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Supplementary Information: Multiphysics Simulation of Ion Concentration Polarization Induced By Nanoporous Membranes in Dual Channel Devices Mingjie Jia1 and Taesung Kim1,2* *CORRESPONDENCE: Taesung Kim ([email protected]) 1 Department of Mechanical Engineering, Ulsan National Institute of Science and Technology (UNIST), 50 UNIST-gil, Eonyang-eup, Ulsan 689-798, Republic of Korea. 2 Department of Biomedical Engineering, Ulsan National Institute of Science and Technology (UNIST), 50 UNIST-gil, Eonyang-eup, Ulsan 689-798, Republic of Korea. Contents Supplementary Results and Discussion Governing Equations Modeling Numerical Method Ion Concentration Distributions Boundary Conditions Simulation Result of Flow Fields Simulation Result of Electric Fields Effect of Different EPMs and Charge Densities on ICP Supplementary Table Supplementary Figures References S-1 Supplementary Results and Discussion Governing Equations Ionic concentration of species i (ci), fluid velocity ( u ), and electric potential (ϕ) are the main variables in the governing equations, respectively. The generation and development of ICP phenomena depend on the complex effect of those variables on each other. First, the Nernst-Planck equation is in charge of the concentration distribution of different species of ions and trace molecules. The equation is shown as below: ci F ci u D i ci zi ci Di t RT (1) where ci, Di, and zi are the concentration of an ion species i, its corresponding diffusion coefficient, and ionic valence in an electrolyte solution, respectively. T represents the temperature of 25°C and F and R are the Faraday and ideal gas constant, respectively. Second, the Navier-Stokes equation describes the motion of incompressible fluid flow with the continuity equation: u (u )u p 2 u E t u 0 (2) (3) where is the density of the solution, is the dynamic viscosity, and represents the net charge density in the solution. E F i zi ci (4) Third, the Poisson equation relates the ionic concentration of a buffer solution to the electric potential: 2 E (5) where is the permittivity of the medium. Since the governing equations are coupled with each other, we performed numerical multiphysics simulations to find out CEFs, ionic concentration distributions, ionic currents, electric fields, flow fields including vortex flows, and so on. S-2 Modeling Each channel of the model was 2000 μm long and 100 μm wide so that the configuration of an electric potential (VH = 4 V and VL = 2 V) produced an electric field of 103 V/cm in the anodic channel. Since all the channels were connected with reservoirs, the concentration of ionic species at the ends of the channels was ci = c0, which was the concentration of a buffer solution. The concentration of trace molecules at the high potential end was ctr = 1 nM while ctr = 0 at the other ends. The trace molecules were not permitted to pass through the nanoporous membrane due to size exclusion. The initial concentration of the buffer solution in the channel and the membrane was c(t = 0) = c0 while ctr(t = 0) = 0. The material properties used in the simulations are shown in Table S1 and were obtained from many literatures. H+ and Cl- were used as cation and anion, respectively, without considering the influence of H+ on deprotonation/protonation inside the nanoporous membrane. The diffusivity of ions in the membrane depends on the molecular dimension, pore dimensions, porosity, etc. The value used in the simulations is based on the estimation from other literatures. For example, Shen et al. used one tenth of the bulk value for both cations and anions because the nanoporous membrane used in their work is Nafion membrane whose pore size is around 3~5 nm, leading to a low diffusivity of ions in it[1]. On the other hand, Dhopeshwarkar et al. used one half of the bulk value because the pore size of the membrane is about 100 nm[2]. The values mentioned above were all estimated mainly based on the properties of the relevant nanoporous membrane, so it appears that one tenth of the value used in our work would be reasonable for a small pore-size membrane. As for other membranes, we can adjust the value case by case, but it would not be easy to estimate an exact value without accurate experimental data. Usually, it is believed that the diffusion of ions is not related to their charges and the surface charge density in the membrane, but it indirectly influences its electrophoretic mobility according to the Nernst-Einstein equation (µ i ~ Di/RT). In the Nernst-Planck