Multiphysics Simulation of Ion Concentration Polarization Induced by a

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Multiphysics Simulation of Ion Concentration Polarization Induced by a Surface-Patterned Nanoporous Membrane in Single Channel Devices Mingjie Jia† and Taesung Kim*,†,‡ †

Department of Mechanical Engineering, Ulsan National Institute of Science and Technology (UNIST), 50 UNIST-gil, Eonyang-eup, Ulsan 689-798, Republic of Korea ‡ 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: Microfluidic devices utilize ion concentration polarization (ICP) phenomena for a variety of applications, but a comprehensive understanding of the generation of ICP is still necessary. Recently, the emergence of a novel single channel ICP (SC-ICP) device has stimulated further research on the mechanism of ICP generation, so that we developed a 2-D model of an SC-ICP device that integrates a nanoporous membrane on the bottom surface of the channel, allowing bulk flow over the membrane. We solved a set of coupled governing equations with appropriate boundary conditions to explore ICP numerically. As a result, we not only showed that the simulation results held a strong qualitative agreement with experimental results, but also found the distribution of ion concentrations in the SC-ICP device that has never been reported in previous studies. We confirmed again that the electrophoretic mobility (EPM) of counterions in the membrane is the most dominant factor determining the generation and strength of ICP, whereas the charge density of the membrane was dominant to the ICP strength only when a high EPM value was assumed. From the viewpoint of practical applications, an SC-ICP device with a long membrane under low buffer strength showed enhanced performance in the preconcentration of charged molecules. Therefore, we believe that the simulation results could not only provide sharp insight into ICP phenomena but also predict and optimize the performance of SC-ICP devices in various microfluidic applications. the biased transport flux of counterions over co-ions through them.13,14 However, the phenomena of ion transport through the ion-permselective structures are not so easily explained. For example, Duan and Majumdar estimated that the electrophoretic mobility (EPM) of monovalent ions in 2 nm hydrophilic nanochannels is 4 times as large as that in a bulk solution. The increased mobility was attributed to the geometrical confinement, which not only makes the water in the nanochannels well organized due to hydration interaction, but also makes the ions exhibit higher mobility than in bulk water.15 They also asserted that the increased mobility in hydrophobic nanochannels is caused by an enhanced electroosmotic flow (EOF) and reduced surface friction. In the nanochannels/nanopores with a length scale of 6−10 nm (e.g., Nafion membrane16), the hydration interaction becomes negligible because of size exclusion. Instead, the electrostatic interactions are a dominant factor in determining ion transport, allowing continuum mechanics to be utilized in analyzing ion transport through the nanochannels/nanopores.17 Additionally,

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n the last two decades, microfluidic devices have been fabricated to generate ion concentration polarization (ICP) by integrating several kinds of ion-permselective structures, such as nanochannels,1 nanoporous membranes,2,3 and even electrodes.4 ICP is known for inducing concentration gradients of ions near the interface between ion-permselective structures and an electrolyte when an electric field is applied.5,6 Although the formation of the ICP usually restricts the operating currents for desalination in electrodialysis,7 the variation of the electric field and a fluidic motion induced by the ICP show a remarkable potential for many microfluidic applications. In particular, dual-channel ICP (DC-ICP) devices integrated with nanochannels or nanoporous membranes have been broadly utilized, and their applications include preconcentration of target molecules at very low initial concentrations,8 micromixers using vortices generated on the ion depletion zone (IDZ),9,10 and desalination using the IDZ.11,12 However, it seems that the generation mechanism of ICP has not been thoroughly characterized for the most part. One hypothesis/ description is based only on ion-permselectivity from the overlap of the electric double-layer that originates from the fixed charge density of the ion-permselective structures (e.g., nanochannels and nanoporous membranes). In other words, the charge density of the ion-permselective structures causes © 2014 American Chemical Society

Received: July 22, 2014 Accepted: September 30, 2014 Published: September 30, 2014 10365

