# Investigation on the Stability of Random Vortices in an Ion Concentration Polarization Layer with Imposed Normal Fluid Flow

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## Abstract

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## 1. Introduction

^{−2})–O(10) of Peclet number (this value represents characteristic magnitude of normal flow), we set stronger normal flow for theoretical investigation of nonlinear interactions between the normal flow and electroconvection.

^{−2}) to O(1) in the Peclet number (Pe) in general [30]. Recently, ultrafast molecule sieving nanosheet membranes was fabricated with Pe~O(10) [31]. Therefore, water permeability through an ion-selective membrane is a non-negligible factor for the consideration of membrane process.

## 2. Numerical Formulation

#### 2.1. Domain Description

#### 2.2. Governing Equations and Boundary Conditions

_{D}is the dimensionless Debye length, c

^{+}was cation concentration, c

^{−}was anion concentration, t was the time,

**u**was velocity of fluid, Sc was the Schmidt number, p was pressure, and κ was the electrohydrodynamic coupling constant defined as

^{−9}m

^{2}/s for K

^{+}and 2.03 × 10

^{−9}m

^{2}/s for Cl

^{−}[37] so that the diffusivities of two ionic species were treated as 2.00 × 10

^{−9}m

^{2}/s. The κ was typically O(1) so that we set κ = 0.5, which was the same value in Druzgalski’s work [36]. In order to consider the effect of finite thickness of EDL and SCL, λ

_{D}was fixed to be 0.001. This case was corresponded with O(100 μm) gap filling with O(10 μM) of electrolyte.

_{reservoir}= 25 was obtained from t = 0 to t = 2 and other cases of reservoir potential were calculated from t = 0 to t = 0.2 [38]. The above governing equations and boundary conditions were solved by the commercial software, COMSOL 4.4.

_{mem}is the cross-sectional area of the membrane, d

_{mem}is the membrane thickness, and water permeability is normalized by ${{\tilde{L}}_{y}}^{4}$. Note that the Δp represents the pressure difference across the membrane. Thus, the Peclet number (or flow rate) used in this work can be directly related to membrane properties, such as A

_{mem}, d

_{mem}, and k

_{w}.

## 3. Results and Discussions

#### 3.1. The Effect of Space Charge Layer on Electro-Convection with Admitted Normal Flow

^{+}–c

^{−}) in Figure 2a,b, the SCL was extremely close to the ion-selective surface so that it could be assumed to be near zero thickness, inferring much smaller thickness of EDL even we considered it as finite length. This explained the reason why we obtained the comparable consequences with Khair’s ones which employed second-kind electro-osmotic slip velocity [8]. When the direction of imposed flow was inversed (i.e., Pe = 12.5) but still in low voltage (ϕ = 25), the SCL appears as spaciously as the half of the domain as shown in Figure 2c. Accordingly, electro-convective vortices grew and expanded in the middle of the domain. Such deformation and thickness of SCL is non-negligible so that we received a hint that a finite SCL plays a crucial role in electro-convective instability.

_{y}). Note that the time steps in Figure 2a,b vs. Figure 2c were different due to a growth rate of instability which will be discussed in next section.

#### 3.2. Growth Rate of Electro-Convective Instability

**u**) was decomposed as

**u**=

**U**

_{app}+

**u**

_{EC},

**U**is the imposed normal flow, and

_{app}**u**is the flow velocity solely by the electro-convective instability. Note that the space- and temporal-average of ${u}_{\mathrm{EC}}$ in the entire domain was equal to be zero as similar as turbulence analysis [39]. Thus, we defined the domain-averaging root-mean-square velocity (U

_{EC}_{rms}) of the electro-convective instability as

**U**described the normal flow away from the ion-selective surface (i.e., upward normal fluid flow), while negative

_{app}**U**does the normal flow into the ion-selective surface (i.e., downward normal fluid flow). Utilizing above decomposition and integration, the intensity of the chaotic fluid flow can be extracted apart from the imposed normal flow. The dynamics of electro-convective instability was investigated by the observation on the growth rate determined as the exponent of fitting curve of U

_{app}_{rms}as a function of time.