equation, we can see that the charge of ions influences the electrophoresis in the bulk solution or in the membrane, while the influence of the surface charge density of the membrane on the EPM of the ions in the membrane seems not to be formulated to our best knowledge. The factor of 4 was obtained by measuring current and then calculating conductance to get an equivalent value for the electrophoretic mobility of proton in a 2-nm wide nanochannel. It is certain that the factor varies, depending on the kind of ions and the properties of the membrane, such as surface charge density, pore size, etc. It could be larger or S-3 smaller according to different conditions. That is another reason why we used different EPMs in the membrane for simulations. Numerical Method COMSOL Multiphysics (ver 4.3b), which is based on a finite element method, was utilized to solve the governing equations. Correspondingly, the microfluidics module, chemical reaction engineering module, and AC/DC module were utilized. High and low electric potentials were applied at the right (VH) and the left (VL) end of the anodic channel, respectively, whereas the electrodes at the cathodic channel were connected to an electric ground (VG). Both the anodic ends were set up as an open boundary without external pressure. As a result, the electroosmotic slip boundary condition significantly reduced the computational load for calculating EOF. That is, the flow velocity (U) at the channel walls was approximated by the Helmholtz-Smoluchowski equation as shown below: U Et (6) where 𝐸𝑡 is the tangential electric field and is the zeta-potential at the charged wall surface. Since the velocity profile of the EOF is fairly uniform across the channel and the electric double layer (EDL) is as thin as 10 nm, the error caused by the electroosmotic slip boundary condition must be negligible. The boundary conditions at the channel walls near the membrane were changed to nonslip ones by repeatedly validating the flow field, because the velocity profile of the EOF would be locally broken due to the existence of extremely fast vortex flows. The reason will be described later. Ion Concentration Distributions Once an electric potential is applied across a nanoporous membrane, ICP starts to develop and an IDZ is generated near the membrane in the anodic channel. The extension of the IDZ is shown in Fig. S1. Initially, the ion concentration in the entire channel is 1 mM. After the application of an electric potential (VH = 4 V and VL = 2 V), a small IDZ (dark blue area) is induced near the anodic interface in a second. As time goes on, the IDZ extends along the channel toward the left end. As reported by Rubinstein and Zaltzman[7], the distribution of ion concentrations S-4 depends on diffusion and convection while the influence of electrophoresis is negligible. Since the EOF flows from the right end (VH) to the left end (VL), the extension of the IDZ is inhibited toward the right end but accelerated toward the left end so that the IDZ continuously propagates only toward the left end. The ion concentrations at both the ends of the anodic channel are assumed to be kept at 1 mM because of the boundary conditions; for experiments huge reservoirs are used so that the assumption is very common in numerical simulations. When the IDZ extends, the ICP relatively reaches a steady state, although the ion concentrations in the cathodic channel continuously increase. The sharp decrease of the ion concentrations in the anodic channel significantly reduces the conductance of the device, while the mild increase of the ion concentrations in the cathodic channel continuously increases the conductance. Therefore, the ionic current shown in Fig. 6c and f in the main text sharply decreases initially and then increases gradually. The pre-concentrated area of the trace molecules is also related to the IDZ. The sharp change of the electric field near the IDZ generates an electrical barrier that prevents the trace molecules from moving forward but helps them to be further accumulated. Boundary Conditions We used a combined boundary condition that divides the anodic channel walls into two regions: near the membrane and far from the membrane (the rest parts) as shown in Fig. 1 in the main text. It is straightforward that the electroosmotic slip boundary condition for the Navier-Stokes equation is applied to the channel walls far from the membrane because it can induce a very similar bulk flow as an EOF across the channel and the error is negligible (EDL ~ 10 nm). However, the electroosmotic