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Yeh et al. demonstrated numerically the variation of ion concentrations in a nanopore with different external environments, such as buffer concentration, surface charge density, and pH.18,19 In their simulation, the boundary condition at the surface of the nanopore was assumed to be no-slip. As a result, the velocity of electrokinetic flow turned out be insignificant, displaying a dramatic difference to experiments. For example, Majumder et al. reported that the flow rate through nanotubes is 4 or 5 orders of magnitude higher than that calculated from conventional fluid mechanics theory and attributed this to the “frictionless” boundary of the nanotubes.20 Although the frictionless characteristic is determined by many factors,21 it is obvious that the flow velocity in a nanoporous membrane is much greater than the calculated value from conventional theory. Thus, the contribution of the inner flow to ion flux through nanopores/nanochannels becomes significant, which seems to be another supplementary hypothesis and to have been overlooked in previous numerical studies of ICP. In fact, in our previous work, we used high EPMs of counterions in the nanoporous membrane to provide theoretical support for the assumptions listed above, and demonstrated more reliable multiphysics simulation results of DC-ICP devices.22 In addition, we found that the variation of the EPM of counterions in the nanoporous membrane makes a greater impact on the strength of ICP than the charge density of the membrane. However, DC-ICP devices are unique in that ions are forced to penetrate through the nanoporous membrane, isolating two channels so that it is necessary to study other general models for comprehensive understanding of ICP phenomena. Recently, ICP was reported to have been induced in single channel devices (SC-ICP)23,24 in which the surface-patterned nanoporous membrane does not completely block the channel. This allows ions to migrate either through or over the membrane along the channel to the electrodes. Consequently, it would be valuable to study SC-ICP to better understand the mechanism of the DC-ICP, which may differ slightly from that of SC-ICP. In this work, we attempt to further explain the multiphysics phenomena of SC-ICP devices and develop a reasonable numerical model. First, we perform numerical simulations to determine the influence of electric potentials on ICP strength. Next, we report on the simulated influence of the EPM of counterions in the membrane and the charge density of the membrane on the strength of ICP. Third, we investigate other factors affecting ICP, such as the charge density of the membrane, the length of the membrane, and buffer strength. Lastly, we introduce the vortex flow in a 2-D model and discuss its limitations, which seem to demand the numerical study of SC-ICP in a 3-D model.

Figure 1. (a) Schematic illustration of an SC-ICP device with a surface-patterned nanoporous membrane located in the center. (b) 2D modeling of the SC-ICP device for multiphysics simulations (symmetric surface at y = 0). (c) The boundary conditions used for simulations.

simulation and avoid the generation of vortices. Additional justifications for this choice are explained in the Results and Discussion section. The governing equations were solved using COMSOL Multiphysics (ver 4.3b). Figure 1b is a schematic illustration of a simulation model showing the side view of the device and set as 2000 μm long and 20 μm high with a negatively charged nanoporous membrane whose length and thickness are 100 and 2 μm, respectively. It is worth noting that the diffusion coefficients of ions in the membrane were significantly reduced due to the local low permeability; they were estimated to be 1/10 (for Nafion)16,25,26 of the values in bulk solution reported in other studies.13,14 This value could be altered because of different types of ions or different membrane properties. 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 wide range of EPMs (μi) of counterions in the membrane were simulated with a fixed membrane charge density (ρfix). Since the surface charge density (C/m2) could not be directly used in simulations, the units were approximated in mM for harmonizing with the concentration of the buffer solution. The detailed description of governing equations and numerical methods is found in our previous work (refer to Supporting Information).22 Figure 1c shows the boundary conditions used in the simulations. Since the channel was connected with reservoirs, the concentration of ion species at both ends of the channel was ci = c0 (buffer concentration). The concentration of trace molecules at the high potential (VH) end was ctr = 1 nM and ctr

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THEORETICAL MODELING AND NUMERICAL METHODS To simulate ICP phenomena, it is necessary to solve the coupled equations of Navier−Stokes, Nernst−Planck, and Poisson simultaneously. Figure 1a shows a schematic illustration of an open SC-ICP device in which a surfacepatterned nanoporous membrane is located at the bottom of the channel. Compared with DC-ICP devices, the channel is called “open” because it allows fluid and ions to pass over the membrane and only two electrodes are used for connecting a high electric potential and a ground. Instead of building a 3-D model which consumes colossal and unaffordable computational resources, we used a 2-D model to simplify the 10366

