Dynamic analysis of the extended space charge layer using chronopotentiometric measurements

In this paper, we experimentally verified the length (LESC) and the concentration (cESC) of the extended space charge (ESC) layer in front of the electrical double layer (EDL) using the chronopotentiometric measurement and the equivalent circuit model analysis. From the experimentation, the coupled-response of the EDL and the ESC layer was discriminated from the contribution of electro-osmotic flow (EOF). In addition, we derived the potential differences across the ESC (VESC) layer using the circuit model of the ICP layer under rigorous consideration of ESC and EDL. As a result, we obtained that VESC was linearly proportional to the square of the applied current (iapplied). Hence, LESC and cESC were quantitatively provided, where LESC is linear to the iapplied and cESC is constant regardless of iapplied. Thus, this experimentation could not only clarify an essential ICP theory but also guide in ESC-based applications.

Therefore, in this study, we suggested an experiment and a circuit analysis for obtaining the potential across the ESC layer (V ESC ). First of all, chronopotentiometric measurement (dc bias with constant current) was used to discriminate the electrical response of both the EDL and the ESC layer out of EOF. Furthermore, we proposed the equivalent circuit model of an ion depletion zone reflecting EDL and ESC layer, where each resistor and each capacitor are serially connected. From those analyses, we finally obtained the relationship between V ESC and the applied current, which has never been proposed before. Finally, we quantitatively derived the ESC layer information such as the length, the total charge and the concentration. Therefore, this study would be one of essential basis for ICP research not only in fundamental aspect but also various applications based on ICP.

Device fabrication
As shown in Fig. 1a, we fabricated a micro/nanofluidic device consisting of the main microchannel (1 cm length, 100 μm height and 15 μm depth), the buffer microchannel (1 cm length, 100 μm height, and 15 μm) and two side microchannels (40 mm length, 15 μm height and 15 μm depth). For the external hydrodynamic injection, the two side microchannels were tangentially connected to the main microchannel, which is 50 μm apart from the end of the main microchannel. The side microchannels on both sides of the main microchannel was installed for easiness of the experiment [44] and preventing ever-increasing ICP layer [45]. By injecting fresh electrolyte solution through the side microchannels, the diffusion length was reduced as an order of a hundred micron, confining the ICP layer as the triangular shape as shown in Fig. 1b, c. The main building block of device were made of a polydimethyl siloxane (PDMS, Sylgard 184 silicone elastomer kit, Dow corning). We followed the general soft-lithographical fabrication method for PDMS [46]. The Nafion nanoporous membrane was patterned on the glass substrate based on the surface patterning method [45,47]. Simply, Nafion was patterned using a straight microchannel (200 μm width × 50 μm depth) on a glass side, and the PDMS piece of the main

Chemical preparation
Potassium chloride 1 mM solution were used for the experimentation. For tracking the electrokinetic flows and visualizing the ion concentration profile around the ion concentration polarization (ICP) layer, the negatively charged particle (d = 0.2 μm, Invitrogen) and the fluorescent dye (Alexa488, Sigma Aldrich) were mixed in the prepared solution [33,37,48].

