# Theory of Chiral Electrodeposition by Chiral Micro-Nano-Vortices under a Vertical Magnetic Field -1: 2D Nucleation by Micro-Vortices

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{9}

^{10}

^{11}

^{*}

## Abstract

**:**

## 1. Introduction

^{−10}s, a solvated ionic vacancy keeps an intrinsic lifetime of 1 s [3,4], which is, compared with the collision time, extraordinarily long. This result strongly suggests that an ionic vacancy behaves as an iso-entropic particle without entropy production during transfer. Plainly, an ionic vacancy plays a role of an atomic scale lubricant, so that a vacancy layer formed on the electrode provides a free surface without friction, and the viscosity of the layer drastically decreases to zero. Such features have been validated by various experiments [3,4,5,6].

^{7}times as large as that of a screw dislocation (~0.1 nm). The first question is—how are such chiral screw dislocations created by the rotation of the VMHDF despite extremely different scales of length? (Q1). Generally, nucleation in electrodeposition is classified into 2D nucleation of the order of 0.1 mm, 3D nucleation of the order of 0.1 μm, and screw dislocations of the order of 0.1 nm. Therefore, the emergence of the chiral activity would be composed of the three generations of chiral nuclei, i.e., chiral 2D nucleus, chiral 3D nucleus, and chiral screw dislocation. A chiral screw dislocation is created on a chiral 3D nucleus, which in turn grows on a chiral 2D nucleus developing under a VMHDF. These three processes form a nesting-boxes structure. As will be clarified later, based on a simple evidence, the fact that the chiral activity arises from the three generations is validated from both theoretical and experimental aspects. As for 2D and 3D nuclei, the nucleation processes under parallel magnetic fields have been established [20,21,22,23], so in the present papers, we should examine how chiral 2D and 3D nuclei emerge under VMHDF rotations.

## 2. Theory

#### 2.1. Vortex Motions in the Stationary Lower Layer

#### 2.2. Amplitude Equations of the Fluctuations in the Lower Layer

#### 2.3. Vortex Motions Induced in the Rotating Upper Layer

#### 2.4. Boundary Conditions

#### 2.4.1. Hydrodynamic Conditions

- (a)
- For the rigid surfaces:

- (b)
- For the free surfaces:

- (c)
- For the upper boundary between the lower and upper layers:

#### 2.4.2. Mass Transfer Conditions

#### 2.5. Solutions of ${W}^{0}$ and ${\Omega}^{0}$ in the Lower Layer

- (a)
- For the rigid surface vortices:

- (b)
- For the free surface vortices:

#### 2.6. Determination of the Velocity Coefficients ${\alpha}_{0}$ and ${\alpha}_{1}$

- (a)
- For the rigid and free surface vortices at the upper boundary:

- (b)
- For the rigid surface vortices in the lower layer:

- (c)
- For the free surface vortices in the lower layer:

#### 2.7. The Solution of ${\Theta}^{0}$ and $D{\Theta}^{0}$at the Electrode Surface

- For the rigid surface vortices:

- (b)
- For the free surface vortices:

## 3. 2D Nucleation

#### 3.1. Asymmetrical Fluctuations in 2D Nucleation Process

^{−1}mol

^{−1}), $T$ is the absolute temperature (K), and $F$ is the Faraday constant ($96,500$ C mol

^{−1}).

^{−3}).

^{3}mol

^{−1}).

#### 3.2. Characteristic Equations of the Vorticity Coefficients ${\beta}_{0}^{a}$ and ${\beta}_{1}^{a}$

- (a)
- For the rigid surface vortices:

- (b)
- For the free surface vortices:

#### 3.3. Nucleation by the Rigid and Free Surface Vortices

#### 3.4. The Rotational Directions of the Micro-MHD Flows on the Rigid and Free Surfaces

- (a)
- For the rigid surfaces:

- (b)
- For the free surfaces:

## 4. Results and Discussion

#### 4.1. Micro-Mystery Circles Formed by the Non-Specific Adsorption of Ions

#### 4.2. Inversion of Chirality by the Specific Adsorption of Chloride Ions

^{−1}C

^{2}m

^{−1}, 25 °C), ${C}_{\mathrm{H}}$ is the electric capacity of the Helmholtz layer ($\approx 10$ μF cm

^{−2}$=0.1$ F m

^{−2}[67]), and λ is the Debye length shown in Equation (35c). ${\left(\partial {Q}_{1}^{\ast}/\partial {Q}_{2}^{\ast}\right)}_{\mathsf{\mu}}$ is the differential charge coefficient, where ${Q}_{1}^{\ast}$ and ${Q}_{2}^{\ast}$ imply the electric charges stored in the Helmholtz and diffuse layers of an electric double layer [68,69]. From our preliminary experiments, we obtained

