Anal. Chem. 2009, 81, 8347–8353
Analyte Concentration at the Tip of a Nanopipette Nils Calander* Department of Physics, Macquarie University, Sydney, NSW 2109, Australia Concentration of molecules within the tips of nanopipettes when applying a DC voltage is herein investigated using finite-element simulations. The ion concentrations and fluxes due to diffusion, electro-migration, and electroosmotic flow, and the electric potential are determined by the simultaneous solution of the Nernst-Planck, Poisson, and Navier-Stokes equations within the water solution containing sodium and chloride ions and negatively charged molecules. The electric potential within the pipette glass wall is at the same time determined by the Poisson equation together with appropriate boundary conditions and accounts for a field effect through the wall. Fixed negative surface charge on both the internal and external glass surfaces of the nanopipette is included together with the field effect through the glass wall to account for the electric double layer and the electroosmosis. The inclusion of the field effect through the pipette wall is new compared to previous modeling of similar structures and is shown to be crucial for the behavior at the tip. It is demonstrated that the concentration of molecules is a consequence of ionic charge accumulation at the tip screening the electric field, thereby slowing down the electrophoretic motion of the molecules, which is further slowed down or stopped by the oppositely directed electro-osmosis. It is also shown that the trapping is very sensitive to the properties of the molecule, that is, its electrophoretic mobility and diffusion coefficient, the properties of the pipette, the ionic strength of the solution, and the applied electric field. Electro-preconcentration with charge-selective nanochannels has been investigated experimentally as well as theoretically.1 The concentration is explained by the oppositely directed electromigration and electro-osmotic flow. High preconcentration has been achieved with a number of experimental schemes by this principle.2,3 Million-fold preconcentration of proteins and peptides has been achieved in a microchannel by means of a generated extended space charge region injected from a nanochannel.4 This is explained in terms of interplay of electro-kinetic phenomena. The injected space charge is causing enhanced electro-osmosis termed electro-osmosis of the second kind. The charge cloud * E-mail:
[email protected]. (1) Plecis, A.; Nanteuil, C.; Haghiri-Gosnet, A. M.; Chen, Y. Anal. Chem. 2008, 80, 9542–9550. (2) Kim, S. M.; Burns, M. A.; Hasselbrink, E. F. Anal. Chem. 2006, 78, 4779– 4785. (3) Foote, R. S.; Khandurina, J.; Jacobson, S. C.; Ramsey, J. M. Anal. Chem. 2005, 77, 57–63. (4) Wang, Y. C.; Stevens, A. L.; Han, J. Y. Anal. Chem. 2005, 77, 4293–4299. 10.1021/ac901142z CCC: $40.75 2009 American Chemical Society Published on Web 09/14/2009
interferes with the electrophoretic migration of the charged molecules and the concentration build up at a potential barrier, reinforced by counteracting electro-osmosis. Trapping of charged molecules at the tip of a nanopipette when applying a DC voltage has previously been shown experimentally.5-7 The concentration enhancement at the tip has been shown to be hundreds to thousands of times. However, the background salt concentration in these experiments is generally higher than in the before-mentioned experiments or simulations of the nanochannels, and the authors explain the effect by dielectrophoresis, that is, electric gradient forces acting on polarizable particles8 which therefore in this case concentrate where the electric field is at its strongest. The nanopipette has successfully been used in a number of applications,9-16 including controlled manipulation of cellular constituents within single cells.17 In many of the applications the possibility to selectively trap and concentrate molecules at the tip region for subsequent delivery would reasonably be of importance. The motivation of this work is to show that the electro-kinetic phenomena electro-migration and electro-osmosis can explain the concentration of molecules at the tip, also at higher salt