Multiphysics Simulation of Ion Concentration Polarization Induced by

Jul 17, 2014 - Department of Biomedical Engineering, Ulsan National Institute of Science and. Technology (UNIST), 50 UNIST-gil, Eonyang-eup, Ulsan, ...
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Multiphysics Simulation of Ion Concentration Polarization Induced by Nanoporous Membranes in Dual Channel Devices Mingjie Jia† and Taesung Kim*,†,‡ †

Department of Mechanical Engineering, ‡Department of Biomedical Engineering, Ulsan National Institute of Science and Technology (UNIST), 50 UNIST-gil, Eonyang-eup, Ulsan, 689-798, Republic of Korea S Supporting Information *

ABSTRACT: Many microfluidic devices have been utilizing ion concentration polarization (ICP) phenomena by using a permselective, nanoporous membrane with electric fields for a variety of preconcentration applications. However, numerical analyses on the ICP phenomena have not drawn sufficient attention, although they are an intriguing and interdisciplinary research area. In this work, we propose a 2-D model and present numerical simulation results on the ICP, which were obtained by solving three coupled governing equations: Nernst−Planck, Navier−Stokes, and Poisson. With improved boundary conditions and assumptions, we demonstrated that the simulation results not only are consistent with other experimental results but also make it possible to thoroughly understand the ICP phenomena. In addition, we demonstrated that the preconcentration of analytes can be simulated and quantified in terms of concentration enhancement factors (CEFs) that were related to many factors, such as ionic concentration distribution, electric fields, and flow fields including vortex flows across the membrane. Furthermore, we demonstrated that a high electrophoretic mobility (EPM) of counterions in the membrane plays the most important role in producing accurate simulation results while the effect of the charge density of the membrane is relatively insignificant. Hence, it is believed that the model and simulation results would provide good guidelines to better develop microfluidic preconcentration devices based on the ICP phenomena.

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multiphysics simulation of ICP phenomena, although among the approaches introduced above, the highest concentration enhancement factors (CEFs) are achieved in ICP-based microfluidic devices. Importantly, Rubinstein and Zaltzman numerically analyzed the ionic concentration distribution and electroosmosis of the second kind in the anodic part in the presence of an ideal ionpermselective structure.16,17 However, it is unlikely that the simulation conditions are realized in an experiment, because no nanoporous structure can perfectly prevent co-ions from passing through it. In addition, Dhopeshwarkar et al. proposed a 1-D model, in which different charge densities in a negatively charged hydrogel were applied, and demonstrated that the higher the charge density in the hydrogel, the stronger is the ICP generated.18 Although the 1-D model and simulation result well supported their experimental results, it seems to be a limitation that the model cannot be applied to typical dual channel-based ICP (DC-ICP) devices in which convection flows must be taken into consideration. Recently, Shen et al. demonstrated numerical simulation results of a vortex and its secondary flow in a DC-ICP device, but the preconcentration of biomolecule analytes was not simulated.7 In their simulations,

he advancement of microfabrication and miniaturization technologies has spawned numerous techniques for amplifying initial biosample concentrations in the past two decades.1,2 In particular, many microfluidic devices have been reported that are potentially capable of preconcentrating target biomolecules at very low concentrations over million-folds.3 For example, nanofluidic channels and/or nanoporous materials were used for generating ion concentration polarization (ICP) phenomena that are achieved by an ionpermselective structure in the microfluidic channel network in conjunction with electric fields.4 Using the ICP phenomena, a number of microfluidic preconcentration devices has been developed.4−7 In addition, many other/similar mechanisms have been utilized for the preconcentration of biomolecules on a chip, such as affinity gradient focusing,8 isoelectric focusing,9 bipolar electrode focusing,10 temperature gradient focusing,11 electrophoresis-based accumulation,12 and ICP-based biomolecule trapping.5,13 In parallel with experimental approaches for fabricating and testing such microfluidic devices, many theoretical approaches have been taken to better understand the multiphysics phenomena and efficiently envision them on a chip. For example, Sommer et al. supported the temperature gradient focusing phenomena with numerical simulation results that were validated by their experimental result.14 Also, both experimental and numerical studies on bipolar electrode focusing were performed simultaneously.10,15 However, it appears that not much attention has been drawn toward the © 2014 American Chemical Society