^{−3}H

_{2}SO

_{4}supporting electrolyte solution, from Equation (35c), we obtain the Debye length λ$=2.47\times {10}^{-10}$ m. Using ${C}_{\mathrm{H}}=0.1$ F m

^{−2}as well as Equation (75), we have the differential potential coefficients as follows:

## 5. Materials and Methods

^{−3}CuSO

_{4}+ 500 mol m

^{−3}H

_{2}SO

_{4}solution. The experimental apparatus was represented elsewhere [6]. Water was prepared by a pure water production system (MERCK KGAA, Darmstadt, Germany). The CuSO

_{4}and H

_{2}SO

_{4}were analytical grades (FUJIFILM Wako Pure Chemical Corporation, Osaka, Japan). The VMHDE was made of a copper disk of 8 mm diameter (oxygen-free copper, 99.99% purity, The Nilaco Corporation, Tokyo, Japan) equipped with a 5 mm-wide fringe of PTFE resin (Flonchemical Co. Ltd., Osaka, Japan). To prevent natural convection, it was set in a downward direction. The counter electrode (oxygen-free copper, 99.99% purity, The Nilaco Corporation, Tokyo, Japan) was a copper plate, 25 mm in diameter, which was placed 30 mm from the VMHDE. A copper rod (1 mm diameter) was used as a reference electrode (oxygen-free copper, 99.99% purity, The Nilaco Corporation, Tokyo, Japan). To stop the VMHDF, a sheath with an 18 mm inner diameter and an 18 mm height was attached to the electrode. By using the limiting diffusion current at an overpotential of −400 mV under a given vertical magnetic field, the experiment was performed. The whole electrode system was settled at the place of a uniform magnetic field selected in the bore space of a 10T-cryocooled superconducting magnet (HF-10-100VH, Sumitomo Heavy Industries Ltd., Tokyo, Japan). The deposited electrode surfaces were observed by a surface roughness analysis 3D scanning electron microscope (ERA-8800, ELIONIX Inc., Tokyo, Japan).

## 6. Conclusions

- 1.
- Chiral screw dislocations under a VMHDF arise from the three generations of chiral nuclei, which constitute nesting boxes. Namely, chiral 2D nuclei are formed by the chiral micro-MHD vortices with rigid surfaces. Then, chiral 3D nuclei are created by the chiral nano-MHD vortices with rigid surfaces on a chiral 2D nucleus. Finally, chiral screw dislocations grow by chiral ultra-micro MHD vortices with rigid surfaces on a chiral 3D nucleus. Such a structure was validated by the fact that the observed enantiomeric excess (ee) ratios are always smaller than $0.125$.
- 2.
- The chiral nucleation system is composed of a rotating upper layer and a stationary lower layer so that vortices in the lower layer can receive the precessions from the upper layer and raise chiral nuclei at fixed places.
- 3.
- For chirality to emerge, two types of vortices are necessary, having rigid surfaces with friction and free surfaces covered with ionic vacancies. Due to the rigid surface with friction, the rigid surface vortices not only work as pins to stop the migration of the vortices in the lower layer but also create chiral nuclei at fixed positions. Which vortex receives the precession depends on whether the growth mode is unstable or stable. Free surface vortices unstably grow faster than the rigid surface vortices, whereas, under stable conditions, rigid surface vortices activated dwindle with time more slowly than free surface vortices. Therefore, when unstable, free surface vortices have the priority of precession, and in stable cases, the precessions are donated to rigid surface vortices.
- 4.
- Due to fluid and vortex continuities, a pair of adjoining vortices are composed of rigid and free surface vortices with opposite rotations. To raise nuclei fixed to a solid surface, chiral nucleation must occur only under the rigid surface vortices. Since in a CuSO
_{4}+ H_{2}SO_{4}solution, simple non-specific adsorption takes place, unstable copper nucleation proceeds. As a result, the rotation of a VMHDF transfers to the free surface vortices as the precessions, so that 2D nuclei with reverse chirality are formed under rigid surface vortices in the rotation opposite to that of the VMHDF. Though this result does not directly explain the chiral activity of the electrode, we can understand the mechanism of the emergence of the opposite chirality to the VMHDF. In accordance with the three-generation model, if such a nucleation process were repeated three times, the opposite chirality would be realized. - 5.
- When a chloride additive is added to a CuSO
_{4}+ H_{2}SO_{4}solution, specific adsorption of the chloride ions takes place, leading to stable nucleation. In this case, the rotation of a VMHDF is bestowed on the rigid surface vortices as precessions. Therefore, 2D nuclei growing under the rigid surface vortices have the same chirality as that of the VMHDF. Namely, due to the stability of the specific adsorption of chloride ions, we can expect a change in the chiral activity of the electrode. However, if the differences between both amplitude factors and their values themselves were sufficiently small, the breakdown would also take place.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${x}_{1}$, ${x}_{2}$, ${x}_{3}$ | Cartesian coordinates corresponding to $x$, $y$, $z$ (m) |