concentration, without considering dielectrophoresis as has been previously suggested.5-7 The importance of the inclusion of the field effect through the pipette wall is particularly important in this respect and is an extension of the method described by White and Bund.18 (5) Clarke, R. W.; White, S. S.; Zhou, D. J.; Ying, L. M.; Klenerman, D. Angew. Chem., Int. Ed. 2005, 44, 3747–3750. (6) Clarke, R. W.; Piper, J. D.; Ying, L. M.; Klenerman, D. Phys. Rev. Lett. 2007, 98, 198102/1–4. (7) Ying, L. M.; White, S. S.; Bruckbauer, A.; Meadows, L.; Korchev, Y. E; Klenerman, D. Biophys. J. 2004, 86, 1018–1027. (8) Pohl, H. A. Dielectrophoresis: the behavior of neutral matter in nonuniform electric fields; Cambridge University Press: Cambridge, 1978. (9) White, S. S.; Balasubramanian, S.; Klenerman, D.; Ying, L. M. Angew. Chem., Int. Ed. 2006, 45, 7540–7543. (10) Piper, J. D.; Li, C.; Lo, C. J.; Berry, R.; Korchev, Y.; Ying, L. M.; Klenerman, D. J. Am. Chem. Soc. 2008, 130, 10386–10393. (11) Vogelsang, J.; Doose, S.; Sauer, M.; Tinnefeld, P. Anal. Chem. 2007, 79, 7367–7375. (12) Bruckbauer, A.; Zhou, D. J.; Ying, L. M.; Korchev, Y. E.; Abell, C.; Klenerman, D. J. Am. Chem. Soc. 2003, 125, 9834–9839. (13) Rodolfa, K. T.; Bruckbauer, A.; Zhou, D. J.; Korchev, Y. E.; Klenerman, D. Angew. Chem., Int. Ed. 2005, 44, 6854–6859. (14) Rodolfa, K. T.; Bruckbauer, A.; Zhou, D. J.; Schevchuk, A. I.; Korchev, Y. E.; Klenerman, D. Nano Lett. 2006, 6, 252–257. (15) Hansma, P. K.; Drake, B.; Marti, O.; Gould, S. A. C.; Prater, C. B. Science 1989, 243, 641–643. (16) Ying, L. M.; Bruckbauer, A.; Zhou, D. J.; Gorelik, J.; Shevehuk, A.; Lab, M.; Korchev, Y.; Klenerman, D. Phys. Chem. Chem. Phys. 2005, 7, 2859– 2866. (17) Bruckbauer, A.; James, P.; Zhou, D. J.; Yoon, J. W.; Excell, D.; Korchev, Y.; Jones, R.; Klenerman, D. Biophys. J. 2007, 93, 3120–3131. (18) White, H. S.; Bund, A. Langmuir 2008, 24, 2212–2218.
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The field effect means attraction/repel of ions in the solution to/ from a surface of a dielectric wall (such as a glass pipette wall) by an electric field caused by an applied voltage across the wall.19-21 The fixed surface charges also attract/repel ions to/ from the surface, and the two effects (i.e., fixed charge and field effect) add together. The layout of the article is as follows. First the numerical models, a two-dimensional cylindrically symmetric and a onedimensional, are presented and validated by applying them in simple cases where the solution is known analytically. The numerical models are then applied to the nanopipette and concentration of a third-party molecule, such as a charged protein, is shown. The concentration is monitored by varying the properties of the molecule, that is, its electrophoretic mobility and diffusion coefficient, the voltage, the pipette geometry, and the salt concentration. The behavior is explained in terms of oppositely directed electrophoretic migration and electro-osmosis, also at higher salt concentration. NUMERICAL MODELS Two-Dimensional Model. Following refs 18, 22, and 23, the Nernst-Planck equations are used for the migration and diffusion of the various ionic species in the solution: Ji ) civ - µici∇φ - Di∇ci
∇ · Ji +
∂ c )0 ∂t i
(1)
(2)
Here, bold face indicates vectors. Ji, ci, µi, and Di, are the currents, concentrations, electrophoretic mobilities, and diffusion coefficients, respectively, for the involved ionic species indexed by i, which are sodium ions, chloride ions, and the charged third-party molecules. ∇ is the gradient operator, φ the electric potential, and v the solution flow velocity. If steady state is reached the time derivative of the concentrations in eq 2 is zero. The electric potential is governed by the Poisson equation:
-εj∇2φ ) NA
∑qc
i i
(3)
η∇2v - ∇p ) NA
(19) Ghowsi, K.; Gale, R. J. J. Chromatogr., A 1991, 559, 95–101. (20) Ghowsi, K. Field-effect Electroosmosis. U.S. Patent 5092972, 1992. (21) Schasfoort, R. B. M.; Schlautmann, S.; Hendrikse, L.; van den Berg, A. Science 1999, 286, 942–945. (22) Vlassiouk, I.; Smirnov, S.; Siwy, Z. Nano Lett. 2008, 8, 1978–1985. (23) Vlassiouk, I.; Smirnov, S.; Siwy, Z. ACS Nano 2008, 2, 1589–1602.