Received: January 20, 2014 Accepted: July 17, 2014 Published: July 17, 2014 7360

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Figure 1. Schematic illustration of a DC-ICP model for numerical simulation. (a) A nanoporous membrane is integrated at the junction of the anodic and cathodic channels. High (VH) and low (VL) electric potentials are applied at the ends of the anodic channel while both ends of the cathodic channel are connected with ground (VG). The ionic concentration in each reservoir is initially the same as c0, and the length and width of each channel are 2000 and 100 μm, respectively. (b) Boundary conditions for the Navier−Stokes equation. Electroosmotic slip boundary conditions were used for the anodic channel walls (solid lines) whereas nonslip ones were used near the membrane as indicated with dashed lines.

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THEORETICAL MODELING AND NUMERICAL METHOD We simulated ICP phenomena by solving the governing equations such as Nernst−Planck, Navier−Stokes, and Poisson simultaneously but did not employ the Brinkman equation that enables the simulation of the flow field inside porous materials/ membranes in this work because the flow rate in the nanoporous membrane by pressure difference was negligible compared to the bulk flow (EOF) because of low permeability (e.g., Nafion: 10−18 m2).7 The detailed description of governing equations, modeling, and the numerical method is found in the Supporting Information (SI). In short, as illustrated in Figure 1a, a nanoporous membrane was located at the junction between the anodic and cathodic channels as an ionpermselective structure. The diffusion coefficients of the ions in the membrane were significantly small due to the local, low permeability through it so that they were estimated to be 1/10 of the values in a bulk solution as used in other work.7,20−22 This value could be different according to different kinds of ions and different properties of the membrane, such as pore sizes and surface charge densities. More discussion is found in Modeling in the SI. In addition, the EPM of co-ions in the membrane was assumed to be proportional to the local diffusivity according to the Nernst−Einstein equation (μi ∼ Di/ RT), while a range of the EPM (μi) of counterions in the membrane and the fixed charge density of the membrane (ρfix) was simulated. Figure 1b shows the boundary conditions for the Navier−Stokes equation. The wall boundary conditions near the membrane were changed to nonslip ones instead of applying electroosmotic slip for the entire channel walls. The reason will be described later. In addition, we only focused on the flow in the anodic channel because the flow in the cathodic channel was less important and even negligible.

although a high charge density was applied to a nanoporous membrane, the electrophoretic mobility (EPM) of counterions in the membrane was still assumed to be proportional to local diffusivity, which was very low due to the low permeability of water through the membrane. In contrast, it has been reported by Duan and Majumdar that the EPM of a proton in a negatively charged 2 nm nanotube is four times as great as the EPM value in a buffer solution.19 The high EPM was estimated to be an equivalent mobility value by measuring current and calculating conductance, but it has not yet been theoretically proved. To date, a very low EPM of counterions that is proportional to local diffusivity in the membrane has been employed in ICP simulations. Therefore, not only was the influence of the extremely fast electroosmotic flow (EOF) in the nanopores neglected, but also the ion-permselectivity of the membrane was assumed to only originate from the concentration difference between counterions and co-ions. For this case, ICP can hardly be generated in a single and open channel (SCICP) device in which an ion-permselective membrane (Nafion) was patterned on the surface of the channel to allow ions to pass over the surface-patterned membrane. Therefore, the biased ion-permselectivity in the membrane can be easily compensated for through a bulk flow. However, SC-ICP phenomena have been well demonstrated in our previous experiment.5 In this work, we created a 2-D model of a DC-ICP device and performed multiphysics simulations to understand the ICP phenomena that are governed by three coupled governing equations. Also, we theoretically characterized the preconcentration performance of the DC-ICP device under many experimental conditions and compared them with experimental results. In particular, we simulated not only a wide range of the EPMs of counterions in the membrane but also the charge densities of the membrane because it seems to be hard to experimentally test the parameters. Since the effects of the EPMs and the charge densities on ICP phenomena were theoretically analyzed in terms of CEFs, ionic concentration distributions, ionic currents, and electric field distributions, the model and simulation results could make it possible to better understand and apply the DC-ICP phenomena to developing microfluidic preconcentration devices with high performance.