$x$, $y$, $z$ | non-dimensional coordinates normalized by $d$ |

$\overrightarrow{r}$ | position vector (m) |

d | representative length (m) |

${d}^{\mathrm{a}}$ | representative length of asymmetrical fluctuations in 2D nucleation (m) |

${k}_{\mathrm{x}}$, ${k}_{\mathrm{y}}$ | wavenumber components in the $x$- and $y$-directions (m^{−1}) |

$k$ | wavenumber defined by ${\left({k}_{\mathrm{x}}^{2}+{k}_{\mathrm{y}}^{2}\right)}^{1/2}$(m^{−1}) |

${a}_{\mathrm{x}}$, ${a}_{\mathrm{y}}$ | wavenumber components of $a$ in the $x$- and $y$-directions |

$a$ | non-dimensional wavenumber ($=kd\mathrm{or}k{d}^{\mathrm{a}}$) |

${a}^{+}$ | autocorrelation distance of the fluctuation, i.e., the average size of the vortices (m) |

$\overrightarrow{U}$ | velocity which an observer feels (m s^{−1}) |

${U}_{\mathrm{i}}^{\ast}$ | i-component of the main flow velocity of the rotation (m s^{−1}) |

$\overrightarrow{u}$ | velocity (m s^{−1}) |

${u}_{\mathrm{i}}$ | i-component of $\overrightarrow{u}$ (i = 1, 2, 3) (m s^{−1}) |

$u$ | $x$-component of the velocity, ${u}_{1}$ (m s^{−1}) |

$v$ | $y$-component of the velocity, ${u}_{2}$ (m s^{−1}) |

w | $z$-component of the velocity, ${u}_{3}$ (m s^{−1}) |

${\omega}_{\mathrm{i}}$ | i-component of the vorticity (s^{−1}) |

${\omega}_{\mathrm{z}}$ | $z$-component of the vorticity (s^{−1}) |

${\varphi}_{\mathrm{s}}$ | $x$-component of stream function (m s^{−1}) |

${\psi}_{\mathrm{s}}$ | $y$-component of stream function (m s^{−1}) |

${P}_{\mathrm{int}}\left({a}_{\mathrm{x}},{a}_{\mathrm{y}}\right)$ | Gaussian-type power spectrum defined by Equation (F5) |

${P}_{\mathrm{xz}}$ | viscous stress tensor defined in Equation (7a) (N m^{−2}) |

${P}_{\mathrm{yz}}$ | viscous stress tensor defined in Equation (7b) (N m^{−2}) |

$\rho $ | density of solution (kg m^{−3}) |

${\mu}_{\mathrm{s}}$ | viscosity of solution (N s m^{−2}) |

$\nu $ | kinematic viscosity (m^{2} s^{−1}) |

${\nu}^{\mathrm{a}}$ | kinematic viscosity of bulk solution in 2D nucleation |

${\mathsf{\Omega}}_{\mathrm{m}}$ | molar volume of deposit metal (m^{3} mol^{−1}) |

P | pressure (N m^{−2}) |

${\mu}_{0}$ | magnetic permeability (4π × 10^{7} N A^{−2}) |

$\eta $ | resistivity defined by Equation (B14) |

$\epsilon $ | dielectric constant of water (6.95 × 10^{−10} J^{−1} C^{2} m^{−1}, 25 °C) |

$R$ | universal gas constant (8.31 J K^{−1} mol^{−1}) |

$T$ | absolute temperature (K) |

$F$ | Faraday constant (96,500 C mol^{−1}) |

$\overrightarrow{B}$ | magnetic flux density (T) |

${B}_{\mathrm{i}}$ | i-component of $\overrightarrow{B}$(T) |

${\overrightarrow{B}}^{\ast}$ | external magnetic flux density in the absence of reactions (T) |