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(4)
i i
i
∇·v)0
(5)
The sum to the right is the negative of the space force, that is, the charge density times the electric field. The electric field is the negative of the gradient of the electric potential. p is the pressure, η is the viscosity of the water solution, and the dot · denotes scalar product. Acceleration forces are neglected as is usually the case in microfluidics and even more justified in nanofluidics. Ji, ci, v, p, and φ depend on the cylindrical coordinates z and r, where r is the distance from the centerline and z the distance along that line and on time t if steady state is not reached. The boundary conditions are that at the solution-glass interface the water solution has zero velocity, the ionic and charged molecule fluxes have no normal component, the potential is continuous across the interface, and the normal component of the electric displacement vector, that is, the electric field times the permittivity, changes abruptly with the amount of the fixed surface charge. A fixed surface charge is usually present at the glass-solution interface due to the ionization of silanol groups. Far outside and far inside the pipette the concentrations, the pressure, and the electric potential are set to prescribed values. Only the differences of the pressure and electric potential between the far outside and the far inside have significance. The equations of this three-dimensional model are cast in cylindrical coordinates, and the finite element method (FEM) is used for the solution of the two-dimensional problem. The FEMLAB program (COMSOL Inc.) is used for the calculations. Validation of the Two-Dimensional Model. The numerical model is validated in two simple cases where exact solutions are known.18 The first case concerns the diffuse double layer adjacent to a flat surface with a fixed surface charge σ. The exact solution for the electric potential and the concentrations of the singly charged ions are18 φ)
1 + K exp(-z/λ) 2kT ln e 1 - K exp(-z/λ)
(
)
exp(-z/λ) ( 11 -+ KK exp(-z/λ) )
-2
c( ) c
i
The sum to the right is the space charge, qi are the respective charges for the ionic species, and εj are the permittivities of the respective media indexed by j, which are the water solution and the pipette glass wall. The glass wall has no space charge; that is, no charge within the wall which means that the righthand side of eq 3 is zero within the wall (it may still have surface charges at the surface of the wall affecting the boundary conditions). NA is Avogadros number. The flow of the water solution is governed by Navier-Stokes equation and the zero divergence since the water is incompressible:
∑ q c ∇φ
(6)
(7)
The parameters K and λ are
K)
√8kTεWc + σ2 - √8kTεWc σ λ)
kTεW 2ce2
(8)
(9)
εW is the permittivity of the medium which for water is 80 · ε0, where ε0 is the vacuum permittivity, k Boltzmann’s constant, T the absolute temperature, c the bulk concentration of the salt ions (Na+ and Cl-), λ the Debye length, and z the distance to the surface. The second case concerns the electro-osmotic flow in a cylinder with radius R and surface charge σ. For a surface potential
behavior under these conditions a one-dimensional model is developed that uses the radial averages of all the variables, keeping only the z-coordinate. Following Plecis et al.1 one-dimensional versions of the Nernst-Planck and Poisson’s equations are used in a way suitable for a channel with varying radius. The variables are assumed to have reached local equilibrium radially and therefore have well-defined radial averages. This works well for the electric field and the various concentrations. The flow profiles of pressure driven and electro-osmotic driven flow are different and have to be considered separately. The electro-osmotic driven flow profile depends on the profiles of the ion concentrations. In ref 1 an apparent zeta potential described by an optimized expression is used. Here an expression for the dependence of the radially averaged flow versus the ion concentrations that has the correct limits for thin and thick double layers compared to the channel radius is used. So, in one dimension eqs 1 and 2 are
Figure 1. (a, b) Validation case 1. A cylindrical container with a surface charge at one of the ends and the potential held at 0 V at the other. The black dots correspond to eqs 6 and 7, respectively. The solid lines are from FEM simulations. (c-f) Validation case 2. A cylindrical channel has reservoirs at the ends with smooth transitions to the channel. The black dots correspond to eqs 10, 11, and 12, and the solid lines are from FEM simulation. The concentrations at the center are [Na+] ) 0.998 mM, [Cl-] ) 1.035 mM, and [M]/[M]0 ) 0.908.