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RESULTS AND DISCUSSION Validation of Numerical Simulation Result. First of all, we validated the numerical simulation results with our previous experimental results.23 Figure 2a demonstrates the preconcentration of trace molecules when t = 0, 10, 20, and 30 min. The trace molecules should be negatively charged, so that the effects of electrophoresis and electroosmosis on these trace molecules 7361

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closest simulation results to the experiment. More discussion is found in Boundary Conditions in the SI. Figure 3 shows the distribution of ion concentrations and electric fields along the y-axis, which was defined across the membrane from the anodic to the cathodic channel in this work (see Figure 1b). First, the ion concentration near the anodic interface sharply decreases from the initial value of 1 mM to 0.1−0.2 mM while it increases to 2 mM near the cathodic interface in 1 s. The variation of the ion concentrations in the membrane seems to be unstable when t < 10 s for which ICP develops and is reaching a steady state; the ion concentration at the anodic interface further decreases down to 0.01 mM while it continuously increases at the cathodic interface, generating an ion enrichment zone (IEZ). The concentration difference between the counterions and the co-ions in the membrane is kept at 0.5 mM, which is the magnitude of the charge density (negative) of the membrane to maintain electroneutrality (solid vs dashed lines); the membrane was regarded as a domain (volume) not a surface in the simulation. Second, the simulation result of the electric field across the membrane is presented in Figure 3b. The dashed line shows the electric field distribution when no ion-permselective membrane is employed in the model (open junction, a reference value = 103 V/m). For this case, the electric field (E) in the domain of the membrane slightly increases because of the narrow crosssection of the junction (Am); the ionic current is expressed as I = σEAm, where σ is local conductivity, which is constant across the membrane because of the continuity of the ionic current. However, when the membrane is employed in the model, the strength of the electric fields in the IDZ becomes much greater than the reference value. It sharply increases up to a magnitude of 106 V/m near the membrane due to the local existence of a nonequilibrium space-charge layer in which counterions dominate over co-ions.25 The increase and decrease of the electric fields along the channel are synchronized with the decrease and increase of the ion concentrations (conductivity), respectively, which obeys the continuity of the ionic current. The conductance in the membrane is increased by the increasing ion concentrations and the application of a high EPM of the counterions, so that the strength of the electric field significantly decreases in the membrane; I = σEAx, where the Ax is an effective cross-section area of the nanopores, and remains constant across the membrane. At the cathodic interface, the strength of the electric field sharply fluctuates because the

Figure 2. Numerical and experimental results of the preconcentration of the trace molecules in the anodic channel. (a) Simulation result of CEFs in the presence of electric potentials (VH = 4 V and VL = 2 V) when t = 0, 10, 20, and 30 min. (b) Fluorescence images of preconcentrated bovine serum albumin as analytes (trace molecules), which were labeled with fluorescein isothiocyanate when the preconcentration was carried out for about 0, 8, and 16 min, respectively.

are in the opposite directions. Near the ion depletion zone (IDZ), the competing electroosmosis and electrophoresis of the trace molecules reach a balance at a certain region in which the trace molecules can be concentrated. In 30 min, the CEFs of the trace molecules continuously increase up to 400 and the location is near the IDZ, showing good consistency with the experiment results shown in Figure 2b (see Figure S1 and Ion Concentration Distributions in the SI). One interesting result is that there exists a concentrated zone of the trace molecules in a ring shape at the entrance of the membrane (in the IDZ when t = 30 min). This seems to be caused by vortex flows that are known to trap some macromolecules.24 It is noted that many different boundary conditions were tested and compared with each other, but the combined boundary condition consisting of electroosmotic slip and nonslip at channel walls produced the

Figure 3. Transient distributions of ionic concentrations and the strength of electric fields across the membrane from the anodic channel to the cathodic channel along the y-axis. Here, VH = 4 V, VL = 2 V, the fixed charge density of the membrane is −0.5 mM, and the EPM of counterions in the membrane is four times as large as that in the bulk solution. (a) The concentration distribution of cations and anions when t = 1, 10, and 100 s. (b) The strength of electric fields when t = 1, 10, and 100 s. The dashed line shows the strength of the electric field in the absence of the membrane. 7362

