Measurement of an eddy diffusivity for chaotic electroconvection using combined computational and experimental techniques

The coupling between ion transport, fluid flow, and electrostatics may give rise to electroconvection, a physical phenomenon in which the buildup of charge near selective surfaces leads to hydrodynamic instability, eventually transitioning to an unsteady and chaotic flow. Though electroconvection contains a wide range of spatiotemporal scales, averaging of the ion concentration, velocity, and electric potential fields may produce a lower-dimensional and smoother representation while still capturing the essential performance metrics: ion current density and mean applied voltage, for example. The Poisson–Nernst–Planck–Stokes equations are known to capture the chaotic dynamics of electroconvection accurately. However, there is as of yet no way to directly compute the mean fields, since the application of the well-known Reynolds-averaging procedure leads to a closure problem. In this work, we combine the macroscopic forcing method, a numerical technique for measurement of closure operators in Reynolds-averaged equations, with high-fidelity experimental data to close the equations for the mean fields in chaotic electroconvection. We show that the unclosed fluxes in the Reynolds-averaged equations may be represented to the leading order as a gradient-diffusion term, with a spatially varying eddy diffusivity that we directly measure from experimental velocity fields. As a result, we are able to directly solve for the mean ion concentration and electric potential fields by supplementing the 1D Poisson–Nernst–Planck equations with an eddy diffusivity that captures the averaged effects of mixing due to chaotic, 3D electroconvection. The resulting current-voltage curve exhibits strong agreement with experiments. Our method allows for the study of electroconvection with orders-of-magnitude cost savings by obviating the need for expensive direct numerical simulations. Additionally, our measured eddy diffusivity profiles provide a benchmark for the development of stand-alone reduced-order models of electroconvection.

Figure 1

Note: Figure not drawn to scale. (a) An exploded schematic of the experimental apparatus, showing two electrolyte reservoirs and the cation-selective membrane sandwiched between copper electrodes. (b) An inset of the anolyte chamber, showing the volume over which PTV is performed and velocity fields are measured in experiments. This optical measurement volume is also the domain for the computational component of this study. (c) A representative sketch of electric potential along the red reference line shown in panel (a), indicating potential drops that occur at various locations in the electrochemical cell. Additional details regarding the apparatus may be found in Ref. [41].

Figure 2

Schematic of the experimental setup for stereo PTV. A laser illuminates the fluorescent tracer particles in the measurement volume, which are recorded by a stereo camera pair. The stereo images of the volume acquired over time are preprocessed to reduce noise and improve the visibility of particles. The spatial boundaries of the volume are then determined in the membrane-normal direction using tomographic PIV. In the last step, volume self–calibration is used to increase the accuracy of the calibration and to correct image distortion due to light refraction at the surface. Additional details may be found in the work by Stockmeier and coauthors [40, 41]. Partially reprinted from Stockmeier et al., “Direct 3D observation and unraveling of electroconvection phenomena during concentration polarization at ion-exchange membranes [40], with permission from Elsevier.

Figure 3

Construction of 3D velocity fields from 3D particle tracks using the fine-scale reconstruction method. The colors indicate the velocity component in membrane-normal direction. Example cases are shown at applied DC current values of $148\mu \mathrm{A}$ and $400\mu \mathrm{A}$, as indicated. Additional details may be found in the work by Stockmeier and coauthors [40, 41].

Figure 4

Eddy diffusivity profiles for different applied potentials, normalized by the ambipolar diffusivity for copper sulfate. The subscript $\mathrm{\Delta }\varphi$ in $D_{\mathrm{\Delta }\varphi }^0\left(x\right)$ signifies that each profile is only valid for the specific $\mathrm{\Delta }\varphi$ value corresponding to the experiment in which velocity fields were measured for that case.

Figure 5

Instantaneous salt concentration fields computed via Eq. (10) for (a) $\mathrm{\Delta }\varphi =0.5347$, (b) $\mathrm{\Delta }\varphi =0.6343$, (c) $\mathrm{\Delta }\varphi =2.1051$, and (d) $\mathrm{\Delta }\varphi =3.6780$. These are not the $c$ fields computed via Eq. (15) as part of the MFM procedure. Rather, we show the unforced fields in order to develop an intuitive connection between the eddy diffusivity and the unaltered system for which the eddy diffusivity is measured.