#### 3.3. Rearrangement of Space Charge Layer and Mixing Layer Depending on Pe

^{−2}which was extremely close to the ion-selective surface. The SCL, where the averaged concentration of cation increased, was ~8.33 × 10

^{−2}thick. The plateau of ion concentration beyond SCL was defined as a ML in which the vortices removed the gradient of ion concentration by strong mixing. The complete picture including all of the layers also confirmed the validity of our numerical scheme which considered the finite thickness of EDL and SCL, as previous literature described [36,40].

#### 3.4. Asymmetric Transition of Electro-Convective Instability

^{1.15}. In the case of negative Pe, the marginal curve followed ~ |Pe|

^{0.45}. Larger absolute value of Pe was demanded to achieve a transition from unstable to stable state in the case of negative Pe than in the case of positive Pe. Figure 5b showed the distribution of ion concentration in the domain along increasing Pe from negative to positive value at V = 50. When normal flow was imposed in the direction from the reservoir into the ion-selective membrane (Pe < 0), the vortices shrank but instability enhanced until the downward normal flow removed mixing layer. The replenished ions provided by the downward normal flow increased overall concentration in the domain, yielding more chances to produce lateral perturbation near ion-selective surface in salt concentration. This helped the instability grow before the disappearance of mixing layer and vortices. This would contribute to the delay of the transition value of negative Pe.

## 4. Conclusions

_{rms}as the magnitude of downward flow increased. The analysis of energy spectrum by Fast Fourier transform revealed the reason that the velocity of vortices increased, while the size of them became small due to the decrease of the size of SCL. On the other hand, the imposed fluid flow from the ion-selective membrane to the reservoir (upward flow) spread out ions in SCL. Then, reduced charge density slowed down the appearance of vortices. As the magnitude of the upward flow became further larger, ions were washed out on a domain into the bulk side, resulting in the suppression of the instability. A marginal stability curve was numerically obtained. It was found that the transition from stable to unstable state was asymmetric depending on the direction of imposed normal flow. In the case of positive Pe, the marginal curve followed ~ |Pe|

^{1.15}. In the case of negative Pe, the marginal curve followed ~ |Pe|

^{0.45}. Those two distinct mechanisms yielded asymmetric transition of chaotic electro-convection near water-permeable ion-selective surface.

