to confront recent published electropreconcentration data to our theoretical results. These results confirm that low ionic strength buffers privilege cathodic preconcentrations of fastmigrating analytes such as fluorescein, while moderate and high ionic strength buffers result in efficient ACGF regime and negligible stacking effects. Beyond the unification of the different preconcentration regimes, this study also pointed out the strong dependence of such processes on the nature of the preconcentrated molecules. Indeed, the preconcentration efficiency depends on both the diffusion coefficient and the valence of the molecule. Finally, our 1D model is applicable to a variety of micro/ nanofluidic systems, which should allow an efficient optimization of such tools for biochemical analyses. Analytical Chemistry, Vol. 80, No. 24, December 15, 2008
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ACKNOWLEDGMENT The authors thank Tom Zangle for fruitful discussions and A.P. acknowledge DGA (French Ministry of Defense) for funding. SUPPORTING INFORMATION AVAILABLE Numerical determination of the initial conditions; apparent ξ determination; Description of the electrophoretic flux; KCl concentration profile at moderate ionic strengths.This material
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is available free of charge via the Internet at http://pubs.acs. org.
Received for review August 26, 2008. Accepted September 11, 2008. AC8017907