Figure 6

Analysis of the shape and the magnitude of the eddy diffusivity profiles measured at different applied electric potentials. (a) $x_{\text{max}}$, the location of the maximum value of $D_{\mathrm{\Delta }\varphi }^0\left(x\right)$. The mean is shown with and without the last point, since the diffusivity profile corresponding to that case is qualitatively different from the other cases due to the effects of confinement. (b) The maximum value of $D_{\mathrm{\Delta }\varphi }^0\left(x\right)$ for different $\mathrm{\Delta }\varphi$. Also shown is a linear fit based on least–squares regression.

Figure 7

Each of the $D_{\mathrm{\Delta }\varphi }^0$ profiles has been scaled by $\alpha \mathrm{\Delta }\varphi +\gamma$, where $\alpha$ and $\gamma$ are found from a linear fit of the maximum of $D_{\mathrm{\Delta }\varphi }^0$ as a function of $\mathrm{\Delta }\varphi$. We may construct a voltage–dependent form of the eddy diffusivity by selecting one of these profiles as the reference profile $D_{\text{ref}}^0$, indicated here as black dashed curve.

Figure 8

Mean concentration and electric potential profiles computed directly from the closed macroscopic equations using $D_{\mathrm{\Delta }\varphi }^0$. Mean ${\mathrm{Cu}}^{2+}$ concentration is shown as a solid line, while mean ${\mathrm{SO}}_4^{2-}$ concentration is shown as a dashed line. Electric potential is shown as a solid line. (a) A view of the entire domain. Due to electroneutrality throughout most of the domain, the two species' concentration curves are indistinguishable. (b) A zoomed-in view of the near-membrane region, highlighting the double layer and extended space charge regions. (c) Electric potential, over the entire domain. (d) Electric potential in the near-membrane region.

Figure 9

(a) Mean electroneutral salt concentration computed using the standard 1D Poisson–Nernst–Planck equations in the absence of $D^0$. Without the representation of mixing due to microscopic vortices, the profiles remain linear. (b) Mean electroneutral salt concentration computed via two different methods. The direct 1D simulations of Eqs. (21) and (22) using $D_{\mathrm{\Delta }\varphi }^0$ are shown as solid curves. The dashed curves are computed via $y, z$, and $t$ averaging applied to the fields resulting from 3D simulation of Eq. (10), in which the measured velocity fields mix the electroneutral salt. Quantitative agreement demonstrates that the leading order eddy diffusivity is capable of capturing the macroscopic transport effects of electroconvection.

Figure 10

Contributions of individual terms to the overall ionic flux in the electroneutral bulk region. Panels (a–c) are the diffusion flux, electromigration flux, and advective closure flux, respectively, for ${\mathrm{Cu}}^{2+}$. Panels (d–f) are the diffusion flux, electromigration flux, and advective closure flux, respectively, for ${\mathrm{SO}}_4^{2-}$. Electromigration and diffusion are the dominant terms throughout most of the electroneutral bulk region at lower applied electric potentials, while electromigration and the advective closure flux are the dominant terms at higher applied electric potentials.

Figure 11

Mean current-voltage curves produced using the eddy diffusivity. In both panels, results of the standard 1D Poisson–Nernst–Planck equations are shown as dash-dot red curves, and the experiment points are shown using black squares. (a) The current density computed using each of the directly measured diffusivity profiles $D_{\mathrm{\Delta }\varphi }^0$ is shown as a blue circle. The dashed curves correspond to j-V sweeps in which the same eddy diffusivity profile was used for the entire range of applied electric potentials. (b) The j-V curves computed using the $\mathrm{\Delta }\varphi$-dependent diffusivity profiles, $D^0(x,\mathrm{\Delta }\varphi )$, for a range of $L_x$ values.

Figure 12

Mean concentration and electric potential profiles computed directly from the closed macroscopic equations using $D^0(x,\mathrm{\Delta }\varphi )$. Mean ${\mathrm{Cu}}^{2+}$ concentration is shown as a solid line, while mean ${\mathrm{SO}}_4^{2-}$ concentration is shown as a dashed line. Electric potential is shown as a solid line. (a) A view of the entire domain. Due to electroneutrality throughout most of the domain, the two species' concentration curves are indistinguishable. (b) A zoomed-in view of the near-membrane region, highlighting the double layer and extended space charge regions. (c) Electric potential, over the entire domain. (d) Electric potential in the near-membrane region.