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Figure 2. Effects of electric potential on ICP phenomena. (a) Preconcentration results for electric potentials ranging from 2.5 to 5.0 V. (b) The variation of the largest CEFs in the channel and the bulk electric fields with increasing electric potential. (c) The distribution of normalized electric fields along the x-axis at z = 10 μm from the bottom surface of the channel with electric potentials ranging from 1.5 to 5 V. (d) The distribution of the normalized concentrations of trace molecules along the same line (at z = 10 μm) for the different electric potentials.

However, in the presence of extremely fast vortex flows, a nonslip boundary condition can be more reliable.22 A fine mesh size (∼10 nm) was applied to the interface between the microchannel and the nanoporous membrane due to the local existence of a space charge layer. A slip boundary condition was applied to the interface for the Navier−Stokes equation because the exact simulation of the fluid flow inside the nanoporous membrane seemed to be currently impossible and its contribution to the perm-selectivity in the membrane could be compensated for by using a high EPM of counterions in the membrane. On the other hand, it is emphasized that the interface was not a boundary for the ions and electric fields (Nernst−Planck and Poisson equations). A solution keeping the conservation of the ions (electric currents) was numerically found by employing the different properties of the microchannel domain and the nanoporous membrane one, resulting in the nonlinear distribution of ion concentrations and electric fields.

= 0 at the grounded (VG) end. The initial concentrations of the buffer solution and the trace molecules were ci(t = 0) = c0 and ctr(t = 0) = 1 nM, respectively. The material properties used in the simulations are provided in the Supporting Information (refer to Table S1). H+ and Cl− were used as cations and anions, respectively, without considering the influence of H+ ions on the deprotonation/protonation inside the membrane. Both ends of the channel were set up as open boundaries without external hydraulic pressure. An electroosmotic slip boundary condition was utilized to simulate the EOF across the channel. That is, the flow velocity (U) at the channel walls was approximated by the Helmholtz−Smoluchowski equation U = −εζEt /η, where ε is the permittivity of buffer solution, ζ is the zeta-potential at the charged wall surface, Et is the tangential electric field, and η is the viscosity of the solution. As a result, the electroosmotic slip boundary condition significantly reduced the computational load for simulating EOF, and errors were negligible.14 It is worth noticing that electric double-layers (EDLs) were neglected at the microchannel walls to which the electroosmotic slip or slip boundary was applied. However, the surface charge inducing EDLs at the membrane/microchannel interface was taken into consideration, which was proved by the nonlinear distribution of ion concentrations and electric fields in the following simulation results. The flow boundary condition near the membrane was changed from an electroosmotic slip to a slip boundary condition (frictionless) along the channel walls to keep the continuity of EOF. In the absence of vortex flows, a slip and a nonslip boundary condition do not make a big difference to flow fields, but the former seems to be slightly more accurate than the latter. This was validated in this study by analyzing flow fields in the same manner as our previous DC-ICP study.

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RESULTS AND DISCUSSION Effect of electric potentials on SC-ICP. The effect of electric potentials on ICP has been established as significant, especially for the preconcentration of trace molecules. Figure 2a shows preconcentration results at t = 30 min under different electric potentials ranging from 2.5 to 5 V (corresponding to the average electric field of 12.5 V/cm to 25 V/cm along the channel). The charge density of the membrane was fixed at ρfix = −0.5 mM. The electric charge of the trace molecules is typically negative, so that the effects of electroosmosis and electrophoresis are in opposite directions when the surface charge of the microfluidic channel is also negative. It is straightforward to assert that preconcentration efficiency 10367