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layer25 are located near the membrane, and the velocities of the vortex flows range from 1 to 20 mm/s, which have been reported in several experimental studies to be much greater than those of the EOF (∼40 μm/s) generated far away from the junction. The extremely fast vortex flows induce the surrounding fluid to generate an additional slow-flow zone where the highest concentration location can be found. It is known that the preconcentration of analytes depends on the balance between electroosmosis and their electrophoresis.26 Thus, the slow-flow zone would be either the highest concentration location (Figure 2a) or a relatively low concentration location (Figure 5a), because the balance between electroosmosis and electrophoresis of the trace molecules is either kept or broken here. The CEFs obtained from the simulation are slightly lower than the experimental result. This can be attributed to the fact that the vortex flows disperse the preconcentrated molecules, resulting in the low CEFs. Since the flow field in a DC-ICP device is not carefully studied, we provide the relevant results based on the improved boundary conditions to offer an insight into the variation of flow field. Figure 4b shows the velocity profiles of the flow in the anodic channel along A−A′ in Figure 4a when t = 0.1, 6, and 10 s and at steady state. Initially (t = 0.1 s), the velocity profile of the flow is not flat but concave and parabolic, because the electric field in the opposite direction to the EOF in the left half of the anodic channel causes hydraulic resistance (see Figure S3 and Simulation Result of Flow Fields in the SI). Then, the formation of a vortex in the IDZ reinforces the flow along the microchannel, showing the flat (t = 6 s) and then the convex and parabolic velocity profile (t > 10 s) gradually. Figure 4c shows the velocity profile along B−B′. Initially (t = 0.1 s), the velocity at the walls is negative (blue line), showing that the direction of flow near the boundary is opposite to the direction of the bulk flow because the local electric field is in the opposite direction of the EOF (from the left end to the junction). Afterward, the electric field in the left half inverses its direction but is still relatively weak (see Figure S4 and Simulation Result of Electric Fields in the SI). Therefore, the velocity of the bulk flow at the left channel walls is slower than that at the right channel walls. Due to the continuity of the bulk flow, the velocity profile along B−B′ shows a more convex shape than that along A−A′. It is noted that the velocities at the wall boundary in Figure 4b,c continue to change such that a local electric field is not constant but varies with time. The convex velocity profile of the EOF has been observed in other literature that used a slip boundary condition at the entire channel walls with a constant velocity.7 However, for that work, the change of velocity generated by the variation of electric fields was not considered. Hence, it is noted that we could obtain a much more convincing result in this work by using the combined boundary conditions of an electroosmotic slip and a nonslip. The electroosmotic slip boundary condition allowed the velocity at channel walls to vary with transient electric fields. On the basis of the result of the flow velocity at the wall boundary, we can also estimate the variation of the electric field with time (see Simulation Result of Electric Fields in the SI). Effect of Electric Potentials on ICP. The electric potential should be the most important parameter affecting ICP phenomena. For this reason, we attempted to apply three different sets of electric potentials to the anode while both the ends of the cathode were connected to an electric ground (VG). Figure 5a shows the preconcentration of the trace molecules for

transition distance from a high-charged zone to a noncharged zone is relatively short (10 nm). The local, fine mesh size of around 10 nm appears not to produce an accurate solution, but it hardly influences the simulation results for the entire domain. The fluctuation does not happen at the anodic interface because the 100 nm-thick space-charge layer is long enough for the transition. In addition, the sudden increase of the electric fields at the cathodic interface caused by the sharp change of the EPM of ions keeps the continuity of ionic currents. Simulation Result of Flow Field. Figure 4a shows the flow field in the anodic channel. Streamlines (shown by the uniform density function of COMSOL) are represented in black lines, and local velocity vectors are shown as normalized arrows. Vortex flows induced by the electrical body force (−ρE∇ϕ in the Navier−Stokes equation) in the space-charge

Figure 4. Simulation results of a flow field in the anodic channel. (a) Vortex flows are observed near the nanoporous membrane, inducing an additional slow-flow zone as indicated with a dashed rectangle. (b, c) Velocity profiles across the line A−A′ and the line B−B′ when t = 0.1, 6, and 10 s and t = 100 s. 7363

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Figure 5. Simulation results of preconcentration and vortex velocity when three different configurations of electric potentials are applied for 30 min, respectively. (a) Qualitative simulation result. (b) Quantitative result of CEFs. (c) The x-component of the velocity of vortex flows along the y-axis when t = 30 min. The arrow indicates a stagnation point.