(zeta potential) smaller than kT/e the exact solution for the flow velocity, the potential, and the concentrations is given by18
vz )
λσEz (I (r/λ) - I0(R/λ)) ηI1(R/λ) 0
(10)
λσI0(r/λ) - Ezz + φ0 εI1(R/λ)
(11)
φ)
(
ci ) c exp -
qiλσI0(r/λ) kTεI1(R/λ)
)
(12)
η is the viscosity of water, Ii the modified Bessel functions of the first kind of order i, z the axial, and r the radial coordinate. The simulated and analytical results are compared in Figure 1. The charged third-party molecules are assumed to have low enough concentration to not have any influence on space charge or potential. The FEM element size is about 1.5 nm in the first model and 10 nm in the second, and quadratic elements are used. The conclusion is that the model is valid in these cases and therefore should be suitable for the pipette. One-Dimensional Model. At higher salt concentrations the Debye length is smaller making it necessary with finer FEM elements near the surface. The number of elements and the workload therefore increase. To calculate the concentration
j - Di ∂ ¯ci ¯J i ) ¯civ¯ - µi¯ci ∂ φ ∂z ∂z
(13)
∂ ∂ ¯ AJ + A ¯ci ) 0 ∂z i ∂t
(14)
A bar over the variable means radial average. Here, A is the cross-sectional area given by πR2, where R is the channel radius which may depend on the longitudinal coordinate z. It has to be included into the continuity eq 14 to account for the fact that the total numbers of ions of the various species have to remain constant. The Poisson eq 3 becomes: -εj
∂ ∂ j A φ ) ANA ∂z ∂z
∑ q ¯c
i i
+ AF
(15)
i
The introduction of the cross-sectional area has the same explanation here, but the total charge has to remain invariant. F represents charges other than the ionic such as the surface charge1 and the capacitive charge by the field-effect through the pipette wall. This is justified by taking the radial average (integration) of eq 3 and using Gauss’ theorem as in Cervera et al.24,25 and is also shown in the Supporting Information. The Navier-Stokes equation is by the discussion above replaced by an equation giving the radially averaged water flow in terms of the pressure gradient and the electric field. v¯ ) -R
∂ j ∂ p¯ - β φ ∂z ∂z
(16)
The water flow velocity v¯ depends on the pressure gradient and the electric potential gradient (negative of electric field) through two coefficients R and β (conductance coefficients), where R depends on R and β on R and the concentrations of the ionic species. Assuming parabolic pressure driven flow (Poiseuille flow) the radial average is found by integration and R becomes
R)
R2 8η
(17)
(24) Cervera, J.; Schiedt, B.; Ramirez, P. Europhys. Lett. 2005, 71, 35–41. (25) Cervera, J.; Schiedt, B.; Neumann, R.; Mafe, S.; Ramirez, P. J. Chem. Phys. 2006, 124, 104706/1–9.
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An expression for β is used that has the correct limits for thin and thick Debye layers as compared to the channel radius. This expression is derived in the Supporting Information and is there shown to be valid for small relative concentration differences of the ionic species.