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depends on two factors: (i) the net flux of the trace molecules under the coaction of electroosmosis and electrophoresis and (ii) the capability of blocking molecules by inducing a strong electric field in the IDZ. A higher applied electric potential results in a higher concentration enrichment factor (CEF). The magnitude of the CEF reached 900 in 30 min at the 5 V condition to which the electric potential was limited. This is due to high electric potentials generating significant vortex flows, leading to inaccurate simulation results; we will discuss vortices again later. Figure 2b shows the maximum values of the CEF in the channel, quantified by using the qualitative results in Figure 2a at steady state (t = 30 min) with a bulk electric field, which was defined as the electric field of the bulk channel with concentration of ions assumed to be uniformly distributed. Basically, the bulk electric field is linearly proportional to the electric potential when the relationship between the current and the electric potential resides in the Ohmic region of the I− V curve.27,28 However, the maximum value of the CEF in the channel is not linearly proportional to the electric fields. This can be explained by the two observations: First, the electric field at the IDZ is not strong enough to cause the trace molecules to accumulate, resulting in a significant leakage of trace molecules over the IDZ. Second, the low electric potentials are unsuitable for preconcentration because the bulk electric field determines the net flux of the trace molecules from the reservoir to the IDZ. For example, the CEFs remain nearly zero when the electric potential is less than 2.5 V. As an additional check on the simulation results, we plotted the distribution of the electric fields at steady state near the membrane along the x-axis (z = 10 μm) in Figure 2c. The values were normalized by the magnitude of the bulk electric field. The amplification of the electric field near the membrane becomes even stronger when a high electric potential is applied. The increased electric field acts as a “dam” to block and then store up the trace molecules. Therefore, the taller the dam, the more the trace molecules are concentrated. In addition, since the average concentration of the solution in the IDZ is not sufficiently low (∼0.5 mM), the additional electric resistance induced by the IDZ has a negligible effect on the current or the bulk electric field. Hence, the bulk electric field in the SC-ICP device can deliver more trace molecules to the “dam” than general DC-ICP devices when the same electric field is applied. This may be seen as an advantage of SC-ICP devices in rapid preconcentration compared to DC-ICP devices (not high); it is noted that DC-ICP is stronger than SC-ICP. Figure 2d shows the normalized concentration distribution of trace molecules at steady state near the membrane along the xaxis (z = 10 μm). Interestingly, the normalized concentrations of the trace molecules above the membrane are raised when the electric potential is reduced, implying that the leakage of the trace molecules is significant when the ICP strength is weak. The leakage effect could be considerably reduced by applying a high electric potential, although leakage is still inevitable. The flux of the trace molecules at the outlet slightly increases with time until it equals the flux of the trace molecules at the inlet. This also implies that the preconcentration in the channel becomes saturated over time. It should be noted that the CEF cannot increase with time indefinitely, because, when the CEF increases to a certain value, the diffusive flux of the trace molecules becomes gradually dominant in the net flux. Consequently, the trace molecules can leak over the “dam”.

Additionally, we found that the positions of preconcentration shown in Figure 2a and 2d agreed qualitatively with our previous experimental results.23 As the electric potential increases, the preconcentration location gets closer to the membrane. This is mainly because the enhanced EOF from a high electric potential shrinks the diffusion region of ions by suppressing the IDZ. We also tested the normalized electric field distribution and normalized concentration distribution of trace molecules along the x-axis at z = 20 μm (top surface) in Figure S1. No obvious differences were observed in this comparison, indicating that the ICP is strong enough to cover the whole channel along the z-direction. This is also consistent with our previous experimental results. When the channel is deep, the ICP cannot cover the entire depth in the z-direction, resulting in serious leakage.23 Effects of the EPM of counterions in the membrane. Figure 3a shows the distribution of ion concentrations from the

Figure 3. Effects of EPMs of counterions in the membrane on the strength of SC-ICP. (a) The distribution of ion concentrations along the center line (z = 10 μm) as shown in the inset when ρf ix = −0.5 mM and the EPM is 0.1, 0.5, 1.0, 2.0, 3.0, and 4.0 times as large as the bulk solution, respectively. (b) The largest CEFs and the bulk electric field strength in cases of (a).