slip boundary condition for the entire channel walls can possibly interfere with the generation of vortex flows and its additional flows (e.g. slow-flow zone); according to experimental results, slow-flow zones exist. Therefore, the electroosmotic slip boundary condition for the entire channel walls potentially holds a critical weakness and in turn it produces less accurate simulation results especially for pre-concentration than the combined boundary condition. Fig. S2 shows the flow field when a slip boundary condition (constant velocity calculated based on the average electric field across the channel) was used near the membrane instead of the nonslip one; here, the elelctroosmotic slip boundary condition is not suitable because the extremely strong, local electric field strength may cause an unrealistic velocity of EOF. Compared with the flow field shown in Fig. 4a, the streamline adjacent to the wall boundary in the rectangle in Fig. S-5 S2 is fairly straight, indicating that the local flow field is dominated by the slip boundary condition (constant velocity) and the slow-flow zone is supposed to exist. Since the continuity of EOF near the vortex is probably broken, it seems that the non-slip boundary condition is capable of better simulating the mobility balance between electroosmosis and electrophoresis, which is very delicate for the pre-concentration of the trace molecules near the IDZ, than the electroosmotic slip boundary condition. In addition, almost no variation in a flow field was found when a slip boundary condition was applied to the channel walls near the membrane instead of the non-slip one, meaning that the inertia effect of the EOF is much weaker than that of the vortex flows. Since the slip boundary condition typically does not generate hydraulic resistances at walls, it seems to be unrealistic for experiments. For this reason, we applied the non-slip boundary condition to the channel walls near the membrane. In real experimental conditions, it appears to be very complicate and difficult to observe the transition from an EOF to vortex flows. Correspondingly, from the view point of simulations, it seems to be intricate to exactly deal with the local boundary condition of experiments. The existence of vortex flows could possibly interfere with the local generation of an EOF so that the electroosmotic slip boundary cannot be valid near the membrane. Thus, the application of the electroosmotic slip boundary condition to the entire channel walls can not only be inappropriate but also eliminate slow-flow zones, leading to less accurate results. Consequently, we validated that the combined boundary condition better compromises with our experimental results from the view point of the formation of the slow-flow zones and vortex flows. In addition, the border of the electroosmotic slip and the non-slip boundary conditions should locate as close as possible to the slow flow zone to minimize the influence of the non-slip boundary condition on the hydraulic resistance in the channel. Simulation Result of Flow Fields Basically, the velocity profile of an EOF across a microchannel is uniform, showing a flat profile when the EDL is much shorter than the channel dimension. But, in the simulations, we found that the velocity profile of an EOF can be flat, concave, and even convex when vortex flows exist. To explore on this, we simulated the same model when no ion-permselective membrane was employed (open junction) but other numerical parameters and boundary conditions were the same. The velocity profile of the EOF along A–A’ at a steady state is shown in Fig. S3 (blue, open square). In addition, the velocity profile of the EOF in a single and straight channel is shown as a reference S-6 value (red, open circle). The velocity profile of the EOF shows a concave shape because the electric field in the left half of the anodic channel is locally applied in the opposite direction (from VL to the junction), which generates an additional EOF against the bulk flow. Without the electric field which is in the opposite direction, the velocity profile would be flat as the reference value. The concave profile of the EOF in Fig. 4b and the negative velocity of the flow at the walls in Fig. 4c at t = 0.1 s are induced by the same reason. Interestingly, the velocity profile for both the cases cannot develop to a convex shape in the absence of vortex flows. Simulation Result of Electric Fields Fig. S4a shows the distribution of the electric potential in the anodic channel at an initial state and a steady