Experimental setup
From the two side microchannels, we pumped the prepared solution with the volume rate, 20 nL/min using a syringe pump (PHD2000, Harvard apparatus) for 30 min until the injected flows were stabilized at the main microchannel. Then, we applied the external current source through the reservoir of the main microchannel utilizing the source measure unit (SMU 236, Keithley) while the two reservoirs of the buffer microchannel were grounded. Note that the reservoir of the two side microchannels were electrically floated during ICP. With a customized LabView program, we performed four experimentations as followed: (1) the chronopotentiometric measurement (V-t) from 1 to 30 nA with an 1 nA interval for each 3 min, (2) the chronoamperometric measurement (I-t) from 0.3 to 9.9 V with an 0.3 V interval for each 3 min, (3) the voltage-current (V-I) responses from 0 to 30 nA with a step current 1 nA for every 60 s per step and (4) the current-voltage (I-V) responses from 0 to 9.9 V with a step voltage 0.3 V for every 60 s per step. In order to capture the optical image of an ICP layer, we used a CCD camera (DP73, Olympus) and the image was obtained through the commercial software program (CellSens, Olympus). Figure 2 showed the representing chronopotentiometric measurement of the ICP system, where the red line and the blue line indicated the electrical response at both the overlimiting current regime and the ohmic current regime, respectively. Previous studies neglected the voltage behavior at the ohmic current regime, while they described the voltage behavior at the overlimiting current regime as: (1) The initial voltage value was ohmic voltage which was subject to the electrodialysis system. (2) The sharp voltage hop (1st hop) appeared and the voltage value depended on the type of membrane. (3) A linear voltage growth (2nd hop) regime was followed, where the electroconvection initiated at this time, and then (4) the voltage value was saturated as the microvortices saturated both the size and speed [49,50] However, the aforementioned steps were insufficient to explain the voltage behavior in chronopotentiometry since the ICP layer model was missing. Furthermore, the internal structures inside ion depletion zone has never been suggested as an electrokinetic circuit model. Thus, we would introduce a unified equivalent circuit model including EDL and ESC as well as 2nd EOF in the following section.

Chronopotentiometric measurement
Especially at the OLC regime, the voltage responses during the chronopotentiometric measurement showed the two voltage hops (V 1st and V 2nd ) as shown in Fig. 3a. When the current was applied at t = 0 (sec) from the main microchannel, the V 1st was followed due to the capacitance of both the ESC and the EDL, which the corresponding image and the circuit was shown in image i) in Fig. 3a and inset of in Fig. 3b, respectively. When the EOF was generated at t = 15 s, the voltage was increasing until the EOF size saturated at t > 50 s with the value V 2nd ~ i applied as shown in image ii) in Fig. 3a [50]. In this experimentation, we applied the various current values from 12 nA to 30 nA so that we can obtain the V 1st -i applied relations as shown in Fig. 3b. Note that the V 1st is not linear to the i applied , indicating that the ohm's law is not valid due to the appearance of the ESC layer as expected by Rubinstein and Zaltzman [40].

Equivalent electrokinetic circuit model of the ICP layer
At the charged membrane surface, the EDL was composed of both resistor (R EDL ) and capacitor (C EDL ) in parallel and they were connected in series to the diffuse layer resistor (R bulk ) as in Fig. 4a. This simple circuit coincided with the voltage-time behavior in the ohmic regime, which showed the gentle slope and the slight voltage hop as in Fig. 2. Once the current was applied exceeding limiting current, the ESC layer grew between the EDL and the diffuse layer, where both resistor (R ESC ) and capacitor (C ESC ) should be additionally employed as in Fig. 4b. This electrical circuit model affected the total RC delay time, converting the gentle slope at ohmic current regime into the sharp one at overlimiting current regime. Normally, time-varying voltage responses existed where the resistance and the capacitance are parallel in the circuit model. Considering that bulk solution was regarded to the quasi-neutral regions, one can ignore the capacitance. In the meantime, the sufficient charge carriers existed inside the electrical double layer (EDL) for compensating the charged surface (e.g. Nafion), thus one should consider the capacitance of the EDL as well as the resistance of one. This means that the voltage responses should be divided into the constant term (for diffusion layer) and the time-varying one (for EDL) as follows: where V 0 is the potential of diffuse layer, V EDL the potential of EDL and the τ EDL is the RC delay time (τ EDL = R EDL C EDL in the circuit model). As shown in Additional file 1: Figure S4, the collapsed data of the V EDL has the linear relations to the applied current density, which lead to the constant resistance values (R EDL -V EDL /I) as 3 MΩ. Each component has the value 240 ± 42 MΩ (for R bulk ), 3 ± 0.7 MΩ (for R EDL ), 6 ± 1.2 μF (for C EDL ), 1.09(I-I lim ) MΩ (for R ESC ) and 2.23(I-I lim ) −1 μF (for C ESC ), respectively. The simple calculation result and the derivations was introduced in supporting materials (Additional file 1: Table S1, Figure S4).
Valenca and co-workers reported that the microvortices by ICP induced the potential difference at V 2nd in the EC dominant regime [50]. This indicated that, in a certain overlimiting current value I OLC > I lim , one can estimate the point conductance at I OLC with a simple calculation as σ OLC = I/V 2nd . We also confirmed the conductivities in EOF regime, where the applied current is ranging from 12 nA to 29 nA, leading to OLC by EOF as the constant value of 0.21 nS in our system. Note that the experimental results and the set of data were provided in supporting materials (Additional file 1: Figure S2). In addition, critical time (T c ) that initiate the EOF has the relation of the OLC conductance(σ OLC ) and its time-derivative one (∂σ OLC /∂t). This means that T c is also subject to the V 2nd and its time-derivative one (∂V 2nd /∂t). The scaling was developed and quantified in supporting materials (Additional file 1: Figure S3).