anodic to the cathodic regions along the x-axis as shown in the inset (z = 10 μm), with EPMs of counterions in the membrane at 0.1, 0.5, 1, 2, 3, and 4 times that of the bulk solution. The electric potential and charge density were fixed at 5 V and −0.5 mM, respectively. As a result, the higher the EPM, the lower the resulting ion concentration near the membrane. This also means that a higher EPM generates a stronger ICP. The ion concentration in the cathodic region is nearly 1 mM without obvious ion enrichment. This can be attributed to the fact that the presence of EOF (∇·(uc)) ̅ in the Nernst−Planck equation) tends to eliminate the concentration gradient. This result was not previously known because it was impossible to visualize the concentration distribution in the cathodic region experimen10368

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Figure 4. Effects of the charge density of the membrane on the strength of SC-ICP. (a) The distribution of ion concentrations along the x-axis at z = 10 μm. The EPM of counterions in the membrane is 0.1 times as great as the bulk solution, and the charge densities are −0.5, −2.0, −3.0, and −5.0 mM, respectively. (b) The largest CEFs and the bulk electric field strength in the cases shown in (a). (c) The distribution of ion concentrations along the x-axis at z = 10 μm. The EPM of the counterions is 3 times as large as the bulk solution and the charge densities are −0.1, −0.25, −0.5, −0.75, and −1 mM, respectively. (d) The largest CEFs and the bulk electric field strength in the cases shown in (c). The vortex-affected data show a high uncertainty in the 2-D model.

membrane may depend on the membrane charge density, which lead us to performing an additional simulation. Effect of membrane charge density on SC-ICP. The charge density of the membrane affects not only the concentration difference between co-ions and counterions but also the velocity of electrokinetic flow in the membrane. First, we performed simulations under low EPM conditions (0.1 × ) and a fixed electric potential of 5 V. Figure 4a shows the distribution of ion concentrations at steady state along the xaxis (z = 10 μm) with the membrane charge densities of −0.5, −2.0, −3.0, and −5.0 mM. It is clearly shown that the distribution of ion concentrations depends on the charge density and decreases near the membrane. When the charge density is −5.0 mM, much larger than the estimated value of an available nanoporous membrane, the normalized concentration drops about 9%. As shown in Figure 4b, the corresponding maximum CEFs are nearly 1 and the bulk electric field strength changes little with the increase in charge density, implying that the device is incapable of producing ICP and preconcentration. Notably, this simulation result is inconsistent with experimental results obtained with the SC-ICP device under similar conditions.23,24 It seems possible that an extremely high charge density may generate strong ICP because the concentration of counterions in the membrane is compulsively increased to satisfy electroneutrality, causing the transport flux of counterions and co-ions to be extremely biased through the membrane. However, a membrane with such a large charge density seems to be unavailable. Therefore, we can conclude that using lowEPM counterions in the membrane is inaccurate for numerical simulations. It is noted that our previous simulation results of DC-ICP devices offered the same conclusion of the effect of the

tally. Figure 3b shows the maximum CEFs in the channel at steady state (t = 30 min) and the bulk electric field strengths for different EPMs of counterions in the membrane. A significant preconcentration result cannot be observed when the EPM is less than 2× that of bulk solution, although the electric potential is relatively high. The bulk electric field strength slightly decreases with the increase of the EPM, demonstrating that a greater potential drop occurs near the membrane. The slope of the bulk electric field strength becomes gradually steeper as the EPM increases. Therefore, a high EPM seems to be favorable for inducing strong ICP. According to the Nernst−Einstein equation, the real EPM of ions in the membrane should be proportional to the local diffusivity. However, it was reported that experimental values for the EPM in an ion-permselective structure are much greater than theoretically estimated values.15 This could be attributed to the extremely fast electrokinetic flow inside the nanochannels/nanopores, with the flow direction favoring the transport of counterions but being adverse to that of co-ions. Unfortunately, the electrokinetic flow can only be simulated at the nanoscopic, single nanochannel level, using either a continuum mechanics method (length scale >5 nm) or a molecular dynamics method (length scale