state, respectively. The normalized arrows indicate the direction of electric fields. As time goes on, the electric potential near the IDZ increases from the initial value of 1.7 V to the steady state value of 2.6 V. This means that the direction of the electric field in the left half of the anodic channel changes during the development of ICP and the strength of the electric field in the right half at the steady state is weaker than the initial value, which could be proved by the velocity profiles shown in Fig. 4b and c. Due to the electroosmotic slip boundary condition, the velocity of flows at the channel walls is proportional to the local strength of an electric field. For this reason, the velocity of flows at the channel walls at the steady state shown in Fig. 4b is smaller than the initial value (t = 0.1 s), confirming that the strength of the electric field at the steady state is relatively weaker than that at the initial state. Since the velocity near the walls shown in Fig. 4b decreases at first and then increases, it could be inferred that the strength of the local electric field initially decreases and then increases gradually. As mentioned in the main text, the initial sharp decrease of ion concentrations in the IDZ and afterward the gradual increase of ion concentrations in the IEZ explains the tendency of the change of electric fields and currents along the channels. Fig. S4b shows the distribution of the electric potential along a–a’ and b–b’ at the steady state, respectively. From this result, we can find two interesting facts. First, from the result along b–b’, the potential drop in the left half is slower than that in the right half, meaning that the strength of the electric field is rather weaker. This can explain why the velocity of the EOF near the walls is relatively slower in the former than that in the latter as shown in Fig. 4b and c. Second, from the result along a–a’, the sharpest potential drop happens near the anodic interface. Therefore, the local strength of the electric field is extremely high as shown in Fig. 3. The potential drop in the cathodic S-7 channel is much smaller than that in the anodic channel and the local strength of the electric field is around 102 V/m. Effect of Different EPMs and Charge Densities on ICP Fig. S5a shows the velocity profiles of flows along A–A’ for the different EPMs of counterions in the membrane at a steady state. For the 0.1x EPM, the velocity profile is concave because vortex flows are not strong enough to increase the EOF in the channel. That is, the electroosmotic mobility (eo) is weaker than the EPM (i) of the trace molecules. Thus, the trace molecules cannot be transported by the EOF, so that pre-concentration does not happen (CEF is nearly zero in Fig. 6a). With the increase of the EPM, the flow rate across the channel increases and causes eo to be much stronger. Consequently, more trace molecules are delivered in the channel so that a higher CEF is obtained. The distribution of the electric field across the membrane for the different EPMs at a steady state is shown in Fig. S5b. As mentioned in the main text, the application of the 0.1x EPM is inappropriate to simulate experiments. Conversely, when a high EPM (µ i > 0.5x) of counter-ions in the membrane is assumed, the distributions of the electric field for the different EPMs are close to each other at the anodic interface. However, the electric field varies significantly in the membrane. This is because the local conductance is remarkably influenced by the EPMs. Similarly, the sharp change of the electric field at the cathodic interface is induced by the sudden change of the EPMs. The velocity profile along A–A’ and the distribution of the electric field across the membrane for the different charge densities of the membrane are shown in Fig. S5c and d, respectively. We can hardly tell the difference of these results except some minor difference of the electric field in the membrane. In other words, the influence of the charge density on ICP is relatively trivial compared with that of the EPM. Since the velocity profile and the distribution of the electric field in the channel are similar, the electroosmosis and electrophoresis effect on the trace molecules are also similar. As a result, the variation of the CEFs shown in Fig. 6d is insignificant. Lastly, as reported by Yeh et al., the surface charge density in nanopores is related to local ion concentrations[8], thus the non-uniform distribution of the ion concentrations leads to the non-uniform distribution of the charge density, making