## Supplementary Materials

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Zabolotsky, V.; Nikonenko, V.; Pismenskaya, N.; Laktionov, E.; Urtenov, M.K.; Strathmann, H.; Wessling, M.; Koops, G. Coupled transport phenomena in over-limiting current electrodialysis. Sep. Purif. Technol.
**1998**, 14, 255–267. [Google Scholar] [CrossRef] - Choi, J.; Huh, K.; Moon, D.J.; Lee, H.; Son, S.Y.; Kim, K.; Kim, H.C.; Chae, J.-H.; Sung, G.Y.; Kim, H.-Y.; et al. Selective preconcentration and online collection of charged molecules using ion concentration polarization. RSC Adv.
**2015**, 5, 66178–66184. [Google Scholar] [CrossRef] - Choi, J.; Baek, S.; Kim, H.C.; Chae, J.-H.; Koh, Y.; Seo, S.W.; Lee, H.; Kim, S.J. Nanoelectrokinetic selective preconcentration based on ion concentration polarization. BioChip J.
**2020**, 14, 100–109. [Google Scholar] [CrossRef] [Green Version] - Baek, S.; Choi, J.; Son, S.Y.; Kim, J.; Hong, S.; Kim, H.C.; Chae, J.-H.; Lee, H.; Kim, S.J. Dynamics of driftless preconcentration using ion concentration polarization leveraged by convection and diffusion. Lab Chip
**2019**, 19, 3190–3199. [Google Scholar] [CrossRef] [PubMed] - Lee, H.; Choi, J.; Jeong, E.; Baek, S.; Kim, H.C.; Chae, J.-H.; Koh, Y.; Seo, S.W.; Kim, J.-S.; Kim, S.J. Dcas9-mediated nanoelectrokinetic direct detection of target gene for liquid biopsy. Nano Lett.
**2018**, 18, 7642–7650. [Google Scholar] [CrossRef] [PubMed] - Son, S.Y.; Lee, S.; Lee, H.; Kim, S.J. Engineered nanofluidic preconcentration devices by ion concentration polarization. BioChip J.
**2016**, 10, 251–261. [Google Scholar] [CrossRef] - Kim, S.J.; Song, Y.-A.; Han, J. Nanofluidic concentration devices for biomolecules utilizing ion concentration polarization: Theory, fabrication, and application. Chem. Soc. Rev.
**2010**, 39, 912–922. [Google Scholar] [CrossRef] [Green Version] - Rubinstein, I.; Zaltzman, B. Electro-osmotic slip of the second kind and instability in concentration polarization at electrodialysis membranes. Math. Models Methods Appl. Sci.
**2001**, 11, 263–300. [Google Scholar] [CrossRef] - Yossifon, G.; Chang, H.-C. Selection of nonequilibrium over-limiting currents: Universal depletion layer formation dynamics and vortex instability. Phys. Rev. Lett.
**2008**, 101, 254501. [Google Scholar] [CrossRef] [Green Version] - De Valença, J.C.; Wagterveld, R.M.; Lammertink, R.G.; Tsai, P.A. Dynamics of microvortices induced by ion concentration polarization. Phys. Rev. E
**2015**, 92, 031003. [Google Scholar] [CrossRef] [Green Version] - Li, L.; Kim, D. Effect of poly-dispersed nanostructures on concentration polarization phenomena in ion exchange membranes. J. Membr. Sci.
**2016**, 520, 639–645. [Google Scholar] [CrossRef] - Kim, S.J.; Ko, S.H.; Kang, K.H.; Han, J. Direct seawater desalination by ion concentration polarization. Nat. Nanotechnol.
**2010**, 5, 297. [Google Scholar] [CrossRef] [PubMed] - Benneker, A.M.; Gumuscu, B.; Derckx, E.G.; Lammertink, R.G.; Eijkel, J.C.; Wood, J.A. Enhanced ion transport using geometrically structured charge selective interfaces. Lab Chip
**2018**, 18, 1652–1660. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gumuscu, B.; Haase, A.S.; Benneker, A.M.; Hempenius, M.A.; van den Berg, A.; Lammertink, R.G.; Eijkel, J.C. Desalination by electrodialysis using a stack of patterned ion-selective hydrogels on a microfluidic device. Adv. Funct. Mater.