Figure 6. Comparison of the strength of ICP for the different EPM of counterions (a through c) and the charge densities of the membrane (d through f). (a) CEFs at t = 30 min when the EPMs are 0.1-, 0.5-, 1.0-, 2.0-, and 4.0-fold of the EPM in a bulk solution. (b) Ion concentration distributions of counterions along the y-axis at t = 10 s. (c) The transient ionic currents across the membrane. (d) CEFs at t = 30 min when the charge densities are −0.1, −0.25, −0.50, −0.75, and −1.0 mM. (e) Ion concentration distributions of counterions along the y-axis at t = 10 s. (f) The transient ionic currents across the membrane.

30 min in the presence of different electric potentials: (i) VH = 3 V and VL = 1.5 V, (ii) VH = 4 V and VL = 2 V, and (iii) VH = 5 V and VL = 2.5 V. The highest concentration locations for each set are slightly different, appearing to depend on the location of an electrokinetic balance. The dispersion of the trace molecules becomes more obvious with the increase of electric potentials, indicating that electroosmosis gets stronger. Interestingly, for the electric potential of set (i), the highest concentration location is not in the slow-flow zone. The concentration at the slow-flow zone appears to be lower than that at the surroundings, implying that the electrokinetic balance is locally broken. For this case, the dispersion of the trace molecules is insignificant, because electroosmosis by the low electric potential is relatively weak. In contrast, when an electric

potential is relatively high, such as in (ii) and (iii), the highest concentration location appears to be slightly different and is found near the slow-flow zone. In addition, the concentrated zone of the trace molecules gets narrower with the increase of the electric potentials. For this case, the migration of the trace molecules is determined by complex phenomena involving electroosmosis, electrophoresis, and diffusion. The influence of diffusion is not related to electric fields while the other two are. The concentration area must be reduced to enhance the diffusion effect so that the electrokinetic balance of those three factors can be reached. We also compared the preconcentration performance as shown in Figure 5b. The CEFs for 30 min are 60-, 400-, and 3000-fold, respectively. Due to the existence of vortex flow, the 7364

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flow rate across the channel is not linear with the applied electric potential; the net flux of trace molecules increases significantly with the increase of the electric potential, leading to an even larger CEF under a high electric potential. The stronger an electric field is applied, the faster the EOF is generated. Therefore, the flux of the trace molecules is consequently enhanced, leading to high CEFs. Although the CEFs seem to be linearly proportional to concentration time at relatively low electric potential, they are nonlinear and much faster under a high electric potential, in which vortex flows induced may make a contribution to additional preconcentration. Basically, not only can the vortex flows induce the dispersion of the trace molecules, but also they can additionally trap trace molecules around the surroundings, which subsequently results in high CEFs. Figure 5c shows the xcomponent of the velocity of vortex flows in the anodic channel along the y-axis. It is obvious that the velocity of the vortex flows increases as the electric potential increases. The zero velocity position indicated with an arrow is the stagnant point of the vortex flows and it is fixed, although electric potentials vary. Presumably, the position of the stagnant point is governed by the geometry of the channel. Effect of EPM of Counterions in the Membrane on ICP. We simulated several different EPMs of counterions in the membrane, which were 0.1, 0.5, 1, 2, and 4 times as large as their EPM in a buffer solution, to investigate the effect of the EPMs on ICP. The electric potential was kept as VH = 4 V and VL = 2 V, and the charge density of the membrane was also fixed as −0.5 mM. In other literature/simulations, only the 0.1× EPM has been used because the EPM is proportional to the local diffusivity of ions according to the Nernst−Einstein equation. Thus, it was assumed that the biased transport between counterions and co-ions is caused only by the ion concentration difference between them. Figure 6a shows CEFs at t = 30 min for the range of the EPMs, showing that the higher the EPM, the stronger are the ICP and vortex flows produced. The CEF is nearly zero for 0.1× EPM. This is because electroosmosis is very weak and the vortex is not strong enough to enhance the EOF (see Figure S5a in the SI). Figure 6b shows the concentration distribution of the counterions across the membrane for the different EPMs at t = 10 s. In addition, it takes a relatively long time for the ICP to form in the case of the 0.1× EPM compared with the other higher EPMs. Consequently, the concentration of the counterions in the IEZ is also low. With the increase of the EPMs (e.g., μi > 0.5×), the difference of the ion concentrations at the cathodic interface is small but distinguishable. The higher the EPM applied, the higher is the ion concentration induced in the IEZ. The profile of the ion concentration distributions in the membrane varies in the same manner as the EPM of the counterion. For the 0.1× EPM, it appears to be very similar to other literature that employed the same EPM for the simulations.7 However, we note that the ion concentration is very linearly distributed in the membrane for higher EPMs, indicating that ion transport by diffusion in the membrane becomes more and more trivial (refer to the Nernst−Planck equation in the SI). In addition, we also compared ionic currents through the membrane for the different EPMs in Figure 6c. The ionic currents decrease sharply at the early stage because the decrease of ion concentrations in the anodic channel significantly reduces the conductance, while they level off soon because the mild increase of the ion concentrations in the cathodic channel