β)
β0(c¯+ - ¯c-) 4c0(4 +
√√¯c+¯c-/c0)
(18)
The parameters β0 and c0 are β0 )
c0 )
εWkT ηe εWkT
2e2NAR2
(19)
(20)
The incompressibility of water gives the following equation: ∂ Av¯ ) 0 ∂z
(21)
The same reasoning concerning A applies here. This means that the total volume of incompressible water remains constant. So, in summary, the one-dimensional equations to solve are eqs 14, 15, and 21, with usage of eqs 13, 16, 17, and 18, in the j , and p¯, only depending on the coordinate z. This variables ¯ci, φ is done by a one-dimensional FEM solver (FEMLAB). This one-dimensional approach extends the approach of Cervera et al.,24,25 where electro-osmotic effects, that is, fluid flow, are not considered. Also, the z-dependence of the channel radius is here generalized. Another extension is the inclusion of the field effect through the pipette glass wall which in eq 15 adds capacitive charge to the fixed surface charge. Validation of the One-Dimensional Model. The onedimensional solution is compared to the two-dimensional cylindrical solution of the previously validated model for a simple test case in Figure 2. The agreement between the two models seems excellent. MODELS APPLIED TO THE PIPETTE Two-Dimensional Model. The model geometry and mesh used in the calculations are shown and described in Figure 3a,b. The boundary conditions are as follows (panel b): The ion and charged molecule concentrations, the electric potential, and the pressure are set to the specified background values at the boundaries of the reservoirs surrounding the pipette tip (blue) and far within the pipette (red). The background values of the ions (Na+, Cl-) are 1 mM. The concentration of the charged molecules at those boundaries should be small enough to not contribute to the charge in the solution (usually < 1 nM). The potential is grounded outside (φ0 ) 0 V) and set to the specified applied voltage inside to mimic the applied voltage between a negative electrode (cathode) outside and a positive inside (anode). The pressure is set to the same value outside and inside, but only the pressure difference counts, which then is set to zero. The ion and charged molecule normal components of their currents and the water flow are zero at the glass-solution 8350
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Figure 2. Validation of the one-dimensional model. A cone of length 1 µm, one end has diameter 100 nm, and the other 200 nm. Boundary conditions: Same pressure at the ends, potential difference 2 V, the right more positive, [Na+] and [Cl-] are 10 mM at the ends. The cone surface has a surface charge of 1 mC/m2. (a) The axial velocity component of the water flow. (b-e) Various variables averaged radially, versus axial distance z. Black is from the two-dimensional cylindrical FEM calculation and red from the one-dimensional FEM calculation. The variables in c and e start at the prescribed values at the ends but make quick changes which may not be seen. The total flow in b is 1.53 × 10-16 m3/s in the two-dimensional case and 1.48 × 10-16 m3/s in the one-dimensional case.
interface (green). The normal component of the electric displacement field (electric field times the permittivity) changes abruptly by the amount of the fixed surface charge at the water glass interface (green). This surface charge is negative and is here set to -1000 · e/µm2, a reasonable value due to ionized silanol groups at neutral pH.26,27 It is noted that for this surface charge the average distance between the charged silanol groups is about 30 nm, and it may be questioned that the surface charge can be assumed smooth enough for the mean-field calculations when this distance is comparable to the radius of the pipette and larger than the Debye length. However, as is found below, the field-effect through the pipette wall contributes more to the surface charge, and it can surely be considered smooth. The centerline (black) is the cylindrical symmetry line. The bulk solution is modeled as described by eqs 1-5. The charge of the molecules is assumed to be -2 · e and the radius 3 nm. This is the apparent charge causing the electrophoretic migration; the actual charge should normally be larger due to the screening by the salt ions. The size of the charged molecule is small compared to the geometrical measures of the pipette and to the distance of concentration variations of the salt which means that the extension of the molecules can be neglected. The concentration of the molecules is then a well-defined property and (26) Behrens, S. H.; Grier, D. G. J. Chem. Phys. 2001, 115, 6716–6721. (27) Berli, C. L. A.; Piaggio, M. V.; Deiber, J. A. Electrophoresis 2003, 24, 1587– 1595.