**2016**, 26, 8685–8693. [Google Scholar] [CrossRef] - Cheow, L.F.; Ko, S.H.; Kim, S.J.; Kang, K.H.; Han, J. Increasing the sensitivity of enzyme-linked immunosorbent assay using multiplexed electrokinetic concentrator. Anal. Chem.
**2010**, 82, 3383–3388. [Google Scholar] [CrossRef] [Green Version] - Cheow, L.F.; Han, J. Continuous signal enhancement for sensitive aptamer affinity probe electrophoresis assay using electrokinetic concentration. Anal. Chem.
**2011**, 83, 7086–7093. [Google Scholar] [CrossRef] [Green Version] - Cheow, L.F.; Sarkar, A.; Kolitz, S.; Lauffenburger, D.; Han, J. Detecting kinase activities from single cell lysate using concentration-enhanced mobility shift assay. Anal. Chem.
**2014**, 86, 7455–7462. [Google Scholar] [CrossRef] [Green Version] - Kim, K.; Kim, W.; Lee, H.; Kim, S.J. Stabilization of ion concentration polarization layer using micro fin structure for high-throughput applications. Nanoscale
**2017**, 9, 3466–3475. [Google Scholar] [CrossRef] - Wang, Y.-C.; Stevens, A.L.; Han, J. Million-fold preconcentration of proteins and peptides by nanofluidic filter. Anal. Chem.
**2005**, 77, 4293–4299. [Google Scholar] [CrossRef] - Jia, M.; Kim, T. Multiphysics simulation of ion concentration polarization induced by a surface-patterned nanoporous membrane in single channel devices. Anal. Chem.
**2014**, 86, 10365–10372. [Google Scholar] [CrossRef] - Ehlert, S.; Hlushkou, D.; Tallarek, U. Electrohydrodynamics around single ion-permselective glass beads fixed in a microfluidic device. Microfluid. Nanofluidics
**2008**, 4, 471–487. [Google Scholar] [CrossRef] - Kwak, R.; Pham, V.S.; Lim, K.M.; Han, J. Shear flow of an electrically charged fluid by ion concentration polarization: Scaling laws for electroconvective vortices. Phys. Rev. Lett.
**2013**, 110, 114501. [Google Scholar] [CrossRef] [PubMed] - Lee, D.; Lee, J.A.; Lee, H.; Kim, S.J. Spontaneous selective preconcentration leveraged by ion exchange and imbibition through nanoporous medium. Sci. Rep.
**2019**, 9, 2336. [Google Scholar] [CrossRef] - Lee, J.A.; Lee, D.; Park, S.; Lee, H.; Kim, S.J. Non-negligible water-permeance through nanoporous ion exchange medium. Sci. Rep.
**2018**, 8, 12842. [Google Scholar] [CrossRef] - Kwon, H.J.; Kim, B.; Lim, G.; Han, J. A multiscale-pore ion exchange membrane for better energy efficiency. J. Mater. Chem. A
**2018**, 6, 7714–7723. [Google Scholar] [CrossRef] [Green Version] - Ko, S.H.; Song, Y.A.; Kim, S.J.; Kim, M.; Han, J.; Kang, K.H. Nanofluidic preconcentration device in a straight microchannel using ion concentration polarization. Lab Chip
**2012**, 12, 4472–4482. [Google Scholar] [CrossRef] - Lee, S.; Park, S.; Kim, W.; Moon, S.; Kim, H.-Y.; Lee, H.; Kim, S.J. Nanoelectrokinetic bufferchannel-less radial preconcentrator and online extractor by tunable ion depletion layer. Biomicrofluidics
**2019**, 13, 034113. [Google Scholar] [CrossRef] - Subramanian, V.; Lee, S.; Jena, S.; Jana, S.K.; Ray, D.; Kim, S.J.; Thalappil, P. Enhancing the sensitivity of point-of-use electrochemical microfluidic sensors by ion concentration polarization—A case study on arsenic. Sens. Actuators B Chem.
**2020**, 304, 127340. [Google Scholar] [CrossRef] - Kim, M.; Jia, M.; Kim, T. Ion concentration polarization in a single and open microchannel induced by a surface-patterned perm-selective film. Analyst
**2013**, 138, 1370–1378. [Google Scholar] [CrossRef] [Green Version] - Van der Bruggen, B.; Vandecasteele, C.; Van Gestel, T.; Doyen, W.; Leysen, R. A review of pressure-driven membrane processes in wastewater treatment and drinking water production. Environ. Prog.