${B}_{\mathrm{j}}^{\ast}$ | j-component of ${\overrightarrow{B}}^{\ast}$(T) |

${B}_{0}$ | $z$-component of ${\overrightarrow{B}}^{\ast}$ with sign (T) |

$\overrightarrow{b}$ | fluctuation of $\overrightarrow{B}$ by reactions (T) |

${b}_{\mathrm{i}}$ | i-component of $\overrightarrow{b}$ (T) |

${b}_{\mathrm{z}}$ | $z$-component of $\overrightarrow{b}$ (T) |

$\overrightarrow{E}$ | electric field (V m^{−1}) |

$\overrightarrow{J}$ | current density (A m^{−2}) |

${j}_{\mathrm{i}}$ | i-component of the current density fluctuation (A m^{−2}) |

${j}_{\mathrm{z}}$ | $z$-component of the current density fluctuation (A m^{−2}) |

${j}_{\mathrm{z}}{\left(x,y,0,t\right)}^{\mathrm{a}}$ | asymmetrical fluctuation of ${j}_{\mathrm{z}}$ at the electrode (A m^{−2}) |

${\sigma}^{\ast}$ | electrical conductivity (S m^{−1}) |

${z}_{\mathrm{i}}$ | charge number of ionic species i including sign |

${z}_{\mathrm{m}}$ | charge number of the metallic ion |

${\lambda}_{\mathrm{i}}^{\ast}$ | mobility of ionic species i (m^{2} V^{−1} s^{−1}) |

${\lambda}_{\mathrm{i}}$ | $\mathrm{i}$-component of unit normal vector |

${C}_{\mathrm{i}}$ | concentration of ionic species $\mathrm{i}$ (mol m^{−3}) |

${D}_{\mathrm{i}}$ | diffusion coefficient of ionic species i (m^{2} s^{−1}) |

${D}_{\mathrm{m}}$ | diffusion coefficient of the metallic ion (m^{2} s^{−1}) |

${\overrightarrow{F}}_{\mathrm{L}}$ | Lorentz force per unit volume (N m^{−3}) |

${F}_{\mathrm{L},\mathrm{i}}$ | i-component of ${\overrightarrow{F}}_{\mathrm{L}}$ (N m^{−3}) |

${\overrightarrow{F}}_{\mathrm{R}}$ | acceleration which an observer feels in a frame of reference rotation with the same angular velocity as the upper layer (N m^{−3}) |

${f}_{\mathrm{Ri}}$ | i-component of the fluctuation of ${\overrightarrow{F}}_{\mathrm{R}}$ (N m^{−3}) |

${f}_{\mathrm{Li}}$ | i-component of the fluctuation of the Lorentz force (N m^{−3}) |

${C}_{\mathrm{m}}$ | concentration of the metallic ion (mol m^{−3}) |

${C}_{\mathrm{m}}^{\ast}$ | concentration of the metallic ion in the absence of fluctuation (mol m^{−3}) |

${c}_{\mathrm{m}}$ | concentration fluctuation of the metallic ion (mol m^{−3}) |

${c}_{\mathrm{m}}{\left(x,y,z,t\right)}^{\mathrm{a}}$ | asymmetrical fluctuation of the concentration of the metallic ion (mol m^{−3}) |

${c}_{\mathrm{m}}{\left(x,y,{0}^{+},t\right)}^{\mathrm{a}}$ | ${c}_{\mathrm{m}}{\left(x,y,z,t\right)}^{\mathrm{a}}$ at OHP (mol m^{−3}) |

${C}_{\mathrm{m}}^{\ast}\left(z=0\right)$ | surface concentration of the metallic ion outside the double layer (mol m^{−3}) |

${C}_{\mathrm{m}}^{\ast}\left(z=\infty \right)$ | bulk concentration of the metallic ion (mol m^{−3}) |

${C}_{\mathrm{j}}^{\ast}\left(z=\infty \right)$ | bulk concentration of ionic species j (mol m^{−3}) |

${L}_{\mathrm{m}}$ | average concentration gradient in the diffusion layer defined by Equation (C8) (mol m^{−4}) |

${\theta}_{\infty}^{\ast}$ | concentration difference between the bulk and the surface (mol m^{−3}) |

$\langle {\delta}_{\mathrm{c}}\rangle $ | average thickness of a diffusion layer (m) |