Conclusions
Recent experiments have been conducted for probing the space charge at the micro-and nano-channel interface device using electrical impedance spectroscopy (EIS), Fig. 4 Schematics of equivalent circuit of ICP layer at a ohmic current regime and b overlimiting current regime. c A unified equivalent electrokinetic circuit model of ICP layer considering EDL, ESC and 2nd EOF employing a conventional equivalent circuit model. However, those literatures revealed out that the EIS method hardly determined the ESC layer response since the multiple electrokinetic responses were tightly coupled during ICP. For example, Yossifon and co-workers probed the diffusion layer(DL) and the electrical double layer(EDL) using EIS at the micro-and nano-channel systems [43]. They found out the detailed components of the EDL by separating the electrode-fluidic interface and microchannel-nanochannel interface. From this experiment, they clearly captured the resistances and the capacitances at both EDL for satisfying the theoretical calculations. However, this demonstration fails to present ESC layer responses at the higher voltage because of the coupling effect where electroconvective flows were involved, thereby arousing another issue for differentiating them, individually. Thus, we emphasized that this equivalent circuit model, for the first time, reflected EDL and ESC layer as well as the convective flows using the micro-/ nano-fluidic systems.
In this paper, we experimentally investigated the ESC layer using chronopotentiometric measurement and the unified equivalent electrokinetic circuit model of internal ICP structure with the consideration of EDL, ESC and 2nd EOF. Each electrical component such as two resistors, two capacitors and dependent current source were included in the new model, confirming the voltage responses in chronopotentiometric measurement. From our rigorous experimentation, we obtained the relationship between the potential across the ESC layer and the applied current, V ESC -i applied 2 . Furthermore, we quantitatively provided the L ESC -j applied and the c ESC ~ constant. Therefore, all this experimental verification of the ESC layer could lead to the further development of ICP theory as well as the ESC/ICP layer related applications.
Additional file 1: Figure S1. In order to obtain the limiting current values, we conducted the voltage-sweeping method in our systems. Under the 20nL/min flows was applied near the Nafion membrane, the limiting current value reaches 12 (nA). Figure S2. The V 2nd from the measurement has been obtained with the applied current, I. This result showed that the slope of V 2nd -I, which is the overlimiting conductance (OLC) by electroosmotic flows (EOF) have the constant values as 0.21 nS. Figure S3. The onset time (τ C ) of electro-convective flows was obtained from the chronopotentiometric measurement. The τ C values are between 10 and 30, which result is coincided our scaling theory, τ C -O (10 1 ). Figure S4. The V 1st , which was time-varying potential reflected by the electrical double layer, was obtained from the chronopotentiometric measurement. From this result, the resistance can be calculated by Ohm's law (R EDL = V EDL /I). (b) The RC delay time caused by the electrical double layer was collected in ohmic current regime. The RC delay times in our experiments were almost constant as the value of 18 s regardless of the applied current. From this result, the capacitance can be calculated by (C EDL = t EDL /R EDL ). Table S1. The electrical components of the equivalent circuit model were calculated by simple calculation. Note that R EDL and C EDL remains same regardless of the applied current (I), while R ESC and C ESC are linearly proportional to the current values (I-I lim ), where I lim is the limiting current values.