relative permittivity of 4.6. The relative permittivity of water is assumed to be 80. One-Dimensional Model. Figure 3, panel c, shows a schematic drawing of how the one-dimensional model is applied to the pipette. This geometry can be compared to the geometry in Figure 3, panels a and b, that is, a conical structure. The interface between the pipette tip and the surrounding medium is modeled as a strong and abrupt widening of the tip, reflecting the much larger volume outside the tip. The pipette wall acts as a spreadout capacitor by the field-effect illustrated in the panel and explained in the Supporting Information. The capacitance per unit length is
C)
2πεG ln(Ro /Ri)
(22)
Ri is the inner and Ro the outer pipette radius, respectively, which depend on z. εG is the permittivity of the pipette wall (borosilicate, εG ) 4.6 · ε0). The F charge density in eq 15 then becomes (Supporting Information equation 10): Figure 3. (a, b) Geometry and mesh for the FEM calculations. The boundary conditions are described in the main text. The mesh is finer near the solution-glass interface as is also described in the main text. (c) Sketch of the principles of the one-dimensional model applied to the pipette. L ) 2 µm, d ) 0.25 µm. In the lower part it is shown as capacitors accounting for the field effect through the pipette wall. (d, e) The space charge without and with applied voltage. Note the build-up of charge at the tip when voltage is applied. A movie showing the build-up of charge versus time is supplied in the online Supporting Information. (f) Concentration enhancement of the molecules.
Figure 4. Ion concentration, electric field, and flow velocity all get stronger within the tip region.
is determined by the differential equations. This is also true when the average distance between the molecules is large compared to the radius of the pipette and to the Debye length, because the molecules are assumed not to interact back to the electric field due to their low concentration. The pipette is slightly conical with an inner half angle of 1° and outer of 3°. The inner tip radius is 50 nm and the outer 150 nm. The mesh is finer near the solution-glass interface. The FEM element size near the glass of the inside of the tip is about 5 nm, and quadratic elements are used. The pipette material is assumed to be borosilicate with a
j) AF ) 2πRiσ + C(φ0 - φ
(23)
σ is the fixed surface charge on the inner pipette wall, φj the potential within the pipette (radially averaged), and φ0 the potential outside, that is, at the other side of the pipette wall, and is taken to be zero. RESULTS Two-Dimensional Model. At the onset of a voltage, a concentration enhancement of the charged molecules may occur. At 2.8 V there is a maximum enhancement near the tip of about 450 times at steady state which is shown in Figure 3, panel f. In panel e the ionic charge build-up is shown. It is seen that the mobile charge layer at the pipette surface, much strengthened by the field effect, causes charge accumulation near the tip. This charge cloud, as the charge cloud created by charge injection from a nanochannel into a microchannel in ref 4, changes the electric potential and therefore the electrophoretic migration of the molecules, causing their concentration build-up, just behind the charge cloud. It is noted that the field effect largely over-rides the effect of the fixed surface charge. This justifies the mean field calculation as noted above. The effect of the fixed surface charge is seen in panel d where the applied voltage is zero. The ionic charge polarization, such as the build-up of charge clouds, has been observed in nanochannels carrying a fixed surface charge both experimentally and theoretically.1,4,22,23,28-30 This charge polarization may cause concentration accumulation of analytes1,4,28 near the ionic charges within or outside the nanochannels. Figure 4, panels a and b, shows the concentration of the sodium and chlorine ions. There is a build-up of both concentrations within the tip region. Panels c and d show the electric field and the electro-osmotic flow. As expected the field is at its strongest near the tip. The electro-osmotic flow is also naturally (28) Wang, Y. C.; Han, J. Y. Lab Chip 2008, 8, 392–394. (29) Pu, Q. S.; Yun, J. S.; Temkin, H.; Liu, S. R. Nano Lett. 2004, 4, 1099–1103. (30) Wei, C.; Bard, A. J.; Feldberg, S. W. Anal. Chem. 1997, 69, 4627–4633.