${U}^{0}$ | amplitude of u (m s^{−1}) |

${V}^{0}$ | amplitude of v (m s^{−1}) |

${W}^{0}$ | amplitude of w (m s^{−1}) |

${\mathsf{\Omega}}^{0}$ | amplitude of ${\omega}_{\mathrm{z}}$ (s^{−1}) |

${W}^{0\ast}$ | real amplitude without i (m s^{−1}) |

${\mathsf{\Omega}}^{0\ast}$ | real amplitude without i (s^{−1}) |

${\mathsf{\Phi}}_{\mathrm{s}}^{0}$ | amplitudes of the stream functions ${\varphi}_{s}$(m s^{−1}) |

${\mathsf{\Psi}}_{\mathrm{s}}^{0}$ | amplitudes of the stream functions ${\psi}_{s}$(m s^{−1}) |

${K}^{0}$ | amplitude of ${b}_{\mathrm{z}}$(T) |

${J}^{0}$ | amplitude of ${j}_{\mathrm{z}}$ (A m^{−2}) |

${\mathsf{\Theta}}^{0}$ | amplitude of ${c}_{\mathrm{m}}$ (mol m^{−3}) |

${W}_{\mathrm{r}}^{0}\left(z,t\right)$ | amplitude of w of the rigid surface vortices (m s^{−1}) |

${W}_{\mathrm{f}}^{0}\left(z,t\right)$ | amplitude of w of the free surface vortices (m s^{−1}) |

${W}_{\mathrm{r}}^{0}{\left(z,t\right)}^{\mathrm{a}}$ | ${W}_{\mathrm{r}}^{0}\left(z,t\right)$ in 2D nucleation (m s^{−1}) |

${W}_{\mathrm{f}}^{0}{\left(z,t\right)}^{\mathrm{a}}$ | ${W}_{\mathrm{f}}^{0}\left(z,t\right)$ in 2D nucleation (m s^{−1}) |

${\mathsf{\Omega}}_{r}^{0}\left(z,t\right)$ | amplitude of ${\omega}_{\mathrm{z}}$ of the rigid surface vortices (s^{−1}) |

${\mathsf{\Omega}}_{\mathrm{f}}^{0}\left(z,t\right)$ | amplitude of ${\omega}_{\mathrm{z}}$ of the free surface vortices (s^{−1}) |

${\mathsf{\Omega}}_{\mathrm{r}}^{0}{\left(z,t\right)}^{\mathrm{a}}$ | ${\mathsf{\Omega}}_{\mathrm{r}}^{0}\left(z,t\right)$ in 2D nucleation (s^{−1}) |

${\mathsf{\Omega}}_{\mathrm{f}}^{0}{\left(z,t\right)}^{\mathrm{a}}$ | ${\mathsf{\Omega}}_{\mathrm{f}}^{0}\left(z,t\right)$ in 2D nucleation (s^{−1}) |

${\mathsf{\Theta}}_{\mathrm{r}}^{0}\left(0,t\right)$ | amplitude of ${c}_{\mathrm{m}}$ at the rigid surface (mol m^{−3}) |

${\mathsf{\Theta}}_{\mathrm{f}}^{0}\left(0,t\right)$ | amplitude of ${c}_{\mathrm{m}}$ at the free surface (mol m^{−3}) |

$Q$ | magneto-induction coefficient defined by Equation (D4c) |

${Q}^{\ast}$ | non-dimensional magneto-induction coefficient defined by Equation (D5a) |

$\overrightarrow{\mathsf{\Omega}}$ | angular velocity vector (s^{−1}) |

$\tilde{\mathsf{\Omega}}$ | angular velocity of the upper layer corresponding to VMHDF (s^{−1}) |

${\tilde{\mathsf{\Omega}}}_{\mathrm{r}}^{\mathrm{a}}$ | representative angular velocity of the rigid surface vortices (s^{−1}) |

${T}^{\ast}$ | rotation coefficient defined by Equation (E20c) (m^{−1}) |

${R}^{\ast}$ | mass transfer coefficient defined by Equation (J2b) (mol m^{−4} s) |

${S}^{\ast}$ | magneto-viscosity coefficient defined by Equation (J9b) (m^{2} A^{−1} s^{−1}) |

${R}^{\ast \mathrm{a}}$ | ${R}^{\ast}$ in 2D nucleation defined by Equation (G5b) (mol m^{−4} s) |

${Q}^{\ast \mathrm{a}}$ | ${Q}^{\ast}$ in 2D nucleation defined by Equation (G5c) |

${T}^{\ast \mathrm{a}}$ | ${T}^{\ast}$ in 2D nucleation defined by Equation (G5d) (m^{−1}) |

${S}^{\ast \mathrm{a}}$ | ${S}^{\ast}$ in 2D nucleation defined by Equation (G5e) (m^{2} A^{−1} s^{−1}) |