continuously increases the conductance. This shows good consistency with our previous experimental result.23 The higher the mobility, the lower is the steady-state value of the ionic current produced, meaning that ICP gets stronger or the ionpermselectivity of the membrane gets stronger. Additionally, the ionic currents for the 0.1× EPM decrease very slowly compared with other results. This means that the generation of ICP is rather slower. The distribution of electric fields across the membrane is shown in Figure S5 (SI) for the EPMs. It is evident that the 0.1× EPM cannot generate a reliable result observed in the experiments. Thus, we concluded that the concentration difference between counterions and co-ions in the membrane cannot completely embody the ion-permselectivity of a nanoporous membrane. Moreover, the 0.5× EPM, which is still smaller than the EPM of a buffer solution, causes the fast generation of ICP, and the ionic current is very similar to that of even higher EPMs (e.g., μi > 1.0×). Basically, the generation of ICP is caused by the biased transport of counterions and co-ions through an ionpermselective structure. In other simulation studies, this bias transport was explained only by the concentration difference of counterions and co-ions in the structure, while in our work, the influence of the EPM of counterions on ICP was revealed to be most dominant. From this simulation result, we confirmed that the ion-permselectivity of the membrane mainly originates from strengthening the EPM of counterions but weakening the EPM of co-ions, implying that the ion concentration difference in the membrane plays a relatively subordinate role. Additionally, the accurate value of the EPM of ions in the membrane may depend on many factors such as the porosity, buffer concentrations, zeta-potentials, temperature, and so on. Therefore, it seems to be difficult to take a suitable value that can exactly simulate experimental conditions. However, it is certain that a relatively high EPM of the counterions in the membrane is much closer to the real experimental condition. Even though the simulation result of ICP or flow in a single nanochannel or nanopore was investigated,27−29 it is inappropriate to directly apply the same mechanism to DC-ICP devices with ionpermselective, nanoporous membranes, because most experimental results were obtained at the level of a membraneintegrated microfluidic device rather than at the nanoscopic, single nanochannel level. In this context, the 2-D model proposed in this work seems to be appropriate to analyze ionpermselectivity of a charged nanoporous membrane in DC-ICP devices. In real conditions, the influence of the flow inside the membrane would make a difference on ion-permselectivity. The flow rate across the membrane by a pressure difference can be calculated using Darcy’s law (Q ∝ k∇p, where k is the permeability of water through the membrane), but it turned out to be negligible due to the extremely low permeability of the membrane and low pressure difference between the anodic and the cathodic channels. However, the influence of the electrokinetic flow in nanopores/nanochannels can be significant,30 and its direction is favorable to the transport of counterions but adverse to that of co-ions (the direction of EOF in a charged nanopore is always the same as the direction of electrophoresis of counterions and opposite to that of co-ions).31 Thus, the ion-permselectivity of the nanoporous membrane is dramatically enhanced, and the adoption of high EPMs (μi > 0.5×) of counterions can generate more reliable simulation results compared with other modeling and simulation. The accurate value of the EPM may vary significantly case by case, depending on the properties of the membrane and the kind of counterions, 7365