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Figure 5. (a) Electrophoresis of the molecules. (b) Electro-osmosis of the water. (c) Combined effect of electrophoresis and electroosmosis. (d) Molecular current as a combination of electrophoresis, electro-osmosis, and diffusion. Molecules are coming in from above and are slowly getting out at the center. Flow lines are shown in red. Note the circular behavior around the point of maximum concentration enhancement. This is an average behavior of the molecules; the track of a single molecule is quite irregular due to Brownian motion and does not follow the lines.
Figure 6. Radially averaged velocities versus the axial distance from the tip. EPM ) electrophoretic migration, EOF ) electro-osmotic flow. The molecules accumulate at the point where they cancel, if the velocity of the molecules is directed toward that point.
concentrated at the tip. The flow profile in the tip region is parabolic due to both the pressure driven flow from pressure buildup within the pipette and electro-osmosis driven by charges not as much localized to the surface (the charge cloud). Figure 5 shows the detailed behavior of the electrophoretic migration, the electro-osmosis, their combination, and the full molecular current, at the tip region. As is seen molecules are preferentially entering the tip and the enhancement region near the glass surface and slowly exiting near the center. Figure 6 shows the electrophoretic migration of the molecules and the electro-osmotic flow of the solution within the pipette averaged radially. The molecules should accumulate where the sum of these two is zero, as illustrated in panel b. By comparing with Figure 3 panel f it is seen to be the case. Figure 7 shows the dependence of the concentration enhancement of the molecules on various parameters, such as applied voltage, time, salt concentration, inner tip radius, molecule charge, and molecule radius. It is seen that the concentration is very sensitive to the properties of the molecules. The properties of the molecules of interest here are their electrophoretic mobility and diffusion coefficient, determined by their screened charge and size. It is noted in the figures how the various peaks will shift by ionic strength, that is, salt concentration. For example, at higher ionic strength the peak appears at a stronger applied voltage. Figure 8 shows how the concentration enhancement is affected by the molecule base concentration, if it is allowed to contribute to the charge. As expected it affects the concentration enhancement when the enhanced concentration of the molecule charge approaches the value of the ionic space charge. One-Dimensional Model. The one-dimensional model is used for higher salt concentrations. The concentration enhancement 8352
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Figure 7. Behavior of the enhancement versus a number of parameters. The salt concentration is generally kept at 1 mM; the arrows show in which directions the peaks move when the salt concentration is increased. (a) The maximum concentration enhancement versus applied voltage. (b) The buildup of concentration enhancement versus time when the applied voltage is suddenly changed from 0 to 2.8 V. To make the time dependent FEM calculation which is more demanding in computing and memory usage, the number of elements in the mesh had to be relaxed as compared to panel a. Therefore the final concentration enhancement does not completely reach the maximal value in panel a. (c) This diagram shows how the position in voltage of the maximal molecule concentration enhancement varies with salt concentration. Higher salt concentration means that higher voltage is needed to reach the maximum. This is also discussed in the section of the one-dimensional model. (d) The maximum molecule concentration enhancement versus pipette radius, when voltage is adjusted for maximal enhancement. (e) The maximum molecule concentration enhancement versus molecule equivalent electric charge. The molecule equivalent charge is here defined to be the screened charge that determines the electrophoretic mobility. (f) Same, but molecule radius is varied.
needs higher applied voltage as already mentioned, the higher the salt concentration, the higher the needed applied voltage. It is also more sensitive to the applied voltage as is seen in Figure 9 panels a, b, and c. The electrophoretic mobility of the molecules is weaker at higher salt concentration due to more screening of their charges. The electro-osmosis is usually even weaker due to the thinner double layer. This is counteracted by the strong field effect through the pipette wall at higher voltages. The field effect makes the dependence of the electro-osmosis on the applied voltage almost quadratic. This is due to the fact that both the electric field and the charge at the surface increase with applied voltage. The electro-osmosis is also made stronger by some electro-osmosis of the second kind, by the charge cloud at the tip mentioned before. Electro-osmosis of the second kind as discussed in ref 4 is due to excess charge injection either from a
Figure 8. If the concentration of the charged molecules is not negligible it affects the concentration enhancement. [M]0 is the prescribed concentration at the boundaries marked blue and red in Figure 3 panel b.