${p}_{\mathrm{r}}^{\mathrm{a}}$ | amplitude factor of the rigid surface vortices in 2D nucleation defined by Equation (46b) (s^{−1}) |

${p}_{\mathrm{f}}^{\mathrm{a}}$ | amplitude factor of the free-surface vortices in 2D nucleation defined by Equation (48b) (s^{−1}) |

${\mathrm{f}}_{\mathrm{r}}^{\mathrm{a}}\left(a\right)$ | amplitude factor function of the rigid surface vortices in 2D nucleation defined by Equation (45b) |

${\mathrm{f}}_{\mathrm{f}}^{\mathrm{a}}\left(a\right)$ | amplitude factor function of the free surface vortices in 2D nucleation defined by Equation (47b) |

${\mu}_{\mathrm{ad}}\left(x,y,t\right)$ | chemical potential of the ad-atom (J mol^{−1}) |

$\mathsf{\zeta}{\left(x,y,t\right)}^{\mathrm{a}}$ | surface morphology of 2D nuclei by the asymmetrical fluctuations (m) |

${\mathsf{\zeta}}^{\mathrm{a}}$ | shortened expression of $\mathsf{\zeta}{\left(x,y,t\right)}^{\mathrm{a}}$ (m) |

$\overline{{\mu}_{\mathrm{m}}}\left(x,y,{\mathsf{\zeta}}^{\mathrm{a}},t\right)$ | electrochemical potential of the metallic ion (J mol^{−1}) |

$\overline{{\mu}_{\mathrm{e}}}\left(x,y,t\right)$ | electrochemical potential of the free electron (J mol^{−1}) |

$\mathsf{\delta}\overline{{\mu}_{\mathrm{m}}}{\left(x,y,{\mathsf{\zeta}}^{\mathrm{a}},t\right)}^{\mathrm{a}}$ | asymmetrical fluctuation of $\overline{{\mu}_{\mathrm{m}}}\left(x,y,{\mathsf{\zeta}}^{\mathrm{a}},t\right)$(J mol^{−1}) |

$\mathsf{\delta}{\mu}_{\mathrm{ad}}\left(x,y,t\right)$ | asymmetrical fluctuation of ${\mu}_{\mathrm{ad}}\left(x,y,t\right)$(J mol^{−1}) |

IHP | inner Helmholtz plane |

OHP | outer Helmholtz plane |

${0}^{+}$ | $z$-coordinate of the outer Helmholtz plane (OHP) |

${\mathsf{\Phi}}_{1}$ | overpotential at IHP (V) |

${\varphi}_{1}{\left(x,y,t\right)}^{\mathrm{a}}$ | asymmetrical fluctuation of ${\mathsf{\Phi}}_{1}$ (V) |

${\mathsf{\Phi}}_{2\mathrm{OHP}}^{\ast}$ | overpotential at the flat OHP without 2D nuclei $\left(z={0}^{+}\right)$ measured from the outer boundary of the diffuse layer $\left(z={\infty}^{+}\right)$ (V) |

${\mathsf{\Phi}}_{2}$ | overpotential of the diffuse layer (V) |

${\varphi}_{2}{\left(x,y,z,t\right)}^{\mathrm{a}}$ | asymmetrical fluctuation of ${\mathsf{\Phi}}_{2}$ (V) |

${\varphi}_{2}{\left(x,y,{0}^{+},t\right)}^{\mathrm{a}}$ | asymmetrical fluctuation of ${\mathsf{\Phi}}_{2}$ at OHP (V) |

${\varphi}_{2}{\left(x,y,{\mathsf{\zeta}}^{\mathrm{a}},t\right)}^{\mathrm{a}}$ | asymmetrical fluctuation at the surface of 2D nuclei in the diffuse layer (V) |

${L}_{{\varphi}_{2}}$ | gradient of the electrostatic overpotential in the diffuse layer defined by Equation (35b) (V m^{−1}) |

$\lambda $ | Debye length equalized to the average diffuse layer thickness defined by Equation (35c) (m) |

${L}_{\mathrm{m}2}$ | average concentration gradient of the metallic ion in the diffuse layer defined by Equation (36b) (mol m^{−4}) |

${\left(\partial \langle {\mathsf{\Phi}}_{1}\rangle /\partial \langle {\mathsf{\Phi}}_{2}\rangle \right)}_{\mathsf{\mu}}$ | differential potential coefficient |

${A}_{\mathsf{\theta}}$ | adsorption coefficient defined by Equation (43b) (s^{−1}) |