the mechanism of ion transport inside the membrane more complicated. We also compared the influence of the charge density of the membrane for a relatively low EPM (µ i = 0.5x). As shown in Fig S5, the variations of CEFs, ion concentration distributions, currents S-8 and the velocity profiles of EOF induced by the change of the charge density of the membrane are relatively insignificant compared with those induced by the change of the EPM of counter-ions in the membrane. Since the influence of the charge density turned out to be trivial even for a relatively low EPM condition, we emphasize that the ion-permselectivity of a nanochannel/nanoporous membrane should be enhanced by a high EPM of counter-ions in it, which generates much more reliable simulation results than the previous simulation results found in other literature. S-9 Supplementary Table Table S1 Properties used in simulations unless otherwise noted[3-6]. Diffusion coefficient of H+ Diffusion coefficient of Cl- 9.36 × 10-9 m2/s 2.032 × 10-9 m2/s Diffusion coefficient of trace molecule 0.45 × 10-9 m2/s Valence of trace molecule -2 Diffusion coefficient of H+ in membrane 9.36 × 10-10 m2/s Diffusion coefficient of Cl- in membrane 2 × 10-10 m2/s Ion concentration in electrolyte solution, c0 1 mM Assumed charge density of the membrane, 𝜌𝑓𝑖𝑥 -0.5 mM -60 mV Zeta-potential at the channel walls, S-10 Supplementary Figures VL = 2 t=1s VH = 4 V t=5s t = 10 s t = 50 s Fig. S1 The distribution of ion concentrations in the anodic channel when t = 1, 5, 10 and 50 s. S-11 Fig. S2 The distribution of flow field when constant-velocity slip boundary condition is applied near the vortex zone. S-12 60 No membrane Velocity (m/s) 50 Straight channel 40 30 20 10 0 0 20 40 60 80 100 Distance across channel (m) Fig. S3 The velocity profile of an EOF in the anodic channel for a DC-ICP device and that of an EOF in a single and straight channel in a SC-ICP device at a steady state when no ionpermselective membrane is employed. The average strength of an electric field is the same for both cases. S-13 a VL = 2 Initial state VH = 4 V VL = 2 Steady state 2.6 V 1.7 V b’ b a Electric potential (V) b VH = 4 V 4 3 Left half Anodic part 2 1 0 a’ a-a' b-b' Right half Near interface 0 1000 2000 Cathodic part 3000 4000 Distance along the line (m) Fig. S4 (a) The distribution of an electric potential in the anodic channel at an initial (t = 0.1 s) and a steady state, respectively. The normalized arrows indicate the direction of the electric field. (b) The distribution of an electric potential along a–a’ and b–b’ at the steady state. S-14 b 40 30 0.1x 0.5x 1.0x 2.0x 4.0x 20 10 0 Velocity (m/s) c Electric Field (V/m) 50 0 20 40 60 80 Distance along A-A' (m) 10 4 10 3 10 2 10 50 40 30 0.10 mM 0.25 mM 0.50 mM 0.75 mM 1.00 mM 10 0 20 40 60 80 Distance along A-A' (m) Nanoporous membrane 10 0 100 50 150 6 10 -0.10 mM -0.25 mM -0.50 mM -0.75 mM -1.00 mM 5 10 4 10 3 10 2 10 Nanoporous membrane 1 100 100 Distance along the y-axis (m) d 20 0.1x 0.5x 1.0x 2.0x 4.0x 5 1 60 0 10 Electric Field (V/m) Velocity (m/s) a 6 60 10 0 50 100 150 Distance along the y-axis (m) Fig. S5 Flow fields and electric fields for the different EPMs of counter-ions and the charge densities in the membrane at a steady state. (a) The velocity profiles along A–A’ at a steady state when the EPMs are 0.1, 0.5, 1.0, 2.0 and 4.0 folds of the bulk solution. (b) The electric fields across the membrane from the anodic to the cathodic channel along the y-axis under the same conditions as (a). (c) The velocity profiles along A–A’ at a steady state when the charge densities are -0.1, 0.25, -0.50, -0.75 and -1.0 mM for the 4x EPM. (d) The electric fields across the membrane along the y-axis under the same conditions as (c). S-15 a b Ion concentration (mM) 200 CEF 150 100 50 0 Current (mA) c -0.10 mM -0.50 mM -0.75 mM -1.00 mM 5 d 3 0 500 1000 1500 Time (s) mM mM mM mM 2 1 Nanoporous membrane 0 50 100 Distance along the y-axis (m) 150 60 50 4 2 3 0 Velocity (m/s) 0.0 0.2 0.4 0.6 0.8 1.0 Charge density of membrane (mM) 6 -0.10 -0.50 -0.75 -1.00 4 40 30 20 10 0 2000 -0.10 mM -0.50 mM -0.75 mM -1.00 mM 0 20 40 60 80 Distance along A-A' (m) 100 Fig. S6 Simulation results for a relatively low EPM (0.5x) when the charge densities are -0.1, -0.5, -0.75 and -1.0 mM. (a) The highest CEF at t = 30 min. (b) Ion concentration distributions along the y-axis at t = 10 s. (c) The transient ionic currents across the membrane. (d) The velocity profiles of the EOF along A-A’ at a steady state. 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