**2003**, 22, 46–56. [Google Scholar] [CrossRef] - Sun, L.; Ying, Y.; Huang, H.; Song, Z.; Mao, Y.; Xu, Z.; Peng, X. Ultrafast molecule separation through layered ws2 nanosheet membranes. ACS Nano
**2014**, 8, 6304–6311. [Google Scholar] [CrossRef] [PubMed] - Tanaka, Y. Concentration polarization in ion exchange membrane electrodialysis. J. Membr. Sci.
**1991**, 57, 217–235. [Google Scholar] [CrossRef] - Hsu, J.-P.; Yang, K.-L.; Ting, K.-C. Effect of convective boundary layer on the current efficiency of a membrane bearing nonuniformly distributed fixed charges. J. Phys. Chem. B
**1997**, 101, 8984–8989. [Google Scholar] [CrossRef] - Khair, A.S. Concentration polarization and second-kind electrokinetic instability at an ion-selective surface admitting normal flow. Phys. Fluids
**2011**, 23, 072003. [Google Scholar] [CrossRef] - Demekhin, E.; Nikitin, N.; Shelistov, V. Direct numerical simulation of electrokinetic instability and transition to chaotic motion. Phys. Fluids
**2013**, 25, 122001. [Google Scholar] [CrossRef] [Green Version] - Druzgalski, C.; Andersen, M.; Mani, A. Direct numerical simulation of electroconvective instability and hydrodynamic chaos near an ion-selective surface. Phys. Fluids
**2013**, 25, 110804. [Google Scholar] [CrossRef] - Masliyah, J.H.; Bhattacharjee, S. Electrokinetic and Colloid Transport Phenomena; Wiley: Hoboken, NJ, USA, 2006. [Google Scholar]
- Karatay, E.; Druzgalski, C.L.; Mani, A. Simulation of chaotic electrokinetic transport: Performance of commercial software versus custom-built direct numerical simulation codes. J. Colloid Interface Sci.
**2015**, 446, 67–76. [Google Scholar] [CrossRef] [PubMed] - Bird, R.B.; Stewart, W.E.; Lightfoot, E.N. Transport Phenomena; John Wiley & Sons: Hoboken, NJ, USA, 2007. [Google Scholar]
- Kim, S.J.; Ko, S.H.; Kwak, R.; Posner, J.D.; Kang, K.H.; Han, J. Multi-vortical flow inducing electrokinetic instability in ion concentration polarization layer. Nanoscale
**2012**, 4, 7406–7410. [Google Scholar] [CrossRef] - Dydek, E.V.; Zaltzman, B.; Rubinstein, I.; Deng, D.S.; Mani, A.; Bazant, M.Z. Over-limiting current in a microchannel. Phys. Rev. Lett.
**2011**, 107, 118301. [Google Scholar] [CrossRef] - Nam, S.; Cho, I.; Heo, J.; Lim, G.; Bazant, M.Z.; Moon, D.J.; Sung, G.Y.; Kim, S.J. Experimental verification of over-limiting current by surface conduction and electro-osmotic flow in microchannels. Phys. Rev. Lett.
**2015**, 114, 114501. [Google Scholar] [CrossRef] - Sohn, S.; Cho, I.; Kwon, S.; Lee, H.; Kim, S.J. Surface conduction in a microchannel. Langmuir
**2018**, 34, 7916–7921. [Google Scholar] [CrossRef] [PubMed] - Huh, K.; Yang, S.-Y.; Park, J.S.; Lee, J.A.; Lee, H.; Kim, S.J. Surface conduction and electroosmotic flow around charged dielectric pillar arrays in microchannels. Lab Chip
**2020**, 20, 675–686. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Schematic of numerical domain. The characteristic length scale was 1 for the y-directional length of the domain, while it was 4 for the x-directional one. The reservoir was assumed to have uniform ion concentration. The ion-selective surface was water-permeable, through which only cations and fluid flow can pass. The periodic boundary condition described an infinite domain case in horizontal direction.