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further confirming that the biased transport between counterions and co-ions through the membrane is most critical for generating ICP. This in turn suggested that the adoption of a high EPM (>0.5×) of counterions is much more reliable and realistic than the 0.1× EPM that has been used in most other simulations, being in good agreement with the recent, single nanochannel experimental result.19 Furthermore, the effect of the charge density of the membrane on ICP was proved to be relatively less important than that of the EPMs. Since the 2-D model and simulation results on the DC-ICP device would be very helpful to understand the mechanism of ICP-based preconcentration, not only could they provide practical guidelines for fabricating ICP-based microfluidic devices (e.g., desalination and micromixing devices)32−34 but also optimize preconcentration performance by manipulating experimental circumstances.

so that further study would be needed to confirm the value. It may require enormous computation resources to fully simulate the electrokinetics of a membrane-integrated microfluidic device at the single nanopore or nanochannel levels. Effect of the Charge Density of the Membrane on ICP. In the same manner, we investigated the effect of the charge density of the membrane on ICP. For this, the EPM of counterions in the membrane was fixed (e.g., 4× and 0.5×, respectively) and the electric potential was kept (VH = 4 V and VL = 2 V). The charge densities of the membrane simulated were −0.1, −0.25, −0.5, −0.75, and −1.0 mM, respectively. Figure 6d shows CEFs at t = 30 min for the different charge densities. Different from our expectation, the CEFs were not simply linear with the charge densities but the variation of the CEFs was insignificant. This can be further supported by the result that the variation of the velocity profile and the electric field caused by the charge densities are negligible, especially when a high EPM (4×) of counterions in the membrane is applied (see Figure S5c,d in the SI). The 2-fold change in CEFs shown in Figure 6d is possibly induced by different leakage rates of trace molecules because the flow field closely adjacent to the junction is highly influenced by the surface charge density of the membrane. Figure 6e shows the ion concentration distribution across the membrane, being consistent with that in Figure 6b. However, it is obvious that the differences of the ion concentration in the membrane are caused by the fixed charge density. Figure 6f shows the ionic current variation. For all the charge densities, the variation was very small regardless of the developing time and strength of ICP (CEFs). In contrast, when a relatively low EPM (0.5×) of counterions in the membrane was used, the influence of the charge densities of the membrane on ICP is still not conspicuous (see Figure S6 in the SI). In short, when the charge densities of the membrane are relatively low compared to the bulk concentration, they seem to be insignificant on ICP (CEF), the distribution of ion concentrations, and ionic currents compared with the EPM of counterions. Conversely, when the charge densities are much higher, we conjecture that their influence on ICP appears to increase but may not be more dominant than the EPM. Unfortunately, simulations using a high charge density of the membrane were unable to be performed because of the limited computational resources; a dramatically increased mesh size is required in the space-charge layer near the membrane.

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ASSOCIATED CONTENT

S Supporting Information *

Detailed description of governing equations, modeling, and numerical method; ion concentration distributions and boundary conditions (Figure S1), simulation result of flow fields (Figure S2 and S3), simulation result of electric fields (Figure S4), effect of different EPMs and charge densities of a membrane on ICP (Figure S5), and simulation result of the charge densities of a membrane for a relatively low EPM (Figure S6). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +82-52-217-2313. Fax: +8252-217-2409. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by a grant from the Next-Generation BioGreen 21 program (SSAC, No. PJ00954905), Rural Development Administration, Republic of Korea and by a National Research Foundation of Korea (NRF) grant funded by the Ministry of Science, ICT and Future Planning (2012R1A1A2044736).

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CONCLUSIONS We have simulated ICP development and vortex generation in a DC-ICP device with improved boundary conditions and a wide range of EPMs of counterions and charge densities in a nanoporous membrane. First of all, we validated the 2-D model and simulation results with experimental ones in terms of CEFs, ion concentrations, electric fields (ionic currents), and flow fields under various electric potentials, and then, we demonstrated that preconcentration is significantly affected by electric potentials and flow fields including vortex flows and a slow-flow zone induced near the nanoporous membrane for the first time to the best of our knowledge. In addition, we found that the EPMs of counterions in the nanoporous membrane play key roles in ICP phenomena. In contrast to the fact that the ion-permselectivity of the nanoporous membrane is only induced by the local dominance of counterions, we revealed that a fast EOF in a nanopore/nanochannel can be beneficial to the transport of counterions but adverse to that of co-ions,

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