${\theta}_{\mathrm{rand}}^{\mathrm{a}}$ | uniform random number between 0 and 2$\mathsf{\pi}$ |

${R}_{\mathrm{d}}^{\mathrm{a}}$ | 2D random number defined by Equation (49) |

${\alpha}_{\mathrm{r}}^{\mathrm{a}}\left(=\sqrt{2}/2\right)$ | initial ratio of the rigid surface component to the total concentration fluctuation |

${\alpha}_{\mathrm{f}}^{\mathrm{a}}\left(=\sqrt{2}/2\right)$ | initial ratio of the free surface component to the total concentration fluctuation |

${\gamma}_{0}^{\mathrm{a}}$ | constant of the vorticity coefficient of the free surface vortex in 2D nucleation defined by Equation (G10b) (s^{−1}) |

${\gamma}_{1}^{\mathrm{a}}$ | constant of the vorticity coefficient of the rigid surface vortex in 2D nucleation defined by Equation (G7b) (s^{−1}) |

${\epsilon}_{\mathrm{screw}}$ | probability that the chiral screw dislocations emerge from all the active points |

${I}_{0}$ | total current of an electrode covered with only achiral active points (A) |

${I}_{\mathrm{active}}$ | total current of the electrode active for either of D- and L-reagents (A) |

${I}_{\mathrm{inactive}}$ | total current of the electrode for the other reagent (A) |

$r\left(ee\right)$ | enantiomeric excess (ee) ratio |

${Q}_{1}^{\ast}$ | electric charge stored in the Helmholtz layer of an electric double layer (A) |

${Q}_{2}^{\ast}$ | electric charge stored in the diffuse layer of an electric double layer (A) |

${\left(\partial {Q}_{1}^{\ast}/\partial {Q}_{2}^{\ast}\right)}_{\mathsf{\mu}}$ | differential charge coefficient |

${C}_{\mathrm{H}}$ | electric capacity of the Helmholtz layer (F m^{−2}) |

${\nabla}^{2}$ | $\equiv {\partial}^{2}/\partial {x}_{1}^{2}+{\partial}^{2}/\partial {x}_{2}^{2}+{\partial}^{2}/\partial {x}_{3}^{2}$ |

${\mathsf{\epsilon}}_{\mathrm{ijk}}$ | transposition of tensor |

$\mathrm{D}$ | operator defined by $\mathrm{d}/\mathrm{d}z$ or non-dimensional operator defined by Equation (D5b) |

$\overline{\mathrm{C}}$ | operator to embed the odd and even functions into a complex space defined by Equation (53a) or Equation (53b) |

rms | operator defining the root mean square value |

${\mathrm{g}}_{1}\left(a\right)$ | function of a defined by Equation (47c) |

${\mathrm{g}}_{2}\left(a\right)$ | function of $a$ defined by Equation (47d) |

${\mathrm{g}}_{3}\left(a\right)$ | function of a defined by Equation (47e) |

${\mathrm{g}}_{4}\left(a\right)$ | function of a defined by Equation (45c) |

${\mathrm{g}}_{5}\left(a\right)$ | function of a defined by Equation (45d) |

${\mathrm{g}}_{6}\left(a\right)$ | function of a defined by Equation (45e) |

${\alpha}_{0}$, ${\alpha}_{1}$ | arbitrary constants of the z-velocity component of vortices (m s^{−1}) |

${\alpha}_{2}$, ${\alpha}_{3}$ | arbitrary constants of the z-velocity component of vortices (m s^{−1}) |

${\alpha}_{0\mathrm{r}}^{\ast}\left(a\right)$ | velocity coefficient of the rigid surface vortices defined by Equation (21b) (m) |

${\alpha}_{1\mathrm{r}}^{\ast}\left(a\right)$ | velocity coefficient of the rigid surface vortices defined by Equation (22b) (m) |

${\alpha}_{0\mathrm{f}}^{\ast}\left(a\right)$ | velocity coefficient of the free surface vortices defined by Equation (25b) (m) |

${\alpha}_{1\mathrm{f}}^{\ast}\left(a\right)$ | velocity coefficient of the free surface vortices defined by Equation (26b) (m) |

${\beta}_{0}$ | vorticity coefficient of the free surface vortices (s^{−1}) |

${\beta}_{1}$ | vorticity coefficient of the rigid surface vortices (s^{−1}) |

${\beta}_{0}^{\mathrm{a}}$ | vorticity coefficient of the free surface vortices in 2D nucleation (s^{−1}) |

${\beta}_{1}^{\mathrm{a}}$ | vorticity coefficient of the rigid surface vortices in 2D nucleation (s^{−1}) |