Figure 9. Results from the one-dimensional model. (a) Concentration enhancement versus applied voltage for some selected parameters. (1) c ) 1 mM, qM ) -2 · e (apparent molecule charge), θ ) 1° (half inner cone angle), (2) c ) 10 mM, qM ) -1.5 · e, θ ) 3°, (3) c ) 100 mM, qM ) -0.9 · e, θ ) 6°, (4) c ) 100 mM, qM ) -1.5 · e, θ ) 3°. (b, c) Concentration profiles and velocity of the molecules at some applied voltages. (1) 19.75 V, (2) 19.85 V, (3) 19.95 V, (4) 20.05 V, and (5) 20.15 V. (d-f) These panels show the impact of the halfcone angle explained in the main text. The salt concentration is 10 mM, and the apparent molecule charge -1.5 · e. Solid line: Half-cone angle ) 3.5°, applied voltage ) 12.5 V. Dashed line: Half-cone angle ) 3.1°, applied voltage ) 12 V.
nanochannel as in ref 4 or as here by the injection from the pipette tip inner wall into the region just outside the pipette. This excess charge within a region of electric field is then causing electroosmosis.
Figure 9 panel d shows the concentration enhancement versus applied voltage for some cone angles, at a background salt concentration of 10 mM, an apparent molecule charge of -1.5 · e, and molecule radius of 3 nm. As is seen higher enhancement is possible at the larger half-cone angles. The peak is shifted toward higher applied voltage. The explanation is that at larger half-cone angle the voltage drop (electric field) is stronger and closer to the tip, which makes the electrophoresis more localized to the tip and stronger there. The electro-osmosis does not catch up fully with the electrophoresis because of a shorter length within the tip region where the electric field is stronger. The less volume where the electro-osmotic forces act, the less electro-osmosis, because it also has to drag water outside that region. Therefore higher applied voltage is needed. Note that as discussed before the electrophoretic migration varies about linearly, whereas the electro-osmosis varies about quadratically with applied voltage. At higher applied voltage the electrophoresis is even stronger, but the electro-osmosis catches up with the electrophoresis. Stronger counteracting electrophoresis and electro-osmosis mean stronger trapping ability at larger half-cone angle. This is seen in Figure 9 panel e. At larger cone angle the velocity from the counteracting electro-osmosis and electrophoresis is larger and steeper, at higher applied voltage, and closer to the pipette tip as expected. In panel f the corresponding concentration enhancements are shown. DISCUSSION That higher voltages are needed for concentration enhancement at higher salt concentrations is in line with experiments.5,6 So is also the fact that the concentration enhancement is more sensitive to the voltage. These are properties well explained by the electrophoresis-electro-osmosis hypothesis and less well by dielectrophoresis. Unknown parameters that may explain the discrepancy between the simulations and the published experimental results may be the thickness of the pipette wall, the cone angle, the exact shape of the tapered structure, and the properties of the molecules. However, it may very well be that all electrokinetic effects, including dielectrophoresis, contribute to the concentration enhancement. One outcome of this investigation is that the field effect through the pipette wall is contributing more to the surface phenomena than the fixed surface charge, that is, enhancing the electro-osmosis and changing the electric field, thereby causing the concentration enhancement. This effect should be useful also in nanochannels and their applications. SUPPORTING INFORMATION AVAILABLE 1. Details of the Gauss integration giving the capacitance charge in Equation 15. 2. Details of the derivation of the conductance coefficient β, i.e., eq 18. 3. A movie showing the buildup of charge versus time, together with a movie legend. This material is available free of charge via the Internet at http:// pubs.acs.org. Received for review May 25, 2009. Accepted August 27, 2009. AC901142Z
Analytical Chemistry, Vol. 81, No. 20, October 15, 2009
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