**Figure 2.**Cation concentration according to each time lapse and the ion concentration of space charge layer (SCL) when (

**a**) the fluid flow was imposed from the reservoir into the ion-selective surface (Pe = −4), and (

**b**) there was no fluid flow (Pe = 0) and (

**c**) the fluid flow was imposed from the ion-selective surface into the reservoir (Pe = 12.5). The white lines represent stream line and the white arrows indicate the direction of the flow. Note that the ion concentrations were rendered in a log scale and denoted time was dimensionless. In order to avoid confusion between dimensional and dimensionless variables, we notated dimensional time ($\tilde{t}$) at the case of t = 2. This value was corresponded with the case of O(100) μm gap filling with O(100) μM KCl electrolyte.

**Figure 3.**The plot of growth rate according to the Pe. The tendency of changes in growth rate was different depending on the direction of normal flow. When the Pe was positive (upward fluid flow), the growth rate decreased monotonically. There was a maximum point of the growth rate when the Pe was negative (downward fluid flow).

**Figure 4.**The different behavior of growth rate depending on the sign of Pe (t = 0.2) was attributed to the distribution of ion concentration in the domain. (

**a**) Representative schematic for electrical double layer (EDL), SCL, and mixing layer (ML) (Pe = -20, ϕ = 50). (

**b**) Averaged concentrations at ϕ = 50 with varying Pe from negative to positive.

**Figure 5.**(

**a**) A stability map was investigated as a function of both applied voltage and the properties of imposed normal fluid flow. A cross sign indicated that there existed vortices (unstable) and a circle sign indicated that there were no vortices in the domain (stable). Marginal stability curve (dotted line) was also obtained. In the case of positive Pe, the marginal curve followed ~ |Pe|

^{1.15}. In the case of negative Pe, the marginal curve followed ~ |Pe|

^{0.45}. (

**b**) The distribution of ion concentration in the domain along increasing Pe from negative to positive value at ϕ = 50.

**Table 1.**Characteristic scales used in this work. The tilde (‘~’) meant dimensional quantity. $\tilde{D}$ is the diffusivity of electrolyte, $\tilde{R}$ is the gas constant, $\tilde{T}$ is the absolute temperature, and $\tilde{\mu}$ is the fluid viscosity.

Physical Quantity | Characteristic Scale | Description |
---|---|---|

Length | ${\tilde{L}}_{y}$ | y-directional length of numerical domain |

Time | ${\tilde{\tau}}_{D}=\frac{{\tilde{L}y}^{2}}{\tilde{D}}$ | Diffusion time scale |

Electric potential | ${\tilde{V}}_{T}=\frac{\tilde{R}\tilde{T}}{\tilde{F}}$ | Thermal voltage |

Concentration | ${\tilde{c}}_{0}$ | Bulk concentration |

Pressure | $\frac{\tilde{\mu}\tilde{D}}{{\tilde{L}y}^{2}}$ | Diffusion-scaled pressure |

Flow velocity | ${\tilde{U}}_{0}=\frac{\tilde{D}}{{\tilde{L}}_{y}}$ | Diffusion-scaled velocity |

Current density | ${\tilde{i}}_{0}=\frac{\tilde{F}\tilde{D}{\tilde{c}}_{0}}{{\tilde{L}}_{y}}$ | Diffusion-limited current density |

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**MDPI and ACS Style**

Choi, J.; Mani, A.; Lee, H.; Kim, S.J.
Investigation on the Stability of Random Vortices in an Ion Concentration Polarization Layer with Imposed Normal Fluid Flow. *Micromachines* **2020**, *11*, 529.
https://doi.org/10.3390/mi11050529

**AMA Style**

Choi J, Mani A, Lee H, Kim SJ.
Investigation on the Stability of Random Vortices in an Ion Concentration Polarization Layer with Imposed Normal Fluid Flow. *Micromachines*. 2020; 11(5):529.
https://doi.org/10.3390/mi11050529

**Chicago/Turabian Style**

Choi, Jihye, Ali Mani, Hyomin Lee, and Sung Jae Kim.
2020. "Investigation on the Stability of Random Vortices in an Ion Concentration Polarization Layer with Imposed Normal Fluid Flow" *Micromachines* 11, no. 5: 529.
https://doi.org/10.3390/mi11050529