Superscript ‘a’ | implies asymmetrical fluctuation |

Subscripts ‘r’ and ‘f’ | mean rigid surface and free surface components, respectively |

Subscripts ‘1′ and ‘2′ | imply the Helmholtz and diffuse layers, respectively |

## Appendix A. Stability by the Non-Specific and Specific Adsorption in 2D Nucleation

**Figure A1.**The 2D nucleation in an electric double layer [22]. (

**a**) Non-specific adsorption. (

**b**) Anionic specific adsorption. (

**c**) Cationic specific adsorption. (

**d**) Schematic view of the relationship between $\langle {\mathsf{\Phi}}_{1}\rangle $ and $\langle {\mathsf{\Phi}}_{2}\rangle $. $z={0}^{+}$, the coordinate of OHP; $z={\infty}^{+}$, the outer boundary coordinate of the diffuse layer; ⊖, anion; ⊕, cation; HL; Helmholtz layer, DL; diffuse layer, ${H}^{\ast}\left(0,t\right)$; the equilibrium concentration overpotentials. Reproduced with permission from Morimoto, R.; Miura, M.; Sugiyama, A.; Miura, M.; Oshikiri, Y.; Kim, Y.; Mogi, I.; Takagi, S.; Yamauchi, Y.; Aogaki, R., The Journal of Physical Chemistry B; published by the American Chemical Society, 2020.

^{−1}), $R$ is the universal gas constant (8.31 J K

^{−1}mol

^{−1}), and $T$ is an absolute temperature (K). ${z}_{\mathrm{j}}$ is the charge number, including the sign, and ${C}_{\mathrm{j}}\left(z=\infty \right)$ is the bulk concentration of ionic species j except for the bulk metallic concentration ${C}_{\mathrm{m}}\left(z=\infty \right)$ (mol m

^{−3}) [66]. Substituting Equation (A3) into Equation (A2a), we have

## Appendix B. Basic MHD Equations in the Stationary Lower Layer

^{−1}) and the magnetic flux density (T), $\overrightarrow{J}$ is the current density (A m

^{−2}), and ${\mu}_{0}$ is the magnetic permeability (4π × 10

^{−7}N A

^{−2}). The overall current density $\overrightarrow{J}$ flows under a magnetic flux density $\overrightarrow{B}$, so that the Lorentz force per unit volume is generated in the following,

^{−1}) (i = 1, 2, 3), and the coordinate (m) $\left(x,y,z\right)$ is expressed by $\left({x}_{1},{x}_{2},{x}_{3}\right).\mathsf{\nu}$ and $\rho $ are the kinematic viscosity (m

^{2}s

^{−1}) and the density (kg m

^{−3}), respectively. In view of an incompressible fluid, the continuity is held.

^{−1}) defined by

^{2}V

^{−1}s

^{−1}), F is Faraday constant (96,500 C mol

^{−1}), ${C}_{\mathrm{i}}$ is the concentration of the ionic species i (mol m

^{−3}), and ${D}_{\mathrm{i}}$ is the diffusion constant (m

^{2}s

^{−1}). Substitution for $\overrightarrow{J}$ from Equation (B2) in Equation (B9) leads to

## Appendix C. Non-Equilibrium Fluctuations Activated in the Stationary Lower Layer

^{−3}) and the concentration fluctuation (mol m

^{−3}), respectively. The mass transfer equation, Equation (B17), is also rewritten as

## Appendix D. Derivation of the Amplitude Equations of the Fluctuations in the Stationary Lower Layer

## Appendix E. Microscopic Vortices Induced in the Rotating Upper Layer

^{−1}), $\overrightarrow{U}$ is the vector of the velocity (m s

^{−1}), and $\overrightarrow{r}$ is the vector of position (m). The term $2\overrightarrow{\mathsf{\Omega}}\times \overrightarrow{U}$ represents the Coriolis acceleration and the term $-\left(1/2\right)\nabla \left({\left|\overrightarrow{\mathsf{\Omega}}\times \overrightarrow{r}\right|}^{2}\right)$ is the centrifugal force.

## Appendix F. Intrinsic Spectrum of the Asymmetrical Fluctuations in 2D Nucleation

^{−3}). With the normalization of ${\theta}_{\infty}^{\ast},$ the intrinsic spectrum of the concentration fluctuation controlled by the micro-MHD flow is represented by

## Appendix G. Amplitudes of the Asymmetrical Concentration and Concentration Gradient Fluctuations in 2D Nucleation

- (a)
- For a rigid surface: